Conditions for Strong Ellipticity of Anisotropic Elastic Materials

We consider the problem of finding the transversely isotropic elasticity tensor closest to a given elasticity tensor with respect to the Frobenius norm. A similar problem was considered by other authors and solved analytically assuming a fixed orientation of the natural coordinate system of the transversely isotropic tensor. In this paper we formulate a method for finding the optimal orientation of the coordinate system—the one that produces the shortest distance. The optimization problem reduces to finding the absolute maximum of a homogeneous eighth-degree polynomial on a two-dimensional sphere. This formulation allows us a convenient visualization of local extrema, and enables us to find the closest transversely isotropic tensor numerically.      

Identification of symmetry type of linear elastic stiffness tensor in an arbitrarily orientated coordinate system

We develop a method through the mirror plane (MP) to identify the symmetry type of linear elastic stiffness tensor whose components are given with respect to an arbitrarily oriented coordinate system. The method is based on the irreducible decomposition of high-order tensor into a set of deviators and the multipole representation of a deviator into a scalar and a unit-vector set. Since a unit-vector depends on two Euler angles, we can illustrate the MP normals of the elastic tensor as zeros of a characteristic function on a unit disk and identify its symmetry immediately, which is clearer and simpler than the methods proposed before. Furthermore, by finding the common MPs of three unit-vector sets using Fortran recipes, we can also analytically recognize the symmetry type first and then recover the natural coordinate system associated with the linear elastic tensor. The structures of linear elastic stiffness tensors of real materials with all possible anisotropies are investigated in detail.     

A stochastic model for elasticity tensors with uncertain material symmetries

In this paper, we consider the probabilistic modeling of media exhibiting uncertainties on material symmetries. More specifically, we address both the construction of a stochastic model and the definition of a methodology allowing the numerical simulation (and consequently, the inverse experimental identification) of random elasticity tensors whose mean distance (in a sense to be defined) to a given class of material symmetry is specified. Following the eigensystem characterization of the material symmetries, the proposed approach relies on the probabilistic model derived in Mignolet and Soize (2008), allowing the variance of selected eigenvalues of the elasticity tensor to be partially prescribed. In this context, a new methodology (regarding in particular the parametrization of the model) is defined and illustrated in the case of transversely isotropic materials. The efficiency of the approach is demonstrated by computing the mean distance of the random elasticity tensor to a given material symmetry class, the distance and projection onto the space of transversely isotropic tensors being defined by considering the Riemmanian metric and the Euclidean projection, respectively. It is shown that the methodology allows the above distance to be (partially) reduced as the overall level of statistical fluctuations increases, no matter the initial distance of the mean model used in the simulations. A comparison between this approach and the initial nonparametric approach introduced in Soize (2008) is finally provided.     

Laws of viscoplastic fluid flow in anisotropic porous and fractured media

In invariant tensor form, the laws of viscoplastic fluid flow are formulated for capillary and fractured media with a periodic microstructure that has orthotropic and transversely isotropic symmetry in the flow properties. An analysis of the laws of viscoplastic fluid flow in transversely isotropic and orthotropic porous and fractured media shows that in formulating the equations it is necessary to distinguish between the permeability tensor and the limiting gradient tensor, which may differ in the symmetry of the flow characteristics, and that the flow law is multivariant and admits one-, two-, and three-dimensional flows.     

Invariant formulation of the electromechanical enthalpy function of transversely isotropic piezoelectric materials

Summary Theoretical and numerical aspects of the formulation of electromechanically coupled, transversely isotropic solids are discussed within the framework of the invariant theory. The main goal is the representation of the governing constitutive equations for reversible material behaviour based on an anisotropic electromechanical enthalpy function, which automatically fulfills the requirements of material symmetry. The introduction of a preferred direction in the argument list of the enthalpy function allows the construction of isotropic tensor functions, which reflect the inherent geometrical and physical symmetries of the polarized medium. After presenting the general framework, we consider two important model problems within this setting: i) the linear piezoelectric solid; and ii) the nonlinear electrostriction. A parameter identification of the invariant- and the common coordinate-dependent formulation is performed for both cases. The tensor generators for the stresses, electric displacements and the moduli are derived in detail, and some representative numerical examples are presented.We thank Dipl.-Ing. H. Romanowski for his support and helpful remarks.     

Penny-shaped crack in transversely isotropic piezoelectric materials

Using a method of potential functions introduced successively to integrate the field equations of three-dimensional problems for transversely isotropic piezoelectric materials, we obtain the so-called general solution in which the displacement components and electric potential functions are represented by a singular function satisfying some special partial differential equations of 6th order. In order to analyse the mechanical-electric coupling behaviour of penny-shaped crack for above materials, another form of the general solution is obtained under cylindrical coordinate system by introducing three quasi-harmonic functions into the general equations obtained above. It is shown that both the two forms of the general solutions are complete. Furthermore, the mechanical-electric coupling behaviour of penny-shaped crack in transversely isotropic piezoelectric media is analysed under axisymmetric tensile loading case, and the crack-tip stress field and electric displacement field are obtained. The results show that the stress and the electric displacement components near the crack tip have (r ^−1/2) singularity. The project supported by the Natural Science Foundation of Shaanxi Province, China     

Coordinate-free Characterization of the Symmetry Classes of Elasticity Tensors

We formulate coordinate-free conditions for identifying all the symmetry classes of the elasticity tensor and prove that these conditions are both necessary and sufficient. Also, we construct a natural coordinate system of this tensor without the a priory knowledge of the symmetry axes.     

Material symmetry and the evolution of anisotropies in a simple material—I. Change of reference configuration

Noll's rule is used to determine the structure of a material symmetry group written with respect to one reference configuration when the representation of the symmetry with respect to another configuration is the traditional material symmetry group associated with isotropy, transverse isotropy or orthotropy, and for an arbitrary deformation gradient relating the two configurations. It is shown that the former symmetry group can contain an orthogonal subgroup. It is determined whether this subgroup is that for isotropic, transversely isotropic, orthotropic, monoclinic, or triclinic response, and the preferred directions of the symmetry are determined.     

Invariant Properties for Finding Distance in Space of?Elasticity Tensors

Using orthogonal projections, we investigate distance of a given elasticity tensor to classes of elasticity tensors exhibiting particular material symmetries. These projections depend on the orientation of the elasticity tensor; hence the distance is obtained as the minimization of corresponding expressions with respect to the action of the orthogonal group. These expressions are stated in terms of the eigenvalues of both the given tensor and the projected one. The process of minimization is facilitated by the fact that, as we prove, the traces of the corresponding Voigt and dilatation tensors are invariant under these orthogonal projections. For isotropy, cubic symmetry and transverse isotropy, we formulate algorithms to find both the orientation and the eigenvalues of the elasticity tensor endowed with a particular symmetry and closest to the given elasticity tensor.      

Nematic polymer mechanics: flow-induced anisotropy

In this paper, we model and compute flow-induced mechanical properties of nematic polymer nano-composites, consisting of transversely isotropic rigid spheroids in an isotropic matrix. Our goal is to fill a gap in the theoretical literature between random and perfectly aligned spheroidal composites (Odegard et al. in Compos. Sci. Technol. 63, 1671–1687, 2003; Gusev et al. in Adv. Eng. Mater. 4(12), 927–931 2002; Torquato in Random heterogeneous materials. Springer, Berlin Heidelberg New York, 2002; Milton in The Theory of Composites. Cambridge University Press, Cambridge, 2002) by modeling the influence of nano-particle volume fraction, flow type and flow rate on nano-composite elasticity tensors. As these influences vary, we predict the degree of elastic anisotropy, determining the number of independent moduli, and compute their values relative to the nano-particle and matrix moduli. We restrict here to monodomains, addressing features associated with orientational configurations of the rod or platelet ensemble. The key modeling advance is the transfer of symmetries (Forest et al. in Phys. Fluids 12(3), 490–498, 2000) and numerical databases (Forest et al. in Rheol. Acta 43(1), 17–37, 2004a, Rheol. Acta 44(1), 80–93, 2004b) for the orientational probability distribution function of the nematic polymer ensemble into the classical Mori–Tanaka effective elasticity tensor formalism. Isotropic, transversely isotropic, orthotropic, monoclinic, and maximally anisotropic elasticity tensors are realized as volume fraction, imposed flow type and flow strength are varied, with 2, 5, 9, 13 or 21 independent moduli for the various symmetries.     

Material symmetry group of the non-linear polar-elastic continuum

We extend the material symmetry group of the non-linear polar-elastic continuum by taking into account microstructure curvature tensors as well as different transformation properties of polar and axial tensors. The group consists of an ordered triple of tensors which makes the strain energy density of polar-elastic continuum invariant under change of reference placement. An analog of the Noll rule is established. Four simple specific cases of the group with corresponding reduced forms of the strain energy density are discussed. Definitions of polar-elastic fluids, solids, liquid crystals and subfluids are given in terms of members of the symmetry group. Within polar-elastic solids we discuss in more detail isotropic, hemitropic, cubic-symmetric, transversely isotropic, and orthotropic materials and give explicitly corresponding reduced representations of the strain energy density. For physically linear polar-elastic solids, when the density becomes a quadratic function of strain measures, reduced representations of the density are established for monoclinic, orthotropic, cubic-symmetric, hemitropic and isotropic materials in terms of appropriate joint scalar invariants of stretch, wryness and undeformed structure curvature tensors.     

Invariant formulation of hyperelastic transverse isotropy based on polyconvex free energy functions

In this paper we propose a formulation of polyconvex anisotropic hyperelasticity at finite strains. The main goal is the representation of the governing constitutive equations within the framework of the invariant theory which automatically fulfill the polyconvexity condition in the sense of Ball in order to guarantee the existence of minimizers. Based on the introduction of additional argument tensors, the so-called structural tensors, the free energies and the anisotropic stress response functions are represented by scalar-valued and tensor-valued isotropic tensor functions, respectively. In order to obtain various free energies to model specific problems which permit the matching of data stemming from experiments, we assume an additive structure. A variety of isotropic and anisotropic functions for transversely isotropic material behaviour are derived, where each individual term fulfills a priori the polyconvexity condition. The tensor generators for the stresses and moduli are evaluated in detail and some representative numerical examples are presented. Furthermore, we propose an extension to orthotropic symmetry.     

Optimization of the strain energy density in linear anisotropic elasticity

The problem considered here is that of extremizing the strain energy density of a linear anisotropic material by varying the relative orientation between a fixed stress state and a fixed material symmetry. It is shown that the principal axes of stress must coincide with the principal axes of strain in order to minimize or maximize the strain energy density in this situation. Specific conditions for maxima and minima are obtained. These conditions involve the stress state and the elastic constants. It is shown that the symmetry coordinate system of cubic symmetry is the only situation in linear anisotropic elasticity for which a strain energy density extremum can exist for all stress states. The conditions for the extrema of the strain energy density for transversely isotropic and orthotropic materials with respect to uniaxial normal stress states are obtained and illustrated with data on the elastic constants of some composite materials. Not surprisingly, the results show that a uniaxial normal stress in the grain direction in wood minimizes the strain energy in the set of all uniaxial stress states. These extrema are of interest in structural and material optimization.     

Exact Lévy-Type Solutions for Plate Bending Exist for Transversely Isotropic but Not for General Monoclinic Materials

An exact three-dimensional Lévy-type solution for the bending of an elastic slab is one in which there are edge loads only and the unknown displacement and stresses have very simple polynomial dependence on the thickness coordinate. Lévy obtained such a solution in 1877 for a linearly elastic, isotropic, plate-like body. The most general material that allows bending and stretching to be uncoupled is monoclinic (13 elastic constants). However, it is shown that only transversely isotropic materials (5 elastic constants) admit exact solutions having polynomial dependence in the thickness direction. Such solutions are listed explicitly.     

Exactly solvable spherically anisotropic thermoelastic microstructures

Microstructures possessing local spherical anisotropy are considered in this paper. An example is a spherulitic polymer which can be modelled by an assemblage of spheres of all sizes in which a radial direction in every sphere is an axis of local transverse isotropy. Our purpose is to construct effectively isotropic microstructures, with spherically anisotropic and thermoelastic constituents, whose effective bulk modulus, thermal stress and specific heat can be exactly determined. The basic microstructure for which this is achieved in the present paper is the nested composite sphere assemblage of Milgrom and Shtrikman (J. Appl. Phys. 66 (1989) 3429) which was originally formulated for isotropic constituents, in the settings of conductivity and coupled fields with scalar potentials. Here, we allow the phases of this microstructure to be spherically thermoelastic with a symmetry class which can be trigonal, tetragonal, transversely isotropic, cubic or isotropic with respect to a local spherical coordinate system. A rich class of new exact results for two-phase composites and polycrystals are obtained.     

Effective elastic moduli of periodic granular composite with transversely isotropic phases

We find a rigorous solution describing the macroscopically uniform stress state of a periodic granular composite with transversely isotropic phases. The structure of the composite is modeled by a cube containing a finite number of arbitrarily arranged and oriented, transversely isotropic spherical inclusions. This provides the model with a flexible means of describing the microstructure. Applying periodic vector solutions and local expansion formulas reduces the initial boundary-value problem to a system of linear algebraic equations. By averaging the solution over the unit cell, we derived exact finite expressions for the components of the effective stiffness tensor. The numerical data presented help to evaluate the efficiency of the method and the limits of applicability of available approximate theories.Translated from Prikladnaya Mekhanika, Vol. 40, No. 9, pp. 123–130, September 2004.     

Strain energy density bounds for linear anisotropic elastic materials

Upper and lower bounds are presented for the magnitude of the strain energy density in linear anisotropic elastic materials. One set of bounds is given in terms of the magnitude of the stress field, another in terms of the magnitude of the strain field. Explicit algebraic formulas are given for the bounds in the case of cubic, transversely isotropic, hexagonal and tetragonal symmetry. In the case of orthotropic symmetry the explicit bounds depend upon the solution of a cubic equation, and in the case of the monoclinic and triclinic symmetries, on the solution of sixth order equations.     

Extrema of Young’s modulus for cubic and transversely isotropic solids

For a homogeneous anisotropic and linearly elastic solid, the general expression of Young’s modulus E(n), embracing all classes that characterize the anisotropy, is given. A constrained extremum problem is then formulated for the evaluation of those directions n at which E(n) attains stationary values. Cubic and transversely isotropic symmetry classes are dealt with, and explicit solutions for such directions n are provided. For each case, relevant properties of these directions and corresponding values of the modulus are discussed as well. Results are shown in terms of suitable combinations of elements of the elastic tensor that embody the discrepancy from isotropy. On the basis of such material parameters, for cubic symmetry two classes of behavior can be distinguished and, in the case of transversely isotropic solids, the classes are found to be four. For both symmetries and for each class of behavior, some examples for real materials are shown and graphical representations of the dependence of Young’s modulus on direction n are given as well.     

Effective thermal conductivity of transversely isotropic media with arbitrary oriented ellipsoïdal inhomogeneities

The objective of this work is to investigate the thermal conduction phenomena in transversely isotropic geomaterials or rock-like composites with arbitrary oriented ellipsoïdal inhomogeneities of low aspect ratio. Based on the evaluation of the Green function, we provide here new expressions for the interaction tensor whose knowledge permits to obtain the concentration tensor of the polarization field used itself to evaluate the effective thermal conductivity tensor by homogenization. Some particular cases of the obtained general solution are equally presented, in order to validate the developed formalism. The obtained results are next used to study the effect of matrix anisotropy, pores systems and microstructure-related parameters on the overall effective thermal conductivity in transversely isotropic rocks. A two-step homogenization scheme is developed for the prediction of the initial anisotropy effects and to test the ability of the proposed model in the evaluation of effective thermal conductivity. With the help of an Orientation Distribution Function (ODF) the anisotropy due to the pore systems is also accounted. Numerical applications and comparisons with available experimental data are finally carried out for a partially saturated Opalinus clay and an argillite which are both composed of an argillaceous matrix and multiple solid minerals constituents.