Invariant formulation of hyperelastic transverse isotropy based on polyconvex free energy functions
Introduction
Anisotropic materials have a wide range of applications, e.g. in composite materials, in crystals as well as in biological–mechanical systems. The study of these different materials involves many topics, including manufacturing processes, anisotropic elasticity and anisotropic inelasticity, and micro-mechanics see e.g. Jones (1975). In this paper we will focus on a phenomenological description of anisotropic elasticity at large strains, for small strain formulations see e.g. Ting (1996). The main goal of this work is the construction of polyconvex anisotropic free energy functions, particularly for transverse isotropic materials. Proposed transversely isotropic free energy functions in the literature are often based on a direct extension of the small strain theory to the case of finite deformations by replacing the linear strain tensor with the Green–Lagrange strain tensor see e.g. Spencer (1987b). Weiss et al. (1996) presented a model for applications to biological soft tissues for fully incompressible material behaviour. They introduced an exponential function in terms of the so-called mixed invariants. In the recent work of Holzapfel et al. (2000) a new constitutive orthotropic model for the simulation of arterial walls has been proposed, where each layer of the artery is modeled as a fiber-reinforced material. In the proposed model the terms in the mixed invariants, with respect to several preferred directions, are additively decoupled. That means the model can be considered as the superposition of different transverse isotropic models. For an overview and a comparative study of several mechanical models in biomechanical systems see also Holzapfel et al. (2000). A model for nearly incompressible, transversely isotropic materials for the description of reinforced rubber-like materials is given in Rüter and Stein (2000); they also developed an error estimator for the measurement of the discretization error within the finite element concept. Anisotropic models for the simulation of anisotropic shells have been proposed by Lürding (2001) and Itskov (2001). A general framework for representation of anisotropic elastic materials by symmetric irreducible tensors based on series expansions of elastic free energy functions in terms of harmonic polynomials was proposed by Hackl (1999). The advantage of this approach is its ability to derive effective schemes of parameter identifications. A set of physically motivated deformation invariants for materials exhibiting transverse isotropic behaviour was developed by Criscione et al. (2001). The authors suggest that this approach is potentially useful for solving inverse problems due to several orthogonality conditions.
In contrast to this, no analysis of general convexity conditions for anisotropic materials, such as polyconvexity, has been proposed in the literature to the knowledge of the authors. We will focus on the case of transverse isotropy at finite strains which automatically satisfy the so-called polyconvexity condition within the framework of the invariant theory. The complex mechanical behaviour of elastic materials at large strains with an oriented internal structure can be described with tensor-valued functions in terms of several tensor variables, the deformation gradient and additional structural tensors. General invariant forms of the constitutive equations lead to rational strategies for the modelling of the complex anisotropic response functions. Based on representation theorems for tensor functions the general forms can be derived and the type and minimal number of the scalar variables entering the constitutive equations can be given. For an introduction to the invariant formulation of anisotropic constitutive equations based on the concept of structural tensors, also denoted as the concept of integrity bases, and their representations as isotropic tensor functions see Spencer (1971), Boehler, 1979, Boehler, 1987, Betten (1987) and Schröder (1996). In this context see also Smith and Rivlin, 1957, Smith and Rivlin, 1958.
The main goal of this paper is the establishment of invariant forms of the stress response function which are derived from a scalar-valued free energy function . These invariant forms automatically satisfy the symmetry relations of the considered body. Furthermore, they are automatically invariant under coordinate transformations of elements of the material symmetry group. Thus the values of the free energy function and the values of the stresses can be considered as invariants under all transformations of the elements of the material symmetry group. For the representation of the scalar-valued and tensor-valued functions the set of scalar invariants, the integrity bases and the generating set of tensors are required. For detailed representations of scalar- and tensor-valued functions we refer to Wang, 1969a, Wang, 1969b, Wang, 1970, Wang, 1971, Smith et al. (1963), Smith, 1965, Smith, 1970, Smith, 1971, and Zheng and Spencer, 1993a, Zheng and Spencer, 1993b. The integrity bases for polynomial isotropic scalar-valued functions are given by Smith (1965) and the generating sets for the tensor functions are derived by Spencer (1971). For the classification of material and physical symmetries see Zheng and Boehler (1994).
The mathematical treatment of boundary value problems is mainly based on the direct methods of variations, i.e. finding a minimizing deformation of the elastic free energy subject to the specific boundary conditions. Existence of minimizers of some variational principles in finite elasticity is based on the concept of quasiconvexity, introduced by Morrey (1952), which ensures that the functional to be minimized is weakly lower semi-continuous. This inequality condition is rather complicated to handle since it is an integral inequality. Thus, a more important concept for practical use is the notion of polyconvexity in the sense of Ball, 1977a, Ball, 1977b (in this context see also Marsden and Hughes, 1983 and Ciarlet, 1988). For isotropic material response functions there exist some models, e.g. the Ogden-, Mooney-Rivlin- and Neo-Hooke-type models, which satisfy this concept. Furthermore, for isochoric–volumetric decouplings some forms of polyconvex energies have been proposed by Dacorogna (1989). Some simple stored energy functions, e.g. of St. Venant–Kirchhoff-type or formulations based on the so-called Hencky tensor, are however not polyconvex (see Ciarlet, 1988, Raoult, 1986 and Neff, 2000). It can be shown that polyconvexity of the stored energy implies that the corresponding acoustic tensor is elliptic for all deformations. The precise difference between the local property of ellipticity and the non-local condition of quasiconvexity is still an active field of research. Polyconvexity does not conflict with the possible non-uniqueness of equilibrium solutions, since it guarantees only the existence of at least one minimizing deformation. It is possible that several metastable states and several absolute minimizers exist, though even so one might conjecture that apart from trivial symmetries the absolute minimizer is unique, at least for the pure Dirichlet boundary value problem. We remark, following Ball, 1977a, Ball, 1977b, that polyconvexity implies unqualified existence for all boundary conditions and body forces, which might be somewhat unrealistic. The proof that some energy is elliptic for some reasonable range of deformation gradients is in general not enough to establish an existence theorem.
This paper is organized as follows. In Section 2 we present the fundamental kinematic relations at finite strains and the reduced forms which automatically fulfill the objectivity condition. After that we focus on the continuum mechanical modelling of anisotropic elasticity based on the concept of structural tensors. Section 3 is concerned with the construction of transversely isotropic material response. The integrity basis is given and special model problems are discussed. One part of this section deals with isotropic free energy terms, where some well-known, as well as some new functions are discussed in detail. The main part of this section is concerned with polyconvex transversely isotropic functions. For all proposed ansatz functions the polyconvexity condition is proved. Furthermore, we give geometrical interpretations of some of the polyconvex polynomial invariants. The representation for the stresses and moduli is given in detail for the Lagrangian description as well as the expression for the Kirchhoff stresses. The problem of the stress-free reference configuration and the linearized behaviour near the natural state is discussed in Section 4. Here we identify the expressions of the material parameters involved in the invariant formulation with the parameters of the classical formulation for the linearized quantities. An extension to orthotropic material response is proposed in Section 5 and a short summary of the variational and finite element formulation and the consistent linearization is given in Section 6. The following section presents two numerical examples: the three dimensional analysis of a tapered cantilever and the two dimensional simulation of the elongation of a perforated plate. In the extensive appendix we have summarized the lengthy proofs of the polyconvexity of the individual terms.
Section snippets
Continuum mechanics: foundations
In the following we consider hyperelastic materials which postulate the existence of a so-called Helmholtz free–energy function ψ. The constitutive equations have to fulfill several requirements: the concept of material symmetry and the principle of material frame indifference, also denoted as principle of material objectivity. Thus, the constitutive functions for anisotropic solids must satisfy the combined material frame indifference and the material symmetry condition, which requires them to
Free energy function for transverse isotropic materials
For the explicit formulation of invariant constitutive equations the representation theorems of tensor functions are used. As discussed in the previous section the governing constitutive equations have to represent the material symmetries of the body of interest a priori. Furthermore, the minimal number of independent scalar variables (the set of independent anisotropic mechanical variables) which have to enter the constitutive expression is required. For a detailed discussion of this topic we
Stress free reference configuration and linearization
In this section we analyse the free energy functions with respect to the natural state condition, i.e. the stresses have to be zero in the reference configuration. Furthermore, we are interested in the linearized stress quantities near the reference configuration in order to identify moduli obtained by the invariant formulation with some well-known linear transversely isotropic moduli. The natural state is characterized by F=1 and the invariants have the values
Extension to orthotropic material response
In this section we discuss the construction of polyconvex orthotropic free-energy functions. Orthotropic materials are characterized by symmetry relations with respect to three orthogonal planes. The corresponding preferred directions are chosen as the intersections of these planes and are denoted by the vectors a, b and c with unit length. Thus (a,b,c) represents an orthonormal privileged frame. The material symmetry group is defined bywhere S1, S2, S3 are the reflections with
Variational formulation and finite element discretization
In the following we give a brief summary of the corresponding boundary value problem and finite element formulation in the material description. Let be the reference body of interest which is bounded by the surface . The surface is partitioned into two disjointed parts with . The equation of balance of linear momentum for the static case is governed by the first Piola-Krichhoff stresses P=FS and the body force in the reference configurationThe Dirichlet
Numerical examples
In this section we analyse a three-dimensional tapered cantilever and a two-dimensional perforated plate with centered hole. In the first example we point out the influence of the anisotropy and in the second example we discuss the influence of the orientation of the preferred direction and compare the results for two sets of material parameters. The corresponding linearized moduli at the reference configuration are given for both material sets within the invariant formulation and in the
Conclusion
In this paper we have proposed the formulation of polyconvex transversely isotropic hyperelasticity in an invariant setting. The constitutive models are based on the Clausius–Planck inequality, so the thermodynamic consistency is guaranteed. The main goal of this work has been the construction of polyconvex anisotropic functions in the sense of Ball in order to guarantee the existence of minimizers of variational principles in finite elasticity. For the free energy we have assumed an additive
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