Representations for isotropic thermoelastic dielectrics

Using Chakrabarti and Wainwright's method, an integrity basis consisting of eighteen invariants is constructed for the energy function of deformation and polarization of Isotropic thermoelastic dielectrics. The minimality of the integrity basis is verified by group theoretic methods and this minimal integrity basis is employed to construct closed form irreducible polynomial representations for the electric tensor, the stress tensor and the electric and heat flux vectors.     

An irreducible polynomial functional basis of two-dimensional Eshelby tensors

The two-dimensional(2D) Eshelby tensors are discussed. Based upon the complex variable method, an integrity basis of ten isotropic invariants of the 2D Eshelby tensors is obtained. Since an integrity basis is always a polynomial functional basis, these ten isotropic invariants are further proven to form an irreducible polynomial functional basis of the 2D Eshelby tensors.     

On Isotropic Invariants of the Elasticity Tensor

Isotropic invariants of the elasticity tensor always yield the same values no matter what coordinate system is concerned and therefore they characterize the linear elasticity of a solid material intrinsically. There exists a finite set of invariants of the elasticity tensor such that each invariant of the elasticity tensor can be expressed as a single-valued function of this set. Such a set, called a basis of invariants of the elasticity tensor, can be used to realize a parametrization of the manifold of orbits of elastic moduli, i.e. to distinguish different kinds of linear elastic materials. Seeking such a basis is an old problem in theory of invariants and seems to have been unsuccessful until now. In this paper, by means of the unique spectral decomposition of the elasticity tensor every invariant of the elasticity tensor is shown to be a joint invariant of the eigenprojections of the elasticity tensor, and then by utilizing some properties of the eigenprojections a basis for each case concerning the multiplicity of the eigenvalues of the elasticity tensor is presented in terms of joint invariants of the eigenprojections. In addition to the foregoing properties, the presented invariants may also be used to form invariant criteria for identification of elastic symmetry axes.     

Dyadic method for tensor functions

In this paper, we discuss tensor functions by dyadic representation of tensor. Two different cases of scalar invariants and two different cases of tensor invariants are calculated. It is concluded that there are six independent scale invariants for a symmetrical tensor and an antisymmetrical tensor, and there are twelve invariants for two symmetrical tensors and an antisymmetrical tensor. And we present a new list of tensor invariants for the tensor-valued isotropic function. The project supported by the Special Funds for Major State Basic Research Project “Nonlinear Science” and the National Basic Research Project “The Several Key Problems of Fluid and Aerodynamics”     

Application of transversely isotropic materials to multi-layer shell elements undergoing finite rotations and large strains

The contribution deals with an extension of a classical Neo–Hookean model for compressible isotropic materials to transverse isotropy. With this enhancement for one preferred material direction there is a possibility to simulate large strains in volume changes of the isotropic basic continuum and supplementary in fiber direction. The integrity basis of polynomial invariants in case of transversely isotropic hyperelasticity consists of three principal invariants of the isotropic basic continuum and additionally of two principal invariants for the preferred material direction. The proposed stored energy function for transverse isotropy contains the classical theory near to the natural state and fulfills the restriction on polyconvexity and coerciveness.By numerical enforcement of the material model into shell kinematics without rotational variables a four-node isoparametric finite element is developed using special concepts to avoid locking. The capability of the algorithms proposed is demonstrated by a numerical example involving large strains as well as finite rotations.     

Initial stresses in elastic solids: Constitutive laws and acoustoelasticity

On the basis of the nonlinear theory of elasticity, the general constitutive equation for an isotropic hyperelastic solid in the presence of initial stress is derived. This derivation involves invariants that couple the deformation with the initial stress and in general, for a compressible material, it requires 10 invariants, reducing to 9 for an incompressible material. Expressions for the Cauchy and nominal stress tensors in a finitely deformed configuration are given along with the elasticity tensor and its specialization to the initially stressed undeformed configuration. The equations governing infinitesimal motions superimposed on a finite deformation are then used to study the combined effects of initial stress and finite deformation on the propagation of homogeneous plane waves in a homogeneously deformed and initially stressed solid of infinite extent. This general framework allows for various different specializations, which make contact with earlier works. In particular, connections with results derived within Biot's classical theory are highlighted. The general results are also specialized to the case of a small initial stress and a small pre-deformation, i.e. to the evaluation of the acoustoelastic effect. Here the formulas derived for the wave speeds cover the case of a second-order elastic solid without initial stress and subject to a uniaxial tension [Hughes and Kelly, Phys. Rev. 92 (1953) 1145] and are consistent with results for an undeformed solid subject to a residual stress [Man and Lu, J. Elasticity 17 (1987) 159]. These formulas provide a basis for acoustic evaluation of the second- and third-order elasticity constants and of the residual stresses. The results are further illustrated in respect of a prototype model of nonlinear elasticity with initial stress, allowing for both finite deformation and nonlinear dependence on the initial stress.     

Hyperelastic dielectrics with saturated polarization2

Variational and invariance principles of modern continuum mechanics are used to establish the field equations, boundary conditions and constitutive relations of a non-linear hyperelastic dielectric with constant magnitude ‘saturated’ polarization. Euclidean invariance places restrictions on the Lagrangian and implies the basic conservation laws. The principles of objectivity and material symmetry restrict the form of the constitutive equations. Four equivalent forms of the free energy functional are listed and for one of these forms the minimal isotropy integrity basis. consisting of eleven invariants, is constructed. The positive definiteness of the energy functional is used to derive various inequalities for the material constants of isotropic dielectrics.     

On thermodynamically consistent constitutive equations for fiber-reinforced nonlinearly viscoelastic solids with application to biomechanics

In this paper, we are interested in developing thermodynamically consistent constitutive equations for fiber-reinforced nonlinearly viscoelastic bodies, in particular for transversely isotropic nonlinearly viscoelastic solids, in isothermal processes. It follows from results in the theory of algebraic invariants that constitutive equations for such materials can be expressed in terms of functions of 18 independent invariants associated with deformation and fiber orientation: 10 of them are isotropic invariants and 8 of them are associated with the deformation and the orientation of the fiber. Among the 8 anisotropic invariants just 6 are related to the viscoelastic response. The terms in the Cauchy stress tensor associated to these 6 invariants are analyzed with respect to thermodynamical consistency, and we obtain restrictions for the corresponding constitutive coefficients. This framework is applied to viscoelastic potentials within the context of biomaterials.     

Hypoelastic form of equations in nonlinear elasticity theory

It is shown that the complete system of equations of elasticity theory for an isotropic medium admits a unique representation in the hypoelastic form (the tensor of the rate of change of stresses is a linear function of the tensor of strain rates with coefficients depending on the invariants of the stress tensor). It is necessary to this end that the hypothesis be satisfied on the determination of strains by stresses which are unknown. Any arbitrariness in the choice of the coefficients of the hypoelastic relation may result in the thermodynamic identity being infringed.     

An iterative data-driven turbulence modeling framework based on Reynolds stress representation

Data-driven turbulence modeling studies have reached such a stage that the basic framework is settled, but several essential issues remain that strongly affect the performance. Two problems are studied in the current research: (1) the processing of the Reynolds stress tensor and (2) the coupling method between the machine learning model and flow solver. For the Reynolds stress processing issue, we perform the theoretical derivation to extend the relevant tensor arguments of Reynolds stress. Then, the tensor representation theorem is employed to give the complete irreducible invariants and integrity basis. An adaptive regularization term is employed to enhance the representation performance. For the coupling issue, an iterative coupling framework with consistent convergence is proposed and then applied to a canonical separated flow. The results have high consistency with the direct numerical simulation true values, which proves the validity of the current approach.     

General invariant representations of the constitutive equations for isotropic nonlinearly elastic materials

This paper develops general invariant representations of the constitutive equations for isotropic nonlinearly elastic materials. Different sets of mutually orthogonal unit tensor bases are constructed from the strain argument tensor by using the representation theorem and corresponding irreducible invariants are defined. Their relations and geometrical interpretations are established in three dimensional principal space. It is shown that the constitutive law linking the stress and strain tensors is revealed to be a simple relationship between two vectors in the principal space. Relative to two different sets of the basis tensors, the constitutive equations are transformed according to the transformation rule of vectors. When a potential function is assumed to exist, the vector associated with the stress tensor is expressed in terms of its gradient with respect to the vector associated with the strain tensor. The Hill’s stability condition is shown to be that the scalar product of the increment of those two vectors must be positive. When potential function exists, it becomes to be that the 3 × 3 constitutive matrix derived from its second order derivative with respect to the vector associated with the strain must be positive definite. By decomposing the second order symmetric tensor space into the direct sum of a coaxial tensor subspace and another one orthogonal to it, the closed form representations for the fourth order tangent operator and its inversion are derived in an extremely simple way.     

A discussion about scale invariants for tensor functions

It is found that in some cases the complete and irreducible scale invariants given by Ref.[1] are not independent. There are some implicit functional relations among them. The scale invariants for two different cases are calculated. The first case is an arbitrary second order tensor. The second case includes a symmetric tensor, an antisymmetric tensor and a vector. By using the eigentensor notation it is proved that in the first case there are only six independent scale invariants rather than seven as reported in Ref.[1] and in the second case there are only nine independent scale invariants which are less than that obtained in Ref.[1].     

Constitutive Equations of an Isotropic Hyperelastic Body

Stress—strain equations for an isotropic hyperelastic body are formulated. It is shown that the strain energy density whose gradient determines stresses can be defined as a function of two rather than three arguments, namely, strain–tensor invariants. In the case of small strains, the equations become relations of Hooke's law with two material constants, namely, shear modulus and bulk modulus.     

Physically motivated invariant formulation for transversely isotropic hyperelasticity

This article discusses an invariant formulation for transversely isotropic hyperelasticity. The work is motivated by the interest of modeling materials such as tendon tissues which may exhibit drastically different characteristics in tensile, shear and volumetric responses. A multiplicative decomposition of the deformation gradient that factors out the dilation and the fiber stretch is proposed. Transversely isotropic strain invariants are constructed on the basis of the multiplicative factors. Within the framework of hyperelasticity theory, these strain invariants generate decoupled stress components in the hydrostatic pressure, the fiber tension and shear terms. An example model is suggested and is assessed against some known features of transversely isotropic solids with strong fibers.     

An Invariant-Based Ogden-Type Model for Incompressible Isotropic Hyperelastic Materials

The Ogden model for an incompressible isotropic hyperelastic material is versatile enough to match complicated data for rubber-like materials at large deformations. However, the tensorial expression for the Cauchy stress in the Ogden model requires determination of the eigenvalues and eigenvectors of the left Cauchy-Green deformation tensor \(\mathbf{B}\). The objective of this paper is to propose an invariant-based Ogden-type model for isotropic incompressible hyperelastic materials. The strain energy function in this new model depends on classical invariants of \(\mathbf{B}\) and the Cauchy stress tensor can be expressed directly in terms of the tensor \(\mathbf{B}\) without need for its spectral form. Examples show that this new Ogden-type model retains the versatility of the original Ogden model in characterizing material response.     

Optimal Orthotropy for Minimum Elastic Energy by the Polar Method

The polar method is a minimal invariant representation in plane elasticity. A plane orthotropic elastic behaviour is expressed by five polar invariants related to the elastic symmetries. In this paper, considering the orthotropy orientation and the polar invariants as optimisation parameters, we discuss the problem of minimising the elastic energy for a given state of stress. The minimisation with respect to the orientation is solved in order to find the associated optimal elastic energy for given polar invariants. Then, this quantity is minimised with respect to the polar invariants which characterise the magnitude of the anisotropic components of the elastic stiffness tensor. Optimal uncoupled composite laminates corresponding to this optimum are presented for membrane and bending loadings.     

Polyconvexity of generalized polynomial-type hyperelastic strain energy functions for near-incompressibility

In this article we investigate several models contained in the literature in the case of near-incompressibility based on invariants in terms of polyconvexity and coerciveness inequality, which are sufficient to guarantee the existence of a solution. These models are due to Rivlin and Saunders, namely the generalized polynomial-type elasticity, and Arruda and Boyce. The extension to near-incompressibility is usually carried out by an additive decomposition of the strain energy into a volume-changing and a volume-preserving part, where the volume-changing part depends on the determinant of the deformation gradient and the volume-preserving part on the invariants of the unimodular right Cauchy–Green tensor. It will be shown that the Arruda–Boyce model satisfies the polyconvexity condition, whereas the polynomial-type elasticity does not. Therefore, we propose a new class of strain-energy functions depending on invariants. Moreover, we focus our attention on the structure of further isotropic strain-energy functions.     

WAVE PROPAGATION IN ELECTROSTRICTIVE MATERIALS UNDER BIASED FIELDS

I. INTRODUCTION Di?erent from piezoelectricity which is a linear coupling between mechanical and electric ?elds andcan only exist in anisotropic materials[1], electrostriction refers to the quadratic dependence of strainor stress on electric ?elds[2,3] …