Giuseppe Grioli ( E d.)
Propriet à di media e teoremi di confronto in fisica matematica Lectures given at the Centro Internazionale Matematico Estivo (C.I.M.E.), held in Bressanone (Bolzano), Italy, June 30- July 9, 1963
C.I.M.E. Foundation c/o Dipartimento di Matematica “U. Dini” Viale Morgagni n. 67/a 50134 Firenze Italy
[email protected]
ISBN 978-3-642-11017-7 e-ISBN: 978-3-642-11018-4 DOI:10.1007/978-3-642-11018-4 Springer Heidelberg Dordrecht London New York
©Springer-Verlag Berlin Heidelberg 2011 Reprint of the 1st ed. C.I.M.E., Ed. Cremonese, Roma, 1963 With kind permission of C.I.M.E.
Printed on acid-free paper
Springer.com
CENTRO INTERNATIONALE MATEMATICO ESTIVO (C.I.M.E)
Reprint of the 1st ed.- Bressanone, Italy, June 30-July 9, 1963
PRIOPRIETÀ DI MEDIA E TEOREMI DI CONFRONTO IN FISICA MATEMATICA
B. D. Coleman:
On global and local forms of the second law of thermodynamics ......................................................... 1
J. Serrin:
Comparison and averaging methods in mathematical physics ....................................................... 43
H. Ziegler:
Thermodynamic aspects of continuum mechanics ............... 133
C. Agostinelli:
Un teorema di media sul flusso di energia nel moto di un fluido di alta conduttività elettrica in cui si genera un campo magnetico........................................ 165 Su alcuni teoremi di media in magnetofluidodinamica nel caso stazionario............................................................... 171
D. Graffi:
Principi di minimo e variazionali nel campo elettromagnetico ................................................................... 181 Teoremi di reciprocità nei fenomeni non stazionari ............. 189
G. Grioli:
Proprietà generali di media nella meccanica dei continui e loro applicazioni ............................................ 201 Problemi di integrazione nella teoria dell’equilibrio elastico .......................................................... 217
CENTRO INTERNAZIONALE MATE MATICO ESTIVO (C.I.M.E.)
B. D. COLEMAN
ON GLOBAL AND LOCAL FORMS OF THE SECOND LAW OF THERMODYNAMICS
ROMA - Istituto Matematico dell'Universitl 1
Preface
The mathematical methods used here were set forth in the following two articles: (1) "Thermodynamics of elastic materials with heat conduction and viscosity", B. D. Coleman and W. Noll, Archive for Rational Mechanics
-
and Analysis 13, 167-178 (1963). (2) "Thermodynamics and departures from Fourier's Law of heat conduction", B. D. Coleman and V. J. Mizel, Archive for Rational Mecha-
-
nics and Analysis 13, 245-261 (1963). Parts of the present text have been taken, with alterations and elaborations, from (1). These lectures are concerned, however, mainly with some new research to be published shortly by B. D. Coleman and V. J. Mizel in an article entitled "Existence of caloric equations of state in thermodynamics".
3
- 2B. D. Coleman
Lecture I
f 1. Introduction The basic physical concepts of classical continuum mechanics are body, configuration of a body, and force system acting on a body. In a formal rational development of the subject, one first tries to state precisely what mathematical entities represent these physical concepts. In rough language, a body is regarded to be smooth manifold whose elements are the material points; a configuration is defined as a mapping of the body into a three-dimensional Euclidean space, and a force system is defined to be a vector-valued function defined for pairs of bodies.
~
Once these concepts are made precise one can
proceed to the statement of general principles, such as the principle of objectivity or the law of balance of linear momentum, and to the statement of specific constitutive assumptions, such as the assertion that a force system can be resolved into body forces with a mass density and contact forct's with a surface density, or the assertion that the contact forces at a material point depend on certain local properties of the configuration at the point. While the general principles are the same for all work in classical continuum mechanics, the constitutive assumptions vary with the application in mind and serve to define the material under consideration. When one has stated the mathematical nature of bodies, configurations and forces, and has laid down the ways in which these concepts occur in the general principles and the constitutive assumptions, then the properties of these concepts are fixed, and one can present rigorous arguments without recourse to "operational definitions" and other metaphysical paraphernalia, which may be of some use in deciding
'* For more extensive discussions of the foundations of continuum mechanics see references
(1] -
[4J .
5
- 3B. D. Coleman
on the applicability of a theory to a specific physical situation but seem to have no place in its mathematical development. Albeit the problem of the formulation of a detailed list of axioms for mechanics still has, even for the experts, some troublesome open questions, we can still assume in these lectures that we have sufficient familiarity with continuum mechanics to use the basic concepts and principles of the subject without continual reference to such a list. To discuss the thermodynamics of continua, it appears that to the concepts of continuum mechanics one must add five new basic concepts: these are temperature, specific internal energy*, specific entropy*' ~ heat flux, and heat suppl/**(due to radiation}. Once mechanics is axiomatized, it is easy to give the mathematical entities representing the thermodynamic concepts: temperature, specific internal energy, specific entropy and heat supply are scalar fields defined over the body, while heat flux is a vector field over the body. I believe that in presenting thermodynamics one should retain all the general priciples of mechanics but add to them two new principles: the first law of thermodynamics,
.
1. e.
~jU'if
the law of balance of energy
,and the second
law, which for continua takes the form of the Clausius·-Duhem inequalitl'" *~* Of course in thermodynamics one must make constitutive assumptions which involve some of the new variables which the subject introduces. The main
* Sometimes called "internal energy density".
_* Sometimes called "entropy density!'. -If*'* Sometimes called "denSity of absorbed radiation". jHU..
* Cf.
,~241 and 242 of (4) •
If/ftt.
* Cf.
,257 of [4J .
6
- 4B.D. Coleman
purpose of these lectures will be to examine the restrictions which the second law places on constitutive assumptions.
if
Generalizing some earlier work of Truesdell pin
(4J
[6] ,
Truesdell and Tou-
have formulated the following principle of equipresence: "a varia-
ble present as an independent variable in one constitutive equation should be so present in all". In other words, one should start a theory by assuming that all causes contribute to all effects. If one suspects a certain separation of effects one should not assume it a priori but should rather prove that general physical principles or assumed material symmetries require the separation. In their
quali~ ative
explanation of their original formulation of this
principle, Truesdell and Toupin emphasized the separation of effects due to the invariance requirements of material objectivity and symmetry. I at first found myself unable to believe in the usefulness of equipresence, but a study of t~e consequences of thermodynamics restrictions
[7J
on constitutive equa-
tions has changed my viewpoint. Here we shall use equipresence and assume that an independent variable present in one constitutive equation is so present in all, unless its presence is in direct contradiction with the assumed symmetry of the material, the principle of material objectivity or the laws of thermodynamics. One of the things which we shall do here is to show that it is possible to use equipresence to motivate the classical linear thery of viscous fluids with heat conduction, although a cursory examination of the constitutive equations of that theory can yield the specious conclusion that the theory does not allow every cause to contribute to every effect.
*' This concept of the structure of thermodynamics is explained in more detail in
[5J .
7
B. D. Coleman
On Notation
We shall use the direct, as distinguished from the component, tensor notation, dei10ting vectors and points in Euclidean space by boldface Latin minuscules and tensors by lightface Latin majuscules. Tensors of order higher than two will not occur. We shall denote the transpose of a tensor F by
FT. The tensor
Q will be said to be orthogonal if QQT =QTQ=I,
where I is a unit tensor. The symbol but
o
will always denote the zero vector,
0 may denote (ither the scalar zero or the zero tensor.
8
- 6B. D. Coleman
92.
Thermodynamic Processes
Consider a body consisting of material points
X. A thermodynamic
process for this body is described by eight functions of
X and the time t,
with physical interpretations as follows: (1) The spatial position
~
= X (X, t); here the function
X,
called the deformation function, desc ribes a motion of the body. (2) The symmetric stress tensor (3) The body force
b = b(X, t) ..........
T = T(X, t).
per unit mass (exerted on the bo-
dy by the external world). (4) The specific internal energy (5) The heat flux vector (6) The heat supply
E= €
(X, t).
!l, = !l,(X,t).
r = r(X, t)
per unit mass and unit time
(absorbed by the material and furnished by radiation from the external world). (7) The specific entropy (8) The local temperature always positive,
1=,
(X, t).
e = e(X, t)
,which is assumed to be
e > O.
We say that such a set of eight functions is a thermodynamic process
[5J
if the following two consevation laws'*' are satisfied not only for the body
but for each of its parts
~:
(A) The law of balance of linear momentum:
(2. 1)
*'
A thorough discussion of these conservation laws is given in -205, 240, 241. 9
(4J '
H196-
- 7-
B. D. Coleman (B) The law of balance of energy
(2.2)
1 d "'2 dt
)(b( ~• ~ dm + ~e dm = '0
(.
l'
(·
d\ (~.b + r)dm + ~~ (~.T~ - ~~)ds.
In (2.1) and (2.2) ,dm denotes the element of mass in the body, ~ @) the surface of
6) , ds the element of surface area in the configuration at time t ,and n the exterior unit normal vector to a~ in the configuration at time
-
t; a superimposed dot denotes the material time derivative, i.
e. the derivative with respect to t
keeping X fixed.
-
X( 6, t)
assumed to be such that the region,
~ and
't
-
are
, occupied by .~ is, for
each t , the closure of a bounded open connected set possessing a piecewise smooth surface. The assumed symmetry of the stress tensor
T
insures that the mo-
ment of momentum is automatically balanced. Couple stresses, body couples and other mechanical interactions not included in
-
T or b are assumed
to be absent.
Under suitable smoothness assumptions the balance equations (2.1) and (2.2) in integral form are equivalent to the following two balance equations in differential
form~ : div T -
(2.3)
e denotes the mass density; ,
-
L = grad x; tr if
x=-
tr \TLl - diva -
(2.4)
Here
t)
't ......
1\ b "",,'
~
ei
=-
er .
L is the velocity gradient, i. e.
is the trace operator; and the operators grad and div refer
See the sections of [4J cited above. 10
- 8 -
B. D. Coleman
to spatial derivatives, i. e. the gradient and divergence with respect to
~
keeping t fixed. We note that in order to define a thermodynamic process it suffices to
X
prescribe the six functions ctions
~
and
r
....
T,
J
e, a, ~,
and
• The
(J
remaining fun-
are then determined by (2.3) and (2.4).
It is often convenient to identify the material point
X
with its position
X in a fixed reference configuration R and to write ..... (2.5a)
'" The gradient
F
X(~,
of
with respect to
t)
....X
, i. e .
F = F(~, t) = 'VJ.(~, t) , '"
(2.5b)
is called the deformation gradient at X (i, e. at X) relative to t! le configuration
R
. It is well known that
,
(2.6)
= LF
F
We assume that
X(~,
t)
. 1.
.
0f
We consistently use the symbol
V
configuration
L
= F' F- 1
.
is always smoothly invertible in its first variaF,-l
bl e, i. e. t hat t he inverse
e.
F
exits, or, equivalently, that det F .l-T O.
to indicate a gradient in the reference
R ,i. e. a gradient computed taking
variable, whereas grad is used when the position
~
as the independent
....x in the present configu-
ration is taken as the independent variable. For a scalar field over as (} ,it is easily shown that
(2.7)
v
(J
= F T grad
(J
11
•
&J
,such
- 9B. D. Coleman
Since grad
e
occurs often in our subject, it is convenient to have a single
symbol for this vector. Let use the abbreviation
( =grad 9 .
(2.8) The mass density
e=
(2.9)
where
e is determined by
er
1 Idet FI
F
through the equation
Pr
is a poritive number, constant in time and equal to the mass
density in the reference configuration lue of the determinant of
F.
12
R
• and Idet
FI
is the absolute va,.
- 13 -
H. Ziegler
U = feM.dV ,
(3.5)
where
.»..
denotes the specific intrinsic energy, dependent on the mechani-
cal state of the element, i. e., on its deformation, and on the temperature. The influx of heat into the volume
V is
(3.6)
where the vector qk denotes the heat flux. Starting from (1. 2) and observing that, in a continuum, .the energy of an element is composed of its kinetic and intrinsic energies, we state the first fundamental theorem for the volume
V in the following form:
The material rate of increase of the sum of the kinetic and intrinsic energies in equal to the rate of work of the
exf~ior
forces plus the heat in-
flux. The analytical form of this statement is
(3.7)
On account of (3.4) and the symmetry of
(3.8)
where (3.9) 13
~kl
(3.7) reduces to
- 10 -
B. D. Coleman
Lecture II
93.
Admissible Processes and Constitutive Assumptions
We assume that the material at the point functions
,..
E(X)'
1(X)' 1\
which give l " , T, 9.
,..
1\
T (X)' ~(X) at
X is characterized by four
which we call response functions and
•
X when
6, {, F, F
are known at
X:
(3. 1)
(3.2)
.
"
(3.3)
T = T(X)(6, ~ F, F) ,
(3.4)
We say that a thermodynamic process in dynamic process
[5)
~
is an admissible thermo-
if it is compatible with the constitutive equations
(3.1)-(3.4) • In dealing with response functions it is often important to distinguish between them and their values. Here a symbol with a superimposed 1\,""', -, or = always denotes a function. Since, for a given process, the values of
•
F and F must depend on the choice of the reference configuration the response functions
"
,..
,..
,..
£ (X)' " (X)' T(X)' 9.(X)
will depend on
R R
, . As
the notation of (3.1)-(3.4) indicates, in general, these functions can also de15
- 11 B. D. Coleman
pend on the material point R
~
all
X. If there exists a reference configuration
of
n which makes £A" /I " '''J (X)' , (X)' T (X)' !l(X) independent of
X
in
~
,then we say that
~
R ff- is a homogeneous configuration of R tt of
~ ,then
e
X
for
is materially homogeneous and that
p.>
;if there is no such configuration
is materially inhomogeneous. For ease in writing,
we shall drop the subscript
(X)
on response functions; however, all the ar-
guments we shall give here are valid equally for materially homogeneous and materially inhomogeneous bodies. In an admissible thermodynamic process, the arguments
and the values £. , "
T,!l
of the response functions
I.)
"
9,~..
,. 1\
",,,, T, q
course, depend on the time t . We assume that the functions
F, F•
will, of
" "T, !l i\ £, "I,
f!\
are themselves independent of t. The constitutive equations considered here are not the most general imaginable; for example, they do not allow for all the long range memory effects covered in the purely mechanical theory of simple materials*'. ()ur assumptions are, however, sufficiently general to cover many applications; in particular, they include as special cases the constitutive equations of the classical theories of thermoelastic phenomena and the hydrodynamics of viscous fluids with heat conduction. In contradistinction to the usual presentations of these classical theories, we here, in Eqs. (3.1)-(3.4), start with constitutive assumptions that are compatible with the principle of equipresence. We do not lay down constitutive equations for body force density ..... band the heat supply
r
due to absorbed radiation. The quantities
~
and
r
are regarded as assignable; they can be assigned any values compatible with the balance equations (2.3) and (2.4) . Let
*' Cf. [2] &. [8]
. 16
us
elaborate on the physical
- 12 -
B. D. Coleman
significance of this assumption. Let
~
X be a material point in
. In
the present theory we are following standard procedure and are ignoring mutual body forces and self-radiation within depend not only on the "local state"
~
. Here band
• (8.~.' F. F)
at
band ....
r
~
so that rand
X
. Our mathemati-
are assignable has the physical meaning
that we suppose that for each local state at outside of
at
X but also on the "ex-
ternal world". i. e. on the state of regions outside of ~ cal assumption that
r
b....
X one can adjust the conditions
take on arbitrary values compatible
with balance of momentum and energy. That an experimenter might prefer to fix the outside conJitions and thus lose freedom in assigning thermodynamic fields should not affect our proofs: the theorist can consider processes which the experimenter finds difficult to realize. provided only that they are not impossible to realize. We assume that for allty fixed set of values of (. F. is smoothly invertible in its first variable
(3.5)
'Jt
F the function
8; i. e ••
•
~(8.(.F.F) +0.
This implies that there exist functions. 8J ~.
1'.
i
also called response.
functions. which can be used to rewrite (3.1)-(3.4) in the forms (3.6) (3.7)
£
8
.... =8
•
(L (. F. F)
#ow • , =, (£. [. F. F)
• = T ( €. (. F. F) fV
(3.8)
T
(3.9)
S. = s.( £. i. F. F)
•
17
- 13 B.D. Coleman
For each set of the quantities
~.'
"'.
the function
•
1\
inverse function of
.
F, F,
8 (., It, F, F)
is the
-
£ (. ,It, F, F) ,and '\ is defined by
(3. 10)
T
and
9.
are defined by formulae analogous to
To every choice of the deformation function distribution
8, as functions of
X and
admissible thermodynamic process in ~ are known for all
9.
and
are known,
-
rand b
Let Of (t) pendent vector;
(b
throughout
and the temperature
, there corresponds a unique
.
F, F,
. Once the
-
'X,
• For, when
8 . The constitutive equations
throughout
1 ' T,
and t ,clearly
X
t
(3. 10).
and
(X, t)
and 8(X, t)
8 are determined
(3.1)-(3.4) then determine fields
1.'
-
T, E ,
9.
I
and
E, 8
are determined by the balance laws (2.3) and (2.4).
be any time-dependent positive scalar; A(t)
-
t
~(t)
any time-de-
any time-dependent invertible tensor; and Y any ma-
terial pOint of ~ whose spatial position in the reference configuration R is
-
Y • We can always construct at least one admissible thermodynamic process
in
lues
~ such that 8(~, t) , It(~, t) , F(~, t) have, respectively, the va-
q(t),
~(t),
A(t)
at
~
=
y .An example of such a process is the one
determined by the following deformation function and temperature distribution: (3.l1a)
?t
=
X(X, t) = Y+·A(t) [~ - y] ,
-
(3.11b)
18
- 14 B. D. Coleman
i. e.,
(3.11b ' ) where
t =Xry, t) = Y .Thus,
-
cify ant only at a point
-
0, {
and
F
at a given time
t
. , .,
, we can arbitrarily spe~
but also their time derivatives 0, {, F, F, etc.
Y and be sure that there exists at least one admissible thermo-
dynamic process corresponding to this choice. Furthermore, it follows from this, (3.1) and (3.5) that
C, {, F,
•
•
•
If
and the time-derivatives £ ,{, F, F
also form a set of quantities which can be chosen independently at one fixed point and time.
19
- 15 -
B. D. Coleman
Lecture III
§4.
The Clausius-Duhem Inequality and Its Consequences
We regard
s./9
to be the vectorial flux of entropy due to heat flow and
r/9 to be a scalar supply of entropy from radiation. In other words, for each process we define the rate of production of entropy in the part
S
to
be
r · Jcr, '1
(4.1)
where
:1
dm
dm
';) @.,
of
d;
dm +
~
is the element of mass in ~
to the surface
-1 ; f. -
,n
~,
1 - q I nds 9 - -
the exterior unit normal
,and ds the element of surface area in the
configuration at time t. Under appropriate smoothness assumptions we can write
r
(4.2)
where
r
(4.3)
,
= , - r/9 +
e
-1
div s./9
is the specific rate of production of entropy. One way
(5]
of giving the Second Law of Thermodynamics a precise
matheinatical meaning is to lay down the following postulate. 21
- 16 -
B. D. Coleman Postulate: For every admissible thermodynamic process in a body, the following inequality must hold for all
t
and all parts
(B
of the body:
r ~O.
(4.4)
The inequality (4.4) is called the Clausius-Duhem inequality. Our postulate places restrictions on constitutive equations of the type (3.1)-(3.4). ( or (3.6)-(3.
9U . We now attempt to find necessary and sfficient set of such
restrictions.
~
In order that (4.4) holds for all parts
of a body, it is necessary
and sufficient that
t
(4.5) at all material points
X
~O
of the body.
For each thermodynamic process, the energy balance equatilm (2.2) permits us to rewrite (4.3) as follows
r
(4.6)
•
=, -
• E.
"8 +
In an admissible process !i and
"l•
1
and
[oJ
tr 1TL
-
ee 2 1
!i. (
T must be given by (3.8) and (3.9),
must be given by
, ='t '" •
(4.7)
where
ee
~E
is the (scalar)
•
va~ue
+
'Vi { + tr I~FF 1+ tr \'FF} , f)~
of
~
;
"I{
is the (vector)value of
the gradient of the function ." with respect to its second variable (; while
'!I F
and ~F
are, respectively, the (tensor) values of gradients of 22
- 17 -
B. D. Coleman
"
with respect to its third variable
F
and its fourth variable
•
F
. It fol-
lows from (4.6), (4.7), and (2.5) that
+ tr
{'0I' -I} e TFF
1 02
-
.9. [
On looking at (4.8), (3.6)-(3.9), and (2.7) we see that f
the values of the seven quantities,
f. ,[, F ,
•
,
r
depends on only
,.
e,[ , F ,F
at
X
and t .
According to the remarks made at the end of Section 3 ,these seven quantities can be independently and arbitrarily chosen at
X
and t ,and there
will always exist an admissible thermodynamic process corresponding to the choice. Our postulate (4.4) is equivalent to the assertion that
r
be
~O
for all such choices. To find the necessary conditions for the validity of our postulate first observe that (4.8) can be written in the form
. .. -
(4.9)
If we assign
F, ", F, F)
f., [,
..
F, £, F, F
,
any fixed values,
will be fixed at some finite value, say
a
f( E. [, F, Eo, F, F)
,and the postulate will require
that
"l[ (£,
(4.10)
•
for all values of [
•
[, F, F), [
+ a }O
. But clearly this is possible only if 23
- 18 B. D. Coleman
(4.11 )
,
r ( ,",
[
I
po
F, F) = Q
1<.'-
Futhermore, this equation must hold for all values of
"l.
E, [,
•
F ,F
;i. e.,
in (3.7) cannot depend on [ . It follows from (4. 11) that (4. 8) can be written in the form
(4.12)
Hence for any fixed values of
t. , [,
•
f , F, F,
the postulate (4.4) requi-
res that (4.13)
where b is a finite number. The inequality (4.13) can hold for all choices of
.
F
only if
•
(4.14)
~F(£' F, F)
=0
where 0 is the zero tensor. Since (4.14) must hold for all values of
F,
and
,
. F , •
but also
F
ction
of
E,
we have proved that our postulate requires that not only [ , ,must drop out of (3.7) ,i. e. that ,
E.
and
F
Of course, the function
"
must be given by a fun-
alone:
(4.15) "-
depends on the point
It follows from what we have done so far that
24
X under consideration.
- 19 -
B. D. Coleman
(4. 16)
and a now familiar argument yields the result that if (4.4) is to hold for all
•
admissible processes, then the coefficient of
l. in (4.16) must be zero
,
for each choice of f , {, F, F • Thus, the postulate implies that 9 must
-
also be given by a function
9 of
(4.17) and that
-
f
and
alone,
F
9 = 9(£, F) ;
-
must be related to the function
9
"
through the temperature re-
lation : (4.18)
9(£,F)
' t h e equl'l'b ' 1 rlum s t ress glve
Let
E
= [~f( t,F)]-1 •
and
T(O)
correspond'109 t 0
F, i. e., the stress when the temperature gradient {
velocity gradient
L = FF-l -(0)
(4.19)
T
The value
T(e)
(4.20)
-(e)
T
is called the
and the
both vanish : _(i , F) - T( l, 0, F, 0).
of the function
-
T(e) defined by
, _• -(0) (E,{,F,F)-T(',,i,F,F)-T (i,F)
extra~tress;
it is the contribution to T "caused" by the gra-
dients of temperature and velocity. Using (4.15), (4.17), (4.18), and (4.20) we can write (4.8) in the form: 25
- 20 H.
•
- i (f, I!!
F)FF-1--'''-1 O( f , F)
!).I ,J\lli:Hli
F,
In writing (4.21) we have made use of the fact that the trac(, i!> ;; ction with the property that
'" T
.
is continuous in its fourth variable
'" T(f, [, F,
(4.22)
where
•
~
f
tr {TFF-1} = tr F- 1TF
ct
F
at
.
F =0
J.
'il1\.I:
ASl>tlll: Illg'
fu!:-
~h:tt
,the foji,J\' illL: l/l,ids:
-
F) = '" T(l, [, F, 0) + 0(1) ,
is a real number and
o( 1)
is such that for fixed '.
[, F, F,
~OO(1)
oI~
=0 .
It follows from (4.22) and the definitions (4.19), (4.20) that
(4.23 )
"'(e)
T
In (4. 21) let us now put
• _
(£,0, ..... F, c{F) - 0(1) [=
Q.
and replace
.
•
F
by
ci F•
(4.23), that our postulate requires that
+ o(e{)) 0 , where 26
'Vv-{
J'
1",
,~s;~·:
.
- 21 B. D. Coleman
~o o(ot)/q
c(..,
Equation (4.24) must hold for all values of
=0
.
E,
F, F, and
dering the behavior of (4.24) for small values of coefficient of
cf.
~
ct.
On consi-
,we conclude that the
must be zero; i. e., for each value of the pair
C,
F ,
we must have
•
for all values of F . but this is possible only if the stress-relation (4.26)
holds. Equations (4.18) and (4.26) tell us that the equilibrium stress defined by (4.19) is determined when the caloric equation of state (
(4.27)
e-9(f.,F)t = tr l-T
J-
.s
(e) (E,SJ F,F)L' -
It follows from (4.27) and our postulate that when
- (t'l-l<',F)'g. • _ 1 9 (i , F)
.& = ~
chanical dissipation inequality
(4.28)
and when (4:29)
L = 0 we have the heat conduction inequality
-a( f, (.
F, 0)' { ~ O. 27
we have the me-
- 22 -
B. D. Coleman
e9
We note, however, that in general a resolution of the inequality into a mechanical dissipation inequality valid for non-zero { conduction inequality valid for non-zero
L
if the following conditions were met: that {
and that
5l.' (
be independent of
r·~
0
and a heat
is not valid. It would be valid tr
L
t
J be independent of
T(e)L
These conditions do not follow
from the assumptions we have made so far, although they hold when one adds certain special assumptions of linearity and symmetry. It is clear, from (4.27), that if (4.15), (4.,18), and (4.26) are assuI.lled, the inequality 1
(4.30)
q. (
~
9 -
,
0
which we call the general dissipation inequality, is not only necessary, but also sufficient for our postulate. In writing (4.30) we have used the fact that since
T(e)
is symmetric,
symmetric part of
tr {T(e)L]
= tr {T(e)D
1 where
D is the
L.
We summarize in the following Theorem: Consider a body
c:B
and assume only that constitutive equations
of the general form (3.1) - (3.4) hold at each
X
b .Under this as-
in
sumption, a necessary and sufficient condition that the inequality (4.4) hold for all admissible thermodynamic processes in statements be true at each
X
~
is that the following
in ~ .
1. There exists a caloric equation of state (4. 15). II. The temperature is given by the relations (4.17) and (4.18). III.The equilibrium stress, defined in (4.19), is given by the stress-relation (4.26). IV. The functions
-(e)
T
of (4.20) and 28
-
q
of (3.9) obey the general dis-
- 23 -
B. D. Coleman
sipation inequality (4.30) for all values of
E ,F, grad 9,
and
D.
The inequality (4.4) implies the mechanical dissipation inequality tr {T(e)Ll fIJ and the heat conduction inequality such as those shown in
a' {
~0
only in special circumstances,
(4.28) and (4.29) •
Addenda on Other Forms of the Temperature and Stress Relations
It follows from our assumption that
-
the function ,
-
is smoothly invertible in its first variable
there exists a functiC:l function of
'1.
9 > 0 and the relation (4.18) that
and
F
£
(depending on
at
X:
X ) giving
£ . Hence,
E
at
X as a
(4.31)
Using this function, one can rewrite (4.17) and (4.18) in the form
(4.32)
Equation (4.31) can also be used to express
1 ' {,
F
and
,
F:
(4.33)
--
T
and
as functions of
.
.,
T=T(&(F,'1),{, F, F)='r(1' {, F,' F)
(4.34)
and to express
~
T
(0)
"l
as a function of
29
and
F
- 24 B. D. Coleman
Using the chain rule we can cast (4.26) into the simpler form
(4.36)
-
It follows from our assumptions (3.1) and (3.5) that the function 8 of (4.17) is smoothly invertible in a function
•£,
of
8 and
(4.37)
F
at
i '
i.e., that
at
X
is given by
X:
t. = l
(8, F);
and, by (4.15) we have (4.38)
E
,., -= ., ="1 (E.(8,F),F)
=
-..
'Yl
(8, F)
Using (4.37) and (4.38) we can define a (Helmholtz) free-energy function c in the usual way:
r
(4.39)
and it is not difficult to show that (4.18) and (4.26) [or, equivalently, (4.32) and (4.36)] yield
(4.40)
(4.41)
30
- 25 B. D. Coleman Lecture IV
§5.
Objectivity, Fluids
The discussion of the present section will not require a separate notation for response functions and their values. The theorem of Section 4 tells us that the second law of thermodynamics implies that constitutive equations of the type (3.1)-(3.4) reduce to the following equations (S.la)
E
(5. 1b)
8 = 8(" F)
(5.lc)
T =
(5. ld)
9-
=
E.
(~, F)
T(O)(~,
F) + T(e)("
~/
= a(~, (. F, L)
In writing (S.lc) and (5.ld) we have used the fact that have chosen
1
F, L)
L
• -1 = FF
and we
as an independent variable, in accord with (4.31)-(4.35),
to obtain simpler formulae. The reduced constitutive eq'uatibns (5. 1) mUst obey the principle of material objectivity (2], [3] . In r.ough language, this principle states that an admissible process remains admissible after. a change of frame (or oQserver). We now: consider the limitations placed on the Eqs. (5.1) by objectivity. The results will be intuitively plausible, and since the rigorous arguments which lead to them differ only in minor details from related proofs which in the last ten years have been frequently given in continuum ... Cf. (2) , [7] ,
[8] , e.g] , (10) . 31
phySiCS~ I give here only
- 26 -
B.D. Coleman
a descriptive outline of a method which is in essence that used by Noll
C2J '
(8] in different contexts. A change of frame is defined by a time-dependent orthogonal tensor Q. The scalars
,
F, L, T
"l ,E.,
and (} are unaffected by a change of frame, but F,
and { = grad (} F
,
F
-+
transform as follows : QF
. ,.
~ QF=QF+QF
(5.2)
The tensor
- T is always skew. QQ
The constitutive equations (5. 1) are compatible with material objectivity if and only if the functions in (5. 1) obey the following identities for each orthogonal tensor
!l..
and
Q
,each skew tensor
L:
32
W , and all values of f
, F.
- 27 B. D.Coleman
f (1, F)
= E (~, QF)
8(~, F) = 8(1' QF)
QT(O)("
(5.3) QT
(e)
('I' {,
F,L)Q
F)QT
= T(O)(~, QF)
T _ (e) - T ("
Q!l.(~' {, F, L) =
.s (1'
Q{, QF, QLQ
Q{, QF, QLQ
T
T
+ W)
+ W)
These identities can be used to derive the following reduced forms:
(5.4a) (5.4b)
(5.4c)
(5.4d)
Here
C = F TF
is the left Cauchy-Green tensor; D=
the stretching tensor; and gradient of
r:; 8
8 with respect to
1-
,which is related to {
(L + L T)
is again
by (2.,6) ,is the
X. A material obeying (5.4) automatically
obeys (5.1) and the identities (5.3) . In other words, the existence of the reduced forms (5.4) is not only a necessary, but also a sufficient, condition that the material under consideration be compatible with the principle of material objectivity. The restrictions on the equations (5. 1) caused by the symmetries which a material might possess will not be discussed here. A mathematical defini33
- 28 B.D. Coleman
tion of material symmetry (i. e. the "isotropy group") in materials obeying N-
only slightly less general constitutive equations is given in reference [7J • We say that a material obeying (5.1) is a fluid if the tensor in (5.1) only through
I det F I
F
occurs
,or, by (2.8) ,only through the specific vo-
l ume V (= 1/! ) . It is not difficult to show that Eqs. (5. 3) imply that for a fluid the Eqs. (5.1) must reduce to
(5.5a) (5.5b)
(5. 5c)
(5.5d)
where
T(o)
is a hydrostatic pressure,
(5.6)
and
T(e)
and
!l are, for fixed
'1
and V- ,isotropic functions of
!l and D ,1. e., obey the following identities for every orthogonal tensor Q
(5.7a) (5.7b)
QT
(e)
("I' fl/If', D)Q
Qs.("1' [.
T _ (e) T - T (,,]' Qi, ", QDQ ),
'J", D)
= !l(',
T
Q[. "\7, QDQ ).
.,. The definition given there is, in turn, analogous to that given in discussed in detail for elastic materials in [10J . 34
C2J
and
- 29 B. D. Coleman
It follows from the definition (4.20) that
(5.8)
T
(e)
(',
2.>1r,
0) = O.
For a fluid, Eq. (4.36) reduces to the following familiar expression for the equilibrium pressure function p in (5. 6)
(5.9a) and (4.32) becomes (5.9b)
Let us return to the identities (5.7) . Representation theorems for such. tensor-valued and vector-valued isotropic functions exist
*' ,but there is no
need for us to state them in full generality here. Some special cOllsequences of the identities of (5.7) may, however, be of interest. If in (5.7) we put Q = -I , then we obtain the identities
(5. lOa)
(5. lOb)
*'For (5.7b) one can use directly the representation theorem for isotropic vector-valued functions of a vector and a symmetric tensor, proved by Pipkin and Rivlin [l1J in a different context. In (5.7a) one can replace l by l and then use the representation theorems of Rivlin and Ericksen [12) ·for symmetric-tensor-valued functions of two symmetric tensors.
®[
35
- 30 B. D. Coleman
Thus, for any fixed values of ction, and ..3
'1
,v-;
an odd function, of
D, T(e)
must be an even fun-
In particular, we have
~
!i(' , ..2, tr ,
(5.11)
and
D)
=0 ;
i. e. in a fluid, regardless of the motion, there can be no heat flux under
11- . ' zero temperature gra dlent We now assume that differentiable at T(e)
and
D
= 0,
T(e) [
=..2
and
!i
,as functions of
Q
and
~
• are
and consider approximation formulae for
S. for small D and [ . Since D and { are independently
variable, and of different dimension, there is no intrinsecally preferred way of making precise the concept of a "first-order term in D and gil. It appears to me that the physicists I usual concept of a "linearized theory" corresponds to considering the space of vectors
I
D9 {
,using the "natural" norm
1/ of that space,
(5.12)
and saying that an approximation is "complete to first-order" if it includes all terms
~0
dl D(±)[ /I).
Now, using smoothness and known representation
theorems for isotropic functions one can prove that Eqs. (5.7) imply
(5.13a)
110 The present argument can be generalized to yield the same result for any
material with the central inversion, -I, in its symmetry group. Note that this argument does not use the general dissipation inequality (4.30); cf. (5) and [7J . 36
- 31 B. D. Coleman
(5.13b)
The scalars
A 'f'
and k in (5. 13) are functions of ,
8 and V' ) alone. We notice that to within terms is independent of I{,
and
!i is independent of
od\ D $
and "Ir (or
gil)
,T(e)
D . This is an obvious
consequence of Eqs. (5.10). Since (5.13) holds if (5.12) be replaced by
(5.12)'
where ~ 1
and ~ 2 are any two positive constants, the Eqs. (5.13) are
invariant under changes of units, albeit they do not appear so at first glan-
ceo The constitutive equations of the classical linear theory of viscous fluids with heat conduction are (5. Sa)
( I) (5.6) and (5.9a)
(II)
-p(~
,V-)1 ,
(5.5b) and (5.9b)
(III)
and the equations obtained by striking out the terms (5.13) : 37
o(
\\D e I{ \\)
in Eqs.
- 32 -
B. O. Coleman
T(e) =
(IV)
2fD + A(tr
0)1 ,
(V)
When terms
o(
l,00glll
are omitted from Eqs.(5.15) the general
dissipation inequality (4.30) requires that both mechanical dissipation inequality (5.14a) and the heat conductiJn inequality (5.14b)
hold. Given that T(e)
has the form (IV), a necessary and sufficient condi-
tion that (5.14a) hold for all
)4- ~ 0,
(VI)
Of course, when
a
L
is that both
and
3
A+ 2/ ~ 0
.
has the form (V), (5. 14b) holds if and only if
(VII)
k ~ 0
The inequalities (VI) and (VII) are just as basic to classical fluid dynamics as the equations (I)-(V). Thus using general physical principles and starting from our constitutive equations (3.1 )-(3.4) which reflect equipresence, we can derive the constitutive assumptions of classical fluid dynamics by adding only two specializing assertions: (1) that (3.1)-(3.4) do describe a fluid, i. e., that 38
F
enters
- 33 -
B. D. Coleman
only through
I det F ~
in expressions for
• and (2) that terms
T and
q.
39
0("
De { II)
can be neglected
- 34 -
~
6. References
[1] Noll, W: The foundations of classical mechanics in the light of recent advances in continuum mechanics. In: The Axiomatic Method with Special Reference to Geometry and Physics. Pp. 266-281. Amsterdam: North Holland Co. 1959. [2] Noll, W: Arch. Rational Mech. Anal.
!,
197 (1958).
[3]
Noll, W.: La m~canique classique, bas6e sur un axiome d 1objectivit6. A paper read at the Colloque Internationale sur la M~thode Axiomatique in M6canique Classique et Moderne, Paris, 1959 (to be published by Gauthier- Villan, Paris).
(4)
Truesdell, C., & R. A. Toupin: The Classical Field Theories. In: Encyclopedia of Physics, Vol. III/I, edited by S. F1Ugge. Berlin-~ttingen Heidelberg: Springer 1960.
[5] Coleman, B.D.,& W.Noll:Arch.RationalMech.Anal.
!l..
167 (1963).
[6] Truesdell, C.: J. Math. pures Appl.l2., 111(1951). [7] Coleman,B.D.,& V.J.Mizel: Arch. Rational Mech.Anal. .!~,245 (1963).
(8J Green, (9]
A.E.,& R.S.Rivlin: Arch. Rational Mech.Anal.
!.'
1(1957).
Noll,. W.: J.Rational Mech. Anal. !" 3 (1955).
~O] Coleman, B. D., & W. Noll: Material symmetry and thermostatic inequalities in finite elastic deformations, Arch. Rational Mech. Anal. (in press).
~ 1] Pipkin, A. C., & R. S. Rivlin: J. Math. Phys. !.' 127 (1960). (12] Rivlin, R. S., & J. L. Ericksen: J. Rational Mech. Anal.!"
41
323 (1955).
CENTRO INTERNAZIONALE MATEMATICO ESTIVO ( C.I.M.E. )
J.SERRIN
COMPARISON AND AVERAGING METHODS IN MATHEMATICAL PHYSICS
ROMA - Istituto Matematico dell'Dniversita
43
COMPARISON AND AVERAGING METHODS L'J MATHEMATICAL PHYSICS by .JAi\IES
SEi{H~~~
It may be worth saying a few words about the general subject of the lectu-
res before beginning with the actual work. I understand that methods of mathematical physics is a subject far too large for anyone person to encompass. To the mathematician, on one hand, the subject may mean that part of mathematics which is of immediate or probable value in the study of problems suggested by physics. To the physicist, on the other hand, mathematical physics certainly means the theoretical methods actually used to study the phelnomena of mechanics, heat, electromagnetism, field theory, and so forth. These points of view are not entirely separate, of course, but the present lack of communication between mathematician and physicist is sufficient evidence that there is a real difference of emphasis. I think that my point of view in these lectures will contain something of both sides, but will also be rather restricted in its scope. This specialization is essential if one wishes in a few days to come to grips with real problems. The physical side of the lectures will be confined almost completely to continuum mechanics, and I
even more specially to fluid mechanics,a subject in which I hope then' is considerable interest. The particular topic of comparison methods, which will occupy the first four or five lectures, is itself a large subject. In its most frequent meanin,l!:. the phrase "comparison methods" is used in connection with certain problems in the calculus of variations, and with techniques in partial differelltial equations involving application of the well-known maximum prinCiple. In both cases, the object is to derive inequalities relating quantities of primary physical or mathematical Significance. Now to study compari>wn methods in the calculus of variations would easily require a conferellce ill Therefore, in these lectures we shall concentrate on the comparison
;t~elf.
111l't:
as it appears in connection with the maximum principle in partial differential equations. The particular problems selpcted for discussion ha'"t· been 45
,d
- 2J. Serrin
chosen for their physical interest and in order to exemplify main ideas. Although there are many other problems of importance, it is nevertheless hoped that the techniques illustrated here will be an adequate representation of the field. I will not comment here on the subject of averaging methods, for these will be discussed in later lectures.
1. THE MAXIMUM PRINCIPLE This is the generic name for a useful set of theorems in partial differential equations (and hence in mathematical physics) of which perhaps the simplest is the result that a solution of Laplace's equation which assumes an extreme value in the interior of its domain of definition must be a constant Consider more generally elliptic partial differential equations of second order, of the form Lu = a'k u'k + b, u, = f(x). 1
1
1
1
Here the coefficients a ik , b i are bounded functions of
x
= (xl'
... ,
xnt
in some domain D of n dimensional Euclidean space, and we have used thE; abbreviations u
= u(x)
,
~u
u. = - 1 r;) Xi '
as well as the summation convention. The ellipticity of the operator L is expressed by the condition (m> 0
for xED and all real vectors
Y= (J l' ... ,
~ n)' Under these conditions,
we have the following results, due to Eberhard Hopf. In three dimensions we shall frequently write x, y, z instead of xl' x 2' x3' 46
- 3J. Serrin
THEOREM (Boundary point lemma). Let
S, be an ·open sphere
in D. Let u = u(x) be twice continuously differentiable (class C2) in S , and continuously differentiable (class C1) in
t 1 ,where
S+ p
P is
a point on ~ S . Assume that Lu ~ 0 u
Then dU/~ n
< u(P)
< 0 at P, where rt is any direction into S at P •
The proof is based on a comparison argument, and goes as follows (cf. Courant-Hilbert, Vol. II, pp. 327-38). Let K be an open sphere internally tangent to S at P (see figure). Then P is the only maximum point of u in K. the closure of K. Let the center of K be chosen for the origin of coordinates, and let robe the radius of K . Construct a sphere with center P and radius less then r
,and let C deo nots the intersection of this sphere with K, (shaded in the figure).
We now introduce the auxiliary comparison function
h
- of r =e
which is positive in shows that for
2 - e
- ~r
2 0
K and vanishes on its boundary. An easy calculatill!!
c( sufficient large and x e C
47
- 4J. Serrin
e
a(
r2
~
,
Lh = 4~
4~
2
2
a .k x. xk - 2 Cl( (a.. + b. x.) 1 1 11 1 1
2 mr - 2 C( (a 11 .. + b. x.) 1 1
> O.
Now on the lower spherical boundary C1 of C the function u is bounded
E> 0
away from u(P). Hence there is an v =u
+ £ h ~ u(P)
such that on
On the upper spherical boundary
C2
C1 •
of C we have v = u . Hence
v ~ u(P)
Consider the function
v =u+
Eh
-
in
C. We assert that
v
~
u(P).
For if not, then v would take on an interior maximum at someipoint Q C. Then
v. = 0 at Q, 1
while
Hence
Lv = a .. v .. IJ IJ
struction
~
Lv = Lu +
Now since
(v .. ) is negative definite at Q. IJ 0 at Q. On th~ other hand, by hypothesis and by con-
C. Lh > 0 in C. This contradiction proves the assertion.
v ~ u(P) in C, and v(P) = u(P) , it follows that at P dv = du + dn dn
Since
dh/ dn
£
dh
f('
0
dn~'
> 0, we have du/ dn <0 • and the lemma is proved.
Theorem (Maximum principle). If u satisfies Lu
~
0
in an open con-
nected set R, and has a maximum at an interior point of R , then 48
-5J. Serrin
u
constant in R .
~
Proof. If u
j
constant, and has an interior maximum, it is clear that
we can find a sphere S, with closure in R, such that u has a maximum point P on the boundary of S but not interior to S . The hypotheses of the boundary point l~mma are satisfied, hence
;) u/ ;)n < 0
at P, contra-
dicting the fact that the first derivatives of u must vanish at P. We may restate the result as follows: any non constant function u satisfying
Lu
~
0 in R takes its maximum only at a boundary point; and
if there is an open
'J u/0 n < 0
then
sp~ere
in R which is tal1gent to the boundary at this point,
at this point. (Similar results hold wen Lu
~
0, provided
we replace "maximum" with "minimum". Finally, if Lu = 0, any nonconstant solution u can have neither an interior maximum nor an interior minimum. ) As is well-known, this result allows one to give elementary proof of the uniqueness of the various boundary vall!e problems associated with elliptic partial differential equations (cf. the discussion in Courant-Hilbert). A corresponding comparison theorem is this: Let
T 1(x)
and
T 2(x)
be stea-
dy state temperature distributions in a bounded domain R . Suppose that T1
~
T 2 on R. Then
T1
~
T2
inside R . To see this, we note that
T 1 - T 2 is a solution of Laplace's equation ~u
Hence
= u 11..
=0
T 1 - T 2 takes on its maximum and minimum at the boundar::, and
the result follows. Although this example is elementary, it illustrntes till' fact that the maximum principle can often lead to results of physical sign:ficance by entirely simple means. Let us illustrate this further with 49
exampll~;
- 6-
J. Serrin
from fluid mechanics. Consider the Navier-Stokes equations for a voscous incompressible fluid, namely
{
. ~
d1V V
1. = - grad
(l
=0
?
- dv d - =-(a = dt 'dt Qt
+ r2) + 6':{)
+ ~v.grad) .
I assume, of course, that you are all acquainted with this equation. The ivllowing equations are consequences of (1).
6 ( .., + r2) = (W 2 - 1) b:t
(2)
e
V
A ~
(3)
~
H - v . grad H
d~
..
dt = w •
(4)
=yw2 ,
grad? +~4i1
.., . . w = cur1 ~. v 1S th e vortic1ty,
Here
w2... = W. ..W,
deformation tensor (Tile components D .. 1J
1
~ ~
() v i ? v· )
=-2 (~ + ~). OJ X.
J
D: D
II X.
ticity measure, and
J H
(steady flow).
"+D
. th e we 11 known ra t e 0 f IS
Dij of D are defined by
= D ..
D .. , W 1J 1J
.~ =lwi ~2D:
D is Truesdell's vor-
= ~ q2 + : + r2 is the Bernoulli function.
Proof. By vector analysis
-: =
? 1 + '1 • \)t
grad
1 = <J'i)tif + ~x~ grad....!..2
Hence by an easy calculation, using the fact that
and also
50
.,
q2 .
div v = 0 , we have both
- 7But from (1) we have diva
= - /j
J. Serrin
(1- + n)
,and (2) is there fore proved.
Equation (3) requires the identity -+ ~ 2 4 ~ div (w x v) = w - v. curl w
= w2 + ~-1~.
= w2 +,~I
-1
{t + grad (;
+ n~
~
v • grad H.
The last result is the well-known vorticity equation in a viscous fluid (cf. [3]). Let us now apply the maximum principle to the three equations just derived. The results are as follows: TH. 1. In a region where the vorticity measure
en P + en p+
W, 1
,the "pressure"
cannot have an interior maximum, while in a region where W ~ 1 , cannot have an interior minimum. In case
form gravitational field,
p+
en
fl n = 0 , as in a uni-
can be replaced by
TH.2. In steady flow, the Bernoulli function
palone.
H cannot have an interior
maximum. TH.3. In steady, plane flow the vorticity
?=
(u, v, w,)
can have neither an interior maximum nor an interior minimum. Consider next steady irrotational flow of an inviscid incompressible fluid. It is well known that the motion is governed by a velocity potential
~
In stating Theorem 1 through 5 we shall for simplicity omit the necessar~. qualifying phrases concerning the connectivity of the region in quest ion and the non-constancy of the functions involued. The reader may easily supply these things for himself. 51
- 8-
J. Serrin
such that
v = grad
~
and
By differentiating with respect to
x, y,
Z,
in order, we obtain
Av = 0,
A u = 0, where u, v, w, are the velocity
compon~nts.
results: TH. 4. The velocity potential
~ w = 0, Hence we have the following
f cannot take an interior maximum nor
an interior minimum in the region of flow. In steady plane flow, the same , since
holds for the streamfunction
=
TH. 5. The velocity components u, v, w,
o.
cannot take either an interior
maximum or an interior minimum value in the region of flow. The velocity magnitude q cannot take an interior maximum (since the component of velocity in the direction
1
at the maximum point, could not have a maximum).
Similar results hold when the fluid is compressible, as was shown by Gilbarg. Consider, in particular, steady isentropic, irrotatiunal flow of compressible non-viscous fluid. The velocity potential ;
satisfies the' equa-
tion
where c is the speed of sound, and a .. =c 2
IJ
1 = (v I' v 2' v 3)'
For this equation
JIJ.. -v.v.1 J
and
1
a ij ) i
2,2
~j = c ~
52
-'~2 - (v.
t)
~(c
2
2~2 - q )
5 .
- 9J. Serrin Therefore, if the flow is subsonic (q < c) then the equation is elliptic and the maximum principle applies. Since a similar equation holds for the stream function in plane flow, we have thus proved Theorem 4. Now differentiate the velocity potential equation with respect to x. It is not difficult to see (cf.
[3J or (6J)
that the result is an equation of the form
a .. lJ
~u
+ linear terms in
? x.1 ~ x.J
=0
f) x.
1
Th'ls the velocity components u, v, w,
also satisfy the maximum principle,
and Theorem 5 is proved.
A somewpat deeper result, which at the same time well illustrates the comparison method, is the following. TH. 6. Let u be a solution of the Schroedinger type equation Lu = in some region stant
r
~
r
Au + au = 0
o m > 0 , and that
(n = 3)
. Suppose that
e
a = a(x, y, z)
~
-m
2
for some con-
mr as
u = 0(--) r
r
~
00.
Then -mr e u = 0(--
as r
r
~ 00.
The proof goes in essence as follows. The fact that -mr -mr 2 e L ( r ) = (m + a) - - 1 ' - ~ 0
e
allows one to construct by standard methods (cL differential equation, such that
53
C4J) a solution
v of the
- 10 -
J. Serrin
v =u v=O(e
-mr
/r)
=r
on
r
as
r~ 00 •
o
Now consider the function mr w=v_~_£_e_. r
(t>
0).
We have mr Lw = - £ L ( - r - ) ) 0 f'
while also on r = r
o
for r sufficiently large. By the maximum principle, Hence
w
~
v
~
u
£ -+ and
cannot have an interior positive maximum,
0 everywhere. Thus
v~ u
Letting
w
0 yields
e
mr
+ E- - r v
~
at each point
u. By a similar argument v
u has the same asymptotic behavior as v as
~
u . Hence. r ..... oo, This
completes the proof. Other examples will be found in references
[4J - [7J,
as well as in subsequent lectures. I should like to close this lecture with a statement of the maximum principle as it applies to parabolic partial differential equations. We consider in particular equations of the form
~
u = a .. u .. 1J
1J
+ b. u, + c 1
1
2 II
tU = f(x, t)
where a .. and b. are bounded functions of (x, t) in a domain D of space1J
1
54
- 11 -
J. Serrin
time, and
u
~
u u =-i ~ x, '
= u(x, t),
u
1
ij
=
The fact that the equation is parabolic is expressed by the conditions m
~ 2 < a" 'f , 'f ' ~ lJ ~ 1 1J
, and
0 ~
0 .
Exactly as in the case of elliptic equations we have the fundamental
,..,
Theorem (Boundary flbint lemma). Let S denote an open sphere in the space of the variables (x, t), and let S be a set of the form S with
class
extremity of C 1 in S
+
S . Suppose that
fp} , and that
:t.
u~O < u(P)
u
Then
? ul In < 0
(to}
P be a point of the spherical boundary of S ,not at ei-
SeD. Let
ther t
=S n { t
, where
it
J
in
u is of olass
C2 in S , and of
S
is any vector directed into S at P .
The proof of this is essentially the same as that of the earlier boundary point lemma, and lemma, and will be omitted. #'OJ
Theorem (Maximum principle). Let R be an open set in and let R be a set of the form
%u ~ 0 c
~ -~
where
space,
i't" [t ~ to J. Suppose that
in R ,
and that u takes a maximum at If
(x, t)
PER. Then u
X is a positive constant,
(see figure);
55
then u
~
~
constant in the set C(P). constant in R(P),
- 12 .J. Serrin
Figure difining the sets C(P) and R(P) ; specifically
C(P)
denotes
that component of the set {t=tp\nR which contains the point P . This result is uue to Nirenberg
(8] . The
first statement is a consequence of the boundary point lemma, as in the elliptic case. Indeed supposing that P' on
C(P) such
ment AP' PIA
and
me side of pI
on
C(P), there exists some point
u(P') < u(P) , and indeed there is even a vertical seg-
on which P'B
u, u(P)
u <. u(P) . Consider a quarter ellipsoid with semi axes
as shown in the accompanying figure, with B on the saas P .
C(fJ_
\~
•
e
R'
)
Now a::; B moves from pI to P thf're will be a first point where the boundary of the ellipsoid meets a point P" where u(P") = u(P) • At this point· we construct an internal tangent sphere to the ellipsoid. Then by the boundary point lemma
J ul ~n
< 0 at P" , contradicting the fact that P" is a
spatial maximum of u . (This proof does not take into account all possible configurations of P and pI, but does contain the essence of the argument). To prove the second part of the theorem, we suppose that u
+- u(P)
R(P) . Then using the first part of the proof it is clear that we can find a point pI
such that u;. u(P)
on C(P') while u(x, t) < u(P) 56
when
in
- 13 -
J. Serrin
t p ' -~ ~ t
< tp '
' By an argument similar to that of the boundary point lem-
ma, except that the comparison function is of the form h
= ~ (t p ' - t) -
vanishing on some paraboloid tangent to that
Ju/.)t")-
u ;. constant on
Eo ;)h/ ~t
2
C(P') at P' , it can be shown
> 0 at P' . But on the other hand, since
C(P') , the conditions
c
fu~o, imply that
r
Ju/~t ~ 0
at P'
This contradiction proves the theorem.
Applications of these theorem to the uniqueness of various boundary value problems for parabolic equations are easily given, exactly as in the case of elliptic equations, and we need not dwell on this. In our third and fourth lectures we shall see several less trivial applications. Other applications, in particular to asymptotic behavior, to singular perturbation problems, and to the Stefan problem, are noted in the references, numbers
[10J - [1~
.
Finally, attention should be drawn to an alternate form of the maximum principle due to Westphal and Nagumo. Its conclusion is somewhat weaker than Nirenberg's but this is compensated for by the greater generality allowed the differential operator ~u , which may in particular be strongly nonlinear.
57
-14J. Serrin
References: Chapter I ••
••
E. Hopf, Elementare Bemerkungen uber die Losungen Partieller Differentialgleichungen Zweiter Ondnung vom Elliptischen Typus. Sitz. Preuss. Akad. Wiss . [2]
.!.£
(1927), 147-152.
E. Hopf, A remark on linear elliptic equations of second order. Proc. Amer. Math. Soc.
~
(1952), 791-793.
Applications of Comparison Methods to Elliptic Parital Differential Equations
[3]
J. Serrin, Mathematical Principles of Classical Fluid Mechanics, Handbuch der Physik, V::>l. 8/1. 1959. Esp. , ; 22, 28, 45, 77.
[41
N. Meyers and J. Serrin, The exterior Dirichlet problem for second order elliptic partial differential equations. Journ. Math. Mech. 9 (1960), 513-538.
(5]
•
..
D. Gilbarg, The Phragm~n-Lindelof theorem for":elliptic partial differen-
..
tial equations. J. Rat. Mech. Anal. 1 (1952), 411-417.
..
cr. also J. Serrin,
J.Rat. Mech. Anal. 3 (1954), 395-413 •
[6]
D. Gilbarg, Comparison methods in the theory of Subsonic flows. J. Rat. Mech. Anal.
t7 J
!
(1953), 233-251.
D. Gilbarg and J. Serrin, On isolates singularities of solutions of second
..
order elliptic differential equations. J dlAnal. Math. 4 (1956), 309-340 .
58
- 15 -
.r. Serrin Maximum Principles for Parabolic Partial Differential Equations [8] L. Nirenberg. A strong maximum principle for parabolic equations. Comm. Pure Appl.Math. !(1953). 167-177. d. also A.Friedman. Pacific J.Math.
--
8 (1958). 201-211.
[9J H. Westphal. Zur Abschatzung der Losung nichtlinearer parabolischer Differentialgleichungen. Math. Z. 51 (1949). 690-695. :'
Applications of Comparison Methods to Problems Involving Parabolic Par.tial Differential Equations [1OJM. Krzyzanski. Sur les solutions de I 'equation lin~are du type parabolique d~termin~es
--
par les conditions initiales. Ann. Soc. Polon. Math. 18 (1945).
145-156.
9
[1 D. Aronson. Linear Parabolic differential equations containing a small parameter. J.Rat. Mech. AnaL.§. (1956), 1003-1014.
-
(1~R. Narasimhan. On the asymptotic stability of solutions of parabolic dif-
-
ferential equations. J. Rat. Mech. Anal. 3 (1954). 303-314.
[1 ~A. Friedman.
Free boundary problems for parabolic equations 1. II,
·~ber
III. Journ. Math. Mech. 8 (1959). 499-517; 9 (1960). 19-66. 327-346.
(l~R. Redheffer. Bemerkungen
Monotonie und Fehlerabsch'~tzung bei
nichtlinearen partiellen Differentialgleichungen. Arch. Rat. Mech. Anal.
-
10 (1962). 427-457.
Maximum Principles and Comparison Methods for Other Types of Equation,; l1~S. Agmon. L. Nirenberg. and M. Protter. Comm. Pure Appl. Math. ~ (195:l),
445.
59
- 16 J. Serrin
Q6)C. Morawetz, Note on a maximum principle and a uniqueness theorem for
--
an elliptic hyperbolic equation. Proc. Roy. Soc. London A 236 (1956), 141-144 IT~ C. Pucci, Propriet~ di massimo e minimo delle soluzioni di equazioni a
--
derivate parziali del secondo ordine di tipo ellittico e parabolico. Rend. Lincei. Ser. 6, 23, 24.
In this as in later chapters, the reference list is mainly restricted
to papers actually quoted. Further references may be found by consulting the bibliographies of the papers quoted, as well as the second volume of "Method of Mathematical Physics", by Courant and Hilbert.
60
- 17 J. Serrin
II HYDRODYNAMICAL COMPARISON THEOREMS In this lecture we shall be concerned almost exclusively with steady, plane, irrotationat flows of an incompressible fluid. As we have already remarked, such flows are characterised by a stream function
'II
such that
the velocity components are given by
v
=-
rx '
and
\JJ T xx +UJ , yy
=0
.
Moreover, by its very definition, the streamlines of the motion are the curves
tf = constant. Our main interest will be in the well konwn Helmholtz-Kirchhoff free
streamline theory, and most particularly in the case where a constant pressure wake or cavity forms behind an obstacle in uniform flow, (the speed on the bounding streamline is then constant by Bernoulli's theorem). I assume that you are familiar with the fundamental theory of this flow model, as found for example in the books of Lamb and Milne-Thomson, including the hodograph method for the determination of explicit solutions. Here we shall be concerned with the associated comparison theory, which yields a variety of interesting qualitative results and adds greatly to the understanding'of the phenomena involved. It should be emphasized also that the comparison the(\ry applies to more or less general obstacle shapes, and is not confined to the more special flows where an explicit solution is available. Unfortunately, a 61
- 18 -
J. Serrin
similar body of results has not yet been developed for other classes of free boundary problems, say those involving free surfaces under gravity. The teory is ultimately based on two fudamental hydrodynamical comparison theorems, which we now present. THEOREM (Speed comparison) Let Rand R be flow regions for two plane flows having uniform non-zero velocities q and q at infinity, whe0 o re ~q . Let Rand R be bounded by smooth streamlin~s extending to o 0 x = ,!,oO, as shown. If R C and the boundaries touch at P, then
q
R
q(P) ~q(P),
- -
the equality holding only if R = Rand qo = qo .
r
r· Proof. Let
'f'
and
If
r . -
be the respective streamfunctions for the
flows in question, assumed to be zero on rand
'f
is from left to right, we have both
and
If
Clearly since the flow
positive in their respective
flow regions (technically this can be obtained as a consequence of the maximum principle for the streamfunction). Consider first the case when
'J (~- '1') 'a
it is clear that on
r.
y
Y-
-
-
=u-u~q
-,
0
Cio > qo
. Since
-q>O
as (x, y)
Y> 0 outside some large circle. But also
Therefore, again by the maximum principle (since 62
4(
-+ co,
.ip - ~ ,,0
r- r) = 0),
- 19 -
J. Serrin
we find
'I' -r> 0
in R.
r-.r --'1') "'~dn (r --
By the boundary point lemma (since
--
q(P) - q(P) =
Next suppose
-
pose then that
> O.
qo = qo • If R = R then R
#it.
= 0 at P) we have
q(P) = q(P) , and we are done. Sup ...
We observe that. the stream function
(t( > 1) "'" with velocity at infinity describes a flow in R vious argument
d Letting
Df.., 1 yi~lds
"
'-II - r> o. -
r
~
qo
> qo • Hence by the pre-
i' - r is not constant, hence it
,0 . But
cannot take its minimum in R • That is
The conclusion
-
'P' -'I' > 0
q(P)
> q(P)
ved. For references, see
in R.
now follows as bef()re, and the theorem is pro-
(3) , [5] ,and [6) .
THEOREM (Interchange theorem). Let two non-zero plane flows be defined in regions Rand
r- have an arc MN in common, but in-
R bounded by smooth streamlines
extending to infinity. Suppose rand
rand
F
terchange their' positions on either side of MN , as shown. Then 9(M) > ~ q(N) ~ q(N) ,
-
the equality holding only if R = R.
r 63
- 20 J. Serrin
-
Proof. If R = R the two flows must be multiples of one another, and the equality is obvious. Suppose then that
R
IR
,and assume also that
q(N) = q(N) , (this can always be attained by multiplication of one of the flows by a suitable factor, a process which leaves the conclusion invariant.) It is thus necessary to prove under these circumstances that
q (M) > q(M) . Consider the function
n
="- -'II .
Since
?n (N) = -q(N) - q(N) = 0 , 7JIi' it is clear that a level line
n=0
n
of
issues from
N into
R
-
nR
.
Some rather annoying difficulties are avoided if we assume that this level line R
C extends directly to
nR
n~ 0
Al
and
A2
n<0
n~0
such that
on thE1 finite boundary of A2
principle in the separate regions and
without intersecting the boundary of
at any time. It is then the case that the set
two portions and
00
in
Al
A2 . In particular
and
n >0
R
nR
is divided into
on the finite boundary of Al
. An application of the maximum A2 near
yields then
n >0
in Al
M, hence by the boun-
dary point lemma
Q.E.D. There are several points of rigor in the above proof which require additional effort. For these, one may consult reference [6] or
[8J .
Both of these preceding comparison theorems remain true ( without essential alteration of the proof) for axially symmetric flows, and for subso64
- 21 -
J. Serrin
nic flows of a compressible fluid. The reason is that the stream function
'f' satisfies an elliptic equation in either of these more general situations,
while the proofs were based only on comparison arguments involving
r.
Of course, the proofs above require the maximum principle only in its simple form for solutions of Laplace's equation, while in the more general situations it is necessary to have the maximum principle in its general form for elliptic equations. The application of comparison methods in subsonic flow is due particularly to Gilbarg. Applicatiorn of the speed comparison theorems. As a first extremely simple observation consider irrotational flow past a symmetric obstacle, as shown in the figure.
Then certainly the maximum flow velocity is greater than the speed at infinity (apply the speed comparison theorem at
P
). Suppose next that the for-
ward part of the obstacle is in the form of a wedge:
r Comparing the given flow with that in the wedge bounded by 65
fi
,and using
- 22 -
J. Serrin
the interchange theorem, we have .q(M) > q(M) q(N) q(N)
Letting
s
be the arc length from .
q(N) = q(s. ),
0 ,setting
0
II(
and observing that then q(M) / q(N) = const.
s"-ec
q(M) =q(s),
,we get
a result which is not directly obvious. From here on, let us turn our attention to the main issue of free boundary flows. Consider a symmetric infinite cavity flow (the upper symmetric part is sufficient), and let
T
denote the curve consisting of the upstream
axis of aymmetry together with the obstacle up to the detachment point A.
7' Also let
~
be the corresponding free streamline, along which the velocity
is assumed to have a constant value. Since
I
extends down stream to :nfi-
nity, it is apparent that this value is precisely the denote here by U . The
corre~ponding
~tream
speed, which we
flow is of the type to which the pre-
66
- 23 J. Serrin ceding speed comparison theorems apply. This will in fact the basis for the following results. Single intersection theorem. Any straight line which does not cut T
L
can intersect
in at most om: point
[7J.
Proof. Suppose the contrary, for example as in the following
figu~
reo
r --."
By applying the interchange theorem to this situation one sees that
q(M)
q(M)_
~(N) > q(N) - 1.
However, by the speed comparison theorem it is evident that q(M)
and
This yields q(M) / q(N) < 1 ,in contra-diction with the preceding inequality. Since other cases of multiple intersection can obviously be handled by a similar technique, the theorem is proved. Uniqueness theorem. Consider symmetric flows past an obstacl,' C (in the upper half plane) with the free streamline detaching from a fixed endpoint
A on
C . Suppose that the corresponding curve 67
T
is star-
- 24 -
J. Serrin
like with if
J.
respe~t
r
and
to some point .Q on or below the axis of symmetry. T!.en are two corresponding free streamlines, we have
(5J ' (6 J, [7J .
and the flows are identical
T +1.. =
Proof. The single intersection theorem shows that and
T+
I = f'
=
r
r
are both starlike with respect to Q. In the standard theo-
ry of free boundary problems it is shown that the f~ee boundaries
r
I
are asymptotic to a parabola downstream. Now suppose
l
one of the curves, say
J.
and
ft . Then
l
,lies above the other at infinity, or else they
are both asymptotic to the same parabola. In either case, a similarity transformation (contractioll) about
-, rand
'H'
r
bounded by
have a point
f'
I
contains
.r
Q takes P
into a new curve
r'
such that
in common, and such that the flow region
R
r
_____-.Lr
A~ /.----
/'~
-, ======::':..J~(~ ... -.. :. . .f -.. . _. . . . r Moreover the flow in the original flow in
-R
it'
~/
has the same velocity at corresponding points as The speed comparison theorem applied to these
two fows thus yields
This contradiction shows that we must actually have rem is proved. Interchange Theorem. Let
-
C1 and
68
l
='
I
,and the theo-
C2 be two obstacles for the
- 25 -
J. Serrin
--
symmetric infinite cavity problem, with the same detachment point each flow. Suppose that and that both T
lies above
T
and
T,
T
T
(or T) is starlike with respect to a point
-
lie to the left of the extended line AQ.
~ below ~
we have
Proof. We know that It will first be shown that ~ (if
l:
la) then
lies above
l:
l
and ~
Q,
Then if
,[51, [7).
r
lies below
are asymptotic to parabolas. at infinity. For otherwise
or if they are both asymptotic to the same parabo-
R can be contracted so that
daries touch at a point
A for
R'
R, and their boun-
contains
P. The speed comparison theorem., applied exactly
as in the preceding proof, supplies a contradiction. Having shown this, it follows that if the theorem were not true the two flows would be as shown in the figure.
-'
r~
-.-- --
\11"
I
-
r
r
We construct three auxiliary flow regions. First by an expansion we conSLt'.lct
R' . Next by tl4e contraction we get
by IOMNI'.
-R II
•
Finally
R' II
.Applying the speed comparison theorem at 69
is the region boullu,·d M one obtains
- 26 J. Serrin q'''(N)
< q"(M) = U,
q"' (N)
> q'(N) = U •
and similarly
-
Next applying the interchange comparison theorem to ve
-
>.~
q "'(M) q"'(N)
., q(N)
Rand
-
R'"
we ha-
= 1.
The last three conditions are in contradiction, and the theorem is proved. The proof is entirely the same if the curve
T
is starlike.
Similar results hold for axially symmetric flows and for subsonic flows of a compressible fluid
(t 7J - [II),
and for free boundary pro-
blems involving jets. The proofs in thesE' .cases, although similar· 11 their basi<: structure, tend to be technically more difficult and for this reaSOIJ have not been given in detail. The preceding theorems allow an interesting application to the problems of determining the symmetric obstacle of given dimensions with least cavity drag. For the purposes of the discussion, the class of obstacles allowed into competition will be described by smooth starlike curves confined to the rectangle point on
x =a
I
0~x
~
a
0 ~ Y~ b
I
and touching the line
y =b
I
C,
joining the origin to a
in at least one point. The as-
sociated free streamline may detach at any point of the obstacle
C provi-
ded that i ) it does not reintersect
C
ii) the max flow speed is
U
(This condition reflects the assulpption that cavitation will immediately occur at pressures lower than that in the free stream. ) 70
- 27 -
J. Serrin
We consider a particular curve
K in this class constructed by
choosing the Kirchhoff infinity cavity flow past a vertical segment in such a way that the free streamline passes through the point
We assert that the curve
K has less cavity drag then any other curve
in the class described above. Proof. If
LC
LK
is above
L
(2] ),
C lies partially above
L:
K ,then by the interchange theorem,
. But it is well known that the drag increases as the
whose free streamline
must cross
C
[5], [7] .
C lies below
cavity opening increases (see If
(a, b) ; see figure.
i:
whence Drag (C) > Drag (K). K ,then there is a vertical segment
is tangent to
C , say at
P
. Clearly
rc
for otherwise the speed comparison theorem gives a contra-
diction to ii) at
P
. But if
LC
crosses
L
it then stays above
L
from then on, because of the interchange theorem. It follows then, as before, that
Drag (C) > Drag (K L ) > Drag(K). The result obv'iously applies similarly to axisymmetric flow. Thus
the design of an obstacle of least cavity drag involves, rather remarkably, a flat leading profile. As a final application we notice a remarkable relation between free boundary problems and the problem of determining the symmetric obstaclt' of given dimensions for which the maximum flow speed is least. To state tile problem quite definitely, consider smooth, symmetric obstacles with a fixed 71
- 28 -
J. Serrin
ratio of width to length, placed in a uniform stream with velocity
U at in-
finity. Among such obstacles, the problem is to determine one for which the maximum flow velocity (necessarily attained on the profile) is least. We assert that the solution of this problem is a profile
E consi-
sting (above the axis of symmetry) of two equal vertical segments joined by a convex arc
S ,such that the resulting profile has the prescribed dimen-
sions and has the property that the corresponding flow is of constant speed on S . (Thus
S is the solution of a certain free boundary problem. It is
easy to show by the hodograph method that there exists exactly one solution of this problem; inde( d, the problem and solution are identical with that of the celebrated Riabouchinsky finite cavity flow. Curiously, although this free boundary model is physically unrealistic in its original setting, as a solution of a cavitation problem, here it proves to have a genuine physical importance.) To prove that the profile
E
is the required solution of the given problem,
consider any obstacle
fE
with the same width to Length rati<'. We may
suppose that
B
B has the same length as
E by an appropriate normaliza-
tion, (leaving the speeds at corresponding points unchanged.) Then since B it must either touch or cross the arc
has the same width as
E
me point. If it touches
S ,say at
P
S at so-
,then by the speed comparison theo-
rem qB(P) > qE (P) = max speed in flow past Thus. the maximum speed in the flow past past
E
•
72
E.
B is greater
~han
that of the flow
- 29 -
J. Serrin
>
)
-
On the other hand if B
into a smaller set
S at a point
B
crosses
E
B we can manage that
-
..
, then by a contraction of
BeE
while
B
touches
P. The speed comparison theorem then applies exactly as
before, and the assertion is proved. This result was first obtained by Gilbarg and Shiffman, in a slightly different physical situation. It is als'o easy to show (see S
[4]
that as the length-width ratio is increased the speed on
must decrease. Thus the longer the obstacle is allowed to be, for a gi-
ven width, the more one can reduce the maximum flow speed (toward its limiting value
U
).
73
- 30 J. Serrin
REFERENCES:
CHAPTER 2
Texts and Monographs
(1]
H. Lamb, Hydrodynamics, Dth editiort, Cambridge, 1932, pp. 94-105.
(2)
L. M. Milne-Thomson, Theoretical Hydrodynamics, 3rd Ed., New York, 1950; Chapter XII. G. Birkhoff and E. Zarantonello, Jets, Wakes and Cavities, New York, 1957. D. Gilbarg, Jets and cavities, Handbuch der Physik, Vol. 9. Springer, 1960. Esp. 368-:87, 407-499.
Papers
[5]
M. Lavrentieff, Sur certaines propri~t~s des fonctions univalentes et leur applications ~ la tMorie des sillages. Mat. Sbornik 46 (1938), 391-458. (Russian, with French summary). J. Serrin, Uniqueness theorems for two free boundary problel J. Math. 74 (1952), 492-506.
lS.
Amer.
J. Serrin, On plane and axially symmetric· free boundary problems. J. Rat. Mech. Anal. ! (1953), 563-575.
[8]
J. Serrin, Comparison theorems for subsonic flows, J. Math. Physics, ~ (1954), 27-45. D. Gilbarg, Uniqueness of axially symmetric flows with free boundaries. J.Rat. 1\IIech. Anal. .! (1952), 309-320. D. Gilbarg, Comparison methods in the theory of subsonic flows. J. Rat. Mech. Anal. 2 (1953), 233-251. Cf. also M. Schiffer, Analytical theory of subsonic a;d supersonic flows: Handbuch der Physik, VoL 9, Springer 1960; Especially pp. 106-107.
D. Gilbarg and M. Shiffman, On bodies achieving extreme values of the critical Mach number. J.Rat. Mech. Anal. ~ (1954), 209-230. 74
- 31 -
J. Serrin
III. THE COMPARISON I\IETHOD IN BOUNDARY LAYER THEORY A. More than 60 years have passed since Prandtl formulated the main concepts of boundary layer theory in 1904. But today we are still far from having a complete knowledge of those equations. Even the simplest case of a steady laminar two dimensional boundary layer in' an incompressible fluid presents many mathematical problems, for example, existence, stability, and asymptotic behavior, which are still only partially solved. At the same time, among many mathematicians there is the feeling that boundary layer theory, being based on an approximation, is in some way not quite respectable. Perhaps this is partly due to the lack of established techniques for treating the boundary layer equations as a mathematical system, or in other words, to the fact that the equations themselves are even further approximated in most of the known solution processes. However, in 1958 Karl Nickel, using comparison methods ill the theory of parabolic differential equations, was able to obtain a number of valuable results concerning the exact theory. These results I wish to discuss in this and the following lecture, though the presentation itself owes much to a paper of Velte. I begin by reminding you that classical boundary layer theory is based on the assumption that in viscous fl?w past a fixed wall, the flow region can be conveniently divided into two parts: 1) A thin layer adjacent to the bopy, ill which the fluid velocity rapidly adjusts from zero to its general streaming value. In this region the gradients of the tangential velocity component will be large and the viscou:;:; shearing stresses will exert appreciable influellce on the motion. 2) The remaining region of flow in which the effects of viscosity arE' of much less immediate importance. In this region the fluid may be 75
COll-
- 32 J. Serrin
sidered inviscid, and potential theory is usually applied to determine the flow. It should be observed that this flow in the exterior region automatically provides the streaming velocity required in I}. By these assumptions, the flow problem is thus divided into two different and separate parts. We observe that an approximation process is explicitly postulated for the second region, the full ,Navier..stokes equations
...
div
'1=0
dv = _ grad (l... + n) + "I dt Q
.
~1
being replaced by their inviscid counterparts div ~= 0,
~
curl v = 0
(i.e.
6;= 0).
In the thin boundary layer region a different approximation is necessary, which, while greatly simplifying the Navier-Stokes equations, still retains the effects of viscosity. This approximation lies at the heart of boundary layer theory, and consists in neglecting certain terms in the Navier-Stokes equations. If coordinates are introduced as shown (I confine myself from
here on to steady plane flow, though the ideas in many cases carryover to the unsteady three dimensional case), the approximation is given by 76
- 33 -
J. Serrin
u
uu
where
x
+ vu
p = p(x)
x
+ v = 0, y
e dpdx + "
1..
Y
u yy
is the pressure (1) in the streaming flow, determined ac-
cording to Bernoulli's equation 1
e
dp = UU dx x
U(x) = streaming velocity.
We thus have the familiar Prandtl system of equations
uu + vu = UU + ~ u x y x yy
u +v
y
x
= 0,
which I assume that you have all seen at one time or another. The associated problem of partial differential equations is that of the downstream evaluation of the u-velocity profile. We thus suppose that at some initial position, say
x=O
,we have N
u(O, y) = u(y) given, together with the well-known Prandtl boundary conditions
(1)
u = v =0
when
y =0
u=U
when
y = 00
More precisely,
p here corresponds to 77
(at edge of the boundary
p
+,.n.
previously.
la~'er).
- 34 J. Serrin In part
B I shall discuss this latter condition further.
The problem as thus phrased bears a striking resemblance to the theory of parabolic differential equations, with the coordinate x playing the role of t. That is, the downstream evolution of the velocity profile is ana-
the time
logous to a time evolution problem. In the following results we shall apply the maximum principle (ef. lecture 1) to the study of this problem. I shall assume throughout that. u for any solution in question, and also that the streaming velocity class
C1 and positive for
~O
U is of
x ~ 0 . Finally, we shall use the letter
R to
denote a region in the l:: oundary layer of the form
R={ 0< x
~ Xo
THEOREM 1. Suppose. Ux uy (x, 0)
> 0 for 0 < x ~ Xo
O
'
~O
• Then
u > 0 in
R
. Moreover,
•
Proof.- For a given solution u = u(x, y), boundary layer equations, let
J
v = v(x, y)
for the
~ denote the linear differential operator
(c = -u
~
0).
Then one sees at once that
1, u = -UUx ~ O. We can thus apply the maximum principle in the form stated in the opening lecture. That is, suppose for contradiction that R
. Then
u has a minimum at
P
u = 0 at some point
, and consequently 78
P of
u is constant on
- 35 -
J. Serrin
C(P) • That is
u = 0
on
C(P)
This violates the condition at infinity, and the first part of the theorem is proved. The seI
,
cond part of the theorem now follows from the
r'
boundary point lemma in an obvious way. This result shows that in a flow with and
x. uy(x, 0) = 0
Ux
~
0
u
~0
,no incipient backflow can de-
velop, and the separation condition
can never arise. (This theorem is in fact the theoretical justifi-
cation for the term favorable p"ressure gradient. ) THEOREM 2. The shear component tremum at any point of
f
uy
cannot assume an ex-
R.
Proof. This is based on the calculation
L (y
o=~
y
u + UU - uu - vu ) yy x x y
=" u - u u - uu - v u - vu xy y y yy yyy y x
=Vuyyy Thus if on
u
y
- uu
yx
- vu
yy
should assume an extremum at
PER, then
C(P). Integration then yields u
= ay + b
on 79
C(P),
u
y
= constant
- 36 -
J. Serrin
which is in contradiction to the assumed boundary conditions. THEOREM 3. Suppose that Suppose also that the initial velocity
,.,
u(y)
where
e
~
0
u(x, y)
~
lim u(x, y) = U(x)
t~e
1(y)
uniformly in
x .
satisfies the condition
U(O) + e
is the initial overvelocity. Then for
(x, y) • R
< {U(X)2 + 2eU(O) + e2•
In paticular, if there is no initial overvelocity, then u < U , and no overvelocity will developat tl later time. Z = U2 - u2 , in order to compare
Proof. Set
= y (-2uu) - 2u(UU - uu ) - 2uvu x
YY
x
U and
u
• Then
Y
= -2u(Y u + UU - uu -vu) - 2 Yu 2 = -2 yy x x y y
~u2y
•
Thus
Z.
and the maximum principle may be applied to minimum at a point
P
of R
, then
Now we observe that 80
Z const.
;that is, if on
C(P).
Z takes a
- 37 -
J. Serrin 2
0.1
Z
= U(O) - u(y)
Z
= U2 > 0
2
~-2eU(O)
- e
2
lim Z = 0
Since it is impossible for that
Z
Z
on
x =0
on
y
as
y",
to be constant on any line
cannot have a minimum in
==
0 00,
uniformly in x.
C(P), we infer
R. Consequently, the boundary con-
ditions require that
Z > -2eU(O) - e
2
Q.E.D.
We remark that a similar result holds in three dimensional boundary layer theory. Consider in particular flow over a plate in the x, y plane, the coordinate
,
z
being taken in the direction normal to the plate. Then if
u 2 + v 2 < U2 + V2 have
v
u2 + 2 <
"
on some arc
V2 + V2
AB
of the initial curve, as shown, we
in the corresponding shaded region.
The next theorem shows another interesting property of the boundary layer equations. THEOREM 4 . Suppose that
lim uy = 0
uniformly in
y .... oo
the initial profile
u(y)
has exactly
N
proper extrema.
~A proper extremum is a point where a function takes on
maximum or a strict local minimum. 81
x, and t.hat
••
Then any dowlJ-
C\ strict local
- 38 J. Serrin
stream proflle cal' have at m ~~:
u, u
y
>0
N proper extrema. If
,.,
u y
~
then
0
n R.
Proo:' of 11 eorem. T lE' last statement follows from Theorem 3 and the condition
lim
y. .oc
=0 ,
y
c:
part suppose that j1:e grS;)'.l Then we can find
+1
~
u
(~inee
y = 0). To prove the first
x = Xo
p< ir ~s
Pi
on the line
11
ras
11
I'as;l min at
P2
on
at
y,
P 1 and
, 0
u
respect to the variable
Between
Y
,l .
on
N extrema.
x· Xo such that, with
PI' P 3' •••..
lax at
}.. Xo
had more than
P 2, P 4' •••... there is a point
P 12
where
takes a negal ive rr, ;nLn~m wLh respect to its values on the segment But since
u
ca:llot take a n inimum on 'f
P 12
issues from
R, it is clear that a level line
-
irto
R.
a maximum princif) le, and
DE
('his line cannot end in it'ler can it go to
fore finds itE way 'J:lck to the iLitialline
t~~ 't /---:;1 t flo
x = o.
1,
I
'., ••
o
I
'.~ ~IJ, {~
[!.
)x 82
y
z
0
R, since or
y
= co
uy
has
• It there-
- 39 -
J. Serrin
Apllying the same process between some point
P 23
P2
three issues a level line
which likewise must find its way back to
and
P3
we see that from
C2
x = 0 . Continuing in this way we
obtain N level curves each lying above the proceding, and each extending back to the initial line. We construct l".slty the curve there will be some point respect to
x
= Xo
P N+l,
00
CN+1 . Since
on
o
at which
u
y
aesumes with
an extremum opposite in sign from that at
We can then construct the level curve We now have
x =x
uy{P N+l) = Uy{oo) = 0,
P N, N+l
CN+1 exactly as before.
+ i points on the initial profile at which u
is y alternately negative, positive, negative, etc, It follows easily that there are at least
N
N
+ 1 points on the initial profile at which it has proper extrema.
This contradiction demonstrates the theorem, In order to summarize the preceding results we may illustrate them schematically in the following way, where the curved lines are velocity pro. files,
83
- 40 J. Serrin
r". 3
)t
B. We continue our discussion of the Prandtl boundary layer theory by taking up the question of the uniqueness of the evolution problem. The proof here is based on the maximum principle and associated comparison arguments and involves, moreover, a preliminary change of variable due to von Mises. It is this change of variables which effectively limits the proof to the plane flow case. A consideration of three dimensional boundary layers here would be of the greatest interest, but has not yet been obtained. The von Mises transformation. If we consider a specific solution u in
= u(x, y), v = v(x, y) of the boundary layer equations, such that u > 0 R: 0 < x ~ x0 '
0 < Y < 00
J=
,
then we can introduce new variables
x ,
84
- 41 -
J. Serrin
where
r
geover the domain
f <1~
R':
0
Xo
vious one-to-one connection between u() "
)
(f' '1,)
is the streamfunction of the flow. The variables '
0<
,and there is an
,< co
Rand
R'
ranOl>-
. The function
= u(x, y) satisfies, as is easily checked, the equation 2 (U)l
=
(U2
,
+
~ rr
2 u(u \ ,
or equivalently
The analogy with the heat equation is apparent,
. tr me.
THEOREM 1. If u lim u = y-+ co
--
U
!
playing the role of ti-
is positive and bounded in
uniformly in x,
R , then
[6)
The proof uses a typical comparison argument. It is first of all evi-
dent that our re,l?ult is equivalent to
lim
,~
*
co
!
(= t)
For simplicity in writing further equations, we shall assume from here on that the kinematic ~iscosity ""J = 1 • This clearly involves no loss of-generality, since it can always be attaineo by a simple change of variables. The von Mises equations now take the form (U 2
and Zt
Jt*
uniformly in
Z = 0
)S
!
'= (U 2
= u Z"t'l
+ U(U 2\ \ Z
2
=V
This requires a Heine-Borel type argument, which we omit. 85
2
- u •
- 42 J. Serrin
Now we know that
Z-+O
as
~ ....
at
QO
t = 0 . Thus for any
C >0
we have
if Now set
~ = constant.
Then
and so, introducing a new linear differential operator
4 . 2..2
2 c( t+1
- F = { u(.!!L!L - - ) t
(t+1)2
:
provided
«= 1/4M ,where
f
!,2 (t + 1)
2 (41(u - 1) -
eXP(~ + et) 1 - dt
so that 86
JF
",2 (t+l)2, ..
2C(u } t+T
M = Max u . Similarly, set
2
G =
t '
F
~ 0,
- 43 J.Serrin
'.
=
f
~'--u~_ e}
@!t2(4IlU_d)+ ----12 (1 _ dt) 2 r"' 1 - dt
G<0
if
1 0< t < 2M .
e=d=M,
Finally set
H=
where
'6
~+ ~ F + Z G
is a positive constant, which we shall fix later. Since
1- (F + Z) ~O
we find that
o~ ~ (GH)
+ 2G"\. H, + GH'l" ) - GtH - GHt
= U(G"
H
= G~H
+ 2u G1 H-,. + H JeG.
Finally, this may be written in the form
~ H + ( 2u
~
G1\.) H ( _ G 'Y\..'"
~ G) H G
.
Now let us apply the maximum principle to this differential inequality. To t: i j s end, note that from the definition of
H we have 87
- 44 J. Serrin
H" 0
uniformly in
as
x.
Furthermore
H >0 (since
\
z \ < ~ ),
1z \ =
t
= 0,
and
H >0
(because
on
o ~ t < 1/2M
on
u 2 - u21 ~ U2 + u2 ~ {F
' follows from the maximum principle,
ve, that
H
if
r
is suitably large). It
since the coefficient of
cannot have a negative minimum. Hence
H
is negati-
H > O. Therefore,
we must have
± z ~ E.+ IF. But
F..., 0
uniformly as
1\. 7
00
hence for all sufficiently large
we have
fzl<2£
o ~ t < 1/2M.
Repeating the process a finite number of times proves the theorem.
(.)
The preceding argument is a typical example of the application of comparison methods to determine the behavior of solutions of elliptic or parabolic equations at a singular point. Of course, it is clear that the application determines the choice of the comparison function, and that considerable care must occasionally be taken in finding a suitable comparison function. We co(f()
The reader will observe that we have not used the condition u = U at y = 00 ,except at the initial line x = 0 • It follows that the condition at inlim ~y) = (0) finity is superfluous pro~ided y~oo .
U
88
- 45 -
J. Serrin
me now to the culminating theorem in our investigation of comparison methods in fluid mechanics, a proof that the time evolution of a velocity profile is uniquely determined. THEOREM 2. Let
u
and
u
be two solutions of the boundary la-
yer equations corresponding to the same initial velocity profile lr(y). Suppose that
u u
yy
and
~
satisfy the hypothesis of Theorem 1 and that
<0 ,
or more generally that
u (x, 0) > 0 y for some constant
u
yy
~
B .
,
u ~ u.
B. Then
Proof. For simplicity we shall consider only the case when u < O. yy This case includes most of the known exact solutions. We observe that since u
> 0, 7i> 0, the von Mises transformation is applicable, and that by Theo-
rem 1 both u and the region
(u
Now set
R' : 2
~
)y
~
u
tend uniformly to
0<~
u
= 2 - u2
0<,<00
'
2
2 = (V )
,~
~ Xo
~2
+ u(u )
(u
"'
.
l
~
(5)
)
1
'\~oo.
Moreover, in
we have
= (V 2)
~
- -2
+ u(u \ ,
,whence by subtraction of the preceding equations there
arises
fl
as
V
- -2
= U(U)n - ,L
= u .""
=u
1"
2
- u(u ~,
2
-
+ (u - u) (u \, +a
~
89
- 46 J. Serrin
where
2u yy
<0
= u(u+ii)
•
Now obviously
,~ 0 as
f
= 0 when
~ -+
~
= 0
~
From equation 5 it is clear that mqm or a negative minimum in
00,
RI
.~ uniformly in
and when
,
) = o.
cannot have either a positive maxi,
hence
¢ = 0 . Thus
To conclude the proof, note that the inverse von Mises transformation is given by
y
=
Hence for a given value of "
x=
f.
the corresponding values y and
y
for
the two solutions are the same. Consequently,
This complets the proof. As we have already observed, an argument which serves to prove a uniqueness theorem can frequently be used to obtain a corresponding comparison theorem. This is the case here, with the following result. THEOREM 3. Let u and
u be solutions of the boundary layer equa90
- 47 J. Serrin
_ tions corresponding to initial profiles ties
U and
U. Suppose that u and u <0 yy
rem 1 and that either
or
':t
u
and
u
,and streaming veloci-
u satisfy the hypotheses of Theo-
uyy < 0
. Suppose also that, in the von
Mises variables,
and that
Then u(x, y) ~ u(x, y). The proof is almost the same as before except that we are led to the
< 0)
equation (Supposing u yy
( rJ..2 =u
,J... - ~ J.. -2 2 ,." = u r,'l. + a 'f +(U - U)y ,
(5')
In addition, , ~
satisfies the boundary conditions ~
r=U
-U
2
)0
r=u':t2 -u..,2 >0
~
¢
=0
Consequently, since
at infinity at
~
=0
at ~ = 0
¢
cannot have a negative minimu!.we have
and u( ~
'~ ) ~ 7I( ~, , ).
To conclude the proof, we have for any fixed value of
~ At
2 - u ).
such a point, we would have
u1Tf ~J < O. 91
'l. '
~ ~O
- 48 -
J. Serrin
Thus it is evident that
U(, '~ )~ ~( l ', )
u(x, y) =
(6)
= u(x,
y)
(i)
u > 0 , hence u(x, y) ~ u(x, y) , and the required inequality y is proved. If Uyy '< 0 , we prove in almost the same way that (6) holds.
Now we have
The final result the follows since
...uy > 0
and
-u(x. -y)
~
.
u(x. y) .
Remark. Thi:. theorem is analogous to certain results of Nickel but neither contains his results nor is contained in them.
L~t
us note
t2J.
Si\"e-
ral consequences which have some physical intereat •
..!.:.
Suppose the external speed is constant. U; U0
•
Let
u
be the
corresponding Blasius solution
~
=U
Y
fl (
r~v.ITJ.
o
Then for any other solution u
u(x, y) ~ U fl ( o
)
such that
Y
V2 vr/u.
~
u
~
U0
we have
).
2. Suppose the external flow is of the form
U = Cx m • m > 0 • Sup-
pose also that initially IV
O~u~U(O)
Since
uyy < 0
necessar.ily
,an£l
u >0 y
for
,
u
= 0 'at
y . = O..
0 < Y < 00 92
At
u
= U at y = 00, 'one sees that
- of!! -
.1. Se rr- in
ie,
"" u; 0
u =0 , 3, If we exam il; P t hi:. so - ,'ali l·e! cut,,], \J()unda ry condition it must be
admit~eli
that this
j:::;
u
= l:
at y
= cc,
only an artifice tu obtain a solutic'n with the
more realistic beha"'.(ll' u~
e
at
"\'
= Ol'~) \ l-W
u ~O
Y
Let us now consider the bou.ndary
layl~r
equations with this (more realistic)
boundary condition, By t.he change of variaiJl( s
v = (iv,
1I
= u
our problem becomes
uu + ~U .. y
x
U
x
;;
-:-vy =0
with conditions
u = :r y=-O(l)
at ll_'::
y
0
Since both these bOlUHlal',\ cuntiit iot:.s (a:lllot b,' simultaneously met, let
IlS
keep just the first, ,u:d cOllsider sul'.l1.iollS Cit' tIl'." !)(lundary layer equation'> such that on a
("l1've
To compare this with a S(,lllLolJ of th(:
u
=U
f(x)
1,:51'0.:
.. y
0(:,
93
problem
• 50 •
J.Serrin we use the ideas of the preceding theorem. In particular, let us suppose that
< 0 for the solution u, and that the initial conditions are such that
u
yy
In particular, this will be the case if the initial conditions tor both u and the solution
u
are identical. Then applying the methods of Theorem 3 one
is easily led to the inequality u(x, y) ~ ;(x, y), which is illustrated below.
u.
u
-
o
.-
--
y
..
·Although there is no compelling reason that u need be cloae to u, the graph does i'Adicate that in order to guarantee
u-y V 0 ,
edge of the boundary layer in a region where then
u differs only slightly from
we should take the outer
u.U. If thia is done, however,
u, and we might as well take u itself
for the boundary layer profile. Thus we see rather clearly the reason for the success of Prandtl's artifice.
94
- 51 J. Serrin
REFERENCES: CHAPTER III
[11
K. Nickel, Einige Eigenschaften von Lgsungen der Prandtlschen Grenzschi-
chtdifferentialgleichungen. Arch. Rat. Mech. Anal.
(2) K. Nickel, chiv, [
3]
2!.
!.
(1958), 1-31.
Eine einfache Absch~tzung flrr Grenzschichten. Ingenieur Ar(1962), 85-100. ~.
.t
K. Nickel, Ein Eindeutigkeitssatz rur instationare Grenzschichten. Math.
Zeit. 74 (1960), 209-220.
.
W. Velte, Eine Anwendungen des Nirenbergschen Maximumprinzips fur parabolische Differentialgleichungen in der Grenzschichttheorie. Arch. Rat. Mech. Anal.
I'
"
~
(1960), 420-431.
••
H. Gortler, Uber die Losung nichtlinearen partieller Differentialgleichurt:" gen yom Riebungsschichttypus. ZAMM
~
(1950), 265-267.
[6] J. Serrin, Mathematical Aspects of Boundary Layer Theory. Univ. of Minnesota, 1963.
95
- 52 J. Serrin
IV. STRESS, VORTICITY, AND ENERGY AVERAGES A second classical method for studying the behavior of a mechanical system is through vadous averaging procedures, in which the primary interest is in the space or space-time average of some physical variable. These methods are valuable because they allow us to concentrate on overall behavior, rather than on unimportant local variations. Statistical mechanics is perhaps the best example which can be offered of the extreme importance of the averagingidea, but even so, averaging can play an important part in field theories and in continuum mechanics, and it is this side of the picture which we discuss here. It should be added that even if mathematicians were not gifted with physical insight, they would still be led to the idea of averaging, for as we shall see, it is an extremely natural process at the elementary level; while at the advanced level it allows one to bring into play the powerful techniques of functional analysis. ~
Let us begin our discussion by setting down, in differential form,
the fundamental equations of motion of a continuous medium, namely
f where
.
-+ T
• -+ .. (a - f) = div T ,
is the stress tensor, assumed to be symmetric,
leratlon, and
,.f
~
a
is the acce-
the external force, As with the special case of the Navier-
Stokes equations; I assume that you are also familiar with this equation. For the reasons indicated earlier, it is natural to multiply this equation by a wei:ghting function or vector
n,
¢
and average over a(possibly
thus =
1~ n
96
div
Tdv.
movin~volume
- 53 -
J, Serrin
The right hand side may be rewritten in the form
Combining the two proceding equations then leads to the important formula
where
~
t
is the stress vector on the surface, Here for simplicity we have
omittes the conventional infinitesimals bols
nand'
Jn
dv
and
ds, as well as the sym-
denoting the set of integration. These things will be ap-
parent from the contest in all future formulas, The standard averaging technique which leads to formula (7) can be summarized by the following steps: 1) Multiply b9th sides of the basic equation in question by a test (or weight) function
¢.
2) Integrate over an appropriate region
n
3) Simplify the result by using the divergence theorem (i. e, integration by parts) 4) Choose an appropriate test function
~
,
We intend to illustrate the last step by several particular choices of ction
~
th~
fun-
, It will be convenient, however, first to recall a simple result, the
so-called, TRANSPORT THEOREM. Let
n
be a (possibly moving) volume in
the interior of a region of fluid motion, and let variable. Then 97
F
denote some physical
- 54 J. Serrin
where
G denotes the normal outward speed of the boundary of
ticular, if
n
is a volume
n.
In par-
V moving with the fluid, then
The transport theorem in the form noted above is prove~ in
[1 OJ
'
as well
as in articles in the Fandbuch der Physik by Serrin and by Toupin and True ... sdell. Let us now obtain our first application of (7) by setting and assuming that
..
..,
1 2
v , a = -
where grad
n=V d
dt
q
=
~
,
is a volume moving with the fluid. Since
2
~ is the tensor
~
grad
'1:
T =T : D
J~,/ ~)( j . this gives
(8)
the important energy transfer formula. We will return to the notion of energy averages later from a slightly different standpoint. A second example arises if one sets
~ =? (the position vector)in
order to examine the first moments of the Cauchy equation. Then grad. is the identity matrix, and we obtain the interesting formula
f[ et Ii -~ +1) •
f11. 98
- 55 -
J. Serrin
Signorini has pointed out several applications to the static case, when the preceding formula reduces to the simple stress average identity
Thus in particular the stress average is completely determinate from the external load. To take a third example, suppose Confining our attentic I to the static case
¢ !
is the dyadic tensor
1t
.
= 0 , and using tensor notation,
the result is
the
a ike being a constant tensor which is determined by the given external
loading. Now we have (x k XI)'· T .. = T" ~ . x + T .. X k 1 1J 1J.n t 1J
{D. = T k · Xi ~1 J ~
+ T•. X k r.J
By appropriate permutation of the indices, and combination of the results, we then get the remarkable formula
that is, the first moments of the stress are likewise determined by the external loading. The preceding two italicized results imply that the four quantities (9)
99
- 56 -
J. Serrin
are determinate from the loading. Now the functions
are orthogonal over
V, provided that the origin is chosen at the centroid
of V and the axes are oriented in the principal directions. Assuming this to be the case we can apply Bessel's inequality to obtain bounds for the square of individual stress components in terms of the loading. Indeed the quantities (9) are just Fourier coefficients of T 1 which we denote by bo"'" b3 • Thus if
1:' is some particular stress component we have
where
~o
=
/1
~ 1. =
dv = V
f x~ 1
dv •
This lower bound for internal stresses is due to Signorini, who gives a number of examples. Grioli has similarly investigated higher mements. We indicate here, without any particular motivation, the more important known vorticity"average theorems.
fw 1: -I 't.~= 2/0: ti - 2f:· t , I (~~ 1) =f [~(t.it) - -+ if] , • 2 = /grad
1)
grad -:
q2
2)
! r~·grad += f[:'~
3) ~
where div v = 0 in
-
•
~t (:x'~).grad ~J
1) and 2). These. are all essentially kinematic. The
second one is due to Lamb ; the third is basically due to Ertel. (For proof 100
- 57 -
.T. Serrin
of the above results, see At a fixed wall
it. it
-:. 'it = 0
[5] }. , and if the fluid adheres at the wall then
It,
and
are also zero. In these cases, then, the surface, integrals in the abo-
ve identities all vanish, resulting in extremely simple formulas. For example, if we take
1='1
in the last formula, and assume that the motion takes
place within a rigidly bounded region
I~.
that
V we obtain the remarkable theorem
constant
V
Similarly if
~ = S = entropy,
and we consider non-conducting, inviscit:! fluids,
there results
11, V
~
grad S
~
constant, following motion.
We have not yet turned to the energy average, which is in many
respects the most interesting and most useful, especially when used to estimate the energy of a difference motion. We shall treat viscous incompressible fluids first, then insert a section on compressible fluids, and finally return to incompressible fluids, where we shall discuss the general theory of the initial value problem. Our interest here will be in the application of averaging methods to the initial value problem, and in particular to the corresponding questions of uniqueness and stability. Consider a bounded region
Q = Q(t}
occupied by
a viscous incompressible fluid, with prescribed velocity distribution on the boundary
~Q
. In the case of greatest interest
Q is bounded by (pos-
sibl}' moving} material walls and th,e boundary conditions arise from the alil,I'rencE' condition at the walls. We now ask, is the fluid motion under these cumstances uniquely. determined by the initial velocity distribution 101
C:l'-
~J(X},
- 58 -
J. Serrin
and if so, is the motion stable with respect to perturbations of the initial state? The basic technique in this study is an identity expressing the rate of change of total energy. In studying this and related problems, it is convenient to begin with the Navier-Stokes equations in
linear space of vector functions basic region
11
,=,
appropria~y averaged form. Let ~ denote the (x, t)
which are divergence free in the
and vanish on the boundary of
11. Taking the Navier-Sto-
kes equation in the form
.a. . - f~= -grad PIe.
+ YAv ,
~
multiplying through by raging over
¢and forming the scalar product, and finally ave-
11, we get
( 10)
which is, of course, nothing more than our original identity expressed for the Navier-Stokes equation. Now let
v
and
each satisfying the
tfIW
v
~iven
be two
possib~y
different velocity fields in
boundary conditions. Let
~
u =v - v
11 ,
be the pertur-
*'
bation velocity field, and
K
1 =K(t) ="2
the perturbation energy over (11)
*
dK dt
=
f
I
u2
11. Then we have the important formula
(u. grad u • v -" grad u : grad u).
From here on, we shall generally omit writing the arrow over a vector quantity. 102
- 59 -
J. Serrin
,...,
Proof. Writing (10) also for
v, we have
Hence by subtraction
{[ +·(": -
aj +"lgrad
Now suppose that ;
f
=u
IV u • (a - a)
~
grad u
J
"0.
. Then one shows easily that
=
r
1 )u 2 ("2 ~ -
u • grad u , v)
and the required result follows at once, using the transport theorem. #\J
THEOREM 1. Let v and v be two continuously differentiable solutions of the Navier-Stokes equations in
n
,both assuming the given boun-
dary data. Then
Jl:t
K~ Ko exp(y
Here
K o the speed
- Cl(
V
2
2 /d ) t/-j
.
= K(O)
is the initial perturbation energy,
tv,
of the basic flow in the time interval
meter of a sphere containing
n
,and
V
is the maximum of
(0, t) , d is the dia-
ot is a pure number,
Remarks. Before proving this result we observe that it implies simu1taneous1~ Q,
uniqueness theorem, and a stability theorem. 103
- 60 -
J. Serrin
1) If the flows
v and
Ko = 0 • Consequently
-
v have the same initial data, then obviously
K=0
and so
""
u=v-v=O, that is, the two flows must be identical for all ved by
t
~O
• This was first pro-
E. Fo~ in 1930.
2) Suppose that in the time interval dV
.
(0,00)
we have
r:::
Re = ~ < V 80 :: 8,98 then where ~. > 0 Consequently
K .... 0
as
t
-+ 00
and the basic motion
v
is stable in the
mean with respect to arbitrary disturbances in the initial data. We have called 8.98 the Reynolds number for universal stability." 3) Small change in data causes small changes in the solution; the problem is thus well-set in the sense of Hadamard (C - H, pp. 226 ff. ) Proof of theorem. This hinges on the inequality
ot d
-2J u2
~
Jlgrad u:
grad u ,
01
~80 ,
which was recently demonstrated by Payne and Weinberger. Although we shall
*'
assume this inequality without proof, it should be clear that it holds for
~
It should be emphasized here that the stability here is with respect to arbitrary disturbances, whereas in the usual linearized theory one obtains stability only with respect to infinitesimal disturbances. 104
- 61 -
J. Serrin
at
> 0 , according to the well known fact that one can estimate the L2 norm
of a function in terms of the
L2
norm of its first derivatives (Poincar6's
inequality). Now it is clear from the Cauchy inequality that
hence we obtain easily from (11) dK dt
1
~·2~
1.
~ 2"1
f
2 2 2 (u v - ~ grad u: grad u)
r
2..2. 2 -2 2 (u-V- - ~~ d u)
1 2 2 2 - ~ (V - \l('Y / d ) K •
Integrating this differential inequality yields the required estimate. I observed in a paper several years ago that the stability of the basic flow
v
number
could be reduced to the variational problem of determining the least 4IV
""'J
such that
f(~
grad u : grad u - u-grad u·v)
~O
for all divergence free vector fields u which vanish on the boundary of
n.
Since the integral is homogeneous in u , and since
f where
D
u, grad u • v = -
r
u • D - u
,
is the rate of deformation tensor, we can obviously reformulate 105
- 62 J, Serrin
the problem as that of determining the maximum of the integral
subjedt to the side conditions
f
div u = 0,
grad u : grad u = 1.
The corresponding Euler-Lagrange equations for this variational problem have the remarkably simple form u • lJ = - grad ~ + -oJ (12)
*~
A.\,
div u = 0, Here
~ = ~ (x)
and
u = 0 on
-V.
·;an.
are the "eigenvalues" of the problem. We
now have the basic
t'J
I
THEOREM 2. The motion is stable if V) ~ ,where ~
is the grea-
test eigenvalue of the problem (12). Proof.
By standard methods of the calculus of variations we know
~
there is a maximizing vector N
U
is a solution of (12). Let
...,
suppose
~
for the integral
-, U' D. u,
be the eigenvalue corresponding to
and that ttl
u .
-V is the greatest eigenvalue,
We show first that Indeed let
IW
u be any other solution of (12) with eigenvalue "'tI., and
u (as well as
I
grad u : grad u = 1 •
J
~.,
~) normalized so that
Then
.f
u· D· u'
I
u • (grad
X....,+ tJ
Similarly we find 106
u) • ...,
grad.: grad u'
- 63 -
J. Serrin N
Hence since
';f
,.
...,? ""I
solves the variational problem, we have
,and
"oJ
V is the greatest eigenvalue. Now we shall show that the motion is stable if ~
se, since
~
is the maximum of
-
f
u • D. u
N
> ~ . But in this ca-
under the side conditions,
it is clear that
f(~grad u : grad u -
u • D· u) > O.
-v The motion is therefore stable if II> 11. If one wishes tu consider the stability of a specific flow, it is clear that
the variational approach will provide in general a better criterion than that qf Theorem 1. The only flow for which this program has been carried out,
however, is that of C01.lette motion between circular cylinders, and even here the results are fragmentary. Further research into this problem, especially with regard to the dependence of the stability on the spacing between the cylinders woold seen to me of the greatest importance. The preceding uniqueness theorem (c. f. Remark 1) applies only to a bounded region
n.
There is a corresponding result for an unbounded re-
gion which is available under either one of the following hypotheses: I. The velocity contained in the set
V
0~ t
~
T
,and also is
V (d. Chap. V).
II. The quantities
o~ t ~ T
is uniformly bounded for
v, and grad
v, are uniformly bounded for
,and in addition the pressure
p
tends to a limit
p
o
at infiniti-
ly in such a way that
I p - pol where
A and
e
~ Const.
Ix I
-j. -f
are fixed positive constants.
More precisely, we have the following theorem: Jet 107
n = n t~)
deno-
- 64 -
J. Serrin ~
te the exterior of a bounded region in space. Let
v and v be two solutions
of the Navier-Stokes equations in {) ,having the same initial data and the same boundary data on
~ {)
. Suppoae also that
hypothesis I or hypothesis II. Then
ItI
v for
'i ~
v and 0 ~
~
satisfy ~
t~T •
This theorem, whose proof we omit, is due to Leray (Hypothesis I) and to Graffi (Hypothesis II). fa C. We shall conclude our discussion of energy averages by considering the uniqueness and stability of the initial value problem for a compressible inviscid fluid. Except that the physical properties of the fluid are now quite altered, this is preci"';ely the problem which we have just treated. Moreover, the mathematical techniques involved in the present discussion are essentially the same as before, though the proofs do take a more complicated appea" rance. The important point to be made is that the method applies almost equally well, in spite of the added physical complications. The appropriate differential equations governing the motion are well known, namely dt + dt ( 13)
e div v = 0'
E' (a - f) + grad p = 0, dS dt = 0,
together with an equation of state
e
.,
(S
~
entropy)
p = P (€I, S) • We assume that the denSity
is always positive, and that
It is my opinion that an even stronger result should hold, probably requiring no hypotheses either on grad v or on p. For a similar result concerning parabolic equations, one may consult a paper of Rosenbloom in Contributions to Partial Differential Equations, Princeton, 1953 108
- 65-
J. Serrin
corresponding to a real speed of sound in the fluid. Now suppose that
e : e (x, t),
tions for
v = v(x, t),
S = S(s, t)
is a given solution of the above equa-
It is W.ell known that the system (13) is hyperbolic. For this
t~O.
reason, if K denotes a come-like region ir.. space time, with cross section Q(t) , such that the mantle (lateral boundaY'y) of
K is a characteristic sur-
face, we expect that any other soluticn ,..,
~
N
.....,
, S, v
of the flow equations which takes the same initial values on
Q(O)
,will be
identical with the original solution. The situation is shown, for two space dimension~,
, in the accompanying figure. The charaderistic condition at the
boundary of
Q(t)
is
von
=G + C
,
where G denotes the outward
_-.JJ. (e)
Jl(o) normal velocity of the boundary, thinking of this as a moving surface in space." Thus the boundary of the basic motion
e,S, v
Q has the property that at each point the fluid il
n
is leaving
respect to the velocity of the boundary. 109
with precisely sonic speed with
- 66 J. Serrin
THEOREM. Consider a solution
f, s, v
of the system (13) in a region
K of space-time bounded by the plane t = 0 and a mantle ~ on which v.n = G + c . Suppose that for
ItIN'lI
f
,S, v is another solution in K which assumes
t = 0 the same values as the first one. Then the two solutions are identi-
cal in K,
[14]
We may remark that this theorem is the analogue of the well known uniqueness theorem for the wave equation, where the solution is determined by its initial values in a certain retrograde light cone. In fact, this latter theorem is also proved by an application of the energy averaging method, albeit an ex'· tremely simple one, d. Courant-Hilbert, Vol. II, pp.642-646. In this connection I should also point out the large body of work on symmetric linear hyperbolic systems initiated by the papers of K. O. Friedrichs, where the idea of energy identities again plays a crucial role. I believe that some of this material was covered in the CIME lecture of R. S.Phillips earlier this summer. Although this work has considerable generality it must be noted that it does not aplly to the present case, since the equations (13) are neither symmetric nor linear. We may now turn to the proof of the theorem. We observe, to begin with, N
,.,
..,
that the alternate flowe, S, v satisfies (13), that is
'X (~ yo + ~ , grad ~ - f) + grad ~ ~t
p= 0
?s ~ t + v • grad S = O. N
,.,
Let us subtract from each of these the corresponding equation for the basic flow, multiply the resulting equations respectively by 110
- 67 -
J. Serrin N
v' = v - v, and integrate over a cross section
S' = S - S,
)2(t). (In essence, this is the procedure
already successfully applied in the incompressible case.) The resulting equations are unfortunately fairly complicated, so we shall treat in detail only the last one. In this case, subtraction yields
d~ S' Multiplying by
S'
+ v • grad S' + v' • grad S = O.
then gives
1 ~ S,2 2 - ( - - + v , grad S' ) + S'v'. grad S = 0
2
'j) t
or simply dS,2 dt
Multiplying both sides by
:t
f@ 5,2 ~
-/
?
d
(ill = ~t + v
+ 2S'v' • grad S = 0,
e and integrating over
~5'v'
• grad 5 -
)2(t)
fe
f I ~ -Ie f
~
=
~
dt
(v • n - G)f •
But 2 E? S'v' • grad S
~
then gives
(v • n - G)S,2
where we have used the transport theorem in the form
~ dt
• grad).
2
2
const. (v' + S' ),
whencE: with the help of the given boundary conditions, 111
- 68 -
J. Serrin
Carrying out the same procedure for the equations of motion, and incidentally using the continuity equations at a crucial point, yields
-dt f e d
2
c (v' + -
2
e2
- f[e where
~
=
(~pl J S)e
2
2(1( +-
2
c
D'
~
c(v'
• e Sf) < Const. e2'"
2
2
+ ~ ("
f .
(b' t
2~,
+~
2
~ Sf) + 2(c
2 2 + v' + S' )
e
2 I
+ ~ S')v' • nJ '
is a thermodynamic variable. Now multiply the
S'
equation by a large constant A and add it to the one just written, to obtain
-
d dt
f
tl(v' \;
2
2
e
c 2 2q 2 +-e' e2 + -2 e'S'+AS')
2 1.[ 2 c -T ec(v' + e2
ee
e
2 2t ] ~' 2 + 2~' 2 S' + AS' ) + 2(c +CI(S')v'. n ,
where the Const. is perhaps different than before, but a constant nonetheless. Define
2 J = ecJ + 2v' • n(c We assert that
A
e +t,f Sf). I
can be chosen so large that both 112
- 69 -
J. Serrin
eJ
~
2 2 2 Const. (,I + VI + SI )
(Const. >0)
-
and
J ~O •
Assuming for the moment that this can be done, the proof is completed by observing that then
df eJ
Cit Integrating from
0
f(e
~ Const.
to
t
I2 + VI 2 +2 SI ) ~ Const. /
eJ.
yields
J~eJ ~ (}eJ)t=O
e
Const. t,= 0
.
eJ } 0 ,it follows that eJ = 0 . , e= VI = SI , and the two flows are identical.
Thus since
This in turn obviously implies
The proof will thus be completed as soon as we verify the assertion. Since ,
and
c
are both continuous positive functions they have positive
upper and lower bounds in
K during the time interval
assertion is then clear, for the constant
2
eel c
2
+
T201
,
~ SI
+
PAS'
2
A can certainly be chosen so that
~ Const. (E>'
2
e, r
2
+ S' ).
For the second assertion we observe that'
Now consider the two quadratic forms (in 113
0 to t • The first
VI ( , Sf)
- 70 J. Serrin
Certainly and
J
equals either
J1
or
J2
J2 are non-negative for suitably large
If we can show that both
A • then
J1
J} 0 and we are
done. But by a direct calculation one finds that the eigenvalues of both J1 and
J2 are
Hence if J 1 }. 0
A is large enough all the eigenvalues will be non-negative. Thus
and
J2 ~ 0
for large
A • and the proof is complete.
114
- 71 J. Serrin REFERENCES
( 1]
CHAPTER IV
A.Signorini, Sopra alcune questioni di static a dei sistemi continui. Ann.
-
Scuola Norm. Pisa 2 (I933), 231-257. ( 2]
A. Signorini, Alcune proprieta di media nella elastostatica ordinaria.
-
Rend. Lincei (6) 15 (1932), 151-156. [3]
G. Grioli, Limitazioni pE'r 10 stato tcnsionale di un qualunque sistema con-
-
tinuo. Ann. di Mat. 39 (1955), 255-266. [4]
C. Truesdell, Kinematics of Vorticity. Indiana University Press, 1954.
(5)
J. Serrin, Mathematical principles of classical fluid mechanics. Handbuch der Physik,. Vol. 8/ 1. ~pringer, 1957. Especially~t17, 26, 28.40,72.73,74.
[6]
E. Foa, Sull'impiego dell'analisi dimesionale nello studio del moto turbolento. L'Industria (Milan) 43 (1929), p.426.
[7]
-
.J. Sel'rin, On the stabU ity of viscous fluid motion. Arch. Rat. Mech.
Anal. 3 (1959), 1-13. [8]
L. Payne and H. Weinberger, An exact stability bound for Navier-Stokes flow in a sphere. Nonlinear Problems, p.311-312, edited by R. E. Langer. Univ. of Winsconsin Press,I963.
(9]
D. Graffi, Il teorema di unicita nella dinamica dei fluidi compressibili.
-
Journ. Rat.l\1ech. Anal. 2 (1953), 99-106. [10]
D. Graffi, SuI teorema di unicita nella dinamica dei fluidi, Annali di Mat.
-
50 (1960), pp.379-388. [111
D. Graffi, SuI teorema di unicita per Ie equazioni del moto dei fluidi
COllJ-
pressibili in un dominio illimitato, Atti Acad. Sci. dell'Istituto Bologna,
-
Sci-Fis. Classe, Series XI, 7 (1960), pp.1-8 • 115
- 72 J. Serrin
[12]
D. Graffi, Ancora suI teorema di unicit~ per Ie equazioni del moto dei fluidi, Atti Acad. Sci. dell'Istituto Bologna, Sci-Fis. Classe, Series XI, ~
[13J
(1961), pp. 7-14.
D. Edmunds, On the uniqueness of viscous flows, Arch. Rat. Mech. Anal. (to appear 1963).
[14]
J.Serrin, On the uniqueness of compressible fluid motions. Arch.Rat.
-
Mech. Anal. 3 (1959), 271-288.
(15J
K. O. Friedrichs, Symmetric Hyperbolic linear differential equations.
-
Comm. Pure App:. Math. 7 (1954), 345-392. Cf. also J. Leray, Hyperbolic differential equations. Lecture notes, Institute for Advanced Study, Princeton, 1952.
116
- 73 -
J. Serrin
V. THE INITIAL VALUE PROBLEM FOR THE NAVIER-STOKES EQUATION
As has been evident throughout this course, a selection of material frequently had to be made. In illustrating the comparison method it was necessary to restrict discussion to two or three main areas, with just passing mention being given to other applications of the method. Nevertheless, an attempt was mad,e to include examples which ulluminated the more important ideas, while still staying near some physical problem of interest. In the same way, in discussing averagin;, methods a choice has to be made. Because of the remarkable amount of interest in the initial value problem for the Navier-Stokes equations, and because it illustrates very well the application of space averages to a difficult nonlinear problem, we have chosen this as the main object of st'ldy in the final lectures. In what follows I shall confine myself for the most part to a somewhat restricted version of the problem. Consider a fixed (bounded or unbounded) domain
n
in
En
,where
n = 2 or 3 . The domaln
n
may be thought
of as a rigid vessel filled with incompressible fluid; the fluid is initially setinto motion, and (as in the preceding lectures) we are interested in the subse;. quent motion, subject to the ;..J'avier-Stokes equations and the condition of adherence at the boundary of ty field for
v
t >0
n. More precisely, it is required to find a veloci-
= v(x, t) and a pressure field p = p(x, t) which for x e nand satIsfy the differential equations
vt + v
grad v = -grad p +
/J.v
div v = 0,
and obey in come sense the initial condition 117
- 74 J. Serl'in
v(x, 0)
= vO(x) ,
and the boundary condition x E ;; n
v(x, t) = 0 , In these equations we have set the external force
f =0
and the density and
viscosity equal 1 for simplicity. Historically, it was C. W. Oseen and J. Leray who, some thirty years ago, first interested mathematicians in this problem; their work, moreover, still retains its importance. More recently, in 1951 Eberhard Hopf di ... covered that the problem could be studied using
method~
of functional analysis, and several years later Kiselev and Ladyzhenskaya published a further paper which set the modern trend in the subject. Many papers have appeared since then, extending the work of these authors. We shall begin our discussion by considering the notion of ~?ak solution, which is fundamental to all the modern work. Next we shall reviev. the fundamental existence theorems of Hopf, and of Kiselev and Ladyzhensk'.ya. In the final lecture we shall consider the more delicate problems of regularity ,"1d uniqueness naturally associated with the concept of weak solution. In this wilri\, which is in many ways rather abstract, I think you will see that the
undt~rlying
idea is just that discussed in our opening lecture on averaging, namely, that
it
is valuable to consider the differential equation in an integrated form, and that the energy average is a natural tool to use in the theoretical investigation of a dynamical problem. ~
The program outlined above is fairly sophisticated, and requires a
certain degree of preparation. We shall therefore begin with some definit ions. Let
u
= u(x, t) , v = v(x, t) be vectors defined in R = n K lO,
We write 118
ex)
J.
- 75 J, Serrin
for the spatial inner product of t
u and
v, Clearly
;when we wish to make the dependence on
the product in the form
t
°
(u(t), v(t)) • A vector
(u, v)
is a function of
more explicit we shall write
v will be called weakly diver-
gence free if (v, grad for every function
r) = 0
r
'I' =
shes near the boundary of div v
(x)
0 ~ t < Q() which is continuously differentiable and vani-
n,
Clearly if v
is differentiable and satisfies
= 0 then it will satisfy the preceding condition; however, it is also clear
that a vector can satisfy the condition without having any derivatives in the ordinary sense, Now let ble vectors dary of
n,
:;n (R)
denote the family of all twice continuously differentia-
+:: 0 ,
~ (x, t) in R such that div A vector
v
and
4> = 0
near the boun-
= v(x, t) \till be called a weak solution of the initial
value problem if it is weakly divergence free and if
= (v(T), for each
T >0
and each vector
+
t/> 0)
(T)) - (v 0'
+~ €
(R
o )
:
°
,
Here
f
0
= ~ (x, 0) ,
It is easy to see that this equation results from a direct averaging pro-
cess on the Navier-Stokes equations, where
"*
t
(x, t)
.
is the weighting funt:iion,
d, Chapter IV, Part A • Thus any solution,of th\llTavier-Stokes equations
*
In particular, one multiplies the Navier·Stokes -eqUation by the vector ~ , integrates over n , and then finally integrates with respect to t from 0 to T ,The integral form then results at once if we observe that, sillce , / ' 119
- 76 J. Serrin
also satisfies this integral form of the equation. On the other hand, as already remarked previously when we discussed vectors which are weakly divergence free, it is clear that a vector may be a weak solution of the Navier-Stokes equations without being an ordinary solution. Finally, if v is a weak solution and if v has continuous derivatives, then
v is an ordinary solution. To
see this, we merely have to reverse the steps by which the integral equation was obtained, and use the well known device of the calculus of variations by which the Euler equation is obtained from the variational condition. The important point observed originally by Leray is that it is easier to prove the existence of a weak solution then to prove the existence of an ordinary solution. Of course, the problem remains whether a weak solution can be considered as a genuine fluid motion, but at least the original problem is now reduced to two parts, each of which can be considered separately. One final definition is necessary before we can state then major results of Hopf and of Kiselev and Ladyzhenskaya. Definition. A vector and only if for each
v = v(x, t)
T, 0 < T < 00
,
will be said to be in the clas!:; it is in the closure of ~ (R)
V if under the
norm
(grad p"
) = 0,
(v. grad v,
t) ='(v. grad ~, v),
(A v,
f) =(v, 4~),
and
(T
T
10 (vt , +)dt ." f 0 (v, ~ tId! + (v(T), ~ (T)) " (v0' ; 120
0)'
- 77 -
J-. Serrin
where D
+ =I 2
grad
+: grad f dv
We observe that qny vector in
V
has zero boundary data in the genera-
lized sense, that is, it is the limit in norm of continuously differentiable fun-
•
ctions which are zero on the boundary of
n. _Moreover
if
v 4i V
then v
has a generalized gradient, denoted by grad v , which is defined to be the 1imit in the norm of the corresponding tensors grad
t.
It can be shown that such
generalized gradients obey the ordinary rules of calculus, though we shall not need this fact here. (The calculus of generalized derivatives is discoussed in many places; the reder may be referred specifically to references
[7J - [8].)
THEOREM 1 (Hopf). For any weakly divergence free initial vector field of
v0
which vanishes near
n
there exists a weak solution
vE V
of the
initial 'lalue problem. Moreover,
1 'v ,2 + It, 2 1, 2 0 Dv I dt ~"2 v I
"2
0
that is, the sum of the kinetic energy and dissipated energy is less than or equal to the initial kinetic energy. The proof is a beautiful application of the technique of Fourier approximation, unfortunately too long to include here. Nevertheless, we may observe that the process succeeds precisely because the original problem admits a formal energy identity
[A proof, assuming that
v
is a continuously differentiable solution, follo\\"s 121
- 78 -
J. Serrin
by integrating the energy transfer formula (8)
that is dtd
('21
I
12 = - Dv 12 . tV)
]
Two things should be noted. First, that is the a priori boundedness of the energy and stress averages which makes the proof work (it is exactly the quantities on the left side of the energy identity which are the building blocks of the space V ). And second, that the rigorous proof given by Hopf yields only an energy
inequality, not an energy identity. The fact that the energy identity leads to a weak solution in the space V, leads us to expect that if we can obtain stronger a priori estimateF for the norms (averages) of a solution, then we can correspondingly obtain the existence of a solution with more nearly classical behavior, that is, one which is not as "weak" as the solution found· by Hop!. This is, in fact, exactly what Kiselev
•
and Ladyzhenskaya did. Their r,esult is as follows. THEOREM 2. For any twice differentiable initial vector field vanishes near
n
,there exists a weak solution
problem, and a positive number formly bounded in the interval
T such that
Vo which
v E V of the initial value
I vt ,
and
I Dv I
are uni-
0 ~ t < T . Moreover
The proof depends on Fourier approximation techniques, exactly as before. We can, however, present the formal procedure by which the necessa:122
- 79 -
J. Serrin
f-
ry a priori estimates are obtained. This is the important part of the proof . By differentiating the Navier-Stokes equation with respect to . t vtt
+ vt • grad v + v. grad vt = -grad Pt + AVt .
Next multiplying by
since
we get
(grad, Pt' v t )
vt
n
and integrating over
= -(Pt'
div vt )
=0
yields
. The previous equation can obviously
be written
We next estimate the size of the first term on the right; thus -(v • grad v, v )
t
t
..
using Holder's inequality, where a theorem of Sobolev the
L4
<'vt , 42 'f
14
=(
'Dv
f
f
4
I' dx)
1/4
• Now according to
norm of a function can be estimated in terms
of the norm of its first derivatives. In particular (cf.
ff= If ..L
3%'
I -/
[12 J)
Of'
If ,ttJ InrI ~t
'* .Indeed, as we have already indicated, it is just these a priori estimates which allow us to carry through the solution process, and which determine the type of weak solution which we obtain. 123
- 80 -
J, Serrin
Hence
..i \Vtl'lbl~I'lbvl
n
1
3»~
\Vtl~ It>"t~ Ib~'
J.-) \ 'It' I~ I bv I~ +\ b~t Jt
Substituting into the earlier formula yields now
~-l \Dvlt ''It-It
i' \bv\1 \vl
•
This is the inequality which in the work of Kiselev and Ladyzhenskaya takes the places of the energy transfer formula in the paper of Hopf. The next step is to integrate this differential inequality. In case
n=2
the result is easily seen to be
Since also
the result for T =
00 ,
n = 2' is completely proved. One even sees that we can take
since the estimates are uniformly valid for all time. 124
- 81 -
J. Serrin
When
n
d , Vt
f
= 3 ,we have using the pret!eding inequality fnv ~ Iv I· ~tl '
I
dt
Dividing both sides by
I ,- 2
_ -1 ( v 2 t
and integrating now gives
-Iv
to
I -2)<2 -8 fv , 2 " 0
t
that is
)
Thus
Ivt I is bounded as long as
Since
IDv 12 ~ 1v I '
Iv t , ,the required estimates are thus proved, with T
being any number less than
7, v .-2 1vto1-2
2
t,
A rigorous proof of the theorem of Kiselev and Ladyzhenskaya remains a technical matter which can be handled by Hopf's method. The main idea of the proof, however, is the a priori estimate which we have just derived. In conclusion, we whish to point out an important open problem. In two dimensions, the a priori estimate holds with
T
= 00 ,and correspondingly,
as we shall see in the final lecture, a genuine solution of the initial value problem exists for all
t > 0 . However, in three dimensions, the a priori esti-
mates last only until some finite time
T 125
,and correspondingly a classical
- 82 -
J. Serrin
solution of the ipit,ial
o~ t < T
• It is (;f:!J!'
t~:e
the~r.f, 02
answer will invoi:lc
physical and theoretical importance to know
ir"L'a~;:st
whether this breakdc ..':;1 0f to an incomplete
problem is only known to exist in the interval
va>(~
elth~r
solution process at the time
T
is simply due
whether it is an inherent part of the problem. The
a new existence theorem (ie. stronger a priori
estimates), or else a C0i.lllterexample to show that classical solutions need not persist for all time.
:"'ej~ay
proposed the problem of finding such an ex-
ample (cf. p. 225 of [4J ~, aEd in fact gave a possible method for its construction. Moreove:., the problem has not yet been solved, and no one knows which way the anSWICI
1v~1l
go.
B. The solutions r-btained by Hopf and by Kiselev and Ladyzhenskaya are of course not necessarilY
soJ.u~ions
of the initial value probleJ? in the sense
origiOally intended. It is thus necessary to say a few words clarifying the relation between tbo::se slJh:tior.s and a classical solution. Let us aSSL!nl'2 to t2gi:l with that
vE V
is a weak solution of the ini-
tial value problem, tilat if:;, that
= (V(T), For each T > 0 u(x, t)
and each
+ dJ E
, (T)) - (Vo) ¢o ),
(R) . Now corresponding to any function
let us introduce the space-time average
uh = uh (x, t) = where the kernel
f
K(
f ,t ) u (x +, ' t +1:) d ~ d 1:' is a smooth non-negative function with the pro126
- 83 -
J. Serrin
perty that
K("
Thus
is a function whose values are averages of
uh
radius
h
1:) ; 0
centered at
outside a sphere of radius h about
u
0, while
over a sphere of
(x, t) • It is clear that the translation of a weak so-
lution is again a weak solution, hence, in abbreviated notation,
f{
{V(x+J
t+'Ir)~t(x.t)+ ••••••• }
Multiplying both sides by
K(
~
dxdt· ....
, 1:') , integrating with respect to
l' ?: ,
and finally reversing the orders of integration, then yieds
provided that Now since and
t
~
vanishes when
vh
t
=0
and
t
=T
•
is continuously differentiable with respect to both x
, the preceding equation can be integrated by parts to give
Thus by standard techniques of. the calculus of variations, since arbitrary function in
fJJ (R)
, we obtain the differential equation
"ht + (tJ. grad V)h = - grad Ph + 127
fJ
Vh .
~ is an
- 84 -
J. Serrin
This shows that the averaged velocity
Vh
is approximately a solution of
the Navier-Stokes equations, or looked at in another way, the velocity v satisfies the Reynolds average form of the Navier-Stokes equation. If we form the curl of the preceding equation, and set
w = curl v , we
obtain the averaged vorticity equation
where the right hand side is defined by its components . (w. v. - w. v')h . . 1 J J 1 ,1 Now let
k(x, t)
denote the fundamental solution of the heat equation. Then
clearly we have the following integral representation
where
Bh (x, t)
is a solution of the heat equation. Finally letting h ~ 0
yields the formula
('*' )
w(x, t) =
where
B(x, t)
f
grad k(x -
~,t - T:)
• (wv - vw) d
l
d t" + B(x, t) ,
is again a solution of the heat equation. We can now state the
fundamental regularity theorem for weak solutions of the Navier-Stokes equation [9] • THEOREM. The weak solution of Kiselev and Ladyzhenskaya is continuously differentiable in the space variables, and Lipschitz continuous in time, and satisfies the Navier-Stokes equation almost everywhere in
R = n x (0, T
J.
The proof is too long to include here, but the main idea is to consider (if) as a linear integral equation for
w ,the function v being conside-
red fixed. For the details of the proof the reader is referred to reference 128
[8] .
- 85 -
J. Serrin
It may be added that the same process fails for the Hopf solution because in
this case the kernel of the integral equation (which involves
v
) is not suf-
ficiently regular. Thus the additional properties of the Kiselev-Ladyzhenskaya solution are seen to be of crucial importance in finally establishing the existence of a differentiable solution of the Navier-Stokes equation. If the boundary of
n
is smooth then stronger conclusions can be ob-
tabed. In particular, Ito has shown by quite different methods, the existence of a classical solution continuously taking on the given boundary values. However, his proof arc extremely complicated and one w{)uld like to obtain his results by means of the relatively simpler methods outlined here. To conclude the course, it may be of interest to review some of the open problems which we have noticed. 1) To extend the comparison method in free boundary theory to non-symmetric flows 2) To exploit more fully the variational approach to the stability of la minar fluid motions 3) To obtain stronger uniqueness theorems for the initial value problem in e;.."j:erior domains. 4) To obtain the existence of a suitably regular solution of the initial value problem for the Navier-Stokes equation in 3 dimensions which persir;ts for all
t >0 •
5) It would finally be worth while to be able to prove the existence theorems of Hopf and of Kiselev and Ladyzhenskaya. using fixed point methods (cf.
(10] ) rather than Fourier approximation. There are of course countless other problems still open in the applica-
tion of comparison and averaging methods, and it is hoped that some of at least will find this a fruitful field of study. 129
yl)~l
- 86 J. Serrin REFERENCES : CHAPTER V [ 1]
C.Oseen, Neuere Methoden und Ergebnisse der Hydrodynamik, Leipzig, 1927
,
J. Leray, Etude de diverses equations intt!grales non-;tin6ares et de quelques Appl • [ 3]
probl~mes
..!!.
que pose I'Hydrodynamique, Journ. Math. Pures
(1933), pp. 1-82.
J. Leray, Essai sur les mouvements plans d 'un l,iquide visqueux que limitent des parois, Journ. Math. Pures Appl.l!. (1934), pp.331- 418.
[4]
J.Leray, Sur Ie mouvement d'un liquide visqueux emplissant l'espace, Acta Math • .!!. (1934), pp. 193-248.
[5]
E. Hopf; Uber die Anfangswertaufgabe fUr die hydrodynamischen Grundgleichungen, Math. Nachrichten! (1951), pp.213-231.
[6]
A. A. Kiselev and O. A. Ladyzhenskaya, On existence and uniqueness of the solution of the nonstationary problem for a viscous incompreSt '
rJ 7
sible fluid, Izvestiya Akad. Nauk SSSR
.!! '
(1960).
L. Nirenberg, Remarks on strongly elliptic partial differential equations. Comm. Pure Appl. Math.
[9]
(1957), pp. 655-680.
C. B. Morrey, Multiple integral problems in the calculus of variations, Ann. di Pisa
(8]
!!.
!. (1955).
J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rlt-tional Mech. Analysis!. (1962), pp.187-195. 130
• 87 J. Serrin
[10]
D. Gilbarg, Boundary value problems for nonlinear elliptic equations in n variables. Nonlinear Problems, edited by R. E. Langer. Univ. of Wisconsin Press, 1963.
[11]
S. Ito, The existence and uniqueness of regular solution of nonstatio .. nary Navier-Stokes equation. Journ. Fac. Science, Univ. of Tokyo,!.. (1961), pp. 103-140.
~ 2]
J. Serrin, The initial value problem for the Navier-Stokp.s equations. Nonlinear problems, edited by R. E. Langer. Univ. of Wisconsin Press, 1963. (This paper contains a large bibliography.)
131
CENTRO INTERNAZIONALE MATE MATICO ESTIVO (C. I. M. E. )
H.ZIEGLER
THERMODYNAMIC ASPECTS OF CONTINUUM MECHANICS
ROMA - I6Itituto Matematico dell'UniversiU 133
THERMODYNAMIC ASPECTS OF CONTINUUM MECHANICS by
HANS ZIEGLER
1. Classical thermodynamics, Recent developments have made it clear , that continuum mechanics cannot be separated from thermodynamics. In the second half of the last century statistical mechanics has been created in order to provide a mechanical basis for thermodynamic phenomena. Today the process is being reversed: we are turning to thermodynamics and statistical mechanics for the explanation of certain aspects of continuum mechanics, The borderline is inevitably reached in any attempt to give a complete outline of the fundamental laws of continuum mechanics, In this first section we shall formulate the laws of
class~cal
thermodyna-
mics in a manner fit for use in connection with mechanical problems
[IJ .
Let us consider a system the state of which is completely described by the mechanical coordinates
xk (k = 1,2, , , , '.' n)
and the tempf'rature
8 > 0, (The state of an infinitesimal element of 'an elastic or a ri~id/perfectly plastic body, e, g., is completely described by the strain components and 8). The
xk
and
8
£kl
are the independent state variables; any function
of them will be called a state function. If the work done on the system is given by (1)
(1. 1)
the
Xk
are the forces corresponding to the mechanical state variables x k'
(For a volume element under infinitesimal strains the forces corresponding to the
Ekl
are the stress components
CSkl) .
The first fundamental theorem states that there exists a state function
U (x k '
8),
called the intrinsic energy of the system, such that
(1) We shall use the summation convention. 135
- 2H. Ziegler
(1. 2)
where
dQ
is the influx of heat.
The second fundamental theorem states that there exists another state function
S(X k , 6),
called the entropy of the system, such that
> dQ OdS ..
(1. 3)
.
If (1. 3) holds with the equality sign, the process is referred to as reversi-
ble,
otherwis~
as irreversible. The theorem ca.n also be stated in the form
(1. 4)
due to Carnot and Clausius, where
(1. 5)
is called the influx of entropy and
(1. 6)
the entropy production inside the system, zero for reversible processes and positive for irreversible ones. The last statement justifies the use of the superscripts rand i for the reversible and irreversible parts of the process. From (1. 2), (1. 5) and (1. 4) we deduce (1. 7)
dW = dU - dQ = dU - Od(r)S = dU - 6dS + 6d(i)S 136
- 3-
H. Ziegler On account of (1. 1) and the fact that
U and
S are state functions, (1. 7)
is equivalent with the relation
(1. 8)
which thus is a direct consequence of the fundamental theorems and hence is valid for any process. For pure heating or cooling dXk:ll O. In this case (1. 8) reduces to
(1. 9)
On account of (1.6) the second therm is non-negative, while the quantity between brackets is a state function and hence is independent of dB (1. 9)
must hold for positive and negative values of dB
. Since
,it follows that, in-
dependent of the type of process,
(,I. 10)
and that, for the process
consider~d here, d(i)S = 0,
i. e., that heating
and cooling are reversible phenomena. The differential equation (1.10) establishes a connection between the intrinsic energy and the entropy of the system. Making use of (1. 10) and of the notations
(1.11) 137
- 4H. Ziegler
and x
(l. 12)
_ X(r): X(i) k
k
k
we obtain, instead of (1. 8) , 9d(i)S': X(i) dx
(1. 13)
k
k
~ 0
where the statement concerning the sign follows from (1. 6) • Inserting (1.5) and (1.13) in (1.4), we also have (1.14)
Accordin~
x~r)
and
to (1.12) each force
~).
~
appears as the sum of two terms
On account of (1.13) the entropy production inside the sy-
x~i). It is therefore reasonable to refer to the X~i) as the irreversible forces and to the X~) as the reversible ones.
stem is completely determined by the
If, as an additional state function, we introduce the free energy
(1. 15)
F : U - 9S,
we obtain from (l. 1l) and (1. 10) (1. 16)
OF _X(r)
~F -
~ --lc
~
9
=-S
.
Apart from its sign, the free energy thus serves as a potential function for the reversible forces and the
neg~tive
entropy. It follows that the reversible
forces are state functions. Sometimes a process is conducted in such a way that 138
9 is a given
- 5H. Ziegler xk . (In an isothermal process
function of the process
S = const ). In such cases
-
F
(3
= const
,in an isentropic
has the properties of a mechanical
potential. It follows from (1. 13) that any
sponding dX k
x~)
changes sign whenever the c9rre-
is reversed. In consequence the
but depend on the velocities
xk
x~)
are not state functions,
. Besides, they may depend on the state of
the system and on its history. Classical thermodynamics does not provide any clue as to the dependence
X~i) (~j)
• For linear relationships between the velocities and the irre-
versible forces,
(1.17)
J
Onsager [2 has established the symmetry relations
(1. 18)
They are based on a statistical treatment of systems moving freely in the vicinity of an equilibrium configuration. Onsager1s demonstration makes use of a principle of microscopic reversibility and of some additional assumptions. In continuum mechanics many processes are irreversible (particularly
on account of interior friction). However, we are usually not concerned with infinitesimal free motions about an equilibrium position, but rather with finite processes taking place under given forces (e. g., with the deformation of an element of a plastic body under prescribed stresses). In the more interesting cases the relationship between velocities and irreversible forces is not linear. Sometimes (e. g., in a plastic body) it is even impossible to linearize it. There exists thus a definite need for a generalization of Onsager1s theory. 139
H. Ziegler
In fact, classical thermodynamics is little more than a theory of thermostatic equilibrium, restricted to certain special cases, and a really dynamic theory does not exist in this field. This has been emphasizend by Truesdell
[3]
in the following wards: "It is suggested that an attempt be made to crea-
te and organize the logical structure of a true thermodynamics of irreversible processes along the lines successfully employed two hundred years ago by Euler and others in converting the unorganized special methods and principles of seventeenth-century mechanics into t,he general theory we know today. "
2. Additional principles. In this section a possibility of realizing Truesdell's program will be described. It consists in a generalization of Onsager's principle [1,4,5, 6J ,limited to processes which are slow compared with the motions of the molecules involved. In a system of the type considered in Section 1 the rate of entropy produ-
ction
d(i)S/ dt
depends on the velocities
~k'
on the state of tIle system
and possibly also on its history. On account of (1. 13) the rate of dissipation work is
(2.1 )
In a given state of the system, preceded by a given history, ction of the velocities
x•k
p(i)
is a fun-
alone and hence can be written
(2.2)
The function
D(x•k ) is referred to as the dissipation function of the system.
It must be considered as the primary quantity in an irreversible process and 140
- 7-
H. Ziegler
is of similar importance for the irreversible part of the process as the state
U
F
functions or are for its reversible part. The irreversible forces X(i) k are secondary quantities, connected with D by the relation
(2.3)
following from (2.1) and (2.2) • Let us interpret x•k (Fig. 1) of
D(x• k) as a
fu~ction
in a euclidean
velocity space
n dimensions, and let us assume, for convenience, that
(6]).
D(X k ) be sufficiently regular. (For a more exact treatment see
dissipation function may be visualized by means of (hyper- )surfaces where
~k
M
and the
D(xk)=M,
belongs to a set of non-negative constants. Furthermore, the
X~i)
define two vectors in velocity space.
x•k and
What we are looking for is a connection between the vectors
X~i)
The
. From the first fundamental theorem we obtain no statement ci. use for
this purpose. The second one yields the inequality (2. 1) ,implying that the scalar product of the two vectors is non-negative. In order to establish a more definite connection, let us stipulate the following Principle of least irreversible force: If the value pat ion function
M >0
of the dissi-
D(x k) and the direction of the irreversible force
prescribed, the actual velocity
xk minimizes the magnitude of-
subject to the auxiliary condition (2.3).
,
In other words: Among all vectors
xk with end points
P
X(i) are k X(i) k
on a given
D-surface, the projection of the real one (or ones) in the direction of
X~)
is a maximum. It follows immediately that
X~)
is normal to the D-surface at
P.
A great deal of additional implications can be derived from the principle of 141
- 8-
H. Ziegler
least irreversible force. In the remainder of this section some of them will be discussed without proof. (For the proofs see [6] ). Provided the prir1ciple is valid, the surfaces
D(X k)
Each one of them contains those with smaller values of
X~i)
the origin. It follows that the vector normal at
P. The function
= M are convex. M and hence also
has the direction of the exterior
D increases monotonically on any radius
from 0 . If the increase of
of
X~)
D on any radius from
0 is sufficient, the projection
on the radius also increases. Let us restrict ourselves to systems
subject to this condiLon and let us denote them as stable, for it can be shown that, whenever the condition is violated, self- sustained oscillations are apt to develop. In a stable system the last principle is equivalent with the following. Principle of maximum rate of dissipation work: If the irreversible force
X~i)t 0 is prescribed, the actual velocity
xk '
subject to ;.he auxilia-
ry condition (2.3), maximizes the rate of dissipation work. On account of
(2.1) this principle can also be formulated as a princi-
ple of maximum rate of entropy production. In this form it appears as a natural and physically particularly plausible extension of the second fundamental theorem. Another consequence of the principle of least irreversible force is the inequality (2.4)
X(i) (x' _~ff') k
k
k
>0
:: ,
valid for the actual irreversible force any other velocity
x:
with
of the connection between
xk
X~)
,th: ~ctual velocity
~k
and
D(~:); D(x k) . Still another representation and
X~i), based on the assumed regularity 142
- 9H. Ziegler of the function
.
D(x k )
is
(2.5)
Once the relation between
~k
~i)
and
is established, the dissipation
function can be expressed, by
(2.6)
in terms of the irreversible forces. It can then be shown that each one of the results stated above has a corollary, obtained by interchanging the roles of
xk
and
X~i)
. Thus the three principles can be reformulated, the first
one as a principle of least velocity. The inequality corresponding to (2.4) is (2.7)
(X(i) _ X(i)--) x' k
k
>0
k a
.
It holds for the actual velocity . i k ' the actual ~rreversible. force X~) and any other irreversible force X~l)", with DI(X~)*) ~ D'(X~l)) . Finally the corollary of (2.5) reads
(2.8)
If
D(x k) satisfies the functional equation 143
- 10 -
H. Ziegler
dD (1.IL
where
•
-:r;:- xk = f
(2.9)
f (D)
(D)
k
is arbitrary, the D-surfaces are similar and similarly situa-
ted with respect to the origin in velocity space. Let us refer to a dissipation function of this type as quasi-homogeneous. In this case (2.5) reduces to
~i)
(2. 10)
-K
D
= f(D)
dD ~xk
and this is equivalent with
where
(2. 11)
With
=
f
DdD
f(n)
f (D) = r D ,(2.9) takes the form
JD •
x = rD ~xk k
(2.12)
A function
cP
D satisfying (2.12) is called homogeneous of degree r . Here
(2.10) yields
(2. 13)
In the particular case
X(i) = 1 k r
';)D
~Xk
r = 2 the dissipation function is given by the
quadratic form 144
- 11 H. Ziegler
(2.14) the generality of which is not restricted by setting (2.15) On account of (2.13) (2.16) Thus Onsager's relations (1.17), (1.18) are obtained as a special case of the present theory. 3. Thermodynamics and continuum mechanics. It has been pointed out in Section 1 that it is impossible to separate continuum Illechan:\,s from thermodynamics. The reason fro this is the fact that, in continuulll meci'ianics, the microstructure of the material under consideration remains indefined. In consequence it is impossible to formulate the work of the interior forces, entering the energy theorem of mechanics. This makes it necessary to replace this theorem by the first fundamental law, and thus thermodynamics is brought in even in cases where heat effects are negligible. In this section we shall formulate the basic mechanical and thermodynamic equations for a continuum, using cartesian tensor notation. Let
y. denote the cartesian coordinates and t the time. Let partial J derivatives with respect to y. or t be indicated by the subscript j or.O J respectively, preceded by a comma, and let the material derivative be denoted by a dot. If
e
represents the density and 145
V.
J
the velocity field, the principle
- 12 H. Ziegler
of conservation of mass for an arbitrary volume
f ee
+ v.
(3. 1)
V requires that
.)dV = 0 .
J, J
(For a detailed derivation of this result and the next ones up to (3.9) see [7J ). The differential form of (3.1), i.e., the principle of conservation of mass for an element, is
e+e v J,. .J = eo + (e v.).J J = 0 •
(3.2)
The momentum theorem for the volume
where
S is the surface of
V,
~
V is given by
its exterior unit normal, f k the
specific body force (i. e., the body force per unit mass) and <SkI tensor. The differential form of (3.3), i. e., the momentum
the stress
theorem for
a single element, reads
(3.4)
The angular momentum
theorem is similar and, in its differential form,
establishes the symmetry of the stress tensor. In order to replace the energy theorem of mechanics by the first funda-
mental theorem of thermodynamics, we note that the intrinsic energy contained in the volume
V is 146
- 13 -
H. Ziegler
U = feM.dV ,
(3.5)
where
.»..
denotes the specific intrinsic energy, dependent on the mechani-
cal state of the element, i. e., on its deformation, and on the temperature. The influx of heat into the volume
V is
(3.6)
where the vector qk denotes the heat flux. Starting from (1. 2) and observing that, in a continuum, .the energy of an element is composed of its kinetic and intrinsic energies, we state the first fundamental theorem for the volume
V in the following form:
The material rate of increase of the sum of the kinetic and intrinsic energies in equal to the rate of work of the
exf~ior
forces plus the heat in-
flux. The analytical form of this statement is
(3.7)
On account of (3.4) and the symmetry of
(3.8)
where (3.9) 147
~kl
(3.7) reduces to
- 14 H. Ziegler
is the rate of deformation. Thus the differential form of the theorem, i. e., the first fundamental law for the element, reads
,
e AJ..=~kl Vkl - qk, k
(3.10)
In this form the analogy with (1. 2) is complete. Once the first fundamental theorem has been accepted, it is a matter of consequence to proceed to the second one. In order to do so, ·we introduce the entropy contained in the volume
s of esdV
(3.11)
where
V ,
s
se where
denotes the specific entropy. Let us restrict ourselves to the cas
depends on the same variables as No, i. e., on the deforma-
tion and the temperature of the element. Comparing (3.6) with (1. 5) ,it is easy to see that the influx of entropy into the volume
(3.12)
2. (s) =
~
-f ~ (J
V is
Y k dS •
Combining (1. 3) and (1. 5) ,we state the second fundamental theorem for the finite volume as follows : The material rate of increase Clf entropy is equal to or greater than the entropy influx.
In other words: The entropy production inside analytical form of this statement is (3.13) 148
V is non-negative. The
- 15 H. Ziegler
or
(3.14)
fe· I sdV :> -
qk" u dS 8 k
=-
f
qk (-) 8 ,k dV .
If (3.14) holds with the equality sign, the process is reversible, otherwise
it is irreversible. In the last case entropy is produced inside V. The differential form of (3.14) is
(3.15)
8, k
In order to interpret this inequality, let us transfer the results of Section 1 to a single element of the continuum considered here, restricting ourselves, for convenience, to infinitesimal deformations. (For finite deformations see
(6J ).
Here the infinitesimal strain components
mechanical state variables, and the stress components S' kl
Ckl
are the
are the cor-
responding forces for the unit volume. According to (1. 12) the stress tensor can be represented, by
(3. 16)
as the sum of a reversible and an irreversible part. e ~ _ (r) With u(" kl ,8) and s ( Ii kJ. ,8) the reversible stress tensor U kl is a state function. The relations (1.11) and (1.10) take the form
(3.17)
'J s
- 8~) kl 149
,
- 16 H. Ziegler Instead of (1. 16) we now have
,. , kl
(r)
(3.18)
v
Jf -s=If) 9
'\ f
=e~ ~Ekl
where (3. 19)
f = u - 9s
is the specific free energy.
(i)
6' kl depends on the rate of deforma.
The irreversible stress tensor
tion and possibly also on the state of the element and on its history. Any component of () ~i
changes sign together with the corresponding component of
• A comparison with (1. 2) and (3.10) kl becomes
\t
1
(i)
eS = -S'kl 9
(3.20)
shows that the relation (1.14)
, kl -qk k
V
9
Here the first term on the right -hand side represents the entropy production inside the element, due to the work of the irreversible stress tensor, while the second one describes the entropy influx, due td heat exchange with the (i)
environment. The sum e"kl Vkl
is the rate of dissipation work and indi-
cates the rate at which the work done on the element is transformed into heat. Writing (3.20) in the form (3.21 )
•
1
(i)
e s = -9 $' kl
and integrating over
qk
qk
Vkl - (-) k - 9, 92
V, we obtain 150
9, k
- 17 -
H. Ziegler
(3.22)
· J~.e f"9
S=
sdV =
1
~l(i)
/ Vkl dV -
!
qk -;; e,k dV -
qk "8 YkdS.
On account of (3.12) the last term on the right-hand side represents the entropy influx, due to heat exchange with the environment. It follows that the two other terms describe the entropy production inside V. The first one obviously represents the entropy production due to the work of the irreversible stress tensor, the second one the entropy production due to heat exchange inside
V.
It is interesting to compare (3.20) and (3.22) . In (3.20) the term representing intrinsic heat exchange is not present, and this is clea'rly due to the fact that the element is characterized by a single temperature. Thus any kind of heat exchange inside the finite volume
V is indeed an irrever-
sible process, accompanied by an entropy production. For the sillgle element, however, the same process appears as reversible, since no entrupy is produced in its interior. The apparent paradox is easily solved by cbnsidering the boundaries between the elements as the sources of entropy production due to heat exchange. It follows, however, that staterpents concerning entropy production must be handled with caution: any such statement, although valid for the single elements, need not necessarily hold for a finite volume, and vice versa . . With (3.21) the inequality (3.15) reduces to
(3.23)
The left hand side is the rate of entropy production per unit volume. It consists of the entropy production within the element and the element's share 151
- 18 -
H. Ziegler of the entropy production in the boundaries. On account of the presence of the second term,
(3.23) cannot be considered as the expression of the se-
cond fundamental theorem for the single element, although it is the differential form of this theorem for the finite volume. However, since the two terms in (3.23) represent entropy productions of entirely different sources, it is to be expected that they are independent of each other and that, in consequence, each one of them must be non-negative. In fact, it is clear that the irreversible stress tensor 6"k\i)
,as a function of the deformation rate
Vkl '
the state of the element and possibly its history. is independent of the surrounding elements and hence of the temperature gradient sible that the heat flux
9, k . It is equally plau-
qk depends solely on the differences between the sta-
tes of adjacent elements but not on the instantaneous deformation rate of a single element. It follows that (3.23) must be split up into (3.24)
Q'" (i) > kl Vk1 = 0
and
The first inequality represents the second fundamental theorem for the element. It states that the entropy production within the element, due to the work of the irreversible stress tensor, is non-negative. The second inequality may be considered as the expression of the same theorem for the boundaries between the elements. It states that any entropy production due to heat exchange is non-negative. 4. Constitutive equations. The basic equations formulated in the last section are valid for arbitrary continua. For any specific material they must be supplemented by the proper constitutive equations, connecting the kinematic variables (such as strain, rate of deformation, etc.) with the static ones 152
- 19 H. Ziegler (stress, stress rate, etc.). It is clear that these constitutive equations must be consistent with the general laws, in particular with the fundamental theorems of thermodynamics. In this section we shall discuss some implications of this postulate.. In elasticity some authors (see, e. g ..
[8] ) distinguish between three
different types of material. Although the definitions are usually given in terms of finite deformations, it seems possible without loss of any essential feature to discuss them in terms of infinitesimal strains ture
£ kl
and the tempera-
8 as state variables. In order to get rid of the temperature and of the
necessity to take heat exchange into consideration, one usually assumes that the process is conducted in such a lI\anner that
9 is either constant (iso-
thermal process) or a given function of the strain history (as, e. g., in an adiabatic process). For an anisotropic material the definitions then are essentially the following ones : The hypoelastic body is defined by a linear relation,
(4.1 )
between the increments of strain and stress. The elastic body is defined by a relation (4.2)
between strain and stress. If thi~ relation has the form
(4.3)
G""IJ 153
- 20 H. Ziegler
where
- f
denotes the specific potential energy, the body is called hypcre-
lastic. It is evident that, with these definitions, any hyperelastic body is elastic, and that any elstic body is hypoelastic. It is usually maintained that the reverse is not true, and from a purely mathematical point of view this statement is clearly correct. By simple thermodynamic reasoning, however, it is easy to see that any hypoelastir body is elastic, and that any elastic body is hyperelastic, so that there is no point in distinguishing between the three types of material. From the viewpomt of thermodynamics it is reasonable to retain
0
as an indipendent state variable and to generalize the definitions (4. 1) through (4. 3) accordingly. Let the hypoelastic body be defined by the genralization
(4.4)
dG'".. IJ
= CIJ"kl(fJ"mn ,0) de kl + GIJ.. (6'" mn ,0) de
of (4. 1) . If the sign of
d ~kl
is changed, this affects
d
u..IJ
but does
not reverse the sign of any finite part of () ., . It follows from Section 3 IJ that the stress tensor is reversible, i. e. that
(It. ) (4.5)
G'
ij
(i)
tr ..
=
(t ij = 0
IJ
On account of (3.18) and (3.19)
(4.6)
G" IJ.. =
(E kl ,6) ~ £ IJ..
() f
e
where 154
H. Ziegler (4.7)
is the specific free energy. Equation (4.6) il the natural thermodynamic generalization of (4.3). It il obvioua that (4.4) allo followa from (4.6) . Moreover, for ilothermal proceal'l, (4.4) and (4.8) reduce to (4.1) and (4.3) respectively. Thul hypoel.IUC, elaltic and hyperelaltic materials are
identical. So far we have dilcuased implications of the tundamental theorems. If the principles of Section 2 are valid, it becomes pOllible, e, g., to simplify the general const1tuti~e eqUation I eltablished by lome authorll [9, 10, 11] for non-newtonian fiuids
(4.8)
p
II
(13J .The rate ot work per unit volume il (r) O'jk Vjk • ('jk
(1)
+ O'jk )
Vjk
In a fluid the reversible stress tensor is given by the hydrostatic pressure palone. Thus
r
(r)
6" jk
(4.9)
= -p <J jk
The rate of reversible work is theretore (4. 10)
p(r)
= G' (r) V •• V jk
jk
P it
and the rate of dissipation work is given by (4.11) 155
- 22 H. Ziegler
(i) where a" jk
is the stress tensor due to viscosity and
D(V Jk )
denotes the
dissipation function per unit mass. Comparing (4.11) with (2.3) we find (i) that, for the unit volume, the (S" jk are the irreversible forces correspolluillg" Vjk • Thus the principle of least irreversible force
to the velocities
(2 ° : ; )
requires that
(4. 12)
(i)
dD
J
0"
~k = ~ D ("V
1m
In an isotropic fl..lid the dissipation function has the form
(4.13)
where
v
(1)
=V
ii'
(4.14)
= _1_ (2V V V
jk ki -
3V V V ij ji kk
+V
are the basic invariants (see, e. g.,
[7] , p.22)
of thl' ddul'mation rate. Com-
V
(3)
6
ij
\T V ) kk
i i ' jj
bining (4.12) and (4.14) ,one obtains
(4.15)
(i )
G" J'k =
{
';) D
"V IJ (1)
dok + 7\V(Vo 'J D (' '.) k - V(l) () 'k)+1"\ J
IJ
(2)
156
J
J"
1)
T
J~
(Vo,v'k - V(l)v'jo - \rJ\ 'k) y(.~ Jll J\ \'"'1 J
F
- 23 -
H. Ziegler
with
~D
('J
(4.16)
K = \ D('f)V(1) V(1)
+
2()D 't)V(4
3r;)D
V~ + ~V(3)
V)-1
(3)
This is the most general constitutive equation of a fluid as defined above. It contains a single, physically significant, function D ,while the equations of Reiner, Prager and Rivlin dQpend on two or three functions without apparent physical meaning. The constitutive equation (4.15) can be simplified only certain powers of
Vjk
or by assuming that
(12] by retaining
D depends on a restric-
ted number of fundamental invariants. The simplest special case, obtained by linearization, is Stokes' equation. If we write the dissipation function in the form (2.6), i. e., in terms of
the forces
'it can be visualized, in stress space (Fig.2) ,by means
6' ~t
of the surfaces
D'(
S-f1) = const.
On account of (2.7)
(4.17)
where
~k
stress and
is the actual deformation rate,
*
the actual irreversible
B" ~ any other irreversible stress with D'("~.) ~D'(.~t).
It follows that the vector
mal of the
$" ~i~
~k
in Fig.2 has the direction of the exterior nor-
D' -surface at P.
The plastic body is obtained as a limiting case by assuming that all D,-surfaces coincide, thus forming the yield surface, which still may depend on the state of the element and on its history. Here (4. 17) implies the convexity of the yield surface and supplies v. Mises' theory of the plastic poten157
- 24 H. Ziegler tial [13J . Thus v. Mises' hypothesis, which is fundamental for the theory of plasticity, can be justified by thermodynamic considerations. There are many more instances where thermodynamics plays an essential part in the formulation of constitutive equations. In the first one of the exampl~s
treated here we have merely made use of the fundamental theorems.
In the last two cases a more recent and still hypothetical theory has been used. Maybe the results obtained in these examples and in similar cases will contribute to justify this theory.
158
- 25 REFERENCES 1
H. Ziegler, Zwei Extremalprinzipien der irreversiblen Thermodynamik,
-
Ing.Arch. 30, 410 (1961). 2
L.Onsager, Reciprocal Relations in Irreversible Processes, Phys. Rev.
-
-
37, II, 405 (1931) and 38, II, 2265 (1931). Compare also H. B. G. Casi-
mir, On Onsager's Principle of Microscopic Reversibility, Rev. mod.
-
Phys. 17, 343 (1945) or S. R. de Groot, Thermodynamics of Irreversible Processes (North-Holland Publishing Co., Amsterdam 1952). 3
C. Truesdell, Reactions of the History of Mechanics upon Modern Research, J. Appl.
4
~ech. ~,
Series E, 229 (1962).
H. Ziegler, Die statistischen Grundlagen der irreversiblen Thermodyna-
-
mik, Ing. Arch. 31, 317 (1962). 5
H. Ziegler, Ueber ein Prinzip der grossten spezifischen Entropieproduktion und seine Bedeutung fur die Rheologie, Rheol. Acta~, 230 (1962).
6
H. Ziegler, Some Extremum Princ;ples in Irreversible Thermodynamics, with Application to Continuum Mechanics, in I. N. Sneddon and R. Hill,
•
Progress in Solid Mechanics, vol. IV (North-Holland Publishing Co. , Amsterdam), in print. 7
W. Prager, Introduotion to Mechanics of Continua (Ginn and Co., Boston 1961) .
8
C. Truesdell, The Classical Field Theories, in S. FlUgge, Encyclopedia of Physics, vol. III/ 1 (Springer-Verlag, Berlin 1960), p.723, 725,731.
9
-
M.Reiner,A Mathematical Theory of Dilatancy, Amer.J.Math. 67,350 (1945).
10
W. Prager, Strain Hardening under Combined Stresses, J. Appl. Phys. 16, 837 (1945).
159
H. Ziegler 11
R. Ri vlin, The Hydrodynamics of Non- Newtonian Fluids I Proc. Roy. Soc.
-
A 193, 260 (1948).
12
Ch. Wehrli and H. Ziegler, Emige mit dem Prinzip von der grassten Dissipationsleistung vertrlgl1che Stoffgleichungen,Z. angew. Math. Phys.
-
13, 372 (1962).
13 R. v. Mises, Mechanik der plaatilchen FormHnderung von Kristallen, Z. angew. Math. Mech. 8, 161 (1928).
160
- 27 H. Ziegler
o Fig. 1 : Connection between velocity and irreversible force in velocity space,
o
D'a(01\st,
Fig.2 : Connection between irreversible stress and deformation rate in stress space. 161
CENTRO INTERNAZIONALE MATEMATICO ESTIVO ( C.!. M. E.
)
CATALDO AGOSTINELLI
1.
UN TEOREMA DI MEDIA SUL FLUSSO DI ENERGIA NEL MOTO DI UN FLUmO DI ALTA CONDUTTIVITA' ELETTRICA IN CUI SI GENERA UN CAMPO MAGNETICO.
2. SU ALCUNI TEOREMI DI MEDIA IN MAGNETOFLUIDODINAMICA NEL CASO STAZIONARIO.
ROMA - Istituto Matematico dell'UniversitS.
163
CATALDO AGOSTINELLI
UN TEOREMA DI MEDIA SUL FL USSO DI ENERGIA NEL MOTO DI UN FLUmO DI ALTA CONDUTTIVITA' ELETTRICA IN CUI SI GENERA UN CAMPO MAGNETICO.
1. Se si considera il mota di un fluido non viscoso, di alta conduttivita elettrica, in cui si genera un campo magnetico, sussiste una notevole relazione relativa al flusso di energia totale attraverso una superficie fissa chiusa qualsiasi appartenente al campo del mota del fluido. Se poi gli elementi del campo magnetico e del moto sono periodici rispetto al tempo, si ha l'equivalenza in un periodo del flusso totale di energia attraverso la superficie considerata e del flusso dello stress magnetico e di pressione attraverso la stessa superficie.
2. Le equazioni magnetodinamiche per un fluido perfetto di alta con(iuttivita elettrica, tale da poterla ritenere infinita, scritte nella metrologia gaussiana razionalizzata, si riducono,
aB
~
com'~
~ + rot (B
.
noto, alle seguenti
..
1\ v)
=0
div B = 0 (1)
d'"
P dt
V
~
B=fH
.. = rot H II
it - grad p + f
grad U
+ d.\...,. (r v) = 0
t> = ~(~), in cui i simboli hanno il solito significato e dove la permeabilita magnetic a ~
~
supposta costante. Se ora nel campo in cui si muove i1 fluido consideriamo una superficie
chiusa
6' qualsiasi, che limita un volume S , moltiplichiamo quindi ambo i 165
- 2C. Agostinelli
membri dell'equazione del moto scalarmente per i1 vettore velociU -: e integriamo sopra tutto i1 volume S. abbiamo
Is :: xV.
(2)
dS
+
Is
gradp)(
~. dS - J'- fs rotH ii. v. dS A
-Lf
grad U )( ;. dS = 0
S
Tenendo conto dell'equazione di continuiU risulta
f
~
~ .. = ~ dt )( v 2
f~ dt 2
["l
= ~ [~
~
2 ~ 2 = ~ _Cl 2. 2 ~ 2 dt ( Pv ) - dt v ] 2 ~ t (f v )+ dlV( f v . v)
e integrando rispetto al volume S. applicando il teorema della divergenza. si ha
1
(3)
S
essendo
it
dV....
d
f dt )( v. dS = dt
112' f S
1(
~
2 2.. v ;dS + 2" ~ P v . v X n. dO' •
il versore della normale esterna alla superficie 6"" •
Analogamente. essendo la pressione p funzione della densiU. se poniamo
C?( ~)
(4)
..J
r
=
f dE .l
r
-- l'1 ~ dp
do J
•
possiamo scrivere grad p)(
~ =f
grad
-
~!P))( 1 = diV[ p(p). f ~J p(P)' div( f -;l
= diV[
ma
p(]' l
r ~]+
g'(fl :~ ;
~ = ~t rf~(f l] - f ~~ = ~t [r~(f lJ - f:~ = ; t
[r ~ (f lJ-
*
166
;{
=
=
- 3C. Agostinelli
ne segue
e
:t IJp i?(p) - p]
L
~ xil. d6'.
(5)
fsgrad
pd. dj;'
dove
f (j>( f
)-p ~ l'energia di pressione riferita all'uniU di volume.
dS +
p(p ),1'
Nel caso di un fiuido in condizioni adiabatic he in cui p = C f'{ risulta
0(p) = _O_..E..
O~D)_P=-L.
1-1 f'
.J
I
I
r-l
Si ha inoltre, per la prima delle (I) rotH"
H l( ~ = rotH )( (H " 1) - HX
fi"..
= diVlHA(H I\~)
-.
[rot(H 1\ 1) +
] - 2"~ 1 dH2 ,
e quindi
I ..
-t
...
S rotH 1\ H X v. dS = -
(6)
~~ J =
dI12 dt S 2" H . dS +
L. . .
~
..
H/dH 1\ v) X n. d6" .
Infine, se i1 potenziale Udelle forze di massa non elettromagnetiche non dipende esplicitamente dal tempo, si ha
f grad U X 1 = div(U. p~) - U. div( P~) = div(U. f't) + ~ t (f U) e
(f gradU l( ~. dS = :t
(7)
Js
JS ~ U. dS + I0 f
u. ~ x it. d6"
Sostituendo (3), (5), (6), (7) nella (2) si ottiene
d~ +
U!
L{! f
p.2 + [p :p( P) - p
v2.
l ! ".. H2 - f 1 +
U dS
+
~ )( rt +
- 4C. Agostinelli ehe si puo scrivere
!
J {~ f v 2 +[p~( f ) - pJ+ ~ l' H2 -
PU}
S
dS +
+tHpv2+[r
181
i fl
+ r;; 1P ....
~
dove 6" (H, H)
1
p +2
e una
f
H
2
-
f
'U ...
..,J""'v x ...n. do
~ (H, H)
,...,; ..... ...,
= 0 ,
. . . . . .",
diade tale ehe 01. (H, H)v x ] = H X
V. 11)(
.. n .
Osserviamo ehe la qu<..ntita
rappresenta l'energia totale riferita all'unita di volume (1), e ehe l'espressione
J 't'
(10)
e 10 stress
...
= -
{
p
1
2
+ 2 )J.. H -
r
~ ~ 'U (H,., H) 5
dovuto agli sforzi magnetiei e di pressione idrodinamica. Invero questi sforzi derivano dai termini dell'equazione del mote
....,
p.rotH A H - gradp, ehe rappresentano l'azione eombinata della forza di Lorentz e del gradiente di pressione. Ora risulta
....
"'*
...
dH -+ 1 2 rot H" H = dP H - 2 grad H
....
dH" LI ~ .. (2) ; percio dove dpH e uguale al gradiente della diade tf~ (H, H)
(1) cfr. C. Agostinelli , Sulla stabilita dei moti magnetofluidodinamici stazionari. "Rendieonti Aeead. Naz. dei Lineei", Serie VIII, vol. XXIX, fase.6 e vol. XXX,. fase, 1. (2) Infatti, eon riferimento a un sistema di assi eart€siani ortogonali, si ha: ~ dB ~ '() ~ ,.,; ... ~ H. r \ =L~ . . . X' (H. H) = grad ('0 (H, H) dP H = L~ • 1 dXi 1 Q 1 1
dB ...
168
- 5-
C. Agostinelli
e quindi (11)
p,rot
it A H- grad p = - grad{
tt Lt.
p+
~ fH2
- , . ~(H,
'H)]
=
grad
t.
L'equazione (8) si puo scrivere allora in forma piu semplice (12) che
:1 ~
dS +
\l Xit db
•
Lq. ~. it
d6'
una notevole relazione relativa al flusso di energia totale attraverso u-
na superficie chiusa qualsiasi. Ora
~
noto che i1 sistema di equazioni (1) ammette delle soluzio-
ni in cui gli elementi del moto e del campo magnetico sono funzioni
period~
che rispetto al tempo. Se percio consideriamo una soluzione periodic a di pe-
r
riodo T e integriamo ambo i membri della (12) rispetto a un periodo, poi-
ch~ ri,ulta
[f ~t.lS
= 0 , si deduce
it. ~
TSOT
(13)
J
o
dt
rr
X
nd~
=f
dt 0
f. ~ 6
~ x It d6"
la quale esprime il seguente teorema di media: In un moto periodico di un fluido compressibile, non viscoso, di alta conduttivitil elettrica, i1 flusso di energia totale in un periodo attraverso una super:.. ficie chiusa qualsiasi fissa, immersa nel campo del moto,
~
uguale al flusso
dello stress magnetico e di pressione attraverso la stessa superficie e nella stesso periodo.
169
- 6C. Agostinelli
SU ALCUNI TEOREMI DI MEDIA IN MAGNETOFLUIDODINAMICA NEL CASO ST AZIONARIO SUNTO, Si stabiliscono delle formule che esprimono altrettanti teoremi di di media per i moti magnetofluidodinamici stazionari.
===== 1. Le equazioni della magnetofluidodinamica, quando si trascura la corrente di spostamento in confronto della corrente di conduzione, col ben noto si, b 0 l'1 sono (1) gm'f'1cat 0 d' e1 Slm
.....
rot H = t
~
...
8B
rot E = - - -
at
...,
div B = 0 (1)
dV f"dt = I 1\ B + r F - grad p+( A'+ t-')graddiv-;J+ -t...
-.
'
~+ div ( f 1) = 0 , dove la conduttiviU elettrica coefficienti di
viscosit~
).',
..
+fl' !:::.2 v
r e la permeabiliU magnetic a f.. ' nonche i JA-',
si suppongono costanti. -t
Nel caso di moti magnetofluidodinamici stazionari si ha rot E = 0, (1) Cfr, C. Agostinelli, Problemi di Magnetofluidodinamica, ecc. "Atti del Simposio sulla Magnetofluidodinamica, Bari 10 -14 gennaio 1961" (Edizioni Cremonese, Roma),
171
- 7C. Agostinelli
e dalle prime cinque delle equazioni (1) si ricava per i1 campo magnetico l'equazione
A -+ u2H =
(2)
or
~ rot (H II v) .
Cosi pure l'equazione del mota e quella di continuitl\ diventano 4
(3)
fJ
J
ddVP -+v
=
(4)
...
dH'" 1 H2) + fF+ -. ("I,,'+,)4'graddivv+ ) ... ~dP H-grad ( P+"2f
div ( p~)
dV essendo dP
=
0
-+
dH e dP omografie vettoriali che rappresentano Ie derivate ri...
...
spetto al punta P dei vettori v ed H .
2. Cia premesso consideriamo, nell'interno del carrp occupato dal fluido, un dominio sferico S, con centro in un punta qualsiasi P , limitato da una suo perficie sferica f{ di raggio r", arbitrario, e integriamo ambo i membri del·la (2) rispetto al volume sferico S. Si ha cosi
JS
(5)
/j 2H. dS =
rr JS
rot
(H A ~). dS .
Ora, per Ie formule preliminari di Green, risulta
VI
2
H. dS ~
U~ dO ~ d~
Js rot(H II ~). dS = J~ Ii" (if
nell'ultima delle quali sferica (6)
~
rt
/I,
L ...
Hdb
v). d6"'
~ i1 versore della normale esterna aHa superficie
. Sostituendo nella (5) si ottiene
d~ fo Ii d6'
=
rr 16' rt
I\.
(H "
~). d6" .
Applichiamo ora alla sfera S la formula di Green 172
- 8 C. Agostinelli 1 1 dU d -;( - - U) d 10 6' r dn dn
I
4 It U( P ) = o
(7)
dove U
e funzione
f
1 S r
A
LI
U. dS 2 '
finita e continua colle derivate prime e seconde in tutto
i1 dominic che si considera. Essa sussiste anche nel caso in cui U
tore, e poiche r
e ora la distanza di un punta
e un vet-
P dal centro P della sfera o
S , quella formula diventa (71)
4 7t U(P ) = _1__d_
~
o
( Ud6"
k-
drcr
~
Ponendo in luogo di U il vettore H che rappresenta il campo magnetico, e tenendo canto della (2), si deduce (8)
~
1 = (r.-
41C H(P 0)
6'
d 1 l~ r 1 -+ err + z-) Hd6" - '(tL -;- rot (H" s 6'
~
v)dS.
6'
rEi'
Ma
fSr~ rot(HA 1)dS
=
=
J.
S
rot(
+- I G"
~r HA~)dS -
f
S
it A (HA V)d6' -
G'
grad
~r A rH A ~). dS =
JS grad ~A (HA ~). dS , r
Sostituendo nella (8), e semplificando, avendo riguardo alIa (6), si ha infine (9)
-+ i
41di(p ) = o r.
10
Hd6" +
G'
0ILt-fS grad ~r A (it" ~)dS
,
che esprime un teorema di media per il campo magnetico.
3.
Integrando ora ambo i membri dell1equazione (3) del moto rispetto al vo-
lume della sfera S, otteniamo (10)
... fS dH dP H. dS ~
+ ( AI + f-I)
Is
-
JS
grad (p + 21
grad div 173
V. dS + fl
t-' H2). dS +
Is ~
2
J~
~. dS
S
f
F. dS +
-9C. Agostinelli Avendo riguardo aU'equazione (4) di continuita, se si indicano con 1[1' x 2 ' x3 Ie coordinate cartesiane del punta P, e con vI' v 2' v 3 Ie componenti del vettore
v,
risulta
d\t...
P dP
v=
~
f
£.;,
1
a\t
V. -1::11 aXi
~
9 ax.
= ~~
.. (f v .. v)
t I l
e quindi, per Ie formule di Gauss,
...,
Analogamente, essencfo div H = 0, si ha
L'equazione (10) diventa pertanto
L
L
~f' Ii X itH. do -
(11 )
~' +1"')
+I
I" HZ)itdfl' +
Ip + :
L
div ;. itdis' + f'
d~
i
Applicando d'altra parte la formula (7') di Green al vettore
~(
1 dr_ d + -2-) 1 4ltv P ) = (r.--o 6" .. 16-
(12)
Ma sostituendo in luogo di (13)
f
f
A2~
d-:
- -, f f
1
Sr
d~ -dP tdS
.do •
7, si ottiene
-1 u"., 2v. dS . Sr
JL
dH...H -+- - 1
- L... -
; ' dP
, ,
p'
grad (p + -1 JL H2) 21
-+]
F - A+fo ,grad div v dS .
..,.
Con facile calcolo si vede che risulta
J-. r
\!1
dS +
i1 valore che si ricava dalla (3), si ha
1{ 1 ~ -1 Il ~v. dS = - - " ---:-='V S r 2 S r )Ai J dP 1
1....vdG" -
L
lfF.
=r1G'
jJ.
f"
-I> ~ pvxn.v.dG' -
b 174
L.... 1.... fvxgrad-.v.dS S
r
- 10 -
C. Agostinelli
L"
~
f
1 dB 1 S -;dP H. dS = 1;-
I
H
x ~... n. H. d6'"
I~
1 . -* - S H X grad -;H. dS
I 12 11 12 S -;- grad (p + 2" dS = r" 6" (p + 2" fH lit. d6' -
tH ).
J
1 2 1 S (p + 2" ~H )grad -;- . dS
-
J1..
grad div
V. dS =
Sr
-f- i div ;t. it d6'" -1S div~. grad 1..r dS EO
•
""
Sostituendo allora nella (13), e quindi nella (12}, e semplificando, tenendo conto della (11), si ottiene (14)
4~~(Po) = t
,. J.,
- ---; ,.
S
+1, [J }I-
r6"
1~dEr
+.i,
jJ.
f)
1 .. 1 Hxgrad - H. dS + I r
1.. f r dS
Sr
}
-
fS f~
X grad! .\t.dS-
J S
1 2 1 (P+2" jJJI ). grad - . dS + r
~~ Jr fF. dS S
l' +
;-'1
;-
S
div -:. grad 1... dS , r
che esprime un teorema di media per la velociU in un punto P del campo o di moto.
f'0
Nel caso di un fiuido incompressibile di densiU ...
I
e in assenza di forze
di natura non elettromagnetica (F = O),·la (14) diventa piu semplicemente (15)
~
1 4JCv(P ) = -2-
o
~
L~v de-
+
Po tl r
1 +,.
J. . S
v X grad -1. -t v dS -
r
~f 1 -HdS+ , ~ HX grad-.
1s
J (p + -21 f'H 2) grad -1 . dS
f' S
r
175
r
.
- 11 C. Agostinelli
4.
Osserviamo che la (11), risoluta rispetto all'
( 1 2 ~ ,k-(p +'2,u,H ) n db , for-
nisce il risultante della pressione totale sulla superficie E) . Essa
e valida
qualunque sia il dominio S , salvo la modifica dell'ultimo integrale, e puo comprendere anche tutto il campo in cui si muove i1 fluido. In questo caso, se supponiamo che il fluido elettricamente conduttore sia contenuto in un recipiente limitato da una parete rigida 6"", perfettamente conduttrice, alli-
-
mite su 6' dovremo avere -+ ... vx n=O ,
~
Hx n=O ,
e in tal caso la (11) porge
Se piu in particolare il fluido
e non viscoso e Ie forze
di massa non elettro-
magnetic he sono nulle, si ha che il risultante delle pressioni totali sulla superficie limite 6"
e nullo.
E' facile dimostrare ancora che insieme alla (16) sussiste l'equazione dei . (1)
momenh (17)
I(p-O) " (p + ~
+ (;, '+r') dove 0
~ P-H2) It d~
J~p-O) "
J
=
(P-O) A
1. dS +
S
diW. ad6'
+t'
[ ~(P-O) A : : d,; -in,,~. d.-J '
e un punto fisso.
(1) Si osservi che risulta (P-O) "
~ (P-O) I\!J2~' dS = ~ d~
-J!/\ ~d6" .
[(P-OlA
A~V = ~ J(P-O)A -;] -2rot~,
e quindi
~ JdS -2L Ii" v. dE) =~P-O)" ~! 176
do
-
- 12 -
C. Agostinelli
5. Un altro teorema di media si pub stabilire per il vettore vortice. Invero I' equazione (3) del moto si pub scrivere -+
-t
..,
...
rot v 1\ f v = ~ rotH 1\ H - grad p + ( ~' + fo
1
"2 f
') grad div 1 + ;,'
2-+ grad v + F +
P
~2~ ~
..
Prendendo i1 rotore di ambo i membri, e ponendo rot v = W , 81 r1cava A.... 1 ~... ~ ~ -+ 1 2 1 LJ 2W = )Cfrot(WApv) - ;.,rot(rotH/\ H)+ 2j1 gradf"gradv -fI',rot(f F ). Con procedimento analogo a queUo dei numeri precedenti, sempre con riferimento a una flfera S di centro Poe raggio r e (18)
1'+
1 Wd6" + 2 1' 41t -+ W(P ) = -:y o rG" 6= ~r6""
'
si deduce
J gradt') " grad v2. dS S
JI
'
Ii f
1 1 II (w...... - -2' - gradf/\ grad v 2. dS + -, gradI\F)dS ,. Sr ~ S r
P"l grad -1 " (rotH~ 1\
,.,S
- -;
1 H). dS - -,
-of
r
Ps
.., dS grad -1 II f F. r
che esprime un teorema di media per il vortice. Nel caso di un fluido incompressibile e in assenza di forze di massa non elettromagnetiche si ha piu semplicemente (19)
~ 1 r..... fo 4:JCW(Po)=2J:Jdtf + y r6" ~
f
1
~ ~
grad~/I. (wAv)dS-
S
-~J , grad -1 II (rot ..... H" 1H). dS . P. S r 6. Nel caso di piccoli movimenti di un fluido incompressibile soggetto a un campo magnetico uniforme
iio, supposto che il campo magnetico indotto h
sia molto piccolo, trascurando i termini di or dine superiore al primo rispetto a
v ed'! ,
e supponendo nuUe Ie forze di massa non elettromagnetiche, Ie
equazioni (2) e (3) si riducono aIle seguenti 177
- 13 -
C. Agostinelli J.
(20)
~
LJ 2n=
-+ " ()Lrot(H o
'"
V)
)trot~"ito-gradP+,.,' A2~=0
(21)
...
con div ~ = 0 e div h = O. Dalle (20) e (21) con procedimento analogo a quello dei numeri precedenti si ricava (22)
(23)
4~h(P 4ft
o
)=
-+.k-rh t 1~ d~
dEr +(f' ( grad.lA )s r
r6'"
~(PO ) = rb r
+
(il
A t)dS 0
~ Jr p grad.lr dS - f')J.j(grad.lA h) "it ,dS S r S
)A-
0
Inoltre, prendendo ora la divergenza di ambo i membri della (21) si ha (24)
cio~ la (25)
p +~
h X ito ~ una funzione armonica, (p
'-t + ~" h X H )p =
o
0
1
4ft
2
l-
e si ha 6enz!altro
L + .. (p
b
come conseguenza del teorema della media di Gauss,
178
-+
f"h X H ) db 0
CENTRO INTERNAZIONALE MATEMATICO ESTIVO (.C. I. M.E. )
DARIO GRAFFI
1. PRINCIPI DI MINIMO E VARIAZIONALI NEL CAMPO ELETTROMAGNETICO. 2. TEOREMI DI RECIPROCITA' NEI FENOMENI NON STAZIONARI.
ROMA - Istituto Matematico dell'Universitl 179
PRINCIPI DI MINIMO E VARIAZIONALI NEL CAMPO ELETTROMAGNETICO
D. Graffi
1. In questa conferenza non intendo esporre tutti i principi di minima e varia-
zionaU che si incontrano neUa teoria del campo elettromagnetico: mi limitero soltanto a considerare alcuni principi che, a mio avviso, presentano qualche novitA.. Comincero col richiamare due teoremi di minimo per 11 campo elettrostatico dovuti, in sostanza, a Lord Kelvin. SI consideri un sistema di n conduttori, elettrizzati, immersi in
un dielettrico, che supporremo perfetto, e, solo per sempUcltl di esposizione, neutro.
n primo teorema di Lord Kelvin afferma che l'energia del campo e-
lettrico compatib11e con assegnate cariche sui conduttori l! minima iri condizioni di equilibrio, clol! quando il campo l! elettrostatico. PiO precisamente,
-
.~
considerato un campo elettrico E (anche fittizio). infinitesimo all'infinito, ~
col corrispondente vettore spostamento D soluzione dell'equazionc : I· l:-W
(1)
-D 1
= 0
...,
-
e con valori aS8egnati (uguali aUe cariche) per 11 nusso di D attraverso la superficie di ogoi conduttore, l'energia che compete al campo E diventa minima quando
E
deriva da un potenziale V continuo, costante sulla superfi-
cie del conduttore e soddisfacente alle solite condizioni di convergenza all'infinito.
n secondo teorema di Lord Kelvin afferPla che fra tutti i campi elettrici (in generale fittizi) che derivano da un potenziale con valore costante assegnato sui conduttori, e infinitesimo all'infinito, compete l'energia minima al campo elettrostatico, clol! a quello in cui il corrispondente vettore spostamento soddisra la (1).
181
- 2D. Graffi I teoremi ora enunciati si provano ammettendo fra vettore spo-
__
--I>
stamento D e il campo elettrico E (s'intende nell0 stesso punto dell0 spazio) la relazione :
- c:.E-
(2)
D:::
dove C. (costante dielettrica)
~
-
una grandezza scalare (0 al pill un tensore
doppio simmetrico) funzione, al pill, del punto in cui si considerano D ed ~ E. In sostanza la (2) presuppone, come si suol dire, i1 mezzo lineare. Perb, come
~
noto, in tempi abbastanza recenti sono stati introdotti, anche nella tecni-
ca, dielettrici non-lineari,
cio~
tali che per essi la (2) viene sostituita da una
relazione pill complicata non-lineare di cui diremo fra breve. Ci proponiamo di ricavare una estensione dei teoremi di Kelvin ai dielettrici non lineari (1); dovremo perb introdurre Ie seguenti ipotesi che sembrano plausibili dal punto di vista fisico.
-
a) L'isteresi sia trascurabile,
sicch~
~
.
i1 vettore D sia funzione univoca,anche
non-lineare,del vettore E. In formule :
--
(3)
relazione che generalizza la (2). La funzione D(E) verra supposta differen-
-..
ziabile, per ogni valore di E
b) Esista un'energia elettrica, pill precisamente, i1 lavoro del campo elettrico -->
-
per una vari'azione dl) del vettore spostamento sia i1 differenziale esatto di una funzione scalare (dens ita di energia del campo elettrico) p(E) del --.;.
campo elettrico E
(4)
->
cio~
~
sia (D espresso da (3)) : ~
E - d f):::
-..
d p( E )
(1) Cfr. D. Graffi - Aleuni teoremi di elettrostatica dei dielettrici non lineari Scritti matematici in onore di Filippo Sibirani, Bologna, 1957, pag.143. 182
- 3 -
D. Graffi nel caso particolare in cui la (3) si riduca a (2) .:.l
fj = L,j E;»J rlE);:
come i! ben noto.
t:..
E l"
.b
--.
-
c) Sia D(O) = 0, p(O) = 0 d) La D sia una funzione crescente di ..... E. Questa nozione pub precisarsi nel modo seguente. Se Ea ed \
(Ea rEb) e
I5'a e
sono due valori qualsiasi del campo elettrico,
l\ i corrispondenti valori del vettore spostamento vale
la disuguaglianza :
- Dc!. ) > :) \ -,' - 7'c' \I . (D, \ - ::>
( -:"l
( 5)
e) La(3) sia invertibile e, per la (5), in un solo modo sicchi! :
(6) -~
--;0
ossia E i! funzione univoca di D. Notiamo che Ie ipotesi enunciate potrebbero sostituirsi con altre meno restrittive ,rna su cib non insisteremo. 2. Cib posta passiamo a provare che il primo teorema di Kelvin rimane vali~
do anche in presenza di dielettrici non lineari. Sia E
il campo elettrostatico o corrispondente a cariche assegnate sui conduttori e siano Q r e 51" rispet-
tivamente la carica e la Buperficie del conduttore erresimo, il flusso del vet-,..
.-.~
-»
-..",.
tore Do = D (E 0 ) attraverso,-/.... sarA uguale a QI"' . Sia El un altro cam_ _ -7 po elettrico il cui corrispondente valore dello spostamento Dl = D (E l ) soddisfi la (1) e inoltre il flusso di ~ su 01'" valga ancora
41" .
-#
Confrontiamo ora, in un punta generico del dielettrico, p(E l) con ~
-
->
->
p(E ). A questa scopo osserviamo che per ottenere la differenza p(E l ) - p(E ) o 0 basterA integrare la (4) facendo variare E in modo qualunque. Noi supporremo che E'" vari in modo che
D
soddisfi la relazione : 183
- 4D, Graffi -->
~
~-->
dove D' = D1 - Doe A e un parametro variabilefra 0 e 1 sicche per ~\ = 0
-..
-
-..
-,)-)
D = Do' per A = 1 D = D1' In altre parole immaginiamo che E vari da
---;t.
_>
Eo a E1 con la legge che si ottiene sostituendo (7) in (6), Si ha
d p (t) ~ E'
(8)
2 D=
?, 0
--
COS! :
:LA
quindi integrando e poi aggiungendo e togliendo E . D' o (1.
piE, \ - \.ilEo \ ~ \ -,
(9)
->
t
l-/ -+ AD. -~I) -D'
1\
Q 1\
[/Q
=
/J
0'+ ("flo (D~:_~J)')::- t ~~ \, A I)' dA
~ ~-:
.~
~
~
-
-
,
->
'::'.
~
-,)-...,
Ora per la (5) (posto D = D, D = D + >. u' E = E(D ), E = E(D + AD') a 0 b 0 cl 0 b 0 si ha che l'ultimo integrale di (9) e positivo quindi :
~(E,' -
(10)
flE:)
-;>
-;;.
~ E:· 0 '
->
--,
-j>
(2)
f -.Ec' -,0') d
Integriamo ora la (10) su tutto i1 volume v esterno ai conduttori. Si ottiene: I
I
Ora
~
E'o e un
I, r -::-> ) I
ld v - i : ~t ~ '-'
-1,
,
, I tv '
(11)
V
>
campo elettrostatico, quindi
E0
V
V
'V
-
= - grad V e V ha suI condutto-
re erresimo un valore costante che indicheremo con V., ; inoltre E
e ino finitesimo all'infinito del secondo ordine. Quindi tenendo conto che D' e pu->-> re,per Ie nostre. ipotesi,infinitesimo all'infinito e che D', come DeDI' sod0 ~
disfa la (1) si ha : (11' )
t' ~ },. E(,' D' dV: -:>
~
-
(
I
- I /v
.{
Ma, come si e visto, i flussi di D1 e di
!d'\.V (-t')d ~v. V r; V:- l- t'
1/, D' J\I ~J
n:
--
WI.
'
Iv
attraverso
i
j-D·" -"'d -",----0'
h
Y CS ..
6,.. sono gli stessi per-
cia il flusso di D' attraverso la medesima superficie e nullo. Risulta quindi
-
nullo il secondo membro di (11). Allora poiche gli integrali al primo membro di (11) sono rispettivamente l'energia U1 e Uo del campo E1 e del campo elettrostatico
Eo si ha
:
(2) Notiamo che se Eo e uguale a zero si ha p(~) ~ 0, cioe la densita di energia e sempre positiva, come del resto e intuitivo. 184
{$
"
- 5D. Graffi
( 12)
come dovevasi dimostrare (3). 3. 11 secondo teorema di Kelvin si estende anch 'esso ai dielettrici non linea-
ri sostituendo all'energia la cosiddetta energia complementare la cui dens ita q(E) l! definita dalla formula:
- --
ite )'" E· D- p\E-,; )
(13)
Nel caso lineare, come si verifica racUmente, l'energia complementare
coin~
cide con l'energia ordinaria. Quindl i1 secondo teoremadl Kelvin ai esprime mediante la formula:
/'-1IE~)JV> (~lE:\dv
(14)
\' ,
)v
Per dimostrare la (14), si oaaervi che in un punto generico della spazio 8i ha: (15)
e posto (
).
l! ancora un parametro variabile tra 0 e 1) -)-;
(16)
-
E" t:tAE'
~
~->
t';E-Eo
si ricava ( 17)
(3) Nel testo si ~ supposto p(E;) nulla, come p(E~), nell'interno del conduttore. Se rosse nel .;.2nduttore p(~) dlversa da zero, la U1 sarebbe uguale all'integrale di p(Et) esteso a tutto 10 spazio e la (12) sarebbe valida a maggior ragione. 185
- 6D. Graffi Ora ragionando come nel numero precedente si trova che l'integrale
~ positivo (4). Quindi integrando la (17) eu tutto i1 volume v esterno ai condutto-
ri si ha : ( 18)
) -",E c V f 1(g~ I J y - i lEo) dv> j -'Do'
l
.v
I
=.>
v
~
......
.
-
-
"::::>
MaE o=-gradV o ,E 1 = - grad VI' El - Eo = - grad V', (V' = VI - V)Jdiv vo = 0, Allora, come nel caso precedente ricordando VD' infinitesimo all'infinito di ordine maggiore di due si ha : (19)
1.E>:. E'dv :-fyo..dV'.O:dv·-ldW(V'D,)Jv·lrv:i~·~¥~
e poich~ VIe V0 hanno su 6...
10
stesso valore ne segue, sempre su
V' = 0, quindi i1 secondo membro di (19)
~
ta estensione del secondo teorema di Kelvin
0;... ,
nullo. Si ha cOSI la (14); l'enuncia~
completamente provata.
Notiamo che i teoremi ora ottenuti potrebbero applicarsi per raggiungere i teoremi di esistenza per Ie soluzioni delle equazioni dell'elettrosta-
-
tica non line are,
cio~
dell I equazione che si otterrebbe ponendo in (1) la (3)
con E = - grad V . Notiamo che teoremi analoghi sono stati ottenuti per i1 campo ma-
gnetostatico in presenza dei corpi terromagnetici, ma su
(4) Supponendo E e quindi D = 0 si ha o 0 mentare ~ anch 'essa positiva.
q(~I)
'7 0,
ci~ non insisteremo(5).
cio~ l'energia comple-
(5) D. Graff! - Su una legge del minimo della magnetostatica - Annali Universitl Ferrara, (7) III (1954) - 25. D. Graffi - Alcuni problemi non-lineari della Fisica matematica - Rendiconti Seminario Matematico dell'UniversiU e del Politecnico di Torino 14 (1954-55) - 75.
186
- 7D. Graffi 4. Passiamo ora a una questione ben diversa, ma di maggiore interesse pratico;
cio~
a un principio variazionale a cui soddisfa l'impedenza di una anten-
na radio. Si abbia un1antenna
f~rmata
da un conduttore cilindrico che sup-
porremo perfetto. Al solito immaginiamo che l'azione del sistema eccitatore sull'antenna sia equivalente ad una forza elettromotrice inserita nell'antenna stessa
meglio ad un campo elettrico impresso
0
Ef.1 diverso dallo zero in una
piccola regione dell'antenna, detta regione di alimentazione. Ora sia I l'intensitl di corrente in una sezione ben determinata nella regione di alimentazione; notiamo che se, come avviene spesso in pratica, quella regione
~
abbastanza piccola ,in essa si pub 8upporre I identic a
in ogni sezione. E' nato che l'impedenza Z dell'antenna si puc esprimere mediante la formula:
Z.=
(20)
dove
f
:Jt! I_' r d6 I"
-
~ la densiU di corrente 8uperficiale nell'antenna, E i1 campo elet-
trico prodotto dalla corrente 1, (1,la superficie dell'antenna (6). Determiniamo ora la proprletl variazionale per la Z a cui si accennato. Supponiamo di variare la corrente in un'antenna re la corrente zione sia
~
,I
'f
~.
T'
cio~
~
di considera-
,supponiamo perc che nella regione di alimenta-
j> = 0, quindi (I
~ i1 nU8S0 attraverso una sezione della re-
E:
(6) E' bene notare che sia je I sono vettori 0 numeri complessi che rappresentano grandezze alternative. EI bene anche notare che essendo l'antenna cilindrica J si pub supporre parallelo all1asse dell'antenna, e se la sezione (a ragglo della sezione). dell'antenna ~ circolare rvale in grandezza.L ~ii,t
187
- 8 D, Graffi
, d'1 al'Iment ' )rlmarrCl ' >. mvana ' ' t a (7) . Q um 'd'1 Sl'h a: glOne aZlOne
(21) ~
Ora se j varia di
.
-,
........
( ((22)
(J
Ora si noti che
-)0.
~ j , E varia di ~ E , Sara percio a menD di infinitesimi: ~
.!~ E· ) J ~
~ ~ vale
,--.. . ~ d
=J../' j J .
{ (- ~
~'d ~
,~T.!l (j t . j ()
il campo' generato dalla
r 1t ~ r
corrente~ r;
per il teorema
di reciprocita del campo e1ettromagnetico si puo scrivere : (23)
/;
~ E· dG:
J;,
Ora $ si puo dividere in due parti, Ia regione di alimentazione gione rimanente .:' ", Nella prima
r
J
-
= 0; nella seconda,
r.5 ' e 1a re-
~r Ie proprieta
dei conduttori perfetti, e poiche il campo impresso e nullo, E e normale al-
J
la superficie dell' antenna cioe E~ S = 0; ~uindi i termini al secondo membro di (22) sono nulli e si ha la relazione cercata : (24) Questo risultato e importante anche dal punta di vista pratico. Poiche la stribuzione della corrente in antenna e nota solo in modo approssimato
di~
potre~
mo calcolare Z solo assumendo un val ore approssimato della corrente che differira dal valore esatto per termini dell'ordine di derando
1?
1.'
Per la (24) consi-
come infinitesimo si deduce che nel calcolo di Z si commette
un errore dell' ordine di
q2 ,
sicche la formula (20) e la piu adatta per il
caIc010 dell'impedenza dell' antenna,
(7) Non sara inutile notare che se I e identic a nella regione di alimentazione e se l'antenna e a sezione circolare per quanta si e detto nella nota (6) l'ipotesi ~ = 0 equivale a supporre invariata la I .
J
188
- 9D. Graffi TEOREMIDIRECIPROCITA'NEIFENOMENINON STAZIONARI
1. E' ben noto il teorema di reciprocitA di Betti nella elastostatica ordinaria.
Esso si esprime mediante la formula:
r pF\t" oIv
(1)
t
Il'
J t~.;l)eL7
-.; \ \::""~'ClV'" \ {"',:" cAv
"6"
6"
'v
dove v indica il volume del corpo elastico, 6' la superficie che 10 limita,
S'
ed ~, sono gli spostamenti dovutt rispettivamente, aUe forze di massa e
r it, f', f p." t" ( f densita) a~enti suI corpo.
superficiali,
11 teorema espresso dalla (1), di notevole importanza pratica, e in certo modo, i1 capostipite di una serie di teoremi di reciprocita validi in diversi campi della fisica matematica. Ora e ovvio che i1 teorema di Betti si puo estendere alla elastodinamica aggiungendo nella (1) alle
...
", . . ", i
'.
-f. -'
,-. ~.
I.
,-
r· r: 2/1 " ~-
.'
-
f -F', PF"
rispettivamente l termini
(t indica i1 tempo) cioe Ie forze d'inerzia cambiate di
,/'
segno. Si ottiene pero in tal modo una relazione, di poco interesse pratico, perche in essa compaiono termini, Ie forze d'inerzia, che sono, in generale, incogniti. E' percio naturale domandarsi se neU'elastodinamica vale, almena sotto certe condizioni, un teorema analogo a quello espresso da (1), ma in cui non compaiono termini d'inerzia. Alcuni risultati in quest'ordine di idee sono stati da me ottenuti applicando la trasformazione di Laplace (1). In questa conferenza intendo ritrovare i detti risultati evitando la trasformazione di Laplace (essa introduce infatti ipotesi suI comportamento per t tendente all'infini(1) D. Graffi "Sui teoremi di reciprociU, nei fenomeni dipendenti dal tempo"
Annali di Matematica (IV) XVIII, (1939), 173. "SuI teorema di reciprocita nella dinamica dei corpi elastici" Memorie Accademia Scienze Bologna (10) 189
. /.
- 10 -
D. Graffi to degli spostamenti che almeno, dal punto di vista concettuale,
e opportuno
evitare rna, mediante un procedimento adoperato da Gurtin e Sternberg (2) nei loro studi sulla viscoelastic ita. Inoltre farb vedere, con un esempio preso dalla meccanica ondulatoria, come quel procedimento possa condurre e relazioni di reciprocita valide in altri campi della fisica e per fenomeni non stazionari. 2. Converra, per ottenere la preannunciata estensione della (1) (3) scrivere Ie equazioni della elastodinamica quando suI corpo agiscono Ie forze di massa e , ..... superfichlli f f (or) J ~ (1:") funzioni, oltre che delluogo, anche del tempo che,
...
come vedremo in seguito, ,converra indicare ora con t' . Si ha cosi, con Ie " t ensOrla . l'1 (4) ,In 'd'lcan d0 con sol 1't e notaZlOnl
sore degli sforzi all'istante
'f~ (7:)
,fF,;' ('7:)
1;'
r
I~ ( 't')
I ' dent e 1' corrlspon
e in un punta generico del corpo, con
Ie componenti (covarianti) delle forze di massa e su-
perficiali, con ni Ie componenti (contravarianti) del versore nornlale G""
Ii a
e diretto verso 1'esterno del volume v:
Ij
fF.~ (t:) + rb,·LJ (1:) =JP J
(2)
t en-
I
p2~~ tt) '() -z;-l
(3)
" ./. IV (1946-47), 103. Uber den Reziprositatsatz in der Dynamik der Elastischen KHrpen - Ingenieur Archiv 22, 1954, 45. (2) M. E. Gurtin e Eli Sternberg - On the linear theory of viscoelasticity - Archive of Rational Mechanics and Analysis, 11 (1962) 29i. (3) Ammetteremo che Ie forze, gli spostamenti e Ie derivate prime e seconde di queste ultime grandezze soddisfino Ie solite condizioni di regolarita della Fisica matematica. (4) Cfr. per esempio B. Finzi ed M. Pastori - Calcolo tensoriale - Zanichelli Bologna, cap. IX. 190
- 11 D. Graffi Ovviamente la (2) vale in ogni punto di v, la (3) in ogni punto di 6' Ora fra il tensore di deformazione :
1:.. (t) I
(4)
I
I
.j lli
or;.
~J
... ~i It. 2
e il tens ore degli sforzi passa la relazione di Hooke : (5)
dove i coefficienti
Cii'ti
sono costanti
0
al piu funzioni solo del posto. 1-
noltre ammessa l'esistenza del potenziale elastico si ha Cit> posto,se t
.
-1'~ (t - T:)
Cy'S4 ::: Cu~i
~.
un istante superiore a 7: , moltiplichiamo Ia (2) per
(-4 (l.)
cor•• ponente controvariante delle spostamento
It,
"'11
generato dalle forze
p ph:) ,
rl(t)
; ovviamente ~"("t)
. J
soddisfa Ie
equazioni analoghe aIle (2), (3), (5)). Sommiamo rispetto a i e integriamo suI volume v. Si ha : (6)
Jr ri
(t:)
lilt -1:) ~ n
v
/
P/f.d ~ .ilt -i-J oW -If u~~~~) /it-..)~.
.,
Ma:
Sostituendo nella (6), appUcando il teorema della divergenza e ricordando Ia (3) si ha :
(8)
.
./I./i
jf Fi'(t l4 'it-,,) tAv"'I.(Ir:-H 'It-t) ao~ f~ I'ql.H (I-r) .Lv.
1'()24it'l"~ 1j''tt-7:J
~
... ./ ?'7;1.
Integrando ora la (8) da 0 ate osservando che
(5) Il simbolo .
fra due vettori
(5)
oLv
:
~ il notissimo simbolo di prodotto scalare. 191
- 12 -
(1:)
Si ha poi ricordando d.c p~.
D. Graffi
~ un tensore simmetrico :
i' ~ "0' "ti-f r;'(7:J~II(t_'tj - P0,(t) f (t-t); c.)1.1 T ('t) (t-r:)
1
(11)
-1
sostituendo nella (9) si ottiene (facendo anche uso di formule analoghe a (8): (12)
r!1:( rf F'(~). 7{t -t) 0.' 1f{1:),?(t-tJ cI., t{ ~:'(/J, ;j,(,) - . )0 j." -~... . rt It'Y j j'f)
'P t
v
r }o
"I
't
0
Z;
Ora ricordiamo che se A(t) e B(t) sono funzioni integrabili vale la relazione
(6)
( 13)
:
1 t
)' tA (t)
o
B (t - 't J d 7.:
=
A (t - t-) Bet) d:c
0
relazione che si ricava, assai facilmente, eseguendo al primo membro di (13) il cambiamento di variabili v = t - \' e poi cambiando v con "C
modo analogo si ha : (13')
o
c(A(,r) C(8(t-r;) cit;
d. to
c( 1:
In
t
(t dA(1:) B(t-'C)olt::i dAU-f) Jo at: 0 c(1:
"it
(13 )
•
B(7:)dt:
-jt: d-A(.t-r:) ote -
0
.0- e('t} dl: ol. t-
(6) Cfr. per esempio: Ghizzetii, Calcolo simbolico, Zanichelli, Bologna 1944
pag.30.
192
- 13 D. Graffi
Tornando ana (12) si ha (in base ana (13) e (13')) scambiando la posizione degli apostrofi che il secondo membro invariato (nel termine tensoriale occor-
).
re scambiare gli indici r,s con i,j e tener presente che Ci'[~:" Ct"~i
Percio scambiando gli apostrofi rimane invariato anche il primo membro di
i
(12) e si ha cosi la relazione : (14)
t
~
i,
t
I "
c1;t(:p!hJ.~·(t t)clv+lf(rJ.i'(t·1:)ol)]f~()t;--·1
-;-
~
o(l~'
'0
-
~t ~ el(tJ --=()
•.•
~
~
~) ~
f),/(o)
0 -
'Qt,·-1
'ft) Ici
-z,.j ;(0)
:(t) -;
( =,olt rF(t).,~(t'TJCh'+/~tr-~)·1 (t-r)clj)J ~t-- .~/-J)_ ·~t I ;.
(,.:
J
.;J
V
iJ.} d.1I
7
k
v " v E questa relazione esprime l'estensione alla dinamica del teorema di Betti. 3. Indichiamo alcuni casi particolari della (14): Supponiamo Ie condizioni iniziali nulle
cio~ ~~(o) = ~~ ( , ) = 0 /
Supponiamo poi r'
o'
[,'0
;.J ,. 'It
I
~/"
/;) ;'(0) = ? "1"(0) 'P t
-.
-,..
( 15)
dove
I
,,1'1
.. !
•
0 •
-~
f'!t) ~
Git)b'
r' t ". I ':"
I.' { ~
,
-, .
:I<
'Qt
4'
-.. 1I,
I
II
•. .1
sono vettori funzioni del posto, rna non del tempo. In al-
tre parole supponiamo che Ie forze si possano scomporre nel prodotto di una funzione del posto per una funzione del tempo; la funzione del tempo identica
t
per tu!te Ie forze. Si ha allora da (14) (16) f
(17)
~
.... _
~i I, )r\ L( : f~ ~;(t <Jcl.v .;i",'tt -r) ~+JG(t )dt1ri'11/-tJ«,{1:A (t-~)dO)
,
Ora
, ._
f
0-
0
6'"
noto che se per ogni t : t
J A(t) S(t-r-) eLt ~"
o e se A(t) non
~
identicamente nulla, ne segue B(t)
=- 0(7) quindi dalla (14)
si
(7) Un modo relativamente semplice per provare questa teorema ~' il seguente. Sia T un numero positivo; si ponga f( (t) = A(t) per t < T, A·(t) = 0 per 193
. /.
- 14 D. Graffi
ha : (18)
l
jfo.~ .IJ-:(tJ d,v ~J. b'. ~-r.{t) at,b" "'lpr". 4--(t) d,v + b" .~'i( f) dsv
6'
yo
r;-
e moltiplicando per G(t) si trova che la (1),
cio~
forma originaria, vale, istante per istante,
purch~
no nulle e Ie forze soddisfino Ie (15}.
il teorema di Betti' nella
Ie condizioni iniziali sia-
-
Supponiamo ancora, piu in particolare, che i1 primo sistema di
-
forze si riduca rispettivamente ad una rorza R(A, t) concentrata in A , il secondo a una forza R(B, t) concentrata in B e che Ie due forze mutino col tempo
..
in modo identico. Supposto, per fissare Ie idee, A e B interni al corpo, Ie
-
-,.
forze superficiali fl ed f" saranno nulle e si potra scrivere :
-/?'(A,t) -1(t) -m'rA)
pF'(t) - Hp-A) R (A,t)
~Q(A,t)= ~(t) ~"(B) -. ....,. dove m' ed mil sono vettori costanti)
-
i(P-A)
~ la funzione di Dirac del
punto generico P. Sostituendo in (14) e indicando con ;;(A, t),
:;-"(B, t) gli
~
spostamenti s" ed s' all'istante t e rispettivamente in A e B , si ottiene (scambiando t, cont-1::, tenendo presente (13) e ricordando inoltre Ie proprie-
ta della funzione di Dirac) :
[~(t-~) (;';'(Al· 1"(A, tJ - ';;·(B}.1'( B. t)) 0(1; = 0 o
. /. t ;;? T e analoga definizione sia valida per B*(t), T si pub scegliere in modo arbitrario purche A(t) non sia identicamente nulla per t c:: T. Poiche la trasformata di Laplace i (A-(t)) (8W(t)sono assolutamente convergenti si ha (dr. Ghizzetti loc. cit. pag. 33) I
1.
i 1~ y~) 1;It{t-t) ctt: ..t(A*(tl) t( S"'(t))
da cui poiche A (t) non ~ identicamente nulla ne segue l(B'l(t)) = 0 quindi B·(t)=.O percib per t < T B(t): O. Dall'arbitrarieta di T segue B(t)=.O per ogni t. 194
- 15 D. Graffi
e per quanta si e affermato poco fa : -,.
.....
-">
R'(A , tJ'.-1(A,tJ:: R"(B,t).:;'(r~,t)
(19)
relazione che estende alIa elastodinamica il teorema di
reciprocit~
di Maxwell.
Indichiamo un'altra conseguenza della (14). Siano nulle Ie forze superficiali, Ie forze di massa soddisfino Ie (15) inoltre sia :
](D) =>.~,
();,1'(o) ., ~;r, ~t
(20)
1'(0) ., dove
A
e
fA'
~ ;"(0)
AQ,~
--~-I;-- -.:.
"til
fJ'" 0-
sono costanti. La (14) si pub scrivere in questo caso tenendo
presente (13) (21)
f.~ (t .t") 0\ t I( p;i'. ;"~) - pii", A' (~))c:A t - ~if( ~(t),;;;,(.) -:;;ro).{ltl )dv,
j
~N(t) ,i"(o) +/ ( ()~'((J) ____ ~
- - I
=0
()t
l)t
Ossia ricordando (20) (22)
;'~(+:))
~
l~lt-.)ct.t ((rit".·(t)-fQ;"·A'ltJ) o.v .).. ~j(i',:;O(tJ -a.', A 'It)) cl.v + o l v + r ~(i"~~'{tJ - a.". 2(1:)) dN =0
Quindi per Ie proprieta delle equazioni integro differenziali di Volterra si ottiene (23)
j f F"{t) '-1"1(t) (Lv "'1; r'(t) '4~;'(t) till v
t
Cioe Ia (l) pub essere valida in elastodinamica anche quando Ie condizioni iniziali non sono nulle, Per altre conseguenze della (14) rimando aIle memorie citate.
195
- 16 D. Graffi
4. Come si
e detto in principio applichiamo ora Ie considerazioni precedenti
all'equazione temporale di SchrHdinger (8):
in cui i simboli hanno i1 ben noto significato, in particolare U indipendente da r; . Siano
ftc) e
II
e i1 potenziale
due soluzioni della (24) corrispon-
1JI('t")
denti a diversi valori iniziali. Si consideri l'espressione :
'f''(t-7:) d;..vr-ad. '1/(7:) - ~~: m U\fh'l fH-~) ;:
(25)
~ 4 rrl- iK
Iv
I
'P tf(-C:) o't"
'1'( t -7:)
Integriamo i1 primo termine al primo membro di (25) su tutto 10 spazio. Si ottiene (v indica ora l'estensione dello spazio infinito) : (26)
1'f't t
-'t)
v
d-iv~'t{(.tt y/ft'J oW ;: ) o~v ({t t --c) r-a-d.. ~I('t") ) dN V
_I.v (~r(t-7;). ~ 1/,)rVolv
y' (t· () /I-a,.d.. 'f'r?::) e infinitesimo all'in,I
Ora se, come si ammette di solito,
finito di ordine maggiore di due, i1 primo termine al secondo membro di (26)
e nullo.
Si integri ora la (25) da 0 a t e poi su tutto i1 volume della spazio.
Tenendo presente la (26) si ha : (27)
-1
J. t
cl v 'f,ed. {( t t J '~..",.(, ~t, J 01 t
= -47rL'r1ll [.Iv Jt -'Y Ie) h,
v
,)
"'<:;
j ;1:~- ,u rMi , r'lr) r"/t -t)ct t ~ l
-
r (t· l~) ;1 c
Ora cambiando la posizione degli apostrofi si ha, ricordando (13), che il primo membro di (27) non cambia. Si ottiene
COSl
la relazione :
(8) Cfr. E. Persico, Fondamenti della meccanica atomica, Zanichelli, Bologna, 1936, pag.169. 196
- 17 -
D. Graffi
Ora integrando per parti si ha :
•
t
(29)
it
[~r'(7:) 'I'(t-'C) cJ.t ·IJI'(t) '1'(0) - c({°N (t) - /{.. I
)
'U 'let ot") tie i 1:'
o ~'r; sostituendo in (28) e tenendo presente (13') ai ha dopo aemplici calcoli :
che t! la cercata relazione di
reciprocit~.
Come appE.::azione della (30) indichiamo i1 seguente teorema di cui. per breviU.; ometteremo la dimostrazione. Se da una certa misura risulta che un corpuscolo t! inizialmente nel punto A , la probabiliU di trovarlo all'istante t nel punto B. vale la
probabilit~
di trovarlo, nello stesso istan-
te. in A qualora risultasse inizialmente in B .
197
CENTRO INTERNAZIONALE MATE MATICO ESTIVO (C.I.M.E.)
G.GRIOLI
I : PROPRIETA' GENERALI DI MEDIA NELLA MECCANICA DEI CONTINUI E LORO APPLICAZIONI. II : PROBLEMI DI INTEGRAZIONE NELLA TEORIA DELL'EQUILIBRIO ELASTICO.
ROMA - lstituto Matematico dell'UniversiU 199
PHOPRIETA I GENERAL! DJ MEDIA NELLA l\IECCANICA DEI COXTl\,CI E LORO APPLICAZIONI
GIUSEPPE GRIOLI
Quanto diro ha 10 scopo di stabilire alcune limitazioni per 10 state tens ionale di un corpo continuo e - quando possibile - per la sua deformazione. Le limitazioni stabilite per 10 stress mantengono la lora validitll. qualunque sia la natura del corpo continuo, anche rigido, rna la lora concreta applicabilita presuppone la
conoscen~a
dello stato attuale. Se questa
caso statico come pure in quello dinamico,
~
~
noto, sia nel
nota il campo d'integrazione
che in modo essenzic:.:e interviene nelle espressioni delle limitazioni stabilite ed esse acquistano si gnificato concreto quando siano conosciute tutte Ie forze esterne, in esse comprese quelle d'inerzia. Pertanto, dalle considerazioni svolte nei primi tre paragrafi non esula neppure il caso delle deformazioni finite se la configurazione attuale equilibrio -
~
0
di
nota.
Vi rientra sempre il caso dei corpi poco deformabili, in quanto per essi nella valutazione degli integrali il campo d'integrazione va classicamemc so~ stituito da queUo di una vicinissima configurazione nota di
riferim(~nto
nei
cui punti si ritengono applicate, senza lora alterazione, Ie forze estelne. Le difficolta che nascono se una parte della sollecitazione esterna superficiale non
~
nota in quanto abbia carattere di reazione vincolare sono supera-
hili in base al contenuto dell'Osservazione riportata alla fine del quarto paragrafo.
201
- 2G. Grioli
1. Premesse di carattere generale
C
Denotero con
la configurazione attuale di un qualunque corpo con-
tinuo riferito ad una terna trirettangola Ievogira
x
X
=
rs punto P
xl x2 x3
,con
Ie sei caratteristiche della stress, funzioni del gene rico
sr
di
0
C. , con .j.
la
densit~ in
t Fr
e con
P
il vettore
che esprime la forza di massa specUica, compresa in essa la torza d'inerzia (caso dinamico). Detto
~
il cpntorno completo di
Ia forza superficiale esterna agentesu ciale
dEe
l'interno di
C
C
,sia frdZ
attraverso I 'elemento 8uperfi-
il vettore unitario normale a L e orientato verso
Nr
C
Le equazioni fondamentali sono
X
/s =
rs
~F r
( in
C) ,
(1 )
Xrs N
S
=r
r
ove Ia sbarretta denota derivazione rispetto aHa
x s
e vale la consueta con-
venzione della somma. Per ogni scelta dei numeri interi, positivi
0
nulli, "
'l',
Ie coordinate iperastatiche della sollecitazione totale. (1) (l)Facilmente generalizzabili in presenza di forze concentrate. 202
j. siano Ie
- 3-
G. Grioli Le equazioni cardinali della Meccanica si scrivono b(r) 0 , 000 =
(r = 1, 2, 3)
(3) b(l) 100
b( 1) 010
(3) (1) _ b 100 - b 001 - 0,
= 0,
Per una qualunque funzione
e
-f = 1
Da
(1)
f e.
fd
f(P)
integrabile in
b(3) _ b(2) = O. 010
001
C si ponga
C
si trae facilmente
(r = 1,2,3).
Le (5) generalizzano note relazioni di media di Signorini [1 ] coincidono con Ie (5) per tit~
n
=." +t
che
+ ~ = 1,2 . Anzi, in tal caso Ie quan-
a primo membro di (5) 80no tutte determinate dalle (5) stesse. Invece,
per a
> 2 Ie equazioni (5) sono in numero inferiore a quello dei valori medi che contengono. Tuttavia, ~ facile riconoscere che per ogni n > 2 il sistema (5) permette di determinare quindici dei valori medi in eSiO contenu-
ti. Precisamente, s'i ha
203
- 4G. Grioli
~
X
J
,,¥to- =b (r) r?t! D0""
J
(6)
ecc.
Se
P
~ un qualunque polinomio neUe
t
Y
~r
,Ie quantitA
X·
P
rs t
risultano combinazioJ'li lineari di soluzioni dei sistemi (5) corrispondenti ad opportuni valori di
n
nate da tali sistemi se· P t
. Naturalmente, tali quantitA risultano determi~
combinazione lineare di quei soli monomi che _
Xrs Pt
intervengono neUe (6) . In tal caso dirb che
~ di classe
M.
2. Limitazioni per 10 s.tato tensionale Per Ie componenti della stress userb spesso notazioni ad un solo indice. secondo la convenzione
X X
(7)
r+3 =
Dette
at' (t =0, 1, ...... , m)
r+lr+2
e
stanti arbitrarie, risulta
(8)
da cui segue 204
0
s ' (s
= 1,2; ..... ,6) delle co-
- 5G. Grioli
(9)
ove si
~
omesso il segno di uguaglianza che pub capitare in casi veramente
eccezionali e di facile individuazione. Da (9) segue l'interessante limitazione
( 10)
Le
at
possono determinarsi in modo ~he il secondo membro di (9)
[ e di (10) ] sia massimo rna cib pub riuscire laborioso. Una scelta conveniente delle
at
~
, -
a =
(11)
t
2:
s=1
con
n 2 = 1.. (
(12)
\:'t
come
risulter~
t
C
chiaro nel
Identificando il polinomio
~
p~ d e t
3.
~ A\'s P s
::i
205
successivamente con quello che cspri-
- fi -
G. Grioli
me
10
sforzo normale, di taglio
bili - con quello che e::;pl'ir.I<"
~e
0 -
nel caso dei corpi elastici poco deforma-
cr.rattl'L'istiche di defol'mazione
0
Ie dilata-
zioni, gli scorrimenti, Ie variazbni di temperatura nel caso adiabatico, ecc., la (9) da Ie corrispolll!emi limitazioll! pe{ i massimi moduli. Ad es., posto
( 13)
10
sforzo normale
vettore unitario
~
V
soddisfa alla linlitaz ionl:
I (~ \
( 14)
,
l'ela: i VlJ alb d L l"l. ionc or ientata caratterizzata dal
1»11
r..'
> '-
• ~ " H11aX
qualunque siano Ie ('ostanti
(-'
Si supponga che ,_
a
t
sia un corpo dJ indl'icC) eli sezione
ad una terna centrale III cui
A
,riferito
sia parallelo aIle generatrici. Si
1'~lsi:;e
assuma (15 )
p o ='4,
p =0 1
1)
'
J:
-
p =x 3 . 3'
X
2 . 2'
sultante delle fo,ze esterl'(' ag i.'lIti sulla parte di ba~e COli
se t >3.
Ie componenti del ri-
Fissata una sezi()ne
ne considerata e la
P =0 t "
:\1 <,0
('
206
I
;\I i (X 1 )
('ompresa tra la sezioquellc del
101'0
momenta
~
7-
G. Grioli risultante rispetto al baricentro della sezione considerata, siano i valori medi di R'i (x 1) , M~ (xl)
R., M. 1
al variare di xl
tra Ie due basi di
la retta del piano
~l
1
e.
Da (10) segue
(16)
max
Supposto Rl
~
0 e detta
r
= 0 di equazione
(17)
sia AlIa parte di A ove il primo membro di (17) 1'intera A se r non taglia A,
d' la sua distanza da
A . Inoltre, siano
r e
Si pub dimostrare che se IE:
M* at
6J.
~ positiv~
oppure
il baricl~ntro di
M M
il vettore di componenti 0, ';,-{.
~l
el.
sono espresse da (11) sussiste una
delle due limitazioni :
J IXIII rna. >
(18)
l\X
II \ rna. >
delle quali vale la prima. se 1'antipolo Q di 207
r
rispetto all 'ellisse centra-
- 8-
G.Grioli Ie di A no
interno al nocciolo centrale di A , la seconda se
~
eIISO
l! ester-
[2 J . Si riconosce che
r
e Q generalizzano i concetti di asse neutro e
di centro di pressione nel claasico problema di De Saint Venant della pre soflessione.
Si supponga ora che
C sia un prisma rettangolare la cui sezione abbia
b2 e bS ~ b2 e inoltre risulti R1 • 0 • In tal caso,posto a O = 0, non ~ difficile determinare aI' a 2 in modo da rendere massimo il secondo membro di (18) • Ne segue llseguente risultato i lati di
ri~pettive
[2] . Posto q = Ie
~1
'
e2
lunghezze
bS
(I b2 '"
,nella sezione xl = 0 si considerino Ie parabo-
di rispettlve equazioni
( 19) x
2 2
2 b3
+3q2
xs
e la retta di equazione (20)
Siano : BI
la porzione di A delimitata dalla parabola
retta di equazione (20) e dalle condizioni delimitata da
~ 2 ' da x2 ~ 0,
Xs
X2 ~
0 ,
Xs ~
0;
@1
' dalla
B2 queUa
~ 0 e dalla retta di equazione (20); 208
- 9G, Grioli
B
la porzione della sezione
metriche rispetto agli assi
A x
costituita da
BI + B2
e dalle sue sim-
x ' Si ha la seguente condizione di sicu3 rezza: condizione necessaria affinch~ IXIII non superi una quantita pomax M M sitiva h2 ~ che il punto di coordinate 0, ~2 ' -1.2 sia interno a Ah Ah B , Si riconosce facilmente che B ~ interna alIa ellisse centrale della se2'
zione A dal che si deduce che la condizione enunciata migliora un 'analoga condizione gifi data da Signorini [3],
3, Altre limitazioni fondamentali per 10 stress, Le relazioni integrali (5) possono sfruttarsi per dare concretezza anche a certe nuove limitazioni per 10 stress, Tali limitazioni possono riuscire menD vantaggiose di quelle precedentemente stabilite rna sono di maggiore por'tata, di piu facile calcolo e permettono di stabilire, a volte, limitazioni per Ie deformazioni. Siano: q
i coefficienti costanti di una form& quadratica defil"\ita 0 aIrs menD semidefinita positiva in sei variabili; Ort delle costanti arbitrarie; Qo ' ' , " Qm , m+ I funzioni di x I' x 2, x3 definite in
gonali, Siano, inoltre, (21)
)': "
~
f
'i2 dC
C
e (22)
f=
Da (21) segue
6
(23)
'f =
rf
"I,~I
C
, qrsXrXsdG + C
L
It,): I
209
e
e ivi tra lora orto-
- 10 -
G. Grioli che per (24)
diviene (25)
r
Si riconosce facilmente [4]che Ie
rt espreue da (24) minimizzano 1a
'+'
se la forma quadratica di cui i q sono i coefficienti ~ proprio detinita pOltrs tiva. Anzi, si riconosce che nella (25) vale 11 leino di uguaiUanza Ie e 1010 se risulta
=!
XQ r 2 t Qt
IW\
(26)
Xr
t:o
It
.
-
Ne segue: tra tutti gli stress corrispondenti ai medesimi valori medi XrQt' quello che minimizza in media ogni forma quadratic a definita positiva nelle componenti dello stress
-
~
espresso da (26).
La limitazione acquista interesse concreto ogni qualvolta ili
X r Qt siano conoscibili, come accade, in particolare, se Ie funzioni Qt coincidono con i polinomi Pt di cui nei paragrafi precedent!. In tal cuo la (25) si scrive (27)
,
~
','\11
L'applicazione della
disugu~g1ianza
di Schwarz permette dl dimo-
J
strare [4 che ogniqua1volta Ie (9), (27) sono confrontabiU in quanto 8u88ilte l'uguaglianza (28)
\~X ~R. I, V~, qrsXr} s =) ~,.-8
la limitazione (9)
X
l,~'
~
-.It
II
piu efficace della (27) per opportuna sce1ta delle at' Cit> 210
- 11 -
G. Grioli capita, in particolare, quando si assumano per Ie at proprio i valori espressi da (11). Da (27) segue immediatamente
I Xrlmax~ f. ~ ;~t ~
(29)
X'P2
-
con i1 secondo membro noto se Ie forze esterne sono note e gli XrPt di classe
M. Tra Ie limitazioni concrete che possono dedursi da (27) segnalo soltanto una condizione necessaria di plasticita. Precisamente, tenendo presente la nota condizione di von Mises, da (27) si deduce che condizione necessaria affinch~
un corpo sia al dis otto della sOElia plastic a
(30)
~
~
_1_
t- 0
\-
IJ 2
t
\'
L
4=t
[(XP _s+ls+l ss t X
e che risulti
P )2 + b X
t
p2 ]
ss+l t
<
b2
ove b 2 caratterizza illimite elastico. La condizione (30) da in effetti una condizione per Ie forze esterne.
4. Limitazioni per gli spostamenti dei corpi elastici poco deformabili. In questo paragrafo supporro che C sia la configurazione di equilibrio di un corpo elastico per cui vale la teoria line are classica. La deformazione si pensera, pertanto, valutata a partire da una configurazione naturale vicinissima
C*.
Le componenti dello stress X possono pensarsi, COrnIe abituale rs nella teoria classica, quali dirette funzioni delle coordinate del generico punto P" di C*, net senso che esse - che in una teoria con deformazioni finite hanno i1 significato di componenti euleriane dello str.ess - si devono ritenere coinci-
,delle caratteristiche di tensione. In rs definitiva, in armonia con la teoria lineare, ai fini della valutazione degli intedenti con Ie componenti lagrangiane, Y
211
- 12 G, Grioli grali il campo d 'integrazione si con Ie coordinate
y
y.
1
identifichr~
,del corrispondente
..
con
C ,le coordinate di P
II P in
II
C ,Ie
X
rs
con Ie
rs Identificando i coefficienti
q
con quelli,
rs
m
rs
, della forma qua-
dratica
W
(31)
1
,=-
2
6
L r, s:=1
m
ra
YY
r s
che esprime la dens itA di .energia potenziale elastica, il teorema di Clapey-
,
ron e la (27) permettono di dedurre per illavoro delle forze esterne nella spostamento
C4
C e per 1'energia elastica totale la limitazione 6
~
(32)
r, s =1 Varie applicazioni possono farsi della (32) . Ad es., nel caso di un cor-
po incastrato
0
appoggiato senza attrito au una parte del suo contorno e sog-
getto a forze esterne attive di direzione e verso invariabili, come nel cuo del peso, il lavoro delle reazioni vincolari durre per la componente
~
nullo e la (33) permette di
de~
u dello spostamento nella direzione della fona
la limitazione
(33)
ove
c*' I Imax ) ~ R-
6
h.l
u
R indica il modulo del risultante della sollecitazione attiva. Interes-
santi limitazioni si deducono da (32) per elementi caratteristici della defor212
- 13 G. Grioli
mazione nel caso che la sollecitazione esterna attiva agisca solo su due porzioni
ct
I ,
cr II
non ci sono forze
del contorno n~
fS" del corpo mentre sulla rimanente parte
vincoli.
Se Ie forze agenti singolarmente sulle due parti ". I , llaltra sy
II
gole coppie, si trovano limitazioni per un certo allungamento
0
cibili a due singole forze agenti una su
(f"
I
,
sono ridu-
«,,"II 0
a due sin-
per una certa
rotazione che richiamano risultati noti della teoria di De Saint Venant e che spesso mostrano come Ie deformazioni cui porta quell a teoria sono pitl piccole di quelle volute dalla teoria lineare esatta
[5]. [6] .
Qualche interessdnte limitazione pub dedursi dalla (32) anche per Ie variazioni di volume dellt> singole cavita nel caso di corpi con cavita (involucri)
[7J . Se si denota con
1:'
la forma lineare nelle
Y
r
=
(34)
che esprime la variazione di temperatura di un corpo isotropo nel pas saggio adiabatico da
C* a
C
, da (32) si deduce
m
L
-
L2
3(;',+-)+2}1. I.. c
L
J '3( f + - ) + 2\.t c .-
~
L2 c
'
L
c
t =0
\~+t V 11
213
t=1
1
~2 t
- 14 -
G. GrioIi
ove II = Y1 + Y2 + Y3 • ). e
)J.. sono Ie
costanti di
Lam~
I
L il coefficien-
te di tensione termica e c dipende dal cal ore specifico a configurazione costante
[4
I
pag
.86] .
OSSERVAZIONE Tutte Ie limitazioni sin qui stabilite presuppongono per la Ioro concreta valutazione Ia completa conoscenza delle ·forze esterne. In presenza di vincoli intero,engono Ie reazioni vincolari generalmente incognite e quasi sempre non eliminabili dai secondi membri delle disuguaglianze stabilite. Tuttavia in tale eventualitA Ie disuguaglianze stabilite avranno i secondi membri noti se essi si minimizzano rispetto a tutti gIi elementi indeterminati che essi contengono [valori medi V t
e reazioni] • minirnizzazione che
va fatta tenendo conto non solo delle (5) rna anche della classe di sisterni di reazioni consentite dai vincoli.
214
- 15 G. Grioli
BIBLIOGRAFIA
[ 1] A. Signorini IISopra alcune questioni di Statica dei sistemi continui II Ann. Scuola Norm. Sup. Pisa; Ser.II,2,3 - (1933).
[2]
G. Grioli "Limitazioni per 10 state tensionale di un qualunque sistema continuo ll Ann. Mat. Pura Appl. ser. IV, 39 , 225-266 (1955).
[3]
A. Signorini IISopra un'estensione della teoria linearizzata delllelasticitfi " Rend.Sem. Univ.Pol. Torino,
[4)
g,
83-93 (1952-53).
G. Grioli I'Mathematical Theory of Elastic Equilibrium (Recent Results)1I Springer- Verlag. Berlin Gottingen Heidelberg (1962).
[ 5] G. Grioli "Sullo state tensionale dei continui in equilibrio e sulle deformazioni nel caso elastico" Conferenze Sem. Mat. Univ. Bari 35-36 (1958).
t 6]
E. Bentsik
II
Sulle deformazioni elastiche dovute ad una sollecitazione
riducibile a due coppie in equilibrio IIRend. Sem. Mat. Padova V. XXXIII; Pag.297 (1963).
t 7]
G. GrioE IISulle
deformazioh~ elastiche di un involucro omogeneo sogget,.
to a pressione
trazione" Rend. Sem. Mat. Univ. Padova 20, 278-285,
0
(1951).
215
- 16 G. Grioli
PROBLEM I DI INTEGRAZIONE NELLA TEORIA DELL'EQUILIBRIO ELASTICO
Lettura I
SuI teorema di Menabrea. Considerazioni sulla questione di esistenza
1. Considerazioni L'applicazione delle
~J1troduttive. propriet~
Richiamo del teorema di Menabrea.
di media precedentemente indicate permette
di costruire un metodo d'integrazione del problema di equilibrio elastico valido non solo nel caso lineare rna addirittura in quello delle deformazioni finite. II metodo assume quali dirette incognite
anzich~
Ie componenti della sposta-
mento queUe dello stress le quali, del resto, quaSi sempre sonG gli elementi di maggiore interesse concreto. Nella prima lezione saranno svolte delle considerazioni suI teorema di Menabrea che da un lato sonG essenziali per la costruzione del metodo d'integrazione mentre dall'altro permettono di tradurre 10 stesso teorema di esistenza della soluzione in quello dell'esistenza del minima di un certo funzionale operante sullo stress e qui si notertl l'analogia della circostanza con quanta ha giA stabilito Signorini [~2J operando sulle componenti di spostamento e dando luogo ad interessanti risultati di G. Fichera
(3J
a proposito
di un problema dallo stesso Signorini definito con la.locuzione "dalle ambigue condizioni al contorno"
[1) .
Le prime due letture si riferiscono a corpi elastici poco deformabili per i quali vale la teoria classica linearE\. E' pertanto lecito,
coml~
abituale, con-
fondere 10 state attuale con quello nato di riferimento ai fini del calcolo degli 217
- 17 -
G. Grioli integrali che s'incontrano e della valutazione delle forze esterne. L'energia elastica
~
espressa da una forma quadratica definita positiva con coefficien-
ti indipendenti dana temperatura il che derazione trasformazioni isoterme
0
~
quanto dire che si prendono in consi-
adiabatiche. Per semplicitA, si conside-
reranno deformazioni a partire da uno state naturale -
cio~,
esente da stress -
di corpi elastici anche anisotropi e - limitatamente ana prima lettura - inomogenei. Si suppone l'assenza di vincoli interni, rna
~ nota [2, pag. 363 ... ] che
il caso dei solidi elastici incomprimibili si pub far rientrare in queUo dei solidi comprimibili semrllicemente particolarizzando i valori dei coefficienti dell' espressione de1l' energia elastica.
Al solito, denoto con
C
* la configurazione di riferimento (~tato naturale)
do che in corrispondenza aHa parte
r
di
supponen-
siano note Ie forze superfi-
~II. vi siano dei vincoli per i quali si sa ca-
ciali mentre sull'altra parte, ratterizzare l'insieme
!,* la sua frontiera,
r) r*
di un solido elastico poco deformabile e con
di tutti i possibili sistemi di reazione di cui essi
sono capaci. 11 sistema di Cauchy esplicitamente si scrive y (1)
y
*
~ 's k F = rs r
rs
NS
(in C ),
(su
l)
(su
r)" ,
I
~
= f"r'
(2) 218
,
- 18 G. Grioli
con
f
r
Sia
r·
appartenente a
AY l.J rs
unlarbitraria variazione dello stress, soddiifacente al si-
sterna
J.
=0,
(in
~
A Y NS = 0 , U rs
(su
I')
IS
(uY) rs (3)
,
~II
e sia su L
(4)
La corrispondente variazione della densitA di energia potenziale elastica espressa nelle
Y
rs
_
~(1)
~w !J L1w=L: 'fY.:' s s =1 6
(5)
Detto
Y
+
r
W(/Jy).
r
*
10 stress effettivo 0 - in assenza di unicit~ - uno stress conrs gruente verificante Ie equazioni di Cauchy e uit il corrispondente spostarnen-
Y
r
to, da (3), (4), (5) segue
(6)
(1) Si usano notazioni ad un solo indice analoghe a queUe usate nella Lettura 1.. 219
- 19 G. Grioli
valida per ogni scelta delle
IJ. Yrs
non tutte nulle, verificanti Ie (3) e ta-
li da dar luogo, in base a (2) ,(4) , a reazioni
af
. La (6) esprime,
terizza 10 (
0
com'~
P tJ f r +
r
appartenenti
ben noto, il teorema di Menabrea e carat-
uno) stress congruente.
2. Qualche conseguenza del teorema di Menabrea. Questione di esistenza. Se
ull ~ assegnato su ~" il teorema di Menabrea caratterizza la soluzior
ne come quella che rende minima [ vedi ({) ]
nella classe degli stress verificanti Ie (I). In tale caso dell'i.lcastro su assegnate Ie
~"
modalit~
bile definire l'insieme
,per
r
u; = 0 • Interessante
~
~
compreso quello
il caso che su
~"
siano
di realizzazione dei vincoli in modo che rio sca possie il tipo di spostamento da essi consu1tito, pur
rimanendo incognito quello effettivamente realizzato. Trattasi, in effetti di . problemi sostanzialmente non lineari a causa di disuguaglianze nelle condizioni al contorno e l'assumere come dirette incognite Ie componenti di spostamento pub non riuscire conveniente e dar l\logo a
difficolt~
che con vantaggio pos-
sono eliminarsi riferendosi direttamente aIle caratteristiche della stress, sulla base del sistema (I), (2) , associato al teorema di Menabrea. Coslope-
r.
rando, trattasi di determinare, tra tutti gli stress verificanti Ie (I) , (2) , quello che minimizza 10 stress trovato
~
A ,lasciando
P r
in
Si
~
cosi sicuri che
congruente rna rimane legittimo il dubbio se 10 sposta-
dedotto per integrazione di ben note espressioni nelle Y ~ e r rs che contiene di arbitrario al piu uno spostamento rigido, verifichi Ie condimento
u
zioni geometrico-cinematiche volute dai vincoli. 220
- 20 G. Grioli Se per il problema considerato negli spostamenti esiste un teorema di esistenza e di
unicit~
il dubbio naturalmente cade,
r.
com'~
che sia stata caratterizzata in modo corretto la classe
evidente, sempre
Cib accade nel caso di un appoggio liscio con un supporto non cedevole quando Ie forze esterne attive appartengono ad una certa classe, come di recente ha dimostrato G. Fichera [ 3] studiando un interessante problema semilineare posto da
A. Signorini [1] .
A me qui pare interessante osservare che il teorema di Menabrea, oppor-
tunamente inteso, permette di prescindere dal teorema'di esistenza e di unicit~
nel problema sugli spostamenti nel senso che esso permette di tradurre
la condizione di esistenza in quella di esistenza del minima di un certo funzionale che opera direttamente sullo stress. Denominando stress matemati-
££ ogni soluzione delle
(1) , si deve precisamente ritenere vero il teorema:
!! A ammette un minimo nell'insieme degli stress matematici che danno luogo a reazioni appartenenti alla classe
r,
di equilibrio esiste. Cib nel senso che 10 stress
la soluzione del problema
y'*rs
nell'insieme degli stress matematici tenuto conto della classe solo
~ (com'~
ben noto) congruente rna
d~
r
minimizzante
A
,non
luogo a spostamenti verificanti
quasi ovunque Ie condizioni imposte dai vincoli. 11 teorema di esistenza
~
pertanto tradotto in quello dell 'esistenza del minimo di A in un 'opportuna classe. 3. Precisazione del teorema precedente. Dimostrazione in casi fisicamente interessanti. I casi di maggiore interesse concreto sono indubbiamente quelli in cui
~
presente il vincolo d'incastro
terale (con supporto rigido
0
0
di appoggio bilaterale
0
unila-
cedevole) e in tali casi darb la dimostrazione
del teorema enunciato. 11 primo di tali casi rientra, 221
com'~
stato gil osserva-
- 21 G. Grioli
~ II. Sia ur
to, in quello di assegnato spostamento su
10 spostamento as-
segnato su
L"
colo
contiene qualunque vettore] come accade nel caso dlincastro. 11
[r
e si supponga che qualunque reazione sia eplicabile dal vin-
teorema del numero precedente diviene ora: Se su
~ assegnato 10 spo-
ul' ,la soluzione esiste se e solo se esiste il minimo di
stamento
r
I
AI =
(8)
ZU
..
W(Y
C
rs
)
dC~
fA..Yr 'iir d ~If I'I
nella classe di tutti gli stress matematici. Lo stress minimante coincide con quello effettivo. La dimostrazione allo stress
*
~
immediata (2). BasterA far vedere che
minimizzante AI nella classe di tutti i possibili stress rs matematici corrisponde uno spostamento u' coincidente (quasi ovunque) con Y
r
Ur
su ~" . A tal fine si osservi che la condizione di minimo di
AI
si
esplicita, in base a (5) e alle note relazioni tra stress e strain nella disuguaglianza
che, per Ie (3) ,diviene
( 10)
i*
* '*+ £:t1 ~ f
W(~ Y
rs
C
)dC
2,"
r
(VI r -
valendo il segno di uguaglianza solo per
~'I~ UI' ) d G
~Y
=0
•
0,
rs ~ Fissata una sestupla A Y = ,sia q = "t N s . Se ~ U rs ~ rs r ) rs 5 rs verifica Ie (3) ,anche con A costante Ie verifica e la (10) si traduce S rs ~
A'
(2) La necessitA
~
r
evidente. 222
- 22 -
G. Grioli in
Dato che
*
Y mmlmlzza A nella classe di tutti gli stress matemars tici, la (11) ,fissato ,deve valere per ogni e qualunque sia il suo 1 ra segno. eib implica l'annullarsi di q (u,r _ir) quali ovunque IU e
A
r
poich~ la larga arbitrarietl d i ' dei vettori qr
con direzioni non
X·
r
. rende certo pOlsibile avere nella (11)
:U~te complanart~),
Ii deduce che su
III
risulta quasi ovunque u,r - U'r ,. O. Nel caso di vincolo d'incastro
~
"ii ,. r
0
e conseguentemente u' = o. r
~II
Si supponga ora che il vincolo presente au I:. Ie liscio cedevole elasticamente
0,
in particolare. rigido. L'insieme
comprende tutti i vettori paralleli e concordi a
r
sia un appoggio unilatera-
Neil vettore nullo. Supporr
rb che 10 spostamento corriapondente alIa reazione esplicata dal vincolo sia esprimibile nella forma se
~ > O.
Ie
f,.
(12)
O.
(3) Si tratta in effetti di ammettere .1'esistenza di soluzioni del siltema (1), (2) con r fissato in modo opportuno.
t
223
- 23 -
G. Grioli
ove
vr
~ un arbitrario spostamento tangenziale e
jJ
una costante da
supporsi infinita nel caso di appoggio. su supporto rigido. II teorema del n.2 si esplicita nei seguenti termini: Condizione necessaria e sufficiente
affinch~
esista la soluzione del problema di equilibrio
~
che esista il minima di
np'inSieme di tutti La
necessit~ ~
eli stress matematici che laseiano
ove risulta di
A II
p1l'
uI
nella classe
10 spostamento indotto dalle
Y
,II
mini-
,II
1a corrlspon , r den t e reazlOne. ' S'la L 1 1a parrtsed'l L
cP*r +r. 0 2:1 quella ovle
J...*
~r
J
,in base a (5)
fc*
( 14)
r
una conseguenza immediata di (6), (12) . Per dimostrare
la sufficienza, si denoti con ' t '1 e con mlzzan
f
J
= 0 • La condizione di minima
si esplicita nella disuguaglianza
'* W(~ Yrs)dC
*. i +
1
L r)
(~ J.If 2d I
"+
I"
+ (Ll~ :(U,r +; Nr )d i.; +1 tl .: u,rd l:i )0 . ~II1
J't
~
I",
r
da riten~rsi verificata per ogni scelta ~ ytf soddisfacente aIle (3) e ana rs condizione che il vettore + ~ appartenga a . Precisamente,
¢'*" !J .. r
dato che considereremo i vettori
rr
~f~
piccoli di fronte a
.£1
dovranno soddisfare aIle (3) e inoltre dar luogo a vettori ad
N~ r
,con verso arbitrario su
~ II
~.
224
,
concordi ad
w* r
*-<
<prJ Ie ~ Y +~
su
paralleli
~
II
~1
•
*
rs
- 24 -
G.. Grioli
y'rs* : ~f rs
Per una prefissata seelta A vettore
.J,J
J,~
tra Ie possibili, sia
corrispondente. Lo iupporrb eoncorde ad
una costante, anche
\
l t 1 rs
e ~ q
e sussiste, per ipotesi, per ogni sceUa delle
~
seguenza
soddisfi alIa ad
N
(12)
positiv~ 0
quasi ovunque
* . Pertanto in (15) manea
iU
I
"
A~
rs ,tra quelle eonsentite.
negativo e Is tutta
~;'
il terzo
(15)
L
II 2
colo su glianza .
x"2
,la (15) impliea proprio
•
D'
implies che
dato ehe
qr
~
1
u~
N*r
10
con-
u' r
paraUelo
termine. Supposto invece
I
A> 0
~
2 • La (14) diviene
r., it \ non nullo su l. e eoncorde ad Nr ,si deve ritenere ,,~O
do proprio
il
r . Detta
Nr
'e in modo che sia(4) q I 0 su tutta C rs r pub essere
q
soddisfano aHe condizioni richieste,
t'
pureh~ sia ,. ~ 0 se qr non ~ n~Ho iU tutta
Si assumano Ie
*
q
r . Supponen-
ehe l'ispetta
i1 vin-
ed esprime distaeco dall'appoggio nel easo di effettiva disugua-
Osservazione In quanta ii
~
precedentemente esposto si constata ehe in
effetti dall'esistenza del minimo di A nei vari casi considerati segue ehe 10 spostamento
u'
r
indatto dallo stress minimizzante verifiea su
I
" certe
condizigni integrali da cui segue ehe Ie eondizioni imposte dal vincolo sono verificate su
LII a me no di un insieme di misura nulla. Si deve osservare ehe
il soddisfare in media queUe condizioni ~ in armonia con una visione integrale
della Meeeanica dei continui - in particolare della teoria matematiea dell 'Elasticitll - che, forse, solo ha senso fisico. Cib d'altronde da vari Autori in modo pit! (4) Vedi nota (3) .
0
meno esplieito.
225
~
stato gill r,imarcato
- 25 G. Grioli
BIBLIOGRAFIA
[1 )
A. Signorini "Questioni di elasticit~ non linearizzata e semilinearizzata II Rend. di Mat. Roma, vol.1S ; 95-139 (1959)
t 2J
A. Signorini "Trasformazioni termoela8tiche finite - Solidi vincolati' I Ann. Mat.pura ed applicata; S,IV T.LI, 329-372 (1960)
[3]
G. Fichera Presentato per la stampa in "Atti delllAccademia Nazionale dei Lincei" (Memorie).
~
(ltalia).
227
- 26 -
G. Grioli Lettura II Integrazione del problema fondamentale dell 'Elastostatica lineare.
1. Premesse analitiche. Sia {Wt 1la successione di tutti i possibili
monomi formati con Ie coordinate crescente. Sia
{pt}
y l' Y2' Y3' ordinati per grado non de-
la successione di polinomi costruita aggiungendo a
quella combinazione lineare di ,W 0 ' W1 , .•••••• ."
Wt
Wt _1 c~ rende
P t ortogonale a Wo' WI' ......... , Wt _1' nella regione ta dal corpo nel suo stato di riferimento. La successione
{pa
mi ortogonali in
C* cd
La supporrb inoltre, E' ben nota una
~
~ di polino-
costruibile in base ad un nota teorema di Gram.
com'~
certo possibile, normalizzata.
propriet~ di completezza di
[pt} . In base ad essa,~
dizione necessaria e sufficiente affinch~ due funzioni la prima in
C occupa-
'II.!..
definite
C .., la seconda sulla sua frontiera ~. siano quasi ovunque
nulle rispettivamente in
C'* e su
~* ~ che siano soddisfatte Ie infinite e-
quazioni integrali [lJ (t = 0, 1, .... ),
(2.1)
Da tale
propriet~
di completezza segue che ogni funzione di quadrato somma-
coW" ~ rappresentabile in serie dei P t con convergenza in media. In particolare, ogni soluzione delle equazioni di equilibrio di Cauchy, che bile in
nelle notazioni ora adoperate si scrivono
229
- 27 G. Grioli
,s
y
rs
=k
(2.2)
NS =
Y
rs
~
F
~
tt-
(in C),
r
f*r
(su
esprimibile mediante una serie dei
tezza di
~
•
f
Wt
J
Rt
• In effetti, la
(identica a quella di [ptJ ) assicura
•
I ),
propriet~
Ch~ i1
di comple-
sistema (2.2)
equivalente al sistema delle infinite equazioni integrali( )
(2.3)
(r = 1,2,3),
nel senso che ogni soluzione delle (2.2) di quadrato sommabile in
C..t'sod-
-
disfa aUe (2.3) , mentre ogni soluzione delle (2 ~ 3) per la quale gli Y P r t - che si costruiscono quali combinazioni lineari delle indeterminate del sistema (2.3) - siano tali che Ie serie ~
(2.4)
Y = r
~ t=o
-
y P • P r t t
risultino convergenti in media, fornisce attraverso Ie (2.4) uno stress soddisfacente aUe (2.2).
(5) Per Ie
b1'i A valgono definizioni analoghe a queUe della Lettura 230
I.
- 28 -
G. Grioli
2. Appossimazioni polinomiali della stress. Si consideri la successione dei soli
m
+ 1 polinomi P, PI' ...... , P
e delle equazioni (2.3) soo m 10 quelle corrispondenti a tutti e soli i monomi Wt contenuti nei polinomi presi in esame. In corrispondenza ad una qualunque soluzione di tale sistema - che chiamerb sistema
Sm
- si costruisca la successione [q(m) _ rt
r = 1,2 ....... ,6; t = 0,1. .... , m)
dei valori medi
J'
YrPt'
Detta al solito, 6
(2.5)
Ia
~ ~
W =
densit~
m
1',8=1
rs
Y Y r
s
di energia potenziale elastica, si ponga
(2.6)
V
m
e, inoltre, y(m) =
(2.7)
r
% t=O
Detta y(m) r
(2.8)
W cib che diviene m si ha evidentemente
V
m
= -1
da cui si vede che
~
W
2 ~ m
V
m
W quando si identificano Ie
Y r
con Ie
de....
rappresenta l'energia totale corrispondente allo stress
espresso da .y(m) r
Al variare della considerata soluzione del sistema
{q~~)J
varia in un insieme che dirb
1m' Sia 231
S
fq~~)~J
m
Ia successione la successione
- 29 G. Grioli di
I
v!
che minimizza
m
determinazioni di
V e siano y((m) e Ie corrispondenti m r y~m) e V m La sestupla ~(m) rappresenta uno
stress congruente.
(k~F"
Per dimostrarlo si denoti con
-m
na corrispondente allo stress polinomiale soluzione delle (2. 2)
, fmll-) -
la sollecitazione ester-
y. (m) in base alle (2.2) • Ogni r
corrispondente a tale sOllecitazione esterna
~
espriini-
bile nella forma y
(2.9)
r
=y~(m)+yl
r
r
al variare di
y' nella classe di tutte Ie soluzioni del sistema omogeneo r associato aUe (2.2) • Stabilito per y' uno sviluppo del tipo (2.4) , si ponr
ga y' = y' (m) + y"
(2.10)
r
r
r
~ i1 gruppo dei primi m + 1 termini della sviluppo, sicch~ y II ~ ortogonale a yf (m) e a yf • L'energia totale corrispondente a
ove
yl (m) r
r
I'
r
(2.10)~
6
~
(2.11)
r,s=1
ove q'(tm ), q"t si riferiscono a y,(m) , Y" , rispettivamente(6). r
r
r
r
Poidh~
1 due tel'mini'espressi dalle sommatorie1in (2. U) sono singolarmente defini(6) SI suppone, cio~, ,(m) = y,(m)p q" = Y"P . qrt r t ' rt rt 232
- 30 •
G, Grioli
.,
ti positivi e Ie
ql (m) , qrt sono vincolate a soddisfare ad equazioni omort genee e pertanto possono assumere valori tutti nulli, si riconosce che il mi_1imo di
V
q'rt(m) '
si consegue quando Ie
Cib significa che 10 stress
q" sono proprio tutte nulle, rt minimizza 1'energia elastica totale nella
,,(m) r
classe di tutte Ie soluzioni di (2.2) corrispondenti ad una speciale sollecitazione esterna e pertanto
~
congruente, in base al teorema di Menabrea, c,d,d"
E' ben naturale, dopo quanta si
i'(m)
(2, 12)
r
=
It m
,.
detto, definire la sestupla
~
l(m) P rt
(r = 1,2", ,6),
t'
quale approssimazione polinomiaie di ordine
m
della stress effettivo.
Speciali cautele vanno osservate se su una parte
r
ll di
I*' sono pre-
senti dei vincoli. Per breviU. mi riferirb al caso dell'incastro e dell 'appoggio liscio unilaterale con supporto rigido, In tali due casi, come risulta dalla lettura precedente, 10 stress effettivo
proprio
qu~llo
che minimi.lza l'energia
2:" a re~:ioni appartenenti all'insieme r
potenziale elastica - espressa nelle tematici che danno luogo su
~
Y
'
.. nella classe di tutti gli stress ma-
Si ponga b (r)
(2, 14)
"t"~
con (r)
(2.15)
mentre Ie
c,,~
1
• "C
e (ri
(~
~. 'fr
, 1" ~ ~Il Y1 Y2 Y3 d " '
sono costruite in base alle (2) della lettura I in modo
"'f'
eV1'd ente e con r1 enmento a 11a parte
1 4. ' ave ~I d'~.
~
te Ie forze esterne, 233
Sl'
suppongono assegna-
- 31 -
G. Grioli
n minimo di V e delle
e.~;~
di
in
1 r r
va cercato al variare delle
m
nell'insieme
q(m) nell'insieme I rt m definito in base a (2'.15) al variare
im
Ad es., se il vincolo ~ quello d'incastro Ia classe ti i possibili vettori e Ie
per Ie quali "
r
e tenuto conto delle equazioni cardinali della Statica.
e"l(~)~
+,. + A. = 0, 1
comprende tut-
sono tutte libere ad eccezione di quelle
• Per esse risulta
(r) coon
(r) - - e OOO
(2) clOD
COlO
(2.16) (1)
(1)
(2)
eec.
= e OlO - e lOO
Invece nel caso di un appoggio unilaterale rigido e liscio I'insieme comprende tutti e soli i vettori
il Tr
r
soddisfacenti alIa condizione
(2. 17)
con
+~
0
.
n metodo presenta
delle complcazioni se non ci si riferisce
ad un opportuno sistema di coordinate curvilinee, tranne nel caso d'interesse concreto di un appoggio piano -
0,
anche formato di piu parti piane -. Se
unico, si pub fare in modo che l'appoggio piano appartenga al piano con l'asse
y3
concorde ad
y3 = 0
!:! . In tale ipotesi valgono Ie equazioni di
condizione
c
(3)
(3)
= - e OOO 000
(2.18)
(3)
(3)
(1)
C100 = e OOl
e lOO ' 234
(3)
(2)
COlO = e OOl
(3)
e OlO
- 32 -
G. Grioli mentre Ie altre r
c ~;)A si devono ritenere nulle se
=3 •
d~
r = 1, 2 ,libere se
In generale, se 10 stress yf(m) luogo, come puO accadere, ad ~ rs" It un vettore T r che su una parte non so:disfa alla (2.17) 1 di
i.
in quanto risulti
~,
; < 0 , si deve ritenere che su
supporto di appoggio. In tale va ripetuto supponendo ancora secondo Ie l'altro,
r
Lt
f
il procedimento di minimizzazione
Ii .
eventualit~
r = 0 su
ci sia distacco dal
c ~~A vanno allora definite
Le
I-
(2.15) ma sostitutendo al campo d'integrazione
3. Sull'integrazione del problema d'equilibrio. Uno dei motivi per cui le
y*(m) r
espresse da
costituiscono una effettiva approssimazione
(2.13)
della stress reale l! che sussiste il teorema: detto V* il minima di V m m in I e, per ogni m , detti q"t(m) i valori delle q(m) minimizzan-
-m---
-r
ti, condizione necessaria e sufficiente
rt., affinc~ la sestupla \"m) definita da r
(2 . .13) sia convergente in media verso la soluzione
librio l! che la
~uccessione
f mJ V
y: del problema di equi-
sia convergente.
La necessita della condizione segue dal fatto che se il sistema delle equazioni di Cauchy ammette soluzioni di quadrat a sommabile in
*-
C esse sana
esprimibili, in base a (2.4), (2.7), nella forma
y
(2.19)
e se
r
=
lim m .. ao
Iy~m) J converge la successione
[Vm}
al divergere di
m
conver-
ge anch'essa, per una nota proprieta del prodotto integrale. Del resto, per convincersene basta osservare che se per ogni
E
fy~)}
converge ad
positivo e a partire da un certo intero 235
n
Yr
' si ha,
- 33 G. Grioli
(2. 20)
Segue
(2.21)
1 V - V =n 2
~ J ~1 c ..
m
(Y
rs
-
r
<
1
[ Vm
che assicura la convergenza di
La convergenza di
1
*c
C 2"
~
2
V •
a
Vm} assicura quell a di [V ~1 che
~ non decrescente.
Per dimostrare la sufficienza della condizione eel teorema enunciato, supposta convergente la successione
(2.22)
It V mt
1
="2
f J
6
~
L
V:
al valore
V~ 0
, si ponga
*(m) *(m) mrs qrt qst
r,s=l e, inoltre,
(2.23)
*
't'(m) = t
qrt
(m) = ~(m) z
t
r
Da (2.23) segue
(2.24) 6 Per cose ben note, la forma quadratic a
,
m
r,s=l 236
z z
L- rs r s
ammette, sot-
G. Grioli
to la condizione (2.24) , un minima positivo
2 ( . Di conseguenza, in base
a (2.22), (2.23). si ha
(2.25)
Da (2.23) , (2.25) si deduce .. (m) qrt
(2.26)
2
< t(m)
2
t
<
V'mt
r
e quindi
(2.27)
lim m.,.ao
m
~
qif (m) rt
2
< -1
r
l'1m m-tao
m
I: V"mt t =0
V*' -r
=-
1
C
Posto, formalmente.
(2.28)
~ = lim q'rt
la (2.27) si scrive
(2.29)
la quale assicura che comunque si scelga un piccolo esiste un intero positivo
*2
q rt
positivo arbitrariamente
n tale che per ogni intero positivo p si
ha (2.30)
f
- 35 -
G. Grioli
La (2.30) pub scriversi
(2.31 )
La (2.31) ,in base ad un noto criteriodi Cauchy, assicura la convergenza in media di
~ L t =0
(2.32)
~ q~ (m)
q- P = lim rt t m-+oo
~
t=O
rt
P = lim
t
m-+oo
l'(m) r
Posto ~
(2.32)
Y
= lim r m .... oo
dall'essere,per ogni m
Poich~
Y
*
r
r
* < Vm ,
V m
segue
~ I/:' V < lim V = V V = lim m m .... oo m m ... oo
(2.33)
a
~(m)
Y
n~
;to V non pub differire dall'energia totale elastica corrispondente
V da quella corrispondente alIa generica sestupla
Y
r
di qua-
drato sommabile, la (2.33) mostra che la sestupla
Y* definita da (2.32) r soddisfa al teorema di Menabrea nella classe di tutti gli stress matematici di quadrato sommabile che lasciano quanta si
~
1> r nella classe
r
e quindi congruente €, per
dimostrato nella lett'ura' precedente, rispetta i vincoli, c. d. d ..
238
- 36 G. Grioli
Osservazione. Da quanta sopra si deduce: condizione necessaria e sufficiente affincM i1 problema fondamentale dell'Elaatostatica lineare co9 forze assegnate 81) una parte della fromtiera e con vincolo d 'incastro 0 d 'appouio Iiscio con liIupporto rigido iulla rimanente parte.abpia una ed una sola 8oluzione di quadrato sommabile t! che il sistema fondamentale di ta soluzioni di quadrato sommabile e che la successione
Cauch~
[v! J
ammet-
aia conver-
gente,
BIBLIOGRAFIA
[lJ
L: Amerio •'SuI calcolo delle soluzioni dei problemi al contorno per Ie
equazioni lineari del secondo ordine di tipo ellittico" Am. J. Math. LXIX,
-
3, 447 (1947)
239
- 37 -
G. Grioli
Lettura III
Sull1integrazione del problema fondamentale delllElastostatica nel caso delle deformazioni finite.
1. Premessa generale. Siano al solito
C e
*'
C Ie configurazioni at-
tuali e di riferimento, quest1ultima supposta di equilibrio naturale,
r
if Ie loro frontiere,
ex, y r
r
PeP
*punti corrispondenti di
L
C +! e
e
Ie loro coordinate rispetto alIa medesima terna trirettangola
levogira. Sia
(3. 1)
D
~II xr,I II '
e
(3.2)
Le Y
rs
ve
X sono, comll! ben noto, Ie componenti euleriane della stress, Ie rs quelle lagrangiane. n sistema fondamentale dell 'Elastostatica si scri-
[l'
(3.3)
(3.4)
pag.
3~
.
(Y
x
,.
,. *
*'
(in C)
= Ok F ) r r,.I ,m
ylm x
N
*
rIm
f' r
= Of
241
- 38 -
G. Grioli ~
,~
la fona specifica di massa riferita aUo stato C1# e f -1- la r f r densiU deUa forza superficiale esterna rife rita a Le componenti dello ove
8k F
spostamento
l .
I-
C ... C sono espresse da u = x .. y
(3.5)
r
r
r
mentre Ie caratteristiche di deformazione(7),
Ers
(3.6)
=-
1
2
(u
~ rs sono, notoriamente,
Ie
+ us, r + U.1, r r, s
Limitandosi aUe trasformazioni isoterme
0
i
U
,s
).
adiabatiche, aUe equazioni
precedenti vanno associate Ie relazioni costitutive
(3.7)
(r = 1,2, .... ,6),
~ la densit~ di energia potenziale elastica (8). Sar~ bene tenere presente che, essendo C if stato naturale, per 8 = 0 10 stress ~ nullo: ove
W
y(9 = 0)
(3.8)
rs
n problema (3. 7)
~
~
I 0
(in C) •
analitico connesso aIle equazioni
(3.3), (3.4), (3.5), (3.6),
notevolmente comples8o. Mancano in generale teoremi di uniciU,
esistenza e
sviluppabi1it~
in serie di potenze del parametro 9 della soluzio-
(7) A rigore Ie caratteristiche di deformazione sono Ie
,(r
f s),
£ rs notazioni ad un solo indice analoghe a queUe gil usaXrs [vedi Ie (7) in Letfura: IJ.
(8) Si usano per Ie
te per Ie
Err ,2 [ rs
242
- 39 -
G. Grioli ne, tranne nel caso in cui il vettore
..
f'* sia nota sull'intera
r*
In tal caso, sotto varie ipotesi concernenti i vettori tF~ f* la front!: - tiera Leper , sufficientemente piccolo, teoremi siffatti sono stati
f) I
stabiliti da F. Stoppelli
[1,2,3,4,5, 6J sia nel caso in cui non visi'ano as-
si di equilibrio come pure in quello (eccezionale) in cui ve ne siano. Qui non posso che rinviare ai lavori originali, limitandomi solo a ricordare che, almeno nel caso di forze superficiali ovunque note su
r*
,f),
e per
suffi-
cientemente piccolo si deve ritenere, salvo casi eccezionali (che possonb presentarsi solo quando ci sono assi di equilibrio) che per larghi tipi di sollecitazioni esterne e di forma di frontiera la soluzione esiste ed Nel caso, invece, di spostamenti assegnati su
~
unica.
Lite nel problema mi-
sto mancano teorema generali del tipo di quelli di Stoppelli. Lo scopo di quanto seguirtl
~
di mostrare come il metodo d'integrazione
precedentemente esposto sia adattabile al caso delle deformazioni finite, supposta esistente la soluzione. La validita del metodo abbraccia, pertanto, il caso di forze
'*
~* e, formalmente, anche gli altri due f ovunque note su £.
-
(problema di assegnati spostamenti su
r*
e problema misto) e poicM !lin-
tuizione fisica fa presumere l'esistenza della soluzione anche in tali casi e la possibilita di stabilire teoremi di esistenza e di unicita analoghi a quelli di Stoppelli, si
~
indotti a ritenere che la validittl del metodo possa essere pit!
che formale, almeno nel futuro, anche in riguardo agli altri due casi sopra menzionati. 2. Esposizione del metoda d'integrazione. La sviluppabilita in serie di potenze della soluzione - se
c'~
- del problema fondamentale si traduce nell'esi-
stenza degli sviluppi
243
- 40 G.Grioli
(3.9)
(i) ur
= l: QC)
u r
1=1
E ~
frm
,i
~ (1)
if'
~r
i Y (1) 9 t rf:i rif
y =
(3.10)
18'
da ritenersi validi per Poich~
(3.11)
,i
11 '
8ufficientemente piccolo(9).
W non dipende esplicitamente da
~W
(~r) ""r
(1) _
f
~2W
- , ~'.k ) s=l~'" (~O
8 ,81 ha
's (1)
_ 'i)W(2) f\
.,
(1) r
con ((1) = .1. (u(l)
(3.12)
2
8
r, s
+ u(l)
s, r
)
e
(3.13)
In generale ai ha, anzi,
(3.14)
r;) w (r)
Er
(n)
~
l:6 (IT) £; + termini che dipendono ,\2W
( )
s =1 " . , (=0
(1)
fl
244
8010
('" -1)
, .... ,f l
•
da
- 41 G. Grioli
La (3.14) pub scriversi () (3.15)
(? w ) n ~fr
'M f:.l 6
=
E(n) + termini che dipendono solo da
rs
s
(1)
~
C.. ·.)
, ...... ,E.,
.
Risulta
!
(3. 16)
(n) = e(n) rs rs
+ l(n)
,
rs
con e(n) =1.. (u(n) rs 2 r, s
(3.17)
+ u (n)
s, r
)
(n) (1) (n-l) e 1 dipendente solo dalle derivate di u , ••••• , u • ra s s Da quanto si ~ detto segue che l'uguaglianza
(3.18)
ha come conseguenza che y(n) '"
(3.19)
r
ove
(n) pr
per
n=1
1r
(n)
+
(n)
Pr
dipende solo dalle derivate di mentre
(3.20) 245
u, ,....... ,u, (1)
(n-1)
ed
~
nullo
- 42 G. Grioli
Da (3.19). (3.20) si riconosce che la dipendenza di ~
dalle e(n) 1
y(n) r
proprio la stessa che nel caso lineare.
Nel caso del problema misto. da1 sistema (3.3). (3.4),per derivazione rispettoa fJ
per: 6=0 .• sideduce(10) y(n). s rs
(3.20)
/
S (3.21 ) Y (n) N = rs
"
*
ove ne
(4)
k *F(I) • f(l)
r r • mentre per
*'
= k F(n) r f(n) r
f(n) r
*'
( in
su
+ ;(n)
su
r
coincidono con Ie forze assegnate e n > 1 si deve ritenere
~ ~1}
C )
if-
I
1
~
..
2
con 1a reazio-
~. pag, 34J .
(3.22)
(n=2 •...• ). n-l fr(n) = _ ,
L-
q=1
.
(n) y(q) (n-q). m q 1m ur
~.
r<.
* '
(10) Si suppone che su ~ 1 siano assegnate 1a forze, mentre au sono pre senti nei vincoli. 246
~.
'2
- 43 -
G. Grioli
Le (3.22) sono formate con Ie soluzioni dei sistemi successivi del tipo (3.20), (3.21) corrispondenti ai primi (3.10) ed
~
ma d'indice ad
n - 1 termini degli sviluppi (3.9),
interessante osservare che condizioni d 'integrabilit~ del sisten
u 1, •.. un_I'
agenti sui secondi membri delle (3.20), (3.21)
impongono
condizioni integrali che possono non essere verificate in
casi eccezionali.. In particolare,la teoria lineare, se considerata come primo termine di unosviluppo del tipo
(3.9) perde di significato quando db
accade (11). Tali casi eccezionali, facilmente individuabili, non si presentano in assenza di assi di equilibrio. E' ormai chiaro .:::he la determinazione del termine di ordine
n
della
soluzione pub farsi con il metodo esposto nella lettura II se per il sistema (3.20), (3.21) associato alle (3.19) sussiste un teorema del tipo di quello di
Menabrea. Cib effettivamente
~,
come si riconosce effettuando sulle (3.20), (3.21)
un'opportuna trasformazione.
osservare che, in base a (3.19), Ie
Baster~
(3.20), (3.21) possono porsi nella forma
(3.22)
Ih (n), s ( rs
_ *w(n) - k Tr
'
~ (in C )
*
( su
~1)
(su
t ~2 ) ,
(3.23)
(11) Almeno come teoria statica
[7,
8, 9, 1~ • 247
- 44 G. Grioli
con
kf't' ~) = k *F (n)r
-
(n\s P rs
(3.24)
f~n) = r(n) r
(n) - P rs
~
I secondi membri di (3.22)1 (3.23) sono espressi mediante Ie soluzioni .. (n) u(1) , ......• , u (n-1) d' el prlml no:' 1 sistemi successivil mentre r r °l rs dipende soltanto dalle e(n) e nel modo stesso con cui 10 stress ~ legato rs allo strain nel caso lineare. 8i pub, anzi, scrivere
rn
e(n) __
(3.25)
r
con
(3.26)
w(2)
(~(n)) = l
La
t
6
L
m
r,s=1
11
(n) Ib (n) (S
rs l r
>0
'
W esprime evidentemente Ia dens itA di energia potenziale elastica
- in una teoria lineare - corrispondente aUo stress 'W) (n)
l rs
che il sistema (3.22), (3. 23) segnate k
"'t' ~n)
ni espresse dalle
'f~n)
~
.E' chiaro orrnai
analogo a queUo del caso lineare con forze as-
,reazioni
~ ~n)
e caratteristiche di deformazio-
e(n) e legale alle ", (n) r
(r
dalla (3.25) . Cib basta per
assicurare la validitA del teorema di Menabrea per 10 stress
,~n)
.
Osservazione. Nelle applicBzioni concrete 10 sviluppo (3.9,1) va fermato ad un certo termine di ordine P nel senso che si ritiene trascurabile il contributo dei termini successivi. A tale proposito 248
~
opportuno fare una pre-
- 45 -
G. Grioli
cisazione, ad evitare equivoci. sviluppo (3. 9)
~
Precisamente si dovrtl. ritenere che nella
lecito fermarsi al termine di grado p se e solo se sono
trascurabili tutti i prodotti
u
(3.27 )
(1)~1 (2)f2 u r,B t ,n
.......... u
(q)t
y,6"
interi positivi
0
aq qq-, q.
nulli e soddisfacenti alIa condizio-
'1 '1 2
(3.28)
+ ......... + ~q = q ~p + 1.
+
Cib implica la trascurabilittl. di mente 10 sviluppo
£ (q) aq
per q>p . Conseguenter qJ (3.10) va arrestato al termine di grado p . Cib si giu-
stifica ammettendo la naturale ipotesi che Ie derivate della
€ rs '
to alle
valutate per
dezza comparabile il che
~
0
a= 0
e di ordine maggiore di
W (q p
rispet-
abbiano gran-
e= 0)
trascurabile rispetto alle derivate seconde (per
quanta dire che i coefficienti elastici della teoria line are (coefficien-
ti di Lam~ nel caso lineare) non sono trascurabili rispetto a quelli della teoria non lineare. Sotto tale ipotesi, la trascurabilittl. dei prodotti (3.27) per q >p
implica, in base a (3.19), (3.25), quella dei termini
y~q) :~
•
nello sviluppo (3.10).
249
- 46 G. Grioli
BIBLIOGRAFIA
[1]
G. Grioli "Mathematical Theory of Elastic Equilibrium (Recent Results)" Springer- Verlag. Berlin Gottingen Heidelberg (1962).
[2]
F. Stoppelli "Un teorema di esistenza ed unicita relativo alle equazioni dell' elastostatica isoterma per Ie deformazioni finite" Ric erche Mat. 3, 247-267 (1954) .
•
[ 3]
F. Stoppelli "Sulia svil uppabilita in serie di potenze di un parametro delle soluzioni delle equazioni dell'EIastostatica isoterma" Ricerche Mat. 4,
-
58-73 (1955).
[4J
F. Stoppelli "Sull 'esistenza di soluzioni delle equazioni dell 'Elastostatica isoterma nel caso di sollecitazioni dot ate di assi di equil ibrio" Me-
-
moria I; Ricerche di Mat. 6, 241-287 (1957).
[5J
F.Stoppelli: Titolo come! Memoria II,2, 71-101 (1958).
[6]
F.Stoppelli: Titolo come 4 Memoria III, 7, 138-152 (1958).
-
-
G. Capriz "Sopra Ie deformazioni elastic he finite di un solido tubolare"
-
Rend. Mat. Roma; 15, 228-262 (1956).
[8J
G. Capriz "Alcune osservazioni su problemi di instabilita delle travi ela-
-
stiche" Rend. Mat. Roma; 16, 23-42 (1957)
[9J
G. Capriz "Alcune osservazioni sulla instabilita di una trave sollecitata a torsione" Riv. Mat. Univ. Parma, 8, 145-160 (1957) 251
• 47 G. Grioli
[10J G. Ca~riz
"Sui casi di
"incornpatibilit~"
tra l'elastostatica classica e
la teoria delle deformazioni elalitiche finite" Riv, Mat. Univ. Parma,
-
10, 119-129 (1959),
252
