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A class of constitutive relations for elastic bodies has been proposed recently, where the linearized strain tensor is expressed as a nonlinear function of the stress tensor. Considering this new type of constitutive equation, the initial boundary value problem for such elastic bodies has been expressed only in terms of the stress tensor. In this communication, this new type of nonlinear wave equation is studied for the case of a one-dimensional straight bar. Conditions for the existence of the travelling wave solutions are given and some self-similar solutions are obtained.

To investigate complex physical phenomena, bi-dimensional models are often an interesting option. It allows spatial couplings to be produced while keeping them as simple as possible. For linear physical laws, constitutive equations involve the use of tensor spaces. As a consequence the different types of anisotropy that can be described are encoded in tensor spaces involved in the model. In the present paper, we solve the general problem of computing symmetry classes of constitutive tensors in [Formula: see text] using mathematical tools coming from representation theory. The power of this method is illustrated through the tensor spaces of Mindlin strain-gradient elasticity.

For a new class of elastic bodies, where the linearized strain tensor is given as a function of the Cauchy stress tensor, the problem of considering unsteady motions is studied. A system of partial differential equations that only depends on the stress tensor is found from the equation of motion, which is a system of six partial differential equations for the six components of the stress tensor. A simple boundary value problem is solved for a 1D bar using exact and numerical methods.

A transversely isotropic visco-hyperelastic constitutive model is provided for soft tissues, which accounts for large deformations, high strain rates, and short-term memory effects. In the first part, a constitutive model for quasi-static deformations of soft tissues is presented, in which a soft tissue is simulated as a transversely isotropic hyperelastic material composed of a matrix and reinforcing fibers. The strain energy density function for the soft tissue is additively decomposed into two terms: a neo-Hookean function for the base matrix, and a polyconvex polynomial function of four invariants for the fibers. A comparison with existing experimental data for porcine brain tissues and bovine pericardium shows that this new model can well represent the quasi-static mechanical behavior of soft tissues. In the second part, a viscous potential is proposed to describe the rate-dependent short-term memory effects, resulting in a visco-hyperelastic constitutive model. This model is tested for a range of strain rates from 0.1 /s to 90 /s and for multiple loading scenarios based on available experimental data for porcine and human brain tissues. The model can be applied to other soft tissues by using different values of material and fitting parameters. U.S. Army

Shape memory polymers commonly experience both finite deformations and arbitrary thermomechanical loading conditions in engineering applications. This motivates the development of three-dimensional constitutive models within the finite deformation regime. In the present study, based on the principles of continuum thermodynamics with internal variables, a three-dimensional finite deformation phenomenological constitutive model is proposed taking its basis from the recent model in the small strain regime proposed by Baghani et al. (2012). In the constitutive model derivation, a multiplicative decomposition of the deformation gradient into elastic and inelastic stored parts (in each phase) is adopted. Moreover, employing the mixture rule, the Green–Lagrange strain tensor is related to the rubbery and glassy parts. In the constitutive model, the evolution laws for internal variables are derived during both cooling and heating thermomechanical loadings. Furthermore, we present the time-discrete form of the proposed constitutive model in the implicit form. Using the finite element method, we solve several boundary value problems, that is, tension and compression of bars and a three-dimensional beam made of shape memory polymers, and investigate the model capabilities as well as its numerical counterpart. The model is validated by comparing the predicted results with experimental data reported in the literature that shows a good agreement.

In this paper, a hyperelastic plastic damage-induced anisotropic constitutive model is proposed based on the logarithmic strain in finite deformations. The evolution equations are written in terms of the material time derivative of the stress tensor conjugate to the logarithmic strain. The dissipation inequality is considered in the intermediate configuration with some appropriate internal variables. The Helmholtz free energy is considered as combination of three parts including elastic, plastic and damage. The exponential isotropic plasticity hardening and linear isotropic damage hardening are considered. For taking into account the unilateral effect in the damage growing, a modified stress tensor is considered. A return mapping two-step operator split algorithm, elastic predictor and plastic-damage corrector, is adopted for the integration of the evolution equations. To present some numerical results, the proposed constitutive model has been implemented into a finite element implicit code. For validating the model and also showing its capabilities, several benchmark examples are solved. In these examples, the deterioration in the material properties due to the propagation of the damage is well simulated.

The determination of the equivalent (homogenized) constitutive tensor is one of the most important steps in multiscale models as well as in the classical homogenization theory. In this article, a procedure for determining the homogenized instantaneous (tangent) constitutive tensor of elastic materials containing growing cracks is proposed. The primary purpose of this procedure is its use in two-way coupled multiscale finite element algorithms that can model crack formation and propagation at the local microstructure. The procedure is basically developed by relating the local displacement field to the global strain tensor at each location and using first-order homogenization techniques. The finite element formulation is developed and some example problems are presented in order to verify and demonstrate the model capabilities.

A wave-like dynamic solution of elastic materials is studied. It is shown that if the material is not hyperelastic, the amplitude of the wave can grow exponentially. As a result, the kinetic energy in the material would increase indefinitely even when the applied forces do zero work on the material.

The deformation gradient admits a multiplicative decomposition into a purely volumetric component and a purely distortional component. For a hyperelastic material, based on this decomposition, the elastic strain energy potential, the stress, and the elasticity tensor can be expressed in general as a function of both the volumetric deformation and the distortional deformation. However, the volumetric—distortional decomposition of deformation has often been employed in a fully decoupled form of the elastic strain energy potential, which is expressed as the sum of a term depending solely on the volumetric deformation and a term depending solely on the distortional deformation. This work has three main objectives. First, to derive the elasticity tensor in the general (non-decoupled) case, in its material, spatial, and linear forms; this is achieved by extensive use of fourth-order tensor algebra, and in particular of the properties of the so-called spherical operator, which is largely used, but very seldom given the dignity of being assigned a symbol and a name, in the literature. Second, to show that a fully decoupled potential gives rise to an elasticity tensor which may be inconsistent with its linearized counterpart, as some components of the linear elasticity tensor in general do not have a corresponding term in nonlinear decoupled elasticity tensor. Third, to obtain the conditions under which a purely hydrostatic stress causes a purely volumetric deformation, by means of the developed theory; the results show that this condition is satisfied if and only if the elastic potential is fully decoupled. While the whole approach is completely independent of the material symmetry, the cases of isotropy and transverse isotropy are shown as an example.

Elastic—plastic constitutive equation of hard metal for cold forging tools such as WC-Co cemented carbide with anisotropic damage is proposed to predict a precise service life of cold forging tools. A second rank symmetric damage tensor is introduced in order to express the anisotropic material damage and damage-induced stress unilaterality, namely a salient difference in uniaxial behavior between tension and compression. The conventional framework of irreversible thermodynamics is used to derive the constitutive equation. The Gibbs potential is formulated as a function of stress tensor, damage tensor, isotropic hardening variable, and kinematic hardening variable tensor. The elastic-damage constitutive equation, conjugate forces of damage, isotropic hardening, and kinematic hardening variable is derived from the potential. For the kinematic hardening variable, the superposition of three kinematic hardening laws is employed in order to improve the cyclic behavior of the material. For the evolution equation of the damage tensor, the damage is assumed to progress by fracture of the Co matrix—WC particle interface and by the mechanism of fatigue, i.e., the accumulation of microscopic plastic strain in matrix and particles. By using the constitutive equations, calculation of uniaxial tensile, and compressive test is performed and the results are compared with the experimental ones in the literature. Furthermore, finite element analysis on the behavior of cemented carbide as a die-insert of cold forward extrusion die set was carried out. The proposed constitutive equation was implemented to commercial FE software MSC.Marc2005 using User Subroutines. It is found that the anisotropic damage component can describe the different failure modes of the die-insert, namely fatigue crack and forced rupture.

Rivlin's observation that elastic stress response that does not derive from a strain energy (Cauchy elasticity) admits negative work in closed cycles of deformations is demonstrated for specific isotropic stress responses and for a simple closed cycle of deformation. It is concluded that elastic materials must be hyperelastic.

The MCrAlY bond coats (BCs) used in thermal barrier coatings (TBCs) undergo severe microstructural changes that affect their creep behaviour. One method to take into account the effect of the high-temperature degradation in the creep properties of these alloys is by means of unit cell calculations that reproduce the microstructure present in the alloy. However, this method is not suitable to be included in large-scale calculations and a self-consistent constitutive model, based on Eshelby's inclusion technique, is presented for that purpose, showing good numerical agreement. Results are compared with experimental data obtained for several MCrAlY BCs.

A viscoelastic constitutive model is proposed. The strain energy density function is introduced in the constitutive model that is validated by biaxial tensile experimental data of nylon cord-rubber composite. Results show that the large deformation and viscoelastic constitutive model is effective for cord-rubber composite.

Constitutive equations for the back stress in amorphous glassy polymers based on extended forms of the non-Gaussian James—Guth 3-chain and Arruda—Boyce 8-chain models of rubber elasticity are derived from the extended Wu and van der Giessen non-Gaussian full-network model. A simple and invariant constitutive equation for the back stress tensor is then derived from the Wu and van der Giessen model by an average-stretch approximation. Although the average-stretch model of the full-network amorphous microstructure is more general than other chain cell models, the constitutive equation is the same as the 8-chain back stress relation. Back stress equations for a class of extended phenomenological models, including the Gent material model, are described.

We study material deformations of polycrystals, rocks and other elastic materials that display different types of imperfections with a typical size of 1 μm. Stress-strain relationships of these materials depend on the processing history and exhibit common behavior, including nonlinearity, hysteresis, etc. We focus our study on the continuous distribution of singularities in the strain field, which we describe in terms of surface densities and fluxes. We define the mass mesodensity tensor and deduce the constitutive relationships between the strain singularity current and the linear mesomomentum. Based on the modification of Peach-Koehler formula we consider the constitutive relation between the line mesostress tensor and the strain singularities density. These constitutive relationships allow us to model stresses in mesoelastic materials.

We derive basis-free formulae for the derivative of general isotropic tensor functions of symmetric tensors, and the formulae for elastic moduli and stress rate for compressible isotropic elastic materials. Part I of this paper concerns the tensor functions with the representation of spectral decomposition. The solution is obtained through the directional derivatives on certain subspaces that span the set of symmetric tensors.

The relation is established between the constitutive equations for the stress in an isotropic elastic material based on strain-energy functions that are functions of the right Cauchy-Green strain matrix and the right stretch matrix. It is shown that results to problems obtained on the basis of the latter can be simply obtained from those derived on the basis of the former.

A constitutive equation is proposed to describe the steady-state and dynamic rheological properties of fiber-filled viscoelastic media. The constitutive equation used to describe the viscoelastic effects is an integral equation of the K-BKZ type, suitable for polymer solutions and melts. The contribution to the total stress of the composite due to the presence of fibers is accounted for by the extra stress tensor proposed by Dinh and Armstrong for fiber-filled Newtonian liquids. Steady-state predictions of the constitutive equation are compared with experimental measurements of fiber-filled viscoelastic media, namely polypropylene and nylon-66, for different sized fibers and different volume fractions. The constitutive equation predicts qualitatively the rheological properties of fiber-filled viscoelastic media.

A three-dimensional constitutive model is proposed for anisotropic inelastic deformed continuous medium, and further adapted for unidirectionally fiber-reinforced composites. In the constitutive relation, the material anisotropic elastic, anisotropic yielding and anisotropic hardening behavior are considered. The model accommodates both compressible and incompressible deformation properties. With all six stress and strain components retained in the constitutive relation, the model can be applied to predict three-dimensional nonlinear deformation response of composites under different loadings.

Formulation of constitutive and damage evolution equations of elastic-plastic damage materials are first discussed. In order to facilitate the description of the experimental damage surface in axialtorsional stress space, Gibbs thermodynamic potential is employed. The damage surface is expressed as a function of the damage conjugate force which is easily related to stress tensor. Then, the resulting equations are applied to the experimental results of the spheroidized graphite cast iron. The change in elastic moduli due to damage development and the initial and the subsequent damage surface expressed in stress space are described well by the proposed theory. Finally, experimental verification of the existence of a damage potential and the corresponding normality law is performed. By noting that the normality law must be discussed in the space of damage conjugate, the direction of the damage rate vector on each loading path is identified from the change in the elastic modulus due to damage. The resulting direction of the damage rate at each loading path coincides sufficiently well with the normal of the damage surface, and hence, the proposed damage surface is certified to be identical to the damage potential.

Constitutive equations have been developed to describe the creep behaviour of a 0.5Cr-0.5Mo-0.25V ferritic steel under uniaxial and multi-axial states of stress, as well as over the temperature range of 600–675°C. This material has widespread application as the parent steam pipe in the power generation industry. An accurate description of creep behaviour has been achieved by formulating constitutive equations that describe the physical mechanisms of deformation and rupture. To this end, two state variables were introduced to model intergranular creep cavitation and the coarsening of carbide precipitates respectively. The influence of temperature on the constitutive equations is also considered and activation energies are identified for each of the material constants that show a notable temperature dependence. Numerical optimization methods are discussed for the determination of the material constants and of the corresponding activation energies. The constitutive equations provide a good representation of the experimental creep data. The limitations on the range of validity of the equations are explained with reference to the mechanisms of material deformation and rupture. The description of the creep behaviour is completed by the determination of the constant that defines the multi-axial stress rupture criterion of the material. It is argued that the constitutive equation set is suitable for the analysis of engineering components.

A one-dimensional constitutive model for the thermomechanical behavior of shape memory alloys is developed based on previous work by Liang and Tanaka. An internal variable ap proach is used to derive a comprehensive constitutive law for shape memory alloy materials from first principles without the assumption of constant material functions. This constitutive law is of such a form that it is well suited to further practical engineering applications and calculations. A separation of the martensite fraction internal variable into temperature-induced and stress- induced parts is presented and justified, which then allows the derived constitutive law to accurately represent both the pseudoelastic and shape memory effects at all temperatures. Several numerical ex amples are given that illustrate the ability of the constitutive law to capture the unique thermome chanical behavior of shape memory alloys due to their internal phase transformations with stress and temperature.

A reasonable microstructural model of fabrics is adopted, and an explicit. constitutive equation for such materials is derived. At infinitesimal strains the bulk stress tensor is directly proportional to the area fraction and the Young modulus of the fiber strands. Anisotropy is a direct consequence of the distribution of fiber strands. Two cases are considered: the completely random distribution, and a regular pattern of fiber strands. In the former case the material is not Hookean, in the sense that normal stresses are generated during a shear deformation; in the latter case material properties are strongly dependent on the pattern distribution itself.

When a composite body consisting of two isotropic and elastic phases is subjected to prescribed surface tractions, generally the stress field depends on three parameters involving the elastic con stants. If the composite is in a state of plane deformation, and there are no net forces on internal boundaries, however, the stress depends on only two combinations of elastic constants. Convenient constants for the composite material are proposed.

In this paper a material region is considered to be an aggregate of point particles which move in accordance to Newton's Laws and inter act by means of forces that depend only on the relative distance between the particles and are derivable from a potential U. The follow ing propositions are, then, established: the continuum field equations are obtained by a smoothing assumption that allows the representation of sums by integrals; the classical constitutive equations of elastic materials are obtained when forces between particles are of infini tesimally short range; when the range of interaction between particles is of short but finite range, higher order stresses and strains are neces sary to describe the mechanical state of the material.