On spatial and material covariant balance laws in elasticity - CiteSeerX

JOURNAL OF MATHEMATICAL PHYSICS 47, 042903 共2006兲

On spatial and material covariant balance laws in elasticity Arash Yavaria兲 School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332

Jerrold E. Marsden and Michael Ortiz Division of Engineering and Applied Science, California Institute of Technology, Pasadena, California 91125 共Received 6 January 2006; accepted 23 February 2006; published online 28 April 2006兲

This paper presents some developments related to the idea of covariance in elasticity. The geometric point of view in continuum mechanics is briefly reviewed. Building on this, regarding the reference configuration and the ambient space as Riemannian manifolds with their own metrics, a Lagrangian field theory of elastic bodies with evolving reference configurations is developed. It is shown that even in this general setting, the Euler-Lagrange equations resulting from horizontal 共referential兲 variations are equivalent to those resulting from vertical 共spatial兲 variations. The classical Green-Naghdi-Rivilin theorem is revisited and a material version of it is discussed. It is shown that energy balance, in general, cannot be invariant under isometries of the reference configuration, which in this case is identified with a subset of R3. Transformation properties of balance of energy under rigid translations and rotations of the reference configuration is obtained. The spatial covariant theory of elasticity is also revisited. The transformation of balance of energy under an arbitrary diffeomorphism of the reference configuration is obtained and it is shown that some nonstandard terms appear in the transformed balance of energy. Then conditions under which energy balance is materially covariant are obtained. It is seen that material covariance of energy balance is equivalent to conservation of mass, isotropy, material Doyle-Ericksen formula and an extra condition that we call configurational inviscidity. In the last part of the paper, the connection between Noether’s theorem and covariance is investigated. It is shown that the DoyleEricksen formula can be obtained as a consequence of spatial covariance of Lagrangian density. Similarly, it is shown that the material Doyle-Ericksen formula can be obtained from material covariance of Lagrangian density. © 2006 American Institute of Physics. 关DOI: 10.1063/1.2190827兴

I. INTRODUCTION

Invariance plays an important role in mechanics and in physics. In any continuum theory one has some conservation laws; i.e., quantities that are constant in time, such as mass and energy or balance laws, such as balance of linear and angular momentum. One way of building a continuum theory is to postulate these conservation or balance laws. On the other hand, as we shall recall later, conservation laws and even balance laws can be obtained as a result of postulating invariance of a quantity such as energy or Lagrangian density, under some group of transformations. Traditionally, continuum mechanics is developed using Euclidean space as the ambient space. This has been motivated by the engineering applications of continuum mechanics and the general tendency of the engineering community to work with the simplest possible spaces. This is of

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course useful and the implicit simplifying assumptions of continuum mechanics have made it applicable to many problems of practical importance. However, being restricted to the misleading and rigid structure of Euclidean space, one should expect to lose important geometric information. For example, for many years there were debates on different stress rates and whether one stress rate is “more objective” than the other one. Putting continuum mechanics in the right geometric setting, one can clearly see that different stress rates in the literature are simply different representations of the same Lie derivative.28 Another basic example of the lack of geometry in the traditional formulation of continuum mechanics is the dependence of the well-known balance of linear and angular momenta on the linear structure of Euclidean space. These laws are written in terms of integrals of some vector fields. Of course, integrating a vector field has no intrinsic meaning and is dependent on a linear structure or a specific coordinate choice. One can argue that a geometric point of view has proven useful in, for example, building systematic numerical schemes as well as in bridging length and time scales. For example, geometry has proven useful in Refs. 25, 6, and 3, although much remains to be done in the future. Following Einstein’s idea that physical laws should not depend on any particular choice of coordinate representation of ambient spaces, Marsden and Hughes28 developed a covariant theory of elasticity building on ideas originated from the work of Naghdi, Green and Rivilin.19 This work starts from balance of energy, which makes sense intrinsically as it is written in terms of integrals of scalar fields 共or more precisely 3-forms兲. Then they postulate that balance of energy is invariant under arbitrary diffeomorphisms of the ambient space. They observe that this invariance assumption gives all the usual balance laws plus the Doyle-Ericksen formula that relates the stress and the metric tensor. Our motivation for studying spatial and material covariant balance laws was to gain a better understanding of the geometry of configurational forces, which are forces that act in the reference configuration. One may ask the following question. What are the consequences of postulating that balance of energy is materially covariant? In the process of answering this question we discovered that such invariance cannot hold in general and this led us to obtain formulas for the way in which balance of energy transforms under material diffeomorphisms. In this paper we also study the connection between spatial and material covariance with Noether’s theorem. It will be shown that spatial and material covariance of a Lagrangian density lead to the spatial and material forms of the Doyle-Ericksen formula, respectively. As was mentioned, one of our motivations for this study was to initiate a geometric study of configurational forces. These forces and their balance laws are important in formulating the evolution of defects in solids in the setting of continuum mechanics. Driving 共configurational, material and so forth兲 forces in continuum mechanics were introduced by Eshelby,13–15 and many researchers have studied them from different points of view. We mention the work of Knowles,22 Abeyaratneh and Knowles1,2 on driving force on a phase interface, Gurtin’s work20,21 on configurational forces by postulating new balance laws, the work of Maugin31,32 and Maugin and Trimarco33 on pull-back of balance of standard linear momentum to the reference configuration, etc. However, even after more than five decades after Eshelby’s original work there does not seem to be a consensus on the nature of configurational forces and their exact role in continuum mechanics and there are still some controversies in the literature. We believe that the geometric ideas in this paper may be helpful in this direction. This paper is organized as follows. The geometry of continuum mechanics is reviewed in Sec. II. The Lagrangian field theory of elastic bodies with evolving reference configurations is presented in Sec. III, where deformed bodies and their reference configurations are treated as Riemannian manifolds. Using this setting, the classical Green-Naghdi-Rivilin theorem and a new material version of it are discussed in Sec. IV. Spatial covariant energy balance is revisited in Sec. V. In Sec. VI we obtain the transformation 共push-forward兲 of energy balance under an arbitrary material diffeomorphism. Then, we investigate the consequences of material covariance of energy balance. Section VII studies the connection between covariance and Noether’s theorem. It is


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Covariant balance laws in elasticity

shown that spatial and material covariance of a Lagrangian density result in spatial and material versions of the Doyle-Ericksen formula, respectively. Conclusions and future directions are given in Sec. VIII. II. GEOMETRY OF CONTINUUM MECHANICS

This section recalls some notation from the geometric approach to continuum mechanics that will be needed. It is assumed that the reader is familiar with the basic ideas; refer to, for example, Marsden and Hughes28 for details. See also Refs. 30 and 29. If M is a smooth n-manifold, the tangent space to M at a point p 苸 M is denoted T pM, while the whole tangent bundle is denoted TM. We denote by B a reference manifold for our body and by S the space in which the body moves. We assume that B and S are Riemannian manifolds with metrics denoted by G and g, respectively. Local coordinates on B are denoted by XI and those on S by xi. The material body B is a subset of the material manifold, i.e., B 傺 B. A deformation of the body is, for purposes of this paper, a C1 embedding ␸ : B → S. The tangent map of ␸ is denoted F = T␸ : TB → TS; in the literature it is often called the deformation gradient. In local charts on B and S, the tangent map of ␸ is given by the Jacobian matrix of partial derivatives of the components of ␸, which we write as F = T␸:TB → TS,

T␸共X,V兲 = 共␸共X兲,D␸共X兲 · V兲.

共2.1兲

If F : B → R is a C1 scalar function, X 苸 B and VX 苸 TXB, then VX关F兴 denotes the derivative of F at X in the direction of VX, i.e., VX关F兴 = DF共X兲 · V. In local coordinates 兵XI其 on B, VX关F兴 =

⳵F I V. ⳵ XI

共2.2兲

For f : S → R, the pull-back of f by ␸ is defined by

␸* f = f ⴰ ␸ .

共2.3兲

If F : B → R, the push-forward of F by ␸ is defined by

␸*F = F ⴰ ␸−1 .

共2.4兲

If Y is a vector field on B, then ␸*Y = T␸ ⴰ Y ⴰ ␸−1, or using the F notation, ␸*Y = F ⴰ Y ⴰ ␸−1 is a vector field on ␸共B兲 called the push-forward of Y by ␸. Similarly, if y is a vector field on ␸共B兲傺 S, then ␸*y = T共␸−1兲 ⴰ y ⴰ ␸ is a vector field on B and is called the pull-back of y by ␸. The cotangent bundle of a manifold M is denoted T*M and the fiber at a point p 苸 M 共the vector space of one-forms at p兲 is denoted by T*pM. If ␤ is a one form on S 共that is, a section of the cotangent bundle T*S兲, then the one-form on B defined as 共␸*␤兲X · VX = ␤␸共X兲 · 共T␸ · VX兲 = ␤␸共X兲 · 共F · VX兲

共2.5兲

for X 苸 B and VX 苸 TXB, is called the pull-back of ␤ by ␸. Likewise, the push-forward of a one-form ␣ on B is the one form on ␸共B兲 defined by ␸*␣ = 共␸−1兲*␣. We can associate a vector field ␤ to a one-form ␤ on a Riemannian manifold M through the equation 具␤x,vx典 = 具具␤x ,vx典典x ,

共2.6兲

where 具,典 denotes the natural pairing between the one-form ␤x 苸 T*x M and the vector vx 苸 TxM and where 具具␤x , vx典典x denotes the inner product between ␤x 苸 TxM and vx 苸 TxM. In coordinates, the components of ␤ are given by ␤i = gij␤i.


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Traditionally force is thought of as a vector field in the deformed configuration. For example, body force B per unit undeformed mass is a vector field on S and its associated one-form can be defined as 具␤x, ␦w典 = 具具B, ␦w典典x

共2.7兲

for all ␦w 苸 TxS. The pull-back of ␤ is defined as 具共␸*␤兲X, ␦W典X = 具␤x,F␦W典X = 具具B,F␦WX典典X = 具具FTB, ␦WX典典X .

共2.8兲

Therefore FTB is the vector field associated with the pull-back of the one-form associated with B. p A type -tensor at X 苸 B is a multilinear map, q

共兲

共2.9兲 T is said to be contravariant of order p and covariant of order q. In a local coordinate chart, T共␣1, . . . , ␣ p,V1, . . . ,Vq兲 = Ti1¯ip j1¯jq␣i1 ¯ ␣ip V1j1 ¯ Vqjq , 1

where ␣k 苸 TX* B and Vk 苸 TXB. Suppose ␸ : B → S is a regular map and T is a tensor of type denoted ␸*T and is a

共兲

p -tensor on ␸共B兲 defined by q

p

共2.10兲

共兲

p . Push-forward of T by ␸ is q

共␸*T兲共x兲共␣1, . . . , ␣ p,v1, . . . ,vq兲 = T共X兲共␸*␣1, . . . , ␸*␣ p, ␸*v1, . . . , ␸*vq兲,

共2.11兲

where ␣k 苸 T*x S , vk 苸 TxS , X = ␸−1共x兲 , ␸*共␣k兲 · vl = ␣k · 共T␸ · vl兲 and ␸*共vl兲 = T共␸−1兲vl. Similarly, pullback of a tensor t defined on ␸共B兲 is given by ␸*t = 共␸−1兲*t. In the setting of continuum mechanics push-forward and pull-back of tensors will have the following forms: 共␸*T兲i1¯ip j1¯jq共x兲 = Fi1I1共X兲 ¯ FipIp共X兲TI1¯IpJ1¯Jq共F−1兲J1 j1共x兲 ¯ 共F−1兲Jq jq共x兲, 共␸*t兲I1¯IpJ1¯Jq共X兲 = 共F−1兲I1i1共x兲 ¯ 共F−1兲Ipip共x兲ti1¯ip j1¯jqF j1J1共X兲 ¯ F jqJq共X兲. A two-point tensor T of type

共兲

q q⬘ at X 苸 B over a map ␸ : B → S is a multilinear map, p p⬘

共2.12兲 where x = ␸共X兲. Let w : U → TS be a vector field, where U 傺 S is open. A curve c : I → S, where I is an open interval, is an integral curve of w if dc 共r兲 = w共c共r兲兲 dt

" r 苸 I.

共2.13兲

If w depends on time variable explicitly, i.e., w : U ⫻ 共−␧ , ␧兲 → TS, an integral curve is defined by dc = w共c共t兲,t兲. dt

共2.14兲


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Let w : S ⫻ I → TS be a vector field. The collection of maps Ft,s such that for each s and x, t 哫 Ft,s共x兲 is an integral curve of w and Fs,s共x兲 = x is called the flow of w. Let w be a C1 vector field on S, Ft,s its flow, and t a C1 tensor field on S. The Lie derivative of t with respect to w is defined by

冏

d * 共F t兲 dt t,s

冏

d * 共F t兲 dt t,s

L wt =

冏

.

共2.15兲

,

共2.16兲

t=s

If we hold t fixed in t then we denote £ wt =

冏

t=s

which is called the autonomous Lie derivative. Hence L wt =

⳵ t + Lwt. ⳵t

共2.17兲

Let v be a vector field on S and ␸ : B → S a regular and orientation preserving C1 map. The Piola transform of v is V = J␸*v,

共2.18兲

where J is the Jacobian of ␸. If Y is the Piola transform of y, then the Piola identity holds, Div Y = J共div y兲 ⴰ ␸ .

共兲

共2.19兲

0 -tensor. The space of k-forms on M is k k denoted ⍀ 共M兲. If ␸ : M → N is a regular and orientation preserving C1 map and ␣ 苸 ⍀k共␸共M兲兲, then A k-form on a manifold M is a skew-symmetric

冕

␸共M兲

␣=

冕

␸ *␣ .

共2.20兲

M

Geometric continuum mechanics: We next review a few of the basic notions of continuum mechanics from the geometric point of view. A body B is a submanifold of a Riemannian manifold B and a configuration of B is a mapping ␸ : B → S, where S is another Riemannian manifold. The set of all configurations of B is denoted C. A motion is a curve c : R → C ; t 哫 ␸t in C. For a fixed t, ␸t共X兲 = ␸共X , t兲 and for a fixed X, ␸X共t兲 = ␸共X , t兲, where X is position of material points in the undeformed configuration B. The material velocity is the map Vt : B → R3 given by Vt共X兲 = V共X,t兲 =

⳵␸共X,t兲 d = ␸X共t兲. ⳵t dt

共2.21兲

Similarly, the material acceleration is defined by At共X兲 = A共X,t兲 =

⳵ V共X,t兲 d = VX共t兲. ⳵t dt

共2.22兲

In components Aa =

⳵ Va a b c + ␥bc VV , ⳵t

共2.23兲

a is the Christoffel symbol of the local coordinate chart 兵xa其. where ␥bc


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Here it is assumed that ␸t is invertible and regular. The spatial velocity of a regular motion ␸t is defined as vt:␸t共B兲 → R3,

vt = Vt ⴰ ␸−1 t ,

共2.24兲

and the spatial acceleration at is defined as a = v˙ =

⳵v + ⵜvv. ⳵t

共2.25兲

In components aa =

⳵ va ⳵ va b a b c + v + ␥bc vv . ⳵ t ⳵ xb

共2.26兲

Let ␸ : B → S be a C1 configuration of B in S, where B and S are manifolds. Recall that the deformation gradient is denoted F = T␸. Thus, at each point X 苸 B, it is a linear map F共X兲:TXB → T␸共X兲S.

共2.27兲

If 兵xi其 and 兵XI其 are local coordinate charts on S and B, respectively, the components of F are FiJ共X兲 =

⳵␸i 共X兲. ⳵ XJ

共2.28兲

The deformation gradient may be viewed as a two-point tensor, F共X兲:T*x S ⫻ TXB → R;

共␣,V兲哫具␣,TX␸ · V典.

共2.29兲

Suppose B and S are Riemannian manifolds with inner products 具具,典典X and 具具,典典x based at X 苸 B and x 苸 S, respectively. Recall that the transpose of F is defined by FT:TxS → TXB,

具具FV,v典典x = 具具V,FTv典典X

共2.30兲

for all V 苸 TXB , v 苸 TxS. In components, 共FT共X兲兲Ji = gij共x兲F jK共X兲GJK共X兲,

共2.31兲

where g and G are metric tensors on S and B, respectively. On the other hand, the dual of F, a metric independent notion, is defined by F * 共x兲:T*x S → TX* B;

具F*共x兲 · ␣,W典 = 具␣,F共X兲W典

共2.32兲

for all ␣ 苸 T*x S , W 苸 TXB. Considering bases ea and EA for S and B, respectively, one can define the corresponding dual bases ea and EA. The matrix representation of F* with respect to the dual bases is the transpose of FaA. F and F* have the following local representations: F = F jK

⳵ ⳵xj

丢

dXK,

⳵ . ⳵xj

共2.33兲

C共X兲 = F共X兲TF共X兲.

共2.34兲

F* = F jK dXK 丢

The right Cauchy-Green deformation tensor is defined by C共X兲:TXB → TXB, In components,


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CIJ = 共FT兲IkFkJ .

共2.35兲

It is straightforward to show that C = ␸*共g兲,

i . e . ,CIJ = 共gij ⴰ ␸兲FiIF jJ .

共2.36兲

From now on, by C we mean the tensor with components CIJ. The Finger tensor is defined as b = ␸t*G, where G is the metric of the reference configuration. To make ideas more concrete, a comment is in order. In the geometric treatment of continuum mechanics one assumes that the material body is a Riemannian manifold 共B , G兲. Here B is an embedding of the material body, i.e., material points are identified with their positions in the reference configuration. A deformation of the material body is represented by a mapping ␸ : B → S, where 共S , g兲 is the ambient space, which is another Riemannian manifold. If ␸ = Id, the reference configuration is a trivial embedding of the material body in the ambient space. Physically, in the deformation process the relative distance of material points change in general. In other words, in terms of material points X , X + dX and their positions in the deformed configuration x , x + dx we have dx · dx = C dX · dX ⫽ dX · dX.

共2.37兲

g ⫽ ␸t*G.

共2.38兲

This means that in general

The following identities will be used frequently in this paper.

⳵ gab d d = gad␥bc + gbd␥ac , ⳵ xc

共2.39兲

⳵ GAB D D = GAD⌫BC + GBD⌫AC , ⳵ XC

共2.40兲

d D and ⌫BC are the Christoffel symbols associated to the metric tensors g and G, respecwhere ␥bc tively. The covariant derivative of two-point tensors will also be used frequently in this paper. The following two examples would be useful to clarify the idea. For definition for an arbitrary twopoint tensor the reader may refer to Marsden and Hughes,28

PaA兩B =

⳵ PaA A a + PaC⌫CB + PbAFcA␥bc , ⳵ XB

共2.41兲

QaA兩B =

⳵ Q aA A b + QaC⌫CB − QbAFcA␥ca . ⳵ XB

共2.42兲

Let ␸t : B → S be a regular motion of B in S and P 傺 B a k-dimensional submanifold. The transport theorem says that for any k-form ␣ on S, d dt

冕

␸t共P兲

␣=

冕

␸t共P兲

L v␣ ,

共2.43兲

where v is the spatial velocity of the motion. In a special case when ␣ = f dv and P = U is an open set, d dt

冕

␸t共P兲

f dv =

冕冋 ␸t共P兲

册

⳵f + div共fv兲 dv . ⳵t

共2.44兲

We say that a body B satisfies balance of linear momentum if for every nice open set U 傺 B,


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d dt

冕

␸t共U兲

␳v dv =

冕

␸t共U兲

␳b dv +

冕

共2.45兲

t da,

⳵␸t共U兲

where ␳ = ␳共x , t兲 is mass density, b = b共x , t兲 is body force vector field and t = t共x , nˆ , t兲 is the traction vector. Note that Cauchy’s stress theorem tells us that there is a contravariant second-order tensor ␴ = ␴共x , t兲共Cauchy stress tensor兲 with components ␴ij such that t = 具具␴ , nˆ典典. Note that 具具,典典 is the inner product induced by the Riemmanian metric g. Equivalently, balance of linear momentum can be written in the undeformed configuration as d dt

冕

U

冕

␳0V dV =

U

␳0B dV +

冕

⳵U

ˆ 典典dA, 具具P,N

共2.46兲

where, P = J␸*␴ 共the first Piola-Kirchhoff stress tensor兲 is the Piola transform of Cauchy stress tensor. Note that P is a two-point tensor with components PiJ. Note also that this is the balance of linear momentum in the deformed 共physical兲 space written in terms of some quantities that are defined with respect to the reference configuration. As was mentioned before, balance of linear momentum has no intrinsic meaning because integrating a vector field is geometrically meaningless. As is standard in continuum mechanics, this balance law makes use of the linear 共or affine兲 structure of Euclidean space. A body B is said to satisfy balance of angular momentum if for every nice open set U 傺 B, d dt

冕

␸t共U兲

␳x ⫻ v dv =

冕

␸t共U兲

␳x ⫻ b dv +

冕

⳵␸t共U兲

x ⫻ 具具␴,nˆ典典da.

共2.47兲

As with balance of linear momentum, balance of angular momentum makes use of the linear structure of Euclidean space and this does not transform in a covariant way under a general change of coordinates. One says that balance of energy holds if, for every nice open set U 傺 B, d dt

冕冉

冊冕

1 ␳ e + 具具v,v典典 dv = 2 ␸t共U兲

␸t共U兲

␳共具具b,v典典 + r兲dv +

冕

⳵␸t共U兲

共具具t,v典典 + h兲da,

共2.48兲

where e = e共x , t兲 , r = r共x , t兲 and h = h共x , nˆ , t兲 are internal energy per unit mass, heat supply per unit mass and heat flux, respectively. The geometry of inverse motions: The study of inverse motions in continuum mechanics was started by Shield38 and further extended by Ericksen10 and Steinmann.43,42 Here the idea is to fix spatial points and look at the evolution of material points under the inverse of the deformation mapping. It is known that in inverse motion, Eshelby’s tensor has a role similar to that of stress tensor in direct motion. One should note that formulating continuum mechanics in terms of the inverse motion is simply a change in describing the same physical system and so, in general, cannot have any profound consequences. However, in the general relativistic setting, in which it is desireable to have the fields to be defined on space-time and take values in a bundle over spacetime, inverse configurations are preferred; see Ref. 5 and references therein. III. LAGRANGIAN FIELD THEORY OF ELASTIC BODIES WITH EVOLVING REFERENCE CONFIGURATIONS

Suppose the reference configuration evolves in time and assume that this evolution can be represented by a one-parameter family of mappings that map B 傺 B 共reference configuration at t = 0兲 to Bt 傺 B 共the reference configuration at time t兲, ⌶t:B → Bt .

共3.1兲

We call these maps the configurational deformation maps. Note that this is not the most general form of reference configuration evolution. In general, one should look at the reference configura-


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FIG. 1. Configurational and standard deformation maps.

tion evolution locally 共see Refs. 9 and 8 for some discussions on this兲. For the sake of simplicity, we assume a global reference configuration evolution. The configuration space for the evolution of the reference configuration is Cconf = 兵⌶兩⌶:B → Bt其.

共3.2兲

An evolution of the reference configuration is a curve c : I → Cconf in Cconf. It is important to put the right restrictions on ⌶t. It does not seem necessary for ⌶t to be invertible, in general. Here, we assume that ⌶t is a diffeomorphism. A standard deformation is represented by a one-parameter family of mappings,

␸t:Bt → S.

共3.3兲

The standard configuration space is defined by C = 兵␸兩␸:Bt → S其.

共3.4兲

Again, a standard deformation is a curve in the standard configuration space. The total deformation map is the composition of standard and configurational deformation maps,

␾t = ␸t ⴰ ⌶t:B → S;

共3.5兲

that is, the following diagram commutes:

Figure 1 below shows the same idea schematically.


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J. Math. Phys. 47, 042903 共2006兲


In terms of mapping the material points, xt = ␸t共Xt兲 = ␸t ⴰ ⌶t共X兲, as is shown in the following commutative diagram:

The configuration space for the total deformation is defined as Ctot = 兵␾兩␾ = ␸ ⴰ ⌶, ␸ 苸 C,⌶ 苸 Cconf其 = C ⴰ Cconf .

共3.6兲

A deformation is a curve c : I → Ctot in the total configuration space. Note that ⌶t = Id 共identity map兲 in most of classical continuum mechanics. Notice that there are two independent deformation mappings ␸t and ⌶t when reference configuration evolves in time 共see Fig. 1兲. These separate mappings represent independent kinematical processes and hence may correspond to two separate systems of forces, in general. Definition 3.1 (configurational velocity): The configurational velocity is defined by V0共X,t兲 =

⳵ ⌶t共X兲 . ⳵t

共3.7兲

Definition 3.2: The total material velocity is defined by

冏

˜ 共X,t兲 = ⳵ ␾t共Xt兲 V ⳵t

冏

= X fixed

⳵␸t + FV0 = V + FV0 , ⳵t

共3.8兲

where, as before, F = ⳵␸t / ⳵Xt is the deformation gradient 共holding t fixed兲. Note that T␾t = T␸t ⴰ T⌶t

˜F = FF . 0

or

共3.9兲

Thus F0 = F−1 ⴰ ˜F .

共3.10兲

Now we may think about postulating the conservation of configurational mass and balance of linear and angular configurational momenta. Conservation of mass is defined in terms of conservation of mass for deformation mappings ⌶t and ␸t separately or equivalently for ⌶t and ␾t separately. This makes sense as ⌶t and ␸t correspond to configurational and standard deformations and should preserve the mass of an arbitrary sub-body. Definitiion 3.3 (conservation of mass): Suppose B is a body and ␾t = ␸t ⴰ ⌶t is a deformation map. We say that the deformation mapping is mass conserving if for every U 傺 B, d dt

冕

⌶t共U兲

␳0共Xt,t兲dV = 0

and

d dt

冕

␾t共U兲

␳共x,t兲dv = 0,

共3.11兲

where ␳0共Xt , t兲 is the mass density at point Xt 苸 Bt and ␳共x , t兲 is the mass density at the point x 苸 S. Localization of the above equations gives the local form of conservation of mass, namely ˜, R0共X兲 = ␳0共Xt,t兲J0 = ␳共x,t兲J

共3.12兲

where J0 = det共F0兲共冑det G / 冑det G0兲, F0 = T⌶t is the configurational deformation gradient, G0 is ˜兲 the fixed metric of B, G is the metric of Bt and R0 is the mass density at X 苸 B and ˜J = det共F ⫻共冑det g / 冑det G兲 = JJ0. Note that this is equivalent to


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J. Math. Phys. 47, 042903 共2006兲


R 0 = ␳ 0J 0

and

␳0 = ␳J.

共3.13兲

One may be tempted to postulate a balance of configurational linear momentum as follows. A body B satisfies the balance of configurational linear momentum if for any U⬘ 傺 Bt, d dt

冕

U⬘

␳0V0 dV =

冕

U⬘

␳0B0 dV +

冕

⳵U⬘

P0N dA.

共3.14兲

Localization of this balance law and using Cauchy’s theorem gives the following local form of the balance of configurational linear momentum Div P0 + ␳0B0 = ␳0A0 .

共3.15兲

Thinking of configurational deformation mapping ⌶t as a deformation of a fixed reference configuration, this balance law is similar to the standard balance of linear momentum written in the deformed configuration. Note that postulating such a balance law requires the introduction of two new quantities, namely P0 and B0 and does not seem to be of any use at this point. It should be noted that a configurational change need not be volume preserving. An example is a phase transformation from cubic to tetragonal which has the following configurational deformation gradient 共this is called Bain strain or matrix in martensitic phase transformations兲,

冢冣 1 0 0

F0 =

0 1 0

共3.16兲

c 0 0 a

where a = b and c ⬎ a are the tetragonal lattice parameters. The Lagrangian may be regarded as a map L : TC → R, where C is the space of some sections 共for technical details see Ref. 28兲, associated to the Lagrangian density L and a volume element dV共X兲 on B and is defined as L共␸, ␸˙ 兲 =

冕

B

L共X, ␸共X兲, ␸˙ 共X兲,F共X兲,G共X兲,g共␸共X兲兲兲dV共X兲.

共3.17兲

Note that here we have assumed an explicit dependence of L on the material and spatial metrics G and g. Let us first revisit the classical Lagrangian field theory of elasticity using the above Lagrangian density with explicit dependence on material and spatial metrics. The action function is defined as S共␸兲 =

冕

t1

L共␸, ␸˙ 兲dt.

共3.18兲

t0

Hamilton’s principle states that the physical configuration ␸ is the critical point of the action, i.e., dS共␸兲 · ␦␸ = 0.

共3.19兲

Note that variation in ␸ leaves the material metric unchanged. The statement of Hamilton’s principle can be simplified to read

冕冕冉 t1

t0

B

⳵L ⳵␸

· ␦␸ +

⳵L ⳵␸˙

· ␦␸˙ +

⳵L ⳵F

:␦F +

⳵L ⳵g

冊

:␦g dV共X兲dt = 0.

共3.20兲

After some manipulations the above integral statement results in


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J. Math. Phys. 47, 042903 共2006兲


⳵L ⳵␸

a

−

冉冊冉冊冉冊

⳵ ⳵L ⳵ t ⳵␸˙

−

a

⳵L

⳵L

A

−

⳵F

⳵F

a 兩A

A b FcA␥ac +2 b

⳵L ⳵ gcd

b gbd␥ac = 0.

共3.21兲

Noting that

冉冊

d ⳵L dt ⳵␸˙

c b d = ␳0共gabAb + gbc␥ad ␸˙ ␸˙ 兲,

冉冊 ⳵L ⳵F

2

共3.22兲

a

⳵L ⳵ gcd

A

= − P aA ,

共3.23兲

a

= ␳0␸˙ c␸˙ d − J␴cd ,

共3.24兲

Eq. 共3.21兲 can be written as PaA兩A +

⳵L ⳵␸a

b + 共FcA PbA − J␴cdgbd兲␥ac = ␳0gabAb .

共3.25兲

Note that if L depends on F and g through C, then the term in the parentheses would be zero and hence PaA兩A +

⳵L ⳵␸a

= ␳0gabAb ,

共3.26兲

which is nothing but the familiar equations of motion. 关Also note that in 共3.25兲 use was made of Doyle-Ericksen formula 共3.24兲. However, for arriving at 共3.26兲 there is no need for using DoyleEricksen formula.兴 Now suppose that during the process of deformation the continuum undergoes a continuous material evolution. This means that the deformation mapping ␸ is the composition of a total deformation mapping and a referential mapping, i.e.,

␸ = ␾ ⴰ ⌶−1

or

␾ = ␸ ⴰ ⌶.

共3.27兲

Note that defining such a composition is ambiguous because there are infinitely many possibilities for decomposing a given deformation mapping ␾ into two mappings ␸ and ⌶. The new mappings can represent part of the standard deformation and material evolution. To make sure that ␸ is the standard part of total deformation mapping, the Lagrangian is written as an integral on the current reference configuration Bt L共␸, ␸˙ 兲 =

冕

Bt

L共X, ␸共X兲, ␸˙ 共X兲,F共X兲,G共X兲,g共␸共X兲兲兲dV共X兲.

共3.28兲

It would be more convenient to write the Lagrangian as a functional on B 共the fixed initial reference configuration兲. Let us denote points on B by U. Note that ˙ 共U兲 ␾˙ 共U兲 = 共␸˙ ⴰ ⌶兲共U兲 + T␸共⌶共U兲兲 · ⌶

or

˙ 共U兲. ˙ 共U兲 − F共⌶共U兲兲 · ⌶ 共␸˙ ⴰ ⌶兲共U兲 = ␾ 共3.29兲

Also F共⌶共U兲兲 = F␾共U兲F−1 ⌶ 共⌶共U兲兲.

共3.30兲

Thus,


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J. Math. Phys. 47, 042903 共2006兲


˙ 共U兲,F 共U兲F−1共⌶共U兲兲,G共⌶共U兲兲,g共␾共U兲兲兲J 共U兲, ˙ 共U兲 − F␾共U兲F−1共⌶共U兲兲 · ⌶ L = L共⌶共U兲, ␾共U兲, ␾ ␾ ⌶ ⌶ ⌶ 共3.31兲 where

J⌶ = det共T⌶兲

冑det G 冑det G0 ,

共3.32兲

and where G0 is the fixed metric of the fixed reference configuration and G is the metric of Bt. As before, the action is defined as S共⌶, ␾兲 =

冕

t1

˙ , ␾, ␾ ˙ 兲dt. L共⌶,⌶

共3.33兲

t0

Hamilton’s principle states that the physical configurations ⌶ and ␾ are the critical points of the action, i.e., dS共⌶, ␾兲 · 共␦⌶, ␦␾兲 = 0.

共3.34兲

For the sake of clarity, we look at the two independent variations separately.

A. Vertical variations

Let us first look at vertical variations; that is, we assume that ␦⌶ = 0 and see if we can recover the classical Euler-Lagrange equations. Proposition 3.4: Allowing only vertical variations in Hamilton’s principle, one obtains the following equations of motion

⳵L ⳵␸

a

−

冉冊冉冊

⳵L d dt ⳵␸˙ ⴰ ⌶

−

a

⳵L

B

⳵F

a 兩B

b − FcB␥ac

冉冊 ⳵L ⳵F

B

+ b

⳵ L ⳵ gbc = 0. ⳵ gbc ⳵ xa

共3.35兲

Proof: The derivative of the action with respect to vertical variations is computed as follows: dS共⌶, ␾兲 · 共0, ␦␾兲 =

冕冕再 t1

B

t0

+

⳵L ⳵F

⳵L ⳵␸ ⴰ ⌶

· ␦␾ +

⳵L ⳵␸˙ ⴰ ⌶

:␦关F␾共U兲F−1 ⌶ 共⌶共U兲兲兴 +

˙ 共U兲兴兲 ˙ − ␦关F␾共U兲F−1共⌶共U兲兲 · ⌶ · 共␦␾ ⌶

⳵L ⳵g

冎

:␦g ⴰ ␾ J⌶共U兲dV共U兲dt = 0.

共3.36兲

Note that −1 −1 −1 ␦共F␾F−1 ⌶ ⴰ ⌶兲 = ␦共F␾F⌶ 兲 ⴰ ⌶ = T共␦共␾ ⴰ ⌶ 兲兲 ⴰ ⌶ = T共␦␾ ⴰ ⌶ 兲 ⴰ ⌶

= 共T␦␾T⌶−1兲 ⴰ ⌶ = D␦␾F−1 ⌶ ⴰ ⌶.

共3.37兲

Let us assume coordinates 兵U␣其, 兵XA其, and 兵xa其 and basis vectors E␣, eA, and fa on B, Bt, and S, respectively. Thus, in coordinates D␦␾ =

⳵ ␦␾a fa 丢 E␣ . ⳵ U␣

共3.38兲

The first part of the second term is simplified as


042903-14

冕冕 t1

⳵L

˙ ⴰ⌶ B ⳵␸

t0

J. Math. Phys. 47, 042903 共2006兲


␦␾˙ J⌶共U兲dV共U兲 = −

冕冕冋冉

⳵L d dt ⳵␸˙ ⴰ ⌶

t1

B

t0

冊

+ a

⳵L ⳵␸˙ ⴰ ⌶

B aW 兩B

册

␦␾aJ⌶ dV共U兲dt, 共3.39兲

where W共U兲 =

d ⌶共U兲. dt

共3.40兲

The second part of the second term in 共3.36兲 can be simplified to

冕冕冕冕冋冉冊冕冕冋冉冊 ⳵L

t1

−

B

t0

J⌶

⳵␸˙ ⴰ ⌶

D␦␾F−1 ⌶ ⴰ ⌶ · W dV共U兲dt

⳵L

t1

=

B

t0

␤ B J⌶共F−1 ⌶ ⴰ ⌶兲 BW

⳵␸˙ ⴰ ⌶ J⌶

+

B

t0

a

⳵L

t1

⳵␸˙ ⴰ ⌶

册

兩␤

␦␾a dV共U兲dt

册

b 共F ⴰ ⌶兲cA␥ac WA ␦␾a dV共U兲dt. b

共3.41兲

Using the Piola identity we have

冋冉冊 ⳵L

⳵␸˙ ⴰ ⌶

␤ B J⌶共F−1 ⌶ ⴰ ⌶兲 BW a

册冋冉冊册兩␤

⳵L

= J⌶

WA

⳵␸˙ ⴰ ⌶

a

共3.42兲

. 兩A

Also

冋冉冊册 ⳵L

WA

⳵␸˙ ⴰ ⌶

a

冉冊冉冊

⳵L ⳵ A ⳵ X ⳵␸˙ ⴰ ⌶

= 兩A

WA +

a

⳵L

⳵␸˙ ⴰ ⌶

WA兩A − a

冉冊 ⳵L

⳵␸˙ ⴰ ⌶

b c WA␥ac FA . b

共3.43兲 Therefore 共3.41兲 is simplified to

冕冕冋冉 t1

B

t0

⳵L ⳵ A ⳵ X ⳵␸˙ ⴰ ⌶

冊冉冊册 ⳵L

WA +

⳵␸˙ ⴰ ⌶

a

WA兩A ␦␾aJ⌶dV共U兲dt.

共3.44兲

a

Note that

冉冊冉冊

⳵ ⳵L ⳵ t ⳵␸˙

=

a

⳵L ⳵ ⳵ t ⳵␸˙ ⴰ ⌶

ⴰ ⌶−1 − a

冉冊

⳵L ⳵ ⳵ XA ⳵␸˙ ⴰ ⌶

ⴰ ⌶−1WA .

共3.45兲

a

˙ is simplified to Hence adding 共3.39兲 and 共3.44兲 the term corresponding to ␦␾

冕冕冉冊 t1

−

t0

Bt

⳵ ⳵L ⳵ t ⳵␸˙

␦␾a ⴰ ⌶−1 dV共X兲 dt.

共3.46兲

a

After some lengthy manipulations, the third term in 共3.36兲 can be written as

冕冕冋冉冊 ⳵L

t1

−

t0

B

⳵F ⴰ ⌶

B a 兩B

+ F cB ⴰ ⌶

冉冊册 ⳵L

⳵F ⴰ ⌶

B

b ␥ac ␦␾aJ⌶共U兲dV共U兲dt.

共3.47兲

a

The last term is simplified as


042903-15

冕冕

⳵L

t1

⳵g ⴰ ␾

B

t0

J. Math. Phys. 47, 042903 共2006兲


:␦g ⴰ ␾J⌶ dV共U兲dt =

冕冕冕冕 t0

ⴰ ⌶−1 dV共X兲dt = −

⳵L

⳵ gbc a a ␦␾ J⌶ dV共U兲dt = bc B ⳵g ⴰ ␾ ⳵x

t1

⳵L

t1

bc Bt ⳵ g

t0

冕冕

⳵ L ⳵ gbc a ␦␾ bc ⳵ xa Bt ⳵ g

t1

t0

b c 共gcd␥ad + gbd␥ad 兲␦␾a ⴰ ⌶−1 dV共X兲dt.

共3.48兲

Therefore, adding the above four simplified terms, we obtain dS共⌶, ␾兲 · 共0, ␦␾兲 =

冕冕冋 t1

Bt

t0

⳵L ⳵␸

a

−

冉冊冉冊

d ⳵L dt ⳵␸˙

−

a

⳵L

B

⳵F

a 兩B

b − FcB␥ac

冉冊 ⳵L ⳵F

B

+ b

册

⳵ L ⳵ gbc ␦␾a ⳵ gbc ⳵ xa

ⴰ ⌶−1 dV共X兲dt.

共3.49兲

As ␦␾a is arbitrary we conclude that

⳵L ⳵␸

a

−

冉冊冉冊

⳵L d dt ⳵␸˙ ⴰ ⌶

−

a

⳵L

B

⳵F

a 兩B

b − FcB␥ac

冉冊 ⳵L ⳵F

B

+ b

⳵ L ⳵ gbc = 0, ⳵ gbc ⳵ xa

共3.50兲䊐

which gives the stated result. B. Horizontal variations

Now let us try to find the Euler-Lagrange equations resulting from horizontal variations; that is, variations of the configurational deformation mapping ⌶. Proposition 3.5: Allowing only horizontal variations in Hamilton’s principle, one obtains the following configurational equations of motion:

⳵L ⳵X

A

+

⳵ ⳵t

冋冉冊册冋 ⳵L

F aA − L ␦ BA −

⳵␸˙

a

冉冊册冉冊 ⳵L ⳵F

B

F aA

a

+

兩B

⳵L ⳵F

B C D FaC⌫AB + 2GCD⌫AB a

⳵L ⳵ GBC

= 0, 共3.51兲

is the Christoffel symbol of a local chart in Bt. where Proof: The derivative of the action with respect to horizontal variations is computed as follows: C ⌫AB

dS共⌶, ␾兲 · 共␦⌶,0兲 =

冕冕冉冋 t1

t0

+

B

⳵L ⳵G ⴰ ⌶

⳵L ⳵⌶

· ␦⌶ −

⳵L ⳵␸˙ ⴰ ⌶

˙ · ␦共F␾F−1 ⌶ ⴰ ⌶ · ⌶兲 +

冊

⳵L ⳵F ⴰ ⌶

册

:␦共F␾F−1 ⌶ ⴰ ⌶兲 J⌶

:␦G ⴰ ⌶ + L␦J⌶ dV共U兲dt = 0.

共3.52兲

Note that −1 ␦共F␾F⌶ ⴰ ⌶兲 = F␾␦共F−1 ⌶ ⴰ ⌶兲.

共3.53兲

−1 −1 ␦共F−1 ⌶ ⴰ ⌶兲 = − F⌶ D共␦⌶兲F⌶ ⴰ ⌶.

共3.54兲

−1 −1 −1 ␦共F␾F−1 ⌶ ⴰ ⌶兲 = − F␾F⌶ D共␦⌶兲F⌶ ⴰ ⌶ = − FD共␦⌶兲F⌶ ⴰ ⌶.

共3.55兲

But

Thus

Similarly


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J. Math. Phys. 47, 042903 共2006兲


−1 ˙ ␦共F␾F−1 ⌶ ⴰ ⌶ · ⌶兲 = − FD共␦⌶兲F⌶ ⴰ ⌶ · W + F ⴰ ⌶ ·

d 共␦⌶兲. dt

共3.56兲

In coordinates, D共␦⌶兲 =

⳵ ␦⌶A eA 丢 E␤ . ⳵ U␤

共3.57兲

The second term in 共3.52兲 has two parts which are simplified as follows. The first part is

冕冕冉 t1

B

t0

⳵L ⳵␸˙ ⴰ ⌶

冊

F aA ⴰ ⌶ a

⳵ ␦⌶A −1 共F ⴰ ⌶兲␤BWBJ⌶ dV共U兲dt = − ⳵ U␤ ⌶

ⴰ ⌶−1 dV共X兲dt −

冕冕冉冊 ⳵L

t1

˙ B⬘ ⳵␸

t0

冕冕冋冉冊册 t1

t0

⳵ B B⬘ ⳵ X

⳵L ⳵␸˙

F aA W B␦ ⌶ A

a

FaAWB兩B␦⌶A ⴰ ⌶−1 dV共X兲dt.

共3.58兲

a

Similarly, the second part is simplified as

冕冕 t1

−

t0

B

⳵L ⳵␸˙ ⴰ ⌶

冕冕冉冊 ⳵L

t1

+


d 共␦⌶兲J⌶ dV共U兲dt = dt

·Fⴰ⌶·

⳵␸˙

Bt

t0

Bt

t0

d dt

⳵L ⳵␸˙

FaA ␦⌶A ⴰ ⌶−1 dV共X兲dt

a

FaAWB兩B␦⌶A dV共X兲dt.

共3.59兲

a

Adding 共3.58兲 and 共3.59兲, the second term of 共3.52兲 can be written as

冕冕 t1

−

t0

B

⳵L ⳵␸˙ ⴰ ⌶


˙ ␦共F␾F−1 ⌶ ⴰ ⌶ · ⌶兲J⌶ dV共U兲dt =

Bt

t0

d dt

⳵L ⳵␸˙

FaA ␦⌶A ⴰ ⌶−1 dV共X兲dt.

a

共3.60兲 After some lengthy manipulations, the third term of 共3.52兲 is simplified to

冕冕

⳵L

t1

t0

B ⳵F ⴰ ⌶

:␦共F␾F−1 ⌶ ⴰ ⌶兲J⌶ dV共U兲dt

冕冕冋冉冊册冕冕冉冊 ⳵L

t1

=

⳵F

Bt

t0

F aA

a

⳵L

t1

+

t0

B

Bt

⳵F

兩B

␦⌶A ⴰ ⌶−1 dV共X兲dt

B

C FaC⌫AB ␦⌶A ⴰ ⌶−1 dV共X兲dt.

共3.61兲

a

The fourth term of 共3.52兲 is simplified to

冕冕 t1

t0

B

⳵L ⳵G ⴰ ⌶

:␦G ⴰ ⌶J⌶ dV共U兲dt =

冕冕 t1

t0

Bt

D 2GCD⌫AB

⳵L ⳵ GBC

dV共X兲dt.

共3.62兲

Note that J⌶ = 共det F⌶兲

冑

det G , det G0

共3.63兲

where G0 is the fixed Riemannian metric of the fixed reference configuration. Thus,


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J. Math. Phys. 47, 042903 共2006兲


␦J⌶ = ␦共det F⌶兲

冑

B ⳵ 冑det G 1 det G ␦ det G ␤ ⳵ ␦⌶ + 共det F⌶兲 = J⌶共F−1 ␦⌶. ⌶兲 B ␤ + 共det F⌶兲冑 ⳵U det G0 det G0 det G0 ⳵ X

共3.64兲 Note that

⳵ 冑det G 1 ⳵G B = 冑det GG−1 = 冑det G⌫AB ␦⌶A . 2 ⳵X ⳵X

共3.65兲

Hence ␤ ␦J⌶ = J⌶共F−1 ⌶兲 B

⳵ ␦⌶B B + J⌶⌫AB . ⳵ U␤

共3.66兲

Thus the last term of 共3.52兲 is simplified to

冕冕 t1

B

t0

冕冕 t1

L␦J⌶ dV共U兲dt = −

Bt

t0

共L␦AB兲兩B␦⌶A ⴰ ⌶−1 dV共X兲dt.

共3.67兲

Now substituting the above five simplified terms into 共3.52兲, we have dS共⌶, ␾兲 · 共␦⌶,0兲 =

冕冕再 t1

t0

ⴰ⌶

−1

Bt

⳵L ⳵X

A

+

⳵ ⳵t

冋冉冊册冋冉冊册冎 ⳵L ⳵␸˙

冕冕再冉冊 a

⳵L

t1

dV共X兲dt +

F aA − L ␦ AB −

⳵F

Bt

t0

⳵L ⳵F

B

F aA

a


⳵L ⳵ GBC

冎

兩B

␦⌶A

␦⌶A

ⴰ ⌶−1 dV共X兲dt = 0.

共3.68兲

Because ␦⌶A is arbitrary, we conclude that

⳵L ⳵X

A

+

⳵ ⳵t

冋冉冊册冋冉冊册冉冊 ⳵L ⳵␸˙


a

⳵L ⳵F

⳵L

B

F aA

+

⳵F

兩B

a

⳵L


⳵ GBC

= 0. 共3.69兲

䊐 We now show that this is equivalent to the classical Euler-Lagrange equations and does not give us any new information. After some lengthy manipulations, it can be shown that

⳵L ⳵ XA

+

−

冋冉冊册冋冉冊册冋冉冊册冉冊 ⳵ ⳵t

⳵L ⳵␸˙

⳵L ⳵F


a

A

b FcA␥ac +2

b

⳵L

⳵ gcd

⳵L ⳵F

B

F aA

a

b gbd␥ac F aA −

=

兩B

⳵L ⳵F

⳵L ⳵␸a

−

冉冊冉冊

⳵ ⳵L ⳵ t ⳵␸˙

−

a

B

C D FaC⌫AB − 2GCD⌫AB

a

⳵L

A

⳵F

a 兩A

⳵L ⳵ GBC

.

共3.70兲

䊐关It will be seen in Sec. VI that material covariance of internal energy density implies that the sum of the last two terms is zero. In Sec. VII, it will be shown that material covariance of Lagrangian density results in the same identity. However, at this point there is no such relation and the variational principle does not give us any new information.兴 This result is known for the case where the underlying metrics are trivial.25 In conclusion, we have proved the following proposition. Proposition 3.6: In the absence of discontinuities, i.e., when all the fields are smooth, the configurational and the standard equations of motion are equivalent, even if one is allowed to vary the referential and spatial metrics.


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J. Math. Phys. 47, 042903 共2006兲


IV. THE GREEN-NAGHDI-RIVILIN THEOREM

Green, Rivilin, and Naghdi19 realized that conservation of mass and balance of linear and angular momenta can be obtained as a result of postulating invariance of energy balance under isometries of R3, i.e., rigid translations and rotations in the deformed configuration. Later Marsden and Hughes28 extended this idea to Riemannian manifolds and diffeomorphisms of the deformed configuration showing that this covariant approach gives the Doyle-Ericksen formula for Cauchy stress as well as conservation of mass and balance of linear and angular momenta. In another relevant work, Šilhavý39 considered all the densities in the energy balance to be volume densities and assuming 共i兲 invariance of energy balance under Galilean transformations and 共ii兲 boundedness of energy from below, proved the existence of mass, its conservation, balance of linear and angular momenta, transformation of body forces and the splitting of total energy into internal and kinetic energies. Before discussing the covariant approach to elasticity, let us first discuss the classical GreenNaghdi-Rivilin 共GNR兲 theorem and a nonconventional material form of it. We consider two cases: 共i兲 material energy balance invariance under spatial isometries of R3 and 共ii兲 material energy balance invariance under material isometries of R3. We call 共i兲 and 共ii兲 the spatial-material and material-material GNR theorems, respectively. A. The spatial-material GNR theorem

Consider the material energy balance for a nice subset U 傺 B, d dt

冕冉

冊冕

1 ␳0 ⌿ + V · V dV = 2 U

U

冕

␳0共B · V + R兲dV +

⳵U

共T · V + H兲dA,

共4.1兲

where ⌿ = ⌿共t , X , F兲 is the free energy density per unit mass of the undeformed configuration. Now consider an isometry ␰t : R3 → R3 of R3. We postulate that the material energy balance is invariant under ␰t. For the sake of simplicity we consider translations and rotations separately. 共i兲

共Rigid translations兲 A spatial rigid translation is defined by

␰t共x兲 = x + 共t − t0兲c,

共4.2兲

where c is some constant vector field. We now postulate that the material balance of energy holds for the deformation mapping ␸t⬘ = ␰t ⴰ ␸t as well. This balance law is still written on U but with different fields 共primed fields兲 in general, d dt

冕 ⬘冉

冊冕

1 ␳0 ⌿⬘ + V⬘ · V⬘ dV = 2 U

U

␳0⬘共B⬘ · V⬘ + R⬘兲dV +

冕

⳵U

共T⬘ · V⬘ + H⬘兲dA. 共4.3兲

Using Cartan’s space-time theory, the primed fields are related to the unprimed quantities through the following relations:

␳0⬘共X兲 = ␳0共X兲, V⬘兩t=t0 =

R⬘共X兲 = R共X兲,

冏冏 ⳵ ␸⬘ ⳵t t

t=t0

H⬘共X兲 = H共X兲,

= 共T␰tV + c兲t=t0 = V + c,

T⬘共X,N兲 = T共X,N兲.

共4.4兲

Also because


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J. Math. Phys. 47, 042903 共2006兲


b⬘ − a⬘ = ␰t*共b − a兲

and

B − A = 共b − a兲 ⴰ ␸t .

共4.5兲

We have B⬘ − A⬘ = ␰*t 共b − a兲 ⴰ ␸t⬘ .

共4.6兲

共B⬘ − A⬘兲t=t0 = 共b − a兲 ⴰ ␸t = 共B − A兲.

共4.7兲

Hence

It can be easily shown that F⬘共X兲 = F共X兲.

共4.8兲

The free energy density would have the following transformation: ⌿⬘共t,X,F⬘共X兲兲 = ⌿共t,X,F共X兲兲.

共4.9兲

⳵⌿ d ⌿⬘共t,X,F⬘共X兲兲 = . dt ⳵t

共4.10兲

Thus,

Balance of energy for U 傺 B for the new deformation mapping at t = t0 can be written as

冊冕冉冕

冕冉冕

1 ⳵␳0 ⌿ + 共V + c兲 · 共V + c兲 dV + 2 U ⳵t =

U

␳0共B⬘兩t=t0 · 共V + c兲 + R兲dV +

U

⳵U

␳0

冊

⳵⌿ + 共V + c兲 · A⬘兩t=t0 dV ⳵t

共T · 共V + c兲 + H兲dA,

共4.11兲

where Div c = 0 was used. Subtracting the material energy balance of the deformation ␸t for U 傺 B from the above equation and using 共4.7兲 we obtain

冕冉

冊冕

1 ⳵␳0 c · V + c · c dV + 2 ⳵ t U

U

␳0A · c dV =

冕

U

␳0B · c dV +

冕

T · c dA.

⳵U

共4.12兲 Because U and c are arbitrary one concludes that

共ii兲

⳵␳0 = 0, ⳵t

共4.13兲

Div P + ␳0B = ␳0A.

共4.14兲

共Rigid rotations兲 Now let us consider a rigid rotation in the ambient space, i.e., ␰t : S → S, where

␰t共x兲 = e共t−t0兲⍀x,

共4.15兲

for some constant skew-symmetric matrix ⍀. Note that T␰t兩t=t0 = e共t−t0兲⍀兩t=t0 = Id

and

冏冏 ⳵ ⳵t

␰t共x兲 = ⍀x.

共4.16兲

t=t0

Also


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J. Math. Phys. 47, 042903 共2006兲


V⬘共X兲兩t=t0 = V + ⍀x共X兲.

共4.17兲

Subtracting the balance of energy for U for deformation mapping ␸t from that of ␸t⬘ = ␰t ⴰ ␸t at time t = t0 results in

冕

U

␳0⍀x共X兲 · 共A − B兲dV =

冕

T⍀x共X兲dA.

⳵U

共4.18兲

But

冕

T⍀x共X兲dA =

⳵U

冕

U

共Div P · ⍀x + PFT:⍀兲dV.

共4.19兲

Thus PFT = FPT ,

共4.20兲

where use was made of balance of linear momentum.

B. The material-material GNR theorem

To our best knowledge, there is no study of invariance of energy balance under isometries of the reference configuration in the literature. It turns out that such an invariance does not hold in general, even in Euclidean space. In this section we study the transformation of balance of energy under rigid translations and rotations of the reference configuration in the Euclidean space context. It will be shown that balance of energy is invariant under translations and rotations of the reference configuration for isotropic materials that satisfy an internal constraint that we call material inviscidity. Again we consider rigid translations and rigid rotations of the reference configuration separately. 共i兲

共Rigid translations兲 Consider a time-dependent rigid translation of the reference configuration ⌶t : B → B⬘. Let X⬘ = Xt = ⌶t共X兲 = X + 共t − t0兲W,

共4.21兲

for some constant vector field W. Note that T⌶t = Id,

X = ⌶−1 t 共Xt兲 = Xt − 共t − t0兲W.

共4.22兲

Deformation gradient with respect to the new reference configuration is denoted F⬘ and, dx = F dX = F⬘ dX⬘ .

共4.23兲

But, dX⬘ = dX and hence F dX = F⬘ dX

" dX.

共4.24兲

This means that F⬘共Xt兲 = F共X兲

or

F⬘ = F ⴰ ⌶−1 t .

共4.25兲

In the differential geometry language this means that F⬘ = ⌶t*F = F ⴰ ⌶−1 t .

共4.26兲

The material velocity with respect to the new reference configuration is


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J. Math. Phys. 47, 042903 共2006兲


V⬘共Xt兲 =

⳵ −1 ␸t ⴰ ⌶−1 t 共X⬘兲 = V ⴰ ⌶t 共X⬘兲 − FW. ⳵t

共4.27兲

Thus at t = t0, V⬘ = V − FW.

共4.28兲

Free energy density is assumed to have the following transformation: ⌿⬘共X⬘,F ⴰ ⌶−1 t 兲 = ⌿共X,F兲.

共4.29兲

⌿⬘共X⬘,F兲 = ⌿共X,F ⴰ ⌶t兲.

共4.30兲

Or

关Note that this does not put any restrictions on the material properties as here all we assume is that under a change of frame the 3-form ␳0⌿ dV is transformed to a 3-form ␳0⬘⌿⬘ dV⬘ = ⌶t*共␳0⌿ dV兲.兴 More precisely, ⌶*t ⌿⬘共X⬘,F兲 = ⌿共X,F ⴰ ⌶t兲.

共4.31兲

⳵⌿ ⳵⌿ ⳵F d ⌿⬘共X⬘,F兲 = + : . W. dt ⳵ t ⳵ 共F ⴰ ⌶t兲 ⳵ ⌶t共X兲

共4.32兲

⳵⌿ ⳵⌿ ⳵F d ⌿⬘共X⬘,F兲 = + : . W. dt ⳵t ⳵F ⳵X

共4.33兲

Thus

Hence at t = t0

Material balance of energy for U 傺 B reads

冕冉冕

冊冕冉冕

1 ⳵␳0 ⌿ + 具具V,V典典 dV + 2 U ⳵t =

U

␳0共B · V + R兲dV +

U

⳵U

␳0

冊

d ⌿ + 具具V,A典典 dV dt

共T · V + H兲dA.

共4.34兲

Let us assume that material balance of energy for U⬘ 傺 B⬘ reads d dt

冕 ⬘冉

冊冕冕

1 ␳0 ⌿⬘ + V⬘ · V⬘ dV⬘ = 2 U⬘

U⬘

+

␳0共B⬘ · V⬘ + R⬘兲dV⬘ +

⳵U⬘

共T⬘ · V⬘ + H⬘兲dA⬘ ,

冕

U⬘

B0⬘ · Wt dV⬘ 共4.35兲

for some vector field B0⬘ which will be determined shortly. Note that thinking of the integrand of the left-hand side of balance of energy as a 3-form ␣, we have d dt

冕

U⬘

␣⬘ =

冕

U

d * 共⌶t ␣⬘兲. dt

共4.36兲

But ⌶*t ␣⬘ = ␳0共X兲⌿共X , F ⴰ ⌶t兲dV, thus material balance of energy for U⬘ 傺 B⬘ at t = t0 reads


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J. Math. Phys. 47, 042903 共2006兲


冕冉冕

冊冕冉冏冏兲冕

1 ⳵␳0 ⌿ + 具具V − FW,V − FW典典 dV + 2 U ⳵t =

U

␳0共B⬘兩t=t0 · 共V − FW兲 + R dV +

U

⳵U

␳0

d dt

t=t0

冊

⌿⬘ + 具具V − FW,A⬘兩t=t0典典 dV

共T · 共V − FW兲 + H兲dA +

冕

U

B0 · W dV, 共4.37兲

where B0 is an unknown vector field at this point. Note that 共B⬘ − A⬘兲t=t0 = B − A.

共4.38兲

Now subtracting the material balance of energy for U 傺 B from that of U⬘ 傺 B⬘ at time t = t0 yields

冕冉 U

P:

冊

⳵F + ␳0FT共B − A兲 − B0 · W dV + ⳵X

冕

FTT · WdA = 0

⳵U

" W.

共4.39兲

Localization leads to the following conclusion: B0 = Div共FTP兲 + ␳0FT共B − A兲 + P:

⳵F . ⳵X

共4.40兲

Note that P:

⳵F ⳵⌿ = Div共⌿I兲 − , ⳵X ⳵X

共4.41兲

and Div共FTP兲 = FT Div P + P:

⳵F . ⳵X

共4.42兲

Thus 共4.40兲 is equivalent to B0 = FT关Div P + ␳0共B − A兲兴 + 2P:

⳵F ⳵F = 2P: . ⳵X ⳵X

共4.43兲

Therefore, the transformed balance of energy is 共4.35兲 with B0⬘ = ⌶t*共B0兲. Invariance of balance of energy under rigid translations of the reference configuration is equivalent to B0 = 0, i.e., P:

⳵F = 0, ⳵X

共4.44兲

which is equivalent to Div共FTP兲 = FT Div共P兲.

共4.45兲

Obviously, if F is independent of X, i.e., if the deformation gradient is uniform then this condition is satisfied but as we will see in the sequel this is not necessary. Note that 共4.43兲 is independent of balance of linear momentum. It is seen that an additional constraint must be satisfied for the material energy balance to be invariant under time-dependent rigid referential translations. This shows the very different natures of material and spatial manifolds. We will show at the end of Sec. VI that 共4.45兲 implies that configurational stress tensor is hydrostatic. For this reason we call 共4.45兲 the configurational inviscidity constraint. Example: Consider a Neo-Hookean rod in uniaxial tension. The deformation gradient is


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J. Math. Phys. 47, 042903 共2006兲


冢

␭−1/2 0 ␭

F= 0 0

0 −1/2

0

冣

共4.46兲

0 . ␭

It can be easily shown that the first Piola-Kirchhoff stress tensor has the following representation:

P=

冢

0 0

0

0 0

0

␮ 0 0 ␮␭ − 2 ␭

冣

共4.47兲

,

where ␮ = ␮共X兲. It is now an easy exercise to show that 共4.45兲 is satisfied only if ␭ is constant, i.e., only if the deformation gradient is uniform. Thus in this case the only possibility would be a uniform deformation gradient for balance of energy to be invariant under rigid translations of the reference configuration. Example: We know that for an isotropic material SAB = ␣0GAB + ␣1CAB + ␣2CADCDB ,

共4.48兲

where ␣0 , ␣1, and ␣2 are scalar functions of X and SAB are components of the second Piola-Kirchhoff stress tensor. For the sake of simplicity, suppose ␣1 = ␣2 = 0. In terms of P and F we have PaA = ␣0GABFaB .

共4.49兲

When the reference configuration and ambient space are Euclidean the condition Div共FTP兲 = FT Div共P兲 is equivalent to

␣ 0F aB

⳵ F aB = 0. ⳵ XA

共4.50兲

Or F aB

a ⳵ F aB a ⳵F A = F = 0. B ⳵ XA ⳵ XB

共4.51兲

Note that, in general, this does not imply that the deformation gradient is uniform and it is simply an internal constraint. Example: Consider an incompressible perfect fluid 共ideal fluid兲 for which

␴ab = − pgab

and

共4.52兲

J = 1.

Thus PaA = − J共F−1兲Ab pgab .

共4.53兲

Using Piola identity we have 共Div共FTP兲兲A = 共− pJGAB兲兩B = − J

⳵ p b AB F G . ⳵ xb B

共4.54兲

Also 共FT Div共P兲兲A = − gabFbBGABJ共pgad兲兩d = − J

⳵ p b AB F G . ⳵ xb B

共4.55兲

Thus 共4.45兲 is satisfied for an ideal fluid.


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共ii兲

J. Math. Phys. 47, 042903 共2006兲


共Rigid rotations兲 Consider a time-dependent rigid rotation of the reference configuration ⌶t : B → B⬘ defined as X⬘ = Xt = e共t−t0兲⍀X,

共4.56兲

for some constant skew-symmetric matrix ⍀. Note that V⬘ = V − F⍀X,

F⬘ = F ⴰ ⌶−1 t .

共4.57兲

Let us assume that material balance of energy for U⬘ 傺 B⬘ has the following form: d dt

冕 ⬘冉

冊冕冕

1 ␳0 ⌿⬘ + V⬘ · V⬘ dV⬘ = 2 U⬘

U⬘

+

␳0共B⬘ · V⬘ + R⬘兲dV⬘ +

U⬘

冕

⳵U⬘

共T⬘ · V⬘ + H⬘兲dA⬘

共B0⬘ · ⍀X + C0⬘:⍀兲dV⬘ ,

共4.58兲

where C0⬘ = ⌶t*C0 and C0 is an unknown vector field at this point. Material balance of energy for U⬘ 傺 B⬘ at t = t0 reads

冕冉

冊

1 ⳵␳0 ⌿ + 具具V − F⍀X,V − F⍀X典典 dV 2 ⳵ t U +

=

冕

冕冉冏冏 U

U

+

␳0

d dt

t=t0

␳0共B⬘兩t=t0 · 共V − F⍀X兲 + R兲dV +

冕

U

冊

⌿⬘ + 具具V − F⍀X,A⬘兩t=t0典典 dV

冕

⳵U

共T · 共V − F⍀X兲 + H兲dA

共B0 · ⍀X + C0:⍀兲dV.

共4.59兲

Subtracting the material balance of energy for U 傺 B from that of U⬘ 傺 B⬘ at time t = t0 and considering the relation for B0 coming from rigid translations of the reference configuration yields

冕

共FTP − C0兲:⍀ dV = 0.

共4.60兲

FTP − C0 = 共FTP − C0兲T .

共4.61兲

U

This means that

Thus C0 = −PTF + S for some symmetric tensor S. This symmetric tensor does not contribute to balance of energy and we can choose it to be S = 0. Thus the transformed balance of energy under rigid rotations of the reference configuration is 共4.58兲 where C0⬘ = ⌶t*共C0兲 and C0 = −PTF. In conclusion, we have proved the following proposition. Proposition 4.1: Balance of energy is invariant under time-dependent translations and rotations of the reference configuration if B0 = C0 = 0, i.e., if the reference configuration is both configurationally inviscid and isotropic. Thus, balance of energy is invariant under material isometries of the reference configuration only under some constraints. As an example, it is seen that balance of energy is invariant under material isometries in the case of ideal fluids.


042903-25

J. Math. Phys. 47, 042903 共2006兲


V. COVARIANT SPATIAL ENERGY BALANCE

In this section we start by a reappraisal of the concept of covariance in elasticity and its consequences. We revisit Marsden and Hughes’ theorem28 and clarify some details in their proof. We then show that the same conclusions can be reached if one assumes that mass density is a 3-form instead of a scalar. A proof is then given for converse of Marsden and Hughes’ theorem, i.e., assuming conservation of mass, balance of linear and angular momenta and Doyle-Ericksen formula, balance of energy is invariant under arbitrary spatial diffeomorphisms. At the end of this section, we show that assuming spatial covariance for material energy balance yields results that are identical to those obtained by assuming spatial covariance for spatial energy balance. A. Covariance and the Doyle-Ericksen formula

First recall that the general notion of covariance of a set of equations is as follows. Definition 5.1 (Covariance): Suppose a theory has some tensor fields U , V , . . . defined on a space A and the governing equations of the theory have the form F共U , V , . . . 兲 = 0. These governing equations are called covariant if for any diffeomorphism ␰ : A → A, ␰*共F共U , V , . . . 兲兲 = F共␰*U , ␰*V , . . . 兲. A theory is covariant if all its governing equations are covariant. The Doyle-Ericksen formula: Doyle and Ericksen7 showed the following interesting relation:

␴ = 2␳

⳵e , ⳵g

共5.1兲

i.e., Cauchy’s stress tensor is proportional to the partial derivative of the free energy density with respect to the Riemannian metric in the deformed configuration. 关Note that 共see Ref. 28, p. 198兲

⳵e ⳵␺ = . ⳵g ⳵g

共5.2兲

In other words, in Doyle-Ericksen formula internal energy density can be replaced by free energy density because e = ␺ + ␪s,

共5.3兲

where ␪ is absolute temperature and s is entropy density. Thus

⳵e ⳵␺ ⳵␺ ⳵␪ ⳵␪ ⳵␺ + s= = + , ⳵g ⳵g ⳵␪ ⳵g ⳵g ⳵g

共5.4兲

as ⳵␺ / ⳵␪ = −s兴. Doyle and Ericksen7 looked at changes of spatial frame passively, i.e., as changes of coordinates while Marsden and Hughes28 chose the active point of view. The Doyle-Ericksen formula is known to be the essential condition for covariance of energy balance. Later Simo and Marsden40 found a material version of Doyle-Ericksen formula, which we discuss next. Here by “material version” they mean an analogue of the usual Doyle-Ericksen formula that ensures covariance of material energy balance under spatial diffeomorphisms. 关An interesting question to ask would be the condition共s兲 that ensures covariance of material energy balance under diffeomorphisms of the reference configuration. This will be discussed in Sec. VI.兴 Simo and Marsden consider a general form of polar decomposition theorem by first associating two Riemannian metrics G0 and G to B, where G0 does not change under spatial diffeomorphisms while G does change. The polar decomposition theorem states that F = RU,

共5.5兲

where


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J. Math. Phys. 47, 042903 共2006兲


U共X兲:共TXB,G0兲 → 共TXB,G兲

共5.6兲

is the material stretch tensor 共a positive-definite symmetric linear map with respect to the given metrics兲 and R共X兲:共TXB,G兲 → 共T␸t共X兲S,g兲

共5.7兲

is, for each X 苸 B, a 共G , g兲-orthogonal linear transformation. The metric G is arbitrary and can change under spatial diffeomorphisms, G = R*共g兲.

共5.8兲

The internal energy density per unit mass of the deformed configuration is e = e共x,t,g共x兲兲.

共5.9兲

E共X,t,G兲 = e共␸t共X兲,t,R*共G兲兲.

共5.10兲

Now define

Simo and Marsden40 show that ⌺ = 2␳

⳵E , ⳵G

共5.11兲

where ⌺ is the rotated stress tensor defined as ⌺ = R *␴

⌺AB = 共R−1兲Aa␴ab共R−1兲Bb .

or

共5.12兲

In this paper we prove a similar theorem by postulating a balance of energy for an arbitrary reframing of the reference configuration for a special class of materials. It should be noted that there are four possibilities for a covariant energy balance law. 共i兲共ii兲共iii兲共iv兲

Spatial energy balance law for any reframing of the deformed configuration: This gives the usual Doyle-Ericksen formula. Material energy balance law for any reframing of the deformed configuration: This gives the Doyle-Ericksen formula in terms of Kirchhoff stress tensor. Material energy balance law for any reframing of the reference configuration: This should give a material form of Doyle-Ericksen formula for Eshelby’s stress tensor. Spatial energy balance for any reframing of the reference configuration: This should give a spatial form of Doyle-Ericksen formula for Eshelby’s stress tensor.

Note that cases 共i兲 and 共ii兲 and also cases 共iii兲 and 共iv兲 are equivalent as the important thing here is the type of the diffeomorphism. B. Revisiting Marsden and Hughes’ theorem

Let us first revisit Marsden and Hughes’ covariant energy balance theory.28 These authors postulate a covariant spatial energy balance, i.e., they consider a motion ␸t : B → S and postulate that balance of energy still holds for any spatial change of frame. Marsden and Hughes consider arbitrary changes of frame for the deformed configuration and postulate that energy balance is invariant under these framings. For a given nice subset U 傺 B, the 共spatial兲 balance of energy reads d dt

冕冉

冊冕

1 ␳ e + 具具v,v典典 dv = 2 ␸t共U兲

␸t共U兲

␳共具具b,v典典 + r兲dv +

冕

⳵␸t共U兲

共具具t,v典典 + h兲da,

共5.13兲

where e , r, and h are the internal energy function per unit mass, the heat supply per unit mass and the heat flux, respectively. Marsden and Hughes then consider an arbitrary reframing of the deformed configuration, which can be regarded as a motion of S in S, i.e., ␰t : S → S. Postulating


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J. Math. Phys. 47, 042903 共2006兲


the balance of energy 共5.13兲 for such a reframing and considering it for t = t0 they obtain 共i兲 conservation of mass, 共ii兲 balance of linear momentum, 共iii兲 balance of angular momentum, and 共iv兲 the Doyle-Ericksen formula. Conversely, if 共i兲, 共ii兲, 共iii兲, 共iv兲 and balance of energy hold, then balance of energy would hold for any change of spatial frame. We will give a proof for the converse of the theorem in the sequel. Proposition 5.2 (Transport theorem in a reframing of the deformed configuration): Suppose f ⬘ = ␰t* f is a scalar quantity defined on ␸t⬘共U兲, i.e., f ⬘ : ␸t⬘共U兲 → R and f : ␸t共U兲 → R. 关Marsden and Hughes have the following transport theorem on p. 166 of Ref. 28 in the second equation after their Eq. 共2兲, which needs to be corrected: d dt

冕

␸t⬘共U兲

f dv =

冕

␸t⬘共U兲

共f˙ + f div v兲dv⬘ .

共5.14兲

In fact, the first dv should read dv⬘.兴 Then,

冏冏冕 d dt

t=t0 ␸t⬘共U兲

f ⬘ dv⬘ =

冕

␸t共U兲

共f˙ + f div v兲dv .

共5.15兲

Proof: The usual transport theorem can be written as d dt

冕

␸t⬘共U兲

f ⬘ dv⬘ =

冕

␸t⬘共U兲

共f˙ ⬘ + f ⬘ div⬘ v⬘兲dv⬘ ,

共5.16兲

where ˙f ⬘ = ⳵ f ⬘ + ⳵ f ⬘ · v⬘ = ⳵ f ⬘ + df ⬘ · v⬘ , ⳵ t ⳵ x⬘ ⳵t

共5.17兲

v⬘ = ␰t*v + w.

共5.18兲

and

Therefore, d dt

冕

␸t⬘共U兲

f ⬘ dv⬘ =

冕冋 ␸t⬘共U兲

册

⳵ f⬘ + df ⬘ · 共␰t*v + w兲 + f ⬘ div⬘ v⬘ dv⬘ . ⳵t

共5.19兲

Note that

⳵ ⳵ = ⴰ 共T␰t兲 ⳵ x ⳵ x⬘ This means that

⳵ ⳵ . = 共T␰t兲−1 ⴰ ⳵ x⬘ ⳵x

or

冏冏 ⳵ ⳵ x⬘

= t=t0

⳵ . ⳵x

共5.20兲

共5.21兲

Lemma 5.3: If ␰t : S → S is a diffeomorphism with the properties,

␰t兩t=t0 = Id,

T␰t兩t=t0 = Id.

共5.22兲

Then 共div⬘ v⬘ dv⬘兲兩t=t0 = div v dv .

共5.23兲

Proof: We prove the lemma when S is equipped with an arbitrary volume form ␮. This will imply the particular case of a Riemannian manifold with the volume form induced by the Rie-


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J. Math. Phys. 47, 042903 共2006兲


mannian metric. Recall that the divergence of a vector field X with respect to ␮ is defined as LX␮ = 共div␮ X兲␮ .

共5.24兲

Under the spatial change of frame v⬘ = ␰t*X + w, ␮⬘ = ␰t*␮. Thus, 共div␮⬘ v⬘兲␮⬘ = Lv⬘共␰t*␮兲 = ␰t*共Lv␮兲,

共5.25兲

where use was made of Theorem 6.19 of Marsden and Hughes28. Therefore, 共div⬘ v⬘ dv⬘兲兩t=t0 = div v dv .

共5.26兲

䊐 One should be careful with partial time derivatives as ⳵ f ⬘ / ⳵t is not equal to ⳵ f / ⳵t at t = t0 because the former is partial time derivative for fixed x⬘ while the latter is a partial time derivative for fixed x. Note that

冏冏冏冏 ⳵ f⬘ ⳵t

⳵ f⬘ ⳵ t x⬘

=

x fixed

+ df ⬘ · wt .

共5.27兲

fixed

Hence,

冉冏冏冊 ⳵ f⬘ ⳵ t x⬘

=

fixed t=t 0

⳵f − df · w. ⳵t

共5.28兲

Therefore 共5.19兲 is simplified to

冏冏冕 d dt

t=t0 ␸t⬘共U兲

f ⬘ dv⬘ =

冕


␸t共U兲

共5.29兲

䊐 Now let us take a more natural approach and assume that we are transporting a 3-form. Note that this is more general in the sense that we have not chosen a volume form dv a priori. Proposition 5.4: Suppose ␣⬘ = ␰t*␣ is a 3-form defined on ␸t⬘共U兲. Then,

冏冏冕 d dt

t=t0 ␸t⬘共U兲

␣⬘ =

冕

␸t共U兲

L v␣ .

共5.30兲

Proof: Using the usual transport theorem for forms we have d dt

冕

␸t⬘共U兲

␣⬘ =

冕

␸t⬘共U兲

L v⬘␣ ⬘ .

共5.31兲

Assuming that ␣ transforms objectively, i.e., ␣⬘ = ␰t*␣, using Theorem 6.19 of Marsden and Hughes28 we have Lv⬘␣⬘ = ␰t*Lv␣ .

共5.32兲

Thus, d dt

冕

␸t⬘共U兲

␣⬘ =

冕

␸t⬘共U兲

␰t*Lv␣ .

共5.33兲

Therefore,


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J. Math. Phys. 47, 042903 共2006兲


冏冏冕 d dt

t=t0 ␸t⬘共U兲

␣⬘ =

冕

L v␣ .

␸t共U兲

共5.34兲䊐

Now substitute ␣ = f dv, where f is a scalar. Note that f ⬘ dv⬘ = f ⬘ Ù dv⬘ = 共␰t* f兲 Ù 共␰t* dv兲 = ␰t*共f Ù dv兲 = ␰t*共f dv兲.

共5.35兲

The above proposition now reads

冏冏冕 d dt

t=t0 ␸t⬘共U兲

f ⬘ dv⬘ =

冕

␸t 共U兲

Lv共f dv兲.

共5.36兲

0

Note that L is a derivation and hence Lv共f dv兲 = 共Lv f兲dv + f共Lv dv兲 = 共f˙ + div v兲dv .

共5.37兲

Therefore

冏冏冕 d dt

t=t0 ␸t⬘共U兲

f ⬘ dv⬘ =

冕

␸t 共U兲


共5.38兲

0

Thus, this approach recovers the same transport equation 共5.15兲. C. Energy balance in terms of differential forms

In this section we regard ␳ as a 3-form and write the energy balance equation as d dt

冕冉

冊冕

1 ␳ e + 具具v,v典典 = 2 ␸t共U兲

␸t共U兲

␳共具具b,v典典 + r兲 +

冕

⳵␸t共U兲

共具具t,v典典 + h兲da.

共5.39兲

关Traction can be thought of as a covector-valued 2-form. There are some technical details involved and we choose to stick to the usual definition of traction.兴 Under a spatial diffeomorphism ␰t : S → S we postulate that d dt

冕 ⬘冉 ⬘

冊冕

1 ␳ e + 具具v⬘,v⬘典典 = 2 ␸t⬘共U兲

␸t⬘共U兲

␳⬘共具具b⬘,v⬘典典 + r⬘兲 +

冕

⳵␸t⬘共U兲

共具具t⬘,v⬘典典 + h⬘兲da⬘ . 共5.40兲

Let f be the scalar multiplying the density 3-form in the first integrand, i.e., d dt

冕

␸t共U兲

␳f =

冕

␸t共U兲

Lv共␳ f兲 =

冕

␸t共U兲

f ª e + 21 具具v , v典典.

共␳Lv f + fLv␳兲.

Thus

共5.41兲

But Lv f = Lve + Lv共 21 具具v,v典典兲 = e˙ + = e˙ +

具具典典

冉

冊冉

冊

⳵ 1 1 具具v,v典典 + d 具具v,v典典 · v 2 ⳵t 2

⳵v ,v + 具具v,ⵜvv典典 = e˙ + 具具v,a典典. ⳵t

共5.42兲

Also,


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d dt

冕

␸t⬘共U兲

␳⬘ f ⬘ =

冕

L v⬘共 ␳ ⬘ f ⬘兲 =

␸t共U兲

冕

␸t⬘共U兲

共␳⬘Lv⬘ f ⬘ + f ⬘Lv⬘␳⬘兲.

共5.43兲

Note that v⬘ = ␰t*v + wt and thus Lv⬘␳⬘ = ␰t*共Lv␳兲.

共5.44兲

Also,

具具

典典

⳵ v⬘ + ⵜv⬘v⬘ = e˙⬘ + 具具v⬘,a⬘典典. Lv⬘ f ⬘ = e˙⬘ + v⬘, ⳵t

共5.45兲

Thus 共Lv⬘ f ⬘兲兩t=t0 = e˙ +

⳵e :Lwg + 具具v + w,a⬘兩t=t0典典, ⳵g

共5.46兲

共Lv⬘␳⬘兲兩t=t0 = Lv␳ .

共5.47兲

Therefore

冏冏冕 d dt

t=t0 ␸t⬘共U兲

␳⬘ f ⬘ =

冕冉冕冉 ␸t共U兲

⳵e :Lwg + 具具v + w,a⬘兩t=t0典典兲 ⳵g

␳ e˙ +

+

␸t共U兲

冊

1 f + 具具v,w典典 + 具具w,w典典 Lv␳ . 2

共5.48兲

Now subtracting the balance of energy equation for ␸t共U兲 from that of ␸t⬘共U兲 at t = t0 we obtain

冕冉冕 ␸t共U兲

␳

冊冕冉

⳵e :Lwg + 具具v,a⬘兩t=t0 − a典典 + 具具w,a⬘兩t=t0典典 + ⳵g

=

␸t共U兲

共具具v,b⬘兩t=t0 − b + 具具w,b⬘兩t=t0典典兲 +

冕

⳵␸t共U兲

␸t共U兲

冊

1 具具v,w典典 + 具具w,w典典 Lv␳ 2

具具w,t典典da.

共5.49兲

Using the identity 共b⬘ − a⬘兲兩t=t0 = b − a we have

冕冉 ␸t共U兲

␳

冊冕冉

⳵e :Lwg + 具具w,a − b典典 + ⳵g

␸t共U兲

冊冕

1 具具v,w典典 + 具具w,w典典 Lv␳ = 2

⳵␸t共U兲

具具w,t典典da. 共5.50兲

We know that

冕

⳵␸t共U兲

具具w,t典典da =

冕冉 ␸t共U兲

冊

1 具具div ␴,w典典 + ␴: Lwg + ␴:␻ dv , 2

共5.51兲

where ␻ has the coordinate representation ␻ab = 21 共wa兩b − wb兩a兲. Let us replace ␳ by ␳ dv in the first integral of Eq. 共5.50兲,


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J. Math. Phys. 47, 042903 共2006兲


冕冉冕冉 ␸t共U兲

冊冕冉冊

⳵e :Lwg + 具具w,a − b典典 dv + ⳵g

␳

=

␸t共U兲

␸t共U兲

冊

1 具具v,w典典 + 具具w,w典典 Lv␳ 2

1 具具div ␴,w典典 + ␴: Lwg + ␴:␻ dv . 2

共5.52兲

Since w is arbitrary we conclude that Lv␳ = 0,

␴ = 2␳

共5.53兲

⳵e , ⳵g

共5.54兲

div ␴ + ␳b = ␳a,

共5.55兲

␴T = ␴ .

共5.56兲

D. Proof of the converse of Marsden and Hughes’ theorem

Marsden and Hughes28 do not give a proof for the converse of the covariant energy balance theorem, i.e., when Eqs. 共5.53兲–共5.56兲 are satisfied then energy balance is invariant under ␰t : S → S. Such a proof is nontrivial and is given here. Let us assume that Eqs. 共5.53兲–共5.56兲 are satisfied and define ⌬E共␰t兲 =

d dt

冕 ⬘冉 ⬘

冊冕

1 ␳ e + 具具v⬘,v⬘典典g − 2 ␸t⬘共U兲

␸t⬘共U兲

␳⬘共具具b⬘,v⬘典典g + r⬘兲 −

冕

⳵␸t⬘共U兲

共具具t⬘,v⬘典典g + h⬘兲da⬘ . 共5.57兲

Note that balance of energy for ␸t共U兲 can be written as ⌬E共Id兲 = 0. We need to prove that for any diffeomorphism ␰t, ⌬E共␰t兲 = 0. We know that e⬘共x⬘,t,g兲 = e共x,t, ␰*t 共g兲兲.

共5.58兲

Let us denote wt ª

d ␰ t, dt

Wt = ␰*t 共wt兲,

gt = ␰*t 共g兲.

共5.59兲

Note that by definition

冕

␸t⬘共U兲

␳ ⬘r ⬘ =

冕

␸t共U兲

␳r,

冕

⳵␸t⬘共U兲

h⬘ da⬘ =

冕

h da.

共5.60兲

具具t,v + Wt典典gt da.

共5.61兲

⳵␸t共U兲

Also note that

冕

⳵␸t⬘共U兲

具具t⬘,v⬘典典g da⬘ =

冕

⳵␸t⬘共U兲

具具␰t*t, ␰t*v + wt典典g da⬘ =

冕

⳵␸t共U兲

A straightforward computation shows that


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J. Math. Phys. 47, 042903 共2006兲


冕冉 ␸t⬘共U兲

冊冕冕

1 ␳⬘具具v⬘,v⬘典典g − ␳⬘具具b⬘,v⬘典典g = 2

␳⬘具具␰t*共a − b兲, ␰t*v + wt典典g =

␸t⬘共U兲

+

␸t共U兲

冕

␸t共U兲

␳具具a − b,v典典gt

␳具具a − b,Wt典典gt ,

共5.62兲

where use was made of Lv␳ = 0. Note that d dt

冕

␸t⬘共U兲

␳ ⬘e ⬘ =

冕

␸t⬘共U兲

关␳⬘Lv⬘e⬘ + e⬘␰t*共Lv␳兲兴 =

冕

␸t⬘共U兲

␳ ⬘L v⬘e ⬘ =

冕

␸t⬘共U兲

␳␰*t 共Lv⬘e⬘兲. 共5.63兲

But

␰*t 共Lv⬘e⬘兲 = e˙ + Therefore ⌬E共␰t兲 = ⌬E共Id兲 +

冕冋冉 ␸t共U兲

2␳

⳵e :LWtgt . ⳵ gt

共5.64兲

册冕

冊

⳵e 1 − ␴ : L Wtg t + ␴ : ␻ t d v − 2 ⳵ gt

␸t共U兲

具具div ␴ + ␳共b − a兲,Wt典典gtdv 共5.65兲

= 0.

䊐 E. Spatial covariant material energy balance

Let us consider the material balance of energy d dt

冕冉

冊冕

1 ␳0 E + 具具V,V典典 = 2 U

U

␳0共具具B,V典典 + R兲 +

冕共 ⳵U

具具T,V典典 + H兲dA,

共5.66兲

where we have assumed that ␳0 is a 3-form. Physically this is equivalent to the spatial energy balance; material energy balance is simply the spatial energy balance expressed in terms of quantities defined with respect to the reference configuration. Let us postulate that the material energy balance is invariant with respect to diffeomorphisms ␰t : S → S. This is physically equivalent to the postulate of covariant spatial energy balance. The material energy balance for ␸t⬘共U兲傺 S is written as d dt

冕 ⬘冉 ⬘

冊冕

1 ␳0 E + 具具V⬘,V⬘典典 = 2 U

U

␳0⬘共具具B⬘,V⬘典典 + R⬘兲 +

冕共 ⳵U

具具T⬘,V⬘典典 + H⬘兲dA.

共5.67兲

Note that for both deformations balance of energy is written for the same subset U 傺 B. The material velocity V⬘ is related to V by the following relation: V⬘共X兲 = T␰t ⴰ Vt + wt ⴰ ␸t共X兲.

共5.68兲

兩V⬘兩t=t = V + w ⴰ ␸t0 . 0

共5.69兲

R⬘ = J␸⬘r⬘ ⴰ ␸t⬘,

共5.70兲

Thus

We know that R = J ␸tr ⴰ ␸ t,

t

r = J ␰tr ⬘ ⴰ ␰ t .

Hence


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J. Math. Phys. 47, 042903 共2006兲


J␸⬘r⬘ ⴰ ␸t⬘ = 共J␰tr⬘ ⴰ ␰t兲 ⴰ ␸tJ␸t = J␸tr ⴰ ␸t .

共5.71兲

R⬘ = R.

共5.72兲

H⬘ = H.

共5.73兲

t

Thus

Similarly

Note that looking at densities as 3-forms

␳0共X,t兲 = ␸*t ␳共x,t兲,

␳0⬘共X,t兲 = 共␸t⬘兲*␳⬘共x⬘,t兲.

共5.74兲

But 共␸t⬘兲*␳⬘共x⬘,t兲 = 共␰t ⴰ ␸t兲*␳⬘共x⬘,t兲 = 共␸*t ⴰ ␰*t 兲 ⴰ ␰t*␳共x,t兲 = ␸*t ␳共x,t兲.

共5.75兲

␳0⬘共X,t兲 = ␳0共X,t兲.

共5.76兲

Thus

Because balance of energy is written for the same subset U 傺 B the same equality holds for densities as scalar fields, i.e., one can replace ␳0⬘ and ␳0 by ␳0⬘ dV and ␳0 dV, respectively. Define E共X,t,g兲 = e共␸t共X兲,t,g ⴰ ␸t共X兲兲.

共5.77兲

e⬘共x⬘,t,g兲 = e共x,t, ␰*t g兲.

共5.78兲

E⬘共X,t,g兲 = e⬘共x⬘,t,g兲 = e共x,t, ␰*t g兲 = E共X,t, ␰*t g兲.

共5.79兲

We know that

Thus

Therefore

冏冏 d dt

E⬘ = t=t0

⳵E ⳵E + :Lwⴰ␸t共g ⴰ ␸t兲. ⳵t ⳵g

共5.80兲

Now the material energy balance for the motion ␸t⬘ at t = t0 can be written as

冕冉

冊冕冋冉

1 ⳵␳0 E + 具具V + w ⴰ ␸t0,V + w ⴰ ␸t0典典 dV + 2 U ⳵t

册

U

␳0

⳵E ⳵E + :Lwⴰ␸t共g ⴰ ␸t兲 ⳵t ⳵g

冊

+ 具␳0具具V + w ⴰ ␸t0,A⬘兩t=t0典典 dV =

冕

U

␳0共具具B⬘兩t=t0,V + w ⴰ ␸t0典典 + R兲 dV +

冕共具具 ⳵U

T,V + w ⴰ ␸t0典典 + H兲 dA.

共5.81兲

Subtracting the balance of energy for the motion ␸t from 共5.81兲 one arrives at the following identity:


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冕冉具具 ⳵␳0 U ⳵t

V,w ⴰ ␸t0典典 + =

冕

U

冊冕冉

1 具具w ⴰ ␸t0,w ⴰ ␸t0典典 dV + 2

具具␳0B,w ⴰ ␸t0典典dV +

冕

⳵U

U

␳0

冊

⳵E :Lwⴰ␸t共g ⴰ ␸t兲 + 具具w ⴰ ␸t0,A典典 dV ⳵g

具具T,w ⴰ ␸t0典典 dA.

共5.82兲

Let us denote W = w ⴰ ␸t and note that W is a spatial vector field with components Wa. Lemma 5.5: The surface integral term in 共5.82兲 is transformed to a volume integral as

冕具具 ⳵U

T,W典典dA =

冕共具具 U

Div P,W典典 + ␶ :␻ + ␶ :k兲dV,

共5.83兲

where, ␶ab = PaBFBb is the Kirchhoff stress and ␻ and k have the coordinate representations kab = 21 共Wa兩b + Wb兩a兲 and ␻ab = 21 共Wa兩b − Wb兩a兲. Proof: The integrand has the following component form: TagabWb = PaCGCDNDgabWb = 共PaCgabWb兲GCDND .

共5.84兲

Now using divergence theorem the surface integral is transformed to an integral on U with an integrand with the following component form: 共PaCgabWb兲兩C = PaC兩CgabWb + PaCgabWb兩C ,

共5.85兲

where use was made of the fact that gab兩C = 0. Note that Wb兩C =

⳵ Wb b b b + ␥cd WcFdC = Wb,dFdC + ␥cd WcFdC = 共Wb,d + ␥cd Wc兲FdC = Wb兩dFdC . ⳵ XC

共5.86兲

Therefore, 共PaCgabWb兲兩C = PaC兩CgabWb + PaCWa!dFdC = PaC兩CgabWb + PaCFdC关 21 共Wa!d + Wd!a兲 + 21 共Wa!d − Wd!a兲兴 ,

䊐

which proves the lemma. Substituting 共5.83兲 into 共5.82兲 yields

冕冉具具典典冕具具 ⳵␳0 U ⳵t

V,W

−

U

+

共5.87兲

冊冕冉

1 具具W,W典典 dV + 2

U

2␳0

Div P + ␳0B − ␳0A,W典典 dV = 0.

冊

⳵E − ␶ :k dV − ⳵g

冕

U

␶ :␻ dV 共5.88兲

As W and U 傺 B are arbitrary we conclude that,

⳵␳0 = 0, ⳵t

␶ = 2␳0

⳵E , ⳵g

␶T = ␶ ,

共5.89兲

共5.90兲

共5.91兲


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Div P + ␳0B = ␳0A.

共5.92兲

In conclusion, these computations result in the following proposition. Proposition 5.6: Energy balance written in material form, but still with the assumption of spatial covariance yields results that are identical to those of energy balance written in spatial form, also with covariance under spatial diffeomorphisms. The converse can be proved similar to what was done in the previous subsection. VI. TRANSFORMATION OF ENERGY BALANCE UNDER MATERIAL DIFFEOMORPHISMS

As was seen in the preceding section, invariance of balance of energy under an arbitrary change in spatial frame is equivalent to 共1兲 balance of linear momentum, 共2兲 balance of angular momentum, 共3兲 conservation of mass, and 共4兲 Doyle-Ericksen formula. To our best knowledge, there is no material version of this theorem in the literature. Our motivation for studying the possibility of material invariance of energy balance was to gain a better understanding of configurational forces as they are believed to be related to rearrangements of the reference configuration. It turns out that, in general, energy balance cannot be invariant under diffeomorphisms of the reference configuration and what one should be looking for instead is the way in which energy balance transforms under material diffeomorphisms. In this section we first obtain such a transformation formula under an arbitrary time-dependent material diffeomorphism 关see Eq. 共6.51兲兴 and then obtain the conditions under which balance of energy is materially covariant. A. The energy balance material transformation formula

We begin with a discussion of how energy balance transforms under material diffeomorphisms. Define E共X,t,G兲 = ⌿共X,t,C共F共X兲,g共␸t共X兲兲兲,G兲,

共6.1兲

where ⌿ = ⌿共X , t , G , C兲 is the material free energy density. Material 共Lagrangian兲 energy balance can be written as d dt

冕冉

冊冕

1 ␳0 E + 具具V,V典典 dV = 2 U

U

␳0共具具B,V典典 + R兲dV +

冕

⳵U

共具具T,V典典 + H兲dA,

共6.2兲

which can be simplified to read

冕冋冉 U

1 d ␳0 E + 具具V,V典典 2 dt

冊册冕 dV =

U

␳0共具具B,V典典 + R兲dV +

冕

⳵U

共具具T,V典典 + H兲dA,

共6.3兲

where U is an arbitrary nice subset of the reference configuration B, B is body force per unit undeformed mass, V共X , t兲 is the material velocity, ␳0共X , t兲 is the material density, R共X , t兲 is the ˆ 兲 is the heat flux across a surface with normal heat supply per unit undeformed mass, and H共X , t , N ˆ N in the undeformed configuration 共normal to ⳵U at X 苸 ⳵U兲. It is to be noted that this is balance of energy for a deformed part of the body written in terms of quantities that are defined with respect to the undeformed 共reference兲 configuration. Here we assume that we have a material manifold which is a Riemannian manifold 共B , G兲 and a given reference configuration B 傺 B. Change of reference frame: In this paragraph we consider a change of frame for the reference configuration and look at the transformed quantities for the new reference configuration. A reframing of the reference configuration is a diffeomorphism ⌶t:共B,G兲 → 共B,G⬘兲.

共6.4兲

A change of frame can be thought of as a change of coordinates in the reference configuration 共passive definition兲 or a rearrangement of microstructure 共active definition兲. Under such a framing, a nice subset U is mapped to another nice subset U⬘ = ⌶t共U兲 and a material point X is mapped to


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J. Math. Phys. 47, 042903 共2006兲


FIG. 2. A material reframing and the corresponding deformation maps.

X⬘ = ⌶t共X兲. Note that X is the position of a particle in the reference configuration, i.e., material points are identified with their positions in the reference configuration 共which is arbitrary兲. The change of frame is mathematically a mapping between two manifolds and one would expect to define an object on 共B , G⬘兲 as push-forward of the corresponding object on 共B , G兲. The deformation mapping for the new reference configuration is ␸t⬘ = ␸t ⴰ ⌶−1 t . This can be clearly seen in Fig. 2. The material velocity in U⬘ is V⬘共X⬘,t兲 =

⳵ ⌶−1 ⳵ ⳵␸t t ␸t⬘共X⬘兲 = 共X 兲 + T ␸ ⴰ ⴰ ⌶−1 共X⬘兲. ⬘ t t ⳵t ⳵t ⳵t

共6.5兲

We assume that ⌶t兩t=t0 = Id,

⳵ ⌶t 共X兲 = W共X,t兲. ⳵t

共6.6兲

Note that W is the infinitesimal generator of the rearrangement ⌶t. It can be shown that at t = t0,

冏

⳵ ⌶−1 t 共X⬘兲 ⳵t

冏

= − W共X,t兲.

共6.7兲

t=t0

Thus, at t = t0, V⬘ = V − FW.

共6.8兲

To find the relation between G and G⬘ we note that the Finger tensor b = ␸t*G is a spatial tensor and hence independent of framing of the reference configuration. Thus, b = ␸t*G = 共␸t⬘兲*G⬘ .

共6.9兲

That is, −1 * * −1 * * −* −1 G⬘ = 共␸t ⴰ ⌶−1 t 兲 ⴰ ␸t*G = 共⌶t 兲 ⴰ ␸t ⴰ ␸t*G = 共⌶t 兲 G = ⌶t*G = 共T⌶t兲 G共T⌶t兲 .

共6.10兲 Note that for an arbitrary X0 苸 B,


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F共X0兲:TX0B → T␸t共X0兲S

and

F⬘共X0⬘兲:TX⬘B → T␸⬘共X⬘兲S. 0

t

0

Given dX 苸 TX0B, dx = F共X0兲 · dX

and

dX⬘ = T⌶t · dX.

Hence, dx = F⬘共X0⬘兲 · dX⬘ = F⬘共X0⬘兲 ⴰ T⌶t · dX = F共X0兲 · dX for all dX 苸 TX0B. Thus, F⬘ = ⌶t*F,

共6.11兲

⌶t*F = F ⴰ 共T⌶t兲−1 .

共6.12兲

where

The easier way of proving this is the following: −1 −1 F⬘ = T␸t⬘ = T共␸t ⴰ ⌶−1 t 兲 = T␸t ⴰ 共⌶t兲 = F ⴰ 共⌶t兲 .

共6.13兲

The material internal energy density is assumed to have the following tensorial property: E⬘共X⬘,t,G⬘兲 = E共X,t,G兲.

共6.14兲

Note that this is different from assuming local covariance for internal energy density. This is simply the material analogue of 共5.78兲; all that 共6.14兲 says is that internal energy density at X⬘ evaluated by the transformed metric G⬘ is equal to the internal energy density at X evaluated by the metric G. We know that G⬘ = ⌶t*G, thus E⬘共X⬘,t,G兲 = E共X,t,⌶*t G兲.

共6.15兲

This means that

冏冏 d dt

E⬘共X⬘,t,G兲 = t=t0

⳵E ⳵E + :LWG. ⳵t ⳵G

共6.16兲

Remark: Marsden and Hughes28 defined covariant constitutive equations by looking at isometries of TX0B at a given point X0 苸 B. This is why they did not need to consider an explicit dependence of ⌿ on G. Another more general way of defining material covariance for the strain energy function ⌿ is to assume that for any local diffeomorphism ⌳ : TX0B → T⌳共X0兲⌳共B兲 that leaves X0 fixed, ⌿共X0,G,C兲 = ⌿共X0,⌳*G,⌳*C兲.

共6.17兲

Note that this is different from the implication of Cartan’s space-time, e.g., ⌿共X⬘,G⬘,C⬘兲 = ⌿共X,⌶*G⬘,⌶*C⬘兲, for an arbitrary diffeomorphism ⌶ : B → B. We emphasize that this relation and similarly 共6.14兲 do not put any restrictions on material properties. Ju and Papadopoulos26,27 proved that a consequence of 共6.17兲 is the following infinitesimal covariance condition: G

⳵⌿ ⳵⌿ +C = 0, ⳵G ⳵C

共6.18兲

which is equivalent to


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J. Math. Phys. 47, 042903 共2006兲


⳵⌿ 1 1 ⳵⌿ = − G−1CS = − FTP. = − G−1C 2 2 ⳵C ⳵G

共6.19兲

We will obtain this condition in the sequel as a consequence of assuming material covariance of energy balance. Example: Consider a 共materially uniform兲 Neo-Hookean material with the following energy density ⌿共X,G,C兲 = ␮关tr共C兲 − 3兴 = ␮共CIJGIJ − 3兲.

共6.20兲

We now show that this is an example of a materially covariant material. Note that

冉冊 C

⳵⌿ ⳵C

J

= ␮CIKGKJ .

共6.21兲

I

Also

冉冊 G

⳵⌿ ⳵C

J

= ␮GIKC MN I

⳵ GMN ␮ = − C MNGIK共G MKGJN + G MJGKN兲 = − ␮CIKGKJ , ⳵ GKJ 2

共6.22兲

i.e.,

C

⳵⌿ ⳵⌿ +G = 0. ⳵C ⳵G

共6.23兲

䊐 Spatial covariance of strain energy function 共material-frame-indifference兲 can be defined similarly 共see Ref. 41兲. However, one should note that this is different from Marsden and Hughes point of departure for developing a covariant theory of elasticity; in Marsden and Hughes’ theory28 balance of energy is assumed to be covariant and not the energy function. In covariant energy balance, a global diffeomorphism is considered and energy balance is assumed to be invariant under this global diffeomorphism. Balance of energy for reframings of the reference configuration: One way to obtain the governing balance equations of a continuum is to use the homogeneity of the ambient space and postulate that if a deformed body satisfies the balance of energy, any framing of it should satisfy the balance of energy as well. This is a postulate and cannot be proved. But, one can justify it 共or motivate it兲 by the fact that the ambient space S is homogeneous. Invariance of energy balance under framings of the reference configuration is less obvious and, in general, it turns out not to hold. The following is the main conclusion of this section. Under referential diffeomorphisms, material energy balance has some extra terms in it. The extra terms correspond to some forces that contribute to the rate of change of energy when the reference configuration evolves. Consider a deformation mapping ␸t : B → S and a referential diffeomorphisms ⌶t : B → B. The mapping ␸t⬘ = ␸t ⴰ ⌶−1 t : B⬘ → S, where B⬘ = ⌶t共B兲, represents the deformation of the new 共evolved兲 reference configuration. We are interested in understanding the form of material energy balance for ⌶t共U兲傺 B⬘ for any nice U 傺 B. In addition to contributions from the mapping ␸t⬘, in general, one should expect to see contributions from the referential mapping ⌶t as well, i.e., evolution of reference configuration may, in general, contribute to the energy balance. Now the balance of energy should include the following two groups of terms: 共i兲共ii兲

Looking at ␸t⬘ as the deformation of B⬘ in S, one has the usual material energy balance for ⌶t共U兲. Transformation of fields from 共B , G兲 to 共B , G⬘兲 follows Cartan’s space-time theory. Nonstandard terms may appear to represent the energy associated with the material evolution.


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Here a comment is in order. The mapping that represents all the physical processes is ␸t. This mapping is the composition of ␸t⬘ and ⌶t and hence it is expected that, in general, both ␸t⬘ and ⌶t represent part of the physical processes. This means that standard deformation represented by ␸t⬘ and evolution of microstructure 共or any other material evolution兲 represented by ⌶t should contribute to balance of energy. 共This is similar to Gurtin’s idea20,21 of including both standard and nonstandard terms in the expression of working. However, here we consider the full balance of energy.兲 This rough picture should be enough to convince the reader that the lack of invariance of energy balance under ⌶t should not be surprising. Lack of invariance implies the appearance of some new terms that are work-conjugate to Wt = 共⳵ / ⳵t兲⌶t. Let us denote the volume and surface forces conjugate to W by B0 and T0, respectively. Instead of looking at spatial framings, we fix the deformed configuration and look at framings of the reference configuration. We postulate that energy balance for each nice subset U⬘ has the following form: d dt

冕冉⬘

冊冕冕

1 E + ␳0⬘具具V⬘,V⬘典典 dV⬘ = 2 U⬘

␳0⬘共具具B⬘,V⬘典典 + R⬘兲dV⬘ +

U⬘

+

U⬘

具具B0⬘,Wt典典dV⬘ +

冕

⳵U⬘

冕

⳵U⬘

共具具T⬘,V⬘典典 + H⬘兲dA⬘

具具T0⬘,Wt典典dA⬘ ,

共6.24兲

where U⬘ = ⌶t共U兲 and B0⬘ and T0⬘ are unknown vector fields at this point. Using Cartan’s space-time theory, it is assumed that the primed quantities have the following relations with the unprimed quantities dV⬘ = ⌶t*dV 共J共⌶t兲dV⬘ = dV兲,

␳0⬘共X⬘,t兲 = ␳0共X兲,

R⬘共X⬘,t兲 = R共X,t兲,

ˆ ⬘,t兲 = H共X,N ˆ ,t兲, H⬘共X⬘,N

共6.25兲

ˆ ⬘,t兲 = T⌶ 共X兲 · T共X,N ˆ ,t兲. T⬘共X⬘,N t We know that B⬘ − A⬘ = ⌶t*共B − A兲.

共6.26兲

共B⬘ − A⬘兲兩t=t0 = B − A.

共6.27兲

Thus

Note that if ␣ is a 3-form on U, then

冏冏冕冕冏冏 d dt

t=t0 U⬘

␣⬘ =

U

d dt

共⌶*t ␣⬘兲,

共6.28兲

t=t0

where U⬘ = ⌶t共U兲. Thus

冏冏冕 d dt

t=t0 U⬘

E⬘dV⬘ =

冕冏冏 U

d dt

共⌶*t E⬘兲dV = t=t0

冕冉 U

冊

⳵E ⳵E + :LWG dV. ⳵t ⳵G

共6.29兲

Material balance of energy for U⬘ 傺 B⬘ at t = t0 reads


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冕冉冕冕

冊冕冉冕

1 ⳵␳0 E + 具具V − FW,V − FW典典 dV + 2 U ⳵t =

U

␳0共具具B⬘兩t=t0,V − FW典典 + R兲dV +

+

⳵U

U

⳵U

␳0

冊

⳵E ⳵E + :LWG + 具具V − FW,A⬘兩t=t0典典 dV ⳵t ⳵G

共具具T,V − FW典典 + H兲dA +

冕

U

具具B0,W典典dV

具具T0,W典典dA.

共6.30兲

Note that T0 and B0 are defined on B and T0⬘ and B0⬘ are the corresponding quantities defined on ⌶t共B兲. Here we assume that T0⬘ = ⌶t*T0 and B0⬘ = ⌶t*B0. Subtracting balance of energy for U from this and noting that 共A⬘ − B⬘兲t=t0 = A − B we obtain

冕冉冕

冊冕冉冕冕

1 ⳵␳0 − 具具V,FW典典 + 具具FW,FW典典 dV + 2 U ⳵t =−

U

具具␳0B,FW典典dV −

⳵U

U

␳0

具具T,FW典典dA +

冊

⳵E :LWG − 具具FW,A典典 dV ⳵G

U

具具B0,W典典dV +

冕

⳵U

具具T0,W典典dA. 共6.31兲

Cauchy’s theorem implies that ˆ 典典典典, 具具T,FW典典 = 具具FW,具具P,N

共6.32兲

where P is the first Piola-Kirchhoff stress tensor. Similarly ˆ 典典. T0 = 具具P0,N

共6.33兲

Lemma 6.1: The surface integral in material energy balance has the following transformation:

冕

⳵U

具具FTT,W典典dA =

冕

Div具具FTP,W典典dV =

U

冕

U

关具具Div共FTP兲,W典典 + FTP:⍀ + FTP:K兴dV, 共6.34兲

where ⍀IJ = 21 共GIKWK兩J − GJKWK兩I兲 = 21 共WI!J − WJ!I兲, KIJ = 21 共GIKWK兩J + GJKWK兩I兲 = 21 共WI!J + WJ!I兲,

K = 21 LWG.

共6.35兲共6.36兲

Proof: In components the integrand can be written as 共FT兲AaTaGABWB .

共6.37兲

Ta = PaCGCDND .

共6.38兲

But

Hence in components the integrand reads 共共FT兲Aa PaCGABWB兲GCDND .

共6.39兲

Using the divergence theorem the surface integral is transformed to an integral on U with the following integrand in components:


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共共FT兲Aa PaCGABWB兲兩C = 共共FTP兲ACGABWB兲兩C = 共FTP兲AC兩CGABWB + 共FTP兲ACGABWB兩C = 共FTP兲AC兩CGABWB + 共FTP兲AC关 21 共GABWB兩C + GCBWB兩A兲 + 21 共GABWB兩C − GCBWB兩A兲兴 ,

共6.40兲䊐

where use was made of the fact that GAB兩C = 0. Similarly,

冕

⳵U

具具T0,W典典dA =

冕

U

Div具具P0,W典典dV =

冕

U

关具具Div共P0兲,W典典 + P0:⍀ + P0:K兴dV.

共6.41兲

By definition, at time t = t0 the transformed balance of energy should be the same as the balance of energy for U. Subtracting the material balance of energy for U from the above balance law and considering conservation of mass, we obtain

冕

U

␳0

⳵E :LWGdV + ⳵G

冕

U

具具␳0FT共B − A兲,W典典dV −

冕

U

具具␳0B0,W典典dV +

冕

⳵U

具具FTT − T0,W典典dA = 0. 共6.42兲

Thus

冕冉 U

2␳0

冊

1 ⳵E + FTP − P0 : LWGdV + 2 ⳵G

冕

U

共FTP − P0兲:⍀dV +

冕

U

具具␳0FT共B − A兲 − B0 + Div共FTP兲共6.43兲

− Div P0,W典典dV = 0. Now using the balance of linear momentum the identity 共6.43兲 simplifies to

冕冉 U

2␳0

冊

1 ⳵E + FTP − P0 : LWGdV + 2 ⳵G

冕

U

共FTP − P0兲:⍀dV +

冕

U

具具Div共FTP − P0兲 − FTDiv P 共6.44兲

− B0,W典典dV = 0. Because U and W are arbitrary P0 = 2␳0

⳵E + FTP, ⳵G

共6.45兲

共FTP − P0兲T = FTP − P0 ,

共6.46兲

B0 = Div共FTP − P0兲 − FT Div P.

共6.47兲

Note that 共6.46兲 is trivially satisfied after having 共6.45兲. Thus we have P0 = 2␳0

⳵E + FTP, ⳵G

B0 = Div共FTP − P0兲 − FT Div P.

共6.48兲共6.49兲

Note that P0 is a measure of anisotropy 共deviation from material Doyle-Ericksen formula兲. This is an interesting result that in a natural way shows the contribution of some nonstandard terms to balance of energy when reference configuration evolves. Thus we have proven the following theorem. Theorem 6.2: Under a referential diffeomorphism ⌶t : B → B, and assuming that material energy density transforms tensorially, i.e.,


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J. Math. Phys. 47, 042903 共2006兲


E⬘共X⬘,t,G兲 = E共X,t,⌶*t G兲,

共6.50兲

material energy balance has the following transformation: d dt

冕冉⬘ ⌶t共U兲

冊冕冕

1 E + ␳0⬘具具V⬘,V⬘典典 dV⬘ = 2

⌶t共U兲


+

⌶t共U兲

具具B0⬘,Wt典典dV⬘ +

冕

⳵⌶t共U兲

冕

⳵⌶t共U兲

共具具T⬘,V⬘典典 + H⬘兲dA⬘

具具T0⬘,Wt典典dA⬘ ,

共6.51兲

where T0⬘ = ⌶t*

冋具具

2␳0

⳵E ˆ + FTP,N ⳵G

典典册

共6.52兲

,

B0⬘ = ⌶t*关Div共FTP − P0兲 − FT Div P兴 ,

共6.53兲

and the other quantities are already defined. B. Consequences of assuming invariance of energy balance

This section shows the consequences of assuming material covariance of energy balance. It turns out that energy balance, in general, cannot be materially covariant. Material energy balance is invariant under material diffeomorphisms if and only if the following relations hold between the nonstandard terms: P0 = 0 B0 = 0

or

2␳0

⳵E = − FTP, ⳵G

共6.54兲共6.55兲

Div共FTP兲 = FT Div P.

or

Equation 共6.54兲 is the material Doyle-Ericksen formula and 共6.55兲 is the configurational inviscidity constraint, which will be defined in the sequel. Let us now start with the “naive” assumption that energy balance is materially covariant and see what its consequences are. Material covariance of energy balance: Let us postulate that the balance of energy is invariant under a diffeomorphism ⌶t : B → B, i.e., d dt

冕 ⬘冉 ⬘

冊冕

1 ␳0 E + 具具V⬘,V⬘典典 dV⬘ = 2 U⬘

U⬘


冕

⳵U⬘

共具具T⬘,V⬘典典 + H⬘兲dA⬘ . 共6.56兲

Proposition 6.3: If material energy balance is invariant under arbitrary material diffeomorphisms ⌶t : B → B, then

2␳0

⳵␳0 = 0, ⳵t

共6.57兲

⳵E = − FTP, ⳵G

共6.58兲

FTP = PTF,

共6.59兲

Div共FTP兲 = FT Div P.

共6.60兲


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J. Math. Phys. 47, 042903 共2006兲


Conversely, if the above four conditions hold, then material energy balance is invariant under any material diffeomorphism. Proof: Material balance of energy for U⬘ 傺 B⬘ at t = t0 reads

冊冕冉冕

冕冉冕

1 ⳵␳0 E + 具具V − FW,V − FW典典 dV + 2 ⳵ t U =

U

␳0共具具B⬘兩t=t0,V − FW典典 + R兲dV +

U

⳵U

␳0

冊

⳵E ⳵E + :LWG + 具具V − FW,A⬘兩t=t0典典 dV ⳵t ⳵G

共具具T,V − FW典典 + H兲dA.

共6.61兲

Subtracting balance of energy for U 傺 B from this and noting that 共A⬘ − B⬘兲t=t0 = A − B we obtain

冕冉冕

冊冕冉

1 ⳵␳0 − 具具V,FW典典 + 具具FW,FW典典 dV + 2 U ⳵t =−

U

具具␳0B,FW典典dV −

冕

⳵U

U

␳0

冊

⳵E :LWG − 具具FW,A典典 dV ⳵G

具具T,FW典典dA.

共6.62兲

We know that

冕

⳵U

具具FTT,W典典dA =

冕冋 U

册

1 具具Div共FTP兲,W典典 + FTP:⍀ + FTP: LWG dV. 2

共6.63兲

Thus, 共6.62兲 simplifies to

冕冉冕

冊冕冉

1 ⳵␳0 − 具具V,FW典典 + 具具FW,FW典典 dV + 2 U ⳵t +

⳵U

U

2␳0

冊

1 ⳵E + FTP : LWG dV + 2 ⳵G

具具Div共FTP兲 + ␳0FT共B − A兲,W典典 dV = 0.

冕

FTP:⍀ dV

U

共6.64兲

As U and W are arbitrary, we have

2␳0

⳵␳0 = 0, ⳵t

共6.65兲

⳵E = − FTP, ⳵G

共6.66兲

FTP = PTF,

共6.67兲

Div共FTP兲 + ␳0FTB = ␳0FTA.

共6.68兲

Equation 共6.65兲 is nothing but material conservation of mass. Equation 共6.66兲 is the material Doyle-Ericksen formula. This is what Lu and Papadopoulos26 call infinitesimal material covariance. Equation 共6.67兲 is balance of configurational angular momentum or isotropy of the material. 关Note that if 共6.66兲 holds then 共6.67兲 holds trivially.兴 Finally, Eq. 共6.68兲 is a condition that must be satisfied for the balance of energy to be invariant under material diffeomorphisms. This constraint is equivalent to Div共FTP兲 = FT Div P.

共6.69兲

Assuming the above four conditions, it is easy to show that material energy balance is invariant under arbitrary material diffeomorphisms.


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Ideal fluids do satisfy all these conditions. In fact, their transformation properties under material diffeomorphisms gives rise to Kelvin’s circulation theorem and it is a key ingredient in the geometric approach to fluid mechanics; see the introduction to the Marsden and Ratiu book29 for a discussion and references to the literature. C. Material energy balance and defects

We now make a connection between 共6.69兲 and Eshelby’s idea of force on a defect. The idea of a driving force in continuum mechanics goes back to Eshelby13–15 and this notion is important in developing evolution laws for the movement of defects, including dislocations, vacancies, interfaces, cavities, cracks, etc. Driving forces on these defects cause climb and glide of dislocations, diffusion of point defects, migration of interfaces, changing the shape of cavities and propagation of cracks, to mention a few examples. Eshelby defined the force on a defect as the generalized force corresponding to position of the defect 共in the reference configuration兲, which is thought of as a generalized displacement. Eshelby studied inhomogeneities in elastostatic and elastodynamic systems by considering the explicit dependence of the elastic energy density on position in the reference configuration. Defect forces: Suppose the elastic energy density has an explicit dependence on X 共the position of material points in the undeformed configuration兲, i.e., W = W共␸,F,X兲,

共6.70兲

where ␸ and F are the deformation mapping and the deformation gradient, respectively. Consider an open neighborhood ⍀ of an isolated defect. Force on the defect in the sense of Eshelby is defined as Fdefect =

冕冉冊 ⍀

⳵W ⳵X

dV =

冕

Div E dV =

冕

ˆ dA, EN

⳵⍀

⍀

explicit

共6.71兲

where E = WI − FTP is Eshelby’s energy-momentum tensor. It turns out that for a crack 共thought of as a defect兲 Fdefect is related to the celebrated J-integral;37 J is the component of Fdefect in the direction of crack propagation. The following proposition makes an explicit connection between 共6.69兲 and Eshelby’s idea of force on a defect. Proposition 6.4: Suppose an elastic material in an isothermal and quasistatic deformation satisfies the internal constraint Div共FTP兲 = FTDiv P. In the absence of body forces, force on a defect in the sense of Eshelby would be Fdefect =

冕冉冊 ⍀

⳵W ⳵X

dV = explicit

冕

Div E dV =

冕

ˆ dA. WN

⳵⍀

⍀

共6.72兲

Proof: Note that Fdefect =

冕

⳵⍀

ˆ dA = EN

冕

⳵⍀

ˆ dA − WN

冕

⍀

FTDiv P dV =

冕

ˆ dA. WN

⳵⍀

This means that the configurational traction on ⳵⍀ is normal to ⳵⍀ at all points, i.e., the configurational stress is hydrostatic. For this reason we call the internal constraint Div共FTP兲 = FT Div P, the configurational inviscidity constraint. If there is a stationary surface S across which deformation gradient and other quantities have jump discontinuities, the balance of standard forces reads ˆ = 0. 冀P冁N

共6.73兲

Now let us look at the normal jump in Eshelby’s energy-momentum tensor,


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J. Math. Phys. 47, 042903 共2006兲


ˆ = 冀⌿I − FTP冁N ˆ = 冀⌿冁N ˆ − 冀FT冁具P典N ˆ − 具FT典冀P冁N ˆ, 冀E冁N

共6.74兲

where 具·典 denotes average of inner and outer traces. Using Hadamard’s compatibility equations, 冀F冁tˆ = 0

ˆ = 0, for all tˆ such that tˆ · N

共6.75兲

it can be easily shown that ˆ · 冀E冁N ˆ = 冀⌿冁 − 冀F冁N ˆ · PN ˆ, N

ˆ = − Ftˆ · 冀P冁N ˆ. ˆt · 冀E冁N

共6.76兲

Now if the balance of standard forces hold one concludes that ˆ = 0. tˆ · 冀E冁N

共6.77兲

This means that jump in configurational traction on ⳵⍀ is always normal to ⳵⍀. However, the previous remark shows that in the absence of body forces the condition Div共FTP兲 = FT Div P implies that the configurational traction itself is normal to ⳵⍀. Are configurational forces newtonian?: There have been doubts and discussions concerning the nature of configurational forces in the literature already starting from Eshelby himself. Eshelby strongly believed that force on a defect is fictitious and is different from the usual forces in mechanics. He defined force on a defect to be the thermodynamic force conjugate to the generalized coordinates defining the defect, for example, the crack tip position in the case of a crack. Eshelby16 observed that the configurational force on a disclination in a liquid crystal is a real force. A similar observation was made by Nabarro34 for dislocations. Kröner23 and Ericksen11,12 have similar discussions. Batra4 argues that force on a defect is equal to the standard force exerted on the boundary of a subbody embracing the defect. Steinmann42 introduces the spatial signature of a material force. One should note that this viewpoint is not in agreement with Gurtin’s theory in which standard and configurational forces have their own balance laws. Batra4 proves a theorem that states that force on a defect is equal to the resultant of tractions on the boundary of any region enclosing this single defect. This seems to be a very surprising result. First of all, if body forces are considered resultant of tractions on different regions embodying the defect cannot be independent of the region as in this case stress tensor is not divergence free. Barta suggests that problems involving the J-integral could be reinterpreted using his theorem. As a matter of fact, the J-integral can serve as a counter example for Batra’s theorem. The reason is that in the case of a linear elastic material in mode I fracture, for example, the J-integral is quadratic in KI while the stress is linear in KI and hence the resultant of tractions acting on the boundary of a small region enclosing the crack would be linear in KI. This means that the J-integral, which is the component of configurational force in the direction of crack growth, cannot be a real force. The incorrectness of Batra’s theorem is because of the way he defines force on a defect. Force on a defect in the sense of Eshelby is the rate of change of potential energy of the elastic body with respect to changes in the position of the defect in the reference configuration. Batra defines force on a defect to be the rate of change of energy with respect to changes of position of the defect in the current configuration. This is the source of his surprising result. One should note that direct and inverse motions describe the same physical process and cannot lead to different conclusions regarding forces. Having the duality picture is useful but one should note that positions of defects in the reference and current configurations are not related by the standard deformation mapping as the evolution of defects is an independent kinematical process. Standard forces in continuum mechanics are one forms in the deformed configuration, i.e., at each point x 苸 S, force is an element of T*x S. Configurational forces on the other hand are one forms in the reference configuration, i.e., at each point X 苸 B, configurational force is an element of TX* B. Therefore, geometrically it is meaningless to ask if a configurational force is a real force


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very much like asking whether the deformation gradient 共a two-point tensor兲 is symmetric. This is why arguments like the one proposed by Steinmann43 where he defines a spatial signature for a material force do not make sense from the geometric standpoint. Plasticity and embeddings: A traditional means of introducing configurational forces is based on remapping the reference configuration of the body. However, this approach tacitly assumes that the reference configuration can be embedded in Euclidean space. This approach fails when there is no natural embedding of the reference configuration. A case in point is provided by multiplicative plasticity,24 where the total deformation gradient at a point x has the representation: F共x兲 = Fe共x兲F p共x兲, where Fe共x兲 and F p共x兲 are the elastic and plastic deformation gradients, both of which fail to be a gradient in general. The plastic deformation mapping F p共x兲 defines an intermediate configuration that defines the reference configuration for the elasticity of the material. In particular, the elastic energy density is assumed to be of the form W共Fe , x兲. Since F p共x兲 is not the derivative of a mapping, the intermediate configuration cannot be embedded in Euclidean space. Therefore, remapping cannot be applied to deriving configurational forces directly from W共Fe , x兲. By contrast, the present approach can be applied for that purpose, for example, by equipping the intermediate configuration with a constant metric. The derivation of certain conserved integrals, such as the L-integral that gives the configurational torque on isotropic subbodies, relies on the metric structure of the embedding Euclidean space. In addition, the conventional formulation of material symmetry also presumes the existence of an Euclidian embedding. Such an embedding may not be natural or available in certain models of materials, such as liquid crystals or smectic polymers, where the reference configuration may include a unit director field.

VII. NOETHER’S THEOREM AND BALANCE OF CONFIGURATIONAL FORCES

As is well known, there is a strong connection between conservation laws and symmetries. If the Euler-Lagrange equations are satisfied and the Lagrangian density of a system is invariant under a group of transformations, Noether’s theorem gives the corresponding conserved quantity. In this sense, conservation laws are related to symmetries of a given system. Marsden and Hughes28 consider material invariance in elasticity 共in the absence of body forces兲 and show that invariance of Lagrangian density under rigid translations in the reference configuration results in the following conservation law

⳵ 共⳵␾˙ L · D␾ · W兲 + DIV共⳵FL · D␾ · W − LW兲 = 0. ⳵t

共7.1兲

This has been obtained assuming that the flow of W is volume-preserving and that L does not explicitly depend on X. For a constant W, this equation in our notation reads

冋冉

Div

冊

册

1 ⳵ ⌿ − ␳0兩V兩2 I − FTP = − 共␳0FTV兲. 2 ⳵t

共7.2兲

It is seen that this is identical to balance of configurational linear momentum if ␳0 and ⌿ are independent of X 共note that this is stronger that homogeneity of L兲. Ignoring the inertial effects, Noether’s theorem results in Div共⌿I − FTP兲 = 0.

共7.3兲

Roughly speaking, Noether’s theorem states that when the Euler-Lagrange equations are satisfied, any symmetry of the Lagrangian density corresponds to a conserved quantity. Here we revisit Noether’s theorem for nonlinear elasticity assuming that undeformed and deformed configurations are Riemannian manifolds. Writing action in the reference configuration, Lagrangian density has the following explicit independent variables:


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J. Math. Phys. 47, 042903 共2006兲


L = L共XA, ␸a, ␸˙ a,FaA,GAB,gab兲.

共7.4兲

For the sake of clarity, we consider spatial and material symmetries of the Lagrangian density separately. A. Spatial covariance of Lagrangian density

Theorem 7.1: If the Lagrangian density is spatially covariant, then the following hold: (i) spatial homogeneity of the Lagrangian density and (ii) the Doyle-Ericksen formula. Proof: Suppose ␺s is a flow on S generated by a vector field w, i.e.,

冏冏 d ds

␺s ⴰ ␸ = w ⴰ ␸ .

共7.5兲

s=0

Invariance of L means that

冉

L XA, ␺sa共␸兲,

冊

⳵ ␺sa b ⳵ ␺sa b ⳵ ␺sc ⳵ ␺sd ˙ ␸ , F ,G ,− gcd = L共XA, ␸a, ␸˙ a,FaA,GAB,gab兲. AB A ⳵ xb ⳵ xb ⳵ xa ⳵ xb

共7.6兲

关This reminds us of the definition of covariance for internal energy density. So, it would be very natural to expect some connection between Noether’s theorem and covariant balance laws.兴 Now differentiating the above relation with respect to s and then evaluating it at s = 0 关This is somewhat similar to subtracting two balance relations and evaluating the result at t = t0兴, one obtains

⳵L ⳵␸

aw

a

+

冉冊

⳵ L ⳵ wa b ⳵L ␸˙ + a ⳵ xb ˙ ⳵F ⳵␸

A a

⳵ L ⳵ wc ⳵ wa b gbc = 0. bF A−2 ⳵x ⳵ gab ⳵ xa

共7.7兲

Note that

冉冊冉冊

⳵ L ⳵ wa b ⳵ ⳵ L a ⳵ ⳵L a ␸˙ = w − w . a ⳵ xb a ˙ ⳵ t ˙ ⳵ t ⳵␸˙ a ⳵␸ ⳵␸

共7.8兲

After some manipulations, it can be shown that

冉冊 ⳵L ⳵F

A a

⳵ wa b F A= ⳵ xb

冋冉冊册冉冊 ⳵L

A

wa

⳵F

−

兩A

a

⳵L ⳵F

A

wa − a 兩A

冉冊 ⳵L ⳵F

A b c ␥ac F Aw a .

共7.9兲

b

Also

−2

冋

⳵ L ⳵ wc ⳵L gbc共F−1兲aAwc a gbc = − 2 ⳵ gab ⳵ x ⳵ gab

册冋

+ 2

兩A

⳵L ⳵ gab

gbc共F−1兲aA

册

wc

+2 兩A

⳵L ⳵ gab

d c w . gbd␥ac

共7.10兲 Therefore, symmetry of L implies that

冋

⳵L ⳵␸

a

+

冉冊冉冊冉冊冋冉冊册冋册冋

−

⳵ ⳵L ⳵ t ⳵␸˙ ⳵L ⳵F

−

a

⳵L

A

⳵F

a 兩A

−

A

wa

a

− 2

兩A

⳵L

⳵ gab

⳵L ⳵F

A

b FcA␥ac +2

b

gbc共F−1兲aAwc

⳵L

⳵ gcd

+ 2

兩A

册冉冊

b gbd␥ac wa +

⳵L ⳵ gab

gbc共F−1兲aA

⳵ ⳵L a w ⳵ t ⳵␸˙ a

册

wc = 0. 兩A

共7.11兲

Note that the term multiplied by wa is zero if the Euler-Lagrange equations are satisfied. Thus, Noether’s theorem states that


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J. Math. Phys. 47, 042903 共2006兲


冉冊冋冉冊册冋

⳵ ⳵L a w + ⳵ t ⳵␸˙ a

⳵L

⳵L

A

wa

⳵F

− 2

兩A

a

⳵ gab

gbc共F−1兲aAwc

册冋

+ 2

兩A

⳵L ⳵ gab

gbc共F−1兲aA

册

wc = 0. 兩A


冋冉冊册冉冊 ⳵L

A

wa

⳵F

=

兩A

a

⳵L

A

⳵F

a 兩A

冉冊 ⳵L

wa +

⳵F

A

共7.13兲

FbAwa兩b . a

Also

冋

2

⳵L ⳵ gab

gbc共F−1兲aAwc

册冋

= 2

兩A

⳵L ⳵ gab

gbc共F−1兲aA

册

wc + 2 兩A

⳵L ⳵ gab

共7.14兲

gbcwc兩a .

Therefore 共7.12兲 is simplified to

冋冉冊冉冊册冉冊冋冉冊 ⳵L ⳵ ⳵L + ⳵ t ⳵␸˙ a ⳵F

A

wa +

a 兩A

⳵ L ⳵ wa + ⳵␸˙ a ⳵ t

⳵L ⳵F

A

F cA − 2 a

⳵L ⳵ gbc

册

gab wa兩c = 0.

共7.15兲

Note that

⳵ wa ⳵ wa c a d c = c ␸˙ = ␸˙ cwa兩c − ␥cd w ␸˙ . ⳵t ⳵x

共7.16兲

Therefore statement of Noether’s theorem, Eq. 共7.12兲 can be rewritten as

冋冉冊冉冊 ⳵L ⳵ ⳵L + a ⳵ t ⳵␸˙ ⳵F

⳵L

A

− a 兩A

␥d ␸˙ c ˙ d ac

⳵␸

册冋冉冊 ⳵L

wa +

⳵F

A

F cA + a

⳵L ⳵␸

˙a

␸˙ c − 2

⳵L ⳵ gbc

册

gab wa兩c = 0. 共7.17兲

Note that

⳵L ⳵␸˙ a

␸˙ c = ␳0gab␸˙ b␸˙ c .

共7.18兲

If Lagrangian density is covariant, i.e., if w is arbitrary then 共7.17兲 implies that

2

⳵L ⳵ gab

= gbc

冉冊 ⳵L ⳵F

冉冊冉冊

⳵L ⳵ ⳵L + a ⳵ t ⳵␸˙ ⳵F

A

FaA + gbc c

A

− a 兩A

⳵L ⳵␸˙ d

⳵L ⳵␸˙ c

␸˙ a ,

d c ␥ac ␸˙ = 0.

共7.19兲

共7.20兲

Equation 共7.19兲 can be rewritten as 2

冉冊

⳵W ⳵W = gbc ⳵ gab ⳵F

A

F aA ,

共7.21兲

c

which is nothing but the Doyle-Ericksen formula. 关Note that this includes balance of angular momentum.兴 Note that


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J. Math. Phys. 47, 042903 共2006兲


冉冊冉冊

⳵L ⳵ ⳵L + a ⳵ t ⳵␸˙ ⳵F

A

− a 兩A

⳵L

⳵L

⳵␸˙

⳵␸

␥d ␸˙ c = d ac

a

−

冉冊 ⳵L ⳵F

⳵L

A b FcA␥ac +2 b

⳵ gcd

b gbd␥ac −

⳵L ⳵␸˙ b

b ␸˙ c␥ac .

共7.22兲 But

−

冉冊 ⳵L ⳵F

A

F cA + 2 b

⳵L ⳵ gcd

gbd =

⳵L ⳵␸˙ b

␸˙ c .

共7.23兲

Thus

冉冊冉冊

⳵L ⳵ ⳵L + a ⳵ t ⳵␸˙ ⳵F

A

− a 兩A

⳵L

␥d ␸˙ c ˙ d ac

⳵␸

=

⳵L ⳵␸a

.

共7.24兲

Hence 共7.20兲 implies that

⳵L ⳵␸a

共7.25兲

= 0.

䊐 Note that this theorem implies that arbitrary flows and in particular rigid translations cannot be transitive 共in the sense of Gotay et al.17,18兲 for arbitrary Lagrangian densities.

B. Material covariance of Lagrangian density

Let us first consider the case of Euclidean spaces. Consider a flow ⌳s on B generated by a vector field W. Invariance of L with respect to this flow means that

冉

L ⌳sA共X兲, ␸a, ␸˙ a,

冋冉冊册冊 ⳵ ⌳s ⳵X

−1 B

FaB = L共XA, ␸a, ␸˙ a,FaA兲.

共7.26兲

A

Differentiating the above relation with respect to s and evaluating the result at s = 0, one obtains

⳵L ⳵X

A AW −

⳵ L ⳵ WB a F B = 0. ⳵ F aA ⳵ X A

共7.27兲

If W is a constant, then

⳵L ⳵ XA

共7.28兲

= 0,

i.e., the Lagrangian density must be materially homogeneous. This is also what Nelson35,36 obtains. After some manipulation and assuming that Euler-Lagrange equations are satisfied 共7.27兲 can be rewritten as

冉

冊冉

冊

⳵ ⳵ a B ⳵ ⳵L a A ⳵ WA A − F W − L = 0, A A LW − a F BW ⳵X ⳵F A ⳵ t ⳵␸˙ a ⳵ XA

共7.29兲

where use was made of the fact that W is time independent. For a volume-preserving flow this gives us


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J. Math. Phys. 47, 042903 共2006兲


冊冉

冉

冊

⳵ ⳵ a B ⳵ ⳵L a A A − F AW = 0, A LW − a F BW ⳵X ⳵F A ⳵ t ⳵␸˙ a

共7.30兲

which is what Marsden and Hughes28 obtain. Now let us consider the general case of Riemannian manifolds and assume that ⌳s is a flow on the Riemannian manifold 共B , G兲 generated by a vector field W, i.e.,

冏冏 d ds

⌳s共X兲 = W共X兲,

X 苸 B.

共7.31兲

s=0

Theorem 7.2: If the Lagrangian density is materially covariant then the following hold: 共i兲 material homogeneity of the Lagrangian density and 共ii兲 material Doyle-Ericksen formula. Proof: Invariance of L with respect to ⌳s means that

冉

L ⌳sA共X兲, ␸a, ␸˙ a,

冋冉冊册 ⳵ ⌳s ⳵X

−1 B

FaB,gab, A

冋冉冊册冋冉冊册冊 ⳵ ⌳s ⳵X

⳵ ⌳s ⳵X

−1 C

A

−1 D

GCD

B

= L共XA, ␸a, ␸˙ a,FaA,gab,GAB兲.

共7.32兲

Differentiating the above relation with respect to s and evaluating the result at s = 0, one obtains

⳵L ⳵X

A AW −

⳵ L ⳵ WB a ⳵L ⳵ WC F − 2 G = 0. DC B ⳵ XK ⳵ F aA ⳵ X A ⳵ GDK

共7.33兲

Note that

冉

冊冉

⳵L ⳵ WC GDCWC K =− 2 ⳵X ⳵ GDK

冊冉

⳵L B ⳵ L ⳵ WB a a W AF B=− F B a ⳵ F aA ⳵F A ⳵X

−

兩A

+ F aB

⳵L ⳵F

a

冊

A 兩A

W B + F aC

⳵L ⳵ F aA

C ⌫AB WB

共7.34兲

and

⳵L

−2

⳵ GDK

GDC

冉

+ 2

兩K

⳵L ⳵ GDK

GDC

冊

WC + 2 兩K

⳵L ⳵ GDK

B GBD⌫CK WC .

共7.35兲 Also

冉

F aB

⳵L ⳵F

a

冊冉 WB =

A 兩A

冊

⳵L B ⳵ F aB C a W + F aA A − F C⌫BA ⳵X ⳵ F aA

冋冉冊 ⳵L

⳵F

a

+

B 兩B

⳵L ⳵ F bB

册

b FcB␥ac WA .

共7.36兲 Assuming that Euler-Lagrange equations are satisfied and using the above identities after a lengthy series of simplifications, one obtains

冉

LWA −

⳵L ⳵ F aB

WB

冊冉 −

兩A

冊

冉

⳵L ⳵L ⳵ FaA a WA − LWA兩A − 2 GDCWC ˙ ⳵t ⳵ GDK ⳵␸

冊冉

+ 2

兩K

⳵L ⳵ GDK

GDC

冊

WC = 0. 兩K


冉

− 2

⳵L ⳵ GDK

GDCWC

冊冉

+ 2

兩K

⳵L ⳵ GDK

GDC

冊

WC = − 2 兩K

⳵L ⳵ GDK

C GDCW兩K .

共7.38兲

Therefore in this case Noether’s theorem states that


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J. Math. Phys. 47, 042903 共2006兲


冉

⳵L

LWA −

⳵F

a B

WB

冊冉 −

兩A

冊

⳵L ⳵L ⳵ FAa a WA − LWA兩A − 2 GDCWC兩K = 0. ⳵t ⳵ GDK ⳵␸˙

共7.39兲

Note that

冉

F aB

⳵L ⳵F

a

WB

A

冊

= 兩A

冉

冊

⳵L ⳵L ⳵L ⳵ B C a WB + FaB a WB + FaB a 共WB⌫AC − WC⌫CA 兲. a A F B ⳵X ⳵F A ⳵F A ⳵F A 共7.40兲

Using the above relation and some lengthy simplifications, one can rewrite 共7.39兲 as

冋冉 ⳵L

⳵X

A

+

⳵L ⳵F

a

F aB + 2

C

⳵L ⳵ GCD

冊册冉

B GBC ⌫AC WA −

⳵L ⳵F

a

F aB + 2

A

⳵L ⳵ GAC

冊

GBC WB = 0.

共7.41兲

If L is materially covariant, i.e., if W is arbitrary, then

⳵L ⳵ XA

+

冉

⳵L ⳵F

a

⳵L ⳵F

a

F aB + 2

C

F aB + 2

A

⳵L ⳵ GCD ⳵L

⳵ GAC

冊

B GBC ⌫AC = 0,

GBC = 0.

共7.42兲

共7.43兲

Or equivalently

⳵L ⳵ XA

= 0,

⳵W a ⳵W GBC = 0, a F B+2 ⳵ GAC ⳵F A

共7.44兲

共7.45兲

where W is the material potential energy density. Note that 共7.45兲 is nothing but the material Doyle-Ericksen formula 共6.66兲. 䊐 Remarks: There are some differences between covariant energy balance 共CEB兲 and Lagrangian density covariance 共LDC兲: 共i兲共ii兲共iii兲

CEB is global while LDC is local. In CEB the arbitrary vector fields w and W are time-dependent 共being velocities兲, in general, while in LDC they are time independent. In writing balance of energy in CEB for a material diffeomorphism spatial quantities contribute to energy balance. But in LDC a material flow does not affect the spatial quantities.

VIII. CONCLUSIONS AND FUTURE DIRECTIONS

The results of this paper can be summarized as follows. 共i兲

We studied continuum mechanics of bodies with global referential evolutions by enlarging the configuration manifold to two Riemannian manifolds with their own metrics. A deformation is then a pair of referential evolution, i.e., a motion in the referential manifold, and a standard motion. We showed that in the absence of discontinuities, configurational and standard equations of motion are equivalent even if the metrics are allowed to vary.


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共ii兲共iii兲

共iv兲

共v兲


J. Math. Phys. 47, 042903 共2006兲

The classical theorem of Green, Naghdi, and Rivilin19 was revisited and a material version of it was investigated. We showed that under a referential isometry balance of energy cannot be invariant, in general, and obtained its transformation. The idea of covariance in elasticity was reviewed. We revisited a theorem by Marsden and Hughes28 and some of the details of its proof were clarified and a proof was given for its converse. It was also shown that spatial covariance of material energy balance leads to identical results. We posed the question that whether energy balance can be materially covariant. It was shown that, in general, energy balance cannot be invariant under referential diffeomorphisms. We obtained the transformation of energy balance under arbitrary material diffeomorphisms. We found conditions under which energy balance is materially covariant. It was shown that in the absence of body forces the nontrivial condition for material covariance of balance of energy is equivalent to configurational stress tensor 共Eshelby’s stress tensor兲 being hydrostatic. It was shown that for ideal fluids energy balance is materially covariant. An explicit relation between covariance and Noether’s theorem was found. We showed that spatial covariance of a Lagrangian density implies spatial homogeneity of the Lagrangian density and the Doyle-Ericksen formula. Similarly, material covariance of a Lagrangian density implies its material homogeneity and the material Doyle-Ericksen formula.

In summary, spatial covariance is reasonable and holds for most materials. The transformation properties of energy balance under material reframings was obtained. However, material covariance of energy balance only holds for special materials, such as ideal fluids. The main application of the ideas presented in this paper will be in gaining a better understanding of the continuum theory of defects. In particular, if one repeats some of the developments presented in this paper in a space-time setting, one should, in principle, be able to obtain dynamic equations for evolution of defects. Another important relevant problem would be the study of covariance and its meaning in discrete systems. This may lead to a better understanding of “stress” in discrete systems. ACKNOWLEDGMENTS

The authors have benefited from discussions with M. Arroyo, K. Bhattacharya, M. Desburn, E. Kanso, J. K. Knowles, J. Lu, P. M. Mariano, P. Papadopoulos, Y. Tong, and M. Zielonka. The research of one of the authors 共J.E.M兲 was partially supported by the California Institute of Technology and NSF-ITR Grant No. ACI-0204932. 1

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