Plastic Flow and Fracture in Solids. (Mathematics in science and engineering, Volume 2)

Plustic Flow AND

Fmctnre in Sokds TRACY Y. THOMAS Graduate Institute for Mathmatics and Mechanics lndiana University, Bloomington, Indiana and

Applied Mathematics Staf US. Naval Research Laboratory, Washington, D.C.

1961

New York

ACADEMIC PRESS

London

COPYRIGHT @ 1961, BY ACADEMIC PRESSIsc. ALL RIGHT8 RESERYED

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3. N. Y.

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Preface In the following book* we have given an account of plastic flow and fracture in solids, based to a large extent on work which has been done over the last five years on the growth and decay of discontinuities in continuous media. We believe that this book will be suitable for a course at the graduate level for students of applied mathematics and engineering whose interest is in the mechanics of solids. A convenient source of reference to the basic results in tensor analysis and differential geometry, which we have presupposed, is offered by our recent survey Concepts f r o m Tensor Analysis and Diferential Geometry, Volume 1 of this Series. Specific references to this survey have been made when it was felt that such references might be helpful to the reader. In the first chapter we have discussed the tensor invariants of the continuous medium and have derived the fundamental equations of continuity and motion as well as the general dynamical conditions for discontinuity in density, velocity and stress over a surface. The theory of geometrical and kinematical conditions of compatibility which is developed in Chapter I1 underlies most of the following work and it would appear advisable to master this chapter thoroughly before proceeding to the other chapters. As a first application of the conditions of compatibility we have treated the problem of the decay of waves in elastic media in Chapter 111. We believe that Chapter IV will be of considerable interest to most readers. In this chapter we have derived the various constitutive equations for a perfectly plastic solid from a set of assumptions characterizing the plastic deformations. This presentation of the theory automatically provides a yield condition involving an arbitrary material function from which the usual Tresca and the quadratic or von Mises yield conditions are obtainable as special instances. To satisfy the basic dynamical requirement of the invariance of constitutive equations under transformations relating arbitrarily *The writing of this book has been supported in part by the Office of Naval Research under Contract Nonr-908(09), Indiana University, NR 041 037 and in part by the National Science Foundation through Grant NSF-G14506 to Indiana University. V

vi

PREFACE

moving coordinate systems, we have developed a theory of combined time and coordinate differentiation, analogous to the ordinary covariant differentiation of differential geometry, by which invariance under such transformations is preserved; use of this theory will permit the direct derivation of constitutive equations of the correct dynamical form, i.e., those which possess the required property of invariance under time dependent coordinate transformations. Chapter V deals mainly with characteristic surfaces in perfectly plastic solids, the interpretations of these surfaces as wave surfaces, and problems involving the propagation of plastic into elastic regions. The problem of the determination of possible surfaces of fracture in a perfectly plastic solid is treated in the final chapter on the basis of the concept of the fracture surface or surface of instability over which, by hypothesis, an initial slip of the material particles will fail to be damped out as a consequence of the equations governing the behavior of the medium and the pertinent symmetry and boundary conditions. Fracture surfaces are determined for flat plates and round bars under tension, for round bars under pure torsion, and for circular cylinders subjected to both tension and internal pressure. In all cases good qualitative agreement has been found between the observed and predicted surfaces of fracture. The work on fracture surfaces in closed circular pipes under internal pressure and axial tension, which was carried out by my former student Dr. T. W. Ting, enables us to locate the point on the load diagram at which the fracture surfaces ~ i lchange l abruptly from longitudinal to transverse planes or vice versa; this work should be well received because of the excellent agreement between the calculated and the measured fracture loads for pipes of various materials and dimensions as shown by the graphs at the end of Chapter VI. It is hoped that the discussion in this chapter may remove any of the misconceptions which may have arisen from the publication of the individual results at various times during the formative period of this theory. We have attempted to give an objective account of the topics treated and to meet a proper mathematical standard. Engineers and serious students of the theory of plasticity may therefore find this volume of genuine interest and possibly of some aid in the solution of their problems. T. Y. THOMAS Los Angetes April, 1961

I. Basic Invariants in the Mechanics of Continuous Media. Equations of Continuity and Motion 1. CONTINUOUS MEDIA

A solid, liquid, or gas is considered from the present standpoint as forming a continuous medium. The position of the particles of this material medium can be represented by equations of the form xi = 4y?t,t)l (1.1) where the xi and ziare coordinates of the same rectangular system and t denotes the time; a particle, initially a t the point P with coordinates zt,will be located a t the point P with coordinates xi a t time t in accordance with the relations (1.1). It will be assumed that the relations (1.1) have a unique inverse a t any time t, and also that the 4iare continuous and differentiable functions of the initial coordinates ziand the time t. Since the correspondence x t-f 2 defined by the above equations is (1,l) it follows that the inverse functions lLi(x,t), by which the coordinates 2*are expressed in terms of the coordinates xi and the time t, must have similar properties of continuity and differentiability. Moreover, the functional determinants 1 axi/dPl and lazi//axkl must be different from zero everywhere; hence these determinants must be positive if we assume that the correspondence x t-f Z reduces to the identical transformation at the initial time. The deformation P + P with components x i - zi can be represented by xi - = Ui(Z,t), (1.2) when we replace the T iby their values in terms of xi$ as given by the relations inverse to (1.1). Similarly the velocity v has components u i given by v*.

=

a4i - Vi(X,t), at

1

(1.3)

2

I. BASIC INVARIANTS. EQUATIONB OF CONTINUITY AND MOTION

in terms of the coordinates xi and the time t. Use of the instantaneous or Eulerian coordinates xi by which we have expressed the deformation u and velocity ZJ in (1.2) and (1.3) will be continued in the following discussion, i.e., all quantities which enter into consideration will be referred to these variables rather than the Lagrangian coordinates 2' which give the initial position of the material particles. We shall use Latin letters in the following, as in the above discussion, for indices which have the values 1,2,3. Greek letters will be assigned the values 1,2 only and will usually appear as indices of quantities intrinsically associated with surfaces in the space. The summation convention will be employed with respect to both Latin and Greek indices. 2. RIGIDDISPLACEMENTS AND THE DEFORMATION TENSOR

Let Pl and Pz be two points of a material medium, referred to a system of rectangular coordinates x iand let xi and xi denote the coordinates of these points. Now the expression $, =

(xi - d)(% - xi)

(2.1) gives the square of the distance between the points PI and Pf. Using this determination of distance let us say that a deformation P --+ P of the medium (see Sect. 1) is a rigid displacement if the distance between the points P1and Pz into which arbitrary points PIand PI are and Pz. displaced, is the same as the distance between the points Suppose that a point P goes into a point P as the result of a rigid displacement; denote by ziand xithe coordinates of F and P respectively. If we consider the coordinate axes to be fixed in the medium during this displacement, then the coordinates of P will evidently have the values Zirelative to the displaced coordinate system. Denoting this latter system by the relation between the original coordinates x i and ziwill be that due to a proper orthogonal transformation x t-)3 between the coordinates of the x and z systems. Hence we can write =

.:xi

+ bi,

(2.2)

where

la:\ = 1, (2.3) in which the quantity lajl denotes the determinant of the coefficients u!uk =

6jh;

2.

RIGID DISPLACEMENTS AND DEFORMATION TENSOR

3

a: in these relations. The coefficients a: and bi in (2.2) can be considered to be functions of the time t in which case the equations (2.2) will describe a rigid motion of the medium (see Sect. 4); however, in the immediate discussion we shall assume the a: and bi to be constants SO that the equations (2.2) give the relation between the coordinates xiand ~i of the final and initial positions of points in a medium which has been subjected to a rigid displacement. From the equations (2.2) we now have u1

=

x 2'

-

-z x. - (6; -

- bi.

(2.4)

Hence, by differentiation of (2.4), we find that - 63i - u;. -

(2.5) Substituting these values of the a's into the first' set of equations (2.3), t.he resulting equations are seen to reduce to ad j

ui,j

+ uj,i -

Um,iUm,j =

0,

(2.6)

where we have taken the liberty of lowering the index on the components uisince there is no distinction between covariant and contravariant indices relative to rectangular coordinate systems; such changes in the position of indices will hereafter be made in the following discussion, when desired, without special mention. Conversely, suppose that the deformation u has components ui(x) which satisfy the equations (2.6). Differentiating (2.6) we obtain Ui,jk

+

uj,ik

- uk.il - Um,ikUm,j

Uj.
- Um,ikUm,j - um,SuB.jk

Hence

+

Um,ij%n,k

=

0-

(2.7)

=

0,

(2.8)

when we interchange j and k in (2.7) and subtract corresponding members of the resulting equations. Again, interchanging i and k, the relations (2.7) become uk,ji

+

Uj,ik

- u m , k u m , j i = 0.

(2.9)

- U m , j ) U m , i k = 0.

(2.10)

- Um,ikUm.j

Adding (2.8) and (2.9) we find that uj.ik

- %m.jUm,ik =

(&nj

But since u defines a deformation by hypothesis it follows, from the assumptions in Sect. 1, that the determinant ) 6 , j - u m , , l is different > 0 since the deformation from zero; in fact we must have /arnj -

4

I. BASIC INVARIANTS. EQUATIONS O F CONTINUITY AND MOTION

under consideration can be regarded as the end result of a motion for which urn,?= 0 initially. Hence from (2.10) we obtain U i , j k = 0. (2.11) Integrating (2.11) we can write ui

=

(8; - a:).'

-

bi,

(2.12)

where the a: and bi are constants. Substituting these values of the ui into (1.2) we have 2' = a '.: + bi, (2.13) as the equations relating the initial and final positions of the particles. But by differentiation of (2.12) we obtain (2.5) and hence we find t,hat

a:ai

=

8jk

- uj.k - u k , j

+

U;,jUi,k.

Hence %at t t -

8jk,

(2.14)

on account of (2.6). Finally, we see from (2.14) and the fact that u,., = 0 initially, as above mentioned, that la:\ = 1 for the deformation (2.13). Hence the conditions (2.3) are satisfied and it follows that (2.13) gives a rigid displacement of the medium. To state the above result let us put Dij

=

t(ui,j

+

uj.i

- uk.iuk,j).

(2.15)

The tensor D whose components are defined by (2.15) is called the deformation tensor. W e have thus shown that a deformation u will be a rigid dispaaeement i f , and only i f , its deformation tensor D i s equal to zero. In arriving at this result we have introduced the second partial derivatives of the components u;(x) of the deformation; obviously the explicit assumption of the existence of these derivatives is not necessary and has been made only for the purpose of giving a simple formal demonstration of the result. If the deformation u is sufficiently small, products involving the squares of the components ui(z) and their derivatives can be neglected; in this case we can replace the tensor D by the tensor e defined by e 13. . =

1(u. . 2

tsJ

+ uI.. *.).

The tensor e is called the strain tensor and is one of the fundamental invariants of the linear or classical theory of elasticity.

3.

5

THE DISTORTION TENSOR

3. THEDISTORTION TENSOR Let v1 and v 2 be two unit vectors a t a point P of the material medium and let t9 be the angle determined by these vectors. We may think of v1 and v 2 as tangent vectors to curves C1 and Cz passing through P. When the medium is deformed this configuration will be transformed into a corresponding configuration which can be indicated by writing

P.+ P';

c1.+

v1-9

c;;

v;;

v2.+

c2.+ c;;

v;,

t 9 4 t9'.

Now if we assume that the deformation is sufficiently small (see statement a t end of Sect. 2), it can be shown that the angular change At9 = 8' - t9 is given by sin t9 At9

(e$Yid

+ eGv(2d)cos t9 - 2e$v\d,

(3.1) where v: and v(2 are the components of the above vectors v1 and vz respectively and e& = ei, - &&j. (3.2) =

It is readily observed that the relation (3.1) is equivalent to the relation obtained by replacing the quantities ezj by the components eij of the strain tensor defined in Sect. 2. For a derivation of this latter relation the reader is referred to A. E. H. Love, The Mathematical Theory of EZasticity, Cambridge, 4th Ed., 1927, p, 62; also S. Timoshenko, Theory of Elasticity, McGraw-Hill, New York, 1934, p. 192. The tensor e*, defined by (3.2), is called the distortion tensor. If this tensor vanishes in the medium, then At9 = 0 for arbitrary directions v1 and v 2 a t points P and hence any geometrical configuration C will be deformed into a configuration C' without local change of shape; i.e., without distortion. Conversely, suppose the deformation produces no distortion of geometrical configurations, i.e., At9 = 0 for arbitrary directicns v1 and v 2 a t points P of the medium. Then, a t any selected point P , we have (e;v\yfi

+ e:p\d)

cos e

=

2e$d,

(3.3)

for arbitrary unit vectors v1 and vz a t P. Taking v1 perpendicular to so that cos 0 vanishes we obtain ec;v:yi = 0,

v2

6

I. BASIC INVARIANTS. EQUATIONS OF CONTINUITY AND MOTION

from (3.3). If we now choose V I to be a vector in the direction of the xk axis, and v2 a vector in the direction of the xm axis, ( k # m ) , it follows from the above equation that

eEm = 0, ( k # m). (3.4) Next choose v1 in the direction of the positive x1 axis and take v2 not a t right angles to a coordinate axis; then cos e = v:vi= v+ # 0. (3.5) Also for this selection of the vectors v1 and v2 we readily see that the above equation (3.3) reduces to e;pid = e f l , (3.6) Similarly, taking V I to lie when account is taken of (3.4) and (3.5). along the positive x2 and x3 axes in turn, we obtain

&id

e&,

(3.7) ~ Z ~ V Z VZ e&. (3.8) Adding corresponding members of the three equations (3.6), (3.7), and (3.8) it now follows that * f j esjv2v~ - 0. (3.9) But the relation (3.9) must obviously hold for arbitrary vectors v2 if it is valid for unit vectors v2 as above selected. Hence we must have eij = 0, i.e., the distortion tensor e* must vanish at arbitrary points of the medium. The above result can now be stated as follows. A small deformation u oaf the medium will produce n o distortion i f ,and only i f ,the distortion tensor e* i s equal to zero. =

* i j -

4. RATEOF DEFORMATION AND RATEOF DISTORTION TENSORS Consider a continuous rigid displacement of the medium as represented by t.he equations (2.2) in which the coefficients a: and bi are functions of the time t, i.e., a: = aj(t) and bi = bi(t). We assume the functions aj(t) and bi(t) t.o be differentiable and such that bye)

0, (4.1) i.e., the displacement begins a t time t = 0. Now differentiate the relations (2.2) with respect to t ; this gives aj(0) = 8;:

+

+ l;i

=

=

0,

(4.2)

4.

RATES OF DEFORMATION AND DISTORTION TENSORS

7

where the dot denotes differentiation with respect t o the time. Then, differentiating (4.2) with respect t o the coordinates xk, we find that

+ ci;

0. (4.3) Hence, multiplying (4.3) by ah and summing on the repeated index m, we have ajvj,k

=

on account of (2.3); the second set of these equations is obtained by interchanging the indices Ic and m in the first set of the equations. Now, when the first set of equations (2.3) is differentiated with respect t o t, we obtain ciiu', a:& = 0. (4.5) Adding the two sets of equations (4.4) and using (4.5) it therefore follows that vi,j vj,i = 0. (4-6) Conversely, consider a continuous deformation of the medium for which the velocity components vi(z,t) of the material particles satisfy the equations (4.6). Then, from (4.6), we obtain

+

+

+

0,

(4.7) by coordinate differentiation. Interchanging the indices j and k and subtracting corresponding members of the resulting equations and the equations (4.7), we find that Visjk

d (vj,k

vj,ilc

=

- 2)k.j)

=

0.

(4.8)

Hence the quantities in the parentheses in (4.8)are independent of the coordinates, and we can therefore write

v . . - v . . = Zw..(t) %I 7 332

293

(4.9)

where the w i j are skew-symmetric quantities which depend only on the time t. Combining (4.6) and (4.9) we now have vi,j

=

Uij(t).

(4.10)

Hence, by integration, it follows from (4.10) that ~i

= wi3xi

+ bi,

(4.11)

8

I.

BASIC INVARIANTS. EQUATIONS O F CONTINUITY AND MOTION

where the bi are at most functions of the time. Now differentiate the quantity .$,defined by (2.1), to obtain

in which the xi and ziare considered, for the moment, to be the coordinates of two arbitrary moving points P and P in the medium. When we eliminate the vi in (4.12) by the substitution (4.11) and the Viby the corresponding equations for the velocity of the material point P , we find that

in view of the skew-symmetric character of the quantities w i j . But this means that, at any time, the deformation under consideration is a rigid displacement (see Section 2) and hence can be represented by the equations (2.2) in which the coeffcients a: and bi are functions of the time. Let us say that the equations (2.2) define a rigid motion of the material medium when the coefficients a: and bi are functions of the time t as in the above discussion. Writing €.. 2)

= "z 8 . C J .

+ 21. .), 3.t

(4.13)

the following result can now be stated. A deformation of the material medium will be a rigid motion i f , and only i f , the components vi(x,t) of the particle velocity, produced by the deformation, are such that the tensor E i s equal to zero. The symmetric tensor E , whose components are defined by (4.13), is called the rate of deformation tensor or the rate of strain tensor. In proving the above italicized statement we have assumed, for simplicity in the demonstration, that the velocity components v;(x,t) have partial derivatives of the second order with respect to the coordinates. The deformation u produced in the medium in a small time interval At, i.e., as t varies from t to t At? has components given by vi(z,t)At to a first approximation. Replacing the components ui which occur implicitly in the approximate relation (3.1) by the quantities vi(x,t)At, dividing through by At, and passing to the limit as At -+ 0, we thus obtain the exact equation

+

d6

+

sin 8- = ( ~ ; ~ l l d ~;,&4) cos e - 2 & : 4 dt

(4.14)

5.

FUNDAMENTAL DYNAMICAL ASSUMPTIONS

9

where E:j

=

-

$Ekk6ij1

(4.15)

for the time derivative dO/dt of the angle 0 determined by the vectors v1 and v 2 in Sect. 3. On the basis of our previous discussion of the relation (3.1) we can immediately state the following result. A necessary and sufieient condition for a deformation of the medium to produce n o distortion i s the vanishing of the tensor E* whose components are determined by the components vi(x,t) of the particle velocity, caused by the dejormation, i n accordance with the equations (4.15). The tensor E* is called the rate of distortion tensor. Remark. The above tensors e* and E* are sometimes referred to as deviators; more precisely the tensors e* and E* are called the deviators of the tensors e and e respectively. In general, the deviator T* of a symmetric tensor T is defined by

T'ij - IT 4.. - T.. (1 3 k an where the T i j are the components of the tensor T. Deviators will be found to play an important role in the theory of plasticity.

5. FUNDAMENTAL DYNAMICAL ASSUMPTIONS Consider a finite volume V of a material medium and denote by S the boundary or surface of V . We suppose that V is a continuous (1,l) map of a cube and that S is composed of a finite number of surface elements each of which is regular in the sense that it can be represented by an equation of the form I$(x*,x2,x3) = 0,

where +(x)is a continuous and differentiable function of the coordinates; t,hese conditions will be retained under the deformations of the medium which enter into consideration (see Sect. 1). We now make the following dynamical assumptions. (a) The rate of change of the momentum of V in a n y fixed direction i s equal to the component, in this direction, of the total external force acting o n

V. (b) T h e rate of change of the angular momentum of V about a n y fixed line i s equal to the moment of the external forces acting o n V about this line. We also assume the law of the conservation of mass which states that

10

I. BASIC INVARIANTS. EQUATIONS OF CONTINUITY AND MOTION

the mass of the moving volume V remains unchanged under deformations of the medium. This condition can be expressed by writing

$

PdV = 0,

where p is the density or mass per unit volume at points of the medium and the integration is over the volume V . The assumptions (a) and (b) will be satisfied in full if the conditions involved are expressed for each of three fixed and mutually perpendicular directions in space. We take these perpendicular directions to be the axes of the rectangular coordinate system x to which the position of the material particles is referred. Relative to these coordinates let X ibe the components of an external or applied force per unit mass acting on V . Also let T ibe the components of a surface force per unit area, acting on V , over the surface S. The momentum per unit volume in the medium has components pvi where v denotes the velocity of the material particles. We also recall that the moment of a force F about the xi axis is given by Mi = eijkXiFk, where the Fkare the components of F and the quantities eilk are defined by the following two requirements. First, e123 = 1 and second the ei,k are skew-symmetric; it is well known that these quantities e i , k are the components of a tensor under proper orthogonal transformations of the coordinates. Using these designations and assuming that X = 0, i.e., that no applied force acts in the medium, the conditions (a) and (b) have their formal expression in the following two sets of equations

Iv

pvi

$

dV

=

TidS,

peijkxivkdV =

6. EQUATION OF CONTINUITY

It can be shown that

(5.3)

6.

11

EQUATION OF CONTINUITY

under deformations of the medium, where f is a continuous and differentiable function of the coordinates xiand the time t, and G is the coordinate velocity, i.e., the velocity relative to the fixed coordinate system, of the surface S along its outward normal. In the particular case for which the deformation of the volume V is determined solely by the motion of the material particles we have G = vivi where the v i are the components of the outward unit normal to S. For a derivation of the equation (6.1) the reader may refer to J. Hadamard, Cours d'dnalyse, Hermann and Co., Paris, vol. 1, 1927, p. 504. Take f = p and G = vivi in (6.1); then

on account of (5.1), when use is made of Green's theorem and the comma in the last integrand denotes coordinate differentiation. Since V is arbitrary it follows that

The equation (6.2) is called the equation of continuity; it is equivalent to the condition (5.1) for arbitrary volumes V . A useful result can be derived from (6.1) and the equation of continuity (6.2). We have

Is

fvivi

dS

=

IV

(fvi),i

dV

=

Iv

f,ivi

dV

+ IV

fvi.i

dV.

(6.3)

But

from the equation of continuity (6.2). Hence from (6.1) we obtain

when we assume that G = vivi and make use of the above relations (6.3) and (6.4). Now, if we replace the function f in (6.5) by pf we see that

T h i s equation i s satisfied in a medium when the deformation of the volume V i s produced entirely by the motion of the material particles.

12

I. BASIC INVARIANTS. EQUATIONS OF CONTINUITY AND MOTION

Remark 1 . P u t f

=

1 in (6.1); then

provided the motion of the material particles is the sole cause of any change in the volume V . It is seen from the above equation that the rate of increase of V per unit volume a t points of the medium is given by the divergence u i , ; of the velocity. Volumes V will be unchanged by the deformation if the equation u i , i = 0, (6.7) is satisfied. The equation (6.7) is referred to as the equation ofincompressibility. If (6.7) is satisfied the equation of continuity (6.2) becomes dp/dt = 0 and hence the density p will remain constant following the motion of the individual particles. I n particular, if (6.7) holds and the density is constant throughout t,he medium a t some initial time t = 0, i t will remain constant over the medium during the deformation. Analogous results can be stated for small deformations u produced during a time interval dt. Thus, representing the components ui of such a deformation by vi(z,t) dt, we have u%., I. - vi,; at.

The equation (6.7) therefore implies = 0,

16.8) and conversely. Hence the deformation u will leave volumes unchanged, to the approximation under consideration if, and only if, the condition (6.8) is satisfied. Also if (6.8) holds the equation of continuity (6.2) shows that d p = 0, i.e., there is no change of density as a result of the deformation. Hence if u represents a deformation ojthe unstrained medium in which the density p i s constant and if the divergence of the deformation u is equal to zero, then the density throughout the medium will be unaltered, i.e., will have the same constant value, after the deformation. Remark 2. To analyze the instantaneous state of the motion of the medium in the immediate neighborhood N of an arbitrary point P , let us choose the origin of coordinates at P , for simplicity, and then consider the equations ui,i

+

U i ( 2 ) = Ui(0) fJi,](O)Zj, (6.9) which will give the velocity components u i ( z ) approximately in the neighborhood N . Now put (6.10) Ui,j(O) = A ; , Bi, j where (6.11) A t,i . - " 2 oi,,(O) ~j,i(O)l, Ai, = $uk.k(O)aij, (6.12) (6.13) Bii = t [ v i . j ( O ) - ~j,i(O)].

+

+ +

6.

13

EQUATION OF CONTINUITY

The quantities Aij and B;j are defined by the equations (6.11) and (6.13); the equations (6.12) can be considered as equations expressing the Aij in terms of the deviator components ATj (see Remark in Sect. 4). Making these substitutions the equations (6.9) can be written

+ Aljx'

+

&,k(0)xi Bipi. Hence the velocity of points in the neighborhood N can be regarded as the resultant of the following four types of velocity wi'ch components, denoted by u, in each case, given by the equations Ui(X)

= Ui(O)

(a) u;

=

u;(O);

(c)

ui

=

+Vk.k(0)xi,

(d) U; = B i j ~ i . (b) U; = A;#; The velocity of type (a) is a translational velocity of the particles in the neighborhood N and has as its components the constant values vi(0) appearing as the first term in the right member of (6.9). For the velocity of type (b) we have v;.j = ATj in N ; hence v;,i = 0 and consequently this velocity produces no change in volume by the result in the above Remark 1; however, the rate of distortion tensor E* will fail to vanish unless A t = 0 and hence any such motion will produce distortion in the neighborhood (see Section 4). In the case of the velocity of type (c) we have u1., j.

2

l3 yk,k(0)8ij;

Eij

= guk,k(O)&i;

E:j

=

0.

From the result in Sect. 4 this velocity produces no distortion in the neighborhood N . Volume changes, however, will occur since the divergence of the velocity has the value u k , k ( O ) which is necessarily different from zero in the case of an actual motion of this type. With regard to the velocity of type (d) let us observe first that the determinant IBijl = 0 because of the skew-symmetry of the quantities Bij. Hence B..ai = 0, at for some set of constants aj not all of which are equal to zero; all points on the straight line L defined by the equations = ais, in which s is a parameter, will therefore have zero velocity. Without loss of generality we may assume that the 2 3 axis lies along the line L since this condition can be imposed, if not initially satisfied, by a rotation of the axes about the origin. But then the quantities a1,aa,a3can be selected to have the values 0, 0, 1 and hence Bi3 must vanish. The velocity u is therefore given by equaitions of the form UI = Bia2; ~2 = --BI~x'; ~3 = 0, which represent, as is readily observed, an angular velocity of the neighborhood

N about the x 3 axis, i.e., the line L , of magnitude (Blzl. We have thus proved the following result.

14

1. BASIC INVARIANTS. EQUATIONS OF CONTINUITY AND MOTION

T h e instantaneous velocity of the particles in the immediate neighborhood of a point P of a material medium, undergoing a deformation, can be regarded as the resultant of (a) a translational velocity which i s the same for all points in the neighborhood, (b) a velocity producing distortion without change of volume, (c) a velocity producing change of volume without distortion, and (d) a n angular velocity of rotation about a straight line L through the point P. It is easily seen that the magnitude of the angular velocity of rotation about the line L is given by the scalar

d

m 42 ’

in which the quantities B,, are not restricted by the special choice of coordinates for which the z3axis lies along the line L, but can have the general values given by the equations (6.13).

7. THE STRESSTENSOR AND

THE EQUATIONS

OF

MOTION

One can readily establish that the above vector T giving the force per unit area or stress over the surface S , bounding the volume V in Sect. 5, has components of the form

Ti = ~ i j v ’ , (7.1) where the aij are, in general, functions of position and time; it is to be understood specifically that the unit normal v, whose components appear in (7.1), is directed outward from the volume V and that correspondingly the stress T , whose components are given by the above equations, is the stress on the outer surface S as opposed to the inner surface S which is contiguous to the volume V (see Remark 1). We can infer f r o m the relations (7.1) and the vector character of T and v that the quantities a,j are the components of a tensor. Making the substitution (7.1) the equations (5.2) and (5.3) become

(7.3) when use is made of (6.6) and Green’s theorem. But from (7.2) we have

7.

THE STRESS TENSOR AND THE EQUATIONS OF MOTION

15

since the volume V is arbitrary. When we perform the differentiations indicated in the equations (7.3) and take account of the skew-symmetric character of the quantities eijk it is found that these equations can be written

But the first integral in (7.5) vanishes on account of (7.4); hence the second integral in (7.5) must vanish and consequently we obtain

eijkujjk = 0, (7.61 when it is considered that the volume V is arbitrary. Taking i = 1,2,3 successively in (7.6) we find that these relations are equivalent to the condition that the quantities ui, be symmetric. We have thus proved the following result. The dynamical conditions (5.2) and (5.3) can be replaced by the equations (7.4) in which the quantities ail are the components of a symmetric tensor. The equations (7.4) are called the equations of motion and the tensor t~ having the components aij is called the stress tensor of the medium. Remark 1. The usual proof of the relations (7.1) involves the assumption of the equality of action and reaction in consequence of which the stresses Tl and T z on the two sides of the surface S are equal in magnitude but opposite in direction. How- Q 2 s4 FIG.1 ever, this can be shown by the use of the condition (5.2). I n fact let TI and T1 be the stresses on the two sides of a surface element 2 and let V be the volume of a small sphere which is divided by the element Z into volumes V1 and VP. (See Fig. 1.) Then, applying (5.2), we have

(7.7)

where S1 and SPare the portions of the spherical surface S bounding VI and V2 respectively, and the area of integration Z obviously refers to the portion of the surface element enclosed by the spherical surface; it is assumed, of course, that

16

I. BASIC INVARIANTS. EQUATIONS OF CONTINUITY AND MOTION

the stresses are continuous over the spherical surface S and on each side of the element Z. But, when corresponding members of (7.8) and (7.9) are added and use is made of (7.7), we obtain Since the area of integration Z is arbitrary, it follows that TI; = - Tti which is the relation under consideration. The remainder of the derivation of the equation (7.1) can be made by considering the stresses over the surface of a tetrahedron three of the faces of which are perpendicular to the coordinate axes. (See P. Appell, Trait6 de Micunique Rationnelle, Gauthier-Villars, Paris, vol. 3, 1928, p. 134.) Remark 2. The work done per unit time on a moving volume V of the material medium by the stresses acting over its surface S , when the motion of V is produced solely by the deformation of the medium, is given by

w=

Is

UijViYj

dS =

Iv

(UijVi),i

dV,

(7.10)

in which use is made of Green's theorem. Or, expanding the integrand in the right member of (7.10), we have

w=

IV

uii, jvi

dV

+ Iv

aijvi9 j dV.

(7.11)

Now, when we make the substitution (7.4) we find that (7.12) in view of the relation (6.6). Also putting p = -Ukk/3 and introducing the deviators a* and t* we see that

1-

uijvi,jdV

=

jV

€?jUtj

dV -

1

pvi,i dV.

(7.13)

'It may be observed that the integrand

in the right member of this equation can be replaced by the quantity &rii as a direct consequence of the definition of the deviator of a tensor. Making this substitution and also the substitutions (7.12) and (7.13), we find that the equation (7.11) for W becomes

The equation (7.14) is called the energy equation. The first term in the right member of (7.14) gives the rate of increase of the kinetic energy of the moving volume V . Because of the association of the deviator t* with distortion (see Sect. 4) and the association of the divergence vi,, with volume changes in the medium (see Remark 1 in Sect. 6), it is customary to interpret the second and third terms in the right member of (7.14) as the rates a t which work isdone by the stresses, acting over the surface S, in changing the shape and volume of

8.

DISCONTISUITIES

17

IN DENSITY, VELOCITY AND STRESS

V respectively. In particular the quantity +,, can be thought of as giving the rate of distortional work per unit volume and the quantity -pv;,, as giving the rate per unit volume at which work i s expended to produce a n actual change of volume. Remark 3. The quantity p which enters in the above Remark 2 is called the hydrostatic pressure; correspondingly a stress field is said to be due entirely to hydrostatic pressure if its components, relative to a system of rectangular coordinates, are given by cr,, = -pa,, where p is a scalar function of the coordinates and the time. Now suppose that the stress field is modified by the addition of a purely hydrostatic pressure 9, i.e., u,,-+ c?,~where (7.15)

as is seen immediately when use is made of the relations (7.15). Hence we can state the following result. The addition of a purely hydrostatic pressure to a stress Jield produces n o change in the stress deviator.

8.

DISCONTINUITIES DENSITY, VELOCITY,. AND STRESS

DYNAMICAL CONDITIONS FOR

IN

Suppose that a volume V , whose motion is determined by the deformation of the material medium, is divided by a moving surface Z ( t ) into two volumes V1 and 7% (see Fig. 2 ) . Denote by XI and SZ the portions of the surface S of V which form parts of the boundaries of V1 and Vz respectively; the remaining part of the furnished boundary by the of Vl surface and VZ Z(t).willThe be Qz(t, normal component of the velocity of V a t points of its surface X is given by v, = vivi since the variation in the volume V is produced by the moving particles of the medium by hypothesis. FIG.2 Let G denote the normal velocity of Z(1) along the outward normal to Z(t) considered as part of the boundary of V1; then -G will be the corresponding normal velocity when I; is taken as a part of the boundary of V z . From the relation (6.1) we now have

18

I. BASIC INVARIANTS. EQUATIONS OF CONTINUITY AND MOTION

where the designation Z as the area of integration in these equations (as well as in following equations) naturally refers to the part of the surface Z bounding the volumes V1 and V z . We shall be concerned essentially in this section with functions f(x,t) admitting possible discontinuities over the surface Z(t). Thus, let us denote by fl the value o f f on the side of Z bordering V1 and by fi the value of this function on the side of 2 bordering Vz. Combining the above equations (8.1) and (8.2) we can therefore write

For convenience of terminology we shall refer to the side of Z bordering V1 as the side 1 and to the side bordering Vz as the side 2 of Z. Then, putting f = p(x,t) in (8.3) and taking account of (5.1), we have

where p1 and p z denote the values of the density on the sides 1 and 2 of 2. Now let V approach zero a t a fixed time t i n such a way that it will pass, in the limit, into a part Zo of the surface Z. The volume integral in the above equation will evidently approach zero; but

where vln and vzn denote the normal components of the particle velocities on the sides 1 and 2 of Z along the normal direction from side 1 to side 2. Hence we obtain /a P1(Vln - G) dS -

1%

P ~ ( U Z-~

G) dS

=

0,

and since this condition is independent of the extent of the surface of integration 20it follows that Pl(V1n - G) = Pz(Vzn - G),

(8.4)

8.

DISCOXTIKUITIES I N DENSITT, VELOCITY .4SD STRESS

19

over X ( t ) . This is the first of our discont.inuity conditions. It is seen t,o originate entirely from the principle of the conservation of mass. We now consider the relations (5.2), i.e.,

when account is taken of the equation (7.1). Allowing the volume V to approach zero as in the derivation of the condition (8.4), it is easily seen that

where ulij and mi, are the components of the stress tensor on the sides 1 and 2 of Z and the vi in the limit integral are the components of the unit normal v directed from the side 1 to the side 2 of 2 . If we put f = pvi in (8.3) we see that the left member of (8.5) becomes

where vli and v2i are the velocity components on the sides 1 and 2 of X . On passage to the limit the above sum of integrals reduces to

Hence, substituting the limit in (8.6) and the sum of integrals (8.7) for the terms comprising (8.5), and making use of (8.4),we are led to the following conditions

- 6‘)[oil,

(8.8) where the quantities [uij] and [vi] represent the discontinuities in the stress and velocity across the surface Z(t), i.e., [ ~ i j l v ’= ~l(v1n

[Uij]

= u2ij

- Ulij,

[Vi]

= v2j

- Vli.

It can be shown that the relations (5.3) give no conditions beyond those given by (8.4) and (8.8). Hence (8.4) and (8.8) constitute all the conditions, which follow from the general dynamical assumptions in Sect. 5, for discontinuities in the density, velocity, and stress across the surface 2. Remark. The fact that (5.3) provides no new conditions can be shown very simply as follows. Using the relation (8.3) we find, as in the above discussion, that the left and right members of (5.3) can be reduced to

20

I. BASIC INVARIANTS. EQUATIONS O F CONTINUITY AND MOTION

respectively. Equating (8.9) and (8.10) and making use of the condition (8.4), we obtain eijk%’{pi(Ui,

- G)[vr] - [ukm]Vm}

=

0,

on the surface2. But these equations are satisfied automatically in view of the conditions (8.8).

9. PRINCIPAL STRESSES AND DIRECTIONS

Let 2: be a plane surface element containing an arbitrary point P of the material medium and denote by v the unit normal vector to Z a t P. The side of departure from Z when one moves in the direction of the normal v at P, will now be called its positive side. Then the components T i of the stress T on the positive side of 2 (hereafter referred to simply as the stress on 2 for brevity), due to the action of the material in the immediate vicinity of this side of the surface, will be given by a i , v j in conformity with the designation of such stress in Sect. 7. The scalar aijvivj will give the value of the normal stress N on Z a t the point P. Also, using the letters N , T , and S to denote the magnitude of the normal stress N , the stress T and the shearing stress S, i.e., the projection of the stress T on the plane Z, we have the relation S2

=

T2 - N2,

(9.1)

where

N = L T ~ ~ v ~ v ~T, 2 = ~ i j ~ i & v k . (9.2) The direction determined by the above vector Y (or the opposite direction) is called a principal direction a t the point P if the stress T is normal to the plane 2 , i.e., if the shearing stress S vanishes on 2. When v determines a principal direction the normal stress N is called a principal stress a t P and the plane 2 is called a principal plane. Assuming that v determines a principal direction we have a& = T V where ~ 7 is now used to denote the principal stress. Hence we can write (aij - 76ij)v’ = 0; (9.3) these are three linear and homogeneous equations in the components v j and since not all of these components can vanish, we must have

21

9 . PRIXCIPAL STRESSES AND DIRECTIONS 611 /Qij

- T & ~ /=

- 7-

~2~ Q31

612

422

-7

6 32

813 623

683

-

= 0.

(9.4)

22

I. BASIC IKVARIANTS. EQUATIOXS O F

COATINUITY AiSD MOTIOX

Use will be made of these equations in the problem of showing that if the principal directions V I , V Z , V ~are not necessarily mutually perpendicular at a point P of the medium, they can always be selected so as to be mutually perpendicular. The discussion of this problem will be based on the following three cases.

Equation (9.4) has simple roofs, Case 11. Equation (9.4) has a double root, Case 111. Equation (9.4) has a triple root. Case 1.

Case

I

For this case the equations (9.6) give

vtvk = 0,

(k # na).

But this means that the three vectors vl, v2, and v3 are mutually perpendicular. :Iforeover, the components Y:, v i and v\ of the unit vectors v1,v 2, and v3 giving the principal directions at P are determined to within algebraic sign. To show this let us denote by w:,wi, and w\ the components of any other set of three mutually perpendicular unit vectors which give principal directions at the point P. Then from the equations (9.5) and the corresponding equations for the components w i we readily find that U,,Wkd, . I ' = T"v~,) =

!w,"vjiu

TkVLWk.

Hence, subtracting corresponding members of these equations, we obtain (7k

- T,)WZ,V:,

=

0,

from which it follows that the quantities wivk vanish for k # m. Thus if k = 1we see that the vector w1is perpendicular to the vectors vz and va. Hence w\ = &v:. Similarly wi = =tvi and w$ = f v $ ; this completes the proof of the above italicized statement. Case I1

To treat this case let us put 6.. 23 -

where

T

g.

- T6.

(9.7) is considered, for the moment, as an arbitrary tariable; let us 11

111

9.

PRINCIPAL STRESSES AND DIRECTIONS

also denote by Oii the cofactor of the element loij!. Then

eii

23

in the determinant

0 =

thiefk = es:; (uij - d i i j ) e j k = es:, (9.8) where the second set of these equations results from the first set by the substitution (9.7). Now differentiate the equations (9.8) with respect to r ; representing such differentiation by the use of a bar for brevity, we thus obtain 8.z, Bik - eik = &tL. (9.9) Then, multiplying the two members of these equations by eim, summing on the repeated index i, and taking account of (9.8), we deduce @k@m

= eekm

- &km.

(9.10)

e

Now suppose 71 = r 2 # r3for definiteness; then 0 = = 0 for r = 7 1 since r1 is a double root of (9.4). Hence the left member of (9.10) must vanish for r = 7 1 and hence eii = 0 for r = rl. All second-order determinants in the matrix l\fliJl!therefore vanish for r = r1 and hence, for this value of r, the rank of the matrix \l&,lj i s 5 1. But the rank of the matrix lleijll cannot be zero for r = T~ since this would imply that 71 is a triple root of the determinantal equation (9.4) contrary to the hypothesis of this case. Hence 1 j&,I 1 has rank 1 and the equation (9.4) has two independent solutions v: and vk for r = r1 in terms of which any other solution is linearly expressible; any vector v in the plane of the vectors VI and v2 will therefore give a principal direction corresponding to the root r1 of the equation (9.4). Denote by v: a nontrivial solution of (9.4) corresponding to r = 7 3 . We now see that v:v;

=

0;

viv; = 0,

from the equations (9.6). Hence the vector v3 having the components v: is perpendicular to the plane determined by the vectors v1 and v 2 ; this implies that the components v; of the vector v3 are uniquely determined to within algebraic sign under the condition that v 3 is a unit vector. The following result has now been proved. T h e unit vector v 3 determines a principal direction, corresponding to the principal stress 7 3 , and its components v(3 are uniquely determined to within algebraic sign. A n y vector v perpendicular to the vector v3 determines a principal direction associated with the principal stress 71, which i s the double root of the equation (9.4).

24

I. BASIC INVARIANTS. EQUATIONS OF CONTINUITY ARID MOTION

Hence it is always possible to choose a set of three mutually perpendicular unit vectors v1,v2, and v3 giving priiicipal directions at the point P. Case 111

Using the same notation as in the discussion of the preceding case we observe (a) that O i j is linear in 7 if i # j , and (b) that Oii is a quadratic expression in 7 starting with the term T~ if i = j. Hence 8 = 26'i where we have used the double bar to denote the second derivative with respect to the variable 7 . Differentiating (9.9) we now obtain 2eik = 28'k

+ 8s:.

Multiplying both members of these equations by 0 t h and summing on the repeated indices, we find that the resulting equation can be written

2oikeik= 2eii

+ 68 + kii.

(9.11)

= 7 2 = 7 3 for this case. Hence 8 = e' = 0 for 7 = 7 1 . Also 0 for 7 = T~ as shown in the discussion of Case 11. It follows, therefore, that the right member of (9.11) must vanish for 7 = 7 1 , and hence all quantities %ij = 0 for this value of 7. H e w e a n y vector v determines a principal direction at the point P. Any set of three mutually perpendicular unit vectors will therefore give principal directions a t P and each of these vectors will have the triple root 71 of (9.4) as its associated principal stress.

But

V

71

=

Remark

I . Expanding the determinantal equation (9.4) we have (9.12)

where Zi, is the cofactor of the element uij in the determinant In<,/. Since the coefficients in this equation are scalars under proper orthogonal transformations, it follows that the principal stresses 7,determined as solutions of (9.12), do not depend on the rectangular coordinate system employed; more precisely we can say that the principal stresses T ~ r2, , and -r3 are scalar invariants of the stress tensor. I n particular, it may be observed in this connection that the sum of the normal stresses on any three mutually perpendicular planes at the point P is equal t o the scalar u k k , and hence equal t o the sum of the principal stresses on three mutually perpendicular principal planes. Remark 2. More generally the principal d u e s rl, . . . T,, of a symmetric tensor u in a Riemann space R of n dimensions and the vectors v1, . . . , vn giving the associated principal directions are obtained as solutions of the system of equations ~

10. CANONICAL $1

25

COORDINATES

- T g $1. . ) Y i = 0

(9.13)

where uij are the components of the tensor CT and the g i j are the components of the fundamental metric tensor of R ; the above indices i and j and all indices in the remainder of this Remark have values 1, . . . ,n and are to be summed over this range of values in accordance with the usual convention unless the contrary is stated. The treatment of the equations (9.13) is similar to that of the above equations (9.3) and the results obtained from (9.13) are analogous to those derived in our discussion of the principal stresses and directions in the material medium. Thus the principal values 71, . . . , T . are determined as solutions of the equation (9.14) IU,i - Tgijl = 0.

As so defined the principal values r are scalar invariants of the tensor u in the space R. All solutions T of the equation (9.14) are real and it is always possible to find a set of mutually perpendicular unit vectors vl, . . . , v, corresponding respectively to the solutions T I , . . . , T,, of (9.14) a t any point P of the space R ; i.e., such that (Uij

- 7kgij)Vtf = 0,

(9.15)

in which k is not summed, and where 1 1

gijvtvm =

(9.16)

8km.

Any vector v k of the set v1, . . . , vn is said to determine a principal direction of the tensor u. If T k is a simple root of (9.14) the components & of the associated unit vector v k will be determined to within aIgebraic sign; if is a multiple root of (9.14), e.g., a root of order m, the set of vectors vlJ. . , v, (chosen to be mutually perpendicular) will contain m vectors v each of which is associated with , any linear combination of these m vectors will also be a the multiple root T ~ and vector associated with ~h and will determine a principal direction of the tensor u. Finally, it is of some interest to observe that if we multiply the relations (9.15) through by vi and sum on the repeated index i we obtain, when account is taken of the orthogonality condition (9.16), the following equation for the principal value T k , namely

.

Tk

= ( ~ i ~ v i v i , ( k not summed).

(9.17)

10. CANONICAL COORDINATES

Let v1,vz,v3 be a set of three mutually perpendicular unit vectors giving principal directions at an arbitrary point P of the material medium (see Sect. 9). It is evidently possible t o choose these vectors SO t h a t the vector triad v1,vZ,v3will have the same orientation as the positive directions of the x1,x2,x3 axes. Then a rectangular coordinate

26

I. BASIC INVARIANTS. EQUATIONS OF CONTINUITY AND MOTION

system y having its origin at P and such that its positive y1,y2,y3 axes have the directions of the vectors v1,v2,v3 respectively, will be related t o the underlying x coordinate system by a proper orthogonal transformation x .++ y which is readily seen t o be given by the equations

+

(10.1) where the pi are the x coordinates of the point P and the coefficients vi are the components of the vector V k a t P relative to the x system; also the coefficients v i in (10.1) satisfy the relations &?:,= &m; \vil = 1. (10.2) The coordinates y k will be called canonical coordinates and the transformation (10.1) t o these coordinates will be called a canonicaE transformation for the purpose of reference. The equations (9.5), in which the principal values r m are scalars as observed in the Remark 1 in Sect. 9, are invariant under transformations of the type (10.1). Hence if we denote the components of the stress tensor by rij relative t o the canonical system y and take cognizance of the fact that the components v', have the values S', a t the point P , when referred t o the y systems, it follows immediately from the equations (9.5) that 7zl. . = r.8.. %,, (i not summed), (10.3) a t the origin of the canonical system. When dealing with problems involving the principal values of the stress tensor the use of canonical coordinates will frequently be found t o simplify our formulae.

xi = pi

viyk,

Remark 1. In the case of the Riemann space R (see Remark 2 of Sect. 9) we shall say that a system of coordinates yi having its origin at a point P is a canonical system for the tensor u if a set of mutually perpendicular vectors vl, . . . , vn which determine principal directions of u at P, are such that the vector vk has the direction of the positive y k axis at P fork = 1, . . . ,n. Denoting the components 2, by p i relative to the canonical system y it follows that 4

t

(10.4)

P t = 6t,

at the origin of the y system. Hence, if the components of the tensor u are denoted by r i i , and the components of the fundamental tensor of R are denoted by hij in the canonical coordinate system y, we see from the equations (9.15) and (9.16), when taken relative to the y system, that h%. I. = 6 .1.1.)

T . . --r . 6 . .

at the origin of the canonical system.

(i not summed),

(10.5)

10.

CANONICAL COORDINATES

27

Remark 2. I n any investigation of differential character involving principal directions it will be advantageous to have a formula which expresses the derivatives of the components v i of the vectors vk which give these directions in terms of the derivatives of the components uil of the basic tensor u. We shall now derive such a formula when the vectors vk determine principal directions of a tensor u in an n dimensional Riemann space (see Remark 2 in Section 9). The formula obtained will apply in particular to the special case for which the principal directions are associated with the stress tensor u of a material medium. -4ttention will be confined to the general case for which the roots T of the equation (9.14) are distinct; it will be assumed, for definiteness, that the roots T are labeled SO that Ti

> Tz > . . . > 7 ,

as is evidently possible. It will be assumed, furthermore, that the components uii and the components gii of the metric tensor are continuous and have continuous partial derivatives with respect to the coordinates xl,. . . ,zn of the allowable coordinate systems in R . Then the scalars ~k will be continuous and , ~ the are simple roots of an will have continuous partial derivatives T ~since algebraic equation, i.e., the determinantal equation (9.14), whose coefficients are continuous and have continuous first partial derivatives. Since the components 2, of the n mutually perpendicular unit vectors v1, . . . , v n which give the principal directions are determined as solutions of the equations (9.15), in which the uii and gij are continuous by hypothesis, it is evident that each vector v k can be chosen so as to form a continuous field in the immediate neighborhood N of an arbitrary point P of R ; indeed, i t is readily seen that this determination of the vectors v1, . . . , vn will be unique after the selection of these vectors has been made at the point P due to the fact, as mentioned in the Remark 2 of Sect8ion9. that the components v i of each vector v k are determined to within algebraic sign when the roots T of (9.14) are distinct. Let us now assume the existence of the partial derivatives of the components of the vectors v1, . . . , v, in the above neighborhood N ; this assumption will be justified later. Then, from (9.15) and (9.16), we obtain 1

(uij - Tkgij)Vt,m = (Tk.mgii

- b i j , n r ) Vjt ,

(10.6) (10.7)

where the comma denotes covariant differentiation based on the metric of the space R. Equations (10.6) do not suffice for the determination of the quantities v ; , since ~ the determinant of the coefficients of these quantities is equal to zero. But there will be, at most, one determination of the v ; , a~ t each point Q in the neighborhood 1V when the equations (10.7) are adjoined t o the equations (10.6). To show this, select a canonical coordinate system with origin at the point Q. Then, limiting our attention to those equations (10.7) for which k = s, we see that

28

I. BASIC INVARIANTS. EQUATIONS

(71

- 7k) 0 0

0 6;

0 (72

0 0

- 7k) 0 0 6;

OF CONTINUITY AND MOTION

(73

... ...

- 7k) 0 6:

...

0 0 0

...

...

.., ...

(7,

- Tk) 6;

.

11.

MAXIMUM NORMAL STRESS

29

when use is made of (9.16);hence the f f p k m must be skew-s-ymmetric in their first two indices. Now, substituting the above expression for the vi,, into (10.6) and taking account of the equations (9.15), we obtain (

~

-p T k ) g i i v f , a p k m =

(Tk.mgij

j - gij.rn)vt.

Multiplying these latter equations through by v6 and summing on the index i, it follows that (79

- 7k)aqkm = T k , r n s k q - f f ' i j , m Vs pjv t j

in which there is no summation on the indices k and q. Hence, replacing the index q in the above equations by the letter p , we have (10.11) Now, in the summation on the index p in the right member of (10.10) we can omit the value p = k since the quantities apkmhave been shown to be skewsymmetric in their first two indices. Hence, making the substitution (10.11) for the apkmin (lO.lO), we obtain (10.12) where, for any value of k, the summation on p is over all values of p from 1, . . . n with the exception of the value p = k. Conversely, it is easily seen from the derivation of the formula (10.12) that the values of the v ; , ~given by (10.12) will satisfy the equations (10.7) and also (10.6) when account is taken of the relations (10.9). It can be stated furthermore that the determination (10.12) of the components v;,, is unique since, as observed above, the equations (10.6) and (10.7) have a t most one solution.

11. MAXIMUM NORMAL STRESS Consider a plane element Z: a t an arbitrary point P of the material medium and let v be the unit vector normal to B at P. The element (or elements) B for which the normal stress N assumes its maximum value are of especial interest in the mechanics of continuous media. To simplify the problem of finding these planes B let us choose a system of canonical coordinates yi with origin at the point P (see Sect. 10). At the origin of the canonical system the stress N on Z becomes (11.1) N = T ~ ( v ~ )T~~ ( Y ' ) ~ T3(V3)2, from equations (9.2) and (10.3), where rl, r2, and 7 3 are the principal values of the stress at the point P and v1,v2, and v 3 are the components

+

+

30

I. BASIC INVARIANTS. EQUATIOXS OF CONTINUITY AND MOTION

of the unit normal to Z relative to the canonical system; also, since v is a unit vector, we have

+ (v’)Z + (v”)” = 1.

(v’)Z

(11.2)

Denoting by 6N the variation of the normal stress produced by a small variation 6vi of the components of the unit normal V , we find that

6N

=

2[71V16V1 V

W

+

+

TZV’~V’

v26v2

+

+

T3V36V3]

=

0,

(11.3) (11.4)

= 0,

v36v3

from (11.1) and (11.2). Specifically, the condition (11.3) must be satisfied for any variation 6vi which satisfies (11.4) if L: is an element of maximum normal stress. Conversely, if (11.3) holds for the plane 2 whenever (11.4) is satisfied, then the normal stress N on Z has a stationary value; from among these stationary values one can readily select the absolute maximum value of N . Combining the equations (11.3) and (11.4) we can write (71

-X)vW

+

(72

- X)v26v2

+

(73

- X)v36v3 = 0,

(11.5)

where X is an arbitrary variable; obviously the conditions (11.4) and (11.5) can replace the conditions (11.3) and (11.4). Now suppose v 3 # 0 and take X = r3. Then we see that (71

- 73)Y16V1 f

(72

- 73)V26V2

=

0,

for arbitrary variations 6v1 and 6 9 . Hence (71

-

73)V’

=

0;

(72

- 73)V’

0,

(V3

#

o),

(11.6)

for any plane 2 such that v 3 # 0 and for which the normal stress N has a stationary value. Similarly we have (71

-7

(72

- 71)V’

0;

(73

- n ) v 3 = 0,

(v? # 0),

(11.7)

0;

(73

- 71)V3

(V’

# 0).

(11.8)

2 ) = ~ ~

=

=

0,

Any plane 2 admitting a stationary value of the normal stress a t the point P must have a unit normal vector v satisfying at least one of the above conditions (11.6), (11.7), and (11.8); hence an examination of these conditions will lead to all such planes 2. I n determining the planes L: for which the normal stress has a stationary value, we shall suppose that 71

2

72

2

73

11.

MAXIMUM NORMAL STRESS

31

for definiteness. Then one of the following four cases must occur, namely

Case C.

> T Z > 73, 71 = 7 2 > 73, 7 1 > T Z = 73,

Case D.

71

Case A. Case B.

71

= 72 =

73,

depending on whether the determinantal equation (9.4) has distinct roots, a double root or a triple root. If Case A holds it follows from (11.6) that v l = v 2 = 0. Hence N has a stationary value for the principal plane 2 whose normal coincides with the y 3 axis, i.e., the principal direction corresponding to the principal stress 7 3 ; also N = 7 3 for this plane Z as we see from (11.1). Similarly, it follows from (ll.l), (11.7), and (11.8) that N will have the stationary values r 2and 71 €or the principal planes perpendicular to the principal directions vz and v1 respectively. For Case B the condition (11.6) shows that v l = v 2 = 0 and hence the plane perpendicular to the y3 axis, i.e., the principal plane associated with the principal stress 73, has a stationary normal stress N ; from (11.1) the value of this normal stress is 7 3 . It follows from (11.7) and (11.8) that v 3 = 0 which shows that any plane Z through the y3 axis will have stationary normal stress N ; such a plane Z will be a principal plane corresponding to the principal stress 7 1 which is, of course, the value of its normal stress N . Case C, which also corresponds to a double root of the equation (9.4), is not essentially different from Case B. Here we see from (11.6) and (11.7) that v i = 0, and hence any plane 2 through the y' axis will have stationary normal stress N ; the stress N will have the value 7 2 from (11.1) and the plane 2 will be a principal plane corresponding to the double root T~ of (9.4). Also it follows from (11.8) that v 2 = v 3 = 0, i.e., the plane Z perpendicular to the y1 axis has a stationary normal stress N which will have the value r1 from (11.1); this plane Z is the principal plane associated with the principal stress 71. Finally for Case D all conditions (11.6), (11.7), and (11.8) are satisfied identically. Hence the normal stress N will have a stationary value for any plane 2 at the point P and the value of the normal stress N will be equal to the triple root 7 1 of the equation (9.4).

32

I. BASIC INVARIANTS. EQUATIONS OF CONTINUITY AND MOTION

The following result can now be concluded from the above discussion. The greatest normal stress N on any plane Z at the point P will be equal to the greatest of the roots r of the determinantal equation (9.4), and this greatest normal stress will occur on a principal plane whose unit normal v is associated with the maximum root r . An analogous result can, of course, be stated for planes Z at P for which the value of the normal stress is an absolute minimum. Remark 1. In the above discussion we were concerned with the plane elements Z of maximum normal stress a t a specific point P of the material medium. Now consider the plane elements Z of maximum normal stress over a region R (open set) of the medium, and let us suppose that the elements Z can be chosen so that their unit normal vectors w form a continuous and differentiable vector field in R. It can be shown that such elements Z can be united to form a one parametry family of surfaces of maximum normal stress if, and only if, the condition ei'kw;w;,k = 0, (11.9)

is satisfied, where the wi are the components of the vector w. A derivation of the geometrical condition (11.9) can be found in the recent book by T. Y. Thomas, Concepts from Tensor Analysis and Digerential Geometry, Academic Press, New York, 1961, p. 105. In the general case for which the roots r of the determinantal equation (9.4) are distinct, i.e., Case A of the above discussion, the vector w in the condition (11.9) can be identified with the vector v1 of the set of mutually perpendicular unit vectors vl,v2,v3 giving the principal directions; also, for this case, we can avail ourselves of the formula (10.12) for the derivatives of the components v i of the vectors vk. Hence (11.9) becomes (11.lo)

in which we have writ,ten e i j k instead of eiik and all repeated indices are summed over the permissible range 1,2,3. Equation (11.10) gives the general condition for the existence of surfaces of maximum normal stress in the material medium.

12. MAXIMUM SHEARING STRESS The shearing stress S on a plane element Z at a point P of the medium has been observed t o be given by the equation (9.1). We now seek those planes Z at P for which S has its maximum numerical value. Choosing a system of canonical coordinates y' with origin at the point P , the equation (9.1) becomes

12.

8'

+

33

MAXIMUM SHEARING STRESS

+

-

f

~ Z ( Vf ~ )7 3~( V 3 ) 2 ] ,

(12.1) in which the v1,v2,v3are the components of the unit normal v to Z a t P relative to the canonical system. Now eliminate the component v3 from the right member of (12.1) by means of the relation (11.2), and then differentiate the resulting expression with respect to vl and v2. Equating these derivatives to zero we obtain the conditions, namely (71

(72

= T?(V1)2

."2(V2)'

- 7 3 ) v 1 [ ( 7 1 - 73)(V1)' - 7 3 ) v 2 [ ( 7 1 - 73)(V1)'

7;(V3)'

+ +

(72 (72

[71(V')2

- 7 3 ) ( V 2 ) 2 - $(TI - T 3 ) ] - ' T 3 ) ( V 2 ) 2 - a(72 - T a ) ]

1 1 r

= 0, =

0,

(12.2)

(component v3 # 0 ) for S2to have a stationary value when v3 # 0 for the plane Z under consideration. Similarly we obtain (71 (72

- 7 2 ) v 1 [ ( 7 1 - 7z)(V1)' - 7 3 ) V 3 [ ( 7 i - 72)(V1)'

+ +

(73 (73

~ ) $(Ti ~ -~ Z ) ( V- T Z ) ] = 0, 7 z ) ( V 3 ) ' - i ( 7 3 - 7 2 ) ] = 0,

(component v2 # 0 ) and (Ti (71

+

- 73)V3[(73 - 71)(V3)2 - 7 2 ) V 2 [ ( n - T ~ ) ( Y -/-~

(72

- 7i)(V2)'

)( 7~2

-T ~ ) (

- $(73

- 7111 V ~ ) $(Q ~ - 71)]

=

0,)

=

0

(12.3)

(12.4)

(component v l # 0 ) as the conditions for S2 to have a stationary value when v2 # 0 and vl # 0 respectively. From among these stationary values of S2 the maximum (and minimum) values of the shearing stress S can readily be selected by inspection. Our discussion of this problem will be based on the four Cases A, . . . ,D considered in Section 11. Case A

Observe first that vl and v2 in (12.2) cannot both be different from zero. For if vl # 0 and v2 # 0 each bracket expression in (12.2) must vanish. Hence, subtracting these expressions, we have the contradiction 7 1 = 72. Similarly we cannot have v1 # 0 and v 3 # 0 in (12.3) nor can we have v2 # 0 and v 3 # 0 in (12.4). A solution of (12.2) is given by vl = v 2 = 0; hence S2has a stationary value when Z is the y 1 , y 2 plane. Correspondingly from (12.3) and (12.4) we see that S2has a stationary value on the y1,y3 plane and the y 2 , y 3

34

I. BASIC INVARIANTS. EQUATIONS OF CONTINUITY AND MOTION

plane. These are the three principal planes at the point P on which the shearing stress vanishes as we know from the discussion in Section 9. Taking v 1 = 0 and v2 # 0 in (12.2) we see that v2 = -I1 / 4 5 . Hence v 3 = =tl/dZ. Hence X2 has a stationary value for the two planes which pass through the y' axis and bisect the angles between the y? and y 3 axes; for these planes the value of S2 is ( T ~- T3)2/4 from (12.1). If v l # 0 and v 2 = 0 in (12.2) we find that v l = =tl/dz; hence v 3 = f1/4/2, and it follows that S2has a stationary value on the two planes through the y2 axis which bisect the angles between the y' and y3 axes. The value of S2on these latter planes is ( T ~- 73)*/4. Finally we deduce from (12.1) and (12.3) that S2 has the stationary value ( T ~- ~ ~ ) on ~ /each 4 of the planes through the y3 axis which bisect the angles between the y l and y2 axes. The conditions (12.4) give no additional information and hence this completes the discussion of this case. Case B

Taking two of the components vl, v 2 and v 3 equal to zero we again find the stationary value S2 = 0 when 2 is any one of the y coordinate planes, i.e. the principal planes at the point P. Next suppose v 1 = 0, v 2 # 0 and v 3 # 0. Then from (12.2) we have v2 = &l/* and hence v 3 = =tl/dZ; this gives the two planes passing through the y' axis which bisect the angles between the y2 and y3 axes and for which S2 has the stationary value ( T Z - T 3 ) 2 / 4 . If v 2 = 0, v l # 0 and v 3 # 0 we may use (12.2) to find that 'v = f 1 / 4 / 2 and .hence v 3 = =tl/d/2; hence the two planes which pass through the y2' axis and bisect the angles between the y1 and y3 axes give stationary ~ / 4(12.1). I n the revalues of S2 which are equal to (q - ~ ~ ) from maining case for which a single one of the components v i vanishes, i.e. when v3 = 0 with 'Y # 0 and v2 # 0, the conditions (12.3) and (12.4) are satisfied identically. Hence S2 has a stationary value for all planes through the y3 axis; this stationary value is equal to zero from (12.1). There remains the case for which no component vl, v 2 and v 3 is equal to zero. But then we see from (12.2) that (v')2

Hence v 3

=

+

(v"2

=

5.

& 1 / 4 5 and S2has a stationary value on any plane Z; whose

12.

MAXIMUM SHEARING STRESS

35

normal makes an angle of f45" with the y 3 axis. If C is the circular cone whose generators make an angle of A45" with the y 3 axis, the planes Z can be described as those planes which are tangent to t'he cone C. It is seen from (12.1) that S2has the value (71 - T3)2/4 for each of these tangent planes 2. Case

C

We again obtain the principal planes at the point P as planes Z for which S2has the stationary value zero. If v l = 0, v 2 f 0 and v 3 # 0 the conditions (12.2) and (12.3) are satisfied identically while (12.4) is not applicable; hence for any plane Z through the y 1 axis the value of S2 is stationary and equal to zero from (12.1). Taking v2 = 0,v l # 0 and v 3 # 0 it follows from (12.2) that v 1 = f1/& Hence v 3 = fl/d/2and S2 is stationary for the two planes Z which pass through the y 2 axis and bisect the angles between the other two axes; 4 these planes 2. For v 3 = 0,v l # 0 the value of S2is (71 - ~ 3 ) ~ /for and v 2 # 0 the conditions (12.3) show that v l = f 1 / & Hence v2 = =tl/dzand S2has a stationary value of (n - T2)2/4 on the two planes Z which pass through the y 3 axis and bisect the angles between the y 1 and y 2 axes. Finally if none of the components v i are equal to zero we have v l = f l / h f r o m (12.2). Since this is the only condition on the quantities v i it follows that S2 has a stationary value on any plane Z for which the normal v makes an angle of f45' with the y' axis. Hence any plane 8,tangent to the circular cone C whose generators make an angle of f45" with the y1 axis, will provide a stationary value of S2equal to ( T ~- T2)2/4 from (12.1). This completes the discussion of Case C. Case

D

For this case we see immediately that S2has the stationary value zero for every plane element Z a t the point P. Examining the results obtained in the above Cases A, . . . ,D we find that the maximum stationary value of S2 is equal t.0 (71 - ~ 3 ) ~ / 4 and that this value occurs on the planes Z which pass through the y 2 axis and bisect the angles between the other two coordinate axes; this

36

I. BASIC INVARIANTS. EQUATIONS OF CONTINUITY AND MOTION

involves a special instance of the case for which the stationary values of S2occur on the planes Z tangent t o the above circular cones C mentioned in Case C and Case D. Hence the following result can be stated. The greatest numerical value of the shearing stress S on a n y plane element Z at a point P is equal to (TI - 7J/2 where r1 and r 3 are the greatest and least of the principal values of the stress. This shearing stress S occurs on the two planes at P which pass through the principal direction associated with the intermediate principal stress 72 and bisect the angles between the other two principal directions. As shown in Sect. 9 the three principal directions at P are mutually perpendicular when the determinantal equation (9.4) has distinct roots T, i.e. for the above Case A; otherwise it is assumed that mutually perpendicular principal directions are selected in making this italicized statement. Remark. Assuming that plane elements Z of maximum numerical shearing stress can be chosen so that the unit normal vectors w to the elements 2: form a continuous and differentiable field in a region R, then the necessary and sufficient condition for the elements Z to unite to form a one parameter family of surfaces in R is that the equation (11.9) be satisfied. Any surface of this family will be called a surface of maximum numerical shearing stress. In the general case, i.e. the Case A, there will be exactly two such plane elements Z a t each point P and they will be perpendicular. In fact it is readily seen that the unit normal vectors w to these two elements 2: have components wigiven by 1;t (12.5) wi = Walt*,

a t any point P, where the v\ are the components of the mutually perpendicular unit vectors v k which determine the principal directions and the coefficients w,” are the constants defined by one or the other of the following two sets of equations, namely

There are thus a t most two one parameter families of surfaces of maximum numerical shearing stress in the general case and the conditions for their existence are obtained by substituting the values of the wigiven by (12.5) into (11.9) and making use of the formula (10.12) for the derivatives of the components v i ; if these two families of surfaces exist they will obviously be perpendicular.

II. Conditions of Compatibility 1. BASICFORMULAE FROM

THE

THEORY OF SURFACES

Consider a moving surface z(t) defined by equations of the form (1.1) where u1,u2are curvilinear coordinates of the surface, t denotes the time and x1,x2,xaare the coordinates of an allowable rectangular system for the dynamical problem. It is assumed that the functions 4i are continuously differentiable and that the surface Z is regular in the sense that the functional matrix Ilt3$i/auall always has rank two; this condition is sufficientfor the existence of a tangent plane at each point of Z(t) and for the local representation of the surface, a t any time t, by means of an equation involving only the x coordinates. Beyond this all necessary continuity and differentiability requirements will be assumed, as needed, without special mention. With regard to the position of the indices on the quantities in the equations (l.l),and more generally on other quantities which will enter the discussion, we shall use either the superscript or subscript position for indices associated with the x coordinate system since there is no distinction between covariance and contravariance under the orthogonal transformations relating rectangular coordinate systems. We must, however, distinguish between covariant and contravariant indices when the indices are associated with the curvilinear or surface coordinates of Z(t) since these coordinates are subject to arbitrary differentiable transformations. The coefficients of the first fundamental form of the surface Z(t) are given by gas = 4!=4:ol (1.2) where alp= 1,2 and the comma in the right member of these equations denotes partial differentiation with respect to the curvilinear coordinates; the derivatives will also be denoted by xi without the use of the comma to indicate differentiation. Latin indices have the range xi = #i(U*,U2,f),

37

38

11. CONDITIONS OF COMPATIBILITY

1,2,3 and Greek indices the range 1,2 in (1.2) and throughout the remainder of this chapter. I n equations (1.2) and in the following discussion, as in the foregoing work, we have adopted the convention that a n index which occurs twice in a term is t o be summed over the proper range of values of this index. From the covariant components gap of the metric tensor we can construct the contravariant components p@of this tensor in the usual manner. An essential fact to be observed is that the quantities 4;. or xk are the components of a covariant vector X iunder transformations of the curvilinear coordinates. On the other hand these quantities have the vector transformation, i.e. the xi are the components of a vector X u , under orthogonal transformations of the z coordinates. Moreover this space vector X u is tangent to the surface Z and has the direction of the corresponding urncurve. Hence we can write

1; x y = 0. (1.3) The first of these equations expresses the fact that the normal vector v to the surface Z is taken to be a unit vector and the second equation expresses the condition that the above mentioned vectors X u are tangent to the surface 2 . The relations (1.3) will have frequent application in the following derivations. Two basic formulae from the theory of surfaces will be especiallyuseful in our work. These are yivi =

x L , ~= b a p i ;

v!= = - g @ ~ b a a x ~ ,

(1.4) where the xh.@are the components of the covariant derivative of the are the comvector X i based on the metric of the surface Z and the ponents of the second fundamental form of this surface. The quantities x h , ~are symmetric in the indices ar,p and hence the b8. are likewise symmetric in the indices a,p from the first set of equations (1.4). I n addition the surface scalar 12 defined by 22

=

p6,ol

(1.5)

will appear in certain of the following formulae. The scalar 12 is the mean curvature of the surface 2. Another useful relationship to which we desire to call particular attention is furnished by the following identity pSx&$ =

gii

- vivi

(1.6)

2.

GEOMETRICAL CONDITIONS O F TH E FIRST ORDER

39

in which Ziii is the Kronecker delta. The derivation of the identity (1.6) and the other geometrical formulae in this section can be found in the book by Thomas, Concepts from Tensor Analysis and Digerential G e m etry, Academic Press, New York, 1961, 119 pages; for brevity, specific references to this volume will be made hereafter by the single term Cowepts followed by the designation of the appropriate page or section. 2. GEOMETRICAL CONDITIONS OF COMPATIBILITY OF THE FIRSTORDER Let Z ( x , t ) be a function of the rectangular coordinates xi and the time t. Assuming that Z is a continuous and differentiable function of the variables xi and t on each side of the moving surface Z ( t ) we shall now consider the possible restrictions on the “jumps” or discontinuities in the function Z ( x , t ) and its derivatives which can arise when we traverse this surface. We define the discontinuity [ Z ] in Z by writing [ Z ] = Zz - 2, where the subscripts 1 and 2 refer to the sides 1 and 2 of the surface Z ( t ) ; the discontinuities in the derivatives of Z are, of course, defined in a corresponding manner. It will be supposed, for definiteness, that the unit normal vector v to Z ( t ) is directed from the side 1 to the side 2 of this surface. Now we shall find that the discontinuity [ Z ] and the combinations [Z,;]viand [ Z , i j ] ~ iwhere ~ i , Z,i and Z,ij denote the first and second partial derivatives with respect to the coordinates, occur more or less frequently in the following formulae. Hence we shall use the abbreviations

c.

[Z] = A ; [Z,i]Vi = B ; [Z,iJViVi = (2.1) From the above definitions and with due regard to the necessary assumptions concerning the existence of derivatives of Z in space and on the surface, it is easily seen that we have

A,u = [Z,J = [Z,ili, (2.2) in which the index (Y denotes partial differentiation with respect to the curvilinear coordinates ‘u of the surface Z ( t ) . Now multiply the left and right members of (2.2) by fad and sum on the repeated indices. But this gives [Z,i](Ziii- v~v’) = $LBA,u~$, (2.3) on account of the relations (1.6). Hence, carrying out the indicated

40

11. CONDITIONS O F COMPATIBILITY

summation in the left member of (2.3) and introducing the symbol B defined by (2.1), we have

[z,il = Bvi

+

gQBA,Uxip,

(2.4)

in which we have taken the liberty to lower the position of the index i in the right member to conform to the position of this index in the left member of these relations. The equations (2.4) express the discontinuities [Z.;] of the coordinate derivatives of the function in terms of the two quantities A and B defined on the surface Z ( t ) ; in this connection it may be observed, e.g. by multiplying the two members of (2.4) by vi and summing on the repeated index i, that the conditions (2.4) imply that B has the value given by (2.1). In case the function Z is continuous across Z ( t ) , so that A = 0 from (2.1), the equations (2.4) reduce to relations of the restricted type [Z,i] = B v ~ .

(2.5)

Equations (2.4) are called the geometrical conditions of compatibility of the first order for the function 2, the designation geometrical being used for those compatibility relations which do not depend on the actual motion of the surface Z ( t ) . Conditions which involve time derivatives and which depend on the motion of Z are called kinematical conditions of com.patibility. In the next section we shall derive the corresponding kinematical conditions of compatibility of the first order for the function Z(x,t). Conditions of compatibility of the second order will be constructed in Sect. 5 and Sect. 6 of this chapter.

3. THE6 TIMEDERIVATIVE AND THE KINEMATICAL OF THE FIRSTORDER CONDITIONS

+

Consider two positions Z(t) and Z(t At) of the surface Z. Erect the unit normal v to Z ( t ) a t a point P and suppose that this normal intersects the surface Z ( t At) at the point P'. Let P have coordinates x i and PI the coordinates x i Axi so that Axi represents the coordinate differences of these two points. Then, denoting by AA the difference of the values of the quantity A a t the points P and PI, we can write

+

+

3.

KINEMATICAL CONDITIONS O F THE FIRST ORDER

41

where the subscripts 1 and 2 denote evaluation on the sides 1 and 2 of the surface Z and the prime indicates evaluation at the point P'. Let us now define the 6 time derivatives of the quantities A , Z1 and Z2 as follows 6A - = lim -7 61

~t-0

At

Using this definition and allowing At to approach zero in (3.1) we now have 6A - -6t

"-%=[GI. at

(3.3)

6t

It is clear that the process of 6 time differentiation is, quite generally, applicable to any function defined over the moving surface Z ( t ) . The normal velocity G of the moving surface Z(t) is defined by As At)

(3.4)

G = lim At-0

where As denotes the distance of the point P' from the point P. Hence we have

Now consider the two approximate equations

relative to the sides 1 and 2 respectively of the surface X ( t ) . Allowing At to approach zero it follows that

6Zr 6t = (Z,i)lGv' + ,322

6t = (Z,&Gv'

(z)

az + (at)

9

when use is made of the relations (3.5). Subtracting corresponding members of the two equations (3.6) we see from (2.1) and (3.3) that 6A (3.7) = -BG+

[z]

42

11. CONDITIONS OF COMPATIBILITY

T h e relation (3.7) is called t h e kinematical condition of compatibility of the first order for t h e function Z(z,t). It is evident from its derivation that the equation (3.7)expresses the condition for the discontinuity A in t h e function 2 to persist during the motion of the surface Z(t). In particular if 2 is continuous across the surface Z(t), so that A = 0, the condition (3.7) reduces to the special compatibility condition

[$]

=

-BG.

Remark 1. The geometrical and kinematical conditions of compatibility of the first order for the velocity components vi, the pressure p and the density p are given explicitly by the following equations, namely

+ ga’[viI,azj~,] p?!?]= -xiG + 6t [v,,~I

1

= Xivi

(3.9)

(3.10)

and

r$] =

-{G

+

1

(3.11)

in which the quantities Xi, 5 and { are functions defined over the surface z‘(t). When the functions vi, p and p are assumed to be continuous across the moving surface Z(t), the above conditions (3.9), (3.10) and (3.11) reduce respectively to

(3.14) Remark 2. As illustrated in the preceding Remark the symbol Z(z,t) will frequently represent any function of a set of functions, e.g. the components vi of the velocity, which enter into the problem under consideration. A surface Z ( t ) over which there exists a discontinuity in any one of such functions Z ( z , t ) or their derivatives will be called a singular or wme surface in a general sense. More precisely we shall say that Z ( t ) is a singular surface of order zero relative to the functions Z(z,t) if at least one of these functions is discontinuous in the

4.

VARIATION O F THE U N I T NORMAL VECTOR

43

spat,ial coordinates xi over X ( t ) . A shock wave in the usual theory of compressible gases is an example of a singular surface of order zero relative to the set of functions consisting of the pressure p , the density p and the velocity components ut in the gas. The surface Z ( t ) will be said to be singular of order one relative to the functions Z(z,t) if (a) the functions Z(z,t) are continuous over Z(t), i.e. [Z(z,t)] = 0, and (b) at least one of the coordinate derivatives Z,i(z,t) is discontinuous over Z(t). The singular surface of order n 2 2 is defined in an analogous manner. It follows from the equations (2.5) and (3.8) that there is a discontinuity in the time derivatives aZ(z,t)/dt at points of the singular surface of order one if there is a discontinuity in the coordinate derivatives Z,i(z,t), provided the velocity G of the surface is different from zero; a corresponding remark can be made concerning the time derivatives of the functionsZ(z,t) in the case of singular surfaces of higher order.

4. VARIATIONOF

THE

UNIT NORMAL VECTOR

In this section we shall derive a rather interesting and useful formula for the 6 time derivative of the components vi of the unit normal vector v to the moving surface Z ( t ) . We begin by differentiating the two relations (1.3) to obtain v'-.6v' - 0; 6t

x i -6Vi 6t

=

-y'-.

.6XL 6t

Now multiply both members of the second set of equations (4.1) by sum on the repeated indices. This gives

~ 8 x and 6

Then, eliminating the coefficients of the derivatives 6vi/6t by the substitution (1.6) and using the first equation (4.1), it follows that

To eliminate the derivatives 6x;/6t in (4.2) we consider the following two sets of relations (4.3) (4.4)

44

11. CONDITIONS O F COMPATIBILITY

Equations (4.3) are a direct consequence of the definition of the 6 time derivative while (4.4), in which the C p are the components of the affine connection of the surface Z(t), is the usual formula for the components x& of the covariant derivative of the vector X i(see Sect. 1). When the equations (4.3) and (4.4) are combined and use is made of the first set of relations (1.4), we have

We must now derive a formula for the derivatives 6u@/6twhich are introduced when we eliminate the quantities 6xL/6t from (4.2) by the substitution (4.5). For this purpose let us observe that the equations (3.5) can be expressed as

Multiplying both members of (4.6) by v i and again by xk and in each case summing on the repeated index i, we obtain

The first relation (4.7) will not concern us immediately. In the remaining relations (4.7) the coefficients of 6ua/6t are equal to the components gap from (1.2) and hence these relations can be solved to give

When we eliminate the derivatives 6xL/6t from (4.2) by means of the relations (4.5) and then eliminate the quantities 6up/8t, which are thus introduced by the substitution (4.8), we find that the term in the symbols r vanishes on account of the second set of equations (1.3). Hence (4.2) becomes

Now consider the identity

But, using the second set of equations (1.4) and the first equation (4.7), these relations can be written

5.

GEOMETRICAL CONDITIONS OF THE SECOND ORDER

45 (4.10)

Combining (4.9) and (4.10) we arrive at the desired formula, namely

(4.11) Remark. If G = const. on the moving surface Z(t), then G , , = 0, and it follows from (4.11) that 6vi/6t vanishes on B ( t ) ; conversely, if 6vi/6t = 0 on Z ( t ) it is easily seen from (4.11) that G , , = 0, i.e. G = const. on the surfaceZ(t). Now the vanishing of 6vi/6t on 2(t)implies that the normal trajectories to the family of spatial surfaces Z ( t ) are straight lines and hence that the surfacesZ ( t ) form a family of parallel surfaces (see Concepts, p. 108). We can therefore state the following result. The successive positions of the wave surface Z(t) in space will form a family of parallel surfaces if,and only if,the velocity G i s constant over the individual surfaces Z (t).

5. GEOMETRICAL CONDITIONS OF

THE

SECOND ORDER

Replacing the function 2 in (2.4) by one of its derivatives 2,; we obtain a relation of the form

[z,ijl = Bivj

+

guBAi,n2,p,

(5.1)

where, in accordance with (2.1), the quantities A , and Bi are given by

[Z,,]; Bi = [ Z , i j ] V ’ . (5.2) But the expressions in the right members of (5.1) must be symmetric in the indices i and j since the left members of (5.1) have this property. Hence Biv3 4- ~’Ai,axj:,a = Bjvi 4- P’Aj,aXiB. (5.3)

Ai

=

Multiplying (5.1) by vi and vi and summing on the repeated indices

i and j , we find that Bavi= C as defined by (2.1). Next multiply (5.3) by vi and sum on j to obtain

+

Bi = CV; yBAj,,vi~i:,a.

(5.4)

But

Aj,avi =

- [z,j]$a.

[Z,j],avi = ([Z,j]vi>,a

Hence, when use is made of the second set of equations (1.4) and the second equation (2.1), it follows that

Aj.nv’

=

B,a

+

[Z,j]gp’bpnd-

15.5)

46

11. CONDITIONS O F COMPATIBILITY

Replacing the quantities [ Z , j ] in (5.5) by the values given by (2.4), we find that the resulting equations can be written in the form

+

A j p d = B,a g”b#aA,v. (5.6) Then, making the substitution (5.6) in the right members of (5.4), we have Bi = Cvi f’(B,a g’YbpaA,v)xib(5.7) in the right members of It remains to consider the quantities (5.1). But Ai,, = [Z,i],a= (Bvi g’’A,pziu) ,a) (5.8) on account of (2.4). Now the indicated differentiation of the parenthesis expressions in the right members of (5.8) may be assumed to be either partial or surface covariant differentiation. Assuming surface covariant differentiation the above relations (5.8) become

+

+

+

+

+

+

Ai,a = B,avi Bvi.a gpvA,paxiv gpvA,pxiv,a) are the components of covariant where the quantities A,,,a and derivatives. Hence, using (1.4)) the above equations for Ai,a can be written A i p = B , a ~i Bg”ba~zi, guT(A,auXir A,ubarvi). (5.9)

+

+

Making the substitutions (5.7) and (5.9) in the relations (5.1) and combining terms we now obtain [Z,ij]

=

Cvivj

+ f’(B,a + gUrbauA,T)(vi~j~ +

vjxia)

I

(5.10) f’gur(A,au - Bbau)XiBXir. These relations are the geometrical conditions of compatibility of the second order for the function Z ( x , t ) ;we observe that the right members of (5.10) are symmetric in the indices i and j as required. If, in particular, the function Z is continuous across the surface Z(t), the relations (5.10) reduce to [Z,ij] = Cvivj $’B,a(vixjp vjxij3) (5.11) - BgQBgurbauxipxjT. If, furthermore, the first derivatives of the function Z with respect to the coordinates x i are also continuous, the relations become merely [Z,,] = cvavj. (5.12)

+

+

I

Remark. Certain compatibility conditions which can be derived from (5.10) will sometimes be found to be useful in the applications of this formula. Thus

6.

KINEMATICAL CONDITIONS OF THE SECOND ORDER

47

if we contract the indices i and j in (5.10) and take account of (1.2) and (1.5)' we obtain [Z,iiI = S"@A,"@ - 2BQ.

c+

6. KINEMATICAL CONDITIONS OF

THE

SECOND ORDER

Let us now replace Z by the derivative aZ/at in the conditions (2.4) and (3.7). This leads to relations of the form

where

We can derive an equivalent expression for the quantity B' by considering the 6 time derivative of B. In fact we have 6B 6 6[Z .] . 6Vi - = - ([Z,i]Vi)= [Z,i]X ' 6t 6t 6t v' Now from the first equation (1.3) it follows that

+

&)'-

. 6vi 6t

- 0.

Hence the last term in (6.4) becomes

on account of (2.4) and (6.5). Making this substitution in (6.4) and expanding the first term in the right member of the equation we have

+

6B

- = [Z,ii]G~ivi 6t

[zt] + vi

6VC

pBA,axig- 9

6t

or

B'

=

-CG

6B +-f@A,di~-i 6t 6t 6Vi

(6.6)

when use is made of the third equation (2.1), and the second equation (6.3); finally hhe equation (6.6) becomes 6B B' = -CG - V@A,~G,B, (6.7) 6t

+ +

48

11. CONDITIONS OF COMPATIBILITY

when we eliminate the derivatives 6vi/6t by means of the substitution (4.11). Making the substitution (6.7) in the equations (6.1) and (6.2) we now obtain = (-CG 6B fBA,aG,~ v, f@Abx;~, (6.8)

I"[

+ st +

+

axiat

s] (

[

2 2

=

+

6B 6A' CG - st - g 4 8 ~ , ~G~ , ~6t)

(6.9)

The relations (6.8) and (6.9) are the kinematical conditions of compatibility of the second order for the function Z(x,t). We observe that these conditions contain derivatives of the quantity A' in addition to the surface scalars and their derivatives which appear in the compatibility conditions of the first order and in the geometrical conditions of the second order; however it is possible to eliminate these derivatives as we shall show in the following Remark 1. If the function 2 is continuous over Z(t), then A = 0, and the equation (3.7) reduces to A' = -BG. Hence, making these substitutions, the kinematical conditions (6.8) and (6.9) take the form

(6.11) Finally, if both the function 2 and its first coordinate derivatives 2,i are continuous across X ( t ) , the compatibility conditions (6.8) and (6.9) reduce to

[gt] =

-CGvi;

[atp] a22

=

CG2.

(6.12)

Remark 1. It is possible to eliminate the derivatives A(, and 6A'/6t of the quantity A' from the relations (6.8) and (6.9) and thus to decrease the number of basic surface scalars needed to express the kinematical conditions of the second order. I n the case of the time derivative 6A'/61 the elimination is immediate since

A ' = - B G + - 6A 6t

(6.13)

from (3.7) and (6.3). Making the substitution (6.13) for A' in (6.9), we find that

6.

KINEMATICAL

CONDITIONS

OF THE SECOND ORDER

49

To eliminate the quantities A,; from (6.8) we can proceed as follows. First consider the relations

A ( , = [=]xi, a x i at

(6.15)

which follow directly from (6.8). Now from the definition of the 6 time derivative we see that (6.16) when use is made of (3.5). Hence from (6.15) and (6.16) we have (6.17)

It follows readily that

[z.ild = A,,, [z,ijl~hp' = B,a + g"'ba.,A,r,

(6.18) (6.19)

when we perform the indicated operations on the equations (2.4) and (5.10). Making the substitutions (6.18) and (6.19) the equations (6.17) now become (6.20)

Hence, in place of (6.20), we can write

(6.21)

To simplify the above expression for A(P let us consider the following formulae of covariant differentiation, namely

(6.22)

where the r's are Christoffel symbols based on the metric of the surface X ( t ) .

50

11. CONDITIONS O F COMPATIBILITY

Hence the first parentheses expressions in the right members of (6.21) can be written

A.aB

+

- [~,iIxh,9- [z,~Iz',~&.

(6.23)

But the second and last terms in these expressions are seen to cancel when we make the substitution (2.4). Taking account of this fact, and eliminating the quantities x : , ~in (6.23) by means of the first set of equations (1.4), it follows that (6.21) becomes

+

- G(B,a gufbauA,r). Finally, when we eliminate the quantities [ZJ and 6uS/6t from (6.24) by the substitutions (2.4) and (4.8), we obtain

The relations (6.8) in which the quantities A,: are given by the above equations (6.25), and the relations (6.14), furnish the desired form of the kinematical compatibility conditions of the second order. Remark 2. Suppose that the surface Z is stationary; then the equations (1.1) defining this surface do not involve the time and the velocity G = 0 as is formally evident from the first equation (4.7). Moreover it is obvious that there is no distinction between 6 and partial time differentiation in the case of a stationary surface Z. It is therefore seen immediately from the equations (6.8) in which the A:, are given by (6.25), and the equations (6.14) that the following result can be stated. If the surface Z i s stationary the general kinematical conditions of compatibility of the second order have the form (6.26) (6.27)

If, in particular, the function 2 is continuous, i.e. if [Z] = 0, over the stationary surface 2 , the kinematical compatibility conditions become (6.28) Finally, when the function Z and its first coordinate derivatives are continuous

7.

51

COMPATIBILITY CONDITIONS IN GENERAL COORDINATES

over a stationary surface2, i.e. when [ Z ] = 0 and [ZJ conditions reduce to

= 0, these compatibility

(6.29)

7. COMPATIBILITY CONDITIONS IN GENERALCOORDINATES In constructing the compatibility conditions in the preceding sections it was assumed that the xi were the coordinates of a rectangular system. Consequently the above form of the compatibility conditions can be said, in general, to be invariant only under orthogonal transformations of the x coordinates although invariance was obtained under arbitrary differentiable transformations of the curvilinear coordinates uU of the surface X ( t ) . We shall now consider the problem of representing the compatibility conditions in an arbitrary coordinate system whose coordinates xi are obtainable by transformation from the coordinates of a rectangular system. Relative to these general coordinates xi. the surface Z(t) will also be defined by equations of the form (1.1). Moreover, the quantities x: defined by

will be the components of a contravariant vector X u in the space under differentiable transformations x f+ Z of the x coordinates and the components of a covariant vector X i on the surface 2 under differentiable transformations u t,ii of the curvilinear coordinates; as in Sect. 1 we observe that the two space vectors X u are tangent to the corresponding u' coordinate curves on the surface Z(t). If the surface Z ( t ) is represented by an equation of the form

f(X1t) = 01 then the covariant and contravariant components of its unit normal vector Y are given respectively by the equations

af

.. a j gt' -.

52

11. CONDITIONS O F COMPATIBILITY

where the quantities g'j are the contravariant components of the metric tensor of the space; these components gcj and the covariant components gij of the metric tensor relative to the x system are obtainable from the corresponding components 6'i and 6 i j of this tensor in a rectangular coordinate system by the usual equations of transformation. We shall continue to denote by gap and ~ " 0the components of the metric tensor for the surface I:(t). No confusion will result, in general, from this notation since the components of the metric tensors of the space and the surface I: can be distinguished from one another by means of their Latin and Greek indices. However in dealing with individual components, e.g. the components gll, g12 and gZ2,we must specify whether these are components of the metric tensor of the space or the surface 2. Let Z(x,t) denote a scalar or any one of the components of a vector or tensor in the space and define the surface quantities A,B,C by (2.1) in which the bracket [ ] stands for the difference of the quantity enclosed a t contiguous points on the two sides of the surface Z and the comma indicates covariant differentiation with respect to the space coordinates. It is seen immediately that the geometrical compatibility conditions of the first order for 2, relative to the general x coordinate system, are given by

[z,iI= Bvi

+ gQBgiJ,&,

(7.2) where the comma in the right members denotes covariant differentiation of the spatial quantity A on Z ( t ) with respect to the surface coordinates (see Concepts, p. 85), since (a) these relations are invariant under differentiable transformations x f-) 33 and u t)Ti of the space and surface coordinates and (b) they reduce to the proper compatibility conditions (2.4) when the xi are the coordinates of a restangular system. To illustrate the construction of the kinematical compatibility conditions of the first order relative to the general x coordinate system, let us consider the particular case of the covariant velocity vector. From the equations of transformation of the components V k of this Vector it follows immediately that the discontinuities [Vk) transform by the equations [Vi] =

a r k

[Vk]

-9

azi

(7.3)

under differentiable transformations x ts Z of the x coordinates. Let us

7.

COMPATIBILITY CONDITIONS IN GENERAL COORDINATES

53

now equate the 6 time derivatives, defined in Sect. 3, of the two members of the equations (7.3). We thus obtain

But, as shown in Sect. 3, we have

where G is the normal velocity of the moving surface Z(t) in the direction of its unit normal v; obviously G is an absolute scalar function on the surface 2(t). Now

where the r’s and T”s are the Christoffel symbols in the x and Z systems respectively. Making the substitution (7.5) and eliminating the second derivatives in (7.4) by means of (7.6), we find that the resulting equations can be written

Putting

and making a corresponding substitution for the quantities in parenthesis in the left members of (7.7), we now have

(7.9) in place of (7.7). But (7.9) expresses the fact that the quantities D[uk]/Dt are vector components under differentiable transformations x t)Z of the x coordinates; moreover, we observe that the D[uk]/Dt reduce to the corresponding 6 time derivatives 6[vk]/6t in a rectangular coordinate system. The quantity having the components D[vk]/Dt will be referred to as the absolute or invariant time derivative of the discontinuity [v] since the use of such components, rather than the corresponding 6 or partial time derivatives, will enable us to express our relations in a form which is invariant under general coordinate transformations. Denoting by v;,k the components of the covariant derivative of the

54

11. CONDITIONS O F COMPATIBILITY

velocity vector with respect to the spat,ial coordinates xk, let us now consider the relat.ions (7.10) in which X i stands for [ v i , k ] v k . The relations (7.10) are invariant in form under general transformations x ts Z of the x coordinates; this follows from (7.9) and the fact that the quantities [ a v i / d t ] and X i enjoy corresponding equations of transformation. Hence the relations (7.10) give the kinematical conditions of compatibility of the first order for the velocity relative to the general x system since these relations reduce to the proper compatibility conditions in a rectangular coordinate system. Corresponding to the conditions (7.2) the kinematical conditions of compatibility of the first order can be written symbolically as

[$]

=

-BG+

DA -7Dt

(7.11)

in which A and B are defined formally by the equations (2.1). The last term in the equation (7.11) is the absolute time derivative of the quantity A ; its components possess the same character, relative to coordinate transformations, as the components of the other quantities in (7.11) and will be given by a formula analogous to the above formula (7.8) for the components of the absolute time derivative of the discontinuity [v] in case A represents a covariant, contravariant or mixed tensor. The conditions (7.11) are therefore invariant under arbitrary differentiable transformations x ts 3 of the x coordinates as well as the transformations u t)ii of the curvilinear coordinates of the surface Z(t). Geometrical and kinematical conditions of compatibility of the second and higher orders relative to a system of rectangular coordinates, can evidently be extended to the general invariant form corresponding to the relations (7.2) and (7.11) for the compatibility conditions of the first order. Remark 1.

and (7.10) that

If the surface 2 is stationary, i.e. if G

=

0, it follows from (7.8)

8.

DYNAMICAL CONDITIONS O F COMPATIBILITY

55

But this relation is obviously an identity since, when G = 0, there can be no distinction between 6 and partial time differentiation. Similarly the general kinematical conditions of compatibility (7.11) are satisfied identically and hence can be disregarded in dealing with stationary surfaces 2. Remark 2. T o construct the general form of the relations (4.11), corresponding to (7.2) and (7.11), we first consider the equations of transformation of the components vi, namely (7.12) Taking the 6 time derivative of (7.12) we have

Now when we eliminate the second derivatives in these relations by means of the equations (7.6) and also eliminate the quantities 6 3 / 6 t by the substitution ( 7 4 , we find that the resulting equations can be written (7.13) where Dv;/Dt and Dp/Dt are the components of the absolute time derivative of the vector v in the x and systems respectively; these components are given by

Hence the required form of (4.11) is (7.14) since these relations are invariant under arbitrary differentiable transformations x f+ on account of (7.13) and moreover reduce to the conditions (4.11) when the xi are the coordinates of a rectangular system.

8. DYNAMICAL CONDITIONS OF COMPATIBILITY

When dealing with such quantities as the density, pressure, velocity, etc. of a dynamical problem other conditions, beyond those deduced in the preceding sections of this chapter, must be imposed on the discontinuities in these quantities and their derivatives over the moving sur-

56

11. CONDITIONS O F COMPATIBILITY

face Z ( t ) . For example, suppose that there is continuity in the pressure p , the density p and the components vi of the velocity across the surface Z ( t ) but that discontinuities occur in the derivatives of these quantities by hypothesis. Then from the equation of continuity, i.e. the equation (6.2) in Chap. I, it follows that

relative to a rectangular coordinate system. Other restrictions on the possible discontinuities in the derivatives will result from a consideration of the equations of motion. Such conditions may be called dynamical conditions of compatibility since they depend primarily on the basic dynamical equations which govern the behavior of the material medium. When these dynamical conditions a.re combined with the above geometrical and kinematical conditions of compatibility important information can be obtained regarding the propagation, growth and decay of various types of discontinuities or waves in material media, e.g. gases and solids. Applications of this theory will be confined to solids in the following chapters.

111. Waves in Elastic Media 1. STRESS-STRAIN RELATIONS We assume the generalized Hooke’s law which states that stress is a linear function of the strain; more precisely this means that the components uij of the stress tensor (see Sect. 7 in Chap. I) are related to the components eij of the strain tensor (see Sect. 2 in Chap. I) by equations of the form

C~~(x)e~m, (1.1) in which the coefficients C$“ are functions of the coordinates xi of the rectangular system which we shall employ in this discussion. Since the uij and ekn in (1.1) are the components of tensors one can evidently regard the quantities Ct,?’ as the components of a tensor C; moreover it can be assumed, without loss of generality, that the C$’ are symmetric in the indices k,m and also in the indices i,j since u and e are symmetric tensors. The medium in which the relation between the stress tensor and the strain tensor is given by (1.1) is called an elastic medium and the relations (1.1)are called the stress-strain relations of this medium. I n using the strain tensor e with components uij =

+

ekm = + ( U E . ~ ~ um.d, (1.2) in terms of the deformations rather than the more appropriate deformation tensor D,defined in Sect. 2 of Chap. I, it is implicitly assumed that the deformations which enter into consideration are so small that the products of the components ui and their derivatives can be neglected without appreciable error. This is a basic assumption of this theory of elasticity; it will be found to yield a certain simplification to our equations in the following work. Attention will be limited to elastic media which are (1) homogeneous and (2) isotropic. Roughly stated the condition of homogeneity means that properties of the medium are independent of position while isotropy 57

58

111. WAVES IN ELASTIC MEDIA

means that its properties are independent of direction. Using rectangular coordinates xi the first of these conditions implies that the coefficients Ciy in the stress-strain relations (1.1) are independent of the coordinates, i.e. the Cty are constants; the second condition requires that the coefficients Ctm retain their values under rotations of the coordinate system. A tensor C whose components possess such numerical invariance is called an isotropic tensor and it can be shown that the most general tensor C of this type has components of the form

+ p(s:s,m + S,tS;"),

cp = Ask"&,

(1.3) relative to a system of rectangular coordinates xi, where the quantities X and p are scalar functions of the coordinates (see Concepts, p. 68). Making the substitution (1.3) the stress-strain relations (1.1) now become cij = Xekdij -l- apei+ (1.4) It is readily observed that the scalar functions X and p in (1.4) must reduce to constants when the medium is both homogeneous and isotropic. Hence the most general set of stress-strain relations, satisfying Hooke's law, has the form (1.4) relative to a system of rectangular coordinates, in the case of a homogeneous and isotropic elastic medium, where the quantities X and p are material constants which are characteristic of the elastic medium under consideration. Remark. Instead of the above material constants X and p , which are usually referred to as Lamb parameters, one frequently employs certain elastic moduli such as Young's modulus El Poisson's ratio v, and the bulk modulus K . These moduli are positive constants and have a direct experimental determination for a given elastic medium. They are defined, in terms of the Lam6 parameters, by equations of the form

E= p

M X+P

X = 2(X

+ PI' 2

R = X + p

l

(Young's modulus),

(1.5)

(Poisson's ratio),

(Bulkmodulus).

(1.7)

In addition it may be stated that the Lamb parameter p is also known as the modulus of rigidity and is usually denoted by G in engineering work. For a discussion of these elastic moduli and a derivation of the above relations (1.5),

2.

DIFFERENTIAL EQUATIONS OF THE ELASTIC MEDIUM

59

(1.6) and (1.7) the reader is referred to any of the standard books on the theory of elasticity. Using (1.5)’ (1.6)’ and (1.7) the following equations X=

Ev

(1

p =

+ v)(l - 2v)’

-. E 2(1

+ v)’

K=

E 3(1

- 2v)’

can readily be verified. The first and second of these relations give the original constants X and /L in terms of Young’s modulus E and Poisson’s ratio v. It follows from the third relation that v < 1/2 since E and K are positive; hence 0 < v < 1/2 since v is positive.

2. DIFFERENTIAL EQUATIONS

OF THE

ELASTIC MEDIUM

The exact relations between the components vi of the velocity and the components ui of the deformation of an elastic medium are given by

in which there is a summation on the repeated index j . But the last term in the right member of (2.1) can be neglected in accordance with the observation in Sect. 1; removing this term from (2.1) and making a similar approximation to the components dvi/dt of the particle acceleration we can write aui vi = -, (approximately), (2.2) at

dt

-

a2ui

(approximately).

at2 ’

These substitutions can be made in the general dynamical equations of continuity and motion, derived in Sect. 6 and Sect. 7 of Chap. I, which we list here for convenient reference, namely

*+ at

+

p , ~ i pvi,i =

dvi

U O‘ .J’

- P dt’

0,

(equation of continuity),

(2.4)

(equations of motion).

The stress-strain relations (1.4) together with the above equations (2.4) and (2.5), in which the components v i and dvildt are given by (2.2) and (2.3) respectively, are the basic differential equations for the determination of the behavior of the elastic medium. If we like we can

60

111. WAVES IN ELASTIC MEDIA

eliminate the stress components uil between the relations (1.4) and (2.5) and thus secure a system of four equations in the four dependent variables u iand p. However, this elimination is not always advisable since one is directly concerned with the stress in many of the problems of the theory of elasticity.

3. THEELASTIC WAVE PROBLEM Consider a moving surface Z ( t ) , propagated into an elastic medium a t rest in its unstrained position, over which there may occur a discontinuity in the density, the stress or the velocity of the material particles. Along Z ( t ) the dynamical relations, derived in Sect. 8 of Chap. I must be satisfied, i.e. we must have pG =

j ( G - Vn),

(3.1)

(3.2) where we have denoted by v i the components of the unit normal vector to 2 ( t ) , directed into the unstrained portion of the medium, and G > 0 is the coordinate velocity of propagation of 2 in this direction. The barred quantities j and Vi denote density and velocity components on the rear side of 2 and the quantity V,, = &vi is the normal component of this velocity; in the following sections of this chapter the use of a bar over a quantity or its derivative will denote a similar evaluation. As in our previous work we use the bracket [ ] to denote the difference in the values on the two sides of Z ( t ) of the quantity enclosed by the bracket. It will be understood, for definiteness, in the above equations (3.2) and in the following equations of this chapter, that the bracket [ 3 is defined as the value of the quantity in question on the rear of Z minus its value on the front of this surface. This latter value will usually be zero due to the fact that the medium on the front side of Z is unstrained and at rest; in such a case the bracket expression will merely give an evaluation of the quantity enclosed on the rear of the surface 2 . Thus the above relations (3.2) are equivalent to the relations [Ui,]Vj =

-p G [ v ; ] = -pGiii,

a.v’ +I = -pGii,. We shall assume that [ui]= 0 or, in other words, that there is no separation or sliding of the medium over the moving surface Z(t). The deformation is therefore continuous over the surface Z(t) ; however it

4.

STRENGTH AND VELOCITY OF WAVES

61

will be assumed explicitly that there is a discontinuity in the spatial derivatives ui,j across Z(t), i.e. that not all of the quantities [ui.j] vanish. Such a surface 2 ( t ) will be said to be singular of order 1 relative t o the deformation, or to be a wave surface of order 1 in the elastic medium (see Remark 2 in Sect. 3 of Chap. 11). I n the following section we shall derive expressions for the velocity G of this wave surface Z ( t ) and in the later sections of this chapter we shall treat the more difficult problem of the variation of the strength of the wave (defined in Sect. 4) during its propagation.

4. STRENGTH AND VELOCITY OF WAVES Since [ui] = 0 by the assumption in the preceding section it follows from the compatibility conditions of the first order in Chap. I1 that we must have relations of the form [Ui,j]

=

[%]

= -Gwi,

W ' V 3" 1

(4.1)

where the wi are functions defined over the surface 2(t). Not all of the functions w ; can vanish from the assumption that Z ( t ) is a wave surface of order 1. It is therefore natural to define the strength W of this wave surface by the equation

w = G. Then W = 0 implies W ; = 0 and conversely; hence if W = 0 the surface Z ( t ) will not be a wave surface of order 1 as postulated. From (2.2) and the second set of relations (4.1) we now obtain

Using these relations (4.2) it follows from (1.4)1 (3.2) and (4.1) that (A

+

+

(4.3) Assuming that o,v; does not vanish on Z ( t ) let us multiply (4.3) by vi and sum on the repeated index i. This gives G2

P ) W ~ V ~ V ~p

=A ___ + 2@, P

~ = i pG'~i.

(WiUi

# 0).

(4.4)

On the other hand if wivi = 0 on Z ( t ) we see directly from (4.3) and the condition that not all the w i can vanish, that

62

111. WAVES I N ELASTIC MEDIA

G 2 = 'I,

(wivi =

P

0).

(4.5)

Substituting the value of G 2 given by (4.4) into (4.3) and taking account of the fact that wivi # 0 the resulting relations become Wi

=

WVi;

W

= W k v k # 0.

Hence from (4.1) we have [u. .] = a.5.3. = w y - y3s. 1.1

with

w

# 0.

(4.6)

7

But from the first set of relations (4.7) we see that

ui*j- aj,j = 0. Hence the rotation (see Remark 2 in Sect. 6 of Chap. I) vanishes immediately behind the wave front in the case of a wave whose velocity is given by (4.4). Correspondingly for waves whose velocity is given by is equal to zero (see Remark 1 in Sect. 6 of (4.5) the dilatation Chap. I) immediately behind the wave front. Adopting the terminology of irrotational and equivoluminal waves in accordance with these observations we therefore have ~2

=

(for irrotational waves), w, P

G2

=

,!

(for equivoluminal waves).

P

It follows immediately from (4.2) that ijn = 0 for the equivoluminal wave. Hence (3.1) yields [ p ] = 0. However in the case of the irrotational wave we find from (4.2) that ij,, = -Gw and hence (3.1) becomes [ p ] = -bw with w # 0. Hence the density i s continuous across the equivoluminal but discontinuous across the irrotational wave. Remark. 1. Let us observe how the above results will be modified when we use the exact relations (2.1) instead of the approximate equations (2.2). Thus, evaluating (2.1) on the rear of the wave surface Z ( t ) , we obtain

4. STRENGTH

AND VELOCITY OF WAVES

63

I n place of (4.3) we now find from (1.4), (3.2), (4.1) and (4.8) that (4.9) and from these relations we obtain 6 2

=

6 2

=

e--x + P P -

" 9

EE,

if

WkVk

# 0,

(4.10)

if

OkVk

= 0.

(4.11)

PP

Substituting the value of G2 given by (4.10) into (4.9) we again deduce the relations (4.6). As above this leads to the irrotational character of the waves whose velocity is given by (4.10) while the formula (4.11) now gives the velocity G of the equivoluminal waves. From (4.8) we obtain G (4.12) =

-%

(wkvk).

P

Hence Vn = 0 for the equivoluminal waves and it follows from (3.1) that [ p ] = 0. Thus p = p and the equation (4.11) reduces to the previous equation (4.5) for the velocity G of the equivoluminal wave. In- the case of the irrotational wave (4.12) becomes

with w # 0, and hence from (3.1), we have

p

= (1

- w)p.

Hence (4.10) can be written G2

= (1 -

x + 21L

W ) *-

(4.13)

P

But when we neglect the small quantity w in comparison with unity the equation (4.13) reduces to the above equation (4.4) for the velocity G of the irrotational wave. Remark 2. Denoting by X i the components of the unit vector in the direction of the flow in the region behind an irrotational or equivoluminal wave Z ( t ) we have

A . - vi.

' - z/o.o,

Immediately behind the wave surface Z ( t ) these equations become (4.14)

64

111. WAVES I N ELASTIC MEDIA

when use is made of the relations (4.2). But from (4.14) we readily obtain

-

xi

=

xivi =

- vi,

(for the irrotational wave),

(4.15)

0,

(for the equivoluminal wave).

(4.16)

Hence the direction of motion of the material particles immediately behind the wave surface Z ( t ) is normal to this surface on account of (4.15) for the case of an irrotational wave and tangent to it from (4.16) when Z ( t ) is an equivoluminal wave. Accordingly the above irrotational and equivoluminal waves are also referred to as longitudinal and shear waves respectively. 5 . WAVESAS PARALLEL SURFACES

Since the velocity G is constant for the irrotational and equivoluminal waves under consideration the following result can be stated. The successive positions of the irrotationd and equivoluminal wave surfaces Z(t) jorm a family of parallel surfaces in space (see Remark in Sect. 4, Chap. 11). By laying off equal lengths along the normals t o the wave surface Z(to) in the direction of the propagation one can therefore construct the surface Z ( t ) a t times t > to.

6. COMPATIBILITY CONDITIONS OF

THE

SECOND ORDER

Since the deformation is continuous over the wave surface Z [ t ) the relations (5.11), (6.10), and (6.11) of Chap. I1 are applicable and we can write [tLi,jk]

=

biViVk

+

f'Wi,a(V$$

+

vkxb)

- W@Bgr'bacx$x~,

(6.1)

(6.3) where the 2 , are new functions defined over t,he surface Z ( t ) and the b . ~are the components of the second fundamental form of this surface. I n constructing the relations (6.2) and (6.3) we have made use of the fact that the velocit!yG is constant for the irrot'ational and equivoluminal waves under consideration. Certain special relations will now be deduced for application in the

7.

65

IRROTATIONAL WAVES

following sections. Thus if we contract the indices j , k and also the indices i,j in (6.1) we obtain [Ui,kk]

where

= iji

- 2Q(Ji,

(6.4)

is the mean curvature of the surface Z ( t ) , and

Other special relations are obtained by multiplying (6.4) and (6.5) by and summing on the repeated index i. This gives

vi

[Ui,kk]vi

=

Gkvk

-2kkvk1

+

(6.6)

t @Ok,ax#?-

(6.7) Finally from the equations of motion (2.5) together with the relations (6.61, (6.7) and (6.8) [gii.j] = (A II) [ ~ k . k i ] ~ [ ~ i , k k l , which follow immediately from the stress-strain relations (1.4), we oan deduce [Uk,ki]vi

=

akvk

+

+

OF THE COMPATIBILITY CONDITIONS 7. APPLICATION TO IRROTATIONAL WAVES

We shall now derive the differential equation for the variation in the strength W of an irrotational wave during its propagation. We have W = ( w j from (4.6) and the definition of the wave strength W i n Sect. 4. Also from (4 6) we have Wk,a

=

m.aVk

+

uvk,a.

Hence the last term in the left member of (6.9) becomes (X

+

P)W$L’vk,ax: =

-(X

+

P)w$ogarbaox;x:,

in view of the equations (1.3) and (1.4) of Chap. 1 1 . But, when use is made of the relations (1.2) and (1.5) of Chap. 11, this expression reduces to - (A P)wp’g‘rbocg’, = - ( A P)W$LBb,’ = -2(X P)Wcl. Making this substitution and also the substitution W h V k = w , the above equation (6.9) now takes the form

+

+

+

66

111. WAVES IN ELASTIC MEDIA

(X

+ 2/.I)(ijkYk- 2wQ) = [ ?-'t ] p

(7.1)

vi.

By differentiation of (2.2) we obtain

Neglecting the last term in (7.2) these relations become identical with the approximate relations (2.3) ; however the retention of this term will not impair the analysis and it will therefore be retained, as is our privilege, in the following discussion. From (7.2) we now find that

when use is made of (4.7), (6.2) and (6.3). Or we can write

[%] (1 + w)G2& - (2 + w)G 6 0 =

vi.

(7.3)

Recalling that the brackets, which are here involved, represent evaluation on the rear of the wave surface, it follows from (7.1) and (7.3) that

+ 2/1)(Opk - 2wa) = (1 + w)G~PO~V, - (2 + U)GP 6t (7.4) But the product (1 + w ) p is equal to the density of the unstrained 6W

(X

-*

p

material as we observed in Sect. 4. Making this substitution and also substituting the value of G2 given by (4.4) the relation (7.4) reduces to

Let us now denote by Q 2 0 the distance measured along the normals to the surface Z(to) mentioned in Sect. 5 . Then

When we make this substitution and again avail ourselves of the relation (4.4), the equation (7.5) becomes

Neglecting the small dimensionless quantity w in comparison with unity and making the substitution w = fW , the equation (7.6) yields

8.

67

EQUIVOLUMINAL WAVES

dW --

- WQ,

da

(7.7)

as the differential equation for the variation of the strength W of the wave during its propagation.

8. APPLICATION OF

THE

COMPATIBILITY CONDITIONS WAVES

TO EQUIVOLTJMINAL

It follows readily from (2.3), (2.5) and (6.8) that

+

(A

P)[Uk,ki]

+

a2Ui

k"ui,kkl

=P

[TI'

(8.1)

when use is made of the fact that the density p is continuous across the equivoluminal wave (see Sect. 4). Now differentiate the relation WkVk = 0,which is valid over the surface of the equivoluminal wave, to obtain = -Wkvk.a

wk,orvk

= OkgUrboaXt.

When this substitution is made in the second term in the right member of (6.5) it is easily seen that the relations reduce to [Uk,ki]

= (bkvk

+

(8.2)

~'Wk,&$)Vi-

Hence from (6.3), (6.4), (8.1) and (8.2) we have (A

+

p)(bkvk

+

gaBWk,ax$)Vi

=p

(G2bi

+

p(bi

- 29Wi))

1

- 2G

When we multiply (8.3) by wi and sum on the repeated index i the resulting relation becomes p(&Ok

- 2QWkWk)

=

pG2bkWk

Wk - 2 p G w k 8-.61

(8.4)

But, making the substitution

where a is the arc length along the normals to the surface 2(to)as in Sect. 7, and also taking account of the relation (4.5), valid for the equivoluminal wave, we find that (8.4) reduces to

68

111. WAVES IN ELASTIC MEDIA

Finally, when we introduce the wave strength W, defined in Sect. 4, the relation (8.5) gives the previous equation (7.7); hence (7.7) i s the diferential equation for the variation in the strength W of equivoluminal as well as irrotational waves.

9. DECAYOF WAVES

If t,he wave surfaces Z ( t ) form a family of parallel planes the mean curvature D vanishes a t all times t and hence it follows from the differential equation (7.7) that plane irrotational and equivoluminal waves will show n o variation in their strength during propagation. Another special case which merits consideration is that for which 2: ( t ) represents a family of concentric spherical surfaces propagated outward, i.e. such that the radius R increases during the propagation. Now we have chosen the unit vector v to have the direction of propagation of the wave surface. Hence v is the outwardly directed normal to the spherical surfaces and in consequence of this fact the mean curvature 52 has the negative value - 1/R rather than the usual positive value 1/R which results when the unit normal is directed toward the center of curvature of the spheres. Evidently u can be replaced by R in the equa.tion (7.7); hence this equation becomes

Integrating we find that the variation in the wave strength W of an irrotational or equivoluminal wave is given by

where WOis the wave strength when the sphere has the radius Ro. It can be shown in general that

where KO and denote the Gaussian and mean curvatures of the wave surface Z ( t o )from which the arc length u is measured (see Concepts, p. 112

9.

69

DECAY O F WAVES

and p. 113). Making the substitution (9.1) in (7.7) and integrating the equation we now find that

W =

wo

dl - 2noa

+ Koa2

1

(9.3)

where W ois the wave strength over the surface Z(t0). It is easily seen from (9.3) that the strength W of an irrotational or equivoluminal wave will decrease monotonically to zero as u -+ m , i.e. the wave will decay if, and only if, one of the following two conditions is satisfied, namely Qo < 0, (a) KO= 0; (b) KO > 0 ; no < 0. The other algebraic possibility, namely KO > 0 and QO = 0, cannot be realized since if Q,, vanishes on Z(t,,) this surface will be a minimal surface and for such a surface the curvature KOmust be negative unless the surface is a plane (see Concepts, p. 114). If the condition (a) is satisfied the surface 2(to)will be a developable surface (not a plane) and when condition (b) holds it will be a surface of positive curvature. Moreover we see from (9.1) that under condition (a) or (b) the mean curvature Q will remain negative during the propagation. Similarly it follows from the equation (9.2) for the variation of the curvature K along the normals to the wave surface z(t,),that K = 0 under condition (a) and K > 0 under condition (b). Hence the wave surface Z ( t ) will be a developable surface or a surface of positive curvature throughout its propagation if the surface 2(to)is a developable surface or a surface of positive curvature. With the above facts in mind we can now state the following result. T h e strength W of a n irrotational or epuivoluminal wave Z ( t ) , propagated into an unstrained elastic solid, wilt approach zero monotonically a s the time of propagation increases indeJinitely f r o m a n initial time to i f , and only i f , the wave surface Z(to) i s ( a ) a developable surface (not a plane) or (p) a surface of positive curvature and ( y ) the direction of propagation of Z(t0) i s away f r o m the centers of curvature at the points of this surface. This last requirement is needed to secure the inequality Oo < 0 in the above conditions (a) and (b) ;it was also used in dealing with the special case of concentric spherical waves.

IV. Perfectly Plastic Solids 1. CHARACTERIZATION OF PERFECTLY PLASTIC SOLIDS

Experimental evidence indicates that plastic deformation is produced in negligible amount, if at all, as the result of hydrostatic pressure. I n constructing a system of relations, analogous to the stress-strain relations of the theory of elasticity, i.e. the equations (1.4) of Chap. 111, for the plastic flow of a solid, it is therefore natural to assume that the previous role of the stress tensor u is played essentially by the stress deviator u* since changes in hydrostatic pressure have no effect on this latter tensor (see Remark 3 in Sect. 7 of Chap. I). With this basic requirement in mind the various special conditions, involving the deviators u* and e* of the stress and rate of strain tensors, for the characterization of the flow of a perfectly plastic solid are embodied in the following assumptions. A,. T h e Jlow i s incompressible, A,. T h e components of the deviator of the stress tensor are proportional to the components of the deviator of the rate of strain tensor, AS. T h e coeficient in the proportionality relating the components of the deviators of the stress and rate of strain tensors i s a positive scalar invariant of the deviator of the rate of strain tensor, ’ Ad. There does not exist a (1,l) correspondence between the components of the deviators of the stress and rate of strain tensors. Assumption A1 appears t o be in good agreement with experiment and is consequently included among the ideal requirements for the perfectly plastic solid; on the basis of this assumption we can write

vi,i = 0, (equation of incompressibility), (1.1) where the vi are the velocity components of the material particles relative to rectangular coordinates xi which will be used exclusively in this chapter (see Remark 1 in Sect. 6 of Chap. I). As a consequence of condition (1.1) the rate of strain tensor e will be identical with its deviator 70

2. e*.

STRAIN RELATIONS AND THE YIELD CONDITION

71

I n accordance with assumption A, we have equations of the form

4 = $4, (1.2) in which the factor of proportionality cp is a positive scalar invariant of the deviator e* from assumption AS. Finally &sumption Ah expresses one of the essential distinctions between the ideal behavior of plastic and elastic solids. It will be seen in the following section that this assumption leads to a determination of the form of the scalar invariant 4. 2. STRESS-RATE OF STRAIN RELATIONS AND THE YIELDCONDITION Denote by 7; the principal values of the stress deviator u* and by qt the corresponding principal values of the deviator e* of the rate of strain tensor. But q; = qi because of the identity of the tensors e and e* as observed in Sect. 1, where the qi are the principal values of the rate of strain tensor E ; hence it follows that

h, (2.1) on account of the relations (1.2) and the fact, which is readily seen, that a system of canonical coordinates yi, defined on the basis of the stress tensor as in Sect. 10 of Chap. I, will also constitute a system of canonical coordinates for the stress deviator u* and the rate of strain tensor e. Hence we will have 7;

=

(inot summed),

(2.2) at yi = 0, where we have used q i j to denote the components of the rate of strain tensor relative to the y coordinate system. Now the principal values T i are given as the three solutions q, which are necessarily real, of the determinantal equation qij = q&,

le..

- $..I

+ + 3r

0, (2.3) in which we have availed ourselves of the simpIicity afforded by the condition that ei; vanishes from (l.l),and where 23

v

= -q3

-

+&I

=

5 = 613. f 13, . 5 = Eikekjeij. (2.4) Hence the principal values q1,q2,t/3 are algebraic functions of the scalar invariants 5 and of the rate of strain tensor. By the process of transforming the scalar +, considered as a function of the components e i j of the rate of strain tensor, to the canonical system

r

72

IV. PERFECTLY PLASTIC SOLIDS

and evaluating at the origin of this system, it is seen from (2.2) that 4 can be expressed as a function of the principal values 7. It follows therefore that 4 is a function of the invariants f and { of the rate of strain tensor from the above italicized statement. Let us now put f* = u:ju;j; {* = u:ka$J:j, (2.5) corresponding to the definition (2.4) of the scalars f and {. Then from the relations (1.2) we have

f*

+YE,{)f; l*= 4 3 ( f , { ) { .

(2.6) Assuming the continuity and differentiability of the scalar 4 as a function of the components cij of the rate of strain tensor, it is easily seen that the above representation $ ( f , Z ) will be continuous and differentiable as a function of the quantities f and {; from the implicit function theorem the equations (2.6) will therefore have a unique solution f ( f * , { * ) , { ( f * , { * ) provided the functional determinant A of the right members of these equations does not vanish. But then 4 becomes a function of f* and {* and the equations (1.2) will have a solution E:~(U*) which contradicts the assumption 114 in Sect. 1. We must therefore have A = 0 to avoid this contradiction; hence, equating A to zero, we find =

To solve the equation (2.7) we first make the transformation f

=

e22;

{ =

e3yl

of the variables f , { and follow this by the transformation x - p. Then (2.7) becomes

= a,

'y = a

and this equation has the general solution

4 = A(P)e-", where A @ ) is an arbitrary differentiable function of p. Returning to the original variables f and 5, we thus find that the general solution of (2.7) is given by

2.

73

STRAIN RELATIONS AND THE YIELD CONDITION

where M is an arbitrary differentiable function of the ratio Hence the relations (1.2) become u:j =

M

(G/G ) cij. d/eabeab

*/G. (2.9)

From the relations (2.9) we can deduce the two relations (2.6) which can be combined to give

Hence, replacing the argument %/dgof the equations (2.9) can be written u;, =

M by the ratio

M ( a / d m )

6

Eij.

+?/*, (2.10)

Kow multiply each member of (2.10) by itself and sum on the repeated indices i and j . We thus obtain (2.11)

The equations (2.9) are called the stress-rate of strain relations and the equation (2.11) is called the yield condition for the perfectly plastic solid. The function M which appears in the yield condition (2.11) and the stress-rate of strain relations (2.9) is a material function which characterizes the perfectly plastic solid under consideration. After the choice of this function has been made we have a system of 10 equations consisting of the equation of continuity (6.2) and the equations of motion (7.4) of Chap. I, the above equation of incompressibility ( l . l ) , the stress-rate of strain relations (2.9) and the yield condition (2.11) for the determination of the 10 dependent variables, namely the density p, the velocity components v i and the components u;j of the stress tensor. Remark 1. I n the above theory of the perfectly plastic solid i t is assumed that the velocity u of the material particles is produced entirely as the result of plastic deformation within the medium. By failing to make provision for the occurrence of elastic deformations in a region of plastic flow we must for consistency, if not from necessity, assume that such deformations likewise vanish in non-plastic regions, i.e. we must assume that the medium under consideration

74

IV. PERFECTLY PLASTIC SOLIDS

is rigid when not deformed plastically. The term riggid-plastic will be used to describe the hypothetical material of this character. Remark 2. Assume M = const. which is the simplest choice one can make for the material function M . Thus, taking M = d z k , where k is a positive material constant for the perfectly plastic solid, the stress-rate of strain relations (2.9) and the associated yield condition (2.11) become (2.12) (2.13)

a?. 51, ?.af = 2p.

The equations (2.12) are the well known stress-rate of strain relations of Saint Venant-Levy and the equation (2.13) is the quadratic or von Mises yield condition. Remark 3. A generalization of the above theory of the perfectly plastic solid is obtained if we retain the assumptions A1 and & b u t replace the assumptions A2 and A3 by the assumption that (2.14) where the 9ii are the components of a tensor invariant 9 of the rate of strain deviator u*; obviously the tensor 9 must be symmetric and such that 9 i i vanishes since the stress deviator u* has these properties. As before the tensors E and E* are identical from assumption A1. The assumption & now asserts that the relations (2.14) cannot be solved uniquely for the components ETj or EC, in terms of the components u;, of the stress deviator. It can be shown that any symmetric tensor of the second order, such as the tensor 9, which is an invariant of some other symmetric tensor of the second order, e.g. the tensor E*, under the group of proper orthogonal transformations, is expressible by relations of the form 9ij = P&j

+

QETj

+ Re&,

(2.15)

where P , &, R are scalar invariants of the tensor E*. See R. S Rivlin, and J. L. Ericksen, Stressdeformation relations for isotropic materials, Jour. Rational Mech. and Anal., 4, 1955, pp. 323-425; also R. S. Rivlin, Further remarks on the stressdeformation relations for isotropic materials, ibid., 4, 1955, pp. 681-701. But, putting i = j in the above relations (2.15) and summing on the repeated indices, we must have 3P

+ RE:&

=

0.

(2.16)

Replacing the 9ij in (2.14) by the expressions given by (2.15) and then eliminating the scalar P by means of (2.16), we now find that (2.17)

2.

STRAIN RELATIONS AND THE YIELD CONDITION

75

But since Q and R are scalar invariants of the tensor e* (or the tensor e) they must be functions of the scalars f and [ as observed in the preceding discussion. Hence the most general relationship (2.14) between the tensors q* and e* i s givesz by (2.17) in which Q and R are functions of the scalars t and {. As a first step in the determination of the explicit form of the scalar inv&riants Q,R and the yield condition associated with the stress-rate of strain relations (2.17) let us define the invariant I , by the equation In =

7;

+ d + 4,

for n 2 2, where 71,12,73 are the principal values of the tensor e. Now by the use of canonical coordinates it is readily seen that we have relations of the type

Iz= f 1 3

= eC. ?. e$. 1. ,

={=

eijejkeik,

(2.18)

... ...

J

Each invariant I,, is a function of the two invariants and { by the result in the above discussion; calculation of these functions for the invariants 14, IS and I6 leads to the following relations (2.19)

For example, from equations (2.3) we have 1,"=

4 f 2 + %%*

Taking i = 1,2,3 successively and adding correspondingmembers of these equations we immediately obtain the first equation (2.19). The second and third relations (2.19) can be deduced in a similar manner. The formal construction of equations corresponding to (2.6) will initially involve homogeneous scalar invariants formed from the components eij. These homogeneous invariants can be identified with the invariants f , {, 14,16 and IS on account of the relations (2.18) and the invariants la, IS and I6 can then be eliminated by means of (2.19). We thus obtain

The procedure is now analogous to that previously employed in deducing the relations (2.9) and (2.11). We first observe that the functional determinant A

76

IV. PERFECTLY PLASTIC SOLIDS

formed by differentiation of the right members of (2.20) must vanish in consequence of assumption A*. This gives a partial differential equation A = 0 for the determination of Q and R as functions of the variables [ and 1. Corresponding to any solution &([,[) and R([,[)of this differential equation, the quantities [* and I*given by (2.20) will be functionally dependent. The equation expressing this functional dependence will be the yield condition associated with the stressrate of strain relations (2.17) in which Q and R are the functions of [ and { satisfying the above differential equation.

3. FORMULATION IN TERMS OF PRINCIPAL VALUES Let us again consider the relations (2.1) between the principal values of the stress deviator CT* and the principal values 9i of the rate of strain tensor e. We suppose these principal values t o be labeled so that 72

7;

2

7;

2

7.3;

71

2 972 2

73.

I n fact if the subscripts on 7; or 7, are chosen so that one of these sets of inequalities is satisfied, as is evidently possible, the other set of inequalities will follow from (2.1) since 4 > 0 by hypothesis. Instead of the two variables C; and [ used in the discussion of the perfectly plastic solid in Sect. 2 we can now select two of the three principal values 7; let us choose these to be 9l and q3. The intermediate principal value q2 in the above inequalities will then be determined by 91 and q3 on account of the relation 91 72 73 = 0, (3.1)

+ +

which is equivalent to the equation of incompressibility (1.1) and follows formally from (2.2). It was observed in Sect. 2 that 4 can be expressed as a function of the principal values q ; hence 6 can be regarded as a function of the two variables q1 and q3 by the remark a t the end of the preceding paragraph. Because of the relation (3.1) and the corresponding relation in the quantities 7; we can confine our attention to the two equations 7;

= 4q1;

7.3

= 473,

(3.2)

obtained by taking i = 1,3 in (2.1). On the basis of the discussion in Sect. 2 we can now immediately state th at assumption A requires the vanishing of the functional determinant of the right members of the equations (3.2), i.e.

3.

FORMULATION I N T E R M S O F PRINCIPAL VALUES

77

Expanding the determinant in the above relation, we obtain the simple equation a4 a4 7717734 = 0, (3.3) a771

+

a773

+

for the determination of 4 as a function of the variables ql and q3. It is readily seen (cf. Sect. 2) that the general solution of (3.3) is given by an equation of the form

where Q is an arbitrary differentiable function of the ratio q3/fll. (1.2) becomes

Hence

But from (3.2) we have

5 = 113, '6

771

Hence, when we subtract corresponding members of the two equations (3.2), eliminate the quantity 4 by means of (3.4) and take account of the above equality (3.6), we find that

- T$

(3.7) The following result has now been proved. When the stress-rate of strain relations of a perfectly plastic solid are written in the form (3.5) the associated yield condition i s (3.7) in which Q i s a material function for the solid. T?

= !P(T$/T;).

Remark 1. If, in particular, we take !P strtnt, the equations (3.5) and (3.7) become

a?.=

'

T:

=

2k, where k is a material con-

771

- 773

- r: = 2k.

The condition (3.9), which is associated with the stress-rate of strain relations (3.8), is known as the Tresca yield condition for the perfectly plastic solid. One

78

IV. PERFECTLY PLASTIC SOLIDS

can readily observe that the difference 7; - 7'3 is equal to the dierence TI - r3 of the corresponding principal values of the stress tensor. Hence the Tresca yield condition (3.9) can also be written as 71

- 7 3 = 2k.

Remark 2. T h e constant k in (2.13) and (3.9) i s the yield stress in simple shear. Thus for simple shear we can choose the coordinate axes so that u12

=k;

uij

=

0 (otherwise),

(3.10)

where we assume k to be the value of ui2for which yield occurs. Hence ufz = k and the other components u;, are equal to zero. Substituting these values of u:, into (2.13) we find that the equation is satisfied identically. Similarly for the simple shear (3.10), producing yield when u1z has the value k , the principal values r* are 7; = k ; 7; = 0 ; r; = -k from which we see that the equation (3.9) is likewise satisfied. Now consider a simple tension s, the coordinate axes being chosen so that uzz= s ;

uij

=

0 (otherwise).

(3.11)

Using the values (3.11) of the u,j we find that the non-vanishing deviator components u:, and the principal values r* are u*1 1 ---.--s3 1

U*zz =

2s

3;

UZa

=

--s 7

(3.12) (3.13)

Substituting the values (3.12) into (2.13) and the values (3.13) into (3.9), we obtain s =fik;

s = 2k,

respectively. Hence the ratio of the yield stress in simple tension and simple under the quadratic yield condition (2.13) and the value 2 shear has the value under the Tresca yield condition (3.9).

4

4. PRANDTL-REUSS EQUATIONS To take account of the elastic as well as the plastic deformations which may occur within a solid we assume that it is legitimate to represent the components v i of the total velocity of the material particles in the form Di = 6 iri, (4.1)

+

4.

79

PRANDTGREUSS EQUATIONS

where the Vi and 0,denote respectively the components of purely elastic and purely plastic contributions t o the total velocity. Accordingly the quantities vi must be derivable from deformations ii which satisfy the stress-strain relat.ionsof the theory of elasticity (see Sect. 1of Chap. 111) while the 0i must sat.isfythe above stress-rate of strain relations for flow in a perfectly plastic solid. This gives us specifically the following two sets of relations, namely (4.2) and 8’ ‘ = 0; 10%

u;* =

&j,

in which the 2ij are the components of the rate of strain tensor based on the b i and the other quantities in these equations have their previous significance. Our problem is now to obtain, if possible, a set of equations which will suffice for the characterization of the flow within the solid and which will involve, as far as velocity is concerned, only the components v i of the total velocity of the material particles since only this velocity is susceptible to actual observation. The first set of relations in (4.2), i.e. the elastic stress-strain relations are equivalent to the two sets of equations u;j

=

2pz;j;

u*; =

(3X

+ 2p)G&<,

(4.4)

where the quantities Zj are the components of the deviator of the strain tensor Z for the elastic deformations G, i.e. -*

eu

=

Z,j -

&&;

eij =

+(iii,j

+

Gj,i).

Differentiating (4.4)with respect to the time and neglecting small quantities of the second order, when these occur, we now find that

80

IV. PERFECTLY PLASTIC SOLIDS

where the Vi are given by (4.2) and t,he Gj are t,he components of the deviator c* of the rate of strain tensor ;for the elastic deformations; the tensors ;* and ;are defined by -*

Elj

=

cij

- +elilisij;

= i(Vi,j

clj

+ q,;).

Now it follows from (4.1) and the first equation (4.3) that 2,.

.-iJ..+s..=c.. 1,Z

1.)

%,1-

2,1*

Hence we can replace the quantity U i , i by v , . ~in the right member of the second equation (4.5). When this substitution is made and we take account of the relations between the elastic moduli given in the Remark in Sect. 1 of Chap. 111we find that the equation can be written

where E is Young’s modulus and v is Poisson’s ratio; the relation (4.6) is one of the desired equations for the determination of the flow in this problem. The remaining relations (4.5), i.e. the first set of these relations, will yield other equations of the required type. Thus from (4.1) we immediately obtain e;j

=

;;j

+

2;j

=

Gj

+

&j,

(4.7)

on account of the first equation (4.3). Eliminating the quantities Gj from the first set of equation (4.5) by means of (4.7) and then eliminating the 2ij by means of the second set of equations (4.3), we now have

in which $ is formally t,he reciprocal of the above proportionality factor 4; hence $ is a positive quantity by hypothesis (see assumption A, in Sect,. 1). I t remains to show that, in addition to the stress, the quantity rl. depends only on the total velocity of the material particles. But, differentiating the last equation (4.3) with respect to the time we have

where M’ denotes the derivative of the material function M with respect to its argument

4.

PRANDTL-REUSS EQUATIONS

81

Carrying out the operation of time differentiation in the right member of (4.9) and eliminating the derivatives da;j/dt which arise by means of (4.8) we find, after some reduction, that the resulting equation can be written in the form

To the equations (4.6) and (4.8) we must now add the last equation in (4.3), i.e. the yield condition (4.11) Then the system comprised of (4.6), (4.8) and (4.11), together with the equation of continuity (6.2) and the equations of motion (7.4) of Chap. I, will give us 10 independent equations for the determination of the 10 variables consisting of the density p , the three velocity components v ; and the six components aij of the stress tensor. The above system of equations (4.8), in which the quantity 4 is given by (4.10) and which is associated with the yield condition (4.11), may be referred t o as the generalized Prandtl-Reuss equations. Since the hypothetical contribution of the elastic deformations was not neglected in constructing the equations of this theory we assume, for consistency, that such deformations may occur in regions of the solid not in a state of actual plastic flow, i.e. that the equations of the theory of elasticity are applicable in such regions. A solid of this character is referred t o as a n elastic-plastic body by some authors. Remark 1. Taking it! = v‘%, where k is a material constant, the equations (4.8) become the usual Prandtl-Reuss equations in connection with which we have

(4.12) The equations (4.12) are obtained immediately from (4.10) and (4.11) as a consequence of the above constant value of the material function M . Remark 2. Using the relations between the elastic moduli given in the Remark in Sect. 1 of Chap. 111, we can write the second equation (4.4) as

82

IV. PERFECTLY PLASTIC SOLIDS ~ i=;

3KTi.8 , s..

(4.13)

Now suppose that the bulk modulus K -+ m ; then Ti<,; -+ 0, assuming that u,i remains finite. But the vanishing of G,i means that the body is elastically incompressible (see Remark 1 in Sect. 6 of Chap. I). Hence the body must be completely incompressible since the condition of plastic incompressibility, i.e. &,; = 0, is imposed by hypothesis. Such a solid is called an incompressible elastic-plastic solid and may be assumed in certain problems in which the elastic compressibility can be neglected. In an incompressible elastic-plastic solid the equations (4.4) must be replaced by u:, = 2&; u;,i = 0, (4.14) where we have omitted the bar over the components u;of the elastic deformations and hence over the components e:, of the deviator, as no longer needed. The equations (4.14)are valid in purely elastic regions within this solid and, as we see, they involve only a single material constant, e.g. the modulus of rigidity I*. Similarly the equation (4.6)must be replaced by the equation va.,* . = 0

(equation of incompressibility),

(4.15)

while the remaining equations governing the plastic flow remain unchanged. 5. STRUCTURE OF CONSTITUTIVE EQUATIONS

Conditions such as those imposed by the stress-strain relations (1.4) of Chap. 111, the above stress-rate of strain relations (2.9) and the set of relations consisting of (4.6) and (4.8), are frequently referred t o by the generic name of constitutive equations. Basically the constitutive equations express relations between the internal stresses and deformations and hence are, obviously, not effected b y arbitrary rigid motions of the body as a whole. Thus any legitimate set of constitutive equations, assuming that the stresses and deformations in the solid are referred to a system of rectangular coordinates, must satisfy the following requirement. T h e constitutive equations are invariant under the group G of transformations relating the coordinates of arbitrarily moving rectangular coordinate systems. It is readily observed that the stress-strain relations (1.4) of Chap. I11 satisfy the above invariance requirement since (1) the quantities uij are the components of a tensor under the group G and (2) the ui,j and hence the eij are tensor components due t o the fact that the deformations ui are the components of a vector under this group of transformations.

6.

KINEMATICALLY PREFERRED COORDINATE SYSTEMS

83

Now it will be shown in the Remark 2 in Sect. 7 that the quantities eii in (2.9) are the components of a tensor under the group G ; hence the stress-rate of strain relations (2.9) also have the required property of invariance. Since the eij are tensor components under the group G as stated above, it follows that the right member of (4.6) is a scalar under the transformations of this group; but the left member of (4.6) is also of scalar character from the following relation (7.11) and hence the equation (4.6) is invariant under transformations of the group G. Similarly the right members of (4.8) are of tensor character under the group G of coordinate transformations. However the left members of these relations do not exhibit a corresponding behavior and consequently the equations (4.8) fail to possess the property of invariance postulated in the above italicized statement. The difficulty regarding the equations (4.8) is not as serious as one might suppose a t first sight, I n fact this difficulty is entirely removed if one adopts the viewpoint, or interpretation of these equations, explained in Sect. 8. Another procedure would be to employ the covariant time derivative, defined in Sect. 7, in the above discussion rather than the total time derivative; one would thus arrive directly at the correct dynamical form of the relations (4.8) without modification of the formal derivation in Sect. 4. Apart from the requirement of invariance which must be imposed on a system of constitutive equations, it is evident that operations involving such systems will be facilitated by the formulation of a general theory of time and coordinate differentiation, analogous to the ordinary covariant differentiation and extension of differential geometry, which will preserve invariance under the group G of coordinate transformations. This problem is treated in the following sections.

PREFERRED COORDINATE SYSTEMS 6. KINEMATICALLY Any transformation of the group G relating the coordinates x i and xi of two moving rectangular systems will have the form

-

xi

= u;(t)zj

+ bi(t),

(6.1) where, as indicated, the coefficients a: and bi depend on the time t . The condition of orthogonality, resulting from the assumption that we

84

IV. PERFECTLY PLASTIC SOLIDS

are dealing with rectangular systems, is expressed by eit,her of the following two sets of equations which are algebraically equivalent, namely -

a?

- 8jk;

]ak

ala6 1 3

s ~ ~ .

(6.2)

Defining the velocity components by 21i

=

d-,x i . dt

-

0.

'

=

dzi -, dt

in the x and z coordinate systems respectively, it follows from the equations (6.1) that 2 ) ; = ai(t)& f$Zi 6i(t), (6.3) where the dot is used to denote differentiation with respect to the time. Also, differentiating the equations (6.3) partially with respect t'o the coordina'te Zk,the resulting equat,ions can be writt'en

+

+

-

vi,j

k m

= ataj Uk.m

- a;aF,

(6.4)

where the comma is employed in the usual manner t o denote partial differentiation. Interchanging the indices i and j in the equations (6.4), we have -vj?; = atayv,,k - az.k ajk . (6.5) Then, subtracting corresponding members of the equations (6.4) and (6.5) and making use of t,he relations obtained by differentiation of the equations (6.2), we can show that a; =

where

af4iit

+ atifit,

(6.6)

yv.. - u . .) 6.. tJ - -(;E i , f - Ej,i).

4 . .= 13

a"

J's ,-I

Now suppose, for definiteness, that the ve1ocit.y field is known and let us represent by xi = g i ( t ) , where t denotes t>hetime, the trajectory of a moving point P of the medium. Let us also consider a moving rectangular coordinate system y related t o the x system by the equations xi = cj(t)yj g i ( t ) ; cj(t)c:(t) = 83k. (6.7) T h i s transformation x ++ y belongs to the group 6' and i s , moreover, such that the y system, at axy time t , will have its origin at the moving point P. It remains t o determine the coefficients d ( t ) so t h a t the y system, defined by the transformation (6.7), will contribute t o t,he solution of the invariance problem discussed in Sect. 5.

+

6.

KINEMATICALLY PREFERRED COORDINATE SYSTEMS

85

Denote by w ; the components of the velocity of points in the medium relative to the above y coordinate system. We now make the assumption that the quantities J/il deJined by J/ij = + ( w . . - w . . a.i 1.J

vanish at the origin of the moving y system. To see the consequence of this condition let us consider the relations (6.6) corresponding to the transformation (6.7), namely 6; = C:+ik

+

ci$'jk.

(6.8)

Referring (6.8) to the origin of the y system, we have 6; = Cjt+ik(t),

(6.9) in which the quantities &b can be regarded, as we have indicated, as functions of the time t. Assuming the continuity of these functions, it follows that the coefficient,sd ( t ) will be det,ermined as a solution of the equations (6.9) when we assign their values at some specified time toW e now impose the condition that cj(t0) = 6.: This determination of the coefficients cj(t) as solutions of the equations (6.9) is such that

in view of the fact that the quantities bij are skew-symmetric. Hence the expression c;ci is independent of the time t, and since this expression has the value 6 ] k a t t = to, by the above assumption, it follows that the required orthogonality condition, i.e. the second set of relations in (6.7), is satisfied. Moreover, by combining the equations (6.8) and (6.9), we see that the quantities vanish a t the origin of the moving y system in accordance with the above requirement. We shall refer to any of the coordinate systems y which are defined by the equations (6.7) as a kinematically preferred coordinate system for the moving point P. In particular, the kinematically preferred system corresponding to the above value of the time t = to will be called the initial system and the time to will be called the initial time. At the initial time to we have

(6.10)

86

N. PERFECTLY

PLASTIC SOLIDS

The first of these relations is the result of the above assumption. The second follows by evaluation of (6.9) a t t = to. The third is obtained by differentiating (6.9) and evaluating a t t = to. Higher time derivatives of the d ( t ) a t t = to can readily be determined by a continuation of this process. Let us now make a transformation (6.1) of the coordinates xi to a system of coordinates Ziand let us then consider the kinematically preferred coordinate system y which is associated with the system Z in the same way that the above system y is associated with the x coordinate system. We suppose in this connection that the y and jj systems have their origins at the same moving point P and also that the initial y and 5 systems are determined for the same value of the time, i.e. the initial time to is the same for the two systems. The relation between the y i and jji coordinates will now be deduced. We observe immediately that the transformation y t-)?j must have the form y i = bj(t)yi, (6.11) since the y and jj systems have the moving point P as their common origin. By comparison with the equations (6.6) we see that

pJ - b!#ik

-k

%$jk

for the transformations (6.11). But qij and ~ J i jvanish a t the origin of the y and jj systems. Hence b$ = 0, i.e. the coeficients bj in equations (6.11) must be constant. To determine these constants we consider the relations (6.12) with reference to the initial y and ji systems. But a t the common origin of these systems the equations (6.12) reduce to

bj = aj(to). We have now proved the following result. W h e n the coordinates x i are subjected to a transformation (6.1) the kinematically preferred system y associated with the x system undergoes a linear homogeneous transformation

aj(k)yi, (6.13) in which the coeficients a: are constant and equal to the corresponding coeficients aj(t) in the equations (6.1) at the initial time to. In the followyi

=

7.

COVARIANT TIME AND COORDINATE

DIFFERENTIATION

87

ing section we shall show how the kinematically preferred systems can be used for the construction of tensors under the transformation group G. Remark. To determine the effect of a change of initial time let y and z be two kinematically preferred systems, associated with the same z system and having their origins at the same moving point P , but with initial times toand tl respectively. Then we have

+ gi(t); x i = dj(t)z’ + gi(t); xi =

cj(t)y’

ck:

= 8jk,

(6.14)

djd;

=

(6.15)

8jk,

where gi(t) defines the trajectory of the point P. Also the coefficients cj(t) and dj(t) in these equations are given as solutions of the following differential equations and initial conditions, namely

i j = C$$ik(t);

cl(t0) =

s,;

(6.16)

d$ = dfbik(t);

d;(ti)

8.;

(6.17)

=

Now the solution cj(t) of (6.16) can be represented by &t)

=

cft(ti)dk(t);

(6.18)

this follows from (6.17) and the uniqueness theorem for a system of equations of the type (6.16). Combining (6.14) and (6.15) and making use of (6.18) we readily see that zm = cj(t)db(t)y’, = ct(ti)di(t)db(t)y’, = cjn(t1)y’.

Hence a n y two kinematically preferred coordinate systems, associated with the same x system and having their origins at the same moving point P , are related by a rotation with constant coeficients. This result was originally shown in a note by G. Margulies, Remark o n kinematically preferred coordinate systems, Proc. Nat. Acad. Sci., 42, 1956, pp. 15S153.

7. COVARIANT TIMEAND COORDINATE DIFFERENTIATION The discussion in this section will be based on a representative tensor having components uil(x,t) relative to the x coordinate system. Between the components r;j and tijof this tensor relative to the preferred systems y and j respectively we have the following relations u

t i j

=

t m

TkmQaj

9

(7.1)

88

IV. PERFECTLY PLASTIC SOLIDS

in which the a's are constants (see Sect. 6). Hence, differentiating (7.1) with respect to the time t and evaluating at the common origin of the initial y and @ systems, we obtain

where we have put

The quantities DUkm/bt and DaijlDt defined by (7.3) can be associated with the x and 3 systems respectively. In fact, we can regard the DUkmlat as functions of the coordinates x i of the point P , i.e. the origin of the initial y system, and the time which may now preferably be denoted by t rather than the special symbol t o used in the above discussion of the preferred coordinate systems; similarly the quantities DFij/Dt can be considered as functions of the coordinates ziof the point P and the time t. But in view of the relations (7.2) in which

as follows from (6.1), the quantities %&Dt and aifij/Dt are the components of a tensor under the group G . We call this tensor the covariant time derivative of the tensor u. By repeated application of this process of covariant time differentiation we may define the second and higher time derivatives of the tensor a; the components of these higher time derivatives will be denoted by

To derive the formula for the components Dui,/Dt of the covariant time derivative of the tensor u we have merely to differentiate the relations 7 a"j

- UkmdC;),

(7.4)

and evaluate at t = to (after which the symbol to is to be replaced by t as stated above). We thus find that

7.

COVARIANT TIME AND COORDINATE DIFFERENTIATION

89

when use is made of the relations (6.10). The formula for the components %Yuij/r3)F of the second covariant time derivative is given by

where

do.. s'.= 2 f '' ' dt

akj$k<

+

Uik#kj-

(7-7)

Substituting the expression for sij given by (7.7) into the right members of (7.6) we obtain

as the explicit formula for the components of the second covariant time derivative. Formulae for higher covariant time derivatives can be found by a continuation of this procedure. If we differentiate (7.1) partially with respect to the coordinate pk and then evaluate a t the origin of the initial y and 7j systems we obtain a set of equations which can be written 3ij.k

p ( l r

= upq.rat~jat,

(7.9)

where (7.10)

It follows from (7.9) that the quantities upq,rand c r j , k defined by (7.10) are the components of a tensor under the transformations of the group G ; this tensor may be called the covariant coordinate derivative or simply the covariant derivative of the tensor u for brevity. But, applying the above procedure to derive the formula for these components, i.e. differentiating the equations (7.4) with respect to y k and evaluating a t t = to, we find that

Hence the components of the covariant coordinate derivative of the tensor u are identical with the partial derivatives of the components of u with respect to the coordinates; the fact that these partial derivatives

90

IV. PERFECTLY PLASTIC SOLIDS

are the components of a tensor under the group G can of course be inferred immediately by coordinate differentiation of the equations of transformation of the components of the tensor u. Remark 1.

Putting i = j in (7.5) and summing on the repeated indices, we

obtain (7.11) on account of the skew-symmetry of the quantities +ij. More generally the following result can be stated as a direct consequence of the definition of the covariant time derivative. The covariant time derivative of a scalar i s equal to its

total time derivative. Remark 2. It is clear from the above discussion that the quantities uil,t. . . mt.. defined as functions of the coordinates xi and the time t by the equations ,

utj.t. . ,

(7.12)

in which the right members are evaluated at the origin of the initial y system, will be the components, relative to the x system, of a tensor under the group G. I n particular if the equations (7.12) involve only differentiation with respect to the coordinates y, or only differentiation with respect to the time t, we have (7.13) (7.14) respectively, provided there are p terms in the set of subscripts k . . .m in (7.13) and n time differentiations involved in the equations (7.14). Differentiating the relations (7.4) with respect to the coordinates yk, . . . ,y" and evaluating, ~ , by (7.13) as required, it follows immediately that the quantities u ~ ~. , , defined are the corresponding partial derivatives of the components uij with respect to the x coordinates, i.e. (7.15) Hence the set of coordinate derivatives of any order of the aij(x,t) are the components of a tensor under the group G as can be seen, of course, more directly by coordinate differentiation of the transformation equations of the components uij of the tensor u. If we transform the equations (7.5) and (7.8) to a preferred y system the uii will be replaced by the components r,, and the quantities &i by the corresponding quantities #;i in accordance with the above notation. But the #,i vanish

7.

COVARIANT TIME AND COORDINATE DIFFERENTIATION

91

a t the origin of the moving y system and consequently the total time derivatives of the #ii of any order will also vanish at the origin of this system. Hence, when we evaluate the above equations at the origin of the init.ia1y system, we obtain (7.16) Now the components of any tensor a t a point P , relative to the x system, are equal to the corresponding components of the tensor when referred to the initial y system a t P and evaluated at the origin of this system. Applying this observation to the components of the first and second covariant time derivatives of the tensor u,we have (7.17) Let us next consider the identity (7.18) in which, we recall, the wk are the components of velocity in the y system. But wk vanishes a t the origin of the moving y system from the construction of this coordinate system and therefore the total time derivatives of the wk must likewise vanish at the origin of the y system. Hence, evaluating (7.18) and also the equations obtained by total time differentiation of (7.18) a t the origin of the initial y system, we have (7.19) when use is made of the equations (7.14). Comparison of (7.16), (7.17) and (7.19) now shows that (7.20) One can see without difficulty, on the basis of the procedure by which the relations (7.20) were obtained, that we must have a general relation of the form (7.21) when the left member contains n subscripts t. T h u s the components of the n-th covariant time derivative of the tensor u are equal to the corresponding partial time ~ the origin of the initial y system. We have derivatives of the components T , at emphasized the above equations (7.21) since they provide a formula for the quantities u ~ ., , ~ t . , However it may also be observed from equations of the

92

IV. PERFECTLY PLASTIC SOLIDS

type (7.19) that the corresponding partial and total time derivatives of any order of the components ~ i are i equal a t the origin of the initial y system. Finally it can be stated that the above quantities uij,*,. .m t . . . I defined by (7.12) are given by the formula Utj.t. . . mc...t

=

D"au.t.

..m

Dt"

(7.22)

'

when the left members involve n subscripts t ; these equations are seen to be essentially a generalization of (7.21) with the quantities u U v t ... replacing the quantities uii in the above discussion. Remark 3. The above definition of the covariant derivative, involving the use of kinematically preferred systems, can be extended t o certain quantities of non-tensorial character under the group G provided that the tensor law of transformation is valid for the linear transformations (6.13) relating the kinematically preferred systems. For example, consider the relations valid under the transformation (6.13); the second set of these relations is of course obtained from the first set by coordinate differentiation. Now both members of the first set of relations (7.23) vanish at the common origin of the y and tj systems so that no significant result is obtained by evaluation at the origin of these systems. However, when the second set of relations (7.23) is evaluated at the origin of the initial y system, we obtain (7.24) where Vi/j

=

(zi,j)o;

uk/m

= (~k.rn)o.

(7.25)

Equations (7.24) express the fact that the quantities Vk/m and V i / j are the components, relative to the x and f systems respectively, of a tensor under the transformations of the group G. Observe, in this connection, that we have used the symbol u i / i rather than u i , j to denote the components of this tensor since the designation u,, has already been employed for the partial derivatives of v i with respect to the appropriate coordinates. To obtain the formula for the components v i / i we consider the equations, corresponding to (6.5), by which the quantities u i , j are transformed from the z to the y system, namely wi,j =

uk.mCltC;(

- cl"i;.

(7.26)

Evaluating (7.26) at the origin of the initial y system, we find u %. /. i -

01.1.

_ 4.. a ) - 3 (ui,i

+

vj.i)

=

eij,

(7.27)

when use is made of the relations (6.10). It follows from (7.27) that the quantities c,, are the components of a tensor under the group G. In other words, the form of the equations

7.

COVARIANT TIME AND COORDINATE DIFFERENTIATION

- -( $ Ui.1

C s1 .. -

+

93

uj,i),

by which the quantities ci; are defined, is preserved not only under orthogonal transformations in the narrow sense but, more generally, under the set of coordinate transformations relating arbitrarily moving rectangular systems. Differentiating (7.26) partially with respect to the coordinates y k , . . .,y" and then evaluating at the origin of the initial y system, we see that (7.28) provided there are at least two indices in the set j , k , . . . , m and where the quantities u i I j t . . . are defined by the equations vi/jt.

. .m

=

(wijt

.. .m)o.

The equations (7.28) show that the coordinate derivatives . . are the components of a tensor under the group G. As another illustration of the application of this method let us transform the identity u2.. 1_ =€..+(#).. 21

11

to the initial y system, differentiate totally with respect to the time t, and then evaluate at the origin of the system. But this gives V %. ./ I t

-

(;it), -

=

?a€ ', -$ '

where the right members are the components of the covariant time derivative of the rate of strain tensor c. Other tensors under the group G can readily be constructed from non-tensorial quantities by this procedure but such tensors will not be needed in the following work. It may be of some interest to observe that, in general, this process of differentiation is not commutative when both coordinate and time differentiation are involved. Thus, defining the quantities uil; as above, and the quantities uil:, uiltli and v i / ; l t in an analogous manner, we see immediately t,hat uilfl; equals zero since uilt is equal to zero. However the quantities u i / ; / t are equal to the components of the covariant time derivative of the rate of strain tensor e; we leave it to the reader to verify this statement. Remark 4. As an illustration of the replacement theorem which appeared in its original form in an article on affinely connected manifolds (see Thomas, A projective theory of afinely connected manifolds, Math. Zeit., 26, 1926, pp. 723733) let us consider a scalar or tensor invariant whose components (7.29) are functions of the components of the tensor CT, the velocity v, and their first partial derivatives with respect to the coordinates and the time. The replacement theorem then states that the components F of this invariant-can be

94

IV. PERFECTLY PLASTIC SOLIDS

expressed by replacing the arguments in the above functions (7.29) as follows:

This result is obtained immediately by transforming the components (7.29) to kinematically preferred coordinates and evaluating a t the origin of the initial system. Hence the components F can be expressed by functions of the type H ( U i j ; Uijtt; a i j . k ; Eii);

(7.30)

in this connection it may be observed that the functions P cannot depend explicitly on the coordinates since such variables cannot enter as arguments of the functions H . This procedure can evidently be extended to invariants whose components involve derivatives of any order. Thus, if the components (7.29) are assumed to depend also on second coordinate and time derivatives of the ui the following substitutions for these derivatives are to be made, namely

In this case the functions H will involve as arguments the quantities v i / i t and in addition to the arguments appearing in (7.30).

tri,jk

8. DYNAMICALLY CORRECT FORM PRANDTL-REUSS EQUATIONS

OF THE

The local motion of the medium in the neighborhood of a point P can be resolved into a n instantaneous rigid motion about a n axis through P , a translation, and motions resulting in distortion and actual change of volume (see Remark 2 in Sect. 6 of Chap. I). This suggests that the invariance requirement, stated in Sect. 5, can be met for the equations (4.8) if we interpret these equations as strictly valid only a t the origin of a kinematically preferred system where the effects of translation and rotation have been eliminated. But if we consider the equations (4.8) relative to the kinematically preferred system y and then evaluate at the origin of the initial y system we are led immediately to the relations

9.

THE HENCKY STRESS-STRAIN EQUATIONS

95

as valid equations in the underlying x coordinate system. These relations (8.1) are invariant under transformations of the group G, i.e. arbitrary rigid motions, and hence constitute a dynamically correct form of the Prandtl-Reuss equations. The above procedure has the merit that it enables us to correct the equations (4.8) in what appears to be a rather natural manner. It would have been possible of course to have derived the equations (8.1) directly by using the covariant time derivative rather than the total time derivative in the discussion in Sect. 4. However this would have required a preliminary treatment of covariant time differentiation and would possibly have put too great a tax on the patience of the reader. The problem of the invariance of constitutive equations under rigid motions has been investigated independently by various authors since the year 1905 when this problem was first treated by G. Janmann. A review of this work has recently been made by W. Prager, An elementary discussion of deJinitions of stress rate, Tech. Report No. 53, Contract Nonr-562( 10), Brown University, NR-064-406. Remark. Assuming that rotational effects are sufficiently small we can neglect the terms involving the + i j in the formula for the covariant time derivative of the tensor u*,i.e.

and hence replace the equations (8.1) by the equations (4.8). The approximate equations (4.8) have sometimes been used in the following work for simplicity or when it is obvious that the rotation can be disregarded in the problem under consideration.

9. THEHENCKY STREWSTRAIN EQUATIONS We shall now consider a system of constitutive equations known as the Hencky stress-strain equations which are generally applicable when the elastic and plastic deformations are of the same order of magnitude. It is assumed in this theory that the components ui of the total deformation can be represented in the form

+

(9.1) where the ti, and the .di are the components of elastic and plastic deforui = tii

.di,

96

IV. PERFECTLY PLASTIC SOLIDS

mations respectively. The elastic deformations ;iiare assumed to satisfy the stress-strain relations of the purely elastic theory, i.e. the conditions (4.2). Correspondingly we assume that the plastic deformations Q are characterized by a set of conditions analogous to the above requirements All . . , Ad for plastic flow, namely BI. T h e deformation .d preserves volume, B2. T h e components of the deviator of the stress tensor are proportional to the components of the deviator of the strain tensor, Bt. T h e coe$cient in the proportionality relating the components of the deviators of the stress and strain tensors i s a positive scalar invariant of the deviator of the strain tensor, Bq. There does not exist a (1,l) correspondence between the components of the deviators of the stress and strain tensors. The assumptions B1,. . . , Ba result from the previous assumptions A l l . . . , A4 when the latter are modified to take into account the fact that deformation, rather than velocity, i s the basic entity in this theory. It follows from B1, . . . , Bq that

.

in which M is an arbitrary differentiable function of its argument. The first condition (9.2) is obtained directly from the assumption B I (see Remark 1 in Sect. 6 of Chap. I) and, in view of this condition, the deviator 6* of the strain tensor 6 is identical with the strain tensor itself, i.e. a;, = 6 2.1. - I(* 2 Ui,J .dj,*).

+

The derivation of the remaining conditions (9.2) from the assumptions B2, BI and Bq is similar to the derivation of the corresponding conditions in Sect. 2. Our problem is now to eliminate the derivatives of the hypothetical displacements ‘iiand Q from the above equations and thus to obtain a set of stress-strain relations which will involve, in addition to the components aij of the stress tensor, only the derivatives of the components

9.

THE HENCKY STRESS-STRAIN EQUATIONS

97

u; of the total deformation.

But, writing the stress-strain relations (4.2) in the equivalent form (4.4), we have immediately

+

(3X 2IL)Ui.i from (9.1) and the first equation (9.2). Also we have ui; =

+

(9.3)

+

e t = st, ej = G& &ij, (9.4) where the e:, are the components of the deviator e* of the strain tensor e whose components eiJ are derived from the total deformation u ; the quantities Szi and &i, are defined in a corresponding manner in accordance with the notation employed. Hence the first set of equations (4.4) becomes ,:a = 2p(e; - dij) = 2p(e; - $uzj), (9.5) when use is made of (9.4) and the second set of equations (9.2); the factor J. in these equations is the reciprocal of the above quantity cb and hence is positive by hypothesis. Equations (9.3) and (9.5) can be written in the following form

where we have introduced Young’s modulus E and Poisson’s ratio Y (see Remark in Sect. 1 of Chap. 111). Let us also observe that if we multiply both members of the second set of equations (9.6) by u?j and sum on the repeated indices, we find that the quantity $ is given by (9.7)

The above equations (9.6) in which 1c. has the value (9.7) are the Hencky stress-strain equations. I n addition to the conditions imposed on the stress components us, and the strain components e;j by these equations we also have the conditions given by the last set of relations (9.2), i.e. the yield condition (9.8)

in which &I is a material function for the solid. Usually the Hencky

98

IV. PERFECTLY PLASTIC SOLIDS

stress-strain equations (9.6) are associated with the quadratic yield condition (2.13); when this is done we have

in place of (9.7) and (9.8). In general there are a t most four independent relations in the second set of equations (9.6) due (a)to the identity resulting from the fact that the 6;and the e t are the components of deviators and (0) to the dependence imposed by the equation (9.7) for the quantity #. Hence there are six independent equations in the set composed of (9.6) and (9.8) and when these are combined with the equations of continuity (6.2) and motion (7.4) of Chap. I we have a system of ten equations for the determination of the ten dependent variables consisting of the density p, the three components ui of the deformation, and the six components a;j of the stress tensor. Regions of the body not in a state of plastic deformation, as described by the system of equations in the above paragraph, are assumed to be subject to purely elastic deformations for which the stress-strain relations (1.4) of Chap. I11 are valid. In this theory, as well as in the theory of plastic flow characterized by the Prandtl-Reuss equations, we are therefore, in general, dealing with an elastic-plastic solid. This solid becomes an incompressible elastic-plastic solid (see Remark 2 in Sect. 4) when the stress-strain relations of the elastic theory are assumed to have the form (4.14) and the first equation in (9.6) is replaced by the equation ui,i = 0,

(equation of incompressibility),

,while the other relations governing the plastic deformations remain unaltered. Finally the solid is of the rigid-plastic type in the highly idealized case for which the elastic deformations are assumed to vanish throughout the material; then the total deformation u in a plastic region of the solid becomes identical with the above plastic deformation fi and the Hencky stress-strain equations, together with the yield condition, are given by the relations (9.2). In particular these equations have the form

9.

T H E HENCKY STRESS-STRAIN EQUATIONS

99

or

ui,i

0;

= 71

u; =

1

&ij,

- 73 = 2k,

(9.10)

when the von Mises or the Tresca yield condition is assumed; the positive factor $C has the value

4

=

-*

d2k

G

j

4 l

=

-1

7?1

2k

- 7?3

in the equations (9.9) and (9.10) respectively, where 71 and 73 are the principal values of the strain tensor e which correspond to the greatest and least of the principal values and 7 3 of the stress tensor. When the deformations and stress in a plastic region of the solid are independent of the time t, the above equations become equations for the determination of a state of plastic equilibrium in the strict sense. I n the following work we have restricted the application of the Hencky stress-strain equations t o the special case of plastic equilibrium and have assumed the Prandtl-Reuss equations (see Sect. 4 and Sect. 8) or one of the sets of stress-rate of strain relations for the perfectly plastic solid (see Sect. 2 and Sect. 3) in discussions involving actual plastic flow. Remark.

Consider a moving regionR(t)in an elastic-plastic solid, the motion

of R being determined by the motion of the material particles which it contains. We suppose that the Prandtl-Reuss equations, or the Hencky stress-strain equa-

tions, with the associated yield condition and the usual dynamical relations, i.e. the equations of motion and continuity, are satisfied in R(t);in particular R may be in a state of plastic equilibrium in which case the particle velocities are zero and hence the region R is stationary relative to the dynamically allowable coordinate system employed. Assuming the quadratic yield condition, for simplicity, we say that unloading from the yield point begins over the region R(t) at a time t = tl if

* *

uUuU = u:ju:j

2k2,

< 2k2,

1

over the region R(tJ, over R(t)for t

> tl.

(9.11)

When this condition is satisfied the value of the scalar u;u; immediately decreases below the value 2k2 necessary for plastic flow or deformation and hence, for t > 11, the equations of the elasticity theory (Sect. 1and Sect. 2 in Chap. 111) must be used to determine the stress and particle velocity in R(t). A sufficient

100

IV. PERFECTLY PLASTIC SOLIDS

condition for (9.11) to hold and hence for unloading from t,he yield point to begin over R(t) at t = tl is that a?.,?. = 11 X I

2p.

'

d - (a:j.:J dt

< 0,

(9.12)

over R(t1). Analogous remarks can undoubtedly be made with regard to unloading from the yield point if we assume one of the more general yield conditions involving the material function M . It is interesting to observe that the above conditions (9.11) and (9.12) are invariant under transformations of the group G. The conditions (9.11) and the first part of the conditions (9.12) are invariant since u ~ , uis~a~scalar under these transformations. Also the inequality in (9.12) has the required property of invariance since the left member of this inequality is equal to the covariant time derivative of the scalar u:p:j (see Remark 1 in Sect. 7). Or we can write

when we made use of the formula (8.2) for the covariant time derivative of the deviator a* and take account of the fact that the +ii are skew symmetric. The above point of view regarding the question of unloading from the yield point appears t o be in general agreement with that usually given in texts on plasticity theory. It seems unlikely t o us however that such unloading will be realized, except in special instances, since it requires, in general, the simultaneous occurrence of discontinuities in velocity, if not in stress, over the entire region R(t1)due to the abrupt change from the non-linear constitutive equations of the plasticity theory to the linear stress-strain relations of the theory of elasticity. We would expect rather that the transition from the plastic to the elastic state, due to unloading, will occur as the result of the growth or propagation, possibly with great rapidity, of an elastic region into the plastic region, the necessary discontinuities involved in this transition being confined to the surface of sepa,ration of these two regions. An analogous situation should develop as the means by which the solid passes from the elastic to the plastic state after the yield point has been reached. Problems of this character will be treated in the following chapter.

APPENDIX

EQUILIBRIUM THEORY OF LUDERS BANDS 1. INTRODUCTION

If a flat bar or plate, such as is commonly used in the ordinary tensile test, is subjected t o a slowly increasing load, a Luders band,

EQUILIBRIUM THEORY OF LUDERS BANDS

101

consisting of plastically deformed material between two inclined parallel planes, will suddenly appear when the tension in the plate reaches the yield stress. A slight decrease in the load will occur a t the instant of formation of the band; in this connection see T. Y. Thomas, A discussion of the load drop and related matters associated with the formation of a Luders band, Proc. Nat. Acad. Sci., 40 (1954), pp. 572-576. With increasing load a second band will be formed, parallel and usually adjacent to the first, and this process can be continued until the plate fractures or is completely covered with these bands (see Fig. 3). To secure the utmost mathematical simplicity in treating this problem we shall employ the Hencky stress-strain equations for rigid-plastic material subject either to the von Mises or the Tresca yield condition, under the assumption that a state of equilibrium occurs after the formation of the Luders band (see Sect. 9 of Chap. IV). FIG.3 Within the Luders band the deformations u, with which we shall be concerned primarily in this discussion, will therefore satisfy the relations

4et

u;j =

u 1.1 ..= 0 u53.3 .. =

(stress-strain equations),

(1.1)

(equation of incompressibility),

(1.2)

(equations of equilibrium),

(1.3)

0

together with one or the other of the following yield conditions, namely =

~f - T;

2k2 =

2k

(von Mises yield condition), (Tresca yield condition),

(1.4) (1.5)

where T; and T: are the greatest and least of the principal values of the stress deviator a*; the notation used in the above and following equations in this Appendix is the same as that employed in our previous work. The proportionality factor 4 in (1.1) is a positive function of position within the band in accordance with the assumption Ba in Sect. 9 of Chap. IV. By making use of the yield condition one can eliminate the factor 4, if desired, in an obvious manner. Along the surface of separation of the rigid and plastic regions the following general dynamical conditions must be satisfied, namely

102

IV. PERFECTLY PLASTIC SOLIDS p(vn

- G)

= P(vw

- G),

- G)[vi],

[ g i j ] ~ ’= ~ ( v n

(see Sect. 8 of Chap. I) where, as usual, the bracket denotes the discontinuity in the quantity enclosed. Also D stands for the velocity of the flow and G is the normal velocity of the surface of separation whose unit normal vector is denoted by v. The quantities p and v, are the density and normal velocity on one side of the surface of separation while p and 0‘, are the corresponding quantities on the other side of this surface. Since we are here concerned with an equilibrium theory we shall have vi = 0; also G = 0 since the surface of separation of the rigid and plastic regions is stationary by hypothesis. Hence the first of the above conditions is satisfied identically and the others reduce simply to

0. (1.6) The conditions (1.6) can be expressed by saying that the stress vector, i.e. the vector having the components aijvi on the surface of separation is continuous across the surface. Account will be taken later of the fact that the stress vector must vanish on the outer boundaries of the plastic band. [U<j]Vj

=

2. THE IDEAL FLATBAR To say that a rectangular bar is thin or flat implies that its thickness is small in comparison with its other dimensions, i.e. its width and length. Let us therefore define the ideal flat bar or plate as one of finite thickness but whose flat sides are formed by two infinite parallel planes. By employing the concept of the ideal flat bar we can limit our attention to the boundary conditions on the two flat sides. Results obtained for such bars may be expected to be valid approximately in the interior, i.e. in a region removed from the narrow sides and ends, of the fiat bar of finite dimensions. In Fig. 4 we have indicated a portion of an ideal flat bar. The lines A B and CD are the intersections, with one of the flat sides of the bar, of parallel planes PI and P z perpendicular to the flat sides; these planes are assumed to bound the Luders band. Two coordinate systems x and y will be used in the following discussion. Both of these systems will be supposed, for definiteness, to have their origins at a point 0 mid-

EQUILIBRIUM THEORY OF LUDERS BANDS

103

way between the flat sides of the bar and also mid-way between the planes PI and P2. The x3 and y3 axes (not shown in Fig. 4) will be taken to be identical and perpendicular to the flat sides. The y2 axis is chosen perpendicular to the planes PI and P2 and hence the y1 axis

FIG.4

will be parallel to these planes. Since the x1,x2 and y1,y2 coordinate axes are in the same plane it can be considered that the y1,y2 axes are obtained from the d 1 x 2axes by rotation through an angle 0 (see Fig. 4). Hence the coordinates of these two systems will be related by the following transformation

x1 = y1 cos e - y2 sin 0, x2 = y1 sin 8 y2 cos 0, 2 3 = y3.

+

1

(2.1)

Now suppose that the ideal flat bar is subjected to a uniform tension Then if r;, denotes the components of the stress tensor relative to the 2 system we shall have r ( > 0) in the direction of the x2 axis. rZ2= r ;

rij

= 0,

otherwise.

Denoting the components of this stress tensor by

p,j

in the y system, the

104

IV. PERFECTLY PLASTIC SOLIDS

quantities p i j can be determined from the values of the ~ i by j the tensor transformation relating these two sets of components. Thus

Computing the derivatives in these relations from (2.1) we find that pll = T

sin28;

plz =

T

pa2 = T

cos28;

pZ3 =

0;

sin 8 cos 8; p33 =

p13

0.

=

}

0,

(2.2)

Let the value of T increase until the yield value is reached. It will then be assumed that the band or layer between the two parallel planes PI and Pz becomes plastic in the sense that the deformations u in this region satisfy the equations ( l . l ) , (1.2), and (1.3) and one or the other of the yield conditions (1.4) or (1.5). In the case of von Mises yield we shall have T = &k and for Tresca yield we shall have T = 2k a t the yield point. These relations between T and k are an immediate consequence of the von Mises or Tresca yield condition and the above values of the stress components T~~ for the uniform tension. Outside this plastic band the material remains rigid in accordance with the assumption that we are dealing with a rigid-plastic solid. We seek a simple solution uiof the equations ( l . l ) , (1.2) and (1.3) in the plastic band satisfying the boundary conditions, i.e. for which the stress vector vanishes (after the deformation) on the free surface of the band, and such that the conditions (1.6) are satisfied over the surfaces of separation of the rigid and plastic regions. It will be shown in fact that, relative to the above y coordinate system, such a solution is given by u1 = a y 2 ; u2 = by2; uq = -by3, (2.3) where a and b are constants; we observe immediately that, regardless of the values of the constants a and b, the equation of incompressibility (1.2) is satisfied by the deformation (2.3). In consequence of (2.3) the planes PI and P2 bounding the plastic band will undergo a parallel displacement. This is illustrated in Fig. 5 where we have indicated the displacement, in a plane parallel to the flat sides of the bar, of two points Q and R in the planes PI and P z respectively. This part of the deformation, which is determined by the first two relations (2.3), involves a shear followed by a movement in the y2

EQUILIBRIUM THEORY OF LUDERS BANDS

105

direction proportional to the y2 coordinate. From the third equation (2.3) it is seen t ha t planes parallel t o the flat sides of the bar remain parallel t o the flat sides. I n particular the quantity -b, which determines the magnitude of the displacement of these latter planes, is the

FIG.5

plastic eEongation in the y3 direction. The exact value of the constant b will not be needed in the following discussion; however b will presumably be positive and hence the distance between the planes Pl and PZwill be increased by the plastic deformation (2.3), resulting from tension, as shown in Fig. 5. Remark. The quantity b in the equations (2.3) represents contraction per unit length in the thickness of the plastic band. Now in the purely elastic case the corresponding contraction is determined for materials of known elastic moduli when the tension is assigned. In the plastic situation, however, the equations at our disposal do not permit a determination of the contraction b which is influenced, in practice, by certain conditions which underlie the test and which would appear to be entirely extraneous from the standpoint of the classical elasticity theory. For example, it is known that the depth of the plastic groove, which is determined by b, will be increased when the length of the bar is increased, and in accordance with this fact it is supposed that a certain amount of the elastic energy in the bar is transmitted to the plastic band in the process of its formation. Unless this and possibly other relevant factors are to be made an inherent part of the theory, we must therefore determine the plastic contraction b by actual measurement.

Since the deformation within the band, as above described, may involve a slip of material particles along the planes of separation P1

106

IV. PERFECTLY PLASTIC SOLIDS

and Pz, it is natural to refer to this band as a plastic slip band. In consequence of the deformation (2.3) within the plastic slip band the material on either side of the band will suffer a rigid displacement which will not effect its internal stresses. It is important to observe that this displacement can occur without separation of the material along the planes P1and Pz. 3. FORMAL CALCULATIONS We shall now calculate the components of the stress tensor u and its deviator u* within the plastic slip band; this calculation will be carried out relative to the y coordinate system. Observe first that by differentiation of the equations (2.3) we have u1.1 =

0;

U1,Z

u Z , ~

=

0;

U Z , Z=

U3,1

=

0;

u3,Z

=

=

0,

b;

U Z ,= ~

0,

0;

u3,3

= a;

u1,3

=

-b.

From these relations we immediately determine the values of the components of the strain tensor e as follows

ell = 0;

a

e12= 2, -*

e13 = 0,

e33 = -b. Hence from (1.1) and the fact that the tensor e is identical with its deviator e" for the incompressible case under consideration, we can write eZ2 = b;

eza =

0;

(3.1) a&? =

br#q

u;3 =

0;

a$3

=

-z+J

It is immediately seen from (3.1) and either of the yield conditions (1.4) or (1.5) that the quantity C#I must be constant in the plastic band. Replacing the components d, in these equations by their values in terms of the components aij we have

c22 = b# - p; 423 = 0 ; 6 3 3 = --b+ - p,J where p is equal to the combination -aaa/3. Substituting the values

EQUILIBRIUM THEORY O F LUDERS BANDS

107

of the aij given by (3.2) into the equilibrium equations (1.3) and taking account of the fact that 4 is a constant, as we have just observed, it follows that p , i = 0 for i = 1,2,3. Hence the quantity p is also constant in the plastic band. On the flat sides, i.e. the outer boundaries of the plastic band, we must have aijvj = 0 where v is a normal vector to these sides after their deformation by (2.3). But v is a vector in the y3 direction, since the deformation (2.3) transforms the sides into parallel planes, and hence this condition becomes ui3 = 0; for i = 1,2 we see that these equations are satisfied identically on account of (3.2). Taking i = 3 it follows from the last equation (3.2) that

P

-b4,

(3.3) over the outer boundaries of the plastic band. Hence (3.3) must hold throughout the interior of the band since the quantities b, p and 4 in this relation are constants. With this choice of the constant p the components ui, are constants, as given by (3.2), and hence the equilibrium equations (1.3) are satisfied identically. Let us now consider the conditions (1.6) which must be satisfied along the two planes PI and Pz separating the plastic band from the rigid portions of the bar. These planes of separation are perpendicular to the yz axis so that the unit vector v in (1.6) has the components (O,l,O); hence (1.6) reduces to =

- pi2

0, (3.4) where the values of the piz and ni2 are given by (2.2) and (3.2) respectively. Making these substitutions and taking account of (3.3), the relations (3.4) become [(Ti21

7

= uiz

sin e cos e =

=

9

2

7

(3.5)

e

2b4. (3.6) Dividing corresponding members of (3.5) and (3.6) we now have COS~ =

tan0

=

a -.4b

(3.7)

This equation will determine the inclination 0 of the slip band after the value of the ratio a/b has been found. To determine this ratio we first set up the relation obtained by elimination of 0 from the above equa-

108

IV. PERFECTLY PLASTIC SOLIDS

tions; this elimination is accomplished quite simply by substitut,ing for tan20 and see20 (= l/cos2 0) in the identity

1

+ tan20 = see20,

the values of these quantities given by (3.6) and (3.7). Thus we obtain

So far the yield conditions (1.4) and (1.5) have not been used explicitly. In the following sections the ratio T/+ is found for each of the two yield conditions; then the equation (3.8) will determine the value of the ratio a/b and hence from (3.7) we can find the inclination 0 of the slip band. 4. INCLINATION OF THE SLIPBANDS UNDER THE VON

MISES YIELD

CONDITION

When we substitute the values of u& given by (3.1) into the condition (1.4) the following expression for cp is immediately obtained

2k

cp =

27

dzjziG = d?6 q - G

(4.1)

where the plus sign is taken before the radical since it is assumed that cp is positive. Substituting the value of Q given by (4.1) into (3.8) we now find a2

+ 16b2 = 4d3b d a 2 + 4b2.

(4.2)

Squaring both members of (4.2) we obtain a fourth degree equation in the quantities a and b which is seen to reduce immediately to the simple relation a = =t&b by which the ratio a/b is determined. Hence equation (3.7) becomes tan 8 = fl/h. I t follows that the slip bands have an inclination of 8 = f35'16' under the von Mises yield condition.

5. INCLINATION OF THE SLIPBANDS TRESCA YIELDCONDITION

UNDER THE

r*

The principal values of the stress deviator are given by the solutions of the determinantal equation

EQUILIBRIUM THEORY OF LUDERS BANDS

Expanding this determinant and solving for

T*

109

we find

Let us denote these three values of T* by 711, 7: and 7: with the understanding that T; 2 T; >= 7:. Assume far the moment that

Then the Tresca yield condition (1.5) gives

- 7: = dXpz$= 2k

=

(5.2) Substituting the value of the ratio T/+ given by (5.2) into (3.81, we obtain a2 16b2 = 8 b w . (5.3) Now when we square both members of (5.3) we find that the resulting fourth degree equation in a and b leads to the following two sets of relations 7;

7.

+

a=*4b,

(5.4)

a = = t a b .

(5.5)

When the values of a given by (5.4) and (5.5) are substituted in turn into the equations (5.1) we obtain T;

= 2b4;

7;

=

3b+;

7:

=

-b+;

7:

=

-b+,

7 ;

=

-b4;

7:

=

-2b4,

respectively. Since b and 4 are positive the required inequalities 7: 2 7; 2 7 : are therefore satisfied for each of the possible relations (5.4) and (5.5).

110

IV. PERFECTLY PLASTIC SOLIDS

Now it is evident that 4 as given by the first equation (5.1) is the greatest of the principal values of the tensor u*. If we suppose that the least of the principal values is given by T; = -b+ and proceed as above, we will obtain only the single set of relations (5.4) between the const,ants a and b. Hence (5.4) and (5.5) give all relations possible between these constants. When we substitute the values of the ratio a/b given by (5.4) and (5.5) into the equation (3.7) we now have

In other words the plastic slip bands have inclinations of f35"16' or &50"46' under the yield Condition of Tresca. Remark. Plastic slip bands having an inclination of approximately 30' have been observed by us personally on specimens of aluminum alloy tested a t the Naval Research Laboratory, Washington, D. C.; this is somewhat less than the angle of 35'16' given by the above theory under the von Mises as well as the Tresca yield conditions. Moreover the angle of 50"46', obtained under the Tresca condition, likewise appears to be slightly in excess of the inclination of the bands which occur in specimens of mild steel. See A. Nadai, Theory of Flour and Fracture of Solids, McGraw-Hill, New York, vol. 1, 1950, p. 278. However closer agreement between theory and observation may possibly be obtained by a consideration of the effect of compressibility, i.e. by assuming an elastic-plastic rather than a rigid-plastic medium. In this connection see T. Y. Thomas, T h e effect of compressibility o n the inclination of plastic slip bands in flat bars, Proc. Nat. Acad. Sci., 39 (1953), pp. 266-273.

V. Characteristic Surfaces and Wave Propagation 1. REDUCED SYSTEMOF FLOW EQUATIONS FOR RIGID-PLASTIC SOLIDS

According as we assume the von Mises yield condition or the Tresca yield condition the stress-rate of strain relations for the rigid-plastic solid (see Sect. 2 and Sect. 3 of Chap. IV) have the form

4% tij, u;r =

2k

u;j

= -eij, 711 - r13

(for von Mises yield), (for Tresca yield),

in which q1 and v3 are the greatest and least of the principal values of the rate of strain tensor e and the quantity k is a material constant (see Remark 2 in Sect. 2 and Remarks 1 and 2 in Sect. 3 of Chap. IV); the other quantities in these equations have entered repeatedly in the foregoing discussion. Actually the relations (1.1) and (1.2) imply the validity of the quadratic or von Mises and Tresca yield conditions respectively, i.e. the conditions (2.13) and (3.9) of Chap. IV. Thus, multiplying corresponding members of (1.1) and summing on the repeated indices i and j we immediately obtain the quadratic yield condition; similarly if we transform the relations (1.2) to canonical coordinates (see Sect. 10 of Chap. I) and evaluate at the origin of the canonical system we find, on account of the relations (2.2) of Chap. IV, that the combination of two of the resulting equations gives the Tresca yield condition. It will be assumed in this section that the relations (1.1) or the relations (1.2) are satisfied. In addition the flow in the rigid-plastic solid satisfies the equation vi,i =

0,

(equation of incompressibility), 111

(1.3)

112

V. CHARACTERISTIC SURFACES AND WAVE PROPAGATION

and the usual dynamical relations

+ + pvi-i = 0,

(equation of continuity),

dt

uil,i

=p

dvi

clt)

(equations of motion).

(1.5)

Combining (1.3) and (1.4) we have dp/dt = 0, i.e. the density p remains constant following the motion of the individual material particles. This condition will be satisfied in particular if the density is constant, i.e. independent of position and time; for simplicity i t will therefore be assumed that the density p i s constant in the following discussion. Let us now differentiate the equations (1.1) with respect to xi and sum on the repeated index j. The resulting equations, after making use of (1.3) and (1.5), can be given the form

where we have put

p =

-AU

..

A2=

(1.7) The relations (1.3) and (1.6) will be called the reduced system of flow equations under the von Mises yield condition. As so defined the reduced system consists of four equations for the determination of the four dependent variables p and vi. If p(z,t) and vi(x,t) furnish a solution of the reduced system, the quantities u;j are determined from (1.1) and hence the components aij of the stress tensor can be obtained from the relations 3 11)

~ : j=

aij

+ p6ij,

&beab

(1.8)

which define the deviator u*. Also it is easily seen that the equations of motion (1.5) are satisfied in view of the relations ( l . l ) , (1.3), (1.6), (1.7) and (1.8). Hence the basic equations (l.l),(1.3) and (1.5) under the above assumption that the density p i s constant, can be replaced by the equations (1.3) and (1.6) of the reduced system for the determination of the flow within the rigid-plastic solid. Analogous remarks apply when the Tresca yield condition is postulated. I n treating this problem we shall assume the general case for which the principal values T l , T z , T 3 of the stress tensor are distinct. Then the components v:,vi,v'3 of the three mutually perpendicular unit vec-

1.

REDUCED SYSTEMS OF FLOW EQUATIONS

113

tors v11v2,v3which give the principal directions are determined to within algebraic sign at each point and hence will be determined uniquely throughout the region of flow, by the requirement of continuity, after the selection of the vector triad vl,v2,v3 at an arbitrary point; moreover the components v:,vi,vi will be differentiable functions if the components of the stress tensor are differentiable as is assumed. Under these conditions it follows from the equations (10.9) of Chap. I that we have relations of the form

aabbl

7i.j

=

cab,]vivij

T:,j

=

c:bJv
qi,j =

a b cab.jvivi,

(1.9)

where the 7: and q i are the principal values of the tensors u* and E and there is no summation on the index i in the right members of these equations. Differentiating (1.2) with respect to xi and summing on the repeated index j we now find, after making use of the relations (1.3), (1.5) and (1.9), that

where

p

= -Iu..' 3

J j ,

Pi' =

V M

- vid.

The equations (1.3) and (1.10) give the reduced system, under the Tresca yield condition, for the determination of the quantities p and vi when the density p is constant. Equations (1.2), (1.3) and (1.5) for the determination of the flow in the rigid-plastic solid can be replaced by the equations (1.3) and (1.10) of the reduced system. The demonstration of this statement is similar to the above demonstration of the corresponding result under the von M i s s yield condition. Remark. From the assumption & in Sect. 1 of Chap. IV for the perfectly plastic solid, the coefficients of the cij in the right members of (1.1) and (1.2) must be positive; more explicitly we understand that these coefficients are also finite except possibly on boundary points of the flow region R or on loci in R of lower dimensionality, e.g. isolated points, curves or surfaces. Now consider the flow in a region R, subject to the relations (l.l),and denote by X the region obtained from R by excluding the above exceptional points, i.e. the points at which ej = 0. Thus e i j c i j > 0 at points P C%. But then 91 - q 3 > 0 in 92 where ql and q 3 are the greatest and least of the principal values of the rate of

114

v.

CHARACTERISTIC SURFACES AND WAVE PROPAGATION

strain tensor E. In fact if ql = 93 at any point P C 92 we have q1 = at P on account of the ordering 91 2

92

2

q2 =

93

93,

(see Sect. 3 of Chap. IV); hence 9; = 0 for i = 1,2,3 from the identity (3.1) of Chap. IV. But this implies eii = 0 at P for i:j = 1,2,3 since the components of the rate of strain tensor E are given by 9,8ii in the canonical system whose origin is a t the point P as we see from the equations (2.2) of Chap. IV. It follows from this contradiction that q1 - v3 > 0 at P, i.e. wl - q3 > 0 in '8. Similarly if the flow in the region R is associated with the equations (1.2) and I is the region obtained from R by excluding points at which 91 = 93, so that q1 - 9 3 > 0 in I,it is readily seen that cijci? > 0 in I. We may observe in this connection that the flow in any region R cannot arise solely from a rigid motion of the medium. For the condition eii = 0, which is seen to be excluded in the region I,is precisely the condition for the flow to be of the rigid type (see Sect. 4 of Chap. I).

2. CHARACTERISTIC SURFACES

Consider a moving surface Z ( t ) defined by an equation

4(21,22,23,t) = 0, (2.1) where 4 is a differentiable function of the rectangular coordinate x i and the time t ; it is assumed that Z ( t ) is regular in the sense that +,i4,i >0 at each point of the surface. As a consequence of this inequality at least one of the derivatives 4,i does not vanish a t any point of the surface and therefore the above equation (2.1) can be solved for one of the variables xi, i.e. for the variable xi which corresponds to the nonvanishing derivative, in accordance with the usual existence theorem. If P is an arbitrary point on the surface 2 ( t ) we can choose the coordinate axes so that 4,3# 0 at P and thus we can represent the surface in the neighborhood of P by an equation of the form 5 3 = f(Z',X2,t).

(2.2)

Now consider the values of the function p , defined in Sect. 1, over the surface Z ( t ). On the part of the surface for which the representation (2.2) is valid the function p is a function of the surface coordinates x1,x2and the time t ; hence we can write

2.

115

CHARACTERISTIC SURFACES

where the derivatives dp/dxiinvolve surface diflerentiation and the letter p , appearing in the symbol of these derivatives, denotes the function p expressed as a function of the surface coordinates x1,x2 and the time t. Equations (2.3) express the partial space derivatives p.1 and p,z in terms of the partial space derivative p ,3. An analogous consideration applies to the components vi of the velocity vector. Thus we have

-dvi,j _

dxk -

vi,jk

f

Vi,j3f,kl

(i,j

=

1,2,3;k = 172)-

(2.5)

Let us now (1) t a k e j = 3 in (2.5) and (2) restrict j to the values 1,2 in (2.5) and then interchange the indices j and k in these equations; we thus obtain dvi 3 (i = 1,2,3;k = 1>2)9 Vi,k3 = dxh - V i , 3 3 f , k , (2.6) vi,jk

=

d0i.k

dzi -

‘#i,k3f,jr

(i = ll2,3;j,k = 1,2).

(2.7)

Combining (2.6) and (2.7) we find that

(i = 1,2,3;j,k = 1 ~ ) .

J

Hence, if the components vi and their first derivatives vi.i are assigned over the surface (2.2), i.e. are given as functions of the surface coordinates x1,x2 and the time t subject to the conditions (2.4), the second derivatives v , . k 3 and V i , j k for i = 1,2,3;j,k = -1,2can be found from (2.6) and (2.8) when the derivatives vi,33are known. To find the values of p , 3 and vi,33which are not determined over the surface (2.2) by the above assignments we must resort to one of the reduced systems of equations derived in Sect. 1. In general the elimination of the derivatives in the left members of (2.3), (2.6) and (2.8) from the reduced system will resuIt in a set of Iinear equations for the determination of the four quantities p.3 and v ; , 3 3 such that the determinant A of the coefficients of these quantities will not vanish at any point of the surface. In this case the determination of the remaining derivatives is reduced to the solution of a set of linear equations. How-

116

v.

CHARACTERISTIC SURFACES AND WAVE PROPAGATION

ever, in particular, the determinant A may vanish at each point of the surface so that the solution of the equations for the quantities p.3 and v;,33 is algebraically indeterminant over the surface. The surface is then called a characteristic surface relative to the assigned data. A modification of the assigned data on a characteristic surface will of course, in general, invalidate the characteristic condition, i.e. the vanishing of the determinant A on the surface. The problem of evaluating A, as above considered, appears to be somewhat complicated, especially for the case of the reduced system associated with the Tresca yield condition. This problem, however, can be simplified materially by the device of choosing the coordinate axes so that one of the coordinate planes is tangent to the surface 2(t) a t the above point P. Let us select the coordinate system so that the origin is a t the point P and the d , x 2 plane is tangent to the surface at this point. Then the above representation (2.2) applies and moreover f , i = 0 for i = 1,2 at P. Hence a t the point P , i.e. the origin of the coordinate system under consideration, the term in p.3 drops out of (2.4) and similarly the terms containing vi,33are removed from (2.6) and (2.8). It follows that in order to determine the quantities p,3 and vi.33 at the origin of coordinates we have merely to solve the reduced system for these quantities without the necessity of first making the substitutions (2.4), (2.6) and (2.8). Let us apply this procedure to determine the quantities p,3 and vi.33 within the flow region $3 (see Remark in Sect. 1 ) by the reduced system, associated with the von Mises yield condition, namely the system composed of the equations (1.3) and (1.6). Expanding (1.6) so as to exhibit the terms p , 3 and vi,33for solution, we have

(2.9)

Conditions on the quantities vi,33due to (1.3) are of course obtained by differentiating this equation with respect to 2 3 ; this gives the relation 211.13 -k v2,P3 %,33 = 0 (2.10)

2.

(A2 - 2ef3) -2623613 -2E33e13 0

117

CHARACTERISTIC SURFACES

-2613623 -2613% (A2 - 2 ~ ; ~ ) -2623e3~ -2633623 ( A 2 - 2&) 0 1

0 0 1 0.

(2.11) as the characteristic condition at the point P of the surface Z ( t ) relative to the above special coordinate system. The equation corresponding to (2.11), under the assumption of the Tresca yield condition, can now be obtained rather quickly. Thus, expanding (1. lo), we have

(11 - 113 - 2%qi3) -2623q13

-2633ql3 0

- 2€i3q23 (Ti - 73 - 2623q23) -22~33q~~ 0

-22~13q~~ - 2 ~ & ~ ~

0 0

(71 - 7 3 - 2 ~ 3 3 q ~ ~1) 1 0.

Equating this determinant t,o zero, we obtain the relation

118

V. CHARACTERISTIC SURFACES AND WAVE PROPAGATION

from which we find the desired equation corresponding to (2.11), namely

+

(2.13) 2ci3q13 2E23qZ3= T i - 71'3. Equations (2.11) and (2.13) express the condition at an arbitrary point P of the surface 2 ( t ) , relative to the special coordinate system whose origin i s at P and whose x1,x2coordinate plane i s tangent to Z ( t ) at P , for this surface to be a characteristic under the von Mises and the Tresca yield conditions respectively. The forms (2.11) and (2.13) of the characteristic conditions would appear at first sight to be of questionable value since the coordinate system, in terms of which these conditions are expressed, is not known in advance of the surface Z ( t ) . However we shall show in the next section how the general invariant formulation of the characteristic conditions can be obtained immediately from (2.11) and (2.13) by a simple application of the principles of tensor algebra, i.e. we shall express these conditions in a form independent of the above special coordinate system.

3. INVARIANT FORMULATION OF THE

CHARACTERISTIC CONDITIONS Denote by w the unit vector a t an arbitrary point P of the surface Z ( t ) . But the components w1,wz,w3or w1,w2,w3of this vector have the values 0,0,1 respectively relative to the coordinate system used to express the characteristic conditions (2.11) and (2.13). Hence we see that €?3

+ €;a

=

Ei3Ei3

= €ijQkWjWk -

and €13fJl3

+

€23qZ3

= €i3qikWiWk

-

t33

(€ijWiWj)2, = €i3fJB

- €33f3

1

- (€i3'wiw')(qkmwkwm).

Substituting these expressions into (2.11) and (2.13) we obtain the desired formulation of the conditions for a surface Z ( t ) to be a characteristic as follows €..€. 13 zkWiwk - (c6wiwi)2 = +A2 (3.1) €ijqikWiwk- (~ij%)iWi)(qkmWkWn) = $(TI - 73), (3.2) under the von Mises and Tresca yield conditions respectively; obviously the conditions (3.1) and (3.2) are independent of the local representa-

4.

119

YIELD CONDITIONS OF VON MISES

tion (2.2) of the surface Z ( t ) used in the demonstration. The unit normal w to the characteristic surface Z ( t ) , whose components enter in the above conditions, will be said to define a characteristic direction a t the point P of the flow region 8. 4. CHARACTERISTIC DIRECTIONS UNDER THE YIELDCONDITION OF VON MISES Consider the conditions (3.1) in a canonical coordinate system (see Sect. 10 of Chap. I) whose origin is a t a point P of the characteristic surface Z ( t ) ; we shall seek to determine the values of the components w1 of the unit vector w, giving a characteristic direction a t P , a t the origin of this canonical system. But, relative to the canonical system and a t the origin of this system? we have

+ +

A' = 7: 7; $, where the v z are the mutually perpendicular unit vectors which determine the principal directions a t the point P (see Sect. 9 of Chap. I). Making these substitutions in (3.1) and expanding we now obtain V; =

6;

~ i= j

7;6ij;

Since w is a unit vector we can write w;

=

1 - w? - w;.

Using this relation to eliminate the quantity w;from (4.1) we find that the resulting equation can be put in the form (71

- 73)'wdi

+ [(w

+

(71

- 73)[2(72

- 7s)W;

1

- (71 - 73)]'d

+ +(7? + 7; +

(4.2) 0. Now the left member of (4.2) is a quadratic expression in w?;hence, when we impose the usual condition that the roots w? of the equation (4.2) must be real, we find that (71

- 73)*(w;

- w;)

- 43)2{[2(7z - 7 d W ;

- 4[(72 - 7]3)2(Wi- W i )

+

-

1

-d I 2 $. 7; + d ) ] )2 0.

(71

+(V?

731 =

(4.3)

But the quantity (71 - ~ 3 is) positive ~ in the flow region % under consideration (see Remark in Sect. 1) and hence can be cancelled from the inequality (4.3). Further simplification, involving the expansion of the

120

V. CHARACTERISTIC SURFACES AND WAVE PROPAGATION

bracket expression and collection of terms, now gives the following reality condition

+

--4(111- rlz)(rlz - 71~)w;- (ql v3I2- 271; L 0. (4.4) But each term in (4.4) is non-positive in consequence of the ordering 91 2

112

2

93

and hence must vanish separately in order for this inequality to be satisfied. Hence

- 712)(7?2 - qdw; = 0; Now the last equation (4.5),i.e. 112 (711

711

+

713

=

0;

T2

= 0.

(4.5)

= 0, implies the second equation of this set on account of the identity (3.1) of Chap. IV. Furthermore it follows that the first equation becomes Tlr/3?-d = 0. (4.6) But the vanishing of either of the quantities v1 or r13 implies the vanishing of the other, since qz = 0, and thus contradicts the result in the Remark in Sect. 1. Hence (4.6)gives wz = 0, and hence the equation (4.2) becomes (111 - 113)2w! - (111 - 93)Ztd B(q9 9 3 = 0.

+

by

The roots of this quadratic equation in Hence

+

td are equal and are given

4 = 1/2.

w1=

1 . *--9

d5

w3

=

1 f--1

&

since w2 = 0. There are thus two independent characteristic directions a t each point and the components of the unit vector w which determine these directions have the values

at the origin of the canonical coordinate system under consideration. But the directions given by (4.7)are normal to the surface elements of maximum numerical shearing stress (see Remark in Sect. 12 of Chap. I); hence we have proved the following result. Under the assumption of the von Mises yield condition there exist at each point of the flow region 93 two, and only two, real characteristic surface elements, which are identical with the surface elements of maximum numerical shearing stress, provided the

5. YIELD

121

CONDITIONS OF TRESCA

invariant qz = 0. I f qz does not vanish real characteristic surface elements do not exist. Remark. It is immediately seen that the unit vectors w which give the characteristic directions have the components

in an arbitrary coordinate system. For the vectors w defined by (4.8) are unit vectors and their components have the above numerical values (4.7) at the origin of the canonical system.

5. CHARACTERISTIC DIRECTIONS UNDER THE YIELDCONDITION OF TRESCA The characteristic directions under the Tresca yield condition can be determined by the process employed in Sect. 4. We first observe that at the origin of the canonical system ql1 = 1, q a 3 = -1 while the remaining quantities qii are equal to zero. Hence, evaluating (3.2) at the origin of the canonical system we obtain

91w: - 9 3 d - (9ld

+ 9zw;+ 93&)(w:

- w3 = 3(91

- 93).

As before we now eliminate the quantity wi and are thus led to the following quadratic equation in w:, namely

+

+

}

2(91 - 9dw; [2(92 - 93)Wi (91 - 9 3 ) d - 2(9l - 9 J1d (5.1) [(VZ - d(w; - Wi) ~ ( V I- 93)] = 0. We next impose the condition for the roots w ! of (5.1)to be real; after some reduction this inequality can be given the form

([2(92 - 93) - (91- v~)]~W;- 4(% - d2)&2 0. (5.2) Let us now assume that wz# 0, which permits us to remove the factor from (5.2). Furthermore 2(92 - 93) - (91 - 13) = 91 - 93 - 2(V1 - 912). Hence the inequality (5.2)becomes [(Ti - 93) - 2(91 - dI2d2 4(9i or

:w[l J-)--(2

711

- 112

B 4.

(5.3)

122

V. CHARACTERISTIC SURFACES AND WAVE PROPAGATION

But 0 5 ? l C ? &5 1, 171

- 173

and hence - 1 5 1 - 2(-)171 - 712 171

5 1.

- 173

Then, since w: 5 1, it is clear that the left member of (5.3)must actually be less than 4 so that this inequality is not satisfied. Hence the above assumption that w2 # 0 i s not valid. Putting w2 = 0 in (5.1) and cancelling the non-vanishing factor ql - 173 we now have 4w: - 4w: 2 = 0,

+

from which it follows that w? = 1/2. Hence the components w,have the values previously found in Sect. 4 at the origin of the canonical system. The following result has now been proved. Under the assumption of the yield condition of Tresca there exist at each point of the flow region 3 two, an.d only two, real characteristic surface elements, which are identical with the surface elements of maximum numerical shearing stress. As before the unit vectors w, which determine the characteristic directions, are given in general coordinates by the relations (4.8);however the previous conditions 712 = 0 for the existence of the characteristic surface elements is not necessary under the yield condition of Tresca. Remark. It is well known (see Concepts, pp. 105-107) that a vector field w will be normal to a family of surfaces in three dimensional space if, and only if, the following condition is satisfied Wl(W2.3

- w3.2)

+

WZ(W3,l

- W1,d

+

WdW1,Z- W2,J =

0.

(5.4)

Hence, when we substitute the components of the vectors w given by (4.8) into (5.4),we obtain the necessary and sufficient conditions for the two families of characteristic surface elements to combine to form two families of characteristic surfaces under the yield condition of Tresca in terms of the components v:, vi, v(3 of the principal directions and their partial derivatives. Replacing the partial derivatives of the components v:, Y:, v$ by their values in accordance with the formula (10.12) of Chap. I, we obtain the required conditions in a form involving the partial derivatives of the components g i f of the stress tensor, the principal directions vi and the principal stresses T, which may, if we like, be expressed in terms of the stress and the principal directions by a substitution of the type (9.17) in Chap. I; the conditions so obtained for the existence of

6.

CHARACTERISTICS

IN PRANDTL-REUSS

THEORY

123

these two families of characteristic surfaces are valid in the general case, i.e. %-henthe principal stresses are distinct. When the von Mises yield condition is assumed the above conditions must be supplemented by the condition qz = 0, as shown above, or by the equivalent condition T; = 0 which is an immediate consequence of the relations (2.1) in Chap. IV. For a detailed discussion of the conditions for the existence of characteristic surfaces see T. Y. Thomas, On the characteristic surfaces of the eon Mises plasticity equations, Jour. Rational &Tech. and Anal., 1, 1952, p. 355. Characteristic surfaces are the three dimensional analogue of the characteristic curves, sometimes referred to as slip lines (Zignes de glissement), which always exist in the two dimensional plasticity problem and which have been treated more or less extensively in the literature. I n this connection the question might be considered as to whether it would be possible to choose the material function M which enters into the general yield condition in Sect. 2 of Chap. IV so that, corresponding to the two dimensional problem, the two families of characteristic surfaces will always exist in any flow region. Beyond this specific question it may well be possible to select the function M so as to obtain a physically more appropriate set of constitutive equations for plastic flow, while remaining within the general framework of the theory of the perfectly plastic solid, if one is willing to depart from the comparatively simple relations afforded by the von Mises or the Tresca yield condition.

6. CHARACTERISTIC SURFACES IN

THE

PRANDTL-REUSS THEORY

We shall now consider t h e characteristic surfaces in a tlow region R of an incompressible elastic-plastic solid (see Remark 2 in Sect. 4 of Chap. IV) which is governed by the Prandtl-Reuss equations (8.1) in Chap. I V under the assumption of the quadratic or von Mises yield condition.

Thus

* u&)! u&u~,=

2k2,

(Prandtl-Reuss equations), (von Mises yield condition),

(6.1) (6.2)

in which %:j/Bt are the components of the covariant time derivative of t h e stress deviator U* (see Sect. 7 of Chap. IV). T h e various symbols in t h e above and following equations are well known from t h e preceding discussion; however we may mention that the components eiJ of the rate of strain t,ensor rahher t h a n t h e corresponding deviator components e*j can be used in (6.1) since this tensor a n d its deviator are identical o n account of the condition of incompressibility. In addition to the above

124

v.

CHARACTERISTIC

SURFACES AND WAVE PROPAGATION

equations, the equation of continuity, the equations of motion and the equation of incompressibility will enter into this consideration; we list these equations here for convenience of reference as follows

+ dt + uij,,

v;,i

pvi,i

=p =

0,

=

dvi dt

-9

0,

(equation of continuity),

(6.3)

(equations of motion),

(6.4)

(equation of incompressibility).

(6.5)

If we sum on the indices i and j in (6.1) the resulting equation is satisfied identicalIy ; hence there are at most five independent equations in the set (6.1). Moreover, the equations (6.1) are not entirely independent of the yield condition (6.2). Thus if we consider the equations (6.1) without regard to the condition expressed by (6.2) we can deduce

The first part of this relation is actually an identity (see Remark 1 in Sect. 7 of Chap. IV). Now write (6.6) in the form

where

and let 2 be a surface, fixed or moving relative to the allowable coordinate system employed, which is intersected by the trajectories of all particles in its immediate neighborhood. We assume that the quantity W vanishes a t all times over 2 in accordance with the yield condition (6.2). It now follows, from the uniqueness theorem for differential equations of the type (6.7), that W = 0 in the neighborhood of 2 at any time t. In other words, we can consider the yield condition (6.2) to be of the nature of a boundary condition over the above surface 2. With the understanding that this boundary condition is imposed we can therefore limit our attention to the set of equations (6.1), (6.3), (6.4) and (6.5). There are, in general, ten independent equations (6.1)1 (6.3), (6.4) and (6.5) and these equations involve the ten quantities p, vi and aij as dependent variables.

6.

CHARACTERISTICS IN PRANDTL-REUSS THEORY

125

Denoting by x1,x2,x3the coordinates of the rectangular system to which the above equations are referred, let us consider a moving surface Z ( t ) defined by an equation F ( x ~ , x ~ , x= ~ ,0, ~) where the function F satisfies the following conditions. First, F(x,t) possesses continuous first partial derivatives with respect to the space variables x and the time t ; second, the (invariant) condition F 9 i F p > i 0 is satisfied on Z(t). When these conditions are satisfied the surface is said to be regular (see Sect. 2). For any assigned value of t the surface Z ( t ) will be assumed to divide the flow region R ( t ) under consideration into a region R l ( t ) and a region R2(t). Now the algebraic sign of F in &(t) will be opposite to its algebraic sign in Rz(t). Otherwise, since F = 0 on Z ( t ) , it would follow that F has a maximum or minimum value on Z ( t ) ; but this implies F,i = 0 on Z (t) in contradiction to the above assumption. We choose this algebraic sign so that F < 0 in R l ( t ) and F > 0 in R z ( t ) ; then if v denotes the unit normal vector to Z ( t ) , this vector being directed from the region R l ( t ) into the region Rz(t), we shall have

where G is the velocity of Z ( t ) in the direction v. In fact we know from the theory of surfaces that the right member of the first set of equations (6.8) is equal to vi to within algebraic sign. To show that this sign is correctly chosen in (6.8) we select a point P with coordinates x i on Z ( t ) and let Q denote the point which is reached by going a positive distance b from P in the direction of the normal Y at P. Then Q will have coordinates xi bvi. Also

+

+

F ( x ~ bvi,t) = bF,i(xlt)vil to within first order terms in the quantity b. But the left member of this relation is positive by the condition that F is positive in the region R z ( t ) and the right member will be positive as a result of the substitution (6.8). Now consider surfaces Z ( t ) and Z(t t’). Let P with coordinates x i be any point on Z ( t ) and suppose that the normal to Z ( t ) at P intersects Z ( t t’) at the point Q . Denote by b the distance of Q from P, this distance being taken positive if Q lies in the region &(t) and negative if it is in the region Rl(t). We now define the velocity G

+

+

126

v.

CHARACTERISTIC

SURFACES AND WAVE PROPAGATION

of the surface X(t) a t P as the Zim (blt’) as t’ -+ 0. Thus G is positive at, a point P of X ( t ) a t which the surface moves from region &(t) into region Ri(t) and negative in the opposite case. To determine the value of G from the function F(z,t) we make use of the fact that the coordinates of the above point Q are xi bvi. Hence since Q lies on X(t

+ t’).

+ F ( z i + bvi,t + t’) = 0, It follows that

to within first order terms in b and t‘. Hence, dividing through by t’ and passing t o the limit, we find that (6.8) gives the velocity G of the surface X(t). Now assign the functions p , vi and uij over the surface z ( t ) subject to the condition (6.2) and obvious differentiability requirements. The moving surface Z ( t ) is said t o be a characteristic surface relative to the assigned data or simply a characteristic surface when the equations (6.1), . . . , (6.5) and the assigned data do not suffice for the determination of all first partial derivatives of p , v;, and uii over Z ( t ) . To obtain the condition for Z ( t ) to be a characteristic surface it will be convenient to introduce a specially selected rectangular coordinate system which may move with uniform velocity relative to the original z system. Denoting by y1,y2,y3the coordinates of this special system, the transformation z ++ y is permissible since this transformation will leave invariant all equations of the set (6.1), . . . , (6.5). Now consider an arbitrary point P on z ( t ) a t some particular time to and choose the y system so that its origin coincides with P and its y 1 and y 2 axes lie in the tangent plane to Z at the time to. In addition we assume that a t time to the velocity G of z(t)is zero at the point P relative t o the y system. We may now represent the surface Z ( t ) locally, i.e., for points Q neighboring P and for times t neighboring to, by an equation of the form Y 3 = f(Y1,Y2,@

(6.9)

Hence a t (P,t,), i.e., a t the point P and time to, we shall have (6.10)

7.

GENERAL CHARACTERISTIC CONDITIONS

127

since the y1,y2 plane is tangent to 2(to)and G = 0 relative to the moving y system. Retaining the above symbols p , vi and uil to denote the components of density, velocity and stress relative to the y system, we suppose that P ( Y ' , Y ~ , Y ~ ,= ~ )P ( Y ' , Y ~ , ~ ) ,

vi(y',y2,y3,t) = vi(y',y2lt>,

~~AY'J~~O,

c ~ i i ( ~ ' , ~ ~=, ~ ~ , t )

on Z ( t ) . The deviator components c& which are determined from the Cij are subject to the condition (6.2). Otherwise the cij as well as the quantities p and Vi are arbitrary differentiable functions of the variables y l , y2 and t. h'ow

)?,k = P , k

fork = 1,2. Hence

+

~.3f,k,J

(6.11) a t (P,to) on account of (6.10). Similar equations hold for the quantities v i and ui,. Hence all derivatives of p , vi and uij are determined at (P,to) from the assigned data on Z ( t ) with the exception of the derivatives of these quantities with respect to the coordinate y3. The determination of these latter derivatives by means of the equations (6.1), . . . , (6.5) will be considered in the following section. 7. GENERALCHARACTERISTIC CONDITIONS Let us suppose that the assignment of data on z(t)is such that the normal velocity of the flow, relative to the velocity of 2, does not vanish on this surface. The yield condition (6.2) can then be disregarded, except as a boundary condition, and our attention confined to the system (6.1), (6.3), (6.4) and (6.5) as shown in Sect. 6. This is the general case and the resulting characteristic condition will be referred to as the general characteristic condition. The special case for which the velocity G of Z ( t ) is equal to the velocity of the flow normal to z ( t ) will be discussed in the Remark 4 in Sect. 8.

128

v.

CHARACTERISTIC

SURFACES AND WAVE PROPAGATION

Since the second term in (6.3) vanishes on account of (6.5) we find that (6.3), (6.4) and (6.5) give the conditions v3p.3 ui3.3

- pv3vi.3 v3.3

:]

.. = .. = .. . =

(7.1)

f

at (P,h) where the dots are used to denote terms not involving differentiation with respect to y 3 . To obtain conditions corresponding to (7.1) from the equations (6.1) we shall need the formula for the covariant time derivative of the tensor u* which is given by

(see Sect. 7 in Chap. IV), where

-

4 t.i . = '(v; a # 3. - v i . a.) Making this substitution in (6.1) and also making the substitutions

we find that the equations (6.1) give the following conditions on derivatives with respect to y3 a t (P,to),namely v3uii.3

1 -5 v3'Jkk,38+j

. 51 u : t v k 8 j + 51 0;kVk.i - 51 Ui3vi.3

1 U .j 3 v i , 3 - P(8i.j -5

+ +$ vj,i)

ufjd3vk,3

=

* *

--

1

(7.2)

The equation (7.2) for which i = j = 3 can be omitted from the set (7.2) since this equation is linearly dependent on the equations for which i = j = 1 and i = j = 2. Hence there are ten independent equations in the system (7.1) and (7.2) for the determination of the ten derivatives v1J;

v2.3;

v3.3;

011,3;

al2.3;

al3,3;

a22,3;

a23.3;

u39.3;

P.3.

Denoting by A the determinant of the coefficients of the above derivatives it can easily be shown that A can be reduced to the following second order determinant

I

6ZT

SNOLLI(IN03 3I&SIZI3&3VBVH3 ‘IVZI3N33

‘L

130

v.

CHARACTERISTIC SURFACES AND WAVE PROPAGATION

nate system, i.e., orthogonal coordinate transformations and uniform translations. We now seek to replace the remaining terms in (7.4) by expressions of corresponding invariant character. It can be shown that

- arzafz =

-

(7.6) In fact if we expand the right member of (7.6) and make use of the fact that a;' is equal to zero, the relation is readily verified. On account of the condition (6.2) we can therefore substitute * * * * Ullu22 - (T12u12 = u;ju&vjvk - k2, (7.7) for the above expression in (7.4). To obtain the corresponding substitution for the last parenthesis expression in (7.4), which we now denote by J for brevity, we first observe that ar1u;z

ur1u;3u;3

= -utzu;3u;3

u;2fJr3ur3

= -u;1u;3u;a

d3d3

- a;3u;3u;3 - a;3ar3ur3

4fJ;jfJ:j.

= -u;zu:3u;3 =

-u;iCrt3u;3

+ +

u;zur3a;3, a;la;3af3.

1

Hence we can write J = -

u;2u;3u;3

- u:lu:3af3

=

-u;ju;3u;3

+

a;3u;3u;s1

and this leads to the desired expression for J , namely J = - U;jff;krS&vkvm

+

(7.8) When we make the substitutions (7.5), (7.7) and (7.8) in the relation (7.4) we obtain the following invariant formulation of the general characteristic condition, namely p2(un - G)4 - 2pp(un - G)3 1 U;jC&CT&#jVkVmVn.

where we have put vn = v2v;for the component of velocity normal to the characteristic surface Z(t). The relation (7.9) can be regarded as giving the condition on the unit normals to the surface elements of characteristic surfaces B(t) in the plastic flow; it may be noted, however, t,hat these surface elements will depend not only on the flow quantities p, v i and u l j but also on the velocity G of the surface Z(t). If we make

8.

CHARACTERISTICS

131

AS SINGULAR OR WAVE SURFACES

the substitutions (6.8) in (7.9) this equation becomes the general differential equation for the determination of the characteristic surfaces in t.he region of plastic flow, i.e. if the equation F(d,29,23,t) =

0

defines a regular surface (see Sect. 6) in the flow region, such that G - v, does not vanish on the surface, then this surface will be a characteristic if, and only if, the function F(z,t) satisfies the above differential equation.

8. CHARACTERISTICS AS SINGULAR OR WAVESURFACES We now return to a consideration of the rigid-plastic solid in which the flow is determined by the reduced system (1.3) and (1.6). Assuming, as before, that the density p is constant for simplicity, let us denote by Z ( t ) a moving surface in the flow region %(t), defined in the Remark a t the end of Sect. 1, such that (a) the variables p , vi and the derivatives vi.j are cont.inuous over Z(t) and (b) there is a discontinuity in a t least one of the derivatives p,i and at least one of the derivatives l ) i s j k over Z( t ) , i.e., Z ( t ) is a wave surface which is singular of order one relative to the function p and singular of order two relative to the velocity v (see Remark 2 in Sect. 3 of Chap. 11). Hence the following geometrical and kinematical conditions of compatibility are valid over Z(t), namely

[ V ; , ~ ~ ] I =Flivjvk;

[st]

= -GKivj;

[s]

=

G2&,

(8.2)

where G is the velocity of the surface Z ( t ) in the direction of its unit normal v and [ and i$are functions defined over this surface; the quantity must be different from zero and not all of the quantities 1, can vanish a t points of Z(t) by hypothesis. To determine the dynamical conditions of compatibility for this problem (see Sect. 8 of Chap. 11)we differentiate (1.3) with respect to xk and then apply the first set of conditions (8.2). But this gives (8.3) which expresses the fact that the vector having the components &i is xzvz = 0,

132

v.

CHARACTERISTIC

SURFACES AND WAVE PROPAGATION

tangent to the surface Z(t). Also from (1.6) and the above compatibility conditions (8.1) and (8.2) we find that

Equations (8.3) and (8.4) constitute dynamical conditions of compatibility for the wave surface Z(t). Since not all the quantities f and hi can vanish, the determinant A of the coefficients of these quantities in the equations (8.3) and (8.4) must be equal to zero. To simplify the evaluation of A let us choose a coordinate system such that the unit normal vector v has components (0,0,1) at a selected point P of the surface Z(t). Relative to this system the equations (8.3) and (8.4) become

at P. Now construct the determinant of the coefficients of the quantities A & ? , A 3 and [ in the equations (8.5). We thus obtain

( A z - 2&)

-2E23E13 -2E33E13

-2E13f23

0

-2e33EZ3

-2 E 1 3 e 3 3 -2E23E.33 (A2 - 26233)

0

1

0

(A2

0

- 2&)

0 1

apart from a non-vanishing factor. Expanding this determinant and equating the result to zero, we find

A' - 2~T3- 2&

=

0;

this equation can be given the following invariant formulation EijEikVjVk

- (E"v'v')2 13 3 = 'A2 2 . 1

(8.6)

Hence the general condition for the wave surface Z(t) to be singular of mixed order, as above described, i s that the equation (8.6) be satisJied over Z(t). Since the conditions (3.1) and (8.6) are identical it follows that these wave surfaces Z(t) are identical with the characteristic surfaces in the flow region % of a rigid-plastic solid under the von Mises yield condition; the special results derived in Sect. 4 for characteristic surfaces are therefore applicable to the above wave surfaces Z(1).

8.

133

CHARACTERISTICS AS SINGULAR OR WAVE SURFACES

Remark 1. When (8.6) is satisfied the quantities X, and [ will be determined over Z ( t ) from (8.3) and (8.4) to within an arbitrary factor. To effect this determination we can proceed as follows. Multiply (8.4) by v; and X i respectively and sum on the repeated index i. We thus obtain

when use is made of (8.3). Write (8.8) in the form EabLVb

=

ftA 4 5dK

(8.9)

and substitute into the right member of (8.7). This gives

5

=

+A

(8.10)

dxax,EijViVj.

When we now substitute (8.9) and (8.10) into (8.4) the resulting equations can be written Xi

=

./z A

a( f E ijvi

(8.11)

7 l&,vavbvi).

In the right members of (8.10) and (8.11) either upper algebraic signs or lower algebraic signs are to be selected. From (8.10) and (8.11) we see that over the surface Z ( t ) the values of XI, A?, & and ,$ are proportional to elivi

- (€abVavb)vl,

- (EabVavb)VZ, Eaivi - ( E a b v a v b ) v 3 ,

EZivi

x,

Remark 2. It is evident that if we can eliminate the quantities and [ from (8.3) and (8.4) by a formal process, the resulting relation must be equivalent to the vanishing of the above determinant A. To eliminate these quantities we first multiply (8.4) by ei,vi and sum on repeated indices to obtain [ E 81 . . v %. Y I.

-

4 %E . . i . V . - d21e (c.e 2A " ' A3 'I

Ik

v.v )€

x v"

4b a

or (8.12)

134

v.

CHARACTERISTIC

SURFACES AND WAVE PROPAGATION

Using (8.12) t o eliminate the quantity ea&b from (8.7) we find that the resulting relation reduces to (8.6). The fact that we can so easily obtain the general invariant condition (8.6) for the wave surface Z ( t ) shows the strength of these formal procedures. As another application of the compatibility conditions let us consider the wave surface Z ( t ) in the flow region R of an incompressible elastic-plastic solid governed by the Prandtl-Reuss equations with the von Mises yield condition. Assuming the continuity of the density, velocity and stress over Z ( t ) , the geometrical and kinematical conditions of compatibility have the form

b,iI = h;

[z]

=

-G{,

(8.13) (8.14) (8.15)

It will be assumed that not all the coefficients {, X i and f i , vanish simultaneously a t points of Z ( t ) ; otherwise no a priori restrictions will be placed on these quantities. The dynamical conditions of the problem are obtained from the equations of the set (6.1), (6.3), (6.4) and (6.5). No conditions result directly from (6.2) due to the continuity of the stress over Z ( t ) ; in fact (6.2) can be excluded from the system of differential equations under consideration since it is of the nature of a boundary condition as observed in Sect. 6. From (6.1), (6.3), (6.4) and (6.5) we now have (8.16)

[$I

+

Vi[P,iI

+

P[Vi,iI = 0;

[Vi.iI

=

0,

(8.17) (8.18)

From the first set of relations (8.14) we see that

[Gjl

= f([Vi.jl

+

[Vj,il)

=

f ( b j

+

XjVi).

Hence (8.16) becomes (8.19)

8.

CHARACTERISTICS AS SINGULAR OR WAVE SURFACES

135

To deduce the expression for the quantity in the left member of (8.19) we first consider the relations

from which it follows that [u&,t]= [uij,k] -

[!q[%I =

1

[uaa,k]sij, 3

-

3 at Application of (8.15) now permits us to write 1 [U:j,.t]

= tijvk

-

3 [aasijvk,

(8.20) (8.21)

Similarly from the definition of the quantities &i and the conditions (8.14) we obtain [ h j ] = $([Vi,j] - [ V J ) = +(XiVj - XjVi). (8.22) But

from the definition of the covariant t'ime derivative and hence, making the substitutions (8.20), (8.21) and (8.22), it follows that

Finally, equating the right members of (8.19) and (8.23), we obtain the following relations

136

v.

CHARACTERISTIC SURFACES AKD WAVE PROPAGATION

It remains to consider the equations (8.17) and (8.18). But from (8.17) we have immediately

- G){

(On

= 0;

X < V ~= 0,

(8.25)

on account of (8.13) and (8.14). Also, using (8.14) and (8.15), it follows from (8.18) that tijvj

- G)Xi.

= p(vn

(8.26)

Equations (8.24), (8.25) and (8.26) give the dynamical conditions of compatibility for the wave surface Z(t). Let us now assume that vn - G # 0 on Z(t). Then if all A, = 0 we have

6.. - 'c3; aa6..13 - 0 .

{ = 0;

21

(8.27)

[..v. 21 I = 07

1

from (8.24), (8.25) and (8.26). Hence c;..r ] ~ ]- 'c; c ; aavi 3 aa6.. p j v ]. - '3

0. It follows that Eaa vanishes and hence all Eij are equal to zero from the second set of relations (8.27). Since { = 0 also from (8.27) we have a contradiction with the assumption that Z ( t ) is singular as postulated. Hence not all X i = 0 when vn - G # 0 o n Z(t). Let us now multiply (8.24) by v j and eliminate the expression Eijvj from the resulting equations by the substitution (8.26). We thus obtain a set of equations which can be written in the form

+

BijXj

=p

+ 51

AXi

f

Q(Vn

=

- G)[kkVi = 0,

(8.28)

where

A B . . = --1 ( 2

f

- p(vn

~ 7 j ~ i ~ j

u;rnvrnvj

+

- G)2,

~;mvmvi)

-P

* *

1

gjrngJnVmvn.

We observe that the second term in the expression for Bij vanishes when multiplied by X i on account of (8.25) ; this term has been added in order to make the quantities B i , symmetric. Now the equations (8.28) and the second relation (8.25) constitute a set of four linear and homogeneous equations for the determination of the variables X1,X2,X3 and t i i . These equations must have a nontrivial solution since we have shown that not all the X i can vanish under the assumption that on - G # 0 on Z(t). Hence we must have

8.

CHARACTERISTICS AS SINGULAR OR WAVE SURFACES

137

I v1 v2 v3 0 1 Corresponding to any nontrivial solution X1,X2,X3 and &k of the equations (8.28) and the second equation (8.25), we can determine the [ i j by (8.24). The values of the t i l so determined will satisfy (8.26) in which the X i are the above solution functions; for if we multiply (8.24) by v j and combine the resulting equations with (8.28) the relations (8.26) are obtained. Since l = 0 from (8.25) we have thus found a solution l l A i l t i j of the system consisting of (8.24), (8.25) and (8.26). The condition for the existence of this solution is of course the vanishing of the above determinant. Hence (8.29) gives the general condition f o r the existence of the singular surface Z ( t ) . The special case for which vn - G = 0 on Z ( t ) will be discussed in the Remark 4 at the end of this section. Choosing a coordinate system relative to which vi = (0,0,1) at an arbitrary point P on Z ( t ) it is immediately seen that the condition (8.29) becomes

where the quantities M and N are defined in Sect. 7; but this is equivalent to the condition expressed by the vanishing of the determinant (7.3) since v3 in (7.3) can be replaced by the quantity v n - G. Hence the procedure in Sect. 7 applies and leads to the relation (7.9) as the condition for Z(t) to be a singular surface. The general differential equation for the singular surface Z ( t ) is of course obtained from (7.9) by the substitutions (6.8) and hence these singular or wave surfaces can be indentified with the characteristic surfaces discussed in Sect. 6 and Sect. 7. Remark 3. From the first equation (8.25) we have l = 0 since vn - G # on Z ( t ) by assumption. Hence the derivatives of the density p are continuous

across the general singular surface Z (t). Remark 4. Assume that the condition v,, - G

=

0 holds on the wave sur-

138

v.

CHARACTERISTIC

SURFACES

AND WAVE PROPAGATION

face Z ( t ) ; then, making use of (6.8) it follows immediately that the differential equation of Z ( t ) is given by

aF at

+ uiF,, = 0.

(8.30)

No condition is imposed on the quantity [ by the first equation (8.25);hence the derivatives of the density can be discontinuous across the surface Z ( t ) . Discontinuities can also occur in the derivatives of the stress components acj since the conditions (8.26) reduce to &jvi = 0 and these equations can obviously be satisfied by non-zero values of the t,2. However the second equation (8.25) and the equations (8.24),which now become (UZVj

+

u;kvi)Xk

- (&Xi

+

Ui*kXi)Vk

1

will have, in general, only the trivial solution X i = 0 in consequence of which the derivatives of the velocity will be continuous across this wave surface.

9. THE PLANE STRESSPROBLEM. CANONICAL COORDINATES

Denoting by aij the components of the stress tensor relative to a system of rectangular coordinates xi, the stress is said to be plane if the coordinate system can be chosen so that

- 4..(x 1, ~ ' , t ) ;

~ i = 3 0, (9.1) for values of the indices i,j = 1,2,3. The first set of equations (9.1) indicates, of course, that the components ~ i are j functions of the two coordinates x1,x2and the time t. On account of the equations (9.1) we have Uij,3 = 0; f f i 3 , k = 0, ( i , j , k = 1,2,3), (9.2) where the comma denotes partial differentiation with respect to the coordinates. Some of the main consequences of the assumption of plane stress will be obtained directly from (9.2), rather than (9.1), in the following discussion. It will be helpful on occasion to employ the canonical system y with origin a t an arbitrary point P of the medium; between the canonical coordinates yi and the above coordinates x i we have relations of the form Q 23 ..

zj

+

xi = p i yZyk, (9.3) where the p i are the coordinates of the point P and the v i are the com-

9.

THE PLANE STRESS PROBLEM. CANONICAL COORDINATES

139

ponents of the three mutually perpendicular unit vectors V k which give the principal directions at P (see Sect. 10 in Chap. I). Now the components vt are determined by the set of equations

- Tk6ij)d

=

- 7 k V f t = 0,

(9.4) where the T k are the principal values of the stress tensor and there is no summation on the index k in these relations; the principal values 7 k are given as solutions of the determinantal equation (Uij

Uijd

which can be written as the product of two factors, as shown, on account of the conditions (9.1). Now assume that the second order determinant in (9.5) vanishes for k = 1,2. In other words by equating this determinant to zero the principal values T~ and r2are determined; the remaining principal value 7 3 i s then determined by the vanishing of thejirst factor in (9.5), i.e. 7 3 = 0. Restricting the free indices i and k in (9.4) to the values 1,2 we see that the summation on the index j in these equations can also be restricted to the range 1,2 on account of the conditions (9.1). These restricted equations (9.4) will determine the components d for j , k = 1,2; moreover the two incomplete vectors defined by these components d can be supposed to satisfy the condition of perpendicularity as we know from the theory of equations of the type (9.4). Hence taking vi = 0 for k = 1,2 we will thus have a set of quantities v i with k = 1,2; j = 1,2,3 which constitute the components of two perpendicular vectors in the x1,x2plane and m-hich are seen to satisfy the equations (9.4) in which the indices i , j have the full range of values. Finally, taking k = 3 in (9.4), we see that the equations are satisfied by v$ = 65 since 7 3 = 0 by the above result. We have thus arrived a t a set of three mutually perpendicular vectors vk with components satisfying the equations (9.4). Either of the two roots of the above second order determinantal equation may be selected as the value of T~ after which the other root will be assigned as the value of r2. However, for definiteness, it may be supposed that the selection of the values of T~ and 7 2 is made in such a way that the associated principal directions v1 and v2, which lie in the x1,x2plane, will have the same orientation as the x1,x3axes. Then the

140

V.

CHARACTERISTIC SURFACES AND WAVE PROPAGATION

jacobian determinant of the canonical transformation (9.3) will be equal to 1 and the transformation will have the property of preserving the orientation of coordinate axes. I n view of the above determinations we can now state the following result. For the plane stress dejined by (9.1) the principal value 7 3 = 0 and the coeflcients a k in the canonical transformation (9.3) satisfy the conditions v; = 1.

v; = v; = v: = v; = 0;

(9.6)

Hence the relations (9.3) decompose into

+

+

xi = p i viyk, (ilk = 112); 2 3 = p3 y3, (9.7) in which the p's and v's are constants and the subscript k is summed over the values 1,2. It may also be observed from this discussion that, apart from the above determinations, the principal values 7 k and the components vt will, in general, vary with the point P but will depend, at any time t, only on the two coordinates x1,x2of this point. Now suppose that the components u,i + T,? as a result of the canonical transformation (9.3). Specifically this means that t m

721

= Ukmvtvj

*

It follows from the first set of conditions (9.1) and the form (9.7) of the transformation, that the T~~ can depend on y1,y2 and t alone. Also from the second set of condit,ions (9.1) and (9.6) we see that r 2 3 = 0. I n other words the plane stress conditions (9.1) are inoariant under the canonical transformation (9.7). On account of this result the special forms assumed by tensorial relations, as a consequence of the conditions (9S)l will be unaltered if these relations are referred to the canonical coordinates yi. When such relations are evaluated at the origin of the y system a simplified or more compact form of the relations is usually obtained; we shall apply this procedure to advantage in the following discussion.

10. COMPATIBILITY CONDITIONS IN

THE

PLANESTRESSPROBLEM

Let z(t)be a wave surface which is singular of order 1 relative to the set of quantities consisting of the density p, the velocity components v i and the st.ress comp0nent.s uij. Then the first order geometrical and kinematical conditions of compatibilit'y given by (8.13), (8.14) and (8.15)

10.

COMPATIBILITY CONDITIONS IN PLANE STRESS

141

will hold over the surface Z(t). In addition we have the following dynamical conditions of compatibility, namely

(v, - G ) [

=

0;

X , V ~ = 0,

(10.1)

f i j v j = p ( ~ n- G)Xi; ~ : j t i j z= 0. (10.2) The above condit.ions (10.1) and the fist set of conditions (10.2) are identical with the previous relations (8.25) and (8.26) and are a consequence of the equation of continuity (6.3), the equations of motion (6.4) and the equation of incompressibility (6.5); the last equation (10.2) results from the relations obtained by coordinate differentiation of the von Mises yield condition (6.2), which we shall assume in this discussion, and the compatibility conditions (8.15). Differentiation of (6.2) with respect to the time t will lead to no additional conditions. The relations (10.1) and (10.2) are part of the dynamical conditions of compabibility; other such conditions will be obtained from the constitutive relations of the problem. There will also be certain special relations arising from the plane stress condition (9.1) which will next be considered. Let us first observe, however, that if the quantity s defined as the difference G - v,, does not vanish, it follows from the condition (10.1) that [ = 0; for convenience of terminology we shall refer to s as the speed of propagation of the wave surface Z ( t ) . Now if s # 0 not all of the quantities f i j can vanish a t any point of X ( t ) . In fact if &j = 0 for i,j = 1,2,3 at a point P of Z(t) then, since s f 0 a t P , we must have all X i = 0 at P from the equations (10.2). But this contradicts the strict condition that Z(t) is singular of order 1. In the following discussion it will implicitly be assumed, unless the cont.rary is stated, that the speed s does not vanish at points of the wave surface Z(t). It follows from the equations (9.2) and (8.15) that 5..**v3 - 0; f i Q V k = 0. (10.3) Hence we must have v3 = 0 from the first set of equations (10.3) since not all of the quantities t i j can vanish a t any point of Z(t) by the above result; from the second set of equations (10.3) we must have t i 3 = 0 over the surface Z(t). Also X3 = 0 as we see by taking i = 3 in the first set of equations (10.2) and using the relations Ei3 = 0 and the above condition that the speed of propagation s is different from zero. For easy reference we express these results by xriting t i 3 = 0; v3 = 0; A3 = 0. (10.4)

142

v.

CHARACTERISTIC

In view of the condition v 3

SURFACES AND WAVE PROPAGATION

=

0 n-e see that the equation of the wave surface

z(t)must be of the fmm +(xf,x?,t) = 0, where the function 4 i s independent of the x3 coordinate. Since the first set of equations (10.2) is satisfied identically when i = 3 on account of (10.4) we can limit i to the values 1and 2 ;expanding these equations we have ~

V

+ +

I~ Z V Z =

p(un

- G)X1,

(10.5)

(10.6) h ~ i Ez2vz = p ( ~ , G)Xz. Combining (10.5) and (10.6) with the last equation (10.2) we can, in general, solve for the three quantities tl1,t12 and &. To obtain this solution let us transform the latter equation to a system of canonical coordinates and evaluate it a t the origin of this system, We thus obtain

dtii

+

(10.7) 0, where the T; are the principal values of the stress deviator u*. In deriving (10.7) we have made use of the first of the equations (10.4) and the fact that these equations are invariant under the canonical transformation (see Sect. 9); the equations (10.5) and (10.6) can also be considered to be valid at the origin of the canonical system because of their invariance under canonical transformations. The determinant A of the coefficients of the variables tll, and &Z in (10.5), (10.6) and (10.7) is given by 7 % ~ =

+

(10.8) If A # 0 a t points of the wave surface Z(t) we can solve the equations (10.5), (10.6) and (10.7) to obtain A = T;V;

T~V?.

(10.9) (10.10) (10.11)

at the origin of a system of canonical coordinates; the remaining components 4 i j vanish from the first of the conditions (10.4). The special case for which A = 0 on Z ( t ) will be discussed in the Remark 3 in Sect. 11.

11.

WAVES IN PLASTIC SOLIDS UNDER PLANE STRESS

143

11. WAVESIN PLASTIC SOLIDSUNDER PLANE STRESS

We shall first consider the case of a rigid-plastic solid subject to the von Mises yield condition. Writing the stressrate of strain relations in the form

we recall that there is no distinction between the components of the rate of strain tensor E and its deviator E* in view of the condition of incompressibility (6.5); the form (11.1) of the stress-rate of strain relations can be obtained immediately from the basic conditions (1.2) of Chap. IV for a perfectly plastic solid by elimination of the proportionality factor 4 in an obvious manner. Hence, from the relations (8.14) and ( l l . l ) , we can deduce xivj

+

X,Vi

1

-a*,aX,vba; ICZ

=

0.

(11.2)

Denoting the left members of the above equations (11.2) by J i j we readily observe that Jii = 0 when use is made of the identity a& = 0. Hence there can be a t most five independent relations in the set (11.2) and we see, in fact, that it is permissible to omit the relation (11.2) corresponding to i = j = 1. But, using (10.4), we can decrease still further the number of independent equations (11.2). Thus, denoting by rij the components of the stress tensor relative to canonical coordinates as in Sect. 9, we have T ; ~= 0 for i # j at the origin of the canonical system; from this fact and (10.4) it follows that Jia vanishes for i = 1,2 when expressed in canonical coordinates y i and evaluated at the origin of these coordinates. Hence the set (11.2) can be reduced to the three equations corresponding to

Jiz

=

J ~ =z 0;

0;

J33 =

0,

(11.3)

at the origin of the canonical system. But at the origin of this system the equations (11.3) can be written XlVZ

(TfX1V1

+

+

XZVl

=

7;XZY2)7:

(11.4)

0,

(11.5)

= 0,

(2k2 - 71T@xZYZ - 7;71X1V1

=

0.

(11.6)

144

v.

CHARACTERISTIC SURFACES AND WAVE PROPAGATION

On account of the equations (10.4) and the fact that { = 0, as mentioned in Sect. 10, we need consider only those equations which give conditions on the three quantities &&,&, the two quantities XI,& and the two components v1,vz of the unit normal vector v to the surface 2(t). The equations giving these conditions are the second equation (lO.l), the equations (10.5), (10.6), (10.7) and the equations (11.4), (11.5) and (11.6); corresponding to the equations (10.7), (11.4), (11.5) and (11.6) the above equations (lO.l), (10.5) and (10.6) will be referred to the origin of a canonical system in the following discussion. Assuming first that Xl,Xz do not both vanish we see from (11.4) that these X’s are proportional t o vl, - vz. Let us express this by writing h1,hzK

Vl,

- vz.

(11.7)

Hence from the second equation (10.1) we have vf = v$ or v l = fvz. Using this relation and the proportionality (11.7) it now follows that (10.8), (11.5) and (11.6) become A =

(7;

+

7;)~:= -T;v?,

- 7;)~:= 0,

(11.9)

+

(11.10)

(7; (77

(11.8)

- 7;)~; 2k2 = 0,

respectively. Hence 7; f 71 from (11.10) since k # 0. Hence 7; = 0 from (11.9) and hence A = 0 on account of (11.8). Now since 7: = 0 we must have 7; = -71 and hence (10.7) yields &I = E z ~ . Making this substitution the equations (10.5) and (10.6) become

+ Envz + 411Vl

[lZVZ

= p(vn -

Eizvi = p(vn

G)h171

- G)Xz.

Now if v1 = vz the left members of these two equations are equal and hence we must have XI = Xz. Hence, from the proportionality (11.7), we have vl = - n and there is a contradiction with the fact that v is a unit vector. Also if we start with the other possibility that v1 = -vz we are led to a similar contradiction. It follows that X1 = Xz = 0 and hence all quantities X i must vanish on the surface z(t). Taking XI = XZ = 0 the compatibility relations under consideration are automatically satisfied with the exception of (10.5), (10.6) and (10.7) which become

11.

WAVES IN PLASTIC SOLIDS UNDER PLANE STRESS

ElIVl E21v1

7Tc;ll

+ + +

1

512v2

=

0,

E22VZ

=

0,

7&2

=

0.

145

(1 1.11)

But the determinant A of the coefficients of the 5's in these equations must vanish if t h e surface Z(t) is t o be singular of order 1. Hence from (10.8) we have (1 1* 12) 7% THY? = 0.

+

Combining (11.12) with the condition for v t o be a unit vector it follows that

(1 1.13) When (11.12) is satisfied, or when the components V ~ , Y are ~ given by (11.13) a t t h e origin of a system of canonical coordinates, the equations (11.11) will admit a nontrivial solution Ell, E l 2 and (22 which will permit a discontinuity in the derivatives of the components of the stress tensor u over the surface Z(t). The following result has now been proved. The derivative of the density and the derivatives of t h velocity components are continuous over a wave or singular surface Z ( t ) of order 1 in the flow region of a rigid-plastic solid, subject to the Don Mises yield condition and the plane stress condition (9.1), provided the speed of propagation of Z ( t ) i s different f r o m zero; however discontinuities will occur in the derivatives of the components of the stress tensor over this surface. Also the normal at a n y point P to the surface Z ( t ) lies in the x1,x2plane (which i s identical with the canonical y1,y2plane at P ) and the direction of the normal makes an angle determined by (11.13) with the two principal directions of the tensor u at P which lie in the x1,x2 plane. The speed of propagation s of the surface Z ( t ) is not determined by the compatibility conditions of the problem. Remark 1. Since we do not assume the existence of second derivatives of the functions p , ui and u i j on the wave surface Z ( t ) the fact that Xi = 0 for i = 1,2,3 does not lead to relations of the form [~i,ik]

=

ftivjvk.

(1 1.14)

The above relations (11.11) therefore constitute all the compatibility conditions which necessarily hold on this wave surface. In fact we easily arrive at a con-

146

V. CHARACTERISTIC SURFACES AND WAVE PROPAGATION

tradiction if (11.14) is assumed to hold with hi = 0. Thus from [ v i , , k ] = 0 and the form (1.1) of the stress-rate of strain relations it follows that [Utj,t]

= tiivk

-

= 0.

$taavksij

(11.15)

But, putting j = k in (11.15) and summing on the repeated indices, we find tau= 0 when use is made of (11.11). Hence tdi= 0 from (11.15) and this contradicts the requirement in Sect. 10 that Z ( t ) is singular of order 1. More generally i t follows from (11.14), in which not all %i are equal to zero, that 7'3 = 0 on Z(t). To show this we first set up the relations (11.16) Then, differentiating (11.1) with respect to (11,16), we find that

( k v $ + %jvi)k2 = tabeab'$j

+

xk

and making the substitutions

+

b b d h ;

'$bEQb(tij

- gtec8ii).

(11.17)

If we consider (11.17) at the origin of a system of canonical coordinates, take i = 1,Z; j = 3, and make use of (10.4), it readily follows that ks = 0. Also if we multiply (11.17) by u& and make use of the last equation (10.2), we find tabE,,b = 0; hence the first term in the right member of (11.17) can be omitted. Differentiating the equation of incompressibility (6.5) and using the relations (11.14), we obtain %iv, = 0 which is equivalent to the proportionality %I, A 2

a

v2, -v1.

Now multiply (11.17) by vihi to obtain

%&k2 = .as is readily seen.

(U&%Qvb)2,

(11.18)

- T;)~V$;,

(1I. 19)

Hence we find k2 =

(7:

when we evaluate (11.18) at the origin of canonical coordinates and use the and %,. Finally when we express k 2 as one half above proportionality for %, the sum of the squares of the scalars T; and use the relations (11.13) for VI and v2 we can deduce, from (11.19), the condition rz = 0 on Z(t) as above stated. Remark 2. A modification of this work is obtained by assuming that not all of the quantities X, and also that not all of the Eii vanish on the wave surface Z ( t ) . I n other words we shall require that first derivatives of the components of the velocity and the stress tensor shall be discontinuous along Z ( t ) ; no assumption will be made as to the vanishing or non-vanishing of the speed of propagation s. For convenience of reference we rewrite the primary set of dynamical conditions of compatibility on which the discussion depends, namely

11.

WAVES IN PLASTIC SOLIDS UNDER PLANE STRESS

(u,- G ) [

Jij

= Xivj

X , V ~= 0,

= 0;

[iivj = p(Vn -

G)Xi;

~t[ii=

147 (11.20) (11.21)

0,

+ Xjvi - -1 uL&v~u!~ 0.

(11.22)

=

k2

From the second equation (10.3) we again have Ei3 = 0 for i = 1,2,3. Also 0 from the first equation (10.3) since not all the tiivanish by assumption. Since the quantity Jii vanishes identically we see, as before, that we can omit the equation JII= 0 from the above set of J-equations (11.22). Omitting this equation let us now consider the set of equations v3

=

JIZ

=

0;

Jl3

=

0;

J23

=

0;

J33

=

0.

(1 1.23)

We shall later take account of the equation corresponding to 5 2 2 which does not appear in the set (11.23). The matrix M of the coefficients of the quantities X1,12,X3 in the equations (11.23), viewed at the origin of a system of canonical coordinates, is given by

when use is made of the fact that v3 = 0. The rank of M must be less than 3 since we have assumed that not all X i are equal to zero. But the upper third order determinant in this matrix is seen t o vanish identically; also the matrix determined by the first two rows of M has rank 2 if v1 # 0 and the matrix determined by the first and third rows has rank 2 if v2 # 0. Hence we see that the condition for the equations (11.23) to have a nontrivial solution X i is that one or the other of the following two equations be satisfied, namely

0

V2

Vl

0 I *

0 * *

73T1v1

T3T2V2

0

V2

v1

0

v1 =

0,

(v1

# O),

(11.24)

0,

(v2

# 0).

(11.25)

or

* *

0 * *

73T1v1

73T&

0

v2

=

0 Assume first that v1 # 0 so that (11.24) holds. Expanding the determinant in (11.24) we readily find that the resulting equation can be reduced t o (7.2

+

Y;T;)T;

=

0.

(11.26)

148

v.

CHARACTERISTIC SURFACES AND WAVE PROPAGATION

Now, since the matrix determined by the first two rows of the determinant in (11.24) has rank 2, we have the following proportionality (1 1.27)

v?, -vlvZ,o.

xI,xZ,x3

Hence X3 = 0. From the second equation (11.20) and the proportionality (11.27) it also follows that v: = v;, or v1 = f v p , and hence (1 1.28) since we can assume, without loss of generality, that v1 is positive for the determination of the normal direction to the surface Z(t). Hence, since v: = 1/2, the above equation (11.26) becomes b

(71

- T Z ) T ~ = 0. b

t

(11.29)

Now consider the relation Jn = 0, which was omitted from the above set (11.23), at the origin of a system of canonical coordinates. This relation, when use is made of the proportionality (11.27), is found to reduce to (7;

- 7 : ) ~ ;+ 2k2 = 0.

(11.30)

Since k # 0 i t follows from (11.30) that 7; # 7:. Hence, on account of the relation (11.29), we must have T: = 0. Since there is no distinction between the subscripts 1 and 2 in the above discussion i t is clear that if we had assumed v p f 0 and used (11.25), instead of (11.24), the same results could have been obtained. No new condition is obtained from the first set of equations (11.21) for i = 3 since t i 3 = 0 and X3 = 0. Expanding the other equations of this set we have VlEll viEzi

+ + vztzz

VZh2 =

p(v, - @XI,

= P(G

- G)h.

(11.31) (1 1.32)

But 7: = -7; since 7; = 0. Hence the last equation (11.21), when considered at the origin of a system of canonical coordinates, yields E l l = [n and so, from (11.32), we have (11.33) ~zEii vitiz = P(V, - G ) L

+

Now consider the following matrix determined by the coefficients in the equations (11.31) and (11.33), namely

l

vi

~2

P(V,

~2

vi

P(V,

I

- ‘3x1 -G)h

Since the second order determinant formed from the elements in the fmt and second columns of this matrix has the value v: - vg, and hence vanishes on account of (11.28), the matrix must have rank 1 in order for the equations (11.31) and (11.33) to be consistent. But this means that

11.

WAVES IN PLASTIC SOLIDS UNDER PLANE STRESS

149

Hence, since p # 0, it follows from this equation and the proportionality (11.27) that 2 (0, - G ) V I V Z = 0. But neither of the quantities v1 and v2 can vanish from (11.28);hence the speed of propagation G - v, i s equal to zero. We have now proved the following result with regard to wave or singular surfaces Z ( t ) of order 1 in the plastic flow region of the rigid-plastic solid under the plane stress condition (9.1). A n y singular surface Z ( t ) of order 1 over which the derivatives of the com.ponents of velocity and stress exhibit discontinuities will (a) have speed of propagation s = 0, (b) be perpendicular to the x1,x2plane and (c) be such that its normal v at a n y point P bisects the angle formed by the two principal directions, determined by the stress tensor, in the x1,x2 plane at P. I t may be observed that the first equation (11.20) is satisfied identically since the factor vn - G = 0 and hence a discontinuity in the derivatives of the density p is possible. We emphasize, however, that the existence of this singular surface Z ( t ) requires the stress tensor to be such that the intermediate invariant T $ = 0 over the surface (cf. Sect. 4). We now turn our attention t o the singular surface Z(t) of order 1 in the flow region of a n incompressible elastic-plastic solid for which the Prandtl-Reuss equations (6.1), subject t o the von Mises yield condition (6.2), are valid. One can then deduce the relations (8.24) as in the Remark 2 in Sect. 8. Corresponding t o the above procedure we observe that t h e equation obtained from (8.24) by putting i = j and summing on repeated indices, is satisfied identically. Similarly, identical relations are obtained from (8.24), at the origin of a system of canonical coordinates, by taking i = 1,2 a n d j = 3; this follows when use is made of the equations (10.4). Hence there are a t most three independent equations in the set (8.24) and these can be written in t h e form (v,

- G)Lz + $(6- 4 ) ( d 1 - d 2 ) 1 -- (v, - G ) L 3

+

- p ( ~ 1 A z ~2x1)= 0,

+ pCL (6Xivi +

&2V2)7.3

=

0,

(11.34) (11.35)

(11.36)

150

v.

CHARACTERISTIC

SURFACES AND WAVE PROPAGATION

at the origin of a system of canonical coordinates. The various conditions which we have established will now be applied to the specific determination of the wave or singular surface Z ( t ) of order 1in the plane stress problem. It will be assumed that the determinant A, given by (10.8), is different from zero over the surface Z ( t ) ; hence we can avail ourselves of the equations (10.9), (10.10) and (10.11) in the following discussion. It was observed in Sect. 10 that not all of the quantities tij can vanish under the hypothesis that the wave surface Z ( t ) has speed of propagation s different from zero. Also if all X i = 0 we see from (10.9), (lO.lO), and (10.11) that till t 1 2 and f22 vanish on Z ( t ) ;hence all ti, vanish in consequence of (10.4) and we have a contradiction with the assumption that Z ( t ) is singular of order 1 (see Sect. lo). T h u s any singuEar surface Z ( t ) of order 1, having speed of propagation s # 0 and over which A # 0, must be associated with discontinuities in the derivatives of the velocity components as well as discontinuities in the derivatives of the components of the stress tensor. However, the derivatives of the density p .will be continuous over such surfaces. This latter statement follows from the first equation (11.20). The above facts will be used in the discussion of these singular surfaces in the remainder of this section. Since both of the quantities X I and Xz cannot vanish the condition given by the second equation (11.20) is equivalent to the proportionality (11.37) X l l X 2 E Y 2 , - Y1. Let us now substitute the expressions (10.9), (10.10) and (10.11) for t 1 2 and 622 into the relations (11.34), (11.35) and (11.36) respectively. When these substitutions are made and when we avail ourselves of the proportionality (11.37) the resulting equations can be written in the following form

- V; = 0,

=

0,

(11.38) ( 11.39)

- 11y l y 2 = 0. (11.40) 2k? In treating these relations we shall divide the discussion into several cases according to whether the quantity 7f - 712 or one of the components

11.

WAVES IN PLASTIC SOLIDS UNDER PLANE STRESS

151

v1,v2 of the unit normal v vanishes, or does not vanish, on the surface

W). Case 1. The quantity 7: - 7; = 0 on Z(t). For this case the relation (11.39) is satisfied identically. Also 6 = -27: from the above assumption. Hence 6 # 0 since otherwise we would have a contradiction with the yield condition. Hence, from (10.8), we have A = ~f and the equations (11.38) and (11.40) reduce to

b(v, - G ) 2 - p]($ - v;) = 0, b(v, - G)* - p ] ~ i ~=z 0.

(11.41) (1 1.42)

But the bracket expression in (11.41) and (11.42) cannot be different from zero since this would lead to a contradiction with the fact that v is a unit vector. Hence p ( ~ ,

G)'

(11.43)

= p.

Hence any wave surface Z(t) with speed of propagation s Z 0 and over which A # 0 and 7; = 7; will have its speed s given by (11.43) in the plane stress problem. The unit normal vector v to this wave surface i s not determined by the dynamical conditions of compatibility. The speed of propagation s, which is equal to from (11.43), is formally identical with the velocity of the so called equivoluminal waves in the classical elasticity theory (see Sect. 4 in Chap. 111). Case II. The component v1 or v2 has the value zero on Z(t). Here the equations (11.39) and (11.40) are satisfied identically. Recourse to the equation (11.38) now permits us to state the following result. According as v1 or v 2 i s equal to zero on the wave surface Z ( t ) the determinant A and the speed s are given respectively by the following two sets of equations

a

A =

7;;

p(V,

A =

7;;

p(V,

+ - d), - G)' = p + $(6- G)2 = p

$(T;

T;),

(1 1.44) (11.45)

in the plane stress problem, where it i s to be understood that ~f # 0 in (11.44) and T; # 0 in (11.45) in accordance with the above assumption that A # 0 on the surface Z(t). I n the j r s t instance the moving surface Z ( t ) i s perpendicular, at a n y of its points P, to the y 2 axis of the canonical coordinate system with origin at P and in the second instance the surface i s perpendicular to the y' axis of this coordinate system.

152

V. CHARACTERISTIC SURFACES AND WAVE PROPAGATION

Case 111. nents

VI,

The quantities T? and

are distinct and neither of the cornpo-

T:

vz vanishes on the surface z(t). From equation (11.39) we now

have p ( ~ ,

-3pArg.

G)’

=

2k2

+ 3777; -

(11.46) 2k2 If we eliminate the speed s = G - vn from equation (11.40) by the substitution (11.46) we find that the resulting equation is satisfied identically when use is made of the yield condition. However elimination of s from (11.38) by means of (11.46) leads to an equation for the determination of the components v 1 and v 2 of the unit normal to Z ( t ) . We shall not give the complete details of this calculation but note merely that we can first deduce the intermediate relation

(4k2 - 37*37;)Y?

=

~

(‘Tr

- T$}k2.

(11.47)

I.(

It can now be shown that 4k2 - 37;7$

=

(7;

-

2k2 f 37;7*3 =

(7;

- 7 ; ) ( 7 $ - ‘T;).J

7f)2,

1

Making these substitutions the equation (11.47) then leads to the following two relations (11.48) (11.49) When we eliminate the quantities v? and vg from (10.8) by means of (11.48) and (11.49) the expression so obtained for the determinant A can be shown to reduce to (11.50)

Hence, eliminating A from the relation (11.46) by means of (11.50), we obtain 3C1(6)2 - 37;. (11.51) d u n - G) kl 2

This case represents the general situation and the results now established will be emphasized by the following italicized statement. 1.f the determinant A, the speed s, the diflerence 71; - 7: and the components v1

11.

WAVES IN PLASTIC SOLIDS UNDER PLANE STRESS

153

and vz do not vanish over the wuve surface Z ( t ) under consideration in the flow region of a n incompressible elastic-plastic solid subject to the plane stress condition (9.1) then, at any point P of the surface, the speed s i s given by (11.51), while V I and vz are given by (11.48) and (11.49) with reference to the s y d e m of canonical coordinates having its origin at P. Remark 3. We now treat the special case for which A = 0 over the wave surfaces Z ( t ) in the elastic-plastic solid under consideration. Since we assume the speed of propagation s to be different from zero the derivatives of the density p will be continuous over Z( t ) from the first equation (10.1). Also we can avail ourselves of the conditions (10.4) ; however the equations (10.9), (10.10) and (10.11) cannot now be used but must be replaced by the equations (10.5), (10.6) and (10.7). Our discussion will be based on the following two mutually exclusive cases. Case A. The quantities XI and Xz are both zero on Z(t). Under this condition on the X’s and the condition s # 0 the equations (11.34), (11.35) and (11.36) reduce to [lZ

= 0;

[ii

= 0;

-gii

fzz

=

0.

(11.52)

From the second and third equations (11.52) we immediately obtain =0 = 0 from the first equation (11.52), we and tZ2 = 0 on Z(t). Hence, since have a contradiction with the assumption that Z(t) is a singular surface of order 1 (see Sect. 10). I n other words there is no wave surface Z(t) which satisfies the conditions of this case. Case B. The quantities XI and XZ are not both zero on Z(t). Let us now consider the matrix determined by the coefficients of the quantities f l l , and in the equations (10.5), (10.6) and (10.7) and the right members of these equations, namely

- G)XI

VI

vz

0

0

vi

VZ

P(V, - G)Xa

r;

0

r;

0

The third order determinant formed from the elements of the first three columns of this matrix vanishes from the condition A = 0, i.e. TfVi

+ r;v:

=

0.

(1 1.53)

Now the equations (10.5), (10.6) and (10.7) will have a solution [if, and only if, the above matrix has rank 2; this follows from the well known theorem on the solutions of systems of linear equations and the fact, that the rank of the third order matrix formed from the coefficients of the E’s in the equations (10.5), (10.6) and (10.7) cannot have rank less than 2 without violating the requirement

v.

154

CHARACTERISTIC SURFACES AND WAVE PROPAGATION

v1

v2

11

0

v1

X2

r;

0

0

=

0, if v1 # 0;

v2

0

A1

v1

v2

hz

0

r;

0

=

0, if v2 # 0.

12.

WEAK STATIONARY DISCONTINUITIES

and p(vn

- G)2 = p +

5.

155

(1 1.62)

Apart from such relations as those given by the equations (11.59) and (11.61) the main result now proved is contained in the following statement. Tf there are discontinuities in the derivatives of the components of the velocity and stress over a wave surface Z ( t ) , uhich i s singular of order 1 relative to the density, velocity and stress, in the $ow region of a n incompressible elastic-plastic solid under the plane stress condition (9.1) and i f , furthermore, the speed of propagation s i s different from zero but A = 0 o n Z ( t ) , then at a n y of its points P the surface Z ( t ) i s perpendicular to the y2 or y1 axis of a system of canonical coordinates with origin at P and has speed of propagation given by (11.60) or (11.62) respectively. The derivatives of the density p will be continuous over this surface Z ( t ) .

12. WEAKSTATIONARY DISCONTINUITIES

In the foregoing discussion we have been concerned with wave or singular surfaces Z associated with weak discontinuities in the density, velocity and stress in the sense that these quantities are continuous and that only discontinuities in the derivatives of their components occur over the surface. We shall now undertake a more exhaustive study, involving compatibility conditions of higher order, of such discontinuities in an incompressible elastic-plastic solid subject to the plane stress condition (9.1). Some simplification will be obtained by assuming, instead of the Prandtl-Reuss equations (6. l ), the approximate relations

dat = 2 4 r i j - $&), at

(12.1)

which are commonly empIoyed (see the Remark in Sect. 8 of Chap. IV), where

In addition to the above relations, the equations of motion (6.4) and the equation of incompressibility (6.5), in consequence of which there is no distinction between the rate of strain tensor E and its deviator E*, will enter into the discussion. We shall be concerned in this section with the special case for which Zis a stationary surface separating a region of plastic flow from a region

156

v.

CHARACTERISTIC SURFACES AND WAVE PROPAGATION

in elastic equilibrium at the yield point; this latter region is assumed to be in a state of simple tension 7 as defined by the conditions 422

=

a.. r3 =

7;

0,

otherwise,

where 7 is a positive constant and the aij are the components of the stress tensor in the elastic region. It follows immediately that a!& =

27 -; 3

= aa ;

= -I-

3’

a2j

=

0,

(i # j ) ,

from the above values of the stress components. The following specific assumptions will be made concerning the order of the discontinuity over 2. The velocity and stress are continuous over I: but there i s a discontinuity in the derivatives of the stress components over this surface. KO assumption will be made as to the continuity of the derivatives of the components of the velocity or as to the continuity of the density and its derivatives along 2. In view of the above assumptions the geometrical and kinematical conditions of compatibility of order 1 for the velocity and stress take the simple form

where the bar is used to denote evaluation of quantities and their derivatives on the flow side of the surface 2. Now consider the equations (6.4), (6.5), (12.1) and the equations which result by coordinate differentiation of the yield condition (6.2) ; evaluating these equations on the flow side of 2 and then combining them with the compatibility conditions (12.2) and (12.3) the following relations are obtained, namely xivi = 0; &,VI = 0 ; a 2 i j = 0, (12.4) (xivj

+

xjv,)

=

1 (‘Jabh’b)‘J:j.

(12.5)

In writing (12.5) we have omitted the bar over the a’s since the quantities iiij are equal to the corresponding stress components ui3 which result from the simple tension in the elastic region. Multiplying (12.5) by v, and summing on the repeated index j , we obtain

12.

WEAK STATIONARY DISCONTINUITIES

1

157 (12.6)

A, = 7 (uubkavb)'J:Jvj,

k

when use is made of (12.4). Similarly, multiplying (12.6) by A, and again using (12.4), we find that Atk,

=

1

(aa&$'b)2. k2

(12.7)

Also we have ( u a b h a p b ) ('J&'d',)

=

(12.8)

0,

when we multiply (12.6) by vI and apply (12.4). It will be seen from the following discussion that the first of the factors in the left member of (12.8) must vanish. One can readily obtain the following conditions 5%3

0;

=

v3

=

0;

A3

= 0.

(12.9)

In fact, putting j = 3 in the first set of relations (12.3) and taking account of the plane stress condition (9.1) we immediately deduce the validity of the first set of relations (12.9). Similarly when we take k = 3 in (12.3) we find that lrJvs = 0 ; hence we obtain the second condition (12.9) since not all of the quantities Ev can vanish from the assumption that 2 is singular of order 1 relative to the stress. I t follows that the surfme Z must be perpendicular to the x1,x2coordinate plane. Finally, taking i = 3 in (12.6) we have h 3

1

=

(flud a V b )'J 3 1 vJ-

(12.10)

But, in view of the plane stress condition, we see that Is;, =

U3J

- +ua,63, = 0,

(12.11)

for j = 1,2. Hence from the second condition (12.9) and the equations (12.11) it follows that the last factor in the right member of (12.10) must be equal to zero; the last condition (12.9) is therefore a consequence of (12.10). Expanding the equations containing the quantities 5,J in (12.4) and taking account of the conditions (12.9), we now have

tush

511Vl

+

512v2

=

0,

521v1

t 522v2

=

0,

+ 2&UL +

5224.2

(12.12) = 0.

158

v.

CHARACTERISTIC

SURFACES AND WAVE PROPAGATION

The determinant of the coefficients of the [ij in (12.12) must be equal to zero since not all of the [’s can vanish by assumption. Hence

+

afzv? - 2 f f ~ Z V l V 2

UflV22

(12.13)

= 0.

Now if we put i = j in (12.5) and sum on the repeated indices we see t,hat the equations are satisfied identically on account of the first equation (12.4). Also (12.5) is seen to be satisfied identically if we take i = 3 and j = 1,2 in view of (12.9) and (12.11). Hence there are at most three independent equations in the set (12.5) and these are obtained by taking i , j = (1,2), (2,2) and (3,3). When account is taken of (12.9) these equations become xlv2

+

X2vl

=

1 (uobXavb)u?2,

(12.14)

But for the elastic region under the above simple tension a& is different from zero. Hence uabxavb

=

0,

7

the quantity (12.15)

from the last equation (12.14). It now follows from (12.6) and (12.15) that xi = 0, (i = 1,2,3). (12.16)

Hence the Jirst derivatives of the velocity components v i are continuous across the surface Z. As a consequence of the conditions (12.16) all equations of the set (12.5) are therefore satisfied. Substituting the above values of the components a; into (12.13) we find that this equation becomes 2v: - v;

=

0.

(12.17)

But since v is a unit vector whose third components v 3 = 0 it follows from (12.17) that v f = 5 and v; = $. Hence the surface Z i s a plane perpendicular to the x1,x2coordinate plane and having a n inclination 6 giren by tan2%= 6. These planes Z therefore have inclinations of f35’16’ and are geometrically identical with the planes bounding the Luders bands (see Appendix to Chap. IV). We now have all of the essential information which can be obtained

12.

WEAK STATIONARY DISCONTINUITIES

159

from the compatibility conditions of the first order. Turning to the compatibility conditions of the second order we see, since the velocity is continuous across 2 by assumption and the first derivatives of the velocity components are continuous by the above result, that

(12.18)

where the ki are functions defined on 2 ;the above equations (12.18) are direct consequences of the geometrical compatibility conditions (5.12) and the kinematical compatibility conditions (6.29) for stationary surfaces in Chap. 11. Since the coefficientsbij of the second fundamental form are equal to zero for the plane 2 it follows from the relations (5.11) and (6.28) of Chap. I1 that

These equations, in which the &j are suitable functions defined on 2, are the geometrical and kinematical conditions of compatibility of order 2 for the stress, Certain simple consequences of the above compatibility conditions can be observed immediately. Thus, putting i = j in the fist set of equations (12.18) and summing on the repeated indices, we have KiVi

= 0,

(12.20)

in view of the condition of incompressibility of plastic material. Also from the plane stress condition (9.1) and the first set of equations (12.19) we readily see that i i 3 = 0; 5 i j . 3 = 0. (12.21)

To obtain additional conditions let us differentiate the Prandtl-Reuss equations (12.1) partially with respect to xk and evaluate at points of 2. We thus have (12.22)

160

V.

CHARACTERISTIC SURFACES AND WAVE PROPAGATION

But, since the Vi,j vanish, we see that $ also vanishes. The equations (12.22) therefore reduce to az-* s= a x k at

2kI(gij.k

-

(12.23)

$ j k ~ ~ l > -

Using the first set of equations (12.18) and the second set of equations (12.19) we now obtain 7 1 t i j , k = - (Xivj X z v i ) ~ , 2

+

I

Similarly, after differentiation of the quantity ,)I we find that

Hence, making these substitutions, the equations (12.23) become

atij

at

1 atmm

at 6ij

=I

P(X
+

- P> k (gnb&vb)(T:j.

(12.24)

Now consider the relations

which result when we differentiate the equations of motion (6.4) with respect to the time t and evaluate on the flow side of 2. But

’Thus the right members of (12.25) vanish. Hence, when we equate the left members of (12.25) to zero, it follows from the second set of conditions (12.19) that -vk atik at

=

0.

(12.26)

It is natural to define the strength S of the discontinuity in the derivatives of the stress components along 2 by the equation

x=

dGj.

(12.27)

To deduce the differential equation for the variation of S pie first put = j = 3 in (12.24). Then from (12.9) we obtain

i

12.

WEAK STATIONARY DISCONTINUITIES

161 (12.28)

Next multiply (12.24) by vivl and sum on the repeated indices. Taking account of (12.20) and (12.26) we now find that (12.29) Hence, combining (12.28) and (12.29), we have (~a&aVb)(~;j;jVi~j - ~ k=) 0. (12.30) But the second factor in (12.30) has the value TV; # 0 for values of the u’s corresponding to the simple tension 7 in the elastic region. Hence the first factor in (12.30) must vanish, i.e.

0. Now from (12.28) or (12.29) and (12.31) we have (Tabkavb

=

(12.31)

(12.32) But from (12.31) and (12.32) the equations (12.24) reduce to

at.at = P(liV,

+

KjVi).

(12.33)

Multiplying (12.33) v, and using (12.20) and (12.26), it now follows that Ki

=

0.

(12.34)

I n other words, the second derivatiEes of the velocity components vi are continuous across the surface 2. From (12.33) and (12.34) the partial derivatives of the f i j with respect to the time are equal to zero. Hence from (12.27) we have

as-- 0. at

A discontinuity of this character, i.e. such that the strength S of the discontinuity does not vary with the time, may be said to be stable. It is possible that the occurrence of such weak but stable discontinuities in the derivatives of the stress components is associated with the dulling or cloud effect which is observed on the flat sides of a polished plate prior to the formation of the Luders band, or fracture of the plate (see Sect. 5 in Chap. VI), after the yield point is reached.

162

v.

CHARACTERISTIC

SURFACES AND WAVE PROPAGATION

13. WEAKDISCONTINUITIES WHOSE VELOCITY OF PROPAGATION IS DIFFERENTFROM ZERO We shall now extend the discussion in Sect. 12 to the case of surfaces of discontinuity or wave surfaces Z whose normal velocity of propagation G is different from zero. Assuming that Z is the boundary of a plastic region which is propagated into an elastic region of an incompressible elastic-plastic solid we shall have G > 0 by taking the unit normal v to the surface Z to be directed from the plastic side of Z into the elastic region, i.e. the direction of v coincides with the direction of propagation of the wave. The following conditions will be imposed relative to a fixed rectangular coordinate system x assumed to be dynamically admissible. (i) The wave surface B i s singular of order 1 relative to the set of functions consisting of the components v i of the velocity and the components cij of the stress, (ii) The velocity components v i = 0 in the elastic region into which the wave i s propagated, (iii) The density p i s constant in the elastic region; also the components uij of the stress are constant in the elastic region and satisjy the plane stress condition (9.1) throughout the entire medium under consideration. Our discussion of these waves will be based on the von Mises yield condition (6.2), the equation of continuity (6.3), the equations of motion (6.4), the equation of incompressibility (6.5) and the above form (12.1) of the Prandtl-Reuss equations. Results obtained can be applied to the problem of the flat plate by taking the stress in the elastic region to be that due to simple tension and identifying the above x coordinate system with the x system used in Sect. 12. Since [v;] = 0 and [a;j] = 0 by assumption (i) the conditions (8.8) in Chap. I are satisfied automatically. However, the condition (8.4) in Chap. I reduces to [ p ] = 0 since G is different from zero, i.e. the density p i s continuous across the surface 2. Hence relations of the form (13.1) hold along the surface Z. Now in the elastic region vi,d = 0 from the above assumption (ii). But v ; , also ~ vanishes in the plastic region from

13.

WEAK MOVING DISCONTINUITIES

163

the condition of incompressibility. Moreover v; = 0 on each side of the surface Z on account of (i) and (ii) and hence, on both the elastic and plastic sides of 2, the equation of continuity reduces to

Since G does not vanish it therefore follows from the second relation (13.1) that { = 0. Hence the first derivatives of the density are continuous across Z. On account of (i), (ii) and (iii) we can write

(13.3)

where the X i and Eij are functions defined on 2 and the bar has its previous significance. From (6.5) and the first condition (13.2) it follows immediately that xiv; = 0. (13.4) Evaluating (6.4) and (12.1) on the plastic side of Z and applying (13.2) and (13.3) we have t;,vj = -GpX;, (13.5)

where p is the density on the elastic side of Z and ffab and doare the components of the stress and stress deviator in the elastic region. Similarly, by coordinate differentiation of (6.2) and use of the condition (13.3) , we obtain gijtij = 0. (13.7) From (i) not all of the quantities Xi and t i jare equal to zero at points of 2. Now if all f i j = 0 it follows from (13.5) that all X i = 0 and we have a contradiction. Conversely, if all Xi = 0 we have from (13.5) and (13.6) that f a. i . - q 3 aa6 ~3 . . = 0. (13.8) , t i j V j = 0; But if we multiply the second of these relations by vj and apply the first relation we obtain Eao = 0. Hence tit = 0 from the second set of

164

v.

CHARACTERISTIC SURFACES AND WAVE PROPAGATION

v1

VZ

0

A= 0

Vl

YZ

2a;z

af2

a;1

=

O&V~

-2

+

~ f 2 ~ 1 ~ U;~Y;. 2

(13.13)

(13.14) 122

=

2GP

7 (u;lxlvl

+ a;zXzvl).

J

A transformat.ion to canonical coordinates y leaves invariant the plane stress condition (13.9), as shown in Sect. 9, and hence the above

13.

165

WEAK MOVING DISCONTINUITIES

relations remain unaltered by this transformation. I n general, the selection of the system of canonical coordinates will depend on the point P of 2 under consideration. However, since the components ag are constant in the elastic region by assumption (iii) we can choose a $xed system of canonical coordinates y which are related to the coordinates x by an orthogonal transformation x t)y independent of the time t. In this system of canonical coordinates the relations (13.13) and (13.14) take the somewhat simpler forms (13.15) A = T ~ V ? r%, and

+

are the principal values of the stress deviator. To avoid where the the introduction of new symbols we use the same letters for the components of the quantities [, X and v in the canonical and x coordinate systems. As observed in Sect. 12 there are only three independent equations (12.5). Similarly, the set of relat,ions (13.6) contains but three independent equations which can be obtained by taking i , j = (1,2), (2,2) and (3.3) ; in canonical coordinates these independent equations are readily seen to be given by

Gtiz

G 3

+ P(XIQ + XZPI)

=

7

0,

i

- &L(7; - r;)r;Xzvz

= 0. 12 Kow when we eliminate the quantities ti3from (13.17) by the substitution (13.16) and make use of (13.11) we find that tau

@ (71XlY2 A

a(6 -

7;)

+ rzxzvl) - p(X1vz + Xzv,) - 3P

3P

-2k2 (Tf - r

p;

=

1

0,

XIVl =

0,

166

v.

CHARACTERISTIC

SURFACES AND WAVE PROPAGATION

But, again using (13.11), the quantities XI and A2 can be eliminated from the above equations. We thus obtain

3

(7;

- 7:) - 2k2 (6 - 6 ) ~-l 3

1

vp2

= 0,

(13.19) (13.20)

The equations (13.18), (13.19) and (13.20) give conditions on the components of the unit normal v and lead t o a determination of the possible wave surfaces 2. The following cases can arise. Case a (v1 = f l , v2 = 0). The determinant A = 6 from (13.15). Equations (13.19) and (13.20) are satisfied automatically and the equation (13.18) reduces to G2

=

.!

(13.21)

P

The wave surface Z i s therefore a plane, perpendicular to the y 1 axis of the system of canonical coordinates, and is propagated with a constant velocity G given by (13.21). From the equations (13.11) we have A1 = 0. Hence A2 # 0 from the last condition (13.10) and the fact that not all of the A’s can vanish by the above result. Consequently the equations (13.16) become ti1 = 0; 512 = FGpXz; 522 = 0, (13.22) with the minus and plus signs in the second of these equations corresponding to v1 = 1 and v1 = - 1 respectively. Hence in the system of canonical coordinates all 5’s vanish except tI2and all A’s vanish with the exception of A2; these non-vanishing quantities are related by the second equation (13.22). Case /3 (v1 = 0, v2 = & l ) . Now A = 7;. Equations (13.19) and (13.20) are again satisfied automatically and the equation (13.18) again reduces to (13.21). From (13.11) it follows that Az = 0. Hence XI # 0 and the equations (13.16) become

+

511

=

0;

(12

=

TGpXl;

&

=

0,

(13.23)

+

in which the minus and plus signs correspond to v2 = 1 and v2 = - 1 respectively. FOT this case the wave surface 2 is a plane perpendicular

13.

WEAK MOVING DISCONTINUITIES

167

to the y 2 axis of the system of canonical coordinates and i s propagated with constant velocity G given by (13.21). All components F vanish in the canonical system except the component f12 which is related to the single non-vanishing component XI by the second equation (13.23). Case y (v1v2 ts 0, ~f = T;). Here A = ~f = T;. Equation (13.20) is satisfied identically. Equations (13.18) and (13.19) become (G2p - p ) ( ~ ?- Y;) G2p(7; - 7;) - 3p7f

=

0,

(13.24)

=

0.

(13.25)

But 7: = -27; for this case. Making this substitution in (13.25) and taking account of the fact that 6 must be different from zero, we see that the equation reduces to (13.21) in consequence of which equation (13.24) is automatically satisfied. Equations (13.16) giving the 5's become fii = 2GpXzvz; [22 = 2GpXlv1, Eiz =

-Gp(XiVz

+ XZVI).

I

From the first two of these relations, the f i s t condition (13.9) and the condition (13.11), we see that the combination faa = 0. Since the components v1 and v2 are not determined by theabove conditions, Z can be anysurface perpendicular to the y1,y2 coordinate plane; hence Z will be perpendicular to the x1,x2coordinate plane since this latter plane is parallel to the y1,y2plane as is immediately observed from the form (9.7) of the transformation relating the coordinates of the z and y systems. Case 6 (vlv2 # 0, T; # 4). On account of (13.11) and the fact that X1 and A2 cannot both vanish we see immediately that neither XI nor Xz can vanish. Equation (13.20) yields (13.26) Eliminating the term G2p/pA by means of (13.26) we find that equation (13.19) can be written (7; - 7 ; ) ~ ;

+

(T;

-

7;)~;

+ 2k2 = 0.

(13.27)

But 2k2 =

+

(T;)~

(T;)'

+

(T:)',

(13.28)

from the yield condition (6.2). Now, replacing 2k2 in (13.27) by the value given by (13.28), it is easily seen that the resulting equation is

168

v.

CHARACTERISTIC

satisfied identically. we obtain

SURFACES AND WAVE PROPAGATION

Using (13.26) and replacing vi by its value 1 - v? (13.29)

from (13.18). But, when use is made of (13.28)) it can readily be shown that 37;~; 2k2 = (7; - T ; ) ( T ; - T ; ) ,

+ 367; + 367; + 2k2 =

(7;

- 71)2.

Making these substitutions in (13.29) and taking account of the fact that v is a unit vector whose third component v3 = 0, we obtain (13.30)

The following result has now been proved. The surface 2 i s a plane, perpendicular to the y1.y2 and hence to the x1,x2coordinate plane, such that the components v1 and v2 of its unit normal v, relatice to the canonical system, are determined from the stress in the elastic region in accordance with (13.30). From (13.30) and the equation (13.15) for A we have (13.31)

But by a n easy manipulation it can be shown that the above relation (13.31) reduces to A = -27s. Hence from (13.26) we obtain

,a s the equation for the determinat.ion of the constant velocity G of the plane 2. 14. FURTHER REMARKSON

THE

TRANSITION PROBLEM

I n the further discussion of the wave surfaces 2 in Sect. 13 we shall limit our attention to the special case of the thin flat plate under simple tension 7 in the x2 direction with the x1,x2plane parallel to the flat sides of the plate. For this case the stress components ui, in the elastic region have the values u22

=

7;

CTij

=

0,

otherwise,

14.

ox THE TRAKSITION PROBLEM

FURTHER REMARKS

169

and it is easily seen that the canonical coordinate system, introduced in Sect. 13, can be identified with the x coordinate system. Thus we have 7;

= a ;'

=

-7 -; 3

7; =

=

27 -* 3'

7f3

=

rJ&

=

-.-7 3

Since 7: # 71: it follows that Case y in Sect. 13 cannot be realized for the problem of the flat plate under simple tension. Also, if we use the above values of the T*, we find that v? = 1from (13.30). Hence Case 6 reduces to Case a. Attention can therefore be limited to Case a and Case P for the flat plate problem. Case a

Let us suppose for definiteness that v1 = 1. Then 2 is a plane, perpendicular to the d axis, which is propagated across the plat,e with a velocity G given by (13.21) in the direction of the positive x1 axis. Explicit equations for the wave surface Z are given by

x' = Gt

+ const.;

x2 = x2;

x3 = x3,

(14.1)

in which the x2 and x3 coordinates are used as the parametric coordinates of 2. By 6 time differentiation (see Sect. 3 in Chap. 11) of t,hese relations we have

Since G = const. and the coefficients bij = 0 for the plane 2, it follows that the geometrical compatibility conditions (5.11) and the kinematical compatibility conditions (6.10) and (6.1 1) in Chap. II give us the following relations [ui,jk]

=

a2vi

[-] ax] at [$]

v'a , j k

=

'

b j v k

a2v.

= -4ax] at = a2vi

=

=

+ xi,ol(vjxt +

(-GIi + 6X. 6t

G2xi - 2 G 4

vj

Vkd),

- GXi,,d,

(14.2) (14.3) (14.4)

where the fii are new functions defined on the plane Z and the comma denotes partial differentiation with respect to the parametric coordinates x2,x3of 2. Correspondingly we have

170

v.

SURFACES AND WAVE PROPAGATION

CHARACTERISTIC

(14.7)

If we put j = 3 in (14.5) and take account of the plane stress condition (13.9) we immediately obtain But

[z3,a

+

+ vmd)

(14.8)

0. vanishes from (13.10). Hence (14.8) gives fd'kvm

(t3,a(Vkx~

gi3

=

=

0.

(14.9)

Next, taking m = 3 in (14.5) and using the fact that v3 #$ij,aVkx: =

0.

=

0, we find that

(14.10)

Now from (14.1) we see that x! = 0 and x! = 1 in the above relations (14.10). Hence (14.10) reduces to [zj,3 = 0. This provides us with a single new relation, namely [12,3 = 0, since all of the E's except 512 have been shown to vanish. Hence the quantity 512 does not vary o n 2 i n the x 3 direction at a n y time. Also putting i = j in (14.2), summing on the repeated indices, and taking account of the incompressibility condition (6.5), we have b&'k Ai,avixt A i , a x b k = 0. (14.11) Since A1 = 0 these relations are satisfied identically when k = 2,3. However, if k = 1 the equations (14.11) yield

+

+

+

(14.12) 0. Additional conditions of a dynamical nature can be obtained from the yield condition (6.2), the equations of motion (6.4) and the PrandtlReuss equations (12.1). Thus, differentiating (6.2) twice partially with respect t o the time t and evaluating on the plastic side of 2, we have Xl

'

A2,z

=

(14.13) where we have omitted the star on the second derivatives and on one of the first derivatives as is evidently possible in view of the identity a;$ = 0. Now from (13.3) we obtain

14.

FURTHER REMARKS ON THE TRANSITION PROBLEM

171

since tao = 0. Making this subst,itution for the first derivatives and eliminating the second derivatives in (14.13) by means of (14.7) we find that (14.14) But when we make the substitution ti2

= -GpXz,

in accordance with (13.22), use the values of uZj appropriate to the simple tension 7,and take account of the condition (14.9), the equation (14.14) is seen to reduce to

(T)

Gzp2

2 h = 611 - 6

(14.15)

A;.

Now differentiate the equations (6.4) with respect to z*, multiply by We thus obtain

vm, and evaluate on the plastic side of 2.

(14.16) when use is made of the conditions p = p and dp/dxm = 0 which follow from the fact that p is constant in the elastic region and that both p and its derivatives are continuous across Z as shown in Sect. 13. But using (13.2), (14.3) and (14.5) we find Vi,,Zij,mvm= XiXl

iiij,jmvm

=

=

8%; V,

0;

iijvj

+

=

ax" at tij,ad

=

iil

-GXi

+

t.1

+ -z,

tiZ.2.

Making these substitutions in (14.16) the resulting equations become (14.17)

If we take i = 1,2 the equations (14.17) give $11

+ bz,z =

(14.18)

-pGL,

(14.19) since X1 = 0 and and (14.9) that

=

0. Similarly, taking i =

0.

=

3 it follows from (13.10) (14.20)

Let us now apply this procedure to the Prandtl-Reuss equations, i.e.

172

V. CHARlCTERISTIC SURFACES &4ND WAVE PROPAGATION

we differentiate the equations (12.1) with respect to the time and evaluate on the plastic side of Z to obtain

where the bar has been omitted from the U; due to the continuity of the stress components along 2. Kow

since vz = 0. Also it readily follows by application of the appropriate compatibility conditions and auxiliary relations from the foregoing discussion that (14.23) 5;j.k

avk -= 0. at ’

’?at$

=

G

2 ~), -, ~ PGX 2k2 ( T X ~ -

(14.24)

when the elastic region is in a state of simple tension 7. Making the substitutions (14.22), . . . , (14.25) the equations (14.21) become

To realize the consequences of the conditions (14.26) we must give the indices i,j their special values 1,2,3 and then reduce the resulting equations by means of the various relations which we have established. Thus taking i = 1, j = 2 we find that, the equat,ion (14.26) can be reduced to give 61 2 = 0, 6t

(14.27)

when use is made of (13.21), (13.22), (14.1) and (14.19); the reduction, of course, also requires the use of the values 1,0,0 of the components

14.

FURTHER REMARKS ON THE TRANSITION PROBLEM

173

vl,v2,v3 and the fact that a h = 0, corresponding to the condition of simple tension in the elastic region. Defining the strength A of the wave by the equation

A

= IX21,

where the symbol I I denotes absolute value, the result expressed by (14.27) can be stated as follows. The strength A of the plane wave 2 does not oary along the normal directions during its propagation. I n view of the above italicized result concerning the variation of t12and the relation between t 1 2 and X2 contained in (13.22) it can also be stated that the strength A does riot vary along in the x 3 direction. Kext take i = 1, j = 3 in (14.26). But when this is done and use is made of the last equation (13.10), (14.9) and (14.20) we find that the equation (14.26) is satisfied identically. When we take i = 2, j = 3 the equation (14.26) yields the condition x2,3

= 0.

(14.28)

But the result expressed by (14.28) has already been observed, namely that the strength A of the wave Z does not vary a t any time along the x 3 direction, i.e. the direction perpendicular to the flat sides of the plate. Let us now put i = j = 1 in (14.26). Then, in the first instance, the equation (14.26) gives

But when account is taken of the equations (13.22), (14.12), (14.15) and (14.18) we find that (14.29) reduces to X2,2 = 0, i.e. the strength of the wave does not vary at a n y time in the direction of the tension. Taking i = j = 2 and i = j = 3 no conditions arise from (14.26) beyond those already a t our disposal. It may be concluded from the above results that points of infinite wave strength are not necessarily associated with the wave Z a t any given time t = to, and it follows also from (14.27) that such points cannot arise as a consequence of the wave propagation. Moreover, the wave will be stable in the sense that its strength does not vary during the propagation (see Sect. 12); we must therefore expect that, for this case, the elastic region of the flat plate will pass into the plastic state without the occurrence of fracture.

174

v.

CHARACTERISTIC

SURFACES

AND WAVE PROPAGATION

Case /3

Due to the general similarity of this case with Case a most of the preceding formulae can be applied to the present discussion without modification. However, there is an essential difference between these two cases which arises from the difference in the orientation of the wave surfaces Z relative to the direction of the applied load. As a consequence of this difference we shall see that there is a tendency for the occurrence of fracture points (see Remark at end of section) on the wave surface Z of Case p which may cause a crack in the direction of the tension in the plate. We recall that for Case p (see Sect. 13) the wave surface Z is a plane perpendicular to the x 2 direction, i.e. the direction of the tension, and to the flat sides of the plate and that it is propagated with a velocity G given by (13.21). For definiteness we assume that the components of the unit normal Y have the values 0,1,0 so that the propagation is in the direction of the positive x2 axis. The equations of 2 are therefore of the form x1 = xl; xz = Gt const.; x 3 = x3, (14.30) in which the coordinates x1,x3are also taken as parametric coordinates on 2. Since v2 = +1 the minus sign applies in the second equation (13.23). Now since all E’s and all X’s vanish except Elz and A1 respectively, let us put tlZ= t and hl = X for brevity. Let us also define the strength A of the wave Z as the absolute value of the quantity X. Hence we have $, = - p G X ; A = 1x1. (14.31) Equations (14.2), . . . , (14.7) hold for this case. Putting j = 3 in (14.5) we recover the condition (14.9). Also taking m = 3 in (14.5) we are led to the relation A13 = 0 from which it follows that the wave strength does not vary in the x 3 direction. Equations (14.11) likewise hold and are found to be satisfied identically for k = 1,3. Putting k = 2 in (14.11) and using the above symbol X we now obtain, in place of (14.12), the equation (14.32) xz X,1 = 0. Equation (14.14) holds but is now seen to reduce to

+

+

$11

=

2$22

+ 6 (y )X2,

(14.33)

14.

FURTHER REMARKS ON THE TRANSITION

175

PROBLEM

instead of (14.15). We now have the equations (14.16) in which

as can readily be observed when use is made of (14.30) and the compatibility conditions. Making these substitutions in (14.16) we obtain fi2

=

til,~= P (-GI,

+ 661). Xi

(14.34)

Taking i = 3 in (14.34) we again find the condition (14.20). However, when we put i = 1,2 in (14.34) we see that (14.35) (14.36) pG(X.1 - 12). These two relations replace (14.18) and (14.19). Combining (14.33) and (14.36) and eliminating the quantity X2 by means of (14.32) we have g22

=

(14.37) E22 = 2pGX,1. (14.38) Equations (14.21) continue t o hold as well as (14.22) since now X2 = 0. Also for this case we have (14.23), the first equation (14.24) and the equations (14.25). The second equation (14.24) must, however, be replaced by

Hence we find that equations (14.21) lead to the conditions

Reduction of (14.39) for i,j = 1,2 gives the condition 6X = 0, 6t

(14.40)

i.e. the strength A of the wave does not vaTy along the normal diTeCtiOnS

176

v.

CHARACTERISTIC

SURFACES AND WAVE PROPAGATION

during its propagation. Taking i,j = 1,3 and i,j = 2,3 in (14.39) we obtain the two relations X,3

= 0;

x.3

= 0,

(14.41)

the first of which expresses the fact that the wave strength i s constant at a n y tirne along the direction perpendicular to the $at sides of the plate. If i = j = 1 in (14.39) the resulting equation is seen immediately to reduce to G(2&1 -

fz2)

+

- PT G2X2 3pX,i = 0. IC2

(14.42)

Then, making the substitutions (14.37) and (14.38) and replacing k2 by its value T2/3 in terms of the tension T, we find that (14.42) gives the condition XJ

+7 PG

A2

= 0.

(14.43)

This same condition is also obtained when we take i = j = 3 in (14.39). Finally, taking i = j = 2 the equation (14.39) is seen to be satisfied identically. The above relation (14.43) gives information regarding the variation of the wave strength in the xi direction, i.e. across the width of the plate. Now from (14.40) and the first equation (14.41) we see that A, or the wave strength A, will be independent of the time and will depend a t most on the single coordinate xi. Hence if X has the finite value XO when xi = 0 we find, by integration of (14.43), that (14.44)

where (14.45)

The algebraic sign of the quantity A will depend on the algebraic sign of the initial discontinuity XO as well as the algebraic sign of r , with T > 0 for tension and 7- < 0 for compression which is equally permissible from the mathematical standpoint in this discussion. Remark. From the form of the relation (14.44)we see that X + f a as a9 + A . Points on the wave surface Z for which 21 = A will be referred to as fracture points and a t such points the wave strength A will be infinite. It is

15.

PLASTIC DISTURBANCES

177

seen from (14.44) and (14.45) that these fracture points will always occur on the wave Z provided the initial discontinuity Xo is sufficiently large. Assuming fracture at points of infinite wave strength A (see Remark 2 in Sect. 1 of Chap. VI) the existence of such fracture points on the wave at the initial time t = 0 will result in a ermk which will presumably be propagated in the direction of the tension or compression with the velocity G, given by (13.21), as a consequence of the propagation of the wave. Such a crack will produce fracture of the plate in a plane perpendicular to the flat sides and parallel to the direction of the tension or compression. Since the thin flat plate can be regarded as the limiting case of a hollow cylinder of indefinitely large radius the longitudinal cracks produced in hollow cylinders, when subjected to compression, can be considered a confirmation of this result. When the above fracture points do not arise the elastic region of the plate will, of course, pass into the plastic state as a result of the wave propagation without the mishap of fracture.

15. PLASTIC DISTURBANCES WHOSESPEEDOF PROPAGATION Is LESS THANTHE VELOCITY OF A SHEARWAVE Let Z be a surface of discontinuity in the plastic region of an incompressible elastic-plastic solid subject to the Prandtl-Reuss equations (12.1) and the von Mises yield condition (6.2). Use will also be made of the equation of continuity (6.3), the equations of motion (6.4) and the equation of incompressibility (6.5) as in the preceding discussion. We assume that [vi] = 0 and [au] = 0 or, more precisely that Z is singular of order 1 relative to the velocity and stress. Hence the first order compatibility conditions for velocity and stress have the simple form (15.1) (15.2) where G ( 2 0) is the coordinate velocity of propagation of L: in the direction of its unit normal v. Not all of the quantities [v;,j] and [ u i l , k ] can vanish simultaneously at any point of Z from the above assumption regarding the order of the singularity. Hence not all of the quantities X i and .$;j can vanish at points of 2. Now consider the relation P(G - un) = P(G - cn),

(15.3)

178

v.

CHARACTERISTIC

SURFACES

AND WAVE PROPAGATION

i.e. the equation (8.4) in Chap. I, where p and p are the densities on the two sides of Z and v, and En denote the corresponding components of normal velocity of the material. But v, = En since [v;] = 0. Hence if we assume that the speed of propagation of Z, defined by the quantity G - vn (see Sect. lo), does not vanish, it follows from the relation (15.3) that [ p ] = 0; i.e. the density i s continuous across Z. We shall be concerned in this section with disturbances or wave surfaces Z whose speed of propagation G - v, is different from zero and numerically less than the velocity of an equivoluminal or shear wave (see Remark 2 in Sect. 4 of Chap. 111), i.e.

0

< (G

-

<

(15.4)

P

A formula for the speed of such disturbances will be derived; however this formula involves the stress components aij and hence the actual determination of the speed of propagation of the disturbance which arises in a particular dynamical situation may constitute a problem of some degree of difficulty. From (6.5) and (15.1) we have immediately XiVi

(15.5)

= 0.

Also from (6.4) and the compatibility conditions (15.1) and (15.2) we readily find that (15.6) [iivj = -p(G - v,)X;, where p denotes the density at points of the surface Z. Similarly, application of the compatibility conditions to the Prandtl-Reuss equa' tions (12.1) leads to a set of relations which can be written in the form

(G - V n ) ( E i j - 31 t a a 8 i j ) -I- ~ ( X i v j If all 4 , j we have

=

XjVi)

= P (UabXaVb)a;j.

(15.7)

0 then all Xi = 0 from (15.6). Conversely if all X i [..v. %a 1 = 0 .

5.. - "5'3 aa8..2 1 21

- 07

=

0,

(15.8)

from (15.6) and (15.7) respectively. But if we multiply the second set of relations (15.8) by v j and make use of the first set of these relations it follows that Ea5 = 0; from the second set of relations (15.8) we then have (ij = 0. Hence, in view of the above italicized statement, it follows

15.

PLASTIC

DISTURBANCES

179

that not all of the quantities X i nor all of the quantities ti, can vanish at any point of Z. We can derive the following two sets of relations ti3viv3 = 0; U&j = 0, (15.9) from (15.6) and (15.7). I n fact the first relation (15.9) is obtained merely by multiplying (15.6) by vi, summing on repeated indices, and making use of (15.5). Similarly, multiplying (15.7) by uZj and making use of the yield condition (6.2), we obtain the second relation (15.9) provided the speed G - vn of Z is different from zero as we assume in this discussion. Let us next multiply (15.7) by Xivi and sum on the repeated indices. Taking account of the formula for the deviator components uZj and making use of (15.5) and (15.6), we now find that [ p ( G - 0,)'

- jL]Xih; =

-< Ir k

((TabXaYb)'.

(15.10)

But XiXi > 0 since, as we have observed, not all of the quantities X i can vanish on Z. Hence the relation (15.10) permits us to make the following statement. The inequality

(G - v,J2 <

P

holds if,and only i f , the quantity U a b X a V b does not vanish. I n other words the speed of propagation of the wave surface Z is numerically less than the velocity of a shear wave if, and only if, the quantity U o b X a V b is different from zero on 2. When U a b X a V b vanishes on 2 the speed of propagation of this surface will be equal to the velocity of a shear wave. Remark 1. The above result can be given a simple dynamical interpretation. We know that the quantity u,b& represents the rate at which work is done by the stress in changing the shape of a unit volume of the material (see Fkmark 2 in Sect. 7 of Chap. I). Now consider the relation [vab&]

=

uab[cab]

=

(Tab[Va,b] = u a b b b ,

which follows from (15.1), the equality of the tensors c and c * under the incompressibility condition (6.5), and the assumption that the stress is continuous across the surface 2 . Hence we can say that the speed of propagation of the surface Z will be numerically less than the velocity of a shear wave if, and only if, the rate at which the stress does work in changing the shape of a unit volume of material i s different at adjacent points on the t w , sides of Z.

180

v.

CHARACTERISTIC

SURFACES AND WAVE PROPAGATION

Remark 2. Since the density p is continuous across Z by the above result it follows that we must have relations of the form

(15.11)

From (6.3), (6.5) and (5.11) we therefore have

(G - vn){

=

0.

Hence b = 0 since the speed G - v,, is not equal to zero by assumption. I n other words the jirst coordinate derivatives p,i and the time derivative dp/at are continuous across the wave surface 2. Remark 3. If we differentiate the relation (6.2) with respect to the coordinates xk we obtain a;ja;,,k = aTjai,,k= 0. Hence, applying (15.2), we find that U;jEijvk

=

0.

Thus we again obtain the second relation (15.9) since not all of the quantities v k can vanish simultaneously. This derivative of the second relation (15.9) does not involve the assumption that the speed G - vn is different from zero which was made in our previous derivation. I n the remainder of this section we shall be concerned with the special case of a thin rectangular plate which we may suppose t o be subjected to uniform tension a t its ends. Choosing the x2 axis of our rectangular system in the direction of the applied tension and the x3 axis perpendicular t o the flat sides of the plate, we assume a state of plane stress (9.1). But under the condition (9.1) it follows from (15.2) t h a t

-

[a;j,3] = EijV.3 = 0. (15.12) 0; Hence ti3 = 0 from the first relations (15.12) since not all Vk can vanish; from the second set of relations (15.12) we have va = 0 since, as we have shown, not all tij can be equal to zero. Hence we see that the surface Z must be a cylinder perpendicular t o the flat sides of the plate. Also, putting i = 3 in (15.6), we have t 3 , ~ j = -p(G - v n ) X 3 = 0. It follows that Xa = 0 for the surfaces Z of non-vanishing speed G - v,, under consideration. To simplify our discussion of the conditions (15.5), (15.6) and (15.7) let us translate the coordinate axes so that the origin of the system falls [ai3,k]

Ei3Vk

=

15.

PLASTIC DISTURBANCES

181

on an arbitrary point P of the surface I:and let us then rotate the x1,x2 axes about the x3 axis through such an angle that the x1 axis will be normal to Z and its positive direction will coincide with the direction of the vector v. All preceding relations will, of course, be left invariant by a coordinate transformation of this character. Relative to the new coordinate system we now have v1 = 1 and vz = v3 = 0 a t P. Hence from (15.5) we see that X 1 = 0 a t P. Hence Xz # 0 a t P ; otherwise we would have all A, = 0, since X3 = 0, and this would contradict the above result. Kow consider the relations (15.6) and (15.7) a t the origin P of our transformed system. Thus, expanding (15.6), we have

in which account' is taken of the fact that XI = 0. From the first of these = 0 and from the second relation that t 1 2 # 0 relations we see that since, as we have shown, Xz # 0 a t P. If we take i = 1, j = 3, or i = 2, j = 3 in (15.7) we find that the relation is satisfied identically as a consequence of preceding conditions. Moreover, if we put i = j in (15.7) and sum on repeated indices, it follows t.hat the resulting equation is satisfied in view of (15.5). It is clear therefore that we can confine our attention to the values

i=l, j=1;

i=l, j=2;

i=3, j=3,

in treating the system (15.7). But for these values of the indices we obtain

(G - ZL)[ZZ

=

(uizXz)caa,

when use is made of the fact that v1 = 1 and the components Ell; XI; AS; VZ; v3 are equal to zero. Hence, replacing Xz by the expression in the second equation (15.13) and taking account of the fact that t12 # 0, we have

182

v.

CHARACTERISTIC

SURFACES

AND WAVE PROPAGATION

(15.15) (15.16)

Kow uI2 # 0 since otherwise the speed G - vn would not be nup as we see from (15.15). Using merically less than the velocity q the fact that u12 # 0 and eliminating the ['s between the two equations (15.14) and (15.16) we find that ull = 0. Expressed invariantly, i.e. independently of the special rectangular coordinate system, this latter condition becomes Ui j Vi Vi = 0. (15.17) The equation (15.15) gives the speed of the disturbance or wave surface Zin terms of the stress components relative to the above special coordinate system. To deduce an invariant relation, corresponding to (15.17)' for the speed we observe first that (6.2) becomes

+ +

k2 = $(& - ~ 1 1 ~ 2 2 a&) &a, under the plane stress condition (9.1). But in the special coordinate system this relation reduces to

+

(15.18) k2 = +a;, u?z, since ull = 0. Substituting this latter expression for k2 into (15.15) we now have

Or we can write (15.19) again using the fact that ull = 0 in the special system. Equation (15.19) is the desired invariant formulation since this equation obviously remains unchanged under orthogonal transformations of the rectangular coordinates. Remark 4. I t may be seen from the relation (15.18) that uy2 must lie between zero and kZ. I n the two limiting cases we have

15.

PLASTIC DISTURBANCES

183

for a:2 = P,) G - un = 0, from (15.15). Conceivably G - u, can have any value satisfying (15.4) under the above assumptions. Remark 5. Representing the wave surface z(t)by the equation F(2*,22,23,t) =

0,

(15.20)

and making the substitutions (6.8) in the relations (15.17) and (15.19),we obtain (1 5.21) (15.22)

Hence we can state the following result. If (15.20) determines a wuvc surface of order 1 relative to the velocity and stress in a $ow region subject to the plane stress condition, the speed of propagation of the wcve being different f r o m zero but numerically less than the velocity of a shear wave, then the equations (15.21) and (15.22) mzLst be satisJied over the surface (15.20). These wave surfaces will obviously not exist in general since the two differential equations (15.21) and (15.22) constitute an over determined system.

VI. Instability and Fracture 1. SURFACES OF STABILITY AND INSTABILITY

A discontinuity [u] in the velocity over a surface in t,he continuous medium will be called a slip discontinuity if its normal component is equal to zero, i.e. [ui]vi = 0, (slip condition), (1.1) where the [vi] are the components of the discontinuity and the vi are the components of the unit normal v to the surface. A surface over which a slip discontinuity exists, or is assumed to exist, will be called a slip surface. When dealing with specific problems in the following discussion it will be assumed, either directly or indirectly, that the slip discontinuity [v] results from a simple slip of the material particles over the surface Z bearing the discontinuity, i.e. without penetration of the surface by the particles; this can obviously restrict the components [vi] beyond the conditions given by (1.1). Thus if the surface Z is h e d or at rest relative to the dynamically allowable coordinate system to which the motion of the medium is referred, the normal components of the velocity must vanish on each side of 2, i.e. we must have v, = U, = 0 where un and V , are the normal velocities on the two sides of the surface, whereas the condition (1.1) merely asserts the equality of the normal velocities. A fixed surface Z will be called a surface of stability if every slip discontinuity [v] over Zis damped out, i.e. if [ui] -+ 0 as the time t + 00, as a consequence of the equations governing the behavior of the medium and the conditions of the problem under consideration e.g. boundary and natural symmetry conditions. In particular the surface Z will also be referred to as a surface of stability if we must necessarily have [vi] = 0 under the conditions of the problem. If Z is not a surface of stability it will be said to be a surface of instability and over such a surface there must exist the possibility of a 184

1.

SURFACES O F STABILITY AND INSTABILITY

185

slip discontinuity [v] which will not be damped out; if this slip discontinuity can be chosen so that at least one of the quantities

[v,] -+ f w , as t -+ T, (1.2) where T is finite, the surface 2 will be called a surface of strong instability. A surface of instability over which a condition of the type (1.2) is not possible for any slip discontinuity [v] will be referred to as a surface of weak instability. Kow it would appear that fracture over a surface of instability would be inevitable if one of its characteristic discontinuities [v] is actually realized. Taking cognizance, in this connection, of the fact that the discontinuity in question depends on the occurrence of an initial slip, a surface of instability will be said to be subject to fracture and will be called a fracture surface in this sense. When a solid has been brought to a state of elastic or plastic equilibrium a t the yield point, which is obviously an unstable condition, it is usually possible to find a surface Z over which a slip of the material particles will produce a slip discontinuity [v] in the velocity satisfying the condition (1.2). In particular this slip may result from plastic flow on one side of the surface while the other side remains in the equilibrium state; one can also sometimes show that the formal condition (1.2)) characterizing the fracture surface, can be met by a surface 2 located entirely in a region of plastic flow. This is not to say that plastic flow is the cause of fracture. Indeed it would appear that such flow has a restraining effect on the occurrence of fracture as is commonly considered. The relationship, if any, between fracture and plastic flow results from the fact that the high stresses which cause one may likewise cause the other, a fact which explains the plastic deformations, which are frequently observed, in test specimens of fractured material, and which enters into the mathematical analysis by which the fracture surfaces are determined. In the following sections of this chapter we have discussed a number of simple problems involving flat plates and round bars under tension, the torsion of round bars, and circular cylinders subject both to tension and internal pressure. In all cases good qualitative agreement has been found between the observed and predicted surfaces of fracture. Remark 1. The above criterion for fracture leaves open the question of whether fracture takes place instantaneously over a surface of instability or

VI.

186

INSTABILITY AND FRACTURE

occurs over such a surface as the result of crack propagation. This latter point of view has considerable appeal from the physical standpoint and appears, moreover, to be valid in certain instances. I n this connection see A. A. Wells and D. Post, T h e dynamic stress distribtuion surrounding a running crackA photographic analysis, Naval Research Laboratory, Washington, D. C., Report 4935, 1957. Remark 2. Various modifications and extensions of the above considerations can obviously be formulated, e.g. one might remove the limitation to fixed surfaces Z to allow the possibility of a moving surface Z(t), bearing a discontinuity [v] in the velocity, over which the condition (1.2) is satisfied; in this case the limiting surface Z ( T ) could be introduced as the theoretical surface of fraeture. Again one might consider the possibility of a fracture surface Z*generated by the motion of a curve C ( t ) on the surface Z ( t ) over which a discontinuity in the velocity components, or their derivatives, becomes infinite on C(t). The development of a fracture surface in this way, which is analogous to the process in gas dynamics by which waves of weaker type culminate in shocks, would correspond more directly to the concept of fracture as the result of crack propagation.

2.

FLATPLATES. P L A N E STRESSAND SYMMETRY CONDITIONS

PROBLEM O F FR.4CTURE I N

As a first application of the criterion for fracture in the preceding section we shall treat the case of the flat plate under tension which was considered in the Appendix to Chap. IV. It will be assumed in this discussion that the material of the plate is of the elastic-plastic type with the plastic flow governed, apart from the equation of contiiiuity which will not be needed, by the following system of equations at

- 2p(ciJ - +&),

uij,j = p

(G+ ui,Jui>' dv,

(Prandtl-Reuss equations), (equations of motion),

(2.1) (2.2)

0, (equation of incompressibility), (2.3) 2k2, (von Mises yield condition), (2.4) in which the various terms have their previous significance; however, for the purpose of reference, we note the relations giving the quantity and the components of the rate of strain tensor E , namely vi.6

=

u:ju& =

+

(2.5)

2.

PROBLEM OF FRACTURE IN FLAT PLATES

187

Use of the approximate equations (2.1), rather than the more accurate form of the Prandtl-RRuss equations involving covariant time differentiation (see Remark in Sect. 8 of Chap. IV), appears to be justified since rotational effects are obviously of minor significance in the problem under discussion. Now consider the possibility of a slip plane 2, perpendicular to the flat sides of the plate, such that the slip discontinuity [v] results from plastic flow on one side of the plane Z while on the other side of Z the material is in elastic equilibrium under uniform tension as assumed in the Appendix to Chap. IV. Denote by 5%the coordinates of a rectangular system whose x 2 axis is in the direction of the applied tension and whose x1,x2plane is parallel to the flat sides of the plate; the x3 axis will then be perpendicular to the flat sides. Also let y$denote the coordinates of a second rectangular system, obtained by rotation of the x system about the x3 axis, such that the y2 axis is perpendicular to the slip plane Z; the y' axis will then be parallel to the plane 8. Because of the similarity of these coordinate systems with the x and y systems used in the discussion of Luders bands in the Appendix to Chap. IV, the equations (1.6) of the Appendix will obviously be valid. Thus, denoting by V~ = (0,1,0) the components of the unit normal to the slip plane 2, relative to the y system, we have [u,,lv] =

3,2

- P%2 = 0,

(2.6)

where the u , ~and ptl are the components of the stress tensor, relative to the y system, in the flow and equilibrium regions respectively and the bar denotes evaluation on the plane 2 ; also the equations (2.2) of the Appendix are applicable, i.e. pll = T

sin26;

pZ2 = 7 cos2

p12

e;

= T

sin 6 cos 8; '13

p23

=

0,

p33 =

0,

'1

(2.7)

in which T is the applied tension and 6 is the inclination of the plane Z relative to the x1axis. It is assumed that T is just sufficient to produce plastic yield in the plate; hence T = d31c from the yield condition (2.4) and the uniform stress field which we have assumed in the equilibrium region. The plane stress condition (see Sect. 9 of Chap. V) is automatically satisfied in the equilibrium region; we shall assume that this condition

188

VI. INSTABILITY AND FRACTURE

is also satisfied in the flow region which is justified since the thickness of the plate is small in comparison with its length and width. Thus the component's u i k of the stress tensor in the flow region are given by equations of the form cik

= fik(y',y2,t);

Ui3

0,

=

(2.8)

relative t o the y system, where the f's are functions of the coordinates y1,y2and the time t alone. Neglecting possible end effects, which is tantamount t o the assumption of the ideal flat bar or plate in the Appendix to Chap. IV, it is clear from symmetry that the various quantities which enter into the discussion, when evaluated on X l will be independent of the y' coordinate. We shall refer to this condition as the symmetry condition. Thus we have, for example, ijdj.l = 0 and V i , l = 0 where uij and v; are the components of stress and velocity, relative to the y system, in the flow region and the bar is used to denote evaluation on the plane 2 . This use of the bar to denote such evaluation will be continued in the following discussion. It is seen immediately that V2 = 0 from the slip condition (1.1) and t'he assumption that the material is in equilibrium on one side of the plane E. Now put B ~ =, wi ~ for i = 1,2,3 and consider the incompressibility condition (2.3), i.e. Gi,i= B1.1

+ + V2,z

213,3

=

0.

But t'he term G1,1 vanishes from the above symmetry condition and hence the equation reduces t o 213,3 = -02. Also, putting v 1 . 3 = w, we see that the quantities G;,k are given by the equations -

v1,1

v2,l

V3,l

= = =

0;

0; 0;

v1,2

v2.2

v3,2

=

Wl;

V1,3

-

=

w2;

=

03;

v2.3

11373

= w, =

0,

=

-w2-

I 1

(2.9)

It follows from the second set of equations (2.5), defining the components of the rate of strain tensor e, that =

0;

& = w2;

€12 -

=

-. w1

2'

= 9.

z13 = ,; z33

=

-w.

(2.10)

3.

INCLINATION OF THE S L I P PLANE

189

3. INCLINATION OF THE SLIP PLANE Our specific task now is to obtain algebraic relations between the components of the stress and rate of strain tensors which will lead to a determination of the inclination 0 of the slip plane in spite of the complication caused by the fact that the above equations (2.1) involve derivatives of the stress components. Let us first take the partial time derivative of the equations (2.6); this gives relations which can be written in the form v j -- 0,

[%],,j=%

since dvi/at = 0 and also api,/at = 0 due t o the fact that the components of the stress tensor are constant in the elastic region. But G2 = 0 and vi = (0,1,0) relative to the y system; also F ~ ,=, ~0 from symmetry and 8 0 . 3 = 0 from the plane stress condition. Hence we see that daij - aaij. dt at

Then from the definition of the stress deviator u* and the equations (3.1) and (3.2), we obtain

where p = -ukk/3. Using the values v i (2.1) and (3.3) that

4 2 6i2 dt

=

=

2p(c;2 - Q&),

relative t o the y system. Hence, taking i we find z12

5%

dt c23

= Q-crtz = 2&2 =

(O,l,O) it now follows from

$853

=

-

(3.4)

1,2,3 successively in (3.4),

Qu12,

(3.5)

- Q&),

(3.6)

=

$823.

(3.7)

To deduce other relations which will have application to the problem at hand we put i = 1,2,3 and j = 3 in (2.1). Taking account of the

190

VI.

INSTABILITY AiVD FRACTURE

plane stress condition (2.8) this leads to the following set of three equations 213 $ 8 f 3 = $513, (3.8) =I

z23

=

=

$a;3

9 at = 2p(&

(3.9)

$523,

-

(3.10)

$63).

Eliminating @/dt between (3.6) and (3.10) and using the plane stress condition we obtain (3.11) z22 - $822 = E33. Now if $ = 0 it follows from (3.5), (3.7), (3.8), (3.9) and (3.11) that Z12

Z23

=

g13

= 0;

Z22

=

z33.

From these relations and (2.10) we see that w = 0 and w i = 0 for i = 1,2,3; hence 5i,j= 0 €or i,j = 1,2,3 from (2.9). I n other words the derivatives V i , j are continuous across the plane 2. Excluding this special case we assume a discontinuity in the derivatives v;,i over the plane Z. Hence $ # 0 and this implies Q > 0 since # > 0 in the flow region by hypothesis (see Sect. 4 of Chap. IV). Now equations (3.7) and (3.9) are identical; moreover 5 2 3 = 0 in these equations from the plane stress condition. Hence it follows that cza = 0 or w3 = 0 from (2.10). Similarly the equation (3.8) reduces to w = 0. Also, when use is made of (2.10), the equations (3.5) and (3.11) give w1 =

2$512;

2w2 =

$822.

(3.12)

A modified form of the equations (3.12) is obtained by replacing the and ~ 2 by 2 plz and pz2 in accordance with (2.6) and then eliminating the latter quantities by the substitution (2.7). Thus (3.12) can be written w1 = 2 4 sine cos 8, (3.13) 312

2wz =

e.

Tg G O S ~

(3.14)

As shown above the quantities w and w3 are equal to zero. It will now be assumed that the flow neighboring the slip surface 2 is such that w1 and w2 do not vanish; this is the general case for which there is a discontinuity in the derivatives ui,f in the transition from the equilibrium to the flow side of the plane 8.

3.

191

INCLINATION O F THE SLIP PLANE

Eliminating the quantities 01 and w2 between (3.12), (3.13) and (3.14) we have (3.15) F~~ = T sin 0 cos 8 ; F~~ = T cos28.

Now expand the left member of the yield condition (2.4) and evaluate on the plane Z; then replace the quantities a t by their values in terms of the components ciI of the stress tensor. Putting 5i3 = 0 for i = 1,2,3 in this latter equation in accordance with the plane stress condition and eliminating the components ~~2 and iiZ2by the substitution (3.15), the resulting equation can be written in the form

& - (T cos2@all + T~ cos20 + 3~~sin20 cos20

=

3k2.

(3.16)

But this equation shows that ull has a constant value on the plane 2. Hence dall,fdt = 0 and hence diill/dt = 0 from (3.2). Similarly dazz/dt = 0 since uZ2is constant on Z from the second equation (3.15).

Using these results we now see that dpldt

=

0 and hence

$&, (3.17) from (3.6) and (3.10). Adding the two equations (3.17) it follows that z22

= ;&i$

633 =

G1

= $81,

and from this relation we obtain & = 0 when account is taken of the first equation (2.10) and the fact that $ # 0. But from this latter equation and the second equation (3.15) we can deduce

cIl = 3.

e.

COS~

(3.18)

Then, making the substitution (3.18) in (3.16) and replacing k2 by its value r2/3, the resulting equation reduces to cos2e = 4. Hence the slip planes Z have inclinations of 35'16' or -35"16' with the x1 axis. Remark 1. Using the above determination of the inclination 6 of the slip planes we see that 811= pll from (3.18) and the first equation (2.7). Hence, taking account of (2.6) and the fact that the d s and p's vanish when one of their indices has the value 3 in accordance with the plane stress condition, it follows that U.' - * 1 - pii,

(i,j = 1,2,3).

I n other words the stress tensor i s continuous over the slip plane 2. Remark 2. For definiteness let us choose the y system so that (1) the y1,y2 plane is the plane Z and (2) the origin of the y system is midway between the flat sides of the plate. Then y2 = 0 is the equation of the plane 2 and we may

192

VI. INSTABILITY AND FRACTURE

take the y1,y3 coordinates as t'he parametric coordinates of points of 2 in the equations giving the conditions of compatibi1it.y. Thus we have y' = u';

?/2

=

0;

y3

=

u2,

(3.19)

as the equations of the plane 2 where the y's in the left members are the coordinates of the y system while the quantities u1 and u2 in the right members of these equations are the parametric coordinates of 2. The relations (3.19) will enter into the evaluation of those terms in the conditions of compatibility, which involve differentiation with respect to the parametric coordinates. Thus the geometrical conditions of compatibility of order 1 for the velocity components u; are

+

(3.20) [u. .] = v,.j = W i Y j ui,myi,a, in which we have used yi rather than yi to denote the y coordinates; also a in these equations has the values 1,2 and the comma denotes partial differentiation with respect to the parametric coordinates u1 and u2. Using (3.19) we see that the conditions (3.20) are identical with (2.9) when account is taken of the fact that v2 = 0 and w = w3 = 0, the above symmetry condition, and the condition of incompressibility (2.3) of the plastic material. It was shown in the above Remark 1 that [crii] = 0 over 21. Since the stress is assumed to be constant in the elastic region this implies that i i i j = const. along 2 and the constant values of t+;i will be independent of the time. Hence the geometrical conditions of compatibility of order 1 for the stress components take the simple form (3.21) [ a i i , k ] = ?ij,k = (ijvk, 191

in which the quantities [ ; j are functions defined over 21. Referring (3.21) t o the y system and putting k = 1 we see that these relations are satisfied identically in view of the symmetry condition; also (3.21) is satisfied identically for k = 3 on account of the condition of plane stress. However, when we take j = 3 the equations (3.21) give t i 3 = 0. (3.22) Since the plane 2 is stationary in this consideration, i.e. its normal velocity G = 0, i t follows that the 6 time derivatives of the components uii and ui are identical with their partial time derivatives. Hence the kinematical conditions of compatibility of order 1 for uij and u; reduce to identities. Remark 3. The above considerations can obviously be applied to the planes Z bounding a Luders band in a plate composed of elastic-plastic material rather than material of the rigid-plastic type as assumed in the Appendix to Chap. IV. Let us now postulate that the Luders band is initiated by the formation of a small plastic region in such an elastic-plastic plate a t its yield point. I n accordance with this point of view the plastic region, so formed, then grows or is propagated with great velocity into the elastic material resulting in the appearance of the band as observed in tensile tests on flat plates.

3.

INCLIXATIOX O F THE SLIP PLANE

193

I n Fig. 6 the above plastic region is represented by ABCD while PCDR and ABQS are the elastic regions which will pass into the plastic state as a result of the propagation. The angle 0 in Fig. 6, which determines the direction of the final band relative to the applied load, is considered t o have its theoretical value 35"16'. This idealized concept of the formation of a Luders band in a flat plate, with PCBQ and RDAS in Fig. 6 representing planes perpendicular to the flat sides of the plate, will furnish the basis of the following investigation in which we shall be concerned primarily with (1) the deter'mination of the portion A B (or C D ) , hereafter denoted by Z* for brevity, of the surface of separation of the plastic material of the band and the elastic medium, and (2) the normal velocity G of propagation of 8*. The stress tensor will be continuous over the planes BC and AD in Fig. 6 from the result in the above Remark 1. I n conformity FIG.6 with this result we make the following assumption. The stress tensor is continuous over 2*. Now consider the dynamica1 conditions pG = $(G -

fin);

[u..]v' *? = - PG[Uil,

(3.23)

which must hold over 8*where p and $ denote the densities on the elastic and plastic sides of Z* respectively, Cn is the normal component of velocity on the plastic side of 2* and the vi are the components of the unit normal vector v t o Z*, assumed to be directed from the plastic to the elastic side of this surface. Also

[.. I

=

8..$1 - c... $11

[Vi]

= oi,

(3.24)

where the uii and Bii are the components of the stress tensor on the elastic and plastic sides of 2*. Finally, the quantities fii are the velocity components on the plastic side of 2*. Account is taken of the fact that the velocity of the material particles is zero in the elastic medium by assumption in writing the conditions (3.23) and (3.24). But the quantities [ o i j ] must vanish from the above continuity assumption. Assuming G # 0 it follows from the second set of conditions (3.23) that [oil = 0 or, in other words, that the velocity i s continuous over 8*. Hence 0% = 0 from the second set of conditions (3.24), and we see from the first condition (3.23) that p = 8, i.e. the density i s continuous over Z*. I n view of the continuity of the velocity and the stress tensor over 2* we can now write

194

VI. INSTABILITY AND FRACTURE

(3.25) and (3.26) relative to the z or g coordinate systems, where the quantities X i and Eii are suitable functions defined over the moving surface 2*. The relations (3.25) and (3.26) constitute compatibility conditions for the problem under consideration (see Sect. 2 and Sect. 3 in Chap. 11);there is of course a corresponding compatibility condition for the density p but this will not be written since it will not be needed in the following discussion. It will be assumed that the surface 2* is singular of order 1 relative t o the functions vi and uii, i.e. not all the quantities Xi and Eij vanish at any point of 2*. I n fact, if X i = 0 and [if = 0 the continuity of the first derivatives of the components of velocity and stress over Z* would follow from (3.25) and (3.26); hence the equations (3.25) and (3.26) could be replaced by analogous equations involving second derivatives of the components of the velocity and the stress tensor. Evaluating (2.3) on the plastic side of 2* and making use of the first set of relations (3.25) we immediately have XiVi

= 0.

(3.27)

Similarly from (2.2), (3.25) and (3.26) we obtain [ijvj

+ pGXi = 0 ,

(3.28)

in which p is the material density in the elastic medium. Finally, when we expand the left member of (2.1) and apply the conditions (3.25) and (3.26), we ' are led t o relations of the form

where ffd, and u t are the components of the stress tensor and its deviator in the elastic portion of the plate; the occurrence of these tensor components in (3.29) results from the above assumption of the continuity of the stress tensor over Z*. We observe immediately that not all of the quantities [ i i can vanish since i t would then follow from (3.28) that all X i must likewise vanish and this would contradict the assumption that the surface Z* is singular of order 1. Conversely, all of the Xi cannot vanish. I n fact, if X i = 0 for i = 1,2,3 the conditions (3.28) and (3.29) become

3.

INCLINATION OF THE SLIP PLANE

it3v3 = 0 ;

[tj

=

+tkk6t3.

195

-

But if we multiply the second set of these relations by v, i t follows that [ k k 0 when account is taken of the first set of the relations. Hence ?,$'. = 0 from the second set of relations and we again have a contradiction with the assumption that Z* is singular of order 1. Putting j = 3 in the first set of relations (3.26) we see that [%3 = 0 on account of the plane stress condition. Hence, putting i = 3 in (3.28), i t follows that X3 = 0. h'ow take i = 1,2 and j = 3 in (3.29). This gives X p 3 = 0 because of the vanishing of the quantities Et3 and X3 and the fact that a& = 0 in view of the condition of plane stress. Hence v3 = 0 since otherwise we would have A, = 0 for i = 1,2,3 and this, as we have seen, would violate the hypothesis that 8* is singular of order 1. T h e surface 8* must therefore be perpendicular to the $at sides of the plnte. Since X3 = v3 = 0 the equation (3.27) becomes

+

XlV,

X2v2

=

0.

Hence we must have the proportionality XJ2

(3.30)

-v2,v1.

To deduce the normal velocity G of the wave front Z* we turn t o the relations (3.29). Multiplying these relations by v, and giving k2 its value r2/3 we obtain (p

- pG2)X, - 31- G [ k k V I

3P CaabXoVabU:,V1 = 0, --

when use is made of (3.27) and (3.28). Also, putting i have 31 [ k k G 314 u,&+,a& = 0.

+

(3.31)

7 2

=j =

3 in (3.29), we (3.32)

72

Now consider the relations (3.31) and (3.32) in the z coordinate system for which the components 7 i l of stress tensor are 722= 7 (yield stress) and rtl = 0 other~ the above relations. Hence wise. But then uil = T , in (3.33) =

while a&

=

-z.3'

g*zz =

$;

= -7

3'

(3.34)

0 for i # j . Hence *[kkG

=

pX2V2,

(3.35)

from (3.32). Making the substitutions (3.33) and (3.35) the relations (3.31) become (y

- pG2)Xi

- p X 2 v 2 v j - ?L! X ~ V ~ U : = ~ V0.~

(3.36)

196

VI. INSTABILITY AND FRACTURE

These relations are satisfied automatically when i = 3 since A3 = v3 = 0. However, taking i = 1,2 and using (3.34) and the proportionality relation (3.30), we find that (3.36) gives (pG2 - P ) ~ = Z 0, (pG2 - p)vi

1

+ 3pViV; = 0.

From these two equations we can immediately deduce that we must have one or the other of the following two possibilities, namely G2 = E,

(v1

=

l),

(3.37)

G2 = e,

(v2 =

1).

(3.38)

P

P

We see f r o m (3.37) and (3.38) that the normal velocity G of Z* i s the same as the velocity of a n equivoluminal wave in the elastic medium (see Sect. 4 of Chap. 111). Moreover, since v1 = 1 or v p = 1 relative to the z coordinate system, and since v 3 = 0, we can also state the following result. T h e surface Z* i s a plane perpendicular to the flat sides of the plate and i s either (1) parallel or (2) perpendicular to the direction of the applied load. The two cases corresponding to the formulae (3.37) and (3.38) are illustrated in Fig. 7 and Fig. 8 respectively; in Fig. 7 the wave surface Z:* is represented by the vertical line A B and in Fig. 8 by the horizontal line AB. The velocity

FIG.7

FIG.8

of propagation of these surfaces in the direction of the Luders band, i.e. the velocity of formation V of the band, is given by (3.39) (3.40)

4.

COMPATIBILITY

CONDITIONS

OF THE SECOND ORDER

197

These formulae determine the velocity V when the velocity G of an equivoluminal wave in the elastic medium is known. The above analysis does not disclose which of the two possible situations, namely that illustrated by Fig. 7 or the one shown in Fig. 8, will arise when the band is produced by tension in the flat plate. Possibly one of these cases is associated in practice with the tension problem and the other with the analytically equivalent problem of compression for which the algebraic value of the yield stress 7 is negative; if this view is correct we would expect the velocity V ,given by (3.39),to correspond to compression and the numerically greater value V , given by (3.40), to be the velocity of formation of the Luders band produced by tension in the plate.

4. COMPATIBILITY CONDITIONS OF

THE

SECOND ORDER

The coefficients bap of the second fundamental form vanish for the slip plane 2; moreover 2 is stationary, i.e. G = 0. Hence the conditions (5.10) and (6.26) of Chap. I1 can be applied t o give vi,jk

=

Givjvk

+

Wi.a(vj!/k,a

+

Vkyj.a)

+

Vi.aBYj,ayk,B,

(4.1)

and

respectively, where we have referred these relations to the y system. Relations (4.1) are the geometrical conditions of compatibility and relations (4.2) are kinematical conditions of compatibility of order 2 for the velocity components v i ; these relations, in which the quant,ities V i , a ~ are second partial derivatives of the components Vi with respect to the parametric coordinates u1 and u2 of the plane Z, supplement the compatibility conditions of order 1 given in the Remark 2 in Sect. 3. The kinematical conditions of compatibility which involve the derivatives d2v;/at2 reduce t o identities since the velocity G = 0 on Z. The analogous conditions of compatibility for the stress components a , j are given by 5ij.km

=

gijvkvm

f

izj,a(Vk?Jrn.a

+

Vm?Jk.a),

(4.3) (4.4)

Terms corresponding to the last terms in the relations (4.1) and (4.2) do not appear in (4.3) and (4.4) due to the fact that the components Z i j have constant values on z as observed in Sect. 3. As for the case of

198

VI. INSTABILITY AND FRACTURE

the velocity, the remaining kinematical conditions of the second order for the ui3 reduce t o identities. Certain simple consequences of the above conditions of compatibility can be observed immediately. Thus from (4.1) and the condition (2.3), expressing the incompressibility of plastic material, we have vi,ikVk = divi

+

wi,ayi,a =

62

+

W1.1

=

0,

when use is made of (3.19) and the fact that w3 = 0. But wl,l the symmetry condition (see Sect. 2) and hence

=

0 from

0. (4.5) Kext put m = 3 in (4.3); also takingj = 3 in these equations and using (3.22) and the condition of plane stress, the resulting equations are seen to reduce to E i j . 3 = 0; g i 3 = 0. (4.6) Now consider the equations of motion (2.2) in the flow region immediately behind the slip plane 2, namely 6 2

=

(4.7) Then from (3.20) and (3.21) we have (4.8) But in the y system the v i have the values (0,1,0) and VZ = 0. When we make these substitutions and also avail ourselves of (3.19) and (2.9), in which w and w3 have been shown to vanish, the equations (4.8) become = i,

Taking i

=

(3 at + %,,v,).

(4.9)

1,2,3 in turn in (4.9) we see that El2

(22

=

= P

avi at'

a52 at

fi - = 0,

(4.10) (4.11) (4.12)

4.

COMPATIBILITY CONDITIONS OF THE SECOND ORDER

199

In obtaining (4.11) we have of course used the fact that VZ = 0 on the plane 2. Also the condition (3.22) and the last of the relations (2.9) are used in the derivation of (4.12). We shall find that the equation (4.12) has an important application in the following discussion. Expanding the left members of the Prandtl-Reuss equations (2.1) we have relations of the form

at + ut,mum = 2P(Eij - pus.

ant

(4.13)

Now differentiate (4.13) with respect to the coordinates yk. Then, when we evaluate the resulting equations on the plane 2 and multiply by the components Vk, we obtain a2-m 3

ayk

at

vk

+

6j,kmvkk

f

zFj,rncm,kVk

2P(zij,kVk - U&$,kvlc - $ z k t v k ) .

=

(4.14)

Application of the compatibility conditions (3.20), (3.21), (4.1) and (4.3) will now lead to various new relations. However, only that relation obtainable from (4.14) by taking i = 1, j = 3 will be needed for our purpose. But for these values of i and j we see that the entire left member and also the second and third terms in the right member of (4.14) are equal to zero on account of the condition of plane stress. Hence, when we replace the v k by their appropriate values, the equation in question becomes (4.15)

0, relative to the y system. But from (4.1) we find &,2

= +(61,32 $. c3,K!)

=

0, when use is made of (3.19) and the condition gives w1.3 = 0. 61,32

=

w1.3;

v33,12

w3 =

0. Hence (4.15)

(4.16)

At any jixed time the quantity w 1 i s therefore constant on the plane 2; this result follows from (4.16) and the fact that w1,1 = 0 from the symmetry condition in Sect. 2. A similar statement can be made with regard to the quantity w2 on account of the relation w1 =

4wZtan 8,

which is obtained by combining the equations (3.13) and (3.14).

200

VI. INSTABILITY AND FRACTURE

5. FRACTURE OF FLAT PLATES

UNDER TENSION

Let us now differentiate (4.12) partially with respect to the parametric coordinate y 3 of the plane 2. This gives the equation

since w2 can depend only on the time by the result a t the end of Sect. 4. Hence, replacing the quantity ?j3,3 by its value - w2 in accordance with the last equation (2.9), we have 8x2 = w2. 2

at

By integration of (5.2) it follows that w2 =

1 - wot

~

(5.3)

1

where wo is used to denote the value of w2 at time t = 0. Now it was observed in Sect. 3 that the quantity J; is positive; also r > 0 for the tension problem under consideration. Hence it follows from the equation (3.14) that w2 > 0; hence wo must be positive since this quantity is the initial value of 02. It follows therefore from (5.3) that w2 -+ 00 as t T where T = l / w o > 0, i.e. the discontinuity [ ~ 3 , ~ becomes ] indeJinitely large over the slip plane L’ within a jinite time; obviously this implies the strict condition (1.2) for i = 3 as shown explicitly in the following Remark 2. The plate is therefore subject to fracture over a plane, perpendicular to its flat sides, which makes an angle of 35’16’ or -35’16’ with the d axis. --f

Remark 1. By coordinate differentiation of the yield condition (2.4) and application of the compatibility conditions (3.21) we readily find that

a;&

=

0,

(5.4)

on the plane 2. Expanding (5.4) and taking account of (3.22) we have Cfltll

+ 282tn +

7i;j2t22

=

0.

(5.5)

But the first term in (5.5) vanishes since iirl = 0, relative to the y system, as was observed in Sect. 3; also the last term in this equation vanishes on account of (4.11). Moreover 572 # 0; this follow from the first equation (3.12) since

5.

FRACTURE OF FLAT PLATES UNDER TENSION

201

= Zl2 and the quantities w1 and $ are different from zero. Hence (5.5) reduces to f l z = 0 and the equation (4.10) therefore gives

37,

Now & , I = 0 and %,3 = 0 from (2.9) since the quantity w has been shown to vanish. Hence the time derivative of cl and also its two derivatives with respect to the parametric coordinates y1 and y3 of I: are equal to zero. In other words, the velocity component v1 i s constant, both with respect to position and time, on the slip surface 2 ; this velocity produces the slipping efect which is commonly associated with the Luders band. Remark 2. It follows from (4.12) and (5.3) that

au3

-=----. wovj

at

1

- wot

Integrating (5.6) we have (1 - mt>m = 4(y3),

(5.7)

where the function 4 cannot involve the parametric coordinate y1 since &,I = 0 from (2.9). Differentiating (5.7) with respect to y3 we obtain

- coot)*

-(1

= 4'(y3).

(5.8)

Hence, replacing the quantity w2 in (5.8) by the value given by the right member of (5.3), the equation (5.8) becomes 4'(y3) = -a.

(5.9)

Now when we integrate (5.9) we find that

cp=-

w0y3

+ C,

(5.10)

where C is an absolute constant independent of the time or position on the plane Z. But from symmetry we must have g3 = 0 for y3 = 0 at all times t on account of the choice of the y system (see Remark 2 in Sect. 3). Hence the constant C must be equal to zero so that (5.7) and (5.10) combine to give ij3

=

-*.1 - wot

(5.11)

I t is seen from (5.11) that the particles in the plastic region contiguous to the slip plane Z (and lying off the central plane y3 = 0) must have a velocity which is directed at all times toward the center of the plate. A flow of this type obviously tends to produce the observed necking of the plastic material in the Luders band. Putting 83 = dy3/dt and integrating (5.11) we see, from the equation 60 obtained, that all particles on the plastic side of Z are compressed into the Y'

202

VI. INSTABILITY AND FRACTURE

axis at time t = l/w; the resulting cut along the plane Z can be interpretated as the cause of fracture. 6. ROUNDBARSUNDER TENSION.CYLINDRICAL AND SYMMETRY CONDITIONS COORDINATES

We shall now consider the surfaces of fracture in a round elasticplastic bar subject to uniform tension at its ends. Cylindrical coordinates r,O,z such that the x axis lies along the central axis of the bar will be employed since these coordinates will enable us to express most conveniently the natural symmetry conditions of the problem; the coordinates r,e,z will be identified with the coordinates x1,x?,x3respectively of an x coordinate system for the purpose of writing equations or formulae in tensor notat,ion. Thus the preceding equations (2.1), . . . , (2.4) for plastic flow in the bar can be written

gatv. % J. = 0,

(6.3)

”

2k’, (6.4) relative to the cylindrical system, where the gii are the contravariant components of the metric tensor and the comma denotes covariant differentiation; also, corresponding to the equations (2.5), we now have 1 \r/ = g ikg ime..,,*e j am; E‘. = - ( V i j vj.i). (6.5) 2 gikgjmu;*u;*

=

+

Between the covariant components vi and the contravariant components vi of the velocity we have relations of the form V *.

-- g..vi. 21 1

vi

= g i i u j,

(6.6)

where the g i j are the covariant components of the metric tensor. It may also be emphasized that the total time derivatives ddj/dt and dvildt appearing in the left and right members of (6.1) and (6.2) are defined by the following relations

dd= % + dt

dt

6.

ROUND BARS UNDER TENSION

203

As so defined these quantities are of tensor character under arbitrary differentiable coordinate transformations. In fact the last set of terms in the right members of (6.7) obviously constitutes the components of a tensor and the f i s t set of terms is easily seen to be of tensor character in view of the fact that the coordinate transformations under consideration do not involve the time; similar remarks, of course, apply t o the relations (6.8). Finally we note, for reference, that the deviators u* and E* of the stress tensor cr and t.he rate of strain tensor E are given by U;j

=

€&

= E<j

Uij

- B(gaaOab)gij, - $(gaa€ab)gij.

(6.9) (6.10)

It follows from (6.3), the second set of equations (6.5), and (6.10) that = ~ i j hence ; we can omit the star on the components e;j as we have done in writing (6.1) and the first set of equations (6.5). The values of the covariant components g i j and the contravariant components gij of the metric tensor are readily calculated for the cylindrical system. Expressed in matrix notation we have

E;;

Specifically the value of the component g i j or gii is given by the element in the i-th row and j-th column of the matrix l\gij\lor \\giil(. We shall also need to consider the Christoffel symbols rlit which are determined by the general formula (6.12)

Using this formula and the above values (6.11) of the components of the metric tensor, we find that the Christoffel symbols are given in the cylindrical coordinate system by the equations =

-r;

I?z =

1 -.

r’

rjt = 0,

otherwise.

(6.13)

In the problem of the round bar acted upon by a uniform tension a t

204

VI. INSTABILITY AND FRACTURE

its ends, the stress components u;j and the velocity components v, are obviously, from symmetry, independent of the angular coordinate 0, i.e. U" 11 - ai,(r,z,t); vi = v,(r,z,t). (6.14) Moreover, it is easily seen from the natural symmetry of the problem that u12 = 0; a 2 3 = 0; 8 2 = 0, (6.15)

in the cylindrical system. The first of the relations (6.15) expresses the condition that the angular component of stress on a surface element, perpendicular to the radial direction, is equal to zero; similarly the second relation states that the angular component of stress vanishes on an element of surface perpendicular to the axis of the bar. Finally the last equation (6.15) gives the condition that the flow takes place in planes through the z axis, i.e. the central axis of the bar; this symmetry condition can also be expressed by writing v 2 = 0 since, from (6.6) and (6.l l ) , we have v1 = vl; v2 = r2v2; v 3 = v3. (6.16)

7. PLANEFRACTURE OF ROUND BARS Let us first investigate the case of a plane surface Z perpendicular to the axis of the bar; for definiteness we assume 2 to be the plane z = 0. Or we may write

e

z = 0, (7.1) as the equations of the plane 2 where we have introduced u1,u2as parametric coordinates. On the basis of the equations (7.1) the components gab of the metric tensor of Z can be found from the relations r = ul;

=

u2;

and we can then obtain the components gab from the gab in the usual manner; thus we find that

7.

PLANE FRACTURE O F ROUND BARS

205

with individual components gas and v p equal to the elements of the a-th row and P-th column of these matrices. It will be assumed (see Sect. 1) that the material in the bar is a t rest within the region z 2 0. Let us now denote by Ei and Vi,j the values of the components u i of the velocity and the components v i , j of its covariant derivative on the plastic side of 2. Since the components v a ol the unit normal v to Z have the values (0,0,1) the slip condition (1.1) gives ih = 0; also i% = 0 from (6.15). Hence the formula for the components Fi .j becomes

where the bar on the first term in the right member denotes evaluation of the partial derivatives on the plastic side of 2. Taking account of (6.13) and the conditions on the v’s given by (6.14) and (6.15), the above formula (7.4) now shows that

i73.1

=

0;

Evidently the quantity aFl/ar in these relations, which we defined as the derivat,ive avl/ar on the plastic side of Z, can also be considered to be the partial derivative of G1 with respect to r . The covariant derivative of the velocity vector v, considered as a vector on 2, will also enter into our discussion. Now the components of this covariant derivative are given by the general formula

in which the Vi are functions of the parametric coordinates u1,u2 and the quantities xt are defined as the derivatives axk/aua of the functions xk(u) giving the surface 2. Taking i = 1,2,3respectively, and observing that x: = :6 for the plane Z under consideration, we see immediately that the relations (7.6) reduce to the following =

av,

-. au-’

~

2

=.

ri7&; ~

~

3

=.

ai?, -*

~

aua

206

VI. INSTABILITY AND FRACTURE

Hence the quantities iii,, have the values

4 . 1

=

0;

vZs2

i&I

=

0;

F3,2 =

=

(7.7)

rzil,

0.

We may put [ u . -3.] =

v. .. a.37

[vi],rr=

Ci,q

since the velocity vanishes in the elastic region, i.e. the region z 2 0, by hypothesis. Hence we have c.% ,.I AiVj + s"8g.3k v. ,p (7.8) *,a 8, from the geometrical conditions of compatibility (7.2) in Chap. 11, where the A's are suitable functions defmed on the plane 2. Using (7.5) and (7.7) we now see immediately that the relations (7.8) reduce to identities when j = 1,2. However, taking j = 3 and i = 1,2,3the equations (7.8) give

av3

av, -, .

Az=O; x3 -- -82(7.9) az Hence when we evaluate (6.3) on the plastic side of Z and make use of the relations (7.5) and (7.9), we find that XI=

(7.10) Assuming that a uniform tension prevails in the elastic region ( z 2 0 ) , the stress components uij will have the values u33

= T;

uij = 0,

otherwise,

(7.11)

in this region, where 7 > 0 denotes the const,ant tension over the ends of the bar. Let us furthermore assume that the components aij and their first derivatives, and hence the components U i j , k of the covariant derivative of the stress tensor, are continuous across the plane 2 ; this boundary condition does not unduly limit the possibility of discontinuities in the velocity and its derivatives with which we are primarily concerned in the investigation of surfaces of instability. Then the values of the stress components uij on the plastic side of L: are also given by (7.11); moreover, it follows that the components U i j , k of the covariant derivative of the stress vanish on the plastic side of Z since these com-

7.

PLANE FRACTURE OF ROUND BARS

207

ponents must vanish in the elastic region on account of the condition of uniform stress. Hence, when we evaluate (6.1) and (6.2) on the plastic side of 2, we obtain cij

(7.12)

= *a;;

where the bar is used to denote this evaluation, as in the foregoing discussion, and the u t are obtained from (7.11) in accordance with the equations (6.9). From (7.5) and the fact that irz = 53 = 0 we see that the second relations (7.12) are satisfied identically for i = 2,3. But for i = 1 these relations yield air, at

av, = 0. + 51 &-

(7.13)

Now from the second set of equations (6.5) and (7.5) we find that the values of the quantities e i j are given by

(7.14)

Also from (6.9) and (7.11) we have or1 =

-4.;

0; a& = 0 ; u;z =

a& = -+rr2;

6 3 = &3

0,

(7.15)

= $r.

Finally it follows readily that (7.16) from (6.3), (6.5) and (6.9), i.e. the quantities a:,,, can be replaced by ukm in the first set of equations (6.5). But k2 = r2/3 a t the yield point, when the u’s have the values (7.11), and hence from (7.16) we obtain (7.17) Making the substitutions (7.14), (7.15) and (7.17) we find that three of the equations in the first set of equations (7.12) are satisfied identically. The remaining three equations become 2 -a@, + - - 0av3 , ar az

(7.18)

208

VI. INSTABILITY AND FRACTURE

av3 + 2C1 = 0,

raz

851

- = 0.

(7.19) (7.20)

dz

Hence XI = 0 from (7.9) and (7.20). Also eliminating &/dz (7.18) and (7.19), we have

between (7.21)

After the component Cl(r,t) has been found by integration of (7.13) and (7.21) we can determine a&/& from (7.18) or (7.19); this determination automatically gives the value of X3 on account of the third equation (7.9). But, integrating (7.21), we obtain Cl = C ( t ) r ,

(7.22)

where the coefficient C depends on the time t alone. Substituting the function F1 given by (7.22) into (7.13), we find that dC -

dt

+

c 2

=

0.

(7.23)

Integration of (7.23) gives

c = t- -1 A’

(7.24)

where A is a constant independent of the time. Hence from (7.22) and (7.24), we have r jjl = (t 2 0). (7.25) t-A I n the case of tension there will evidently be a flow of plastic material toward the central axis, i.e. we shall have Cl < 0 for r > 0. But this implies A > 0, as may be seen by t,aking t = 0 in (7.25). Hence it follows from (7.25) that C1 -+ --43 as t A . I n accordance with the condition in Sect. 1 fracture may therefore occur over the plane Z. ---f

Remark 1. If 4 = 0 we have a&,/& = 0 from (7.17) and hence & = 0 on account of (7.19). This contradicts our assumption of a discontinuity in the velocity. Hence 4 > 0 and since T > 0 for tension we now see that &/dz > 0 from (7.17). Hence from (7.19) it follows that i& < 0, for r > 0, as above as-

sumed.

8.

GENERAL SYMMETRICAL FRACTURE

209

Remark 2. There will be a flow of material away from the central axis, i.e. we shall have El > 0 for r > 0, if the bar is brought to the yield point by compression. I t follows that the constant A in (7.25) must be negative and hence cl 0 as t 4 00. I n other words a n y initial discontinuity Vl udl be damped out ---f

and hence fracture will not occur over the plane 2.

8. GENERALSYMMETRICAL FRACTURE

Let us now consider a surface of revolution about the axis of the t a r as a possible surface of fracture 2 . In terms of the cylindrical coordinates r,e,z such a surface will be given by an equation of the form

F = z - f ( r ) = 0, (8.1) where f is assumed to be an arbitrary differentiable function of the coordinate r . We shall take the variables u1= r and u2 = 8 to be the parametric coordinates of this surface 2. Then, denoting the cylindrical coordinates r,e,z by x1,x2,x3 respectively as in the foregoing discussion, the surface 2 can be represented by x1 = u l ; x2 = u 2 . , x3 = f(u1). (8.2) The components gap of the metric tensor of the surface I: are given by the equations (7.2) in which the derivatives are determined from (8.2) and the quantities gij have their previous values (6.11); we thus find that the values of the covariant components gap and the contravariant components g@, which are obtained from the gap in the usual manner, are given by the following matrices

where we have put

We shall continue to use Greek indices with the range 1,2 for quantities associated wit'h the surface 2 and Latin indices with the range 1,2,3 for quantities associated with the space as we have done in writing the above equations.

210

VI. INSTABILITY AND FRACTURE

The covariant components vi of the unit normal vector to 2 are now given by

where F is defined by (8.1). Thus, using the values of the (6.11), we find that

gik

given by

The vector v is considered to be directed from the plastic into the elastic region (assumed to be in a state of equilibrium) bordering the surface Z. From the relations v i = giivj, we see that v l = v1, v2 = 0 and v 3 = v3. Hence there is no difference in the values of the covariant and contravariant components of this unit normal vector. Denoting by Fi = i%(u,t)the values of the velocity components v i on the plastic side of 2, let us consider the quantities Vi,, defined by (7.6) in which the I”s are Christoffel symbols (6.12) relative to the cylindrical system, and the xt are determined by differentiation of (8.2). Taking cognizance of the fact that iiz = 0 from the symmetry condition (6.15), the explicit values of the components 5;,=are found to be 51.1

=

ai71

-; ar

52.1

=

0;

V3.1

=

aii3

0; 02,z = TG; E3.2 = 0, I, in which u1 has been replaced by r , considered as a surface coordinate of 2. Let us next evaluate the components vi,j of the covariant derivative of the velocity on the plastic side of 2:. These components 5i,j are given by the formula 51.2

=

(8.7)

in which dVi/axj denotes the value on 2: of the partial derivative avi/axi of v i considered as a function of the cylindrical coordinates. Now if iii denotes the velocity components on the plastic side of Z we can write

8.

21 1

GENERAL SYMMETRICAL FRACTURE

where the extreme left members are the derivatives of the Vi with respect to r , considered as a parametric coordinate of 2 , while the other members involve the derivatives dvi/axl and avi/ax3 or avi/z on the plastic side of Z. Calculating the quantities 5;,j by means of (8.7) and eliminating the derivatives aGi/ax' whenever these occur by the substitutions (8.8), we obtain v1.1

=

ac1 avl ar - a-;az

51.2 =

0;

61.3

aVl z))

The components Vi,= and V i , j in (8.6) and (8.9) are related by the geometrical conditions of compatibility (7.8) in which the X i are functions on the surface 2. Examination of the equations (7.8) yields the condit,ion Xz = 0 and the following two relations

By total time differentiation of (8.1) we obtain V3 = a?; or V3 = a&, (8.10) since, there is no difference in the values of the corresponding covariant and contravariant components of the velocity vector. Now assume, as in the foregoing discussion, that the stress components aij have the values (7.11) in the elastic region; also that these components are continuous and have continuous first partial derivatives across the surface 8. We thus arrive again at the relations (7.12). Substituting the values of the V i , j given by (8.9), the second set of relations (7.12) is readily seen to be satisfied identically for i = 2. However for i = 1,3 we obtain

- aF* = 0,

a51 -at + v 1 z

ae3

ac3

at

ar

-+F1--=0

(8.11) (8.12)

212

VI. INSTABILITY AND FRACTURE

when use is made of (8.10). Eliminating U3 from (8.12) by means of @.lo), we find that

av, + 5,- av, + F-: da - = 0.

(8.13) a dr Now if 51 = 0 we have v3 = 0 from (8.10) and hence iTi = 0 for i = 1,2,3 which contradicts our assumption of a discontinuity in the velocity over the surface 2 . Hence El # 0, and from (8.11) and (8.13) it therefore follows that the derivative da/dr vanishes. But this means that j ( r ) = ar b where a and b are constants. The equation of the surface I: can now be given the form z = ar by a proper choice of the origin of coordinates. Hence the surface 2 m w t be a circular cone whose axis coincides with the axis of the bar. Turning now to the first set of relations (7.12), we first note that -

at

ar

+

El2

=

0;

(8.14)

from (6.5) and (8.9). We also observe that the quantity $ is again given by (7.17) and that the u; have the previous values (7.15). Using (7.15), (7.17) and (8.14) we now find that three of the equations in the first set of equations (7.12) are satisfied identically while the remaining equations can be written (8.15) (8.16) (8.17) Kow eliminate dii3/& from (8.15) and (8.16) by means of (8.17). This gives (8.18) (8119)

8.

213

GENERAL SYMMETRICAL FRACTURE

Next, eliminating aFl/az between (8.18) and (8.19), we obtain (8.20) But, since a is a constant, it follows from (8.10) that (8.21) Hence from (8.20) and (8.21) we have 1-. - 2a2 1 a2

+

(8.22)

I

Case

If A # 0 we can integrate (8.22) to obtain vl = C(t)rA, where the coefficient C is a function of the time t. Substituting this expression for Gl into (8.11), the resulting equation can be written 1 dC -=

C2 dt

--.ArA T

But the left member of this relation depends a t most on the time t and the right member at most on the coordinate r ; each member of the relation must therefore reduce to a constant since the variables t and r are mutually independent. Hence we must have

A = - -1- --= 2a2 l. 1 a2 It follows that a = 0 and hence Z must be a plane perpendicular to the axis of the bar. This is the case previously discussed and leads, as we have seen, to fracture of the bar over the plane 2 .

+

Case

II

If A = 0 it follows that a2 = 1/2 and hence Z is a circular cone whose generators make an angle of 35'16' with a plane perpendicular to the axis of the bar. Also aF1/ar = 0 from (8.22) and hence aV3/ar = 0 from (8.21). Moreover it follows from the equations (8.11) and (8.12) that the partial time derivatives of and O3 must vanish. Thus we see

214

VI. INSTABILITY AND FRACTURE

that the magnitude of the velocity vector V, which is obviously directed along the generators of t h e cone I;, is constant both with respect t o position and t h e . T h e cone I; i s therefore a surface of weak instability (see Sect. 1) and, as such, i s a possible surface of fracture within the bar. Remark. When the above cone is combined with the plane I;, or cross section of the bar, we obtain the well known “cup and cone” fracture which is sometimes experienced by round bars under tension (see Fig. 9). It would be expected, from a comparison of the strength of the discontinuities involved, that fracture will actually begin on the plane, or inner portion of the surface, in this type of fracture.

9. FRACTURE IN PLASTIC REGIONOF NECK The situation in which the bar necks down under tension as indicated in Fig. 10 and plastic flow occurs in the region containing the minimum

E (J FIG.9

FIG.10

section of the neck PQ, will now be investigated. Assume that a small discontinuity in the velocity develops across the plane PQ, which will hereafter be denoted by the single letter 2, in conformity with our previous notation. We shall treat the question of whether this diseon-

9.

FRACTURE IN PLASTIC REGION OF NECK

215

tinuity will be damped out or will yield a surface of instability or fracture. We shall employ the cylindrical system, the coordinates r,e,z of which will also be denoted by x1,x2,23respectively, as in our previous discussion, and we shall introduce the variables u1 = T and u2 = 0 as the parametric coordinates of the plane 2. The components of the metric tensors of the space and the plane Z will be given by the matrices in (6.11) and (7.3); also the symmetry conditions in Sect. 6 will be valid. Since the plane Z is stationary in this discussion we shall have the relation Pun = Pun, (9.1) where p and vn denote the density and normal velocity on one side of 2, while p and g,, are the corresponding quantities on the other side of this plane. We assume, for simplicity, that the density p is continuous across the plane Z. Hence vn = En from the above relation (9.1) or, in other words, the discontinuity [us] must vanish; this latter result is also a direct consequence of the slip condition (1.1). Actually we are justified in assuming from symmetry that the component v3, or the numerically equal component us, vanishes on each side of the plane 2 (cp. Sect. 1). Accordingly this assumption will be made and used in the following discussion. But [vz] must also vanish from the symmetry condition (6.15). Hence we must have [vl] # 0 since we have assumed that a discontinuity in the velocity occurs across the plane 2. Now consider the geometrical conditions of compatibility

+

(9.2) in which the various terms have their previous significance, and the following formulae of covariant differentiation, namely [vi,,]

= Lvi

gQBgjk[~iI,uxi,

From (9.3) and (9.4) we find that

(9.5)

216

VI. INSTABILITY AND FRACTURE

and

But from (9.2) we have [VLlI

=

a?€!d* [v2,1] ar '

[v1,21

=

0;

[Z'i,3]

= xi;

=

0;

[ ~ = lr[vlI; [vz,31

= xz;

[v3,11

=

0,

[v3,21

=

0,

[v3,31

=

x3,

1

1

(9.7)

when use is made of (9.5). Hence, equating the right members of corresponding relations in (9.6) and (9.7), it follows that A1

=

[Z];

xz

=

0;

A3

=

[5].

The equation of incompressibility (6.3) now gives 1 gij[vi,iI

=

[u1,1I

7[ ~ z . s ]-I-

[v3,3]

=

0,

and this condition can be written

when account is taken of (9.7). We shall again make the assumption that the u,j and their first partial derivatives with respect to the coordinates are continuous across the plane 2. Beyond this condition we shall consider, for the present, that the a's are subjected only to the symmetry requirement in (6.14) and (6.15). Hence, from (6.1) and (6.2), we obtain (9.10) (9.11) Let us first examine the relations (9.11). Kow the term [vi,jvj] in these relations reduces to [vi,lvl] from the symmetry condition (6.15) and the above assumption that the component v3 vanishes on each side

9.

217

FRACTURE I N PLASTIC REGION OF NECK

of the plane Z; this term can also be written

[vi.1211]

since the components

v1 and u1 have equal values. Hence (9.11) becomes

au+ dt

[Vi,lU1]

=

(9.12)

0.

But u2,1 and 8 3 , l vanish on each side of Z; this follows readily from the formula for the covariant derivative of the velocity vector. For i = 2,3 the equation (9.12) is therefore satisfied identically. However, vl,l = avl/ar on each side of Z and hence, for i = 1, the equation (9.12) becomes (9.13) Taking account of the continuity of the U’S and their derivatives across 2 and the fact that [v2] and [v3] are equal to zero, we observe that

[%I

=

‘Za [&I

+

u:j,k[ffq

=

u:j,l[u1].

Hence (9.10) can be written

- r:j[+l

[~ijl

=

*

utj,1

-.[Vll 2P

(9.14)

Now consider the two relations (9.15)

which we obtain by t,aking i,j

=

1,l and i , j

=

2,2 in (9.14). But

Making these substitut’ionsand eliminating the quantity [+] from the above two relations (9.15) we have (9.16)

It would appear that the values (7.11), in which T is the yield stress, furnish a first approximation for the stress components uij over the minimum section PQ in Fig. 11, i.e. over the plane 2, although significant deviations from these values may possibly occur at other points within the plastic region of the neck. Assuming that the aij can be approxi-

218

VI. INSTABILITY

AND FRACTURE

nated by equations of the form (7.11) over the plane 2,it follows from the formula for the covariant derivative of the stress deviator u* that the components u:l,l and U&?J in (9.16) are equal to zero; moreover we have r-6 1 -

1, r

6 2

from (7.15). Hence (9.16) reduces to

(9.17) Integrating (9.17) we obtain [v11 = C(t)r,

(9.18)

in which the coefficient C depends a t most on the time t. The function C(t) in (9.18) will evidently have a continuous derivative with respect to t since we assume the velocity components vi(z,t) to be continuously differentiable functions; hence it is permissible to differentiate this function as we shall do in the following discussion. Let us now consider the discontinuity [vl] = - w,where we denote by tE and w the values of the component v1 at contiguous points on the two sides of Z. Then we readily see that 2 -- tE2 - w2 = [81]2 [UI] 2w[v1],

+

or, making the substitution (9.18), we have

[u:]

=

C2r2

+ 2Crw.

(9.19)

Hence the equation (9.13) becomes _1 dC _ (9.20) C dt on account of (9.18) and (9.19). But the left member of (9.20) can depend a t most on the time t. Hence (9.20) can be replaced by the equivalent system of equations -dC

C dt

+ c = 2E(t),

+

= -2E(t), dr r where E depends only on the time t. Assuming w # have w = -E(t)r,

@

(9.21) (9.22) a

at r

=

0, we

(9.23)

10.

ROUND BARS UNDER PURE TORSION

219

by integration of (9.22). Now for the tension problem under consideration the 00w is obviously directed toward the central axis of the bar. Hence we must have 8 < 0 and w < 0; it follows therefore from (9.23) that E ( t ) > 0. (9.24) By integration of (9.21) we obtain

where the quantity C(0) in the right member of this equation denotes the value of the function C(t) at time t = 0. It is assumed that the initial discontinuity [s],i.e. the value of [ol] at time t = 0, is the result of a slip of the material particles on one side of the plane Z toward the central axis of the bar (see Sect. 1);let us suppose that the above quantity ti3 was chosen as the velocity on the side of Z on which this slip occurs. But, since tD < 0 and w < 0 as observed, this means that > IwJ and hence [24< 0 at t = 0. Hence from (9.18) the initial value of the function C ( t ) must be negative, i.e. C(0) < 0. Hence, since E ( t ) is positive from (9.24), it will always be possible to find a positive number A such that

It follows from (9.25) that C(t) 3 - w as 2 -+A > 0, i.e. the discontinuity must become inde$nitely large within a finite time. The plane 2 i s

[VI]

therefore a surface of strong instability over which the bar i s subject to fracture.

10. ROUNDBARSUNDER PURETORSION I n treating the round bar under pure torsion it will be assumed, for simplicity, that the bar is composed of rigid-plastic material (see Remark 1 in Sect. 2 of Chap. IV). Thus the stress-rate of strain relations E;j = J.u:j, (10.1) will replace the Prandtl-Reuss equations (6.1) used in the discussion of the tension problem; the other relations, namely (6.2), (6.3) and (6.4) governing the plastic flow in the bar, will remain unchanged. All equa-

220

VI. INSTABILITY AND FRACTURE

tions will be referred to the cylindrical system, previously used, whose coordinates r,B,z are also denoted by x1,x2,x3respectively for the purpose of writing equations in tensorial form. The stress at any point P within the rigid segment of the bar under pure torsion is assumed to result solely from a shearing stress on the cross section through P , the shearing stress being tangent to the circle in the cross section whose center lies on the central axis of the bar; this implies that the components a,j of the stress tensor, relative to the cylindrical system, are equal to zero with the exception of the component 023. Now these stress components must satisfy the following two sets of conditions, namely u..vj $3 = 0, gikUij,k

=

0,

(boundary condition),

(10.2)

(equilibrium equations),

(10.3)

in which the vi in the equations (10.2) are the components of the unit normal to the outer surface and the comma in (10.3) denotes covariant differentiation. But (10.2) is satisfied identically since the v i have the values (1,0,0) on the outer surface of the bar. Using the values of the g j k given by (6.11) and expanding (10.3) we have c11,l

cZl,l

c31.1

+ F1 + + 71 + + 71 +

1

c12,2

c13,3

=

0,

c22,2

c23.3

=

0,

c32.2

c33.3

=

0-

The first of these equations is seen to be satisfied identically when we determine the quantities ail .k from the formula for covariant differentiation in which the r's have the values (6.13); also the second and third equations are found to reduce respectively to

Hence the single non-vanishing component u 2 3 of the stress tensor must be a function of the coordinate r alone. To determine this function of r we have recourse to the yield condition (6.4) from which we readily find that uZ3 = f k r ; for definiteness in the following work we shall

10.

ROUND BARS UNDER PURE TORSION

choose the positive sign in this relation. will be given by ~ 2 3 =

kr;

uij

=0

221

Hence the stress distribution otherwise.

(10.4)

This stress, which corresponds to the applied torque indicated by the arrows in Fig. 11, is just sufficient for the initiation of plastic flow in the bar. Reversing the direction of t.he torque we must introduce the negative sign in the right member of the first equation (10.4). Now assume that a slip discontinuity [v] occurs over a surface

F(r,z,B)

=

0,

(10.5)

,+-

after the above yield stress (10.4) has been reached (see Sect. 1). Such a discontinuity [v] must satisfy the slip condition (1.1) and the associated flow must satisfy the geome__-trical conditions of compatibility (7.8) in which account is taken of the fact that the velocity vanishes on the equilibrium side of the surface (10.5) and the bar denotes evaluation FIG.11 on its plastic side. The problem of the stability or instability of surfaces (10.5) will be treated in the following sections under the assumption that /

-. \r

u33

= 0,

(10.6)

throughout the bar, which would appear to be an acceptable condition because of the purely torsional character of the applied force; in the rigid or equilibrium region of the bar the condition (10.6) is of course automatically satisfied. Actually the strong condition (10.6) will not be needed in the flow region but only certain boundary conditions, derived from (10.6), which will depend on the particular surface (10.5) under consideration. Our discussion of such problems as well as the discussion of the other results contained in Sect. 11 to Sect. 14 is based on the work of Sadia M. Makky, Plastic flow and fracture in round bars under pure torsion, Jour. Math. and Mech., 10, 1961, pp. 199-221.

222

VI. INSTABILITY AND FRACTURE

Remark. To obtain the surface elements of numerically maximum shearing stress for the stress field (10.4) let us first recall from Sect. 9 in Chap. I that if T I , T Z , T ~are the principal stresses in an arbitrary stress field u and if vI,v2,v3 are three mutually perpendicular unit vectors which determine principal directions associated with the principal stresses ?-1,72,73 respectively, then (uij

- 7 k g ; j ) u 1t

=

0,

(not summed on k ) ,

(10.7)

in which the uii are the components of the stress field and the quantities are given as the solutions r of the determinantal equation Id;;

- ~ g i i l= 0.

If we impose the ordering r1 2 r2 2 the gij are given by (6.11), that r1 =

k;

T~ we

T~

(10.8)

find from (10.4) and (10.8), in which

rz = 0 ;

73

=

-k.

(10.9)

Using the vaIues of the u;, given by (10.4) and the above values (10.9) of the T~ we readily find the following two sets of values for the components u ~ , v ~ , vas( , solutions of the equations (10.7), namely

}

va = (1, 0, 01,

and uf = (0,

-9 1 rd2

(10.10)

">'I

dii

I n writing the equations (10.10) and (10.11) we have normalized the values of the components v: so that vl, v2 and v3 will be unit vectors. Since the values of the T'S given by (10.9) are distinct, the vectors vl, v2 and v3 must be mutually perpendicular, as can be verified directly from the equations (10.10) and (10.11) defining these vectors. The plane element of maximum shearing stress at a point P passes through the principal direction v2 and bisects the angle determined by the two principal directions v1 and v3 at P. Now the principal directions given by the vectors v1 and v3 in (10.10) lie in the plane perpendicular to the radial direction at P and make angles of 45" with the cross section of the bar which contains the point P ; also these vectors are on opposite sides of this cross section. Hence a plane

11.

STABILITY O F CROSS SECTIONS

223

element of maximum shearing stress a t P lies in the cross section which passes through P. Obviously these plane elements unite to form cross sections which are therefore planes of maximum shearing stress in the bar. Similarly if we consider the vectors v1 and va in (10.11) we easily see that the other surfaces of maximum shearing stress in the bar are given by planes passing through the central axis.

11. STABILITY OF CROSSSECTIONS

MAXIMUM SHEARING STRESS

OF

The cross sections of maximum shearing stress (see Remark in Sect. 10) would appear a priori to be possible surfaces of fracture in the round bar under pure torsion. Let us suppose therefore that a slip discontinuity [v] occurs over a cross section of the bar after the yield stress (10.4) has been reached; it is specifically assumed that the slip discontinuity is associated with plastic flow on one side only of the cross section (see Sect. 1). Taking this cross section L: to have the equation z = 0, or the parametric equations (7.1), as may be done without loss of generality, the components gno and g@ of its metric tensor are given by the matrices in (7.3). We also note that the slip condition (1.1) reduces to V3 = 0; moreover it follows from the dynamical conditions =

[U&i

0,

since v i = (O,O,l), and the values (10.4) of the components of the stress tensor on the equilibrium side of the plane Z, that &3

=

0;

523

=

kr;

?f33= 0.

(11.1)

Let us now expand the left member of (6.4) and evaluate on the plastic side of Z to obtain (5.;,)2

+ p1 ( 5 a 2 + ( 5 3 2 + 22 (,2)2 +

2(&)2

+7 2

(&)2

=

2k2,

1

(11.2)

when use is made of the values of the gi’ given by (6.11). But the last term in the left member of (11.2) is equal to the right member of this equation on account of the second relation (11.1). Cancelling these two terms the equation (11.2) expresses the vanishing of a sum of non-

224

VI. INSTABILITY AND FRACTURE

negative terms each of which must therefore be equal to zero; this leads to the result that (11.3) a11 = a12 = a22 = 0, when we express the deviator components in terms of the components 8;j of the stress tensor. Comparison of the values of the Zij given by (11.1) and (11.3) with the values (10.4) of the components of the stress tensor on the equilibrium side of Z now shows that [ u i j ] = 0, i.e. the components of the stress tensor are continuous over the cross section 2. Hence the geometrical compatibility conditions of the first order for the stress have the simple form [aij,k] = tijvk. (11.4)

It may also be observed, as a direct consequence of the continuity of the stress tensor across the plane 2, that (11.5)

Now consider the compatibility conditions (7.8). But for j = 1,2 these conditions are readily seen to be satisfied identically; also the conditions (11.4) are satisfied identically for k = 1,2 on account of the above relations (11.5). However for j = 3 and for k = 3 the conditions (7.8) and (11.4) become

aci

= hi, az

aij.3

=

Eij.

(11.6)

It follows from the equations (11.1) and (11.3) that the components 5&and ?ij are identical. Hence, evaluating the stress-rate of strain relations (10.1) on the flow side of 2, we have c $1. . = $Eij.

(11.7)

Before proceeding with the consideration of the individual equations (11.7) we note that the conditions (6.14) may be imposed because of the symmetry of the problem under discussion, i.e. we may assume that the components of the velocity and the stress tensor are independent of the angular coordinate 6. When we now allow the indices i,j to assume their various values, take account of this symmetry condition, the fact that u3 = 0, and finally make use of the equations (11.1) and (11.3) for the components 5 i j we find that the above relations (11.7) yield

11.

STABILITY OF CROSS SECTIONS

225 (11.8) (11.9)

Integrating the second equation (11.8) it follows that i7‘2

(1 1.10)

= r2f(t),

where f(t) is an arbitrary differentiable function of the time t. Also the equations (11.9) give A1 = 0; XZ = 2k$r; A 3 = 0, (11.11) on account of the first set of equations (11.6). The equation of incompressibility (6.3) is readily seen to be satisfied identically when evaluated on the flow side of Z and use is made of the above relations (11.8) and (11.9). Finally when we evaluate the equations of motion (6.2) on Z we have

5

($ + v;*jg% )

(1 1.12)

= g%i,,k.

Taking i = 1,2,3 in turn and expanding the equations (11.12) we obtain

fie+fe+-

-

+ $ $1 = 511,l+ > + r + 522 2 + = + r71 + VlJVl

61,2

-

51.353

512.2

V2

52,353

-

522,2

521,l

=

531,l

+

513.3,

-

V2,lVl

I)

1 7 r 532.2

+

(11.14) 523.3,

(1 1.15) 533.3.

But, from the results of the above discussion, we have -

53

=

511,l

=

512.2

522.2

= ?31,1 =

v1

=

52.2

=

(11.13)

0,

532.2

i

0, = 0.

= 512J =

226

VI. INSTABILITY AND FRACTURE

Hence the equations (11.13), (11.14) and (11.15) reduce respectively to -p

-

-2 82

(1 1.16)

513.3,

a52

P= g23.3, at 533,s

(ii.17j (1 1.18)

= 0.

To determine the value of the quantity 5 2 3 . 3 in the right member of (11.17) let us differentiate the yield condition (6.4) covariantly with respect to the coordinate z ; this gives g"g'"a:p:,,3

=

0.

(1 1.19)

Evaluating (11.19) on the flow side of 2 and substituting the values of 5;j from (11.1) and (11.3), we obtain &3,3

= 0.

(11.20)

Hence the right member of (11.17) is equal to zero and it follows that S is independent of the time, i.e. G2 = Cr2,

(1 1.21)

on account of (ll.lO), where C is a constant. Substituting this value of 02 into the equation (11.16) we now have

-pC2r. (11.22) If we differentiate the equations of motion (6.2) covariantly with respect to the coordinate z we find that 313.3

b

,r

a (vi,3)

+

=

vi,j.3gikvk

+

vi,j&?ykvk,31)

Taking i = 3 in these relations, expanding the various terms, and evaluating on the flow side of Z we can write

11.

STABILITY O F CROSS SECTIONS

227

when account is taken of the fact that the components v , * , , k are symmetric in the last two indices. Making use of the results obtained in the preceding work it follows readily that the entire left member of (11.24) is equal to zero; also the last term in the right member of (11.24) vanishes on account of the condition (10.6). Hence (11.24) reduces to ?13,1,3

+1

>?23,2,3

T

=

(11.25)

0.

Now consider the geometrical conditions of compatibility of the second order for the stress tensor a; since this tensor has been shown to be continuous over the plane 2, the compatibility conditions are given by [aij,k,mI= i i j v k v m

+

€ij,olpB(vkgrna

Taking i,j,k,m = 1,3,1,3 and i,j,k,m ily deduce that

=

+

vmgka)xs.

2,3,2,3 in these relations we eas-

Hence the above equation (11.25) becomes (1 1.26) But

-/iC2r, as we see by taking i , j = 1,3 in the second set of equations (11.6) and comparing the resulting equation with (1 1.22). Making this substitution in (11.26) it follows that C = 0. Hence 5; = 0 from (11.21). Since iTl and B3 have also been shown to vanish we have therefore proved the following result. A discontinuity in the velocity components v i cannot occur over the cross section Z of m a x i m u m shearing stress. Cross sections of the bar are therefore stable under the conditions assumed in this discussion. ti3

=

Remark 1. In the demonstration of the above italicized statement we have used the condition (10.6) only to show that C = 0 in (11.21). Actually this result could have been obtained by assuming the weaker condition

over the surface 2 ; such a condition could be regarded as supplementing the above condition (11.18). Without a boundary condition of this character the

228

VI. INSTABILITY AND FRACTURE

constant c in (11.21) may be diflerent from zero and hence cross sections of the bar may be surfaces of weak instability over which the bar may fracture in accordance with the criterion in Section I. Remark 2. Let us now make the following assumption. Not all of the coordinate derivatives of the velocity components v, are continuous over the cross section I:;since XI and XH have been shown to vanish this assumption implies that the quantity X2 must be different from zero. Because of the continuity of the components of velocity and stress over I: i t is obvious that the derivatives of these components with respect to the coordinates r and 0 will also be continuous over I:. It remains t o consider the derivatives of o, and u , ~with respect to the z coordinate. But (1 1.27) (33 = (23 = (13 = 0, from the second set of equations (11.6) and the conditions (11.18)] (11.20) and (11.22) in which the constant C is equal t o zero. The behavior of the quantity and (22 will now be examined. X2 and the other quantities El i.e. Ell, Differentiating the stress-rate of strain relations (10.1) covariantly with respect to the coordinate z and evaluating on the flow side of I:,we have (11.28)

Assigning specific values to the indices i,j in (11.28) we obtain the following set of equations] namely 223 2511.3 - 5- 6-3 3 . 3

rz

c1,2,3

+ g2,1,3

=

2$512,31

(11.29) (11.30) (11.31)

(1 1.34)

when we choose the values of the ci, in accordance with the equations (11.1) and (11.3) and make use of the definition of the eij and u; which, as we have observed, are equal t o the corresponding components uij of the stress tensor on the plane I:. But

since A1 = 0; also we must have

4 > Q fw, if $ = 0, it would follow from (11.11)

11.

STABILITY OF CROSS SECTIONS

229

that A, = 0 and hence all coordinate derivatives of the velocity components oi would be continuous across Z contrary t o our hypothesis #at there is a discontinuity in these derivatives. Hence, using the relations (11.6) and (11.18), the equation (11.29) becomes 2.51

-

9

= 0.

(11.35)

Similarly we find from the equations (11.30) and (11.31) that (11.36) 2 ? TZ

- Ell

= 0.

(11.37)

The other three equations (11.32), (11.33) and (11.34) will not be needed in the following discussion. Combining the equations (11.35) and (11.37), we obtain El1

= En =

0.

Also, using the value of $ given by ( l l . l l ) , the equation (11.36) becomes (1 1.38) Hence all quantities Eii vanish with the exception of tI2which is given by the above equation (11.38). The behavior of E12 will be determined from (11.38) when the behavior of XZis known. To see how XZ varies with the time we choose i = 2 in the equation (11.23); this gives (11.39) Since all components (11.39) reduces to

ijk

=

0 it follows that &/dt vanishes; hence the equation

(11.40) when use is made of the fact that X1 and X, are also equal to zero. Evaluating the first and second terms in the right member of (11.40), we find that

(11.41)

To find a corresponding expression for the last term in (11.40) we take the second

230

VI. INSTABILITY AND FRACTURE

covariant derivative of the left member of (6.4) with respect to z and then evaluate on the flow side of Z to obtain g'kg'mF:;,3Ffm,3

+ g"g'mFfjFfm,3,3 = 0.

(11.42)

But, as we have observed, all [ i j vanish except tI2;using this fact and also the fact that all F i j vanish except 5%we see that the equation (11.42) reduces to (1 1.43)

Making the substitutions (11.41) and (11.43) in (11.40) the latter equation becomes

ax,

(w.

a h+ ar

r

Finally, substituting the expression for can be given the form

kr

t12given by (11.38) the above equation (1 1.44)

To solve the equation (11.44) let us put A2

=

(11.45)

g(t)Nr),

where g and h are differentiable functions of t and r respectively. substitution the equation (1 1.44) can be written

Making this

Now assume, for simplicity, that the density p is constant in conformity with the equation of incompressibility (6.3) ; then the above equation yields

where C is a constant independent of t and r. By integration of the two equations (11.46) we find h(r) =

in which p is a constant of integration.

-12 -9

Cr

Hence from (11.45) we have (1 1.47)

12.

TRAXSITION FROM EQUILIBRIUM

TO PLASTIC FLOW

231

To determine the algebraic sign of the constant p in (11.47) we observe first that Xt is positive from the second equation (11.11) since k and $ are positive. Hence, taking t = 0 in (11.47), we have

from which it follows that p is positive. Hence Xz + 0 as t + p from (11.47) and we have proved the following result. If a discontinuity in the derivatives of the velocity compon.ents v; develops over the cross section 2 it will be damped out in ajinite time. We emphasize that this result depends on the condition (10.6) or on the weaker boundary condition mentioned in the preceding Remark.

12. TRANSITION FROM THE EQUILIBRIUM STATETO PLASTIC FLOW Let us now remove the restriction that the cross section 2 , separating the regions of equilibrium and plastic flow, is fixed in the bar. Specifically it will be assumed that the position of the plane 2 a t any time t is given by an equation of the form z = f ( t ) so that the velocity of propagation G of the plane is at most a function of the time t. We suppose, for definiteness, that G > 0 and that the unit normal v with components (0,0,1) is directed into the equilibrium region; hence the plastic region is spreading throughout the bar with a velocity of propagation equal to G. Instead of the stress-rate of strain relations (10.1) used in Sect. 10 we shall now assume the Prandtl-Reuss equations (6.1) as more appropriate for the treatment of this type of problem; the other equations which will enter into the discussion are (6.2), (6.3) and (6.4). It will be supposed that the stress field in the equilibrium region is given by the relations (10.4). Strictly speaking this latter region should be considered as elastic rather than rigid in character in accordance with the viewpoint adopted in Sect. 4 of Chap. IV; however it is immaterial whether the equilibrium region is composed of elastic or rigid material since we shall be concerned in the following discussion only with the stress field in this region. It will be assumed that the components uiJ of the stress tensor are continuous over the moving plane Z; thus the relations (11.1) and (11.3) are applicable in which the bar denotes evaluation on the flow side of 2. Now consider the general dynamical conditions

232

VI. INSTABILITY AND FRACTURE

(12.1) P(G - vn)[vi], where p and v, refer to the density and normal component of velocity on the side of Z bordering the equilibrium region. But v, = 0 since the velocity vanishes in the equilibrium region and [cij] = 0 by hypothesis; hence the above conditions (12.1) give [vi] = 0, i.e. the velocity i s continuous over the moving plane Z. The density p is also seen to be continuous over Z as a consequence of the continuity of the velocity and the condition P(G - vn) = P(G - Vn), [ ~ i j ] ~= ’

in which the bar denotes evaluation on the flow side of the plane 2. Since velocity and stress are continuous over Z we have compatibility conditions of the form

Now evaluate (6.1) on the flow side of 2 ; taking account of the fact that the velocity components iTi = 0 from the continuity of the velocity over 2 , we thus obtain (12.3) Applying the compatibility conditions (12.2) we readily find that

- i33

2.51 - 7 b 2

(13

2522 T2

=

bz

=

=

-G A17

(11

- ia3

0,

(12.4)

0,

(12.5)

P

(12.6)

=

0,

(12.7) (12.8)

-G(23

= p(A2

- 2k$~),

(12.9)

when we allow the indices in (12.3) to assume their various values and

12.

TRANSITION

FROM EQUILIBRIUM

TO PLASTIC FLOW

233

$,

choose the values of the in accordance with the relations (11.1) and (11.3). Similarly we obtain X 3 = 0, (12.10) from the equation of incompressibility (6.3). Combining (12.4), (12.7) and (12.8) and taking account of (12.10) we see that

bl

=

f22 r2 =

533.

(12.11)

Also by differentiation of (6 4), as in Sect. 11, we obtain (11.20); hence it follows that EZ3 = 0 from the above compatibility conditions for the stress. Hence (12.9) reduces to Xz = 2k$r.

(12.12)

If we evaluate the equations of motion (6.2) on the flow side of Z and then take i = 1,2,3 in turn in the resulting equations we are led t o the following conditions -pGXi = f f i , (12.13) = 523, (12.14) -pGXz = E33. (12.15) Eliminating the quantity ,$I3 between the equations (12.6) and (12.13) we have (12.16) (pG2 - p)X1 = 0. Also from (12.14) and the fact that f23 = 0 we see that Xz = 0; this implies that $ = 0 on account of equation (12.12). Finally from (12.10) and (12.15) it follows that f 3 3 must vanish. Hence t1land f22 must also vanish on account of (12.11). We now make the following assumption. There exists a diwontinuity in the derivatives of the velocity components v, over the moving plane 2 . It follows from this assumption that X1 # 0 since X2 and A 3 have been shown t o vanish. Hence, from (12.16), the velocity G of the plane Z is given by

Remark. By recourse to compatibility conditions of the second order and the various relations obtained in the above discussion, i t can be shown that the time derivative 6X1/6t is equal to zero. Hence the discontinuities in the deriva-

234

VI. INSTABILITY AND FRACTURE

tives of the velocity components over 2, as represented by the quantities A, will remain constant during the propagation of the plastic into the elastic region of the bar.

13. HELICOIDAL SURFACES OF FRACTURE Suppose that the slip surface defined by (10.5), which is considered to separate the rigid portion of the bar from the region of plastic flow, has the special form F(r,O,z) = ae - z = 0,

(13.1)

where a is a constant different from zero. Assuming that the slip is associated with a discontinuity in the derivatives of the contravariant components of the velocity we shall show in this section that the helicoidal surfaces (13.1) are surfaces of strong instability and hence are possible surfaces of fracture in the bar. The covariant components of the normal Y to the surface (10.5) are given by

8F -

dqii

axi

vi =

aF aF'

axi 8x1

where the g i j are the contravariant components of the metric tensor for the space relative to the cylindrical system; hence for the surface Z defined by (13.1) we have v1=0;

ar

v2=ds,

.

-r

Y3=2/a2+T2'

(13.2)

Representing the surface Z by the equations r = u';

B

=

u2;

(13.3)

z = au2,

we readily find that the covariant and contravariant. components of its fundamental metric tensor, relative to the parametric coordinates u,are given by the following matrices I1S~SlI=

1 0

0

a2

+ r2 ;

0 1

1

119"811=o a2

+ r2

.

(13.4)

13.

HELICOIDAL SURFACES OF FRACTURE

235

Since the velocity vanishes on the equilibrium side of the surface Z it immediately follom-s from the slip condition (1.1) that

-

v, =

ap,

(13.5)

when the values of the contravariant components vi, which appear in ( l . l ) , are determined from the above covariant components (13.2) in the usual manner. Similarly it follows from the dynamical conditions (12.1), the right members of which are equal to zero for the problem under consideration, that [(Ti31

a r

=7

[Uizl.

(13.6)

Taking i = 1,2,3 in (13.6) and using the values (10.4) of the stress components in the equilibrium region, we have 313

=

2a 312,

(13.7) (13.8)

a z33= 7 (cz3- kr). r

(13.9)

But 8 3 3 = 0 since the component, u33vanishes identically by the assumption in Sect. 10; hence (13.9) yields 5 2 3 = kr, (13.10) since a # 0 by hypothesis. Evaluating the condition (6.4) on the flow side of 2 we find, as in Sect. 11, that all other components c;, are equal to zero; hence (13.7) is satisfied identically and (13.8) reduces to the above equation (13.10). These results can be expressed by saying that the components ail of the stress tensor are continuous over the surface 8. By differentiation of the components v i and evaluation on the flow side of the surface Z, we obtain (13.11) where we have used x 2 and e respectively to denote the angular coordinate in the cylindrical system and in the above system of parametric coordinates for the surface 2. Evaluating the stress-rate of strain equations (10.1) on the flow side of 2 we now find that

236

VI. INSTABILITY AND FRACTURE

(13.12) (13.13) (13.14) (13.15) (13.16) (13.17) when the values of the components a:* are assigned in accordance with the relations (11.1) and (11.3) and when the derivatives azi;/W are eliminated by means of (13.11). Assuming Cl = 0 for r = 0, thus insuring that there is no separation of material along the central axis of the bar, it follou-s from (13.12) that 51 = 0 along the radial lines which generate the surface 2 ; hence El = 0, (13.18) over 2. We now readily find that (13.19) from t.he equations (13.5), (13.13), (13.14) and (13.18). I n t e g r a h g (13.19) we obtain vz = r2f(t,e), (13.20) where f is any differentiable function of the time t and the coordinate 8. It also follows from the relations (13.5), (13.15), (13.16), (13.17) and (13.18) that (13.21) The equation of incompressibility (6.3) is satisfied identically over B on account of the equations (13.12), (13.15) and (13.16). However when we evaluate the equations of motion (6.2) on the flow side of Z we obtain 5

($+

ciejg?kck)

=

g i k -U i j ,k.

(13.22)

13.

HELICOIDAL SURFACES OF FRACTURE

237

Choosing i = 3 and taking account of (13.16) and (13.18), the above equations (13.22) give

p (aii, = 831,1 + at + 5%) T Z ae

+ a33,3*

(13.23)

Examining the various terms in the right member of (13.23) we see that

when use is made of (11.1); also by covariant differentiation of (6.4) with respect to the coordinate x2 and evaluation on the flow side of 2, it can be shown that 8 3 2 , 2 is equal to zero; finally 833,3 vanishes on account of the condition (10.6). Hence, making use of (13.5), the equation (13.23) becomes

av2 - + - - -oz= av2 o, at T Z ae

(13.24)

To solve the equation (13.24) we first replace Vz by the value given by (13.20); we thus obtain % + f - = af O. (13.25)

ae

at

Assuming a solution of (13.25) of the form

f

(13.26) g(t)Q(e), where g and Q are functions of t and e respectively, we must have =

4l (13.27) where C is a constant independent of the time t and the coordinate 0. If C # 0 we can integrate the above two differential equations (13.27) to obtain

(13.28) in which p and CY are constants of integration. Obviously we can take = 0 by a suitable choice of the polar axis; hence from (13.5), (13.20), (13.26) and (13.28) we have

CY

(13.29)

238

VI. INSTABILITY AND FRACTURE

If $ = 0 it follows that aG2/a0 = 0 from (13.21); hence aF2/at = 0 from (13.24) and (13.20) reduces to e2 = br2 where b is a constant. Also v1 = 0 from (13.18) and V3 = ab on account of (13.5). The contravariant components of the velocity 5 are thus given by -1 v -- 0.9 52 = b ; vs = ab. (13.30) Now we see that a&,/az = 0 from (13.15); hence W / a z = 0. Using this condition and the second relation (13.30) it follows that aV2/ax2= 0 from a relation of the type (13.11). Also we see that av2/dr = 0 from the second equation (13.30). Similarly we can show quite easily that a51---avl - -- a51 = o ; ar

ax2

az

aiia a 3 - a53 -ar---ax2 - - -az - 0.

In other words the vanishing of $ implies the vanishing of all partial derivatives of the contravariant components of velocity contradicting our hypothesis that there is a discontinuity in these derivatives over the surface 2. Hence we must have $ > 0 since #J > 0 in the region of plastic flow by hypothesis. Similarly if the constant C = 0 in (13.27) it follows that the functions g ( t ) and Q(0) reduce to constants. Hence vz = br2, where b is a constant, from (13.20) and (13.26). Since 9 = 0 and 9 3 = ab from (13.18) and (13.5) respectively we again arrive a t the relations (13.30) and can proceed, as above shown, to a contradiction with the hypothesis of a discontinuity in the derivatives of the components u i on 2. Hence we must take C # 0 in (13.27) as we have assumed in the derivation of the relations (13.29). To determine the algebraic sign of the above constant p we first R Q differentiate (13.29) with respect to 0 and evaluate at t = 0 to obtain

(2)t=o-5’ r2

=

(13.31)

But from (13.21) we see that a&/a0 and a must have the same algebraic FIG.12 sign since k and $ are both positive. Hence it follows from (13.31) that the constant p will be positive if a has a negative value. Choosing a < 0 it is clear from (13.29) that VZ -+00 and & + -a as t + p (> 0). I n other words the surface L: defined by (13.1) m’th a < 0 is a surface of P

A

14.

HELICOIDAL SURFACE OF FRACTURE

239

strong instability and i s therefore a possible surface of fracture in the bar. Such a surface 2 is represented by OPQR in Fig. 12; the radial lines, such as OP and RQ, are the generators of the surface Z and the curve PQ is the intersection of Z with the surface of the bar; finally the line OR in Fig. 12 is the cent.ra1 axis of the bar and OA denotes the polar axis of the cylindrical coordinate system. Reversing the direction of the applied torque, the constant a in the equation (13.1) defining the fracture surface Z will, of course, have a positive value. Remark. It suffices, for the demonstration of the above results on fracture, to assume the boundary conditions azz (13.32) = 0; - 0,

az

rather than the strong condition given by the equation (10.6). By limiting our boundary assumption t o the comparatively weak conditions (13.32) unnecessary restrictions may be avoided in any investigation of these fracture surfaces involving compatibility conditions of higher order. 14. MOST PROBABLE HELICOIDAL SURFACE OF FRACTURE IN THE ROUNDBARUNDER PURETORSION

We have shown in Sect. 13 that possible suFfaces of fracture Z are given by the equation (13.1) in which a is any non-zero constant whose algebraic sign depends on the direction of the applied torque. Now fracture over the surface Z would appear (a)most likely to start on the outer boundary of the bar where the material restraints on the motion of particles are less severe than on interior particles (see Remark 1 in Sect. 1) and (p) to take place over a surface element on which the normal stress has its greatest value since this stress is of the nature of a tensile force tending to produce separation of material over the element. Let us therefore seek to determine the constant a in (13.1) so that the surface elements of the resulting surface Z along the curve of intersection of Z with the surface of the bar, e.g. the curve PQ in Fig. 12, will be elements of maximum normal stress in the bar. A fracture surface Z satisfying this boundary condition will be referred to as the most probable surface of fracture. The maximum normal stress a t a point P is along the principal direction v, associated with the greatest principal stress T , and its value is

240

VI. INSTABILITY AND FRACTURE

equal to the value of the greatest principal stress 7 a t the point P. The contravariant components of this direction or vector Y are given by the first set of equations in (10.10). Hence the covariant components of the vector have the values (14.1)

The required boundary condition for I: to be the most probable surface of fracture will be satisfied if we choose the components of the vector, normal to the surface 2 , to be proportional to the components of the above vector Y when r = R, where R is the radius of the bar. We thus obtain

from the equations (13.2) and (14.1); it follows from this proportionality that a = -R. Hence the most probable surface of fracture Z: i s given by z = -Re,

when the applied torque has the direction indicated in Fig. 11. When the direction of the torque is opposite to that shown in Fig. 11 the surface z = Re will be the most probable surface of fracture. It is easily seen that each of these surfaces intersects the surface of the bar in a helix, e.g. the curve PQ in Fig. 12, which makes an angle of 45" with the generators of the surface of the bar. Remark. Fracture over cross sections of bars subject to torsion, which is predicted by the italicized result in the Remark 1 in Sect. 11, occurs in bars of ductile material. Bars composed of brittle material, on the other hand, frequently fracture along a helicoidal surface. See A. Naidai, Theory of Flow and Fracture of Solids, Eoc. cit., 1950, p. 243. Such fracture would appear to be a confirmation of the above result on fracture over helicoidal surfaces when it is recalled that the concept of the rigid-plastic bar makes no provision for the large angular deformations which precede the fracture of ductile bars under torsion and in this respect conforms closely to the behavior of brittle materials.

15. ELASTICDEFORMATION OF CIRCULAR PIPES DUE TO INTERNAL PRESSURE

Consider a circular pipe subject to uniform internal pressure, insufficient to cause plastic deformation. In treating this problem we shall

15.

ELASTIC DEFORMATION OF CIRCULAR PIPES

24 1

employ the cylindrical system whose coordinates T , ~ , zare also denoted by x1,x2,x3respectively; it is assumed, of course, that the z axis lies along the central axis of the pipe. Relative to this system the stressstrain equations of elasticity theory and the equilibrium equations have the form uij = X(g'*ua,a)gij ~ ( u i , j uj.i), (15.1) pa.. a3.k = 0, (15.2) respectively (see Sect. 1 and Sect. 2 of Chap. 111), where X and p are the Lam6 parameters, a;j the components of the stress tensor and the quantities u1, u2 and u3 represent the covariant radial, angular and longitudinal components of the deformation; also the g;j and g'j are the covariant and contravariant components of the metric tensor and are given by the matrices in (6.11) relative to t,he cylindrical system; finally the comma in the equations (15.1) and (15.2) denotes covariant differentiation based on this metric tensor. From the relations between the covariant and contravariant components of the deformation u, namely u z. = g..ui aj 2 we see that u1 = u'; u2 = r2u2; u3 = u3. (15.3) The natural symmetric conditions of this problem can be expressed most conveniently when use is made of the above cylindrical coordinates. Relative to the cylindrical system it is obvious, from symmetry, that the components ui will be independent of the angular coordinate 6. Also the angular component u2must vanish and the radial component UZ must be independent of the z coordinate. Hence (15.3) becomes u3 = u3, 241 = u'; uz = u2 = 0; and there is therefore no distinction between the values of the covariant and contravariant components of the deformation. Moreover it is clear from symmetry that a plane section of the pipe, perpendicular to the central or z axis, must be displaced into a plane section perpendicular to this axis. Hence u3 must be independent of the coordinate T . In the case of the equilibrium problem, for which the quantities ui are independent of the time, we can summarize these symmetry requirements by writing U1 = U l ( T ) ; U2 = 0; U3 = U j ( Z ) . (15.4)

+

+

242

VI. INSTABILITY AND FRACTURE

Similarly we see that the stress components uij must be independent of the 0 and z coordinates. In addition it follows from symmetry that the two components u12and uz3 must vanish as in Sect. 6. Corresponding to (15.4) we can therefore write 0; ui, = u i , ( r ) , otherwise. (15.5) Making use of (15.4), (15.5) and the equations (6.13) giving the Christoffel symbols in the cylindrical system, it follows from the formula for covariant differentiation that 612

=

U1,l

U23

=

=

au1

-; dT

u1,2

=

0;

and am1

u22

r

=

0,

0;

m3,3

=

0,

ru13;

u33,3

=

0.

=

-; dT

u12,2

=

TU11

U21,l

=

0;

U22,Z

=

(T31,l

=au13*

u32,a

=

aT

7;

u13,3

U11,l

-

(15.7)

Now consider the equilibrium equations (15.2). For i = 2 we see immediately from (15.7) that (15.2) is satisfied identically. However for i = 1 and i = 3 the equations (15.2) yield (15.8) Similarly, using (15.6), we can readily calculate the divergence of the deformation u. Thus we find that

au, + u-1 + au3. (pU. .=-

aT

T

a2

(15.9)

Taking i , j = 1,2 and i,j = 2,3 we see that (15.1) is satisfied identically. For i,j = 1,3 the equations become (TI3

=

0.

(15.10)

Hence the second equilibrium condition (I 5.8) is satisfied automatically and there is a slight simplification in the last row of the set of equations

15. (15.7).

ELASTIC DEFORMATION OF CIRCULAR PIPES

243

The remaining conditions resulting from (15.1) are obtained by

taking

i,j = 1,l;

i,j = 2,2; i,j = 3,3. Making these substitutions in turn, and using (15.9), it follows that u11

= (X

+ 2p) 2 +

(z +

(15.11) (15.12) (15.13)

It is seen from (15.4), (15.5) and (15.11) that the derivative du,/az must be a constant. Hence, since u3can depend at most on the variable z, we have 263 =

CZ + D,

(15.14)

where C and D are constants. Substituting the values of u11 and 8 2 2 given by (15.11) and (15.12) into the first relation (15.8) we find, after some reduction of the resulting equation, that (15.15)

where we have now used the symbol of ordinary differentiation since u1can depend only on the coordinate T from (15.4). Integrating (15.15) we have =

~1

AT

B +7

(15.16)

where A and B are constants. Hence from (15.14) and (15.16) we find that (15.11), (15.12) and (15.13) can be written (15.17) (15.18) a33

= 2XA

+ (A + 2p)C.

(15.19)

Denoting by a the inner radius and by b the outer radius of the pipe in its strained position, we have u11

= --p

Ull

= 0

for T for T

= a, =

b,

(15.20) (15.21)

244

VI. INSTABILITY AND FRACTURE

where p is the internal pressure. Hence from (15.17) we obtain

+ p)A - 2p-a2B + XC - p , B 2(X + p)A - 2p + XC = 0. 2(X

(15.22) (15.23)

Subtracting corresponding members of (15.22) and (15.23) we find that a2b2 B = 2p(b2 -'a2)*

(15.24)

A condition involving A and C alone is obtained from (15.22) or (15.23) by elimination of B by means of the substitution (15.24). Thus (15.25) Another equation in A and C is obtained from the fact that, by hypothesis, no load is applied on the ends of the pipe. Since the ends of the pipe are assumed to be planes perpendicular to the central axis, i.e. cross sections, and since (r33 is a constant according to (15.19), it follows that the load is given by as3Swhere S is the area of a cross section of the pipe. Hence u33= 0 and hence from (15.19) we have

+ +

2XA (A 2p)C = 0. Solving (15.25) and (15.26) for A and C we now find that

(15.26)

(15.27) (15.28)

It is seen from the above equation (15.28) that the constant C must be different from zero. Hence the constant D in (15.14) can be reduced to zero by a proper choice of the origin of our coordinate system and we can therefore write u3 = (15.29)

cz,

in place of (15.14), Hence the equation uz = 0 and the equations (15.16) and (15.29) in which A, B and C are given by (15.27), (15.24) and (15.28) respectively will determine the deformation u. Moreover, elimination of the constants A, B and C from (15.17) and (15.18) by means of the substitutions (15.27), (15.24) and (15.28) provides us with

16.

245

YIELD CONDITION

the determination of the components ull and a22 of the stress tensor; since the remaining components of this tensor have been shown to vanish we thus arrive at the complete solution of the purely elastic problem. For reference in the following discussion we observe that the above elimination of the constants A , B and C from (15.17) and (15.28) leads to the following relations Qll =

(1

- b2 >) b 2 a2p- a2,

(15.30) (15.31)

16. YIELDCONDITION.BEGINNING OF PLASTIC DEFORMATION

Ull

- 7-

0

0

u22

0

- 7r2 0

72

=

0 0 = 0,

(16.1)

-7

0,

I-

(16.3)

W e shall assume the Tresca yield condition in accordance with which yield may occur whenever the equation 71

- 7 8 = 2k,

(16.4)

is satisfied, where k i s a material constant for the solid (see Sect. 3 in

246

VI. INSTABILITY AND FRACTURE

Chap. IV). It is considered here that the inequalities (16.2) hold so that the left member of (16.4) represents the difference between the greatest and least of the principal values of the stress. Making the substitution (16.3) in (16.4) we obtain a2b2p = k, (16.5) (bZ - a2)r2 as the explicit form of the yield condition for the circular pipe under elastic deformation. If the internal pressure p is continually increased from a value for which the pipe is entirely in a state of elastic equilibrium, we see immediately from (16.5) that yield will begin on the inner surface of the pipe, i.e. for r = a. Hence we can state the following result. Yield will Jirst occur o n the inner surface of the pipe for a value of the internal pressure given by

p = (1 -

$) k.

(16.6)

17. PLASTIC DEFORMATION OF THE PIPE.

BOUNDARY CONDITIONS Suppose the internal pressure p is allowed to increase to such an extent beyond the value given by (16.6) that the pipe will enter a state of complete plastic equilibrium. I n treating this problem we shall use the previous symbols u;and uij to denote the components of the deformation u and the components of the stress tensor u in the plastically deformed pipe. It is assumed that the components ui of the plastic deformation and the stress components uii satisfy the general symmetry and equilibrium conditions in Sect. 15. To secure the utmost mathematical simplification in the following discussion we assume the pipe to be composed of rigid-plastic material, subject to the Tresca yield condition; hence the above deformation u and the stress tensor u satisfy the equations (9.10) in Chap. IV, i.e. the equations q'iui,, = 0, equation of incompressibility, (17.1) eij = \I.&, Hencky stress-strain equations, (17.2) r1 r3 = 2k, Tresca yield condition, (17.3) relative to the cylindrical system, where the comma in (17.1) denotes

-

17.

PLASTIC DEFORMATION OF PIPE

247

covariant differentiation, the proportionality factor 9 in (17.2) is positive by hypothesis, and the quantities n and 73 in (17.3) are the greatest and least of the principal values of the stress tensor in the plastically deformed pipe; the components of the tensors e and u* in (17.2) are defined explicitly by the equations eij

= 3(ui,j

+

Uj,;);

u;j

=

aij

- +(g"bcoa)gij.

As stated in Sect. 15 it follows from symmetry that a plane section of the pipe, perpendicular to the central axis, must be displaced into a plane which is likewise perpendicular to this axis. I n particular, the flat ends of the pipe will remain flat, i.e. perpendicular to the central axis of the pipe. Hence, since the components u;j depend a t most on the coordinate r by (15.5), the condition that there is no load on the ends of the pipe can be expressed as (17.4)

where the integration is over a plane section of the pipe perpendicular to the central axis. Or, in place of (17.4), we can write more simply

1."

1u33dr =

(17.5)

0,

where a and b are the inner and outer radii of the pipe in its strained position. Beyond this condition there are the boundary conditions given by (15.20) and (15.21) which will enter into the discussion of this problem. Using (15.6) and the definition of the quantities e;, we can write el1

e22

=

=

du1 -;

dr

rul;

e12 = 0 ; eZ3= 0;

e13

= 0,

era =

I

z'

du3

(17.6)

Similarly from (15.5) and the definition of the u; we have

(17.7)

248

VI. INSTABILITY AND FRACTURE

The condition (17.1) can be written dul UI dug -+-+--0, dr r dz

(17.8)

on account of (15.9). But the last term in the left member of this equation can depend at most on the coordinate z while the first two terms depend a t most on the coordinate r ; hence (17.8) must decompose into the following two relations

dul+: dr

=2

~ ;

dz

=

-2M,

(17.9)

where M is a constant. Integrating (17.9) we obtain UI=

Mr

+ N--;

u3 = -2Mz,

(17.10)

in which N is an additional constant. The constant of integration in the second relation (17.10) has been taken equal to zero, as was done in Sect. 15, which is possible by a proper choice of the origin of the cylindrical coordinate system. For i,j = 1,2 and i,j = 2,3 the equations (17.2) are seen to be satisfied identically. However for i,j = 1,3 we find that (17.2) yields 413

(17.11)

= 0.

If we multiply (17.2) by gifandsum on the repeated indices, the resulting equation is satisfied identically on account of (17.1). Hence we may omit the equation corresponding to i,j = 3,3 and consider only the equations obtained by taking i,j = 1,l and i,j = 2,2 in (17.2). But, when we take account of (17.10) as well as (17.6) and (17.7), we then have M

-r

=

$ ( 2 0 , ~- %5r2 -

Q,>.

(17.12) (17.13)

Subtracting corresponding members of (17.12) and (17.13)) we obtain (17.14)

Let us now assume that the Tresca condition (17.3), which by hy-

17.

PLASTIC DEFORMATION OF PIPE

249

pothesis is satisfied in the plastically deformed pipe, can be represented by writing

azz - 411 = 2k. T2

(17.15)

After the stress components uij have been determined it will be verified that (17.15) is equivalent to the condition (17.3) in which T~ and T~ are the greatest and least principal values of the stress (see Sect. 18). Hence from (17.14) and (17.15) we have

#

=

N --kr2

(17.16)

The constant N is therefore positive since k > 0 and # is positive by hypothesis. Expanding the equilibrium equations (15.2) we obtain (17.17) (17.18) (17.19) But from (15.7) and (17.11) we observe that (17.18) and (17.19) are satisfied identically. However, the equation (17.17) gives us

Or, we have (17.20) in view of (17.15). Integrating (17.20) we obtain ull = 2 k l o g r const. To evaluate the constant in this relation we need only avail ourselves of the boundary condition (15.21). We thus find

+

(17.21) Then, combining this equation with (17.15), we have Qzz = t 2

2k (1

+ log

a>.

(17.22)

250

VX. INSTABILITY

AND FRACTURE

Moreover we obtain the relation (17.23) when we eliminate the quantity $ from (17.12) by the substitution (17.16) and the stress components ull and uZ2by the substitutions (17.21) and (17.22). Equations (17.21), (17.22) and (17.23) provide us with a determination of the components ull, uZ2and m3of the stress tensor ; the remaining components ulz, u13 and of this tensor vanish on account of (15.5) and (17.11). Utilization of the boundary condition (15.20) leads to an equation for the internal pressure p in the plastically deformed pipe. Thus from (15.20) and (17.21) we obtain p

=

b 2klog-. a

(17.24)

Finally, the condition (17.5) becomes (17.25) on account of (17.23). Performing the various integrations indicated in (17.25) and reducing the resulting equation, we find that a2

b = 3M log - (b4 - a4). a 4N

(17.26)

If the plastic deformations which enter into this problem are sufficiently small for us to identify the radii a and b in the above relations with the known values of the corresponding radii of the unstrained pipe, we see that the ratio M I N is determined by (17.26). Hence equations (17.21), (17.22) and (17.23) together with (15.5) and (17.11) give a complete determination of the stress in the pipe. However, only the ratio of the components ul and u3 of the plastic deformation is determined by (17.10). The failure to determine such deformations uniquely is one of the characteristic features of this theory of the plastic behavior of solids. We observed above that the constant N is positive. Hence, since b > a, it is seen from (17.26) that the constant M is also positive. I n view of the second equation (17.9) this fact permits us to state the following result. There is a contraction of the plastic matem'al in the pipe in

18.

251

TENTATIVE YIELD CONDITION

the direction of its central axis. This result will have an important bearing in our discussion of the fracture problem in Sect. 20. 18. JUSTIFICATION OF THE TENTATIVE YIELDCONDITION

411

-7 0

422

0

0

0

- 7r2

0

0

433

=

0,

-7

But it is obvious that if (17.15) is to be a consequence of (17.3) we must take 71

=

422 . r2,

72

= 433;

73

=

411.

(18.1)

Our problem is therefore to show that if the principal stresses T ~ ,7 2 and 73 are selected in accordance with (18.1) the inequalities (16.2) will actually be satisfied, or, in other words, that

for the components ul1,4 2 2 and u33determined in Sect. 17. Substituting the values of the components ull,m2 and m3 given by (17.21), (17.22) and (17.23) the above inequalities become (18.2)

Since it was shown in Sect. 17 that M and N are both positive the first inequality (18.2) is automatically satisfied. Now the second inequality (18.2) will be satisfied for all required values of r, i.e. for a 5 r 6 h, if, and only if, we have M 1 - g -. (18.3) N 3b2

252

VI. INSTABILITY AND FRACTURE

But, eliminating the ratio M / N by the substitution (17.26), the inequality (18.3) can be written log-b 6 -1 b2 (18.4) a 4 a2 To show that (18.4) is a valid inequality for b > a let us put x = b / a so that the inequality becomes

(- $).

logx 6 4J ( x 2 -

$),

x 2 1.

(18.5)

Now consider the curves Cl and Cz which are the graphs respectively of the functions appearing in the left and right members of the inequality (18.5). Denote by a1the slope of the curve Cl and by a2the slope of the curve C2. Thus

The curves Cl and Czpass through the same point and have the same slope for 2 = 1. Hence if we can show that a1 < a2 for x > 1 it will follow that curve Cl lies beneath curve C2 for values of x > 1, or, in other words, that the strict inequality (18.5) is satisfied. But the condition

for x > 1 reduces immediately to the condition (x2 - 1)2 > 0 which is automatically satisfied. The principal values Tk given by (18.1) therefore satisfy the inequalities (16.2) and hence the Tresca yield condition is furnished by the equation (17.15) as assumed in Sect. 17. Remark. An interesting and also a shorter demonstration of the above inequality (18.5) was pointed out to me by Dr. T. W. Ting who observed that (18.5) could be written in the form

(18.6)

To prove (18.6) it suffices to show that l l ( x - - ; + 2- $ ) d x for x

5

=~[(&+&)’dx20,

1. But (18.7) is obviously a valid inequality.

(18.7)

19.

VIOLATION OF CIRCULAR SYMMETRY

253

19. PLASTIC FLOWAND VIOLATION OF CIRCULAR SYMMETRY

Denote by Z the intersection of the pipe with a half plane through its central axis. It will be shown, subject to the assumed symmetry and boundary conditions, that Z is a surface of strong instability in the plastically deformed pipe discussed in Sect. 17. In carrying out the demonstration of this result it will be convenient, although not necessary, to assume that the stress components aij are continuous over the plane 2 a t all times; accordingly this assumption will be made. Thus we can write I.[ 11 = 3.. r j - aij = 0, (19.1) where the are the stress components defined in Sect. 17 and the B;, are the stress components at points on the side of Z facing the region of plastic flow (see Sect. 1). From the conditions (19.1)) the equations for the aij in Sect. 17, and the equations defining the atr we now find that

(19.2)

where the bar denotes evaluation on the flow side of the plane X. With regard to the equations for the determination of the plastic flow we assume, in the first instance, that the Tresca yield condition (16.4) is satisfied; however no explicit use will be made of this full condition. Indeed, it will suffice for our purpose to observe that '22 -

l.2

- all

=

2k,

(19.3)

which follows from (17.15) and the above continuity assumption (19.1)In addition we assume the following equations for flow in rigid-plastic material, namely g%i.j

=

0,

equation of incompressibility,

(19.4)

254

VI. INSTABILITY AND FRACTURE eij

= #u:,,

gjkaij,k = p

stress-rate of strain relations,

(2+

equations of motion,

vi,pj),

(19.5) (19.6)

in which p denotes the density, the v i and v i are the covariant and contravariant components of velocity, the quantity # is a positive factor of proportionality and the comma represents covariant differentiation; corresponding to the definition of the components 6;in Sect. 17 we also recall the definition of the components eij of the rate of strain tensor, namely eij = $(vi,j

+

uj,r).

The existence of the plane L: separating the regions of equilibrium and plastic flow evidently permits a violation of the condition of circular symmetry used in the foregoing discussion. I n other words we are not justified in assuming, from symmetry, that the velocity and stress components in the region of plastic flow are independent of the angular coordinate 0. However, the stress components ai3and the two velocity components v1 and v 2 can be assumed independent of the longitudinal coordinate x from the symmetry of the problem. Taking the stress components utj to be functions of the form (19.7)

ail = aij(r,6,t),

in the region of plastic flow, let us now calculate the values on the plane 2 of certain of the components U r j , k of the covariant derivative of the stress which we shall need in the following discussion. In carrying out this calculation account must be taken of the values 5ij of the stress components which result from the continuity assumption (19.1). But these values are given by the equations 311

=

2k log -; r b

5 2 2 - 2k (1 r2

+ log

$1 (19.8)

from which we may infer the above relation (19.3). Hence from (19.7), (19.8) and the formula for covariant differentiation we find that

19.

VIOLATION OF CIRCULAR SYNMETRY

255

It will be assumed that 52 = 0 ;

av, ae

- = 0.

(19.10)

Actually the first of these boundary conditions follows directly from the general slip condition (1.1); the second condition (19.10) may be expressed by saying that the flow remains in planes 5 const. to a first approximation in the immediate neighborhood of 2. Thus the combination of the two conditions (19.10) can be described as a strong slip condition over the plane 2. Some further simplification can be made in our equations by the assumption that cross sections of the pipe in the region of plastic flow are displaced into cross sections in this region. This latter restriction on the type of flow obviously implies that the component 2’3 shall be independent of the coordinates T and 8 ; hence, in view of the above symmetry assumption, it follows that the components v i will be functions of the form

-

01 = Vl(T,e,t)t

v2

=

I

(19.11)

v2(r,~,t),

Va = v3(z$). By covariant differentiation of the velocity we find that

vz,l

=

0;

i ~ z= , ~ rcl;

-

vz,s = 0,

}

(19.12)

when account is taken of the relations (19.10) and (19.11). Hence

256

VI. INSTABILITY AND FRACTURE

from (19.12) and the formula for the components E+ More generally ib can readily be observed that the quantities e13 and eZ3 vanish in the region of plastic flow in consequence of the relations (19.11). It follows 3 vanish in the flow from (19.5) that the components 413 and ~ 2 likewise region; hence the quantity i ~ ’ 3 2 . 2= 0 in the relations (19.9). This fact will have application in the following section. 20. FRACTURE OF CIRCULAR PIPESBY INTERNAL PRESSURE

We are now in a position to determine the nature of the plastic flow bordering the plane Z. First, expanding the relation (19.4), evaluating on Z, and using (19.12), we obtain airl

-

+ irl + air3

- - = 0. ar T a2 But the third term in this equation depends at most on the variables z and t while the other terms depend only on T and t from (19.11). Hence we must have

av, +- -

ac3

-

= 2Q(t); = -2Q(t), ar where Q is a function of t. Integrating (20.1) we find

(20.1)

(20.2) 53 = -2Q(t)~

+ R(t),

(20.3)

in which S and R are functions of the time t. Now evaluate the equations (19.5) a t points of the plane 2. Using (19.2) and (19.13) we see that these equations are satisfied identically for i , j = 1,3 and i,j = 2,3. For i,j = 1,2 we obtain air,- 0.

ae

Taking i,j = 1,l and i,j = 2,2 the equations (19.5) lead to the relations (20.4) (20.5)

20.

FRACTURE OF CIRCULAR PIPES

257

Finally, if we put i , j = 3,3 in (19.5) and make use of the second equation (20.1) we find that (20.6)

Let us us now eliminate IJ from (20.4) and (20.5) by the substitution (20.6). This gives (20.7) (20.8) Combining (20.2) and (20.8) we obtain the condition

It may be observed that (20.7) can be obtained by partial differentiation of (20.8) with respect to r, i.e. (20.7) is a consequence of (20.8). T h u s we see that, in addition to the equation (20.6) for $, the equations (19.4) and (19.5) provide us only w i t h the two independent relations (20.9) for the behavior of the components v1 and v 3 o n the plane 2. We now consider the equations of motion (19.6) a t points on 2. Observing that the values of the covariant components 5i are equal to the values of the corresponding contravariant components Vi on account of the first equation (19.10), we find that (19.6) gives

aall

ar

aa,, + 71 (as - 2kr)

=

p

(2+ cl z),

(20.10)

for i = 1, when use is made of the equations (19.9) and (19.12). Substituting the values of the all and the & given by the first equation (19.8) and the first equation (20.9), the above equation (20.10) becomes

% = p [ ( r 3 + - ) - +N(rr 3 dQ -E)Q2]. (20.11) ae M dt M2r Assuming that j is a constant in view of the condition of incompressibility (19.4) of the plastic material, the equation (20.11) is an equation

258

VI. INSTABILITY AND FRACTURE

for the determination of the partial derivative dalz/dO on the plane 2 , after the determination of the function Q(t) in the following discussion. Similarly, taking i = 2, the equation (19.6) reduces to

ae - 0, on the plane 2. Finally, for i

=

3, we obtain (20.12)

Substituting the value of V3 given by the second equation (20.9) into the equation (20.12), we have

(2-

2Q2)z = 2 1 -dR g - QR.

But, due t o the independence of the variables z and t, this equation reduces to the following two differential equations for the determination of the functions Q(t) and R ( t ) , namely (20.13) Integrating the first of these equations (20.13) and substituting the function Q(t) so obtained into the second equation (20.13), prior to its integration, we find -2Q

=

1 t - G’

-

0

R=- H t - G’

(20.14)

where G and H are constants independent of the time t. I n writing the equations (20.14) i t i s assumed that the plastic flow over the plane Z i s initiated at time t = 0 so that t 2 0 in the following discussion. Substituting the values of Q(t) and R ( t ) given by (20.14) the equations (20.9) become (20.15)

To determine the algebraic sign of the constant G in the equations (20.15) we assume that J,I > 0 as a boundary condition; actually this assumption, as is easily seen from (19.5), is equivalent to the assumption that there is a discontinuity in the coordinate derivatives of the velocity components V i over the plane 2. Hence we must have Q(t) > 0 from

20.

259

FRACTURE O F CIRCULAR PIPES

equation (20.6) since the constants M and N were shown to be positive in Sect. 17. Putting f = 0 in the first equation (20.14) it therefore follows that

G=-

’

(20.16) 2Q& O. Allowing the time t to increase from its initial value f = 0 it follows from the first equation (20.15) that iil +. +a as t + G (> 0); similarly we see from the second equation (20.15) that the component will, in general, become indehitely large as the time t approaches the value G. Hence 2 i s a surface of strong instability over which the pipe i s subject to fracture when the internal pressure p has the value given by the equation (17.24). The bursting of water pipes in winter is a well known example of this type of fracture (see Fig. 13).

FIG. 13

21. FRACTURE OF CLOSED CIRCULaR PIPES UNDER INTERNAL PRESSURE AND AXIALTENSION The problem of the fracture of closed circular pipes under combined internal pressure and axial tension has been investigated by Dr. Tsuan Wu Ting on the basis of the dynamical relations used in Sect. 15 to Sect. 20 for the case of the open pipe subjected only to internal pressure. We shall limit ourselves in this section to a brief presentation of some of the main results obtained and refer the reader for complete details to the original article by T. W. Ting, Fracture of closed circular pipes under internal pressure and axial tension, Jour. Math. and Mech., 9, 1960, pp. 821-867. Assuming that the internal pressure p and axial tension r are such that the pipe has attained a state of complete plastic equilibrium (see Sect. 9 of Chap. IV) it can be shown that the plastic deformations are given by the equations u1 =

Mr

+ N-;r,

u2 = 0 ;

uq = -2M.2,

260

VI. INSTABILITY AND FRACTURE

where M and N are constants. If this pressure p and tension T are maintained by the continued application of work upon the pipe, plastic flow and fracture may occur. Let us say that a fracture surface Z is of type 1 if it separates a region of plastic equilibrium from a region of plastic flow; a fracture surface Z will be said to be of type 2 if the surface is contained entirely within a region of plastic flow. If both types of fracture surfaces are possible for a given stress distribution, fracture over the surface of type 2 will require the expenditure of greater total work; hence we conclude that fracture over a surface of type 1 will have precedence over fracture along a surface of type 2. Denoting by Z1 the intersection of the pipe with a half plane through its central axis and by Z2 a plane perpendicular to the central axis, i.e. a cross section, the following result can now be stated. Fracture will occur over a plane 21 of type 1 or a plane 22 of type 2 according as the first or second of the following two inequalities i s satisfid, namely 3Ma2 3Ma2 -2 -1. -< -1, N N w h r e a i s the inner radius of the deformed pipe. Hence the fracture surface may change abruptly from a plane Z1 to a plane 22 when 3Ma2 - - -1. N

If we approximate the inner radius a and the outer radius b of the plastically deformed pipe by their initial values, denoted by a0 and bo respectively, it can be shown that the interior pressure p and axial tension T are given by equations of the form (21.1)

where k is the material contsant occurring in the Tresca yield condition (17.3); the functions g and h are simple elementary functions of their arguments although their exact determination depends upon which of three possible types of stress distribution prevails within the pipe. Putting J = 2k and considering the ratio M / N as a parameter in the above equations (21.1) we can plot the quantity p / J against the quantity T / J when the radii a and bo of the pipe are known. Thus, choosing a0 = .625" and bo = .725", we obtain the graph in Fig. 14. As the

20.

FRACTURE OF CIRCULAR PIPES

0

0. I

261

0.2

P/J

FIG.14

value of the parameter M / N increases the associated point on the curve moves in the direction indicated by the arrow. At points H and G the parameter M / N has the values

respectively. The portion FGH of the curve corresponds to values of p and r which produce fracture over a plane Z1 while fracture over a plane Zz is produced by values of p and r corresponding to points on the portion H K of the curve. In this connection it may be mentioned that the segment GH of the curve represents values of p and r which permit fracture either over a plane Z1 of type 1 or a plane Zz of type 2; we have 0.2

T/J

0 .I

0 0

0.1 P/J

FIG.15 Lessels and Gregor Ni-Cr-Mo steel

0.2 P/J

FIG.16 Thomsen and Dorn J-1 Mg. alloy

262

VI. INSTABILITY AND FRACTURE

P/J

P/J

FIG.17 Grassi and Cornet Gray cast iron

FIG.18 Marin and Kotalik Alcoa 24s-T

2

P /J FIG.19 Marin and Hu Aluminum alloy 14ST4

therefore assigned the segment GH to fracture over a plane ZIin accordance with the above criterion. For a perfectly plastic solid without strain hardening the pressure p and stress T which are just sufficient to produce complete plastic equilibrium in the pipe, i.e. which are given by the above equations (21.11, will be the maximum loads before fracture. The curves in Figs. 15-19

20.

FRACTURE OF CIRCULAR PIPES

263

show these theoretical loads and the dots the measured maximum loads before fracture from tests on pipes of isotropic, or nearly isotropic, material as assumed in the theory. See J . M. Lessells and c. W. MacGregor, Combined stress experiments on a nickel-chrome-molybdenum steel, Jour. Franklin Inst., 230, 1940, pp. 163-181; E. G. Thomsen and J. E. Dorn, The efect of combined stresses on the ductility and rupture strength of magnesium-alloy extrusions, Jour. Aero. Sci., 11, 1944, pp. 125-136; R. C. Grassi and I. Cornet, Fracture of gray cast iron under biaxial stress, Jour. Appl. Mech., 71, 1949, pp. 179-182; J. Marin and B. J. Kotalik, Plastic biaxial stress-strain relations for Alcoa 24S-T subjected to variable-stress ratio, Jour. App. Mech., 72, 1950, pp. 372-376; and J. Marin and L. W. Hu, Plastic stress-strain for biaxial tension and variable stress ratio, Proc. of ASTM, 62, 1952, pp. 1098-1123.

Subject Index Absolute time derivatives, 53-55 Bulk modulus, 58

Decay of waves, 68, 69 Deformations, 1 elastic, 57, 78, 95 plastic, 73, 78, 95 Deformation tensor, 4 rate of, 8 Derivatives, absolute time, 53-55 covariant coordinate, 89, 90 covariant time, 83, 88, 90, 91, 93, 95 6 time, 41 Developable surface, 69 Dev.iators, definition of, 9 Dilation, 62 Discontinuities, 18, 39 conditions for, 19, 40, 42, 46, 48, 50, 52, 54, 56 slip, 184 Displacements, rigid, 2 Distortional work, 17 Distortion tensor, 5 rate of, 9 Dynamical compatibility conditions, 56

Canonical coordinates, 26 for plane stress, 140 Characteristic directions, 119, 121 surface elements, 120, 122 Characteristic surfaces, 116, 118, 123, 126, 132 conditions for, 118, 127, 130 Circular pipes, elastic deformation of, 240 fracture loads for, 259, 262 fracture surfaces in, 259, 260 plastic deformation of, 246, 259 Cloud effect, 161 Compatibility conditions, dynamicall 55 geometrical, 40, 46, 52 kinematical, 42, 48, 50, 54 Conservation of mass, 9 Constitutive equations, 82, 95 Continuity, equation of, 11 Coordinates, canonical, 26, 138, Elastic media, 57 140 homogeneous, 57 Eulerian, 2 isotropic, 57 Elastic moduli, 58 kinematically preferred, 85 Elastic-plastic solid, 81, 98 Lagrangian, 2 incompressible, 82, 98 Covariant derivatives, coordinate, 89, 90 Energy equation, 16 Equation of continuity, 11 time, 83, 88, 90, 91, 93, 95 Equation of incompressibility, 12 Crack propagation, 177, 186 265

266

SUBJECT INDEX

Equations of motion, 15 Equivoluminal waves, 62 velocity of, 62 Eulerian coordinates, 2

Load drop, 101 Longitudinal waves, 64 Luders bands, 100, 192 velocity of formation of, 196

Flat plates, fracture of, 200, 202 Fracture points, 174, 176 Fracture surfaces, general, 185 in circular pipes, 259, 260 in flat plates, 200, 202 in round bars, 228, 238, 240 Fundamental form, first, 37 second, 38

Mass, conservation of, 9 Material constants, 58, 74, 77, 81 functions, 73, 77, 80, 97 Maximum normal stress, 29, 239 surfaces of, 32 Maximum shearing stress, 32 surfaces of, 36, 223 Mean curvature, 38, 68 Mixed time and coordinate derivatives, 90, 92, 93

Gaussian curvature, 68 Geometrical compatibility conditions, 40, 46, 52 Hencky stress-strain equations, 97 Homogeneous elastic media, 57 I-Iooke’slaw, generalized, 57 Hydrostatic pressure, 17 Ideal flat bar, 102 Incompressibility, equation of, 12 Instability, surface of, 184 Invariant time derivative, 53 Irrotational waves, 62 velocity of, 62 Isotopic media, 57 Isotropic tensors, 58 Kinematical compatibility conditions, 42, 48, 50, 54 Kinematically preferred coordinates, 85 transformation of, 86, 87 Lagrangian coordinates, 2 Lam6 parameters, 58

Normal stress, 20 maximum, 29, 239 surfaces of, 32 Normal vector, unit, 11,38,51, 125 variation of, 43, 45, 55 Normal velocity, 41 Parallel surfaces, 45 waves as, 64 Perfectly plastic solids, 70 generalization of, 74 Plane stress, 138, 187 Plastic elongation, 105 Plastic equilibrium, 99 Plastic slip band, 106 inclination of, 108, 110 Poisson’s ratio, 58 Prandtl-Reuss equations, 81 generalized, 81, 95 Principal directions, 20, 24 covariant derivatives of, 29 Principal planes, 20 Principal stress, 20 Principal values, 24, 71, 76

SUBJECT INDEX

Rate of strain tensor, 8 transformation of, 92 Reduced systems, 112, 113 Replacement theorem, 93 Riemann space, 24, 26, 27 Rigid displacements, 2 Rigid motion, 3, 8 Rigid-plastic solids, 74, 98 Rotation, 62 Round bars under tension, conical fracture of, 214 plane fracture of, 208,213,219 Round bars under torsion, helicoidal fracture of, 238, 240 plane fracture of, 227 Scalar invariants, 24, 25 positive, 71 Shearing stress, 20 maximum, 32 surfaces of, 36, 223 Shear waves, 64, 178 Simple shear, 78 Simple tension, 78, 156 Singular surfaces, 42, 131,137, 149, 150, 155 Slip condition, strong, 255 Slip discontinuity, 184 Slip plane, inclination of, 191 Slip surface, definition of, 184 Speed of propagation, 141 Stable discontinuity, 161, 173 Stability, surface of, 184 Strain tensor, 4, 57 rate of, 8 Stress-rate of strain relations, 73, 76, 77 Stress-strain relations, 57, 97 Stress tensor, 15 vector, 102

267

Strong instability, surface of, 185 Time derivatives, absolute, 53-55 covariant, 83,88,90,91,93,95 6, 41 Tensor, deformation, 4 distortion, 5 isotropic, 58 rate of deformation, 8 rate of distortion, 9 rate of strain, 8 strain, 4 stress, 15 Unloading, 99, 100 Vectors, unit normal, 11, 38, 51 stress, 102 variation of, 43, 45, 55 Waves, decay of, 68, 69 equivoluminal, 62 longitudinal, 64 irrotational, 62 shear, 64,178 Wave strength, 61, 69, 173, 174, 175 equation for, 67, 68 Wave surfaces, 42, 61, 131, 132, 134, 137 in plane stress problem, 145, 149, 151, 153, 155, 183 Weak discontinuities, 155, 162 Weak instability, surface of, 185 Work, distortional, 17 Yield conditions, general, 73, 76, 77, 81, 97 Tresca, 77, 99 von Mises, 74, 99 Young's modulus, 58