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A finite strain constitutive model to predict the deformation behaviour of orthotropic metals is developed in this paper. The important features of this constitutive model are the multiplicative decomposition of the deformation gradient and a new Mandel stress tensor combined with the new stress tensor decomposition generalized into deviatoric and spherical parts. The elastic free energy function and the yield function are defined within an invariant theory by means of the structural tensors. The Hill’s yield criterion is adopted to characterize plastic orthotropy, and the thermally micromechanical-based model, Mechanical Threshold Model (MTS) is used as a referential curve to control the yield surface expansion using an isotropic plastic hardening assumption. The model complexity is further extended by coupling the formulation with the shock equation of state (EOS). The proposed formulation is integrated in the isoclinic configuration and allows for a unique treatment for elastic and plastic anisotropy. The effects of elastic anisotropy are taken into account through the stress tensor decomposition and plastic anisotropy through yield surface defined in the generalized deviatoric plane perpendicular to the generalized pressure. The proposed formulation of this work is implemented into the Lawrence Livermore National Laboratory-DYNA3D code by the modification of several subroutines in the code. The capability of the new constitutive model to capture strain rate and temperature sensitivity is then validated. The final part of this process is a comparison of the results generated by the proposed constitutive model against the available experimental data from both the Plate Impact test and Taylor Cylinder Impact test. A good agreement between experimental and simulation is obtained in each test.

This paper is concerned with the effect of residual stress on the elastic behaviour of materials undergoing finite elastic deformations. The theory is based on a general constitutive framework for hyperelastic materials with residual stress. Several simple problems, whose solutions are known in the situation where there is no residual stress, are analyzed in order to elucidate the influence of residual stress, and the results are illustrated for two prototype constitutive laws.

In this short note we articulate the need for a new approach to develop constitutive models for the non-linear response of materials wherein one is interested in describing the Cauchy–Green stretch as a non-linear function of the Cauchy stress, with the relationship not in general being invertible. Such a material is neither Cauchy nor Green elastic. The new class of materials has several advantages over classical elastic bodies. When linearized under the assumption that the displacement gradient be small, the classical theory leads unerringly to the classical linearized model for elastic response, while the current theory would allow for the possibility that the linearized strain be a non-linear function of the stress. Such bodies also exhibit a very desirable property when viewed within the context of constraints. One does not need to introduce a Lagrange multiplier as is usually done in the classical approach to incompressibility and the models are also more suitable when considering nearly incompressible materials. The class of materials considered in this paper belongs to a new class of implicit elastic bodies introduced by Rajagopal [19] , [20] . We show how such a model can be used to interpret the data for an experiment on rubber by Penn [18] .

There are two approaches that can be used to model the large strain mechanical response of material systems in which elastic fibers are embedded in an elastic matrix. In the first approach, a fiber reinforced material undergoing large deformation is homogenized in the sense that it is assumed to act as an equivalent single material that is transversely isotropic and hyperelastic. Both constituents then share a common reference configuration, which is typically assumed to be a natural or stress-free configuration for the equivalent single material. The stress depends on a single deformation gradient defined with respect to the natural configuration. In the second approach, the fiber/matrix system is treated as a mixture, with the matrix and the fibrous constituents having their own reference configurations and material symmetries. The total stress depends on the deformation gradients and material symmetries for both constituents, defined with respect to their reference configurations. Under appropriate assumptions, the constitutive theory developed using mixture theory can coincide with the constitutive theory assuming an equivalent single material that is transversely isotropic and hyperelastic. This paper explores the connection between the two approaches by considering the various reference configurations and material symmetries.

A constitutive framework for electro-sensitive materials in the context of non-linear elasticity is analyzed. Constitutive equations are given in terms of energy functions that depend on several invariants. The study includes both the analysis of the invariants, which are present in the energy functions, and the analysis of constitutive restrictions that have to be obeyed by the constitutive functions. Isotropic as well as non-isotropic electro-sensitive elastomers are studied. The set of invariants that describe each material model is analyzed under two homogeneous deformations: (i) an uniaxial elongation and (ii) a simple shear deformation. These deformations are chosen since they are relevant to specific experiments, from which one may try to fit constitutive equations. The constitutive restrictions developed are based on classical ones used for isotropic non-linear elastic materials, in particular, are based on the Baker–Ericksen inequality and the ellipticity condition.

The paper presents a procedure for the formulation of constitutive equations for rate-independent pseudoelastic SMA material models. The procedure applies a rheological scheme representing mechanical properties of the material. An additive decomposition of strains into two parts is proposed. The first part describes strains of a perfectly elastic body while the second part may be represented by a combination of a rigid perfectly elastic body and a rigid perfectly plastic body. It is demonstrated that the key problem of formulation of constitutive relationships is to derive the 1st order differential equation with respect to the tensor describing the second part of the strain field. This equation may be obtained in explicit form starting from the variational inequalities defining non-elastic parts of rheological model. The uniqueness of the obtained differential equation has been proved. A numerical implementation of the constitutive relationships of SMA material was done through the user subroutine module VUMAT within the FE commercial code ABAQUS/Explicit. As an example we analyzed the problem of vibration of a simple 3D structure made of SMA.

This paper describes experiments and the development of constitutive equations to predict the steady-state response of filled rubber under cyclic loading. An MTS servo-hydraulic machine was used to obtain the dynamic hysteresis curves for a filled rubber compound in uniaxial tension-compression. The material tests were performed with mean strains from −0.1 to 0.1, strain amplitudes ranging from 0.02 to 0.1, and strain rates between 0.01 and 10 s−1. Temporary material set, the Payne effect and rate-dependence were observed from the experimental results. A hyper-viscoelastic constitutive model was developed to characterize the dynamic response of the rubber. A cornerstone of this constitutive modeling was to devise a scheme to evaluate material set and a finite strain, non-linear viscoelastic law from the test data. Predictions of the dynamic hysteresis curves using the proposed constitutive equation were found to be in good agreement with the uniaxial test results.

Some simple boundary value problems are studied, for a new class of elastic materials, wherein deformations are expressed as non-linear functions of the stresses. Problems involving ‘homogeneous’ stress distributions and one-dimensional stress distributions are considered. For such problems, deformations are calculated corresponding to the assumed stress distributions. In some of the situations, it is found that non-unique solutions are possible. Interestingly, non-monotonic response of the deformation is possible corresponding to monotonic increase in loading. For a subclass of models, the strain–stress relationship leads to a pronounced strain-gradient concentration domain in the body in that the strains increase tremendously with the stress for small range of the stress (or put differently, the gradient of the strain with respect to the stress is very large in a narrow domain), and they remain practically constant as the stress increases further. Most importantly, we find that for a large subclass of the models considered, the strain remains bounded as the stresses become arbitrarily large, an impossibility in the case of the classical linearized elastic model. This last result has relevance to important problems in which singularities in stresses develop, such as fracture mechanics and other problems involving the application of concentrated loads.

In this paper we discuss the computational implementation of a new constitutive model that describes the muscle properties in a soft-bodied arthropod. Qualitatively, the muscle tissues behave similar to particle-reinforced rubber and are capable of large non-linear elastic deformations, show a hysteretic behavior, and display stress softening during the first few cycles of repeated loading. Such behavior can be described by the framework of pseudo-elastic transversely isotropic hyperelasticity. The computational model assumes compressible overall response, and is based upon a multiplicative split of the deformation gradient tensor into volumetric and isochoric parts. Details regarding the implementation of the computational model in the context of an implicit finite element solution procedure are presented. In particular, an explicit expression is provided for the material tangent stiffness tensor. Results obtained utilizing the new implementation are also presented.

In the light of recent and growing interest in the applications of magneto-sensitive elastomers and the corresponding theoretical analysis of their properties, this paper is devoted to the derivation of universal relations for these materials, that is connections between the components of a stress tensor and the components of the magnetic (induction) field vector that hold independently of the choice of constitutive law within a considered class of such laws. Here, attention is focussed on isotropic magnetoelastic materials. In particular, within this framework, it is shown that in general there is only one possible universal relation for these materials, but for particular classes of constitutive laws or for special deformations there can be more than one. The theory is exemplified by application to the problem of homogeneous triaxial deformation combined with a simple shear.

Kinematic hardening models describe a specific kind of plastic anisotropy which evolves with the deformation process. It is well known that the extension of constitutive relations from small to finite deformations is not unique. This applies also to well-established kinematic hardening rules like that of Armstrong–Frederick or Chaboche. However, the second law of thermodynamics offers some possibilities for generalizing constitutive equations so that this ambiguity may, in some extent, be moderated. The present paper is concerned with three possible extensions, from small to finite deformations, of the Armstrong–Frederick rule, which are derived as sufficient conditions for the validity of the second law. All three models rely upon the multiplicative decomposition of the deformation gradient tensor into elastic and plastic parts and make use of a yield function expressed in terms of the so-called Mandel stress tensor. In conformity with this approach, the back-stress tensor is defined to be of Mandel stress type as well. In order to compare the properties of the three models, predicted responses for processes with homogeneous and inhomogeneous deformations are discussed. To this end, the models are implemented in a finite element code (ABAQUS).

We use four linear constitutive relations to study "nite deformations of a biaxially loaded elastic membrane, triaxially loaded cube, and simple extensional and simple shearing deformations of an elastic body. In each case, the body is assumed to be isotropic and homogeneous. It is shown that only the neoHookean relation (a linear relation between the Cauchy stress tensor and the left Cauchy}Green tensor) and the Signorini’s relation (a linear relationship between the Cauchy stress tensor and the Almansi}Hamel strain tensor) predict load}deformation curves that qualitatively agree with most of the test observations. A similar conclusion holds when the body is assumed to be incompressible. ( 2000 Elsevier Science Ltd. All rights reserved.

The standard formulation of a constitutive theory for isotropic, hyperelastic solids involves the prescription of the Cauchy stress as a function of the left Cauchy-Green strain field through an elastic potential. For certain applications, however, it is desirable to express a constitutive law by giving the left strain field as a function of Cauchy stress. In this paper, conditions on the form of such an inverted constitutive law are given that ensure the hyperelasticity of the material. Both compressible and incompressible materials are treated, and in the incompressible case a general representation for a strain-stress relation is given in terms of a generating scalar potential.

Expressions are developed for the strain energy and stress resultants in the case of large strain-large bending cylindrical deformations of shells of non-linear materials. The deformation variables are the extension and curvature of the material reference surface. The Mooney material is used as an example where appropriate. The influence of the large deformations on the constitutive laws is evaluated. Of special interest is the effect of cross terms and of the ability to express the stress resultants as partial derivatives of the strain energy with respect to appropriate deformation variables.

An approximate stress-strain relation is derived for a granular material composed of spherical elastic granules in contact. The material is assumed to be statistically homogeneous so that the effective stress tensor can be obtained by a volume average. The resulting stress-strain relation is markedly non-linear and begins with the term ∥e∥ 3 2 , where e is the classical infinitesimal strain tensor. Some simple deformation fields are worked out.

From the viewpoint of irreversible thermodynamics an admissible form of rate-type constitutive equation of inelastic materials is given. The displacement gradient tensor F referred to the temporarily fixed reference frame which coincides with the Euler frame at the instant of the reference time is decomposed linearly into elastic and inelastic parts so that the procedure of formulation is simplified and clarified. The inelastic deformation rate is directly related to the internal production rate of entropy. The existence of an inelastic potential of the usual sense is not assumed, though the result can be understood to include the conventional flow theory based on an inelastic potential. An example of an elastoviscoplastic constitutive equation is given and some properties of yield surfaces are discussed.

Assuming the free energy function to depend on the deformation gradient tensor, the polarization vector, the polarization gradient tensor and the temperature, material and spatial forms of the constitutive equations are obtained which involve the invariants constituting the minimal isotropy integrity basis for the free energy function. A special form of stored energy function, involving material descriptors, is used for anisotropic dielectrics. Upon suitable linearization. the results obtained here reduce to the constitutive equations for Voigt's piezoelectricity theory and Mindlin's theory for elastic dielectrics with polarization gradient.

The non-linear elastic constitutive equations proposed by Kauderer, [1], are a particular case of a class of quasi-linear equations. They cannot give an account of some specific non-linear effects of the Poynting and the Kelvin types. It is more useful in two and three dimensional problems to start with general non-linear forms proposed initially by Brillouin, [2].

The non-linear stress relation is derived from the general theory of incompatible deformations of elastic media. It is assumed that the thermal and elastic deformations do not satisfy the compatibility conditions, that the gradients of the total deformation are a composition of the thermal and elastic distortions, that the total deformation satisfies the compatibility conditions and that the strain energy is a function only of the elastic distortions. For the known distribution of temperature in a body and for the known thermal distorsions it is possible to eliminate the elastic distortions from the stress relation and the components of the total deformation tensor are the only unknown quantities in the stress relation. For the isotropic media the strain energy is expanded into a power series in terms of the principal invariants of the total strain tensor. The components of the stress tensor are calculated in the linear and in the second approximation.

Behavior of a Tresca type plastic dielectric is investigated theoretically from a continuum mechanical point of view. The optical constitutive equations are defined as special cases of a hypo-elastic dielectric of grade two. The singularity condition of the constitutive equations satisfies the Tresca yield criterion. The index deviator tensor is proportional to the stress deviator tensor and, then, the birefringence and the extinction angle are expressed by the stress deviator. Their numerical variations with the angle of shear in simple shear deformation are shown.