Differential Complexes in Continuum Mechanics

Transcription

1 Archive for Rationa Mechanics and Anaysis manuscript No. (wi be inserted by the editor) Arzhang Angoshtari Arash Yavari Differentia Compexes in Continuum Mechanics Abstract We study some differentia compexes in continuum mechanics that invove both symmetric and non-symmetric second-order tensors. In particuar, we show that the tensoria anaogue of the standard grad-curdiv compex can simutaneousy describe the kinematics and the kinetics of motions of a continuum. The reation between this compex and the de Rham compex aows one to readiy derive the necessary and sufficient conditions for the compatibiity of the dispacement gradient and the existence of stress functions on non-contractibe bodies. We aso derive the oca compatibiity equations in terms of the Green deformation tensor for motions of 2D and 3D bodies, and shes in curved ambient spaces with constant curvatures. Contents 1 Introduction Differentia Compexes for Second-Order Tensors Compexes Induced by the de Rham Compex Compexes for 3-manifods Compexes for 2-manifods Compexes Induced by the Caabi Compex The Linear Easticity Compex for 3-manifods Arzhang Angoshtari Schoo of Civi and Environmenta Engineering, Georgia Institute of Technoogy, Atanta, GA 30332, USA. E-mai: arzhang@gatech.edu Arash Yavari Schoo of Civi and Environmenta Engineering & The George W. Woodruff Schoo of Mechanica Engineering, Georgia Institute of Technoogy, Atanta, GA 30332, USA. Te.: Fax: E-mai: arash.yavari@ce.gatech.edu

2 2 Arzhang Angoshtari, Arash Yavari The Linear Easticity Compex for 2-manifods Some Appications in Continuum Mechanics Compatibiity Equations Bodies with the Same Dimensions as the Ambient Space Shes Linearized Easticity on Curved Manifods Stress Functions Further Appications Introduction Differentia compexes can provide vauabe information for soving PDEs. The ceebrated de Rham compex is a cassica exampe: Let B be a 3-manifod and et Ω k (B) be the space of smooth 1 k-forms on B, i.e. α Ω k (B) is an anti-symmetric ( 0 k )-tensor with smooth components α i 1 i k. The exterior derivatives d k : Ω k (B) Ω k+1 (B) are inear differentia operators satisfying d k+1 d k = 0, where denotes the composition of mappings. 2 Using the agebraic anguage, one can simpy write the compex 0 Ω 0 (B) d Ω 1 (B) d Ω 2 (B) d Ω 3 (B) 0, to indicate that d is inear and the composition of any two successive operators vanishes. Note that the first operator on the eft sends 0 to the zero function and the ast operator on the right sends Ω 3 (B) to zero. The above compex is caed the de Rham compex on B and is denoted by (Ω(B), d). The compex property d d = 0, impies that im d k (the image of d k ) is a subset of ker d k+1 (the kerne of d k+1 ). Compex (Ω(B), d) is exact if im d k = ker d k+1. Given β Ω k (B), consider the PDE dα = β. Ceary, β im d, is the necessary and sufficient condition for the existence of a soution. If (Ω(B), d) is exact, then dβ = 0, guarantees that β im d. In genera, the de Rham cohomoogy groups H k dr (B) = ker d k/im d k 1, quantify the deviation of (Ω(B), d) from being exact, i.e. this compex is exact if and ony if a H k dr (B) are the trivia group {0}. If H k dr (B) is finite-dimensiona, then the ceebrated de Rham theorem tes us that dim H k dr (B) = b k(b), where the k-th Betti number b k (B) is a purey topoogica property of B. For exampe, if B is contractibe, i.e. does not have any hoe in any dimension, then b k (B) = 0, k 1, or if B is simpy-connected, then b 1 (B) = 0. On contractibe bodies, dβ = 0, is the necessary and sufficient condition for the sovabiity of dα = β. If B is non-contractibe, then the de Rham theorem tes us that β im d if and ony if dβ = 0, and β = 0, (1.1) c k where for the purposes of this work, c k can be considered as an arbitrary cosed (i.e. compact without boundary) k-dimensiona C 0 -manifod inside 1 Throughout this paper, smooth means C. 2 When there is no danger of confusion, the subscript k in d k is dropped.

3 Differentia Compexes in Continuum Mechanics 3 B. 3 Thus, we observe that the de Rham compex together with the de Rham theorem provide the required conditions for the sovabiity of dα = β. Suppose that B is the interior of another manifod B. One can restrict (Ω(B), d) to ( Ω( B), d ), where Ω k ( B) is the space of smooth forms in Ω k (B) that can be continuousy extended to the boundary B of B as we. If B is compact, then ( Ω( B), d ) induces various Hodge-type decompositions on Ω k ( B). Such decompositions aow one to study the above PDE subject to certain boundary conditions, e.g. see Schwarz [25], Gikey [15]. On the other hand, it has been observed that compexes can be usefu for obtaining stabe numerica schemes. By propery discretizing the de Rham compex, one can deveop stabe mixed formuations for the Hodge-Lapacian [3, 4]. Generaizing the above resuts for an arbitrary differentia compex can be a difficut task, in genera. This can be significanty simpified if one can estabish a connection between a given compex and the de Rham compex. The grad-cur-div compex of vector anaysis is a standard exampe: Let C (B) and X(B) be the spaces of smooth rea-vaued functions and smooth vector fieds on B. From eementary cacuus, we know that for a 3-manifod B R 3, one can define the gradient operator grad : C (B) X(B), the cur operator cur : X(B) X(B), and the divergence operator div : X(B) C (B). It is easy to show that cur grad = 0, and div cur = 0. These reations aow one to write the compex 0 C (B) grad X(B) cur X(B) div C (B) 0, that is caed the grad-cur-div compex or simpy the gcd compex. It turns out that the gcd compex is equivaent to the de Rham compex in the foowing sense: Let {X I } be the Cartesian coordinates of R 3. We can define the foowing isomorphisms ı 0 : C (B) Ω 0 (B), ı 0 (f) = f, ı 1 : X(B) Ω 1 (B), (ı 1 (Y )) I = Y I, ı 2 : X(B) Ω 2 (B), (ı 2 (Y )) IJ = ε IJK Y K, ı 3 : C (B) Ω 3 (B), (ı 3 (f)) 123 = f, (1.2) where ε IJK is the standard permutation symbo. Simpe cacuations show that ı 1 grad = d ı 0, ı 2 cur = d ı 1, ı 3 div = d ı 2. These reations can be succincty depicted by the foowing diagram. 0 C (B) grad X(B) cur X(B) div C (B) ı 0 0 Ω 0 (B) d ı 1 ı 2 Ω 1 (B) d Ω 2 (B) ı 3 0 d Ω 3 (B) 0 (1.3) 3 In fact, c k is a singuar k-chain in B that can be identified with (a forma sum of) cosed k-manifods for integration, see standard texts such as [6, 22] for the precise definition of c k.

4 4 Arzhang Angoshtari, Arash Yavari Diagram (1.3) suggests that any resut hoding for the de Rham compex shoud have a counterpart for the gcd compex as we. 4 For exampe, diagram (1.3) impies that ı 0,..., ı 3 aso induce isomorphisms between the cohomoogy groups. This means that Y = grad f, if and ony if ı 1 (Y ) = d(ı 0 (f)), and simiary Y = cur Z, if and ony if ı 2 (Y ) = d(ı 1 (Z)). By using the de Rham theorem and (1.1), one can show that Y is the gradient of a function if and ony if cur Y = 0, and ı 1 (Y ) = G(Y, t )ds = 0, B, (1.4) where is an arbitrary cosed curve in B, t is the unit tangent vector fied aong, and G(Y, t ) is the standard inner product of Y and t in R 3. Simiary, one concudes that Y is the cur of a vector fied if and ony if div Y = 0, and ı 2 (Y ) = G(Y, N C )da = 0, C B, (1.5) C C where C is any cosed surface in B and N C is the unit outward norma vector fied of C. If B is compact and if we restrict C (B) and X(B) to smooth functions and vector fieds over B, then by equipping the spaces in diagram (1.3) with appropriate L 2 -inner products, ı 0,..., ı 3 become isometries. Therefore, any orthogona decomposition for Ω k ( B), k = 1, 2, induces an equivaent decomposition for X( B) as we, e.g. see Schwarz [25, Coroary 3.5.2]. Moreover, one can study soutions of the vector Lapacian = grad div cur cur, and deveop stabe numerica schemes for it by using the corresponding resuts for the Hodge-Lapacian [3, 4]. In summary, the diagram (1.3) aows us to extend a the standard theories deveoped for the de Rham compex to the gcd compex. The notion of a compex has been extensivey used in inear easticity: Motivated by the mechanics of distributed defects, and in particuar incompatibiity of pastic strains, Kröner [21] introduced the so-caed inear easticity compex, aso caed the Kröner compex, which is equivaent to a compex in differentia geometry due to Caabi [7]. Eastwood [11] derived a construction of the inear easticity compex from the de Rham compex. Arnod et a. [3] used the inear easticity compex and obtained the first stabe mixed formuation for inear easticity. This compex can be used for deriving Hodge-type decompositions for inear easticity as we [14]. To our best knowedge, there has not been any discussion on anaogous differentia compexes for genera (noninear) continua. Contributions of this paper. Introducing differentia compexes for genera continua is the main goa of this paper. We can summarize the main contributions as foows. We show that a tensoria anaogue of the gcd compex caed the GCD compex, can address both the kinematics and the kinetics of motions of continua. More specificay, the GCD compex invoves the dispacement 4 More precisey, isomorphisms ı 0,..., ı 3 induce a compex isomorphism.

5 Differentia Compexes in Continuum Mechanics 5 gradient and the first Pioa-Kirchhoff stress tensor. We show that a diagram simiar to (1.3) commutes for the GCD compex as we, and therefore, the noninear compatibiity equations in terms of the dispacement gradient and the existence of stress functions for the first Pioa-Kirchhoff stress tensor directy foows from (1.1). Another tensoria version of the gcd compex is the gcd compex that invoves non-symmetric second order tensors. This compex aows one to introduce stress functions for nonsymmetric Cauchy stress and the second Pioa-Kirchhoff stress tensors. By using the Cauchy and the second Pioa-Kirchhoff stresses, one obtains compexes that ony describe the kinetics of motions. It has been mentioned in severa references in the iterature that the inear easticity compex is equivaent to the Caabi compex, e.g. see [11]. Athough this equivaence is trivia for the kinematics part of the inear easticity compex, in our opinion, it is not trivia at a for the kinetics part. Therefore, we incude a discussion on the equivaence of these compexes by using a diagram simiar to (1.3). Another reason for studying the above equivaence is that it heps us understand the reation between the inear easticity compex and the GCD compex. In particuar, the inear easticity compex is not the inearization of the GCD compex. The Caabi compex aso provides a coordinate-free expression for the inear compatibiity equations. Using the above compexes one observes that on a 3-manifod, the inear and noninear compatibiity probems, and the existence of stress functions are reated to H 1 dr (B) and H2 dr (B), respectivey. Using the ideas underying the Caabi compex, we derive the noninear compatibiity equations in terms of the Green deformation tensor for motions of bodies (with the same dimensions as ambient spaces) and shes in curved ambient spaces with constant curvatures. Notation. In this paper, we use the pair of smooth Riemannian manifods (B, G) with oca coordinates {X I } and (S, g) with oca coordinates {x i } to denote a genera continuum and its ambient space, respectivey. If B R n, then B denotes the cosure of B in R n. Uness stated otherwise, we assume the summation convention on repeated indices. The space of smooth rea-vaued functions on B is denoted by C (B). We use Γ (V) to indicate smooth sections of a vector bunde V. Thus, Γ ( 2 T B) and Γ ( 2 T B) are the spaces of ( 2 0 )- and (0 2 )-tensors on B. The space of symmetric (0 2 )-tensors are denoted by Γ (S 2 T B). It is customary to write X(B) := Γ (T B), and Ω k (B) := Γ (Λ k T B), i.e. Ω k (B) is the space of anti-symmetric ( 0 k )-tensors or simpy k-forms. Tensors are indicated by bod etters, e.g. T Γ ( 2 T B) and its components are indicated by T IJ or (T ) IJ. The space of k-forms with vaues in R n is denoted by Ω k (B; R n ), i.e. if α Ω k (B; R n ), and X 1,..., X k T X B, then α(x 1,..., X k ) R n, and α is anti-symmetric. Let ϕ : B S be a smooth mapping. The space of two-point tensors over ϕ with components F ii is denoted by Γ (T ϕ(b) T B).

6 6 Arzhang Angoshtari, Arash Yavari 2 Differentia Compexes for Second-Order Tensors In this section, we study some differentia compexes for 2D and 3D manifods that contain second-order tensors. These compexes fa into two categories: Those induced by the de Rham compex and those induced by the Caabi compex. Compexes induced by the de Rham compex incude arbitrary second-order tensors and can be considered as tensoria versions of the gcd compex. Compexes induced by the Caabi compex invove ony symmetric second-order tensors. In the foowing, we study the appications of these compexes to some cassica probems in continuum mechanics. 2.1 Compexes Induced by the de Rham Compex Compexes for second-order tensors that are induced by the de Rham compex ony contain first-order differentia operators. We begin our discussion by considering 3-manifods and wi ater study 2-manifods separatey Compexes for 3-manifods Let B R 3 be a 3-manifod and suppose {X I } is the Cartesian coordinates on B. We equip B with metric G, which is the standard metric of R 3. The gradient of vector fieds and the cur and the divergence of ( 2 0 )-tensors are defined as grad : X(B) Γ ( 2 T B), cur : Γ ( 2 T B) Γ ( 2 T B), div : Γ ( 2 T B) X(B), (grad Y ) IJ = Y I,J, (cur T ) IJ = ε IKL T JL,K, (div T ) I = T IJ,J, where,j indicates / X J. We aso define the operator cur T : Γ ( 2 T B) Γ ( 2 T B), (cur T T ) IJ = (cur T ) JI. It is straightforward to show that cur T grad = 0, and div cur T = 0. Thus, we obtain the foowing compex 0 X(B) grad Γ ( 2 T B) curt Γ ( 2 T B) div X(B) 0, that, due to its resembance with the gcd compex, is caed the gcd compex. Interestingy, simiar to the gcd compex, usefu properties of the gcd compex aso foow from the de Rham compex. This can be described via the R 3 -vaued de Rham compex as foows. Let d : Ω k (B) Ω k+1 (B) be the standard exterior derivative given by (dβ) I0 I k = k ( 1) i β I0 Î i I k,i i, i=0

7 Differentia Compexes in Continuum Mechanics 7 where the hat over an index impies the eimination of that index. Any α Ω k (B; R 3 ) can be considered as α = (α 1, α 2, α 3 ), with α i Ω k (B), i = 1, 2, 3. One can define the exterior derivative d : Ω k (B; R 3 ) Ω k+1 (B; R 3 ) by dα = (dα 1, dα 2, dα 3 ). Since d d = 0, we aso concude that d d = 0, which eads to the R 3 -vaued de Rham compex ( Ω(B; R 3 ), d ). Given α Ω k (B; R 3 ), et [α] i I 1 I k denote components of α i Ω k (B). By using the goba orthonorma coordinate system {X I }, one can define the foowing isomorphisms ı 0 : X(B) Ω 0 (B; R 3 ), [ı 0 (Y )] i = δ ii Y I, ı 1 : Γ ( 2 T B) Ω 1 (B; R 3 ), [ı 1 (T )] i J = δ ii T IJ, ı 2 : Γ ( 2 T B) Ω 2 (B; R 3 ), [ı 2 (T )] i JK = δ ii ε JKL T IL, ı 3 : X(B) Ω 3 (B; R 3 ), [ı 3 (Y )] i 123 = δ ii Y I, where δ ii is the Kronecker deta. Let T T be the transpose of T, i.e. ( T T) IJ = T JI, and et {E I } be the standard basis of R 3. For T Γ ( 2 T B), we define TN to be the traction of T T in the direction of unit vector N = N I E I S 2, where S 2 R 3 is the unit 2-sphere. Thus, T N = N I T IJ E J. By using (1.2), we can write ( TE1 ) ( TE2 ) ( TE3 )) ı k (T ) = (ı k, ı k, ı k, k = 1, 2. (2.1) It is easy to show that ı 1 grad = d ı 0, ı 2 cur T = d ı 1, ı 3 div = d ı 2. Therefore, the foowing diagram, which is the tensoria anaogue of the diagram (1.3) commutes for the gcd compex. 0 X(B) grad Γ ( 2 T B) curt Γ ( 2 T B) ı 0 ı 1 ı 2 div X(B) 0 Ω 0 (B; R 3 ) d Ω 1 (B; R 3 ) d Ω 2 (B; R 3 ) d Ω 3 (B; R 3 ) 0 ı 3 0 (2.2) Remark 1. Diagram (1.3) is vaid for any 3-manifod, see Schwarz [25, 3.5] for the definitions of grad, cur, and div on arbitrary 3-manifods. However, we require a goba orthonorma coordinate system for defining cur and the isomorphisms ı k. Thus, the gcd compex and diagram (2.2) are vaid merey on fat 3-manifods. The contraction T, Y of T Γ ( 2 T B) and Y X(B) is a vector fied that in the orthonorma coordinate system {X I } reads T, Y = T IJ Y J E I. Ceary, if N C is the unit outward norma vector fied of a cosed surface C B, then T, N C is the traction of T on C. Suppose H k gcd (B) is the k-th cohomoogy of the gcd compex. Diagram (2.2) impies that ı k aso induces the isomorphism H k gcd (B) 3 i=1h k dr (B) between the cohomoogy groups. By using this fact and (1.4), we can prove the foowing theorem.

8 8 Arzhang Angoshtari, Arash Yavari Theorem 2. An arbitrary tensor T Γ ( 2 T B) is the gradient of a vector fied if and ony if cur T T = 0, and T, t ds = 0, B, (2.3) where is an arbitrary cosed curve in B and t is the unit tangent vector fied aong. Proof By using (2.1) and diagram (2.2), we concude that T = grad Y, if and ony if ı 1 (T ) = d(ı 0 (Y )), if and ony if ı 1 ( TEI ) = d Y I, I = 1, 2, 3. The condition (1.4) impies that in addition to cur T T = 0, T shoud aso satisfy ( TEI ) ı 1 = G( T EI, t )ds = 0, B, I = 1, 2, 3, which is equivaent to the integra condition in (2.3). Simiary, one can use (1.5) for deriving the necessary and sufficient conditions for the existence of a potentia for T induced by cur T. The upshot is the foowing theorem. Theorem 3. Given T Γ ( 2 T B), there exists W Γ ( 2 T B) such that T = cur T W, if and ony if div T = 0, and T, N C da = 0, C B, (2.4) C where C is an arbitrary cosed surface in B and N C is its unit outward norma vector fied. Remark 4. If the Betti numbers b k (B), k = 1, 2, are finite, then it suffices to check (2.3) and (2.4) for b 1 (B) and b 2 (B) independent cosed curves and cosed surfaces, respectivey. In particuar, one concudes that if B is simpyconnected, then any cur T -free ( 2 0 )-tensor is the gradient of a vector fied and if B is contractibe, then any div-free ( 2 0 )-tensor admits a curt -potentia. If B R 3 is compact, i.e. B is cosed and bounded, then a bk (B) s are finite. The cacuation of b k (B) for some physicay interesting bodies and the seection of independent cosed oops and cosed surfaces are discussed in [2]. We can aso write an anaogue of the gcd compex for two-point tensors. Let S = R 3 with coordinate system {x i } that is the Cartesian coordinates of R 3. Suppose ϕ : B S is a smooth mapping and et T X ϕ(b) := T ϕ(x) S. Note that athough ϕ is not necessariy an embedding, the dimension of T X ϕ(b) is aways equa to dim S. We can define the foowing operators for two-point tensors that beong to Γ (T ϕ(b)) and Γ (T ϕ(b) T B): Grad : Γ (T ϕ(b)) Γ (T ϕ(b) T B), Cur T : Γ (T ϕ(b) T B) Γ (T ϕ(b) T B), Div : Γ (T ϕ(b) T B) Γ (T ϕ(b)), (Grad U) ii = U i,i, (Cur T F ) ii = ε IKL F il,k, (Div F ) i = F ii,i.

9 Differentia Compexes in Continuum Mechanics 9 We have Cur T Grad = 0, and Div Cur T = 0. Thus, the GCD compex is written as: 0 Γ (T ϕ(b)) Grad Γ (T ϕ(b) T B) CurT Γ (T ϕ(b) T B) Div Γ (T ϕ(b)) 0. By using the foowing isomorphisms I 0 : Γ (T ϕ(b)) Ω 0 (B; R 3 ), [I 0 (U)] i = U i, I 1 : Γ (T ϕ(b) T B) Ω 1 (B; R 3 ), [I 1 (F )] i J = F ij, I 2 : Γ (T ϕ(b) T B) Ω 2 (B; R 3 ), [I 2 (F )] i JK = ε JKL F il, I 3 : Γ (T ϕ(b)) Ω 3 (B; R 3 ), [I 3 (U)] i 123 = U i, one concudes that the foowing diagram commutes. 0 Γ (T ϕ(b)) Grad Γ (T ϕ(b) T B) CurT Γ (T ϕ(b) T B) Div Γ (T ϕ(b)) 0 I 0 0 Ω 0 (B; R 3 ) I 1 d Ω 1 (B; R 3 ) I 2 d Ω 2 (B; R 3 ) I 3 d Ω 3 (B; R 3 ) 0 The above isomorphisms aso induce an isomorphism H k GCD (B) 3 i=1h k dr (B), where H k GCD (B) is the k-th cohomoogy group of the GCD compex. Let {E I } and {e i } be the standard basis of R 3. For F Γ (T ϕ(b) T B), and n = n i e i S 2, et F n = n i F ij E J X(B). Then, one can write ( Fe1 ) ( Fe2 ) ( Fe3 )) I k (F ) = (ı k, ı k, ı k, k = 1, 2. Let F, Y := F ii Y I e i. The above reations for the GCD compex aow us to obtain the foowing resuts that can be proved simiary to Theorems 2 and 3. Theorem 5. Given F Γ (T ϕ(b) T B), there exists U Γ (T ϕ(b)) such that F = Grad U, if and ony if Cur T F = 0, and F, t ds = 0, B. Moreover, there exists Ψ Γ (T ϕ(b) T B) such that F = Cur T Ψ, if and ony if Div F = 0, and F, N C da = 0, C B. C Remark 6. Note that for writing the GCD compex, ony S needs to be fat and admit a goba orthonorma coordinate system. This observation is usefu for deriving a compex for motions of 2D surfaces (shes) in R 3. By using the natura isomorphism : X(B) Ω 1 (B) induced by G and the Hodge star operator : Ω k (B) Ω n k (B), where n = dim B, we can write I 1 (F ) = ( (F (dx 1 )), (F (dx 2 )), (F (dx 3 )) ), I 2 (F ) = ( (F (dx 1 )), (F (dx 2 )), (F (dx 3 )) ).

10 10 Arzhang Angoshtari, Arash Yavari Compexes for 2-manifods Let B R 2 be a 2-manifod and suppose {X I } is the Cartesian coordinate system. For 2-manifods, instead of cur T, we define the operator c : Γ ( 2 T B) X(B), (c(t )) I = T I2,1 T I1,2, that satisfies c grad = 0. Aso consider the foowing isomorphisms j 0 : X(B) Ω 0 (B; R 2 ), [j 0 (Y )] i = δ ii Y I, j 1 : Γ ( 2 T B) Ω 1 (B; R 2 ), [j 1 (T )] i J = δ ii T IJ, j 2 : X(B) Ω 2 (B; R 2 ), [j 2 (Y )] i 12 = δ ii Y I. It is straightforward to show that the foowing diagram commutes. 0 X(B) grad Γ ( 2 T B) j 0 j 1 c X(B) 0 Ω 0 (B; R 2 ) d Ω 1 (B; R 2 ) d Ω 2 (B; R 2 ) 0 j 2 0 (2.5) The compex in the first row of (2.5) is caed the gc compex. This diagram impies that H k gc(b) 2 i=1h k dr (B), where Hk gc(b) is the k-th cohomoogy group of the gc compex and we obtain the foowing resut. Theorem 7. A tensor T Γ ( 2 T B) on a 2-manifod B R 2 is the gradient of a vector fied if and ony if c(t ) = 0, and T, t ds = 0, B. For 2-manifods, we can write a second compex that contains div. In an orthonorma coordinate system {X I }, the codifferentia operator δ k : Ω k (B) Ω k 1 (B) reads (δβ) I1 I k 1 = β JI1 I k 1,J. We have δ δ = 0, that gives rise to the compex (Ω(B), δ) with the homoogy groups H k co(b) := ker δ k /im δ k+1. By using the Hodge star operator : Ω k (B) Ω n k (B), it is straightforward to show that H k co(b) H n k dr (B). One can aso write the compex ( Ω(B; R 2 ), δ ), where δα = (δα 1, δα 2 ). By defining the operator s : X(B) Γ ( 2 T B), (s(y )) IJ = δ 1J Y I,2 δ 2J Y I,1, we obtain the foowing diagram. 0 X(B) j 0 div Γ ( 2 T B) j 1 s X(B) 0 Ω 0 (B; R 2 ) δ Ω 1 (B; R 2 ) δ Ω 2 (B; R 2 ) 0 j 2 0 (2.6)

11 Differentia Compexes in Continuum Mechanics 11 We ca the first row of (2.6) the sd compex and denote its homoogy groups by H k sd (B). We have Hk sd (B) 2 i=1h n k dr (B). Let {E I} be the standard basis of R 2 and et N be a unit vector fied aong a cosed curve, which is norma to tangent vector fied t, such that {t, N } has the same orientation as {E 1, E 2 }. The foowing theorem is the anaogue of Theorem 7 for the sd compex. Theorem 8. On a 2-manifod B R 2, there exists Y X(B) for T Γ ( 2 T B) such that T = s(y ), if and ony if div T = 0, and T, N ds = 0, B. (2.7) Proof We know that T = s(y ), if and ony if j 1 (T ) = δ(j 2 (Y )), if and ony if ( T EI ) = δy I, I = 1, 2. The Hodge star operator induces an isomorphism between the cohomoogy groups of (Ω(B), d) and (Ω(B), δ), and therefore, the ast condition is equivaent to ( T EI ) = ( T E I ) = dy I, where T E I = (T I2, T I1 ), I = 1, 2. Since G( T E I, t ) = G( T EI, N ), one obtains (2.7). Next, suppose ϕ : B R 2 is a smooth mapping and et {x i } be the Cartesian coordinates of R 2 with {e i } being its standard basis. Consider the foowing isomorphisms J 0 : Γ (T ϕ(b)) Ω 0 (B; R 2 ), [J 0 (U)] i = U i, J 1 : Γ (T ϕ(b) T B) Ω 1 (B; R 2 ), [J 1 (F )] i J = F ij, J 2 : Γ (T ϕ(b)) Ω 2 (B; R 2 ), [J 2 (U)] i 12 = U i, together with the operators C : Γ (T ϕ(b) T B) Γ (T ϕ(b)), (C(F )) i = F i2,1 F i1,2, S : Γ (T ϕ(b)) Γ (T ϕ(b) T B), (S(U)) ii = δ 1I U i,2 δ 2I U i,1. Repacing j 0, j 1, j 2, c, and s with J 0, J 1, J 2, C, and S, respectivey, in diagrams (2.5) and (2.6) yieds the corresponding diagrams for two-point tensors. The associated compexes are caed the GC and the SD compexes and we concude that: Theorem 9. Let ϕ : B R 2 be a smooth mapping and F Γ (T ϕ(b) T B). We have F = Grad U, if and ony if C(F ) = 0, and F, t ds = 0, B. Moreover, we can write F = S(U), if and ony if Div F = 0, and F, N ds = 0, B.

12 12 Arzhang Angoshtari, Arash Yavari As we mentioned earier, the compexes introduced for two-point tensors do not require B to be fat. This aows us to obtain a compex describing motions of 2D surfaces (shes) in R 3. Let (B, G) be a 2D surface in R 3 with an arbitrary oca coordinate system {X I }, I = 1, 2, and et {x i } and {e i }, i = 1, 2, 3, be the Cartesian coordinates and the standard basis of R 3, respectivey. The oca basis for T B induced by {X I } is denoted by {E I }. Suppose ϕ : B R 3 is a smooth mapping and consider the foowing isomorphisms J 0 : Γ (T ϕ(b)) Ω 0 (B; R 3 ), [J 0 (U)] i = U i, J 1 : Γ (T ϕ(b) T B) Ω 1 (B; R 3 ), [J 1 (F )] i J = G JI F ii, J 2 : Γ (T ϕ(b)) Ω 2 (B; R 3 ), [J 2 (U)] i 12 = det G IJ U i, where G IJ are the components of G and det G IJ is the determinant of matrix [G IJ ] 2 2. Let G IJ be the components of the inverse of [G IJ ] 2 2. We define the operators Grad : Γ (T ϕ(b)) Γ (T ϕ(b) T B) and C : Γ (T ϕ(b) T B) Γ (T ϕ(b)) by ( G2K F ik) (Grad U) ii = G IJ U i,j, (C(F )) i,1 = ( G 1K F ik),2. det GIJ By using the above operators, one obtains the foowing diagram for the GC compex. 0 Γ (T ϕ(b)) Grad Γ (T ϕ(b) T B) C Γ (T ϕ(b)) 0 J 0 0 Ω 0 (B; R 3 ) J 1 d Ω 1 (B; R 3 ) J 2 d Ω 2 (B; R 3 ) 0 Thus, the foowing resut hods. Theorem 10. Let B be a 2D surface and et ϕ : B R 3 be a smooth mapping. Then, F Γ (T ϕ(b) T B) can be written as F = Grad U, if and ony if C(F ) = 0, and F, t ds = 0, B, where F, t = G IJ F ij (t ) I e i. 2.2 Compexes Induced by the Caabi Compex A differentia compex suitabe for symmetric second-order tensors was introduced by Caabi [7]. It is we-known that the Caabi compex in R 3 is equivaent to the inear easticity compex [11]. In this section, we study the Caabi compex and its connection with the inear easticity compex in some detais. As we wi see ater, this study provides a framework for writing the noninear compatibiity equations in curved ambient spaces and comparing stress functions induced by the Caabi compex with those induced by

13 Differentia Compexes in Continuum Mechanics 13 the gcd or the GCD compexes. Moreover, the Caabi compex provides a coordinate-free expression for the inear compatibiity equations. The Caabi compex is vaid on any Riemannian manifod with constant (sectiona) curvature (aso caed a Cifford-Kein space). These spaces are defined as foows: Let be the Levi-Civita connection of (B, G) and et X i X(B), i = 1,..., 5. The curvature R and the Riemannian curvature R induced by G are given by R(X 1, X 2 )X 3 = X1 X2 X 3 X2 X1 X 3 [X1,X 2]X 3, and R(X 1, X 2, X 3, X 4 ) = G(R(X 1, X 2 )X 3, X 4 ). Let Σ X be a 2-dimensiona subspace of T X B and et X 1, X 2 Σ X be two arbitrary ineary independent vectors. The sectiona curvature of Σ X is defined as K(Σ X ) = R(X 1, X 2, X 2, X 1 ) (G(X 1, X 1 )G(X 2, X 2 )) 2 (G(X 1, X 2 )) 2. Sectiona curvature K(Σ X ) is independent of the choice of X 1 and X 2 [9]. A manifod B has a constant curvature k R if and ony if K(Σ X ) = k, X B and Σ X T X B. If B is compete and simpy-connected, it is isometric to: (i) the n-sphere with radius 1/ k, if k > 0, (ii) R n, if k = 0, and (iii) the hyperboic space, if k < 0 [19]. An arbitrary Riemannian manifod with constant curvature is ocay isometric to one of the above manifods depending on the sign of k. For exampe, the sectiona curvature of a cyinder in R 3 is zero and it is ocay isometric to R 2. Discussions on the cassification of Riemannian manifods with constant curvatures can be found in Wof [29]. One can show that (B, G) has constant curvature k if and ony if R(X 1, X 2 )X 3 = k ( G(X 3, X 2 )X 1 G(X 3, X 1 )X 2 ). (2.8) Simiar to the de Rham compex, the Caabi compex on n-manifods terminates after n non-trivia operators. For n = 3, these operators incude the Kiing operator D 0, the inearized curvature operator D 1, and the Bianchi operator D 2 that are defined as foows. The first operator in the Caabi compex on (B, G) is the Kiing operator D 0 : X(B) Γ (S 2 T B) defined as (D 0 U)(X 1, X 2 ) = 1 2 ( ) G(X 1, X2 U) + G( X1 U, X 2 ). Note that D 0 U = 1 2 L U G, where L U is the Lie derivative. The kerne of D 0 coincides with the space of Kiing vector fieds Θ(B) on B. 5 If an n-manifod B is a subset of R n with Cartesian coordinates {X I }, any U Θ(B) at X = (X 1,..., X n ) R n can be written as U(X) = v + A X, where v R n, A so(r n ) := {A R n n : A + A T = 0}, with R n n being the space of rea n n matrices. 6 Therefore, we concude that dim Θ(B) = n(n + 1)/2. The second operator of the Caabi compex can be obtained by inearizing the Riemannian curvature. Let A be a Riemannian metric on B and et 5 A Kiing vector fied U Θ(B) is aso caed an infinitesima isometry in the sense that its fow F U induces an isometry F U t := F U (t, ) : U B B [9]. 6 This impies that Θ(B) is isomorphic to euc(r n ), which is the Lie agebra of the group of rigid body motions Euc(R n ).

14 14 Arzhang Angoshtari, Arash Yavari A and R A be the corresponding Levi-Civita connection and Riemannian curvature, respectivey. The symmetries of R A are induced by the identities R A (X 1,X 2,X 3,X 4 ) + R A (X 2,X 3,X 1,X 4 ) + R A (X 3,X 1,X 2,X 4 ) = 0, R A (X 1,X 2,X 3,X 4 ) = R A (X 2,X 1,X 3,X 4 ) = R A (X 1,X 2,X 4,X 3 ). (2.9a) (2.9b) Equivaenty, the components R A I 1I 2I 3I 4 of R A satisfy R A I 1I 2I 3I 4 + R A I 2I 3I 1I 4 + R A I 3I 1I 2I 4 = 0, R A I 1I 2I 3I 4 = R A I 2I 1I 3I 4 = R A I 1I 2I 4I 3. The identity (2.9a) is caed the first Bianchi identity. The above symmetries impy that R A has n 2 (n 2 1)/12 independent components [26]. The reations (2.9a) and (2.9b) aso induce the symmetry R A (X 1, X 2, X 3, X 4 ) = R A (X 3, X 4, X 1, X 2 ), (2.10) i.e. R A I 1I 2I 3I 4 = R A I 3I 4I 1I 2. For n = 2, 3, (2.9b) and (2.10) determine a the symmetries of R A, and therefore, the space of tensors with the symmetries of the Riemannian curvature is Γ (S 2 (Λ 2 T B)). 7 Let e Γ (S 2 T B) be an arbitrary symmetric ( 0 2 )-tensor. The inearization of the operator A RA is the inear operator r A : Γ (S 2 T B) Γ (S 2 (Λ 2 T B)) defined by r A (e) := d t=0 R A+te [13, 12]. One can write dt with 2r A (e)(x 1, X 2, X 3, X 4 ) = L A (e)(x 1, X 2, X 3, X 4 ) + e(r A (X 1,X 2 )X 3, X 4 ) e(r A (X 1, X 2 )X 4, X 3 ), L A (e)(x 1, X 2, X 3, X 4 ) = ( A X 1 A X 3 e ) (X 2, X 4 ) + ( A X 2 A X 4 e ) (X 1, X 3 ) ( A X 1 A X 4 e ) (X 2, X 3 ) ( A X 2 A X 3 e ) ( ) (X 1, X 4 ) A A X X 3 e (X 2, X 4 ) ( ) ( ) ( 1 ) A A X X 4 e (X 1, X 3 ) + A A 2 X X 4 e (X 2, X 3 ) + A A 1 X X 3 e (X 1, X 4 ), 2 where A T for ( 0 k )-tensor T is defined as ( A X0 T ) (X 1,..., X k ) =X 0 (T (X 1,..., X k )) k T (X 1,..., A X 0 X i,..., X k ). i=1 7 Tensors in Γ (S 2 (Λ 2 T B)) have (n 2 n + 2)(n 2 n)/8 independent components. For n 4, (2.9b) and (2.10) do not impy (2.9a), and thus, tensors with the symmetries of the Riemannian curvature beong to a subspace of Γ (S 2 (Λ 2 T B)). If T B is induced by a representation, i.e. it is a homogeneous vector bunde corresponding to an irreducibe representation, the representation theory provides some toos to neaty specify tensors with compicated symmetries such as those of the Riemannian curvature [5, 24].

15 Differentia Compexes in Continuum Mechanics 15 Note that r A (e) inherits the symmetries of the Riemannian curvature. If (B, G) has constant curvature k, by using (2.8), one obtains the operator D 1 : Γ (S 2 T B) Γ (S 2 (Λ 2 T B)), D 1 := 2r G, that can be written as (D 1 e)(x 1, X 2, X 3, X 4 ) = L G (e)(x 1, X 2, X 3, X 4 ) { + k G(X 2, X 3 )e(x 1, X 4 ) G(X 1, X 3 )e(x 2, X 4 ) } G(X 2, X 4 )e(x 1, X 3 ) + G(X 1, X 4 )e(x 2, X 3 ). (2.11) One can show that D 1 D 0 = 0. The Caabi compex for a 2-manifod B reads 0 X(B) D0 Γ (S 2 T B) D1 Γ (S 2 (Λ 2 T B)) 0. The ast non-trivia operator of the Caabi compex for 3-manifods is defined as foows: Let Γ (V 5 T B) be the space of ( 0 5 )-tensors such that h Γ (V 5 T B) admits the symmetries h I1I 2I 3I 4I 5 = h I2I 1I 3I 4I 5 = h I1I 3I 2I 4I 5, h I1I 2I 3I 4I 5 + h I1I 3I 4I 2I 5 + h I1I 4I 2I 3I 5 = 0, h I1I 2I 3I 4I 5 = h I1I 3I 2I 4I 5 = h I1I 2I 3I 5I 4, i.e. h is anti-symmetric in the first three entries and has the symmetries of the Riemannian curvature in the ast four entries. For n = 3, h has 3 independent components that can be represented by h 12323, h 21313, and h The operator D 2 : Γ (S 2 (Λ 2 T B)) Γ (V 5 T B) is defined by (D 2 s)(x 1,..., X 5 ) = ( X1 s) (X 2, X 3, X 4, X 5 ) + ( X2 s) (X 3, X 1, X 4, X 5 ) + ( X3 s) (X 1, X 2, X 4, X 5 ). By using D 2, the second Bianchi identity for the Riemannian curvature R can be expressed as D 2 (R) = 0. We have D 2 D 1 = 0. Thus, the Caabi compex on 3-manifod B is written as 0 X(B) D0 Γ (S 2 T B) D1 Γ (S 2 (Λ 2 T B)) D2 Γ (V 5 T B) 0. Caabi [7] showed that there is a systematic way for constructing operators D i, 2 i n 1, for n-manifods. Let H k C (B) := ker D k/im D k 1, be the k-th cohomoogy group of the Caabi compex. Caabi aso showed that the Caabi compex induces a fine resoution of the sheaf of germs of Kiing vector fieds. Thus, the dimension of Θ(B) determines the dimension of H k C (B). In particuar, if an n-manifod B R n has finite-dimensiona de Rham cohomoogy groups, one can write dim H k C(B) = n(n + 1) 2 dim H k dr(b) = n(n + 1) b k (B). (2.12) 2 Next, we separatey study 2- and 3-submanifods of the Eucidean space.

16 16 Arzhang Angoshtari, Arash Yavari The Linear Easticity Compex for 3-manifods Let B R 3 be a 3-manifod and et G and {X I } be the standard metric and the Cartesian coordinates of R 3, respectivey. Consider the foowing operator grad s : X(B) Γ (S 2 T B), (grad s Y ) IJ = 1 2 ( ) Y I,J + Y J,I. It is straightforward to show that cur cur grad s = 0. If T Γ (S 2 T B), then cur cur T is symmetric as we. Therefore, one obtains the foowing operator cur cur : Γ (S 2 T B) Γ (S 2 T B), (cur cur T ) IJ = ε IKL ε JMN T LN,KM. We have div cur cur = 0. Let ι 0 : X(B) X(B) be the identity map. The goba orthonorma coordinate system {X I } aows one to define the foowing three isomorphisms: ι 1 : Γ (S 2 T B) Γ (S 2 T B), (ι 1 (T )) IJ = T IJ, the isomorphism ι 2 : Γ (S 2 T B) Γ (S 2 (Λ 2 T B)) defined by (ι 2 (T )) 2323 = T 11, (ι 2 (T )) 3123 = T 12, (ι 2 (T )) 1223 = T 13, (ι 2 (T )) 1313 = T 22, (ι 2 (T )) 2113 = T 23, (ι 2 (T )) 1212 = T 33, and ι 3 : X(B) Γ (V 5 T B)) given by (ι 3 (Y )) = Y 1, (ι 3 (Y )) = Y 2, (ι 3 (Y )) = Y 3. Simpe cacuations show that ι 1 grad s = D 0 ι 0, ι 2 cur cur = D 1 ι 1, ι 3 div = D 2 ι 2, and therefore, the foowing diagram commutes. 0 X(B) grads Γ (S 2 T B) cur cur Γ (S 2 T B) ι 0 ι 1 ι 2 div X(B) 0 X(B) D0 Γ (S 2 T B) D1 Γ (S 2 (Λ 2 T B)) D2 Γ (V 5 T B) 0 ι 3 0 (2.13) The first row of (2.13) is the inear easticity compex. Therefore, we observe that usefu properties of this compex foow from those of the Caabi compex. In particuar, (2.12) impies that the dimensions of the cohomoogy groups H k E3 (B) of the inear easticity compex are given by dim H 1 E3(B) := dim (ker cur cur/im grad s ) = 6b 1 (B), dim H 2 E3(B) := dim (ker div/im cur cur) = 6b 2 (B). It is aso possibe to cacuate the cohomoogy groups of the inear easticity compex without expicity using its reation with the Caabi compex [17]. Note that the Caabi compex is more genera than the inear easticity

17 Differentia Compexes in Continuum Mechanics 17 compex in the sense that the Caabi compex is vaid on any Riemannian manifod with constant curvature, however, the inear easticity compex is ony vaid on fat manifods that admit a goba orthonorma coordinate system. Yavari [30, Proposition 2.8] showed that for T Γ (S 2 T B), there exists Y X(B) such that T = grad s Y, if and ony if cur cur T = 0, ( T IJ X k (T IJ,K T JK,I) ) dx J = 0, ( ) T IK,J T JK,I dx K = 0, B. (2.14) Let N C be the unit outward norma vector fied of an arbitrary cosed surface C B. Gurtin [16] showed that the necessary and sufficient conditions for the existence of cur cur-potentias for T are div T = 0, T, N C da = 0, ε KIJ X I T JL (N C ) L da = 0, C B. (2.15) C The Linear Easticity Compex for 2-manifods C Next, suppose B R 2 is a 2-manifod and et {X I } be the Cartesian coordinates of R 2. Let D c : Γ (S 2 T B) C (B), D c T = T 11,22 2T 12,12 + T 22,11. Then, we have D c grad s = 0. Aso consider isomorphisms γ 0, γ 1, and γ 2 that are defined as foows: γ 0 : X(B) X(B) and γ 1 : Γ (S 2 T B) Γ (S 2 T B) are defined simiar to ι 0 and ι 1 introduced for 3-manifods and γ 2 : C (B) Γ (S 2 (Λ 2 T B)), (γ 2 (f)) 1212 = f. Using these operators, one obtains the foowing diagram. 0 X(B) grads Γ (S 2 T B) γ 0 γ 1 D c C (B) 0 X(B) D0 Γ (S 2 T B) D1 Γ (S 2 (Λ 2 T B)) 0 γ 2 0 (2.16) Then, (2.12) impies that the dimension of the cohomoogy group H 1 E2 (B) := ker D c /im grad s is 3b 1 (B). Moreover, the necessary and sufficient conditions for the existence of potentias induced by grad s for T Γ (S 2 T B) is D c T = 0, together with the integra conditions in (2.14). For 2-manifods, it is aso possibe to write the compex 0 C (B) Ds Γ (S 2 T B) div X(B) 0, (2.17) where (D s f) 11 = f,22, (D s f) 12 = f,12, and (D s f) 22 = f,11. The kerne of D s is 3-dimensiona, which suggests that the dimension of H 1 E2 (B) := ker div/im D s, is 3b 1 (B). By repacing C with arbitrary cosed curves in (2.15), one obtains the necessary and sufficient conditions for the existence of D s -potentias.

18 18 Arzhang Angoshtari, Arash Yavari 3 Some Appications in Continuum Mechanics Let B R n, n = 2, 3, be a smooth n-manifod. Note that B can be unbounded as we. For 3-manifods, the inear easticity compex (2.13) describes both the kinematics and the kinetics of deformations in the foowing sense [21]: If one considers X(B) as the space of dispacements, then grad s associates inear strains to dispacements, Γ (S 2 T B) is the space of inear strains, and cur cur expresses the compatibiity equations for the inear strain. On the other hand, one can consider Γ (S 2 T B) as the space of Betrami stress functions and consequenty, cur cur associates symmetric Cauchy stress tensors to Betrami stress functions, and div expresses the equiibrium equations. We observed that for 2-manifods, the kinematics and the kinetics of deformations are described by two separate compexes: The former is addressed by the compex (2.16) and the atter by the compex (2.17). Let a smooth embedding ϕ : B S = R 3, be a motion of B in S. Let C := ϕ g Γ (S 2 T B), and F := T ϕ be the Green deformation tensor and the deformation gradient of ϕ, respectivey. Aso suppose σ Γ ( 2 T ϕ(b)), P Γ (T ϕ(b) T B), and S Γ ( 2 T B) are the Cauchy, the first, and the second Pioa-Kirchhoff stress tensors, respectivey. If σ is symmetric, then the ast two operators of the inear easticity compex on (ϕ(b), g) address existence of Betrami stress functions and the equiibrium equations for σ. The first operator in this compex does not have any apparent physica interpretation. If σ is non-symmetric, then the ast two operators of the gcd compex on (ϕ(b), g) describe the kinetics of ϕ: The operator cur T associates stress functions induced by cur T to σ and div is reated to the equiibrium equations. Simiar concusions aso hod for S if one considers the inear easticity compex and the gcd compex on the fat manifod (B, C). On the other hand, by using P, one can write a compex that contains both the kinematics and the kinetics of motion. Let U Γ (T ϕ(b)) be the dispacement fied defined as U(X) = ϕ(x) X T ϕ(x) S, X B. 8 Then, Grad U is the dispacement gradient and Cur T expresses the compatibiity of the dispacement gradient. On the other hand, we can assume that Cur T associates stress functions induced by Cur T to the first Pioa-Kirchhoff stress tensor, and that Div expresses the equiibrium equations. Hence, the GCD compex is the anaogue of the inear easticity compex in the sense that both contain the kinematics and the kinetics simutaneousy. Note that the inear easticity compex is not the inearization of the GCD compex. In particuar, the operator cur cur is obtained by inearizing the curvature operator, which is reated to the compatibiity equations in terms of C and not the dispacement gradient. In the foowing, we study the appications of the above compexes to the noninear compatibiity equations and the existence of stress functions in more detai. Cassicay, the inear and noninear compatibiity equations are 8 Dispacement fieds are usuay assumed to be vector fieds on B. The choice of Γ (T ϕ(b)) instead of X(B) is equivaent to appying the shifter T XB T ϕ(x) S to eements of X(B), see [23, Box 3.1].

19 Differentia Compexes in Continuum Mechanics 19 written for fat ambient spaces. We study these equations on ambient spaces with constant curvatures. 3.1 Compatibiity Equations We study the noninear compatibiity equations for the cases dim B = dim S, and dim B < dim S (shes), separatey. It is we-known that compatibiity equations depend on the topoogica properties of bodies, see Yavari [30] and references therein for more detais. More specificay, both inear and noninear compatibiity equations are cosey reated to b 1 (B). The noninear compatibiity equations in terms of the dispacement gradient (or equivaenty F ) directy foow from the compexes we introduced earier for second-order tensors Bodies with the Same Dimensions as the Ambient Space Suppose dim B = dim S. Since motion ϕ : B S is an embedding, it is easy to observe that the Green deformation tensor C = ϕ g is a Riemannian metric on B. The mapping ϕ is an isometry between (B, C) and (ϕ(b), g). Thus, the compatibiity probem in terms of C reads: Given a metric C on B, is there any isometry between (B, C) and an open subset of S? Note that a priori we do not know which part of S woud be occupied by B. This suggests that a usefu compatibiity equation shoud be written ony on B. Let R g and R g be the curvature and the Riemannian curvature of (S, g) that are induced by the Levi-Civita connection g. Let X 1,..., X 4 X(B). By using the puback ϕ and the push-forward ϕ, one concudes that the inear connection g on T S induces a inear connection ϕ g on T B given by (ϕ g ) X1 X 2 = ϕ ( g ϕ X 1 ϕ X 2 ). The definition of the Levi-Civita connection g impies that C(X 3, (ϕ g ) X1 X 2 ) = g(ϕ X 3, g ϕ X 1 ϕ X 2 ) = 1 { X 2 (C(X 1, X 3 )) + X 1 (C(X 3, X 2 )) X 3 (C(X 1, X 2 )) 2 } C ([X 2, X 3 ], X 1 ) C ([X 1, X 3 ], X 2 ) C ([X 2, X 1 ], X 3 ), and therefore, ϕ g coincides with the Levi-Civita connection C on (B, C). Since (ϕ R g )(X 1, X 2 )X 3 = ϕ (R g (ϕ X 1, ϕ X 2 )ϕ X 3 ) = C X 1 C X 2 X 3 C X 2 C X 1 X 3 C [X 1,X 2] X 3, we aso concude that ϕ R g is the curvature R C of (B, C) induced by C. Hence, if ϕ : B S is an isometry between (B, C) and (ϕ(b), g), then we must have R C (X 1, X 2, X 3, X 4 ) = R g (ϕ X 1, ϕ X 2, ϕ X 3, ϕ X 4 ), (3.1)

20 20 Arzhang Angoshtari, Arash Yavari where R C is the Riemannian curvature of (B, C). It is hard to check the above condition on arbitrary curved ambient spaces. However, if S has constant curvature, (3.1) admits a simpe form. The foowing theorem states the compatibiity equations in terms of C on an ambient space with constant curvature. Theorem 11. Suppose dim B = dim S, and (S, g) has constant curvature k. If C is the Green deformation tensor of a motion ϕ : B S, then (B, C) has constant curvature k as we, i.e. C satisfies R C (X 1, X 2 )X 3 = kc(x 3, X 2 )X 1 kc(x 3, X 1 )X 2. (3.2) Conversey, if C satisfies (3.2), then for each X B, there is a neighborhood U X B of X and a motion ϕ X : U X S, with C UX being its Green deformation tensor. Motion ϕ X is unique up to isometries of S. Proof If C = ϕ g, then by using R C = ϕ R g, and (2.8), one obtains (3.2). Conversey, consider arbitrary points X B and x S and et {E i } and {e i } be arbitrary orthonorma bases for T XB and T x S, respectivey. Choose the isometry i : T X B T x S such that i(e i ) = e i. Then, by using a theorem due to Cartan [9, page 157] and (3.2), one can construct an isometry ϕ X : U X ϕ X (U X ) S, in a neighborhood U X of X such that T X ϕ X = i. This concudes the proof. Remark 12. Theorem 11 impies that there are many oca isometries between manifods with the same constant sectiona curvatures. Formuating sufficient conditions for the existence of goba isometries between arbitrary Riemannian manifods is a hard probem. Ambrose [1] derived such a condition by using the parae transation of Riemannian curvature aong curves made up of geodesic segments. In particuar, his resut impies that (3.2) is aso the sufficient condition for the existence of a goba motion ϕ : B S, if B is compete and simpy-connected. For the fat case B S = R n, Yavari [30] derived necessary and sufficient conditions for the compatibiity of C if B is non-simpy-connected. Remark 13. The symmetries of the Riemannian curvature determine the number of compatibiity equations induced by (3.2), i.e. the number of independent equations that we obtain by writing (3.2) in a oca coordinate system. Thus, the number of compatibiity equations in terms of C ony depends on the dimension of the ambient space and is the same as the number of inear compatibiity equations induced by the operator D 1 in the Caabi compex. Next, suppose B S = R n, n = 2, 3, and et {X I } and {x i } be the Cartesian coordinates of R n. Any smooth mapping ϕ : B R n induces a dispacement fied U Γ (T ϕ(b)) given by U(X) = ϕ(x) X. One can use the GCD and the GC compexes for writing the compatibiity equations in terms of the dispacement gradient. Note that ϕ is assumed to be specified for writing the above compexes. Let Υ Ω 0 (B; R 3 ) and κ Ω 1 (B; R 3 ). If κ = dυ, then I 1 1 (κ) = Grad I 1 0 (Υ ), where I 1 0 (Υ ) and I 1 1 (κ) are twopoint tensors over any arbitrary smooth mapping ϕ. In particuar, by using

21 Differentia Compexes in Continuum Mechanics 21 Fig. 1 A mapping with a compatibe dispacement gradient, which is not an embedding. the inear structure of R 3, one can choose ϕ to be ϕ(x) = X + Υ (X). Thus, we obtain the foowing theorem, cf. Theorems 5 and 9. Theorem 14. Given κ = (κ 1,..., κ n ) Ω 1 (B; R n ) on a connected n- manifod B R n, there exists a smooth mapping ϕ : B R n with dispacement gradient I (κ) (or J 1 (κ) if n = 2) if and ony if dκ = 0, and κ(t )ds = 0, B. The mapping ϕ is unique up to rigid body transations in R n. Remark 15. This theorem does not guarantee that the dispacement gradient is induced by a motion of B, i.e. ϕ is not an embedding, in genera. For exampe, consider the mapping depicted in Fig. 1 which is not injective. This mapping is a oca diffeomorphism, its tangent map is bijective at a points, and its dispacement gradient satisfies the above condition. Aso note that in contrary to Theorem 11, ϕ is unique ony up to rigid body transations and not rigid body rotations. This is a direct consequence of the fact that H 0 dr (B) R, for any connected manifod B. If H 1 dr (B) is finite-dimensiona, then the integra condition in the above theorem merey needs to be checked for a finite number of cosed curves and Theorem 14 is equivaent to Proposition 2.1 of [30]. In contrary to the compatibiity equations in terms of C, by using the notion of dispacement, we are expicity using the inear structure of R n for writing the compatibiity equations in terms of the dispacement gradient. Remark 16. The Green deformation tensor does not induce any inear compex for describing the kinematics of ϕ. Let (S, g) have constant curvature k and et C(B, S) and ΓM (S 2 T B) be the spaces of smooth embedding of B into S and Riemannian metrics on B, respectivey. Consider the operators D M : C(B, S) Γ M (S 2 T B), D M (ϕ) := ϕ g, and D R : Γ M (S 2 T B) Γ (S 2 (Λ 2 T B)) given by ( DR (C) ) (X 1, X 2, X 3, X 4 ) = R C (X 1, X 2, X 3, X 4 ) kc(x 3, X 2 )C(X 1, X 4 ) + kc(x 3, X 1 )C(X 2, X 4 ). The compatibiity equation (3.2) impies that D R D M = 0. However, note that the sequence of operators C(B, S) D M Γ M (S 2 T B) D R Γ (S 2 (Λ 2 T B)),

22 22 Arzhang Angoshtari, Arash Yavari is not a inear compex as the underying spaces and operators are not inear. The operator cur cur of the inear easticity compex is reated to the noninear compatibiity equations in terms of C. The kinematics part of this compex is not the inearization of the kinematics part of the GCD compex Shes Let (S, g) be an orientabe n-manifod with constant curvature k. We wi derive the compatibiity equations for motions of hypersurfaces in S, i.e. motions of (n 1)-dimensiona submanifods of S. We first tersey review some preiminaries of the submanifod theory, see [9, 27] for more detais. Suppose (B, G) is a connected orientabe submanifod of S, where G is induced by g. Let and g be the associated Levi-Civita connections of B and S, respectivey, and et X 1,..., X 4 X(B). We have the decomposition T X S = T X B (T X B), X B, where (T X B) is the norma compement of T X B in T S. Any vector fied X 1 on B can be ocay extended to a vector fied X1 on S and we have X1 X 2 = ( g X1 X2 ) T, where T denotes the tangent component. The second fundamenta form B of B is defined as B(X 1, X 2 ) = g X1 X2 X1 X 2. Let X Γ (T B ) =: X(B). The shape operator of B is a inear sef-adjoint operator S X : T B T B defined as G(S X (X 1 ), X 2 ) = g(b(x 1, X 2 ), X). One can show that ( g X 1 X ) T = S X (X 1 ). It is aso possibe to define a inear connection on T B by X 1 X = ( g X 1 X) N, where N denotes the norma component. The norma curvature R : X(B) X(B) X(B) X(B) is the curvature of. Thus, there are two different geometries on T B and T B. The reation between these geometries is expressed by the foowing reations: R g (X 1, X 2, X 3, X 4 ) = R(X 1, X 2, X 3, X 4 ) +g(b(x 1, X 3 ), B(X 2, X 4 )) g(b(x 1, X 4 ), B(X 2, X 3 )), (3.3) G([S Y, S X ]X 1, X 2 ) = g(r g (X 1, X 2 )X, Y) g(r (X 1, X 2 )X, Y), (3.4) g(r g (X 1,X 2 )X 3,X) = ( X1 B ) (X 2, X 3, X) ( X2 B ) (X 1, X 3, X), (3.5) where X, Y X(B), [S Y, S X ] = S Y S X S X S Y, and B(X 1, X 2, X) = g(b(x 1, X 2 ), X), with ( X1 B ) (X 2, X 3, X) = X 1 ( B(X2, X 3, X) ) B( X1 X 2, X 3, X) B(X 2, X1 X 3, X) B(X 2, X 3, X 1 X). The equations (3.3), (3.4), and (3.5) are caed the Gauss, Ricci, and Codazzi equations, respectivey. These equations generaize the compatibiity equations of the oca theory of surfaces. Let dim S dim B = 1. By using (2.8) and the fact that the second fundamenta form of hypersurfaces can be expressed as B(X 1, X 2 ) = g(b(x 1, X 2 ), N)N, where N is the unit norma