Forces and the existence of stresses in invariant continuum mechanics

Reuven SegevDepartment of Mechanical Engineering, Pearlstone Center for Aeronautical Engineering Studies, Ben GurionUniversity, Beer Sheva, Israel

(Received 12 November 1984; accepted for publication 21 August 1985)

In an invariant formulation of pth-grade continuum mechanics, forces are defined as elements ofthe cotangent bundle of the Banach manifold of C p embeddings of the body in space. It is shown

that forces can be represented by measures which generalize the stresses of continuum mechanics.The mathematical representation procedure makes the restriction of forces to subbodies possible.The local properties of the stress measures are examined. For the case where stresses are given interms of smooth densities, it is shown that the structure offorces agrees with the form offorces oneassumes in the traditional formulation, and the equilibrium differential equations are obtained.

I. INTRODUCTION

It is well known that the laws of continuum mechanics,the mechanics of deformable bodies, cannot be deducedfrom the laws of mechanics of material points and rigid bo-dies. Additional assumptions are introduced and new no-tions such as internal forces, external forces, stresses, and the

equilibrium equation emerge.The geometric framework in which the classical theory

of continuum mechanics is developed is the three-dimen-sional Euclidean space. The following paragraphs review thebasic structure of the theory .

The first basic assumption made in continuum mechan-ics regarding the nature of forces is that the total force actingon a body Bis of the form

./=(bdv+( tda, (I)JB JaB

where b is a continuous vector field, called the body force,defined in the body, and t is a continuous vector field, calledthe surface force, defined on the boundary of the body. Thebasic problem of continuum mechanics is encountered whenwe try to restrict a given force on B to a subbody. Consider-ing a subbody P of the body B, the total force f p acting on itshould also be given in terms of a body force and a surfaceforce as in Eq. ( 1 ). In general, the fields b and t associatedwith the subbody P are different from those given on B. Inparticular, physical experience shows that even if P is dis-joint from the boundary of B, a surface force acts on theboundary of P. This newly emerged surface force is tradi-tionally termed internal force or traction as it may be inter-preted as the force that is applied on P by its complement inthe body. Thus, the values of b and t at a point X E P willdepend in general on the subbody Punder consideration andwe write

b=b(X,P), t=t(X,P).

The next assumption, called Cauchy's postulate, dealswith the dependence of b and Ion P. It states that b does notdepend on p so that b = b (X ), and that the surface force de-pends on p only through the unit vector n perpendicular tothe boundary of p at X, i.e., I = I (X,n). Clearly, this last hy-

pothesis does not provide all the necessary information need-ed in order to determine I.

Assuming that the total force on each subbody of thebody B vanishes, it is possible to prove the following results.

There exists a tensor field u in the body such that

t(X,n) = u(X)(n(X)). (2)

The tensor field u is the stress field, and it has to satisfy thedifferential equation

div u + b = O in B. (3)

If we assume in addition that the total moment on eachsubbody of B vanishes, we find that u is symmetric.

From Eqs. ( I) and (2) it is clear that if u is given, one canassociate a unique body force field and a unique surface forcefield with each subbody. However, the differential equation(3), known as the equlibrium equation, and the boundarycondition (2) cannot determine the stress uniquely for given band t on B. This lack of uniqueness in the determination ofthe stress field means that the force on a body cannot berestricted to subbodies in a unique fashion. In order to deter-mine the stress field, constitutive relations are introduced.The constitutive relations, obtained by physical experi-

ments, relate the stress with the configuration of the bodyand supply all the necessary information so that the stresscan be determined uniquely. Clearly, using Eq. (2), u can bedetermined uniquely if t is given for every subbody p of B.

Modem attempts to axiomatize the theory of forces andstresses can be found in Gurtin and Williams! Gurtin andMartins, 2 and Truesdell.3 The authors postulate a system of

axioms describing the properties of forces in general. In ad-dition, they assume equilibrium and they assume that exter-nal forces are composed of body forces that are absolutelycontinuous with respect to the volume measure, and surfaceforces that are absolutely continuous with respect to the sur-face area of the body. With these assumptions the authorsprove that forces are given in the form of Eq. ( 1) and thatCauchy's postulate holds. Marsden and Hughes4 have gen-eralized the theory to Riemannian manifolds using an invar-iance principle for an assumed form of a balance of energywhere they assume the transformation rules for the variousvariables including b and t.

During the 1960's, in an attempt to formulate theoriesthat would account for interactions that are more complicat-ed than those afforded by the classical theory , the theories ofcouple stresses and the theories of materials of grade p> I

163 J. Math. Phys. 27 (1 ). January 1986 0022-2488/86/010163-08$02.50 @ 1985 American Institute of Physics 163

quence of the choice of a topology on the set of configura-tions.

(g) A generalized form of the equilibrium equation isobtained as a result of the mathematical procedure, and theorigin of the nonuniqueness in the relation between stressesand forces is explained.

(h) A simple constitutive theory is suggested in whichbody self-determinism and continuity imply jet locality.

were developed. A historical account of the subject togetherwith a review of the various approaches can be found inTruesdell and No115 (pp. 389-401). For a variety of applica-tions one can consult Mindlin6 and references cited therein.Unlike Cauchy's theory, the theories of materials of grade pare based on energy principles in which the potential energydensity is assumed to depend on derivatives of order p of thedeformation.

In this paper we propose a theory 'of forces and stressesbased on the principle that forces should be defined as ele-ments of the cotangent bundle T *Q of and appropriate con-figuration manifold Q. Specifically, we show that pth-ordercontinuum mechanics corresponds to the case where theconfiguration space is the set of all p-times differentiableembeddings of the body in space equipped with the C p topol-

ogy. It turns out that in this case forces can be represented bymeasures on the pth jet bundle over the body where therepresenting measures generalize the stresses. For example,it follows that in first-order continuum mechanics and thecase of three-dimensional Euclidean geometry [where thefirst jet bundle can be identified with B XR 3 EBL (R 3, R 3)]

any force can be represented in the form

II. THE BASIC STRUCTURE

Definition 2.1: A body is a compact differentiable mani-fold with comers. A typical body will be denoted by B and itsdimension will be denoted by m.

Definition 2.2: The physical space is a differentiablemanifold S without a boundary .

Definition 2.3: A configuration of class p is a C p embed-

ding of a body B in the physical space for p> [The requirement that a configuration of a body into

space is an embedding is a result of two traditional princi-ples: the principle of impenetrability stating that one portionof the matter never penetrates within another, and the prin-ciple of permanence of matter stating that no region of posi-tive finite volume is deformed into one of zero or infinitevolume (cf. Truesdell and Toupin,9 pp. 234-244).

For a fixed body B and a given p, the configuratio~ spaceQ is the set of all configurations of class p of the body in

space.We recall10-13 that the set CP(B,s) ofCP mappings of B

into Scan be given the structure of a Banach manifold. Forany K E C P(B,S ), the tangent space TC P(B,S )K can be identi-fied with C P(K*1" s ), the Banachable space of C p sections of

the pullback of the tangent bundle 1" s by K. The Banach spacetopology of C P(K*1" s ) is given as follows. Let v:K -+- R n be a

C p mapping defined on a compact set K. We use the notation

IIvIlpmax sup { IDjv(x)I },

j xEKO<j<p.

Clearly, 1111 pis a norm for the space of all such C p mappings.

Now, let B1,...,Br be a covering ofB by compact sub-manifolds of the same dimension as B such that each B i iscontained in the domain of a vector bundle chart 1/1 i of K*'T s .Then for u E CP(K*'Ts) define

lIuJI =max lI~ill,i

i ,...,r,

f(u) =Lu;d,u; +LU~j d,uj; ,

where ,u;, and ,uj; are the components ofa measure over Bvalued in R 3 ffiL(R 3, R 3). The first three components

vanish if a Euclidean symmetry requirement is imposed andthe ,u j; correspond to the stress. If these measures are differ-entiable with respect to the volume measure, their densitiesare the components. of the stress field. In the more generalcase where a connection is specified on the space manifold,any force can be represented in the form

p r kf(u) = k~OJBV u dUk'

where vk is the k th covariantderivative and the { u k J are therepresenting measures.

The resulting structure has the following features.(a) The theory applies in the general geometry of differ-

entiable manifolds.(b) The definition of a force extends the definition given

in the case of finite-dimensional classical mechanics by Ar-nold7 and TulczyjewS to the infinite-dimensional case. Thus,it clarifies the point of departure of continuum mechanicsfrom analytical mechanics.

(c) Some assumptions made in the classical construc-tion, such as the form ( 1) of the forces on bodies, are obtainedmathematically as results of the definition of forces.

(d) The theory links the properties of forces and stresseswith the axiom of impenetrability.

(e) The theory allows stresses which are as irregular asmeasures, while the classical theory deals with continuousstresses only.

(1) The theories of materials of grade p are generalized todifferentiable manifolds. The suggested formulation is freeof any energy consideratir>ns and the relation between thetheory of materials of grade one and materials of a highergrade is clear and simple. The grade of a material is a conse-

where ~; is the local representative of u in the chart f/!;.Again, 1111 is a norm on CP(K*'Ts) and any other norm in-duced by another covering will induce an equivalent topol-ogy on 'C P(K*'T s ).

The tangent space TC P(B,s IK can also be identified withthe vector space of vector fields along K, i.e.,{u E CP(B,TS); 'TsoU =KJ.

In addition, we recall that since p>l, the set ofCP em-beddings is an open subsetl4 of CP(B,S). Hence, Q is a Ban-ach manifold and we have

164 J. Math. Phys., Vol. 27, No.1, January 1986 Reuven Segev 164

TQK ~CP(K*'TS)~ {U E CP(B,TS); 'TSOU = K}.

An element of TQ is a virtual displacement, a term moti-vated by the second interpretation we gave of TQK .

Definition 2.4: Aforce (ofgrade p) is an element of thecotangent bundle T*Q.

Let fE T*QK and u E TQK for some configuration K.The evaluation f(u) is traditionally called the virtual workperformed by the force f on the virtual displacment u .

The basic structure, as defined in this section, has beengiven for the finite-dimensional configuration space by Ar-nold7 and Tulczyjew.8 In the infinite-dimensional case con-sidered here, the specification of the class of admissible con-figurations and the topology chosen will determine thenature of forces. It is our aim to study the consequences ofthese choices and to show that the basic properties of forcesand stresses in continuum mechanics can be obtained natu-rally in the suggested framework.

w. IB.nB. =w. IB. nB. }, , J .J , J

is isomorphic to CO(1T). The isomorphism is given byW f--+ (WJBl'...'wIBr) and its inverse given by(W1,...Wr) f--+ }:jWj, where Wj E CO(1T) is given byWjlBj = t/1jWj andwj = OoutsideBj (for a complete proof see

Palais,l°pp. 10 and 11).Given vector bundle chartsI{/j: 1TIBj --.K XR N, KCR m

(assuming that B is m-dimensional and that the fiber of 1T isN-dimensional) and denoting by t/lj the first component ofI{/j, CO(1TIBj) can be identified with CO(t/lj(Bj))N, the spaceof N-tuples of continuous real valued functions on t/lj(Bj ) foreach i. Thus, CO(1T) is isomorphic with1°

{(Wl'...'Wr) E j~ I CO(t/lj(Bj))N ;

'lI.-1ow.o.,.. = 'lI-1oW .0.1.. onB.nB .}I -1'1'1 1 -1'1'1 I 1

via w ~(W;,...,Wr ), where W; = 'lI;owIB;°f/!;- 1, and the in-verse is given by W = ~;W;, where W;E GO( 11") are given by

w. IB. =.I.. ( 'll.-1ow.o0'.. ) andw. =OoutsideB ,.. I I 'I'" -I 'I', IWe conclude, therefore, that given a partition ofunity, a

vector bundle atlas, and I" E GO( 11") *, there exists a collec-tion {I!:;}, i = 1,...,7, I!:; E GO(f/!;(B;»)N*, such that

,.U(W) = L /!;('lI;owIB;°J/!;-I).

;=1

Identifying CO(J/!;(B;))N* with CO(J/!;(B;))*N and observingthatCO(J/!;(B; ))* is the space of Radon measures on J/!;(Bi), weconclude that each.u; is a collection of N measures on J/!;(B; ).

Let { U a' 'lI a' J/! : J be a vector bundle atlas of 17", and letC~(J/!a(Ua)) denote the Banach space of continuous func-tions with compact support in J/!a(Ua) equipped with theusual topology so that C~(J/!a(Ua))* is the space of Radonmeasures on J/! a ( Ua ). Assume that for each a there is a given/!a E C~(J/!a(Ua))*N, such that for each pair of indices,/!a('lIaoWOJ/!a-l)= /!p('lIpoWOJ/!pl), for each WECO(17")whose support is contained in UanU p. We now define.u E CO(17")* by

III. THE REPRESENTATION OF FORCES BY STRESSESAND THE PRINCIPLE OF VIRTUAL WORK

Given K E Q, the identification of TQK with C P(K*T s )allows us to identify the forces in T*QK with section distri-butions in CP(K*Ts)*. Thus, the problem of restriction offorces from a given body to its subbodies means mathemat-ically that we have to study the restrictions of CP sectiondistributions.

Consider the jet extension mapping

jp: CP(K*Ts) --+ CO(JP(K*Ts)).

We note that jp is linear, injective and if we use naturalcharts on both K*Ts and JP(K*Ts) and norms induced onCP(K*Ts) and CO(JP(K*Ts)) by these charts, we observe thatjp is also norm preserving. It follows that every force inCP(K*Ts)* is of the form 1;(0-) for some o-E CO(JP(K*Ts))*,where

1;: CO(JP(K*Ts))* --+ CP(K*Ts)*

is the adjoint of the jet extension mapping. The elements ofCO(JP(K*Ts))* are called stresses. Hence, if f = 1;(0-), wehave f(u) = 0- (jp(u)) for every virtual displacement u and

we say that the stress 0- represents the force f. This is ageneralization of the principle of virtual work in continuummechanics which states that the virtual work performed bythe force on a virtual displacment is equal to the virtual workperformed by the stress on the derivative of the virtual dis-

placement.

Jl(W) = ~ 1!;;('1';Ol/J;W°t/l;-I), W E CO(1T),

IV. LOCAL PROPERTIES OF STRESSES

By their definition, forces are special types of sectiondistributions or currents (Choquet-Bruhat et al.15 pp. 400-406, DeRham,16 Schwartzll and stresses that belong to asimpler class of distributions (measures) represent them. Inthis section we consider the local properties of stresses.

Let B1,...,Br be compact submanifolds of B of the samedimension as B whose interiors cover B and let t/JI'...'t/Jr be aCoo partition of unity such that supp t/Jj Cinterior Bj. It canbe shown that if 1T is a vector bundle over B, then

where { t/Jj J is a finite partition of unity such thatsuppt/Jj C Ua. It can be shown that p, is independent of thepartition of unity so that any collection of local measuresthat satisfies the transformation rule define an element of

CO(1T)*.Having reviewed the local properties of elements of

CO(1T)*, we extend them to a wider class of sections, the inte-

grablesections.Wesaythatafunction!lj: r/tj(Uj)_RN isintegrable with respect to the collection of measures{ I!:jk J, k = 1.,...,N, if each component is integrable with re-

spect to all the I!:jk' i.e., if

lIu;jIL' = sup-j.k .

I'!ij j d Il!:ik I < 00 .h(u,)

165 J. Math. Phys., Vol. 27. No.1, January 1986 Reuven Segev 165

Thus, in case connections are specified on'TB and 'Ts,forces can be represented by tensor valued measures. Wenote that the case p = I corresponds to classical continuum

mechanics, where U1 is the tensor measure which corre-sponds to the stress tensor. In this case we do not need theconnection on 'T B , as it is not required for the first covariantderivative.

VI. THE RELATION TO PREVIOUS WORKS

In this section we review some ideas suggested in pre-vious works,I8-20 and relate them to the formulation gi~enhere.

It can be shown that for a section u of 17" with support inU;nUj and measures I!:; and I!:j on f/J;(U;)and f/Jj(Uj)' re-spectively, satisfying the compatibility condition givenabove, 'I/ ; °uof/J;- 1 is I!:; integrable if and only if 'I/ j ouof/J j- 1

is I!: j integrable. Thus, we say that a section of 17" is integrablewith respect to ,u E CO(17").ifitslocalrepresentativesareinte-grable with respect to the local representatives of ,u; of ,u.

Let X T be the characteristic function of a subset T of B .If TnU; is I!:; measurable, X TnU, is integrable with respect toeach of the ~;k' k= 1,...,N, and we can restrictI!:; to TnU; by I!:; jTnU; = XThU, I!:;. In case the family { I!:; }

satisfies the compatibility condition on the intersections ofdomains of charts so that it contains local representatives ofsome ,u E CO(17")., the same holds for the collection{ I!:; I TnU; } which will represent ,u I T = X T ,u, the restric-

tion of ,u to T. In particular, if p is a subbody of B, i.e., acompact m-dimensional submanifold of B, ,u can be restrict-ed to P.

Applying the foregoing results to the case where 17" is thevector bundle JP(K.(Ts)), we conclude that a stress is repre-sented lacally by a collection of N Radon measures thattransform according to the rule given above, where N is thedimension of the fiber OfJP(K.(Ts)). Conversely, any suchcollection of measures satisfying the transformation rulerepresents a stress.

We denote the evaluation of the stress measure0" E CO(JP(K.Ts )). on a section w by S B w dO", and for a sub-body P, we denote the evaluation of O"IP on a section u ofJP((KJP).(Ts)) by S pU dO".

v. THE CASE OF A CONNECTION

We now assume that connections are specified both onthe vector bundle Ts: TS-+S and the vector bundle TB:TB -+ B. The connection on Ts induces a connection onK-TB and we recall that given a connection on both TB andK-T B we have an induced connection on the vector bundle of

p-multilinear mappings LP(TB,K-Ts): LP(TB,K-TS) -+B,such that we have covariant derivatives

ViueCP-i(Li(TB'K-Ts)), O<i<p,for u e C P(K-T s ) (see Eliasson II for details).

Consider the mapping

In Refs. 19 and 20 it was suggested that vector bundlesover B and S can serve as mathematical models for the localproperties of both body and space so that the vector spaceattached to each point represents mathematically the neigh-borhood of this point. A local configuration was defined as avector bundle morphism between the two vector bundles.The local configuration space, local virtual displacements,and local forces were defined for this new model, termed thelocal model, in analogy with the previous set of definitionswhich will be referred to henceforth as the global model. Thelocal configuration space is the Banach manifold of all localconfigurations, local virtual displacements are elements ofthe tangent bundle, and local forces are elements of the co-tangent bundle of the local configuration space. It wasshown that local forces generalize the stresses of continuummechanics, and the principle of virtual work was obtained asa result of a requirement for compatibility between these twomodels. The particular case where T B and T s represented thebody and space in the local model was studied. In this casethe local configuration space is the collection of vector bun-dle morphisms TB -Ts which can be identified with thecollection of sections of the jet bundle 1Tl: J I(B, S) -B.

Using the language of jet bundles and the properties ofmanifolds of sections of jet bundles, 10 the following obvious

generalization can be made. A local configuration of order pis a continuous section of 1TP: J P(B, S) -B. The local config-uration space is the manifold of sections C °(1TP). A local vir-tual displacement is an element of the tangent bundleTCO(1TP), and a local force 0- is an element of the cotangentbundle T*CO(1TP).

Since both the global model and the local model repre-sent the same physical phenomenon they are related by com-patibility conditions in the following way.

Consider the jet extension mapping

jp: CP(~).- CO(1TP).

We say that a local configuration X E CO(4TP) is compatiblewith a global configuration KE CP(~) if X = j p(K). A local

virtual displacement w E TCO(1TP) is compatible with a globalvirtual displacement u E TCP(~) if w = T(jp)(u). We say

that a global force f E T *C P(~) is compatible with the localforce o-E T*CO(1TP)limagejp if f= T*Up)(0-). These defini-tions can be summarized by saying that the two models arerelated by the jet functor J p.

The relation between the formulation given in this sec-tion and the rest of this paper is established in the following

proposition.

given by

u ~ (u,Vu,...,VPu).

Again, this is a linear continuous injection with a closedimage, and since

Co( i~OL i(TB'K*TS)). = ( i~O CO(L i(TB'K*Ts)) ).

p= EB CO(Li(TB'K*Ts))*,

i~O

we have a representation of forces by collections of tensormeasures (0"°'O"I'...'O"p), O"i E CO(L i(TB'K*Ts))*, in the form

f(u)= i i ViudO"i.

i=O B

166 J. Math. Phys., Vol. 27, No.1, January 1986 Reuven Segev 166

Propqsition 6.1: (i) For any global configuration,K, T*CO(1TP)jp!K)' the space of local forces at the local config-uration compatible wjth K, can be identified with the space ofstresses representing forces at K.

(ii) A global force f is compatible with a local force u ifand only if the stress that Can be identified with u by (i) repre-sents f .

Proof The proof of the proposition becomes obviousonce the following results of Palaislo on sections ofjet bun-dles are used.

(a) Given K E CP(~), there is a natural isomorphism

TCO(1TP)jp!K) ~CO(JP(K*'Ts)).

(b) For KE C P(1TO), the tangent to the jet extension map-

pIng

We denote by ui the components of the local representativeof u E CP((KIP)*1"s) and for the multi-indexa = (al,...,am ), we recall that the local representative of

jp(u) is{Daui}, lal =al + ...+am<p. Let .Uia and Via bethe measures on ~-the image of p under the chart-thatrepresent 0"1 and 0"2' respectively. By the representation offorces by stresses we have

jp(U)= i f Dauid.uia=i i DaUidvia1! la~O 1! lal =0

for everyu E CP((KIP)*1"s) and every subbody P.In particular, for j E { 1,...,n } , where n is the dimension

of S, let u satisfy ui = t5ij. By the equation above we have

.Uja(~) = Vja(~)' la] = 0,

for every subbody p of A. Since the two measures agree onevery subbody we have .Uja = V ja' la I = 0.

Now, given j, /3, with 1/31 = 1, let u satisfy ui = t5ijxP,

where (Xk) are the local coordinates in the given chart. Wehave

TU p)K: TCP(1TO)K -TCO(1TP)jp(K)

is given by UI-+ j p{U), where U E CP(K.TS)~TCP(1fO)K and

the identification of (i) is used.

The assertions follow immediately.

as the higher-order derivatives vanish. Since Jlia = Via forlal = 0, and since D aui#O only for i =j, a = .8, we haveJl jp(!) = v jp(!), for every subbody P and arbitrary j, .8,1.8 I = I. We conclude that that Jlia = Via for all i and a,with lal == I.

We can continue the process evaluating the virtual workperformed on the virtual displacements u such thatui = t5ijxa = t5ii(xl)a,(X2)a,...(Xn)an, with lal = 2,3,...,p to ob-tainJlia = Via for all a with lal<p.

Proposition 7:2: Let a force system [ f p I which is consis-tent with a stress 0", be given. Then, if A is a subbody con-tained in the domain of a chart on K*'T s with coordinates(Xk,uj), the local representatives Jlia of 0" are given by thefollowing inductive process.

Let (t5ijxa)' be the section of CP((KIA )*'Ts) wh9se localrepresentatives satisfy ui = t5ijxa for given j and a. Then,

0 < lal < p,

where {3 <a means that {3; <;;a; and 1.8 1< lal.

Proof: By hypothesis

f p(b';jxQ)' = i ')" DP(b';jxQ) d.u;p.!' I pk p

VII. FORCE SYSTEMS

The representation of forces by stress measures providesan answer to the basic problem of restriction of forces tosubbodies. Given a stress measure u, a unique force fp isinduced on every subbody P by

fp(u) = L jp(u) du, u E CP((KIP)*rs),

or in other words, the force on P is represented by the restric-tion of the stress measure to P.

We will use the termforce system for a set funct~on as-signing a force f p E C P((KIP )*r s 1* to every subbody P of B.We will say that a force system is consistent if there exists astress representation u such that the force given on any sub-body P is represented by the restriction of u to P.

Since the jet extension map is not surjective we cannotexpect that the representation of forces by stresses will beunique. This feature is well known in continuum mechanicsand it is referred to as static indeterminacy. It is the staticindeterminacy which forces the use of material properties orconstitutive relations in order to be able to restrict forces tosubbodies. However, as the next proposition shows, a forcesystem can be consistent with at most one stress, i.e., if weknow the force acting on each subbody we can determine thestress uniquely. This statement is a generalization of theprinciple in continuum mechanics according to which thestress at a point can be determined uniquely if the tractionacross every surface is given. In the classicatcase however,the result is stated for the case p = 1 only, and the stresstensor measure is given in terms of a tensor field whose valueat a given point we want to determine.

Proposition 7: 1: If a force system is consistent with thestresses u1 and u2 then, u1 = u2.

Proof In order to show that u1 = u2, it suffices to showthat their local representatives in any given chart are equal.Let A be a subbody contained in the domain ofa chart in B.For any subbody P of A , let f p be the force acting on P in thegiven force system which is consistent with both u land u2.

Since,D{3xa.= [a!/(a- /3)!]xa- {3

for /3 < a, and D{3xa = 0, for a < /3, we have

167 J. Math. Phys., Vol. 27, No.1, January 1986 Reuven Segev 167

Jlja(~) = jp(t5i1', lal = 0;

f p(b'ij~)' = b'ija! 'Uia (~J

~ a'i+""' .xa-Pd

p<a(a-.B)! !' ftjp.

,!The proposition suggests a procedure which enables oneto determine whether a given force system is consistent withany stress, and to obtain the local representatives of thisstress if it exists. Given any vector bundle atlas on K*1" s , onehas to evaluate fp (8ij)' for all subbodies P contained in thedomain of charts. Then, if for every chart, the set functionPI-+ fp (8ij)' for all subbodiescontained in the domain of thechart can be extended, to a measure on the domain of thechart, we can identify 'Uja (~), lal = 0, with fp(8ij)'forthevarious charts. We proceed by evaluatingfp (8ijxa)', lal = 1, and we use the relations of the last pro-

position and the previously obtained 'Uja (~), la] = 0, to ob-tain 'Uja (~), lal = 1. We check that 'Uja, lal = 1, can beextended to measures and we continue the process forfp(8ijxa),!lal>l, until we reach apth step such that'Uja = ° for alllal > p. Next, we have to check that the 'Uja

satisfy the transformation. rules on the intersections ofcharts. If the compatibility conditions are satisfied, we con-clude that the { ,u ja } , lal, p, obtained are the local repre-sentatives of a stress which is consistent with the given force

system.

Given a force fE T.Q and a constitutive relation 'liBsuch that f = T.V p)('liB (K)), the measure 'liB(K) induces a

unique force on any subbody and the problem of the restric-tion of the force is immediately solved. The general problemof continuum mechanics can be formulated now as follows.Given a loading F B of B and a constitutive relation 'li B , de-termine the configuration K such that 'li B (K) represents F B (K),i.e., FB(K) = T.(j p)('liB (K)).

It should be noted that in the general geometric frame-work we use, any "force" is a "follower force" in the sensethat a force has meaning only when it is associated with a

configuration. Thus, rather than looking for an equilibriumconfiguration under a given force, a meaningless problem,one has to find the equilibrium configuration for a given

loading.We can examine now the way in which the principle of

local determinism restricts the constitutive relations. Let Pbe subbody of B and let 'lip, 'li B be constitutive relations on Pand B, respectively. Since for any K, the principle of bodydeterminism implies that the force on P and any of its subbo-dies is determined by KIP, we have 'lip(KIP) = 'liB(K)IP.Thus, we will omit the suffix and we will write 'I' when noconfusion can arise. We also note that this principle impliesthat it is sufficient to examine the case where B is in Rm .Moreover, assuming that the constitutive relations are con-tinuous,we can show that the constitutive relations are p-jetlocal in the following sense.

Proposition 8.3: Let 'li be a continuous constitutive rela-tion and let x E B. Then, for any> 0, there exists a !5>0such that if asubbody P is contained in a ball of radius !5 (inthe Rm Euclidean metric) centered at X, then,

1I'lI(K)IP- 'li(j p(K)(x))IPIl <,

where j p (K)(X) denotes the pth-order Taylor expansion of Kabout x.

Proof" Given any > 0, the continuity of'll implies thatthere exists a !51 >0 such that if IlK -j p(K)(x)llcp <!51, then

1I'1'(K) -'li(j p (K)(X))II <. By Taylor's theorem, given !51 > 0,there is a !5 > ° such that IIKIP- j p(K)(x)IP Ilcp < !51 if P iscontained in a ball of radius !5 about x. Thus, by locality

1I'li(K)IP- 'li(j p(K)(x))IP II= 1I'lI(KIP) -'li(jp(K)(X)IP)II <.

Since there is no meaning to the value of a stress at apoint, the classical locality assumption that the value of thestress at a point depends on the value of the deformationgradient at that Pbint cannot be obtained or even conjec-tured. If stresses were continuous sections and if the space ofstresses were given the C 1 topology, then the continuity ar-

gument of the previous proposition together with the twoprinciples would imply that the value of the stress at a pointdepends only on the value of the pth jet at that point.

VIII. CONSTITUTIVE RELATIONS

As we mentioned in the introduction, the problem of therestriction of forces to subbodies, which was transformedinto a problem of nonunique relation between forces andstresses, leads to the specification of the material propertiesas additional information. The material properties are intro-duced via the so called constitutive relations, which in classi-cal continuum mechanics, associate the stress at. a point withthe deformation gradient at that point. In this section wesuggest a way by which constitutive theory may be incorpo-rated in the structure that we developed.

We assume that the following two principles hold incontinuum mechanics.

Axiom 8.1. (the principle ofbody self-determinism): Theforce acting on a body is determined by the configuration ofthe body, i.e., for any body B there is a section

FB: Q~ T*Qwhich we call the loading of B.Axiom 8.2 (the principle of consistency): Given any con-

figuration K of the bodyB, the force system {Fp(KIP); Pis asubbody of B } , is consistent.

Thus, by Proposition 7.1, the principle of consistencyimplies that any configuration of B determines a uniquestress representation in T*CO(1TP). The mappingII' B : Q ~ T*CO(1TP) that associates stresses with the variousconfigurations is called a constitutive relation for B.

IX. STRESSES GIVEN BY SMOOTH DENSITIES

In this section, in order to complete the analogy withclassical continuum mechanics, we obtain the representa-tion of forces by surface forces and body forces, the equilibri-um differential equations, and the boundary conditions.

168 J. Math. Phys., Vol. 27. No.1, January 1986Reuven Segev 168

= a!.u ja(l!)

Since the procedure involves integration by parts, we assumethat the stresses are given in terms of smooth densities. Wealso assume that a connection is specified on S and that B andS have the same dimension. It follows that the connection onS induces a connection on B. Keeping K fixed during thediscussion, we identify the body with its image under K.

We saw that if connections are given on B and S, anyforce can be represented in the form

f(u) = L so(u) + Sl(VU),

f(u) = f r VkU dO"k'

k=OJB

where O" k is the k th-order stress measure. Consider the vec-m

tor bundle L (L k(1"B'K*1"s), A T*B ). Assuming that B is

orientable, a smooth section Sk of this vector bundle inducesa k th-order stress measure O"k by

i Vku dO"k = iSkOVkU,

where Sk oVku is the m-form whose value at x E B isSk(X)(Vku(x)). In particular, ifa volume element 8is given onB, the collection of sections {.:Jk I, .:JkE C~(L k(1"B'K*1"s)*)will induce a stress representation

where ,jk (VkU) is the real function whose value at x E B is

,jk(X)(Vku(x)). More geometric structure is available in thecase where both the connection and the volume element arederived from a Riemmanianmetric.

In order to perform the integration by parts in the gen-eral geometric framework, we generalize the definition of thedivergence of a tensor field as follows. We have the isomor-

phism

m

L(Lk(TB'K*TS)' " T*B)

mwhere s;EC~(L(L;(Tb'K*TS)' I\ T*B)). Using the defini-

tion of the divergence and Stokes' theorem, one can showthat f can be represented by two sections

m m-1bEC~(L(Tb' I\ T*B))andtEC~(L(i*Tb' I\ T*aB))(i

is the embedding aB -+B ) in the form

f(u)= r b(u)+ r t(u),Jb Jab

where b and t satisfy div SI + b = So and t = i*oCO(Si). (Weuse i* for both the pullback of differential forms and thepullback of vector bundles.)

In the case of Riemannian geometry we obtain for the

three-dimensional case the usual result, i.e., if

f(u) = L (SOjuj + ~j u{;) dv

(the vertical bar denotes covariant derivative), we have

f(u) = i b-juj dv + r l'juj da,b Jab

where J~jl; + b- j = JOj' l'j ,= n; J~j' and n is the unit nor-mal to the boundary .

Remark: The term Jo vanishes and the term Ji can beshown to be symmetric in the Euclidean geometry if we re-quire that the force is invariant with respect to the Euclideangroup (cf. Refs. 4, 21, and 22).

For the case p = 2, the case of second-grade continuummechanics, we assume that the stresses are given in terms ofthe densities so, Si' ands2 such that

.f(u) = L (so(u) + Si(VU) + siV2u)). ,

It can be shown that f can be represented in the form

f(u) = r b (u) + r (t (u) + t '(Vu)),Jb Jab

where b, t, and t ' are in

m m-iC~(L(Tb' I\ T*B)), C~(L(i*Tb' I\ T*aB.)),

m

~ I\ T*B~TB ~L k-I(TB,K*TS)*

and we define

mco: L(Lk(TB,K*TS).1\ r*B) and

m-1

c W(L (L(i*1"B'i*(K*1"s)), /\ T*aB )),m-l

-L(Lk-l('TB'K*'TS)' 1\ T*B),

respectively, and they satisfyb = div2 S2 -div s J + so,t = j*oco(sJ) -j*oco(div S2)' .

t' = j*oCO(S2).

A further integration by parts of the term involving t ' is pos-

sible only if we have additional geometric structure. Again,for the three-dimensional Riemannian geometry, the classi-cal results (see, e.g., Refs. 23 and 24) can be obtained.

to be the mapping induced by the contraction of the firsttwo factors in the tensor product above. Then, for

mskeCOO(L(Lk(1"B'K*1"s), A T*B)),wedefinethedivergence

mdivskeCOO(L(Lk~I(1"B'K*1"s), A T*B))by

, divSk(Vk-lu)=d(co(sk(Vk-1U)))-Sk(VkU).

Using local expressions it can be shown that the diver-gence is well defined and that it agrees with the usual defini-tion in the case of a Riemannian manifold.

For the case p= I, let the force f be represented bysmooth densities in the form

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