A covariant constitutive theory for anisotropic. hyperelastic solids with initial strains

Transcription

1 A covariant constitutive theory for anisotropic hyperelastic solids with initial strains Jia Lu Department of Mechanical and Industrial Engineering, The University of Iowa, Iowa City, IA , USA Dedicated to the memory of Professor Jerrold E. Marsden, Abstract Soft tissue systems typically do not have a global stress-free configuration, and the configuration taken as the reference normally contains a pre-existing strain field which we refer to as the initial strain. This article presents a constitutive theory for hyperelastic solids with initial strains. The notion of material metric is introduced. A constitutive description with the material metric describing the natural geometry is developed. Covariant conditions are introduced into the constitutive equation. It is found that the covariant conditions embody the notion of local natural configuration and imply a well-known invariance principle in elasticity. A representation theorem for covariant anisotropic functions is established. This theorem enables the derivation of covariant anisotropic functions using existing Euclidean representation theorems. Examples of covariant constitutive representations for isotropic, transversely isotropic, and general anisotropic materials are presented. Keywords: Initial strain, soft tissue, covariant theory, anisotropic material, inverse problems. address: jia-lu@uiowa.edu. Tel: Fax:

2 1 Introduction In the analysis of soft biological organs, one often encounters systems in which a global stress-free configuration either does not exist or is not accessible. A prominent example is the vascular system. Vascular organs are eternally loaded in their service life and therefore the intact (i.e. load-free) state is not accessible from in vivo measurement. The intact configuration, even if reachable, is generally residually stressed. To address these peculiarities, some analyses not commonly seen in the traditional field arise. The first is the inverse elastostatic analysis. This deals with problems in which a deformed configuration of an elastic body is known while the stress-free configuration is not. The goal is to find the stress-free configuration and at the same time the stress in the given deformed state. Problems like this are common in vascular mechanics, for example the analysis of human aneurysms [1, 2, 3]. It is known that, if a globally attainable stress-free configuration exists, it is possible to solve for this configuration using the standard equilibrium equation but parameterizing the equation in terms of the inverse motion [4, 5]. At the constitutive level, this requires representing the stress function with the current configuration as the reference. The second analysis relates to the non-invasive characterization of material parameters. Researchers have been developing methods for quantifying vascular tissues using organ-level pulsation data [6, 7, 8, 9, 10, 11]. For in vivo studies, a major challenge lies in that the stress-free configuration is unknown. A possible strategy, as investigated in [6, 8, 12, 13], is to incorporate the unknown stress-free configuration locally by geometric parameters, and identify them simultaneously with material properties. Of course there is no guarantee that such inverse problems are well-posed in general, but as a starting point a constitutive representation with unknown initial strains is imperative. At first glance, it may seem trivial to incorporate initial strains in a hyperelastic constitutive equation; the classical theory of simple material [14, 15, 16] provides a possible framework for this extension. According to the theory, the stress at a point depends on the local motion of the material, namely, the motion of material particles in the vicinity of the 2

3 point. It is meaningful to regard an infinitesimal material volume as a material element, whichrespondstoits own motion whiletherest of thebodyacts as its environment. Hence, it is possible to identify a local natural (i.e., undistorted) configuration, the one at which the stress is zero, for each material element individually (For soft tissues this identification better be local because a global stress-free configuration may not exist). A local natural configuration can be obtained, for example, by cutting an infinitesimal material volume out of the material body and releasing the external load. If K is the local deformation that brings a material element from its natural configuration to the chosen reference configuration, then, in a regular motion, the local configuration changes by FK where F is the deformation gradient relative to the reference configuration. Thus, the stress in the material element depends on FK. As outlined in Section 2, one can proceed to develop a constitutive description for initial strains using this approach with K representing the initial strain. Historically, the multiplicative composition FK has been adapted in a variety of constitutive theories, including finite strain plasticity [17, 18, 19], material inhomogeneity [20, 21, 22], tissue growth [23, 24, 25], residual stress [26, 27, 28], and initial strains [6, 7, 8]. In finite plasticity, K corresponds to F 1 p, the inverse of the plastic part of the deformation gradient. In the theories of tissue growth, K corresponds to the inverse of the growth strain F g. However, to be used as a constitutive framework for possible application in parameter identification, this approach has a limitation related to the rotational indeterminacy of the natural configuration. If K 1 brings a material element to a zero-stress state, so does QK 1 for any rotation tensor Q because a superposed rotation does not change the zero stress. If K is to be inversely determined, the rotational indeterminacy may render the inverse problem ill-posed. The rotational indeterminacy of K 1 further implies that the energy function ε = ε(fk) must be necessarily isotropic. As such, an anisotropic function must contain tensorial arguments to represent the orientation of the medium. If one follows the classical recipe of defining the material symmetry group in the stress-free state, the orientation tensors must be specified in a local natural configuration. But the 3

4 rotational indeterminacy of K 1 would leave this specification questionable, if not entirely meaningless. In [29], Noll further refined the theory of simple material noting that the concept of local configuration can be replaced by a metric tensor - a first fundamental form - that characterizes the local geometry [29, Section 3]. As the material deforms, the metric tensor evolves and thus, the deforming geometry is described by a one-parameter family of metric tensors. Moreover, Noll advocated to characterize the symmetry group intrinsically using transformations that leave the metric tensor invariant. In this manner, the symmetry group is defined irrespectively of configurations. Noll s new theory is advantageous for describing pre-strained elastic body because(1) the rotational indeterminacy of the natural configuration is circumvented; and (2) the orientation of the medium can be specified using tensors in any accessible configuration, not necessarily the stress-free one. In Section 3, we lay out a constitutive framework for pre-strained hyperelastic solids based on Noll s new theory. An accessible configuration is taken as the reference. The configuration is endowed with a metric tensor that characterizes the natural geometry. Material symmetry is characterized by isometries of the metric. For tissue-like materials, the symmetry of the material is determined by the underlying microstructure, in particular the distribution and orientation of protein fibers. It is reasonable to expect that information of fiber orientation in the reference configuration can be acquired, and thus the anisotropy of the material can be described without resorting to the local stress-free configuration. In Sections 4 and 5, we further develop the theory by introducing covariant conditions into the constitutive equation. The covariant theory of elasticity was developed by Marsden and Hughes [30]. The theory contains two essential postulations: one is that the law of balance of energy be invariant under arbitrary spatial diffeomorphisms, and the other is that the energy function itself be invariant under arbitrary spatial diffeomorphisms and a class of referential diffeomorphisms. The motivation for introducing the covariant construction is to answer the question as to what is the appropriate mathematical form for a function of the deformation gradient and the material metric to encode the assumption 4

5 that stress depends on FK. In the course of the study, we derive two implications of the referential covariant condition not previously reported in the literature. We show that the referential covariance embodies exactly the foregoing assumption. In Section 5, we establish a representation theorem for covariant anisotropic functions. This is an extension of the classical isotropicization theorem by Boehler [31] and Liu [32]. Based on this theorem, we propose a practical method for deriving covariant representation of anisotropic functions. The method makes use of covariant conditions to transform the energy function into an Euclidean setting, invokes classical Euclidean representation theorems to derive the functional bases, and transforms the functions back. In this manner, one can readily convert a regular constitutive equation into a covariant one with initial strains. This procedure is demonstrated using examples of isotropic and anisotropic material models. The remainder of the paper is organized as follows. Some infinitesimal covariant conditions are recorded in Section 6. These are necessary conditions for the constitutive equation to be covariant, and are useful in revealing the kinetic implication of the material metric [33]. The spatial form of constitutive equation and its application in the inverse analysis is briefly outlined. The article is concluded with remarks in Section 7. 2 Background As is customary in continuum mechanics, a configuration of a material body is a smooth embedding to the three-dimensional Euclidean space S. Let B be a configuration chosen as the reference. Since there is a one-to-one correspondence between a material particle and the position it occupies in a configuration, we can use the the position X B to label the material particle. A motion is a one-parameter family of configurations ϕ : B S. The tangent space T X B is a vector space describing a small neighborhood of X. The deformation gradient is the tangent map F = Tϕ : TB S. The co-tangent space at X is denoted by TX B. There is a natural paring between a co-vector (i.e. a one form) β T X B and a vector v T XB, defined by β v = β A v A. For an inner product space 5

6 V, V = V, and the operator ( ) coincides with the standard inner product. Tensors are multi-linear mappings or multi-linear forms. For example, a ( 0 2 )-tensor T on B is a linear mapping T : T X B T X B, or a bilinear form T XB T X B R. The adjoint of a tensor T : T X B V is the tensor T : V B T X B defined by β (Tv) = (T β) v for all β V and v T X B. In component, if T = T i A e i E A, then T = T i A EA e i. Here V denotes a generic vector space. As alluded in the introduction, the notion of configuration can be understood in a local sense. With reference to Figure 1, let K 1 : T X B Z, detk 1 > 0 be the local mapping that maps a material element in B to its natural configuration Z S (we prefer to use K 1 because in this way, K : Z T X B naturally corresponds to an initial strain). Note that K does not have to be produced by a prior elastic deformation, nor does it have to be the gradient of a global motion. During a regular motion ϕ : B S, the local motion is described by the linear mapping F : Z TS, F = FK. If ε is the strain energy function (per unit mass) characterized relative to the natural configuration, then ε = ε(fk,x), (1) here X is included to indicate material heterogeneity; in what follows X will be suppressed unless otherwise needed for clarity. The energy function must satisfy the fundamental requirement of invariance under superposed rigid body motions, which renders ε = ˆε(K CK), (2) where C = F F is the Cauchy-Green deformation tensor. The symmetry of the material is characterized by orthogonal transformations in the natural state [14, 15]. The symmetry group contains orthogonal tensors Q : Z Z that leave the energy function invariant: ε(fkq 1 ) = ε(fk). (3) If G = Orth, the group of Euclidean orthogonal tensors, the material is isotropic. If G is a subgroup of Orth, the material is anisotropic. Additional tensorial arguments are 6

7 FK Current Configuration Local natural configurations K 1 F Reference configuration Figure 1: Local configuration and local mappings. needed to describe the orientation of the medium for an anisotropic material. For tissuelike materials, these tensors represent the directional distribution of protein fibers. Let λ = {λ i i = 1,N} be the orientation tensors. The strain energy function ε(fk,λ), now including λ in the arguments, must satisfy an additional requirement, the principle of isotropy of space [31]: ε(qfkq 1,Q λ) = ε(fk,λ) Q Orth +, (4) or ˆε(Q K CKQ 1,Q λ) = ˆε(K CK,λ) Q Orth +, (5) where Orth + denotes the group of rotation tensors. Here Q λ denotes the transformation of λundertherotation. Ifλconsists of( 2 0 )-tensors, then Q λ = QλQ. Eq. (4) stipulates that if the external agencies and the medium are rotated, the constitutive equation remains invariant. Eq. (5) follows from Eq. (4) upon invoking the invariant requirement under superposed rigid body motions. Eq. (4) (or Eq. (5)) is the cornerstone of the representation theorem of Boehler [31] and Liu [32]. They proved that an anisotropic function of a certain symmetry can be represented as an isotropic function with the structural tensors as additional argument. The structural tensors are tensors that are invariant under the symmetry group [31, 32]. Using this theorem, one can represent the constitutive equation 7

8 as an isotropic function of K CK and the structure tensors on Z. This approach will not be adopted in the present work due to reasons outlined before. The material is included to motivate the constitutive theory introduced below, which builds upon the notion of natural configuration but employs a different mathematical formulation. 3 Metric approach 3.1 Material metric The configuration B chosen as the reference is regarded as a Riemannian manifold. A metric tensor on B is a symmetric, positive-definite bilinear form T X B T X B R + that characterizes the distance between two material particles in the neighborhood of X. We assume that there is a metric tensor G refereed to as the material metric that describes the natural geometry of B. To illustrate this idea, let us momentarily fall back to the notion of local natural configuration. Let K 1 : T X B Z be the mapping that maps a material element in T X B to the natural configuration Z. For a line element dx T X B, the image in the natural configuration is dx = K 1 dx. (6) Its length is given by ds 2 = (K 1 dx) (K 1 dx) = dx (K K 1 )dx := dx GdX. (7) From (7), we can make the connection that G = K K 1, and clearly G characterizes the undistorted geometry of the material element. This connection is helpful for understanding the nature of G, but is not necessary for the theoretical development. In what follows G is regarded as a primitive variable. Since the reference configuration is embedded in the Euclidean space, it is naturally endowed with the Euclidean metric, the one describing the referential geometry: ds 2 = dx dx. (8) 8

9 As the reference configuration is pre-strained, the referential geometry does not designate the natural state and for this reason, it plays no special role in constitutive development. The Cauchy-Green tensor C = F F furnishes another metric in B (the pullback of the Euclidean metric) that characterizes the current geometry: ds 2 = dx dx = (FdX) (FdX) = dx CdX. (9) The change of geometry from the natural configuration to the current configuration is completely described by functions of C and G. For example, the stretch λ of a line element dx T X B is given by The area strain and volume ratio can be derived accordingly. λ 2 = ds2 ds 2 = dx CdX dx GdX. (10) Remark 1. If there is a global stress free-configuration, i.e., an embedding Ξ : B S for which Ξ(B) is everywhere stress-free, then the tangent map K 1 = T X Ξ defines a local natural configuration and G = (T X Ξ) (T X Ξ). In this case, G is derived from a global deformation and satisfies the compatibility condition (the Riemann-Christoffel tensor vanishes identically) [34, 35]. On the other hand, if G satisfies the compatibility condition a priori, there exists a global stress-free configuration which can be determined from G to within a rigid-body motion. For the sake of generality, we don t require G to be compatible. This implies that the material body in general does not attain a global stressfree configuration. In particular, the intact (load-free) configuration is not necessarily stress free. 3.2 Constitutive equations As a starting point, we assume that the constitutive equation depends on the deformation gradient F, the material metric G, and possibly the position X but the explicit reference to which is suppressed: ε = ε(f,g). (11) 9

10 Invariance under superposed rigid body motions further requires that F enters the constitutive equation through C: ε = ˆε(C,G). (12) For G to represent the natural geometry, we impose the following zero stress condition: ˆε C = 0, (13) C=G orequivalently ε F=K 1 = 0whereK 1 isamultiplicative factorsuchthatg = K K 1. F K 1 can be computed, for example, by Cholesky decomposition of G modulo an arbitrary left rotation. Let ρ r be the material density in the natural configuration, and let W = ρ r ε(c,g) be the energy density per unit undistorted volume. If ρ 0 is the material density in the reference configuration, then, by conservation of mass ρ 0 = Gρ r where G = detg. (14) The energy function per unit volume in the reference configuration is W = GW(C,G). By balance of energy, ( ) d GW 1 + dt B 2 ρ 0v v dv = ρ 0 b vdv + p vda, (15) B B where p is the nominal traction (contact force per unit reference surface area), b is the body force, and v is the particle velocity. A standard derivation leads to the local form where S denotes the 2nd Piola Kirchhoff stress on B. Hence W G C : Ċ = 1 S : Ċ, (16) 2 S = 2 G W C. (17) It follows that Jσ = 2 GF W C F, JP = 2 GF W C, (18) where σ is the Cauchy stress, P is the first-piola-kirchhoff stress, and J = detf. 10

11 Remark 2. The present theory provides a way to describe residual stress in hyperelastic solids. The residual stress is the non-vanishing stress field in the intact (load-free) configuration. Within this constitutive framework, if B is taken to be the intact configuration, the residual stress is given by S = 2 G W C. (19) C=I As an example, for the neo-hookean material introduced later in (47), if residually stressed, the residual stress follows S = µ G(G 1 I), whereiisthe( 2 0 )-identity tensor. Itshouldbenotedthat, theuseofahyperelasticfunction does not necessarily mean that the residual stress results from a prior elastic deformation. The equation (19) merely indicates that the residual stress depends elastically on the local motion relative to the natural state. 3.3 Material symmetry In classical continuum mechanics, the symmetry group of a material is characterized by orthogonal transformations on the stress-free configuration. With regard to a local natural configuration, the symmetry group G contains orthogonal tensors Q : Z Z that leave the constitutive equation invariant. Let K : Z T X B be the local motion that brings the natural material element to that in the reference configuration. By Noll s rule [15, Section 31], the material symmetry group in T X B is G = KGK 1. (20) Let G = K K 1 be the material metric. Tensors Q : T X B T X B having the property Q GQ 1 = G are called G-orthogonal tensors. G-orthogonal tensors obviously form a group, which we refer to as the G-orthogonal group, Orth(G). The conjugate relation Q = KQK 1 (Noll s rule)inducesanisomorphismbetween theeuclidean orthogonal group Orth and the G-orthogonal group Orth(G). By direct computation, Q GQ 1 = (K Q K )(K K 1 )(KQ 1 K 1 ) = K (Q Q 1 )K 1. 11

12 Obviously, Q Q 1 = I if and only if Q GQ 1 = G. This observation suggests the following manifold definition of the material symmetry group. Definition: The material symmetry group G is a group of G-orthogonal tensors Q having the property ε(fq 1,G) = ε(f,g), or ˆε(Q CQ 1,G) = ˆε(C,G) (21) A material is isotropic if G = Orth(G), otherwise anisotropic. This definition was initially contained in[29]. It is fully equivalent to the classical notion of symmetry, but is intrinsic in the sense that it does not require an a priori identification of the local natural configuration. As alluded before, the symmetry of the material is eventually characterized by some orientation (or structural) tensors. In situations in which G is unknown, it is reasonable to expect that type of symmetry (isotropy, transverse isotropy, etc.) and the orientation tensors are known. We can formulate the constitutive equation using orientation tensors Λ available in the reference configuration: ε = ε(f,g,λ). (22) The mathematical representation of such functions will be discussed later in the context of the covariant theory. 4 Covariant theory The covariant theory by Marsden and Hughes [30] was motivated by the relativistic view that a physical law should be invariant under different observer frames. They postulated that the law of balance of energy be invariant under spatial diffeomorphisms, and that the constitutive equations be invariant under spatial and referential diffeomorphisms. In this work, we focus on the invariance of constitutive equations, particularly the invariance under referential diffeomorphisms. A referential diffeomorphism, in a passive way, may be thought as a change of coordinates in the reference configuration, but we take it as an actual change of local configurations. 12

13 The pushforward/pullback notations will be used to describe the tensorial transformations induced by diffeomorphisms. Some basic operations are recalled here; interested readers are referred to [30] for a full exposition. We describe these definitions in the context of the motion ϕ : B S. For a scalar function f : S R, the pullback of f is defined as ϕ f = f ϕ. The pushforward of a scalar function F : B R is ϕ F = F ϕ 1. For a vector v field in TS, the pullback of v gives a vector field in TB: ϕ v = (Tϕ) 1 v ϕ = F 1 v ϕ. (23) Similarly, for the vector field V TB, the pushforward by ϕ is a vector field in TS: ϕ V = (Tϕ)V ϕ 1 = FV ϕ 1. (24) For an one-form β S, the pullback is defined intrinsically by the relation (ϕ β) V = β (ϕ V), which gives ϕ β = F β ϕ. (25) Similarly, the pushforward of a one-form on B by ϕ is defined by (ϕ B) v = B (ϕ v), thus ϕ B = F B ϕ 1. (26) A ( p q)-tensor T on B is understood as a multi-linear form, which takes p slots one-forms and q slots of vectors to produce a scalar. The pushforward by ϕ, in component, is (ϕ T) i 1 i p j 1 j q = F i 1 I1 F ip I p T I 1 I p J 1 J q (F 1 ) J 1 j1 (F 1 ) Jq j q ϕ 1. (27) Similarly, for a ( p q) tensor t on S, the pullback is (ϕ t) I 1 I p I 1 J q = (F 1 ) I 1 i1 (F 1 ) Ip i p t i 1 i p j 1 j q F j 1 J1 F jq J q ϕ. (28) In a similar manner, we can define the pushforward and pullback under other diffeomorphisms applied to B or S. Some relevant results are recorded here. Let ξ : S S be a spatial diffeomorphism that maps ϕ(b) to ξ ϕ(b). The pushforward of the spatial metric tensor g (g is taken to be the metric tensor of the Euclidean space) is 13

14 ξ g = (Tξ) g(tξ) 1 ξ 1. In particular, (ϕ 1 ) g = F gf := C. Here C is the Cauchy-Green deformation tensor, which we also write C = F F as g is identity and S = S. Since (ϕ 1 ) g = ϕ g, we say that C is the pullback of the Euclidean metric. Under a referential diffeomorphism Ξ : B Ξ(B), the material metric G transforms according to Ξ G = (TΞ) G(TΞ) 1. The Cauchy-Green tensor C transforms to Ξ C = (TΞ) C(TΞ) 1. Under the simultaneous action of a spatial diffeomorphism ξ and a referential diffeomorphism Ξ, the deformation gradient F, as a two-point tensor, transforms to (Tξ)F(TΞ) 1. Readers not familiar with these notions may think them as the usual tensor transformation rules under a change of coordinates. 4.1 Spatial covariance The covariant theory of Marsden and Hughes starts by writing the strain energy as a function of the deformation gradient F, the material metric G, and the spatial metric tensor g. With extension to anisotropic materials, we also include orientation tensors Λ B in the argument list: ε = ε(f,g,g,λ). (29) As advocated by Marsden and Hughes [30], spatial covariance requires the energy function to be invariant under any superposed spatial diffeomorphism ξ : S S: ε(ξ F,ξ g,g,λ) = ε(f,g,g,λ). (30) Spatial invariance is a strong version of the classical requirement of invariance under superposed rigid body motions. The latter is restricted to Euclidean isometries that leave the spatial metric g invariant. In the covariant theory, the isometries are replaced by arbitrary diffeomorphisms, but the metric g is required to change tensorially. Marsden and Hughes proved that, spatial invariance implies that the energy function depends on C and G. This can be showed simply by letting ξ = ϕ 1 in (30): ε = ε((tϕ) 1 F,(Tϕ) g(tϕ),g,λ) = ε(1,c,g,λ) = ˆε(C,G,Λ), (31) 14

15 where 1 is a mixed identity tensor on B, which does not change under referential diffeomorphisms and therefore can be dropped out. It is clear that spatial covariance induces the same constitutive reduction as the invariance under superposed rigid body motions. Next, we discuss the referential covariant condition and its implications. 4.2 Referential covariance If the material is heterogeneous, the constitutive equation can not be form-invariant under an arbitrary change of referential coordinates. For this reason, we restrict referential diffeomorphisms to local diffeomorphisms, those anchored at the point of interest, that is, Ξ(X) = X. AnexampleoflocaldiffeomorphismisthelinearmappingΞ(Y) = H[Y X]+X for material points Y in a small neighborhood of X. Referential covariance requires the energy function to be invariant under arbitrary local referential diffeomorphisms: ε(ξ F,g,Ξ G,Ξ Λ) = ε(f,g,g,λ). (32) Alternatively, if one starts with the function ε = ˆε(C, G, Λ), referential covariance means ˆε(Ξ C,Ξ G,Ξ Λ) = ˆε(C,G,Λ). (33) Here, Ξ F = F(TΞ) 1, Ξ C = (TΞ) C(TΞ) 1, (34) Ξ G = (TΞ) G(TΞ) 1. The pushforward of the orientation tensor Λ depends on its type. Without loss of generality, let us assume that Λ contains ( 2 0 )-tensors, then Ξ Λ = (TΞ)Λ(TΞ). Referential covariance is clearly an additional restriction placed on the constitutive equation. Marsden and Hughes showed that, if the strain energy depends only on C and G, then, referential covariance implies isotropy, namely ˆε(Q CQ 1,G) = ˆε(C,G) Q = 15

16 Orth(G). This is the direct consequence of setting TΞ = Q in the energy function. Below, we prove two additional implications of this condition. Proposition 1. Let ε = ε(f,g,g,λ), and let K be a multiplicative factor such that G = K K 1 (we call K 1 : T X B S a local Euclidean chart). Referential invariance implies the reduction ε = ε(fk,λ), where λ = K 1 ΛK. Proof: Given the Euclidean chart K 1, construct a local diffeomorphism Ξ such that TΞ = K 1. Under this diffeomorphism, the tensors transform as F F(TΞ) 1 = FK, G (TΞ) G(TΞ) 1 = K (K K 1 )K = I, (35) Λ (TΞ)Λ(TΞ) = K 1 ΛK = λ. By covariance, ε(f,g,g,λ) = ε(fk,g,i,λ). (36) In the traditional setting, the Euclidean metric g and the ( 0 2 )-identity tensor I can be omitted from the arguments, leading to the desired reduction. Here, λ corresponds to the orientation tensors in Z introduced immediately before Eq. (4). If the function ε(f,g,g,λ) obeys the zero stress condition (13), one can readily show that the reduced form has the property that ε F=K 1 = 0. Consequently, K 1 defines a local natural configuration. The significance of the covariant condition, therefore, lies in that it automatically embodies the fundamental assumption underlining the constitutive equation (1). Proposition 2. Covariant conditions (30) and (32) imply the principle of isotropy of space. Proof: Let K 1 : T X B S be an Euclidean chart of G (that is G = K K 1 ), and let Q : S S be an Euclidean rotation tensor. One can construct a local diffeomorphism Ξ such that TΞ = QK 1, and a spatial diffeomorphism φ such that Tφ = Q. Under the F 16

17 simultaneous action of these diffeomorphisms, F (Tφ)F(TΞ) 1 = QFKQ 1, g (Tφ) g(tφ) 1 = Q gq 1 = g, G (TΞ) G(TΞ) 1 = Q K GKQ 1 = Q Q 1 = I, (37) Λ (TΞ)Λ(TΞ) = QK 1 ΛK Q = QλQ. Covariance implies ε(f,g,g,λ) = ε(qfkq 1,g,I,QλQ ). (38) Meanwhile, setting Q to be identity, we obtain ε(f,g,g,λ) = ε(fk,g,i,λ). (39) Equalities (38) and (39) conclude (4), as g and I are invariant under Euclidean rotations. The referential covariant condition (32) or (33) is a stronger condition than the principle of isotropy of space. The latter is restricted to Euclidean rotations while the former applies to all local diffeomorphism. In particular, the referential covariance requires the constitutive equation to be invariant under the isometries that leave G invariant, implying that a covariant function is necessarily G-isotropic. Remark 3. It is worthnoting that the orientation tensor λ defined in (35) 3 is a derived instead of a primitive variable. Both K 1 and λ are computed to within a left rotation. However, since λ co-rotates with K 1 (that is, K 1 QK 1 implies λ QλQ ), the rotational indeterminacy is immaterial due to the principle of isotropy of space (or covariance). This is in sharp contrast with the local configuration approach whereby λ is a primitive variable and needs to be unambiguously specified. 5 Covariant representation of anisotropic functions 5.1 Covariant representation It has been long established that an anisotropic function of a certain symmetry group can be represented as an isotropic function with structural tensors as additional arguments 17

18 [31, 32, 36, 37]. Structural tensors are tensors that are invariant under the action of the symmetry group. Zheng and collaborators [38, 36] showed that most of the compact groups can be represented by a single structural tensor, albeit a higher order one. In [39], the definition of the structural tensor was extended to the manifold setting. The structural tensors A of the symmetry group G are tensors that are invariant under the action of the group: Q A = A Q G. (40) Note that G contains G-orthogonal tensors. The operation Q A depends on the type of A. If A is a ( p 0 )-tensor, then, in component, (Q A) I 1I 2 I p = Q I 1 J 1 Q I 2 J 2 Q Ip J p A J 1J 2 J p. In [39], it was also shown that the symmetry group defined in Section 3.3 and the structural tensors are fully covariant quantities, in the sense that if G is the material symmetry group relative to the metric tensor G with the structural tensor A, then Ξ G is the symmetry group relative to Ξ G with the structural tensor Ξ A, and vice versa. Here Ξ G = (T Ξ )G(T Ξ ) 1, by Noll s rule. The pushforwardof A depends on its type. For ( p 0 )-tensors, (Ξ A) I 1I 2 I p = (TΞ) I 1 J 1 (TΞ) I 2 J 2 (TΞ) Ip J p A J 1J 2 J p. The covariant nature of the structural tensors motivates us to extend the isotropicization theorem to material manifold. Let us begin with the constitutive equation ε = ε(f, g, G, A), and impose full covariant conditions on the function. Spatial covariance leads to the reduction ε = ˆε(C,G,A). (41) Referential covariance further requires ˆε(Ξ C,Ξ G,Ξ A) = ˆε(C,G,A). (42) In particular, for Ξ such that TΞ = Q with Q G, we have Ξ A = A because A is the structural tensor of G, and also Ξ G = G because by construction Q Orth(G). Then, Eq. (42) implies ˆε(Q CQ 1,G,A) = ˆε(C,G,A). (43) 18

19 ε = ˆε(C,G,A) ε = ε(i 1,I 2, ) I i = I i (C,G,A) K 1 K ε = ˆε(K CK,I,K 1 A) ε = ε(i 1,I 2, ) I i = I i (K CK,I,K 1 A) Figure 2: Derivation of covariant representation for anisotropic functions Recalling the definition of the material symmetry (21), it is clear that the condition (43) indicates that the function ˆε is anisotropic with symmetry group G. We have thus proved the following manifold version of Boehler s theorem: Theorem. An anisotropic function of ε = ε(c,g) of a symmetry group G can be represented as a covariant function ε = ˆε(C, G, A) with structural tensors A as additional arguments. Thenextquestionishowtoconstructcovariant functionsofc, G, anda. Weproposea practical method to build covariant basis functions using classical Euclidean representation theorems. Beginning with the energy function (41), we utilize the covariant condition (33) to reduce (41) into the form ε = ˆε(K CK,I,K 1 A). (44) where K 1 is a local Euclidean chart, K K 1 = G. Again by covariance, the function (44) must be isotropic in the classical sense (invariant under Euclidean orthogonal transformations). Then, we can apply existing Euclidean representation theorems to establish the functional basis for (44). Note that, the basis functions are made out of scalarvalued isotropic combinations of K CK, I, and K 1 A. By construction, the basis functions are automatically covariant under local diffeomorphisms. Applying the mapping K : Z T X B, the basis functions can be transformed back to the referential form, as functions of C, G, and A. This approach is illustrated in Figure 2. 19

20 5.2 Examples Isotropic material As discussed before, the constitutive equation of an isotropic material is a covariant function of Cand G. Starting fromɛ = ˆε(C,G), referential covariance implies ε = ˆε(K CK,I), and the latter is an Euclidean isotropic function. It is well known that an Euclidean isotropic function can be represented as a function of the following basis functions I 1 = tr(k CK), I 2 = 1 2 ( I 2 1 tr(k CKK CK) ), (45) I 3 = det(k CK). The trace operator is defined by tr(β v) = β i v i for any v V and β V. The basis functions are invariant under the transformation (Ξ : T X B V, K 1 K 1 Ξ 1, C Ξ CΞ 1 ), or (H : Z V, K 1 HK 1, I H H 1 ), or the combination of both. Setting H = K, we obtain I 1 = tr(cg 1 ), I 2 = 1 2 (I2 1 tr(cg 1 ) 2 ), (46) I 3 = detc detg. Alternativelyonecouldusethecyclicpermutationpropertytr(A 1 A 2 A k ) = tr(a 2 A k A 1 ) and the identity det(ab) = (deta)(detb) to arrive at the same result. Example (neo-hookean material). Consider a material described by the energy function W = µ 2 (I 1 logi 3 3), (47) where, in the classical sense, the deformation tensor C is computed relative to the natural state, and I 1 = trc, I 3 = detc. Relative to a pre-strained state described by the material metric G, I 1 and I 3 are defined in Eq. (46). Invoking the stress formula (17), S = µ G(G 1 C 1 ). In the derivation the identities I 1 C = G 1 and I 3 C = I 3C 1 are used. 20

21 5.2.2 Transverse isotropy As alluded in the introduction, our perspective is that the orientation tensors (or the structural tensors) on the reference configuration B are known. For a transversely isotropic material, the symmetry is characterized by rotations and reflections with respect to a preferred direction N T X B, again assumed known. In the manifold setting, transverse isotropy means ε(q CQ 1,G) = ε(c,g) Q Orth(G) QN = ±N). (48) By covariance, the above condition is equivalent to ε(q K CKQ 1,I) = ε(q K CKQ 1,I) Q Orth QN = ±N), (49) where K K 1 = G and N = K 1 N. N is the preferred direction (not necessarily unit length) in the local natural configuration Z. N is defined to within a left rotation, but the rotational indeterminacy is immaterial by construction. According to a well-known result [31], the function (49) can be represented by the following strain invariants: { I 1 (K CK), I 2 (K CK), I 3 (K CK), N (K CK)N N N, N (K CK) 2 N N N }. (50) Again the basis functions are covariant. They remain invariant under arbitrary transformation (Ξ : T X B V, K 1 K 1 Ξ 1, C Ξ CΞ 1 ), or (H : Z V, K 1 HK 1, N HN, I H H 1 ), or the combination of both. Setting H = K, the basis functions are transformed into { I 1 (CG 1 ), I 2 (CG 1 ), I 3 (CG 1 ), We can also write the last two bases as N CN N GN, N (CG 1 } C)N. (51) N GN tr(ca) tr(ga), tr(cg 1 CA). (52) tr(ga) where A = N N is the structural tensor, type ( 2 0 ). It can be readily verified that every basis function in (51) is covariant in (C, G, A). 21

22 5.2.3 Structural models - Holzapfel function Holzapfel et al. [40, 41] established a family of structure-motivated constitutive models for arterial tissues. Arterial tissues consist of a random fiber network and multiple layers of protein fibers each with a distinct fiber orientation. Holzapfel et al. assumed that the random network is described by an isotropic function while the laminar fibers contribute to an anisotropic function of the fiber stretches λ i : ε = ε iso (C)+ε aiso (λ 1, λ k ). (53) Here k is the number of fiber families. In the present framework fiber directions N i are specified in the reference configuration B. Recalling Equation (10), the square stretch of the i-th fiber is λ 2 i = N i CN i N i GN i. (54) This is obviously a covariant function of C, G, and Λ i where Λ i = N i N i. The isotropic term can be readily represented by covariant basis functions (46). In this manner, the constitutive equation (53) is converted into a covariant function of C, G, and Λ i. Note that this is a model for which the material anisotropy is represented by orientation tensors (i.e., Λ i ), not structural tensors. Thematerial may not have a non-trivial symmetry group if fiber angles and stiffness properties are not symmetrically distributed. Example. Consider a particular Holzapfel function (as energy per unit undistorted volume) W = µ 2 (I 1 3)+ k i=1 µ ( ) i e α(λ2 i 1)2 1. (55) 8α In the covariant setting, the strain invariants I 1 and λ i s are defined in (46) 1 and (54), respectively. Invoking the stress formula (17), the 2nd Piola-Kirchhoff stress is found to be S = µ GG 1 + k i=1 µ i G e α(λ2 i 1)2 (λ 2 i 1) N i N i. 2 N i GN i 6 Infinitesimal covariance conditions. Spatial form The invariant conditions imply the following infinitesimal covariance conditions: 22

23 1. For the energy function ε = ε(f,g,g), ε F F 2g ε g = 0. (56) 2. For the function ε = ˆε(C,G) that depends only on C and G, C ˆε ˆε +G = 0. (57) C G 3. For the anisotropic function ε = ˆε(C,G,A), if A is a set of ( 2 0 ) tensors, then C ˆε ˆε +G C G ˆε A = 0, (58) A Here, ˆε A A means k i=1 ˆε A i A i. Equalities (57) and (58) are proved in [39]. Equality (56) can be proved in the following way. Consider a one-parameter family of spatial diffeomorphism ξ s, with ξ 0 being the identify mapping and let l = d ds Tξ s. From the covariance s=0 condition, 0 = d ds ε(ξ s F,ξ s g,g) = ε s=0 F [ ε ε (lf)+ g ( l g gl) = F F 2g ε ] l. (59) g Since the identity holds for arbitrary l and the tensor inside the bracket is independent of l, we conclude (59). In the derivation, we used the identity d ds (Tξ) 1 = l, and ) s=0 the symmetry conditions g = g,( ε g = ε g. If one recalls that the Cauchy stress is ρg 1 ε F F, Equation (56) readily implies the Doyle-Ericksen formula [30, Section 2.4] σ = 2ρ ε g. (60) By covariance, the energy function can be readily transformed into the spatial form. Starting from ε = ˆε(C, G, A), using the covariant condition (33) and setting T Ξ = F pointwise so that C (TΞ) C(TΞ) 1 = g, G (TΞ) G(TΞ) 1 = F GF 1 := c, (61) A (TΞ) A := a. 23

24 We obtain ε = ˆε(g,c,a). (62) Here if A consists of ( 2 0 )-tensors, a = FAF. When G = I, c becomes the classical Finger tensor. In the spatial form (62), the Cauchy stress can be directly computed using the Doyle-Ericksen formula (60). In addition, the function (62) satisfies the infinitesimal covariant condition g ˆε +c ˆε g c ˆε a = 0. (63) a At the absence of anisotropy tensors, g ˆε g +c ε = 0. (64) c As an example, consider the energy function (55). In spatial form the strain invariants are represented as I 1 = tr(gc 1 ), λ 2 i = n i gn i n i cn i, i = 1,2,,k where n i, i = 1,2,,k are the fiber directions in the current configuration, n i = ϕ N i = FN i. We have (65) I 1 g = c 1, λ 2 i g = n i n i n i cn i. The Cauchy stress is worked out to be σ = 2 c W g = [ c µc 1 + k i=1 ] µ i 2 eα(λ2 i 1)2 (λ 2 i 1)n i n i. (66) n i cn i where c = detc. The spatial form has a distinct advantage in formulating numerical procedures for inverse elastostatic analysis. This deals with problems in which the system has a (yet unknown) global stress-free configuration, and this configuration is to be determined from the given current configuration and the applied load. Let Φ : S B, X = Φ(x) be the inverse motion that brings the material body back to the stress-free state when the external load is released, then, K 1 := T x Φ constitutes a local natural configuration. The material metric is G = (T x Φ) (T x Φ), and it coincides with the Finger tensor c. The 24

25 Cauchy stress, as a function of (g,c,a), is automatically a function of the inverse motion. In this manner, the equilibrium equation gives rise to a nonlinear differential equation for the inverse motion, which is to be solved numerically. The tangent tensor σ c, which is needed in a Newton-type iterative procedure, can be readily derived. A finite element inverse formulation based on the spatial form of elasticity was described in [42]. 7 Concluding remarks We presented a constitutive framework for describing pre-strained hyperelastic materials. A metric-based theory, which is free from the limitations of the ones built upon the notion of local natural configuration, was developed. A major contribution of the work lies in the introduction of the covariant construction in the constitutive theory. We showed that the constitutive equations with and without initial strains can be represented in exactly the same form, provided that the covariant construction is in place. We found that the referential covariance condition, traditionally regarded as a generalized manifestation of material isotropy, embodies the assumption of local natural configuration and the principle of isotropic of space. We also established a representation theorem for covariant anisotropic functions. This theorem is instrumental in converting a regular constitutive equation into a pre-strained form. We have briefly commented on the advantage of this constitutive description in the inverse elastostatic analysis. Another potential application is in the area of parameter identification. Using the present theory, one can represent an unknown stress-free configuration locally by a metric. For some systems, it may be possible to identify the metric tensor together with the material parameters using carefully formulated inverse methods. Research in this area remains open. 25

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