Mathematical Elasticity Theory in a Riemannian Manifold Nastasia Grubic, Philippe G

Mathematical elasticity theory in a Riemannian manifold Nastasia Grubic, Philippe G. Lefloch, Cristinel Mardare

To cite this version:

Nastasia Grubic, Philippe G. Lefloch, Cristinel Mardare. Mathematical elasticity theory in a Rieman- nian manifold. 2013. hal-00918075v1

HAL Id: hal-00918075 https://hal.archives-ouvertes.fr/hal-00918075v1 Preprint submitted on 12 Dec 2013 (v1), last revised 11 Apr 2014 (v2)

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Mathematical elasticity theory in a Riemannian manifold1

Nastasia Grubic, Philippe G. LeFloch, and Cristinel Mardare UniversitéPierre et Marie Curie & Centre National de la Recherche Scientifique 4 Place Jussieu, Laboratoire Jacques-Louis Lions, 75005 Paris, France. Email: [email protected], contact@philippelefloch.org, [email protected]

Abstract We study the equations of nonlinear and linearized static elasticity in a Riemannian manifold, which generalize those of classical elasticity in the three-dimensional Euclidean space. Our approach relies on the principle of least energy, stating that the deformation of the elastic body arising in response to given loads minimizes the total energy of the elastic body, defined as the difference between the strain energy and the potential of the loads, over a specific set of admissible deformations. Assuming that the strain energy is a function of the metric tensor field induced by the deformation, we first derive the principle of virtual work and the boundary value problem of nonlinear elasticity from the total energy of the elastic body, then we show that the latter equations possess a solution if the loads are sufficiently small in a specific sense. Resum´ e´ Nous etudions´ les equations´ de l’elasticit´ e´ nonlineaire´ dans une variet´ e´ riemannienne, qui ge-´ neralisent´ celles de la theorie´ classique de l’elasticit´ e´ dans l’espace euclidien tri-dimensionnel. Notre approche repose sur le principe de moindre action, affirmant que la deformation´ du corps elastique´ sous l’action des forces externes minimise l’energie´ totale du corps elastique,´ definie´ comme la difference´ entre l’energie´ de deformation´ et le potentiel des forces externes, sur l’en- semble des deformations´ admissibles. Sous l’hypothese` que l’en´ ergie´ de deformation´ est une fonction du champ de tenseurs metriques´ induit par la deformation,´ nous derivons´ dans un pre- mier temps le principe des travaux virtuels et le probleme` aux limites de l’elasticit´ e´ non lineaire´ a` partir de l’energie´ totale du corps elastique,´ puis nous demontrons´ que ce probleme` admet une solution si les forces externes sont suffisamment petites (en un sens que nous precisons).´ Key words: Nonlinear elasticity, Riemannian manifold, least action principle, existence theory, implicit function theorem

1Completed in December 2013 1. Introduction

In this paper we study the deformation of an elastic body immersed in a Riemannian manifold in response to applied body and surface forces. We show how the equations of static nonlinear elasticity can be derived from the principle of least energy, then establish existence theorems for these equations. Alternative approaches to modeling elastic bodies in a Riemannian manifold could be found elsewhere in the literature; see, for instance, [8, 9, 11, 12, 13, 14, 15] and the references therein. Our approach is akin to the one in Ciarlet [5], but is formulated in a Riemannian manifold instead of the three-dimensional Euclidean space and favors simplicity over generality. As such, our results can be easily compared with their counterparts in classical elasticity and can be used therein to model the deformations of thin elastic shells that preserve the shape of its middle surface. More specifically, letting (N, gˆ) be the three-dimensional Euclidean space and ϕ0 : M → Mˆ ⊂ N be a global local chart (assuming it exists) of the reference configuration Mˆ := ϕ0(M) of an elastic body immersed in N reduces our approach to the three-dimensional = ˆ = classical elasticity in curvilinear coordinates, while letting M M ⊂ N and ϕ0 idMˆ reduces our approach to the classical three-dimensional elasticity in Cartesian coordinates. An outline of the paper is as follows. Section 2 describes the mathematical framework and notation used throughout the paper. Basic notions from differential and Riemannian geometry are briefly discussed. It is important to keep in mind that in all that follows, the physical space is a differential manifold N endowed with a single metric tensorg ˆ, while the abstract configuration of the elastic body (by definition, a manifold whose points label the material points of the elastic body) is a differential manifold M endowed with two metric tensors, one g = g[ϕ]:= ϕ∗gˆ induced = = ∗ by an unknown deformation ϕ : M → N, and one g0 g[ϕ0]: ϕ0gˆ induced by a deformation ϕ0 : M → N of reference. The connection and volume on N are denoted ∇ˆ and ωˆ (induced by gˆ), respectively. The connections and volume forms on M are denoted ∇ = ∇[ϕ] and ω = ω[ϕ] (induced by g = g[ϕ]) and ∇0 and ω0 (induced by g0). Tensor fields on M will be denoted by plain letters, like ξ, and their components in a local chart will be denoted with Latin indices, like ξi. Tensor fields on N will be denoted by letters with a hat, like ξˆ, and their components in a local chart will be denoted with Greek indices, like ξˆα. Tensor fields on M × N will be denoted by letters with a tilde, like ξ˜ or T˜, and their components ˜α ˜ i in local charts will be denoted with Greek and Latin indices, like ξ or Tα. Functionals defined over an infinite-dimensional manifold, such as C1(M, N) or C1(TM):= {ξ : M → TM; ξ(x) ∈ Tx M}, will be denoted with letters with a bracket, like T[·]. Functions p defined over a finite-dimensional manifold, such as M × N or Tq M, will be denoted with letters with a paranthesis, like T˙( ). Using the same letter in T[ ] and T˙( ) means that the two functions are related, typically (but not always) by

(T[ϕ])(x) = T˙(x, ϕ(x), Dϕ(x)) for all x ∈ M, where Dϕ(x) denotes the differential of ϕ at x. In that case, the function T˙( ) is called the constitutive law of the function T[ ] and the relation above is called the constitutive equation of T. Letters with several dots denote constitutive laws of different kind, e.g., ...... (T[ϕ])(x) = T˙(x, ϕ(x), Dϕ(x)) = T¨(x, g[ϕ](x)) = T (x, E[ϕ0, ϕ](x)) = T (x, ξ(x), ∇0ξ(x)) for all x ∈ M, where E[ϕ , ϕ]:= 1 (g[ϕ] − g ) and ξ := exp−1 ϕ (the mapping exp is defined 0 2 0 ϕ0 ϕ0 below). The Gateauxˆ derivative of a function f [ ] at a point ϕ in the direction of a tangent vector η at ϕ will be denoted f ′[ϕ]η. 2 In Section 3, we define the kinematic notions used to describe the deformation of an elastic body. The main novelty is the relation

ϕ = ξ = ϕ ξ ◦ ϕ expϕ0 : (exp( 0∗ )) 0

1 between a displacement field ξ ∈ C (TM) of a referenced configuration ϕ0(M) of the body and the corresponding deformation ϕ : M → N of the same body. Of course, this relation only holds if the vector field ξ is small enough, so that the exponential maps of N be well defined at each point ϕ0(x) ∈ N, x ∈ M. We will see in the next sections that the exponential maps on the Riemannian manifold N replace, to some extent, the vector space structure of the three-dimensional Euclidean space appearing in classical elasticity. The most important notions defined in this section are the metric tensor field, or the right Cauchy-Green tensor field,

g[ϕ]:= ϕ∗gˆ, induced by a deformation ϕ : M → N, the strain tensor field, or Green-St. Venant tensor field, 1 E[ϕ, ψ]:= (g[ψ] − g[ϕ]) 2 associated with a reference deformation ϕ and a generic deformation ψ, and the linearized strain tensor field 1 e[ϕ, ξ]:= L (g[ϕ]), 2 ξ associated with a reference deformation ϕ and a displacement field ξ˜ = (ϕ∗ξ) ◦ ϕ of the configuration ϕ(M). In Section 4, we translate the assumption that the body is made of an elastic material into mathematical terms. The assumption underlying our model is that the strain energy density associated with a deformation ϕ of the body is of the form ... = Λn (W[ϕ])(x): W(x, (E[ϕ0, ϕ])(x)) ∈ x M, x ∈ M, where ϕ0 : M → N denotes a natural configuration of the body. The stress tensor field associated with a deformation ϕ is then defined in terms of this density by ... ∂W Σ[ϕ]:= (·, E[ϕ , ϕ]). ∂E 0 Other equivalent stress tensor fields, denoted T[ϕ], T˜ [ϕ], Σˆ [ϕ], and Tˆ [ϕ], are defined in terms of Σ[ϕ] by lowering and pushing forward some of its indices; cf Remark 4.3. The novelty are the tensor fields Σˆ [ϕ]:= ϕ∗Σ[ϕ] and T[ϕ]:= g[ϕ] · Σ[ϕ], where · denotes the contraction of one single index, which are not needed in classical elasticity because of the particularity of the three- dimensional Euclidean space. The three other tensor fields Tˆ [ϕ], T˜ [ϕ], and Σ[ϕ], correspond in classical elasticity to the Cauchy, the first Piola-Kirchhoff, and the second Piola-Kirchhoff, stress tensor fields, respectively. Here and in the following, boldface letters denote n-forms with scalar or tensor coefficients; the corresponding plain letters denote components of such n-forms over a fixed volume form. For instance, if ω := ϕ∗ωˆ denotes the volume on M induced by a deformation ϕ, then

W = W ⊗ ω, Σ = Σ ⊗ ω, T = T ⊗ ω, T˜ = T˜ ⊗ ω, Tˆ = Tˆ ⊗ ωˆ , Σˆ = Σˆ ⊗ ωˆ . 3 ω = ∗ω In the particular case where the volume form is 0 : ϕ0 ˆ , where ϕ0 deﬁnes the reference conﬁguration of the body, we use the notation

W = W0 ⊗ ω0, Σ = Σ0 ⊗ ω0, T = T0 ⊗ ω0, T˜ = T˜0 ⊗ ω0.

Incorporating the volume form in the definition of the stress tensor field might seem redundant (after all, only W0, Σ0, T˜0, Tˆ are defined in classical elasticity), but it has three important advan- tages: First, it allows to do away with the Piola transform and use instead the more geometric pullback operator. Second, it allows to write the boundary value problem of both nonlinear and linearized elasticity (equations (1.1) and (1.2), resp. (1.3) and (1.4), below) in divergence form, by using appropriate volume forms: ω in nonlinear elasticity and ω0 in linearized elasticity, so that ∇ω = 0 and ∇0ω0 = 0. Third, the normal trace of T = T[ϕ] on the boundary of M appearing in the boundary value problem (1.1) is independent of the choice of the metric used to define the unit outer normal vector field to ∂M, in contrast to the normal trace of T = T[ϕ] on the same boundary appearing in the boundary value problem (1.2); see the relation (2.3) and the subsequent comments. Section 5 is concerned with the modeling of external forces. The main assumption is that the densities of the applied body and surface forces are of the form

= ˙ ∗ Λn ( f[ϕ])(x): f(x, ϕ(x), Dϕ(x)) ∈ Tx M ⊗ x M, x ∈ M, = ˙ ∗ Λn−1Γ Γ (h[ϕ])(x): h(x, ϕ(x), Dϕ(x)) ∈ Tx M ⊗ x 2, x ∈ 2 ⊂ ∂M, where the functions f˙ and h˙ are sufficiently regular. In Section 6, we combine the results of the previous sections to derive the model of nonlinear elasticity in a Riemannian manifold, first as a minimization problem (Proposition 6.1), then as variational equations (Proposition 6.3), and finally as a boundary value problem (Proposition 6.5). The latter asserts that the deformation ϕ of the body must satisfy the system

− div T[ϕ] = f[ϕ] in intM,

T[ϕ]ν = h[ϕ] on Γ2, (1.1)

ϕ = ϕ0 on Γ1, or equivalently, the system

− div T[ϕ] = f [ϕ] in intM,

T[ϕ] · (ν[ϕ] · g[ϕ]) = h[ϕ] on Γ2, (1.2)

ϕ = ϕ0 on Γ1, where div = div[ϕ] and ν[ϕ] respectively denote the divergence operator and the unit outer normal vector field to the boundary of M induced by the metric g = g[ϕ], and where Γ1∪Γ2 = ∂M denotes a partition of the boundary of M. Note that the divergence operators appearing in these boundary value problems depend themselves on the unknown ϕ. In Section 7, we deduce the equations of... linearized elasticity from those of nonlinear elasticity, by linearizing first the constitutive law Σ of the stress tensor field as a function of the strain tensor field (small strains nonlinear elasticity), then by linearizing the constitutive law E˙ (see Remark 3.3) of the strain tensor field as a function of the gradient of the displacement field of the reference configuration of the body (linearized elasticity). Thus the unknown in the linearized 4 1 elasticity is the displacement field ξ ∈ C (TM) of the reference configuration ϕ0(M) of the body, assumed to be a natural state (that is, an unconstrained configuration of the body). ... The elasticity tensor field of an elastic material whose (nonlinear) constitutive law is W is ∈ defined at each x M by ... ∂2W A(x):= (x, 0). ∂E2 The linearized stress tensor field associated with a displacement field ξ is then defined by

lin T [ξ]:= (A : e[ϕ0, ξ]) · g0, where g0 = g[ϕ0] and : denotes the contraction of two indices (the last two contravariant indices of A with the two covariant indices of e[ϕ0, ξ]). The aﬃne part with respect to ξ of the densities of the applied forces are deﬁned by

aﬀ ′ aﬀ ′ f [ξ]:= f[ϕ0] + f [ϕ0]ξ and h [ξ]:= h[ϕ0] + h [ϕ0]ξ,

′ d 1 2 where f [ϕ0]ξ := f[exp (tξ)] = f · ξ + f : ∇0ξ for some appropriate tensor fields dt ϕ0 t=0 1 0 0 Λn h 2 0i 1 Λn ′ f ∈ C (T2 M ⊗ M) and f ∈ C (T2 M ⊗ M) (a similar relation holds for h [ϕ0]ξ). It is then shown that, in linearized elasticity, the unknown displacement field of the reference = 1 configuration ϕ0(M) is the vector field ξ˜ (ϕ0∗ξ) ◦ ϕ0, where ξ ∈ C (TM) satisfies the boundary value problem lin aff −div0 T [ξ] = f [ξ] in intM, lin = aff Γ T [ξ]ν0 h [ξ] on 2, (1.3)

ξ = 0 on Γ1, or equivalently, the boundary value problem

lin = aff −div0 T0 [ξ] f0 [ξ] in intM, lin = aff Γ T0 [ξ] · (ν0 · g0) h0 [ξ] on 2, (1.4) ξ = 0 on Γ1, where div0 and ν0 respectively denote the divergence operator and the unit outer normal vector field to the boundary of M induced by the metric g0 = g[ϕ0]; cf. Proposition 7.5. It is also shown that these boundary value problems are equivalent to the variational equations

aff aff (A : e[ϕ0, ξ]) : e[ϕ0, η] = f [ξ] · η + h [ξ] · η, (1.5) ZM ZM ZΓ2 for all sufficiently regular vector fields η that vanish on Γ1. In Section 8, we establish an existence and regularity theorem for the equations of linearized elasticity in a Riemannian manifold (eqns (1.3)-(1.5)). We show that the variational equations 1 (1.5) have a unique solution in the Sobolev space {ξ ∈ H (TM); ξ = 0 on Γ1} provided the ′ ′ elasticity tensor field A is uniformly elliptic and f [ϕ0] and h [ϕ0] are sufficiently small in an appropriate norm. The key to this existence result is a Riemannian version of Korn’s inequality, due to [7], asserting that, if Γ1 , ∅, there exists a constant CK < ∞ such that

1 2 kξkH (TM) ≤ CKkLξg0kL (S 2 M), 5 1 for all ξ ∈ H (TM) that vanish on Γ1. The smallness assumption mentioned above depends ′ ′ on this constant: the smaller CK is, the larger f [ϕ0] and h [ϕ0] are in the existence result for linearized elasticity. Furthermore, when Γ1 = ∂M, we show that the solution to the equations of linearized elasticity belongs to the Sobolev space Wm+2,p(TM), m ≥ 0, 1 < p < ∞, and satisfies the boundary ′ value problems (1.2) and (1.3) if the data (∂M, ϕ0, f[ϕ0], and f [ϕ0]) satisfies specific regularity assumptions. In Section 9, we study the existence of solutions to the equations of nonlinear elasticity (1.1) and (1.2) in the particular case where Γ1 = ∂M and the applied forces and the constitutive law of the elastic material are sufficiently regular. Under these assumptions, the equations of linearized lin lin ′ elasticity define a surjective continuous linear operator A [ξ]:= div0 T [ξ]+ f [ϕ0]ξ : X → Y, where = m+2,p 1,p = m,p ∗ Λn X : W (TM) ∩ W0 (TM) and Y : W (T M ⊗ M), for some exponents m ∈ N and 1 < p < ∞ that satisfy the constraint (m + 1)p > n, where n denotes the dimension of the manifold M. ϕ = ξ Using the substitution expϕ0 , we recast the equations of nonlinear elasticity (1.2) into ξ C1 TM → C1 M, N an equivalent (when is small enough so that the mapping expϕ0 : ( ) ( ) be well-defined) boundary value problem, viz.,

− T ξ = f ξ M, div [expϕ0 ] [expϕ0 ] in int ξ = 0 on ∂M, whose unknown is the displacement field ξ. Then we show that the mapping A : X → Y defined by A ξ = T ξ + f ξ ξ ∈ X, [ ]: div [expϕ0 ] [expϕ0 ] for all satisfies A′[0] = Alin. Thus proving an existence theorem for the equations of nonlinear elasticity amounts to proving the existence of a zero of the mapping A. This is done by using a variant of Newton’s method, where a zero of A is found as the limit of the sequence

′ −1 ξ1 := 0 and ξk+1 := ξk − A [0] A[ξk], k ≥ 1.

+ m+1,p 1 Note that the constraint (m 1)p > n ensures that the Sobolev space W (T1 M), to which ∇0ξ belongs, is an algebra. This assumption is crucial in proving that the mapping A : X → Y is ff di erentiable, since .... (A[ξ])(x) = A(x, ξ(x), ∇0ξ(x)), x ∈ M, .... for some regular enough mapping A, defined in terms of the constitutive laws of the elastic material and of the applied forces under consideration; cf. relations (9.5) and (9.6). Thus A is a nonlinear Nemytskii (or substitution) operator, which is known to be non differentiable if ξ belongs to a space with little regularity. ′ In addition to the regularity assumptions, we must assume that f [ϕ0] is sufficiently small in an appropriate norm, so that the operator A′[0] ∈ L(X, Y) is invertible; cf. Theorem 8.1 establishing the existence and regularity for linearized elasticity. Finally, we point out that the assumptions of the existence theorem of Section 9 are slightly weaker than those usually.... made in classical elasticity, where either p > n is imposed instead of + m+1 1 (m 1)p > n (cf. [5]), or f is assumed to belong to the smaller space C (M × TM × T1 M) (cf. [17]). 6 2. Preliminaries

Throughout this paper, M and N denote smooth oriented diﬀerentiable manifolds of dimension n. In addition, M is compact and N is endowed with a smooth Riemannian metricg ˆ. Generic i n α n points in M and N are denoted x and y, respectively, or (x )i=1 and (y )α=1 in local coordinates. To ease notation, the n-tuples (xi) and (yα) are also denoted x and y, respectively. = ∗ = ∗ The tangent and cotangent bundles of M are denoted TM : x∈M Tx M and T M : x∈M Tx M, p = respectively. The bundle of all (p, q)-tensors (p-contravariant andFq-covariant) is denotedFTq M : (⊗pTM) ⊗ (⊗qT ∗M). Partial contractions of one or two indices between two tensors will be denoted · or : , respectively. The bundle of all symmetric (0, 2)-tensors is denoted

= 0 S 2 M : S 2,x M ⊂ T2 M, xG∈M and the bundle of all positive-deﬁnite symmetric (0, 2)-tensors is denoted by + = + S 2 M : S 2,x M ⊂ S 2 M xG∈M 2 = 2 Analogously, the bundle of all symmetric (2, 0)-tensors is denoted by S M : x∈M S x M. The bundle of all k-forms (that is, totally antisymmetric (0, k)-tensors ﬁelds)F is denoted Λk = Λk M : x∈M x M; volume forms (that is, n-forms on M and (n − 1)-forms on the boundary of ω ω M) will beF denoted by boldface letters, such as and iν . Fiber bundles on M × N will also be used with self-explanatory notation. For instance, ∗ = ∗ T M ⊗ TN : TxM ⊗ TyN, (x,yG)∈M×N

∗ where TxM ⊗ TyN is canonically identiﬁed with the space L(Tx M, TyN) of all linear mappings from Tx M to TyN. The set of all mappings ϕ : M → N of class Ck is denoted Ck(M, N). Given any mapping 0 p ϕ ∈ C (M, N), the pullback bundle of Tq N by ϕ is denoted and deﬁned by ∗ p = p ϕ Tq N : Tq,ϕ(x)N. xG∈M

p p ∗ 0 The pushforward and pullback mappings are denoted ϕ∗ : T0 M → T0 N and ϕ : Tq M → 0 = = Tq N, respectively. By way of example, if p 1 and q 2, then ∂ϕα ∂ϕα ∂ϕβ (ϕ ξ)α(ϕ(x)) := (x)ξi(x) and (ϕ∗gˆ) (x):= (x) (x)ˆg (ϕ(x)), x ∈ M, ∗ ∂xi i j ∂xi ∂x j αβ where the functions yα = ϕα(xi) describe the mapping ϕ in local coordinates (xi) on M and (yα) on N. p p ∗ p The Lie derivative operators on Tq M, Tq N, and ϕ (Tq N), are denoted L, Lˆ, and L˜, respectively. By way of example, the Lie derivative ofg ˆ along a vector field ξˆ ∈ C1(TN), and ofg ˜ along a vector field ξ˜ ∈ C1(ϕ∗TN), are defined by

ˆ = 1 ∗ ˜ = 1 ∗ Lξˆgˆ : lim (γξˆ(·, t) gˆ − gˆ) and Lξ˜g˜ : lim (γξ˜(·, t) g˜ − g˜), t→0 t t→0 t 7 ˜ ˆ ffi where γξ˜ and γξˆ denote the flows of ξ and ξ, respectively. These flows are defined, for a su - ciently small parameter (whose existence follows from the compactness of M), by the mappings = (x, t) ∈ M ×(−ε, ε) → γξ˜(x, t): γξˆ(ϕ(x), t) ∈ N and (y, t) ∈ (−ε, ε) → γξˆ(y, t) ∈ N, where γξˆ(y, ·) is the unique solution to the Cauchy problem d γ (y, t) = ξˆ(γ (y, t)) for all t ∈ (−ε, ε), and γ (y, 0) = y. dt ξˆ ξˆ ξˆ

The notation ξ|Γ designates the restriction to the set Γ of a function or a tensor field ξ defined over a set that contains Γ. Given any smooth fiber bundle X over M and any submanifold Γ ⊂ M, we denote by Ck(X) the space of all sections of class Ck of the fiber bundle X and

k k C (X|Γ):= {S |Γ; S ∈ C (X)}. If S ∈ Ck(X) is a section of a ﬁber bundle X over M, S (x) denotes the value of S at x ∈ M. k The tangent at x ∈ M of a mapping ϕ ∈ C (M, N) is a linear mapping Txϕ ∈ L(Tx M, Tϕ(x)N). The section Dϕ ∈ Ck−1(T ∗M ⊗ ϕ∗TN) deﬁned at each x ∈ M by

Dϕ(x) · ξ(x):= (Txϕ)(ξ(x)) for all ξ ∈ TM is the diﬀerential of ϕ at x. In local charts, ∂ϕα ∂ Dϕ(x) = (x) dxi(x) ⊗ (ϕ(x)), x ∈ M. ∂xi ∂yα

Let ∇ˆ : Ck(TN) → Ck−1(T ∗N ⊗ TN) be the Levi-Civita connection on the Riemannian manifold N induced by the metricg ˆ. Any immersion ϕ ∈ Ck+1(M, N) induces the metrics = ∗ k + = ∗ k + ∗ g : ϕ gˆ ∈ C (S 2 M) andg ˜ : ϕbgˆ ∈ C (S 2 (ϕ TN)), where ∗ = ∗ ˆ = ˆ (ϕ gˆ)(ξ, η): gˆ(ϕ∗ξ, ϕ∗η) ◦ ϕ and (ϕbgˆ)(ξ ◦ ϕ, ηˆ ◦ ϕ): gˆ(ξ, ηˆ) ◦ ϕ, and the corresponding connections ∇ : Ck(TM) → Ck−1(T ∗M ⊗ TM), ∇˜ : Ck(ϕ∗TN) → Ck−1(T ∗M ⊗ ϕ∗TN). In local coordinates (see [1] for details), we have ∂ϕα ∂ϕβ g := g˜ , g˜ := gˆ ◦ ϕ, i j ∂xi ∂x j αβ αβ αβ and ∂ξˆβ ∇ˆ ξˆβ = + Γˆ β ξˆγ, α ∂yα αγ ∂ξ j ∇ ξ j = + Γ j ξk, i ∂xi ik ∂ξ˜α ∂ϕβ ∇˜ ξ˜α = + Γ˜ α ξ˜γ, i ∂xi ∂xi βγ Γˆ β Γβ Γ˜ α = Γˆ α ﬀ where αγ, αγ, and βγ : βγ ◦ ϕ, denote the Christo el symbols associated with the metric tensorsg ˆ, g, andg ˜, respectively. 8 Remark 2.1. The metric tensors g and g˜ and the connections ∇ and ∇˜ all depend on the immersion ϕ. When necessary, the notation g[ϕ]:= g, ∇[ϕ]:= ∇ and g˜[ϕ]:= g,˜ ∇˜ [ϕ]:= ∇˜ will be used to indicate this dependence. The above connections are related to each other by the relations

∇˜ ξ˜ = Dϕ · ∇ξ = Dϕ · ((∇ˆ ξˆ) ◦ ϕ) (2.1)

k for all ξ ∈ C (TM), ξˆ := ϕ∗ξ, ξ˜ := ξˆ ◦ ϕ, which in local coordinates read:

∂ϕα ∂ϕβ ∇˜ ξ˜α = ∇ ξ j = ((∇ˆ ξˆα) ◦ ϕ), (2.2) i ∂x j i ∂xi β

ˆα = ∂ϕα i ˜α = ˆα where ξ : ∂xi ξ and ξ : ξ ◦ ϕ. For further use, we note that

∗ ∇θ = ϕ (∇ˆ θˆ) and ∇˜ ηξ˜ = (∇ˆ ηˆ ξˆ) ◦ ϕ,

∗ k−1 k ∗ k withη ˆ = ϕ∗η, θ = ϕ θˆ and ξ˜ = ξˆ ◦ ϕ, for all η ∈ C (TM), θˆ ∈ C (T N) and ξˆ ∈ C (TN). The connection ∇, resp. ∇ˆ , is extended to arbitrary tensor fields on M, resp. on N, in the usual manner, by using the Leibnitz rule. The connection ∇˜ is extended to arbitrary sections ˜ k p ∗ r = S ∈ C (Tq M ⊗ ϕ (Ts N)) by using the Leibnitz rule and the connection ∇ ∇[ϕ]. Specifically, ˜ ˜ k−1 p ∗ r = = = = the section ∇ηS ∈ C (Tq+1 M ⊗ϕ (Ts N)) is defined by (we set p q r s 1 for simplicity)

(∇˜ ηS˜ )(ξ, σ, ζ,˜ τ˜):= η(S˜ (ξ, σ, ζ,˜ τ˜)) − S˜ (∇ηξ, σ, ζ,˜ τ˜) − S˜ (ξ, ∇ησ, ζ,˜ τ˜)

−S˜ (ξ, σ, ∇˜ ηζ,˜ τ˜) − S˜ (ξ, σ, ζ,˜ ∇˜ ητ˜), for all sections η ∈ Ck−1(TM), σ ∈ Ck(T ∗M), ξ ∈ Ck(TM), ζ˜ ∈ Ck(ϕ∗TN)andτ ˜ ∈ Ck(ϕ∗T ∗N). The divergence operators induced by the connections ∇ = ∇[ϕ], ∇˜ = ∇˜ [ϕ], and ∇ˆ , are respectively denoted div = div[ϕ], div = div[ϕ], and div. If T˜ = T˜ ⊗ω with T˜ ∈ C1(TM ⊗ϕ∗T ∗N) and ω ∈ Λn M, then f f c ˜ = ˜ ˜ i α (div T)(x): (∇iTα)(x)dy (ϕ(x)), i T˜ x = ∇˜ T˜ x dx j1 x ⊗ ... ⊗ dx jn x ⊗ dyα ϕ x (divf )( ): ( i j1... jn,α)( ) ( ) ( ) ( ( )) for all x ∈ M. If inf addition ∇ω = 0, then

∇˜ ηT˜ = (∇˜ ηT˜) ⊗ ω and div T˜ = (div T˜) ⊗ ω.

0 n f∗ ∗ f 0 n−1 ∗ ∗ The interior product iη : T˜ ∈ C (Λ M ⊗ TM ⊗ ϕ T N) → iηT˜ ∈ C (Λ M ⊗ TM ⊗ ϕ T N) is deﬁned by (iηT˜ )(ζ1, ..., ζn−1; θ, ξ˜):= T˜ (η, ζ1, ..., ζn−1; θ, ξ˜) 0 0 ∗ 0 ∗ for all η, ζ1, ..., ζn−1 ∈ C (TM), θ ∈ C (T M), and ξ˜ ∈ C (ϕ TN), or equivalently by

iηT˜ = T˜ ⊗ iηω if T˜ = T˜ ⊗ ω.

The normal trace of a tensor field T˜ ∈ C0(Λn M⊗TM⊗ϕ∗T ∗N) on the boundary ∂M is defined by 0 n−1 ∗ ∗ T˜ ν := (iνT˜ ) · (ν · g) ∈ C (Λ (∂M) ⊗ ϕ T N), 9 or equivalently by T˜ ν = (T˜ · (ν · g)) ⊗ iνω if T˜ = T˜ ⊗ ω. (2.3) where ν denotes the unit outer normal vector field to ∂M defined by the metric g. Note that the definition of T˜ ν is independent of the choice of the Riemannian metric g, since ˜ = ˜ (iν1 T) · (ν1 · g1) (iν2 T) · (ν2 · g2) for all Riemannian metrics g1 and g2 on M (νi denotes the unit outer normal vector field to ∂M defined by the metric gi, i = 1, 2). Indeed, ˜ = ˜ (iν1 T) · (ν1 · g1) g2(ν1, ν2)[(iν2 T) · (ν1 · g1)] and g2(ν1, ν2)(ν1 · g1) = ν2 · g2. Integration by parts formulas involving either connection ∇, ∇˜ and ∇ˆ will be needed in Sec- tion 6. We establish here the formula for the connection ∇˜ , since it might not be recorded elsewhere in the literature. Letting M = N and ϕ = idM in the lemma below yields the integration by parts formulas for the other two connections ∇ and ∇ˆ , which otherwise are classical. Recall that · , respectively : , denotes the contraction of one, respectively two, indices. Lemma 2.2. Setting ξ˜ ∈ C1(ϕ∗TN) and T˜ ∈ C1(Λn M ⊗ TM ⊗ ϕ∗T ∗N), one has

T˜ : ∇˜ ξ˜ = − (div T˜ ) · ξ˜ + T˜ ν · ξ.˜ ZM ZM Z∂M f Proof. Let ω be the volume form induced by the metric g and let T˜ ∈ C1(TM⊗ϕ∗T ∗N) be deﬁned ˜ = ˜ ω ˜ ˜ ˜ = ˜ i ˜ ˜α = ˜ i ˜α ˜ ˜ i ˜α by T T ⊗ . Since T : ∇ξ Tα∇iξ ∇i(Tαξ ) − (∇iTα)ξ , it follows that

(T˜ : ∇˜ ξ˜) ω = div(T˜ · ξ˜) ω − (divT˜ · ξ˜) ω ZM ZM ZM f = ω ˜ ˜ ω LT˜·ξ˜ − (divT · ξ) , ZM ZM f where L denotes the Lie derivative on M. The first integral of the right-hand side can be written as ω = ω = ω LT˜·ξ˜ d(iT˜·ξ˜ ) iT˜·ξ˜ . ZM ZM Z∂M Let ν be the unit outer normal vector field to ∂M defined by the metric g. Since T˜ · ξ˜ = g(T˜ · ξ,˜ ν)ν + {T˜ · ξ˜ − g(T˜ · ξ,˜ ν)ν} and since the vector field {T˜ · ξ˜ − g(T˜ · ξ,˜ ν)ν} is tangent to ∂M, the integrand of the last integral becomes ω = ω = ˜ ˜ ω = ˜ ˜ ω iT˜·ξ˜ ig(T˜·ξ,ν˜ )ν g(T · ξ, ν)iν (T · ξ) · (ν · g) iν

= [(T˜ · (ν · g)) · ξ˜] iνω. Therefore,

(T˜ : ∇˜ ξ˜) ω = − (div T˜ · ξ˜) ω + [(T˜ · (ν · g)) · ξ˜] iνω. ZM ZM Z∂M This relation is equivalent to the integrationf by parts formula announced in the theorem since (T˜ : ∇˜ ξ˜) ω = T˜ : ∇˜ ξ˜,(div T˜ ·ξ˜) ω = div T˜ ·ξ˜ and [(T˜ ·(ν·g))·ξ˜] iνω = (iνT˜ ·(ν·g))·ξ˜ = T˜ ν ·ξ˜. 10 f f All functions and tensor fields appearing in Sections 3-7 are of class Ck over their domain of definition, with k sufficiently large so that all differential operators be defined in the classical sense (as opposed to the distributional sense). Functions and tensor fields belonging to Sobolev spaces on the Riemannian manifold (M, g0) will be used in Sections 8-9 in order to prove existence theorems for the models introduced in Sections 6 and 7. Following [2], the Sobolev space Wk,p(TM) is defined for each k ∈ N and 1 ≤ p < ∞ as the completion in the Lebesgue space Lp(TM) of the space Ck(TM) with respect to the norm

k p ℓ p 1/p kξkk,p = kξkWk,p(TM) := |ξ| + |∇0 ξ| ω0 , Z n M Xℓ=1 o where ℓ ℓ ℓ 1/2 |∇0 ξ| := {g0(∇0 ξ, ∇0 ξ)} 1/2 = i1 j1 iℓ jℓ i j (g0)i j(g0) ...(g0) (∇0)i1...iℓ ξ (∇0) j1... jℓ ξ . n o k,p k,p The Sobolev space W0 (TM) is deﬁned as the closure in W (TM) of the space k = k Cc(TM): {ξ ∈ C (TM); supp(ξ) ⊂ intM}. k = k,2 k = k,2 We will also use the notation H (TM): W (TM) and H0(TM): W0 (TM).

3. Kinematics

Consider an elastic body with abstract configuration M undergoing a deformation in a Rie- mannian manifold (N, gˆ) in response to external forces. All the kinematic notions introduced below are natural extensions of their counterparts in classical elasticity. Specifically, if (N, gˆ) is the three-dimensional Euclidean space and if the reference configuration of the elastic body is described by a single local chart with M as its domain of definition, then our definitions coincide with the classical ones in curvilinear coordinates. A deformation of the body is a C1-immersion ϕ : M → N that preserves orientation and satisfies the impenetrability of matter axiom. This means that

det Dϕ(x) > 0 for all x ∈ M,

ϕ|intM : intM → N is injective, where intM denotes the interior of M. Note that ϕ needs not be injective on the whole M since self-contact of the boundary may occur. A displacement field of the configuration ϕ(M) of the body is a section ξ˜ ∈ C1(ϕ∗TN). It is often convenient to identify displacement fields of ϕ(M) with vector fields on ξ ∈ C1(TM) by means of the bijective mapping ξ → ξ˜ := (ϕ∗ξ) ◦ ϕ. When no confusion should arise, the vector field ξ will also be called displacement field of the configuration ϕ(M). Remark 3.1. An example of displacement field is the velocity field of a body: If ψ(t): M → N is = ˜ = dψ a time-dependent family of deformations and ϕ : ψ(0), then ξ : dt (0) is a displacement field of the configuration ϕ(M). 11 Deformations ψ ∈ C0(M, N) that are close in the C0-norm to a given deformation ϕ ∈ C1(M, N) are canonically related to the displacement fields ξ˜ ∈ C0(ϕ∗TN) of the configuration ϕ(M) of the body by the relation ψ = (exp ξˆ) ◦ ϕ, ξˆ ◦ ϕ = ξ,˜ ˜ = where exp denotes the exponential mapsd on N. When ξ (ϕ∗ξ) ◦ ϕ is defined by means of a vector field ξ ∈ C0(TM) on the abstract configuration M, we denote d = = ψ expϕ ξ : (exp ϕ∗ξ) ◦ ϕ. (3.1)

Of course, these relations make sense only if |ξˆ(ϕd(x))| = |(ϕ∗ξ)(ϕ(x))| < δˆ(ϕ(x)) for all x ∈ M, where δˆ(y) denotes the injectivity radius of N at y ∈ N. Let δˆ(ϕ(M)) := miny∈ϕ(M) δˆ(y) be the injectivity radius of the compact subset ϕ(M) of N, and define the set 0 0 = 0 ˆ Cϕ(TM): {ξ ∈ C (TM); kϕ∗ξkC (TN|ϕ(M)) < δ(ϕ(M))}. (3.2) It is then clear from the properties of the exponential maps on N that the mapping = 0 0 expϕ exp ◦ Dϕ : Cϕ(TM) → C (M, N) 1 ff −1 ff is a C -di eomorphism onto its image.d Together with its inverse, denoted expϕ , this di eomorphism will be used in Sections 7-9 to transform the equations of elasticity in which the unknown is ϕ into equivalent equations in which the unknown is the displacement field ξ of a reference configuration ϕ0(M) of a body; of course, the two formulations are equivalent only for sufficiently ξ = −1 ϕ small : expϕ0 . = Remark 3.2. (a) The relation ψ expϕ ξ means that, for each x ∈ M, ψ(x) is the end-point of the geodesic arc in N with length |ξ(x)| starting at the point ϕ(x) in the direction of (ϕ∗ξ)(ϕ(x)). = −1 (b) The relation ξ expϕ ψ means that, for each x ∈ M, ξ(x) is the pullback by ϕ of the vector that is tangent at ϕ(x) to the geodesic arc joining ϕ(x) to ψ(x) in N and whose norm equals the length of this geodesic arc. The metric tensor field, or the right Cauchy-Green tensor field, associated with a deformation ϕ ∈ C1(M, N) is the pullback by ϕ of the metricg ˆ of N, i.e., g[ϕ]:= ϕ∗gˆ. Note that the notation C := g[ϕ] is often used in classical elasticity. The strain tensor field, or Green-St Venant tensor field, associated with two deformations ϕ, ψ ∈ C1(M, N) is defined by 1 E[ϕ, ψ]:= (g[ϕ] − g[ψ]). 2 The first argument ϕ is considered as a deformation of reference, while the second argument ψ is an arbitrary deformation. The linearized, or infinitesimal, strain tensor field associated with a deformation ϕ ∈ C1(M, N) and a displacement field ξ ∈ C1(TM) of the set ϕ(M) is the linear part with respect to ξ of the mapping ξ 7→ E[ϕ, expϕ ξ], i.e.,

= d e[ϕ, ξ]: E[ϕ, expϕ(tξ)] . "dt #t=0 Explicit expressions of e[ϕ, ξ] are given in Theorem 3.4 below. 12 1 = = ∗ Remark 3.3. Let ϕ0 ∈ C (M, N) be a deformation of reference and let g0 : g[ϕ0] ϕ0g.ˆ Given ∗ ∗ 0 0 any (x, y) ∈ M × N and any F, G ∈ TxM ⊗ TyN, let (F, G) : T2,yN → T2,x M denote the pullback 0 0 mapping defined by the bilinear mapping (F, G): T2,x M → T2,yN, that is ∗ = 0 ((F, G) τˆ)(ξ, η): τˆ(Fξ, Gη) for all (ξ, η) ∈ Tx M × Tx M and all τˆ ∈ T2,yN, and let 1 g˙(x, y, F):= (F, F)∗gˆ(y) and E˙(x, y, F):= (˙g(x, y, F) − g (x)). 2 0 Then g[ϕ](x) = g˙(x, ϕ(x), Dϕ(x)) and E[ϕ, ϕ0](x) = E˙(x, ϕ(x), Dϕ(x)). The mappings g˙ and E˙ defined in this fashion are the constitutive laws of the right Cauchy-Green tensor field g[ϕ] and the Green-St Venant tensor field E[ϕ0, ϕ] associated with a deformation ϕ. Linearized strain tensor field can be expressed either in terms of the Lie derivatives on M and on N, or in terms of either of the connections defined in the previous section, as we now show. Recall that · denotes the partial contraction of one single index of two tensors. Proposition 3.4. Given any immersion ϕ ∈ C1(M, N) and any vector field ξ ∈ C1(TM), define the vector fields

1 1 ∗ ξˆ = ξˆ[ϕ]:= ϕ∗ξ ∈ C (Tϕ(M)) and ξ˜ = ξ˜[ϕ]:= (ϕ∗ξ) ◦ ϕ ∈ C (ϕ TN), and the corresponding 1-forms ξ♭ = ξ♭[ϕ]:= g[ϕ] · ξ, ξˆ♭ = ξˆ♭[ϕ]:= gˆ · ξˆ[ϕ], and ξ˜♭ = ξ˜♭[ϕ]:= g˜[ϕ] · ξ˜[ϕ], = = ∗ = = ∗ = ˜ = ˜ ˆ where g g[ϕ]: ϕ gˆ and g˜ g˜[ϕ]: ϕbg.ˆ Let ∇ ∇[ϕ], ∇ ∇[ϕ], and ∇, respectively be the connections induced by the metric tensors g, g,˜ and g,ˆ and let L, L˜, and Lˆ, denote the Lie p ∗ p p derivative operators on Tq M, ϕ (Tq N), and Tq N, respectively. Then, one has

1 1 1 ∗ e[ϕ, ξ] = L g = L˜ ˜g˜ = ϕ (Lˆ gˆ) 2 ξ 2 ξ 2 ξˆ and 1 1 e[ϕ, ξ] = (∇ξ♭ + (∇ξ♭)T ) = (g · ∇ξ + (g · ∇ξ)T ) 2 2 1 1 = ϕ∗(∇ˆ ξˆ♭ + (∇ˆ ξˆ♭)T ) = ϕ∗(ˆg · ∇ˆ ξˆ + (ˆg · ∇ˆ ξˆ)T ) (3.3) 2 2 1 1 = (Dϕ · ∇˜ ξ˜♭ + (Dϕ · ∇˜ ξ˜♭)T ) = (˜g · Dϕ · ∇˜ ξ˜ + (˜g · Dϕ · ∇˜ ξ˜)T ). 2 2 In local charts, equations (3.3) are equivalent to the relations 1 1 e [ϕ, ξ] = (∇ ξ + ∇ ξ ) = (g ∇ ξk + g ∇ ξk) i j 2 i j j i 2 jk i ik j 1 ∂ϕα ∂ϕβ = (∇ˆ ξˆ + ∇ˆ ξˆ ) ◦ ϕ (3.4) 2 ∂xi ∂x j β α α β 1 ∂ϕβ ∂ϕβ = ∇˜ ξ˜ + ∇˜ ξ˜ , 2 ∂xi j β ∂x j i β i α α where ξ(x) = ξi(x)dx (x), ξˆ(y) = ξˆα(y)dy (y) and ξ˜(x) = ξ˜α(x)dy (ϕ(x)). 13 Proof. For each t in a neighborhood of zero, deﬁne the deformations = = ϕ(·, t): expϕ(tξ) and ψ(·, t): γξˆ(·, t) ◦ ϕ,

ˆ 1 ˆ 1 where ξ ∈ C (TN) denotes any extension of the section ξ ∈ C (Tϕ(M)) and γξˆ denotes the ﬂow of ξˆ (see Section 2). By deﬁnition,

d ϕ(·, t)∗gˆ − ϕ∗gˆ e[ϕ, ξ] = E[ϕ, ϕ(·, t)] = lim . t→0 "dt #t=0 2t Since ∂ϕ ∂ψ (x, 0) = (x, 0) = ξ(x) for all x ∈ M, ∂t ∂t it follows from the above expression of e[ϕ, ξ] that ψ(·, t)∗gˆ − ϕ∗gˆ e[ϕ, ξ] = lim . t→0 2t Then the deﬁnition of the Lie derivative yields γ (·, t)∗gˆ − gˆ = ∗ ξˆ = 1 ∗ ˆ e[ϕ, ξ] ϕ lim ϕ (Lξˆgˆ) t→0 2t 2 1 1 1 = ϕ∗(Lˆ gˆ) = L (ϕ∗gˆ) = L g. 2 ϕ∗ξ 2 ξ 2 ξ ˆ ˆ Expressing the Lie derivative Lξˆgˆ in terms of the connection ∇ gives

1 ∂ϕα ∂ϕβ e [ϕ, ξ] = gˆ ∇ˆ ξˆγ + gˆ ∇ˆ ξˆγ ◦ ϕ i j 2 ∂xi ∂x j αγ β βγ α 1 ∂ϕα ∂ϕβ = ∇ˆ ξˆ + ∇ˆ ξˆ ◦ ϕ. 2 ∂xi ∂x j β α α β This next implies that 1 e [ϕ, ξ] = (g ∇ ξk + g ∇ ξk) i j 2 ik j jk i 1 = (∇ ξ + ∇ ξ ) 2 j i j j and 1 ∂ϕα ∂ϕβ e [ϕ, ξ] = g˜ ∇˜ ξ˜γ + g˜ ∇˜ ξ˜γ i j 2 αγ ∂xi j βγ ∂x j i n o 1 ∂ϕα ∂ϕβ = ∇˜ ξ˜ + ∇˜ ξ˜ . 2 ∂xi j α ∂x j i β n o

Remark 3.5. A vector field ξ ∈ C1(TM) defines two families of deformations, ϕ(·, t) and ψ(·, t), both starting at ϕ with velocity ξ˜ = (ϕ∗ξ) ◦ ϕ; cf. Proof of Theorem 3.4. Note that ϕ(·, t) depends on the metric tensor field gˆ of the manifold N, while ψ(·, t) depends only on the differential structure of the manifold M. 14 We conclude this section with the definition of the strain tensor field in terms of the displacement fields ξ˜ and ξˆ, which will be needed in the next section. Definition 3.6. Let ϕ ∈ C1(M, N) be a deformation and let ξ ∈ C1(TM) be a displacement field of the abstract configuration M. The linearized strain tensor fields associated with the displacement fields ξ˜ := (ϕ∗ξ) ◦ ϕ and ξˆ := ϕ∗ξ of the configuration ϕ(M) of the body are defined by

1 0 1 0 e[ϕ, ξ˜]:= L˜ ˜g˜ ∈ C (S M) and eˆ[ϕ, ξˆ]:= Lˆ gˆ ∈ C (S N), (3.5) 2 ξ 2 2 ξˆ 2 respectively, and are related to each other by (see Theorem 3.4):

e[ϕ, ξ] = e[ϕ, ξ˜] = ϕ∗(ˆe[ϕ, ξˆ]). (3.6)

4. Elastic materials

The behavior of elastic bodies in response to applied forces clearly depends on the elastic material of which they are made. Thus, before studying this behavior, one needs to specify this material by means of a constitutive law, usually a relation between deformations and stresses inside the body. Note that a constitutive law is usually given only for deformations ϕ that are close to a reference deformation ϕ0, so that plasticity and heating do not occur (are negligible). We assume in this paper that the body is made of a hyperelastic material satisfying the axiom of frame-indifference, that is, an elastic material whose behavior is governed by a stored energy function W˙ := W˙ 0ω0 satisfying the relation (4.1) below. The stress tensor field associated with a deformation ϕ : M → N of the body will then be defined in terms of W by either of the sections Σ[ϕ], Σˆ [ϕ], T[ϕ], T˜ [ϕ], and Tˆ [ϕ], which are related to each other by the formulas (4.7) of Proposition 4.4 below. 2 Let a reference configuration of the body be given by means of an immersion ϕ0 ∈ C (M, N). ∗ The metric tensor fields and the connections induced by ϕ0 on TM and on ϕ0TN are denoted (see Remark 2.1)

g0 := g[ϕ0], g˜0 := g˜[ϕ0], ∇0 := ∇[ϕ0], ∇˜0 := ∇˜ [ϕ0].

The volume form induced by ϕ0, or equivalently by the metric tensor ﬁeld g0, on the manifold M ω = ∗ω is denoted 0 : ϕ0 ˆ . The strain energy corresponding to a deformation ϕ of an hyperelastic body is deﬁned by

I[ϕ]:= W[ϕ] = W0[ϕ]ω0, ZM ZM

1 n where the ﬁeld of n-forms W[ϕ] = W0[ϕ]ω0 ∈ L (Λ M) is of the form

(W[ϕ])(x):= W˙ (x, ϕ(x), Dϕ(x)) = W˙ 0(x, ϕ(x), Dϕ(x))ω0(x), x ∈ M, ˙ = ˙ ω ∗ Λn for some given mapping W(x, y, ·) W0(x, y, ·) 0(x): TxM ⊗ TyN → x M,(x, y) ∈ M × N, called the stored energy function of the elastic material constituting the body. We say that the stored energy function satisﬁes the axiom of material frame-indiﬀerence if

′ W˙ 0(x, y, F) = W˙ 0(x, y , RF) 15 ′ ∗ ∗ = for all x ∈ M, y ∈ N, y ∈ N, F ∈ TxM ⊗ TyN, and all isometries R ∈ TyN ⊗ Ty′ N L(TyN, Ty′ N). In that case, the polar decomposition theorem applied to the... linear mapping F implies that, for ¨ + R R each x ∈ M, there exist mappings W0(x, ·): S 2,x M → and W0(x, ·): S 2,x M → such that ... ˙ = ¨ = ∗ W0(x, y, F) W0(x, C) W0(x, E) for all F ∈ TxM ⊗ TyN, (4.1) where the tensors C and E are deﬁned in terms of F by (see Remark 3.3) 1 C = g˙(x, y, F) = (F, F)∗(ˆg(y)) and E = E˙(x, y, F) = (C − g (x)). 2 0 Hence the axiom of material frame-indiﬀerence implies that, at each x ∈ M,

(W[ϕ])(x):= W˙ (x, ϕ(x), Dϕ(x)) = W˙ 0(x, ϕ(x), Dϕ(x))ω0(x)

= W¨ (x, (g[ϕ])(x)) = W¨ 0(x, (g[ϕ])(x))ω0(x) (4.2) ...... = W(x, (E[ϕ])(x)) = W0(x, (E[ϕ])(x))ω0(x),

= = ∗ = ˙ = 1 where g[ϕ] g˙(x, ϕ(x), Dϕ(x)) ϕ gˆ and E[ϕ] E(x, ϕ(x), Dϕ(x)) 2 (g[ϕ] − g0). ˙ ∗ Λn Fix (x, y) ∈ M × N. The Gateaux derivative of the mapping W(x, y, ·): Tx M ⊗ TyN → x M ∗ ∗ at F ∈ Tx M ⊗ TyN in the direction G ∈ TxM ⊗ TyN is deﬁned by

∂W˙ 1 (x, y, F): G = lim W˙ (x, y, F + tG) − W˙ (x, y, F) . ∂F t→0 t n o The constitutive law of an elastic material whose stored energy function is W˙ is the mapping = ∗ associating to each (x, y) ∈ M × N and each F ∈ L(Tx M, TyN) Tx M ⊗ TyN the tensor

∂W˙ ∂W˙ T˜˙ (x, y, F) = T˜˙ (x, y, F) ⊗ ω (x):= (x, y, F) = 0 (x, y, F) ⊗ ω (x) (4.3) 0 0 ∂F ∂F 0 in (T M ⊗ T ∗N) ⊗ Λn M. x y x ... The constitutive law of an elastic material whose stored energy function is W is the mapping associating to each x ∈ M and each E ∈ S 2,x M the tensor ...... ∂W ∂W Σ(x, E) = Σ (x, E) ⊗ ω (x):= (x, E) = 0 (x, E) ⊗ ω (x) (4.4) 0 0 ∂E ∂E 0 in S 2 M ⊗ Λn M. x x ... ˜˙ Σ The next lemma establishes a relation between the two constitutive laws T and... , which derives from the relation between the corresponding stored energy functions W˙ and W. ... Lemma 4.1. The constitutive laws T˜˙ and Σ satisfy ... 1 T˜˙ (x, y, F) = gˆ(y) · F · Σ(x, E), E = E˙(x, y, F) = {(F, F)∗gˆ(y) − g (x)}, 2 0 ∗ = for all linear operators F ∈ Tx M ⊗ TyN L(Tx M, TyN).

16 ˙ ...... Proof. It suﬃces to prove that T˜0(x, y, F) = gˆ(y) · F · Σ0(x, E). Since W˙ 0(x, y, F) = W0(x, E), the ∗ chain rule implies that, for each G ∈ TxM ⊗ TyN, ... ∂W˙ ∂W ∂E˙ T˜˙ (x, y, F): G = 0 (x, y, F): G = 0 (x, E): (y, F): G . 0 ∂F ∂E ∂F Besides, ∂E˙ 1 (y, F): G = lim (F + tG, F + tG)∗gˆ(y) − (F, F)∗gˆ(y) ∂F t→0 2t n o 1 = (F, G)∗gˆ(y) + (G, F)∗gˆ(y) . 2 n ... o Σ = ∂W0 Since the tensorsg ˆ(y) and 0(x, E) ∂E (x, E) are both symmetric, combining the last two relations implies that ˙ ... ∗ T˜0(x, y, F): G = Σ0(x, E):(F, G) gˆ(y), which is the same as ˙ ... T˜0(x, y, F): G = gˆ(y) · F · Σ0(x, E) : G.

We are now in a position to define the stress tensor field associated with a deformation ϕ ∈ C1(M, N) of an elastic body, a notion that plays a key role in all that follows. = ∗ ω = ∗ω Definition 4.2. Let g0 : ϕ0gˆ and 0 : ϕ0 ˆ be the metric tensor field and the volume form 1 ∗ ∗ induced by a deformation of reference ϕ0 ∈ C (M, N), let g[ϕ]:= ϕ gˆ and ω[ϕ]:= ϕ ωˆ denote the metric tensor field and the volume form induced by a generic deformation ϕ ∈ C1(M, N), and let 1 E[ϕ , ϕ]:= (g[ϕ] − g[ϕ ]) 0 2 0 ... denote the strain tensor field associated with the deformations ϕ0 and ϕ. Let Σ be the constitutive law defined by (4.4). (a) The stress tensor field associated with a deformation ϕ is either of the following sections ... Σ[ϕ]:= Σ(·, E[ϕ0, ϕ]), Σˆ [ϕ]:= ϕ∗(Σ[ϕ]), T[ϕ]:= g[ϕ] · Σ[ϕ], Tˆ [ϕ]:= gˆ · Σˆ [ϕ], T˜ [ϕ]:= g˜[ϕ] · Dϕ · Σ[ϕ], where · denotes the contraction of one index (no ambiguity should arise). Σ Σ 0 2 0 1 ˜ ˜ (b) The tensor fields [ϕ], 0[ϕ] ∈ C (S M) and T[ϕ], T0[ϕ] ∈ C (T1 M) and T[ϕ], T0[ϕ] ∈ 0 ∗ ∗ Σˆ 0 2 ˆ 0 1 C (TM ⊗ ϕ T N) and [ϕ] ∈ C (S N|ϕ(M)) and T[ϕ] ∈ C (T1 N|ϕ(M)), defined by

Σ[ϕ] = Σ[ϕ] ⊗ ω[ϕ] = Σ0[ϕ] ⊗ ω0, Σˆ [ϕ] = Σˆ[ϕ] ⊗ ωˆ ,

T[ϕ] = T[ϕ] ⊗ ω[ϕ] = T0[ϕ] ⊗ ω0, Tˆ [ϕ] = Tˆ[ϕ] ⊗ ωˆ ,

T˜ [ϕ] = T˜[ϕ] ⊗ ω[ϕ] = T˜0[ϕ] ⊗ ω0, are also called stress tensor fields. (c) The first Piola-Kirchhoff, the second Piola-Kirchhoff, and the Cauchy stress tensor field associated with the deformation ϕ are T˜0[ϕ], Σ0[ϕ], and Tˆ[ϕ], respectively. 17 Remark 4.3. (a) The stress tensor fields Σ[ϕ] and Σˆ[ϕ] are symmetric. (b) The stress tensor fields Σ[ϕ],T[ϕ], T˜[ϕ], Σˆ[ϕ], and Tˆ[ϕ] are obtained from each other by lowering and raising indices in local charts. Specifically, if ∂ ∂ ∂ ∂ Σ[ϕ](x) = Σi j(x) (x) ⊗ (x), Σˆ[ϕ](ϕ(x)) = Σˆ αβ(x) (ϕ(x)) ⊗ (ϕ(x)), ∂xi ∂xi ∂yα ∂yβ ∂ ∂ T[ϕ](x) = T i(x) (x) ⊗ dx j(x), Tˆ[ϕ](ϕ(x)) = Tˆ α(x) (ϕ(x)) ⊗ dyβ(ϕ(x)), j ∂xi β ∂yα ∂ T˜[ϕ](x) = T˜ i (x) (x) ⊗ dyβ(ϕ(x)), β ∂xi Σαβ = Σˆ αβ α = ˆ α at each x ∈ M, and if : ◦ ϕ, and Tβ : Tβ ◦ ϕ, then

α β i = Σik Σαβ = ∂ϕ ∂ϕ Σi j i = Σik T j g jk , i j , T j g jk , ∂x ∂x (4.5) ∂ϕβ T˜ i = g Σi j, T α = g Σασ, Tˆ α = gˆ Σˆ ατ, α αβ ∂x j β βσ β βτ where gi j and gαβ := g˜αβ = (ˆgαβ ◦ ϕ) respectively denote the components of the metric tensor ﬁelds g[ϕ] = ϕ∗gˆ and g˜[ϕ]:= gˆ ◦ ϕ. (c) The components of the stress tensor ﬁelds Σ[ϕ], T[ϕ], and T˜ [ϕ], over the volume forms ω[ϕ] and ω0 are related by

Σ[ϕ] = ρ[ϕ] Σ0[ϕ], T[ϕ] = ρ[ϕ] T0[ϕ], T˜[ϕ] = ρ[ϕ] T˜0[ϕ], (4.6) where the function ρ[ϕ]: M → R is deﬁned by ω0 = ρ[ϕ]ω[ϕ]. In local charts,

α ∂ϕ0 det( i (x)) ρ(x) = ∂x for all x ∈ M. ∂ϕα det( ∂xi (x))

The next proposition gathers several properties of the stress tensor ﬁelds T[ϕ], T˜ [ϕ] and Tˆ [ϕ], for further use. Proposition 4.4. (a) Let T˜˙ be the constitutive law deﬁned by (4.3). Then

˜ = ˜˙ ∗ Λn (T[ϕ])(x) T(x, ϕ(x), Dϕ(x)) ∈ (Tx M ⊗ Tϕ(x)N) ⊗ x M, x ∈ M. (b) The stress tensor ﬁelds appearing in Deﬁnition 4.2 are related to each other by

Σ[ϕ]: e[ϕ, ξ] = T[ϕ]: ∇ξ = T˜ [ϕ]: ∇˜ ξ˜ = ϕ∗(Tˆ [ϕ]: ∇ˆ ξˆ) = ϕ∗(Σˆ [ϕ] :e ˆ[ϕ, ξˆ]), (4.7) Σ[ϕ]: e[ϕ, ξ] = T[ϕ]: ∇ξ = T˜[ϕ]: ∇˜ ξ˜ = (Tˆ[ϕ]: ∇ˆ ξˆ) ◦ ϕ = (Σˆ[ϕ] :e ˆ[ϕ, ξˆ]) ◦ ϕ for all vector ﬁelds ξ ∈ C1(TM). As above, the vector ﬁelds ξ, ξ˜ and ξˆ appearing in these relations are related to each other by means of the formulas

ξ˜ = (ϕ∗ξ) ◦ ϕ and ξˆ = ϕ∗ξ.

18 Proof. The relation of (a) is an immediate consequence of Lemma 4.1. The relations of (b) are equivalent in a local chart to the relations

Σi j = i k = ˜ i ˜ ˜α = ˆ β ˆ ˆα = Σˆ ατ ei j[ϕ, ξ] Tk ∇iξ Tα ∇iξ (Tα ∇βξ ) ◦ ϕ ( eˆατ) ◦ ϕ. Σi j = Σ ji i = Σi j = 1 k + Using the relations and Tk g jk (cf. Remark 4.3), and that ei j[ϕ, ξ] 2 (g jk∇iξ k gik∇ jξ ) (cf. Theorem 3.4), we ﬁrst obtain Σi j = Σi j k = i k ei j[ϕ, ξ] g jk∇iξ Tk ∇iξ .

i = ∂ϕα ˜ i ˜ ˜α = ∂ϕα k We next infer from the relations Tk ∂xk Tα (cf. Remark 4.3) and ∇iξ ∂xk ∇iξ (cf. relations (2.2)) that ∂ϕα T i ∇ ξk = T˜ i ( ∇ ξk) = T˜ i ∇˜ ξ˜α. k i α ∂xk i α i ˜ i = ∂ϕτ Σik Στβ = ∂ϕτ ∂ϕβ Σik ˆ β = β = Στβ Then, since Tα gατ ∂xk and ∂xk ∂xi and Tα ◦ ϕ Tα gατ (cf. Remark 4.3), ˜ ˜α = ∂ϕβ ˆ ˆα and since ∇iξ ∂xi ((∇βξ ) ◦ ϕ) (cf. relations (2.2)), we have ∂ϕτ ∂ϕβ T˜ i ∇˜ ξ˜α = g ( Σik)((∇ˆ ξˆα) ◦ ϕ) = g Στβ((∇ˆ ξˆα) ◦ ϕ) α i ατ ∂xk ∂xi β ατ β β α β α = Tα((∇ˆ βξˆ ) ◦ ϕ) = (Tˆα ∇ˆ βξˆ ) ◦ ϕ.

ˆ β = Σˆ ατ Σˆ ατ = Σˆ τα ˆ = 1 ˆ ˆα + Finally, since Tα gˆατ and (cf. Remark 4.3), and sincee ˆατ[ϕ, ξ] 2 (ˆgατ∇βξ α gˆαβ∇ˆ τξˆ ) (cf. relations (3.4) and (3.5)), we also have 1 Tˆ β ∇ˆ ξˆα = Σˆ ατ(ˆg ∇ˆ ξˆα + gˆ ∇ˆ ξˆα) = Σˆ ατ eˆ [ϕ, ξˆ]. α β 2 ατ β αβ τ ατ

5. Applied forces

We assume in this paper that the external body and surface forces acting on the elastic body under consideration are conservative, meaning that they are deﬁned by means of a potential P : C1(M, N) → R of the form

P[ϕ]:= F[ϕ] + H[ϕ] = Fˆ [ϕ] + Hˆ [ϕ], (5.1) ZM ZΓ2 Zϕ(M) Zϕ(Γ2) where the volume forms

= ∗ ˆ 2 Λn = ∗ ˆ 2 Λn−1Γ F[ϕ] ϕ (F[ϕ]) ∈ C ( M) and H[ϕ] (ϕ|Γ2 ) (H[ϕ]) ∈ C ( 2)

1 on M and Γ2 ⊂ ∂M, respectively, are given for each admissible deformation ϕ ∈ C (M, N) of the elastic body. Let ϕ ∈ C1(M, N) be a deformation of the elastic body. The work of the applied body 0 and surface forces corresponding to a displacement field ξ˜ = (ϕ∗ξ) ◦ ϕ, ξ ∈ C (TM), of the configuration ϕ(M) of the body is denoted V[ϕ]ξ and is defined as the Gateauxˆ derivative of

19 the functional P : C1(M, N) → R at ϕ in the direction ξ˜. Hence there exist sections ˆf[ϕ] ∈ 0 Λn ∗ ˆ 0 Λn−1 ∗ C ( N ⊗ T N|ϕ(M)) and h[ϕ] ∈ C ( N ⊗ T N|ϕ(Γ2)) such that

V[ϕ]ξ := P′[ϕ]ξ˜ = f[ϕ] · ξ + h[ϕ] · ξ = ˜f[ϕ] · ξ˜ + h˜ [ϕ] · ξ˜ ZM ZΓ ZM ZΓ 2 2 (5.2) = ˆf[ϕ] · ξˆ + hˆ [ϕ] · ξ,ˆ Zϕ(M) Zϕ(Γ2)

0 for all ξ ∈ C (TM), where ξˆ := ϕ∗ξ, ξ˜ := ξˆ ◦ ϕ, and = ∗ ˆ = ∗ ˆ f[ϕ] ϕ ( f[ϕ]), h[ϕ] (ϕ|Γ2 ) (h[ϕ]), ∗ ∗ (5.3) ˜ = ˆ ˜ = Γ ˆ f[ϕ] ϕb( f[ϕ]), h[ϕ] (ϕ| 2 )b(h[ϕ]).

In these relations, the notation · denotes the contraction of one index (for instance, ˜f[ϕ] · ξ˜ := ˜ ˜α ∗ Λk ⊗ ∗ → Λk ⊗ ∗ ∗ Λk ⊗ ∗ → Λk ⊗ ∗ ∗ f[ϕ]a1...an,αξ ), whereas ϕ : N T N M T M and ϕb : N T N M ϕ T N denote the (usual) pullback and the “bundle pullback”, respectively. = ∗ = ∗ Remark 5.1. Let g[ϕ]: ϕ gˆ and g0 : ϕ0gˆ be the metric tensor fields on M induced by the 1 1 deformations ϕ ∈ C (M, N) and ϕ0 ∈ C (M, N), let ν[ϕ] and ν0 denote the unit outer normal vector fields to the boundary of M with respect to g[ϕ] and g0, respectively, and let νˆ[ϕ] denote the unit outer normal vector fields to the boundary of ϕ(M) with respect to the metric g.ˆ Let

ω = ∗ω ω = ∗ω 0 Λn ω ∞ Λn [ϕ]: ϕ ˆ , 0 : ϕ0 ˆ ∈ C ( M) and ˆ ∈ C ( N) be the volume forms induced by these metrics on M and on N, respectively, and let

ω ω 1 Λn−1Γ ω 0 Λn−1 Γ iν[ϕ] [ϕ], iν0 0 ∈ C ( 2), and iνˆ[ϕ] ˆ ∈ C ( ϕ( 2)) denote the corresponding volume forms on the hypersurfaces Γ2 ⊂ M and on ϕ(Γ2) ⊂ N. (a) The tensor fields f[ϕ], ˜f[ϕ], and ˆf[ϕ], respectively h[ϕ], h˜ [ϕ] and hˆ [ϕ], are called the densities of the applied body, respectively surface, forces. 0 ∗ 0 ∗ ∗ 0 ∗ (b) The 1-form fields f [ϕ], f0[ϕ] ∈ C (T M), f˜[ϕ], f˜0[ϕ] ∈ C (ϕ T N), fˆ[ϕ] ∈ C (T N|ϕ(M)) 0 ∗ ˜ ˜ 0 ∗ ∗ ˆ 0 ∗ and h[ϕ], h0[ϕ] ∈ C (T M|Γ2 ), h[ϕ], h0[ϕ] ∈ C (ϕ T N|Γ2 ), h[ϕ] ∈ C (T N|ϕ(Γ2)), defined in terms of the densities of the applied forces by = ω = ω = ω = ω f[ϕ] f [ϕ] ⊗ [ϕ] f0[ϕ] ⊗ 0, h[ϕ] h[ϕ] ⊗ iν[ϕ] [ϕ] h0[ϕ] ⊗ iν0 0, ˜ = ˜ ω = ˜ ω ˜ = ˜ ω = ˜ ω f[ϕ] f [ϕ] ⊗ [ϕ] f0[ϕ] ⊗ 0, h[ϕ] h[ϕ] ⊗ iν[ϕ] [ϕ] h0[ϕ] ⊗ iν0 0,

ˆf[ϕ] = fˆ[ϕ] ⊗ ωˆ , hˆ [ϕ] = hˆ[ϕ] ⊗ iνˆ[ϕ]ωˆ , are related by ∗ f [ϕ] = ρ[ϕ] f0[ϕ] = ϕ ( fˆ[ϕ]), f˜[ϕ] = fˆ[ϕ] ◦ ϕ, = = ∗ ˆ ˜ = ˆ h[ϕ] ρ[ϕ|Γ] h0[ϕ] (ϕ|Γ2 ) (h[ϕ]), h[ϕ] h[ϕ] ◦ ϕ. where the (scalar) functions ρ[ϕ]: M → R and ρ[ϕ|Γ]: Γ → R are deﬁned by ω = ω ∗ ω = ω [ϕ] ρ[ϕ] 0 and (ϕ|Γ2 ) (iνˆ[ϕ] ˆ ) ρ[ϕ|Γ] iν0 0.

20 (c) The components of the external body and surface forces in a local chart, deﬁned at each x ∈ M by i i f [ϕ](x) = fi(x) dx (x), h[ϕ](x) = hi(x) dx (x), α α f˜[ϕ](x) = f˜α(x) dy (ϕ(x)), h˜[ϕ](x) = h˜ α(x) dy (ϕ(x)), α α fˆ[ϕ](x) = fˆα(ϕ(x)) dy (ϕ(x)), hˆ[ϕ](x) = hˆ α(ϕ(x)) dy (ϕ(x)), satisfy ∂ϕα ∂ϕα f := f˜ , f˜ = fˆ ◦ ϕ, and h := h˜ , h˜ = hˆ ◦ ϕ. i ∂xi α α α i ∂xi α α α Remark 5.2. If the volume forms Fˆ [ϕ] = Fˆ and Hˆ [ϕ] = Hˆ are independent of the deformation ϕ, the densities of the body and surface forces appearing in (5.2) are given explicitly by ˜ ˜ = ∗ ˆ ˆ ˜ ˜ = ∗ ˆ ˆ f[ϕ] · ξ ϕ (Lξˆ F) and h[ϕ] · ξ ϕ (Lξˆ H), for all ξ˜ = ξˆ ◦ ϕ ∈ C1(ϕ∗TN). Indeed, in that case we have

= ′ ˜ = d = ∗ ˆ ˆ + ∗ ˆ ˆ V[ϕ]ξ P [ϕ]ξ P(γξˆ(·, t) ◦ ϕ) ϕ (Lξˆ F) ϕ (Lξˆ H). dt t=0 Z ZΓ h i M 2 We assume in this paper that the applied body and surface forces f[ϕ] and h[ϕ] are local, so that their constitutive equations are of the form:

( f[ϕ])(x) = f˙(x, ϕ(x), Dϕ(x)), x ∈ M,

(h[ϕ])(x) = h˙ (x, ϕ(x), Dϕ(x)), x ∈ Γ2,

˙ = ˙ ω ∗ ∗ Λn for some (given) mappings f(x, y, ·) f0(x, y, ·) ⊗ 0(x): Tx M ⊗ TyN → T M ⊗ M,(x, y) ∈ ˙ = ˙ ω ∗ ∗ ΛnΓ Γ M × N, and h(x, y, ·) h0(x, y, ·) ⊗ 0(x): Tx M ⊗ TyN → T M|Γ2 ⊗ 2,(x, y) ∈ 2 × N. Note that the constitutive laws f˙ and h˙ of the applied forces are of the same form as the constitutive law T˙ of the elastic material constituting the body, but do not necessarily satisfy the axiom of frame-indiﬀerence.

6. Nonlinear elasticity

In this section we combine the results of the previous sections to derive the model of nonlinear elasticity in a Riemannian manifold, first as a minimization problem, then as variational equations, and finally as a boundary value problem. The principle of least energy states that the deformation ϕ : M → N of the body under conservative forces independent of time should minimize the total energy of the body over the set of all admissible deformations. The total energy is defined as the difference between the strain energy I[ϕ] and the potential of the applied forces P[ϕ], viz.,

J[ϕ]:= I[ϕ] − P[ϕ] = W[ϕ] − F[ϕ] + H[ϕ] , Z Z ZΓ M M 2 the dependence on ϕ of the densities W[ϕ], F[ϕ] and H[ϕ] being that specified by the constitutive laws of the material and applied forces (cf. Sections 4 and 5). 21 An admissible deformation is an immersion ϕ ∈ C1(M, N) that preserves orientation at all points of M, satisfies the impenetrability of matter axiom at all points of the interior of M, and verifies the boundary condition of place imposed on a subset Γ1 ⊂ ∂M of the boundary of the body. Thus the set of all admissible deformations is defined by

1 Φ := {ϕ ∈ C (M, N); ϕ|intM injective, det Dϕ > 0 in M, ϕ = ϕ1 on Γ1},

1 where ϕ1 ∈ C (Γ1, N) is an immersion that specifies the position in N of the points of the body = 2 that are kept fixed. In this paper we assume that ϕ1 ϕ0|Γ1 , where ϕ0 ∈ C (M, N) is the reference deformation of the body. Therefore, the principle of least energy that constitutes the axiom underpinning the nonlinear elasticity theory developed in this paper asserts that the deformation ϕ : M → N of the body under consideration is a solution to the following minimization problem: Proposition 6.1. Let the total energy associated with a deformation ϕ ∈ C1(M, N) of an elastic body be defined by

J[ψ]:= I[ψ] − P[ψ] = W[ψ] − F[ψ] + H[ψ] , (6.1) Z Z ZΓ M M 2 and let the set of all admissible deformations of the body be deﬁned by

1 Φ := {ψ ∈ C (M, N); ψ|intM injective, det Dψ > 0 in M, ψ = ϕ0 on Γ1}. (6.2)

Then the deformation ϕ of the body satisﬁes the following minimization problem:

ϕ ∈ Φ such that J[ϕ] ≤ J[ψ] for all ψ ∈ Φ. (6.3)

The set Φ defined in this fashion does not coincide with the set of deformations with finite energy, but is dense there. Therefore, minimizers of J are usually sought in the completion Φ of Φ in the set of deformations with finite energy. However, the existence of such minimizers is still not guaranteed, since the functional J[ϕ] is not convex with respect to ϕ for realistic constitutive laws; cf. [3, 4] and the references therein. One way to alleviate this difficulty is to apply the strategy of J. Ball [3], in which case W˙ must be polyconvex. Another way is to study the existence of critical points instead of minimizers of J. It is the latter approach that we follow in this paper. To this end, we next derive the variational equations of nonlinear elasticity, or the principle of virtual work, in a Riemannian manifold from principle of least energy (see Proposition (6.1)). The principle of virtual work states that the deformation of a body should satisfy the Euler- Lagrange equations associated to the functional J appearing in the principle of least energy. We will derive below several variants of the principle of virtual work, corresponding to the different stress tensor fields, Σ[ϕ], T[ϕ], T˜ [ϕ], Tˆ [ϕ], or Σˆ [ϕ], defined in Section 4. The set of admissible deformations Φ being that of (6.2), the space of all admissible displacement fields associated with a deformation ϕ ∈ Φ of the body is defined by

1 ∗ Ξ˜ [ϕ] = {ξ˜ ∈ C (ϕ TN); ξ˜ = 0 on Γ1}

= {ξ˜ = (ϕ∗ξ) ◦ ϕ; ξ ∈ Ξ} (6.4) = {ξ˜ = ξˆ ◦ ϕ; ξˆ ∈ Ξˆ [ϕ]}, 22 where 1 Ξ := {ξ ∈ C (TM); ξ = 0 on Γ1}, 1 (6.5) Ξˆ [ϕ]:= {ξˆ ∈ C (TN|ϕ(M)); ξˆ = 0 on ϕ(Γ1)}. Note that Ξ˜ [ϕ] and Ξˆ [ϕ] depend on the deformation ϕ, whereas Ξ does not. Remark 6.2. The spaces Ξ and Ξˆ [ϕ] are called the space of all admissible displacement fields of the abstract configuration, and the space of all admissible displacement fields of the configuration ϕ(M) of the body, respectively. ... In what follows we assume that the stored energy function W of the elastic material constituting the body is of class C1. In this case, a solution ϕ ∈ C1(M, N) to the minimization problem (6.3) is a critical point of the total energy J = I − P, defined by (6.1), that is, it satisfies the variational equation J′[ϕ]ξ˜ = 0 for all ξ˜ ∈ Ξ˜ [ϕ]. This equation is called the principle of virtual work. The next proposition states this principle in five equivalent forms, one for each stress tensor field appearing in Definition 4.2. Note that the first two equations are defined over the abstract configuration M and are expressed in terms of the stress tensors fields T[ϕ] and Σ[ϕ], also defined on M. The third equation is defined over M, but is expressed in terms of the stress tensor field T˜ [ϕ], which is defined over both M and the deformed configuration ϕ(M). The last two equations are defined over the deformed configuration ϕ(M) and are expressed in terms of the stress tensor fields Tˆ [ϕ] and Σˆ [ϕ], also defined over ϕ(M). Proposition 6.3. Let the sets Ξ, Ξ˜ [ϕ] and Ξˆ [ϕ] of admissible displacement fields associated with a deformation ϕ ∈ Φ of an elastic body be defined by (6.4) and (6.5). If a deformation ϕ ∈ C1(M, N) satisfies the principle of least energy (Proposition 6.1), then it also satisfies one of the following five equivalent variational equations:

Σ[ϕ]: e[ϕ, ξ] = f[ϕ] · ξ + h[ϕ] · ξ for all ξ ∈ Ξ, ZM ZM ZΓ2 T[ϕ]: ∇ξ = f[ϕ] · ξ + h[ϕ] · ξ for all ξ ∈ Ξ, ZM ZM ZΓ2 T˜ [ϕ]: ∇˜ ξ˜ = ˜f[ϕ] · ξ˜ + h˜ [ϕ] · ξ˜ for all ξ˜ ∈ Ξ˜ [ϕ], ZM ZM ZΓ2 Tˆ [ϕ]: ∇ˆ ξˆ = ˆf[ϕ] · ξˆ + hˆ [ϕ] · ξˆ for all ξˆ ∈ Ξˆ [ϕ], Zϕ(M) Zϕ(M) Zϕ(Γ2) Σˆ [ϕ] :e ˆ[ϕ, ξ] = ˆf[ϕ] · ξˆ + hˆ [ϕ] · ξˆ for all ξˆ ∈ Ξˆ [ϕ]. Zϕ(M) Zϕ(M) Zϕ(Γ2) Proof. The right-hand sides appearing in the above variational equations are equal when the vector ﬁelds ξ, ξ˜ and ξˆ are related by

ξ˜ = (ϕ∗ξ) ◦ ϕ and ξˆ = ϕ∗ξ, since they all define the same scalar, P′[ϕ]ξ˜ ∈ R, representing the work of the applied forces; cf. Section 5. Likewise, the left-hand sides appearing in the same equations are equal for the same vector fields, since Σ[ϕ]: e[ϕ, ξ] = T[ϕ]: ∇ξ = T˜ [ϕ]: ∇˜ ξ˜ = ϕ∗(Tˆ [ϕ]: ∇ˆ ξˆ) = ϕ∗(Σˆ [ϕ] :e ˆ[ϕ, ξ]) ; 23 cf. Proposition 4.4. Therefore, it suffices to prove the first equation. Let ϕ ∈ C1(M, N) be a solution to the minimization problem (6.3). Given any admissible displacement field ξ ∈ Ξ, let ξ˜ and ξˆ be defined as above, and let ξˆ ∈ C1(TN) also denote any ˆ = 1 ˆ extension to N of the vector field ξ ϕ∗ξ ∈ C (TN|ϕ(M)). Let γξˆ denote the flow of ξ (see Section 2) and define the time-dependent family of deformations = ψ(·, t): γξˆ(·, t) ◦ ϕ, t ∈ (−ε, ε). Note that there exists ε > 0 such that ψ(·, t) ∈ Φ for all t ∈ (−ε, ε). Since J[ϕ] ≤ J[ψ(·, t)] for all t ∈ (−ε, ε), we deduce that d J[ψ(·, t)] = 0, dt t=0 h i which next implies that d d I[ψ(·, t]) = P[ψ(·, t)] = P′[ϕ]ξ.˜ dt t=0 dt t=0 h i h i It remains to compute the first term of this relation. = ... Using the Lebesgue dominated... convergence... theorem, the chain rule, and the relations W[ϕ] ∂W = Σ W(·, E[ϕ0, ψ(·, t)]) and ∂E (x, E) (x, E) (cf. Section 4, relations (4.2) and (4.4)), we deduce on the one hand that d d ... I[ψ(·, t]) = W(·, E[ϕ0, ψ(·, t)]) dt t=0 dt t=0 h i ZM h i ... d = Σ(·, E[ϕ0, ϕ]) : E(ϕ0, ψ(·, t)) . dt t=0 ZM h i On the other hand, we established in the proof of Theorem 3.4 that d e[ϕ, ξ] = E(ϕ0, ψ(·, t)) . dt t=0 ... h i Besides, Σ[ϕ] = Σ(·, E[ϕ0, ϕ]); cf. Definition 4.2. Therefore, the above relations imply that d I[ψ(·, t]) = Σ[ϕ]: e[ϕ, ξ], dt t=0 h i ZM and the first variational equations of Proposition 6.3 follow. Remark 6.4. The principle of virtual work can be extended to functionals J defined in terms of stored energy functions W˙ that may not satisfy the axiom of frame indifference. We conclude this section by formulating the equations of nonlinear elasticity in a Rieman- nian manifold. These equations are defined as the boundary value problem satisfied by a a sufficiently regular solution ϕ of the variational equations that constitute the principle of virtual work (Proposition 6.3). We derive below several equivalent forms of this boundary value problem, each involving a different stress tensor field, as does the principle of virtual work. The divergence operators induced by the connections ∇ = ∇[ϕ], ∇˜ = ∇˜ [ϕ], and ∇ˆ , are denoted div = div[ϕ], div = div[ϕ], and div, respectively; cf. Section 2. We emphasize that the differential operators ∇, ∇˜ , div, and div, depend on the unknown deformation ϕ, while the f f c differential operators ∇ˆ and div do not; see Remark 2.1. f 24 c Proposition 6.5. A deformation ϕ ∈ C2(M, N) satisfies the principle of virtual work (Proposition 6.3) if and only if it satisfies one of the following six equivalent boundary value problems:

− div T[ϕ] = f[ϕ] in intM, − div T[ϕ] = f [ϕ] in intM,

 T[ϕ]ν = h[ϕ] on Γ2, ⇔  T[ϕ] · (ν[ϕ] · g[ϕ]) = h[ϕ] on Γ2,    ϕ = ϕ0 on Γ1,  ϕ = ϕ0 on Γ1,     −div T˜ [ϕ] = ˜f[ϕ] in intM, −div T˜[ϕ] = f˜[ϕ] in intM, ⇔  T˜ [ϕ] = h˜ [ϕ] on Γ , ⇔  T˜[ϕ] · (ν[ϕ] · g[ϕ]) = h˜[ϕ] on Γ ,  f ν 2  f 2    ϕ = ϕ0 on Γ1,  ϕ = ϕ0 on Γ1,     −div Tˆ [ϕ] = ˆf in int(ϕ(M)), −div Tˆ[ϕ] = fˆ[ϕ] in int(ϕ(M)), ⇔  Tˆ [ϕ] = hˆ [ϕ] on ϕ(Γ ), ⇔  Tˆ[ϕ] · (ˆν[ϕ] · gˆ) = hˆ[ϕ] on ϕ(Γ ),  c νˆ 2  c 2    ϕ = ϕ0 on Γ1,  ϕ = ϕ0 on Γ1,     where ν := ν[ϕ] and νˆ := νˆ[ϕ] denote the unit outer normal vector fields to the boundaries of M and ϕ(M), respectively, defined by the metric tensor fields g[ϕ]:= ϕ∗gˆ and g,ˆ respectively. Proof. A deformation ϕ ∈ C2(M, N) satisfies the principle of virtual work if and only if the associated stress tensor field T[ϕ] satisfies the variational equations

T[ϕ]: ∇ξ = f[ϕ] · ξ + h[ϕ] · ξ, ZM ZM ZΓ2 for all vector fields ξ ∈ Ξ. The standard integration by parts formula on the Riemann manifold (M, g) (or Lemma 2.2 with N = M and ϕ = idM) applied to the left-hand side integral yields the first boundary value problem. Likewise, since the principle of virtual work satisfied by the stress tensor field T˜ [ϕ], respectively Tˆ [ϕ], is equivalent to the variational equations

T˜ [ϕ]: ∇˜ ξ˜ = ˜f[ϕ] · ξ˜ + h˜ [ϕ] · ξ˜ for all ξ˜ ∈ Ξ˜ [ϕ], ZM ZM ZΓ2 respectively to the variational equations

Tˆ [ϕ]: ∇ˆ ξˆ = ˆf[ϕ] · ξˆ + hˆ [ϕ] · ξˆ for all ξˆ ∈ Ξˆ [ϕ], Zϕ(M) Zϕ(M) Zϕ(Γ2)

Lemma 2.2, respectively Lemma 2.2 with M = N and ϕ = idN , yields the second boundary value problem, respectively the third boundary value problem.

7. Small strains nonlinear elasticity and linearized elasticity

The objective of this section is to define two useful approximations of nonlinear elasticity, viz., small strains nonlinear elasticity and linearized elasticity. Small strains nonlinear... elasticity is deduced from nonlinear elasticity (Section 6) by replacing the constitutive law Σ of the stress tensor field Σ[ϕ] as a function of the (nonlinear) strain tensor field E[ϕ0, ϕ] with its linear part, 25 ... Σss denoted , while linearized... elasticity is deduced from nonlinear elasticity by linearizing both the constitutive law Σ of the stress tensor field Σ[ϕ], and the constitutive law E˙ (see Remark 3.3) of the strain tensor field E[ϕ0, ϕ]. Thus, in nonlinear elasticity the stress tensor field is a possibly nonlinear function of the strain tensor field, namely ... Σ[ϕ] = Σ(·, E[ϕ0, ϕ]), in small strains nonlinear elasticity the stress tensor field is a linear function of the strain tensor field, namely ss ...ss Σ [ϕ] = Σ (·, E[ϕ0, ϕ]), and in linearized elasticity the stress tensor field is a linear function of the displacement field, namely ... Σlin ϕ = Σss ·, e ϕ , ξ , ξ = −1 ϕ. [ ] ( [ 0 ]) where : expϕ0 By way of example, Saint Venant - Kirchhoff’s constitutive law (see Remark 7.3) belongs to the small strains nonlinear elasticity, while Hooke’s constitutive law (see Remark 7.4) belongs to linearized elasticity. Consider an elastic body which occupies in a reference configuration a subset ϕ0(M) ⊂ N ... 2 of the physical space and whose stored energy... function W ...is of class...C . There is no loss of generality in replacing... the stored energy density W(x, E) by (W(x, E)−W(x, 0)), so we henceforth = assume... that...W(x, 0) 0 for all x ∈ M. If the reference... configuration is unconstrained, then Σ = ∂W = (x, 0) ∂E (x, 0) 0 too, so the Taylor expansion of W(x, E) as a function of E ∈ S 2,x M in a neighborhood of 0 ∈ S 2,x M starts with the second derivative. This justifies the following definition of the elasticity tensor field, a notion which plays a fundamental role in both small strains nonlinear elasticity and linearized elasticity. ... Definition 7.1. The elasticity tensor field of an elastic material with stored energy function W ∈ 2 n 1 2 2 C (S 2 M ⊗ Λ M) is the section A = A0 ⊗ ω0,A0 ∈ C (S M ⊗sym S M), defined at each x ∈ M by ...... ∂2W ∂2W A(x):= (x, 0) ⇔ A (x):= 0 (x, 0). (7.1) ∂E2 0 ∂E2

Note that the components of A0 in a local chart satisfy the symmetries

i jkℓ kℓi j jikℓ i jℓk A0 = A0 = A0 = A0 . (7.2)

In small strains nonlinear elasticity,... the stored energy function is deﬁned as the quadratic part in E of the stored energy function W, that is, by

...ss 1 1 W (x, E):= (A(x): E): E = (A (x): E): E ⊗ ω (x) 2 2 0 0 n o = i jkℓ for all... x ∈ M and E ∈ S 2,x M, where (A0(x): H): K [A0(x)] HkℓKi j. Hence the constitutive law Σss of an elastic material in small strains elasticity is linear, since it is deﬁned by (see (4.4)) ...ss Σ (x, E):= A(x): E for all E ∈ S 2,x M.

But the constitutive equation relating the stress tensor field Σ[ϕ] to the strain tensor field E[ϕ0, ϕ] is still nonlinear, since the latter tensor field is nonlinear in ϕ. Specifically, the stress tensor fields associated with a deformation ϕ ∈ C1(M, N) is defined in small strains nonlinear elasticity by

ss ss ss Σ [ϕ]:= A : E[ϕ0, ϕ] and T [ϕ]:= g[ϕ] · Σ [ϕ], 26 = 1 = ∗ = ∗ (compare with Deﬁnition 4.2), where E[ϕ0, ϕ]: 2 (g[ϕ] − g0) with g[ϕ]: ϕ gˆ and g0 : ϕ0gˆ. Note that the nonlinearity of these tensor ﬁelds is not arbitrary, but multilinear in Dϕ. Using the above constitutive equations in Propositions 6.1, 6.3, and 6.5 yields the minimization problem, the variational equations, and the boundary value problem of small strains nonlinear elasticity. We record them below for completeness:

Proposition 7.2. (a) The deformation ϕ ∈ C2(M, N) of an elastic body satisﬁes in small strains nonlinear elasticity the following boundary value problem:

− div Tss[ϕ] = f[ϕ] in intM, ss T [ϕ]ν = h[ϕ] on Γ2, (7.3)

ϕ = ϕ0 on Γ1.

(b) The deformation ϕ ∈ C1(M, N) of an elastic body satisﬁes in small strains nonlinear elasticity the following variational equations:

Σss[ϕ]: e[ϕ, ξ] = f[ϕ] · ξ + h[ϕ] · ξ for all ξ ∈ Ξ, (7.4) ZM ZM ZΓ2 or equivalently,

Tss[ϕ]: ∇ξ = f[ϕ] · ξ + h[ϕ] · ξ for all ξ ∈ Ξ, (7.5) ZM ZM ZΓ2 where (see (6.5)) 1 Ξ := {ξ ∈ C (TM); ξ = 0 on Γ1}. (c) If the external forces f[ϕ] and h[ϕ] are conservative (cf. Section 5), then the deformation ϕ ∈ C1(M, N) of an elastic body satisﬁes in small strains nonlinear elasticity the following minimization problem:

ϕ ∈ Φ, and Jss[ϕ] ≤ Jss[ψ] for all Ψ ∈ Φ, (7.6) where (see (6.2)) Jss[ϕ]:= Wss[ϕ] − F[ϕ] + H[ϕ] Z Z ZΓ M M 2 is the total energy of the body associated with a deformation ϕ, and (see (6.2))

1 Φ := {ϕ ∈ C (M, N); ϕ|intM injective, det Dϕ > 0 in M, ϕ = ϕ0 on Γ1} is the set of all admissible deformations. Remark 7.3. Saint Venant - Kirchhoﬀ’s constitutive law models elastic materials belonging to small strains nonlinear elasticity. Its interest in practical applications is due to the fact that it depends on only two scalar parameters, the Lam´econstants λ ≥ 0 and µ > 0 of the elastic material constituting the body (which are determined by experiment for each elastic material), by means of the relations

i jkℓ i j kℓ ik jℓ iℓ jk A0 := λg0 g0 + µ(g0 g0 + g0 g0 ), 27 defining the elasticity tensor field A = A0 ⊗ ω0. The stored energy function of a Saint Venant - Kirchhoff material is then defined by

...svk λ W (x, E):= (tr E)2 + µ |E|2 ω (x) 2 0 = i j 2 = ik jℓ = ∗ for all x ∈ M and all E ∈ S 2,x M, where tr E : g0 Ei j, |E| : g0 g0 EkℓEi j, and g0 : ϕ0g.ˆ Hence the strain energy of a body made of a St Venant - Kirchhoﬀ material is given by

svk ...svk I [ϕ]:= W (x, (E[ϕ0, ϕ])(x)), ZM which, in the particular (but signiﬁcative) case where λ = 0 and µ = 1, becomes

svk = 2 I [ϕ] kE[ϕ0, ϕ]k 2 . L (S 2 M) The theory of linearized elasticity states that, under suitable conditions of smallness on the applied forces, the deformation ϕ : M → N of a body that occupies a subset ϕ0(M) ⊂ N in a reference configuration, assumed to be a natural state (that is, unconstrained), can be approached by ϕ = ξ ξ ∈ Ξ the deformation : expϕ0 induced by a displacement field of the abstract configuration M of the body. The vector field ξ : M → TM, which becomes the new unknown in linearized elasticity instead of the deformation ϕ, should then satisfy the equations of linearized elasticity, which can take the form of a minimization problem, of variational equations, or of a boundary value problem; cf. Proposition 7.5 below. The actual deformation and displacement field of the body are then defined in terms of ξ by the mapping and section

ϕ = ξ M → N ξ˜ = ϕ ξ ◦ ϕ M → TN, : expϕ0 : and : ( 0∗ ) 0 : respectively. 2 Let ϕ0 ∈ C (M, N) be a deformation of reference such that the configuration ϕ0(M) ⊂ N is ω ω a natural state and let 0, iν0 0, and ν0, respectively denote the volume form on M, the volume form on Γ = ∂M, and the unit outer normal vector field to the boundary of M, induced by the = = ∗ metric g0 g[ϕ0]: ϕ0 gˆ; see Section 2...... In linearized elasticity, the stored energy function W and the constitutive law Σ of the elastic ...lin ...ss ...... material are the same as in small strains nonlinear elasticity, that is W = W and Σlin = Σss, but the definition of the stress tensor fields is modified in such a way that they become linear ξ = −1 ϕ with respect to the displacement field expϕ0 . Hence 1 Wlin[ξ]:= (A : e[ϕ , ξ]) : e[ϕ , ξ], Σlin[ξ]:= A : e[ϕ , ξ] and Tlin[ξ]:= g · Σlin[ξ]. (7.7) 2 0 0 0 0 Remark 7.4. Hooke’s constitutive law models elastic materials belonging to linearized elasticity. Its elasticity tensor field and its stored energy functions are the same as Saint Venant - Kirchhoff’s, that is (see Remark 7.3)

i jkℓ i j kℓ ik jℓ iℓ jk A0 := λg0 g0 + µ(g0 g0 + g0 g0 ), ...Hooke λ W (x, E):= (tr E)2 + µ |E|2 ω (x), x ∈ M, E ∈ S M, 2 0 2,x 28 but its strain energy is different, since it is defined in terms of the linearized strain tensor field 1 e[ϕ0, ξ], ξ ∈ C (TM), by

Hooke ...Hooke I [ξ]:= W (x, (e[ϕ0, ξ])(x)). ZM In the particular (but signiﬁcative) case where λ = 0 and µ = 1, this relation becomes

2 Hooke = 2 = 1 2 = 1 ♭ + ♭ T I [ξ] ke[ϕ0, ξ]k 2 Lξg0 2 (∇0ξ (∇0ξ ) ) . L (S 2 M) L (S 2 M) 2 4 2 L (S 2 M)

Note that the Lam´econstants appearing in Saint Venant - Kirchhoﬀ’s and Hooke’s constitutive laws are the same. The applied body forces f[ϕ] and h[ϕ], which are given in nonlinear elasticity by constitutive equations of the form (see Section 5)

( f[ϕ])(x) = f˙(x, ϕ(x), Dϕ(x)) and (h[ϕ])(x) = h˙ (x, ϕ(x), Dϕ(x)), are replaced in linearized elasticity by their affine part with respect to the displacement field ξ = −1 ϕ f aff ξ haff ξ : expϕ0 , that is, by [ ] and [ ], respectively, where

aff ′ aff ′ f [ξ]:= f[ϕ0] + f [ϕ0]ξ and h [ξ]:= h[ϕ0] + h [ϕ0]ξ. (7.8) More specifically, in view of the constitutive equations of the applied forces (see above), there 1 0 n 0 2 0 n 1 1 0 n−1 0 ∈ C Λ ⊗ ∈ C Λ ⊗ ∈ C Λ Γ ⊗ |Γ exist sections f ( M T2 M), f ( M T2 M), h ( 2 T2 M 2 ), and 2 0 n−1 1 ∈ C Λ Γ ⊗ |Γ h ( 2 T2 M 2 ), such that

′ = 1 = 1 + 2 f [ϕ0]ξ : lim f[expϕ (tξ)] − f[ϕ0] f · ξ f : ∇0ξ, t→0 t 0 (7.9) ′ = 1 = 1 + 2 h [ϕ0]ξ : lim h[expϕ (tξ)] − h[ϕ0] h · ξ h : ∇0ξ. t→0 t 0 Likewise, the densities F[ϕ] and H[ϕ] in the deﬁnition of the potential of the applied forces (see (5.1)) are replaced in linearized elasticity by their quadratic part with respect to the displace- ξ = −1 ϕ Fqua ξ Hqua ξ ment ﬁeld : expϕ0 , that is, by [ ] and [ ], respectively, where

qua ′ 1 ′′ F [ξ]:= F[ϕ0] + F [ϕ0]ξ + F [ϕ0][ξ, ξ], 2 (7.10) 1 Hqua[ξ]:= H[ϕ ] + H′[ϕ ]ξ + H′′[ϕ ][ξ, ξ], 0 0 2 0 where ′′ = 2 ′ F [ϕ0]ξ : lim F[expϕ (tξ)] − F[ϕ0] − tF [ϕ0]ξ . t→0 t2 0 lin 1 1 aff 0 ∗ aff 0 ∗ ∈ C ∈ C ∈ C |Γ Finally, define the tensor fields T0 [ξ] (T1 M), f0 [ξ] (T M) and h0 [ξ] (T M 2 ), by lin = lin ω aff = aff ω aff = aff ω T [ξ] T0 [ξ] ⊗ 0, f [ξ] f0 [ξ] ⊗ 0, h [ξ] h0 [ξ] ⊗ iν0 0. (7.11) We are now in a position to derive the boundary value problem, the variational equations, and the minimization problem, of linearized elasticity from the corresponding problems of nonlinear elasticity: 29 Proposition 7.5. Let 1 Ξ := {ξ ∈ C (TM); ξ = 0 on Γ1} be the space of all admissible displacement fields of the abstract configuration M of the body (see Section 6). (a) The displacement field ξ ∈ C1(TM) of the abstract configuration M satisfies in linearized elasticity the following two equivalent boundary value problems:

lin = aff lin = aff −div0 T [ξ] f [ξ] in intM, −div0 T0 [ξ] f0 [ξ] in intM,  Tlin = haff Γ ⇔  T lin ξ · ν · g = haff ξ on Γ , (7.12)  [ξ]ν0 [ξ] on 2,  0 [ ] ( 0 0) 0 [ ] 2    ξ = 0 on Γ1.  ξ = 0 on Γ1.     (b) The displacement field ξ ∈ C1(TM) of the abstract configuration M satisfies in linearized elasticity the following variational equations:

aff aff ξ ∈ Ξ and (A : e[ϕ0, ξ]) : e[ϕ0, η] = ( f [ξ]) · η + (h [ξ]) · η for all η ∈ Ξ. (7.13) ZM ZM ZΓ2 (c) If the external forces f[ϕ] and h[ϕ] are conservative (cf. Section 5), then the displacement field ξ ∈ C1(TM) of the abstract configuration M satisfies in linearized elasticity the following minimization problem:

ξ ∈ Ξ, and Jlin[ξ] ≤ Jlin[η] for all η ∈ Ξ, (7.14) where lin 1 qua qua J [η]:= (A : e[ϕ0, η]) : e[ϕ0, η] − F [η] + H [η] (7.15) 2 Z Z ZΓ M M 1 denotes the total energy of the body in linearized elasticity. Proof. (a) The boundary value problem of linearized elasticity is the aﬃne (with respect to ξ) approximation of the following boundary value problem of nonlinear elasticity (see Proposition 6.5) − div T[ϕ] = f[ϕ] in intM,

T[ϕ]ν = h[ϕ] on Γ2, (7.16)

ϕ = ϕ0 on Γ1, ϕ = ξ ffi satisfied by the deformation : expϕ0 . It remains to compute this a ne approximation explicitly. T ϕ ξ = −1 ϕ The dependence of the stress tensor field [ ] on the vector field expϕ0 has been specified in Section 4 by means of the constitutive law of the elastic material, namely,

(T[ϕ])(x) = (g[ϕ])(x) · (Σ[ϕ])(x) ∗ ... = (ϕ gˆ)(x) · Σ(x, (E[ϕ0, ϕ])(x)), x ∈ M.

... Since the reference configuration ϕ0(M) has been assumed to be a natural state, we have Σ(x, 0) = 0 for all x ∈ M. The definition of the elasticity tensor field A = A0 ⊗ ω0 next implies that ... ∂Σ (x, 0)H = A(x): H, x ∈ M, H ∈ S M. ∂E 2,x 30 Besides (see Section 4), d d E ϕ , tξ = e ϕ , ξ g tξ = g . [ 0 expϕ0 ( )] [ 0 ] and [expϕ0 ( )] 0 "dt #t=0 "dt #t=0 Combining the last three relations yields

T[ϕ] = g[ϕ] · Σ[ϕ] = g0 · (A : e[ϕ0, ξ]) + o(kξkC1(TM)), which next implies lin div T[ϕ] = div0 T [ξ] + o(kξkC1(TM)), (7.17) lin since T [ξ]:= g0 ·(A : e[ϕ0, ξ]) is linear with respect to ξ. As above, div0 denotes the divergence operator induced by the connection ∇0 := ∇[ϕ0]. f ϕ h ϕ ξ = −1 ϕ The dependence of the applied force densities [ ] and [ ] on the vector ﬁeld expϕ0 has been speciﬁed in Section 5 by means of the relations

( f[ϕ])(x) = f˙(x, ϕ(x), Dϕ(x)) and (h[ϕ])(x) = h˙ (x, ϕ(x), Dϕ(x)).

Thus, using the notation (7.8) above, we have

aff aff f[ϕ] = f [ξ] + o(kξkC1(TM)) and h[ϕ] = h [ξ] + o(kξkC1(TM)). (7.18) The boundary value problems (7.12) of linearized elasticity follow from the boundary value problem (7.16) of nonlinear elasticity by using the estimates (7.17) and (7.18). (b) The variational equations of linearized elasticity are the affine part with respect to ξ of the variational equations of nonlinear elasticity (see Proposition 6.3)

S ξ η = η ∈ Ξ, [expϕ0 ] 0 for all where S[ϕ]η := Σ[ϕ]: e[ϕ, η] − f[ϕ] · η + h[ϕ] · η ZM ZM ZΓ ... 2 and Σ[ϕ]:= Σ(·, E[ϕ0, ϕ]). Thus the variational equations of linearized elasticity satisﬁed by ξ ∈ Ξ read: d Slin ξ η = S ϕ η + S tξ η = η ∈ Ξ. [ ] : [ 0] [expϕ0 ( )] 0 for all "dt #t=0 lin lin S ξ η Σ ξ = Σ ξ + o kξk 1 It remains to compute [ ] explicitly. Using that [expϕ0 ] [ ] ( C (TM)) (see part lin Σ ξ e ξ, η = e ϕ , η +o kξk 1 (a) of the proof), that [ ] is linear, that [expϕ0 ] [ 0 ] ( C (TM)), and the relations (7.18), in the above deﬁnition of S[ϕ]η, we deduce that

lin aff aff S [ξ]η = (A : e[ϕ0, ξ]) : e[ϕ0, η] − ( f [ξ]) · η + (h [ξ]) · η . Z Z ZΓ M M 2 Using this relation in the equation Slin[ξ]η = 0 yields (7.13). (c) The minimization problem of linearized elasticity consists in minimizing the functional Jlin : Ξ → R over the set Ξ, defined as the quadratic approximation with respect to the parameter t J tξ of the total energy [expϕ0 ( )], where

J[ϕ] = W[ϕ] − F[ϕ] + H[ϕ] ; ZM ZM ZΓ2 31 cf. Proposition 6.1. Thus 1 Jlin[ξ]:= J[ϕ ] + J′[ϕ ]ξ + J′′[ϕ ][ξ, ξ] for all ξ ∈ C1(TM), 0 0 2 0 where ′ = 1 J [ϕ0]ξ : lim J[expϕ (tξ] − J[ϕ0] , t→0 t 0 ′′ = 2 ′ J [ϕ0][ξ, ξ]: lim J[expϕ (tξ] − J[ϕ0] − tJ [ϕ0]ξ . t→0 t2 0 Using that W[ϕ0] = 0, that the reference configuration is a natural state, and the definition of the elasticity tensor field A, we deduce that 1 1 Jlin[ξ]:= (A : e[ϕ , ξ]) : e[ϕ , ξ] − P[ϕ ]ξ + P′[ϕ ]ξ + P′′[ϕ ][ξ, ξ] 2 0 0 0 0 2 0 ZM for all ξ ∈ C1(TM), where P[ϕ]:= F[ϕ] + H[ϕ] denotes the potential of the applied M Γ2 forces. Then the explicit expression (7.14)R of theR functional Jlin follows from the definition (7.10) of Fqua[ξ] and Hqua[ξ]. Remark 7.6. (a) The variational equations of linearized elasticity of Proposition 7.5 are extended by density to displacement fields ξ ∈ H1(TM) in order to prove that they possess solutions; cf. Theorem 8.1. (b) If the forces are conservative, then the three formulations of linearized elasticity are equivalent. If the forces are not conservative, then the equations of linearized elasticity cannot be expressed as a minimization problem, but only as boundary value problems (cf. (7.12)), or as variational equations (cf. (7.13)).

8. Existence and regularity theorem in linearized elasticity

Throughout this section, the manifold M is endowed with the Riemannian metric g0 = = ∗ g[ϕ0]: ϕ0gˆ, so that ϕ0 : M → N becomes an isometry. As in the previous sections, ∇0, div0, and ω0 denote the connection, the divergence operator, and the volume form on M induced by g0. The solutions to the boundary value problem of linearized elasticity will be sought in Sobolev spaces whose elements are sections of the tangent bundle TM; cf. Sect. 2. The existence of solutions in linearized elasticity relies on the following Riemannian version of Korn’s inequality, due to Chen & Jost [7]: Assume that Γ1 ⊂ ∂M is a non-empty relatively open subset of the boundary of M. Then there exists a constant CK such that

1 2 kξkH (TM) ≤ CKkLξgkL (S 2 M) (8.1)

1 for all ξ ∈ H (TM) satisfying ξ = 0 on Γ1. The smallest possible constant CK in the above inequality, called the Korn constant of M and Γ1 ⊂ ∂M, plays an important role both in linearized elasticity and nonlinear elasticity (see assumptions (8.3) and (9.14) of Theorems 8.1 and 9.2, respectively) since the smaller the Korn constant is, the larger the applied forces are in both existence theorems. To the best of our knowledge, the dependence of the Korn constant on the metric g0 of M and on Γ1 is currently unknown, save a few particular cases; see, e.g., [10]. 32 One such particular case, relevant to our study, is when Γ1 = ∂M and the metric g0 is close 1 to a flat metric, in the sense that its Ricci tensor field satisfies the inequality kRic0kL∞(S M) ≤ , 2 CP where CP is the Poincare´ constant of M, that is

2 2 1 kξkL2 TM ≤ CPk∇0ξk 2 1 for all ξ ∈ H0 (TM). ( ) L (T1 M) To see this, it suﬃces to combine the inequality

2 + 2 = 1 2 + ω k∇0ξk 2 1 kdiv0 ξk 2 kLξg0k 2 Ric0(ξ, ξ) 0 L (T M) L (M) L (S 2 M) 1 2 ZM 1 2 + 2 ≤ kLξg0k 2 kRic0kL∞(S M)kξk 2 , 2 L (S 2 M) 2 L (TM)

1 which holds for all ξ ∈ H0 (TM), with the above assumption on the Ricci tensor field of g0 to deduce that 2 1 2 k∇0ξk 2 1 ≤ kLξg0kL2 S M . L (T1 M) ∞ ( 2 ) 2(1 − CPkRic0kL (S 2 M)) −1 = ∞ Hence the constant CK 2(1 − CPkRic0kL (S 2 M)) can be used in Theorems 8.1 and 9.2 when 1 Γ1 = ∂M and kRic0kL∞(S Mn ) ≤ . Interestingly enough,o particularizing these theorems to a flat 2 CP metric g0 yields existence theorems in classical elasticity with CK = 1/2. The next theorem establishes the existence and regularity of the solution to the equations of linearized elasticity under specific assumptions on the data. Recall that in linearized elasticity the applied body and surface forces are of the form

aﬀ ′ f [ξ] = f[ϕ0] + f [ϕ0]ξ = f[ϕ0] + ( f1 · ξ + f2 : ∇0ξ), aﬀ ′ h [ξ] = h[ϕ0] + h [ϕ0]ξ = h[ϕ0] + (h1 · ξ + h2 : ∇0ξ), cf. relations (7.11).

Theorem 8.1. Assume that Γ1 ⊂ ∂M is a non-empty relatively open subset of the boundary of M, that the elasticity tensor ﬁeld A = A0 ⊗ω0 is essentially bounded and uniformly positive-deﬁnite, that is, there exists a constant CA0 > 0 such that 2 2 = (A0(x): H(x)) : H(x) ≥ CA0 |H(x)| , where |H(x)| : g0(x)(H(x), H(x)), (8.2) for almost all x ∈ M and all H(x) ∈ S 2,x M, and that the applied body and surface forces satisfy the smallness assumption

′ ′ 1 2 ∗ n + n−1 k f [ϕ0]]kL(H (TM),L (T M⊗Λ M)) kh [ϕ0]kL 1 2 ∗ |Γ ⊗Λ Γ ≤ CA0 /CK, (8.3) (H (TM),L (T M 2 2)) where CK denotes the constant appearing in Korn’s inequality (8.1). 2 ∗ Λn 2 ∗ Λn−1Γ (a) If f[ϕ0] ∈ L (T M ⊗ M) and h[ϕ0] ∈ L (T M|Γ2 ⊗ 2), there exists a unique vector 1 ﬁeld ξ ∈ H (TM), ξ = 0 on Γ1, such that

aﬀ aﬀ (A : e[ϕ0, ξ]) : e[ϕ0, η] = f [ξ] · η + h [ξ] · η (8.4) ZM ZM ZΓ2

1 for all η ∈ H (TM), η = 0 on Γ1. 33 (b) Assume in addition that Γ1 = ∂M and, for some integer m ≥ 0 and 1 < p < ∞, the m+2 m+2 m+1 4 Λn m 0 Λn boundary of M is of class C , ϕ0 ∈ C (M, N), A ∈ C (T0 M⊗ M), f1 ∈ C (T2 M⊗ M), m 1 Λn m,p ∗ Λn m+2,p f2 ∈ C (T2 M ⊗ M), and f[ϕ0] ∈ W (T M ⊗ M). Then ξ ∈ W (TM) and satisfies the boundary value problem lin aff −div0 (T [ξ]) = f [ξ] in M, (8.5) ξ = 0 on ∂M. Furthermore, the mapping Alin : Wm+2,p(TM) → Wm,p(T ∗M ⊗ Λn M) defined by

lin lin ′ m+2,p A [η]:= div0 T [η] + f [ϕ0]η for all η ∈ W (TM), (8.6) is linear, bijective, continuous, and its inverse (Alin)−1 is also linear and continuous. Proof. (a) Korn’s inequality, the uniform positive-definiteness of A, and the smallness of the linear part of the applied forces (see (8.1), (8.2), and (8.3)), together imply by means of Lax- Milgram theorem that the variational equations of linearized elasticity (8.4) possess a unique 1 solution ξ in the space {ξ ∈ H (TM); ξ = 0 on Γ1}. (b) It is clear that the solution of (8.4) is a weak solution to the boundary value problem (8.5). Since the latter is locally (in any local chart) an elliptic system of linear partial differential equations, the regularity assumptions on A and f aff and the standard theory of elliptic systems of partial differential equations imply that this solution is locally of class Wm+2,p. Furthermore, the regularity of the boundary of M together with the assumption that Γ1 = ∂M imply that ξ ∈ Wm+2,p(TM). The mapping Alin defined in the theorem is clearly linear and continuous. It is injective, since Alin[ξ] = 0 with ξ ∈ Wm+2,p(TM) implies that ξ satisfies the variational equations (8.4), hence m,p ∗ n ξ = 0 by the uniqueness part of (a). It is also surjective since, given any f0 ∈ W (T M ⊗ Λ M), ∈ 1 = · ∈ 1 there exists ξ H0 (TM) such that M(A : e[ϕ0, ξ]) : e[ϕ0, η] M f0 η for all η H0 (TM) (by part (a) of the theorem), and ξ ∈ WmR+2,p(TM) by the regularity resultR established above. That the inverse of Alin is also linear and continuous follows from the open mapping theorem. m+1 4 Λn Remark 8.2. The regularity assumption A ∈ C (T0 M ⊗ M) can be replaced in Theorem m+1,p 4 Λn + = 8.1(b) by the weaker regularity A ∈ W (T0 M ⊗ M), (m 1)p > n : dim M, by using improved regularity theorems for elliptic systems of partial differential equations; cf. [16].

9. Existence theorem in nonlinear elasticity We show in this section that the boundary value problem of nonlinear elasticity (see Propo- sition 6.5) possesses at least a solution in an appropriate Sobolev space if Γ2 = ∅ and the applied body forces are sufficiently small in a sense specified below. The assumption that Γ2 = ∅ means that the boundary value problem is of pure Dirichlet type, that is, the boundary condition ϕ = ϕ0 is imposed on the whole boundary Γ1 = Γ of the abstract manifold M. Thus our objective is to prove the existence of a deformation ϕ : M → N that satisfies the system (see Proposition (6.5)): − div T[ϕ] = f[ϕ] in intM, (9.1) ϕ = ϕ0 on Γ, where (T[ϕ])(x):= T˙ (x, ϕ(x), Dϕ(x)), x ∈ M, (9.2) ( f[ϕ])(x):= f˙(x, ϕ(x), Dϕ(x)), x ∈ M, 34 the functions T˙ and f˙ being the constitutive laws of the elastic material and of the applied forces, respectively (see Sections 4 and 5). Recall that the divergence operator div = div[ϕ] depends itself on the unknown ϕ (since it is induced by the metric g = g[ϕ]:= ϕ∗gˆ; cf. Remark 2.1) and that ω[ϕ]:= ϕ∗ωˆ denotes the volume form induced by the metric g[ϕ]. ϕ = ξ ϕ M → N The idea is to seek a solution of the form : expϕ0 , where 0 : denotes a natural configuration of the body and ξ : M → TM is a sufficiently regular vector field in the set

0 0 C (TM):= {ξ ∈ C (TM); kϕ0 ξkC0 | < δˆ(ϕ0(M))}, ϕ0 ∗ (TN ϕ0(M)) where δˆ(ϕ0(M)) denotes the injectivity radius of the compact subset ϕ0(M) of N; see (3.2) in ϕ = ξ ξ ∈ C1 TM ∩ C0 TM Section 3. It is then clear that the deformation : expϕ0 , ( ) ϕ0 ( ), satisfies the boundary value problem (9.1) if and only if the displacement field ξ satisfies the boundary value problem = − div T[expϕ ξ] f[expϕ ξ] in intM, 0 0 (9.3) ξ = 0 on Γ. Remark 9.1. The divergence operator appearing in (9.3) depends itself on the unknown ξ, since it is defined in terms of the connection ∇ = ∇[ϕ] induced by the metric g = g[ϕ]:= ϕ∗g,ˆ ϕ = ξ. : expϕ0 ξ ∈ C1 TM ∩ C0 TM Given any vector field ( ) ϕ0 ( ), let A ξ = T ξ + f ξ . [ ]: div( [expϕ0 ]) [expϕ0 ] (9.4) Proving an existence theorem to the boundary value problem (9.3) amounts to proving the existence of a solution to the equation A[ξ] = 0 in an appropriate space of vector fields ξ : M → TM satisfying the boundary condition ξ = 0 on Γ. This could be done by using Newton’s method, which finds a zero of A as the limit of the sequence defined by

′ −1 ξ1 := 0 and ξk+1 := ξk − A [ξk] A[ξk], k ≥ 1. This sequence converges to a zero of A under the assumptions of Newton-Kantorovich theorem (see, e.g., [6]) on the mapping A, which turn out to be stronger than those of Theorem 9.2 below, which uses a variant of Newton’s method, where a zero of A is found as the limit of the sequence defined by ′ −1 ξ1 := 0 and ξk+1 := ξk − A [0] A[ξk], k ≥ 1. The key to applying Newton’s method is to find function spaces X and Y such that the mapping A : U ⊂ X → Y be differentiable in a neighborhood U of ξ = 0 ∈ X. The definition (9.4) of A can be recast in the equivalent form

A ξ = T ◦ ξ + f ◦ ξ , [ ]: div(( expϕ0 )[ ]) ( expϕ0 )[ ] (9.5) T ◦ f ◦ where the mappings ( expϕ0 ) and ( expϕ0 ) are defined by the constitutive equations .... = = ˙ ((T ◦ expϕ )[ξ])(x) T (x, ξ(x), ∇0ξ(x)) : T(x, ϕ(x), Dϕ(x)), x ∈ M, 0 .... (9.6) f ◦ ξ x = f x, ξ x , ∇ ξ x = ˙f x, ϕ x , Dϕ x , x ∈ M, (( expϕ0 )[ ])( ) ( ( ) 0 ( )) : ( ( ) ( )) ξ ∈ C1 TM ∩ C0 TM ϕ = ξ T˙ ˙f for all vector fields ( ) ϕ0 ( ), where expϕ0 and and are the mappings appearing in (9.2). 35 T◦ f ◦ Relations (9.6) show that ( expϕ0 ) and ( expϕ0 ) are Nemytskii (or substitution) operators. It is well known that such operators are not differentiable between Lebesgue spaces unless they are linear, essentially because these spaces are not stable under multiplication..... Therefore.... ξ must belong to a space X with sufficient regularity, so that the nonlinearity of T and f with respect to (ξ(x), ∇0ξ(x)) be compatible with the desired differentiability of A. Since we also want ξ to belong to a reflexive Sobolev space (so that we could use the theory of elliptic systems of partial differential equations), we set = m+2,p 1,p X : W (TM) ∩ W0 (TM), (9.7) for some m ∈ N and 1 < p < ∞ satisfying (m + 1)p > n. Note that the space X endowed with the norm k · kX := k · km+2,p is a Banach space, and that the condition (m + 1)p > n is m+1,p 1 needed to ensure that the Sobolev space W (T1 M), to which ∇0ξ belongs, is stable under X ⊂ C1 TM ϕ = ξ multiplication. It also implies that ( ), so the deformation expϕ0 induced by a ξ ∈ X ∩C0 TM C1 vector field ϕ0 ( ) is at least of class ; hence the results of Section 7 about modeling ξ ∈ X ∩ C0 TM nonlinear elasticity hold for ϕ0 ( ). Define U = BX(δ):= {ξ ∈ X; kξkX < δ} ⊂ X (9.8) X ϕ = ξ as an open ball in centered at the origin over which the exponential map expϕ0 is well- defined. It suffices for instance to set

δˆ(ϕ0(M)) δ = δ(ϕ0, m, p):= , (9.9) C (m + 2, p)kDϕ k 0 ∗ ∗ S 0 C (T M⊗ϕ0TN) m+2,p 0 where CS (m + 2, p) denotes the norm of the Sobolev embedding W (TM) ⊂ C (TM), since kϕ0 ξkC0 | = sup |Dϕ0(x) · ξ(x)| ≤ kDϕ0kC0 ∗ ⊗ ∗ CS (m + 2, p)kξkX < δˆ(ϕ0(M)) for ∗ (TN ϕ0(M)) x∈M (T M ϕ0TN) all ξ ∈ BX(δ). We assume that the reference configuration ϕ0(M) ⊂ N of the elastic body under consideration is a natural state, and that the deformation of reference, the constitutive laws of the elastic material constituting the body, and the applied body forces defined by (9.6), satisfy the following regularity assumptions: m+2 ϕ0 ∈ C (M, N), .... T ∈ Cm+1(M × TM × T 1 M, T 1 M ⊗ Λn M), (9.10) .... 1 1 m 1 ∗ Λn ( f − f[ϕ0]) ∈ C (M × TM × T1 M, T M ⊗ M), and m,p ∗ n f[ϕ0] ∈ W (T M ⊗ Λ M), (9.11) for some m ∈ N and p ∈ (1, ∞) satisfying (m + 1)p > n. Under these assumptions, standard arguments about composite mappings and the fact that Wm+1,p(M) is an algebra together imply that the mappings T ◦ ξ ∈ B δ → T ξ ∈ Wm+1,p T 1 M ⊗ Λn M , ( expϕ0 ): X( ) [expϕ0 ] ( 1 ) f ◦ ξ ∈ B δ → f ξ ∈ Wm,p T ∗M ⊗ Λn M , ( expϕ0 ): X( ) [expϕ0 ] ( ) C1 B δ X A ξ = T ξ + are of class over the open subset X( ) of the Banach space . Since [ ] div [expϕ0 ] f ξ ξ ∈ B δ A ∈ C1 B δ , Y [expϕ0 ] for all X( ), this next implies that ( X( ) ), the space Y := Wm,p(T ∗M ⊗ Λn M)) (9.12) 36 being endowed with its usual norm k · kY := k · km,p. = ω ω = ∗ω Finally, we assume that the elasticity tensor field A A0 × 0, 0 : ϕ0 ˆ , of the elastic material constituting the body under consideration is uniformly positive-definite, that is, there exists a constant CA0 > 0 such that 2 2 = (A0(x): H(x)) : H(x) ≥ CA0 |H(x)| , where |H(x)| : g0(x)(H(x), H(x)), (9.13) for almost all x ∈ M and all H(x) ∈ S 2,x M (the... same condition as in linearized elasticity; cf. ∂Σ (8.2)). Note that the elasticity tensor field A := is defined in terms of the constitutive law T˙ ∂E appearing in (9.2) by means of the relation ... T˙ (x, ϕ(x), Dϕ(x)), = g(x) · Σ(x, (E[ϕ0, ϕ])(x)), x ∈ M, = = ∗ ∗ where E[ϕ0, ϕ]: (g[ϕ] − g[ϕ0])/2 (ϕ gˆ − ϕ0gˆ)/2; cf. Section 7. We are now in a position to establish the existence of a solution to the Dirichlet boundary ′ value problem of nonlinear elasticity if the density f[ϕ0], resp. the first variation f [ϕ0], of the applied body forces acting on, resp. nearby, the reference configuration ϕ0(M) are both small enough in appropriate norms...... Theorem 9.2. Suppose that the deformation of reference ϕ0 and the constitutive laws T and f satisfy the regularity assumptions (9.10) and (9.11), let the elasticity tensor field A = A0 ⊗ ω0 satisfy the inequality (9.13), and let the abstract manifold possess a non-empty boundary of class m+2 C . Let A : BX(δ) ⊂ X → Y be the (possibly nonlinear) mapping defined by (9.4)-(9.9) and (9.12). (a) Assume that the first variation of the density of the applied body forces at ϕ0 satisfies the smallness assumption: ′ 1 2 ∗ n k f [ϕ0]kL(H (TM), L (T M⊗Λ M)) ≤ CA0 /CK, (9.14) where CK denotes the constant appearing in Korn’s inequality (8.1). ′ Then the mapping A is differentiable over the open ball BX(δ) of X, A [0] ∈ L(X, Y) is bijective, and A′[0]−1 ∈ L(Y, X). Moreover, A′[0] = Alin is precisely the differential operator of linearized elasticity defined by (8.6). (b) Assume in addition that the density of the applied body forces acting on the reference configuration ϕ0(M) of the body satisfies the smallness assumption:

= A′ −1 −1 A′ A′ k f[ϕ0]kY < ε1 : sup r k [0] kL(Y,X) − sup k [ξ] − [0]kL(X,Y) , (9.15) 0

A′ −1 −1 A′ A′ k f[ϕ0]kY < δ1 k [0] kL(Y,X) − sup k [ξ] − [0]kL(X,Y) . (9.16) kξkX <δ1 Moreover, the mapping ϕ = ξ satisﬁes the boundary value problem - . : expϕ0 (9.1) (9.2) (c) Assume further that ϕ0 is injective and orientation-preserving. There exists ε2 ∈ (0, ε1) such that, if k f ϕ k < ε , the deformation ϕ = ξ found in (b) is injective and orientation- [ 0] Y 2 : expϕ0 preserving. 37 1 Proof. (a) It is clear from what precedes the theorem that A ∈ C (BX(δ), Y). Fix some ξ ∈ BX(δ) ϕ = ξ and : expϕ0 . We have seen in the Section 8 that

lin aff div T[ϕ] + f[ϕ] = div0 T [ξ] + f [ξ] + o(kξkC1(TM)), where div and div0 denote the divergence operators induced by the connections ∇ := ∇[ϕ] and ∇0 := ∇[ϕ0], respectively; cf. relations (7.17) and (7.18). Using the definitions of the mappings f aff, Alin, and A (see (7.8), (8.6), and (9.4), respectively) in this relation, we deduce that

lin A[ξ] = f[ϕ0] + A [ξ] + o(kξkC1(TM)).

This relation shows that A′[0] = Alin. Since Alin is precisely the diﬀerential operator appearing in Theorem 8.1(b), and since assumption (9.14) of Theorem 9.2 is the same as assumption (8.3) ′ of Theorem 8.1 when Γ2 = ∅, Theorem 8.1(b) implies that A [0] ∈ L(X, Y) is bijective and A′[0]−1 ∈ L(Y, X). (b) The idea is to prove that the relations

′ −1 ξ1 := 0 and ξk+1 := ξk − A [0] A[ξk], k ≥ 1, define a convergent sequence in X, since then its limit will clearly be a zero of A. This will be done by applying the contraction mapping theorem to the mapping B : V ⊂ BX(δ) → Y defined by B[ξ]:= ξ − A′[0]−1A[ξ]. The set V has to be endowed with a distance that makes V a complete metric space and must be defined in such a way that B be a contraction and B[V] ⊂ V. ′ Since the mapping A : BX(δ) → L(X, Y) is continuous, it is clear that ε1 > 0. Hence there exists δ1 ∈ (0, δ) such that A′ −1 −1 A′ A′ k f[ϕ0]kY < δ1 k [0] kL(Y,X) − sup k [ξ] − [0]kL(X,Y) . (9.17) kξkX <δ1 Note that this definition is the same as that appearing in the statement of the theorem; cf (9.16). So pick such a δ1 and define

V = BX(δ1]:= {ξ ∈ X; kξkX ≤ δ1} as the closed ball in X of radius δ1 centered at the origin of X. As a closed subspace of the Banach space (X, k·kX), the set BX(δ1] endowed with the distance induced by the norm k·kX is a complete metric space. Besides, the mapping B : BX(δ1] → X is well deﬁned since BX(δ1] ⊂ BX(δ). It remains to prove that B is a contraction and that B[BX(δ1]] ⊂ BX(δ1]. Let ξ and η be two elements of BX(δ1]. Then

′ −1 ′ kB[ξ] − B[η]kX ≤ kA [0] kL(Y,X)kA[η] − A[ξ] − A [0](ξ − η)kY.

1 Applying the mean value theorem to the mapping A ∈ C (BX(δ), Y) next implies that

kB[ξ] − B[η]kX ≤ CBkξ − ηkX,

38 where ′ −1 ′ ′ CB := kA [0] kL(Y,X) sup kA [ζ] − A [0]kL(X,Y). kζk<δ1 But the inequality (9.17) implies that

= A′ −1 A′ −1 −1 A′ A′ CB 1 − k [0] kL(Y,X) k [0] kL(Y,X) − sup k [ζ] − [0]kL(X,Y) kζk<δ1 ′ −1 k f[ϕ0]kY < 1 − kA [0] kL(Y,X) ≤ 1, δ1 which shows that B is indeed a contraction on BX(δ1]. Let ξ be any element of BX(δ1]. Since

′ −1 kB[ξ]kX ≤ kB[0]kX + kB[ξ] − B[0]kX ≤ kA [0] f[ϕ0]kX + CBδ1, using the above expression of CB and the inequality (9.17) yields

′ −1 ′ ′ kB[ξ]kX ≤ kA [0] kL(Y,X) k f[ϕ0]kY + δ1 sup kA [ζ] − A [0]kL(X,Y) < δ1, kζk<δ1 which shows that B[BX(δ1]] ⊂ BX(δ1]. The assumptions of the contraction mapping theorem being satisfied by the mapping B, there exists a unique ξ ∈ BX(δ1] such that B[ξ] = ξ, which means that ξ satisfies the equation A[ξ] = 0. ϕ = ξ This equation being equivalent to the boundary value problem (9.3), the deformation : expϕ0 satisfies the boundary value problem (9.1)-(9.2). k (c) The contraction mapping theorem shows that the rate at which the sequence ξk = B [0], k = 1, 2, ..., converges to the solution ξ of the equation A[ξ] = 0 is

k (CB) kξk − ξkX ≤ kB[0]kX. 1 − CB In particular, for k = 0,

′ −1 1 kA [0] kL(Y,X) kξkX ≤ kB[0]k ≤ k f[ϕ0]kY ≤ CAk f[ϕ0]kY, (9.18) 1 − CB 1 − CB where −1 = A′ −1 −1 A′ A′ CA : k [0] kL(Y,X) − sup k [ζ] − [0]kL(X,Y) . n kζk<δ1 o The Sobolev embedding Wm+2,p(TM) ⊂ C1(TM) being continuous, the mapping

η ∈ B δ → ψ = η ∈ C1 M, N → Dψ ∈ C0 M X( 1] : expϕ0 ( ) det( ) ( ) is also continuous. Besides minz∈M det(Dϕ0(z)) > 0 since ϕ0 is orientation-preserving and M is compact. It follows that there exists 0 < δ2 ≤ δ1 such that

kηkX < δ2 ⇒ k det(Dψ) − det(Dϕ0)kC0(M) < min det(Dϕ0(z)), z∈M which next implies that

kηkX < δ2 ⇒ det(Dψ(x)) > 0 for all x ∈ M. (9.19) 39 Assume now that the applied forces satisfy k f[ϕ0]kY < ε2 := δ2/CA. Then the relations ϕ = ξ ξ ∈ B δ (9.18) and (9.19) together show that the deformation : expϕ0 , where X( 1] denotes the solution of the equation A[ξ] = 0, satisﬁes

det(Dϕ(x)) > 0 for all x ∈ M, which means that ϕ is orientation-preserving. Moreover, since ϕ = ϕ0 on ∂M and ϕ0 : M → N is injective, the inequality det Dϕ(x) > 0 for all x ∈ M implies that ϕ : M → N is injective; cf. Ciarlet [5, Theorem 5.5-2].

Remark 9.3. (a) The mapping F : BX(δ) ⊂ X → Y deﬁned by

F [ξ]:= A[ξ] − f[ϕ0] satisfies the assumptions of the local inversion theorem at the origin of X if the assumption (9.14) is satisfied. Hence there exist constants δ3 > 0 and ε3 > 0 such that the equation F [ξ] = − f[ϕ0], or equivalently A[ξ] = 0, has a unique solution ξ ∈ X, kξkX < δ3, if k f[ϕ0]kY < ε3. Using the Banach contraction theorem instead of the local inversion theorem in the proof of Theorem 8.1 provides (as expected) explicit estimations of the constants δ3 and ε3, namely δ3 = δ1 and ε3 = ε1 (see (9.15) and (9.16) for the definitions of ε1 and δ1). (b) Previous existence theorems for the equations of nonlinear elasticity in Euclidean spaces (see, e.g., Ciarlet [5] and Valent [17]) can be.... obtained from Theorem 9.2 by making additional = ....assumptions on the applied forces: either f − f[ϕ0] 0 in the case of “dead” forces, or m 1 ∗ Λn f ∈ C (M × TM × T1 M, T M ⊗ M) in the case of ”live” forces. (c) Theorem 8.1 (a) and (b) can be generalized to mixed Dirichlet-Neumann boundary conditions provided that Γ1 ∩ Γ2 = ∅, since in that case the regularity theorem for elliptic systems of partial differential equations still holds.

Acknowledgments

The authors were supported by the Agence Nationale de la Recherche through the grants ANR 2006-2–134423 and ANR SIMI-1-003-01. This paper was completed when the second author (PLF) was in residence at the Mathematical Sciences Research Institute in Berkeley, Cal- ifornia, during the Fall Semester 2013 and was supported by the National Science Foundation under Grant No. 0932078 000.

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