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On the consistency conditions for force systems
oo?O 7462 915300 * w Pcrgamon Press pie
ON THE CONSISTENCY CONDITIONS SYSTEMS
FOR FORCE
REUVENSEGEVand GAL DE BOTTON The Pearlstone Center for Aeronautical Engineering Studies, Department of Mechanical Engineering, Ben-Gurion University of the Negev. Beer-Sheva. Israel (Received 6 August 1989; accepted 4 finery
1990)
Abstract-In analogy with the classical Cauchy conditions, this work presents conditions so that a force system, the assignment of a force to each subbody of a given body, can be represented by a stress. The setting in which the theory is formulated is more general than that of classical continuum mechanics as stresses can be. as irregular as measures, equilibrium is not assumed, it applies to continuum mechanics of order higher than one and it may be extended to the case where the body and space are modelled by general differentiable manifolds. The consistency conditions presented are that of additivity of the force system on pairs of disjoint subbodies, continuity and boundedness.
I. INTRODUCTION
The present paper is concerned with the conditions so that a force system on a body can be represented by a stress. Here, by a force system, we mean an assignment of a force to each subbody of a given body. In the classical formulation of continuum mechanics, it is assumed that force systems on bodies satisfy the following conditions: (i) The force& on any subbody P is given in terms of a body force field b, and a continuous surface force field t, in the form fp = fP b, dv + lBpt, da. (ii}The body force on a subbody is independent of the subbody and the surface force on a subbody depends on the subbody only through its unit normal n to the boundary. It is also assumed that the surface force depends on the normal continuously. (iii) The total force on each subbody vanishes. On the basis of these conditions it is proven that there is a unique tensor field (7 over the body, called the stress, so that the forces on the various bodies are given by b = - div cr and tp = a(n). Thus, we may say that the stress o represents the force system (f,,j with these two equations and that the aforementioned conditions are coff~i~te~c~ conditions for the existence of a stress representing the given force system. Modern formulations of stress theory replace some of these assumptions by other, more general assumptions. In [l J, which summarizes the works of Nell, Gurtin, Williams and others (see references therein), the relevant structure is as follows. Forces are defined as interactions for pairs of subbodies so that f(P, C) represents the force, a vector in some inner product space, that the subbody C exerts on the subbody P. The consistency conditions in this formulation are shown to be the following: (i) f(p, C, u C,) = f(P, C,) +f(P, C,) for separate Ci, C,, and f(P, u P,, C) =f(Pi, C) +f(P2, C) for separate PI, Pz. (ii) The forcef( P, C) for separate P and C is bounded by both the volume of P and the area of the common boundary of P and C. (iii) The total force on each subbody vanishes. (iv) Proving the existence of a surface force field on the basis of the previous assumptions it is assumed that it is continuous. In previous works (see [Z, 3]), an alternative framework for the theory of forces in continuum mechanics was suggested. A force on a body was defined as a continuous linear functional on a tangent space to an appropriate configuration Banach manifold. Assuming that bodies are compact submanifolds with boundary of space, and using the axiom of impenetrability of continuum mechanics as a guideline, it was shown that it is natural to take the collection of differentiable embeddings of the body in space equipped with the C’ topology as the configuration manifold. Restricting ourselves for simplicity to the case where the space is modelled by g3 and a body B is a 3-dimensional submanifold of W3, it follows that the various tangent spaces can be identified with C’(B, #) and that forces can
Contributed by K. R. Rajagopal. 47
48
R. SEGEV
and G.
DE
Borros
be identified with elements of the dual space C’fB, s3)*. It is shown that such forces can be represented in the form f(u) =
c
P.~~,lr.a,
sB
d?L+zz+Z’n ’ da PSlZZ’, ax;1 dx;‘ax;
’
where p = 1,2,3, xi, [x2, ~1~are non-negative integers with xi + a2 + a3 I r, and CJ,,~~_~, are Bore1 measures to which we will refer as stress measures. From the construction it follows that the collection of stress measures that represent the force f is not unique. The representation presented in the previous equations corresponds to high-order stresses, those appearing in rth order continuum mechanics. For the simplest case r = 1 the equation can be rewritten as
where here, rsw are measures corresponding to the components of the usual stress tensor and Q,, which are not present in usual formulations, appear in the equation because no equilibrium hypothesis was made (i.e. if one assumes that the total force on each subbody vanishes it follows that dP = 0). If these stress measures are given by differentiable densities in terms of the volume measure in g3, it can be easily shown that forces can be represented by body forces and surface forces. Since, unlike forces, stress measures can be restricted, a given collection of stresses (Tag__., induces a force system on B in which the force_& on a subbody P is represented by
In addition to the fact that it does not require the assumption of equilibrium and permits stresses that are irregular as Bore1 measures, this formulation of the theory of stresses in continuum mechanics has the advantage that it applies in the general geometrical setting of differentiable manifolds. While the suggested formulation models the representation of a single force on a body by nonunique stress measures, the aforementioned papers do not contain the appropriate consistency conditions on a force system that guarantee that there is a collection of stress measures that induce it in the form presented in the last equation. In this paper, it is shown that the following conditions on the force system (fpe c’(P, L@~)*;P is a subbody of B>, which are clearly necessary, imply that there is a unique collection of Bore1 stress measures LT parEtZx,p = 1, 2%3, xi + a2 -t a3 I r, that represent it as in the last equation. (We use the convention that the empty set is a subbody and the force on it is zero.) (i) Additiuity. If PI and P2 are disjoint subbodies of B then for any UEC’(B, Z3), fP,UPAUlP,UP,) =h,(ul/y)
+h*(&,).
(ii) Continuity. We recall that the set A is the Iimit of the sequence of sets 4 if and only if for every element x in A there is an integer i,, so that x E pi for every i 2 i,, and every point that is contained in an infinite number of sets in the sequence is contained in A. It is required that if A E @],where Q,is the minimal field containing the open subsets of the body, and 4 is a sequence of subbodies whose limit is A, then, for any UE C’(B, a3), the sequencefp,(ufP,) converges and its limit is independent of the particular sequence of subbodies Pi. (iii) Boundedness. u E C’(B, W3),
There is a finite bound K such that for any subbody
P and any
I_MUlP)I< Kllul,II * We note that as the traditional consistency conditions hold only for continuum mechanics of order one, and while it is only very recently that No11and Virga were able to present consistency conditions for second order continuum mechanics E43, the consistency conditions presented in this paper hold for continuum mechanics of any order.
Consistency conditions for force systems
49
The reason why the representation of a force system by a stress is unique while the representation of a force by stresses is not unique may be described roughly as follows. In the expression for the representation of a given force by stress measures we know the values of the integrals only for “compatible” collections of continuous functions w,,.,._, , p = 1, 2, 3, zi + x2 -t x3 s r, i.e. collections for which there are differentiable functions ug such that PI +az+lJu n Hence, the measures cannot be determined uniquely. However, we can approximate any collection )tipa,_, by a family of compatible collections such that the members of the family are defined on a family of disjoint subbodies. If we know the force on each subbody we can approximate the integrals for “non-compatible” collections and determine the measures uniquely. Section 2 discusses the approximation by subbodies of sets that belong to the minimal field of subsets containing the open subsets. For each set A in the minimal field, a sequence of subbodies whose limit is A is constructed and some useful properties of this sequence are presented. Section 3 uses the construction of Section 2 in order to give sufficient conditions (which are also necessary conditions) so that a real valued set function defined on the collection of subbodies can be extended to a Bore1 measure. In addition to its use as a tool for the proof of the sufficiency of the consistency conditions, this result is of some interest as it specifies the conditions under which quantities such as electric charge can be extended to measures if they are given for the various subbodies. Section 4 presents the consistency conditions and proves that they are su~cient (again, these conditions are also necessary conditions). 2. THE
APPROXIMATIONS
OF SETS
BY BODIES
As was mentioned a body is defined as a three-dimensional compact submanifold with boundary of s3. We now add formally the empty set to the collection of bodies. Given a body B, a subset D of B which is a body is a subbody of B. Clearly, the complement of a subbody in the body is not a subbody and the same holds in general for the unions and intersections of subbodies. In this section it will be shown that although our collection of subbodies is small, any set in the smallest field of subsets containing the open subsets of B can be approximated by a sequence of subbodies of B. This approximation has some additional properties that will be used in the following sections. Proposirion 2.1. For any compact subset C of .9P3and a bounded open subset U of .?J?~ containing C, there is a body B which is contained in U whose interior contains C. ProoJ By a standard theorem of differential topology (see [S, p. 123) it is possible to construct a smooth real function h: g3 + 9 with the following properties: (i) its value at any point in C is 1; (ii) its value at any point in the complement of U is 0; and (iii) its value at any other point is in the interval [O, 11. By Sard’s theorem (see [6, p. 2041) any neighborhood of the value l/2 contains a point x such that h is not singular at any point in the inverse image of x. From the implicit function theorem it follows that h-‘( f.x)) is a smooth submanifold of &I‘3that is clearly compact and which is the boundary of h- ‘([x, I]). It follows that h-‘f[x,l]) is a body. i Proposirion 2.2. Let D be a body, LJ a bounded open subset of a3 and C a closed set contained in U. There exists a body B contained in U whose interior contains C such that B u D and B n D are bodies. Proof Let us construct a body Bz contained in U whose interior contains C as in the previous proposition. In the case where U n D is empty, or D I> U it follows from the construction that B2 LJ D and B, n D are bodies. However, in general, when the boundaries of B, and I) interest, B, u D and B, A D will not be bodies. We will modify B2 so that the modified body will have the required properties.
50
R. SEGEVand G. DE BOTTOK
We recall that if X and Y are submanifolds of the differentiable manifold Z, any C’ neighborhood, r 2 1, of the inclusion of X in Z contains an embedding g: X + Z which is transversal to the inclusion of Y in 2 (see [7, p. 781). It follows that we may modify B, slightly to obtain a diffeomorphic body B, so that the properties guaranteed by the previous proposition still hold, and in addition, dB, is transversal to ZD. Hence, the intersection dS1 n dD is a one-dimensional submanifold of both dB, and SD. Next we modify B, to obtain a diffeomorphic body B so that the intersection between the boundary of B and the boundary of D contains an open subset. In other words, the two boundaries will overlap on an open set so that Bu 5 and Bn D are bodies. This modification will take place in a small neighborhood of the intersection of the transversal submanifolds dB, and 8D. In constructing the neighborhood in which the modification will take piace we use the following theorems of differential topology (see [7, pp. 109- 11.53). (i) If Y is an orientable submanifold of the differentiable manifold Z of codimension one, then, there is an embedding g: Y x ( - 1, 1) -+ Z, called a tubular neighborhood of Y in Z having the following properties: (a) g Ir yto) is the inclusion of Yin Z, and (b) the image of g is an open neighborhood of Yin 2. We will also use the term tubular neighborhood for the image of the mapping g, and we wiil identify a point in the image with the “coordinates” given to it by g. (ii) If Y is a neat submanifold of Z, then, there is a tubular neighborhood of Y in Z. (iii) If Y is a neat submanifold of Z, then, every tubular neighborhood of d Y in dZ is the intersection of dZ with a tubular neighborhood of Y in Z. We first construct tubular neighborhoods Vi and V, of dB, A 1?D in dB, and in dD, respectively, with the convention that the parameters in ( - I, 1) are positive in D and the complement of B1, respectively. Next, considering the neat submanifold dB, n D in D, we can use theorem (iii) above to show that there is a tubular neighborhood V;, of ?B, n D in D such that V, = VJ n dD, where we set the values of the parameter SE( - 1, 1) to be positive outside B,. Let
By definition, every point in V has three coordinates (x, t, s) such that XESB, n alI, t, s E [O, 1), where t = 0 for points on aD and s = 0 for points on as,. Since there is a finite positive distance between 8B, and the complement of II, there is a number 5 > 0 such that a point in V is in U if the coordinate s is less than <. Consider the folIowing functions: a:&?-+9
a(r) = 0
for r I 0
a(r) = exp( - l/r) /!I:&?-+_,
for r > 0,
p(r) = r(r - W4S
- r)
D(r) dr 8: .a! --) [O, n/2]
d(t) = ; P(r) dr .
The function 0 is smooth and has the following properties: (i) e(t) = n/2 if t s 5/4 (ii) 0 2 e(t) s n/2 if 5/4 ~5t s 35/4 (iii) s(t) = 0 if t 2 3t/4. We will now modify the portion of dB, that is a subset of Y (which is clearly in I)) as follows. Let g: Y, + Y be defined by gfx, t) = (x, t case(t), g(.% t) = (x, t, 0)
t sin O(t))
if 0 I t < (, otherwise.
Consistency
conditions
for force systems
Clearly, g is an embedding and has the following properties: (i) Im(g) n dD = {p = g(.x, t); 0 s t I 5/4) contains an open set of SD, (ii) Im(g) n dB, = {p = g(x, t); 35/4 I t I 1) contains an open set of SB,, (iii) Im(g) is a subset of U and is disjoint from C. In a similar way we will define a function h that will modify a neighborhood of dS, n c?D in c?B, A (DO)’ whose image will be contained in a neighborhood of (?B, n dD in B, n (DO)‘. (The superscript “c” denotes the complement of the corresponding set.) This mapping will have the analogous properties to those of g, i.e. it is an embedding, its image is contained in U-C, Im(h) n dD contains an open set of c?Dand Im(hf n ZB, contains an open set of 2B,. We can define now the mapping @: ZB, + U-C by i&(p) = g(y) if y~domain(g), $(y) = h(y) if yEdomain and i&y) = y otherwise. Clearly, $ is an embedding and its image overlaps with aD on an open set. By the Jordan-Brower Separation theorem (see [6, p. 891) Im($) is a boundary of a body 5 and it follows from the construction that B n D n and B u D are bodies as their boundaries are smooth. We can now state the analog of proposition subsets of the body B.
2.1 for the case where both C and U are
Proposition 2.3. If U is an open subset of the body D and C is a closed subset of D contained in U, there is a subbody P of D contained in U whose interior contains C. Proof: It is clear that C is closed in g3 and that it is contained in an open subset l’of Z3 such that Vn D = U. By proposition 2.2 we can construct a body B contained in V whose interior contains C such that B n D is a body, hence, a subbody of D which clearly has the required properties. m
Henceforth, unless otherwise stated, we will refer by open and closed sets to open and closed subsets of the body under consideration. Proposition 2.4.Let 0 be the smallest field of subsets containing the open subsets of a body B. For each A in CDthere is a sequence fl of subbodies of B such that pi + A. Proof: It follows from a standard representation theorem for the members of the smallest field containing a given collection of sets (see [S, p. 73) that a subset A of B belongs to 4 if andonlyifA=
0
Uk n C,,where U, and C, are open and closed sets, respectively, with
k=l Cj) n (& n C,)= @ for j # k. Let A be a member of CDand assume that such a representation of A is given. For each k let cki be defined by
(V’n
where B(d, x) denotes the open ball of radius d centered at x. Clearly, for a given i the sets Cki are disjoint. In case all Cki = 0 we will set pi = 0 and henceforth we may assume that at least for one k, Cki # 0. In the case where there is only one Cki # 0, we will set 6, = 1/2i. Otherwise, since for any given i the sets C,, are disjoint and compact there is a number 6, > 0 smaller than half the distance between any two of them and lj2i. For each cki # 0, let V,i = U B(6,, X). XECki Chdy, any vki is a subset of Ukand for a fixed i the sets qi are disjoint. Using proposition 2.3 it is possible to construct for a fixed i and each k such that C,i # 0, a subbody Pki contained in uki whose interior contains Ckj. In case Cki = 0 we just set Pki = 0. Define
Since the various Pki are disjoint, 4 is a body.
R.
52
SEGEV
and G. DE BOTTOK
Lemmu 2.4.1. The sequence pi satisfies lim infiG 3 A.
~~oo~o~~e~~u 2.4.1. We first note that for each k the limit of the increasing sequence Cki is C, n Uk. Clearly, each C,i is a subset of C, n U, hence their union is also a subset of Ck n Uk. On the other hand, assuming that XE C, n U,, there is a minimal distance d > 0 between x and CJE.Choosing an i such that l/i < d, we have
and it follows that We conclude that
Since, 4 3 E, for each i we have lim inriP{ 1 lim infiEi = A. Lemma 2.4.2. The sequence pi satisfies A 13 lim
SUpiP,.
Proof of lemma 2.4.2. Let us construct the sequence
which is decreasing to Ck. By our construction Phi is a subset of V,i for each k and i, hence, C, = lim sup, Gi 3 lim SUp,P,i. On the other hand, U, 3 Pkj for each i and it follows that U, n C, I3 lim SUpi Pki. Assume that x $ A, then, for each k, x +!U, n Ct and x 4 lim supi Pki. In other words, XE Pki for a finite number of elements, say, Nkelements of the sequence only. Thus, x E (J si = 4 k=l
for a finite number of elements and hence, x # lim Supi 4. From lemma 2.4.1 and lemma 2.4.2, we have lim infiq =, lim Supi~::, and by the definitions of the limit of the sets it follows that lim,P, = A. H Coro~~ury 2.5. Let A = fi U, n C,. Then for each k the sequence of subbodies k=l
Pki
constructed in the previous proposition converges to Ukn C, and it has the following property: for each E > 0 there is an N independent of k such that for each i > N and each .YE Pki there is a point in &. n C, A Pki whose distance from x is less than E. ProofI This corollary proposition.
follows immediately
from the construction
of the previous
Proposition 2.6. If A,, A,, . . . , A,, are disjoint members of 0, then, there are n sequences of subbodies PIi, Ptit. . . , P,i converging to A,, AZ,. . . , A,, respectively, such that for any i, PIi, Pzi,. . . , Pni are mutually disjoint and the sequence of subbodies ~=Pp,iuP,iu... Proof:
uP,,convergestoA,vA,u...uA,.
for the case n = 2. Let A, and A, be
It is sufficient to prove the proposition
represented in the form A, = ,c, Ulj A Clj, A, = k()l Ule n C,, as in proposition 2.4. We can represent A, u A2 in the form ii j=l
U,j”Clj
I[ U
; k=l
U,knC,k
1
so that we can apply to it proposition 2.4 and corollary 2.5. It follows that for each j and k there are sequences P,ji and P,kj of subbodies converging to Ulj n C,j and u,i, n Cztr
Consistency
conditions
for force systems
53
respectively, such that for any fixed i the various subbodies are disjoint. Hence, the K
J
sequences P,i = u P,ji, Pti = u P2Li have the required properties. j=
H
k=l
1
Proposition 2.7. Let A = fi
wkn Ek, A’ = fi
I$ n Fj be members of Q represented as in
j=l
k=l
2.4 (i.e. wk, T are open and E,, FJ are closed) such that A’ 3 A. Then, it is
proposition
possible to represent A in the form A = fi Ui n Ci such that the following conditions hold: i=l
(i) The subsets Ui are open, the subsets Ci are closed and the various Ui n Ci are mutually disjoint. (ii) For each i there is a unique j(i) such that ~(i, z) Ui and Fj,, 2 Ci so that vjfi, n Fjti, 3 CJin Ci. Proof: Define the closed sets ckj = E, n Fj and the open Sets yj = wk n f$. We note that A = u Ukjn C,, and that the various Ukjn Ckj are mutually disjoint. In addition, by the k. j
for each pair j, k there is a unique j(k) such that qfk, 3 Ukj and Fjck,I Ckj so
construction,
that 5(k) n Fj(k) 3 Ukjn
ckj.
Finally, we obtain the required representation
A = 6
Ui n Ci
i=l
by re-enumerating
the subsets Ukj and Ckj.
n
Proposition 2.8. Let A, be a decreasing sequence of members of 0,. Then, there are sequences Pki of subbodies such that Pki+ A,, and for a fixed i, Pki 2 Pck+l,i. Proof: We first construct the sequence PIi converging to A, as in proposition 2.4. Inductively, we construct the sequence Pki on the basis of the sequence qk_ l)i. Hence, we assume that the sequence qk_ l)i + A,_ I has the following properties as in proposition 2.4:
(a)
Ak-I
(b)
et-l)i
(c)
For each subbody PC,_ r )ni we have interior (ok _ I )ni) =) ctk _ 1)ni where
=
6 II=1
=
6 n=1
c(k
~k-IjnnC(k-ljn.
qk-l)ni.
- 1)ni =comp{
uB(+,x):
xEcomp(u(k-l)n)
}
nc(k-l)n.
(d) For each subbody qk_ i)ni we have V,k_l)ni 2 ok_ I,ni where qk-.l)ni=
u x
{B(6~k-,,i,x);xEC~k-,,,i}.
Using the fact that A,_ 1 I A, and proposition 2.7 we can represent A, in the form A, = fi j=
Ukjn Ckj such that the following hold: I
(i) The subsets ukj are open, the subsets Ckj are closed and the various Ukjn Ckj are mutually disjoint. (ii) For each j there is a unique n(j) such that U,,_ l,nt,353 Ukj and C,,_ ijn(,? 2 Ckj so that U,k-l)n(j)n
Cck-l)n(j)
3
yjn
ckje
We now construct the subbody Pki such that 4k_i,i 3 Pki. For each Ukjn Ckj let
Lemma 2.8.1. For each j there is one n(j) such that interior ( 4k_ lIntj)i) 3 Ckji. Proof of lemma 2.8.1. By (ii) above U,,_ l)nc,, 3 Ukj, it follows that
= @ we will set Pki = Qr and henceforth we may assume that at least for In the case where there is onfy OSI~ Ckji # a, we wilt set 6,i = (A?;. Otherwise, since for any given i the sets C,ji are disjoint and compact there is a number Ski > 0 smaller than half the distance between any two of them and 1/2i. For each Crji $ QY fn case ail Caji
VIE j.
Ckji # a.
let
Clearly, any E!,&is a subset of Ukjand for a fixed i the sets i&i are disjoint. Using proposition 2.3 it is possibfe ta construct for a fixed i and each j such that Ckjz# @, a subbody Pkji contained in Fuji whose interior contains C~ji- fn case 6, =; 0 we justset I&; = @If Define
Since the various ~~~~o~ies Pkji are disjoint, their union Pki is a body and in addition, as interior (P,,_ l)n(j)i ) 3 Ukji =) Pkjiit is clear that ek_ l)i 3 Pk(. The PrOOf that si -+ A, follows from lemma 2.4.1 and lemma 2.42. R
3. THE
EXTENSION
OF A SET FUNCTION
DEFINED
UN SUBBODIES
TO A MEASURE frr the next section we wilE need a condition such that a set fun~t~on defined over the collection of subbodies of a given body can be extended to a unique measure. Such a condition is of interest in general because it can be applied to various physical instances such as mass, electric charge, etc. Just as we formafly added the empty set as a subbady, we now set formally the value of the set function for the empty set to be zero.
Propusition 3. t. Let I( be a bounded real valued set function defined on the collection of subbodies of a body L3that satisfies the foIlowing conditions: (i) if Pi and P2 are disjoint subbodies, then, @(PI LJ P2) = &PI) -f &P,). {ii) ff A E Q1and I$ -, il, then, the sequence p(e) converges and its Emit is jnde~nde~t
of
the ~ar~j~u~ar sequence pi. Then, there is a unique Bore1 measure Y on B such that v(P) = p(P) for all subbodics P of B. We will refer to the first and second conditions as additivity and continuity conditions, respectively. Proof. Define the real valued set function v on Q by v(A) = limip(&), where iq: is a sequence of subbodies whose limit is A. By proposition 2.4 v is well defined, and in addition, it follows from the continuity assumption that for any subbody P+ v(P) = g(P). To show that t’ is additive, consider the disjoint sets A, and A, in Cp.We now construct two sequences Pki and Pzi converging to A, and A,, respectivety, satisfying the properties guaranteed by proposition 2.6, so in particulars P,,r, Pzi = $Zj for a fixed i. Since A, w A, = fim,
So far, we obtained a finitely additive set function on a field. We next show that P is countably additive. We recali (see [9, p. lo]) that in order to prove that an additive set function v on a field is countably additive it is sufficient to show that v is continuous from above at the empty set, i.e, if the decreasing sequence A; converges to the empty set, then ~imiV~~i~ = 0.
~~~~a 3.1.1. The set function v is continuous
from above at the empty set.
Consistency
conditions
for force systems
55
Proof of lemma 3.1.1. Let A, be a sequence of sets in Q,decreasing to the empty set and for each k let Pki be the sequence converging to A, as in proposition 2.8, i.e. for a fixed i, Si I3 k such that x E Pj,,(jj. From the construction of the double sequence it follows that P,,i(j, 1 Pj,i,jj. Hence, XE Pn.i(j) for some j > k for each k so that x E lim SUpjP,,,( jJ. Since P,.i(j) is a subsequence of P”iit follows that x E lim supi Pni= A,. We can conclude that lrm sup, P~.i(t,c n A, = 0 since A,, I lim SUp,P,.i(~, for any n.
By the continuity assumption on ; we have lim,p( Pk,ick,)= 0. In addition, our construction implies that the sequence v( A,) - P(P~,~,~,)converges to zero so that we conclude that lim,v(A,) = 0. Now, the proof of the proposition follows from the fact (see [8, p. SO]) that a bounded countably additive set function on a field, such as v in our case, can be represented as the difference of two positive bounded countably additive set functions each of which can be extended uniquely (see [9, p. 131) to a positive measure on the smallest o-field containing the original field. In our case this a-field is the collection of the Bore1 sets and the difference between the two positive measures gives us the required Bore1 measure. n
4.
THE
REPRESENTATION
OF
A FORCE
SYSTEM
BY A STRESS
Definition 4.1. A force system {fP; P is a subbody of B} is consistent if the following conditions hold. (i) If PI and Pz are disjoint subbodies of B, then, for any u E C’(B, a3),
fP,"Fb(UIP,"‘? )=fP,(4P,) +_tP2(uIP*). (ii) If A E 0 and Pi -t A, then, for any u E C’( B, W3) the sequencef,,(u IP,)converges and its limit is independent of the particular sequence of subbodies pi. (iii) There is a finite bound K > 0 such that for any subbody P and any u E C’(B, W3),
IM4P)I -=IK IIUlPlL We will refer to these conditions as the additiuity, continuity and boundedness conditions, respectively. In the sequel we will use the following notation of multi-indices. Let r = (a,, a2, z,), where zP is a positive integer. We will write x2 for x:*x;lx7, Izl for g1 + a2 + a3, r! for cc,!a,!a,!, and given the real function w, we use w,, for a’=’w
axyaxyaxy We say that a function w on P is a piecewise rth order polynomial over a subbody P if the restriction of w to any connected component is a polynomial of order r, i.e. if the restriction is of the form c a,x’, a,tz%?. Ialar
In proving that a consistent force system can be represented by a unique stress, we will need the following proposition. Proposition 4.2. For each UE C’(B, .GT3)and &o> 0 there is a sequence of subbodies Pi + B, a sequence of piecewise rth order polynomials S,E Cr( pi, a3) and an integer i, such that if i > i,, then, IIulP, - sillpi < cc,. Proof: Let min,..{inf,.,{
UE C’(B, ~3’~) and .sO> 0 be given. Set lIuII* = IIuII + 1, a = u~.~}} and b = maxP,_ {supxse{up,~)} + lbw_here p = 1, 2, 3, Ial I r. We
choose the positive number
E
and the integer N so that E = -
a
N
s 2 where m is a constant m
J6
Borrow
and G. DE
R. SEGEV
depending only on r that we will specify later. We define the index i. = {A,; p = 1, 2, 3, 1x1I; r, i., = 0, . . . , N - 1). Clearly, for each component i., of 2 we have u I a + &+s czb. We also set L = min,{a, inf,,,{
IIuII*x”)},
H’ = max, {b, supXEB( /I u II*-T’> >,
H = L + NIE, where iv, is the smallest integer such that Nr > (H’ - L),k, and finally 8, = 0, . . * ) Iv, - I.
P = (D13821B3It
With these definitions we have L ( L + PP.c < H. Divide the body B into the subsets A,, defined by A,, =
((upll)-’ [a + &E, a + (j.,, + 1)E)f n
n P= 1.2.3 /z/ _
L
+
&E
I
IltlI/*Xp
<
L + (8, -t- I)&}.
By definition, if XE A,, then for each p and a we have a + &E I np,Jx) < a + (i.,, + I)&, L t &E 2 I/ulI*x;p < L + (p, + I)&.
Clearly, the various A,, are disjoint and their union is 8. Moreover, each A,, belongs to 0 and can be represented as the intersection of an open set and a closed set. Using the constructions of Section 2, for each pair of indices i. and /I there is a sequence of subbodies Pipi -+ A,, so that for a fixed i the various P,,i are disjoint and their union is a subbody that we denote by I?. We choose a point yrS in each of the sets A,, and define sApE C’(B, B3) by its pth component as (sA&o~) = C :(a Izf5r 3.
f Q)(x
- Y,$.
Set SlaiE Cr( P~~i, W3) by siBi = sIp/plsi and the piecewise rth order polynomial siEC’(pi, g3) by fi = 1 SA~X~~~,, where %Adenotes the characteristic function of the i,B subset A. We recall that
= S”p{lup,a(x)IIl4lpisi- slpi IIPAP,
(slai)p,.(x)l;xEPnSi,
Ix1 1 rr P = I, 2, 3)-
By adding and subtracting u,,,(z,,(x)), a + &E and by using the triangle inequality we obtain for any choice of a point SAKE A,,,
II4PiSi - Sij?iIIP,#,5 sup &&) X.l,P
+ sup (I~,,&&)) x.z.p + SUP flfQ + +I X.&P
- ~,&&4)l> - (a f &&)I 1 - tsl@i)p,z(x)l>*
We now examine the terms on the right hand side of this inequality. (i) The such that an integer there is a
functions up,a I ,,, are uniformly continuous for each p and r. Hence, there is a 6 > 0 Ix - yl c S implies that Iu,,~J~~(x) - ~~,~l~~(y)Jc E, for each p and 2. We choose i, such that i, > l/6 and it follows from corollary 2.5 that for each XE Pj,Bi, i > i,, ziS(.u)~AIP n P+ with 1.~- z,~(.u)~ < 6. Clearly, for i > i,, sup t I up.z(s) - up,x(Zlaf.X))f; -YEPi@it /rj I r, p = 1,2, 3) < 8,
and from that corollary i, is independent of the values of il and /3.
Consistency conditions for force systems
57
(ii) Since zlg( x) E A,, we have by our construction sup{ ln,,,(Z~~(x)) - (0 + ims)I; XEPlfli, Ial 5 r, p = 1,2,3} < s. Clearly, the last inequality holds for each L and 8. (iii) Consider the term
where we used
and where a < y means c(~< yP for p = 1, 2, 3. By the definition of Iju/I* we have
Ita + ‘pa’) -
(S*pi)p,a(X)I5 Ilull*
1
(X - _VAfi)y-a
IYISr Y’l
22
Ilull*
2
(x-
z(x)
+
44
-
yIfl)Y-=
IYl5r Y’a
where z(x)EA~~ will be specified later. The last sum contains a finite number of binomials raised to powers smaller or equal to r in the variables x-z and z-yLg. Let us denote the number of all such powers by ml(r). It follows that
c tx- z(x) + z(x) -
y&dy-= I ml(r) ;y
Ivlsr
Ix - z(x) + z(x) - yqy-=* I
Y'M
Y'a
Now, if mz( r) is the maximal number of terms in the binomial expansion for powers smaller or equal to r and mj(r) is the largest binomial coefficient for such expansions we have I(x - z(x)) + (z(x) - Yq?)r 5 ~2W~3Ww{l~
- z(x)l”; 0 52 lrll
I r} sup{Iz(x) - yApI*; 0 2 Iti1 5 r}.
We can choose by corollary 2.5 an integer i, independent of L and /I such that for each x E pibi, i > i, there is a z(x)~A;~ n Prbi with lx - z(x)1 I; E/IIuII*. In addition, by our construction, for any $, IJ/I 5 r, [z(x) - yAsl* I E/IIuI[*. It follows that sup {I(a + &d) - Cs&9i)p.mCx>l} S ~~~W~20W3(r)~. X.&P
Finally, the proof of the proposition is obtained by choosing m = Zm,(r)m,(r)m,(r) and choosing an i, which is greater than i, and i,. N
+ 2
Corollary 4.3. Let ( fP) be a consistent force system on B, then, for each u E C’(B, W3) and e0 > 0, there is a sequence of subbodies P;:--, B, a sequence of piecewise rth order polynomials SiE Cp(Pi, B3) and an integer i, such that if i > i,, then, Ifs(u) -fpi(Si)( < E,,. Proof. Let u E C’(B, g3) and a0 > 0 be given. By proposition 4.2 there is a sequence of subbodies Pi --+B, a sequence of piecewise rth order polynomials SiE C’(Pi, .@) and an
integer i, such that if i > i, then, 11ulp, - Sillpi < 3 where K is the bound in definition 2K’ 4.1 (iii). Using the linearity of the forces and the boundedness property we have for each i > i,
58
R.
SEGEV
and G. DE BOTTON
By the continuity of the force system there is an integer i, such that if i > i, then, Ifs(u) fP,(ulP,)j < so/2. Now, if i, is greater than i, and i2 we have for each i > i, I_f*(u)
-h,(si)I
s
IfBt”)
-_hP,(“lP,)l
+
lfP,(“lP,)
-fP,(si)I
<
&O.
Remark. We note that the sequences e and si depend only on u and B and do not depend on the force system. Proposition 4.4. Let {fP} and {gP} be two consistent force systems on B. Assume that gP(s) =fP(s) for each subbody P and each piecewise rth order polynomial s on P, then, gP = fP for each subbody P. Proof: Assume that there is a subbody P and a ueC’(B, W3) such that gp(ulp) #fp(uIp) and let so = Igp(uIP) -fP(ujP)l. Since gP(s) =f.(s) for each subbody P and each piecewise rth order polynomial s on P we have
IgPC”lP) -fP(“lP)l s ISPt”lP)-fPi(si)l +
IfP,(si)
-fP(“lP)l
= IgPt”lP)- CJPi(si)l + IfP,(si) -fp(“lP)l. Since the sequence si is independent of the force system, by corollary 4.3 we may choose an i, so that (gP(uIP) - gP,(si)I c so/2 for i > i, and an i, so that IfP,(si) -fP(ulP)/ < co/2 for i > i,. A choice of an i, greater than both i, and i, will result in a contradiction. n Proposition 4.5. A force system {fP} on B is consistent if and only if there is a unique collection of bounded Bore1 measures {op.}, p = 1, 2, 3, 1x1< r that represents the force system in the form
fp(n) =
1 1 np.edapa I.l
UE UP,
W3).
p=1.2,3
Proof: It is clear that a force system which is represented by a collection of bounded stress measures as in the previous representation equation is consistent. We now show that consistency is indeed a sufficient condition for the representation. Define the functions utq., E C’(B, 9t3) by their pth components as ucqan,(x) = b4,,xa,p, q = 1,2,3, I rj < r, where 6,, denotes the Kronecker 6. For Irl = 0 we define the set functions pqa on the collection of subbodies of B by L(J P) = fp( ucqajI p ). Clearly, the set functions pqa satisfy the conditions of proposition 3.1 and it follows that they can be extended to Bore1 measures aq2, Irj = 0, on B. Next we define inductively
for 0 < I@/I r. Since the conditions of proposition 3.1 hold for both terms in the previous 0 < 1~1I r as well, and hence, we can extend equation, they hold for the set functions r(lqor, them to the Bore1 measures aqz. Construct the force system {gp} on B by gp(u)
=
1
lzl5r
P’1.2.3
iP
up.a da,=t
u E C’( P, 33).
It is obvious that (gp} is a consistent force system. We have
Consistency conditions for force systems
59
By using cr,,(P) = g,,(P) and the definition of pq3, in the Iast line we obtain gP(u6,_)lP) = ~,,(z+~.)I~).By the linearity of forces it foIlows that the force systems {fP) and {gp) have equal values for any rth order polynomial, by the additivity of the force systems they have equal values for any piecewise rth order polynomial and it follows from proposition 4.4 that they are equal. It remains to show that if {u+} and {gzpa}, p = 1,2,3, Irl I r represent the force system (.Mr then blP. = czP.- Let A be an arbitrary set in CDand let e be a sequence of subbodies converging to A. We have for 1~1= O,j&(u& = al,( pi) = a& 4) for each i and p. Using =0 the continuity of the measures alPa and bzPSwecon~ludethat~~~(~~=~~~~(A)for~~l and every AE@. We recall that if two measures are equal on a field they are equal on the minimal e-field containing it, hence, brPz = ~~~~for Iri = 0. To use an induction process, assume that blPE = b2P=for all [ix1c r0. Again, for an arbitrary AE@, pi + A and 7 with I’il = r0, we have by using the definition of uCP.),
Hence,
and from the induction hypothesis it follows that clPy( pi) = 02,,( pi) for each i and p. Again, the properties of measures imply that dlP., = u2Py. n Acknowledgements-This work, which is based on the second author’s Master’s thesis, was supported by the BathSheva de Rothchild Foundation for the Advancement of Science in Israel and some of it was done during the first author’s visit to Rational Mechanics at Johns Hopkins University.
REFERENCES 1. C. Truesdell, A firsr
in Rutjonuf Co~rin~u~ ~echun~cs, Vol. I. Academic Press, New York (1977). 2. R. Segev, On the definition of forces in continuum mechanics, in ~~~a~ic~~ Systems and ‘microphysics, Conrrol COUW
Theory and Mechanics (Edited by A. Blaquiere and G. Leitmann), pp. 341-357. Academic Press, New York (1984). 3. R. Segev, Forces and the existence of stresses in invariant continuum mechanics. J. Marh. Phys. 27. 163- 170 (1986).
4. 5. 6. 7. 8. 9.
W. Noll and E. Virga, On Edge lnteracrions. To appear. J. Eells, Jr, Singulariries ofSmooth Maps. Gordon and Breach, New York (1967). V. Guillemin and A. Poilack, Diflerential Topology. Prentice-Hall, Englewood Cliffs, New Jersey_ (1974). W. M. Hirsch, Differential Topoibgy. Spring&, B&in (1976). K. P. S. Bhaskara Rao and M. Bhaskara Rao, Theory of Charges.Academic Press, New York (1983). R. B. Ash, Measure Inregrarion and Functional Analysis. Academic Press, New York (1972).
ON THE CONSISTENCY CONDITIONS SYSTEMS
FOR FORCE
REUVENSEGEVand GAL DE BOTTON The Pearlstone Center for Aeronautical Engineering Studies, Department of Mechanical Engineering, Ben-Gurion University of the Negev. Beer-Sheva. Israel (Received 6 August 1989; accepted 4 finery
1990)
Abstract-In analogy with the classical Cauchy conditions, this work presents conditions so that a force system, the assignment of a force to each subbody of a given body, can be represented by a stress. The setting in which the theory is formulated is more general than that of classical continuum mechanics as stresses can be. as irregular as measures, equilibrium is not assumed, it applies to continuum mechanics of order higher than one and it may be extended to the case where the body and space are modelled by general differentiable manifolds. The consistency conditions presented are that of additivity of the force system on pairs of disjoint subbodies, continuity and boundedness.
I. INTRODUCTION
The present paper is concerned with the conditions so that a force system on a body can be represented by a stress. Here, by a force system, we mean an assignment of a force to each subbody of a given body. In the classical formulation of continuum mechanics, it is assumed that force systems on bodies satisfy the following conditions: (i) The force& on any subbody P is given in terms of a body force field b, and a continuous surface force field t, in the form fp = fP b, dv + lBpt, da. (ii}The body force on a subbody is independent of the subbody and the surface force on a subbody depends on the subbody only through its unit normal n to the boundary. It is also assumed that the surface force depends on the normal continuously. (iii) The total force on each subbody vanishes. On the basis of these conditions it is proven that there is a unique tensor field (7 over the body, called the stress, so that the forces on the various bodies are given by b = - div cr and tp = a(n). Thus, we may say that the stress o represents the force system (f,,j with these two equations and that the aforementioned conditions are coff~i~te~c~ conditions for the existence of a stress representing the given force system. Modern formulations of stress theory replace some of these assumptions by other, more general assumptions. In [l J, which summarizes the works of Nell, Gurtin, Williams and others (see references therein), the relevant structure is as follows. Forces are defined as interactions for pairs of subbodies so that f(P, C) represents the force, a vector in some inner product space, that the subbody C exerts on the subbody P. The consistency conditions in this formulation are shown to be the following: (i) f(p, C, u C,) = f(P, C,) +f(P, C,) for separate Ci, C,, and f(P, u P,, C) =f(Pi, C) +f(P2, C) for separate PI, Pz. (ii) The forcef( P, C) for separate P and C is bounded by both the volume of P and the area of the common boundary of P and C. (iii) The total force on each subbody vanishes. (iv) Proving the existence of a surface force field on the basis of the previous assumptions it is assumed that it is continuous. In previous works (see [Z, 3]), an alternative framework for the theory of forces in continuum mechanics was suggested. A force on a body was defined as a continuous linear functional on a tangent space to an appropriate configuration Banach manifold. Assuming that bodies are compact submanifolds with boundary of space, and using the axiom of impenetrability of continuum mechanics as a guideline, it was shown that it is natural to take the collection of differentiable embeddings of the body in space equipped with the C’ topology as the configuration manifold. Restricting ourselves for simplicity to the case where the space is modelled by g3 and a body B is a 3-dimensional submanifold of W3, it follows that the various tangent spaces can be identified with C’(B, #) and that forces can
Contributed by K. R. Rajagopal. 47
48
R. SEGEV
and G.
DE
Borros
be identified with elements of the dual space C’fB, s3)*. It is shown that such forces can be represented in the form f(u) =
c
P.~~,lr.a,
sB
d?L+zz+Z’n ’ da PSlZZ’, ax;1 dx;‘ax;
’
where p = 1,2,3, xi, [x2, ~1~are non-negative integers with xi + a2 + a3 I r, and CJ,,~~_~, are Bore1 measures to which we will refer as stress measures. From the construction it follows that the collection of stress measures that represent the force f is not unique. The representation presented in the previous equations corresponds to high-order stresses, those appearing in rth order continuum mechanics. For the simplest case r = 1 the equation can be rewritten as
where here, rsw are measures corresponding to the components of the usual stress tensor and Q,, which are not present in usual formulations, appear in the equation because no equilibrium hypothesis was made (i.e. if one assumes that the total force on each subbody vanishes it follows that dP = 0). If these stress measures are given by differentiable densities in terms of the volume measure in g3, it can be easily shown that forces can be represented by body forces and surface forces. Since, unlike forces, stress measures can be restricted, a given collection of stresses (Tag__., induces a force system on B in which the force_& on a subbody P is represented by
In addition to the fact that it does not require the assumption of equilibrium and permits stresses that are irregular as Bore1 measures, this formulation of the theory of stresses in continuum mechanics has the advantage that it applies in the general geometrical setting of differentiable manifolds. While the suggested formulation models the representation of a single force on a body by nonunique stress measures, the aforementioned papers do not contain the appropriate consistency conditions on a force system that guarantee that there is a collection of stress measures that induce it in the form presented in the last equation. In this paper, it is shown that the following conditions on the force system (fpe c’(P, L@~)*;P is a subbody of B>, which are clearly necessary, imply that there is a unique collection of Bore1 stress measures LT parEtZx,p = 1, 2%3, xi + a2 -t a3 I r, that represent it as in the last equation. (We use the convention that the empty set is a subbody and the force on it is zero.) (i) Additiuity. If PI and P2 are disjoint subbodies of B then for any UEC’(B, Z3), fP,UPAUlP,UP,) =h,(ul/y)
+h*(&,).
(ii) Continuity. We recall that the set A is the Iimit of the sequence of sets 4 if and only if for every element x in A there is an integer i,, so that x E pi for every i 2 i,, and every point that is contained in an infinite number of sets in the sequence is contained in A. It is required that if A E @],where Q,is the minimal field containing the open subsets of the body, and 4 is a sequence of subbodies whose limit is A, then, for any UE C’(B, a3), the sequencefp,(ufP,) converges and its limit is independent of the particular sequence of subbodies Pi. (iii) Boundedness. u E C’(B, W3),
There is a finite bound K such that for any subbody
P and any
I_MUlP)I< Kllul,II * We note that as the traditional consistency conditions hold only for continuum mechanics of order one, and while it is only very recently that No11and Virga were able to present consistency conditions for second order continuum mechanics E43, the consistency conditions presented in this paper hold for continuum mechanics of any order.
Consistency conditions for force systems
49
The reason why the representation of a force system by a stress is unique while the representation of a force by stresses is not unique may be described roughly as follows. In the expression for the representation of a given force by stress measures we know the values of the integrals only for “compatible” collections of continuous functions w,,.,._, , p = 1, 2, 3, zi + x2 -t x3 s r, i.e. collections for which there are differentiable functions ug such that PI +az+lJu n Hence, the measures cannot be determined uniquely. However, we can approximate any collection )tipa,_, by a family of compatible collections such that the members of the family are defined on a family of disjoint subbodies. If we know the force on each subbody we can approximate the integrals for “non-compatible” collections and determine the measures uniquely. Section 2 discusses the approximation by subbodies of sets that belong to the minimal field of subsets containing the open subsets. For each set A in the minimal field, a sequence of subbodies whose limit is A is constructed and some useful properties of this sequence are presented. Section 3 uses the construction of Section 2 in order to give sufficient conditions (which are also necessary conditions) so that a real valued set function defined on the collection of subbodies can be extended to a Bore1 measure. In addition to its use as a tool for the proof of the sufficiency of the consistency conditions, this result is of some interest as it specifies the conditions under which quantities such as electric charge can be extended to measures if they are given for the various subbodies. Section 4 presents the consistency conditions and proves that they are su~cient (again, these conditions are also necessary conditions). 2. THE
APPROXIMATIONS
OF SETS
BY BODIES
As was mentioned a body is defined as a three-dimensional compact submanifold with boundary of s3. We now add formally the empty set to the collection of bodies. Given a body B, a subset D of B which is a body is a subbody of B. Clearly, the complement of a subbody in the body is not a subbody and the same holds in general for the unions and intersections of subbodies. In this section it will be shown that although our collection of subbodies is small, any set in the smallest field of subsets containing the open subsets of B can be approximated by a sequence of subbodies of B. This approximation has some additional properties that will be used in the following sections. Proposirion 2.1. For any compact subset C of .9P3and a bounded open subset U of .?J?~ containing C, there is a body B which is contained in U whose interior contains C. ProoJ By a standard theorem of differential topology (see [S, p. 123) it is possible to construct a smooth real function h: g3 + 9 with the following properties: (i) its value at any point in C is 1; (ii) its value at any point in the complement of U is 0; and (iii) its value at any other point is in the interval [O, 11. By Sard’s theorem (see [6, p. 2041) any neighborhood of the value l/2 contains a point x such that h is not singular at any point in the inverse image of x. From the implicit function theorem it follows that h-‘( f.x)) is a smooth submanifold of &I‘3that is clearly compact and which is the boundary of h- ‘([x, I]). It follows that h-‘f[x,l]) is a body. i Proposirion 2.2. Let D be a body, LJ a bounded open subset of a3 and C a closed set contained in U. There exists a body B contained in U whose interior contains C such that B u D and B n D are bodies. Proof Let us construct a body Bz contained in U whose interior contains C as in the previous proposition. In the case where U n D is empty, or D I> U it follows from the construction that B2 LJ D and B, n D are bodies. However, in general, when the boundaries of B, and I) interest, B, u D and B, A D will not be bodies. We will modify B2 so that the modified body will have the required properties.
50
R. SEGEVand G. DE BOTTOK
We recall that if X and Y are submanifolds of the differentiable manifold Z, any C’ neighborhood, r 2 1, of the inclusion of X in Z contains an embedding g: X + Z which is transversal to the inclusion of Y in 2 (see [7, p. 781). It follows that we may modify B, slightly to obtain a diffeomorphic body B, so that the properties guaranteed by the previous proposition still hold, and in addition, dB, is transversal to ZD. Hence, the intersection dS1 n dD is a one-dimensional submanifold of both dB, and SD. Next we modify B, to obtain a diffeomorphic body B so that the intersection between the boundary of B and the boundary of D contains an open subset. In other words, the two boundaries will overlap on an open set so that Bu 5 and Bn D are bodies. This modification will take place in a small neighborhood of the intersection of the transversal submanifolds dB, and 8D. In constructing the neighborhood in which the modification will take piace we use the following theorems of differential topology (see [7, pp. 109- 11.53). (i) If Y is an orientable submanifold of the differentiable manifold Z of codimension one, then, there is an embedding g: Y x ( - 1, 1) -+ Z, called a tubular neighborhood of Y in Z having the following properties: (a) g Ir yto) is the inclusion of Yin Z, and (b) the image of g is an open neighborhood of Yin 2. We will also use the term tubular neighborhood for the image of the mapping g, and we wiil identify a point in the image with the “coordinates” given to it by g. (ii) If Y is a neat submanifold of Z, then, there is a tubular neighborhood of Y in Z. (iii) If Y is a neat submanifold of Z, then, every tubular neighborhood of d Y in dZ is the intersection of dZ with a tubular neighborhood of Y in Z. We first construct tubular neighborhoods Vi and V, of dB, A 1?D in dB, and in dD, respectively, with the convention that the parameters in ( - I, 1) are positive in D and the complement of B1, respectively. Next, considering the neat submanifold dB, n D in D, we can use theorem (iii) above to show that there is a tubular neighborhood V;, of ?B, n D in D such that V, = VJ n dD, where we set the values of the parameter SE( - 1, 1) to be positive outside B,. Let
By definition, every point in V has three coordinates (x, t, s) such that XESB, n alI, t, s E [O, 1), where t = 0 for points on aD and s = 0 for points on as,. Since there is a finite positive distance between 8B, and the complement of II, there is a number 5 > 0 such that a point in V is in U if the coordinate s is less than <. Consider the folIowing functions: a:&?-+9
a(r) = 0
for r I 0
a(r) = exp( - l/r) /!I:&?-+_,
for r > 0,
p(r) = r(r - W4S
- r)
D(r) dr 8: .a! --) [O, n/2]
d(t) = ; P(r) dr .
The function 0 is smooth and has the following properties: (i) e(t) = n/2 if t s 5/4 (ii) 0 2 e(t) s n/2 if 5/4 ~5t s 35/4 (iii) s(t) = 0 if t 2 3t/4. We will now modify the portion of dB, that is a subset of Y (which is clearly in I)) as follows. Let g: Y, + Y be defined by gfx, t) = (x, t case(t), g(.% t) = (x, t, 0)
t sin O(t))
if 0 I t < (, otherwise.
Consistency
conditions
for force systems
Clearly, g is an embedding and has the following properties: (i) Im(g) n dD = {p = g(.x, t); 0 s t I 5/4) contains an open set of SD, (ii) Im(g) n dB, = {p = g(x, t); 35/4 I t I 1) contains an open set of SB,, (iii) Im(g) is a subset of U and is disjoint from C. In a similar way we will define a function h that will modify a neighborhood of dS, n c?D in c?B, A (DO)’ whose image will be contained in a neighborhood of (?B, n dD in B, n (DO)‘. (The superscript “c” denotes the complement of the corresponding set.) This mapping will have the analogous properties to those of g, i.e. it is an embedding, its image is contained in U-C, Im(h) n dD contains an open set of c?Dand Im(hf n ZB, contains an open set of 2B,. We can define now the mapping @: ZB, + U-C by i&(p) = g(y) if y~domain(g), $(y) = h(y) if yEdomain and i&y) = y otherwise. Clearly, $ is an embedding and its image overlaps with aD on an open set. By the Jordan-Brower Separation theorem (see [6, p. 891) Im($) is a boundary of a body 5 and it follows from the construction that B n D n and B u D are bodies as their boundaries are smooth. We can now state the analog of proposition subsets of the body B.
2.1 for the case where both C and U are
Proposition 2.3. If U is an open subset of the body D and C is a closed subset of D contained in U, there is a subbody P of D contained in U whose interior contains C. Proof: It is clear that C is closed in g3 and that it is contained in an open subset l’of Z3 such that Vn D = U. By proposition 2.2 we can construct a body B contained in V whose interior contains C such that B n D is a body, hence, a subbody of D which clearly has the required properties. m
Henceforth, unless otherwise stated, we will refer by open and closed sets to open and closed subsets of the body under consideration. Proposition 2.4.Let 0 be the smallest field of subsets containing the open subsets of a body B. For each A in CDthere is a sequence fl of subbodies of B such that pi + A. Proof: It follows from a standard representation theorem for the members of the smallest field containing a given collection of sets (see [S, p. 73) that a subset A of B belongs to 4 if andonlyifA=
0
Uk n C,,where U, and C, are open and closed sets, respectively, with
k=l Cj) n (& n C,)= @ for j # k. Let A be a member of CDand assume that such a representation of A is given. For each k let cki be defined by
(V’n
where B(d, x) denotes the open ball of radius d centered at x. Clearly, for a given i the sets Cki are disjoint. In case all Cki = 0 we will set pi = 0 and henceforth we may assume that at least for one k, Cki # 0. In the case where there is only one Cki # 0, we will set 6, = 1/2i. Otherwise, since for any given i the sets C,, are disjoint and compact there is a number 6, > 0 smaller than half the distance between any two of them and lj2i. For each cki # 0, let V,i = U B(6,, X). XECki Chdy, any vki is a subset of Ukand for a fixed i the sets qi are disjoint. Using proposition 2.3 it is possible to construct for a fixed i and each k such that C,i # 0, a subbody Pki contained in uki whose interior contains Ckj. In case Cki = 0 we just set Pki = 0. Define
Since the various Pki are disjoint, 4 is a body.
R.
52
SEGEV
and G. DE BOTTOK
Lemmu 2.4.1. The sequence pi satisfies lim infiG 3 A.
~~oo~o~~e~~u 2.4.1. We first note that for each k the limit of the increasing sequence Cki is C, n Uk. Clearly, each C,i is a subset of C, n U, hence their union is also a subset of Ck n Uk. On the other hand, assuming that XE C, n U,, there is a minimal distance d > 0 between x and CJE.Choosing an i such that l/i < d, we have
and it follows that We conclude that
Since, 4 3 E, for each i we have lim inriP{ 1 lim infiEi = A. Lemma 2.4.2. The sequence pi satisfies A 13 lim
SUpiP,.
Proof of lemma 2.4.2. Let us construct the sequence
which is decreasing to Ck. By our construction Phi is a subset of V,i for each k and i, hence, C, = lim sup, Gi 3 lim SUp,P,i. On the other hand, U, 3 Pkj for each i and it follows that U, n C, I3 lim SUpi Pki. Assume that x $ A, then, for each k, x +!U, n Ct and x 4 lim supi Pki. In other words, XE Pki for a finite number of elements, say, Nkelements of the sequence only. Thus, x E (J si = 4 k=l
for a finite number of elements and hence, x # lim Supi 4. From lemma 2.4.1 and lemma 2.4.2, we have lim infiq =, lim Supi~::, and by the definitions of the limit of the sets it follows that lim,P, = A. H Coro~~ury 2.5. Let A = fi U, n C,. Then for each k the sequence of subbodies k=l
Pki
constructed in the previous proposition converges to Ukn C, and it has the following property: for each E > 0 there is an N independent of k such that for each i > N and each .YE Pki there is a point in &. n C, A Pki whose distance from x is less than E. ProofI This corollary proposition.
follows immediately
from the construction
of the previous
Proposition 2.6. If A,, A,, . . . , A,, are disjoint members of 0, then, there are n sequences of subbodies PIi, Ptit. . . , P,i converging to A,, AZ,. . . , A,, respectively, such that for any i, PIi, Pzi,. . . , Pni are mutually disjoint and the sequence of subbodies ~=Pp,iuP,iu... Proof:
uP,,convergestoA,vA,u...uA,.
for the case n = 2. Let A, and A, be
It is sufficient to prove the proposition
represented in the form A, = ,c, Ulj A Clj, A, = k()l Ule n C,, as in proposition 2.4. We can represent A, u A2 in the form ii j=l
U,j”Clj
I[ U
; k=l
U,knC,k
1
so that we can apply to it proposition 2.4 and corollary 2.5. It follows that for each j and k there are sequences P,ji and P,kj of subbodies converging to Ulj n C,j and u,i, n Cztr
Consistency
conditions
for force systems
53
respectively, such that for any fixed i the various subbodies are disjoint. Hence, the K
J
sequences P,i = u P,ji, Pti = u P2Li have the required properties. j=
H
k=l
1
Proposition 2.7. Let A = fi
wkn Ek, A’ = fi
I$ n Fj be members of Q represented as in
j=l
k=l
2.4 (i.e. wk, T are open and E,, FJ are closed) such that A’ 3 A. Then, it is
proposition
possible to represent A in the form A = fi Ui n Ci such that the following conditions hold: i=l
(i) The subsets Ui are open, the subsets Ci are closed and the various Ui n Ci are mutually disjoint. (ii) For each i there is a unique j(i) such that ~(i, z) Ui and Fj,, 2 Ci so that vjfi, n Fjti, 3 CJin Ci. Proof: Define the closed sets ckj = E, n Fj and the open Sets yj = wk n f$. We note that A = u Ukjn C,, and that the various Ukjn Ckj are mutually disjoint. In addition, by the k. j
for each pair j, k there is a unique j(k) such that qfk, 3 Ukj and Fjck,I Ckj so
construction,
that 5(k) n Fj(k) 3 Ukjn
ckj.
Finally, we obtain the required representation
A = 6
Ui n Ci
i=l
by re-enumerating
the subsets Ukj and Ckj.
n
Proposition 2.8. Let A, be a decreasing sequence of members of 0,. Then, there are sequences Pki of subbodies such that Pki+ A,, and for a fixed i, Pki 2 Pck+l,i. Proof: We first construct the sequence PIi converging to A, as in proposition 2.4. Inductively, we construct the sequence Pki on the basis of the sequence qk_ l)i. Hence, we assume that the sequence qk_ l)i + A,_ I has the following properties as in proposition 2.4:
(a)
Ak-I
(b)
et-l)i
(c)
For each subbody PC,_ r )ni we have interior (ok _ I )ni) =) ctk _ 1)ni where
=
6 II=1
=
6 n=1
c(k
~k-IjnnC(k-ljn.
qk-l)ni.
- 1)ni =comp{
uB(+,x):
xEcomp(u(k-l)n)
}
nc(k-l)n.
(d) For each subbody qk_ i)ni we have V,k_l)ni 2 ok_ I,ni where qk-.l)ni=
u x
{B(6~k-,,i,x);xEC~k-,,,i}.
Using the fact that A,_ 1 I A, and proposition 2.7 we can represent A, in the form A, = fi j=
Ukjn Ckj such that the following hold: I
(i) The subsets ukj are open, the subsets Ckj are closed and the various Ukjn Ckj are mutually disjoint. (ii) For each j there is a unique n(j) such that U,,_ l,nt,353 Ukj and C,,_ ijn(,? 2 Ckj so that U,k-l)n(j)n
Cck-l)n(j)
3
yjn
ckje
We now construct the subbody Pki such that 4k_i,i 3 Pki. For each Ukjn Ckj let
Lemma 2.8.1. For each j there is one n(j) such that interior ( 4k_ lIntj)i) 3 Ckji. Proof of lemma 2.8.1. By (ii) above U,,_ l)nc,, 3 Ukj, it follows that
= @ we will set Pki = Qr and henceforth we may assume that at least for In the case where there is onfy OSI~ Ckji # a, we wilt set 6,i = (A?;. Otherwise, since for any given i the sets C,ji are disjoint and compact there is a number Ski > 0 smaller than half the distance between any two of them and 1/2i. For each Crji $ QY fn case ail Caji
VIE j.
Ckji # a.
let
Clearly, any E!,&is a subset of Ukjand for a fixed i the sets i&i are disjoint. Using proposition 2.3 it is possibfe ta construct for a fixed i and each j such that Ckjz# @, a subbody Pkji contained in Fuji whose interior contains C~ji- fn case 6, =; 0 we justset I&; = @If Define
Since the various ~~~~o~ies Pkji are disjoint, their union Pki is a body and in addition, as interior (P,,_ l)n(j)i ) 3 Ukji =) Pkjiit is clear that ek_ l)i 3 Pk(. The PrOOf that si -+ A, follows from lemma 2.4.1 and lemma 2.42. R
3. THE
EXTENSION
OF A SET FUNCTION
DEFINED
UN SUBBODIES
TO A MEASURE frr the next section we wilE need a condition such that a set fun~t~on defined over the collection of subbodies of a given body can be extended to a unique measure. Such a condition is of interest in general because it can be applied to various physical instances such as mass, electric charge, etc. Just as we formafly added the empty set as a subbady, we now set formally the value of the set function for the empty set to be zero.
Propusition 3. t. Let I( be a bounded real valued set function defined on the collection of subbodies of a body L3that satisfies the foIlowing conditions: (i) if Pi and P2 are disjoint subbodies, then, @(PI LJ P2) = &PI) -f &P,). {ii) ff A E Q1and I$ -, il, then, the sequence p(e) converges and its Emit is jnde~nde~t
of
the ~ar~j~u~ar sequence pi. Then, there is a unique Bore1 measure Y on B such that v(P) = p(P) for all subbodics P of B. We will refer to the first and second conditions as additivity and continuity conditions, respectively. Proof. Define the real valued set function v on Q by v(A) = limip(&), where iq: is a sequence of subbodies whose limit is A. By proposition 2.4 v is well defined, and in addition, it follows from the continuity assumption that for any subbody P+ v(P) = g(P). To show that t’ is additive, consider the disjoint sets A, and A, in Cp.We now construct two sequences Pki and Pzi converging to A, and A,, respectivety, satisfying the properties guaranteed by proposition 2.6, so in particulars P,,r, Pzi = $Zj for a fixed i. Since A, w A, = fim,
So far, we obtained a finitely additive set function on a field. We next show that P is countably additive. We recali (see [9, p. lo]) that in order to prove that an additive set function v on a field is countably additive it is sufficient to show that v is continuous from above at the empty set, i.e, if the decreasing sequence A; converges to the empty set, then ~imiV~~i~ = 0.
~~~~a 3.1.1. The set function v is continuous
from above at the empty set.
Consistency
conditions
for force systems
55
Proof of lemma 3.1.1. Let A, be a sequence of sets in Q,decreasing to the empty set and for each k let Pki be the sequence converging to A, as in proposition 2.8, i.e. for a fixed i, Si I3
By the continuity assumption on ; we have lim,p( Pk,ick,)= 0. In addition, our construction implies that the sequence v( A,) - P(P~,~,~,)converges to zero so that we conclude that lim,v(A,) = 0. Now, the proof of the proposition follows from the fact (see [8, p. SO]) that a bounded countably additive set function on a field, such as v in our case, can be represented as the difference of two positive bounded countably additive set functions each of which can be extended uniquely (see [9, p. 131) to a positive measure on the smallest o-field containing the original field. In our case this a-field is the collection of the Bore1 sets and the difference between the two positive measures gives us the required Bore1 measure. n
4.
THE
REPRESENTATION
OF
A FORCE
SYSTEM
BY A STRESS
Definition 4.1. A force system {fP; P is a subbody of B} is consistent if the following conditions hold. (i) If PI and Pz are disjoint subbodies of B, then, for any u E C’(B, a3),
fP,"Fb(UIP,"‘? )=fP,(4P,) +_tP2(uIP*). (ii) If A E 0 and Pi -t A, then, for any u E C’( B, W3) the sequencef,,(u IP,)converges and its limit is independent of the particular sequence of subbodies pi. (iii) There is a finite bound K > 0 such that for any subbody P and any u E C’(B, W3),
IM4P)I -=IK IIUlPlL We will refer to these conditions as the additiuity, continuity and boundedness conditions, respectively. In the sequel we will use the following notation of multi-indices. Let r = (a,, a2, z,), where zP is a positive integer. We will write x2 for x:*x;lx7, Izl for g1 + a2 + a3, r! for cc,!a,!a,!, and given the real function w, we use w,, for a’=’w
axyaxyaxy We say that a function w on P is a piecewise rth order polynomial over a subbody P if the restriction of w to any connected component is a polynomial of order r, i.e. if the restriction is of the form c a,x’, a,tz%?. Ialar
In proving that a consistent force system can be represented by a unique stress, we will need the following proposition. Proposition 4.2. For each UE C’(B, .GT3)and &o> 0 there is a sequence of subbodies Pi + B, a sequence of piecewise rth order polynomials S,E Cr( pi, a3) and an integer i, such that if i > i,, then, IIulP, - sillpi < cc,. Proof: Let min,..{inf,.,{
UE C’(B, ~3’~) and .sO> 0 be given. Set lIuII* = IIuII + 1, a = u~.~}} and b = maxP,_ {supxse{up,~)} + lbw_here p = 1, 2, 3, Ial I r. We
choose the positive number
E
and the integer N so that E = -
a
N
s 2 where m is a constant m
J6
Borrow
and G. DE
R. SEGEV
depending only on r that we will specify later. We define the index i. = {A,; p = 1, 2, 3, 1x1I; r, i., = 0, . . . , N - 1). Clearly, for each component i., of 2 we have u I a + &+s czb. We also set L = min,{a, inf,,,{
IIuII*x”)},
H’ = max, {b, supXEB( /I u II*-T’> >,
H = L + NIE, where iv, is the smallest integer such that Nr > (H’ - L),k, and finally 8, = 0, . . * ) Iv, - I.
P = (D13821B3It
With these definitions we have L ( L + PP.c < H. Divide the body B into the subsets A,, defined by A,, =
((upll)-’ [a + &E, a + (j.,, + 1)E)f n
n P= 1.2.3 /z/ _
L
+
&E
I
IltlI/*Xp
<
L + (8, -t- I)&}.
By definition, if XE A,, then for each p and a we have a + &E I np,Jx) < a + (i.,, + I)&, L t &E 2 I/ulI*x;p < L + (p, + I)&.
Clearly, the various A,, are disjoint and their union is 8. Moreover, each A,, belongs to 0 and can be represented as the intersection of an open set and a closed set. Using the constructions of Section 2, for each pair of indices i. and /I there is a sequence of subbodies Pipi -+ A,, so that for a fixed i the various P,,i are disjoint and their union is a subbody that we denote by I?. We choose a point yrS in each of the sets A,, and define sApE C’(B, B3) by its pth component as (sA&o~) = C :(a Izf5r 3.
f Q)(x
- Y,$.
Set SlaiE Cr( P~~i, W3) by siBi = sIp/plsi and the piecewise rth order polynomial siEC’(pi, g3) by fi = 1 SA~X~~~,, where %Adenotes the characteristic function of the i,B subset A. We recall that
= S”p{lup,a(x)IIl4lpisi- slpi IIPAP,
(slai)p,.(x)l;xEPnSi,
Ix1 1 rr P = I, 2, 3)-
By adding and subtracting u,,,(z,,(x)), a + &E and by using the triangle inequality we obtain for any choice of a point SAKE A,,,
II4PiSi - Sij?iIIP,#,5 sup &&) X.l,P
+ sup (I~,,&&)) x.z.p + SUP flfQ + +I X.&P
- ~,&&4)l> - (a f &&)I 1 - tsl@i)p,z(x)l>*
We now examine the terms on the right hand side of this inequality. (i) The such that an integer there is a
functions up,a I ,,, are uniformly continuous for each p and r. Hence, there is a 6 > 0 Ix - yl c S implies that Iu,,~J~~(x) - ~~,~l~~(y)Jc E, for each p and 2. We choose i, such that i, > l/6 and it follows from corollary 2.5 that for each XE Pj,Bi, i > i,, ziS(.u)~AIP n P+ with 1.~- z,~(.u)~ < 6. Clearly, for i > i,, sup t I up.z(s) - up,x(Zlaf.X))f; -YEPi@it /rj I r, p = 1,2, 3) < 8,
and from that corollary i, is independent of the values of il and /3.
Consistency conditions for force systems
57
(ii) Since zlg( x) E A,, we have by our construction sup{ ln,,,(Z~~(x)) - (0 + ims)I; XEPlfli, Ial 5 r, p = 1,2,3} < s. Clearly, the last inequality holds for each L and 8. (iii) Consider the term
where we used
and where a < y means c(~< yP for p = 1, 2, 3. By the definition of Iju/I* we have
Ita + ‘pa’) -
(S*pi)p,a(X)I5 Ilull*
1
(X - _VAfi)y-a
IYISr Y’l
22
Ilull*
2
(x-
z(x)
+
44
-
yIfl)Y-=
IYl5r Y’a
where z(x)EA~~ will be specified later. The last sum contains a finite number of binomials raised to powers smaller or equal to r in the variables x-z and z-yLg. Let us denote the number of all such powers by ml(r). It follows that
c tx- z(x) + z(x) -
y&dy-= I ml(r) ;y
Ivlsr
Ix - z(x) + z(x) - yqy-=* I
Y'M
Y'a
Now, if mz( r) is the maximal number of terms in the binomial expansion for powers smaller or equal to r and mj(r) is the largest binomial coefficient for such expansions we have I(x - z(x)) + (z(x) - Yq?)r 5 ~2W~3Ww{l~
- z(x)l”; 0 52 lrll
I r} sup{Iz(x) - yApI*; 0 2 Iti1 5 r}.
We can choose by corollary 2.5 an integer i, independent of L and /I such that for each x E pibi, i > i, there is a z(x)~A;~ n Prbi with lx - z(x)1 I; E/IIuII*. In addition, by our construction, for any $, IJ/I 5 r, [z(x) - yAsl* I E/IIuI[*. It follows that sup {I(a + &d) - Cs&9i)p.mCx>l} S ~~~W~20W3(r)~. X.&P
Finally, the proof of the proposition is obtained by choosing m = Zm,(r)m,(r)m,(r) and choosing an i, which is greater than i, and i,. N
+ 2
Corollary 4.3. Let ( fP) be a consistent force system on B, then, for each u E C’(B, W3) and e0 > 0, there is a sequence of subbodies P;:--, B, a sequence of piecewise rth order polynomials SiE Cp(Pi, B3) and an integer i, such that if i > i,, then, Ifs(u) -fpi(Si)( < E,,. Proof. Let u E C’(B, g3) and a0 > 0 be given. By proposition 4.2 there is a sequence of subbodies Pi --+B, a sequence of piecewise rth order polynomials SiE C’(Pi, .@) and an
integer i, such that if i > i, then, 11ulp, - Sillpi < 3 where K is the bound in definition 2K’ 4.1 (iii). Using the linearity of the forces and the boundedness property we have for each i > i,
58
R.
SEGEV
and G. DE BOTTON
By the continuity of the force system there is an integer i, such that if i > i, then, Ifs(u) fP,(ulP,)j < so/2. Now, if i, is greater than i, and i2 we have for each i > i, I_f*(u)
-h,(si)I
s
IfBt”)
-_hP,(“lP,)l
+
lfP,(“lP,)
-fP,(si)I
<
&O.
Remark. We note that the sequences e and si depend only on u and B and do not depend on the force system. Proposition 4.4. Let {fP} and {gP} be two consistent force systems on B. Assume that gP(s) =fP(s) for each subbody P and each piecewise rth order polynomial s on P, then, gP = fP for each subbody P. Proof: Assume that there is a subbody P and a ueC’(B, W3) such that gp(ulp) #fp(uIp) and let so = Igp(uIP) -fP(ujP)l. Since gP(s) =f.(s) for each subbody P and each piecewise rth order polynomial s on P we have
IgPC”lP) -fP(“lP)l s ISPt”lP)-fPi(si)l +
IfP,(si)
-fP(“lP)l
= IgPt”lP)- CJPi(si)l + IfP,(si) -fp(“lP)l. Since the sequence si is independent of the force system, by corollary 4.3 we may choose an i, so that (gP(uIP) - gP,(si)I c so/2 for i > i, and an i, so that IfP,(si) -fP(ulP)/ < co/2 for i > i,. A choice of an i, greater than both i, and i, will result in a contradiction. n Proposition 4.5. A force system {fP} on B is consistent if and only if there is a unique collection of bounded Bore1 measures {op.}, p = 1, 2, 3, 1x1< r that represents the force system in the form
fp(n) =
1 1 np.edapa I.l
UE UP,
W3).
p=1.2,3
Proof: It is clear that a force system which is represented by a collection of bounded stress measures as in the previous representation equation is consistent. We now show that consistency is indeed a sufficient condition for the representation. Define the functions utq., E C’(B, 9t3) by their pth components as ucqan,(x) = b4,,xa,p, q = 1,2,3, I rj < r, where 6,, denotes the Kronecker 6. For Irl = 0 we define the set functions pqa on the collection of subbodies of B by L(J P) = fp( ucqajI p ). Clearly, the set functions pqa satisfy the conditions of proposition 3.1 and it follows that they can be extended to Bore1 measures aq2, Irj = 0, on B. Next we define inductively
for 0 < I@/I r. Since the conditions of proposition 3.1 hold for both terms in the previous 0 < 1~1I r as well, and hence, we can extend equation, they hold for the set functions r(lqor, them to the Bore1 measures aqz. Construct the force system {gp} on B by gp(u)
=
1
lzl5r
P’1.2.3
iP
up.a da,=t
u E C’( P, 33).
It is obvious that (gp} is a consistent force system. We have
Consistency conditions for force systems
59
By using cr,,(P) = g,,(P) and the definition of pq3, in the Iast line we obtain gP(u6,_)lP) = ~,,(z+~.)I~).By the linearity of forces it foIlows that the force systems {fP) and {gp) have equal values for any rth order polynomial, by the additivity of the force systems they have equal values for any piecewise rth order polynomial and it follows from proposition 4.4 that they are equal. It remains to show that if {u+} and {gzpa}, p = 1,2,3, Irl I r represent the force system (.Mr then blP. = czP.- Let A be an arbitrary set in CDand let e be a sequence of subbodies converging to A. We have for 1~1= O,j&(u& = al,( pi) = a& 4) for each i and p. Using =0 the continuity of the measures alPa and bzPSwecon~ludethat~~~(~~=~~~~(A)for~~l and every AE@. We recall that if two measures are equal on a field they are equal on the minimal e-field containing it, hence, brPz = ~~~~for Iri = 0. To use an induction process, assume that blPE = b2P=for all [ix1c r0. Again, for an arbitrary AE@, pi + A and 7 with I’il = r0, we have by using the definition of uCP.),
Hence,
and from the induction hypothesis it follows that clPy( pi) = 02,,( pi) for each i and p. Again, the properties of measures imply that dlP., = u2Py. n Acknowledgements-This work, which is based on the second author’s Master’s thesis, was supported by the BathSheva de Rothchild Foundation for the Advancement of Science in Israel and some of it was done during the first author’s visit to Rational Mechanics at Johns Hopkins University.
REFERENCES 1. C. Truesdell, A firsr
in Rutjonuf Co~rin~u~ ~echun~cs, Vol. I. Academic Press, New York (1977). 2. R. Segev, On the definition of forces in continuum mechanics, in ~~~a~ic~~ Systems and ‘microphysics, Conrrol COUW
Theory and Mechanics (Edited by A. Blaquiere and G. Leitmann), pp. 341-357. Academic Press, New York (1984). 3. R. Segev, Forces and the existence of stresses in invariant continuum mechanics. J. Marh. Phys. 27. 163- 170 (1986).
4. 5. 6. 7. 8. 9.
W. Noll and E. Virga, On Edge lnteracrions. To appear. J. Eells, Jr, Singulariries ofSmooth Maps. Gordon and Breach, New York (1967). V. Guillemin and A. Poilack, Diflerential Topology. Prentice-Hall, Englewood Cliffs, New Jersey_ (1974). W. M. Hirsch, Differential Topoibgy. Spring&, B&in (1976). K. P. S. Bhaskara Rao and M. Bhaskara Rao, Theory of Charges.Academic Press, New York (1983). R. B. Ash, Measure Inregrarion and Functional Analysis. Academic Press, New York (1972).