Continuum Damage Mechanics and Numerical Applications

v*

=

-------:---=--

1 + 2
(3-221 )

where E and v are Young's modulus and Poisson's ratio for an undamaged material; E * , v* are the corresponding effective values of the material after damage respectively;

(3-222a)

1 - D + v - Dv 'ifJf."(D) = 1 + 3D + v - Dv

(3-222b)

Eq.(3-222) is actually obtained under the assumption that D =
126

3 Basis of Isotropic Damage Mechanics

rate as

.([l) and Wp,([l) instead of Eq.(3-222) should be theoretical expressed by 1/J>.([l) =

(1 - [l~)([l~ + v - [lh)(l + v)(l - 2v) 3 3 3 v(l + 3[l2 + V - [l2v)(1- 2v + 2[l2V)

(3-223a)

1/Jp,([l) =

+v 1 + 3[l2 + V -

(3-223b)

1 - [l~

3

[l~ v 3

[l2 v

For micro-crack damaged materials with 2-D random distributed cracks, somet imes the damage variable defined based on the concept of the effective deducted area of the bearing load has some unexpected points. For example, the phenomenon of crack closing may cause a fuzzy understanding about the concept of effective failure areas. Some researchers[3-41 rv 43] tried to expand the density function of crack orientations into Fourier series, and strictly proved that the damage state of random uniformly distributed micro-cracks can still be expressed by a single scalar damage variable. Assume the damage involved in the material shown in Fig.3-27 consists of micro-cracks, the characteristic length of the ith crack involved in the representative micro-elemental volume is denoted by li,the number of total cracks is N and the volume of the element is Ve , thus the volumetric micro-crack density parameter "(* can be defined as (3-224)

In this sense, the simplest way is to define the damage variable [l directly by the volumetric micro-crack density parameter "(*. If the critical value of the volumetric micro-crack density parameter according to completed failure of the representative micro-elemental volume is denoted by "(~, the standardized damage variable can be defined as (3-225) Benvensite [3-44] modified this concept into the natural sense of the microdamage definition by the effective area density parameter p* of micro-cracks and took into account interactions among micro-cracks, finally finding the solution of the effective modulus using Moil-Tanaka's method as E*

E

1

1 + 1fP*'

p,* [ 1fP* = 1+-p, l+v

-

]-1

(3-226)

where only the above volumetric micro-crack density parameter "(* was replaced by the effective area density parameter p* = ml 2 j ~A of micro-crack

3.8 Generalized Theory of Isotropic Damage Mechanics

127

Fig. 3-27 Micro-scope volumetric element of da maged materials

areas, in which m is the number of micro-cracks on the representative plane of the elemental cubic, l is the averaged half length of micro-cracks, ~A is the area of the representative elemental plane. The damage variable with respect to the random distributed micro-area-cracks can be defined by the method of the standardized micro damage variable as f! = p* / p~ . Similarly it gives f! _

1/;)..

( f!) _

-

(1 + v)(1 - 2v) 2v + 1fp~ f!)

+ v + 1fp~ f!)(l 1+v 1 + v + 1fp~f!

1/;)..( ) - (1

(3-227a) (3-227b)

These expressions are analytical formulations carried out by solutions of micro-damage-mechanics for the two mentioned types of typical damage models. As is shown above, the detailed expressions of these two damage effective functions are related to material properties, such as Poisson 's ratio and characteristics of microscopic geometry. The microscopic geometry of a practical damaged material is often more complicated. In this case the numerical methods of micro damage mechanics or experiments are required for the determination of these damage effective functions 1/;)..(f!) and 1/;,, (f!), or, instead , the dimensionless paramet ers a(n) and (3(n) in Eqs.(3-209) and (3-211). Four curves of damage effective functions 1/;).. (f!) and 1/;" (f!) are depicted in Fig.3-28 for 2-D circular micro-void damage and in Fig.3-29 for 2-D microcrack damage both distributed uniformly and at random, taking v = 1/3 and p~ = 1. In Fig.3-29, the dashed straight line is obtained by the results of strain equivalence hypothesis. A comparison between results obtained by the developed model and the strain equivalence hypothesis is also presented in Fig.3-28. The result based on the strain equivalence hypothesis shows some irrelevance to the material property and the microscopic characteristics of

128

3 Basis of Isotropic Damage Mechanics 1.0

_

- - ___Curve 2: for 1//.(£2)= 1//"(£2)=(1 - £2), v =1/3

~

~ 0.9

cf 0.8

~ 0.7 ell

c: 0.6 .~ u 0.5 c: <8 0.4
~

""'

0.2

~

0.1


8

'"

Curve I : for 1//.(£2)='It,.(£2)=(I-£2) 1(1 +2£2)

OL-____~______~______L-____-L______~

Q

o

0.05

0.10

0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.5

Damage variable £2( or porosity)

Fig. 3-28 The curves of damage effective functions in the case of 2-D circular microvoid damage (curve 1) and a comparison with the result by stra in equivalence hypothesis (curve 2)

~ ,; ,-..

g ~ ell

c:

.g u

1.0 v =1/3, P;=I

0.9

............~urve 4: fOflV).(£2)=4/[(4+31t£2)(I+37t£2)]

0.8 0.7 0.6

--- ---

c:

<8 0.5
.~

0.4

u

~ 0.3 .....

OJ)

0.2

8'" 0.1

'"

Q

0

o

0.05 0.10

0.15

0.20 0.25 0.30

0.35 0.40 0.45 0.5

Damage variable £2 (micro-crack density)

Fig. 3-29 Curves of damage effective functions in case of 2-D micro-crack damage distributed uniformly and at ra ndom

damage, and uncertainties for different mat erials and different microscopic damage characteristics. It always leads to 1/J)..(fl) = 1/J,,(fl) = 1 - fl. As is shown in Fig.3-28, in the case of 2-D random distributed circular micro-void damage when v = 1/ 3, 1/J).. (fl) = 1/J,,(fl) = (1 - fl) /(1 + 2fl) i- 1 - fl (for curve 1) , this is not quite the same as the results obtained from the strain equivalence hypothesis. In the case of 2-D random distributed micro-crack damage, the curves of 1/J)..(fl) and 1/J,,(fl) are significantly different. Furthermore, both 1/J)..(fl) and 1/J,,(fl) are strongly non-linear. It is evident that the damage ef-

3.8 Generalized Theory of Isotropic Damage Mechanics

129

fective functions provide a link between continuum damage mechanics and microscopic damage mechanics. Another two curves of damage effective functions 'lj;)..(D) and 'lj;/lo(D) with regard to the 2-D circular micro-void damage model and the strain equivalence hypothesis damage model are plotted in Fig.3-30 and Fig.3-31 with a comparison between the assumption of D = IfJ where the damage is defined as the volumetric deduction rate corresponding to Eq.(3-222) and the real definition of IfJ = D 3 / 2 , where the damage is defined as the area deduction rate corresponding to Eq.(3-223). Fig.3-30 shows a comparison of damage effective functions with regard to the strain equivalence hypothesis damage models between the damage variable defined by the area deduction rate (1fJ = D 3 / 2 : curve 5) and by the volumetric deduction rate (1fJ = D: curve 2). Fig.3-31 shows a comparison of damage effective functions with regard to the circular micro-void damage models between the damage variable defined by area deduction rate (1fJ = D: curve 1) and by volumetric deduction rate (1fJ = D3/2: curve 6). From comparisons, it is found that using the assumption of IfJ = D (i.e. volumetric deduction rate) this makes the properties of damage effective functions 'lj;).. (D) and 'lj;/lo (D) change both the quantities and shapes. The quantities of damage effective functions 'lj;)..(D) and 'lj;/lo(D) obtained by Eq.(3222) under the assumption of D = Ijf which defines damage as a volumetric deduction rate, are lower than those obtained by Eq.(3-323) under the real definition of IfJ = D 3 / 2 , which defines damage naturally as the area deduction rate.

C .,

0.9

~ ~

.§ '0

.e'"

0.8

.~

0.7

~.., .,

0.6

ell

'6"

A

Curve 2: for 'l'A.(n) = 'I'~(n)=(l- n)

0.5 +-,.....,,...,....-.---r-r..-,.....,,...,....-.---,...,..-,-....-,....,.-,-....-,....,.--.-l 0.2 0.3 0.4 o 0.1 0.5

Damage variable [2

Fig. 3-30 Comparison of damage effective functions between damage defined by the area deduction rate (cJ> = rp/3: curve 5) and defined by the volumetric deduction rate (cJ> = [J : curve 2) with regard to the strain equivalence hypothesis damage model

130

3 Basis of Isotropic Damage Mechanics

6'

~

~'" .ec:; I:

Curve 6: for 'I'.(D)='I'.(.Q) =(I - D ''') 1(1 +2.am)

0.9 0.8 0.7

0:

.,

.E 0.6 .!!;

c:; 0.5

....~.,

..a .. Q)

ell

0.4

Curve 1: for 'I'.(D )='I'.(D ) =(i -Q) 1(l+2D )

0.3

Q

0

0.1

0.2

0.3

0.4

0.5

Damage variable Q

Fig. 3-31 Comparison of damage effective functions between damage defined by a rea deduction rate (

3.8.4 Dissipative Potential and Damage Evolution for Generalized Theory There are two methods used to set up the damage evolution equation, the first one is directly based on the experimental results tested for many types of damage phenomena, such as brittle, ductile, creep and fatigue damage, in order to est a blish an experiential damage evolution equation. The second one employs the ort hogonal flow rule of internal st at e variables to provide a theoretical damage evolution equation. The representative work for the second one was carried out by Lemaitre and Chaboche [3-9, 3-6] and Roussclier [345]. The above developed a damage strain energy release rate model presented in Eq.(3-211) and Eq.(3-216) which can be used to construct the isotropic damage evolution equations. According to [3-19, 3-10]' the expression of the dissipation potential p * can be chosen in the form of exponential functions of Y. p*

= _1_BYs+! 8+ 1

(3-228)

Substituting Eq.(3-227) into Eq.(3-59), we have

f? =

dP*

dY

= BY s

(3-229)

where B, 8 are non-negative material const ants used to represent the characteristic of damage evolution and determined by experimental data. Substituting Eq.(3-211) or Eq.(3-216) into Eq.(3-229) , it gives

3.8 Generalized Theory of Isotropic Damage Mechanics

131

(3-230a) or

(3-230b) Eqs.(3-230a) or (3-230b) are the proposed generalized isotropic elastic damage evolution equations. It should be pointed out that Eq.(3-230a) or Eq.(3-230b) gives a generalized coupled model between damage field and strain field, and therefore it is applicable either to elastic damage problems or to elasto-plastic damage problems (to be described in Chapters 4 and 7). The strain tensor {cij } can be divided into parts of the spherical strain tensor Cm { Oij } and deviatoric strain tensor {eij } as (3-231 ) Employing the quantity of the equivalent strain (3-232) Eq.(3-230a) can be changed into the form

il =

B{ C;q[~tt t, nj3(n) nn-l + 3 (:: 2) t, n(ttj3(n) + ~Aa(n») nn-l]

r

(3-233) Eq.(3-233) shows that the damage evolution of materials is controlled by the quantity of the equivalent strain and the triaxiality ratio (Jm / (Jeq plays an important rule during the process of damage evolution. In addition, the damage rate depends on the current damage state. All these comments conform with the results given in [3-19]. The great numbers of experimental results also confirm the important effects of the real triaxiality ratio (Jm/(J eq on the material damage and fracture. A higher triaxiality ratio makes material become brittle. The validity and the rationality of Eq.(3-233) will be examined by practical and numerical examples in Chapter 4. So far, for isotropic elastic damage problems, the generalized damage stress-strain constitutive Eq.(3-209) (or Eq.(3-215)), the generalized damage strain energy release rate Eq. (3-211) (or Eq. (3-216)) are directly derived from the basic laws of irreversible thermodynamics. Consequently, the damage evolution Eq (3-230) applicable to elastic damage and elasto-plastic damage problems can be implemented using the orthogonal flow rule of internal state varia bles. Obviously, the first group of equations such as Eqs.(3-209), (3-211) etc. is expressed in the form of a series, the second group of equations such

132

3 Basis of Isotropic Damage Mechanics

as Eqs.(3-215), (3-216) etc. is expressed in the generalized form of damage effective functions. These two groups together provide a complete theoretical description of isotropic elastic damage problems. Summarizing the above theoretical treatments shows us that isotropic damage mechanics problems can be studied from a more generalized point of view by thermodynamics. The generalized models of damage stress-strain constitutive equations and damage strain energy release rate expression can be carried out directly from the extended Helmholtz free energy in the form of Taylor's series based on the second law of thermodynamics. The development of generalized damage mechanics does not need to be based on any controversial concepts such as the effective stress and the strain equivalent hypothesis. The generalized method developed in this section can also be applied to study anisotropic and other damage mechanics problems (such as thermo-elastic, visco-elastic damage problems). The generalized damage evolution equations applicable either to elastic damage problems or to elastoplastic damage problems can be obtained using the orthogonal flow rule of internal state variables. The generalized damage mechanics model overcomes the limitations of the classical damage constitutive equation based on the well-known strain equivalence hypothesis. Two damage effective functions in the constitutive equations reflect the different influences of damage on the two independent elastic constants. The detailed expressions of damage effective functions rely on the geometric characteristics of damage and can be determined by micro-mechanics. It is shown that the classical damage constitutive equation based on the strain equivalence hypothesis is only a simplified form of the general expression given in this section and it may fail to describe satisfactorily the damage phenomena of practical materials. The relations and differences between the single and double scalar damage models are revealed for the first time. For finite deformation problems, the third or higher rank terms of strain should be retained in the expansion of the constitutive functional. There will be more damage effect functions in the constitutive equations.

References [3-1] Lemaitre J ., Evaluation of dissipation a nd damage in metals submitted to dynamic loading. In: Proceedings of the ICM-l , Kyoto, Japan (1971) . [3-2] Leckie F .A., Hayhurst D .R. , Constitutive equations for creep rupture. Acta Metall., 25, 1059-1079 (1977). [3-3] Hult J ., Effect of voids on creep rate and strength. In: Shubbs N., Krajcinovic D. (eds.) Damage Mechanics and Continuum Modeling. American Society of Civil Engineering, USA, pp.13-23 (1985) . [3-4] Chaboche J. , Une loi differentielle d 'endommagement de fatigue avec cumulation non Iineaire. Revue Francaise de Mechanique. 50-51,71-82, in French (1974) . [3-5] Lemaitre J ., Chaboche J ., A non-linear model of creep-fatigue damage cumulation and interaction. In: Proceedings of the IUTAM Symposium on Mechanics of

References

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Visco-Elastic Media and Bodies, Gothenburg, Sweden. Springer-Verlag, Berlin, pp.291-301 (1975) . [3-6] Lemaitre J ., A continuous damage mechanics model for ductile fracture. J . Eng. Mater. Tech., 107(1), 83-89 (1985) . [3-7] Lemaitre J., Chaboche J., Mechanics of Solid Materials. Cambridge University Press, Cambridge, UK (1990) . [3-8] Murakami S., Damage mechanics approach to damage and fracture of materials. Rairo, 3 , 1-13 (1982). [3-9] Lemaitre J., Chaboche J., Aspect phenomenologique de la rupture par endommagement. J . Mech. Appl., 2(3), 317-365, in French (1978) . [3-10] Zhang W .H., Numerical Analysis of Continuum Damage Mechanics . Ph.D. Thesis, University of New South Wales, Australia (1992). [3-11] Zhang W .H., Chen Y .M ., Jin Y ., Effects of symmetrisation of net-stress tensor in anisotropic damage models. Int. J . Fract., 106-109, 345-363 (2001) . [3-12] Zhang W.H ., Murti V ., Valappan S., Effect of matrix symmetrization in anisotropic damage model. Uniciv Report No. R-237, University of New South Wales, Australia (1991) . [3-13] Lee H., Peng K., Wang J ., An anisotropic damage criterion for deformation instability and its application to forming limit analysis of metal plates. J. Eng. Fract. Mech ., 21(5), 1031-1054 (1985). [3-14] Zhang W.H. , Valliappan S., Continuum damage mechanics theory and application: Part I. theory ; Part II. application . Int. J . Dam. Mech., 7(3) , 250-297 (1998). [3-15] Valappan S., Zhang W.H ., Murti V ., Finite element analysis of anisotropic damage mechanics problems. J . Eng. Fract. Mech., 35(6), 1061-1076 (1990) . [3-16] Davison L., Stevens A., Thermomechanical constitution of spalling elastic bodies. J. Appl. Phys., 44(2), 667-674 (1973). [3-17] Coleman B ., Gurtin M., Thermodynamics with internal state variables. J . Chem. Phys ., 47(2), 597-613 (1967) . [3-18] Krajcinoic D., Lemaitre J ., Continuum Damage Mechanics: Theory and Applications. CISM Lectures, Springer-Verlag, Berlin (1987) . [3-19] Lemaitre J ., A Course on Damage Mechanics. Springer-Verlag, Berlin Heideberg New York(1992). [3-20] Chaboche J. , Continuum damage mechanics: Part I. general concepts; Part II. damage growth, crack initiation, and crack growth. J . Appl. Mech., 55(1) , 59-72 (1988) . [3-21] Tang X .S., Zhen J .L., Jiang C .P ., Continuum Damage Theory and Application. Chinese Popular Traffic Press , Beijing, in Chinese (2006) . [3-22] McClintock F.A., A criterion for ductile fracture by the growth of holes. ASME J . Appl. Mech., 35(3), 363-371 (1968) . [3-23] Rice J., Tracey D ., On ductile enlargement of voids in triaxial stress fields . J. Mech. Phys. Solids, 17(3), 201-217 (1969). [3-24] Gao Y .X ., Zheng Q.S., Yu S.W., Double-scalar formulation of isotropic elastic damage. Acta Mech. Sin ., 28 (5), 542-549, in Chinese (1996). [3-25] Tang C.Y ., Shen W ., Peng L.H ., et at., Characterization of isotropic damage using double scalar variables. Int . J. Dam. Mech ., 11(1),3-25 (2002). [3-26] Mori T., Tanaka K ., Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall., 21(5), 571-574 (1973) .

134

3 Basis of Isotropic Damage Mechanics

[3-27] Ladeveze P ., Damage and fracture of trid irectional composites. In: Proceedings of the ICCM-4 , Tokyo, Japan, pp.649-658 (1982) . [3-28] Weng G.J., The t heoretical connection between Mori-Tanaka's theory and the Hashin-Shtrikman-Walpole bounds. Int . J . Eng. Sci. , 28 , 1111-1120 (1990). [3-29] Tang C .Y. , Modelling of Craze Damage in Polymeric Materials: A Case Study in Polystyrene and High Impact Polystyrene. Ph.D. Thesis , Hong Kong Polytechnic University, Hong Kong, China (1995) . [3-30] Tang C.Y ., Lee W.B ., Effects of damage on t he shear modulus of aluminum alloy 2024T3 . Script. Metall. Mater ., 32(12), 1993-1999 (1995). [3-31] Tang C .Y., Jie M ., Shen W ., Yung K .C ., The degradation of elastic properties of aluminum a lloy 2024T3 due to strain damage. Script. Mater., 38(2) , 231238( 1998). [3-32] Krajcinovic D., Continuum damage mechanics. Appl. Mech. Rev ., 37(1) , 1-6 (1984). [3-33] Chow C .L. , Wei Y ., Constitutive modeling of material damage for fatigue failure prediction. Int. J . Dam. Mech., 8(4) , 355-375 (1999) . [3-34] Yong W ., Chow C .L., A damage-coupled TMF constitutive model for solder alloy. Int . J . Dam. Mech. , 10(2), 133-152 (2001) . [3-35] Ladeveze P., On a n anisotropic damage theory. In: Proceedings of CNRS International Colloquium No 351, Villard-de-Lans, Balkema, Rotterdam (1983). [3-36] Ganczarski A., Modeling of crack opening/closure effect in low cycle fatigue for AISI 316L stainless steel. Int. J. Strain Anal., (2004) . [3-37] Tang X .S., Jiang C .P. , Zheng J .L., General expressions of constitutive equations for isotropic elastic damaged materials. Appl. Math . Mech ., 22(1 2), 14681475 (2001) . [3-38] Kachanov M ., On the effective moduli of solids with cavities and cracks . Int . J . Fract. , 59, R17-R21 (1993). [3-39] Zhang W.H., Chen Y.M ., Jin Y ., Mechanism of energy release during coal/gas outburst . Chin. J . Rock Mech . Eng., 19(zl), 829-835, in Chinese (2000) . [3-40] Valliappan S., Zhang W .H., Numerical modelling of methane gas migration in dry coal seam . Int . J. Numer. Anal. Methods Geomech., 20 , 571-594 (1996) . [3-41] Onat E .T., Effective properties of elastic materials that contain penny shaped voids. Int . J. E ng. Sci., 22(8-10), 1013-1021 (1984). [3-42] Kanatani K. , Distribution of directional data and fabric tensors. Int . J . Eng. Sci., 22 (2), 149-164 (1984) . [3-43] Lubarda V.A ., Krajcinovic D. , Damage t ensors and the crack density distribution. Int . J . Solids Struct., 30(20) , 2859-2877 (1993) . [3-44] Benvensite Y. , On the Mori-Tanaka's method in cracked solids. Mech . Res. Commun ., 13(4), 193-201 (1986). [3-45] Rosuseer G., Finite deformation constitutive relations including ductile fracture damage. In: Proceedings of the IUTAM Symposium on T hree Dimensional Constitutive Relations and Ductile Fracture, Amsterdam, North-Holland , pp.331-355 (1980) .

4

Isotropic Elasto-Plastic Damage Mechanics

4.1 Introduction Most of the plastic damage models attempt to bring together the theories of plasticity and continuum damage mechanics to yield a unified approach to the damage constitutive model and damage growth model of isotropic damaged materials. The formulation is cast within the generalization of classical plasticity theories by means of the internal variable theory of thermodynamics. Within the framework of general formulation, Dragon and Mroz [4-1], Bazant and Kim [4-2], Krajcinovic and Fonseka [4-3] and [4-18"-'21] treated the continuum models for rock and concrete with brittle-plastic damage behavior. Lubliner et ai. [4-4], Yazdani and Schreyer [4-5], Oller et al. [4-6 ,4-19"-'26] established plasticity and damage-coupled models, which adopt the concepts of plastic surface and damage surface interaction. Frantziskonis and Desai [47] presented a model which combined plastic strain softening and isotropic damage growth. At an early stage Lemaitre [4-8"-'11 ], Chaboche [4-12"-'13] presented a ductile plastic damage model for metals. The model is based on separation of the dissipative inequality into two parts; one is the perfect damaged plastic strain dissipation corresponding to classical plastic theory, the other is damage growth dissipation (i.e. the inequality Eq.(3-50a) is separated into two parts as {(J}T {Ep} - R"y?,; 0 and - Y il ?'; O. Actually, this model is based on independent plastic energy dissipation and damage energy dissipation. The coupling of plastic and damage behavior is only dealt with via the damage variable, and is not associated with the total energy dissipation. The flow rule is an associated flow rule, which does not include the influences of the damage growth. The idea of combining the theory of plasticity and the theory of the internal state variable of continuum damage mechanics via the total dissipations of plasticity and damage was proposed by Simo and Ju [4-14"-'15]. The modified thermodynamic framework developed by Rosuselier [4-16] leads to similar results. The work, based on the homogenization concept, which in the case of W. Zhang et al., Continuum Damage Mechanics and Numerical Applications © Zhejiang University Press, Hangzhou and Springer-Verlag Berlin Heidelberg 2010

136

4 Isotropic Elasto-Plastic Damage Mechanics

ductile fracture gives information about the damage evolution equations, was carried out by Dragon and Chihab [4-17]. The model presented in this chapter is modified and developed from Lemaitre's model for more general cases. This model is based on the full mechanical dissipation inequality {a} T {Ep} - {Y} T {D} - {R} T {"y } ~ 0 without separation, in order to establish a damage-plastic flow potential including the plastic flow and the damage growth flow. The present model yields a non-associated flow rule.

4.2 Associated Flow Rule Model According to the basic relation of Eq.(3-53), Lemaitre [4-19] and Chaboche [4-13] assumed the relations between internal state variables ({ip}, "y ) and the yield function F ({ 5}, il, R}) = 0 based on standard isotropic plasticity and associated flow rule as

dF {Ep} = Ad{a}

(4-1)

da eq d{a}

(4-2)

3 {s} 2 a eq

Further, a dissipation potential which is modified from the definition ofEq.(359) was independently introduced as

D = _ d<1>* = _ A dt} dY

dY

(4-3)

It is evident that, the damage growth law of Eq. (4-3) is independently defined , if the potential of damage dissipation <1>* is chosen independently from the plastic potential, which is associated with the yield function. In this case, the energy dissipation due to damage growth is not fully coupled with that, due to plastic flow and hardening. This means that the normality principle of the associated flow rule is not applicable.

4.2.1 Re-expression of Lemaitre's Model Lemaitre [4-8"-'11 , 4-19] presented an approach for ductile damage constitutive equations and damage kinetic equations in the form of flexibility based on plastic strain space. The standard isotropic plasticity associated with von Mises criterion was modified by Lemaitre through the yield function

F -- a eq

-

R _ a -- 0 s

1 - il

(4-4)

where as is the yield stress, which can be considered to be as the initial variable associated with the accumulative hardening parameter R (0) = a s.

4.2 Associated Flow Rule Model

137

R as the internal state variable is a function of the accumulative hardening parameter, " which was defined [4-11 ] by (4-5) The accumulative hardening rate, which was mentioned in Eqs.(3-53) and (3-59) was defined as

(4-6) Using

P = 0 to

calculate the proportionality factor A

.

F =

a eq

-

aRb).

- -,

a,

1- D

+

a eq - R si = 0 (1 - D)2

(4-7)

substituting Eqs.(4-4) , (4-2) and (4-3) into Eq.(4-7) it gives aR b) .

_ _ ,

F= ~ - A 1- D

A

a, (1 - D)2

- A~ a
1 - Day

= 0

(4-8)

Based on the definition of proportionality factor A for the associated flow rule the multiplier A can be determined as if F ?: 0 (4-9) if F < 0 Substituting Eqs.(4-9) and (4-4) into Eq.(4-1) and using the index of the unit step function H (F), the vector of plastic strain rate can be formulated as

{i } = H (F) P

aR -a,

(req

A

aaeq

a<pa{a} + (1 - D)aSay -

(4-10)

U sing the relation to general plastic theory

aa

eq

a{a}

3 {s} 2 a eq

(4-11)

in Eq.( 4-10), the ductile damage constitutive equation in the form of flexibility can be presented as

138

4 Isotropic Elasto-Plastic Damage Mechanics

{i } p

= H (F) ~ _ _ _{J_.eq- '----_-,- { S } 2 aR a

-a, + (1 -

(4-12)

D) (Js-ay

From (4-13) Eq.(4-12) can be rewritten in the form increments with matrix as (4-14a) where (4-14b)

The inverse of Eq.(4-14a) can be expressed as (4-14c) where

4.2.2 Damage Evolution Equations

Lemaitre suggested [4-11 , 4-19] that the hardening rule can be considered as the power law given by 1 (4-15) R(r) = kr'm where k and m are material constants for defining the strain hardening. Substituting Eqs.(4-4) and (4-6) into Eq.(4-2) , the proportionality factor A can be represented by the plastic strain rate and the damage variable as (4-16) Substituting the expression of Eq.(4-16) into the basic relation Eq.(4-3), the damage growth law (kinetic equation) can be represented in the plastic strain space as (4-17)

4.2 Associated Flow Rule Model

139

Lemaitre also suggested that the potential of damage dissipation i[>* has a form of power function of Y for convenience and linear in 'Y to ensure the non-explicit dependency of D with time

i[>*

=

S (so ~ 1)

(_y) So+l Sa

(4-18)

Based on this suggestion, the function in Eq. (4-17) can be taken 1 Sa i[> = l - D(so + l) A

(_Y)So+l Sa

(4-19)

Thus, the damage growth law can be formulated as

. (_Y) so 'Y. -

D=

Sa

(4-20)

Substituting the expressions of damage strain energy release rate presented in Eqs.(3-87) and (3-88) into Eq.(4-20), for model A it gives (4-21a) for model B it gives

(4-21b) Eq.(4-4) can be rewritten by substituting Eq.(4-15) as (4-22) Substituting Eq.(4-22) into Eq.(4-21) and using the expression for factor presented in Eq.(3-95), the kinetic Eq.(4-21) can be expressed as: for model A

Ie

(4-23a) for model B (4-23b) In order to find a simple integration for the damage growth equation, Lemaitre [4-11, 4-19] and Chaboche [4-12 rv 13] suggested a simple form of von Mises yield function as

140

4 Isotropic Elasto-Plastic Damage Mechanics

(4-24) where no initial variable associated with the accumulative hardening parameter O"y (i. e. Ro) was assumed. Substituting Eq.(4-24) into Eq.(4-21a) , for model A we can obtain (4-25a) and for model B fiB

= (2 E (1 ~ D) Sa k2 ,f:. )

So

'Y

(4-25b)

It is to be noted that it is quite complex and cumbersome to integrate fi given by Eq.( 4-25) in order to obtain the damage varia ble D. However , for all practical purposes, one can use the uniaxial loading sit uation from which it can be easily assumed that the triaxiality ratio 0" m/ 0" eq is constant and hence the integration can be performed using the conditions

< Id (damage threshold) 1 = IR (strain to rupture)

(4-26)

1

(4-27)

to obtain a simple relation between the actual value of damage D and the accumulative hardening parameter I . The integration for model A under the condition of Eq.( 4-26) yields DA

=

(!L~:),o 28

where (x) = x if x > 0, (x) For model B, it yields

= 0 if x

+ 1) o

l;k2 (-

DB = 1 - [1 - (8

2ESa

0

m (

:

/ sa: m_I:sa:=)

(4-28)

o.

~

)8

m

0

28 0

+m

(~ 1 m

-,

~) ] so'+.'

d

(4-29) Eq.( 4-28) for model A can be rewritten in a simpler fashion by introducing t he accumulative hardening paramet er at rupture 1 d as a func t ion of the triaxiality ratio 0" m/O" eq corresponding to the intrinsic value of damage at failure Dc, which had been assumed to be a material property (Eq.(4-27)). The integration of Eq.(4-25a) with conditions of Eqs.(4-26) and (4-27) has t he following form:

_ (!;k )828 m+m\'R / 2 s~:= _ ~) 2ES Id 2

Dc-

Dividing Dc by D yields

a

0

0

(4-30)

4.2 Associated Flow Rule Model

141

(4-31 ) In certain cases of plasticity, the hardening exponent m may be a large value, for example it is equal to 00 in the case of perfectly plastic material. The other coefficient So in Eq.(4-31) has been shown to be nearly equal to unity, from the test results of the one dimensional model [4-11]. In this particular case, (2so + m)/m becomes an unity order. It can also be stated that in the one dimensional situation, (4-32) where Cd and C R are the one-dimensional strain at the damage threshold and at rupture failure. Thus we have (4-33)

'R

The accumulative hardening parameter at rupture can also be expressed as a function of C R , CJ m , CJ eq and Ie. The values in the one-dimensional case are CJmCJ eq = 1/3 and Ie = 1. For the expression of Dc with 8 0 = 1 and assumption of (2so + m)/m = 1 [4-19] follows (4-34) Then in triaxiality it gives

'n

~ ~l ~ [~ (1 + ") + 3 (1 - 2") ( ; : En

rf

(4-35)

Substituting Eq.(4-35) into Eq.(4-33), the final formulation is

D = Dc ( '

[~(1 + v) + 3 (1 - 2v) (~f] -Cd ) = Dc / ,tt - Cd) CR-C d

In the particular case of one dimension (r

\ C R - Cd

= cp), it becomes

(4-36)

(4-37) With the above expression for ,R, the equation for Dc can be rewritten in a very simple way using 80 =1 and assumption of (28 0 + m)/m = 1.

142

4 Isotropic Elasto-Plastic Damage Mechanics

(4-38) This allows the replacement of k 2 I (2ESa) by [2el (cR - cJ in the differential to obtain the final results for any loading path. kinetic equation for The damage growth rate of model A valid for any loading path is

n

(4-39) The damage value of model A valid for radial loading only (4-40) For model B, Eq.(4-29) can be rewritten in the form of

1 - (1 - [2B)So+ l

~) P k2 ) So m / 2 , += = (so + 1) ( 2';; Sa 2s o + m \1-"':;;;;- -1d m

(4-41a) since [2 <1 and So + 1 > 0, (1 - [2) 8 +1 can be approximated by the Taylor series neglecting the high order terms as given below 0

1 - (1 - [2B)"o+ l

= 1 - (so + 1) [2

(4-41b)

Substituting Eq.(4-41b) into (4-41a), for model B,

[2B =

(;L~:ro 2S o:

m \

/ so:m -1:'? nt= )

(4-42)

It can be seen from Eq. (4-42) that this expression of [2 is the same as that for model A expressed in Eq.(4-28). This indicates that all other relevant expressions derived for model A, that is Eqs.( 4-30) to (4-40), are also equally applicable to model B. However, it should be noted that the values of [2 from both models are not the same, even though similar expressions have been obtained. This is due to the fact that the parameters 1, f e, a eq , am are not the same for both models because of different constitutive equations.

4.2.3 Evaluated Damage Variables by Different Hypothesis Models It is also interesting to observe that there exists a natural relationship between the damage variable [2A and [2B. These can be evaluated using Eqs.(3-25) and (3-26). They result in the following form (4-43a)

4.2 Associated Flow Rule Model

143

The relationship and the difference between the quantities of damage evaluat ed from model A and model B are illustrated in Fig.4-1(a) and (b). In Fig.4-1(a), t he curve (A) illustrates that DA is expressed as a function of DB (presented by Eq.(4-43b)) versus D(=DB ) from 0 to 1; the curve (B) illustrates that DB is expressed as a function of DA (presented by Eq. (4-43b)) versus D( =D A ) from 0 to 1. 1.0

,...----------:::::=='~

-

Analytical

1.0,...--------------, --Analytical results

0.8

0.8

Q A 0.6

0.6

and

I!.QA.

Q·O.4

..·.. ···<>·0

F.E. Results

I!.QA.=QA- Q •

(A) I!.QAB= vl-QA-1 +QA

(B) I!.QA.=Q.-Q~

0.4

(B)

0.2

(A)

0.2

A: QA=2Q.-Q.' B: Q.=(I-~)

O~-~-~-~-~~

o

0.2

0.4

Q

(a)

0.6

0.8

1.0

0

0.2

0.4

Q

0.6

(b)

Fig. 4-1 (a ) Relationship of damage va riables between models A and B; (b) Difference between damage variables from models A and B

In Fig.4-1(b) , the curve (A) illustrates that the difference I"l.DAB is expressed as the function of DA (i. e. I"l.DAB = VI - DA - 1 + D A ) versus D( = D A ) from 0 to 1 when substituting the first expression Eq.(4-43b) into the difference I"l.D AB ; and the curve (B) illustrates that the difference I"l.DA B is expressed as the function of DB (i.e. I"l.DA B = DB - D1 ) versus D(= DB ) from 0 to 1 when substituting the second expression Eq.(4-43a) into the difference I"l.D AB . This relationship and the difference between the damage values from models A and B are illustrated in Fig.4-1(a) and(b). In Fig.4-1 (a), the curve (A) illustrates DA as a function of DB (presented by Eq.(4-43a)) versus D = DB from 0 to 1; the curve (B) illustrates DB as a function of DA (presented by Eq.(4-43b)) versus D = DA from 0 to 1. In Fig.4-1 (b) , the curve (A) illustrates that when substituting Eq.( 4-43b) into the difference I"l.DA B = DA - DB, I"l.DAB as a function I"l.DAB = VI - DA - 1 + DA versus D = DA from o to 1 and the curve (B) when substituting Eq.( 4-43a) into the difference I"l.DA B = DA - DB, I"l.DA B as a function I"l.DA B = DB - D1 versus D = DB from 0 to 1. From Fig.4-1, it can be seen that if DA = 0 then DB = 0 and if DA = 1 t hen DB = 1. While 0 < D < 1, there is a difference between DA and DB. It is interesting to note that, when DA = 0.75, the difference between DA and DB that is I"l.DAB (D A ) as a function of DA reaches the maximum value

144

4 Isotropic Elasto-Plastic Damage Mechanics

l1Dmax = 0.25. Whereas, when DB = 0.5, the difference between DA and DB that is l1DA B (DB) as a function of DB also reaches the maximum value l1Dmax = 0.25. This means that the maximum relative error of damage between these two models could reach 25%. Thus, when using model B for the analysis of materials such as metals for which model A may be considered to be suitable, the maximum error from the analysis may reach in the neighborhood of D = 0.5. Otherwise, when using model A to analyze materials such as geological materials for which model B may be more appropriate, the maximum error from the analysis could reach in the neighborhood of D = 0.75, and such a high value would not be acceptable for a geotechnical medium. Therefore, it is necessary to note this difference when one contemplates applying in practice one of these two models. Here it may be suggested that it is better to choose model A for isotropic high strength and hardly cracked materials (for example most metals) and to choose model B for isotropic weak strength materials, cracked materials and as well as anisotropic damage problems (for example geological materials).

4.3 Non-Associated Flow Rule Model 4.3.1 Basic Equations of Elasto-Plasticity for Isotropic Damaged Materials

In the associated flow rule model, it was assumed that there exist two independent potentials, one is the plastic potential associated with the yield function F (Eq.(4-4)) and the other is the damage dissipation potential p * (or tP) (Eq.(4-3)). Because the choice of the function for the potential has some subjectivity, the aspect of damage growth is independent of yield criterion and plastic flow. It should be noted that for this model the normality principle associated with t he flow rule, as outlined by Lemaitre, may not valid. Actually, most of observations from experimental investigations indicate t hat damage growth is accompanied by t he developing of plastic strain and a yield zone in the material. From the point of view of the micro-structure, the plastic flow and damage growth are due to staggering of the crystal lattice, crystal slip and cracking of the grin boundary. Thus, it is only logical to consider the interdependence of damage growth and plastic flow. On the other hand, from the phenomenological point of view, the yield function indicates a failure state, which is only dependent upon the actual stress during the plastic failure process, even if the yield surface is changing while the plastic flow continues. This changing can still be assumed to be only controlled by plastic law, not controlled by damage growth law. Therefore, it can be depicted that when the plastic flow reaches some stage, the phenomenon of damage growth will start and accompany the development of plastic flow. Obviously, the damage growth will also influence the plastic flow.

4.3 Non-Associated Flow Rule Model

145

From the above discussion, it may be considered that there are two kinds of failure flow, one is the classic plastic flow and the other is the damage growth flow. In the damage plastic theory, it is not suitable to simply call the flow function G the (classical) plastic potential or plastic flow function, which must be called a dissipation flow function or damage-plastic flow function. Therefore, (as mentioned in Chapter 3 §3-5) the dissipation flow in material failure includes three parts: • • •

change in plastic strain; change in yield surface (hardening); change in damage state (damage growth or propagation).

As discussed above, the yield function can be assumed to depend on the stress state {a} hardening state R, and damage state il, not dependent on damage rate D. The dissipation flow function G can be assumed to depend on {a}, R, il and the damage strain energy release rate Y. Zhang et al. [4-27'"'-'28] suggested that the general form of the yield function and the damage-plastic flow function of damaged material can be taken as

F({a} , il,R);? 0

(4-44a)

G({a},il,Y, R);? 0

(4-44b)

where the yield function F is a generalized form of the classical yield function. The consideration of this generalization is that, when the material is damaged, the phenomenon of the plastic yield of material is only due to the actual (effective or net) equivalent stress reaching the yield criterion. Therefore, if the undamaged Cauchy stress tensor is replaced by the effective stress tensor in the classical yield function F , the damaged yield function can be presented. The damage-plastic flow function G (i.e. damage-plastic flow potential) must be able to characterize the three kinds of dissipations of damage plasticity. Based on this concept, a method which uses the Lagrange multiplier .\ to minimize the deviation between the mechanical dissipation potential
d{~} [
.\G({a},il,Y, R)] = 0

[
d~ [
.\G ({ a} , il, Y, R)] = 0

(4-45a) (4-45b) (4-45c)

From this, we have (4-46)

146

4 Isotropic Elasto-Plastic Damage Mechanics

(4-47) (4-48) It can be seen that the Lagrange multiplier A defined here is similar to the proportionality factor in plasticity theory.

4.3.2 Static Elasto-Plastic Damage Model without Damage Growth It should be noted that for this particular case with no damage growth. The derivative of Eq.(4-44a) with respect to time is

. (dF)T F = d{ a } {o-}

dFdR

(4-49)

+ dR di l' = 0

Substituting Eq.(4-48) into Eq.(4-49), we obtain

dF dR dG ( dF ) T d{a} {o-} = AdRdidR

(4-50)

From expression {c}={ce }+{cp }, we have

{i} = [D* r l{o-}

+ {i p }

(4-51 )

and substituting Eq.(4-46) into Eq.(4-51),

{i} = [D*rl {o-} + Ad~~} dF By multiplying Eq. (4-52) with ( d{ a}

(4-52)

)T[D*], it yields

dF ) T ( dF ) T ( d{a} [D* ]{i} = d{a} {o-}

( dF ) T

+ A d{a}

dG [D* ] d{a}

(4-53)

Considering the property of the proportionality factor as shown in Eq. (4-9) and substituting Eq.(4-50) into Eq.(4-53), A can be determined as

A = H (F)

dF ( d{a}

)T [D*]{i}

-----'----"----"--'----r;;----

dF dR dG (dF) T dG dRdi dR + d{a} [D*] d{a}

(4-54)

Substituting Eq.(4-54) into Eq.(4-52) , the elasto-plastic constitutive equation for isotropic damaged material without damage growth can be formulated in an incremental form as

4.3 Non-Associated Flow Rule Model

147

where {da}, {de} is the incremental full stress and incremental total strain respectively; [D;p] is the elasto-plastic matrix of isotropic damaged material.

= [D*]- H (F)

[D*] dG ( dF )T [D*] d{a} d{a} dF dR dG (dF) T * dG dR d"( dR + d{a} [D ] d{a}

(4-55b)

Substituting Eq.( 4-54) into Eq.( 4-46), the increment of the plastic strain vector for isotropic damaged materials without damage growth can be calculated from the total strain increment {de} as (4-56a) where

1-

[B* ep

[ d~~} (d~:} )T] [D*]

H (F) ----=-----~---

dF dR dG (dF) T * dG dRd,,( dR + d{a} [D ]d{a}

-

(4-56b)

Substituting Eq.(4-54) into Eq.(4-48), the increment of the accumulative hardening parameter for isotropic damaged material without damage growth can be evaluated by

{ d } - H (F)

"( -

_ dG ( dF )T] [D*] {de} dR d{ a } dFdRdG (dF)T * dG dRd,,( dR + d{a} [D ] d{a}

(4-56c)

4.3.3 Elasto-Plastic Model with Damage Growth

In this case, the rate of the damage vector is not equal to zero, derivative of the yield function Eq. (4-44a) becomes

t2 i= o.

The

(4-57) Substituting Eqs.( 4-47) and (4-48) into Eq.( 4-57) , we have

)T (dFdRdG ( d{dF a} {a} = A dR d"( dR

dFdG)

+ dD dY

(4-58)

148

4 Isotropic Elasto-Plastic Damage Mechanics

From the relation

{c} = [D*rl {a}

+ {cp}

(4-59)

the derivative of Eq.( 4-59) with respect to time is (4-60) Multiplying Eq.(4-60) by [D* ], it gives

{a} = [D*] ({ i } - {i p }) Since [D*][D*r

1

-

[D*][.o*]-l {a}

(4-61 )

= [1], it can be shown that (4-62)

Substituting Eq. (4-62) into Eq.( 4-61), the incremental stress-strain relationship can be given as

{a} = [D*] ({ i } - {i p })

-

[.o* ][D*rl {a}

(4-63)

n

(4-64)

where

[.0*] = d[D* ]

dD Substituting Eq.(4-47) into Eqs.(4-64) and (4-63) , we have

(4-65) and

Substituting Eq.(4-66) into Eq.(4-58) , the proportionality factor ,\ can be defined as Substituting Eq.(4-66) into Eq.(4-58), the proportionality factor'\ can be defined as

,\ = H (F) x

(d~:J T

[D*] {i }

)T[D* ] d{;;} dG + (d F + ( dF )Td[D* ] [D*]- l{a}) dG dn d{;;} dn dY

-------------=------~~--------~----------~--

dF dR dG

dR d:; dR

+(

dF

d{;;}

(4-67)

4.3 Non-Associated Flow Rule Model

149

Su bstituting Eq. (4-67) back into Eq. (4-66), the elasto-plastic constitutive equation for isotropic damaged material with damage growth can be presented in an incremental form as follows:

{da} = ([D*]- [D;]) {de}

(4-68)

where

[D; ] = H (F)

x

[D* ] dG (d F ) T [D* ] + d[D*] [D*r 1 d{O"} d{O"} dn

[{a}( d{O"} dF ) T] [D* ] dG dy

--------------_=----------~----~--_=----~----~----

dF dR dG

dR d:; dR

+ ( dF ) T [D* ] dG + (dF + ( dF ) T d[D* ][D*]-l {a}) dG d{;;}

d{;;}

dn

d{;;}

dY

dn

(4-69)

[Dp ] is the plastic matrix of anisotropic damaged materials with damage growth. Substituting Eq.(4-67) into Eq. (4-46) , the increment of the plastic strain vector for isotropic damaged materials with damage growth can be presented by the total strain increment {de} as (4-70a) where

x

[d~~} (d~:}) T]

[D*]

--------------_=--~------~--~----_=----------~----

dFdRdG dR d'Y dR

+ ( dF ) T [D* ] dG + (d F + ( dF ) T d[D*] [D*r 1{a}) dG d{O"}

d{O"}

dn

d{O"}

dn

dy

(4-70b) 4.3.4 Nonlinear Kinetic Evolution Equations of Elasto-Plastic Damage

The equation of damage evolution (i. e. damage growth equation) and the equation of the accumulative hardening rate can be obtained by substituting the expression of proportionality factor given by Eq. (4-67) into Eqs. (4-47) and (4-48) dD = H(F)

150

x

4 Isotropic Elasto-Plastic Damage Mechanics

_ de ( dF dY d{a}

)T [D*] {dE}

--------------~----~~~--------~~--------~----

dFdRde dR d'Y dR

+ ( dF ) T [D* ] de + (dF + ( dF ) Td[D*l[D*r 1{0"}) de d{a}

d{a}

dn

d{a}

dn

dY (4-71)

d'-y = H (F)

x

_ de( dF ) T[D*] {dE} dR d{a}

--------------~------~~--------~~--------~----

dFdRde dR d'Y dR

+ ( dF )T [D*] de + (dF + ( dF )Td[D*l[D*r1 {0"}) de d{a}

d{a}

dn

d{a}

dn

dY

(4-72) It is evident that, Eqs. (4-67) to (4-72) are presented in the strain space (i. e. in terms of strain increment {dE},. It can be seen that once the stress increment {dO"} has been det ermined , the expression for the proportionality factor A can be significantly simplified , and then these equations can be simply represented in the stress space (i.e. in terms of stress increment {dO" },. Thus, the relationships of Eq.( 4-70) to Eq.( 4-72) can be expressed as follows Substituting Eqs.(4-47) and (4-48) into Eq.(4-57), the proportionality factor A may be represented as

dF ( d{O"} A = H (F) dF dR dG dR d'-y dR

)T{o-} +

dF dG dD dY

(4-73)

Substituting Eq. (4-73) into Eq. (4-46), the increment of the plastic strain vector can be determined by (4-74a) where

(d~~}) (d~~}) T]

*

[

[C ep ]

= H (F) dF dR dG

dF dG

dR d'-y dR

+ dD dY

(4-74b)

Substituting Eq.( 4-73) into Eq.(4-47), the damage growth increment equation can be represented as

)T

dG ( dF - dY d{ 0" } { dO" } dD = H (F) dF dR dG dF dG dR d'-y dR + dD dY

(4-75)

4.3 Non-Associated Flow Rule Model

151

Substituting Eq.( 4-73) into Eq.( 4-48), the increment of the accumulative hardening parameter can be reformulated as

dG ( dF )T - dR d{O"} {dO"} d, = H (F) dF dRdG dF dG dR d, dR + dDdY

(4-76)

where {dO"} was determined from Eq.(4-68). It should be noted that since Lemaitre adopted the associated flow rule, the yield function Fand the plastic pot~nti al G are the same. He further assumed a damage dissipation potential
.

dG

for the damage rate, D = AdY' is different from that assumed by Lemaitre,

.

d
D = dY. However, if it is desired to obtain a form similar to that of Lemaitre's,

dG

d
this can be achieved by equating dY = dY and adopting von Mises criterion. After using this particular criterion, it can be seen that Eq.(4-74) is equivalent to Eq.( 4-14) whereas Eq.( 4-75) is equivalent to Eq.( 4-17). The plastic stiffness matrix for Lemaitre's model corresponding to Eq. (4-69) can be given as [D; ]

= H (F) ( 3(1 - D))2 [D] [{S}{S}T] [D]- 3(1- D) [{O"}{S}T] [D] d
x

20" eq

dR + -:;~

(3) ~~

2

20"eq

3

dY

'

d

a

(4-77) It is interesting to note that for Lemaitre's model, the plastic flexibility matrix Eq.(4-14b) and the stiffness matrix Eq.(4-14d), which are expressed with respect to the plastic strain increment, are symmetric matrices. But the stiffness matrices given in Eq. (4-77) and its associated flexibility matrix, which are comprised with respect to the total strain increment, are non-symmetric matrices. Obviously, the normality principle is only satisfied in the subspace {dEp}, which is a subset of the total strain space {dE}. The reason may be that the damage growth is not coupled with the plastic flow. Usually, for the associated flow rule, the stiffness matrices and flexibility matrices either in the space {dEp } or in {dE} must be symmetric matrices due to the normality principle. However, for the non-associated flow rule problems , the normality principle between the plastic flow and yield surfaces is not necessary, and

152

4 Isotropic Elasto-Plastic Damage Mechanics

hence need not be satisfied. Thus, this theory gives non-symmetric stiffness and flexibility matrices both in space {de: p } and {de:} as shown from Eqs. (4-69) to (4-74). 4.3.5 Model of Combined Dissipation Potential

As mentioned above, even though the normality principle of the associated flow rule model presented by Lemaitre is not satisfied with respect to total strain increment {de:} , the results of damage evolution (growth) computed by Lemaitre's damage potent ial
G ({ O"}, D, Y , R) = d> (Y)

+ F ({ O"} , D, R)

(4-7S)

In this case, the damage growth Eq.(4-7S) and the increment of accumulative hardening parameter given by Eq.( 4-76) are similar to those of Lemaitre's model, but the increment of plastic strain given by Eq.(4-74) and the damage elasto-plastic stiffness matrix [Dep] are not the same as those of Lemaitre's.

dG

Since the damage-plastic flow vector d{ O"} due to Eq. (4-7S) consists of two

dF

dd> dY

parts, one is from the classical plastic flow d{ 0" } , the other, dY d{ 0" } , is due to damage growth (kinetic) flow

dG d{O"}

dF d{O"}

dd> dY dYd{O"}

-- = -- + ---

(4-79)

It should be noted here that the second part of Eq. (4- 79) , which presents the contribution of damage growth to the plastic flow , was not included in Lemaitre's model. Therefore, the influence of damage growth on the plastic flow has been missed in Lemaitre's model due to the independence of plastic potential and damage potential. Using Eq.(3-53b) for models A and B, Eq.(4-79) becomes

dG d{ O"} Thus, we have

dF d[D*rl dd> d{ O"} dD {0" } dY

(4-S0)

4.3 Non-Associat ed Flow Rule Model

153

de ( dF ) T] [dF ( dF ) T] d[D*r l [ (dF ) T] dt} [d{a} d{a} = d{a} d{a} dD {a} d{a} dY

[D*] de ( dF) d{a} d{a}

T[D*] = [D*]

dF ( dF d{a} d{a}

(4-81 )

)T[D*]

_ [D*] d[D*rl [{ } ( dF ) T] [D*] dt} dD a d{a} dY U sing the relation Eq. (4-62),

[D*] de ( dF )T [D*] = [D*] dF ( dF )T [D*] d{a} d{a} d{a} d{a}

+ d1~*][D*rl

(4-82)

[{a}(d~~})T] [D*] ~:

[D*] de ( dF )T [D*] = [D*] dF ( dF )T [D*] d{a} d{a} d{a} d{a}

+ d 1~*] [D*rl [{a} (d~~} ) T] [D*]

(4-83)

~:

Substituting Eqs.(4-80) to (4-83) into Eqs.(4-69) to (4-72), we have

[D; ] H (F) =

[D*]

X

dF dR dF dR d'Y dR

(j~:} ((j~:}) T [D*] + 2d1~*l [D*r

1

[{a} ((j~:}) T] [D*]

*

+ ( dF )T [D*] dF + (dF + 2( dF )T d[D'l [D*]-l {a}) d

~

dn

~

dn

dY

(4-85)

dD = H(F)

154

4 Isotropic Elasto-Plastic Damage Mechanics

_ d ( dF )T [D*] {dE} dY d{a} X

dF dRd F+ (d F ) T [D*] dF dR d'Y dR d{a} d{a}

+ ( dF+ 2(dF) T d[D*l[D*r 1{0"} ) dn

d{a}

dn

d dY (4-86)

dJ' = H (F)

_ dF ( dF )T [D*] {dE} dR~

A modified Lemaitre's model in t erms of the combined potential function Eq.(4-79) and the yield function Eq.(4-4) can be given as (4-88)

dG d{O"}

3{s} 20"eq (1 - D)

(4-89)

Then, in the strain space, we obtain

[D; ] = H (F)

x

( 3(1- D))2 [D] [{ s}{s}T ] [D]- 3(1- D) [{O"}{ s}T] [D] dcP 20" eq 20" eq dY

(- 3) (l - D){s} T [D]{s} + ( (l - D)O"s- -3 {s }T {O"} ) d

dR + -=;O J'

A

20"

~

20"

0

(4-90a)

[S;p] = H (F) (- 3 ) ~

X

dR + -=;~

2

(1 - D) [ {s}{s} T] [D]- - 3 [D]~

1 [ {O"}{s} T]

dcP [D] ~ ~

(- 3) (l - D){s}T [D]{s} + ( (l - D)O"s- -3{s} T {O"} ) -"Yd

2

A

2~

0

(4-90b)

dD = H(F)

4.4 Damage Plastic Criteria for Numerical Analysis

X

dR + -=;0 "(

(3) -

3 2dt:P T - - ( I - D) -{s} [D]{dE} 20"eq dY

2

(I - D){s} [D]{s} + T

20"eq

155

(

3 {O"} (I - D)O"s - -{s} T

A

)

20"eq

dtP ~

oY (4-90c)

dA = H (F)

X

3 T (1 - D){s} [D]{dE} 20"eq

(- 3) (I - D){s} 2

dR + -=;0 "(

20"eq

T

[D]{s} +

(

3 {O"} (I - D)O"s- -{s} T

A

)

20"eq

dtP ~

oY (4-90d)

In the stress space, we obtain

3 ) ( :z;;:eq [B* ] = H (F) ~

2 [

{s}{s}

T]

- 20"

dR d"(

3

eq

- 1 [

(1 - D) [D]

{O"}{s}

T]

dt:P dY

A

dtP + (1 - D)O"sdY (4-91a)

3 dt:P T - (1 - D) - {s} {dO"} dD = H (F) 20"eq dY dR dt:P d"( + (1 - D)O"sdY 3 T - 20"eq (1 - D) {s} {dO"}

d"(

= H (F)

dR d"(

dt:P + (1 - D) O"s dY

(4-91b)

(4-91c)

As can be noted , both stiffness and the flexibility matrix are non-symmetric.

4.4 Damage Plastic Criteria for Numerical Analysis 4.4.1 Damage-Plastic Potential Functions Comparing Eq. (4-69) to Eq. (4-55b) , it can be found that contributions of dG damage growth are employed by terms with dY in equations from Eqs. (4-67) dG . to (4-76). If dY = 0, from Eq.(4-71) it has D = O. As can be expected, in

156

4 Isotropic Elasto-Plastic Damage Mechanics

this case all t he kinetic damage formulations reduce to static damage similar to the formulations presented in subsection 4.3.2. On the other hand, from the experimental results [4-11] redrawn in Fig.42 and the damage threshold condition given by Eq.(4-26) , the evolution of damage basically appears as a linear varying with the plastic strain. The phenomenon of damage growth happens only due to the accumulative hardening parameter r reaching a damage threshold value, rd' (which may be equivalent to r~= cJ . This means that if the accumulative hardening parameter is not higher than the threshold value, r d' the damage state will be at the initial damage value no without damage growth (i.e. if r~ = cp < Cd' then = 0, material will keep a static damage). Thus, formulations in t his case follow the theory as shown in subsection 4.3.2.

n

n,-__________________

n 0.3

£2, =0.23

Alloy

AU4GI

0.3

0.2

0.2

0.1

0.1

0

8 d =0.02

0.1 0.2 (a) 8, (10-')

8 R=0.25

£2, =0.37

8 d=0.5 I

0

•

O~~~~--~--~--~-U

8 d =0.02

0.1 (b)

0.3

steel E24

0.3

~

Alloy INCO 718

8,

0.2 (10-')

8 R=0.29

n,-__________________-.

n 0.6

£2,=0.24

steel30CD4

•

I.

0.25

0.50

0.75

8 R=0.88

(c) 8, (10-')

n....-__________-. 0.3

0.6

0.2

0.3

0.1

OL-~~~~~--~~~~

0.25

0.50 0.75 8.=1.0 (e) 8, (10-')

steel XC 38

O~--L---L--~-~~

0.15

0.30

(f) 8, (10-')

0.45

Fig. 4-2 The measured ductile da mage of 6 typical metals

In subsection 4.2.2, Lemaitre's formulation for damage growth evolution was presented along with his choice for

4.4 Damage Plastic Criteria for Numerical Analysis

157

investigation, since as stated before that Lemaitre's model in the one dimensional case seems to yield comparable results with experimental values (shown in Fig.4-2), it was decided to adopt a similar but a slightly different form for &, such that it will satisfy the formulation presented in Eqs.(4-24) to (4-42). In the general case, the damage-plastic potential function G can be assumed to have the form as

( _Y )So+l +F({CJ},D, R)

S G({CJ} , D,y,R)=ooH('Yf;-Ed)-a- -S So

+1

a

(4-92)

where a can be defined as a material constant, which characterizes the stability of damage growth in a material and can be called a Sensitive Coefficient of Damage Growth. If 00=0 then damage cannot grow in this material at all. If a has a small value, the damage growth in this material is stable and, if a has a high value then the damage growth in this material is unstable. As expected, this model can be applied to a very wide range of damage problems. a should have a small value corresponding to stable damage growth for kinetic damage problems, and a should have a very high value corresponding to unstable damage growth. Substituting Eq.(3-96) into Eq.(4-92), for model A, we have

dG _ dP _ -ooH 2 dY - dY hfc

_

E

- -ooH 2 hfc

_

E d)

A

J

dG _ dP _ -ooH 2 dY - dY hfc A

( (

22 )

CJeqfc 2ESa(1- D)2 geq (1 _ D)2

_

E d)

)

So

geq (1 _ D)3

(

So

)

(4-93a)

So

(4-93b)

where the factor geq is defined by 2 f2 CJeq c geq = 2ESa

(4-94)

4.4.2 Damage-Plastic Yield Function

According to classical plasticity theory, the yield criterion determines the stress level at which plastic deformation begins. The damage plastic yield criterion can also be defined in a similar manner such that the yield condition determines the effective (net) stress level at which plastic deformation begins. This means that it is only necessary to replace the Cauchy stresses in the standard yield function by the effective stresses. The damage plastic yield function can be rewritten in a general form as

158

4 Isotropic Elasto-Plastic Damage Mechanics

F({O"} , Q,R) = F({O"*},R) = 0

(4-95a)

j({O"}, Q) = j({O"* }) = R('y)

(4-95b)

or

where j is a function to be used to determine the effective stress level at which the plastic deformation begins. R('y) is the hardening function associated with the accumulative hardening parameter ,. Commonly, the hardening rule can be considered as the power rule (4-96) The damage yield function can conveniently be expressed in the form of stress invariants as j(I~, J~ ,

J3) = R('y)

(4-97)

where (4-98)

(4-99)

(4-100)

For numerical computations, it is convenient to rewrite the yield function in terms of alternative stress invariants. The present formulation is modified based on Nayak and Zienkiewicz [4-29] since its main advantage is that it permits the computer coding of the yield function and the flow rule in a general form and necessitates only the specification of three constants for any individual criterion, as presented in [4-29]. The effective principal stress vector can be given by summation of the effective deviatoric principal stress vector and the effective mean hydrostatic stress vector [4-30] as

O"l} = __ 2(J* )~ { Sin(~* - 2;)} J* {I} sm 8* + -.!.. 1 {0"2 0"3 J3 sin (8* + 7) 3 1 2_

where 0"1' > 0"2 > 0"3 and to the Lode angle.

-'IT

/6

~

8*

~

'IT

(4-101)

/6 . The term 8* is essentially similar

4.4 Damage Plastic Criteria for Numerical Analysis

159

4.4.3 Different Modeling of Damage Yield Criteria 4.4.3.1 Modification of Tresca Yield Criterion The Tresca yield criterion of isotropic damaged material in terms of effective stresses is

F =

~ (J;) ~

[sin

(8*+ 2;) - sin (8*+ ~)] - R ("() = 0

(4-102)

or expanding, we have 1

F = 2(J;)2 cos

8* - R("() = 0

(4-103)

Substituting Eqs.(4-99) and (4-100) into Eq.(4-103), the modified Tresca yield criterion in t erms of invariants of deviatoric Cauchy stress is represented as

[1

F = 2J221 cos "3 arcsin

( 3V3h)] - (l - Sl)R("() = O -

2J2~

(4-104)

4.4.3.2 Modification of von Mises Yield Criterion The von Mises yield criterion of isotropic damaged material in terms of effective stress is 1

F = (3J;)2 - R("() = 0

(4-105)

Substituting Eq.( 4-99) into Eq.( 4-105), the modified von Mises yield criterion in t erms of the invariant of deviatoric Cauchy stress is represented as (4-106) 4.4.3.3 Modification of Mohr-Coulomb Yield Criterion The Mohr-Coulomb yield criterion of isotropic damaged material is a generalization of Coulomb friction failure law by introducing the effective shearing stress T * and the effective normal stress O"~ on the friction failure surface as T*

= c-

O"~

t an 'P

(4-107)

where c is the cohesion and 'P is the angle of internal friction. Eq.(4-107) can be rewritten in terms of effective principal stress as F

= (O"r - 0"i3 ) - 2ccos 'P + (O"~ + 0"i3 ) sin 'P = 0

Substituting Eq.(4-101) into Eq.(4-108) , we have

(4-108)

160

4 Isotropic Elasto-Plastic Damage Mechanics

. cp + (J*) F = :31 J*1 sm 2 12

COS B* -

(

1 sm . B*· J3 sm cp )

-

C cos

cp = 0

(4-109)

Substituting Eqs.(4-99) and (4-100) into Eq.(4-103), the modified Tresca yield criterion in terms of invariants of deviatoric Cauchy stress is represented as F =

~J~ sin cp+(h)~

( COSB* -

~ sinB* sincp)

-c(1-D) coscp = 0 (4-110)

where . ( - ----y3J3h) B* =:31 arcsm 2J22

(4-111)

The cohesion c can be equivalently expressed by the hardening rule R( ')') [4-30] as

Rh) c= - -

(4-112)

cos cp

and when ,),=0, it gives RI"'I=o = R o, and cl"'l=o = Co = Ro/coscp, we can obtain from Eq.(4-96) C

k 1 = Co + __ ')'m

(4-113)

cos cp

4.4.3.4 Modification of Drucker-Prager Yield Criterion The influence of a hydrostatic stress component on yielding was introduced by inclusion of an additional term in von Mises expression to give F

=

(3oJ~

1

+ (J~p

- Rh) = 0

(4-114)

This yield surface has the form of a circular cone. In order to make the Drucker-Prager criterion with the inner or outer apices of the Mohr-Coulomb hexagon at any section, it can be shown that (30 =

Rh) =

2 sincp

(4-115)

6ccos cp

(4-116)

J3(3 ± sin cp) J3(3 ± cos cp)

where "+"for inner apex, "- " for outer apex. Substituting Eqs.(4-115) and (4-116) into Eq.(4-114), it gives F =

2 sin cp

J3 (3 ± sin cp)

J~ + (J~) ~ _

6c cos cp

J3 (3 ± cos cp)

= 0

(4-117)

4.4 Damage Plastic Criteria for Numerical Analysis

161

Substituting Eqs.(4-98) and (4-99) into Eq.(4-117), the modified DruckerPrager criterion3 in terms of invariants of the Cauchy stress deviator is represented as F =

2 sin 'P h v3 (3 ± sin 'P)

+ (J2)~

-

6ccos'P (1 - D) = 0 v3 (3 ± cos 'P)

(4-118)

where the cohesion C can also be equivalently expressed by the hardening rule R(r) [4-30] as C= v3(3 ± sin 'P)R(r) 6 cos 'P

(4-119)

and whewy=O, it gives R I')'=o = Ro , and 4'1=0 = Co = v3(3±sin 'P)Ro/(6cos'P) , we can obtain C = Co

+ v3(3 ±

sin 'P) k .1. "(m 6 cos 'P

(4-120)

4.4.4 Expression for Numerical Computation 4.4.4.1 Basic Expressions for Three-Dimensional Problems For the purposes of numerical computation, it is required to express the above formulation in the following form For the purposes of numerical computation, it is required to express the above formulation in the following form

dF {b} = d{a}

(4-121)

for model A,

{d* } = [D*]

d~~}

= (1 - D) [D] {b} = (1 - D) {d}

(4-122a)

for model B,

{d* } = (1 - D)2{d}

(4-122b)

dF B = dD

(4-123)

A

A=

dFdRdF dR d"( dR

(4-124)

where (4-125a)

162

4 Isotropic Elasto-Plastic Damage Mechanics

(4-125b)

{d}T = {dl,d2 ,d3, d4,d5,d6}

(4-125c)

The vector {b} can be written as 1

{b} =

ClF Clh Clh Cl{o-}

+

ClF ClJ22 ") ~ Cl{o-} oJ2

ClF ClB*

+ ClB* Cl{o-}

(4-126)

Differentiating Eq.(4-111) with respect to {o-}, it gives

ClB* _ _ J3 Cl{o-} 2 cos 3B*

[J.... Clh

_ 3h

J2~ Cl{o-}

]',1

ClJ2~ 1

Cl{o-}

(4-127)

Substituting Eq.(4-1 27) into Eq.(4-1 26) and using Eq.(4-111) , we can then rewrite the vector in the form of (4-128) where

dI1 T {al} = Cl{o-} = {l , l,l,O,O, O}

(4-129a)

(4-129b)

Clh

{a3} = Cl{o-}

= { (SyS Z - o-;z +

~2) , (S zSx -

o-;x

+ ~2),

(S XSy _ o-;y

+ ~2),

= 2 (o- zx o-xy - Syo-y z ) , 2 (o-xy o-yz - SXO-ZX), 2 (o-y zo-zx _ Szo-xy) } T (4-129c) and ClF

C1 C _ ClF 2ClJ22

= Clh

tan3B* ClF

---r - --l-ClB* J 22

(4-130a) (4-130b)

4.4 Damage Plastic Criteria for Numerical Analysis

G3 = -

V3 2 cos 38*

1 dF 1. d8* J 22

163

(4-130c)

--

Only the constants G l , G2 and G3 are then necessary to define the yield surface. Thus, we can achieve simplicity in programming as only these three constants have to be varied from one yield surface to another. The constants G l , G2 and G3 are given in Table 4-1 for four yield criteria, and if necessary other yield functions can also be expressed in similar forms. Table 4-1 Constants defining the yield surface for numerical analysis Yield Criterion C1 C2 C3 V3sin6l* 2 cos 6l* (1 +tan 6l* tan 36l*) Tresca o h cos36l* von Mises o o V3 V3 sin 6l* + cos 6l* sin

The parameter A in Eq.(4-124) illustrates the influence of the hardening law of the elasto-plastic damaged material on the elasto-plastic matrix [D;p]. Considering the power hardening law given in Eq.(4-96) and substituting different yield functions into Eq.( 4-124), the value for A can be determined as follows (4-131a) where .1.. H ' = - 1 k "(=

(4-131b)

m "(

dF

The derivatives of dD for the four yield criteria mentioned above can be easily obtained from Eqs.(4-104) , (4-106), (4-110) and (4-118) respectively: (1) For the Tresca yield criterion and von Mises yield criterion

dF

B = dD = R("() = Ro A

1

+ k"(-;;;

(4-132)

(2) For the Mohr-Coulomb yield criterion A

B

=

Co

cos
+ k"(-;;; 1

(4-133)

164

4 Isotropic Elasto-Plastic Damage Mechanics

(3) For the Drucker-Prager yield criterion

B'

=

6ccost.p y3(3 ± cos t.p )

+ k "(=l

(4-134)

4.4.4.2 Basic Expressions for Two-Dimensional Problems For two-dimensional problems, the general expressions derived so far can be reduced by deleting the stress (and strain) components, which vanish under the conditions of plane stress, plane strain or axial symmetry. Note that components corresponding to the coordinate independent directions have been included for the plane stress and strain cases. These terms will be excluded for element stiffness formulation and only the first 3x3 portion indicated will be employed. By eliminating the appropriate stress terms the developed expression can be readily modified. The vector {b} becomes (4-135) with x, y and z being replaced by r, z and e respectively for the case of uniaxial symmetry. The specific form of the vector {b} is still given by Eq.(4-128) but in this case it gives by form (4-136a) (4-136b)

{a3} = { (SyS Z +

~2) ,(S zSx + ~2) ,-2s z(Jxy, (S XSY _ (J;y + ~2) }TS

(4-136c) and the deviatoric stress invariants become, from Eqs.( 4-99) and (4-100) (4-137) (4-138) The vector {d} employed in Eq.(4-122) can be expressed for plane strain and axial symmetry as

,

Ml = Ev (b 1 + b2 + b4 ) (1 - v)(1 - 2v)

(4-139)

4.4 Damage Plastic Criteria for Numerical Analysis

165

For plane stress, we have

(4-140)

4.4.4.3 Formulations for Numerical Computation

dF

Using the expressions for {b} = d{a} ' {d} = [D]{b},

dF dD' H

B

A

dFdRdF dR d"( dR and Eq. (4-93), the formulations required can be expressed for both models A and B as follows. Corresponding to the group of Eqs.( 4-84)rv( 4-87), we have

[D;L = H(F) [{d}{d}T]

x

+ 2aHbf~ -Cd ) [{a}{d}T] ( (1 - D)

g eq

H' + {b} T{d} _aHbf~ -cd ) (B _ 2{b}T{a}) ( (1 - D)

(1 - D)

(1 - D)

)8 0

(1 _ D)2

8

)

g eq

0

(1 _ D)2 (4-141a)

[D;lB = H (F)

+ 4aH("(f~ -Cd) [{a}{d}T] C1~e~)3 ro H' + {b}T {d} _ aH ("(f~ - Cd) (B _ 2{b}T {a}) ( )8 (1 _ D)2 [{d}{d}T]

x--------------------~~----=_~~----~--

(1 - D)

(1 - D)

g eq

0

(1 - D)3

(4-141b)

166

4 Isotropic Elasto-Plastic Damage Mechanics

(4-142a)

dD A = H(F)

dD B = H (F) aH(rf; -c d) (I - D) (

g eq

3) 8 {d}T{dc} 0

(1 - D)

x ------------------~----~~~----~~

)(13 _ 2 {b}T {O-})(

H' + {b} T {d} _a H(rf; -c d (1 - D)

(1 - D)

g eq

(1 _ D)3

)8 0

(4-143b)

d

,A= H (F) {df {dc}

x----------------~~~--------------~

H'

+ {b}T {d}

_ aH (rf; - Cd ) (1 - D)

(13 _ 2{b}T {O-}) (1 - D)

(

g eq

(1 _ D)2

)

0

8

(4-144a)

d

,B = H (F)

4.4 Damage Plastic Criteria for Numerical Analysis

167

(1 - D){d}T{dc }

x --------------~--~~~~--~~----~

H' + {b}T {d} _a Hbi; -cd )(iJ _ 2{b}T{O-})( g eq (1 - D) (1 - D) (1 _ D)3

)8 0

(4-144b)

(4-145a)

(4-145b)

(4-146a)

aH dDB

= H (F)

bi; - Cd)

(1 - D)

) 8

g eq

(

(1 - D)3 2

H' _ aH Cdc (1 - D)

cJ iJ (

0

{b} T { do-} 80

g eq

)

g eq

)

(1 _ D) 3

(4-146b)

{b} T {dcr} d

rA

= H (F)

(1 - D)

H' _ aH (2 ric - Cd ) iJ (1 - D)

(

(1 _ D)2

(4-147a) 8

0

168

4 Isotropic Elasto-Plastic Damage Mechanics

{b}T {da} d

rE

= H (F)

(1 - D)

H' _

ooH (2 r i c -Ed ) 13 (

(1 - D)

geq

)

(4-147b) SO

(1 _ D) 3

4.5 Shakedown Upper Bound Model of Elasto-Plastic Damage 4.5.1 Simplified Damage Constitutive Model Let a three-dimensional (3-D) elasto-plastic body, occupying the volume V surrounded by the surface S, be subjected to m-parameter varying loads 6P1 , 6P2 , ... , ~mPm' in which Pa. and ~a. (a = 1,2,···, m) are the oo-th basic load and its load factor, respectively. The load factors change with time in the given load domain R in the m-dimensional space. The strain rate tensor iij' is decomposed into a purely elastic part iij and a purely plastic part ifj'

{tij} = {i~j }

+ {if)

(4-148)

The elastic strain rates are defined by (4-149) in which A ijkl , denotes the fourth-order tensor of elastic properties with respect to rate. The simplified yield condition of material can be written as

F(aij) (; 0

(4-150)

Then the plastic strain rates can be obtained from the yield surface and its associated flow rule by (4-151) in which ;\ is the plastic multiplier. Furthermore, the material is assumed to obey Hill's principle of maximum plastic work (4-152) where, {(Tij } is an arbitrary stress state satisfying the yield condition Eq. (4150). Similarly to the general continuum damage model (CDM) in References [4-19, 4-21], the evolution of damage can be assumed to be related to the

4.5 Shakedown Upper Bound Model of Elasto-Plastic Damage

169

plastic strains. For example, Lemaitre and Chaboche's experimental results showed that the damage varied linearly with the effective plastic strain [4-31 , 4-19]. In a more general form, the damage evolution law can be assumed to be written in the following form (4-153) where c s (= (J s / E) denoting the elastic strain corresponding to the yield stress (J s; the dimensionless two-order tensor components {Cij } are material constants (or functions of plastic strains). In this section, the influences of damage on elastic moduli are not considered. The material behaves like an elastic perfectly-plastic one before the damage [l reaches its critical value [lc (0.2 < [lc < 0.8 for metals [4-19]) , and ruptures suddenly once [l = [lc . So a simplified CDM for elasto-plastic materials is adopted here. Take the case of uniaxial tension as an illustration. Fig.4-3(a) shows the experimental stress-strain curve of low carbon steel.

B

A

C

G H I>

(a)

I>

(b)

Fig. 4-3 The stress-strain curves under uniaxial tension, (a) the experimental curve; (b) two simplified curves, OABC and OABGH

The elastic perfectly-plastic stress-strain curve in the classical shakedown theory is shown as OABC in Fig.4-3(b). The simplified curve in the continuum damage model is shown here as OABGH, in which the damage at point B is equal to [lc . 4.5.2 Upper Bound on Damage of Structures

Consider a point Xo in the 3-D elastic-plastic body, and define the local averaged damage at time T and this point by (4-154)

170

4 Isotropic Elasto-Plastic Damage Mechanics

in which Vo is a prescribed volume element around fictitious stress field

Xo.

Choose the following

for {x} EVo

(4-155)

for {x} E V - Vo with {G ij } being a two-order tensor of material constants written in a vector form, and Geq =( {Gij V {Gij }) 1/2 is an equivalent value of this. Choose a timeindependent residual stress field {gij } satisfying the following inequality for any point on the structure (4-156) From Eq.(4-152), we have (4-157) So from Eqs.(4-154), (4-155) and (4-157), the following inequality holds

(4-158)

The actual stresses in the structure can be decomposed into (4-159) in which {crij } denote the stresses corresponding to the applied loads in a completely elastic structure, {cr ij } denote the actual residual stresses. Then Eq.(4-158) can be rewritten as (4-160) where

f f ({crrj } - {gij }) T {ifj }dVdt, W = f f {crfj }T {ifj }dVdt T

WI

=

T

2

ov

The residual strain rate tensor {irj ing two parts

0 }

v

(4-161)

is assumed to be composed of the follow-

(4-162)

4.5 Shakedown Upper Bound Model of Elasto-Plastic Damage

171

Then, the first integration in Eq.(4-161) is recast as

T

T

f f({a rj } - {gij}) T{irj}dVdt - f f({a rj} - {gij}) T [A ijkl ] {6"kd dVdt

o

V

0 V

(4-163) Here {aij} - {gij } is a self-equilibrated stress field and is a kinem atically compatible strain field. Hence, by using the virtual work principle, the first integral in Eq.(4-163) is equal to zero [4-32]. Then, we have

WI

=

J o

f ({arj } - {gij}f [Aijkl ]:t({aL} - {gkl})dVdt

V

(4-164)

Based on the static shakedown theorem [4-33 , 4-32]' a positive factor m > 1 and a self-equilibrated residual stress field exist for a structure at shakedown such that the following inequality

F (m{a fj } + {aU}) ~ 0

(4-165)

holds for any time t and throughout V. Then from Eq.(4-152),

({ aij } - m{ afj } - {arn

f

{ifj } ~ 0

(4-166)

By substituting Eq.(4-166) into Eq.(4-161), we obtain

W2

~ m~1

Jf o V

({arj } - {arn) T Ufj}dVdt

(4-167)

Using the virtual work principle and taking account of Eq.(4-162), the following inequality is arrived at

W2

~

2 (m1_ 1) f {arn T [Aijkd {akl}dV

(4-168)

V

Thus, from Eqs.(4-160), (4-164) and (4-167) , the upper bound on the damage at point {xo} is obtained as

4 Isotropic Elasto-Plastic Damage Mechanics

172

(4-169) in which

[/* ({gij },{arn,m, {x o })

~;:Vo [[ {g,, }

2a

T

=

[A'Jkl] {gkl }dV + no

~

1[

1

{a;j} T [A'Jkl ] {amdV

(4-170) In order to judge whether the structure is safe, the following mathematical programming problem should be solved (4-171) subject to (4-172) (4-173) Hence, the condition of the safety of the structure at shakedown can be written as (4-174) If the above condition is satisfied, i. e. the damage throughout the elastoplastic structure at shakedown is lower than the critical damage value, then the structure is not only adaptive but also safe. Because Eq. (4-174) is based on the continuum damage model and the classical shakedown theory, it is more reasonable than the previous bounding techniques in references [4-34"-'36], especially for elasto-plastic structures with a high concentration of stresses.

4.6 Gradual Analysis of Double Scale Elasto-Plastic Damage Mechanics 4.6.1 Gradual Constitutive Relation Coupled with Double Scale Damage The elasto-plastic damage constitutive relationship to the exponential hardening rule is taken into account when carrying out the gradual analysis of the double scale damage model under monotonous loadingbringing forward and

4.6 Gradual Analysis of Double Scale Elasto-Plastic Damage Mechanics

173

verifying the scheme of two sub-areas, from which the solution of the gradual field with respect to some typical parameters is obtained. That provides a form of progress area from which we derive the theoretical formulation of cracks developing rate. If the stress state and the strain state are rewritten in the form of deviatoric {Sij}, {eij} and spherical components CT m , Cm as

{CTij} = ({CTij} - {6 ij }CTm ) + {6ij }CTm = {Sij} {Cij } = ({ Cij } -

+ {6 ij }CTm {6 ij }c m ) + {6 ij }c m = {e ij } + {6 ij }cm

(4-175)

From Eq.( 4-175) , the elastic strain energy per unit volume can be expressed as

f {CTij}T d{cij } M

W =

=

o

M

M

M

o

0

0

f {Sij + 6ij CTm }T d{ eij + 6ijCm }- f {S ij }T d{ eij } + 3 f CTmdcm

(4-176)

In the case of the double scale isotropic damage model, the constitutive relations coupled with double scale damage are presented (see section 3.7) by (4-177) Substituting Eq.(4-177) into Eq.(4-176), we have M

W =

f {eij }T d{e ij } + (1 -

M

DK )9K

o

f cmdcm

(4-178)

0

Introducing the equivalent strain Ceq = (2{eij }T {eij }/3)1/2, as well as (4-179) substitut ing it into Eq. (4-178) , the integration results in

W = 3(1 - DJL)Gc;q/2 + 9(1 - DK)K c'?n/2

(4-180)

aw

thus, from Y = - p aD ' Eqs.(4-176) and (4-180)we obtain

aw

2

aw

2

Yc

= - aDc = 3Gceq / 2

YK

= - aD K = 9Kcm/ 2

(4-181) (4-182)

174

4 Isotropic Elasto-Plastic Damage Mechanics

Similarly, introducing the equivalent stress (J' eq = [3{Sij}T {Sij } / 2]1/2 into Eqs.(4-177) and Ceq = [2{eij }T {eij }/3]1/2, we obtain

(J'eq = 3G(1 - ilp,)c eq

(4-183)

as well as from Eq. (4-177), we have

(J'm = 3K(1 - ilK )cm

(4-184)

Therefore, Eqs.(4-181) and (4-182) can be rewritten as Y. p, -

YK =

1 (J'2 6G(1 - ilp,)2 eq 1

2K(1 - ilK)

2(J'

2

m

(4-185) (4-186)

4.6.2 Damage Evolution Criterion Based on Double Scale of Damage In order to model the damage evolution rate, we need not only to set up the formulation of the damage driving force (damage strain energy release rate) Y, but also to establish the damage evolution criterion, which includes the initial damage evolution and subsequent damage growth. The double scale isotropic damage model may be a better scheme for studying these problems. Firstly, let us describe the initial damage evolution rule, using Yp,o and Y K0 to denote the initial values of Yp, and Y K respectively. The criterion of initial damage evolution can be expressed as (4-187) where Yp,o , Y Ko are components of damage driving forces when initial damage grows; ko is the material constant independent to Yp,o and Y Ko . 4>(Yp" Y K) is named the damage evolution function, and the initial damage evolution rule given by Eq.(4-187) can be plotted as a curve in the Yp, rv Y K coordinate systemwhich can be described as the initial damage evolution curve Co as shown in Fig.4-4. The region outside the curved triangle OA 1 A 2 circumfused by the initial damage evolution curve and two coordinates Yp, and Y K is the feasible region of damage growth. When the point (Yp" Y K ) is located in the feasible region, material starts to be damaged. The material constant ko can be considered as the threshold value of initial damage evolution, and can be determined through the simplest experiment , such as the purely shear test of YK = 0, then from Eq.(4-187) we may obtain (4-188)

4.6 Gradual Analysis of Double Scale Elasto-Plastic Damage Mechanics

175

Feasible region of damage growth

A, Initial damage evolution eurve Yktho

Y

o

Y, Y).llho

Fig. 4-4 Initial damage evolution curve

where, YJLtho is the threshold value of YJL in the initial damage evolution. In order to further study the subsequent damage evolution problems, we need to carry out the loading rule. Thus the situation shown in Fig.4-4 should be described first , i. e., when the end point of the damage strain energy release rate vector (damage drive force vector) {Y} = {YJL , YK } is restrained on the initial damage evolution curve Co, the increment {d Y} = { dYJL , dYK } should contain the following three conditions: CD {d Y} is towards the inside of the feasible region of damage growth; ® {d Y} is towards the outside of the feasible region of damage growth; ® {d Y} is located at the boundary of the feasible region of damage growth. The above three conditions are called loading, unloading and neutral variation of loading respectively; the unit vector n is along the outward normal at the point Co on the initial damage evolution curve towards the feasible region. Consequently, the above three conditions can be represented as

CD

p(YJL , YK) = k ,

{n}T{dY} >0

®

p(YJL , YK2 ) = k,

{n}T{dY}
®

p(YJL , YK) = k ,

{n} T {dY} = 0

(4-189)

In the case of neutral loading variation, we have dp

dP

dP

= dYJL dYJL + dYK dYK = 0

(4-190)

Employing the gradient vector VP for P , Eq.(4-190) becomes (4-191) Comparison of Eq.(4-189) to the third equation in Eq.(4-191) presents

176

4 Isotropic Elasto-Plastic Damage Mechanics

'UP

= ~n

(4-192)

and regarding Eq.(4-192) ,Eq.(4-189) can be rewritten as follows

CD

p(YJL , YK )

= k, {V'P}T{dY} > 0

®

P(YJL,YK)

= k,

(V'Pf{dY} < 0

®

p(YJL,YK )

= k,

(V'Pf{dY} = 0

(4-193)

Consequently, the subsequent damage evolution criterion should be taken into account. The mathematical form of this criterion can be assumed as (4-194) where the damage evolution function p(YJL, YK) = k is the same as that of the initial damage evolution,the threshold value k is the material function related to the damage dissipation work down Pn but independent of the ratio of YJL and YK . n

Pn

=

fo (YJLdDJL + YKdDK )

(4-195)

The subsequent damage evolution criterion expressed by Eq.(4-194) can also be plotted as a curve or group of curves in the coordinate system of YJL rv YK , which is named the subsequent damage evolution curve group C shown in Fig.4-5

B, Group curves of subsequent damage growth

A,

Co

C

Y",tb

Fig. 4-5 Curves of subsequent damage evolution

The material function k(Pn) can be considered as the threshold value of the subsequent damage evolution, which can be determined by the simplest experiment, such as a purely shear test. Therefore, from Eq.(4-194) we have

4.6 Gradual Analysis of Double Scale Elasto-Plastic Damage Mechanics

k (Pn)

= P (YILth, 0) = H (YILth)

177

(4-196)

In the condition of a purely shear test, it may further provide (see section 4.6.3) (4-197) Substituting Eq.(4-197) into Eq.(4-196) gives k(Pn)

= H [8(Pn) ]

(4-198)

4.6.3 Damage Evolution Equation- Time Type

For the multi-scale isotropic damage model, the damage evolution rate can also be considered as a vector form of

dDK}T

{ dD} = {dDIL dt dt ' dt

(4-199)

in which the direction of the damage evolution rate vector should be determined first. Based on the second thermodynamics law, we have (4-200) where {YIL O' YKo } is an initial vector independent of time. The end of the vector is located on the boundary or outside the damage evolution region as shown in Fig.4-6. Based on Eq.(4-200), using the reduction to absurdity we may conclude:

{Y-Yo}

\--~.------Y-Yo

YM

Fig. 4-6 Direction of damage evolution rate vector

178

4 Isotropic Elasto-Plastic Damage Mechanics

CD Both curves of initial and subsequent damage evolution are protruding curves; ® The damage evolution rate vector corresponding to the point (YIL , YK ) on the damage evolution curve has the same direction to the outward normal at that point. The above two conclusions are illustrated in Fig.4-6. According to the conclusion ® and Eq.(4-192), we have (4-201 )

(4-202) and that can be used to determine the value of the damage evolution rate. Assuming

_ h d<1> 11 dt

(4-203)

then Eq.(4-202) can be rewritten as

= h{ d<1> d<1> d<1> d<1>}T

{ dillL dilK}T dt ' dt

(4-204)

dYIL dt ' dYK dt

Based on the subsequent damage evolution criterion Eq.(4-194), we have

d<1> = dk (<1>n) d<1>n = k' (<1> ) d<1>n = k' (<1> ) dt

d<1>n

dt

n

dt

n

(Y

IL

dillL dt

+Y

dilK) K dt (4-205)

Substituting Eq.(4-205) into Eq.(4-204) gives

{ dillL dilK}T dt ' dt

= hk' (<1> ) n

(Y

IL

dillL dt

+

Y dilK) {d<1> K dt

d<1>}T dYIL ' dYK

(4-206)

With both sides of the above equation multiplied by {YIL , YK}, we have

Y IL

dillL dt

+

Y dilK K dt

= hk' (<1> ) n

(Y

IL

d<1> dYIL

+

Y d<1» K dYK

(Y

IL

dillL dt

+

Y dilK)

K dt (4-207)

and considering (4-208) Eq. (4-207) becomes

4.6 Gradual Analysis of Double Scale Elasto-Plastic Damage Mechanics

, (dtP hk (tPn) YJL dYJL

dtP ) + YK dY K =

1

179

(4-209)

Thus, the function to be determined is obtained as follows 1

h =, (dtP k (tPn) YJL dYJL

dtP) + Yk dYk

(4-210)

Substituting Eq.(4-21O) back into Eq.(4-204), the component expressions of the damage evolution rate can be determined as

dtP dtP T dtP {dYJL ' dYK } ill

{ dSlJL dSlK}T dt ' dt

,

( dtP k (tPn) dYJL YJL

dtP) + dY K YK

(4-211)

Moreover, when tP is the nth order homogeneous function of YJL and YK, it gives

dtP dYJL YJL

dtP

+ dYK Y K = ntP

(4-212)

Then Eq.(4-211) can be simplified as

dtP

{ dSlJL dSlK} T dt ' dt

dtP T dtP

{~,aY;} ill ntPk' (tPn)

(4-213)

Usually, the damage evolution function can be expressed by damage drive forces YJL and YK as the following homogeneous function

tP (Y. Y ) = _P_ (ym- l 1"

K

m

+1

I'

+ ,.,;yKm- 1)

(4-214)

where, p, m and,.,; are material constants to be determined by the following described experimental method as (4-215)

dtP _ dtP dYJL dtP dYK _ (ym dYJL ym dYK ) dt - dYJL dt + dYK dt - P I' dt +,.,; K dt

(4-216)

The function k(tPn) represents the mechanical behavior of materials and is not related to the ratio of YJL and YK ,w hich can be determined by the purely shear test. In this case it is

180

4 Isotropic Elasto-Plastic Damage Mechanics

nIL

nK

fo Yllthdnll = f YKthdnK

Pn =

(4-217)

0

where Yilt h is the threshold value of the subsequent damage evolution in the case of the purely shear test, and can be determined by Eq.(4-185) as

1)=

CT;qth (

Y ilth

= 6G

1 - nil

2

Y ilth o

(1) 1 - nil

2

(4-218)

where CTeqth is the threshold value of the material equivalent stress in the case of the purely shear test;whereas, Yllth O' is the threshold value of Yll during the initial damage evolution in the case of the purely shear t est. It is evident that (4-219) Substituting Eq.(4-218) into Eq.(4-217) gives

Pn =

Yilt h o

(1 _1nil - 1)

(4-220)

Expunction of (I - nil) from Eqs.(4-218) and (4-220), one gives

Yilth

=

Yilth o [

1+

(~~J 2]

(4-221 )

From Eqs.(4-196) and (4-214), we have

k (p ) = _P_ym+l n m + 1 Ilth

(4-222)

and substituting Eq.(4-221) into Eq.(4-222), the function k(Pn) has the form as follows: (4-223) Since the function k(Pn) represents the mechanical behavior of materials, Eq.(4-223) can be used in the general case of a non-purely shear t est. Thus n IL

Pn =

nK

fo Ylldnll + f YKdnK

(4-224)

0

YIlO and YK 0 are used to present the coordinate of the point on the initial damage evolution curve, which should satisfy Eq.(4-187) as

(4-225)

4.6 Gradual Analysis of Double Scale Elasto-Plastic Damage Mechanics

181

Regarding Eq.(4-214), the above equation becomes

_P_ (ym+ l m + 1 1"0

+ ,.,;ym+l) = k Ko

0

(4-226)

in which (4-227) where O"eqOand O"mO are the equivalent stress and the mean stress respectively during initial damage evolution in the case of the non-purely shear test, i.e. the threshold value. From Eq. (4-226) it can be seen that the quantity of 0" eqO and 0" mO relates to their ratio. Substituting Eq.(4-224) into Eq.(4-223) one can get the result that k(iJ>n) is a function of damage variables.

4.6.4 Basic Equations and Boundary Conditions for Solving Problems (1) Geometrical equations of deformation Under the condition of small deformation, this type equation is similar to that of classical solid mechanics as

{G } = tJ

~ (dUi + dUj) dXj

2

dXi

(4-228)

where {ud and {Gij } are the displacement tensor and the strain tensor written in the form of vectors. (2) Constitutive relationship Different from classical solid mechanics, the constitutive relationship in damage mechanics should involve the damage coupled effects. In the case of the double scale isotropic damage model, the stress tensor {O"ij } can be expressed according to Eqs.( 4-175) to (4-180) as

{O"ij } = 2G(1 - ill") ( {Gij } - {6ij }{ 6kz} T {6kz} /3 + K(l - il K ){ 6ij}{ 6kz} T {Gkz} (4-229) where ill" and ilK are damage states of the shear elastic modulus G and the bulk elastic modulus K respectively. (3) Equilibrium equations Under the condition of small deformation , this type equation is similar to that of classical solid mechanics as

d{O"ij } d{ Xj }

+ {B} = 0

where {Bd is the body force vector. (4) Displacement boundary condition

,

(4-230)

182

4 Isotropic Elasto-Plastic Damage Mechanics

{ud lsu = {1!i}(on Su)

(4-231)

where 1!i is the component of {ud given on the boundary SUo (5) Static force boundary condition (4-232) where T i is the component of the surface force vector given on the boundary ST ; {lj} is the direction cosines vector of the outward normal on the boundary ST.

It should be pointed out that since the damage field is employed in damage mechanics such as the damage field of rlJ.L and rl K , we need to set up an additional equation of damage evolution. (6) Damage evolution equation Taking the case of the minimum stress being less than the threshold value it goves following: Substituting Eqs(4-214) , (4-215) and (4-216) into Eq.(4-213) , the damage evolution equation can be modeled as

{ drlJ.L drlK}T dt ' dt

{pYJ.Lm , pr.;Yl(}T p (YJ.LmyJ.L

+ r.;Yl(YK)

n<1>k' (<1> n)

(4-233)

or in the form of components

drlJ.L dt drlK dt

p2YJ.Lm (YJ.LmyJ.L

+ r.;Yl(YK)

(4-234)

n<1>k' (<1> n)

p2r.;Yl( (YJ.LmyJ.L

+ r.;Yl(YK)

(4-235)

n<1>k' (<1>n)

where k'(<1>n) can be det ermined from Eq.(4-223) as

k' (<1> ) = dk (<1> n) = 2pym- 1 [1 n dfF.. J.Ltho '¥n Regarding Eq.(4-208), tPn = YJ.LsiJ.L <1>n =

YJ.LthO

+ ( y.<1> n

)

J.Lth o

2]

m <1>

n

(4-236)

+ YKsi K and

(1 _1rlJ.L - 1) , k (<1>n) =

m:

1 YJ.L":,":l

(4-237)

Eq.(4-236) can be rewritten as (4-238a)

4.7 Analysis of Coupled Isotropic Damage and Fracture Mecha nics I

k (
=

m

2pYMt h o

[1 - 2flM + 2fl~] m ( (1 _ flM)2

flM ) 1 - flM

183

(4-238b)

Substituting Eq.(4-238) and
P

m +1

(m + 1) 2n

dflK

(m + 1)

dt

2n

(1 - flM) 2m+l Y;' (yMm YM+ "'YK'YK ) m- 1 + "'YlF- 1 ) flM(l - 2flM + 2fl~) m YM~ho (yM

"'YK'

(yMmYM+ "'YK'YK ) m- 1 + "'YlF- 1 ) flM(l - 2flM + 2fl~)m YM~ho (yM (1 - flM) 2m +l

(4-239)

(4-240)

Solving these first order non-linear ordinary differential equations Eqs. (4239) and (4-240) we can obtain the results of double scale isotropic damage evolution rates either theoretically or numerically. In Eqs.(4-239) and (4-240) YM, YK and YMth and YMth o are determined from Eqs.(4-185) and (4-186) by 1 Y. = (J"2 YK M 6G (1 _ flM)2 eq,

=

1

2K (1 _

fl K

)

2(J"

2

m

YMth and YMtho can be det ermined by

YMth =

1 2 [ -----c; 1 +( _ fl 6 1

) 2]

(J" eqth

- 1 Mth

2

an

(J"eqth d Y. MthO = 6G

(4-241)

The above studies belong to the theoretical category of macro-damage mechanics, which has been widely applied in practical engineering problems.

4.7 Analysis of Coupled Isotropic Damage and Fracture Mechanics 4.7.1 Gradual Analysis for Developing Crack under Monotonous Loading 4.7.1.1 Summarize of Damage and Fracture Combination Analysis

The earliest attempt to combine strange defects and distributional defects (i.e. a way to combine fracture mechanics and continuum damage mechanics) was presented by J anson and Hult [4-37], in which they first discussed the simplest

184

4 Isotropic Elasto-Plastic Damage Mechanics

problem-the influence of damage on the tension strength to illustrate the phenomenon where the actual fracture stress is always less than the ideal fracture stress when damage happens within materials. Kachanov [4-21] and Alfaiate et al. [4-38] suggested that the invariant J *integral (in fracture mechanics) can be used to describe the damage dissipative criterion of fracture in the problems of crack growth. Bui and Ehrlacher [4-39] also obtained a quasi-static determinate F. E. solution for the type-I crack in an elastic damaged medium. The results presented showed that the form of the front edge in the damage zone for the type-I crack is more blunt than that for the type-III crack. Based on the above results, Bui and Ehrlacher further discussed the problem of the invariable J-integral, and pointed out that the damage energy release rate Y should be discriminated from the fracture energy release rate G. For an elasto-plastic damaged medium, when C R ---+ 00, h ---+ 0 thus G ---+ O. (that means a falsehood G = 0 presented by Rice [4-40] for a perfect plastic medium can also be obtained). Janson and Hult [4-37] considered that there exists a narrow plastic damage zone along the extended line of the crack in Dugdale's crack model; and assumed that in the narrow plastic damage zone the effective tension stress 0" *, which is perpendicular to the direction of crack (i. e.x direction), keeps a constant value (yield stress of material 0" s). They used the definitions of logarithmic damage presented to show that the effective stress within the narrow plastic damage zone has power-strangeness with a damage parameter [2 as 0" *

= O"ye -S!

Ref. [4-39] studied the rule of a dynamical crack developing equation for the type-III crack model in an elastic and elasto-plastic damaged medium, and obtained an analytical solution. For the elastic damaged medium, using conformal mapping they obtained a solution as shown in Fig.4-7 and Fig.48, where Fig.4-7(a) illustrates the form of the damage zone, and Fig.4-7(b) illustrates the components of shear stress corresponding to the point of the boundary of the damage zone. The undamaged zone in Fig.4-8(a) is mapped from the inside area bounded by the curve shown in Fig.4-7(b). The variation of the path-integral J with the width of the contour for a different model is presented in Fig.4-8(b); the different zones of solution around the crack-tip are illustrated in Fig.4-8(a). An explicit solution of the problem for the quasistatic damaged and plastic zone near the crack in an elasto-plastic medium under anti-plane shear is shown in Fig.4-9. Compared to the static crack field controlled by the traditional stress density factor and the conversation integration,the developing elasto-plastic crack field under monotonous loading has many different characteristics. The reasons leading to these differences are: (1) there is a mechanism of damage evolution in the developing elasto-plastic crack field; (2) The gradual field of developing elasto-plastic cracks may exist in a steady settled condition or unsteady condition. Although some researches studied the elasto-plastic crack developing under monotonous loading [4-37'"'-'4-39], they did not employ any

4.7 Analysis of Coupled Isotropic Damage and Fracture Mechanics

X,

185

O"lI

b

c 1-_ _ _ _--4.=a__O"_]_I_13 c' 0""

b'

(1

V" . -.! -(3' ) '

Fig. 4-7 Steady-state moving damaged zone with the velocity V, the damage front is BB' . (a) Illustration of damaged zone around a crack; (b) Complex mapping to t-plane t = 0'32/ f3 + iO'31 wak"e" zone

Active plastic zone II

Integral J

~~~~~~~~~~?'~------- 1- ·

, ,, I I

<>::'

G G

F Ee D ----- ~ -----~-~-~-~----~ ,

B

"':, ,, I

..."..................................04-........-"""""'-- ______ -'- . I

I

I

I

Damage zone Z P~th r

A 2h

2R

Width

(b)

(a)

Fig. 4-8 Variation of the path-integral J with the width of the contour, for d ifferent models, elasto-plastic damage OABCD, elastic-damage OAED, crack in perfect plasticity BCD, crack in elasticity OFD, the characteristic values are C (fracture energy rate) and C ' (energy-release rate) (a) Damaged zone and mapped area; (b) Path-integral J Cusp cycloid

Damage zone Z

Curly cycloid

Fig. 4-9 The quasi-static damage and plastic zone near a crack in the elasto-plastic medium

186

4 Isotropic Elasto-Plastic Damage Mechanics

concepts of the damage, and divided the gradual field into too complex a su b-area (the maximum divisions are five). The elasto-plastic damage constitutive relationship to the exponential hardening rule is taken into account when carrying out the gradual analysis of the developing crack field with the type-I crack under monotonous loading, bringing forward and verifying the scheme of two sub-areas, from which the solution of the gradual field with respect to some typical parameters is obtained. That provides a form of progress area from which we derive the theoretical formulation of the crack developing rate.

4.7.1.2 Damage Evolution Model of Gradual Field It is pointed out in [4-41 ] that under monotonous loading the area not far from the crack tip can still be considered mainly to be in a proportional loading state. Therefore, the global quantity theory of plastic mechanics is still applicable in that area. According to this, the application of Lemaitre's strain equivalent principle can carry out Ramberg-Osgood 's elasto-plastic constitutive relationship involving the damage effects (4-242) where E and v are the elastic modulus and Poisson's ratio; band n are material constants; 6 ij is the Kronecker tensor and (4-243) Sij -_ O"ij -

6ijO"kk /

3,

O"e q -_

( 30"ijO"ij / 2 )1/2

where {O" ij} and {E ij } are the stress and strain tensors, 'IjJ (0 ~ 'IjJ ~ 1) is the continuity, which describes the area ratio between the remaining effective load bearing area of the damaged material and the initial full load bearing area of the original intact material. In the present problem, assuming {O"i j } with the strangeness at the crack tip (this assumption is collaborated in later results) and considering n > 1, the principal term in Eq.(4-242) is the non-linear one, whereas Eq.(4-242) in the area near the crack tip can be rewritten as (4-244) The damage evolution equation can adopt the following form [4-42] 'IjJ. = -d'IjJ = -cyPy.

dt

'

(.Y = -dY > 0)

dt

(4-245)

where c and p are material constantsthe damage thermodynamic drive force Y is expressed herein as the same as the strain energy release rate

4.7 Analysis of Coupled Isotropic Damage and Fracture Mechanics

y

=

~{(Jij}T [Cijkl ~- 2]{(Jkd = ~{(J:j}T[Cijkl]{(Jkl}

187

(4-246)

where [Cijkd is the flexibility tensor of undamaged materials.

4.7.2 Basic Equation of Gradual Field near Developing Crack 4.7.2.1 Compatibility Equation of D e formations in Gradual Fie ld Let us consider the gradual field of a developing crack of type-I under monotonous loading as shown in Fig.4-10 (in which the area with distributional points represents the damage zone). Assuming Airy's stress function is

Y

...... , . . . ...

...

... #

Q,- princ ip al region

(r ,8)

..

. ,,

...

\ '

x

a

Fig. 4-10 Illustration of regions in damage process

A=

a,·\+2 A( B)

(4-247)

then the stresses can be represented as follows:

(Jrr = a,A(jrr(B) (Jee = a,A(jee(B) (Jre = a,A(jre(B)

(4-248)

188

4 Isotropic Elasto-Plastic Damage Mechanics

d2 A

_

i7rr = (.\ + 2)A + dB2

i7ee = (.\ + 2)(.\ + l)A

(4-249)

dA

i7 r e = - (.\ + 1) dB

Consequently, using Eqs. (4-243) , (4-248) and (4-249), we give the divatoric stress as (4-250)

Srr(B) = zli7rr(B)-Z2i7ee(B),

see(B) = zli7ee(B)-z2i7rr(B),

sre(B) = i7re(B)

(4-251 ) where Zl = 2/3, Z2 = 1/3 are applicable for the plane stress state; Zl = z2=1/2 is applicable for the plane strain state. Substituting Eqs.(4-250) and (4-251) into Eq.(4-243) gives (4-252) where

{

(i7;r

+ i7~e - i7rr i7ee + 3i7;e) 1/2

[3 (i7 rr - i7ee)2

+ 3i7;el 1/ 2

(plane stress) (plane strain)

(4-253)

In the damage field , since near the crack tip the material is fully damaged , the continuity of which is zero, whereas far from the crack tip the damage is smaller and the continuity is better, we may therefore assume (4-254) in which JL > O. Using Eqs.(4-244), (4-250) , (4-252) and (4-254) ,the strain field can be obtained as

{

err = h

(~) nrn().. -/L)srr (B) , eee = b1 (~) nrn()..-/L)see (B)

ere = b1 (~) nrn()..-/L)sre (B)

(4-255)

(4-256) Further substituting Eq.(4-255) into the compatibility equation (4-257) we obtain

4.7 Analysis of Coupled Isotropic Damage and Fracture Mecha nics

el = - 2 (e + 1)

e2 = -e

}

189

(4-259)

e3 = e (e + 1) e = n (A - f-L) Eq.( 4-258) becomes a 4th order ordinary differential equation of involving eigenvalues of A and f-L.

A and 1j;

4.7.2.2 Compatibility Condition of Damage Evolution Now it is possible to discuss the gradual form of the damage evolution equation. From Eqs.(4-246), (4-248) and (4-254) we obtain y

=

~ (~rr2(A-{L)Y(B)

Y(B) =

(4-260)

~E{O'ij f[Cijkl 1P- 2]{O'kl}

y

Fixed element

,,

,,

, ,,

,,

, ,,

,,

, ,,

,,

,

,'''-- dB

B+dB x

a

da

Fig. 4-11 Geometrical relation in crack developing

Regarding t he geometrical relation in Fig.4-11, it is dr } ~e = ~ c~s B

- = -smB da

r

(r > > da)

(4-261 )

190

4 Isotropic Elasto-Plastic Damage Mechanics

as well as considering Eq.(4-260), we have

(4-262)

When [d(a/ ;3)2]/(oJ;3)2 »dalr, the area near the crack but a little way from the crack tip can still be mainly considered to be in a proportional loading state, and taking the principal term in Eq.(4-262) gives

~ r 2(A-IL) Y :t (~ ) 2

y=

(4-263)

Substituting Eqs.(4-260) and (4-263) into Eq.(4-245) gives

d1j; dt

= __ c

EP+l

(~)2Pr2(p+l)(A-IL) YP+1i.(~)2 [i.(~)2 > 0] (3 dt (3 dt (3

(4-264)

Considering the geometrical relation in Fig.4-11 and Eq. (4-254) , we can obtain d1j; da = [d(3 -d1j; = - r IL 1j;- + (3r IL - 1 (d~ - sine - fJ1j;- cos e )] -da dt

da dt

da

de

dt

(4-265)

According to the fact that the closer the point to the crack tip, the greater is the damage and the lower is the continuity, this property needs to keep a hold on the second term in Eq.(4-265) as

-d1j; = (3r IL - 1 (d~ - sin e - fJ1j;- cos e) -da de

ili

ili

(4-266)

which presents the strange property of cJ; when r ---+0 and fJ < 1. Comparing Eq.(4-264) to Eq.(4-266) for independence of terms including e, rand (al(3 ) respectively, may only give a set of compatibility conditions of damage evolution from each term as

d~ - sin

de

2 (p

- +1 e - fJ1j;- cos e = - YP

+ 1) (>. -

fJ)

= fJ - 1

(4-267) (4-268)

4.7 Analysis of Coupled Isotropic Damage and Fracture Mechanics

da 2c d(aj;3) = EP+l (3

P+l (a)2 73

191

(4-269)

where Eq.( 4-268) presents independence between ,\ and f.J,. Therefore Eqs.( 4268) and (4-267) consist of the problem of solving the eigen-functions A and 1[J as well as the eigenvalue f.J,. Eq.(4-269) will be used to determine the crack developing rate. 4.7.3 Boundary Condition and Solution Method of Studied Problem 4.7.3.1 Boundary Condition of Studied Problem Solving Eqs.(4-258) and (4-267) belongs to the boundary problem at the two end points (see B= O and B=7r). Because of the symmetry in the type-I crack field we have

dA(o) d 3 A(O) = d1[J (O) = 0 (4-270) dB3 dB dB Since the angle distribution function A(B) of Ariy's stress function mostly represents the mode of angle distribution, thus we can assume A(O) = 1

(4-271)

Using Eqs.(4-258) and (4-267) we can derive the following

-

7/J (0) =

II

(d f.J"

2 Ao) d21[J" (0) dB2 ' dB2 =

12

(d f.J"

2 Ao) dB2

(4-272)

d2A d2A(0) where dB20 represents dB2 . To save space, the particular form in Eq.( 4272) is no longer presented in detail. On the crack surface, one needs (4-273) (4-274) Since 1[J (0) > 0, and 1[J is gradually decreasing in the region from B= O to B=7r , it is possible to assume that 1[J decreases to zero at a certain angle Bd E (O , 7r). If this case happens, it means that the region [0, 7r] will be divided into two parts: where [0, Bd ] is named the damage developing region or the damage active region,whereas, [Bd , 7r] is named the full damaged region or the damage stop station (see Fig.4-10), which satisfies 1[J ;? 0 and 1[J = 0 correspondingly. Since the fully damaged medium cannot bear any stresses, in the fully damaged region it therefore always has

192

4 Isotropic Elasto-Plastic Damage Mechanics

(4-275) Obviously, Eqs.(4-273) and (4-274) are satisfied together. Next , the situation of the damage active region will be mostly discussed . According to the definition of ed and Eq.(4-275), the interfacial condition on both sides of e = ed can be obtained as follows

1jj(ed- ) = 1jj (ed+) = 0

}

i5"ee(ed+) = i5"ee(ed+) = 0, i5"re(ed- ) = i5"re(ed+) = 0

(4-276)

Applying Eq.( 4-249) and taking a limit, the above equation becomes (4-277) Since the deformation in the full damage region is undetermined (arbitrary), it is not necessary to add the deformational compatibility condition to the interfacial condition. Anyway,the properties of the full damaged region are determined, such as Eq.( 4-275); the control equations of the damage active region are Eqs.(4-258) and (42-267). The functions to be determined are A d2 A(O) and 'IjJ; the parameters to be determined are j.L , ~ and ed ; the boundary conditions are Eqs.(4-270), (4-271), (4-272) and (4-277). 4.7.3.2 Solving Algorithm of Studied Problem

The steps of the solution algorithm are listed as follows: (1) According to an arbitrarily given

j.L (j.L

d2A > 0) and de 20 ' using Eqs.(4-270), (4-271) and (4-

272) to integrate Eqs. (4-258) and (4-267) (it is possible to use the 4th order Runge-Kutta's method with varied steps), it may give a region as e* E [0, 7r] satisfying 1jj (e* ) = 0 (4-278) thus e* can be considered as a function of

_

(2) Since A(e* ) and

j.L

d2 A

and de 2o .

d2 A(e*) de 2 are determined at the same time when solving d2A

e* , thus they can be considered as a function of j.L and de 20 and expressed as A (e* ) = A [e* dA(e* ) =

de

~

de

(j.L,

[A (e* (

d:~o )] j.L,

d2Ao))] de 2

}

(4-279)

4.8 Verify Isotropic Damage Mechanics Model by Numerical Examples

193

d2 A dA(B*) obviously, dB2o , A (B*) and dB usually are not zero for any f-L.

(3) Solving the following non-linear equations

(4-280)

d2 A We can obtain the necessary f-L and dB2o , moreover the corresponding B* actually is Bd, and at the same time the angle distribution functions of different fields are obtained.

4.8 Verify Isotropic Damage Mechanics Model by Numerical Examples In order to verify the formulations presented above, some results of the finite element model have been compared with the available solutions in this section.

4.8.1 Example of Bar Specimen The first example is a simulation of experimental measures for damage evolution presented in Chapter 3. The experimental results of damage evolution were obtained in terms of measures of the effective Young's modulus as shown in Fig.4-12(a) determined for the metals 99.9% Copper, Alloy INCO 718 and Steel 30CD4 in [4-9, 4-11]. The theoretical results are obtained using Eq.(428); the finite element results are obtained by the application of Eq.( 4-68) to Eq.( 4-72). As mentioned by Lemaiture [4-43], it is easier to measure damage variable rl through the variations of the elastic modulus. From the relationship between Eq.(3-25) and Eq.(3-26) , the damage can be evaluated by different definitions, say model A and model B (see Fig.4-12(a)). The finite element mesh for simulation is shown in Fig.4-12(b). In order to obtain more accurate results a higher order Gaussian point scheme has been employed in F. E. analysis. Considering the damage localization, we can assume that the sensitive coefficient of damage growth ex is not equal to zero in the central element only. The damage growth can be observed by processing the results at each stage of the load increment. It should be pointed out that the material constant, ex, is the sensitive coefficient of damage growth. ex can be approximately estimated through experimental results at points (rl=O, C d ) and (rlc, c R ) as shown in Fig.4-2 and Fig.4-12. rlc and c R are the rupture values of rl and c p . The initial (first approximate) value of ex can be taken as the average slope of the experimental

194

4 Isotropic Elasto-Plastic Damage Mechanics il ,= I - E"IE il .= I-v£'iE

,, ,:

E"

(a)

q

p

Local damage b--<;.....:::1r--Gaussian points

p

Unit:mm

q

(b)

Fig. 4-12 (a) Measurement of damage evolution through damaged elastic modulus; (b) Damage evolution test specimen and the F. E. mesh curve by De/(e R - eJ, and substituting the initial value of ex into analysis, the numerical damage evolution curve obtained will be compared with the experimental curve. If there is a significant difference between these two curves, then the value of ex will be adjusted using the numerical curve, thus the analysis will be repeated till the convergence is reached. Consequently, the final value of ex for any materials can be obtained using the above simple iterations model by the bar specimen. In the particular example, after several iterations, the values of ex for Copper, Aluminum alloy and Steel 30CD4 were obtained as 15.5, 8.6 and 6.0, respectively. The result plotted in Fig.4-13 to Fig.4-15 shows the comparisons of theoretical, experimental and finite element solutions of damage evolution in the specimens Copper, Alloy and Steel, respectively. The damage evolution

4.8 Verify Isotropic Damage Mechanics Model by Numerical Examples

195

curves in Fig.4-13(a) and Fig.4-14, Fig.4-15 correspond to model A (i.e. n A = 1 - E * / E); the curve in Fig.4-13(b) corresponds to model B (i.e. n B = 1 - J E* / E). As can be noted , the finite element results compare well with the theoretical and experimental results. 1.0.---------------, - - Eq.(428) .. 2 0 ····0···<> Experiment [4-19] 0.8 '" '" '" F. E. Method

cf

0.6

d

p ...

0.4 0.2

1.0 . - - - - - - - - - - - - - - - , - - Eq.(428) o 0 0 Experiment [4-19] 0.8 '" '" '" F. E. Method

0.4

0.6

0.8

0.4 0.2

Model A 0.2

0.6

ModelB

1.0

0.2

0.4

0.6

&. (x 10-' )

0.8

1.0

(b)

Fig. 4-13 Comparison of theoretical, experimental and F. E. results for damage evolution in the specimen. (a) Model A; (b) Model B • .... ·Experiment [4-19]

0.3

.............. F. E. Results Q<=0 .24

0.2

c:

.. ..

0.1

~

..

O~~~~~~

6 d=0.02

Steel Model A

5 R=0.37

0.2 6.(10 -2 )

__~__~~ .4

0.5

Fig. 4-14 Comparisons of theoretical, experimental and finite element solutions of damage evolution in the steel 0.3

• ..-. Experiment [4-19] .............., F. E. Results

Q<=0.24

0.2

---------------------~~

.

,..

I I

~...,

0.1

•

•

,

~ ... ""

I

I

,.-' Aluminum alloy: Model A I

O~~

__~~~__~--~~ 0 .2

6 d =0 .02

5

=0 .39

R

6.(10 -' )

Fig. 4-15 Comparisons of theoretical, experimental and finite element solutions of damage evolution in the aluminum alloy

196

4 Isotropic Elasto-Plastic Damage Mechanics

4.8.2 Compression of Plastic Damage Behavior Based on Different Hypothesis Figs.4-1 6 to 4-19 show the variation of the displacement , equivalent plastic strain, damage strain energy release rate and the accumulative hardening parameter during damage growth in a specimen, in which it was assumed t hat the damage is initiated at the beginning of the plastic strain (i.e. Cd = 0). It should be noted that the results have been presented for unit loading. 25 0--0--0 ~

20

....../>, ....

Model A ModelB

A

!'

0

x

::1""

15

!' .Il·

sl~ 10

s:::S

5 0.2

0.4

Q

0.6

0.8

1.0

Fig. 4-16 Variation of the displacement per unit load during the damage growth 25 0--0--0

'P

0

x

20

I "'~I""

15

~

10

,6.•••• ,6..•• .6,

.A

Model A ModelB

0

.' .'

.4

5

o~--~--~--~--~--~

o

0.2

0.4

0.6

Q

0.8

1.0

Fig. 4-17 Variation of the displacement per unit load during the damage growth Variation of t he plastic strain per unit load during the damage growth

From Fig.4-16, it can be seen that, as expected, the displacement per unit load for model B is higher than that for model A at the same damage level. When damage grows to a very high level, the displacement distribution per unit load becomes unstable. From Fig.4-17, it can be found that in the stable region of damage growth, the plastic strain for model B is higher than t hat for model A at the same damage level. Fig.4-18 illustrates the variation of the damage strain energy release rate per unit load during the damage growth. As expected , the results of Y/q have the same values for both models A and B at points n = 0 and n = 1

4.8 Verify Isotropic Damage Mechanics Model by Numerical Examples

197

1.5.---------------.

0.9

~

::...1<:>- 0.6

0---0---<> M ode I A ............. ModelB

0.3

OL---~~~~~~~~

o

0.2

0.4

n

0.6

0.8

1.0

Fig. 4-18 Variation of the damage strain energy release rate per unit load during damage growth 10,--------------------,

~ x

,....1<:>-

8

0---0---<> M ode I A .......... .... ModelB

o

6

81~4

8:::E

2

0.2

0.4

n

0.6

0.8

1.0

Fig. 4-19 Variation of accumulative strain hardening parameter per unit load during damage growth

respectively, while with 0 < [? < 1 the value for model A is higher than that for model B. Fig.4-19 presents the accumulative strain hardening parameter per unit load during the damage growth, and it can be seen that the value for model A is higher than that for model B at the same damage level. The difference between these two values is significant when damage grows. In order to verify the conclusion of the relationship between damage models A and B discussed in subsection 4.2.2, a comparison of theoretical and numerical results for the relationship of Eq.(4-43) and difference /).[?AB were presented in Fig.4-l. The solid curves illustrate the theoretical results calculated by Eqs.(4-43a) and (4-43b). Moreover, the circles and triangles in Fig.4-1 indicat e the numerical results evaluated by formulations presented in sections 4.3.4 and 4.4.4 using finite element analysis when substituting different constitutive models A and B into Eqs.(4-141)rv(4-145) , for the example shown in Fig.4-2 and Fig.4-12.

198

4 Isotropic Elasto-Plastic Damage Mechanics

4.9 Numerical Application for Damaged Thick Walled Cylinder 4.9.1 Plastic Damage Analysis for Damaged Thick Walled Cylinder For the application of the elasto-plastic damage (finite element) model, the well known problem of a thick walled cylinder subjected to internal pressure has been analyzed, varying the values of the damage variable (D = 0 to D = 0.5). The problem considered is shown in Fig.4-20 with the relevant input data due to a symmetry. Isotropic damage and plane strain conditions using von Mises yield criterion were assumed for this case. Elemental models 4><4 Gausian schema Elastic modulus £=2100 MPa Poisson's ratio v=0.3 Yield stress 0", =24 MPa von Mises criterion

200mm

Fig. 4-20 Illustration of finite element mesh of thick walled cylinder subjected to internal pressure

Fig.4-21 shows the pressure- displacement curves for three values of the damage variables. As can be noted, when D = O.O the damage finite element solution (undamaged) compares well with the analytical solution. As expected, 20r-----------------------------~

IS

-

o

Analytical Solution F. E. Solution

O~----~----~----~----~----~~

o

5 10 IS 20 25 Displacement ofliner surface(1 0-5mm)

Fig. 4-21 Displacement of inner surface with increasing pressure

4.9 Numerical Application for Damaged Thick Walled Cylinder

199

when the cylinder is damaged under the same pressure, the displacement of the inner surface increases considerably. Fig.4-22 shows the distribution of circumferential stress 0"0 along the radius of the cylinder for three different pressures. It can be seen that for the undamaged situation (D = O) both numerical and analytical solutions compare well.

o

2.0

-

Analytical Solution F. E. Solution

Q=O.O, 0.1,0.3,0.5

0.5 0.0 100

(a):P=8 MPa

120 140 160 180 Distance from centre (mm) 0

2.0

";'

-

200

Analytical Solution F.E.Solution

n =0.0,0.1 ,0.3,0.5

1.5

p.,

~ 1.0 8' 0.5 0.0 100

2.0

(b) :P=12 MPa 140 120 180 160 Distance from centre (mm)

200

n =0.0,0.1,0.2,0.3,0.5

0.5

-

o Analytical Solution F. E. Solution

(c):P=14 MPa O.O'----...I...-----'------'----L------'

100

120 140 160 180 Distance from centre (mm)

200

Fig. 4-22 Distribution of (Te. (a) P =8 MPa; (b) P = 12 MPa; (c) P = 14 MPa

200

4 Isotropic Elasto-Plastic Damage Mechanics

When the pressure is less, the plasticity develops significantly only for

[2

= 0.5 , as shown in Fig.4-22(a) ,whereas when the pressure is increased ,

the plasticity develops for all values of [2 observably. Fig.4-23 illustrates the distribution of the equivalent plastic strain along the radius of the thick walled cylinder for the damage values [2 = 0.0, 0.1,0.3 in the case of the internal pressure P = 12 MPa and P = 14 MPa, respectively. 2.0r------------------, 1.5

I",'

-0.5 -1.0 100

(a) :P=12 MPa

120 140 160 180 Distance from centre (mm)

200

5 4 ." 0

x I",'

3

.0=0 .0,0.1,0 .3,0.5

2

0 -I

100

(b) :P=14 MPa

120 140 160 180 Distance from centre (mm)

200

Fig. 4-23 Distribution of equivalent plastic strain Ep. (a) P = 12 MPa; (b) P = 14 MPa

The effect of damage growth on the distribution of the circumferential stress is illustrated in Fig.4-24. As can be seen from the figure, the results of finite element analysis for the undamaged case compare well with the analytical results. Comparing the distribution for the various cases of damage, which are actually identified in the table shown as an inset, it can be noted that the peak value of stress is shifted away from the center due to the damaged elements. It is interesting to note that case E , which is the completely damaged cylinder, shows a smaller peak value of stress than case C. This could be due

4.9 Numerical Application for Damaged Thick Walled Cylinder

201

to the interaction of plastic yielding near t he inner face of the cylinder and t he highly damaged zone near the outer face of t he cylinder. 2.5r----------------..., o Analytical Solution EDA 8 C F. E. Solution 2.0

2 3 0.00.0 0.0 0.0 B: 0.0 0.0 0.00.1 0.5 0.0 0.0 0.1 0.2 D: 0.0 0. 1 0.2 0.3 E: 0. 1 0.2 0.3 0.4 0.0 100 120 140 160 180 Distance from centre (mm)

0' 1.0

Ele:

A:

c:

200

F ig. 4-24 Distribution of O"e during damage

4.9.2 Analysis for Loca l D a mage Behaviors It is worth pointing out that the thick walled cylinder subj ected to internal pressure is in fact encountered in nuclear installations. If t here are some local initial defects within the material of the thick walled cylinder (for examples, cracks, cavit ies, etc.) then the essential study of localization effects of these defects growth is in the unsymmetrical domain. From the point of view of damage mechanics, these defects can be considered as local damage in t he structure. The damaged zone will grow following the development of the plastic strain locally. On the other hand, the damage growth can also influence the development of the plastic strain locally. As an extension of this example, we assume that there is an initial defect at the central point in an element as shown in Fig.4-25. Using the localization method presented in [4-44'"'-'45], the local damage value can be evaluated by this cavity (for example assumed to be [2 = 0.3) by a unsymmetrical domain. In order to clearly observe the behavior of the evolution of damage and plasticity in the thick walled cylinder, the stress-strain curve, the sensitive coefficient of damage growth, ex, and the accumulative hardening parameter at the damage t hreshold Id for the material of the t hick walled cylinder are assumed to be as shown in Fig.4-26. Some results of local damage growth, damage zone and plastic zone distribution in the thick walled cylinder due to the initial defect are illustrated in Fig.4-27 and Fig.4-2S. Fig.4-27( a) and (b) show contours representing the growth of the local damage due to internal pressure under 16 MPa and 20 MPa. From these results, it can be seen that the damage zone significantly grows from t he initial

202

4 Isotropic Elasto-Plastic Damage Mechanics y

Local defects P

O: =27

<00 E=2) 000 Mh ~~ v=0 .3 : MPa ------X,

Fig. 4-25 The thick walled cylinder with a local defects

48

36

E=21000MPa v=0 .3 0', =24 MPa a =62.5 6,=0.0 O L-~~~~~~~~~~__~~~

0.3

0.6 0.9 e,(x 10" )

).2

).5

Fig. 4-26 Stress and strain relation of the loca l damaged thick walled cylinder

point along the circumferential direction due to an increase in the circumferential tensile stress (J e. Fig.4-28 presents the contours representing the distribution of the equivalent plastic strain due to local damage growth at an internal pressure of 16 MPa and 20 MPa, respectively. Since the localization of the damage, the material is now inhomogeneous and hence equivalent plastic strain contours near the damaged zone are not concentric circles as in the homogeneous case. However, the contours far from the damaged zone still follow the similar pattern as in the homogeneous case. The next section will give an application of analysis for the shakedown theory to a damaged thick walled cylinder and some discussion.

4.9 Numerical Application for Damaged Thick Walled Cylinder

203

Fig. 4-27 Contour of local damage in the case of internal pressure (a) P = 16 MPa and (b) P = 20 MPa

(a) spx 10"

Fig. 4-28 Contour of equivalent plastic strain sp of the local damaged thick walled cylinder. (a) P = 16 MPa; (b) P = 20 MPa

204

4 Isotropic Elasto-Plastic Damage Mechanics

4.9.3 Analysis for Damaged Thick Walled Cylinder Based on Shakedown Theory 4.9.3.1 Problem Illustration Some components in engineering structures, such as pipes in the chemical industry and gun barrels, can be simplified as thick-walled cylindrical tubes (see Fig.4-29). Now consider a long thick-walled cylindrical tube with internal radius a and external radius b (bja <2.22) subjected to the internal pressure P. It is assumed that the pressure changes with time quasi-statically and the material obeys Tresca's yield condition. The elastic stresses corresponding to the internal pressure in a complete tube are

(Jr(P) =

bt~:2

(Jo(P) =

bt~:2

(Jz( P) = v((Jr

(1 -::)

( + ::) 1

(4-281)

+ (Jo)

Fig. 4-29 Thick-walled cylindrical tube subjected to changing internal pressure

in which v is Poisson's ratio. The elastic limit pressure and the plastic limit pressure for the tube are (4-282) (4-283)

4.9 Numerical Application for Damaged Thick Walled Cylinder

205

respectively. Assume the internal pressure changes between 0 and Pmax , and (4-284) Under the changing internal pressure, a plastic zone appears first near the internal wall and the tube may fail by incremental collapse or alternating plasticity. Then the zone with a ~ , ~ ~a, in which ~ is a given constant between 1 and (bja), is chosen as the volume Va. Similar to the area definition of damage and the CDM in [4-11 ], the damage can be assumed to vary linearly with the effective plastic strain i~q , i. e.

i? = C i~q j E = C 8

vi{

ifj } T { ifj } j E 8

(4-285)

From the following Tresca' yield condition

10"0 - 0",.. 1~ 0"8

(4-286)

and its associated plastic flow law, we have i~

=

=0

(4-287)

= v3E 8 li~ - i~1

(4-288)

-i~,

i~

Then Eq.( 4-285) can be rewritten as

.

[2

c

From Eq.(4-155), the stress field {(Tij } in Va is (4-289) After loading the thick-walled tube with the plastic limit pressure P, and unloading the pressure from P to zero, the following residual stress field is obtained

,.. a 2 ln(,jb) ( b2 ) 0",..0 = 0"8 ln (,jb) - 0"8 b2 -a 2 1 - ,2 2 l') b - 0"8a b2ln(,jb) _ a2

0"0 - 0"8 1 + n "'0 _

(

0"~0 (P)

(

1+

b2 )

,2

(4-290)

= v( 0";0 + O"~O )

Choose the residual stress field {gij }, satisfying Eq.( 4-156) by (4-291 ) with (4-292)

206

4 Isotropic Elasto-Plastic Damage Mechanics

Similarly, choose the residual stresses in the following form (4-293) in which the coefficient Ti2 must satisfy the inequality Eq.(4-165) , i.e. (4-294) From Eqs.(4-281) , (4-284) and (4-290), the inequality Eq.(4-294) can be rewritten as (4-295) Then substituting Eqs.(4-291) and (4-293) into Eq.(4-169) and (4-170) results in

4.9.3.2 Analysis for Upper Bound of Damage For any given values of m, Til and Ti2, an upper bound of the local damage will be obtained from Eq. (4-297). For the thick-walled cylindrical tube, the mathematical programming problem of Eq. (4-171) takes the following form

Dmax = 1]lmin D* ,r/2 ,m

(4-297)

subject to (4-298)

(4-299) By solving the mathematical programming problem, the optimal upper bound of damage of the thick-walled cylindrical tube is obtained as for 0 ~ x ~ for in which

1

"2

1

"2 (4-300)

~ x ~ 1

4.9 Numerical Application for Damaged Thick Walled Cylinder

A=

C

2v'3a;a2(~2

207

- 1)

f [(a;O)2 + (a~O)2 + (a~O )2 - 2v(a;O a~O + a~o a~o + a~O a;O ) l rdr b

a

B - Pmax - Pe B _ Pp - Pe 1 Pmax ' 2 Pmax

(4-301 )

The practical example is taken using the internal radius a=25 cm, the external radius b=30 cm, Young's modulus E=21O,000 MPa, Poisson's ratio v=0.3, the yield stress as = 240 MPa, C = 0.1 and ~ = 1.02. Then the elastic limit pressure and the plastic limit pressure are 36.67 MPa and 43.76 MPa, respectively. The damage load factor curve obtained from Eq.(4-300) is given in Fig.4-30. 0.8.----------------r1

...o

t>

...e Q)

~0.4

S

'"

Cl

0.01--0.0

0.5 Load factor X

1.0

Fig. 4-30 T he damage-load factor curve for the tube subjected to internal pressure

4.9.3.3 Some Short Comments

The realistic constitutive relations of materials and the failure conditions of structures are two important problems should be considered in shakedown theory. The evolution of damage is the reason for degradation and failure of materials. Damage exerts a great influence on the mechanical properties of materials and the shakedown load domains of structures. The isotropic elasto-plastic damage constitutive and evolution models established in this section are specified based on shakedown theory for elasto-plastic damaged structures.

208

4 Isotropic Elasto-Plastic Damage Mechanics

Based on a ductile-damage model, an upper bound on the damage factor of elasto-plastic structures ensures that the damage limit is used as an important parameter in the failure criterion of elasto-plastic structures at shakedown. Once the damage for a point on the structure reaches the critical damage value, the structure will fail. Such a failure criterion for structures at shakedown has a clear physical background and can be extended to all cases of materials and loads.

4.9.4 Numerical Results of Gradual Analysis for Developing Crack under Monotonous Loading 4.9.4.1 Parameter Study Order of Strangeness The calculation in this sub-section deals with some typical parameters. Table 4-2 lists the gradual orders of different fields ,in which the gradual orders of the damage field , stress field and strain field are j.L, A and A(n - j.L), respectively, and the interfacial angle between the damage active region and the full damaged region is ed. Fig.4-31 to Fig.4-33 presents angle distributional modes of the dimensionless stress and continuity and shapes of the plastic progress region determined by Eq. (4-303) , (in which d=rp (e)). The meaning of the number of sub-figures from (1) to (6) is illustrated in Table 4-2.

State P lane Plane P lane Plane Plane P lane

Table 4-2 Gradual orders and interfacial a ngle ed (v = 0.3) p n .\ n(.\ - J.l) ed/CO) No. of J.l stress 0.5 3.0 0.5840 0.4453 - 0.4161 119.3 stress o. 9.0 0.8062 0.7368 - 0.5922 116 .5 0.3228 0.2099 - 0.3387 125.6 stress 2.0 3.0 stress l.5 9.0 0.6599 0.5918 - 0.6120 127.5 strain 0.5 3.0 0.9912 0.9898 - 0.0084 90.4 strain 0.5 9.0 0.9092 0.8789 - 0.2727 98.8

subfigures (1) (2) (3) (4) (5) (6)

4.9.4.2 Shapes of Plasticity Progress Region T he shapes of the plastic progress region can be determined through the von Mises criterion modified by the strain equivalent principle as aeqN = as

(4-302)

where as is the yield stress. Substituting Eqs.(4-252), (4-254) into Eq.(4-302) gives the plastic progress boundary as (4-303) Fig.4-33 shows different shapes of the plastic progress region corresponding to different case numbers (1)"-'(6) given in Table 4-2 and F ig.4-32.

4.9 Numerical Application for Da maged Thick Walled Cylinder

'"

1.0

~ "0

§ 0 .8 ._

...~

209

0"00

~ "50.6

:':::::04 «I'" •

§ ~ 0.2 0

o Z

0

140

'"~

'"

~ 1.0 ~ l'loO.8 "0.-

;:: 1'l1.0 ]

co .... 0 • 6 N ;::l

Cd 8

. _.J:J

t;j

·C 0.4

°C

~ 0.4 t;:002 Z .

§ ·~02 0"0 .

Z

'" ·;::0.8 0 a. ~ 0.6 IiIJ -

. _N

0'-'-="'::--'-~"""""~"""""'140

00~....".-';;-~~ (b)

(a)

(c)

Fig. 4-31 Distribution of dimensionless stress

c

.; 1.0

.:: i5

8

."

0.8

0)

N

~ 0.6

~

c:l

'C:

0.4

c:l

o

.; 0.2 ,rJ

"5 5'"

o~

o

__~__~~__~__~~~~~~~~ 20

40

60

en

140

Fig. 4-32 Distribution of normalized continuity

4.9.4.3 Continuous Condition of Regions From Eq.(4-269), it can be seen that once ex and (3 are det ermined,the rate of t he developing crack will consequently be determined. Thus we need to discuss the property of the periphery field near the crack tip in the progress region. Under the precondit ion of a small yield area,assume that t he area surrounding the plastic progress region is an elastic area. Divided sub-areas in the gradual analysis for the crack-t ip field are illustrated in Fig.4-34, which involves elastic and plastic damage progress areas, the damaged area and the undamaged elastic effective area. The stress and strain fields can be represented as follows

210

4 Isotropic Elasto-Plastic Damage Mechanics y/d

y/d

y/d

(2

4

1.5

1.8

1.0

0.6 -1.0 -0.5

0

x/d

(a)

y/d 2.0

0.5 1.0

-0 .6 -0.2 00 .2 0.6 1.0

x/d

x/d

(b)

(c)

y/d 3

y/d 1.6

2

1.2

\

0.8 \ 0.4 -0 .8

-1.0-0.50 0.5 1.0 1.5 2.0

x/d

-0.6 -0.2 00.2 0.6 1.0 1.4

x/d

(t)

(e)

Fig. 4-33 The shapes of progress region Unaffected area

Elastic progress area

Damaged area

Undamaged elastic affected area

Plastic progress area

Fig. 4-34 Illustration of divided sub-areas in the crack tip field

4.9 Numerical Application for Damaged Thick Walled Cylinder

{ CTi"j }

=

Q 1r -

l / q -" (B) { "} CTij ,Cij

=

E - 1Q 1 r -

l /q-" (B) Cij

211

(4-304)

where Q1 is a control factor. The effects of the damage zone on the far field are considered as being like that of a field with a V shape breach (see Fig.4-10). Therefore, q ;?2 [4-46]. As an essential discussion, it is only considered that the interfacial condition between the progress region and Ql-control region at the angle B=O should be (4-305) Substituting Eqs.(4-248), (4-255) , (4-303) and (4-304) into Eq.(4-305) gives

! 0:

- Q (~) nd_1/q_" b (~) ndn(.\-JL)_ Q1 d - 1/ q-" d .\ CTrrO 1 (3 CTrrO 1 (3 cMO E ceeo

d= (

~1jJo

CT

S

)

1/(.\-1')

O:CT e qO

(4-306) The subscription of 0 in the above equation denotes a value taken at B=O. Solving Eq.(4-306) we have _ R - .\ -( l / q)R

0: -

3

n(q.\+l)(Eb )q.\+l Q - q.\ 1 1

lCT s

(4-307) (4-308) (4-309)

where 1

(4-310)

4.9.4.4 Rate of Crack Developing Substituting Eqs.(4-307) and (4-309) into Eq.(4-269) gives da _ -Qq - l

dQ1 -

C

1

(4-311 )

212

4 Isotropic Elasto-Plastic Damage Mechanics

-=

c

2 R 2 (P+l) 3+2p-n(l+q) C

6

CJ s

EP+1 R 5 (Eb 1)l+q

,

R

6

= [R nq (A-P,)+l R- q ] 1

2

A -p,

(4-312)

That is the formulation of the crack developing rate. Since Q1-control field is assumed as elastic, then it can be taken as the form of (4-313) where 7] is a function of the crack length a and geometrical parameters L 1 , L 2 , ... , Lm; KJ is the stress density factor calculated according to the crack length a (of course the control region of K J may not exist). If substituting Eq.(4-313) into Eq.(4-311) , we have (4-314) In particular, if the area surrounding the progress region is the control field of K J, then Q1

= KJ,q = 2,7] = 1

(4-315)

thereby, from Eq.(4-311) or Eq.(4-314), we may obtain da

dK J = cKJ

(4-316)

Eqs.(4-311) , (4-314) and (4-316) have a remarkable referred value for theoretically determining the resistant curve of the monotonous crack developing force. Individually integrating these three equations gives (4-317)

(4-318)

(4-319) where Q1R and KJR are fracture resistant corresponding to Q1 and K J ; Q1Rth and K JRth are threshold values corresponding to Q1R and K JR ; ao and ~a are the initial crack size and increment of crack development, respectively. Eqs. (4317) and (4-318) imply that,if the field surrounded in the progress region is controlled by Q1, then the resistant curve of Q1 is not related to ao, but the KJ resistant curve transformed from its equivalent mapping is related to ao; whereas Eq.(4-319) implies that if the field surrounded in the progress region is controlled by K J , then the K J resistant curve is not related to ao.

References

213

4.9.4.5 Concluding Observations Based on the above analysis for the monotonously plastic developing crack,it can be observed as follows: (1) The stress field in the progress region has no strangeness, but the strain field has. (2) The progress region can be divided into the damage active area and the full damaged area (or damage stop area). The whole progress region can be considered as a field with a V shape breach. (3) The rule curve and resistant curve of crack developing can be carried out in terms of the damage mechanism.

References [4-1] Dragon A., Mroz Z., A continuum model for plastic-brittle behavior of rock and concrete. Int . J. Eng. Sci., 17, 121-137 (1979) . [4-2] Bazant Z. , Kim S., Plastic-fracture theory for concrete. ASCE J . Eng. Mech., 105(3) , 407-421 (1979). [4-3] Krajcinovic D., Fonseka G .U. , The continuous damage theory of brittle materials: Part 1. general theory; Part 2. uniaxial and plane response modes. ASME Trans. J . Appl. Mech., 48(4), 809-836 (1981). [4-4] Lubiner J., Over J ., Onate E., A plastic-damage model for concrete. Int. J . Solids Struct. , 25, 299-325 (1989) . [4-5] Yazdani S., Schreyer H.L., Combined plasticity a nd damage mechanics model for plain concrete. J. Eng. Mech., 116(7) , 1435-1450 (1990). [4-6] Oller S. , Onat e E., Finite element non-linear analysis of concrete structure using a plastic-damage mode\. J. Eng. Fract. Mech ., 35, 219-231 (1990). [4-7] Frantziskonis G., Desai C ., Arizona T., Elasto-plastic model with damage for strain softening geomaterials. Acta Mech., 68(3-4) , 151-170 (1987). [4-8] Lemaitre J ., Dufailly J ., Modernisation and identification of endommagement plasticity of materia\. In: 3 rd French Congress of Mechanics, Grenoble, France, pp.17-21 (1977). [4-9] Lemaitre J ., Da mage modeling for prediction of plastic or creep fatigue-failure in structures. J. Solid Mech ., 4, 1-24, in Chinese (1981) . [4-10] Lemaitre J. , Coupled elasto-plasticity and damage constitutive equations. Comput . Methods App\. Mech . Eng., 51, 31-49 (1985) . [4-11] Lemaitre J ., A continuous damage mechanics model for ductile fracture . J . Eng. Mater. Tech. , 107,83-89 (1985). [4-12] Chaboche J ., Continuum damage mechanics: A tool to describe phenomena before crack initiation. Nuc\. Eng. Des., 64(2) , 233-247 (1981) . [4-13] Chaboche J. , Continuum damage mechanics: Part I. general concepts; Part II. damage growth, crack initiation, and crack growth . J . App\. Mech., 55, 59-86 (1988) . [4-14] Simo J. , Ju J. , Strain- and stress- based continuum damage models: I. formulation; II. computational aspects. Int . J . Solids Struct., 23(7) , 821-869 (1987) . [4-15] Simo J., Ju J. , On continuum damage-elastoplasticity at finite strains: A computation framework. Comput. Mech ., 5(5), 375-400 (1989).

214

4 Isotropic Elasto-Plastic Damage Mechanics

[4-16] Rosuseer G ., Finite deformation constitutive relations including ductile damage. In: Proceedings of the IUTAM Symposium on Three-Dimensional Constitutive Relations and Ductile Fracture, Dourdan , France, pp.331-355 (1981). [4-17] Dragon A ., Chihab A., On finite damage , ductile fracture-damage evaluation . J. Mech. Mater., 4, 95-106 (1985). [4-18] Yi S.M., Zhu Z.D. , Introduction of Damage Mechanics for Cracked Rock Mass. Science Press, Beijing, in Chinese (2005). [4-19] Lemaitre J. , A Course on Damage Mechanics. Springer, Berlin Heideberg New York, (1992). [4-20] Voyiadjis G .Z., Ju J .W. , Chaboche J .L., Damage Mechanics in Engineering Materials. Elsevier, Amsterdam (1998) . [4-21] Kachanov L.M. , Introduction to Continuum Damage Mechanics. Martinus Nijhoff Publishers, Dordrecht, The Nethelands (1986) . [4-22] Hammi Y ., Bammann D.J. , Horstemeyer M .F., Modeling of anisotropic damage for ductile materials in met al forming processes. Int . J. Dam. Mech. , 13(2) , 123-146 (2004) . [4-23] Wang D .A., Pan J. , Liu S.D ., An anisotropic Gurson yield criterion for porous ductile sheet met als with planar anisotropy. Int. J . Dam. Mech., 13(1) , 7-33 (2004). [4-24] Dorgan R .J. , Voyiadjis G.Z., A mixed finite element implementation of a gradient-enhanced coupled damage: Plasticity model. Int. J . Dam. Mech ., 15(3) , 201-235 (2006). [4-25] Omerspahic E., Mattiasson K., Oriented damage in ductile sheets: constitutive modeling and numerical integration. Int. J. Dam. Mech ., 16(1) , 35-56 (2007) . [4-26] Omerspahic E ., Mattiasson K ., Orthotropic damage in high-strength steel sheets: An elasto-visco-plastic material model with mixed hardening. J. Phys. IV France, 110(1), 177-182 (2003) . [4-27] Zhang W .H. , Numerical Analysis of Continuum Damage Mechanics . Ph.D. Thesis, University of New South Wales, Australia (1992). [4-28] Zhang W.H ., Valliappan S., Continuum damage mechanics theory and application : Part I. theory; Part II. application. Int . J . Dam . Mech., 7(3) , 250-297 (1998). [4-29] Nayak G .C ., Zienkiewicz O .C ., Convenient form of stress invariants for plasticity. ASCE J . Struct. Div ., 98 , 949-953 (1972) . [4-30] Owen D., Hinton E., Finite Elements in Plasticity, Theory and Practice. Pineridge Press, Swansea, UK (1980) . [4-31] Lemaitre J ., Chaboche J. , Mechanics of Solid Materials. Cambridge University Press, UK (1990). [4-32] Carpurso M., Some upper bound principles to plastic strains in dynamic shakedown of elastic-plastic structures. J. Struct. Mech., 7(1),1-20 (1979). [4-33] Konig J .A., Shakedown of Elastic-Plastic Structures. PWN-Polish Scientific Publishers, Amsterda m (1987) . [4-34] Polizzotto C ., A unified treatment of shakedown theory and related bounding techniques. S.M. Arch ., 7(1) , 19-75 (1982). [4-35] Panzeca T ., Polizzotto C ., Rizzo S., Bounding techniques and their applications to simplified plastic a nalysis of structures. In: Smith D .L. (ed .) Mathematical Programming Method in Structural Plasticity. Springer , Wien , pp.315-348 (1990) .

References

215

[4-36] Vitiello E., Upper bounds to plastic strains in shakedown of structures subjected to cyclic loads. Meccanica, 7(3) , 205-213 (1972). [4-37] Janson J. , Hult J. , Fracture mechanics and damage mechanics, a combined approach. J . Appl. Mech ., 1, 69-84 (1977) . [4-38] Alfaiate J., Aliabadi M.H ., Guagliano M., et at., Advances in Fracture and Damage Mechanics VI. Key Eng. Met er., 348-349 (2007) . [4-39] Bui H.D ., Ehrlacher A., Propagation of damage in elastic and plastic sods. In: 6 th International Conference on Fracture, Cannes, France, pp.533-551 (1984). [4-40] Rice J.R., Continuum mechanics and thermodynamics of plasticity in relation to micro-scale deformation mechanism. In : Arogon A.S. (ed.) Constitutive Equations in Plasticity. The MIT Press, Cambridge, MA (1975). [4-41] Hhuang K.Z., Yu S.W., Elasto-Plastic Fracture Mechanics. Tsinghua University Press, Beijing, in Chinese (1995) . [4-42] Zhao J. , Study of Damage Mechanics for Fatigue Damage Failure Problems of Metal Components. Beihang University Press, Beijing, in Chinese (1995). [4-43] Lemaitre J ., How to use damage mechanics. Nucl. Eng. Des., 80, 233-245 (1984). [4-44] Lemaitre J ., Local approach of fracture. J. Eng. Fract. Mech ., 25, 523-537 (1986) . [4-45] Janson S., Stigh U., Influence of cavity shape on damage parameter. J . Appl. Mech ., 52(3), 609-614 (1985). [4-46] Wu S.F ., Zhang X., He Q .Z., A new conservative integral with arbitrary singularity and application . Int . J . Fract., 40, 221-233 (1989) .

5

Basis of Anisotropic Damage Mechanics

5.1 Introduction The initial model in damage mechanics was based on a scalar parameter to measure the damage variable under the assumption of isotropic damage within the material. For complex defects and their distribution, it is necessary to use tensorial descriptions of damage. The introduction of second and third dimensions in geometrical modeling clearly indicated the limited of the scalar model, which is strictly speaking geometrically justified only in the case of spherical voids and perfectly random micro-crack fields. Therefore, in more complex cases the damage variable must be considered as a vector or tensor function. For example, in the case of anisotropy, the damage state within the material must be expressed by a set of state variable-tensor functions which had been first suggested by Leckie and Onat [5-1], Cordebois and Sidoroff [5-2], Murakami and Ohno [5-3]. A somewhat different argument leading to similar representation of damage by the second order tensor was suggested by Murakami and Ohno [5-3], and later a mathematically more rigorous formulation was given by Betten [5-4]. The model introduced by Vakulenko and Kachanov [5-5] and later developed by Murakami [5-6] is a symmetric crack density tensor of second rank. This second rank tensorial model was successfully used to describe various damage in metals. Here, the components of the damage tensor were characterized by six independent functions (components). As the damage develops, the material becomes anisotropic and hence the damage can be characterized by a symmetrization tensor of the fourth rank [57, 5-2]' which corresponds to the tensor of elastic coefficients and contains 21 independent components. This is a more general mathematical representation of the fourth rank damage tensor, first proposed by Tamuzh and Lagsdinsh in [5-8]. They describe the elements of the damage tensor by a set of functions given on the surface of a unit sphere surrounding the considered point on the body.

W. Zhang et al., Continuum Damage Mechanics and Numerical Applications © Zhejiang University Press, Hangzhou and Springer-Verlag Berlin Heidelberg 2010

218

5 Basis of Anisotropic Damage Mechanics

Kawamoto et at. [5-9] presented a second rank damage tensor determined by the measurement of density, direction and area of cross section of cracks in a rock specimen sampled from a parent population of rock mass. Zhang et al. [5-lO rv 12] discussed the effects of symmetrizing treatment in an anisotropic damage model and presented a non-symmetrized anisotropic damage model in the analysis of anisotropic damage mechanics [5-13 rv 16].

5.2 Anisotropic Damage Tensor 5.2.1 Micro description of Damage on Geometry If an anisotropic material (for example a rock mass) consists of a number of cracks which are not sufficiently small compared to the structure, the discontinuities within the material can be regarded as damage by a continuum tensor variable. Then the behavior of anisotropic materials involving such cracks can be conveniently treated by describing the geometry of such cracks, and the damage state in the anisotropic material can be approximately evaluated using the methods given by Murakami [5-6] and Kawamoto et al. [5-9]. In the models proposed by Murakami [5-17] and Chaboche [5-18 rv 19]' a characteristic volume V which consists of a lot of lattices and grain boundary cavities is shown in Fig.5-1(a). Denoting the area of a grain boundary element occupied by the kth cavity and its corresponding unit normal by dAk and nk (k =1,2, ... ,N) respectively, the damage tensor may be computed by first considering the kth cavity element as a tetrahedron, O-ABC, as shown in Fig.5-1(b). Generally, if there are a total N cavities within an elemental volume V, the state of damaged materials may be described using a second rank symmetric tensor [5-20rv22]. Thus, the damage tensor can be expressed from the area vector based on this figure as [ill = ;v

tJ

(nk

Q9

nk)dA k

(5-1)

k=IV

where SV is the area of total boundaries of particles in the elemental volume. The second order symmetric damage tensor [il] has three real principal values ilJ with respect to the direction of ni (i=1,2,3), correspondingly. Thus the damage tensor can be represented by the principal damage values as 3

[ill =

2..= ili(ni Q9 ni)

(5-2)

i=1

The area of the triangle ~ABC in Fig.5-1 (b) can be considered as a general infinitesimal cross section area. The area vector of the triangle ~ABC can be written by its principal components AI, A 2 , A3 as

5.2 Anisotropic Damage Tensor

219

The characteristic vo lume V consists of a lot of boundaries of lattices, grain and cavities V

V

\

(a)

-DAn

B ·.-:'777~~ --

-

X,

il.OB'C =(I- D ,)"'-OBC

X,

"'-OCA' =(I- D ,)"'-OCA il.OA'B' =( I- D ,)il.OAB (b)

Fig. 5-1 (a) The damage state in anisptropic materials; (b) The reduction of net area due to damage state on a oblique plane

220

5 Basis of Anisotropic Damage Mechanics (5-3)

Since

(5-4) If the reduced areas of AI, A 2 , A3 due to the principal damaged state are indicated by DiAi (i =1,2,3) then the area vector of the triangle ~ABC is 3

A*n*

=

3

LAini - L DiAini i=l

(5-5a)

i=l

3

A*n*

=

An - L

Di(ni Q9

ni)An

(5-5b)

i=l

From Eq.(5-5b) and Eq.(5-2), the damage tensor [D] can be strictly defined by the area vector as

[D]

=

(An - A*n*)(An)-l

(5-6)

The second rank symmetric tensor [D] expressed diagonally by three real principal damage values, Di along their corresponding principal directions ni fully represents the damaged state of characteristic elemental volume V. This damage tensor described by these crystal cavities can be quantitatively evaluated using the method developed by Kawamoto et al. in [5-9] to observe the discontinuity within an elemental rock volume. If v and V denote the volumes of the net rock element and full rock mass, and both are replaced by the corresponding equivalent cubic with the same original volumes respectively, the total effective areas can be defined as SV = 3V 2 / 3

(

~)

1/2

=

3~

(5-7)

where SV is the total area of the grain boundaries in elemental volume V and l = v l / 3 (see Fig.5-2). The quantified method of an anisotropic rock mass damage tensor presented by Kawamoto et al. [5-9] actually was developed from the basis of the approximate model of micro-damage state in anisotropic materials given by Murakami in [5-6, 5-17]. These works are great contributions to the development of rock damage mechanics. The detail of the major method is as follows. Assuming that there exist N cracks within a rock mass in which the kth crack has area ak with unit normal nk, the damage variable for the kth discontinuity can be written as, (5-8a) and hence its damage tensor is given by

5.2 Anisotropic Damage Tensor

221

v L

L

L

/

/

/

V

v

~ I

1/

V /

~

1/

V

Fig. 5-2 The net volume and effective areas with a damaged elemental volume of rock mass

(5-8b) where"0" indicates the production operation between two tensors. By summing up for all N cracks, the final damage tensor for the rock mass can be obtained as 3

[Sl] = Sv

1

L ak(nk 0 nk) = V L ak(nk 0 nk) N

k=1

N

(5-9)

k=1

Consequently, the damage tensor will be considered in two typical cases. 5.2.2 Damage Tensor Associated with One Group of Cracks

Referring to Fig.5-3(a) and observing the orientation of each crack group on every surface of the elemental cube, if the cracks with length L 1 , L 2 , L3 are oriented in the directions 81 , 82, 83 on each coordinate surface (PI , P 2 , P 3 ) , the unit normal of the set is then specified as

where

II = {cos 28i + sin28isin28j) '/2

(5-11a)

Each oriented angle has the relation (5-11b) Since the unit normal of individual crack groups can be determined by Eq.(5-10), given the number of cracks and length of cracks, the damage tensor

222

5 Basis of Anisotropic Damage Mechanics IX' , 3

X'2

--=--=~ ,

--- ,

,,

, "X'

(b)

I

(a)

8 =0. [ilIJO.OO .O ] LO .OO.312 (a) (b)

8 =30·

[il l =[0.0780.135] 0.1350.234 (a) (b)

HI!

I I

%%%% ~~~~

%~%%

%B% 1111

8 =45· [ilIJo.1560 .156] LO.1560.156

8 =60· (a)

[ill l o.2340.135J 1 0. 1350.078

(b)

8 =90· [ill =[0.3120,OJ 0.00 .0 (c)

Fig. 5-3 (a) Observation of cracks on the three surfaces of cubic rock element ; (b) The distri bution form of cracks on the cu bic element after being transformed into the local principal anisotropic coordinate; (c) Anisotropic damage tensor of cracked specimens(a) regular cracks; (b)zigzag cracks

corresponding to this crack group will be determined in a form of second rank real symmetry.

5.2 Anisotropic Damage Tensor

223

Let three surfaces of the elemental cube involve N l , N 2 , N3 cracks on each plane (PI, P 2 , P 3 ), and assume there is a typical plane in the cube, which involves the least cracks. Rotating this cube to make the unit normal n of this crack group coincide with the outward normal of the typical plane as a new coordinate axis X~, the plane with normal X~ will involve the least cracks as shown in Fig.5-3(b). This procedure is actually equivalent to finding out the transformation of the eigen-direction of the damage tensor associated with this crack group, i.e. the coordinate system to be transformed to the principal coordinate. Three surfaces of the new cube after being transformed will be X~, X~, X: which involve N~, N~, N: cracks respectively, where X: should involve the least cracks. The crack number on the plane along can be estimated as

X:

N12 =

N~ (V;~3)

(5-12a)

where N~ is the number of cracks on plane X~, L~ is the average length of cracks on plane Similarly, the number of cracks on the plane along axis can be obtained as

X;.

X;

N2l =

N~ (V;~3)

(5-12b)

The average number of cracks within the cubic volume can be estimated by (5-12c) Expressing the area of a single crack by may give I

Ni

•

(1_n;)12

N =

/

a=

(i,j

=

L~ L~ with the unit normal n

1,2)

(5-12d)

Since L~ L~ = L l L 2 , the average number of cracks involved in the cubic volume can be rewritten as

N

~ {L'LjJ(1~:;)(l V'h

n;) } '/0

(5-12e)

Substituting Eqs.(5-12c) and (5-12d) into Eq.(5-9),the anisotropic damage tensor associated with this single group of cracks can be evaluated by

224

5 Basis of Anisotropic Damage Mechanics

(no summation for i and j) (5-13) where Ni > N j > Nk for general cases. By introducing an average area of cracks and an average number of cracks within a given volume (of rock mass) V (as shown in Fig.5-3(b)), the anisotropic damage tensor can be estimated by simplified formulation as

l[ill = vNa(n 1?9 n)

(5-14)

where fir is the average number of cracks in the total volume; a = L1L2 is the average area of cracks; l is the space of cracks; Li is the average length of cracks on the ni surface; Ni is the average number of cracks on the plane with the outward normal ni'

5.2.3 Damage Tensor Associated with Multi-Groups of Cracks When a rock cubic involves many crack groups, it is firstly necessary to determine the unit normal vector and associated damage tensor for each crack group individually, then the final total damage tensor can be combined from each individual one. If the orientation angles of cracks e1 , e2 and (h satisfied Eq.(5-11b),they will consist of a group of cracks. For N groups of cracks corresponding to damage tensor [ili] (i = 1, ... , N), in which the ith tensor is associated with the normal ni, the total damaged tensor can be summated as N

(5-15)

[ill = L[ili] i=l

If the damage tensor is a transversal isotropy, that means cracks parallel to an unit normal vector m={ m1, m2, m3}T, then the damage tensor is expressed as 1 - mi -m1 m 2 -m1m3] l 1/2 [ -m2 m 1 1- m22 -m2mi [ill = 2V 2 /a {NiNjLiL j }

-m3m1 -m3m2 1 -

(5-16a)

m3

If cracks are perfectly randomly distributed, three principal damage variables may have the same value and the global damage tensor becomes a scalar such that

(5-16b) For example, using the above formulations, the anisotropic damage tensor of cracked specimens presented in [5-9] can be evaluated for different cracked orientations as Fig.5-3(c).

5.3 Principal Anisotropic Damage Model

225

Since the damage tensor [57] is real and symmetric, it has three real eigenvalues and corresponding orthogonal principal axes. However, if the above damage tensor is not given in the principal damage coordinate system, then it must be transformed to the orthogonal principal coordinate system by solving eigenvalues and eigenvectors of the damage tensor. On the other hand, the anisotropic damage state can also be directly defined in the principal anisotropic coordinate system, in terms of three principal damage variables as shown in Fig.5-4.

Fig. 5-4 Illustration of three anisotropic principal damage variables

5.3 Principal Anisotropic Damage Model 5.3.1 Three Dimensional Space Consider a cubic specimen (for example rock mass) cut along the principal anisotropic coordinate system with a set of parallel discontinuities as shown in Fig.5-5. Ai and Ai (i =1,2), indicating the undamaged and damaged areas with unit normal ni and ni (i =1,2) respectively. The state of damage in this coordinate system can be idealized as an orthotropic damage state. Let (Xl X2 X3) be the principal anisotropic damage coordinate system (XYZ) and be the Cartesian coordinate system (see Fig.55). The principal damage variables are defined by 57i = (Ai - An/Ai, thus the anisotropic damage state can be defined in a simplified way by a vector consisting of the principal damage variables without the loss of any properties either by way of damage or anisotropy. The principal damage vector {57} can be determined by solving eigenvalues and eigenvectors from the damage tensor [57] defined in Eqs.(5-9), (5-12), (5-13), (5-15) and (5-16). It should be noted that sometimes the eigenvalues of the damage tensor may give a value higher than 1. This unexpected value is obviously due to the approximate nature of the treatment defined by Eqs.(5-7) and (5-9). Therefore, it is better to obtain the general form of the anisotropic damage tensor from the principal

226

5 Basis of Anisotropic Damage Mechanics y

Section A , -A,: X,

A , Area of damaged

materials with uni t normal n, A'" Effective area of A, Section A -A : A" Area of damaged material s with unit normal n , A'" Effective area of A , Thus: A',=( l- Q )A , Q ,=(A ,- A'.>IA ,

(i= I ,2,3)

Fig. 5-5 Two dimensional principal anisotropic damage model

anisotropic damage vector (one dimensional tensor) by applying a coordinate transformation. The equations of [2i = (Ai- A:)/Ai may also be written as A i = (1 - [2i )A i . At this point, it is appropriate to introduce different basic assumptions or hypotheses of damage mechanics. Based on isotropy, it is possible to introduce the hypothesis of the strain equivalent , meaning the strain response is governed by damage only through the actual stress[5-20]. In a simplistic form, the hypothesis assumes that the strain before and after occurrence of damage is equivalent (Fig.5-6). This hypothesis is reasonably valid when dealing with isotropic damage modelling. However, in the case of anisotropy, since the effect of the orientation of cracks is significant in the definition of the damage state, the strains are not always equivalent in the undamaged state. Hence, this simple hypothesis is not always valid (Fig.5-6).

Strains are not equivalent

Q =O

·· "

.. "

Strains are I equivalent , I ,

,

I

Fig. 5-6 The strain equivalent hypothesis of damage mechanics

, I I

5.3 Principal Anisotropic Damage Model

227

In order to develop the basic concepts for an anisotropic damaged state, an elemental volume coinciding with the orthotropic direction, as shown in Fig.57, is considered. Assuming that the internal forces acting on any damaged section are the same as the one before damage, the following relationships are obtained;

,X, \ , n,

Fig. 5-7 Illustration of orthotropic damaged element and stresses acting on effective cross section areas

(5-17) where (Jij and (J7j are the components of the Cauchy stress and effective (net) stress tensor, respectively. Oij is the Kronecker tensor. Eq.(5-17) can be rewritten as (J11

(J12

(J1 3

----- ----- -----

I - [h 1 - O2 1 - 0 3 (J21

(J22

(J2 3

----- ----- -----

I - 0 1 1 - O2 1 - 0 3 (J31

(J32

(5-18)

(J33

----- ----- -----

I - 0 1 1 - O2 1 - 0 3

Obviously, the effective stress tenser (J7j is a non-symmetric tensor in the anisotropic damage state. However, since (J ij is a symmetric tensor, the following relation must be true: *

(J ij

1 - 0i *

= 1_ 0

J

(Jj i

(5-19)

The above relation may be called the compatibility relation of the effective shear stress components and hence, considering the non-symmetric nature of

228

5 Basis of Anisotropic Damage Mechanics

the effective stress tensor, Eq.(5-18) may be rewritten in the form of a vector using the tensor continuity factors [w] as

{o-*} = [w]{o-}

(5-20)

where the stress vectors {o-*} and {o-} in the anisotropic coordinate system (Xl X2 X3) are defined as

{ CJ. . . *} = {* * CJ 33' * * CJ23' CJ32' * CJ 31' * * CJ 13' CJ 12' * CJ 21 *}T CJ 11 , CJ 22'

{o-}

(5-21a) (5-21b)

= {CJu, CJ22' CJ33' CJ23' CJ31' CJ12}T

Thus, the transformation matrix [w] in Eq.(5-20) should be defined by a 9 x 6 order matrix as

[w]

=

1 1- [21

0

0

0

0

0

0

1 1- [22

0

0

0

0

0

0

1 1- [23

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1 1- [23 1 1- [22

1 1- [21 1 1- [23

(5-22)

0 0 1 1- [22 1 1- [21

Eq.(5-20) can be considered as the relation that transforms the Cauchy stress vector to the net stress vector in a 3D anisotropic damage model. Some articles call the matrix W] the "damage effective tensor" or "damage effective functions" [5-20",,22]. The inverse of the relation Eq.(5-20) that presents an opposite transform of the net stress vector to the Cauchy stress vector in the 3D anisotropic damage model by the formulation of (5-23) where the matrix of [w- 1 ] is called the matrix of generalized inversion for [w] and defined as a 6x9 rank matrix as

5.3 Principal Anisotropic Damage Model

[w- l ]

229

=

1- fh 0 0 1- [22 0 0 1- [23 0 0

0 0 0 0 0 0 1- [23 1- [22 -----2 2

0

0

0

0

0

0

0

0

0

0

0

0

0

0 0 0

0 0 0

0 0 0

0 0 0

0

0

0

0

0

0

1- [21 1- [23 -----2 2 0

0

1- [22 1- [21 -----2 2 (5-24)

where the product of [W-l]W] = [I]6x6 gives the 6x6 rank unit matrix, whereas [W][W- l ] = [E]9X9 is a 9x9 rank matrix, but which is not a unit matrix. Eqs.(5-20) to (5-24) are presented in the principal anisotropic coordinate (Xl X2 X3). In practical applications, it should be transformed into the global geometric coordinate system (XYZ) of the structure. The coordinate transformation for the Cauchy stress vector can be given as

{O"}

=

[Tu]{o-}

(5-25a)

where the Cauchy stress vector in coordinate system (XYZ) is defined as

{o-} = {O" x' 0"Y' 0"z' 0" yz' 0" zx' 0" Xy} T

(5-25b)

The general transformation matrix of coordinates in 3D space is lr mr nr mInI nIh hml l§ m§ n§ m2n2 n2l2 lzm2 l~ m~ n~ m3n3 n3b l3m3 [Tu] = 2l2 l3 2m2m3 2n2n3 m2n 3 + m3 n 2 n2l3 + n3lz lzm3 + bm2 2l3h 2m3ml 2n3nl m3n l + mln3 n3h + nIb l3 m l + h m 3 2hl2 2mlm2 2nln2 mln2 + m2nl nll2 + n2h hm2 + l2 m l (5-25c) where {li, mi, ni}T are direction cosines of the normal unit vectors ni. The coordinate transformation for the effective stress vector can be given as {O"*} = [T;]{o-*} (5-26a) where the effective stress vector in the system (XYZ) is defined as (5-26b)

230

5 Basis of Anisotropic Damage Mechanics

The transformation matrix [T;] for the effective stress vector is a (9x9) rank matrix. Substituting Eqs.(5-25) and (5-26) into Eq.(5-20), the relationship between the effective stress vector and the Cauchy stress vector in the coordinate system (XYZ) can be presented as

{O"*} = [*]{0"}

(5-27a)

where (5-27b) 5.3.2 Two Dimensional Space In two dimensional space, the effective stress tensor can expressed by the principal anisotropic damage variables as

(5-28a) or

{o-*} = [W]{o-}

(5-28b)

where (5-28c) (5-28d)

W]

=

1 I-Dl

0

0

0

1 I-D2

0

0

0

1 I-D2

0

0

1 I-Dl

(5-28e)

Corresponding to Eq.(5-25a), we have (5-29a) The general coordinate transformation matrix in two dimensional space can be strictly expressed by the inclination angle of cracks as (5-29b)

5.3 Principal Anisotropic Damage Model

231

where 8 is also the angle between coordinates {Xl X2} and {X Y}. Corresponding to Eq.(5-26a), we have

* (J"12, * (J"21 *}T {(J"-* } = {* (J"n,(J"22,

(5-30a) (5-30b)

[Tu]

=

cos28 sin 28 2 28 sin 8 cos [ 0.5 sin 28 -0.5 sin 28 0.5 sin 28 -0.5 sin 28

- sin 28 sin 28 cos 28 cos 28

- sin 28] sin 28 cos 28 cos 28

(5-30c)

For the strain vector, the transformation has a form (5-31a) where

[Te]

=

1

cos28 sin 28 0.5 sin 28 [ sin 28 cos 28 -0.5 sin 28 - sin 28 sin 28 cos 28

(5-31b)

It is obvious that the following relationships can be found

{E} = [Te]-l{c}

=

[Tu]{c}

(5-32a)

and (5-32b) The relationship between the effective stress vector and the Cauchy stress vector, corresponding to Eq.(5-27), can be represented in the two dimensional coordinate system (XY) as follows: For the plane stress case

Fn ~

[p'[

(J"yx

U:J

(5-33)

where

[<1>*]

=

cos 28

sin 28

'l/JI

'l/J2

--+--

0

0

--+--

sin28 (1 -1 ) 'l/JI 'l/J2 2

0

0

(1 -1 ) sin28 'l/JI 'l/J2 2

sin 28

cos 28

'l/JI

'l/J2

sin28 (1 -1 ) 'l/JI 'l/J2 2 (1 -1 ) sin28 'l/JI 'l/J2 2 sin 28 cos 28

--+-'l/JI

'l/J2

cos 28

sin 28

'l/JI

'l/J2

--+--

(5-34)

232

5 Basis of Anisotropic Damage Mechanics

For the plane strain case

(5-35)

where cos 2 (}

sin 2 (}

'l/Jl

'l/J2

--+--

0

0

--+--

0

0

(1 -1 )sin2(} 'l/Jl 'l/J2 2

0

[4>*] =

cos 2 (}

sin 2 (}

'l/Jl

0

2(} (1 -1 )sin'l/Jl 'l/J2 2 ( 1 1 ) sin 2(} 0 ----'l/Jl 'l/J2 2 1 0 0

'l/J2

'l/J2 0

2(} (1 -1 )sin0 'l/Jl 'l/J2 2

(5-36)

cos 2 (}

sin 2 (}

--+-'l/Jl

'l/J2

cos 2 (}

sin 2 (}

'l/Jl

'l/J2

--+--

5.4 Decomposition Model of Anisotropic Damage Tensor 5.4.1 Review of Definition of Damage Variable The principles of continuum damage mechanics are first reviewed in the case of uniaxial tension. In this case, isotropic damage is assumed throughout. Consider a cylindrical bar subjected to a uniaxial tensile force T as shown in Fig.5-8(a). The cross-sectional area of the bar is A and it is assumed that T

T

-

~yo

01 " 0/I \0/0\ 0

CT

0

o \0 0/0 / "0

(a)

Remove both voids and cracks

n A

A

-

CT

(b)

Fig. 5-8 (a) A cylindrical bar subjected to uniaxial tension; (b) Both voids and cracks are removed simultaneously

5.4 Decomposition Model of Anisotropic Damage Tensor

233

both voids and cracks appear as damage in the bar. The uniaxial stress IJ in the bar is found easily from the formula T = IJ A. In order to use the principles of continuum damage mechanics, we consider a fictitious undamaged configuration of the bar as shown in Fig.5-8(b). In this configuration all types of damage, including both voids and cracks, are removed from the bar. The effective cross sectional area of the bar in this configuration is denoted by A * and the effective uniaxial stress is IJ*. The bars in both the damaged configuration and the equivalent undamaged configuration are subjected to the same tensile force T. Therefore, considering the equivalent undamaged configuration, we have the formula T = IJ* A *. Equating the two expressions of T obtained from both configurations, one obtains the following expression for the equivalent uniaxial stress IJ*

IJ*

=

IJA/A*

(5-37)

One uses the definition of the damage variable fl, as originally proposed by Kachanov [5-21]

fl

(A - A*)/A

=

(5-38)

Thus the damage variable is defined as the ratio of the total area of voids and cracks to the total area. Its value ranges from zero (in the case of an undamaged specimen) to 1 ( in the case of complete rupture). Substituting for A/A* from Eq.(5-38) into Eq.(5-37), one obtains the expression for the equivalent uniaxial stress defined in Chapters 3 and 4 as

IJ*

=

1J/(1- fl)

(5-39)

Eq.(5-39) above was originally derived by Kachanov in 1958. It is clear from Eq.(5-39) that the case of complete rupture (fl = 1) is unattainable because the damage variable fl is not allowed to take the value 1 in the denominator. 5.4.2 Decomposition of Damage Variable in One Dimension

The principles of continuum damage mechanics are now applied to the problem of decomposition of the damage tensor in a damaged uniaxial bar subjected to a tensile force T. Isotropic damage is assumed throughout the formulation. It is also assumed that the damaged state is defined by voids and cracks only. Therefore, the cross sectional area A * of the damaged bar can be decomposed as follows: (5-40)

where AV is the total area of voids in the cross-section and AC is the total area of cracks (measured lengthwise) in the cross-section (The superscripts 'v' and 'c' denote voids and cracks, respectively). In addition to the total

234

5 Basis of Anisotropic Damage Mechanics

damage variable D, the two damage variables DV and DC are introduced to represent the damage state due to voids and cracks, respectively. Our goal is to find a representation for the total damage variable D in terms of DV and DC. In order to do this, we need to theoretically separate the damage due to voids and cracks when constructing the effective undamaged configuration. This separation can be performed by two different methods. We can start by removing the voids only, then we remove the cracks separately, or we can start by removing the cracks only, and then we can remove the voids separately. The detailed formulation based on each of these two methods is discussed below and is shown schematically in Fig.5-9 and Fig.5-1O. It is emphasized that this separation of voids and cracks is theoretical in the sense that it is an acceptable method of mathematical analysis and has no physical basis. In fact, the physics of the problem indicates a coupling between the two damage mechanisms, which is apparent in the next section in the general three-dimensional case. T O'Yo

0\0--;;

'- 0/

Remove voids

/0\0 j

D'

10/0\ I "() ,..........0

a--<;.-

T (a) Damaged voids

T A

A+A'

A'

'-\/

\

./

I" II ./

Remove cracks

\\/j a" / '-

.-

-

T

(b ) Undamaged configuration with respect to voids

D'

T (c) Equivalent undamaged configuration

Fig. 5-9 A cylindrical bar subjected to uniaxial tension: voids are removed first then followed by cracks

In the first method, we first remove the voids only from the damaged configuration shown in Fig.5-9(a). In this way we obtain the damaged configuration shown in Fig.5-9(b), which contains damage due to cracks only. This is termed the undamaged configuration with respect to voids. The crosssectional area of the bar in this configuration is clearly A - AV = A * + A C while the uniaxial stress is denoted by (J*v. The total tensile force T in this configuration is then given by T = (J*v(A - AV) = (J*V(A* + AC). This expression is equated to the total tensile force T = (JAin the damaged configuration from which we obtain (5-41 ) The damage variable DV due to voids is defined by the ratio AV fA. Substituting for AV from Eq.(5-40) (A - AC = A* + AV), one obtains (5-42)

5.4 Decomposition Model of Anisotropic Damage Tensor

235

Substituting Eq.(5-42) into Eq.(5-41), one obtains the following relation between IJ*V and IJ (5-43) The similarity between Eqs.(5-43) and (5-39) is very clear. The next step involves removing the cracks from the intermediate configuration in order to obtain the equivalent undamaged configuration shown in Fig.5-9(c). Equating the previous expression for the tensile force T = IJ*V(A - AV) = IJ*V(A* + AC) with the tensile force T = IJ* A * in the equivalent undamaged configuration, one obtains

IJ*

=

IJ*V(A* + AC)/A*

(5-44)

The damage variable DC due to cracks is now defined by the ratio (5-45) Substituting Eq.(5-45) into Eq.(5-44) and simplifying, one obtains the following relation between IJ*V and IJ* (5-46) Finally, substituting Eq.(5-43) into Eq.(5-46) we obtain the relationship between IJ and IJ* (5-47) The above relationship represents a formula for effective stress in terms of the separate damage variables due to voids and cracks. The same result can be obtained by reversing the order of removal of voids and cracks. In the second method, one first removes the cracks only from the damaged configuration shown in Fig.5-10(a). In this way we obtain the damaged configuration shown in Fig.5-1O(b), which contains damage due to voids only. This is termed the undamaged configuration with respect to cracks. The cross-sectional area of the bar in this configuration is clearly A - AC = A* + AV while the uniaxial stress is denoted by IJ*C. The total tensile force T in this configuration is then given by T = IJ*C(A - AC) = IJ*C (A* + AV). This expression is equated to the total tensile force T=IJ A in the damaged configuration from which we obtain (5-48) Tile damage variable DC due to cracks is defined by the ratio AC / A. Substituting for AC from Eq.(5-40), one obtains (5-49)

236

5 Basis of Anisotropic Damage Mechanics T -;;-"Yo 0\0"; '-

T

A'

0/

10/0\ I /0\0 I ci- . . . . 0

(a) Damaged configuration

0

o

Remove cracks

0 0

0

o

[1' d'

o o o

0 0

0

0

TA

A+A'

Remove voids

0 0 0 0

0 0

ft

0

T (b) Undamaged configuration wi th respect to cracks

T

(c) Equivalent undamaged configuration

Fig. 5-10 A cylindrical bar subjected to uniaxial tension: cracks are removed first followed by voids

Substituting Eq.(5-49) into Eq.(5-48) , one obtains the following relationship between cr*C and cr

cr*C = cr / (1 - DC)

(5-50)

The similarity between Eqs.(5-50), (5-43) and (5-39) is very clear. The next step involves removing the voids from the intermediate configuration in order to obtain the equivalent undamaged configuration shown in Fig.5-1O(c). Equating the previous expression for the tensile force T =cr*C(A* + AV) with the tensile force T =cr* A * in the equivalent undamaged configuration, one obtains (5-51 ) The damage variable DV due to voids is now defined by the ratio (5-52) Substituting Eq.(5-52) into Eq.(5-53) and simplifying, one obtains the following relationship between cr* and cr*C (5-53) Finally, substituting Eq.(5-50) into Eq.(5-53) we obtain the relationship between cr* and cr too. (5-54) It is clear that the above relation between the two stresses in the damaged and the equivalent configuration is exactly the same relation obtained using the first method, i.e. Eq.(5-47). Thus both methods of constructing the equivalent undamaged configuration give the same relation between the stresses in the respective configurations. In this way, the decomposition of the damage tensor has been completed in the one-dimensional case. In order to derive the

5.4 Decomposition Model of Anisotropic Damage Tensor

237

final result, we compare either Eq.(5-47) or Eq.(5-54) with the total damage appearing in Eq.(5-39). Equating the denominators on the right-hand-side of these equations, we can easily obtain the formula (5-55) Eq.(5-55) represents the general form for the decomposition of the damage variable into its two respective components, [lV and [lC. The result can be further simplified by expanding Eq.(5-55) and simplifying to obtain (5-56) Eq.(5-56) gives a very clear picture of how the total damage variable [l can be decomposed into a damage variable [lV due to voids and a damage variable [lc due to cracks. It is also clear that Eq.(5-56) satisfies the constraint o ~ [l ~ 1 whenever each of the other two damage variables satisfies it. It is also clear that when damage in the material is produced by voids only ([lc = 0), then [l = [lV. Alternatively, [l = [lC when damage in the material is produced by cracks only ([lv = 0). 5.4.3 Decomposition of Symmetrized Anisotropic Damage Tensor in 3-D

In the anisotropic case of three dimensional deformation and damage, the effective stress tensor, uij can be given by the symmetrized transformation [5-22] as follows, or

(5-57)

where the fourth order tensor Wijkl is given by the following 6x6 matrix based on a symmetrization represented by Voyiadjis et at. [5-22] as 2'
[w] = 2\7

2D13D23 2D13D23

D 13 D 23 D 12 D 23

+ 2D12 '
'
o o

+ + o

0 0 2'
+ 2D13'
D~3

(5-58) where 'V is given by (5-59)

238

5 Basis of Anisotropic Damage Mechanics

and'l/Jij = bij - Dij (bij is the Kronecker delta). Repeating the same procedure that was employed in the one-dimensional case, we can derive the general decomposition in the case of three dimensions. Removing the voids first, then removing the cracks and applying Eq.(5-57), we obtain the following two equations (5-60) (5-61 ) Eqs.(5-60) and (5-61) correspond to Eqs.(5-43) and (5-46) of the onedimensional case. Substituting Eq.(5-60) into Eq.(5-61), we obtain: {O"*} = [!liC][!liV]{O"} or O"\j = !licijmn!liv mnkO"kl

(5-62)

Comparing Eq.(5-62) with Eq.(5-57), we obtain the desired general decomposition (5-63) Alternatively, removing the cracks first, then removing tile voids and applying Eq.(5-57), we obtain the following two equations (5-64) (5-65) Eqs.(5-64) and (5-65) correspond to Eqs.(5-50) and (5-53) in the one-dimensional case. Substituting Eq.(5-64) into Eq.(5-65), we obtain {O"*} = [!liV][!liC]{ O"} or O"\j = !liv ijmn!lic mnkzO"kl

(5-66)

Comparing Eq.(5-66) with Eq.(5-57), we again obtain the desired general decomposition (5-67) It is emphasized that Eqs.(5-63) and (5-67) are equivalent, i.e. both are valid

representations of the decomposition of the damage effect tensor [!liijkd. By carrying out the tensorial multiplications explicitly using Eq.(5-58), one obtains exactly identical results. Using 6x6 rank matrix representation of Eq.(5-58) it is clear that Eqs.(563) and (5-67) reduce to the following form (5-68)

5.4 Decomposition Model of Anisotropic Damage Tensor

239

where both [PV] and [PC] can be represented using the 6x6 rank matrix of Eq.(5-58) by replacing Dij with Dij and Dij respectively. The same thing holds 'l/Jij' The matrix multiplications in Eq.(5-68) are performed using the computer algebra program MAPLE. Performing the first multiplication [PC] [PV] and equating the result with Eq.(5-58), we obtain 36 equations between the components of the damage variables D ij , Dij and Dij . We also obtain another 36 equations (for a total of 72 equations) when we perform the second multiplication [PV][PC]. It is noted that many of these equations are identical or can be shown to be identical. We have been able to reduce this huge set of equations to 18 independent equations. Furthermore these 18 equations have been classified into two categories. The first category includes the following 9 equations which represent the general decomposition of the damage tensor components Dij , into the damage tensor components Dij due to cracks and the damage tensor components Dij due to voids.

~ (nl' nl, _ D2 ) _ _ 1_ (nl'C nl,C _ Dc2) (nl'V nl,V _ Dv2) \7 '/-'11 '/-'22 12 - \7c\7v '/-'11 '/-'22 12 '/-'11 '/-'22 12

(5-69a)

(5-69b)

~(nl' nl, _ D2 ) _ _ 1_(nl'C nl,C _ Dc2)(nl'V nl,V _ Dv2) \7 '/-'22'/-'33 23 - \7C\7V '/-'22 '/-'33 23 '/-'22 '/-'33 23

~ (D13 D23 + D12'I/J33) =

\7c1\7V

[('I/J~2'I/J33 - D~~)(Dr3D~3 + Dr2'I/J33)

+ (D~3D~3 + D~2'I/J33)('I/J~2'I/J33 + 'l/Jr1'I/J33 -

~(D12D23 + D13'I/J22) =

\7c1\7V [('I/J~2'I/J33 -

(5-69c)

(5-69d)

D~l - Drl)]

D~~)(Dr2D~3 + Dr3'I/J~2)

+ (D~2D~3 + D~3'I/J~2)('I/J~2'I/J33 + 'l/Jr1'I/J33 - D~l - Dr~)]

~(D12D13 + D23'I/J11) = \7C~V [('I/J~1'I/J33 - D~~)(Dr2Dr3 + D~3'I/Jr1) + (D~2D~3 + D~3'I/J~1)('l/Jr1'I/J33 + 'l/Jr1'I/J~2 - Drl - Dr~)]

(5-6ge)

(5-69f)

240

5 Basis of Anisotropic Damage Mechanics

1 '\! ('l/Jll 'l/J22 X

where

2

2

1

+ 'l/Jll 'l/J33 - st12 - st23 ) = '\!c'\!v [('l/Jl1 'l/J22 + 'l/Jl1 'l/J33 - stl~ - st2~)( 'l/J11 'l/J2'2 + 'l/J11 'l/J33 - stli - st2'i)]

'l/Jc ij

=

6ij - stC ij and 'l/Jv ij

=

6ij - stV ij·

(5-69i)

It is clear from the components in the above set of equations that the decomposition is not explicit. A set of simultaneous equations needs to be solved for the general three-dimensional decomposition. The second category of equations includes the remaining 9 equations as follows (5-70a) (5-70b) (5-70c) (5-70d) (5-70e) (5-70f) (5-70g) (5-70h) (5-70i)

It is clear that the above equations do not contain any components of the total damage tensor stij . This set of equations relates only to the components of the two damage tensors stij and stJ:j. It is concluded that the above set of 9 equations represents the exact coupling between the two damage mechanisms of voids and cracks. Although this coupling may be obvious based on the physics of the problem, a rigorous mathematical proof has been given for it. Furthermore, the coupling Eq.(5-70) has not appeared before in [5-22].

5.5 Basic Relations of Anisotropic Damage Based on Thermodynamics

241

Finally, it is noted that in the special case of one dimensional damage, Eq.(5-69) reduces to the simple decomposition shown in Eqs.(5-55) and (556) while the coupling Eq.(5-70) reduces to zero.

5.5 Basic Relations of Anisotropic Damage Based on Thermodynamics 5.5.1 First and Second Laws of Thermodynamics of Anisotropic Materials

Similar to Chapter 3, the principle of energy conservation in thermodynamics [5-23] for anisotropic materials is also implied by the same form of the first law of thermodynamics

:t f (~{U}2 + v

E) pdV

=

f {F}T{u}dV + f {Q}T{u}dS+ f rdV - f {q}T{n}dS

v

v

dS 1

(5-71)

dS 2

where {u} is the displacement vector of particles; E is the internal energy per unit mass; r is the heat supplied per unit volume; {Q } is the vector of surface forces acted on body; {F} is the vector of body forces; {q} is the vector of heat flows passing through the unit area per unit times. The same relationship between internal energy and external work in anisotropic damage mechanics is

:t [f ~P{u}2dV + f v

v

{U}T{E}dV]

=

f {F}T{u}dV + f {Q}T{u}dS

v

dS1

(5-72)

since

f {q}T {n}dS f div{q}dV =

(5-73)

v

dS2

From Eq.(5-71) t Eq.(5-72), we have

pE

=

{u}T {i}

+r -

div{q}

(5-74)

The second law of thermodynamics (entropy principle) in the form of Clausius-Duhem inequality implies that [5-24]

:t [f v

pSdV

~ f ~dV]- f {~T {n}dS v

dS2

(5-75)

242

5 Basis of Anisotropic Damage Mechanics

. pTS -

r + div{q} -

{q}

----r-?: 0

T{\7T}

(5-76)

where S is the entropy per unit mass; T is the absolute temperature; \7 is the gradient operator. Let the free energy per unit mass be

W=E-TS

(5-77)

Eq.(5-74) can be rewritten using Eq.(5-77) as,

{O"}T {i} - p(W + s1') - pTS + r - div{q}

=

0

(5-78)

Eq.(5-76) can be rewritten using Eq.(5-78) as,

----r-?: 0

T .. T{\7T} {O"} {E} - p(W + ST) - {q}

(5-79)

5.5.2 Thermodynamic Potential and Dissipation Inequality in Anisotropy

As similarly stated in Chapter 3, the free energy of anisotropic damaged materials with anisotropic accumulative hardening can also be considered as a thermodynamic potential assumed to be

W

=

W({Ee},{il},b},T)

(5-80)

where {il} is the principal damage vector which, as the anisotropic internal state variable, corresponds to the scalar il of the isotropic damage variable, and can be obtained from the anisotropic damage tensor. {,} is the anisotropic accumulative hardening vector which, as the anisotropic internal state variable, corresponds to the scalar I of the accumulative isotropic hardening scale. Substituting Eq.(5-80) and {E} = {Ee} + {Ep} into Eqs.(5-78) and (579), the first law of thermodynamics for anisotropic damaged materials with anisotropic strain hardening can be implied in the form of free energy as

aT ) T+{O"} )T) .{Ee}-P (S+ aw· ({O"} - (Pa{Eae}w T

aW)T. P (( a{il} {il}

T.

{Ep}-

aw ) T + ( ab} h}) - pTS. - r + div{q} =

0

(5-81 )

The second law of thermodynamics for anisotropic damaged materials with anisotropic accumulative hardening can be implied in the form of ClausiusDuhem inequality with the expression of free energy as

5.5 Basic Relations of Anisotropic Damage Based on Thermodynamics

aT ) T+{O"} )T) .{Ee}-P (S+ aw· ({O"} - (Pa{Eae}w T

243

T.

{Ep}-

aW)T. ( aW)T . ) T {VT} P (( a{D} {D} + ab} b} - {q} ----y;-?: 0

(5-82)

Regarding the basic relations Eqs.(3-41), (3-42), from Eqs.(5-81) and (582) we can obtain (5-83) Introducing the heat flow vector {g} = -{VT}/T, following the similar procedure of formulations in Eqs.(3-44) and (3-45) in Chapter 3, Eqs.(5-81) and (5-82) can be rewritten as T

div{q} = {O"} {ip}-p

a wT· aw )T {it}) -pTS-r . (5-84) (( a{D} ) {D} + (ab}

. (( a{D} aW)T. aW)T b} . ) +{q}T {g}?:O {O"} T {Ep}-p {D}+ ( ab}

(5-85)

Similarly to definitions of Eqs.(3-46) and (3-47) in isotropy, definitions of the anisotropic damage strain energy release rate vector, {Y}, and the anisotropic hardening function vector, {R}, associated with the anisotropic accumulative hardening vector {'Y} can be expressed as (5-86) In the anisotropic case, the thermodynamic generalized-deformation vector consists of the observable variables and internal state vector as {E}, {D}, {'Y},TT. The associated (dual) thermodynamic generalized-force vector consists of {{ O"}, {Y}, {R}, S} T. Thus, the basic relationship in anisotropic damage mechanics can be rewritten as

{O"}

{Y} {R}

=

pgradW({Ee},{D},b),T)

(5-87)

S p Corresponding to Eq.(3-49b), the energy dissipation inequality for anisotropic damaged materials with accumulative anisotropic hardening can be represented as

244

5 Basis of Anisotropic Damage Mechanics

5.5.3 Dissipation Potential and Dual Relationship in Anisotropy Similar to Chapter 3, introducing the mechanical potential for anisotropic damaged materials, the rate relationship corresponding to the isotropic case in Eq.(3-53) can be defined as

a([J {O"} = a{Ep} , {Y}

=

a([J -a{D}' {R}

a([J -a{1r}' {q}

=

=

a([J -a{g}

(5-89)

The dual dissipation potential of anisotropic damaged materials with anisotropic accumulative hardening property introduced by the Legendre transformation, can be expressed in the form

([J*({O"}, {Y}, {R}, {q}, {Ep}, {D},{1r}, {g}) {O"}T {Ep} - {Y}{D} - {R}{1r} + {q}T {g} - ([J( {Ep}, {D},{1r}, {g})

=

(5-90) The incremental form of the dual dissipation potential is

d([J * =(

a([J)T. a([J). {O"} - a{Ep} {dcp} - ( {Y} + a{D} {dD} - ( {R}

+ ( {q} -

a([J) a{g} {dg}

. T + {cp} {dO"} -

.

.

a([J). + a{1r} {d,} T

{D}{dY}- {r}{dR} + {g} {dq} (5-91 )

Substituting Eq.(5-89) into Eq.(5-91) gives

d([J*

=

{Ep}T {dO"} - {D}{dY}- {1r}{dR} + {g}T {dq}

(5-92)

Eq.(5-92) shows us that the dual dissipation potential ([J* is related to variables {O"},{Y}, {R},{q} only, independent of variables {Ep}, {D}, {1r}, {g} thus from

([J*

=

([J* ({O"}, {Y}, {R}, {q})

(5-93)

the incremental form of which can be simplified as

d([J*

=

a([J* ) ( a{O"}

T

( a([J* ) ( a([J* ) ( {dO"} + a{y} {dY}+ a{R} {dR}

+

a([J* ) a{q}

T

{dq}

(5-94) Comparing Eqs.(5-94) and (5-91) and in a similar manner Eqs.(3-56) and (3-59), the generalized rate expressions of the dual dissipation potential for

5.5 Basic Relations of Anisotropic Damage Based on Thermodynamics

245

anisotropic damaged materials with anisotropic accumulative hardening can be written as

(5-95) where the dual dissipation potential (P* is defined in a similar procedure for obtaining Eq.(3-55) by application of the Legendre transformation to the mechanical dissipation potential (P. Eqs.(5-89) and (5-95) present the basic dual relationship, which forms the basis of theoretical frames of continuum damage mechanics, in anisotropic damage mechanics.

5.5.4 Damage Strain Energy Release Rate of Anisotropic Damage The internal state variables defined by the anisotropic damage strain energy release rate vector {Y} in the previous section have an important role in the study of the mechanism of damage growth and development. In the case of independence between elasticity and plasticity, the free energy of anisotropic damaged materials during the isothermal process can be represented as

where p is the mass density; W* is the free energy of damaged materials; W; is the elastic part of free energy W*; Wp is the plastic part of W*. The vector of the damage strain energy release rate (i.e. the vector of damage developing force) of anisotropic damaged materials can be defined from the basic relationship of continuum damage mechanics by Eq.(5-86) with {Y} =

aw*

Pa{n} as follows

(5-97) The components of {Y} can be represented either by the strain vector or by the stress vector in the Cartesian co-ordinate system as

y::i = "21{ Ee }T[T ]a[D*][T ani ]T{Ee } or (j

(j

y::•

=

-21{(J}T[T~]Ta[~~i-l[T~]{(J} v

(i = 1,2,3)

v

(5-98)

in which Y; is the component of the damaged strain energy release rate in the ith principle anisotropic direction.

246

5 Basis of Anisotropic Damage Mechanics

Since all components of the damage strain energy release rate have the scalar dimension of the energy unit with non-negative property, the total damage strain energy release rate Y of anisotropic damaged materials should be taken into account by the summation of these three individual components. Therefore, it can be defined as

where 3

[J*]

="

--1

a[D*] ~ aD· i=l

(5-100)

'

Considering that the response sensitivity of damage growth and development of anisotropic materials in different directions is not actually the same, therefore the sensitivity of the damage strain energy release rate of anisotropic materials should not be the same in different directions. In order to emphasize the sensitivity of the anisotropy properties of damage dynamic behavior Zhang et al. [5-11, 5-14"-'15] modified the formulations of the total damage strain energy release rate by introducing the anisotropic damage response sensitive vectors of material property in Eq.(5-99) as (5-101 ) where {Q} is defined as the Anisotropic Sensitivity Coefficient Vector of damage development in anisotropic damaged materials; Qili denotes the actual response of the anisotropic damage strain energy release rate in the ith anisotropic principal direction. A threshold value Ydi of the damage strain energy release rate should be calculated when the anisotropic damage component Di along the ith direction starts growing. The expression of matrix [d*] in Eq.(5-101) is (5-102) The details of elements in the matrix [d*] are described as follows

[d*] =

t

i=l

Qi

a[~~-l '

di1 d Z1 dh 0 0 0

di2 d Z2 d 32 0 0 0

dh dh d 33 0 0 0

0 0 0 9Z3

o 0

0 0 0 0 931

o

0 0 0 0 0 9i2

(5-103)

5.6 Elastic Constitutive Model for Anisotropic Damaged Materials

247

where (5-104a)

(5-104b)

(5-104c)

5.6 Elastic Constitutive Model for Anisotropic Damaged Materials Since the equivalent strain concept is not always applicable in an anisotropic damage state, Ilankamban and Krajcinovic [5-25] and Krajcinovic and Fonseka [5-26] developed an anisotropic damage constitutive relationship for brittle materials in terms of a fourth order tensor. Murakami and Ohno [5-6], using a symmetric second order tensor and symmetrization of the effective stress tensor, presented an anisotropic elastic constitutive relationship for investigation of creep damage problems. Zhang et at. [5-11'"'-'16], using the principal anisotropic damage tensor and the unsymmetric effective stress tensor based on the equivalent internal forces and equivalent complementary elastic energy, developed a complete-orthogonal symmetric elastic damage constitutive matrix as follows. 5.6.1 Elastic Matrix of Damaged Materials in Three Dimensions In the case of uncoupling elasticity and plasticity, the specific free energy of anisotropic damaged materials with anisotropic accumulative hardening is

W

=

W;({Ee}, {il},T)

+ Wp({'Y},T)]

(5-105a)

The complementary energy II* with respect to the isothermal process can be written as

II*

=

{u}T {Ee} - p[W;( {E e}, {il})

+ Wp(b})] =

II; ({E e}, {il}) - pWp(b})]

(5-105b) where II; ({ u}, {il}) is the complementary elastic energy of an anisotropic damaged material. Thus, according to the hypothesis of elastic complementary energy equivalence (see Chapter 3 3.4.3), the complementary elastic energy of an anisotropic damaged material is of similar form to that of an undamaged

248

5 Basis of Anisotropic Damage Mechanics

material under effective stress loading. Thus, the anisotropic damaged elastic complementary energy can be stated in the (principal anisotropic) coordinate system (Xl X2 X3) (similarly see Fig.3-5).

lI:({ce},{Sl}) =

=

lIe({ce},O)

~{O'}T[tli]T[D]-l[tli]{O'} 2

=

1

T

-

2{O'*} [D]

=

-1

{o'*}

~{O'}T[D*]-l {a-}

(5-105c)

2

where {O'*} is the effective (net) stress vector given in the principal anisotropic coordinate system (Xl X2 X3); W] is the transformation matrix of the Cauchy - -1 stress vector to the effective stress vector defined in Eq.(5-22); [D] is the inverse of the undamaged elastic matrix presented in the principal anisotropic coordinate system (Xl X2 X3) with the form of (9x9) order matrix as (5-106a) where 1

V2l

[0]11 =

E2

E3

V12

1

V32

El

E2

E3

--

V13

V23

1

El

E2

E3

----

[Ob=

V3l

----

El

(5-106b)

1 2G 23

0

0

0

0

0

0

1 2G 32

0

0

0

0

0

0

1 2G3l

0

0

0

0

0

1 2G 13

0

°

0

0

0

0

1 2G 12

0

0

0

0

0

0

1 2G 2l

0

(5-106c)

U sing the relation

{f}

=

"dlIe ( {O'}, {Sl)) "d{o-}

(5-107)

the elastic constitutive equations of anisotropic damaged material can be obtained as

5.6 Elastic Constitutive Model for Anisotropic Damaged Materials -

-1

{i} = [D*] {O'} or {O'}

=

-

[D*]{i}

249

(5-108a)

where (5-108b) is the inverse of the damage-elastic matrix of anisotropic damaged material, i.e. the complementary damage-elastic matrix or the effective flexibility elastic matrix. 1 -v21 - -v31 E*1 E*2 E*3 vi2 1 v32 E*1 E*2 E*3 1 vi3 v23 ---E*1 E*2 E*3

[0*] =

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1 G 23

0

0

0

0

0

0

0

0

1

Gh 0

(5-108c)

0 1

Gi2

in which,

E:

=

(1 - fli)2 Ei

(5-109a)

=

(1 - fli) (1 _ flj) v ij

(5-109b)

2(1 - fli)2 (1 - flj)2 + (1- fli )2G ij

(5-109c)

*

v ij G:j -

=

-1

(1- fli )2 -

By inverting the [D*] ,the [D*] matrix may be obtained as below,

ViI Vi2 ViJ 0 [V*] =

V 21 V 22 V 23 0 V 31 V 32 V33 0 0 0 0 G 23 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 G 31 0 0 Gi2

(5-110a)

where (5-110b) (5-110c)

250

5 Basis of Anisotropic Damage Mechanics

(5-110d) In practical applications, the damage-elastic constitutive equations must be transformed into the coordinate system (XYZ). Substituting Eqs.(5-108) and (5-109) into Eq.(5-25a) gives

{oj

=

[D*]{e} or {e}

=

[D*r1{0"}

(5-111a)

where {O"} was defined in Eq.(5-25b); {e} is the strain vector defined in (XYZ) coordinate system as {ex, ey, ez, eyz, ezx, exy }T; [D*] is the damage-elastic constitutive matrix defined in (XYZ) coordinate system as (5-111b)

[D*] = [To-][D*][ToF where [D*]-l is the inversion of [D*], and can be represented as

(5-111c)

5.6.2 Elastic Matrix of Damaged Materials in Two Dimensions -

-

-1

The elastic matrix [D*] and its inverse matrix [D*] of anisotropic damaged materials in the two dimensional case can be represented in a detailed way, as given in [5-14]. (Where each shear term in the undamaged inverse elastic - -1 matrix [D] has been rewritten in the speared form of the two dual shear - -1 terms, i.e. the undamaged inverse elastic matrix [D] has been expressed by a pretend (4x4) order symmetric inverse elastic matrix, in which each shear modulus takes the corresponding two rows and two columns, respectively, as in the case of plane stress.) 1

[D]-l =

or

v21 E1 E2 1 v12 E1 E2 0

0

0

0

0

0

0

0

1

G 12 0

0 1

G 21

(5-112a)

5.6 Elastic Constitutive Model for Anisotropic Damaged Materials E1

E2//12

0

1 - //12//21 1 - //12//21

251

0

E2

E1//21

0 0 [D]= 1 - //12//21 1 - //12//21 0 0 G 12 0 0

0

(5-112b)

0 G 21

in the case of plane strain 1 - //12//21

-

E1

[D]-l =

(1

-

+ //21) E

1

//21

(1

+ //d E

//21

2 1 - //12//21 E2

0

0

0

0 0

0

0

1 G 12

0

0

0

(5-113a) 1 G 21

or E1(1-//12//21)

(1

+ //d (1 -

E2//21

//12 - 2//12 //21 )

E1//12

[D]=

1 - //21 - 2//12//21 0

0 0 1 - //12 - 2//12//12 E 2 (1 - //21//11) 0 0 (1 + //21)(1 - //21 - 2//12 //21 ) 0 G 12 0

0

0

0 G 21

(5-113b) The transformation matrix [w] of continuum factors defined by Eq.(5-20) in the two dimensional case will have the form of (4x3) rank matrix as

[w] =

1 1- D1

0

0

0

1 1- D2

0

0

0

0

0

(5-114)

1 1- D2 1 1- D1 -

-1

Substituting Eqs.(5-112a), (5-113a) and (5-114) into [D*] [0*] in Eq.(5-108b) for plane stress gives

=

T

-

-1

[w] [D]

W]

(5-115)

=

252

5 Basis of Anisotropic Damage Mechanics

in detail as 1 1- [21

0

0

0

0

1 1- [22

0

0

0

0

[D]-l =

1

v 21

E1 v 12

E2

1

E1

E2

X

0

0

0

0

1 1 I- [22 1- [21

----

0

0

1 1- [21

0

0

0

0

0

1 1- [22

0

0

0

0

0

0

1 G 12

1

0

G 21

-

1 1- [22 1 1- [21 -1

The elemental details of the inverse elastic matrix [D*] of anisotropic damaged materials in the two-dimensional case can be obtained by calculation of the matrix product represented in Eq.(5-115) as

1 E 1 (1 - [21)2 -v21 E1(1 -

[21)(1 - [22)

-V12 E2(1 -

o

[21)(1 - [22) 1

E 2 (1 -

o

o

[22)2 (1 - [2i)2

0

+ (1 -

[2j)2

2Gd1 - [2i)2(1 - [2j)2 (5-116a)

Inversing the matrix in Eq.(5-116a), the elastic matrix [D*] of anisotropic damaged materials in the two-dimensional case can be given in a detailed way

[D*] = E1(1 -

[21)2

1 - V12V21 E1(1 -

[21)(1 - [22)V21

E2(1 -

[2d(l - [22)V12

o

1 - V12V21 E2(1 -

[22)2

1 - V12V21

1 - V12V21

o

0

o 2(1 - [2i)2 (1 - [2j )2 (1- [2i)2

+ (1- [2j)2 G12

5.6 Elastic Constitutive Model for Anisotropic Damaged Materials

253 (5-116b)

For the plane strain case, in a similar manner they give (1-!t2)V12+(1-!tl)V21 E2(1-!t!l(1-!t2)

1-V12V21 El(1-!tl)2 (1-!t1)V21 +(1-!t2)V12 E 2(1-!tl)(1-!t2)

o

1-V12V21 E2(1-!t2)2

o

o

(1-!ti)2+(1-!tj)2 2G 12 (1-!ti)2(1-!tj )2

o

(5-117a)

E~V~l (1-v~2-2n*v~l')

o

E~(1-vm (1+v 12 )(1-v;2 -2n*v;t)

o

(1+V~2) (1-v~2 -2n *V~l')

[D*] =

o

o (5-117b)

where Ei =

* V12 =

(1 - fh)2 E 1 , E2 (1 - [21) * (1 _ [22) V12' V21 n

*

Ei

v21

E2

vi2

=-

= =

(1 - [22)2 E 2 , (1 - [22) (1 _ [21) V21 (1 (1 -

[21)2 E1 [22)2 E2

(1 (1 -

[21)2 v21 [22)2 v12

(5-118)

The above elastic constitutive equations are given in the principal anisotropic coordinate system (Xl X2). The two dimensional expression of the anisotropic constitutive equations presented in the coordinate system (XY) can be easily obtained by application of the two dimensional coordinate transformation as defined in Eq.(5-111b) or Eq.(5-111c). After the matrices in Eq.(5-111b) are explicitly multiplied, the final result may be written as

{

~:

Txy

where

Dij

}=

[D*]{E}

=

[~~~ ~~~ ~~:l D31 D32 D33

has the following form [5-11]

(5-119)

254

5 Basis of Anisotropic Damage Mechanics

+ (D- 11 * + D- 22 * -

* D *12 = D*21 = D- 12

2D- 12 * - 4D- 33 * ) sm . 2()cos 2()D-* 213

(5-120) in which fJij has been defined as each element in Eq.(5-116). By inverting the relationship obtained from Eq.(5-119), the strains in terms of stresses are [5-11] "'* _ mx* _Vxy E*1 E*x E*x A* A* 1 _ my _Vyx E*1 E*y E*y A* A * _ my 1 mx

1

{ :; } "(xy

~ [e']{a} ~

A

{;::}

(5-121)

E*1 C~y

E*1 where

E* = _---=-_______ 1_______ ----._ x cos 4() (1 2Vi2). 2() 2() sin4() ~ + C* - E* sm cos + Jjj* 1 12 1 2

(5-122a)

E* = _--;-_----,,---- _ _ _ _ 1-:--_ _ _ _ _ _--,---y cos 4() 2Vi2). 2() 2() sin4() - + (1 -- - -sm cos +-E2 Ci2 Ei Ei

(5-122b)

1

C;y =

----(~-----~*---~)----1 1 1 2v12 1 2 2 -- + 4 Cb

Ei

+ -+ E2

- - -Ei

Ci2

(5-122c)

sin ()cos ()

E* m x* = ( 2 E~ 2

+ 2V~2 -

E* ) sin3() cos () - ( 2 + 2V~2 - C; E*) sin ()cos3() C; 12 12 (5-122d)

E* m* y = ( 2~

+ 2v~2 -

E*) sin ()cos3() - ( 2 + 2v~2 - C; E*) sin3() cos () C; 12 12 (5-122e)

A

A

E2

5.6 Elastic Constitutive Model for Anisotropic Damaged Materials

A* ~* * (1 vxy Vyx v12 = = - - E* E*1 E~ E; 1 A

A

1 2V * 11 ). 2 2 2 + -+ -- -sm 8cos 8 E* E* G* 2

1

255

(5-122f)

12

5.6.3 Property of Anisotropic Damage Elastic Matrix

From the relationships given in Eqs.(5-108c) and (5-109), it can be noted that each anisotropic damage state corresponds to another (new) anisotropic undamaged state with effective elastic properties of an effective module, effective Poisson's ratios and effective shear module given by Eq.(5-109). Using the equations developed in Eq.(5-109) , the relationships between the damage variables and the ratio of certain engineering properties of materials may be presented in a graphical form, as given in Figs.5-11 (a) to (c). The curves plotted in Fig.5-11 can be used in quasi-static damage analysis and safety judgment in a given damage state, which can achieve a general mechanical analysis and safety judgment in a certain damage state in a damaged structure without considering damage growth. That can be used to determine the effective elastic properties of anisotropic damaged materials from a certain damage state (damage variables) and the corresponding undamaged elastic properties of engineering materials. This means that any stress, strain and displacement analysis of a certain static damage but without damage growth can be carried out by substituting the effective parameters of damaged material into the general undamaged anisotropic formulations (or programs). Whereas these substituted effective parameters are determined from Eq.(5109) or Fig.5-11, and assumed to be a fictitious undamaged material state. Therefore the equalent quasi-static mechanical analysis and safety judgment in an anisotropic damaged structure can be employed. As the above formulae imply that the elastic modules in directions other than the principal directions are functions of the damage direction, these elastic modules vary with the angle 8 of anisotropy. The ratios of engineering properties of anisotropic damaged materials versus the angle of anisotropy are shown in Fig.5-12. From Fig.5-12(a), it can be found that the ratio of the effective Young's modulus E;/ Ei between X direction and the principal anisotropic direction Xi decreases with an increasing angle of anisotropy. As expected, the ratio of effective shear modulus 6;y/Gij has a maximum value at angle 8=45°, and this maximum value could reach as high as 1.55. Furthermore, the distribution is also symmetrical around 8=45° as expected. The effective Poisson's ratio f);y may reach a maximum value at an angle within 30° rv 40°. The components m; and m~ shown in Fig.5-12(b) exhibit inverse-symmetrical distribution. Using Fig.5-11 and 5-12(a), values of effective elastic properties E;, f);y, 6;y along the direction 8 can be determined from the undamaged elastic properties E i , Vij, G ij and the anisotropic damage variables [21, [22. For example, take a thin plate made of fiber glass whose damage

256

5 Basis of Anisotropic Damage Mechanics 1.0..--------------------, (a)

0.8

E;

0.6

£;0.4 0.2 o~----~----~~----~----~ o 0.2 0.4 0.6 0.8

fl , 5.---------------. 4

O~--~~--~~-~~--~

o

0.2

0.4

fl ,

0.6

0.8

1.0ro:::------------------, (c)

0.8 Gij 0.6

Glj O.4 r -

__

....

O.21_ _ _ _ _ _-=::~~ 0.2

0.4

fl

0.6

0.8

Fig. 5-11 (a) Variation of modulus with damage variable; (b) Variation of Poisson 's ratio with damage variable; (c) Variation of shear modulus with damage variable variables are, say, ill = 0.12, il2 = 0.07. Its anisotropic module prior to damage is El = 4.69 X 104 (MPa) and E2 = 1.03 X 10 4 (MPa), V12 = 0.26, G 12 = 0.49 X 10 4 (MPa), compared to its anisotropic damage properties after damage obtained from Figs.5-11 (a) and (b) which are = 3.63 x 10 4 (MPa) , E2 ~ 0.89 x 104 (MPa), vr2 = 0.274, Gi2 = 0.41 X 104 (MPa). Thus, the values of E~ , f)~y and G~y can be determined by means of Fig.5-12(a) for the given anisotropic angle.

Er

5.7 Different Models of Damage Effective Matrix 4

2.0 1.8

--

VX)'

1.6 1.4 1.2

257

G

G;

~

~

3

-..

~

mx

mx

-.. 2

my

1

/

V_ /

o o

-....... --- / ---

15

my

t-.,

"" ~1\\

30 45 60 75 Damage angle 90· (b)

90

Fig. 5-12 (a) Ratios of elastic properties of anisotropic damaged materials; (b) Variation of and in anisotropic damaged materials

5.7 Different Models of Damage Effective Matrix 5.7.1 Principal Damage Effective Matrix in Different Symmetrization Schemes The concept of damage effective functions or the damage effective matrix (tensor) is important in modeling the basis of continuum damage mechanics. There are various types of damage effective matrixes based on different hypotheses and different mathematical treatments, which have been widely applied to damage mechanics. In an anisotropic damage model, since the netstress (or effective stress) tensor is non-symmetric, this makes some researchers disdain the non-symmetric nature and they introduce some symmetrization methods into the damage effective matrix. The controvertial symmetrization techniques, which were suggested by Murakarmi [5-3, 5-6]' Voyiadjis and Park [5-22], Cordebois and Sidoroff [5-2], Chow and Yang[5-27], etc. in order to simplify such non-symmetry computationally, are still frequently used in many areas of damage mechanics studies due to the mathematical convenience in analysis. The symmetrization treatment may offer mathematical convenience in analysis, but may also in fact pose a spurious engineering analysis. This is because such an arbitrary treatment does not have a physical basis such that the resultant damage models may implicitly include new properties which originally did not exist and may also exclude some significant properties before symmetrization is done. Zhang et at. [5-11 rv 15] and Lee et at. [5-28] presented some anisotropic damage models using a different approach without applying symmetrization on the net-stress tensor, which is the most general approach. In this section the detailed properties and characteristics of different models of the damage effective matrix will be described in the principal anisotropic coordinate system.

258

5 Basis of Anisotropic Damage Mechanics

5.7.1.1 Damage Effective Matrix Based on Symmetrization Scheme I In an attempt to make the effective stress tensor symmetric, Murakarmi et at. [5-3,5-6]' Voyiadjis et al. [5-22,5-29]' Hayakawa and Murakarmi [5-30] suggested the first type of transformation in an anisotropic damage model form the Cauchy stress tensor [0"] to the effective stress tensor [8-*] as

[8-*] =

~{[0"]([1]-

[0])-1

+ ([1]- [0])-1[0"]}

(5-123)

where [0] and [I] are the damage state matrix and identity matrix, respectively. The function of symmetry in Eq.(5-123) can be observed in the algorithms of the detailed expression of matrixes as follows

[" "] 0"11 0"12 0"13

"'* "'* " "'*0"23 0"21 0"22 A*

"'* A*

=

0"31 0"32 0"33

"21

1[au a" a,,] 0"21 0"22 0"23

0"31 0"32 0"33

1 1-01

0

0

0

1 1-02

0

0

1 1-03

0

+

1 1-01

0

0

0

1 1-02

0

0

1 1-03

0

[au a" a,,] ) 0"21 0"22 0"23

0"31 0"32 0"33

in which all components are defined in the principal anisotropic coordinate system. Thus

(5-124) It is clear that the effective stress matrix expressed by Eq.(5-124) becomes a symmetric one. Eq.(5-124) can be rewritten in the form of stress vectors * * * * * * }T {0"- } = {0"11,0"22,0"33,0"23,0"31,0"12 }T an d 0" *} = {0"11,0"22,0"33,0"23,0"31,0"12 by employing the damage effective matrix or the damage effective transformation expressed as Eq.(5-125) {A

A

A

A

A

A

A

5.7 Different Models of Damage Effective Matrix

259

'* '* 0"22 0"11

&33

'* '* 0"31 '* 0" 12 0"23

1 1- 0

0

0

0

0

0

0

1 1- O 2

0

0

0

0

0

0

1 1- 0 3

0

0

0

0

1

0

0

!(_1_ 2 1- O 2 1

0"11

0

0"22

0

0"33

+1-0)

0

0

0

0

0

0

0"23

!(_1_ 2 1- 0 3 1 + 1- 0 1 )

0

0

0

0"31 0" 12

0

!(_1_ 2 1- 0 1 1 +1-0)

(5-125) In most articles about damage mechanics, the relation of Eqs.(5-124) and (5-125) is always represented in the form of tensor instead of the form of matrix as

&;j =

~ijklO"kl or

&;j = [(c5ikc5jl -

Okc5jl)-lO"kl

+ O"kl(c5j lc5i k

-

01c5i k)-1l/ 2

(5-126) where the damage effective tensor ~ijkl is a 4th order symmetric tensor (5-127) 5.7.1.2 Damage Effective Matrix Based on Symmetrization Scheme II The second symmetrization treatment was carried out by Cordebois and Sidoroff [5-2, 5-7], and Chow and Yang [5-27], etc. to put forward as follows 1

1

[0'*] = ([1]- [0])-2 [0"]([1] - [0])-2

(5-128)

260

5 Basis of Anisotropic Damage Mechanics

The function of symmetry in Eq.(5-128) can be observed from the following algorithms 1

VI - D1 o

o

o

1

o

o

o

1

Vl- D3 1

o

o

o

1

o

o

o

1

0"1\ 0"12 0"13] [ O";h O"h 0"23 = 0"31 0"32 0"33 0" 11

1 - D1 0"21

J(1 - D1)(1 - D2) 0"31

0" 12

0" 13

J(1 - D1)(1 - D2) J(1 - D1)(1 - D3) 0"22

0"23

1 - D2

J(1 - D2)(1 - D3)

0"32

J(1 - D1)(1 - D3) J(1 - D2)(1 - D3)

0"33

1 - D3 (5-129)

It can be seen that the effective stress tensor becomes a symmetric one after the suggested transformation. Similarly, Eq.(5-128) or (5-129) can be rewritten in the form of stress vectors {a-} = {O"ll, 0"22, 0"33, 0"23, 0"31, 0"12}T and [{O"*} = {O"i1, 0"22, 0"33, 0"23, 0"31, O"i2}T by employing the symmetrized damage effective matrix or the damage effective tensor [!P"] as

5.7 Different Models of Damage Effective Matrix 1 0 I-Dl 0 1 0 I-D2 0 0 0 I-1D3 0 0 0

0

0

0

0

0

0

0

0

0

1

0

0

1

0

0

0

0

V(1-D2)(I-D 3 ) 0

0

0

0

0

V(1-D 3 )(I-D1 ) 0

261 0"11 0"22 0"33 0"23 0"31

1

V(1-D1 )(I-D 2 )

0"12

(5-130) If the relation of Eq.(5-130) is represented in the form of tensor instead of the form of matrix, it gives

-* _

O"ij -

,T,.

'PijklO"kl

-* -_ (5:uik or O"ij

-

5: )-1/2 O"kl (5:Ujl

n

Jtkuki

-

n

5: )-1/2

JtlUlj

(5-131)

where the damage effective tensor Ilfi jkl is a 4th order symmetric tensor and in the form of (5-132) 5.7.1.3 Damage Effective Matrix Based on Symmetrization Scheme III Chow and Yang [5-27] assumed the third transformation from the Cauchy stress tensor [0"] to the effective stress tensor [~*] in an anisotropic damage model has the form as follows

[ ~*l __ [('l/Ji+2'l/Jj)-I~O'J'l v

v

0

(no summation £or i and) j

(5-133)

where'l/Ji = 1- Di (i=1,2,3) are the principal continuous factors. The function of symmety in Eq.(5-133) can be observed in the same algorithms as described before, and the detail follows thus: 0"11

20"12

+ 'l/J2

'l/Jl 'l/Jl 0"21 'l/J2 'l/J3

0"22

+ 'l/Jl 'l/J3

+

'l/Jl 'l/J3 0"23

'l/J2 'l/J2 0"32

0"31

+ 'l/Jl

20"13

+ 'l/J2 'l/J3

+ 'l/J3

0"33

262

5 Basis of Anisotropic Damage Mechanics

(5-134)

Eq.(5-133) or (5-134) can be rewritten in the form of vector as {iT}

{0"11' 0"22' 0"33' 0"23' 0"31' 0"12}T and [{o*t = {aj\,a::b,a33,a23,a:h,ai2}T by employing the damage effective matrix [tlf3] or the damage effective tensor as 1 1- [21

0

0

0

0

0

1 1- [22

0

0

0

0

0

0

0

0

0

"* 0"11

0

"* 0"22

0

0

"* 0"33

1 1- [23

0

0

0

"* 0"23

1 1-

"* 0"31

0

0

0

[22+[23

2

1

0 1-

"* 0"12

0

0

0

0

0

[23+ [21

2

1

0 1-

[21 + [22

2

0"11

0"22 X

0"33

(5-135)

0"23 0"31 0" 12

The tensor notation of the damage effective relation in Eq.(5-135) is

"* ='l'ijkIO"kl .Tr O"ij

a7j

or

= [(c5ikc5jl + c5ilc5jk) (1

-

[2j)-l + (c5jkc5il + c5j l c5i k (1

-

[2i)-1)/4]O"kl

(5-136) where the damage effective tensor detail of

tlfijkl

is a 4th order symmetric tensor with

5.7 Different Models of Damage Effective Matrix

263

5.7.1.4 Damage Effective Matrix Based on Unsymmetrization Scheme Because of a physical lack of comprehensive symmetrization processes, in this regard the approach represented by Zhang et al. [5-lO rv I5]' Valliappan et al. [5-16, 5-31 rv 33]' Murti et al. [5-34]' and Lee [5-28] is preferred to cover the symmetrization process. The most natural definition of an anisotropic effective stress tensor should be in the form of (J11

712

713

* *712 * 713 (J11

---------------

23

---------------

[ 721

(J22 7

731 732 733

I - ill 1 - il2 1 - il3 721

722

723

I - ill 1 - il2 1 - il3 731

732

(5-138)

733

---------------

I - ill 1 - il2 1 - il3

Obviously, the anisotropic effective stress tenser [(Jij ] is a non-symmetric tensor with respect to the components of effective shear stresses, which should satisfy the compatibility relation of the effective shear stress components as 7tj = 7ji(1 - ili)/(1 - ilj). In subsection 5.3.1, Eq.(5-138) has been rewritten in the form of a vector using the damage effective matrix in Eq.(5-22) (i.e. the tensor of continuity factors [tli]) is

{o-*} = [tli]{o-}

(5-139)

Because the shear pair law for anisotropic effective shear stresses is no longer satisfactory, due to its non-symmetric nature, the anisotropic damaged effective stress vectors {o-*} should be rewritten in the form of 9 x 1 rank and Cauchy stress vector {o-} in 9 x 1 rank, respectively, according to an anisotropic coordinate system (Xl X2 X3) that should be defined by (5-140) (5-141 ) Thus, the damage effective transformation matrix [tli] (i.e. damage effective tensor) in Eq.(5-139) should be defined by a matrix of (9x6) rank, which has already been given in subsection 5.3.1 by Eq.(5-22) as

264

5 Basis of Anisotropic Damage Mechanics 1

o

o

o

o

o

o

o

o

o

o

o

o

1- [21

o

[tJ.f]

=

o

oo

o

o

oo

o

o

o

o

ooo

o

ooo

o

o

(5-142)

o 1

1- [22

o

1

1- [21

Eq.(5-139) is the almost general relationship that transforms the Cauchy stress vector to the effective (net) stress vector in a 3D anisotropic principal damage model. It should be noted that the matrix (or tensor) [tJ.f] has been given different names in different articles on damage mechanics, such as "damage effective matrix", "damage effective tensor", "damage effective functions", "damage effective transform matrix", "tensor (or matrix) of continuity factors", as well as having been represented by different symbols, such as [tJ.f], [M], [N] and tJ.fijkl, Mijkl, Nijkl, etc. 5.7.1.5 Inverse of Damage Effective Transformations for Different Schemes

The transform matrix [tJ.f] overall symmetrization models are real symmetric square matrixes that can be inver sed as [tJ.f]-1[tJ.f] = [I], and the inversed effective relation of Eq.(5-23) is expressed by the inverse transformation from the effective stress tensor to the Cauchy stress tensor as (5-143) where details of the inverse matrix of the damage effective tensor for symmetrization model I, are

5.7 Different Models of Damage Effective Matrix

[tirr1

=

1- n 1

0

0

0

0

0

0

1- n 2

0

0

0

0

0

0

1- n3

0

0

0

0

0

0

0

0

2

(1-!t2)(1-!t3) (l-!t2)+(1-!t3)

0

0

0

0

0

0

0

0

2

(1-!t3)(1-!tll (1 !t3)+(1 !tll

0 2

0

265

(1-!tll(1-!t2) (1 !tl)+(l !t2)

(5-144) For symmetrization model II, the inverse matrix of the damage effective matrix is [tiF]-l = 1- n 1

0

0

0

0

0

0

1- n 2

0

0

0

0

0

0

1- n3

0

0

0

0

0

0

V(l - n2)(1 - n3)

0

0

0

0

0

0

V(l - n3)(1 - n 1 )

0

0

0

0

0

0

V(l - n 1 )(1 - n 2)

(5-145) For symmetrization model III, the inverse matrix of the damage effective matrix is '" -1

[w]

1- fh

0

0

0

0

0

0

1- O 2

0

0

0

0

0

0

1- 0 3

0

0

0

0

0

0

+ 0 3 )/2

0

0

0

0

0

0

+ 0 1 )/2

0

0

0

0

0

1 - (0 2

1 - (0 3

0

1 - (0 1 + O 2 )/2 (5-146)

When using the inversed relation in Eq.(5-143) to present an opposite transformation from the effective stress vector to the Cauchy stress vector

266

5 Basis of Anisotropic Damage Mechanics

in the general 3D unsymmetric anisotropic damage model, the formulation should be rewritten as ******** *}T { (J" 11, (J" 22, (J" 33' (J" 23' (J" 31, (J" 12 }T['Tr-1]{ '¥ (J" 11, (J" 22, (J" 33' (J" 23' (J" 32, (J" 31, (J" 13' (J" 12, (J" 21

(5-147) where the matrix of [lJf-1] has been defined as the matrix of generalized inversion for [lJf] in subsection 5.3.1 and expressed by Eq.(5-24) with a matrix of (6x9) rank as

[lJf-1] = 1-

D1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1-

0

0

1-

0

0

0

0

0

0

0

0

0

0

0

0

0

D2

D3

1- D3 1- D2 2 2

------

1- D1 1- D3 2 2

------

0

0

1- D2 1- D1 2 2 (5-148)

------

It should be noted again that the matrix symbol W]-l is not a traditional inverse matrix of [lJf] as symbol [lJf]-l, since the product of [lJf]-l[lJf]=[I]6x6 (or [lJf]-l[lJf]=[I]6x6) for overall symmetrization models gives the (6x6) rank unit matrix, but W]W]-l = [E]9X9 for the unsymmetric model is a (9x9) rank matrix, which is not a unit matrix, unfortunately.

5.7.2 Matrix [tl'] Expressed by Second Order Damage Tensor in Different Schemes 5.7.2.1 Principal Damages as Eigenvalues of Second Order Damage Tensor In subsection 5.7.1, all damage effective matrixes in different symmetrization schemes were represented by the principal damage vector {D} = {D1' D 2 , D3}T, which is obtained by solving eigenvalues and eigenvectors with respect to the real and symmetric second order damage tensor [D] (see subsection 5.2). Thus the general form of the real and symmetric second order damage tensor [D] has the form

5.7 Different Models of Damage Effective Matrix

267

(5-149)

Slu,

in which there are only 6 independent components as Sl22, Sl33, Sl23, Sl3I, Sl12, due to symmetric nature, wherein the quantity of these components should be positive and less-equal than 1, as well as the same for principal damage variables SlI, Sl2, Sl3. The principal damage variables SlI, Sl2, Sl3 can be by solving the 3 eigenvalues of the real and symmetric second order damage tensor [Sl] as

det

Slu - Sl

SlI2

Sl3I

SlI2

Sl22 - Sl

Sl23

Sl3I

Sl23

Sl33 - Sl

=0

(5-150)

On this basis, its principal values (i.e. eigenvalues) may be obtained by solving the corresponding eigenequation after expanding the determinate in Eq.(5-157). The eigenequation in Eq.(5-157) is a cubic algebraic equation of Sl written as (5-151) in which (5-152)

h =

=

ISl22 Sl231 + ISlu SlI31 + ISlu SlI21 Sl32 Sl33 Sl3I Sl33 Sl2I Sl22

Sl22Sl33

+ Sl33Slu + SluSl22 -

(Sl~3

(5-153)

+ Sl~1 + Sli2)

Slu SlI2 SlI3 13 =

Sl2I Sl22 Sl23 = Sll1Sl22Sl33+2SlI2Sl23Sl3I-(Sl22Sli3+SluSl~3+Sl33Sli2) Sl3I Sl32 Sl33

(5-154) They are invariables of the damage tensor (invariants). To solve the eigen equation Eq.(5-151),we may directly apply the rooting formulas of the Cardan equation and obtain:

(5-155)

268

5 Basis of Anisotropic Damage Mechanics

the simple notation of which is denoted by I fli=;

J(

+ 2-Vrcos[e + 1200(i-1)]

(i=1,2,3)

(5-156)

i

_~)3, e = arccos( -2~); in which p = h - Ifj3; q = hI2/3where r = 2Ir /27 - 13. Which are functions of damage invariants only, and therefore are invariable quantities too. From Eq.(5-160), the maximum of fl1' fl2' fl3 can be determined. Because roots fl1' fl2' fl3 of cubic Eq.(5-151) should be real positive values, which are within 0 ~ fli ~ 1 these 3 principal damage variables can also be obtained directly by solving the cubic algebra Eq.(5-151) with respect to fl and presented as

(5-157)

where (9 =

[8Ir + 10813 - 36hh

+ 12(12I~ - 3I~Ir -

1

54hhI3

!

+ 81Ij + 12hIl) 2]3

(5-159) Because the three principal anisotropic damage variables fl1' fl2' fl3 have to be real positive values less than 1 based on the physical nature of damage definition, the quantity in a square root must be ? 0, i.e. Ll = 12I~ - 3IiIr 54121113 + 81I5 + 12I3Ir ? O. Thus, Ll > 0 gives three real principal unequal quantities of fl1' fl2' fl3 whereas Ll = 0 gives three real coinciding roots, which reduces into isotropic damage state (fl1 = fl2 = fl3 = fl). Once the principal damage is obtained, then the principal axial directions of the damage tensor can also be determined. 5.7.2.2 Principal Damage Directions as Eigenvactors of Second Order Damage Tensor

The principal damage direction of fli is defined by a set of direction numbers (direction cosine) of the outward normal {ni}={ni1, ni2, ni3}T on the corresponding principal damage plane. The solution of eigenevectors equations

5.7 Different Models of Damage Effective Matrix

269

with respect to the corresponding eigenvalue fli represented by the principal damage variable gives the principal damage direction as fli ) nil + fl12ni2 + fl13ni3 = + (fl22 - fli ) ni2 + fl23ni3 = + fl23ni2 + (fl22 - fl i ) ni3 =

(fl11 fl12ni1 fl13nil

O} (i

0

=

(5-160)

1,2,3)

0

where the direction cosine should satisfy the orthogonal and standardization relationship as

+ ni2 2 + ni3 2 = 1, (i = 1,2,3) nilnj1 + ni2nj2 + ni3nj3 = 0, (i i=- j ni1 2

=

(5-161)

1,2,3)

The corresponding direction cosines with respect to fli can be determined by substituting the appropriate principal value, fl i , into any two of the equations given by Eq.(5-16) and using the relationship ofEq.(5-161). For example, if the principal damage direction (direction cosines) with respect to fli is desired, this can be obtained as follows Li

nil = ( L21,

(i

=

1 ; ni2 = - - - - - - - - - - , - 1 ; ni3 =

+ M2 + N2)2 1,

1,

(L21,

+ M2 + N2)2 1,

1

(L21,

1,

+ M2 + N2)2 1,

1,

1,2,3) (5-162)

where Li = fl31 (fl22 - fli ) - fl12fl23 } Mi = fl23 (fl11 - fl i ) - fl31fl12 , Ni = (flu - fl i ) (fl22 - fl i ) - flr2

(i

=

1,2,3)

(5-163)

From Eqs.(5-151), (5-152) the three components of principal damage can be determined by solution of Eq.(5-151), such as Eq.(5-155) and implied as functions of 6 independent components of the real and symmetric second order damage tensor fl ij as (fl 1 , fl 2 , fl3)

fl1 fl2 fl3

= F1(flu, fl 22 , fl33 , fl 23 , fl31 , fl12)} = F2(flu, fl 22 , fl 33 , fl 23 , fl 31 , fl 12 ) = F3(flu, fl 22 , fl 33 , fl 23 , fl 31 , fl 12 )

(5-164)

From Eqs.(5-162) and (5-163), components of the principal damage direction {ni}={nil, ni2, ni3}T with respect to fli can be determined as a set of nonlinear functions of the principal damage fli and components of the real and symmetric second order damage tensor fl ij .

(5-165)

270

5 Basis of Anisotropic Damage Mechanics

Therefore, all components of the principal damage direction {ni}={ nil, ni2, ni3}T (i=1,2,3) can be determined from Eqs.(5-162) and (5-163) and finally expressed by substituting Eq.(5-164) into Eq.(5-165) and implied as nonlinear functions of components of the real and symmetric second order damage tensor flij only by n11 nl2 n13] [ n21 n22 n23

(5-166)

n31 n32 n33

where the expression Iij ([flkl ]) (i, j =1,2,3) are functions of the 6 independent components of the real and symmetric second order damage tensor in the form of (5-167)

5.7.2.3 Property of Principal Damage Directions with respect to Damage and Stress In the above description, either the effective stress vector **** *}T, {IJ- * } = {* ****** {IJ-* } = {* 1J11,1J22,1J33,1J23,1J31,1J12 1J11,1J22,1J33,1J23,1J32,1J31,1J13, lJi2, 1J21}T or the Cauchy stress vector {iT} = {IJ11, 1J22, 1J33, 1J23, 1J31, IJ 12} Tare expressed in the same principal anisotropic coordinate system with respect to the principal damage vector {fll, fl2 , fl3Y' When substituting Eq.(5-163) or Eqs.(5-155), (5-157) into the damage effective relation {iT*} = [!li]{iT}to eliminate the principal damage vector instead of the invariable functions of the second order tensor of damage, then the damage effective matrix [w] will be defined in the natural (global) coordinate system, from which the principal anisotropic coordinate system was obtained by the transformation of the eigenevector tensor (i.e. direction cosine tensor of principal damage plane) as defined by Eq.(5-166). Consequently both the effective stress vector and the Cauchy stress vector appearing on two sides of the effective equation {iT*} = [w]{iT} should also be transformed into the natural (global) coordinate system. That can be done by

n11 nl2 n 13] T [ n21 n22 n23

(5-168)

n31 n32 n33

(5-169) Spreading the matrixes production in Eq.(5-169) gives

5.7 Different Models of Damage Effective Matrix

T

0- 11 0- 12 a31

n11 n12 n31

5 23

n12 n22 n23

0- 31 0- 23 a33

n31 n23 n33

0"12 0"13

n11 n12 n31

0"21 0"22 0"23

n12 n22 n23

&12 5 22

0"31 0"32 0"33

n31 n23 n33

0"11

271

n~1a-l1+n~2a-22+nila-33+2n31n12a-23+2nl1n31a-31+2nl1n12a-12 nil "120- 11 +"22"12 5 22+"23"31 0-33+(n22"31 +n23"12)0-23+(nl1 "23+"31 "12)0- 31 +(nI2+ n ll "22)0- 12

nil n31 all +"12"23 5 22+"33"31 0-33+(n33"12+ n 23"31 )0-23+(nl1 n33+n~1)a31 +(nl1 "23+"31 "12)0- 12

nil "12 5 11 +"22"120- 22

+ "23"31 a33 +(n22"31 +n23"12)0-23 +(n23"11 +nI2"31)0-31 +(nI2+ n ll "22)0- 12

ni2al1+n~2a-22+n~3a-33+2n22n23a-23+2n23n12a-31+2n22n12a-12 n31 "120- 11 +n22n23a22+n33n23&33+(n22n33+n~3)a23+(n23n31 +n33 n 12)a31 +(n22 n 31 +n23 n 12)u12

nil n31 all +"12"23 5 22+"33"31 0-33+(n33"12+ n 23"31 )0-23+(nl1 n33+n~1)a31 +(nl1 "23+"31 "12)0- 12 n31 n12 a l1 +n22n23a22+n33n23u33+(n22n33+n§3)u23+(n23n31 +n33 n 12)u31 +(n22 n 31 +n23 n 12)u12

n~lu11+n§3u22+n~3u33+2n33n23u23+2n33n31u31+2n23n31u12

(5-170) Rewriting Eq.(5-17) in the form of vector notation 0"11

nil

nI2

nI3

2n13n12

2n11 n13

2n11 n12

0"22

n§l

n§2

n§3

2n22n23

2n23n21

2n22n21

u22

0"33

n~l

n~2

n~3

2n33n32

2n33n31

2n32n31

u33

0"23

n31 n12 n22 n 23 n33 n 23

0"13

n11 n 31 n12 n 23 n33 n 31 (n33 n 12+ n 23 n 31)

0"12

(n22 n 33+ n §3)

u11

(n23 n 31 +n33 n 12) (n22 n 31 +n23 n 12) (n11

n33+n~1)

n11 n 12 n22 n 12 n23 n 31 (n22 n 31 +n23 n 12) (n11 n 23+ n 31 n 12)

u23

(n11 n23+ n 31 n12)

u13

(n11 n 22+ n I2)

U12

(5-171 ) The 2nd order real symmetric anisotropic damage tensor can be transformed into the principal anisotropic damage system using the coordinate transformation, which consists of eigenevectors. Then the transformation relation is represented similarly as

(5-172) Since the matrix of eigenevector coordinate transformation is a unit orthogonal self invertible matrix, the inverse of Eq.(5-172) can be simply obtained as

(5-173)

272

5 Basis of Anisotropic Damage Mechanics

Procedures for spreading the matrixes production in Eq.(5-173) are similarly given by the following details

nu

n12 n13

T

[stu st 12 st 31

nu

n12 n13]

[ n21 n22

n23

st 12 st 22 st 23

[ n21 n22 n23

n31 n32

n33

st 31 st 23 st 33

n31 n32 n33

nIl n 11 +nf2 n22+n~1 n33 + 2n 31 n12 n 23 + 2n l1 n31 n 31 + 2n l1 n12 n 12 n11 n12 n l1 +n22'n12 n 22+ n 23'n31 n 33 +(n22'n31 +n23'n12)n 23 +(nl1 'n23+ n 31 n12)n 31 +(nf2+ n l1 n22)n 12

nIl n31 nIl +n12 n 23 n 22+ n 33 n 31 []33+(n33 n 12+ n 23 n 31)[]23+(nl1 n33+ n

§1)n 31 +(nl1 n23+ n 31 n12)n

12

n11 n12 n 11 +n22 n 12o-22+ n 23 n 31 f.?33+(n22 n 31 +n23 n 12)n 23 +(n23 n l1 +n12 n 31)n 31 +(nI2+ n ll n22)f.?12 nI2 n 11

+n~2 n22+n~3n33 + 2n 22 n 23 n 23 + 2n 23 n 12 n 31 + 2n 22 n 12 n 12

n31 n12.!?11 +n22 n 23 .!?22+ n 33 n 23 .!?33+(n22 n 33+ n §3).!?23+(n23 n 31 +n33 n 12).!?31 +(n22 n 31 +n23 n 12).!?12

nIl n31 nIl +n12 n 23 n 22+ n 33 n 31 []33+(n33 n 12+ n 23 n 31)[]23+(nl1 n33+ n

§1)n 31 +(nl1 n23+ n 31 n12)n

12

n31 n12.!?11 +n22 n 23.fl 22 +n33 n 23 .fl33+(n22n33+n§3).fl23+(n23n3l +n33 n 12).fl 3l +(n22 n 3l +n23 n 12).fl 12

n~l .flll +n~3 .fl22+n~3.fl33 +2n33n23.fl23 +

2n 33 n 3l .fl3l + 2n 23 n 3l .fl12

(5-174) Comparison of each element in the matrix between the two sides of Eq.(5174) and recognizing the symmetry gives st 1 = st 2 = st 3 =

nil stu + ni2 st 22 ni2 st U + n~2st22 n~l stu + n~3st22

+ n~l st 33 + 2n31n12st23 + 2nUn31st31 + 2nUn12st12 + n~3st33 + 2n22n23st23 + 2n23n12st31 + 2n22n12st12 + n~3st33 + 2n33n23st23 + 2n33n31st31 + 2n23n31st12

+ n22 n 12 st 22 + n23 n 31 st 33 + (n22 n 31 + n23 n 12)st 23 + (nu n 23 + n31n12)st31 + (ni2 + nun22)st12 = 0 nU n 31 st U + n12 n 23 st 22 + n33 n 31 st 33 + (n33 n 12 + n23 n 31)st 23 + (nu n 33 + n~1)st31 + (nu n 23 + n31n12)st12 = 0 n31 n 12 st u + n22 n 23 st 22 + n33 n 23 st 33 + (n22 n 33 + n~3)st23+ (n23 n 31 + n33n12)st31 + (n22n31 + n23n12)st12 = 0

(5-175)

nU n 12 st U

(5-176)

Eqs.(5-175), (5-176) can be rewritten in the form of vector and matrix as

5.7 Different Models of Damage Effective Matrix

°°0 °

273

11

22 33

(5-177)

23

°°

31 12

(5-178) and characteristics in the matrix form of Eq.(5-176) or Eq.(5-178) give the compatibility relations, which circumscribe the quantity of all components in the 2nd order real symmetric damage tensor and should be within 0:::::; Oij :::::;l. We should also be concerned with the orthogonal relation of the direction cosine given by ni1nj1 + ni2nj2 + ni3nj3 = 0 (i i= j = 1,2,3) and the unit value relation given by ni1 2 + ni2 2 + ni3 2 = 1 (i = 1,2,3). Only 6 elements in matrix [n] are independent due to the symmetric nature of the direction cosine tensor [n], and we have

n11n12 n11n31 n12n31

+ n12n22 + n31n23 + n12n23 + n31n33 + n22n23 + n23n33

+ n22 2 + n33 2 + n22 2 + n23 2 n31 2 + n23 2 + n33 2

0

n11 2

=

= 0;

n12 2

= 1

=

=

0

=

1 (5-179)

1

5.7.2.4 Rotational Effect of Two-Dimensional Damage Coordinate System The components of stress O"ij strain Cij, and damage Oij are defined with respect to the anisotropic principal coordinate system (Xl, X2, X3), which can be termed as material coordinates Let us now take a second set of the Cartesian coordinates system (X, Y, Z) with the same origin but oriented differently to the global coordinates system. The following linear relations between the two coordinates can be established:

{ ~~} [~~~ ~~~ ~~:l {~} =

X3

n31 n32

n33

(5-180)

Z

where [nij] is the direction cosines between eigenvector directions-anisotropic principal directions (Xl, X2, X3) and the global coordinates-original Cartesian coordinates system (X, Y, Z). Since O"ij, Cij, and Oij are tensors, we

274

5 Basis of Anisotropic Damage Mechanics

can establish similarly the transformation as given by Eqs.(5-168), (169) and Eqs.(171)rv(173). In the state of a two-dimensional plane, the direction cosines between two systems of principal and original rectangular Cartesian coordinates can be expressed in terms of the single angle () of inclination. The transformation of lJij, Cij, and Dij from the material coordinates to the principal damage coordinates can be expressed by the transformation matrix [TO"] as given by Eq.(5-29b) for plane stress and by Eq.(5-30c) for plane strain respectively. The angle () of the principal damage coordinates with respect to the material coordinates can be expressed by components of the two-dimensional damage tensor as

2Dxy

tan2()=D -D x

(5-181)

y

The direction cosine in Eq.(5-181) defines the principal damage directions, and the corresponding normal damages are called the principal damages. The maximum and minimum principal damages are expressed as D

1=

D1 =

Dx

+ Dy + 2

Dx+Dy

(Dx - Dy ) 2

+ D2xy

(5-182a)

(Dx - Dy ) 2

+ D2xy

(5-182b)

2

2

2

5.7.2.5 Damage Effective Matrix Expressed by Second Order Damage Tensor Substituting Eqs.(5-155), (5-157) and (5-177) into different models of the damage effective matrix represented by principal anisotropic damage variables in subsection 5.7.1, respectively, gives a general form of damage effective matrix to be represented by some invariable functions of damage tensor invariants expressed by the six independent components of the 2nd order real symmetric damage tensor (vector). The most general model of the damage effective matrix can be obtained by substituting the above expressions of principal damage variables into the unsymmetrc damage effective matrix Eqs.(5-142) or (5-144) rewritten in the form of 9x6 order non-square matrix W]9X6. All non-zero components in matrix W]9X6 (i.e. Eq.(5-22) or (5-142) can be formulated by using Eq.(5155) as follows:

!Pu =

1 I

1- (i

+ 2~cos())

'

!P22 =

1

1- [i + 2~cos(() I

+ 120°)] '

5.7 Different Models of Damage Effective Matrix

~

_

33 -

1

1- [~+ 2~cos(e

W 54 =

~

_

75 -

1-

+ 240°)]'

W _

1

1

1

+ 240°)]'

1

[t + 2~cos(e + 120°)]

1- [~+ 2~cos(e

1

1- [~+ 2~cos(e

44 -

275

+ 240°)]'

' W65 = - - - - - ; 1 ; - - - - - -

1- (t

~

+ 2~cose) 1

_

1- [~+ 2~cos(e

86 -

+ 120°)]'

1

(5-183)

W96 = - - - - - ; ; - - - - - -

1- (~+ 2~cose)

For the symmetrization model I, all non-zero components in the damage effective matrix [tP] can be formulated as follows:

wn =

1

A

1-

[1"- + 2~cose] A

~ 33 -

1{

1

A

1

'

W22

=

W55 = A

W66 = A

1{

2

1

1- [~+ 2~cos(e

1

1

1- [~+ 2~cos(e

1{

[1"- + 2~cos(e + 120°)] '

-~--------

W - 44 - 2 1- [~+ 2~cos(e + 120°)] A

1-

1

1

2 1- [~+ 2~coseo]

+ 240°)]

1+ } 1}

+ -----;0-------1- [~+ 2~cos(e

+ 240°)]

240°)]

+ -----;0----1- [~+ 2~coseO]

1+ }

+ -~-------1- [~+ 2~cos(e

120°)]

(5-184)

For the symmetrization model II, all non-zero components in the damage effective matrix [l]/] can be formulated as follows: wn =

1

-

1- [~+ 2~cose]' -

~

W22

1

= -----;o---=--.,-----C"":"

1- [~+ 2~cos(e

1 - -----;0-------1- [~+ 2~cos(e + 240°)]

33 -

+ 120°)]'

276

5 Basis of Anisotropic Damage Mechanics (5-185)

For the symm~trization model III, all non-zero components in the damage effective matrix [l]/3] can be formulated as follows: ~

W11

1

=

[%- +

1-

['I¥" +2~cos(e+2400)] '

2~cose]'

1-

1 =

-

W55 =

= ------;--------

1-

~

W33

1

~

W22

0'

2~cos(e + 120°)]'

-

1

[%- - --i- cos(e +

1-

[%- +

'

-

W66 =

1 =

1-

('¥I " -

1

2~cose)

'

1

--------,,-0'=-----

[%- + --i- cos(e +

60°)] (5-186) Using Eq.(5-162) all non-zero components of the unsymmetric damage effective matrix [Wjgx6 can be reformulated in the altern ant form as

W _ 11 l[f.

l[f.

1

1-

_

22 -

1 -{

_

33 - 1 W_ 44 -

l[f.

1 -{

_

86 -

l[f.

1-

_

75 l[f.

1 -{

_

65l[f.

{I-t + 2 12 - e [8

4f - (1 -

3 (1 3 12 -

1-

2)J}

1 gIl

1

A) [;; + ~ (~I2-

{4f - (1 + A)

{4f - (1 + F3)

V?) J}

1

[;; - ~ GI2 - ~I?)J}

1

[;; - ~ (V2 -

V?)J}

1

--~-------------~

54 -

l[f.

1-

120°)]

W44

_

1 -{

4f - (1 -

-t + 2 [812

{I

4f - (1 4f - (1 -

A ) [;; + ~ (V2 -

V?)J}

1

12)J} - e3(13 12 - gIl 1

A ) [;; + ~ (V2 1

A ) [;; + ~ (~I2 1

96- 1 - {I-t + 2 [812 - e3(13 1 gIl 12)J} 2 -

(5-187)

V?)J} V?)J}

5.7 Different Models of Damage Effective Matrix

277

Corresponding to Eq.(5-184), the damage effective tensor Pijkl of symmetrization model I can be represented by using the above expressions as follows:

Pijkl = [

(bikbjl -

+ (bj1bik -

{i +

{i +

2.qrcos[B + 1200(k -1)]} bjl)-l

2.qrcos[B + 1200(l-1)] }bik )

-1] /2

(5-188)

or in the alternant form

(5-189)

Similarly, for the symmetrization model II, the damage effective tensor can be formulated as

(5-190)

or in the alternant form

(5-191)

278

5 Basis of Anisotropic Damage Mechanics

For the third symmetrization model III, the 4th order symmetrized tensor of damage effective functions can be given in detail as

or in the alternant form

(5-193)

Actually, the damage effective matrixes have a payoff function on properties in most quantities in damage mechanics, such as effective stress, effective strain, damage constitutive, damage kinetics, damage energy release, damage energy dissipation and so on. This is thus a basis and key to modeling different damage mechanics models. Consequently, we can unreservedly say that once the damage effective functions are constructed properly, there is the potential for carrying out most aspects of damage mechanics modeling. From the above descriptions, it should in particular be pointed out that unsymmetric anisotropic damage mechanics modeling expressed by the principal anisotropic damage variables (vector) {Ol, O 2, 03}T has produced significant advances not only in mathematical simplicity but also in physical maturity. This is because principle damage modeling can provide all the same functions and rules as those provided equivalently by employing the second order damage tensor in the damage mechanism, which implies that there is no need to use the complex and disadvantageous formulations expressed by second order damage tensor modeling due to equivalent functions. This is ~ecause all components of anisotropic damage effective tensors tirijkl , Ilfi jkl , l}/ijkl and l}/ijkl are invariable functions of damage invariants h, 12 , 13 , which are the natural characteristics of a damage state implied in principal damage values and do not change for any coordinate rotation That creates different forms of the second order symmetric damage tensor but with the same types of eigene-structures. So applying the second order symmetric damage tensor to modeling the damage mechanism does not provide anything new, except mathematical complexity. The best way forward in damage mechanics analysis is that we first make an eigenevalue analysis from the initial second order symmetric damage tensor to obtain the principal damage state 0 1 , O 2 , 0 3 ,

5.8 Different Modeling of Damage Strain Energy Release Rate

279

and then take the values of fh, f22' f23 into account in formulations expressed by unsymmetric principal damage modeling.

5.8 Different Modeling of Damage Strain Energy Release Rate 5.B.1 Overview of the Topic

In subsection 5.5 thermodynamic conjugate Y is defined as the damage strain energy release rate and is expressed by the derivative of strain energy with respect to the damage variable. In the case of isotropic damage, Y is a scalar and its derivation is ~ = Y. Even in the case of anisotropic damage, the damage characterization is often achieved for structures under proportional loading. For this loading condition, the principal coordinate system of damage can be assumed to coincide with that of stress and the shear stress in this principal coordinate system of damage is assumed to be zero too. However, except in a limited number of cases, most service loading conditions are nonproportional. Under such a loading condition, the principal directions of stress no longer coincide with those of damage. Accordingly, shear stress in this kind of principal coordinate system of damage is non-zero and this should be taken into account. This calls for the derivation of a generalized form of the damage strain energy rate. Unfortunately, such a generalized form is not at present available as highlighted recently by Chaboche who stated that "the damage elastic stiffness tensor and the damage energy release rate are difficult to express in the general case" [5-35], It is to remedy the shortcoming in the applicability of different modelings of the damage strain energy release rate in practical engineering problems that a further topic of investigation is planned. In some articles, the concepts of effective strain rather than effective stress are introduced into the formulation of constitutive equations. In general, the damage effect tensor [\]i ( { f2} )1 is a fourth order tensor according to the definition of effective stress, but a second order damage tensor is also employed in damage analysis in many references. The simplification becomes necessary for practical reasons as elements in a fourth order tensor are difficult if not impossible to measure. In subsection 5.7, the representations of the above mentioned.o.three forms of the symmetrization damage effective tensor tIrijkl ,

lffijkl and \]iijkl expressed in terms of second order symmetric damage tensor are discussed in detail. This section is followed by the further representation of the damage strain energy releas; rates corresponding to these three symmetrized forms of tIrij kl' lffij kl and \]iij kl·

280

5 Basis of Anisotropic Damage Mechanics

5.8.2 Modification of [tl'] Based on Different Symmetrization Models When the damage tensor is chosen as a second order symmetric tensor, there are several ways to construct the damage effective tensor [tJ.f({D})] [5-36]. It has been found in subsection 5.7 that as [tJ.f( {D})] is a fourth order symmetric tensor, it could be represented by a 6x6 matrix, but in the principal coordinate system of damage {D}, [tJ.f({D})] was chosen as a second order symmetric tensor for its convenience of application. The details of this were represented respectively for the three different symmetrization cases as shown in subsection 5.7. In order to derive a different form of strain energy release rate {Y} corresponding to the three symmetrization model, an assumed form of [tJ.f([D])] in an arbitrary coordinate system in terms of a second order symmetric tensor [D] was suggested by Chen and Chow in [5-37]. This method is required to deduce the derivative of [tJ.f([D])] with respect to [D] for the alternant formulation of the damage strain energy release rate. For the above mentioned three symmetrization forms of [tJ.f([D])], they can be written in different tensor forms thus: For symmetrization model I, the damage effective tensor can be defined as (5-194)

where (5-195) (5-196)

For symmetrization model II, (5-197)

where (5-198)

For symmetrization model III, A

[~3([D])] =

A

([1] - [0])

-1

(5-199)

where Oijkl =

(8 i k Djl

+ 8il Djk + 8j k Dil + 8j IDi k)/4

(5-200)

From the above representations, it is evident that there are different methods for constructing [!P"l([D])], [~2([D])] and [$3([D])], then [D] is a second order symmetric damage tensor.

5.8 Different Modeling of Damage Strain Energy Release Rate

281

5.8.3 Different Forms of Damage Strain Energy Release Rate The damage strain energy release rate [Y] can be defined for symmetrization models as [5-38, 5-2],

(5-201 ) where "8" indicates taking the symmetrization part. Due to the symmetry of [Y], {O"} and [0], Voigt's notation can be employed. Then [Y] and [0] can be represented in vector form as 0"1

{Y}

; {O}

=

=

0 11 0 22 0 33 0 23 0 31 0 21

fh O2

03 04 05 06 (5-202)

An essential step in the derivation of {Y} is to deduce the derivative of

[W({O})] with respect to the damage variable (vector) {O}. Accordingly, the different models of the damage strain energy release rate can be derived with Eq.(5-201) as follows For symmetrization model I: From Eq.(5-194), since [tP-1([0])] = [&] we have

a[tP-1 ({ O} )] a{O}

a[&] a{O}

a[&] a[q)] a[q)] a{O}

---

(5-203)

and with Eq.(5-196) we have

(5-204) Where [&] is a fourth order symmetric tensor, it can be represented by a 6x6 rank matrix as

q)11 0 0 0 q)31 q)12 0 0 0 q)23 q)12 q)22 0 0 0 q)23 q)31 q)33 [&] = 0 q)23/2 q)23/2 (q)22 + q)33) /2 q)3!/2 q)12/2 q)3!/2 0 q)3!/2 q)3!/2 (q)11 + q)33)/2 q)23/2 q)3!/2 q)12/2 q)12/2 0 q)23/2 (q)11 + q)22)/2 (5-205) Differentiating [&] with respect to [q)]

282

5 Basis of Anisotropic Damage Mechanics

(5-206) The damage strain energy release rate can be similarly obtained to that employed in symmetrization models by first obtaining the derivative of the damage effect tensor to damage tensor. {Y} is then derived with Eq.(5-201) as

(5-207d)

(5-207e)

(5-207f) where

5.8 Different Modeling of Damage Strain Energy Release Rate {/;11 =

of,

_

1 - st 11 ;

_

1 - st 22 ;

2(1 - st 22 ) (1 - st 33 ). 2 - st 22 - st 33

'1-'23 -

of,

{/;22 =

{/;33 =

1 - st33

_

2(1 - st 33 ) (1 - st 11 ) .

,'1-'31 -

'

of,

283

2 - st 33 - st 11

(5-208)

2(1 - st 11 )(l - st 22 ) 2 - st 11

'1-'12 -

-

st 22

which are the diagonal terms of [!P"1 ({ st})] in the principal coordinate system of {st} as shown in Eq.(5-125) of subsection 5.7.l. For symmetrization model II: Based on Eq.(5-197) we have (5-209) -

2

-

2

d[tli2({st})] [P({st})][P({st})]-l= _[~ ({st})]2d[P({st})] [P({st})]-l d{ st} 2 d{ st} d[tli2({st})] d{st}

=

_[~ ({st})]2d[P({st})] [~ ({st})]2 d{st}

2

2

(5-210) From the above equation,

d[tP2({st})]2 d{ st}

=

d[tP2({st})] d{st}

=

d[tP2({st})] d{ st}

d[1({n})]

in}

can be obtained as

2[~ ({st})]d[tP2({st})] d{ st}

2

~[~ ({st})r1d[tP2({st})]2 2

d{st}

2

(5-211)

=

~[tP2({st})r1(-[tP2({st})]2d[~~~})] [tP2({st})]2)

=

_~[~ ({st})]d[P({st})] [~({st})]2 2

2

d{ st}

2

Substituting the above into Eq.(5-201), {Y} is obtained as (5-212) Because [P( {st})] is a fourth order symmetric tensor, it may therefore be represented by a 6x6 rank matrix with the introduction of Voigt's notation. From Eq.(5-198) we have

284

5 Basis of Anisotropic Damage Mechanics

-2(1 - Jt22)Jt23 -2(1 - Jt33)Jt32

[P]=

-(1 - Jtll)Jt12 -(1 - Jt22)Jt12

Jt23Jt31

-2(1 - Jtll)Jt13

-2(1 - Jtll)Jt12

Jt23Jt31 - (1 - Jt33)Jt12

Jt23Jt12 - (1 - Jt22)Jt31

(1 - Jt ll )(l - Jt 33 ) + Jt~l Jt31Jt12 - (1 - Jtll)Jt23 Jt31Jt12 - (1- Jtll)Jt23 (1- Jtll)(l- Jt22)

+ m2

(5-213)

Differentiating it with respect to damage variable {D} gives

aPijkl aDpq [(Oik - Dik)OjpOlq+ (Ojl- Djl)OipOkq+ (Oil- Dil)OjpOkq+ (Ojk - Djk)OipOlq] / 2 (5-214) The above expression is a 6th order tensor. Theoretically, {Y} may be obtained by substituting it into Eq.(5-212), but this is a complex way to derive {Y}. The procedure employed instead in this investigation is to first derive

alP] alP] alP] alP] alP] alP] . . aDu' aD22 , aD33 , aD23 ' aD3I , aD I2 respectIvely, whIch are fourth order tensors. If we assume that the original (undamaged) material property is in isotropy and the anisotropy of materials is induced only by the anisotropic damage, then components of the damage strain energy release rate with respect to the symmetrization model II can be obtained as follows:

5.8 Different Modeling of Damage Strain Energy Release Rate

285

(5-215d)

(5-215e)

(5-215f) where

if;22

1 - Sl22;

if;ll

=

1 - Slll;

if;23

=

J(l - Sl22)(1 - Sl33);if;31

if;12

=

J(l - Slll)(l - Sl22)

=

if;33 =

=

1 - Sl33

J(l - Sl33)(1 - Sln);

(5-216)

which are the diagonal terms in [l]/2( {Sl})] with respect to the principal coordinate system of {Sl} as shown in Eq.(5-130) in subsection 5.7.1. It can be readily observed that Y23 , Y31 and Y12 may not even be zero in the principal coordinate system of damage, when the principal coordinate system of damage is not coincident with that of stress. Although the above expressions of {Y} are presented in the orthotropic property coordinate system of the material due to anisotropic damage, its value in an arbitrary coordinate system could be easily obtained through the tensor transformation rule. For symmetrization model III: From Eqs.(5-199) and (5-200) ,

0([1] - [D]) o{Sl}

-1

=

([I] _ [0])-2 o[D] o{Sl}

=

(5-217)

286

5 Basis of Anisotropic Damage Mechanics

As [ill is a symmetric fourth order tensor, it can be represented by a 6 X 6 matrix with Voigt's notation.

[ill =

Slu

0

0

0

Sl3I

SlI2

0

Sl22

0

Sl23

0

SlI2

0

0

Sl33

Sl23

Sl3I

0

Sl12/2

Sl3d2

0

Sl23/ 2 Sl23/ 2 (Sl22

Sl3d 2

0

Sl12/2 Sl12/2

+ Sl33)/2

Sl3d 2

Sl3d 2

0

Sl3d 2

(Slu

+ Sl33)/2

Sl23/ 2

Sl23/ 2

(Slu

+ Sl22)/2

(5-218)

Differentiating

[ill with respect to {Sl}, we obtain

(5-219) In the case of symmetrization model III, the derivatives of [ill with respect to Slu, Sl22, Sl33, Sl23, Sl3I, Sl12, are first evaluated, Awhich are also fourth order tensors. The damage strain energy release rate {Y} for the symmytrization model III in the principal coordinate system of damage is obtained by first substituting the resulting derivatives)n the principal coordinate system of damage in Eqs.(5-217) and (5-219). {Y} is then obtained with Eq.(5-201)

5.8 Different Modeling of Damage Strain Energy Release Rate

287

(5-220d)

(5-220e)

(5-220f) where ifll =

1 - D l l;

~23

=

1 - (D22

if12

=

1-

if22

=

1 - D22 ; if33 = 1 - D33

+ D33)/2;~31 (Dll + D22 )/2

=

1 - (D33

+ D l l)/2;

(5-221 )

which are the diagonal terms of [l]I([D])] in the principal coordinate system of {D} as shown by Eq.(5-135) in subsection 5.7.1. The above formulations have been incorporated in a finite element program and employed in a failure analysis [5-39].

288

5 Basis of Anisotropic Damage Mechanics

5.8.4 Discussion and Conclusions 5.8.4.1 Discussion

The four different forms of [tJ.f( {D})] have been formulated based on a second order symmetric tensor of {D}. As shown in subsection 5.7.1, the first three diagonal terms in Eqs.(5-120), (5-130), (5-135) and (5-142) are the same for the four cases considered, indicating that the damage effects on the normal stress are equivalent. The other diagonal terms relating to shear stress are, however, different in each of the cases. When the anisotropic principal coordinate system of damage is coincident with that of stress, 0"23 = 0, 0"31 = 0, 0"12 = 0 then Yn , Y 22 and Y 33 in the principal coordinate system of damage are easily obtained. It is evident that the formulations of Yu , Y 22 and Y 33 are the same in all three cases. These formulations are also identical in form to that of {Y} derived for the proportional loading case where the principal coordinate system of stress {O"} and damage coincide. In the case of proportional loading, the principal directions of {0" } do not change, and Y 23 , Y 31 and Y 12 are zero. This leads to the elements dD23 dD31 dD12 . . .... of~, ~' and ~ bemg zero. In thIS way, the prmclpal dlrectlOns of damage vector {D} will not change and always coincide with those of stress during the loading process. When the principal coordinate system of damage {D} does not coincide with that of stress during the non-proportional loading process, 0"23, 0"31, and 0"12, in the principal coordinate system of damage are non-zero. Y 23 , Y 31 and . dD~ Y 12 have correspondmgly real values as well as those of the damage rate ~' dD31

dD12 .. ... . and ~. ThIS IS because the prmclpal dlrectlOn of damage rotates constantly during the loading process. Consequently, the representation of {Y} in general case should be derived first in order to make possible the nonproportional loading analysis, a loading condition that is often observed in real-life structures under service loading. ~'

5.8.4.2 Conclusions

In this section, damage strain energy release rates {Y}, {Y}, fY} with respect to the three symmetrized models are derived from the there corresponding forms of damage effective functions [tP1([D])], [l]/2({D})], [~([D])] based on the second order symmetric tensor of {D}. The relation between the formulation of the three damage effective functions, which are fourth order symmetric tensors, and the second order symmetric damage tensor {D} is elucidated. The generalized tensor representations of [tP1([D])], [l]/2({D})], [~([D])] and [tJ.f( {D})] in terms of {D} developed in this section are important in extending the present applicability of damage mechanics to solve a wide range of engineering problems. This is compared among the unsymmetric [tJ.f( {D})] and

5.9 Effects of Symmetrization of Net-Stress Tensor in Anisotropic Damage Models

289

symmetrized ['h([D])], [tP2({D})], [$([D])] by both matrix and tensor formulations in the principal coordinate system of damage {D} as shown from Eq.(5-183) to Eq.(5-193). It should be noted that the symmetrized models Eqs.(5-120), (5-130) and (5-135) are only special simplified cases of the generalized anisotropic representation and cannot be used to derive unsymmetric anisotropic generalized formulations of {Y}. So far the developed formulations for unsymmetric {Y} in this book have solved what was previously lacking in the symmetrized models, which is necessary for conducting stress analysis of structures under non-proportional loading, when {Y} is chosen to be the controlling thermodynamics force in the damage evolution equation.

5.9 Effects of Symmetrization of Net-Stress Tensor in Anisotropic Damage Models 5.9.1 Review of Symmetrization Models In an anisotropic damage model, the net-stress l (or effective stress) tensor is naturally non-symmetric making the inversion expensive computationally. To simplify such non-symmetry, some investigators suggested different symmetrization techniques, the effects of which on anisotropic damage models will be briefly described in this section. This treatment, although offering mathematical convenience in the analysis, does not have a physical basis such that the resultant damage model may implicitly include new properties which originally did not exist and may also exclude some significant properties before symmetrization is done [5-40]. Such an arbitrary treatment may in fact pose a spurious engineering analysis [5-40] [5-11]. Therefore, it is necessary to examine and investigate the influence of the effects of different symmetrization schema on anisotropic damage mechanics analysis by comparing them. Zhang et al. [5-10rv12] theoretically, numerically and comparatively studied this problem and sequentially developed an anisotropic damage model without applying any symmetrizations [5-13 rv 16]. In this section the effects of various symmetrization techniques on the state of stress, constitutive relations and shear failure criteria will be described along with some illustrative numerical examples. The limitations of symmetrization treatment and a few suggestions will also be presented in this section. The first type of symmetrization treatment to achieve a symmetrically effective stress tensor has been described in subsection of 5.7.1.1 by symmetrization model I, which was often used by Murakarmi et al. [5-3,5-6], Voyiadjis et 1 In damage mechanics literature, the term "effective stress" has been used in place of "net stress" used above. However, since the term "effective stress" has already an established meaning in the geotechnical engineering field, the use of the "net-stress" term is preferred.

290

5 Basis of Anisotropic Damage Mechanics

at. [5-22,5-29] and Hayakawa et at. [5-30] and expressed previously in Eq.(5123), here with a new equation: [&*] =

~{[O"]([1]-

[Sl])-1

+ ([1]- [Sl])-I[O"]}

(5-222)

It is arranged in this section that the symbol of a quantity with a top-hat "1\,, indicates that it was produced from symmetrization model I such as [0'*]. The second type of symmetrization treatment to achieve a symmetrically effective stress tensor has been described in subsection 5.7.1.2 by symmetrization model II, which was frist suggested by Sidoroff and Cordebois [5-2] and applied widely by many researchers, such as Chow and Yang [5-27], with the following form as in Eq.(5-128):

(5-223) Similarly, the symbol of a quantity produced from symmetrization model II is marked by '-' on the top of the symbol. The third type of symmetrization treatment to achieve a symmetrically effective stress tensor has been described in subsection 5.7.1.3 by symmetrization model III, which was assumed by Chow and Wang [5-41] in order to transform the Cauchy stress tensor [0"] to the effective stress tensor [~*] in an anisotropic damage model, the form of which was rewritten by Eq.(5-133) as

[~*]

= [2 _

Sl~ _ Slj O"ij]

(no summation for i and j)

(5-224)

where 'l/Ji =l-Sli (i =1,2,3) are the principal continuous factors. Sidoroff [5-7] also proposed a different type of expression for the symmetrization process, which has a relationship with the material property v, as given below

[i"] =

~{([1] - [Sl])-I[O"]+[O"]([1]- [Sl])-I}+-V-tr{[O"][Sl]([1]-[Sl])-I}[1] 2 1 + 2v

where "tr" indicates the trace of a matrix, v is Poisson's ratio. The function of symmetrization model III defined in Eq.(5-133) or in (5224) can be observed in the similar algorithms as described before, the symbol of a quantity produced from the symmetdzation model III is marked by ,~ , on the top of the symbol. While the basis for the above symmetrization process is lacking, the above is resorted in an attempt to obtain a symmetric form for mathematical convenience. In this regard the approach presented by Zhang et ai. [5-10"-'16] and Lee et ai. [5-28] is preferred over the symmetrization process.

5.9 Effects of Symmetrization of Net-Stress Tensor in Anisotropic Damage Models

291

5.9.2 Effects of Symmetrization on Net-Stress Tensor In this section, the effects of symmetrizational treatment will be described in the principal anisotropic co-ordinate system. When considering only the principal damage state in an anisotropic damage model, the anisotropic damage state variables consist only of diagonal terms as shown below:

1=

'l/J1 0 0 ['l/J] = [1]- [0] = [ 0 'l/J2 0 o 0 'l/J3

1

[1 - 0 1 0 0 0 1 - O2 0 0 0 1- 03

(5-225)

where ['l/J] is the continuity tensor in a diagonal form along with anisotropic principal directions, which transfers the effective stress matrix to the Cauchy stress matrix as [0"]= ['l/J][O"*] and has a completely different form to [tli] that transfers the Cauchy stress vector to the effective stress vector as {O"*}=[tli]{ O"}. It should be clearly mentioned here that all the components of tensors appearing throughout this study are those used with respect to the principal co-ordinate system of the anisotropic damage tensor ['l/J] or [0]. In subsection 5.7.1 we had mentioned that in most general cases, the corresponding net stress tensor of an anisotropic damaged material can be expressed in the principal anisotropic co-ordinate system as Eq.(5-138) which is now reformed by continuity factors

[0"*] = [0"] ['l/Jr1 =

(5-226)

It is obvious that in general, if'l/Ji i=- 'l/Jj (i i=- j), the pair shear stress components in the net-stress tensor are not equal to each other, i.e. O"tj i=- O"tj , for an anisotropic damage case (see Eq.(5-19) in subsection 5.3.1), and the net-stress tensor [0"*] shown above is no longer symmetric, this can be rewritten in a alternative form as

O"tj

=

~; O"ji ( for i i=- j

and nosummation for i, j)

(5-227)

and the above relation may be thought of as the compatibility relation of the net-stress [5-10"" 11] under consideration of an anisotropic damage state. Using Eq.(5-225), Eq.(5-226) can be rewritten as

[0"* ]([1]-[ 0]) =[ 0"]

'*

[0" * ]=[ 0"]+ [0" *] [ 0]

Substituting Eq.(5-226) back into Eq.(5-228) again gives

(5-228)

292

5 Basis of Anisotropic Damage Mechanics

[0"*] = [0"] + [0"][O]['!f;]-l

(5-229)

where the second term in the above equation may be thought of as a deviation of the net-stress from the Cauchy stress tensor. Such a deviation is given a symbol [L10"] such that (5-230) or L1O"ij =

O· 1 - Hj

~O"ij (no summation for index

j)

(5-231)

Using the basic relations just mentioned above, the symmetrization treatment introduced by model I and model II as illustrated in Eq.(5-222) and Eq.(5-223) respectively may now be rewritten as (5-232) (5-233)

] (no summat·lOn Clor z• and') [0""*] = [('!f;i+'!f;j)-l 2 0"ij J

(5-234)

5.9.3 Influence of Symmetrization on Deviatiom Net-Stress Tensor In order to quantify the effects of the symmetrization process for each scheme, a series of non-dimensional ratios will be introduced here. The first family of ratios consists of the following ratios (5-235a) (5-235b) (5-235c) " O ""* ij

77 - (J

-

2

- --:-:-------:::-:-------:--;-:-----:::---;-

O"ij -

(1 -

0i)

+ (1

-

OJ)

(5-235d)

The above ratios can be logically thought of as the magnification factor of the net-stress tensor from the Cauchy stress tensor, since 0 will always be less than unity.

5.9 Effects of Symmetrization of Net-Stress Tensor in Anisotropic Damage Models

293

The second family of ratios consists of the following definitions, and since the first ratio (i.e. Eq.(5-235a)) is treated as the reference point, there are only 3 ratios in this group A

>.

A

'rju

A

u

I

*

(Tij

=-=-= { 1

(Tij

'rju

_

2

(1 + __

I-D.

1)

1- Di

for i = j for i

(5-236a)

=f. j

(5-236b)

for i = j (5-236c) These ratios can be used as a "measure" of deviation of a symmetrization treatment against the non-symmetrization case as given in Eq.(5-226). The third group of ratios will be given below where the variation of the magnitude of the resultant net-stress tensor components under a symmetrization treatment may be assessed by defining the following

(5-237a)

for i = j for i

5u

=

"'* (Tij (Tij

.:,

-

{ (1 - D.)

- Dj

(5-237b)

for i = j

1

2

+ (1

=f. j

)

1 - - - - for i 1- D j

=f. j

(5-237c)

From above Eqs.(5-237a), (5-237b) and (5-237c), it is obvious that all symmetrization schemes do not affect the magnitude of the normal stress components (i.e. i = j case), but shear stress components are affected. The effects on shear stress components due to the symmetrization schemes may be significant and this will be discussed in subsequent sections. A simple conclusion that can be immediately drawn, in view of above equations, would be the concern that the behaviour of the damaged continuum after a symmetrization

294

5 Basis of Anisotropic Damage Mechanics

treatment may be significantly different from what the real behaviour is supposed to be. Furthermore, in granular materials (soils and rocks alike) shear stress components have no-doubt an important role in their failure behaviour, such that when such a symmetrization scheme is not carefully exercised , it may produce a spurious result and/or an irreversible effect in a materially nonlinear analysis. The variation of the magnification factors 7],n fj(n fj(n 7](J for a range of o ~ [l ~ 0.8 is comparatively shown in Figs.5-13(a), (b), (c) where the referenced ratio 1J(J defined by Eq.(5-235a) has been drawn as a dotted line. For the practical range of [l used , all magnification factors do not become larger than 5. Magnification factors fj(J ' fj(J and 7](J differ significantly from the unsymmetrized 1J(J when [li and [lj attain their opposite extreme values (i.e. [li = 0 and [lj = 0.8, or [li = 0.8 and [lj = 0). 5

5~------------'

~

~

17"

T/,,--

1Z •••

4

4 £),=0.0.0.2:0.4.0.6.0.8

3'----

3

2

2

1

0.0

0.2

0.4 ilj

(a)

0.6

0.8

I

0.0

0.2

0.4

ilj (b)

0.6

0.8

I

0.0

0.2

0.4

ilj

0.6

0.8

(c)

Fig. 5-13 Magnification factor of net-stress t ensor based on models I (a), II (b) , III (c) of symmetrization schemes respectively for different damage states

The effects of these symmetrization treatments on the shear stress component may be visualized by plotting Eqs.(5-236a), (5-236b) and (5-236c) , as presented in Figs.5-14(a), (b) , (c). All symmetrization schemes considered here show a significant influence on shear stress components. These effects again are most severe when [li and [lj attain their opposite extreme values.

t)

It should be also noted that in all symmetrization schemes for (~(J ' ),(J ' the shear stress components increase with the damage component [li increasing and decrease with the damage component [lj increasing. By comparing these figures it can be seen that the maximum values of ratios, (~(J' ),(J ' ~(J) ' defined for shear stresses, may reach the values of about 3.0, 2.0 and 1.7 respectively. Since symmetrization schemes considered here lack a physical basis, the significant deviation as demonstrated in Fig.5-14 is not at all surprising. The mean of deviation of stress magnitude may be seen from the diagrams shown in Figs.5-15(a) , (b) , (c) where deviations b(J ' 8(J ' defined in Eq.(5237) are plotted against [l. Although the deviations due to a symmetrization scheme may not be significant for a small value of [li, they can be signifi-

t

5.9 Effects of Symmetrization of Net-Stress Tensor in Anisotropic Damage Models

295

3.0 ,.--- - - - - - - - - , 2.2 t:"""":X:-.- ; "' - j-' . : : : . - - - - - - - , 2.0 i~j •••• ! '~'.j2.5 , ; ; ; J •••• 1. 8 1.6 2.0 1.4 1.2 1.5

t.

1.0~~"""

1.0~~~~. 0.0

0.2

0.4 [2 j (a)

0.6

0.8 0.6

0.8

Fig. 5-14 Effects of symmetrization models I (a) , II (b) , III (c) on deviations of net-stress magnitudes for different damage states cant when a combination of opposite extreme values of [2j are attained, as has been mentioned previously. It is suggested from this assessment that a symmetrization scheme should be a bandoned in a more accurate anisotropic damage model. 3 2

8.

i'#j i =j

-

---

3 2

8.

i'#j i =j

3 2

~

8.

i'#j i =j

----

0

0

0

- I

- I

- I

-2

-2

-3

_ 3 ~L-L-~~~-L-L-J

-2

[2,=0.0.2,0.4,0.6,0.8

-3

0.00. 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.00. 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.0 0. 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 [2; [2; [2, (a) for mode l I

(b) for model \I

(c) for model 11\

Fig. 5-15 Effects of symmetrization models I (a) , II (b) , III (c) on deviations of net-stress magnitudes for different damage states

5.9.4 E ffects of Symmetrization on N et-Stress I nvariant Most failure criteria are usually expressed in t erms of stress invariants because these invariants do not change the magnitude in any stress transformation. Generally, principal stresses and directions of any stress state may be found by considering the following eigenvalue problem (either for an undamaged or a damaged model) as,

(5-238)

296

5 Basis of Anisotropic Damage Mechanics

where lJij is the stress tensor considered; A = {Al,A2,A3} are eigenvalues (in a 3D state of stress), lij is usually called "eigenvector" (direction cosine) corresponding to the eigenvalue Ai, and 6ij is the Kronecher delta. Hence, the corresponding principal stresses and directions of a damaged stress state (lJ:j or &:j' ij:j and j ) may readily be found by substituting an appropriate expression into Eq.(5-238) and solving the resultant cubic equation as given below

-u:

(5-239) where h, 12 , h are the first, second and third stress invariants. For the non-symmetrized model the corresponding stress invariant can be derived as in [5-34]' and only the final results are represented here (5-240a)

(5-240b)

(5-240c) By applying the considered symmetrization treatments, the resultant net stress invariants may be expressed as

j~

=

1~

(5-241a)

(5-241b)

and similarly (5-242a)

5.9 Effects of Symmetrization of Net-Stress Tensor in Anisotropic Damage Models

I~ = I~

297

(5-242b)

0"11

0"13

1- 0 1 1

0"12

I*3 --"2 J(l - 0 )(1 - O ) 1 2

J(l - O2 )(1 - 0 3 )

0"13

0"33

1- 0 3

(5-242c) (5-243a)

0" 110"22

0"220"33

(1 - 0 1 )(1 - O 2 )

(1 - O 2 )(1 - 0 3 )

0"11

0"12

(1 - 0 3 )(1 - 0 1 )

(5-243b)

0"13

1- 0 1

1 - [h+n2

1- nl+ n 3

0"21

0"22

0"23

O2 + 0 1 j* _ 1 3 - - 12 0"31

1-

03

2

2

1- O 2

1 _ n2+n 3

0"32

0"33

2

+01

2

1-

0 3 +02 2

2

(5-243c)

1- 0 3

where 1:::1 in Eqs.(5-241c), (5-242c) and (5-243c) indicate the determinant of the matrix. Because of the complicated nature of stress invariants, it would be easier to consider a two-dimensional case. The first stress invariant in all four cases shown above remains the same, thus this indicates that the normal stresses do not change under any symmetrizations The second stress invariant, however, may be reduced in the following manner: (5-244a) (5-244b) (5-244c)

298

5 Basis of Anisotropic Damage Mechanics (5-244d)

The corresponding deviation ratios of the stress invariant can be defined as follows (5-245a) (5-245b)

(5-245c) (5-245d) The above relations show that the second stress invariant may be significantly different after symmetrization treatment using models I and III whereas the symmetrization treatment using model II does not change the magnitude of the second stress invariant. In order to make a comparison, the deviation of the second stress invariant between the unsymmetrized model and the undamaged model is also presented in this group by Eq.(5-245). 5.9.5 Effects of Symmetrization on Net Principal Stresses and Directions By considering a 2D case again, Eq.(5-239) obtaining principal stresses and directions may be reduced to the formulae

(5-246a) (5-246b) In the damage case, irrespective of whether the symmetrization is applied or not, the governing Eqs.(5-246a) and (5-246b) are still applicable, and hence for unsymmetrization

(5-247a)

5.9 Effects of Symmetrization of Net-Stress Tensor in Anisotropic Damage Models

tan (20: *) =

(-1-_1_fl-1

+ -1-_1_fl-2 )

299

0"12

(5-247b)

--'--0""1::-=1---0""2::':::2-<---

and similarly, when symmetrization schemes are applied, the following equations prevail,

-*1,2 0"

* = 0"1,2

(5-248b)

(5-248c)

tan(2&*) = tan(20:*)

(5-249a)

(5-249b)

40"12

tan(2&*) = (1 - flI)

+ (1 -

fl2 ) ----1 - fl1 1 - fl2 0" 11

0"22

(5-249c)

Using the above equations, the corresponding ratios for the net principal direction may be similarly defined as given below (5-250a) (5-250b)

300

5 Basis of Anisotropic Damage Mechanics

~ rJ

tan(2&*) 4(1 - Sll)(l - Sl2) - ---'-------'---'--------'----.". '" - tan(2a*) - [(1 - Sl2) + (1 - Sld]2

_ _ tan(2&*) _ 2J(1 - Sll)(l - Sl2) _ (1 - Sll) + (1 - Sl2) -

rJ(i - tan(2a*) -

By further introducing parameters ,,(,

9, 1

and

~

VrJ",

(5-250c) (5-250d)

if defined as, (5-251a)

A

"( =

[(1 - Sll) + (1 - Sl2)] 2 2 2(1 - Sll) (1 - Sl2)

2

(5-251b)

1=1 4

A

1

=

(5-251c)

[(1 - Sll)

+ (1

- Sl2)]

2

(5-251d)

Eqs.(5-247a) and (5-248) can be rewritten together in a more compact form as

(5-252)

The maximum shear stress for each model may be obtained as given below

(5-253a)

(5-253c)

5.9 Effects of Symmetrization of Net-Stress Tensor in Anisotropic Damage Models

"*

Tmax =

{

[ 0 " 11

2(1- D 1 )

0"22]

2(1 - D 2 )

-

2

[

+

2

+ (1 -

(1 - D 1 )

D2 )

301

]2 2

0"12

}!

(5-253d) Similarly, ratios rJn fin fiT and ~T' and may be defined below (5-254)

t

Using the expressions of 8'J) 517 , 817 , and rJn fin fin ~T given in Eqs.(5235), (5-237) and (5-251) respectively, the Eqs.(5-253) and (5-254) can be simplified as

rJT

fiT

1

=

((32

+ fi;) '2

1

=

fiT

((32

+ i) '2

(5-255a) 1

=

((32

+ 4fi;) '2

1

=

((32

+ fi;) '2

(5-255c)

= rJT

1

1

+ 1) '2

=

0"11/0"12

_

~T = ((32

(5-255b)

((32

+ 4~;) '2

(5-255d)

where (3 =

2(1 - Dd

0"22/0"12

2(1 - D 2 )

(5-256)

Ratios fin fiT and ~T defined for maximum shear stress by different symmetrization schemes have some relationships with rJn which is the ratio rather than the symmetrized one, and that can be expressed as (5-257a) (5-257b) (5-257c) From the above derived relations, i.e. Eqs.(5-247)rv(5-257), the following observations can readily be made: •

The symmetrization scheme of model I changes the magnitude of the effective principal stress and does not change the effective principal direction, whereas the symmetrization scheme of model II changes the principal direction without changing the magnitude of the effective principal stress But the symmetrization scheme of model III changes both the magnitude of the effective principal stress and the effective principal direction.

302 •

•

5 Basis of Anisotropic Damage Mechanics Both symmetrization schemes of model I and model III change the magnitude of the effective maximum shear stress, whereas model II does not change the maximum shear stress Due to this, symmetrized models I and III are considerably more significant than the symmetrized model II. The effective second stress invariant changes value under the symmetrization schemes of models and III, but remains the same under the scheme of model II.

The deviation ratio for the square of maximum effective-shear-stresses between a symmetrization model and unsymmetrization model can be presented below (5-258a) (5-258b)

(5-258c) 1

1

(5-258d)

Since the pair law of effective shear stresses in the general anisotropic damage case is not compelling due to the nature of Ttj -I- Tji' in order to exam the effects of anisotropic damage of this nature, a square deviation ratio between the difference of these two effective-shear-stresses and the Cauchy shear stress are defined in this group shown by Eq.(5-528). The plots of (J h ,8h ,5h) for Eqs.(237) and (J T =,8T =,8T =) for Eq.(5-258) versus fl are shown in Fig.5-1~ and Fig.5-17 respectively. It should be observed that the deviation of i2 and I2 in the symmetrized models I and III markedly changes with fl. Whereas it can be seen in Fig.5-16 and Fig.5-17, as expected, that the curves representing both deviations of the effective second stress invariant 12 and the effective maximum shear stress T,';,ax due to symmetrized model II are the same as those of components of the effective stress tensor unsymmetrized. In particular" the effects implied in Eq.(5-245d) and Eq.(5259d) are plotted by Fig.5-16(c) and Fig.5-17(c) in comparison with others. It is also interesting to note that some deviation ratios for the square of maximum net-shear stress defined by JT = = (f,';,~x - T,';,~x)/(li2' 8T = = -*2 *2 )/ 2 ~ ("'*2 *2 )/ 2 th . t·IOn ( Tmax - Tmax (l12, u T = = Tmax - Tmax (l12 are e same as th e d eVIa

ratio of the net-shear stress defined by JO" = (a-i2 - (li2) / (l12, etc.; that is, (JT = = JO") and (Jh = J,;) are also true for the symmetrization scheme of model I.

5.9 Effects of Symmetrization of Net-Stress Tensor in Anisotropic Damage Models

303

0.5

4

0.0

20

-0.5

15

I

-1.0

10

0

- 1.5

A

8n

3

2

-I

-2 0.0

0.2

0.4 D, (a)

0.6

-2 .0 8n .... -2.5 0.0 0.2 0.8

5 0 0.4 D, (b)

0.6

0.2

0.4

DJ

0.6

0.8

(c)

Fig. 5-16 Effects of different symmetrization models I (a ), II (b) , III (c) on deviations of second net-stress invariants for different damage states 4

3

A

2

2

8;. or.-

"" 8,.-

....

-I

0.0

0 -10

0 .0,-0,0.2,0.4,0.6,0.8

0.2

0.4 0.6 DJ (a)

0.8

-I 0.0

8,;.

10

8"".0 ••

~ .,

0

20

'" 8,.

-20 0.2

0.4 0.6 DJ (b)

0.8

0.0

0.2

0.4 0.6 DJ (c)

0.8

Fig. 5-17 Effects of symmetrization models I (a) , II (b) , III (c) on deviations of effective maximum shear stress for different damage states

The effect of symmetrization schemes on the principal stress direction can be assessed by plotting i?ex, ria and fl ex versus [2 as shown in Figs.5-18(a) , (b) , (c). The plots in Fig.5-18 show how significantly t he direction may change under a certain symmetrization scheme. The significant effect on the magnitude of the principal stress may be illustrated by plotting parameters of ::;, 'Y and ;Y versus [2 as presented in Figs.5-19(a) , (b), (c). In this plot, effects of "(,1, 'Y and '1 only apply to the shear stresses in Eq.(5-252) , and may reach as high as 25 when both [21 and [22 attain a value of 0.8. Hence, as mentioned before, applying a symmetrization scheme must be carefully exercised to avoid any spurious results in the principal stress and direction analysis. 5.9 .6 Effects o f Sy mme triza tio n o n D a m age C o n stitutive R e la tio n s

The effect of symmetrization treatments on the damage constitutive relationship will be explored in this section. A similar approach with a set of ratios, as done before, will be repeated for various parameters. The net stress tensor given in Eq.(5-226) can be rewritten in t he vector and matrix notation using the transformation matrix [IP] (where [IP] is not the same as the matrix [1/>] defined in Eq.(5-225) but is the same as detailed

304

5 Basis of Anisotropic Damage Mechanics l.lr---------------,

50 r-~~--------.

ij r;. '" 17" ••••

1.0~~~~~~~

40 30 .0,=0,0.2,0.4 ,0.6,0 .8

0.9 0.8 0.7

20

10~~~~:=:= o

0.5 0.4 <--.JL.......I....L...1.-'-'-...........L...L'-'--'-' 0.8 0.0 0.2 0.4 0.6 0.8

~~~~~~~~

0.0

0.2

0.4

0.6

DJ

DJ

(a)

0.6

i7.=~ '7.

_ 11.-

fl•••••

0.5 i7. =tan(2a ')/t.an(2ii) 0.4 L........L..L...t'-'-'................................................, 0.0 0.2 0.4 0.6 0.8

(b)

D,

(c)

Fig. 5-1 8 Effects of symmetrization models I (a) , II (b) , III (c) on effective principal directions for different damage states

1000

A

logr

rA-

100

o.lL.......>....L...1.,.................."............ 0.0

0.2

0.4 OJ

(a)

....L.....r...........J

0.6

0.8

.0 r O,O .2,0.4,0.6,0.8 O.5 '-::-''-"::'-::-'----=-7"'-~:'-'-';:"' 0.0 0.2 0.4 0.6 0.8 0, (b)

(c)

Fig. 5-1 9 Effects of symmetrization models I, II, IlIon the influence parameters of effective principal stress for different damage states shown in contours and surfaces (a) For model I, III; (b) For Model III, II ; (c) Surfaces for model I, III

in [5-12]. For the unsymmetric model in {CT*} = [lP]{CT} , [lP] is a (9 x 6) rank matrix expressed in Eq.(5-22). When this transformation is applied to the damage model with symmetrization treatments, Eqs.(5-222) to (5-224) may also be rewritten as

{a*} = [~]{CT}; {a *} = [q/]{ CT}; {a*} = [q/]{CT}

(5-259)

In order to obtain a corresponding constitutive relationship similar to the non-symmetrized model presented in section 5.6, following a similar manner the complementary elastic energy theorem can also be applied to models with symmetrization treatments. It is reasonable to assume that " The complementary energy of a damaged state is assumed to be equal to that of the undamaged state under effective stress loading under a symmetrization treatment" . Constitutive equations {E} = [D*r1 { CT} or {CT} = [D*]{E} shown in Eqs. (5108)"-'(5-110) , in which [D*r 1 = [lP]T [Dr l [lPJ, can be readily obtained.

5.9 Effects of Symmetrization of Net-Stress Tensor in Anisotropic Damage Models

305

In a similar fashion with the use of Eq.(5-108) , effective elastic matrixes due to symmetrization treatments of models I, II, III have the following forms (5-260a) (5-260b) (5-260c) in which [P]' [l]I], [P] have been expressed by Eqs.(5-125), (5-130), (5-135) in detail. Resultant damaged material constants (anisotropic elastic properties) can be obtained explicitly by inverting [D*]-l, [D*r1 , [V*]-l and hence direct comparison of the unsymmetrized model and models with different symmetrization treatments can be easily made. The explicit expressions for the damaged material in each model are given below: Damaged properties for the unsymmetrized anisotropic model presented by Zhang Wohua et al. [5-10"-'15] have been described in Eqs.(5-108) of section 5.6. However, damaged properties for these three symmetrized anisotropic models are presented as follows: Damaged properties for symmetrized model I (5-261a) A* _

G ij -

[

)]2 ..

2(1- Di)(l- Dj (1 _ D i ) + (1 _ D j

)

G,}

(5-261b)

Damaged properties for symmetrized model II (5-262a)

CT j

=

(1 - Di)(l - Dj)Gij

(5-262b)

Damaged properties for symmetrized model III (5-263a) (5-263b)

It can be readily appreciated that from the above explicit forms the symmetrization schemes do not introduce a deviation in the value of and vij from any unsymmetrized models. However, the damage shear modulus is affected by all symmetrization schemes. In order to illustrate the effects of

E:

306

5 Basis of Anisotropic Damage Mechanics

symmetrization treatments on the constitutive relationship of an anisotrpic damaged material the following ratios are defined (5-264a)

(5-264b)

(5-264c) The plots of Gij/Gij, Cr;j/Gij, C)'; jGij and fic , fic , ~c for various [l are shown with curves and surfaces in Fig.5-20 and Fig.5-21 respectively. An

0.6 0.4 0.2 0.0 L...L............ .L..L..L..L............ .L..L..L..L.L..L.J 0.0 L...L..L..L..L..L..L..L..L..L..L..L..L..L.L..L.J 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8

12) (a)

12j (b)

o.OL-.L..L..L..L.......................................... 0.0

0.2 0.4

12)

0.6

L..L.J 0.8

(c)

F ig. 5-20 T he ratios of effective shear modulus effected by symmet ri za t ion models I (a) , II (b) , III (c) to an unda maged one vers us da mage st a tes shown with curves and surface

0 .0 0

.2

0.4

12 ) (a)

0 .6

0.8 0.0 0.2

0.4 OJ (b)

0.6 0

.8

0 .2

0.4 OJ (c)

0.6 0.8

Fig. 5-21 Effects of symmetrization models I (a) , II (b) , III (c) on effective shear modulus for different damage st a tes shown wi t h curves and surface

5.10 Simple Damage Evolution Modeling

307

effective shear modulus for all models presented in Fig.5-20 decreases with the damage increasing significantly. It is interesting to note from Fig.5-21 that the resultant shear modulus is magnified when a symmetrization scheme is adopted. The symmetrization scheme of model I may magnify the shear modulus 1.5 times, whereas the symmetrization scheme of model II can be as much as 2.5 times the shear modulus obtained from the unsymmetrized model (Fig.5-21). However, model III could be over 28.5 times.

5.10 Simple Damage Evolution Modeling 5.10.1 Damage Kinetic Equations For the complete analysis of dynamic damage problems, besides the appropriate constitutive laws, it is necessary to specify the damage kinetic equation in the form (5-265) where lJij is the state of stress at a particular point and fl is the damage tensor at that point. Also, D represents the rate of damage. Eq.(5-265) implies that the damage growth rate, practically the damage and stress distribution in an element, is a function of time and coordinates (for example in a two dimensional case, lJ(x, y, t), fl(x, y, t). The direct integration of damage kinetic equations in F. E. dynamics analysis is hence difficult. Most investigations consider the damage kinetic equation in the form of a power law, and so far two major damage evolution criteria have been proposed for different kinds of materials, the first one is a power function of tensile normal stress and the other is based on the damage strain energy release rate. Both of the criteria have been applied in this book.

5.10.1.1 Damage Evolution Equation Based on Effective Stress State For some materials, especially ductile materials, the damage kinetic equation based on a function of the tensile stress, which was introduced first by Kachanov, gives a good estimate of the behavior of the damaged material. The above-mentioned function is employed here for unidirectional loading

dfl dt

=

{A( 1 ~ fl)n o

IJ

IJ

>

IJd

(5-266)

~ IJd

where A > 0, n > 0 in Eq.(5-266) are material constants depending on the rate of loading, s is the uniaxial stress, and IJ d is the threshold value of the

308

5 Basis of Anisotropic Damage Mechanics

tensile stress. In the case of brittle materials, and quantity of IJ d is long term or the static fracture strength of the material. A denominator factor (1 - D) is introduced to account for the acceleration of the damage accumulation rate with a reduction in the effective area under load. For a given stress history Eq.(5-266) can be integrated to obtain D. In the case of the multiaxial stress state and isotropic damage, it is better to use the concept of the equivalent stress IJ eq of a Cauchy stress tensor, based on a different criterion, such as the von-Mises criterion. Then, dD = dt

{

A

(

lJeq

1 _ Di

o

)n IJ eq

~

IJ deq

(5-267)

lJeq ~ IJdeq

where 1

lJeq =

{~[(IJX _lJy )2 + (lJy -lJz )2 + (lJ z -lJx )2 + 6(T;y + T;z + T';x)]} "2

(5-268) and IJ deq is the threshold value of the equivalent stress IJ eq corresponding to the Cauchy stress tensor. Kachanov has considered the extension of the kinetic equation to the case of anisotropic materials. The corresponding kinetic equations in the anisotropic principal axes system have the following form: lJii

>

IJdi,

i = 1,2,3 (5-269)

where Ai > 0, ni > 0 (i = 1,2,3) are material constants in the direction of anisotropic principal axes, and IJdi represents the threshold stress in the ith principal direction. It is assumed that the principal axes of damage maintain a prescribed orientation in a material-fixed co-ordinate system and that the damage accumulation on the principal planes depends only upon the applied tensile stress. It should be noted that the material constants Ai and ni may be obtained from experimental results, as will be explained in other section. It may be complex to adopt Eq.(5-269) in a multiaxial state of anisotropic materials for a three dimensional case, because a total of 9 material constants Ai, ni and IJdi(i = 1,2,3) need to be determined by experimental tests. The following damage kinetic equation of anisotropic damaged materials, based on the equivalent stress IJ eq may give a simplified model,

(5-270)

5.10 Simple Damage Evolution Modeling

in which only 5 material constants mined.

O'di

309

(i =1,2,3) and A, n need to be deter-

5.10.1.2 Damage Evolution Equation Based on Damaged Strain Energy Release Rate The concept of an elastic damage strain energy release rate, necessary to propagate the micro-cracks is justified by comparing the stiffness before and after the growth of micro-cracks. In fact, the elastic damage strain energy release rate is an extension of the concept adopted in linear fracture mechanics for the crack strain energy release rate. For the damage evolution, consistent with experimental results, the following kinematics law in isotropic damage is considered [5-21] dO = dt

{HY

k

0

(5-271)

where the parameters B > 0, k > 0 in Eq.(5-271) are material constants that can be determined by experimental measurements in a similar procedure to that involving the parameters of A and n presented in [5-21], and Yd is the threshold value of the damage strain energy release rate at the start of damage o growth. Similarly, for the anisotropic damage, the rate equations can be represented in three anisotropic principal directions, ith (i = 1, 2, 3), as

Yi > Y di (i = 1,2,3) (5-272) in which parameters Bi and k i (i =1,2,3) are constants of anisotropic materials and may be determined by experimental measurements on anisotropic materials of interest. Yi is the damage strain energy release rate in the ith anisotropic principal direction and Ydi is the threshold value of Yi along the ith anisotropic principal damage direction at the start of damage Oi growth. The damage strain energy release rate Y can be determined by Eq.(363) and Eqs.(3-87) and (3-88) or Eqs.(3-146) and (3-147) for Y in isotropic damage, and Eqs.(5-98), Eq.(5-99) or (5-101) for Y (i = 1,2,3) in anisotropic damage. The integration of above damage kinetic equations of all types can be carried out using the Newmark time integration scheme. An accumulation rule of damage increment AD for each time step At at each Gaussian point in each element within the whole analyzed region should be taken into account for a description of the damage state and the damage growth process. The threshold characteristics in all types of damage kinetic equations provide a localization function of damage and damage growth, which makes the damage growth

310

5 Basis of Anisotropic Damage Mechanics

occur only at a point where the threshold condition is satisfied. Consequently, the damage increment AD will be accumulated based on the value of the previous damage state at that point and time where the threshold condition is satisfied. If the threshold condition is never satisfied at a point and time, then the damage would not grow and therefore materials may not be damaged at all if no initial damage exists. It is evident from these comments that the damage growth and propagation only take place at some possible points and in some possible directions where the threshold condition is satisfied, in other words where stress maybe is concentrated, deformation discontinued, material cracked or weakened and so on. This concept actually is based on a micropoint view. 5.10.1.3 Average Integration Scheme for Damage Evolution Equations

In the case of damage growth, the damage and stress distribution in an element are a function of time and coordinates, thus the integration of damage kinetic equations in F. E. dynamic analysis is hence more complex. The rules of time integration and accumulation of damage kinetic equations at each Gaussian point provide more accurate results of damage distribution, but we may spend more computational resources, such as CPU time and memory size as well as plotting capability especially for large dimension structural problems. In order to overcome this difficulty, it is necessary to introduce an average value and an average rate of damage in an element. The average integration scheme for overall damage evolution equations of an element may provide an economic method for integrating the damage rate equation, which will provide an overall mean damage state in the element at each time step. Then the damage growth law in an element can be approximately developed by an average value. Generally, the material parameters A and n (or Band k) are considered as constant within a small element which usually is a localized damage area and is therefore to be discretized by a denser mesh. In a multiaxial stress state it is similar to Eq.(5-269) and Eq.(5-272)

(5-273)

where (5-274) where

v;,

is the volume of the element.

5.11 Verify Anisotropic Damage Model by Numerical Modeling

311

The integration of Eqs.(5-273) and (5-274) can be carried out using a Gaussian integration technique and accumulation of the damage increment. The average damage accumulation in an element at time t = tj+l with respect to the effective stress model can be calculated by the equation (5-275) in which

8 eqn (tj) =

1 ALL Wkz[O"eq(~k' rll, tj)]n H[O"eq(~k' rll, tj) - O"di] 33

e

k=ll=l

(5-276)

where ~k' rJl (k, l = 1,2,3) are Gaussian points, Wkl (k, l = 1,2,3) are weight factors. 0"eq (~k' rJl, t j) is the equivalent stress at the Gaussian point (~k' rJl) and time t = t j . The average damage accumulation in an element at time t = t j +l with respect to the damage strain energy release model can be calculated by the equation

(5-277) in which

Yi(tj) =

1 ALL Wkz[Yi(~k' rJl, tj) ] ki H[ Yi(~k' 33

e

k=ll=l

rJl,

tj) - Ydi ] , (i=1,2,3)

(5-278) where Yi(~k' rJl, tj) are the ith components if the strain energy release rate at the Gaussian point (~k' rJl) and time t = ti.

5.11 Verify Anisotropic Damage Model by Numerical Modeling 5.11.1 Stiffness Matrix of Anisotropic Elastic Damage Model in F.E.M. The general virtual work theorem for the Cauchy stress and strain may be written in the following form, (5-279)

312

5 Basis of Anisotropic Damage Mechanics

where {c}, {u} are the virtual strain and displacement vectors, respectively, {Q} and {F} are the traction and body force vectors, respectively. For anisotropic damage materials, the constitutive equations derived previously in Eqs.(5-111)rv(5-119) must be substituted into Eq.(5-279) to obtain the corresponding stiffness matrix.

Consider finite element discretization in the standard manner, {u}

=

[N]{U}

(5-281a)

{c}

=

[B]{U}

(5-281b)

where {U} is the nodal displacement vector; [ N] is the shape function matrix; [B] is the strain-displacement matrix. Substituting Eq.(5-281) into Eq.(5-280), the following expression is obtained, [K(n)]{U} = {P}

(5-282a)

where (5-282b) which is the required stiffness matrix for a damaged element. The vector {P} is given as, (5-282c) which is known as the general force vector. Whence the nodal displacement vector {U} is obtained by solving Eq.(5282a), and the Cauchy stress of the damaged element may be computed by (5-283) and the effective stress tensor can then be subsequently obtained by the Eqs.(5-27) or Eqs.(5-33)rv(5-35). 5.11.2 Numerical Verifying for Elastic Damage Constitutive Relationship

The first example was taken from an article of Kawamoto et al. [5-9], where the experimental results of the model were provided. The specimens used in the model were plaster and cement mortar, as shown in Fig.5-22. The specimens have regularly embedded cracks formed using steel strips of 0.2

5.11 Verify Anisotropic Damage Model by Numerical Modeling

313

mm. The number of cracks for both specimens varies from 28 to 52. The undamaged material properties for both specimens are taken from [5-9]. The finite element mesh used is shown in Fig.5-23. The results obtained from the method presented in this chapter are shown in Fig.5-24 and Fig.5-25 together with the experimental results from reference [5-9]. The dotted lines indicate the results of isotropic materials with orthotropic damage states in a zigag shape of cracks, whereas the solid lines correspond to the results of orthotropic materials with orthotropic damage states in a regular form of cracks. In the model containing a zigzag shape of cracks, the anisotropic material properties of E2/ El = (1 - Sl2)2 had reasonably been assumed. I

100

I

I

y/

0 0

'"

I

V/

/

/

~-}-.) ~

~ ~

~~ 20~

~~~~

~~~~

~~~~ 0 0

~~~~

~~~~

'"

~~~~

~~~~

~~~~ ~~~~

Applied load

100

~~~~ ~~~~

/

(a) Regular type

1/

(b) Zigzag type

Fig. 5-22 Specimens for uniaxial load ([5-9])

_________

I ________ L I

unit: mm

Fig. 5-23 F. E. mesh for Fig.5-20

J I _________

I __ L

~-

__

~~

I

en

-~ >-< ~---:-=.......,--=I

II

I

"t:I

I

I

I

I

I

I

I

I

I

~

~

(}o

---------r--------~---------r---------

]

~

~

---------r--------,---------r--------I

0 Experimental --Numerical

I

Fig. 5-24 Apparent Young's module of the plaster specimens (regular type)

314

5 Basis of Anisotropic Damage Mechanics o • --

--S1.2 ""0

~ 1.0

rI.l

"'OJ)

I

8d .0 8 --

~

"t:S ~

. :.E

§

~

0.6

...

-!

- -

I ...... I......

0

I ~

I

I I

I

I I

I I

I I

"

'" L ...__

___

I

__ _

~

~'

I

I

___

___

L ---

I

_~~1.

-

....... I

..0.. . .- - A

Exp.zigzag Zxp.regular Num.zigzag Num.regular

'0

...I _ _ _ _ L __ _

"I

- -1;'- ...... I

--

0.4 '-----"'----"'------'-'---' 45 90 0 0 Crack angle (") (a) 28 cracks

+ - --

-1- --I

I

45 90 0 Crack angle (") (b)36 cracks

45 90 Crack angle (") (c) 52 cracks

Fig. 5-25 Comparison of numerical and experimental effective Young's module of cement specimens. (a) 28 cracks; (b) 36 cracks; (c) 52 cracks

5.11.3 Numerical Verifying for Symmetrization Comments The comments about deviation due to symmetrisation schemes will be verified and illustrated by simulation of an experimental test, presented in a reference of Kawamoto et al. [5-9]. In this experiment, a direct shear test is shown schematically in Fig.5-26, where Pn and Pt are the normal force and shear force subject to the specimen, respectively. The simulation model as shown in Fig.5-23 is based on a modified Mohr-Coulomb criterion, which will be employed in the finite element analysis with parameters obtained from P

L

~I

.;

0 N

/

//

/

-:J.///

/

/

/'

///// /

/'

/% /

.;

/

/

/

/70

.;

40

///// /

~t

,-

40

,/

/' /' /' /'

.;

/ /

150

p.

/

~/// \: /

0 "
unit:mm ///////////////////////

Fig. 5-26 Direct shear test and crack pattern of sample

5.11 Verify Anisotropic Damage Model by Numerical Modeling

315

both the unsymmetrized model and two different symmetrized models I and II. Fig.5-27 is the model of finite element mesh for simulation of the test, employing parameters of the material and damage obtained from the reference of Kawamoto et. al. [5-9]. The damaged constitutive matrixes provided by Zhang et al. [5-10""16] or the two symmetrized models and the unsymmetrised model are taken into account in the simulation by substituting different [D*] and [D*], for which explicit expressions are given in Eqs.(5-109)",,(5-111) and Eqs.(5-119)",,(5-120). Different results will be compared to each other and be verified by the experimental results given in [5-9]. The numerical simulations of the test shown in Fig.5-26 are carried out by 8-node serendipitous quadrilateral isoperimetric elements shown in Fig.5-27, where the different damaged constitutive matrixes for different schemes of symmetrization have been employed. In this analysis, the Mohr-Coulomb criterion is modified as an effective stress space in order to obtain the damaged failure load. The modified Mohr-Coulomb criterion in the net-stress space may be rewritten as

T;;'

= C-

(5-284)

tan CPIJ:r,

P,

!! !! -

-

P,

"

; in,;; in,;; in,;;in ,

150mm

Fig. 5-27 Finite element mesh for the direct shear test where 1J':n, T':n are respectively the "failure" normal stress and the "failure" shear stress; c, cP are the cohesion and frictional angle of the material. The shape of the modified Mohr-Coulomb criterion does not change in the netstress space. However, if the net stress is substituted by its corresponding form in terms of damage variables and the Cauchy stress, such that for a unsymmetrized model,

316

5 Basis of Anisotropic Damage Mechanics

{ [

0"11 0"22] 2 2(1 - Sll) - 2(1 - Sl2)

+ 2(1 -

Sll - Sl2 Sll)(l - Sl2) 0"12

+

[

1-

0"11

+ ((1 _

0 1 +0 2

2 (1 - Sl!)(1 - Sl2) Sll)

+

]2

}

!

O"i2

0"22. (1 _ Sl2)) Sllltp

2Rcostp = 0 (5-285) where R is a reduction coefficient of the failure criterion related to the damage state {Sl}, the cohesion and the principal direction are as given in [5-9], and for both symmetrized models I and II we have

{ [

+

0"11 0"22] 2 1 - Sll - 1 - Sl2 (

0"11

(1- Sll)

+

+

40"i2

(1 - Sll)(l - Sl2)

0"22) . (1- Sl2) Sllltp

-

-

{i}}! 1

(5-286)

2Rcostp = 0

where i and 1 have been previously defined in Eqs.(5-251b) and (5-251c). The shape of the Mohr-Coulomb criterion will be observed in the Cauchy stress space for the damaged material. The experimental results from Kawamoto et al. have been reproduced in Fig.5-28(a), whereas the numerical results using the unsymmetrized model are presented by Zhang et al. in [5-10"-'12]. The results in Fig.5-28(b) are drawn for the computed shear failure forces Pt subjected incrementally to the crack-damaged specimens with various dominant crack angles from -90 0 to 90 0 when the modified Mohr-Coulomb criterion was reached under different normal loads of Pn = 25,50,75 N. The numerically simulated results are also represented in a linearized form as shown in Figs.5-29(a) to (c) where each figure corresponds to a particular normal load. It is evident from Figs.5-29(a) to (c) that the unsymmetrised model proposed by Zhang and Valliapan et al. [5-10, 5-16]) agrees reasonably well with the experimental results presented in [5-9]. When symmetrized models are used in place of the unsymmetrized model by replacing [D*] matrix and corresponding parameters in the Mohr-Coulomb criterion, the numerical results are shown in Fig.5-30 to Fig.5-33. The maximum shear stresses (calculated by the three different models of I, II and the unsymmetrized one discussed above under different shearing loads for various crack angles) are presented in Figs.5-30(a), (b) and (c). From Fig.530 it can be predicted that when the crack angle changes from 45 0 to -45 0 , the distribution of the maximum shear stress in the actual specimen should exhibit an inverse-symmetrical nature. However, after these symmetrization treatments, the numerical results for both models I and II in Figs.5-30(b) and (c) show an unsymmetrical distribution, whereas the results in Fig.530(a) from the unsymmetrized model give the correct inverse symmetrical distribution as expected.

5.11 Verify Anisotropic Damage Model by Numerical Modeling

317

-22.5"

o (a)

(b)

Fig. 5-28 Experimental (a) and numerical (b) results for shear failure loads

The plots in Figs.5-31 (a) to (c) show the second stress invariant ofthe middle element in the specimen shown in Fig.5-26. These results are calculated by two different unsymmetrization models I, II, and the unsymmetrized model as mentioned above, under different shear loads for various crack angles. From Fig.5-31, the results prove that the conclusions i2 i=- 12 , 12 = 12 derived from analysis of Eq.(5-244b) and Eq.(5-244c) are correct. An interesting point from Fig.5-31(a) and (c) is that irrespective of using any of the three models discussed above, the numerical results always produce two "node" points for the second stress invariant (i.e. for any loading, 12 and f2 always remain constant at crack angles of e ~ -30 0 , e ~ _10 0 and e ~ -55 0 , e ~ -15 0 respectively) and four "node" points can be found in Fig.5-31(b). Fig.5-32(a) shows a comparison of the maximum shear stress from two different symmetrisation models I, II and the unsymmetrized model under the same loading condition (Pt = 170 N, P n = 75 N). From this figure, it can be seen that for the two different symmetrized models, the maximum shear stress is almost the same and has a distribution that lacks symmetry. For the unsymmetrized model, the distributions of the maximum shear stress possess the correct inverse-symmetry. Also, there are significant differences in

318

5 Basis of Anisotropic Damage Mechanics 300,---------------------------, --0- Experimental ...... Numeriacal

200 p'I

100

o~------~------~----~------~

-90

-45

0

Crack angle 0 (") (a)

45

90

300.-----------------------------, --0- Experimental ...... Numeriacal

p'I

100

0 -90

-45

0

45

90

0

45

90

Crackangl eO (") (b)

300

200 .'

.'

.

p'I

100

0 -90

-45

Crackangle 0 (") (c)

Fig. 5-29 Comparison of experimental and numerical failure loads. (a) P n =25 N; (b) Pn = 50 N; (c) Pn =75 N

5.11 Verify Anisotropic Damage Model by Numerical Modeling

319

P,=120,144,168,192,216,240 N

-~':-0----"""4':-5---~0----4-:'":5:-----::'90 Crack angle () (a)

-45

o Crack angle ()

C)

C)

45

90

(b)

3

Crack angle () C ) (c)

Fig. 5-30 The maximum shear stress determined from various damage models. (a) Non-symmertrization model; (b) Symmertrization model I; (c) Symmertrization model II

320

5 Basis of Anisotropic Damage Mechanics P,=120,144, 168, 192,216,240 (N)

8

/ ,-

4

-4 L-______~------~------~------~ -90 -45 0 45 90 Crack angle fJ (") (a)

/',

4

o

-45

Crack angle fJ (b)

n

45

90

P,= 120, 144, 168, 192,216,240 (N)

-4

~

-90

______

~

-45

________

L -_ _ _ _ _ _

o Crack angle fJ (c)

n

~

45

______

~

90

Fig. 5-31 The second net-stress invariant det ermined from various damage models. (a) Non-symmertrization model; (b) Symmertrization model I; (c) Symmer tri zation model II

5.11 Numerical Application to Analysis of Engineering Problems

321

120

4

t:.

1'_

(a)

-' 1'mn

3

a mu

1'mn

amax

60

a"mu 0 a~ax

-60 P,=170N

0 -90

-45

0

-120 -90 90

45

Crack angle 9(")

P,=170N

-45

0

45

90

Crack angle 9 (")

Fig. 5-32 (a) The maximum shear stress for different damage models (b) Principal stress directions for different damage models 400.-----------------------,

2.0.----------------,

(b)

(a)

]1.5

g 300

]

.£

.£

'"oS

;,§ 1.0 ..... o o

.~

~

0.5

Symmetrization-I

-0-0-0-

Symmetrization-I

~~0-----~45~--~0~---4~5~--~90 Crack angle 9 (")

100

Non-symmetrization -0-0-0Symmetrization-II ...............

~9~0--_745~--70--~475-~90 Crack angle 9 (' )

Fig. 5-33 (a) Effects of different symmetric models on the shear failure loads; (b) Comparison between shear failure loads for two symmetrization models I and II

the numerical results obtained from the unsymmetrized model to those of the symmetrized models. Fig.5-32(b) shows the results of the principal stress direction (the principal angle) versus crack angle e for any loading level. These results again proved that the conclusion (0:* = &*) and (0:* -I- a*) derived from Eq.(5249) is correct. The ratios of shear failure loads between symmetrized and non-symmetrized models with various crack angles are also illustrated in Fig.533(a), which show significantly the effects of different symmetric models on the shear failure loads. Fig.5-33(b) is a plot that presents the comparison between shear failure loads corresponding to the two different symmetrization models I, II and the unsymmetrized model used herein. The results from Fig.5-33(b)

322

5 Basis of Anisotropic Damage Mechanics

show that after syrnmetrization treatments, the peak value of shear failure loads and the crack angle at the peak have significant changes.

5.12 Numerical Application to Analysis of Engineering Problems 5.12.1 Anisotropic Damage Analysis for Excavation of Underground Cavern This example shows damage analysis for the excavation of an underground opening of a large cavern in a hydroelectric power station. The cavern was constructed in a granite rock mass and its 3 dimensional outlook is illustrated in Fig.5-34(a). The cavern was excavated in ten stages, and the cross-section of the cavern is shown in Fig.5-34(b). The pump house cavern is bullet shape in plan 25.4 m long, 31.0 m high and 17.0 m wide. The top of this cavern is 200 m below the surface. The deformation behaviours were monitored by extensometers and convergence meters, thus the initial stress state in the site rock measured at the crown is also shown in Fig.5-34(c) , whereas the principal maximum stress is -55 kgf/cm 2 , and the minimum stress is - 26 kgf/cm 2 , which is inclined at 56.5° to the horizontal. The main supporting members were shotcrete and rock bolts.

0',=-55.0 0',=-26.0 0'.=-45 .5 0',=-35.5 0'.,.=-12.6 y

27m (a) 3D outlook of the cavern

(b) Geometry of the cavern

(kgf/cm' ) (kgf/cm') (kgf/cm') (kgf/cm') (kgf/cm')

x

(c) Tbe initial stress state

Fig. 5-34 Illustrations of states for 3D outlook geometry and initial stress in an underground cavern. (a) 3D outlook of the cavern; (b) geometry of the cavern; (c) The initial stress state

The stability problem of such an underground cavern is usually analyzed by using the finite element method in which rock mass properties are determined by on-site tests. Conventional finite element analysis, however, would not provide sufficient accuracy for the rock mass behaviour if the rock mass

5.12 Numerical Application to Analysis of Engineering Problems

323

involves some sets of cracks and joints. We advance here the idea of applying damage mechanics theory to finite element analysis for simulation of the excavation procedure of the cavern, and then compare this with measured field data. Since the excavation sequence was carried out in 10 stages, it can be assumed that the intact rock is elastic, and the rock and initial stresses are uniformly distributed around the cavern. Young's modulus Er and Poisson's ratio v of the intact rock are determined from triaxial tests by cylindrical specimens as Er = 1.2x105 kgf/cm 2 and v = 0.25. Similarly, the Mohr-Coulomb failure condition at the peak is obtained as

(T = 210 + O"tan600) kgf/cm 2

(5-287)

Young's modulus Em of the rock mass which was determined by in situs plate loading tests is given as Em = 1. 2 X 10 5 kgf/ cm 2 . The Poisson's ratio of the rock mass was assumed to be the same as the intact rock. By block shearing tests, the peak strength for the rock mass was also obtained as (5-288) These material constants of the rock mass are used in conventional finite element analysis to compare with the damage analysis. At each stage of excavation, the state of joints on excavated wall surfaces was carefully observed. If the global co-ordinates and the co-ordinate surfaces are defined as in Fig.5-35, the joint sets on the surfaces are shown in Fig.5-36. Two diagrams introduced earlier (Fig.5-3( a) and (b) in subsection 5.2.2) are obtained through Table 5-1'"'-'5-3, and the size of the unit cell element of the rock mass in Eq.(5-13) is determined as I = 0.89 m by averaging spacing of

Roc k lype : gra ni le Overbu rde n : 200 ni Le ng lh : 25 .410

Posit ions of clC.len s ome lcr$ a nd c o nvergence melen

c..

En

Inilial st re ss ,u e rQw n = - .5.5.0 (kgf/cm1) u, = - 26.0 (kgflcml) O'L

O. - 56 .6 ·

x,

17 m

Fig. 5-35 Dimension and measuring cross-section of the analyzed hydroelectric cavern

324

5 Basis of Anisotropic Damage Mechanics

adjacent cracks on some seam lines. Then the damage tensor of this rock mass is found to be 0.5242 -0.27480 .1326]

[n ] = [ -0.2748 0.2434 0.0966 0.1326

(5-2 9)

0.0966 0.3109

Fig. 5-36 Cracks observed on three independent surfaces of rock specimen, cut off from each excavated wall

Table Rank Cracllength (m) Crack numbers on PI plane Crack angles on P l plane Crack numbers on P 2 plane Crack angles on P 2 plane Crack numbers on P 3 plane Crack angles on P 3 plane

On X I-eo-ordinate surface On X 2-co-ordinate surface X 3-co-ordinate surface

5-1 Data of cracks observed a b c d L 0.125 0.375 0.625 0.875 N 162 48 65 99 e 77.5 77.5 82.5 152.5 N 168 261 126 74 e 177.5 172.5 177.5 107.5 N 297 362 145 72 e 62 .5 62 .5 147.5 127.5

Table 5-2 Arranged data of cracks L 0.401 0.979 N 326 82 80.0 152.5 e L 0.356 0.968 N 91 555 175.0 107.5 e L 0.262 0.625 F 659 145 62.5 147.5 e

1.375 28 57.5 1.125 34 R 0.950 103 125.0

e 1.125 34 152.5 34

R 31 122.5

f 1.375 28 57.5 17 107.5 17

R

1.375 17 R

5.12 Numerical Application to Analysis of Engineering Problems

Xl

X2

(tan81 tan82)-1

325

Table 5-3 Finding combination of angles 80.0 152.5 57.5 175.0 107.5 175.0 107.5 107.0 175.0 -2.02 -0.06 21.96 0.61 -7.28 -0.20 62.5 147.5 125.0 -1.92 0.63 1.43

Since the component Dn is the maximum, it is understood that the damage is dominant in the normal direction to the side wall (Xl-plane), so that the deformation at the side walls will become larger. Convergences measured at C n , C 12 , C 13 , C 14 , C l5 and C l6 are shown in Fig.5-37, together with results of conventional finite element analysis and damage analysis. The damage model is found to agree with the measured data except for C 13 . The reason why the measured convergence at C l3 becomes larger is that there was a fault around the middle of the left-side wall, which is not considered in computations. On the other hand, conventional finite element analysis using the rock mass properties does not give satisfactory results.

30

C 13- ... - - ...... I

- - - - measured - - - damage analysis - - 0 - convenlional analysis

I I I I

I I

I I

I I

I

Excavation step

Fig. 5-37 Accumulated convergences corresponding to each measuring point [5-9] The extenso meter EMl was installed in an access tunnel before excavation. Fig.5-38 shows the measured and computed displacements of EMl. The final displacements computed by damage analysis are mostly the same as the measured data. But in early stages of excavation, measured displacements in the vicinity of EMl move into the cavern, while the computed ones move in the contrary direction. This is because a local loosening zone was formed at the crown shoulder, but it is not taken into account for this elastic analy-

326

5 Basis of Anisotropic Damage Mechanics

sis. The final displacements of extensometers Ell , E12, E16, E17 are shown in Fig.5-39. The damage analysis gives improved results, except for the near surfaces where failure or loosening of the rock may be caused. Deformation modes on the wall surface are computed as shown in Fig.5-40 at the final stage of excavation. It is clear that damage analysis gives much larger horizontal displacement of the side walls than the conventional finite element one. In both analyses the maximum horizontal displacement is found at the center of the left wall, but in damage analysis the value is 1.46 cm, whereas it is 0.95 cm in the conventional one. Next we check the state of failure of each element. The failure criterion used for damage analysis should converted by effective stresses similarly represented in Eq.(5-285). Usually it is difficult to get a comprehensive form of the reduction coefficient R of the failure criterion defined in Eq.(5-286). However, experimental results presented in Fig.5-28 and Fig.5-29 in subsection 5.11.3 suggest that the form of R may be assumed to be

R

(1

1]

= a - b [ -L- 1 - - tr(D) -2 cosB*

(5-290)

l+L 3 where a and b are constants; L is the average spacing of cracks (see Eq.(5-16) ; L is the average trace length of cracks; and B* is the angle between the unit normal vector of the plane damage and the maximum of the principal net stress. The t erm (1 - tr(D) /3) implies a ratio of the mean effective resisting part. For the peak strength of the rock mass can be determined by the least squares method as a = 0.95, b = 0.08.

15

1 m di stance from the wall - - - - Measured - - Ddamage analysis , --<>- Convenlional ," ana lys is ,,

,,

,,

023456789

Fig. 5-38 Accumulated displacements in rock mass measured by cxtensometer EM!

( [5-9])

5. 12 Numcrical Applica t ion to Anal ysis of Engineering P ro blems - - - Measured ___ Damage analysis

E 20

.5

',-o-Conventional

;:;

- - - Measured ___ Damage analysis -o-Conventional F, E. analysis

E20

nE 1I!-

LJ;;;

F. E. ana lysis

327

[}!-6

. E=~~. ,e=:~~ 11 10

-~~

is

11 10 ~

~

0

•

0

.5 0

4

8

12 Distance (m)

16

20

0

- --

-5"0~---:4C-~~8'~~I2"-;1~6~--'>20 Distance (m) (b)

(.)

- - - Measured ___ Dama ge analysis n~2 -0- Conventional F.E. analysis

E 20

.5

- - - Measured ___ Damage _8 n a l YSiS~ 17 -0- Co nventiOnal c', F.E. analysis ~ 1 0 k', __ _

E 20

U--.5

;:

11 10

it o!===!:3C=O~==~,=......____~~ Ci . 5

o

4

8

12 Distanee(m) (e)

16

20

,

0

---

----

0 5 . "0~---:4C-~~8'~~ 1 2"-;1~6~--,i20 Distance (m) (d)

Fig. 5-39 Dis placcmcnt of an extCllsomcter at lhe final excavation stage. (a) Ell ; (b) 812 ;

«) E16 ; (d ) 817 ([5-9j)

Mesh 0

1.46 em

Desp! 0

Scale 10 m

Scale 10

0.95 em

em

00

00

Fig. 5-40 Deforma tions computed a t lhc final cxcava tio n stage. (a) Damage analysis; (b) Conve nUona l fini te elemcnt a nal ysis

328

5 Basis of Anisotropic Damage Mechanics

Failure elements developed at the final stage of excavation are shown in Fig.5-41. It can be found that there are some fractured elements on the leftside wall. This is a typical effect of damage parallel to the wall and the inclined initial stress. In comparison with the observed data of the extensometer E17 in Fig.5-37, it is understood that the loosening zone is developed until we obtain this extent of fractured elements. On the other hand, in conventional finite element analysis, the failure criterion Eq.(5-288) is adapted for the rock mass. However, no element reached failure.

Tured element([5-9])

Fig. 5-41 Fractured element predicted by damage analysis

To summarize this analysis, the theory of anisotropic damage mechanics can effectively analyze the deformation and fracturing behavior of rock mass discontinuities. The finite element method combined with anisotropic damage mechanics provide a developed scheme to simulate the failure problems of rock structures. 5.12.2 Damage Mechanics Analysis for Stability of Crag Rock Slope In order to present more applications and show the acceptability of the developed theory, the stability of a crag rock slope problem was analyzed and judged by numerical results obtained using the developed finite element model of damage mechanics compared with the corresponding results obtained using the traditional elasto-plastic finite element method.

5.12 Numerical Application to Analysis of Engineering Problems

329

5.12.2.1 Summary of Modeling and Geological Conditions The analyzed crag rock mass of Plank Rock Mountain in China is located on the south side of Huangshi Road in Huangshi City less than 100 m from the northern slope of Plank Rock Mountain, where the landform is an abrupt escarpment. There are a total of 9 sets of geological strata from the top to those deep marked by Ptm rv T 1 d y 3-2, in which T 1 d y 3-2, T 1 d y 3-1, T 1 d y 2-2 strata constitute the corpus of the crag rock mass which consist of ash rocks with a thick layer. But Tl dy 2-1, Tl dyl-l are mainly thin layer ash rock interlining shale rock located on the lower part of the crag rock body of the abrupt precipice. P2d and P21 geological strata are mainly shale rock, and some coal seams are interlined in the P 2 1 strata. Because of long-term mining of the coal seams, this has resulted in some intensive excavated empty areas in the lower part. According to the geographical characteristics and distribution of opened cracks on the surface of the earth, the crag rock body is divided into 4 categories as rank I, II, III, IV of crag rock masses. Based on geological investigation and comprehensive research, rank III of a crag rock body is considered to be the most dangerous one, which will be taken into account as the focal point of research. The geological model of the crag rock body in rank III can be overviewed as Fig.5-42.

300

- N E9°

150

x

o

150

300

450

600

(m)

Fig. 5-42 A illustrational sheath of geological model for III category crag rock mass

Based on an investigation of geology and synthesizing various factors, the principal part of underground stress (i.e. earth stress) is considered to be due to gravitational weight. The boundary conditions on both the left and right sides are restricted as hinges, the boundary condition on the bottom is fixed and the top boundary is free. The model of mechanics is represented as Fig.5-43. The finite element mesh in this calculation is divided by a total of 699 elements with 780 nodal points. According to experiments using physical and mechanical properties tests on specimens of the rock block and a shear test on the structural plane of rock pieces, the final physical and mechanical param-

330

5 Basis of Anisotropic Damage Mechanics y

Fig. 5-43 The model of finite element mesh for numerical analysis of the crag rock mass

eters used in the calculation are determined by the estimation method from rock mechanics and t he comprehension method from the insite tests for rock mass given in Table 5-4. Table 5-4 Results Code of Mass geological density strata (g/cm 2 ) P1m 2.0 2.0 P 21 2.0 P 2d Tl dy2 2.4 Tl d y 2 - 1 2.6 Tl d y 2-2 2.6 Tl d y 3- 1 2.6 Tl d y 3-2 2.6 Fractured 2.0 strap

of calculated physical and Deformational Poisson's modulus ratio (CPa) M 2.0 0.25 0.5 0.40 2.0 0.35 2.8 0.30 3.0 0.30 3.0 0.30 13.0 0.25 20.0 0.25 1.0 0.35

mechanical parameters Tensile Cohesion Internal strength (MPa) friction angle (0) (MPa) 0.13 34 0.07 0.05 0.10 25 0.025 0.15 28 0.02 0.15 30 0.09 0.20 33 0.08 0.30 35 0.475 0.40 35 0.80 0.50 37 0.00 0.015 20

5.12.2.2 Traditional Finite Ele m e nt Ana lysis

The calculated results by traditional finite element analysis show that a concentration of the principal stress in the crag rock mass of rank III appears at the crack t ip and in the excavated coal seams and that stress directions within the stress concentration area have deflected. The stress field in the rock mass basically presents the natural stress state due to self-weight. Displacements are mostly in a vertical direction and vary within - 0.01", - 0.25 m. The maximum principal stress 0'1 is about 0.4", -3.2 MPa (tensile as positive, pressure as negative) , and the minimum principal stress O' y is about - 1.6", - 7.6 MPa.

5.12 Numerical Application to Analysis of Engineering Problems

331

The maximum shear stress T max is 0.2rv2.2 MPa. Near the crack tip 0'1 is -1.6 MPa, 0'2 is -2.6 MPa and Tmax is 0.7 MPa. But 0'1 below the crack tip in the nearby region changes to a small value of -0.4 MPa. A tensile stress region appears around cracks on the top of the slope. A certain region of plastic failure area appears on the free empty face and around cracks on the top of the slope. A smaller region of plastic failure area also appears near the excavated area in the coal seams. 5.12.2.3 Finite Element Analysis for Damage Mechanics

As can be known from subsection 5.2.3, if the length, the direction and the area of cracks and joints in a rock mass can be measured from the in-site test, the damage tensor may be calculated. Therefore, the damage tensor of a rock mass should be determined through a standard measurement method of the discontinuous characteristics of structural planes in a rock mass. If there exist multi-groups of joints in a rock mass, the traditional method is to make a summation of them, so that (5-291 ) But practice shows that this kind of simple summation method has a significant disadvantage This is that some components in the total damaged tensors may> 1, which is unreasonable. In order to overcome this, an energy equivalent method can be employed to modify the effect. The commonly used methods for measuring networks of rock structural planes are the statistical windows method and geodetic line method, and the applied conditions of the second one are much wider. In the traditional single geodetic line method, two marked lines should be drawn in parallel with 0.5 m interval from top to bottom, which may approximately determine the plane density (A) and the volumetric density (J) of rock joints. It has already been proved that, as long as the density value in the direction of a geodetic line is tested out, the density value in the normal direction would be determined. In the spatial right angle co-ordinate system as shown in Fig.544, for the geodetic line OD the trend angle is a, the prone angle is 13 and for the geodetic line OE the trend angle is a1, the prone angle is 131. If, along the OD geodetic line, the line density of the rock structural plane is A', the area density of the rock structural plane is A~, whereas, if along the OE geodetic line, the line density of the rock structural plane is A~, the area density of the rock structural plane is As. Then

As

=

" A cosB

=

A sinacosf3sina1 cosf31

+ A, cosacosf3cosa1 cosf31 + Asinf3sinf31

(5-292) From that, the density value of the normal directions of joints is the same as that of the area density. In the calculations of the damage tensor, the

332

5 Basis of Anisotropic Damage Mechanics 0° 330° ;

E

......

)-./

AO=A(COS a)

/

300

0

I I f

.,.,30°, \ .. 60°

.../

/

\

\

\ 90°

i-

270° -t

I \

\

\

f

>-

240° "

/

,

/

';t..

240° . . . (a)

/

r

I

I

120°

>(/

180°

; 150°

(b)

Fig. 5-44 The relationship of rock structural plane densities and arbitrary geodetic lines. (a) Relation of geodetic lines and normal direction of structural planes; (b) The density of structural planes versus directions of that maximum density value of each group of rock structural planes should be generally determined individually. Therefore, a rough scatter of rock structural planes should be carried out. Thus there is a need to plot a diagram of equal density lines (i.e. contours) for joint verteces, and the rough scatter should be divided into combined groups for engineering purposes. The value of the volumetric density J of the rock mass can be determined based mostly on the obtained value of area density. (5-293) The value of J can also be calculated approximately by the engineering quantity of RQD J = (115 - RQD)/3.3

(5-294)

Using the mean values from Eq.(5-16) or (5-291), we may obtain (5-295) where, J i is the volumetric density of the ith joints, (Xi is the average area of the ith set joints. If we assume the form of the joint plane is like a circular dish form, then -k

(X.

•

=

1 J2 -nd·

4

•

(5-296)

where di is the average trace length of the ith set joints. Assume the trend angle is (X, the prone angle is f3 for a set of joints, then the normal vector of these joints can be expressed as

5.12 Numerical Application to Analysis of Engineering Problems

nl = cos (90° - f3)sina } n2 = cos (90° - f3)cosa

333

(5-297)

n3 = sin;3 The volumetric density can be determined based on either Eq.(5-293) or Eq.(5294), and from Eq.(5-294) we have fli = 0.24l i cZT(115 - RQD)(n~

Q9

n~)

(5-298)

From Eq.(5-293) we have (5-299) where

ni Q9 ni =

n1n1 nln2 n1n3] [ n2nl n2n2 n2n 3 n3nl n3n2 n3n 3

(5-300)

Substituting Eq.(5-299) into Eq.(5-291), the effective damage tensor of the fracture-damaged joint rock mass thus can be obtained. The rock structural planes of this region have been measured in geological strata of T1dy3-1 , T 1d y 2-2, T1dy2-1, and T1dyl-l, then the statistical method for rock structural planes was applied to divide groups. The trend angles, prone angles, trace lengths, intervals and density quantity of each group of rock structural planes were statistically worked out. While computing the normal vector of joints based on Eq.(5-297), the relation between the direction of section planes and the production form of rock joints to be calculated should be considered since in a specific investigated region, the characteristics of rock structure development are determinate. Ifthe relation between the direction of section planes and the production form of rock joints is not taken into account, then not only is the calculated damage tensor the same but also their effect on the properties of the rock mass would be the same. In other words, if we do not consider the relationship between these two factors, the influence of the damage tensor on any cross section planes calculated in any directions would be the same in the analyzed region. Obviously it is not true, since the stability of the rock mass is strongly controlled by the structures of the rock mass. Therefore taking different calculated cross sections, the influence of rock structures on their stability has very significant differences. Considering this fact, for a particular calculated plane, Eq.(5-297) should be nl = cos (90° - f3)sin(a - aD) n2 = cos (90° - ;3)cos(a - aD) n3 = sin;3

}

(5-301)

334

5 Basis of Anisotropic Damage Mechanics

where 0: is the trend angle of the calculated cross section plane. The damage values calculated for each geological stratum correspondingly are presented in Table 5-5. Table 5-5 Damage tensor and geometrical parameters of rock structural planes Code of Production geologi- form of cal joints

Trace Int e rval length (m) (m)

T , d y3 T , dy2 T , dy2 T , dy'

1

2

1

Dll

D'2

D'

1.57

1.02 0.64

0.384

0.249

- 0.141 0.210

- 0.088 0.092

1. 39

0.72 0.050

0.049

0.012

0.188

- 0.007 0.005

0.93

0.98

1.50 197° L 87° 1.13 315 0 L 69 ° 0.45 65° L 85° 0. 3 1 280 0 L 67° 0.35

300° L 60 ° 2° L 76 °

Damage t e nsor

1m2

strata

215° L 63 ° 60° L 81 °

Area density (number)

0.35

2.96

0.18

5.56

3

D' 4

D' 5

D16

0.24

4 .17

0.001

- 0.005 0.002

0.026

- 0.011 0.005

0.50

0.62

0.110

- 0.079 0.069

0.139

- 0.092 0.063

0.1 4

0.06

1.61 16.67

Fig.5-45 to Fig.5-49 (the other diagrams are ignored) present a part of the numerical results obtained by a finite element program of damage mechanics. It can be seen from these figures that the principal stress at the crack tip and in the excavated coal seam appears as a significant stress concentration in the area of the crag rock mass with a rank III. The stress directions within the stress concentration area have more deflections. The stress field in the rock mass still presents the field of the natural stress state due to self-weight. Displacements are mostly in the vertical direction and vary within - 0.05rv - 0.40 m and the horizontal displacement increases slightly in some regions of the downstream slope body. The maximum principal stress (}1 is about 0.8rv - 6.6 MPa (tensile as positive, pressure as negative), and the minimum principal stress (}2 is about - 0.13rv - 12.0 MPa. The maximum shear stress T ma x is 0.2rv4.2 MPa. Near the crack tip (}1 is - 3.6 MPa, (}2 is - 7.4 MPa and T max

(unit: MPa)

Fig. 5-45 Contours of the maximum principal stress distributed in crag rock mass with rank III

5.12 Numerical Application to Analysis of Engineering Problems

335

(unit: MPa)

Fig. 5-46 Contours of the minimum principal stress distributed in crag rock mass with rank III (for damage analysis)

(unit: MPa)

Fig. 5-47 Contours of the maximum shear stress max distributed in crag rock mass of rank III (for damage analysis)

(unit: MPa)

Fig. 5-48 Distribution of tensile stress areas in crag rock mass of rank III (for damage analysis)

336

5 Basis of Anisotropic Damage Mechanics

Fig. 5-49 Distribution of failure zones in crag rock mass of rank III (for d amage analysis)

is 2.3 MPa. A larger region of tensile stress areas appears around cracks on the top of the slope. A certain region with quite a few damage failure areas appears on the free empty face and around cracks on the top of the slope. Also some damaged failure zones appear near the excavated area in the coal seams. From the above mentioned numerical results it can be found that a significant stress concentration appears at crack tips, where there exist some damaged failure elements in the crag rock mass of rank III under action of stress due to the self-weight. The results of stress concentration will make the cracks produce continuous tensile failures or shear failures, and hence cause the slope to have a continuous tensile deformation, which agrees with practical observed materials. In the excavated empty area of the coal seam, some obvious stress concentration is also produced Consequently, some damaged failure elements also exist as well as shear failure to occur here. All these results cause the crag rock mass to completely bed down. This then makes the upper part of the crag rock mass a shear slip failure. The empty area on the crest face of the crag rock mass produces some certain areas of tensile stress in the horizontal direction and damaged failure zone. The empty free area of the crag rock mass will be deformed in tension along with the direction of the empty free face under the action of tensile stress, which causes the rock mass to break down. From the above analysis it can be concluded that the crag rock mass of rank III and the crack tips as well as the region of cross boundaries between the empty free area and excavated coal seams are in an unstable situation in this case, which may cause rock mass failure in the form of a breaking down The shear plane slips and cracks are continuously deformed by tension.

5.12 Numerical Application to Analysis of Engineering Problems

337

5.12.2.4 Comparison of Results

From the calculation model and numerical results obtained by the above two kinds of finite element methods, it is found that the results obtained by the method of damage mechanics were changed due to an increase in the overall stress level because of the existence of damage in the rock mass This means the Cauchy stress tensor is replaced by the effective stress tensor within the damaged zone in the rock mass. This kind change is not a simple superposition, having a close relationship to the damage tensor in the rock mass. The obtained damage tensors that are calculated based on different developed situations of rock structural planes in the rock mass have different states, therefore the changed situation of the effective stress is not the same. Usually the effective stress in most damaged elements increases, but may decrease in a few special elements. This may explain why the Cauchy stress has been replaced by the effective stress within the damaged zone in the rock mass because we are really considering the existence of damage in the rock mass, which makes the results more realistic. From contours plotted in the figures of principal stress distribution (see Figs.5.45-Fig.5.47), it has been found that the results calculated by the damage mechanics method are more reasonable and the location of the stress concentration has a more definite response, which can easily explain the rule of deformation and failure in rock mass, and is in good agreement with the real situation. Fig.5-50 to Fig.5-52 show comparison between Cauchy stresses and the effective stresses in the element chosen as the location of the crack tip. From the figures it can be shown that the effective stress calculated by the finite element method of damage mechanics is significantly higher than the Cauchy stress calculated by the traditional finite element method (comparison in absolute quantities). The increased amount of effective stress corresponds with the softening quantity of the elastic matrix weakened by the damage tensor. The response of the effective stress concentration is more obvious at the crack tip. The incremental regulation for normal effective stresses 0'; and 0'; is basically concordant, but for the effective shear stresses of T;y and T;y, the incremental regulation is quite different the one from the other due to the unsymmetrical nature of effective shear stresses (i.e. unequal T;y and T;y). Fig.5-53 shows a comparison of displacements on the top plate of the coal seam at the nodal point 499 which is the cross point of boundaries of excavated and unexcavated coal seams. The results in the figure calculated by finite element method of damage mechanics imply that the roof of the coal seam has no uniform differential settlements under pressure of the blanketed rock mass on the top. The results calculated by the traditional finite element method give a uniform settlement. Generally speaking, different thicknesses and densities of the coal seam roof may cause different pressures in the coal seam, which produce different vertical displacements at different points. The results of displacements calculated based on damage mechanics aptly describe the changed tendency of vertical displacements. Meanwhile,both the displacement results

338

5 Basis of Anisotropic Damage Mechanics 111

109

107

105

103

101

99

0.00

-1.00

Fig. 5-50 Comparison between Cauchy stress 111

109

107

105

O'~

and effective stress

103

101

99

~.--~.---~-,t.;:~--".----..,----.--.----.-~--,

-Traditonal -o-Damaged

O'~

(unit: MPa)

0.00

-3.00 -4.00

Fig. 5-51 Comparison between Cauchy stress

0'; and effective stress 0'; (unit: MPa) 2.00

-

Traditonal

- 0 - Damaged

1.00

111

109

107

105

10 -1.00

Fig. 5-52 Comparison between Cauchy stress T;Y and effective stress T;Y (unit: MPa)

5.12 Numerical Application to Analysis of Engineering Problems 500

498

496

494

492

339

490

~~::;;::::o::=-""-l -0.06 -0.08

--0--

Traditonal Damaged

-0.10 -0.12 -0.14

Fig. 5-53 Comparison of displacements on top plate of the coal seam (unit: mm) and calculated stress results that are based on damage mechanics have a very good consistency, which are in much better agreement with practice than the results calculated by the traditional finite element method. Comparison of results obtained by these two kinds of calculations show that the overall distribution either for displacement or for stress is basically concordant, which illustrates that the traditional finite element method only can give an acceptable result in essential engineering analysis when the required accuracy is not too high. The results of effective stress and displacement that are calculated based on damage mechanics have very good parallelism with the structural characteristics of rock mass. Since the different structural characteristics of rock mass have different damage tensors, their influence on the distribution of Cauchy stress and displacement should be different. The analysis shows that the damage tensor is mostly related to the dimension and the production form of rock structural planes. The damage tensor is directly proportional to the dimension of rock structural planes, and is inversely proportional to the interval between rock structural planes. The more the group number is, the higher the value of the damage tensor. The production form of rock structural planes mainly influences the quantity of components in the damage tensor. From Table 5-5, it can be seen that the average size of geological strata from T 1d y3 to T 1dy3-1 increases. The interval of geological strata between T 1dy2-1 and T 1dy2-2 is in 17.5",,64 cm, but in geological strata of T 1dy2-1 there is only one set of joints to develop. In T1dyl the minimum interval of joints is about 5.6 cm. The interval in the geological strata of T 1dy2-1 is much wider, which means that their relative integrality is much better and corresponds with the property of the rock and the thickness of the rock seam. These structural characteristics of rock mass resolve damage tensors with different deviations, where the quantity of the damage tensor in the geological strata of T 1dy3-1 is the relative maximum and the quantity in T 1dy2-1 is the relative minimum. The influences of the damage tensor on Cauchy stress can be observed in Fig.5-54,where the quantity of the damage tensor for elements 379",,385 in geological strata of P 2 l is 0, which has no effect on the Cauchy stress. Therefore, the two curves coincide; whereas the quantity of the damage tensor for elements 386",,388 in geological strata of P 2 d is relatively bigger and the effects on Cauchy stress are much higher; Further more, elements 389",,392 are in geological strata of

340

5 Basis of Anisotropic Damage Mechanics

T 1d y 2-l, the damage tensor of which is very small, and the effect on Cauchy stress is very lowfor T 1 d y 3-2 geological strata, because the damage tensor is the relative maximum, the effect on Cauchy stress should be the maximum (see Fig.5-54). 391

389

387

385

383

381

379 -0.50

-Traditonal --0- Damaged

-1.50 -2.50

Fig. 5-54 Comparison between Cauchy stress CT; and effective stress CT; in damaged element with different damage tensors (unit: MPa)

5.12.3 Damage Mechanics Analysis for Koyna Dam due to Seismic Event 5.12.3.1 Introduction of Objective Statements The Koyna dam is a 103 m high gravity structure completed in 1963. The dam started impounding water in 1962 and experienced a magnitude 6.5 earthquake, probably reservoir induced, on 11 December 1967 when the reservoir elevation was only 11m below the dam crest. The accelerations of the ground at the site were 0.49 g in the stream direction, 0.63 g in the cross-stream direction and 0.34 g in the vertical direction. The most important structural damage consisted of horizontal cracking on both the upstream and downstream faces of a number of the non-overflow monoliths [5-42"-'43]. A number of 2-D linear analyses have been made to determine the dynamic response of this dam when subjected to the recorded accelerations, while others attempted to include the non-linear features of cracking. A seismic study [5-44] by the finite element method (FEM) considered cracking with stress release once the tensile stress reached a critical value which included a factor to account for strain rate effects. Another study [5-45] used fracture mechanics and a contact/impact model for crack closure within a finite element formulation to analyze the seismic performance of the Koyna dam. Both studies revealed that the formation of cracks on both faces was to be expected during the 1967 earthquake. Experiments have also been conducted to study the dynamic response and cracking pattern. One of these was conducted at the University of California at Berkeley [5-46]. A 1:150 scale model was constructed of a plaster material containing lead powder. During the test run at 1.21 g of the shaking table a crack was initiated on the downstream face at the point of slope change,

5.12 Numerical Application to Analysis of Engineering Problems

341

which then propagated through the dam to the upstream face. Even though the excitation applied to the model was not actual Koyna ground motion and improper gravity scaling for rupture similarity was employed, the results still gave valuable insight into the cracking pattern and location under the test conditions. In spite of the limited field measurements of the pattern of crack-damaging the Koyna dam experience has provided the most complete information todate on seismic crack-damage of concrete gravity dams. Due to the complexity of the problem, analyses made so far have been restricted to simplified models. More sophisticated mathematical models for dealing with the damage process of concrete structures are still needed and model tests for either verification of the mathematical models or simulation of the prototype performance remain imperative. Since the narrow damage zone of the Koyna dam occurred in the upper part of the dam, near the point of slope change where a high stress concentration is to be found, initial crack-damage would be expected to occur at this location even during the early stages of ground shaking during the 1967 earthquake. Once the initial crack-damage has been formed, it is evident that damage mechanics theory should be employed to evaluate the damage growth and damage propagation process and the resulting pattern of the damage state. Based on the above considerations, the authors developed a new procedure for evaluation of the damage evolution process of concrete gravity dams during strong earthquakes based on the article [5-47]. In this procedure the finite element technique, modal analysis and linear elastic damage mechanics theory were combined. The accuracy of the proposed procedure was verified by a bending test of a beam with varied cross section for the dam model presented in [5-47]. The acceptable good agreement obtained between the numerical predictions and test results indicated that the new procedure is relevant for evaluation of the seismic damage process in concrete dam structures. This section is followed by the experimental results for a model of the Koyna dam with initial crack-damage tested on a shaking table under artificial input motion, for which the foregoing procedure is applied to predict the model performance including damage growth and damage propagation. Finally, the damage process of the Koyna prototype dam during the 1967 earthquake is examined, in which the time histories of the dynamic stress distribution and the damage profile during the earthquake are obtained. The results are also consistent with the observed damage of the prototype in terms of damage elevation on both faces and the phenomenon of elemental average damage on the downstream face, the latter confirming the complete penetration of the damage behavior in the dam as predicted by the present analysis. The numerical procedure for seismic damage analysis of concrete structures presented in Reference [5-47] comprises three distinct parts: (1) finite element (FE) analysis of the dynamic response for elastic damage systems; (2) impact simulation in damaged structures and (3) linear elastic damage mechanics theory for simulating the damage extension process.

342

5 Basis of Anisotropic Damage Mechanics

5.12.3.2 Some Results from Model Test of Koyna Dam and Correlation Analysis The objectives of the dam model tests were: (1) to provide, in addition to the beam with varied section test for the dam model reported in reference [5-47], further verification of the proposed numerical procedure for seismic damage analysis and (2) to obtain a qualitative evaluation of the damage process in the Koyna dam under simplified loading conditions. The model scale of the Koyna dam section is 1:200; namely 515 mm in height; 351 mm wide at the base and 80 mm thick as shown in Fig.5-55. The density 'Y of the gypsum material is 480 kg/m 3 and the dynamic modulus of elasticity E = 600 MPa. Acceterometers were installed on the table and at the crest of the model and strain gauges were located on both sides of the model at 5 and 12 mm from the front boundary of the damaged zone. Thus, based on fracture mechanics with strain Cy measured in the y-direction the stress intensity factor KJ was determined from (5-302) where r represents the distance between the damaged tip and the strain gauge. As shown in Fig.5-55, the model was fixed to the shaking table, with no reservoir water included. Harmonic sweeping tests were first conducted to obtain the frequency components of the model. To cause the dam model to rupture, lead blocks with a total mass of 2.76 kg were attached at the crest. A very narrow initial damage zone about 10 mm in length was simulated by a set of small regularly arranged holes drilled through the thickness at the location of slope change on the downstream face. Because of the capacity limitation of the shaking table, the input excitation for the rupture test was comprised of a series of load pulses, which were approximately periodic but not harmonic. For the numerical predictions, 2 percent damping (~ = 0.02) was assumed for the six modes considered and Poisson's ratio v was assumed equal to 0.2. As shown in Fig.5-56, the test model was divided into a finite element mesh assuming the condition of a rigid foundation to simulate the shaking table. The material properties of the elastic modulus E = 1.0 X 10 5 MPa and equivalent density 'Y = 7680 kg/m 3 were specified in the same quantities for all elements. The initial crack-damage is considered as an initial damage zone, which is profiled by very small holes drilled through the thickness of the dam model within the elements surrounding the broken line on the downstream faces distributed at Gaussian points near the horizontal interface between the dam sub-regions I and II as shown in the circle detail in the sub-figure of Fig.5-56. An unequal distribution of elements, with a much denser mesh near the narrow initial damage zone and also at the slope change location on the downstream face, was employed in order to refine the calculation of stress concentration factors and to permit the damage zone to develop following the crack extension.

5.1 2 Numerical Application to Analysis of Engineering Problems

Fig. 5-55 Koyna dam model with lead block on crest 80mm

.....

II>

S

a ........

....

351 mm

Fig. 5-56 FE mesh model of the dam

343

344

5 Basis of Anisotropic Damage Mechanics

The failure process of the dam model was simulated by the damaged constitutive equations and the damage growth equations presented in this chapter. A step-by-step time integration scheme was employed at each Gaussian point to obtain the structural response and the damage growth as well as the damaged zone extension accumulated from the initial damage state. Time step llt = 0.001 sec was used in the calculation. Once the damaged failure criterion is reached at a Gaussian point, the micro-structure to be considered at this Gaussian point comes into the failure process and therefore the damage will grow and accumulate from the previous damage state. Damage zone propagation occurs perpendicular to the maximum circumferential strain direction with infinite velocity. The foregoing value of the failure damage state can also be implied equivalently by the quantity Kid, which represents the dynamic fracture toughness of the concrete of the dam model and can be obtained directly from the theory of fracture mechanics and the test measurements by

E

~

(5-303)

Kid = 4(1 _ v) Ee,cr

The plot shows in Fig.5-57(b) where sudden rupture of the model is seen to occur at time 0.728 sec. It should be noted that different kinds of plaster were used in the tests of the previous cantilever beam [5-47] and the current model dam, resulting in very different values of KId as well as modulus E (600 MPa for the model dam).

N"-"

'"

lc

9 6

.~

~.,

0; u u

< Time(a)(s)

(b) Time(b)(s) - - - Measured

0.8

······ Computed

Fig. 5-57 Rupture test time histories for model dam. (a) Input motion of shaking table; (b) Measured and computed stress intensity factor KJ

5.12 Numerical Applica tion to Analysis of Engineering Problems

345

For the initially crack-damaged model, the measured and calculated frequencies are listed in Table 5-6. The results from the test and calculation are seen to be close. For the rupture t est the table excitation frequency was 6.1 Hz , with the input acceleration exceeding the table capacity and causing a distortion of the intended harmonic motion as is evident in Fig.5-57(a). The time histories of the measured and calculated (force method and assuming no damage zone extension) stress intensity factor KJ at the front tip of the damaged zone are compared in Fig.5-57(b). It is noted that the finite element results and the tested measurements are in good agreement in general. Fig.557(b) also shows that rupture of the model dam occurred at t = 0.728 s, when the strain gauges suddenly broke during the test. Table 5-6 Natural frequencies (Hz) for different modes of the damaged dam model Nat ural frequencies (Hz) Model Measured Calculated

ii

51.2 51.I

252.3 273.9

366 .4 375. 9

604.2 670.2

i5

1119 1127

1233 1187

Fig.5-58 shows a comparison of the crack-damage profiles at rupture observed in the test [5-47] and those obtained by numerical simulation using damage localization techniques with finite element computation. The agreement between these results is outstanding, thereby confirming that the localization damage models for crack-damage simulation are applicable for practical problems.

(a)

(b)

Fig. 5-58 Comparison of rupture profile for model dam between test (a ) and numerical (b) simulation

346

5 Basis of Anisotropic Damage Mechanics

5.12.3.3 Damage Analysis for Practical Koyna Dam in 1967 Earthquake The Koyna dam cross-section and its FE discretization are shown in Fig.5-59. To improve accuracy of local damage behavior by limiting the difference in element size encountered over the domain, a transitional sub-region is introduced in the FE discretization as shown in the detailed subfigure enlarged from the circle in Fig.5-59. The characteristics of the Koyna dam are [542 rv 43]: E = 3.1 X 10 4 MPa, "( = 2640 kg/m 3 , v = 0.2 and ~ = 0.05. y

'V91.80m

Fig. 5-59 FE discretization of Koyna dam

The elastic modulus of the foundation rock (7 x 104 MPa) is approximately twice that of tile concrete. Considering the high stiffness of the foundation rock and the fact that the ground acceleration was obtained directly at the dam base, a rigid foundation and earthquake input applied directly at the base were assumed. The corresponding accelerograms of the ground motion are shown in Fig.5-60. The component in the cross-stream direction was assumed not to affect crack-damage development and was therefore neglected. In the results to follow , seismic loading refers only to the effect of components of the 1967 Koyna earthquake in both the horizontal stream-wise and the vertical directions, whereas static and dynamic loading includes the seismic loading, the dam's self-weight as well as the hydrostatic force.

5.12 Numerical Application to Analysis of Engineering Problems

347

O~ r------------------------------~

e:o

0.4

';;' 0.2 .~

~., oof.~~M'"

U

8-0.2

«

-0.4 - O.O ~--~~--~~--~~--~~--~~--J

o

2

3

4

5 6 Time (s) (a)

7

8

9

10

II

,..,0.4 ~

.g 0.2

t 0.0

-.;

g-0.2

« -0 .4

L...---L.. _ _ L...---L.. _ _ -'-----L.._ _ -'-----L.._ _ -'-----'-_ _ .L-....J

o

2

3

4

5 6 Time (s) (b)

7

8

9

10

11

Fig. 5-60 Ground acceleration of Koyna earthquake, 11/12/1967. (a) Stream direction component; (b) Vertical component

According to field measurements taken after the 1967 earthquake [5-43], most of the downstream cracks occurred at, or near, the location of slope change where the effect of stress concentration is expected to be significant. Similarly according to the test model dam, an initial narrow crack-damage zone was assumed to exist at elevation 66.5 m near the downstream face and the initial damage state of Do = 0.3 is assumed to be put closely at two rows of Gaussian points within the conjoint elements surrounding the broken line on the downstream faces as illustrated in Fig.5-59. This gives a modeling the initial damaged points distributed at these Gaussian points near the horizontal interface between sub-regions I and II of the dam and they are scaled up in a detailed way in the circle sub-figure of Fig.5-59. This has the function of expressing the behavior of damage localization and the narrow damaged zone propagating in order to simulate the crack enlarging. A record over time of the dam's response was computed taking into consideration contributions from different points of view (saturations). As the damage developed (growth and propagation), the frequencies and mode shapes of the dam structure should be changed accordingly. Damping ratio ~ = 0.05 for all elements was assumed. Based on the foregoing assumptions, the following response behavior was predicted for the Koyna dam prototype.

348

5 Basis of Anisotropic Damage Mechanics

The first four modal frequencies of the initially undamaged Koyna damreservoir system are 3.07, 7.98, 11.21 and 16.52 Hz, respectively, which are very close to the results from FE analysis [5-43] and thus serve to verify the accuracy of the present damage finite element discretization. For a load combination consisting of static and dynamic components, time step integration was performed with !1t = 0.005 s and an earthquake duration of 6.0 s. The response of the dam in terms of the crest displacement and acceleration, and also the stresses both on upstream and downstream faces, was examined. The results may be summarized as follows: The horizontal displacement of the dam crest reached 43.6 mm in the downstream direction. The maximum accelerations at the crest were 22.8 m/s 2 and 22.6 m/s [5-43] in the horizontal and vertical directions, respectively, with corresponding amplification factors of 4.8 and 6.6. The computed maximum tensile stress of 6.69 MPa occurred near the point of slope change on the downstream face and already far exceeds the tensile strength of the concrete, thus confirming that the first crack-damage zone is indeed to be expected at the point of slope change. Fig.5-61 shows a comparison of historical displacements at the corner point of upstream faces on the dam top obtained by damage analysis and undamaged elastic analysis. From the results of the damage analysis, it can be found that both the horizontal and vertical displacements ofthe observed point reach the maximum value at time t = 4.4 s and decrease after 6.5 s. Whereas the results of undamaged elastic analysis reach the maximum value at time t=7.0 s and after that the responses of undamaged elastic analysis are always lower than those of the damaged one. These facts state that when the crack-damage is perforated on top of the dam, where the action on the crack-damaged zone due to the earthquake becomes relatively lighter the frequencies of the horizontal and vertical responses are not quite changed. Fig.5-62 presents some numerical results obtained by a Gaussian points time integration scheme and plotted in the form of photo sketches for simulated damage profiles at different rupture times during the 1967 Koyna earthquake computed as damage distribution at all Gaussian points in the observed cross section based on an isotropic damage model for the Koyna dam. The power law equation of a damage strain energy release model presented in Eq.(5-268) was taken into account in the finite element analysis for modeling the damage growth simulation. Sketches (a), (b) and (c) in Fig.5-62 show the procedure of crack-damage growth and crack-damage zone propagation within a cross section of the Koyna dam at the rupture time t = 3.8, 4.0 and 4.5 s due to ground acceleration of the earthquake. It can evidently be seen that the damage localization phenomena was successfully carried out by the damage growth model and the initial Gaussian point damage model. The procedure of narrow damage zone extension presents the crack-rupture profiles due to material damage. As can be expected, the most serious damage zones appear at the tensile stress concentration zones accordingly near the downstream face and horizon-

5.12 Numerical Applica tion to Analysis of Engineering Problems 6.0

8' e

- - EL"'a she ana I"YSIS

4.0 . ~ ~::R\!\Jl\!g~J!I)~IYS1·· i~

:

't

:, i

~

.;t.

349

.

Ii :::~::1¥v~W:f~~~~;~,I ~~t;C · · · · · · · . · ~· [. i ~ .· ;. · ·; ; · I

,•

-2 .0 ............. . . . ... . . . ... . ... ............ .

o -4.0

.

..

................................ ............................................................ : 1........ . ....... , . -6 .0'--_ _-'-_ _--'-_ _---'_ _ _-'--_ _-' 02468 10 Time (s) (a)Horizontal displacement 1.5 - - Elastic analysis ~ 1 Occ~cRamageanalysis

e . -5 0.5

~~

. . ........ ......... ......... ......... ......... . ... ........ ......••. .. ... .......

.

'i' ... . . . . ...

·. 1

1_::~~~,~h~1!.jlt~~i~ir~v1y~~· .~

~

~ I,

J" : '

Cl -I.0 ,, ......'... "", '

I

I. . I

'.

" .... ;....... ............ ......... 1

-1.5 '-----'-----'------':--''---'--'-----' o 2 4 6 8 10 Time (s) (b) Vertical displacement

Fig. 5-61 Comparison of displacement responses for different analysis due to Koyna ea rthquake, 11 / 12/ 1967. (a) Horizontal displacement; (b) Vertical displacement

(a)

(b)

(c)

Fig. 5-62 Sketches of simulated damage profiles at different rupture times during Koyna earthquake computed for damage distribution at all Gaussian points in the observed cross section based on isotropic damage model for Koyna dam. (a) t =3.8 s; (b) t = 4.0 s; (c) t =4.5 s;

350

5 Basis of Anisotropic Damage Mechanics

tal interface between sub-regions I and II of the dam at elevation 66.5 m as well as the regions of the toe corner and heel corner surrounding the bottom of the dam. Fig.5-63 plots the distribution of displacement contours in the damaged Koyna dam due to the Koyna earthquake wherein Fig.5-63(a) shows the horizontal displacement contours, and (b) shows vertical displacement contours respectively. A significant denser gradient both for horizontal and vertical displacement contours appears near the toe corner region of the dam where the less free deformation is restrained at the bottom due to the foundation rock, but does not appear in the crack region, since the crack is opened with freer deformation, which is less restricted due to the crack opening and closing.

(a) Horizon tal di splacemen t

(b) Vertical dis placement

Fig. 5-63 Contours of displacement distribution in damaged Koyna dam due to Koyna earthquake. (a) Horizontal displacement; (b) Vertical displacement;

The plot in Fig.5-64 shows a vector sketch of deformational direction distribution in the cross section of the Koyna dam due to the Koyna earthquake. It can be seen that the tensile form of the deformational direction vector appears in all expected regions surrounding the damage-crack, the toe corner and the heel corner. These tension zones are the most seriously damaged areas in the Koyna dam due to the Koyna earthquake. The profile of the mean damage pattern in damaged elements of the Koyna dam during the Koyna earthquake also has been analysed by the damage finite element method for anisotropic (orthotropic) materials based on the average integration scheme of damage evolution equations expressed in Eqs.(5-

5.12 Numerical Application to Analysis of Engineering Problems

351

11<1,:-----.........

""""4'''''''''''''''''6'''_''-t \~ ~

h''''''''""", U, r/4r"-,

'''\H~\~HU,t.l.t I ~

'~~"'\~~~t~!4JJjJ~

r II i r r direction vector of deformations

~~t--tt'¢.""""""'-";-"~IL/

-

Fig. 5-64 Sketch of deformation direction vector distribution

270)"-'(5-273). It can be seen that the process of damage propagation from the downstream not only towards the upstream face horizontally but also round the downstream surface vertically has been presented from analysed results that are based on the elemental average damage according to the assumption of the initial damage state as being a narrow zone of initial damaged elements with a value of Do = 0.3, which is defined in the narrow distributed initial crack-damaged elements horizontally, as explained before. The pattern sketches of the mean damage may present an observation for the distribution of the damaged element number, location of damaged elements and stages of elemental damage in the dam as shown in Fig.5-65 and Fig.5-66. Fig.5-65 shows the damage patterns at time t =3.80 s for different ratios of anisotropy from Ed E2 = 0.5 to Ed E2 = 4.0. Fig.5-66 shows this at time t = 4.10 s for the corresponding ratios, respectively. It can be seen that the most seriously damaged elements appear on the horizontal interface between sub-regions I and II of the dam near elevation 66.5m and the regions of the heel surrounding the bottom of the dam, where the tensile stress concentration zones are to be found. However the damage state (i.e. the stage and number of damaged elements) at time t = 4.10 s is significantly higher than that at time t =3.80 s for each of the different anisotroic ratios whereas the seriousness of the damage state increases significantly with an increase in anisotropic ratios from Ed E2 = 0.5 to Ed E2 = 4.0. As shown in Fig.5-67, the overall trend of the history over time of the crest displacement is almost similar to that of the crack-damage before t < 0.8 s of

352

5 Basis of Anisotropic Damage Mechanics

(a)

(b)

(d)

(c) II

1.00.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

Fig. 5-65 Damage pattern at time t =3.80 s for different ratios of El /E 2

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

Fig. 5-66 Damage pattern at time t =4. 10 sec for different ratios of EdE2 15

'8

10

,§,.

~ 5

.,8

"os

~O

is

3

- - Tine integration scheme at each Gaussian point - - - - -Time integration scheme ave raged overall an elenent 0.4

0.8

1. 2

1.6

2.0

Time (s)

Fig. 5-67 Comparison of time history of crest displacement for Koyna Dam under different time integration scheme

References

353

the initial growth obtained from the two time integration methods (i.e. damage accumulation in time is integrated at each Gaussian point or is averaged overall an element respectively). Consequently, a significant discrepancy in the magnitude of these two methods of time integration appears from t > 1.56 s, after the crack-damage growth starts. This is clearly due to the spurious average damage distribution in a damaged element and a higher displacement permitted due to the average time integration method. The magnitude of displacement responses obtained by both methods increases significantly with the duration of time after the earthquake started, for 2.0 s due to huge damage growth and propagation occuring in the Koyna dam.

References [5-1] Leckie F., Onate E., Tensorial nature of damage measuring internal variables. In: Proceedings of the IUTAM Symposium on Physical Non-linearities in Structural Analysis, Sens, France. Springer, Berlin, pp.140-155 (1981). [5-2] Cordebois J.P., Sidoroff F., Endommagement anisotropic. J. Theory Appl. Mech., 1(4),45-60 (1982). [5-3] Murakami S., Ohno N., A continuum theory and creep damage. In: Proceedings of the 3rd IUTAM Symposium on Creep in Structures. Springer, Berlin, pp.422444 (1981). [5-4] Betten J., Damage tensor in continuum mechanics. In: Lemaitre J. (ed.) EUROMECH Colloquium 147 on Damage Mechanics. Canhan, France, pp.416-421 (1981). [5-5] Vakulenko A.A., Kachanov L.M., Continuum theory of cracked medium. Mech. Tverdogo Tela, 4, 159-166, in Russian (1971). [5-6] Murakami S., Notion of continuum damage mechanics and its application to anisotropic creep damage theory. J. Eng. Mater. Tech., 105, 99-105 (1983). [5-7] Sidoroff F., Description of anisotropic damage application to elasticity. In: Proceedings of the IUTAM Colloquium on Physical Non-linearities in Structure Analysis, Sens, France. Springer, Berlin, pp.237-258 (1981). [5-8] Tamuzh V., Lagsdinsh A., Variant of fracture theory. J. Mech. Polymer, 4, 457-474, in Russian (1968). [5-9] Kawamoto T., Ichikawa Y., Kyoya T., Deformation and fracturing behaviour of discontinuous rock mass and damage mechanics theory. Int. J. Numer. Anal. Methods Geomech., 12(1), 1-30 (1988). [5-10] Zhang W.H., Murti V., Valappan S., Effect of matrix symmetrization in anisotropic damage model. Uniciv Report No. R-237, University of New South Wales, Australia (1991). [5-11] Zhang W.H., Numerical Analysis of Continuum Damage Mechanics. Ph.D. Thesis, University of New South Wales, Australia (1992). [5-12] Zhang W.H., Chen Y.M., Jin Y., Effects ofsymmetrisation of net-stress tensor in anisotropic damage models. Int. J. Fract., 109(4),345-363 (2001). [5-13] Zhang W.H., Valliappan S., Analysis of random anisotropic damage mechanics problems of rock mass: Part I. probabilistic simulation; Part II. statistical estimation. Int. J. Rock Mech. Rock Eng., 23(2), 91-112, 241-259 (1990).

354

5 Basis of Anisotropic Damage Mechanics

[5-14] Zhang W.H., Valliappan S., Continuum damage mechanics theory and application: Part I. theory; Part II. application. Int. J. Dam. Mech., 7(3), 250-297 (1998). [5-15] Zhang W.H., Chen Y.M., Jin Y., A study of dynamic responses of incorporating damage materials and structure. Struct. Eng. Mech., 12(2), 139-156 (2000). [5-16] Valliappan S., Zhang W.H., Murti V., Finite element analysis of anisotropic damage mechanics problems. J. Eng. Fract. Mech., 35, 1061-1076 (1990). [5-17] Murakami S., Anisotropic damage theory and its application to creep crack growth analysis. In: Desai C. et al. (eds.) Constitutive Laws for Engineering Materials: Theory and Applications. pp.107-114 in Japanese (1987). [5-18] Chaboche J., Continuum damage mechanics: Part I. general concepts; Part II. damage growth, crack initiation, and crack growth. J. Appl. Mech., 55, 59-85 (1988). [5-19] Chaboche J., Anisotropic damage in the framework of continuum damage mechanics. Nucl. Eng. Des., 79, 181-194 (1984). [5-20] Lemaitre J., A Course on Damage Mechanics. Springer, Berlin, Heideberg, New York (1992). [5-21] Kachanov L.M., Introduction to Continuum Damage Mechanics. Martinus Nijhoff Publishers, Dordrecht, The Netherlands (1986). [5-22] Voyiadjis G.Z., Ju J.W., Chaboche J.L., Damage Mechanics in Engineering Materials. Elsevier, Amsterdam (1998). [5-23] Davison L., Stevens A., Thermomechanical constitution of spalling elastic bodies. J. Appl. Phys., 44, 667-674 (1973). [5-24] Coleman B., Gurtin M., Thermodynamics with internal state variables. J. Chem. Phys., 47, 597-613 (1967). [5-25] Ilankamban R., Krajcinovic D., A constitutive theory for progressively deteriorating brittle solids. Int. J. Solids Struct., 23(11), 1521-1534 (1987). [5-26] Krajcinovic D., Fonseka G.V., The continuous damage theory of brittle materials: Part 1. general theory; Part 2. uniaxial and plane response modes. ASME Trans. J. Appl. Mech., 48(4), 809-824 (1981). [5-27] Chow C.L., Yang X.J., A generalized mixed isotropic-kinematic hardening plastic model coupled with anisotropic damage for sheet metal forming. Int. J. Dam. Mech., 13(1), 81-101 (2004). [5-28] Lee H., Peng K., Wang J., An anisotropic damage criterion for deformation instabity and its application to forming unit analysis of metal plates. J. Eng. Fract. Mech., 21, 1031-1054 (1985). [5-29] Geoge Z., Voyiadjis G., Taeayo P., Local and interfacial damage analysis of metal matrix composites using the finite element method. J. Eng. Fract. Mech., 56(4),483-495 (1997). [5-30] Hayakawa K., Murakami S., Thermodynamical modelling of elasto-plastic damage and experimental validation of damage potential. Int. J. Dam. Mech., 6( 4), 333-363 (1997). [5-31] Valliappan S., Zhang W.H., Analysis of structural components based on damage mechanics concept. In: Elarabi M.E. and Wifi A.S. (eds.) Current Advances in Mechanical Design and Production. Pergamon Press, Oxford, pp.265-280 (1996). [5-32] Valappan S., Analysis of anisotropic damage mechanics. Comput. Mech., 1216, 1143-1147 (1991).

References

355

[5-33] Valappan S., Zhang W.H., Elasto-plastic analysis of anisotropic damage mechanics problems. In: International Symposium on Assessment and Prevention of Failure Phenomena in Rock Engineering, Ankara, Turkey (1993). [5-34] Murti V., Zhang W.H., Valappan S., Stress invariants in orthotropic damage space. J. Eng. Fract. Mech., 40, 985-990 (1991). [5-35] Chaboche J.L., Damage induced anisotropy: on the difficulties asscociated with the active/passive unilateral condition. Int. J. of Dam. Mech., 1(2), 148170 (1992). [5-36] Lu T.J., Chow C.L., On constitutive equations of inelastic solid with anisotropic damage. Theor. Appl. Fract. Mech., 14, 187-218 (1990). [5-37] Chen X.F., Chow C.L., On damage strain energy release rate Y. Int. J. Damage Mech., 4(3), 251-263 (1995). [5-38] Chow C.L., Chen X.F., An anisotropic model of damage mechanics based on endochronic theory of plasticity. Int. J. Fract., 55(2), 115-130 (1992). [5-39] Chow C.L., Chen X.F., Failure analysis of a cracked plate based on endo chronic plastic theory coupled with damage. Int. J. Fract., 60(1), 229-245 (1993). [5-40] Lemaitre J., A continuous damage mechanics model for ductile fracture. J. Eng. Mater. Tech., 107, 83-89 (1985). [5-41] Chow C.L., Wang J., An anisotropic theory of elasticity for continuum damage mechanics. Int. J. Fract., 33(1), 3-16 (1987). [5-42] Koyna Earthquake of II December 1967. Report of the UNESCO Committee of Experts, New Delhi (1968). [5-43] Chopra A.K., Chakrabarti P., The Koyna earthquake and the damage to Koyna dam. Bull. Seismol. Society of America, 63(2), 381-397 (1973). [5-44] Pat S.N., Seismic cracking of concrete gravity dams. ASCE J. Struct. Div., 102, 1827-1844 (1976). [5-45] Ayari M.L., Saouma V.L., A fracture mechanics based seismic analysis of concrete gravity dams using discrete cracks. Eng. Fract. Mech., 35, 587-598 (1990). [5-46] Niwa A., Clough RW., Shaking table research on concrete dam models. Report No. CB/EERC 80-85, Earthquake Engineering Research Center, University of California, Berkeley, CA (1980). [5-47] Pekau O.A., Zhang C.H., Feng L.M., Seismic fracture analysis of concrete gravity dams. Earthquake Eng. Struct., 20(4), 335-354 (1991).

6

Brittle Damage Mechanics of Rock Mass

6.1 Introduction and Objective In this chapter, special aspects of brittle damage mechanics are considered, related to the various frameworks by which the behavior of a brittle damaged material can be postulated, still considering the damaged medium as a global continuum. The basic framework of CDM is recalled, stressing the concepts of effective stress and the ways of introducing coupling between damage and brittle behavior. Particular attention is paid to problems relat ed to init ial or induced anisotropy and the unilateral character of the brittle damage. Coupling between damage and plasticity is not discussed herein. The materials concerned are rock, concrete, considered initially isotropic and composite, in particular ceramic matrix composites. Some specific models are developed , taking into consideration scalar damage variables for initially anisotropic brittle mat erials. Chaboche et al. [6-1] considered only small perturbations and isothermal conditions for anisotropic brittle materials. The total strain c is assimilated with the elastic strain and the damage processes are considered timeindependent. We do not discuss the difficult problems related to instabilities by localization and therefore describe only the first stages of damage, before conditions become critical. Determination of elastic parameters of brittle solids in the function of the concentration, shape, size and orientation of micro-cracks is an important part of micro-mechanical damage models. In the last two decades this problem has been addressed by many authors [6-2"-'6]. In all of these cases the elastic brittle paramet ers were estimated within the framework of Mean Field Theories (MFT). The primary objective studied by Krajcinovic et al. [6-7] was to provide additional data defining the influence of micro-cracks on the elastic parameters of brittle solids. More specifically, this study will address the dependence of the elastic parameters on the orientation of micro-cracks, i. e., defect induced anisotropy. This will provide a necessary background for a forthcoming study focused on micro-crack concentrations, for which mean field W. Zhang et al., Continuum Damage Mechanics and Numerical Applications © Zhejiang University Press, Hangzhou and Springer-Verlag Berlin Heidelberg 2010

358

6 Brittle Damage Mechanics of Rock Mass

theories are inadequate. The complementary case of prescribed macro-strains was considered in Nemat-Nasser and Hori [6-5]. Additionally, the considerations will be restricted to two-dimensional continua in [6-7]. The other important investigation of brittle damage mechanics is intended to generalize the damage model, presented earlier by Chow and Yang [6-8, 69], for brittle anisotropic materials to cover the nonlinear damage behaviors of brittle composite laminates. A general form of constitutive relations describing the nonlinear damage response of brittle unidirectional composite lamina is first introduced and then expressed in such a way that it can be readily incorporated in the analysis of brittle laminates. The earliest investigation is limited to symmetric laminates without discontinuity. Brittle laminates are thin and the free edge effects are not considered. Therefore, in the case of in-plane uniform loading, each layer in the brittle laminate, as well as the laminate itself, will be regarded as in a state of plane stress. With the method due to Tsai and Hahn [6-10], the constitutive equation of the brittle laminate with damage is determined from those of its constituent parts. The damage process in the brittle laminate is analyzed and the comparisons between the experimental observation and predictions are discussed in [6-11]. There are classes of quasi-brittle solids where cracks tend to follow tortuous paths when the local stress or strain conditions for the crack propagation are satisfied. Examples are ceramic objects, where micro-cracks are intergranular and concrete, where cracks emanate from weak interfacial bonds, propagate through the mortar phase and go around aggregate particles that act as energy barriers. During the process of micro-cracking, material grains are severed leaving the strained solid as "damaged". Such a process alters the elastic modulus and can lead to a strong material anisotropy. Yazdani and Karnawat [6-12] presented a study which is the effect that damage, in a given direction, might have on the solid's strength and ductility in other orthogonal directions. This becomes of particular interest in ceramics, rocks and concrete since micro-cracks are not planar, nor do they propagate along perfect planes. In [6-13] Sadowski mentioned that: The existence of cracks, pores and other defects within solids diametrically changes the material response due to the applied load. Many effective continuum models have been proposed to estimate mechanical properties of materials [6-5"-'9]. In the case of semibrittle ceramics, a small amount of plasticity also influences the total material response [6-14"-'15]. The aim of his work was to follow the two-dimensional, quasi-static deformation process (tension-compression) of semi-brittle ceramics. The mechanical response of polycrystalline continua, weakened by a set of slits, was modeled by application mean field theories [6-2"-'4]. According to the experimental results of MgO [6-16"-'18], a limited plastic flow is also created by dislocation motion within the range of grains of the representative volume element. Micro-cracks are initiated by Zener-Stroh's mechanism and propagate mainly intergranularly along grain boundaries (zig-zag cracks), leading to final failure of the brittle material.

6.2 General Theory of Brittle Damage Mechanics

359

The study of the part I of Sadowski [6-13] focused mainly on the description of gradual degradation in brittle materials including micro-mechanically based estimation of current elastic properties, whereas the study of the part II [619] was focused on the description of kinetics of considered process specifying crack shapes and their distribution within the unit cell. As the illustration of deformation process in his considered material uniaxial tension, pure shear and uniaxial compression were particularly analyzed. Also limit surfaces for two-dimensional tension-compression were estimated. He found out that the degradation process is related not only to numbers of defects, like in many models, but is also strongly dependent on the real crack shapes and their distribution within the unit cell. Smooth transition from uniaxial tension to uniaxial compression by the two dimensional states was analyzed by him with the specification of particular modes of crack shapes and their distribution within the unit cell, up to the state preceding final failure. In [6-19], Sadowski said that : Experimental results performed on semibrittle ceramics like Mgo [6-18] lead to the conclusion that the deformation process of this material passes through a sequence of phases. The experimental results shown that after a purely elastic response, in some grain of the polycrystalline specimen a conjugate slip system is created. Under the external load, dislocations pile up to the grain boundaries and produce microcracks; These micro-cracks spread along straight segments of grain boundaries creating so-called meso-cracks. The final stage of the deformation process, preceding the material failure, is kinking of meso-cracks and their unconstrained development into macro-cracks.

6.2 General Theory of Brittle Damage Mechanics 6.2.1 Thermodynamic Basic Expression of Brittle Damage Mechanics 6.2.1.1 State Equations The brittle material in the elastic stage may have elastic deformation or inelastic deformation. If the material with an unchanged micro-structure is defined as the undamaged material, then the change in microstructures (such as micro-cracks, micro-cavities and micro-slides of the damage mechanism) may cause inelastic damage deformations. For a given representational material element, the evolution of the existent damage and the generation of newly initialed damage together represent the variation of microstructures. Thus it is possible to use a damage variable for measuring the changes in microstructures macroscopically and to define an internal stat e variable for expressing the thermodynamic state in the representational material element. The formulations developed in this chapter will be expressed in the mixed form of both tensor and matrix notations together as far as possible.

360

6 Brittle Damage Mechanics of Rock Mass

Concretely speaking, for an elastic material with brittle damage behavior (mainly being group micro-cracks), the basic state variables are chosen as the strain tensor {E} and the absolute temperature T as well as using the second order tensor {D} or {1/J } representing the current damage state or the current continuity state, which are named as the damage tensor or the continuity tensor, respectively. Obviously, any inelastic macroscopic deformation in material should depend on changes in {D} or {1/J }. For a brittle damaged elastic material with thermodynamic state variables ({E}, T, {D}), its free energy density function is

(6-1)

W = W({E},T, {D})

The corresponding basic state equations have been presented in Chapter 3 as follows. (1) The equation of the stress-strain relationship

aw

(6-2)

{a} = a{E} (2) The definition equation of entropy

S =_ aw

(6-3)

aT

(3) The equation of the damage expansion force (i.e. damage strain energy release rate)

aw

(6-4)

{Y} = a{D}

As defined in Chapter 3, in the above equation {a} is the Cauchy stress tensor, which is the dual variable of strain tensor {E}, and entropy S is the dual variable of the absolute temperature T. {Y} was defined as the damage strain energy release rate or the damage expansion force tensor, which is the dual variable of the damage tensor {D}. 6.2.1.2 Rate Equations Under the condition of fixed { D} , materials appear thermo-elastic, the variable rate of its free energy density is

.

We =

aw.

aw.

a{E} {E} + aT T

T.

.

= {a} {E} - ST

(6-5)

The variable rate of its free energy density due to changes in {D} is

.

We =

aw.

aw·

a{ E} {E} + aT T

T

.

.

= {a} {E} - ST

(6-6)

6.2 General Theory of Brittle Damage Mechanics

361

Thus, the total changes of the free energy density are summations of these two parts

(6-7) The corresponding change rates of three kinds of general thermodynamic forces can be represented as follows. (1)

. d{ a }. d{ a}. {a} = d{c} {c} + dT T

d{ a}

.

+ d{D} {D}

(6-8)

and

d{a} d (dW) d (dW) dS dT = dT d{c} = d{c} dT = - d{c}

(6-9)

d{ a} d ( dW ) d (dW) d{Y} d{D} = d{D} d{c} = d{c} d{D} = - d{c}

(6-10)

then

.

d{ a}.

dS.

d{Y}

.

{a} = d{c} {c} - d{c} T - d{c} {D}

(6-11)

in which the effective elastic matrix is

[D*] = d{a} d{c}

(6-12)

and the entropy-coupled dual modulus is defined as

{Z* } = _ dS d{c}

(6-13)

and the damage coupled modulus is defined by

[8 *] = d{a} = _ d{Y} d{D} d{c} Since {a} thus we have

= d{a} {i} + d{aLt + d{a} {D} d{c}

dT

d{D}

{a} = [D*]{ i } + {Z* }T + [8 *]{ D}

(6-14)

(6-15)

(6-16)

Regarding Eq.(6-11) and Eq.(6-15), (6-16) gives

{Z* } = _ dS = d{a} d{c} dT and

(6-17)

362

6 Brittle Damage Mechanics of Rock Mass

[8 *] = _ d{Y} = d{a} d{c} d{D}

(6-18)

(2) Rate of entropy S:

. dS dS . S = d{c} {i} + dT T or

. d{ a}. S = - dT {c}

thus we have

dS .

+ dT T -

dS

.

(6-19)

+ d{D} {D}

d (dW) . d{D} dT {D}

s = - d{a} {i} + dS T dT dT

¢=

dW {Y} = - d{D}

{Y} {il} T

(6-20)

(3) rate of damage expansion force {Y}:

{Y} = d{Y} {'} d{c} c

d{Y} T

+ dT

d{Y} {il}

+ d{D}

(6-21)

Since

(6-22)

then

. d{ a}. dS· {Y} = d{D} {c} - d{D} T

d{Y}

.

+ d{D} {D}

{Y} = [C*]{i } - {Z' * }T + [5*]{il}

(6-23)

(6-24)

In which the other damage coupled modulus is defined as

[::;'*] = d{Y} ~ d{D}

(6-25)

and the other entropy-coupled modulus is defined as

{Z'*} =- ~ d{D}

(6-26)

Regarding (6-27) Eq.(6-21)rvEq.(6-24) , gives the relation

[C*] = d{a} = d{Y}

d{D}

d{ c}

(6-28)

6.2 General Theory of Brittle Damage Mechanics

363

and

{Z' * } = _ ~ = d{Y} d{D} dT

(6-29)

In the case of constant damage {D}= O and isotherm-condition 1'=0, from Eq.(6-11), we have

{&} = [D*]{i }

(6-30)

In the case of constant strain {i} =O and isotherm-condition 1'=0, from Eq.(6-25), we have

{Y} = [5*]{D}

(6-31 )

6.2.1.3 Complementary Free Energy Density Definition of complementary free energy density II =1I( {a} , T, {D}) makes (6-32) Applying Legendre's transformation to it gives the basic constitutive equation as

and

dii {c} = d{a}

(6-33)

S = dii dT

(6-34)

dii {Y} = - d{D}

(6-35)

The corresponding three kinds of rate equations are expressed as follows. (1) Rate of complementary free energy density II:

dii . 1I = d{a}{&}

dii .

dii

.

+ dT T - d{D}{D} = {c}T {&} + S1' _ {y}T {D}

(6-36)

II can be divided into two parts as

(6-37) (6-38)

364

6 Brittle Damage Mechanics of Rock Mass

Substituting them into Eqs.(6-5), (6-6) and (6-7) gives T

...

T

.

.

.

{a} {i } + {c} {a-} = W + 11 = W e + W n + 11e + 11n ..

T

..

T

= {a} {i } - ST + W n + {c} {a-} + ST + lIn

We + iIe = {a} T { i } + {c }T { a-}

(6-39)

and (6-40)

From strain we have

{i } = d{c} {a-} d{a} or

+ d{c }t +

d{c} {D} d{D}

(6-41 )

+ {Z// * }t + [17*]{D}

(6-42)

dT

{t'} = [D*r l{a-}

in which, the effective elastic flexibility (compliance) tensor is defined as

[C*] = [D*r l = d{c} d{a}

(6-43)

which is the inverse of t he effective elastic t ensor [D*] and has

[C*][D*] = [D*r l [D*] = [1]

(6-44)

where [1] is t he Kronecker unit t ensor. Regarding Eq.(6-41) and Eq.(6-42) , t he ot her dual modulus is defined as

{ Z* } = d{ cij} tJ dT

(6-45)

[17*] = d{c} d{D}

(6-46)

(2) Rat e of entropy S:

. dS S = d{a} {a-} Eq. (3-39)-Eq. (3-16)

dS .

dS

.

+ dT T + d{D} {D}

(6-47)

6.2 General Theory of Brittle Damage Mechanics

365

or from Eq.(6-47) and Eq.(6-29) with Eq.(6-20) we have

s = a{aTc} {(J. } + aT as t

+

a{y} {ill aT

(6-48)

(3) Rate of damage expansion force {Y}: { Y} = a{y} { . }

a{(J} (J

+

a{y} t aT

a{y} {ill

(6-49a)

+ a{D}

aw

Substituting Eqs.(6-4), (6-29), Eq.(6-25) and regarding to {c}

a{ (J} ,

{ZI*} =_ ~ a{D}

.

a

{Y} = a{(J} or

(aw) . as. a{ a{D} {(J} - a{D} T +[.::: ]{D} = a{D} {(J} - {Z ~*.

{Y} = [17*]{o-}

c} .

~*

'*'

.

}T+[.::: ]{D}

+ {Z"I * }t + [s *]{ il}

(6-49b)

Comparising Eqs.(6-26), (6-46) to Eq.(6-49b), in which the defined dual modulus satisfies

and

{Z"I * } = _ ~ = a{y} = _ {ZI*} a{D} aT

(6-50)

[17*] = a{y} = a{c} a{(J} a{D}

(6-51 )

In the case of constant damage {il} = 0 and isotherm-condition from Eq.(6-42), we have

{i} = [C*]{o-} = [D*r l{o-} In the case of constant stress {o-} = 0 and isotherm-condition Eq. (6-49 ), we also have {Y} = [s *]{ il}

t

= 0, (6-52)

t

= 0, from (6-53)

6.2.1.4 Dissipation Potential

According to dissipation inequality Eq.(5-75), we have

s _.!..- + div{q} _ {q}T {g} ;? 0 pT

pT

pT

and hereafter considering equilibrium Eq.(574) gives

(6-54)

366

6 Brittle Damage Mechanics of Rock Mass

pE = {(J}T {i } + r

- div{q}

(6-55)

Introducing Helmholt 's free energy per unit mass defined in Eq.(5-77) (6-56) Eq.(6-54) gives (6-57) Substituting Eqs.(6-5), (6-6) and (6-9) into the above, the dissipation inequality of elastic-brittle damaged materials can be expressed as (6-58) Simplifying gives (6-59) where the first term on the left side presents the dissipated energy due to changes (damage) of micro-structures, the second term represents the heat dissipations. If not considering the heat dissipation, i.e. the reason for dissipation of damaged mat erials only being the generation and expansion of micro-defects, then according to the theory of irreversible thermodynamics, the evolutional equations can be solved. Assuming the dissipation potential p * defined by Eq.(3-55) or Eq.(5-90) in Chapter 3 or Chapter 5 is a function of damage expansion force {Y} with the form of p* = P* ( {Y}), then the damage rat e equation is

.

ap*

{il} = - Aa{y}

(6-60)

where A is the integral demarcation factor and A = A( {Y}) > o. As a result of that, {Y} is a function of state variables as {Y} = {Y ({ E} ,T, {il})}, thus dil { ill } is a covert function of ({ E}, T , {il}). The meaning of Eq.(6-60) is that in the space of the damage expansion force {Y} , the corresponding damage (or continuity) evolution rate vector dil { ill } is always perpendicular to the surface of the equal dissipation potential p* . Substituting Eq.(6-60) into the non-linear parts of Eqs.(6-11), (6-19), (623) and Eqs.(6-25)rv(6-29) , and according to dual relationships Eqs.(3-53) and (3-59) in Chapter 3 and Eqs.(5-89) and (5-95) in Chapter 5 corresponding to dual dissipation potentials P and p* respectively, the inelastic part of the stress rate can be defined as

6.2 General Theory of Brittle Damage Mechanics dP*

{aD} = Ad{E}

367

(6-61)

the inelastic part of the entropy rate is SD

= - AdP*

dT and the inelastic part of the damage strain energy release rate D

dP*

{Y } = Ad{ DD}

(6-62)

(6-63)

In Eqs.(6-6)rv(6-63), p* = p * ({Y( {E}, T , {D})}) = p * ( {Y( {E}, T , {D})}) implying that p * is the inelastic dissipation potential represented by state variables. In the 10 ranks dimensional space, which consists of ({ E}, T , {D})) , p * = constant defining a super-surface, the gradient of which det ermines the direction of {a*} , S* and {Y* }, which is respected in the second part of Eq.(67). In a special case, {a * } is perpendicular to the equal potential surface in the strain subspace, whereas, {Y*} is perpendicular to the equal potential surface in the damage (or continuity) subspace.

6.2.2 General Constitutive Relationship of Brittle Damage Materials The content of this section is almost the same as that of 5.6 in Chapter 5. The stress-strain relationship presents so called elastic-brittle constitutive equations for characters of elastic deformation in brittle materials. It should satisfy all the basic behavior of elastic deformations. In the case of damage to the elastic-brittle constitutive relationship, , the characteristic of deformation also should satisfy the general basic elastic constitutive relationship between stress and strain, and the behavior of micro or macro failure characters appear only as the brittle fracture. The failure process of macro-brittle damage usually takes a very short time, and produces no residue or permanent deformations. Micro-brittle failure is a local damage process, where the stress level at a micro-location in the material reaches a failure criterion, which makes microstructures at the local failure point crack and fracture , and appears as microdeformations in the form of micro-cracks and micro-defects generating and developing, whereas in the structure (or in materials) no resid ue or permanent deformation is produced. Therefore, the behavior of macro-deformations still appears elastic. The free energy within an elemental anisotropic damaged material taken in the case of an isothermal, infinitely small deformation and guise static loading process can be represented as (6-64)

368

6 Brittle Damage Mechanics of Rock Mass

where [D*]=[D*({Sl})] is the effective elastic tensor written in the form of a 6 x 6 order matrix related to the damage (or continuous) state in the material and satisfies the symmetry conditions.

D* ijmn = D* jimn = D* ijnm = D* mnij

(6-65)

dW* Substituting Eq. (6-64) into the general basic stress-strain relation {CJ} = d{ E}

gives the effective damaged linear elastic stress-strain relationship of brittle materials with an effective form of the generalized Hook's law as follows,

{CJ} = [D*]{E}

(6-66)

Substituting the expression of the corresponding complementary free energy density (alias stress energy density) (6-67) into the definitional expression of the complementary free energy function Eq.(3-23), we have (6-68) Obviously, this equation also can be obtained from the inverse of Eq.(6-66)

6.2.2.1 Constitutive Relationship in the Form of Whole Quantities Assume that the stress-strain relationship Eq.(6-66) of anisotropic brittledamage materials is equivalent to the following virtual undamaged state under effective stress and strain actions

{CJ*} = [D]{E*}

(6-69)

where {CJ*} and {E*} are introduced as the effective stress tensor and the effective strain tensor based on the equivalent condition defined by Eq.(669). Considering the unsymmetrical characters of the effective stress tensor in the case of anisotropic damage presented in Chapter 5, the effective stress vector {CJ*} has been rewritten in the form of (9 xl) rank as s:r { CJ *} = {* CJ 11,CJ * 22,CJ * 33 ,CJ * 23,CJ * 32, CJ * 31,CJ * 13,CJ * 12 , CJ *}T 21 . { E*}.IS th eeuective strain tensor written in the form of (9 xl) rank as {E*} = {E* 11, E* 22, E* 33, . th e vlrgm . . unE* 23 ,E* 32 ,E* 31 ,E* 13 ,E* 12 ,E*}T 21 correspon d·mg t 0 {CJ *} ; h erem damaged elastic tensor [D] should be rewritten in the form of a 9x9 rank matrix. Assume the general strain vector {E} in the form of (6 xl) rank can be transformed into the effective strain vector {E* } with the form of (9 xl) order through a 4th rank transform tensor [tli,,] defined by a 9x6 order transform matrix as

6.2 General Theory of Brittle Damage Mechanics

{E*}(9Xl)

= [IliE](9x6 ) {Elc6xl )

369

(6-70)

where [Ili E] presents the influences of the damage state on the strain state, and can be called the damage effective tensor (function) on strain. In chapter V, the alternative damage effective tensor of damage [Ili] (herein denoted by [Ilia]) defined by Eq.(5-22) defined in the form of a 9x6 rank matrix was applied to stress transformation from the Cauchy stress vector {(J} in the form of a (6x1) rank into the effective stress vector {(J* } with a (9x1) rank as

(6-71) It is evident that both damage effective t ensors [Ili E] and [Ilia ] relate to the damage state tensor {il} (or continuous tensor {7/J}) , i.e., [IliE] = [IliE({il}) ] and [Ilia ] = [Ilia ({ il} )]. Regarding the unsymmetrical property of the effective stress (strain) tensor in the case of anisotropic damage, the function of elements in the matrix (or t ensor) [Ilia ] and [IliE] has the following shape. From Eq.(5-22) in Chapter 5, the detail of elements in the matrix (tensor) [Ili a] is given as 1

0

0

0

0

0

0

1 1 - il2

0

0

0

0

0

0

1 1 - il3

0

0

0

0

0

0

0

1 - ill

[Ilia ] =

0

0

0

0

0

0

1 1 - il3 1 1 - il2

1

0

0

0

0

0

0

0

0

1 - ill 1 1 - il3

0

0

0

0

0

0

0

0

0

0

Substituting {(J* }(9Xl) = into Eq.(6-69) gives

[llia ]( 9X6){(J}(6Xl)

[llia ](9 X6){(J} (6Xl)

=

and

(6-72)

0 0

1 1 - il2 1 1 - ill

{ ¢* }(9Xl)

=

[IliE](9X6){E}(6Xl)

[D ](9X 9) [IliE](9 X6){ E}(6 X1)

in which the virgin undamaged elastic tensor [D] should be rewritten in the form of a 9x9 order matrix. Compared to results of Eq.(5-24) given in subsection 5.3.1, the elements in transpose matrix [IliE] T should be as follows

370

6 Brittle Damage Mechanics of Rock Mass

[!licY = 1- 01 0 0 1 - O2 0 0 1 - 03 0 0

0

0

0

0

0

0

0

0

0

= [!li- I ]

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

1 - 0 3 1 - O2 0 0 0 0 2 2 1 - 01 1 - 03 ---0 0 0 0 2 2 1 - O2 1 - 0 1 ---0 0 0 0 2 2 (6-73)

----

Substituting Eq.(6-70) and Eq.(6-71) into Eq.(6-69), we have (6-74) then obtain {a}(6XI)

= [!lic:] T (6X9 ) [D ](9X9 ) [!lic:] (9x6 ) {Eh6XI )

(6-75)

Similarly to Eq.(5-108), compared with Eq.(6-66) , we obtain

[D* ](6X6)

= [!lic:] T(6X9) [D ](9X9) [!lic:] (9x6) or [D ;jkl ] = [!linmji ]T [Dmnst ][!listkl ]

(6-76) where effective elastic matrix (tensor) [D* ] is still a 6 x 6 rank symmetric matrix. If the free energy W* ([D* ], {E}) of damage materials in the real state is equivalent to the free energy W ([D ], {E*}) of a fictitious undamaged material in the effective state, then (6-77) Similarly to Eq.(5-111) , the damage-elastic constitutive matrix defined in global coordinate system (XYZ) can be expressed by substituting the coordinate transformation matrix [TO" ] in Eq.(5-32) and {E*} = [!lie ] {E} in Eq.(6-70) into Eq.(6-77) as (6-78) It can be seen from the above that if the matrix [D] is symmetric, then the matrix [D *] satisfies the symmetrical condition of Eq.(6-65). Comparison of Eq.(6-78) and Eq.(6-74) may give a relationship between these two damage effective tensors as follows:

6.2 General Theory of Brittle Damage Mechanics

[tlia ][tlic:] T = [1](9x9 }, [tlic:][tlia]T = [1](9x9 }' [tlia ]T [tlic:] = [I ](6 x6}' [tlic:] T [tlia] = [I ](6x6 }

371

(6-79)

Substituting the above relation into Eq.(6-73), the elastic constitutive relation of elastic brittle-damage materials in the form of the whole quantities can be represented as (6-S0) in which detailed elements are the same as presented in Eqs.(5-49)rv(5-52) of subsection 5.6.l. Obviously, since the coefficients in the stress-strain relationship are dependent on the damage state, the relation is in a linear form varying with changes in damage, (actually it is a non-linear relation during damage evolution). This is not the same as the classical generalized Hook's law, which describes only the virgin unchanged material properties in a linear elastic form. 6.2.2.2 Damage Development Force

The definition of a damage development force (alias the thermodynamic force associated with the damage development) defined herein is the same as the definition of the elastic damage strain energy release rate presented in Chapters 3 and 5, , either on a physical basis or by mathematical formulation. The most essential and basically fundamental formulations of the damage development force {Y} will be rearranged below. Substituting Eq.(6-43) into Eq.(6-4) , we have 1 Td [D* ] 1 Td [D*r l {Y} = 2{c} d{D} {c} or {Y} = 2{o-} d{D} {o-}

(6-S1 )

Considering Eq.(6-7S) again, the expression of the damage development force vector can be described in the strain space as (6-S2) Consequently, substituting the inverse equation of Eq.(6-7S) into the above gives the expression of the damage development force vector described in the strain space as

(6-S3a) The detailed properties of the damage development force and formulation of its elements under different conditions have been discussed and analyzed in

372

6 Brittle Damage Mechanics of Rock Mass

subsection 5.5.4, and will not be described any more here. When Eqs.(6-82) and (6-83a) need to be expressed in the global coordinate system (XYZ) , we have to employ the coordinate transformation matrix [TO" ] or [TE]

{Y}

=

~{o-}T[TO"]T (dJf~~ [Dr

l [tlio" ] + [tliO" ]T [Dr l

~~~~) [TO"]{o-} (6-83b)

6.3 Application of Thermodynamic Potential to Brittle Damage Materials 6.3.1 Dissipation Potential and Effective Concepts due to Damage Beyond a purely thermodynamic choice with a potential depending on the elastic strains (Helmholtz) or the stresses (Gibbs) , which choices are generally equivalent, the formal framework of Continuum Damage Mechanics leads to significant differences between the two approaches. This is the case, in particular, when the associated thermodynamic forces are included in the damage criterion. With an approach based on the Gibbs potential, the damage criterion actually involves the stresses as the driving variables of the damage process. On the contrary, the Helmholtz potential naturally introduces a strain criterion. Both cases have their advantages and drawbacks. With a potential based on stresses-involving the compliance tensor, it is relatively easy to construct a theory catering for the compliance variations obtained by simple analyses for varied micro-crack configurations (micromechanical theories developed , for instance, by Kachanov [6-20"-'21] or Horii and Nemat-Nasser [6-22]). However, in this case the damage has to be introduced as an additive term, something like [C*]=[C]:(l - D)] (see, Chapters 3 and 4). We then lose the concept of damage defined through a reduction in a section of resistant material (effective). This approach also has the advantage of facilitating identification, since the mechanical tests are generally conducted under controlled stress, in particular in transverse directions. According to [6-1], the thermodynamic potential, i.e., the Helmholtz or Gibbs free energy that supplies the state equations, including the elasticity equation. we set similarly as in Eqs.(5-89) and (5-95) (T is the t emperature, S is the entropy) <1>

= E - T S or <1>* =

<1> -

{o-} T { E }

(6-84a) (6-84b)

where Dex represents the family of damage variables that are scalars or tensors.

6.3 Application of Thermodynamic Potential to Brittle Damage Materials

373

Conventional processing yields: or {c }

dP*

(6-85)

= d{a}

(6-86) where Y" represents the thermodynamic force associated with the variable

fl" .

The dissipation potential corresponds here to a unique choice, since the only dissipation is due to the damage: Actually we place ourselves in an "associated" framework in which the dissipation potential is identified as the surface limiting the undamaged domain f ~o defined in the space of variables Y" . We set (6-87) (6-88) where the damage multiplier is determined by the consistency condition f = 0,

~~ = o.

Adding the loading/ unloading condition yields

((k) y,,) dg

. fli

T

= H(f)

dT d anI'

if:;

dY

,

(6-89)

Clearly, the heat dissipation associated with the damage growth, which reduces here to I:i dli!.?i is always positive. 6.3.2 Effective Operations for Anisotropic Damage

Initially isotropic materials with no particular symmetry form the most complex case since damage is generally not isotropic. The induced anistropy is defined on the principal axes related to the loading causing the damage (and possibly its history). As mentioned before, the use of a second- or a fourthrank damage tensor is then necessary. Let us first examine the two classical effective stress concepts. The method of strain equivalence in [6-23] states that the effective stress tensor {a* } expresses the net stress that would have to be applied to undamaged material to cause the same strain tensor which is t he one observed in the damaged material submitted to the Cauchy stress {a}. Using the fourthorder "damage effect" operator [(tP-fl,, )] defined by Eq.(5-125) for symmetrized model I, we write

374

6 Brittle Da mage Mechanics of Rock Mass

{a* } = [~(Da )]{a}

(6-90a)

The inverse of [~] was defined as [~]- 1 in Eq.(5-114) and written again

[Ili] A

-1

=

1 - D1

o o o

0 0 1 - D2 0 0 1 - D3 0 0 2

o

0

0

o

o

o

o

0 0 0 (1-D2)(1- D 3) (1-D2)+(1- D 3)

0

2

o o o

(1- D 3)(1-D,) (1- D 3)+ (1-D,)

o

o

o

o o o

2

o (1-D,)(1-D2) (1-D 1 )+(1-D2)

(6-90b) In this case, the damaged material elasticity equation (completely active damage) could be written as: (6-91) Appling symmetrization scheme I to the constitutive matrix, it gives

[b*(D a )] =

~ ( [~]-l[D] + [D]T[~]-T),

[6*(Da )] =

~ ( [~][C] + [C]T[~()

(6-92) The advantage of this solution is that it does pre-define the form of the damaged elasticity operator, i. e., the relation between D and [~] . This method naturally leads to the use of a fourth-rank damage tensor [D] expressed in Eqs.(5-124) or (5-125) and to setting

[~( [D])r1 =

[1] -[D] or

[~ijkl] =

[(Oik Ojl - D k ojl )-l

+ (OjlOik -

Dl Oik )-l ] / 2 (6-93)

where [I ] is the fourth-rank unit tensor (2lij kl = OijOjl + OilOjk). The method of energy equivalence [6-23] states that the effective stress {a * } and the effective strain {c* } are such that the elastic energy of the undamaged material subjected to {a * } and {c* } is the same as for the d amaged material subjected to {a} and {c}. It, of course, can be written as W e= {a* } T {c* } / 2= {a} T {c } / 2 and using symmetrization scheme II to set (6-94)

It is clear that there is a certain indetermination in this case. It revolved around the choice of the form of operator [tlf] . Reference [6-24] used a damage tensor [57] of only the second order and expressed [tlf( [D])]-l as following

6.3 Application of Thermodynamic Potential to Brittle Damage Materials

375

the "diagonal form" in the principal coordinate system of {57}, using the symmetrized Voigt notations (i.e. Eq.(5-145) of model II) as:

[tilrl

=

1- n 1

o o o o o

o

o

I - fh

0 1-fh 0 0 0

o o o o

o

v(1 -

o

0 0 0 0 [.h)(l - fh) 0 0 V(l - fh)(l 0 0

o n1 )

0 0 0 0

yr7"":(1:----=n-,-1)-,-:(1---n-=2--,-) (6-95)

It should be noted right away that this operator is effectively intrinsic. It can be written alternatively like

[tlI({57}) r l = ( [~ - [57])1/2.§2([~ - [57])~

+ ( [~ - [57])~ EEl ( [~

- [57])~ (6-96)

in which the t ensor operations are defined as

(6-97) [C ] [C ]

= [A ] EEl [B ] ---+ Cijkl = AilBjk

(6-98)

= [A] EB [B] ---+ C"ijkl = A ij Bkl

(6-99)

With this damage effect operator, the elasticity tensors are expressed as

The advantage of this method is the use of a tensor of only the second order. The drawback is related to the lack of degree of freedom. For instance, the above expression of [C] cannot account for the compliances of elastic media damaged by parallel cracks (solutions by Hoenig [6-25] and Kachanov [6-20"-'21]). A potential based on strains has three advantages: (1) It is consistent with other analytical solutions in special cases with completely parallel cracks [6-26], at least when we use strain equivalence and a fourth-rank damage tensor; (2) It is the direct expression of an equation between the associated thermodynamic force Y and the strain, again in the same equivalence framework. The potential is linear in 57, which means the Y is independent of 57. This allows significant simplifications in structural computations for which the "local" step is always conducted with a controlled strain ;

376

6 Brittle Damage Mechanics of Rock Mass

(3) The damage criterion based on strains allows a natural distinction between uniaxial tension and uniaxial compression. In the latter case the positive transverse strains can initiate and open cracks parallel to the compression axis.

6.3.3 Progressive Unilateral Character of Damage The damage is obviously irreversible. But its effect on the mechanical behavior can be active or inactive. In practice, depending on the applied loading, the cracks can be open or closed, leading to unilateral behavior, typically with bilinear elasticity behavior. Table 6-1 The elastic compliance ma trix for biaxial principal stra ins with 4 anisotropic and unilateral damage theories

A + 2/1> + 2(C1 + C2)nt

Vectors [6-28]

A + C1 (m + n~)

2nd Order Tensors [6-29] 4th Order Tensors [6-30] New Formulation 2nd Order Tensors [6-31]

hi} + A(l - 0) ht12 + A(1 - 0) (A + 2/1»(1 - £I) A(1 - £I) (A + 2/1»(1 - £1 1)2 A(1 - £1 1)(1 - £12) El

Vectors [6-28]

<

°

A + 2/1>

2nd Order Tensors [6-29] 4th Order Tensors [6-30]

A + C1n~ h 1} + A(1 - 0)

h·; } + A(1 - 0) A + 2/1> A

New Formulation 2nd Order Tensors [6-31] * in w hich A a nd

fJ,

A + 2/1> A(1 - £1 1 )(1 - £12 )

A + C 1 (nt + n~) A + 2/1> + 2(C1 + C2)n~ ht12 + A(1 - 0) ht 22 + A(1 - 0) A(1 - £I) (A + 2/1»(1 - £I) A(1 - £1 1)(1 - £12) (A + 2/1»(1 - £1 2)2 E2

>

°

A + C1n~ A + 2/1> + 2(C1 + C2)n~

ht12 ht12

+ A(1 + A(1 -

0) 0)

A (A

+ 2/1»(1 - £I) A(l - £1 1 )(1 - £12 ) A(l - £1 1 )(1 - n2?

a re the Lame 's coeffici e nts.

This point is crucial for a damage theory with induced anisotropy, as has already been shown by Chaboche [6-27]. In effect, all the theories with anisotropic damage suggested until then had relatively serious theoretical shortcomings (see Table6-1 summarizing these shortcomings for particular examples of references [6-28"-'31]). Either the symmetry of the elasticity operator was not respected or discontinuities of the stress-strain response under non-proportional multiaxialloadings were evidenced. "Progressive" unilateral contact functions make it possible to avoid mathematical discontinuity but in no way settle the underlying physical problem. Solution procedures proposed by Ladeveze [6-32] only partly solve the problem , for initially isotropic materials.

6.3 Application of Thermodynamic Potential to Brittle Damage Materials

377

A unilateral condition respecting continuity was suggested by the first author [6-27]. It consists of writing an "opening/closing" type criterion in the current damage principal axes (axes of the equivalent micro-cracks) based on normal strain or normal stress (normal to these equivalent cracks, and therefore to the corresponding principal directions). The key feature of the criterion is that only the diagonal stiffness term corresponding to the normal strain, which changes sign, is modified unilaterally. For instance, with a strain expression we propose writing the effective elastic stiffness tensor [D ]eff as a function of the stiffness tensor with totally active damage [D *] given by equations such as Eq.(6-92) or Eq.(6-100) in the following form: 3

[D *]eJJ

=

[D *]

+ 7] L

H( -ci )Pi ( [D ] - [D *]) Pi

(6-101)

i=l

where [D ] is the initial elasticity tensor, Pi = n i EEl ni EEl ni EEl n i is the fourth order projection operator on the principal direction ni, H is the Heaviside function and [ci]=[ni]T[c][ni]=tr(Pi : c) is the normal strain associated with direction n i . Coefficient 7] , with values between 0 and 1, plays a phenomenological role to weight the unilateral character (between the initial modulus and a compression modulus of inactive damage, different from the initial modulus). It should be noted that the double projection in the above equation indicates the diagonal term D iiii - Uiiii · A similar formulation is available to express the effective compliance. But it should be noted that it leads to different results, [C:ff ] is not equal to the inverse of [D;ff] . The closing criterion then nat urally concerns the normal stress instead of the normal strain, with corresponding difficulties for evidencing active compression damage with cracks parallel to the compression axis (situation illustrated in Fig.6-1) , as already mentioned above. Obviously, for a damageable elastic medium without residual strain, the Helmholtz free energy as a quadratic function of strain can be expressed by sub-domains as (6-102) where [D;ff] now depends on the current state of the damage and the sign of the normal strains {cd. Recently, it was re-demonstrated that this method was necessary to ensure continuity of the stress-strain response. The damage criterion and the damage growth equations can be expressed as a function of the thermodynamic force Y. This force is dependent on {c} as a continuous function (with a discontinuous derivative when the unilateral condition occurs). The Ref. [6-30] suggested a simple form of the damage criterion, for a fourth-rank damage tensor with only four coefficients depending on the material. Other forms can be used, depending on the choice of the

378

6 Brittle Damage Mechanics of Rock Mass

type of tensor variable, the form of the effective stress chosen, the unilateral criterion selected, etc.

x,

x, X,

(a)

(b)

Fig. 6-1 Two specific cracking arrangements. (a) Planar transverse isotropy, produced by (71 > 0, (72 = (73 = 0; (b) Cylindrical transverse isotropy, produced by (71

> 0,

(72

=

(73

=

°

6.3.4 Case Study of Damage in Anisotropic Materials We now discuss the case of composites, anisotropic materials in which the physical nature of the constituents (oriented and with very different characteristics) means that the micro-racks forming the damage are generally parallel or perpendicular to the fibers (or yams). The development of damage thus can modify the intensity of the anisotropy (differences along the principal axes of the composite) but does not alter the initial symmetries of the material. In this case, it is reasonable to consider a simpler theory: the damage variables are scalars associated with each of the three principal directions of anisotropy of the material (assimilated here in the directions of the constituents, assumed mutually parallel or perpendicular).

6.3.4.1 Expression of Thermodynamic Potential The potential chosen is quadratic as a function of the elastic strain. Actually, an initial strain tensor {co} is introduced to describe irreversible strain (observable in elSie). The Helmholtz free energy is chosen [6-1] with the form tP

= ({c} - {cO})T [D*eff] ({c}-{co})/2+({c} - {co}f [D*R] {co} (6-103)

6.3 Application of Thermodynamic Potential to Brittle Damage Materials

379

where [D;ff ] is defined by the same method as for isotropic material (Eq.(6101) above). This avoids any stress or strain discontinuities for any loading and any damage closing or deactivation time. In this case, directions ni are obviously the principal directions of the material, considered invariable, and the normal strains {Ei} are to be considered in the difference {E} = {E} - {EO}. Moreover, [DR] is taken as the effective stiffness with all damage deactivated; i.e., given by Eq.(6-101) with El < 0, E2 < 0, E3 < o. Thus the stress is obtained by differentiation (6-104) As is shown schematically in Fig.6-2, the behavior obtained by this theory shows the degradation of the elastic modulus in tension, with a unilateral closing effect for a particular strain state {Eo }, to which corresponds the stress state (6-105)


Fig. 6-2 Schematic behavior of SiC/ SiC (a) and C/SiC (b)

The modulus on the compression side ( El < Eod is not necessarily equal to the initial modulus [D]. This depends on the other strains, the transverse and the Poisson's ratios of the damaged material. It should also be noted that the components of tensor {EO } are not necessarily the same. With this choice of potential, the existence of a residual strain ER is described without introducing any additional dissipation. The residual strain is obtained easily as:

([1 ]-[ D* eff ]-1[D*R]){ EO}

(6-106)

380

6 Brittle Damage Mechanics of Rock Mass

It is of course directly related to the damage involved in [D;ff ]. In the framework the damage variables can be assumed by scalar, [h , D2 , D3 (i. e. components of the principal anisotropic damage vector {D}) , corresponding to the crack densities in the three principal axes of the material. The energy equivalence can be used with Eq.(6-100) to define the elasticity operator [D*] applying the damage effect tensor [W-]- lupplied by Eq.(6-93) as

[D* ] = ([1] - [f?]f [D] ([1] - [D])

(6-107)

in which the damage tensor [D], which is a fourth-rank t ensor, does not have a "diagonal" form. In the principal system of the material, it is

[D] =

(6-108) 6.3.4.2 Damage Criterion

The three thermodynamic forces can be associated with the principal anisotropic damage vector {D} (or the three scalar variables {Do,}. These forces are easily expressed from Eqs.(6-103) and (6-101) as 1 _ T Cl [D* ] _ 3 _ -2 ( Cl [D* ] ) _ T Cl [DR] _ {Ya } = - "2{c} Cl{Da } {c}+ 77 ~H( -Ci )Ci Pi :: Cl{Da} - {c} Cl{Da} {co}

(6-109) in which the sign :: denotes the doubly contracted product of two fourth-rank tensors (the result has a scalar value). In addition, Eq.(6-107) allows us to write

:l~~~ where

[d~]

=

= - ([d: ][D]([I ] - [D]) + ([1] -

~~

[f?]f [D][d~])

(6-110)

is easily obtained by differentiating Eq.(6-108).

The damage criterion is expressed in the standard way through a surface defined in the space of variables Ya . Two types of choice are possible (1) Multiple criteria, a priori decoupled:

6.3 Application of Thermodynamic Potential to Brittle Damage Materials

381

in which the threshold functions r" are decoupled. They are scalar functions of D" determined by the variation of D" as a function of Y" and therefore implicitly as a function of the strains. It should be noted that in woven composites, directions 1 and 2 are identical (warp and woof directions), meaning that rl = r2· (2) A combined criterion in which we define a norm of vector {Y} and a norm of vector {D}

11Y11= ((y1 )m + (Y2 )m + a(Y3) m)1/m II D II

f = IIYII- r(IIDII)

(D12 + D22 + D32)1/2 (6-112)

~ 0

(6-113)

As shown in Fig.6-3, in a biaxial representation, exponent m adjusts the amount of coupling and coefficient ex sets the relative magnitude of direction 3 (the much lower strength in this direction leads to taking ex » 1). y ,lr

y,lr

Fig. 6-3 Schematics of the damage criterion

It is always possible to combine these two kinds of criteria. An extension of the multiple decoupled criteria consists of taking multiple combined criteria with functions r,,(D 1, D2, D3). The introduction of undamaged domains f ~ 0 is replaced in certain theories [6-33] by the use of variables such that Y (t) = sup Y (T). It is easy to show that this theoretically amounts to the T ~t

same thing. In the two above criteria, there is no increase in the damage variables associated with negative values of Y" as can also be seen from Fig.6-3. It also induces that the damage rate can never be negative. With the uniaxial compression (along 1 for instance), damage is however possible, since we have

382

6 Brittle Damage Mechanics of Rock Mass

< 0 but C2 > 0 and C3 > 0, which generates Y2 > 0 and Y 3 > o. The corresponding damage is relative to the directions crosswise to the compression direction.

C1

6.3.4.3 Applied Examples for Composite Materials The theoretical model consisting of state Eqs.(6-103) and (6-108) and the second criterion of Eq.(6-113) above was applied to ceramic matrix composites SiC/SiC and C/SiC (with m = 2). In the case of SiC/SiC, the residual strain was considered negligible (with Co = 0). Fig.6-4 shows the results of uniaxial t ension in direction 0° for SiC/SiC and C/SiC. The t ension curves with unloading are correctly reproduced by the model, including the residual strain on C/SiC, and the closing effect is clearly visible (These curves were used for identification). The transverse strain (c2) is also correctly predicted in both cases. The element of compliance C 12 is constant for SiC/SiC and variable for C/SiC (in this case the Poisson's ratio varies with damage). 300

Experimental dita [6-33)

CT

Model dita

(a) Stree

Strain Experimental data [6-33)

(b)

Fig. 6-4 (a) Tensile-compression t est on SiC/ SiC; (b) Tensile-compression test on C / SiC

6.4 Micro-mechanics of Brittle Damage Based on Mean Field Theory

383

Fig.6-5 shows the predictions made for tension and compression tests in directions off the axes for C /SiC and their comparison with experimental data. The predictions are fairly good.

o·

15 '

30' :::====~45'

Experimental data [6-33)

45' 0'

15'

- - - - - 45'

Ii

o· Fig. 6-5 "Off-axis" tests on C/SiC

It is interesting to note that the applied model for composites is a particular case of a more general model of initially fakeisotropic damageable elastic materials such as concrete. The simplification is due to the scalar character of the state variables describing the principal damage, related to the fact that the micro-cracks are considered to be guided by the heterogeneous, oriented structure of materials.

6.4 Micro-mechanics of Brittle Damage Based on Mean Field Theory 6.4.1 Introduction and Objective The mechanical response of solids to a large extent depends on the type, density, size, shape and orientations of defects in its micro-structure. This section focuses on the influence of many atomically sharp micro-cracks on the elastic parameters of materials. This problem attracted a great deal of attention as a result of both its intrinsic importance and its complexity. In general,

384

6 Brittle Damage Mechanics of Rock Mass

the complexities are inherent in the random geometry of the microstructuremicro-defect system and the fact that the in-homogeneities introduce a length scale, rendering the problem non-local. The character of the response essentially depends on the micro-crack concentration. Micro-crack concentration can be considered dilute if the distance separating adjacent micro-cracks exceeds the decay length of fluctuations that they introduce into the stress field. In this case, the direct interaction of the micro-crack has a second order effect on the macro-response. The external stress field of a micro-crack is influenced by the neighboring micro-cracks only indirectly through the contribution to the overall state (effective elastic modulus). The overall (macro) response is, therefore, a function of the orientation weighted micro-crack density and the solid is locally macro-homogeneous rendering local constitutive theories applicable. The effect of direct micro-crack interaction on the macro-response grows with the increase in the micro-crack density, i.e., shrinking distances between neighboring micro-defects. As the micro-crack density is increased further, the micro-cracks self-organize into clusters. The disorder attributed to microcracks randomly scattered over most of the volume decreases. Eventually, the largest micro-crack cluster transects the specimen into two or more fragments, reducing the macro-stiffness of t he specimen to zero. At the incipient failure in a load controlled test , mechanical response of the specimen is dominated by t he largest cluster. Within this phase, the stress and strain fields are strongly inhomogeneous (localized) and the volume averages cease to be meaningful measures of the corresponding random micro-fields. In strain controlled tests, brittle materials exhibit softening as a result of internal stress redistribution, grain (or aggregate) interlocks and bridging, etc. From the percolation point of view [6-34]' the critical state is defined as a state at which the transition from the short- to the long-range connectivity of the defect cluster occurs. In other words, a system percolates at the point at which a cluster of interconnected slits spans the specimen, causing its fragmentation into two or more finite fragments. Since the percolation belongs to the class of the second order phase transitions, these two definitions should define the same state. However, the slit density at which the tangent stiffness vanishes (KT = 0) is not necessarily identical to the slit density at which macro-rupture occurs. One of the objectives of this section is to shed some light on this apparent contradiction. Estimates of the elastic modulus of solids weakened by a large number of slits are commonly provided using Mean Field Theory (MFT) [6-35]. Neglecting spatial correlations (direct interactions) of micro-slits, the mean field theory results are inadequate beyond some undefined micro-slit density. However, as pointed out by Ma [6-36], Cleary et al. [6-37] and many others, significant improvements in mean field theory cannot come from within. At higher densities, the influence of random micro-crack morphology (position, shape, size, etc.) on local stress field fluctuations and macro-properties grows from being important to becoming dominant. The conventional (local) continuum

6.4 Micro-mechanics of Brittle Damage Based on Mean Field Theory

385

theories are, however, inherently unable to replicate the circumstances for which the spatial disorder on the micro-scale becomes the dominant feature of the macro-response. Hence, it seems both sensible and necessary to resort to the methods of statistical physics in order to shed some light on the underlying phenomena. This need is further emphasized by a well-recognized fact that the application of micromechanical models is limited to cracks of very special geometry (rectilinear slits and planar, penny-shaped cracks), periodic arrangements of cracks, etc. The narrow objective of this study is to provide percolation theory estimates of critical slit densities for several different slit configurations emphasizing influence of the loss of isotropy. In a wider sense, these results plotted on the same diagram with the MFT estimates, provide some indication as to whether a particular model exhibits a proper trend for larger slit densities. 6.4.2 Mean Field Theory of Micro-Mechanics 6.4.2.1 Aspects of Mean Field Theory for Brittle Damaged Materials

For low to moderate concentrations of micro-defects, a material is typically assumed to be locally macro-homogeneous. Consequently, the elastic paramet ers of a solid can be estimated using the mean field theories (effective continua). The mean field theories are based on the assumption of equivalence of the strain energy of the actual solid with a disordered microstructure and that of an appropriately defined effective continuum. These methods imply the existence of a small Representative Volume Element (RVE) containing a statistically representative sample of in-homogeneities that can be mapped on a material point of the effective continuum, preserving the equivalence of internal energy density. A configuration space is compiled in the process of mapping the transport properties of the representative volume element on the material point of the effective continuum. This configuration space, attached to each material point of the effective continuum, contains data related to volume averages of micro-in-homogeneities over t he representative volume element , which defines the structure of the material locally, its recorded history and, thus, the macro-response. The most frequently used mean field theories, known as the Self Consistent Method (SCM) and Differential Models (DM), are based on the following assumptions [6-38]: (a) the external field of each micro-defect, assumed to be equal to the ext ernal (far) field, applies to the ent ire representative volume elemen, and (b) the size of the largest defect is much smaller than the linear dimension of the representative volume element. Subject to these assumptions, the problem of many interacting micro-cracks within the actual solid is reduced to a superposition of simpler problems considering isolated cracks embedded in a homogeneous, effective continuum. In the absence of a length parameter and suppressed during volume averaging of in-homogeneities, the

386

6 Brittle Damage Mechanics of Rock Mass

ensuing theory is local. The overall (average) compliance [C*(x , D)] in a material point of the effective continuum is within this approximation and obtained by superimposing contributions of all n cracks within the representative volume element

[C*(x, D)] = [c(x) ] + [C* ([C(x , Dm

(6-114)

where [C(x) ] is the compliance of the virgin matrix, while D denotes a set of parameters used to record history (irreversible changes of the microstructure) , i.e. the damage variable. Also n

[C* (x , D)] =

L C(i) [C(x , D)]

(6-115)

i=1

is the compliance attributable to the presence of all active micro-cracks within the representative volume elemen. In Eq.(6-115), C(i) is the contribution of the ith micro-crack to the specimen compliance, which may be an implicit and/ or explicit function of the overall compliance [C]. Furthermore, depending on the desired degree of accuracy and the selection of the analytical model, the compliance C(i ) of a single crack can be det ermined as a function of the compliance of the virgin matrix [C]=[ C] (self-consistent and differential models) or on the effective compliance of the pristine matrix [O]=[C] (Taylor model for very dilute slit concentrations). Det ermination of the components of the overall (macro) compliance t ensor [C(x , D) ] of a solid weakened by a dilute concentration of micro-cracks proved to be a rather popular topic in the recent past. In general, the effective compliances [C] can be derived from the expressions for crack opening displacements C( i) = C( i) (u) or from the expressions for the stress intensity factors C (i) = C(i) (K). These two approaches were shown to be different only in form [6-39]. Since the stress intensity factors K i are typically more accessible [6-40] the second approach, based on Rice [6-41] has certain, if formal , advantages. The mean field estimates used in this study are derived from Sumarac et al. [6-34]. Thus, even a cursory discussion of the mean field theory seems to be redundant. 6.4.2.2 Dilute Concentration (or Taylor's) Model of Brittle Damage

The most rudimentary brittle damage model is formulated assuming that each defect totally ignores the presence of all other defects. In this case, every defect is assumed to be embedded in the original, undamaged matrix, which is usually assumed to be isotropic. Since the literature devoted to fracture mechanics provides all the necessary formulas for the stress intensity factors (Ki and the elastic energy release rate in the case of isotropic elastic solids containing a single defect of simple geometry) the dilute concentration model of brittle damage is in most cases amenable to a closed form , analytical solution. This

6.4 Micro-mechanics of Brittle Damage Based on Mean Field Theory

387

method provides lower bounds on tensors [C*] and [C] since the weakening effect of adjoining defects on the matrix stiffness is totally ignored. Substituting [G]=[ C] into Eqs.(6-114) and (6-115) and assuming that the original matrix is isotropic and homogeneous, the components of the fourth rank tensor [C* ([CJ) ] can in most cases be determined analytically.

6.4.2.3 Self-Consistent Model of Brittle Damage As the crack concentration increases it becomes both advisable and necessary to incorporate the effect of the interaction between the neighboring cracks into the model. The simplest way in which this can be done is to assume that each micro-defect is embedded in an effective continuum, the parameters of which reflect in some average (smoothed or homogenized) sense the presence of all other micro-cracks within the representative volume element. Thus, the self-consistent estimates for elastic parameters are obtained substituting [6] by [G] in Eqs.(6-114) and (6-115).

6.4.2.4 Differential Scheme of Self-consistent Model The differential method is a clever extension and modification of the selfconsistent scheme. Defects are introduced sequentially in small increments from zero to their final concentration. The overall state is interpreted as being a result of a sequence of dilute micro-crack concentrations. The state containing (,) micro-cracks evolves from the preceding state containing (, - 1) micro-cracks through the addition of a single new micro-crack. Consequently, by its very nature the results obtained using the differential method depend on the sequence in which the micro-cracks are introduced. The simplest method for obtaining the governing equation of the differential method is to follow the above described procedure. A self-consistent estimate for the overall compliance of a representative volume element containing (, - I) micro-defects is from Eqs.(6-114) and (6-115)

[G] = [C] +

r-1

L

j (i) C(i) ([GJ)

(6-116)

i= l

The ,th defect is subsequently introduced , assuming that for, » 1 the increment in the total defect concentration is infinitesimal. As in the selfconsistent method (SCM) the actual location into which the ,th defect is introduced is considered to be irrelevant within this scheme. Under these stipulations, from Eq.(2.3) it follows that r-1

[G]

+ d[G] = [C] + L j (i) C(i) ([GJ) + c(r) ([GJ) dj i= l

(6-117)

388

6 Brittle Damage Mechanics of Rock Mass

It is further assumed that it is possible to write [c(r) ]=[O][H ] (where [H ] is a fourth rank tensor) as an explicit function of the overall modulus (for the effective continuum containing (r - 1) micro-defects). Subtracting Eq.(6-116) from Eq.(6-117) and pre-multiplying both sides of that expression by [0]-\ the original differential equation can be recast into a much simpler form as

d(ln [O]) = [H ]df

(6-118)

The overall compliance [0] can now be obtained as a solution of the differential Eq.(6-118), and the initial condition [O]=[C] when f = O. Analytical quadratures of the differential Eq.(6-118) are possible only if the components of the tensor [H ] are defined as simple, analytical functions of the overall compliances. In general, Eq.(6-118) represents a system of coupled ordinary differential equations, which may, or may not, admit a closed form analytical solution.

6.4.3 Strain Energy due to Presence of a Single Slit To determine the elastic parameters of a brittle solid containing an ensemble of micro-cracks, it is first necessary to derive the expression for the strain energy attributable to the presence of a single slit. The expressions for the strain energy release rate for an open rectilinear slit embedded in an arbitrarily loaded , infinitely extended anisotropic, homogeneous, elastic, two dimensional continuum was derived by Sih et al. [6-42] as

yl = _ KId 1m KI(A~ 2

K

2

YK

=

22

" K2 , TCll Im[K2(AI

+ A~) + K2 X1 X2 ,

(6-119) ,

+ A2 ) + K I AI A2]

yk

where Yk and are the strain energy release rates associated with the slit loading Mode I and Mode II respectively. Also, C;j are components of the anisotropic matrix [C;j] in the local (slit) coordinate system denoted by primes and selected as in Fig.6-6. Additionally

A~ = r~

+ is;

(i = 1,2 and

,\~ =

r; - is;

r; ~ O,s; ~ 0)

(6-120)

are the roots of the characteristic equation [6-42] [6-43] written in the slit coordinate system (6-121) The total strain energy release rate is from Eqs.(6-1 20) and (6-119)

6.4 Micro-mechanics of Brittle Damage Based on Mean Field Theory

389

Fig. 6-6 Rectilinear slit global and local (primed) coordinate system (6-122) The second order matrix Eq.(6-122) is defined as

[cal

(with superscript a standing for anisotropy) in

C~2 2 [ (r~)2

r~ s~

+ r~s~

+ (s~)2][(r;)2 + (S;)2 ] C '1 1 '

,

-2( s 2 + s 1 ) (6-123) For a general anisotropic mat erial, the fourth order algebraic Eq.(6-121) does not admit an explicit analytical solution. Thus, the analytical expressions for the parameters s' and r' in Eq.(6-123) are not available either. For an orthotropic material the compliance tensor in the principal coordinate system simplifies t o (6-124) where in a special case (6-125) The above [C] denotes the global (total) compliances of the solid, which are still to be determined. The corresponding characteristic Eq.(6-131) (in the case of an orthotropic material) is defined by Eqs.(6-124) and (6-125), and when written in the principal coordinates system, it reduces to the form (6-126) where

390

6 Brittle Damage Mechanics of Rock Mass

(6-127) The roots of the bi-quadratic Eq.(6-1 26) are either purely imaginary or complex.

(A 2 )1 ,2 = - 1 ±

VI -

(6-128)

m

For m < 1 the roots in Eq.(6-128) are purely imaginary: (6-129) For m > 1 (which will be considered in this section) the solution of the characteristic equation (6-126) is a complex conjugate (6-130) From the parameters rand s, derived by substituting Eq.(6-130) into Eq.(6-126), we have the following form

r1

=

r2

=r =

J~-

1 ,

Sl

=

s2

J

Vm2 + 1

=S=

(6-131)

The above parameters rand s would suffice for a slit aligned with one of the principal axes. For a slit subtending an angle B with the principal coordinate system of an orthotropic material, it is necessary to find the parameters rand s in the slit (primed) coordinate system for which the compliance matrix is full. Nevertheless, once the solutions of the characteristic equation in principal coordinates are known as Eqs.(6-1 30) and (6-131), the roots of the characteristic equation (6-121), written in an arbitrary (local, primed) coordinate system can be derived using the Lekhnitskii [6-43] transformation

A' = Ak cos B - sin B k cos B + Ak sin B

"\' _ )..k cos B - sin B

/\k -

cos B + Ak sin B

(6-132)

Consequently, the parameters r' and s' in the slit coordinate system can be written in the form [6-44]:

r'l = w1r(rsin2B + cos2B) , r'2 = w2r(rsin2B - cos2B) ,

S'l

,

S 2

= =

SW1

sW2

(6-133)

where W2 ,1

= (ymsin 2B + cos 2B ± rsin2B) - 1

Substituting Eqs.(6-1 33) and (6-134) into Eq.(6-123), the matrix comes

(6-134)

[eij] be-

6.4 Micro-mechanics of Brittle Damage Based on Mean Field Theory

(rm -

391

(rm -

C' 1)cos 2e + 1 C;2 1) sin 2e 221+(m-l)cos 4 e 2 1 + (m - l)cos 4 e

(rm -

(rm -

C~l 1) sin 2e C' 1)sin 2 e + 1 11 1 + (m - 1)sin 4 e 2 1 + (m - 1)sin4 e

(6-135)

The transformation rule for the compliances is

[C' ij] = [T' im] T [Cmn ] [T'jn ] where the transformation matrix

[T~j ]

(6-136)

is

cos2e sin 2 e 0.5 sin 2e

1

[ sin 2 e cos 2e - 0.5 sin 2e

[T;j ] =

- sin e sin 2e

(6-137)

cos 2e

The expressions for the compliances Cb, and Ci1' in the local coordinate system are from Eqs.(6-124), (6-125), (6-136) , and (6-137)

Substitution of Eq.(6-138) into Eq.(6-135) leads to the final expression for the matrix [Cij] a

_

[C pq ] -

Sell

[(rm - 1 )COs2e + l ~(rm - l)Sin2el ~(rm - 1) sin 2e (rm - 1)sin e + 1 2

(6-139)

The path independent M integral can be determined from the expression for the J integral (6-140) Finally, the strain energy due to the presence of a slit in an anisotropic body is

W*=

J~ o

da = 2

J

Jda = 2

0

In the isotropic case (m the solutions are r'l

J

{Kp}T [C;q]{Kq}da

(6-141)

0

= I), Eq.(6-126) (or Eq.(6-121)) is bi-quadratic, and = r' 2 = 0

and

S'l

= S' 2 = 1

(6-142)

To derive an approximate analytical solution for the unknown overall compliances of a macro-orthotropic solid, assume that Ci1 i- C~2 but keep the parameters r' and s' as if the material is isotropic in Eq.(6-142). In this approximation, to be referred to as quasi-isotropic, the matrix [Cij] is

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6 Brittle Damage Mechanics of Rock Mass

(6-143) where 6ij is the Kronecker's delta operator. The ultimate simplicity is achieved assuming that the material is isotropic. In this case, the following equation is commonly used for diluting the concentration of slits (Taylor approximation)

C'l1 = C' 22 = 1/ E'

(6-144)

where (for plane stress) (for plane stress)

(6-145)

Matrix [Cij] in Eq.(6-143) reduces, in this case, to an even simpler expression (6-146) The stress intensity factors for a rectilinear slit embedded in anisotropic two-dimensional continuum (Fig.6-6) , written in the local coordinate system [6-42] are (6-147) The expression for the strain energy attributed to the presence of a crack is then derived substituting Eqs.(6-147) and (6-146) into Eq.(6-141) and performing integration (6-148) In Eqs.(6-147) and (6-148), it was found convenient to use the Voigt's notation

In general, anisotropy may occur as an intrinsic property of the matrix or be induced by slits. In the latter case, which is of interest in this study, the compliances and the elastic modulus are overall or effective properties. Therefore, the formulas derived in this section will be in the sequel used such that [O]=[C] and [O']=[C']

6.4 Micro-mechanics of Brittle Damage Based on Mea n Field Theory

393

6.4.4 Compliances of 2-D Elastic Continuum Containing Many Slits 6.4.4.1 Influence of Cracks Induced Anisotropy The influence of the cracks on the overall (effective) modulus of an elastic solid has been taken as the object of numerous studies in the past. Some of the existing results will be quoted in the sequel, while the expressions listed in the previous section will be used to derive new results needed to assess the influence of crack induced anisotropy on the overall elastic moduli. For convenience, it will be assumed that the undamaged (virgin) matrix is isotropic. Within the mean field theories approximation, the macro-stresses are mapped on macro-strains by means of the fourth rank effective compliance tensor as

{c} = [C]{O"} = ( [C]

+ [C*]){O"}

(6-150)

The expression relating t he overall compliance tensor and the derivatives of the complementary "elastic strain energy" is [6-39]

Since the stress intensity factors in Eq.(6-147) are linear homogeneous functions of stresses, the compliances due to the presence of a single slit embedded in a two-dimensional elast ic continuum can be derived substituting Eq.(6-141) into Eq.(6-151) and performing requisite differentiations

*(k) [Cij ] =

f

d2W *(k) a [d{Kp}T] a [d{Kq}] , ,T = 4 d{ '} [Cpq ] , T da (p,q = 1,2) d{O"Jd{O") 0 O"i d{O"j}

(6-152) The final expression for the compliance tensor attributable to a single slit in an orthotropic two-dimensional continuum is derived substituting Eqs.(6139) and (6-147) into Eq.(6-152) and performing necessary differentiations and integration

[C~(k) ]

= 27ra 2

[Crl {62i}{62j}T + Cr2 ({62i }{i 6j}T + {66i}{62j}T)

+C:fd66i }{66j}T]

(i , j

= 1, 2, 6)

(6-153) The coefficients Cij in Eq.(6-139) are both explicit and implicit (through m) functions of [Cij ]' Hence, they must be determined numerically by iteration.

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6 Brittle Damage Mechanics of Rock Mass

Within the quasi-orthotropic Eq.(6-143) and isotropic Eq.(6-146) approximations, the components of the compliance tensor could be estimated as

[c;?) ] = 27ra 2(C~2 [{ 02iH 02j} T] + C~l

[{ 06iH 06j} T])

(i, j = 1,2,6) (6-154)

and

[C;?) ] = 27ra 2([{02iH02j}T] + [{06iH06j}T])

jE' (i , j = 1, 2,6) (6-155) respectively. Primes in the expression of Eqs.(6-154) and (6-155) (except for E') indicate a reference to the local (slit) coordinate system. The corresponding expression for compliances in the global coordinate system is obtained using the transformation rule (6-156) The transformation matrix in Eq.(6-156) is " [Tij] = [

1

cos28 sin 2 8 sin 28 sin 2 8 cos 2 8 - sin 28 - 0.5 sin 28 0.5 sin 28 cos 28

(6-157)

In Eq.(6-157), 8 is the angle subtended by the axes Xl and X[. From Eqs.(6153) and (6-156) the final expression (in the global coordinate system) for the compliance attributable to a single rectilinear slit is

(6-158) In the quasi-orthotropic Eq.(6-1 43) and the isotropic Eq.(6-146), the approximations are (i ,j

= 1, 2, 6) (6-159)

and

jE' (i ,j= l ,2,6) (6-160) The compliances due to the presence of a single slit are derived substituting Eq.(6-157) into Eq.(6-158) and using Eq.(6-139) below

6.4 Micro-mechanics of Brittle Damage Based on Mea n Field Theory

395

. 2e ; C(k) * = 27m 2 s C- 11 sm 12 = C(k)* 21 =0 ci~) * = C~~) * = 27ra 2 SOlI sin e cos e

(k) * C 11

C~;) * = 27ra2sy'mOllCOS2e

(6-161)

C~~)* = cg)* = - 27ra2sy'm0ll sin e cos e

C~~)* = 27ra 2 SOlI [1 + (y'm - 1)sin2 e] In the quasi-orthotropic Eq.(6-143) , the compliances are estimated substituting Eq.(6-138) into Eqs.(6-159) and (6-160) and using the transformation rule Eq.(6-157)

ci~) * = 27ra 2 [011(1 - 2cos 2 e + 2cos4 e - cos 6 e) C~;)* = 27ra 2 [0 11 (cos 2e - cos8 e -

+ 0 22 (cos 2e - 2cos 4 e + cos6 e)] cos 2 esin 6 e) + 022(COS 8 e + sin 6 ecos 2 e)]

(6-162) The expression for the compliances in the isotropic case is recovered from Eqs.(6-161) and (6-162) setting m = 1 [6-44] (6-163) The effective (overall) compliances for a solid containing many cracks can be derived from Eq.(6-114) once Eqs.(4-161)rv(163) for a single crack are available.

6.4.4.2 Case Study for Two Systems of Aligned Slits Consider the two-dimensional case in which all slits are divided into two systems. Each of these two slit systems consists of N / 2 parallel (aligned) slits of equal length 2a. Slits in these two systems subtend angles +eo and _ eo, respectively, with xi-axis. The compliances attributable to both systems of slits is within the mean field theories approximation (see Eq.(6-114)) [C;j ]

=

~ ( [Ci~ )* (eO)] + [Ci~ )* ( - eo)])

(6-164)

Substituting Eq.(6-161) into Eq.(6-164) leads to

C;l = 27r N a 2 SOlI sin 2eo C;2 C*

22

C 66

= C;6 = C;6 = 0 r:::::: - cos 2 eo = 27rNa 2 symC 11 2 = 27r N a SOlI [1 + (y'm - 1)sin 2eo]

(6-165)

where m in Eq.(6-127) is a function of overall compliances as yet unknown. In the quasi-orthotropic approximation subst ituting Eq.(6-162) into Eq.(6164) gives

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6 Brittle Damage Mechanics of Rock Mass

The overall compliances of the effective continuum can then be det ermined substituting Eq.(6-165) into Eq.(6-114) and solving the system of algebraic equations for unknown effective compliances. In order to compare results of mean field theories estimated by different models of Taylor's, self-consistent and differential methods, superscripts ' tm ' , ' se' and 'dm' are used to denote estimations of Taylor's, self consistent and differential methods respectively. Thus, based on the self-consistent method of Budianski and O'Connell [6-45] we obtained

C SC *

_ C 22 22 - 1 _ 2nD A m

where

All

and

A22

22

(6-167)

are crack state parameters to be described later on, and (6-168)

is the micro-crack density, or damage parameter, which can be considered as the average damage parameter. In Eq.(6-168) , N is the number of slits per unit area. For isotropic matrix, C ll = C 22 , from Eq.(6-167) is obtained m - 1=

27rD(A22 -

All)

(6-169)

In quasi-orthotropic approximation, substituting Eq.(6-166) into Eq.(6-114) leads to the system of equations as

(1 - 27rDB ll )Cll - 27rDB 12 C22 = C ll = liE , - 27rDB 21 Cll + (1 - 27rDB 22 )C22 = C 22 = l i E

(6-170)

The parameters A(m, Bo) and B(Bo) in Eqs.(6-167) and (6-170) are (6-171) and

+ 2cos4Bo - cos6Bo 2COS4Bo + cos6Bo

Bll = 1 - 2cos2Bo B12 B21 B22

= cos2Bo = cos2Bo - cos8Bo - cos2Bosin6Bo = cos8Bo + sin6Bocos2Bo

(6-172)

The Taylor estimate for the compliances can be derived directly from Eq.(6-114) and Eq.(6-166) as C ll = C 22 , =l / E, such that

6.4 Micro-mechanics of Brittle Damage Based on Mean Field Theory

397

E Itm*

1 1 E 1 + 27r.f?(Bll + B 22 ) 1 + 27r.f?sin 2 eo (6-173) E 2tm* 1 1 E 1 + 27r .f?cos 2 eo 1 + 27r.f?( B21 + B 22 ) In a special case, where the two slit systems are mutually orthogonal, bisecting the angle between the global coordinate axes, eo = ± (7r/ 4) , the overall compliances in Xl and X2 directions are equal. In this case from Eq.(6-169) m - 1 = 27r.f? s ( Vm - 1)

(6-174)

The only solution of Eq.(6-174), m = 1 is independent of .f? Thus, from Eqs.(6-167) and (6-171) (7SC* 11

=

(7sc* 22

1

= E(l - 7r.f?)

(6-175)

Consequently, the solid remains macro-isotropic and its overall elastic modulus is jj;sc*

- - = 1 - 7r.f?

(6-176) E An identical result is obtained from Eq.(6-170). In the Taylor approximation for eo = 7r/4 from Eq.(6-173) 1

(6-177) E It can be shown that the isot ropy of the solid is not violated whenever two mutually orthogonal systems contain an equal number of rectilinear slits irrespective of the angle eo This conclusion agrees with the fact that Eqs.(6-174) and (6-176) are independent of eo. The differential method (DM) estimate of the elastic modulus can be obtained by solving the ordinary differential equation derived from Eqs.(6-118) and (6-176) (6-178) Subject to the initial condition that for .f? = 0, jj;* = E, the solution of Eq.(6-178) gives the estimated result by differential method as (6-179) The three different estimates of elastic modulus in a two-dimensional case containing two mutually orthogonal systems of aligned slits are plotted in Fig.6-7 for 0 < .f? < 0.5. Even though the differential method and the selfconsistent method are based on the same, or at least a similar set of approximations, the resulting estimates of the effective elastic modulus are substantially different. In fact , the self-consistent method predicts that jj;sc* = 0 for

398

6 Brittle Damage Mechanics of Rock Mass

[2 = 1/7r, while the differential method predicts that same micro-slit density.

Edm *=0. 368 for the

I~ idm/E i 7E S

0.2 0.0 0.0

0.1

0.2

n

0.3

0.4

0.5

Fig. 6-7 Effective elastic modulus for a two-dimensional continuum containing two orthogonal systems of aligned slits. Superscripts 'em ', 'sc' and 'dm' denote estimations by Taylor's, self-consistent and differential methods respectively

In polycrystalline solids the grain boundaries are often of inferior fracture strength. Consequently, most of the cracks are intergranular. Assuming an idealized two-dimensional case in which all grains are regular hexagons, it is often found desirable to study systems of slits subtending angles of = ± 7r /6 [6-46]. In this case it is difficult to derive the explicit expression for min Eq.(6127) in terms of [2 in Eq.(6-168), from the system of Eqs.(6-167) and (6-169). For [2 = 0.2, using an iterative procedure it is possible to compute

eo

m

= 2.22,

(6-180)

In quasi-orthotropic approximation, the elastic modulus is from Eq.(6-170) 1.85(1.42 - [2)(0.38 - [2) 1 - 97r[2/16 From Eq.(6-181) for [2 m

= 1.97,

= 0.2,

1.85(1.42 - [2) (0.38 - [2) 1 - 77r[2/16 (6-181)

(6-182)

which compares well with Eq.(6-180). Thus, the quasi-orthotropic approximation procedure seems to be reasonably accurate for damage state [2 « 1. Taking the Taylor approximation for Eq.(6-173) we have

6.4 Micro-mechanics of Brittle Damage Based on Mean Field Theory

Ei m * E When D

1

1 1 - 0.51dl '

399

(6-183)

= 0.2, Eq.(6-183) gives (6-184)

Thus, in contrast to the previous case, two systems of aligned slits intersecting at an angle of = ±1f / 6 present the effective orthotropic continuum. The quasi-orthotropic and Taylor estimates for the two effective elastic moduli are plotted in Fig.6-8. The Taylor approximation (upper bound on E) provides rather poor estimates in this case as well.

eo

0.8 0.6 ~

It<j 0.4

£,"IE

0.2

o. 0 '-:-~---:-'-:-~--:-'-:--~--:-'-:----'-''''-1._~-:-' 0.0

0.1

0.2

0.4

0.5

Fig. 6-8 Effective elastic modulus for a two-dimensional continuum containing two syst ems of aligned slits intersecting at an angle of 7f / 6 radians

In a special case, where all N slits are parallel to the Xl axis (and belong to the same system), the coefficients of Eq.(6-172) are Bll = I, Bl2 = B21 = B22 = o. The effective (overall) compliances and the elastic modulus E2 are 1 and 1 - 21fD

E Sc *

_ 2_

E

= 1 - 21fD

(6-185)

The elastic compliance and modulus in the direction of the axis X l remains unchanged. From Eq.(6-185), the estimations of elastic moduli by differential method are (6-186)

400

6 Brittle Damage Mecha nics of Rock Mass

Taking the Taylor approximation for Eq.(6-173) we have 1

E +

tm*

(6-187) =1 E 1 + 2nD In the cases of Eqs.(6-185) to (6-187) the material is orthotropic as deduced by Laws and Brockenbrough [6-46]. The estimations of the effective elastic modulus E2 in a two-dimensional continuum weakened by a system of aligned slits of equal length (parallel to X l axis) are plotted in Fig.6-9. The different discrepancies between the differential and the self-consist ent estimation method are again evident even for very small D. and

1.0 0.8

0.6 ~

I~

0.4

0.2

0.0 L-~_.l....---<.1._L...-~---L_~--'-_~-' 0.1 0.2 0.3 0.0 0.4 0.5

n

Fig. 6-9 Effective elastic modulus £;*2 for a two-dimensional continuum containing a single system of a ligned slits parallel to X l axis

6.4.4.3 Case Study of Fan Type Slits The averaging procedure is somewhat different when the slit orientations are uniformly distributed within a fan + 80 :::; 8 :::; - 80 , (where 8 is t he angle subtended by the slit and Xl axis). If N is the total number of slits per unit area, the compliance attributed to all N slits is within the mean field theory approximation obtained by averaging as (6-188) Applying the procedure of self-consistent method described by Eqs.(6-167) and (6-168)

401

6.4 Micro-mechanics of Brittle Damage Based on Mean Field Theory sc * _ C- 11 -

C ll 1 - 27rDa l l

(6-189)

where the coefficients all and a22 are

all

= 2:0 (eo -

In the case of C ll

~ sin 2eo)

a22 =

,

Sf:; (eo + ~

(6-190)

sin 2eo)

= C 22 as in Eq.(6-169) = 27rD (a22 - all)

m - 1

(6-191)

In the quasi-orthotropic approximation, the overall compliances can be derived solving the system of algebraic equations below

(1 - 27rDbll )Orl - 27rDb 21 0r1

-

27rDb120~2

+ (1 -

= C ll = liE

27rDb22)0~2

(6-192)

= C 22 = l i E

where the coefficients bi j are derived substituting Eq.(6-160) into Eq.(6-188) and performing integration

bl l

171

73

5

9

= eo (16 eo - "6cos eo sin eo + 24 cos eo sin eo - 32 sin 2eo) 111 7 . . 5 3 + "6cos eo sm eo - 24 cos eo sm eo

b12 = eo (16 eo 13

1

7.

1.

+ 32 sm 2eo)

1.7

7

5

.

b21 = eo (16 eo - scos eo sm eo - Ssm eo cos eo - 48 cos eo sm eo 1

+ 48 sin

5

35

3

eo cos eo - 192 cos eo sin eo

15

b22 = eo (16 eo

1

+ scos

1 5 - 48 sin eo cos eo

7. eo sm eo

35

+ 192 cos

3

5

+ 192 sin

1.7

+ S sm

3

17.)

eo cos eO- 128 sm 2eo

eo cos eo 5

7

+ 48 cos

3

5

.

eo sm eo 15.)

eo sin eo - 192 sin eo cos eo+ 128 sm 2eo

(6-1~3)

The results for the Taylor approximation method is derived by taking Or1 =C;2 = l I E * from Eq.(6-114) and using Eq.(6-188) instead of Eq.(6-166)

Etm* 1 E E 2tm* E

1 1 + 27r D(bll 1

1 + 27r D(b21

+ b22 ) + b22 )

1 1 + 7r D ( 1 -

2~o

sin 2eo)

2~o

sin 2eo)

1 1 + 7r D ( 1 +

(6-194)

402

6 Brittle Damage Mechanics of Rock Mass

In the case of slits with orientations uniformly distributed within the limits eo = ± 1f /6, the elastic modulus can be determined numerically by selfconsistent method from Eqs.(6-190) and (6-191). For D = 0.1 we have

m = 1.75,

(6-195)

In the quasi-orthotropic approximation of self-consistent method from Eqs.(6192) and (6-193) it gives

(0.23 - D)(2.19S - D) 0.506 - 2.154D

(0.23 - D)(2.19S - D) 0.506 + 0.355D

(6-196)

For D = 0.1 we have

= 1.S6,

m

(6-197)

which closely fits with the numerical results presented in Eq.(6-195). The Taylor's estimation in the case of eo= ± 1f /6 is according to Eq.(6-194) 1

1

1 + 0.54D '

1 + 5.74D

(6-19S)

Thus, for D = 0.1 we have m

= 1.49,

Eim * /E = 0.950,

E~m * / E

= 0.635

(6-199)

which also compares rather well with the self-consistent method estimation in Eqs.(6-195) and (6-197). The slit induced orthotropic behavior is apparent from the curves in Fig.610. Even though both elastic moduli (in self-consistent approximation) vanish at D = 0.23, their behavior is quite different. The accuracy of the Taylor model is again less than satisfactory. However, in a special case, when eo = 1f/ 2, the two-dimensional continuum remains isotropic. From Eqs.(6-190) and (6-191) it follows that (6-200)

m - 1 = 1fDs(vrn - 1)

The solution of Eq.(6-200) is m = 1 irrespective of the value of D. Thus, the approximate methods based on Eq.(6-154) lead to exact results within the self-consistent method approximation. Coefficients in Eq.(6-193) become

b11

= 7/16, b12 = 1/ 16, b21 = 3/16,

b22

= 5/16

(6-201 )

The overall compliances and effective elastic moduli are derived from Eqs.(6192) and (6-201) as (6-202)

6.4 Micro-mechanics of Brittle Damage Based on Mean Field Theory

403

0.2 0.0 '--~_.L..-~_L.......lI~_.L..-~_L...-~--' 0.0 0.1 0.3 0.4 0.5

n

Fig. 6-10 Effective elastic modulus for a two-dimensional continuum contai ning a syst em of slits with uniformly distributed orientations within (- Jr / 6, Jr / 6) range of a ngles.

The Taylor's estimation for the effective elastic modulus is identical to Eq.(6177) while the differential estimate is equal to Eq. (6-179). Hence, the curves depicting dependence of the effective elastic modulus on the micro-slit density [2 are in this case identical to the ones plotted in Fig.6-7. As the fan opening eo at the limit approaches zero, all cracks become parallel and db n db 12 db 21 db 22 - - - - 1 de o - deo - de o - deo -

(6-203)

The solution in Eq.(6-185) is then recovered from Eq.(6-192). The estimations of Taylor's model lead to Eq.(6-187) and the differential method leads to Eq.(6-186). Thus, in this case, the curves of E* vs [2 are identically to ones plotted in Fig.6-9. 6.4.5 Critical State and Percolation Theory 6.4.5.1 Aspects of Percolation Theory

The geometry of the microstructure-micro-cracks system near the critical state is strongly nondet erministic. Individual crack clusters are irregular in shape and their interaction is too complex to allow application of conventional micromechanical methods. Focusing on the analyses of the critical state, the deformation process may be simulated in the following manner. Defects of desired type and random shape, size and orientation are sequentially and randomly introduced into the specimen. Specifically, it may be interesting to determine the correlation length (related to the size of the largest defect), distribution of defect sizes, macro stiffness [D] (or compliance [C]) of

404

6 Brittle Damage Mechanics of Rock Mass

the specimen, probability that a defect belongs to the infinite cluster, etc. The described simulation is acceptable only if the critical state itself and the above listed parameters are invariant with respect to the dilution sequence. P ercolation theory provides a powerful and efficient framework for the geometrical study of states and systems in the close neighborhood of the critical state (or percolation threshold), defined as the dilution level at which an infinite cluster emerges. The principal advantage of the percolation theory is related to the existence of universal laws and parameters, characterizing an otherwise random process developing within a solid having random microstructure. Processes and systems having identical or nearly identical universal parameters form a universal class. These parameters are universal in the sense that they do not depend on the exact microstructure of the specimen (details of the micro-crack distribution) and dilution sequence. Mechanical response of a homogeneous, elastic matrix diluted by crack-like defects constitutes a universal class . Percolation studies of problems and systems belonging to the center on the determination of the critical volume/area defect density. The scaling laws for the components of the specimen stiffness tensor exhibiting singular behavior in the neighborhood of the percolation threshold. Percolation processes can also be studied on geometrically regular lattices using a disorder-generating variable which defines the probability that a site and/or bond are occupied by a defect . In continuum percolation, the disorder-generating variable is superimposed on a topologically disordered structure. It is notable that, in the case of lattices, the percolation threshold to some extent depends on the microstructure of the material (i.e., lattice geometry or distribution of bonds within the material microstructure). In continuum percolation models, the percolation threshold and the scaling laws are truly universal, i.e., independent of the details of the microstructure. 6.4.5.2 Lattice Percolation Models

Most of the early percolation models were performed on regular lattices. For present purposes, it suffices to consider only the bond percolation problems, denoting by q the probability that a lattice bond is missing and by p = 1 - q that a bond is present. Rectilinear slits (in two-dimensional problems) and sq uare, plaq uette-like, decohesions (in three-dimensional cases) are conveniently modeled by removing the bonds connecting the adjacent nodes in the dual lattice. Each slit in the original lattice (discrete model of the body) is assumed to be perpendicular to the ruptured bond in the dual lattice [6-47]. The bonds of the original .Y' and dual lattice .Y'd intersect at midpoints and are mutually orthogonal. Thus, the dual lattice of a square lattice is also a square lattice. Using symmetry arguments, it was shown [6-48] that the percolation thresholds on the original and dual lattice must satisfy the equality. (6-204)

6.4 Micro-mechanics of Brittle Damage Based on Mean Field Theory

405

In a square lattice, rectilinear slits can be assumed to only have two mutually orthogonal orientations. For a square lattice the bond percolation threshold is Pc = qc = 0.5 [6-49]. Once p~ is available, it becomes possible to compute the critical slit density. The number of broken bonds is denoted by n c and the ratio of the lattice size is denoted by .\ = L / l, where the distance l = 2a separating two adjacent nodes in a square lattice. Thus, for a total number of links in a large lattice 2.\2, the density of ruptured bonds at the percolation threshold is qc = n c/ 2.\2 = 1/2. According to Eq.(6-168) the critical damage deal with the critical slit density is

Dc ~ (Na 2 )c ~ 0.25 In the spirit of micromechanical damage measurement, N = n/ L2 represents the fraction of ruptured bonds (number of slits per unit area), while 2a is the slit length. As suggested by Sornette [6-50], the emergence of a spanning cluster does not always lead to instantaneous fragmentation. Held together by interlocked asperities, two fragments of the specimen will not separate at q = qc. The failure is in this case reached when a percolation path without interlocking parts is established. The formation of such a path occurs at a higher density of ruptured bonds. This type of problem is addressed by the directed percolation models [651]. For a square lattice, the percolation threshold increases to q? = 0.6445 ± 0.0005 such that the critical micro-crack density (damage) is in contrast to the equation of (6-205) 6.4.5.3 Continuum Percolation Models

A much more complicated problem ensues if the micro-cracks are allowed to form a truly random network and when they are allowed to overlap (intersect). The cluster shapes in this case may assume a large spectrum of irregular forms commonly observed in micrographs. The connectivity of defects (microcracks) forming a cluster is determined using the concept of the excluded volume and/or area. The excluded area for a given rectilinear slit (at an angle Bi ) belonging to a system of rectilinear slits of equal length and random inclinations Bj is shown in Fig.6-11. If the center of a slit is within the excluded area of the neighboring slit, they intersect. The probability that a system of slits will form a cluster depends on the mean number of intersections of slits belonging to this cluster. Since the response of a lattice in the neighborhood of the elastic percolation depends only on the infinite cluster, it is reasonable to assume that the connectivity or the mean number of intersections of slits forming the cluster is a universal parameter. Assuming this premise to be

406

6 Brittle Damage Mechanics of Rock Mass

Fig. 6-11 The excluded a rea for the slit "i " intersected by slit "j "

correct, the percolation threshold can be computed using the concept of the excluded volume and/or area for a given class of defects. In a general case of clusters of voids of different shape, size and orientation, the determination of the cluster connectivity is difficult. However, when the distribution of shapes, sizes and orientations is not very broad , the approximate method suggest ed by Balberg et al. [6-52] provides an efficient and conceptually appealing algorithm for the analytical determination of the critical value of the connectivity parameter by approximate estimation, typically denoted by B = B e. The cluster connectivity B can be defined as the average number of slit intersections per slit, i.e., the average number of slit centers locat ed within the excluded area of an observed slit. Thus, the critical connectivity B e (continuum percolation threshold) can be estimated [6-52] as the critical number density N, of slits per unit area multiplied by the averaged excluded area/volume of the slit. (6-206) Thus, the original task of the determination of the defect connectivity in a cluster is reduced to a relatively simple derivation of the average excluded area for a single void and/or crack. Eq.(6-206) is an identity in the case of soft core objects, a reasona bly accurate approximation for t he hard-core continuum case, and is not valid in the case of hard-core objects with centers located on a lattice. As presented in Balberg et al. [6-52]' the excluded area (Aex) is, therefore, more universal than the connectivity parameter Be. In general , the mechanical response of an elastic plate containing slits is anisotropic. In view of the potential dependence of the percolation threshold on the anisotropy (number of vanishing components of the order parameter at critical lac unity) , a study of the extent of universality of the parameters

6.4 Micro-mechanics of Brittle Damage Based on Mean Field Theory

407

Be and the excluded area becomes mandatory. Consider two rectilinear slits of length 2a oriented as sketched in Fig.6-11. The excluded area surrounding the slit "i" (corresponding to a system of potentially intersecting slits at the angle (ei- ej ) is equal to A = (2a)2sin lei-ej l (rhomboid in Fig.6-11). If the angle (e i - ej) is a random variable, the excluded area [6-52] is computed averaging the area A over the entire set of possible realizations of the relative angle (e i - ej) su btended by two potentially intersecting slits

(6-207) In Eq.(6-207) a+

(0: 2 )

f a p(a)da

=

2

(6-208)

a

where p(a) is the probability density function defining the distribution of slit lengths. Also

ot

F(q+ , q- ) = 4(sin lei - ej l) = 4

oj

f p(ei )de f p(ej ) sin lei - ej ldej i

(6-209)

o~ J

In Eq.(6-209) p(e) is the probability density function defining the distribution of slit inclinations. For example, if the slits are uniformly distributed with respect to their orientations p(e) = eo/2 =const. within a fan (- eo < e < eo). Integral Eq.(6-209) allows for a closed form quadrate 2

F(eo) = e2 (2e o - sin 2eo) (6-210) o For the isotropic distribution 2e o = 7r of slits with constant length 2a (6-211) The average number of intersections Be at the percolation threshold was determined numerically by Pike and Seager [6-53] (and corroborated by Robinson, [6-54"-'55] and others) as being Be = 3.6. Thus, the slit density at the elastic percolation limit (loss of specimen stiffness) is from Eq.(6-211) (6-212) The self-consistent, differential and dilute concentration (Taylor's) estimates of the overall elastic modulus of a two-dimensional elastic continuum, containing randomly oriented rectilinear slits, are plotted against the slit density in Fig.6-12. The percolation threshold in Eq.(6-212) seems to be in reasonably good agreement with the differential method estimate, even though

408

6 Brittle Damage Mechanics of Rock Mass

the latter method predicts that the elastic modulus E --+ 0 only if the slit density (Na 2 ) --+ 00. At the percolation threshold in Eq.(6-212), the differential method estimate of the overall elastic modulus is 0.0119 E.

0.6 t.:I

li<j

0.4

0.2 contnuum perc.pt. 0.0 0.00

0.50

0.75 Na'

1.00

1.25

! 1.50

Fig. 6-12 Mean field estimates of the overall elastic modulus for the case of isotropic damage. Percolation threshold is indicated by the arrow

On the basis of Eqs.(6-206) to (6-209) , it becomes possible to estimate the effect of the micro-crack induced anisotropy on the critical state. Consider first the case of slits of equal length 2a = const. From Eqs.(6-206) to (6-209) t he excluded area is in this case (6-213) where F(e o) is defined by Eq.(6-209). Assuming that the critical value of the average number of slit intersections B e = 3.6 is invariant not only in the isotropic case, but also in any other case in which the slit orientations are uniformly distributed within the range (- eo ~ e ~ eo) from Eq.(6-21 3), it follows that (6-214) Eq.(6-214) fits t he results of numerical simulations reported by Robinson [6-54'"'-'55] with remarkable accuracy. The coefficient in the numerator on the right hand side of Eq.(6-214) averages, in Robinson's computations, 3.61±0.03. The width of error bars is well within the accuracy expected from numerical simulations on a grid having only an 80 units square and containing up to 9000 slits. The dependence of the micro-slit density (Na 2 ) on t he parameter eo is depicted in Fig.6-13. A rather dramatic increase in t he crack

6.4 Micro-mechanics of Brittle Damage Based on Mean Field Theory

409

density necessary to form a spanning (infinite) crack as the angle eo , decreases should be intuitively expected. The probability that slits will intersect increases with the angle lei- ej l they subtend. Thus, a much larger micro-slit density is required to form an infinite cluster if the angle eo decreases. At the limit, since the probability that two parallel slits will intersect is zero, the critical micro-slit density is (Na 2 )e --+ 00 for a system of parallel slits. Naturally, in actual materials the parallel cracks coalesce by linking with each other. It is important to note that Robinson's computations indicate that the critical value B e ~ 3.6 is valid not only in isotropic (perfectly random) distribution of orientations (as described by Balberg et al. [6-52]), but also in any case in which the orientations are uniformly distributed within the fan for which o < eo < 7r / 2 (rendering specimen orthotropic). 5,----,-------------------------,

4

2

0.1

0.2

B.l1t

0.3

0.4

0.5

Fig. 6-13 Critical slit density as the function of the fan opening angle The self-consistent, differential and Taylor model estimations of the elastic modulus for eo = 7r / 6 are plotted in Fig.6-14 vs the micro-slit density parameter N a 2 . As in the previous case, the curve for the differential method seems to fit the trend necessary to reach the percolation threshold (Na 2 )e = 2.72 computed from Eq.(6-214). Estimation of the overall effective elastic modulus by the differential method at the above percolation threshold is 3 x 10- 7 E. In materials such as layered composites, it often happens that all, or at least most, slits (or cracks) are embedded in two orthogonal systems of parallel planes. This case was also investigated by Robinson [6-54"-'55], who considered slits of identical length 2a = const., which are with equal probability oriented in two directions (3 and -(3. Consequently, p((3 ) = 0.515((3) and p( -(3) = 0.5J( -(3) while lei - ej I = 2(3. In this case the integral Eq.(6-209) admits a

410

6 Brittle Damage Mechanics of Rock Mass 1.0 r - - - - - - - - - - - - - - - - - - - ,

0.8

ttl

0.6

li.<j 0.4

0.2

perc. pt.

---j 1.0

1.5

2.0

2.5

3.0

Fig. 6-14 Mean field estimates of the overall elastic modulus in the case of slits with orientations uniformly distributed within a fan ±30°, with percolation threshold indicated by the arrow .

form of closed solution F((3) = 2sin2(3. According to Eqs.(6-206) and (6-207) , t he excluded area is (6-215) When the two slit systems are perpendicular to each other, i. e., when t he angle (3 = 71" /4, it follows that (A ) = 2a 2 and (Aex) ~ 2a 2 N. From simulations reported in Balberg [6-56], the excluded area is (Aex) = 3.2 in this case. Assuming that (Aex) ~ 3.2 for arbitrary (3 , the critical value of the micro-crack density can be estimated from Eq.(6-215) as (6-216) which is in excellent agreement with the numerical simulations reported in Robinson [6-54'"'-'55], according to which the ratio of two sides of Eq.(6-215) is 0.998± 0.002. The dependence of the critical slit density on the angle (3 is demonstrated in Fig.6-15. As expected, the critical density of slits is minimal if the two systems are orthogonal and infinite when the two systems are parallel. It is important to reiterate that the numerical simulations performed by Robinson [6-54'"'-'55] indicate that the critical connectivity remains B e = 3.2 for any two systems of parallel slits intersecting at an angle (3 belonging to the range of 0 < (3 < 71"/2. In agreement with Eq.(6-214), Eq.(6-216) predicts that when all slits are parallel ((3 = 0), a spanning cluster becomes possible only when the product (Na 2 )e tends to infinity. The mean field estimations of the elastic modulus in a two-dimensional elastic continuum containing two mutually orthogonal systems of slits ((3 = 71" /4) are plotted in Fig.6-16. Since the effective continuum in this case remains

6.4 Micro-mechanics of Brittle Damage Based on Mean F ield Theory

411

5.-----.---------------------------.

4

2

O .~~~~~~~~~~~~~~~

0.0

0.1

0.2

pin

0.3

0.4

0.5

Fig. 6-15 C ri tical density of two systems of slits subtending a ngle

f3

isotropic, the mean field theory estimations are essentially identical to those computed for the random distribution of slits (as in Fig.6-12). The continuum percolation threshold (Na 2 )c = 1.6 is in good agreement with t he trends of the differential method. The estimation of the overall effective elastic modulus by the differential method at the percolation threshold is 0.0066E. In comparison, t he lattice estimations from Eqs.(6-204) and (6-205) are in excellent agreement with the self-consistent approximation. the two systems are parallel.

1 Lattice perc 2 Directed perc 3 Continuum perc 0.6 t.:I

li
0.2 3 0.0 L--'--...........L---'---'---'----'----'----L----<=:........J'----''--' 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75

Fig. 6-16 Mean field est imates of t he overall effective elastic moduli in t he case of two systems of mutually orthogonal slits . Dot , triangle and square denote ordinary lattice, directed lattice and continuum percolation thresholds, respectively.

412

6 Brittle Damage Mechanics of Rock Mass

In particular, consider again the case from Robinson [6-54'"'-'55] for which the slit lengths L = 2a are uniformly distributed within the range defined by the limits L(l - ry) and L(l + ry). From Eq.(6-210) (L2) = Law2(1 + ry2/3). If orientations of the slits are perfectly random (isotropic), i.e., when eo = 7r / 2, from Eq.(6-209) it gives F(7r/ 2) = 8/7r. The excluded area computed from Eqs.(6-206) and (6-208) becomes (6-217) Following the procedure used in the two previous cases, the magnitude of the critical lacunity is found to be

(N (a 2 ))c = [Na;v

(1

+ ~2)

L~

3.:7r

~ 1.40

(6-218)

Instead of 1.40, according to numerical simulations by Robinson [6-54], (N (a2 ;)c = 1.45± 0.01. The dependence of the unequal slit lengths on the percolation threshold, depicted in the graphs in Fig.6-17, is not as dramatic as the influence of the orientations. The slits of equal length are somewhat more efficient in forming finite clusters, but distributions of the length present a dominant factor in the formation of the infinite cluster. 4.0 3.5 3.0 2.5 2.0 1.5

1.0

0.0

0.2

0.4

Fig. 6-17 Invariance of connectivity Be , slit density n e = In the case of slits of unequal length.

(N (a 2 ))e and

their ratio

Consequently, the density of a system of isotropic oriented penny-shaped cracks at which the specimen ability to transmit additional mechanical stresses vanishes (as a result of the form ation of a cleavage surface formed by cracks)

6.4 Micro-mechanics of Brittle Damage Based on Mea n Field Theory

413

is almost four times higher than the density at which a fluid will percolate through a borehole (formed by intersecting cracks).

6.4.6 Higher Order Models for Rectilinear Slits 6.4.6.1 Two Dimensional Continuum Containing Rectilinear Slits A second order mean field theory allowing for direct interaction of neighboring slits was formulated recently by [6-34"-'35] Direct interaction between the adjoining cracks is introduced using the approximation of Kachanov [6-57]. Despite this simplification, the proposed model requires a substantial computational effort to derive t he second order est imations for t he elastic modulus. Some numerical results in Ju and Chen [6-34"-'35] were provided for the case of an isotropic (perfectly random distribution of slits) and for the case of randomly distributed slits embedded in parallel planes normal to the X2 axis. For the isotropic case Chen and Ju computed that

E*2 E

1

(6-219)

1 + 1T(D + 0.373D2)

In the case of aligned parallel slits randomly located in array from , in addition to [6-34]' estimations of the overall elastic modulus were provided by Dyskin [6-58] and Horii and Sahasakmontri [6-59] In all cases the basic form of the derived expressions was identical

E*2

1

E{ /E =

E

1

(6-220)

However, the coefficient k multiplying the square of the damage parameter in Eq.(6-168) was substantially different. According to [6-34] k = 1.173, while Dyskin [6-58] and Horii and Sahasakmontri [6-59] estimated k as being 4.712 and 9.425, respectively.

6.4.6.2 Doubly Periodic Rectangular Array of Aligned Slits The compliances of a two dimensional continuum perforat ed by a double periodic arrangement of slits with centers forming a rectangular lattice were computed numerically as a solution to the system of singular integral equations with degenerate kernels by Delameter et al. [6-60"-'61]. However, for present purposes, it would be desira ble to derive a simple approximat e in order to have an analytical expression for the overall elastic modulus under the given geometry. Consider first a single row of aligned, equidistant slits with equal length 2a. The exact solution of the SIFs for each slit [6-40] is KIJ

=

, ~ w tan -;;;

76

V

(6-221 )

414

6 Brittle Damage Mechanics of Rock Mass

where w is the distance separating the centers of adjacent slits. The strain energy increase attributed to a single slit can be derived substituting Eq.(6221) into Eq.(6-141) (6-222) The compliance attributed to the presence of a single-slit parallel to the Xl axis can be derived differentiating Eq.(6-221) , taking into account Eq.(6-146) , substituting into Eq.(6-152) and applying the transformation rule in Eq.(6156) (6-223) In the case of slits aligned and centered on nodes of a rectangular lattice, the number density of slits is (6-224) where b = aw 2 is the distance separating two adjacent slit rows. Neglecting the direct interaction of slits belonging to two different rows, the overall compliance is then from Eqs.(6-114) and (6-223)

0 22 =

~

[1 -

a~ In (cos : ) ] = ~

[1 -

a~ In (cos7rVcill)]

0 11 =

1

(6-225) where D = Na 2 as in Eq.(6-168). Expanding the term in the brackets into a series leads to an approximate expression for 0 22 2 C- 22 ~ E1 [1 + 27rD( (17 + r(fDa

1

~ E [1 + 27rD(1

+ 27r4 45 D 2 a 2 + ... )

]

(6-226)

+ 1.645Da + 4.329D 2 a 2 + ... )]

The modulus of elasticity is from Eqs.(6-225) and (6-226)

E tm * 1 1 __2__ ~ ____,-_____________ ~ ______~~--,,----~_,------~~ E

1-

a~ In [cos (7rVcill)]

1 + 27rD (1

+ :2 Da + 2~4 D 2 a 2 + ... )

(6-227) The result of Eq.(6-227) is numerically different from those derived by Dyskin [6-58], Horii and Sahasakmontri [6-59]. The discrepancies are related to consideration of different configurations of slits (periodic vs. random) and interaction of slits belonging to different rows. However, the simple expression in Eq.(6-227) fits the Delameter et al. [6-61] numerical data with surprising

6.5 Non-linear Brittle Damage Model of Porous Media

415

accuracy for 0: = b/w ;::::0.75, i.e. , whenever the adjacent rows are not very close to each other. For example, for 0: = 1.0 the error is below 1% , while for 0: = 0.8 the error increases to a tolerable level of 3.3%. For illustration, curves E vs. [! are plotted in Fig.6-18 for 0: = 0.8, 1 and 2. Numerical results from Delamet er et al. [6-61] are presented by dots in Fig.6-18 to demonstrat e the accuracy of the approximate solution of Eq.(6-227).

0.8

~

1;;S 0.6

0.4

0.2L-~~---'-_~_--'-_~_.L.-~_--'

0.00

0.05

0.10 Q

0.15

0.20

Fig. 6-18 Effective elastic modulus E2 for a two-dimensional continuum containing a single system of interacting slits parallel to Xl axis with centers placed on a regular rectangular lattice. Dots, rectangles and triangles represent results from [6-61 ] for 0: = 0.8, 1.0 and 2.0, respectively

6.5 Non-linear Brittle Damage Model of Porous Media 6.5.1 Relationship between Damage and Porosity Most materials involve different types of defects like caves, pores and cracks, which are the important characteristics of porous media and have a great influence on the physical properties of materials. These kinds of characteristics are generally designated by the porosity of materials and described by a pore rate (porosity), defined by the percentage of the volumetric ratio between the porous volume and total volume of the specimen. Formerly, the pore model for most porous media was assumed to relate to material constants of porosity and permeability independent of time, whereas, in practical porous media there exist many fractures and damage, like the networks of cracks and pores. Therefore, they are considered to be damaged porous media. In fact , they vary with changes in time and coordinate due

416

6 Brittle Damage Mechanics of Rock Mass

to erosion, seepage and damaging. In order to express the kinematic effect of porosity and permeability, it is necessary to study the evolution problems of the networks of cracks and pores caused by damage development in the damage-porous media and to establish the relational model of evolution between the damage growth and porosity changes in the networks of cracks and pores. From the view of damage mechanics, the area deduction rate affects the properties of materials. The definition of area deduction rate ¢ due to variation of porosity in porous media is the same as the definition of damage variable D based on the area deduction rate in damage mechanics. It means that there is an equivalent concept or definition between the damage variable and the area effect of porosity. Thus, the damage variable can be used to express the area deduction rate due to changes in porosity during damage. The anisotropic damage characters can be expressed by the principal damage values (D I , D 2 , D3) of the anisotropic damage state or by the three values (¢ 1,¢2,¢3 ) of the area deduction rate along the anisotropic principal directions. According to this equivalence we have

Di = 'Pi =

ds*

'

d Xj d xk

(6-228)

where Di was defined before as the damage variable in Xi direction, ¢i is the area deduction rate of the section with outward unit normal Xi due to porosity, ds; is the deduced area due to porosity in the section dXjdxk with outward unit normal Xi. Obviously, the relationship between damage variables {D} and volumetric porosity tP can be illustrated by Fig.6-19 and formulated by

Fig. 6-19 The volumetric porosity in a damaged porous medium is described by the area deduction rate induced by damage

6.5 Non-linear Brittle Damage Model of Porous Media

In the isot ropic case, ill

417

= il2 = il3 = il gives (6-230)

Eqs.(6-229) and (6-2301) can be applied to the damage evolutional equation, thus it provides a method for expressing the evolution of porosity from the view point of continuum damage mechanics. The represented relationship between the rat e of porosity evolution and the rate of damage development in the isotropic case is (6-231) The above relationship between porosity and damage is described by the single porous model. However, geotechnical engineering materials are fracturedamaged porous media, called damage porous media or fracture porous media, which actually are models of double (multi) porous media. In order to set up this double porous model, it is assumed that the total (actual) porosities consist of the initial natural micro-porosity and the macro-porosity induced by the fracturing of cracks and/ or by the development of damage, and the combinational relationship [6-62 rv 63] is presented as follows:

¢ = ¢ + il or -

if>

= (¢ + il)"2 -

3

(6-232)

where ¢ is the area deduction rate contributed from the initial natural porosity of the porous medium, il is the damage variable as a function of time and space contributed from the cracking and fracturing of initial cracks and pores in the porous medium. Thus it can be assumed that the evolutional rate of porosity is only induced by developing damage that gives (6-233)

6.5.2 Brittle Damage Model Based on Modified Mohr-Coulomb Porous Media 6.5.2.1 Effective Concepts Due to Damage and Pore-Pressure Water is ubiquitous in various types of porous media, and generally flows through the pores and cracks in the porous medium when there exists a hydraulic gradient, which represents the seepage effects. Seepage water flow is one of the important factors that change the state and property of porous media. The effects of seepage can be implemented due to hydraulic loadings, physical influences and chemical reactions , while porous frames of a solid may provide a certain resistant reaction to seepage flow, and show many characteristics and properties related to that of seepage water, such as permeability,

418

6 Brittle Damage Mechanics of Rock Mass

distensibility, dilapidation and intenerating, etc. The hydraulic property and its index are very important factors in the analysis of stability problems of damaged porous media. The permeability of porous media is defined as the capability of water to flow through the pores and cracks in porous media under seepage pressure. Because many factors of non-uniform distribution, discontinuity and complexity of anisotropic properties and porosities intensely affect the micro-structures of porous media this means that the pore-pressure (seepage pressure) plays an important rule in porous media damage, fracture and variation in structural stability. The dynamics of damage in porous media is a complex damage and failure process of materials due to actions of pore-pressure, which includes coupled effects in the process of deformation, stress concentration, damage development and fracture growth, pore-pressure variation and porosity evolution in porous media [6-62"-'66]. In order to synthetically describe the above complex behavior, this section will use a modified nonlinear brittle damage model for Mohr-Coulomb porous media, based on the author's results [6-62]. The brittle failure mechanism of porous media relates to a transient increase in cracks and a transient change in porosity. If there are some local irregularity zones in the medium, for example, some tensile or shear failure zones, the initial damage may exist in these zones. The initial damage will increase and propagate as the failure process of these irregularity zones increases, which can make the porosity increase. In order to simplify the brittle damage and porosity evolution equations of porous media such as cracked rock, concrete and composite fib er under dynamic loading, we need to assume the following: (1) The shear strength and average dimension of crystal lattices of porous media are time dependent during damage. (2) Once failure criterion is satisfied at any point in the porous medium, damage development and porosity evolution begins at once in the neighboring field to this point in the porous media. (3) There is not any macro-plastic deformation to yield in the failure process of porous media. That means, once the failure criterion at a given point in the materials is satisfied, the local failure (micro-damage or microfracture) at this point happens immediately and the macro deformation resumes at once. From the viewpoint of continu um damage mechanics and thermodynamics, all energies are not dissipated by the plastic deformation but dissipated by micro-structures changing in porous media (i. e. damage development and porosity evolution) when brittle failure occurs. That gives a reasonable physical mechanism for brittle damage in porous media. The concept of effective stress in geotechnical engineering, which is not t he same as that in damage mechanics, was presented firstly by Karl Terzaghi in 1925. He gave the following descriptions to introduce the concept of

6.5 Non-linear Brittle Damage Model of Porous Media

419

effective stress: (1) If both the surrounding pressure Pc and the pore pressure Pf increases by the same quantity, then the volumetric change in the material can be negligible when compared with that in the change produced from the increase in surrounding pressure Pc only. In the shear failure case, if only normal stress increases then shear strength may show a big increase. (2) If both normal stress and pore pressure together increase by the same quantity, then the shear strength will not increase. So we can drew a conclusion that the pore pressure stress Pf has no effect either on deformation or failure behavior, and the deformation and failure are controlled by the effective stress. However , the concept of effective stress in damage mechanics of porous media is more of a physical concept than the mechanism in practice. The function of effectiveness in damaged porous media includes two parts, the first effect is due to the damage associated with the effective area of the cross section, the second effect is due to the reaction force of pore pressure on the associated effective area of the cross section. The effects of effective stress of fracture-damaged porous media depend on 4 factors as follows. (1) the stress caused by external loads (2) the pore-pressure of seepage flow (3) the deduction of the effective bearing area due to fracture and damage in the medium (4) the added action on the effective bearing area caused by pore pressure. The fracture-damaged porous medium can only bear the stress on the effective bearing area (1 - D)A, but the loose area DA, due to damage, cannot receive stress, thus the effective stress tensor on any effective area is expressed by (Til = (Tij/(1 - Di), where superscript "8" denotes the solid frame of the porous medium, and "*,, denotes an effective quantity. According to the relationship of the effective stress tensor (Ti j and the Cauchy stress tensor (Tij in damage mechanics, the effective stress vector in t he case of isotropy is in t he form of

{(T* } = ~ 1- D

(6-234)

From Eq.(6-234) it can be seen that in the case of isotropic damage, all components of the effective stress tensor (Ti j can be expressed by that of the corresponding Cauchy stress tensor divided by the continuity factor 1 - D. For example (Ti j = (Tij /1 - D, when in the isotropic case the directions of (Ti j keep coinciding with (Tij ' If the external water pressure is P, P = pgh, based on the general concept of effects on pore-pressure, the effective stress should be considered as the classical form (T' = (T + P

(6-235)

420

6 Brittle Damage Mechanics of Rock Mass

When considering fracture or damage effects in porous media, the material receives stresses only on the effective area (l-Sl)A of a cross section A , but the pore pressure is distributed only on the surface area of the empties and cracks SlA, which is the outside area of the effective bearing area (1 - Sl)A , in all directions. Therefore, the pore pressure of seepage only has the normal effective stress (see Fig.6-20) aiiw = SlP/(l - Sl) contributed to the effective bearing area (1 - Sl)A , and has no contribution to the effective shear stress = 0, where superscript 'w' denotes water related. Thus, from Fig.6-20 t he final effective stress tensor or stress vector of fractured-damaged porous media with pore-pressure is expressed respectively as

Ttt

pQ,/(I-n) OJ'/(l-.Q)

OJ,J(l-.Q)

Fig. 6-20 Illustration of effective concepts due to damage and pore pressure

* aij aij = 1 _ Sl

SlP {a} Sl ' {a} = 1 _ Sl

+ Oij 1 _

SlP Sl

+ Oij 1 -

(6-236)

where Oij is Kronecker delta (the unit tensor). 6.5.2.2 Brittle Damage Model Based on Effective Shear Strength Mohr-Coulomb criterion is an advanced classical failure criterion, which is widely applied in most of engineering materials and verified by a great number of macro and micro experimental tests. It is especially applicable for porous media. Since most practical engineering materials ineluctably include various micro or macro pores and cracks, thus all physical facts involved in the practical Mohr-Coulomb criterion either affecting or effecting actually are effective quantities and the effective property in nature is influenced due to porosity and cracks. Therefore, it is reasonable either in logical nature or in physical nature to generalize Mohr-Coulomb criterion from the classical model into the effective space as shown in Fig.6-21, where the effective stress action and the effective material strength bearing are taken into account in the modified effective Mohr-Coulomb criterion.

6.5 Non-linear Brittle Damage Model of Porous Media

421

a~ t- ----------

Fig. 6-21 The modified Mohr-Coulomb criterion in damaged effective space

T~

= c* -

(J"~

tan'IjJ*

(6-237)

where T~ is the effective shear stress on the local failure plane of the porous medium ; (J"~ is the effective normal stress on the local failure plane of the porous medium; c* is the effective cohesion of the material property state, which varies with the variation in damage and pore pressure in the porous medium. t.p* is the effective internal friction al angle, which is assumed to vary with damage and pore pressure. From the concepts of the isotropic effective stress tensor described in Eq.(6236) the effective shear stress T~ on the local failure plane can be written as (6-238) The effective normal stress (J" ~ should be determined from the internal force acting on the local failure plane caused by external loads and pore-pressures. According to the concept of equivalence on area deductions induced between damage and porosity (see Eq.(6-236) in previous section), the effective normal stress on the local failure plane can be represented by damage variable [l and pore-pressure P as *

(J" n

1

[l

= -----;:; (J" n + -----;:; P 1 - Jt I - Jt

(6-239)

In the modified effective Mohr-Coulomb criterion Eq.(6-237), due to effect s of damage and pore-pressure, the concept of the effective cohesive strength and the effective internal friction angle must be considered as the state parameter, (c*, t.p* as shown in Fig.6-21), of material property, which consists of the effective shear strength when the crack-damage process is modeled by the brittle damage of modified Mohr-Coulomb porous media. So the effective cohesive strength c* and the effective internal friction angle t.p* are functions of the damage variable and pore pressure concerning porosity. The modified

422

6 Brittle Damage Mechanics of Rock Mass

Mohr-Coulomb criterion as a brittle failure model employing the effective shear strength and pore pressure can be expressed as *

+ [!P

*

(6-240) -----;:) = c + 1 - Jt n tan cp 1 - Jt Applying the relationship of R c = 2c cos cp / (1 - sin cp) between the shear Tn

(Yn

strength (c , cp) and the compressive strength R c for isotropic materials [6-67] into the corresponding effective parameters of damaged porous materials, we have R~

= 2c* coscp* /(1- sincp*)

(6-241)

The ratio of compressive strengths between the effective one and undamaged one is R~ / R c = 1 - [!. Thus, from Eqs. (6-240) and (6-241) we have

n) C coscp 1 - sincp* c* = (1 - Jt 1 - sin cp cos cp*

n) C coscp (sec cp * - tan cp *) = ( 1 - Jt 1 - sin cp

(6-242) The relation between the effective internal friction angle cp* and the undamaged (virgin) internal friction angle cp can be regressively expressed in a non-linear form as the power function by material parameters A and n from the tested data of effective internal friction coefficients, 1* = tancp* , and undamaged internal friction coefficients, f = tan cp, for a damaged porous media as tancp*

= A(l -
[!~) n tancp

The expression of Eq.(6-243) is correct only when [! parameter A has to be defined in the non-linear form as

A- {

I

i-

[! = O

'

constant, [! > 0

(6-243)

0, otherwise the

(6-244)

The relational equations among the effective internal friction angle cp*, the damage variable [! and the effective cohesive strength c* can be obtained combing Eqs.(6-240), (6-242) and (6-343) as the non-linear Eq.(6-245) Tn

1 _ [! = c c*

*

+

+ [!P * 1 _ [! tan cp

(Yn

= (1 - [!)c cos.cp (sec cp* - tan cp* ) 1 - smcp

tan cp*

= A(1_

(6-245a)

~

[! ) n t an cp

The material state variables defined as the effective shear strength (c*, cp* ) and the damage state variable [! can be determined from the non-linear equations

6.5 Non-linear Brittle Damage Model of Porous Media

423

under given pore pressure P and Cauchy stresses Tn , a n on the failure plane with the initial shear strength (c, cp) and tested material parameters A, n of undamaged materials. The evolutionary equations of the effective material state variables (c*, cp* , D) can be obtained by taking a derivative for Eq.(6-245a) with respect to time, which gives

C* = -c

n cos. cp (sec cp * - tan cp *) Jt 1 - SlllCP*

+ (l - D) c sec 2 cp*cp*

=

3

co~cP

1 - SlllCP* 1

(seccp*tan cp*-sec 2 cp* )cp* 3 n-l

- -nD"2(l - D"2) 2

.

Atan cp D

(6-245b) Solving the non-linear ordinary differential equations (6-245b), the evolutionary process rule of the effective material properties D(t), c* (t), cp* (t) during the damage development and the pore-pressure variation in the modified Mohr-Coulomb brittle damaged porous media can be analyzed at an observed point under given undamaged material constants of (c, cp, A , n), and loading conditions of stress state Tn,a n stress rate Tn/Tn and pore-pressure state and rate P , p with the necessary initial conditions, all of which can be obtained from structural analysis effortlessly 6.5.2.3 Brittle Damage Model Based on Effective Cohesive Strength In some cases, the effective cohesive c* is the major factor in damage evolution in the effective shear strengths for the damaged cohesive materials, which means the failure property mostly depends on the evolution of cohesive materials, which means the failure property mostly depends on the evolut ion of cohesive, and the influences of variation of the internal friction angle cp on the damage property can be neglected during material damage failure. On the other hand , sometimes the internal friction angle in cohesive porous medium is an invariant constant during damage or fracture due to the effect of static pore water. The concept of effective cohesive strength must be considered as the principal factor when a crack-damaged porous medium with static pore pressure is analyzed by the modified Mohr-Coulomb brittle damaged porous model. So, the effective cohesive strength c* is the major state parameter of material strength concerning porosity, pore pressure and damage variable. The effect of pore pressure on the effective cohesive strength can be assumed to be a function of porosity