Search

The theory of diffusive stresses based on the time-fractional diffusion equation with the Liouville–Caputo derivative is discussed. The diffusion process is characterized by the chemical potential tensor and the concentration tensor. Eliminating the chemical potential tensor and the concentration tensor from the constitutive equations for the stress tensor, the space–time-nonlocal equations for the mean stress and the deviatoric stress are obtained with the kernel being the Wright function.

In this paper the state of stress and strain close to sharp cracks in bodies subjected to an anti-plane state of stress is studied within the context of a non-monotonic strain-stress relation within the context of a generalization of the Cauchy theory of elasticity, providing an exact analytical solution to the problem. A discussion is provided to highlight the main features of stress and strain distributions, and the implications of the new theory for fracture assessments. In particular, it is proved that the intensity of the complete stress field can be expressed as a function of the Stress Intensity Factor KIII, as in the case of conventional linearized elasticity theory, thus promoting a K based-approach to the fracture of elastic solids obeying a constitutive relation wherein the linearized strain is expressed as a non-linear function of the Cauchy stress.

Within the context of the non-linear theory of Cauchy elastic bodies (hence Green elastic bodies which are a sub-set of Cauchy elastic bodies wherein the stress is derivable from a potential), linearization with regard to the gradient of displacement, in the sense that the squares of the norms of the gradient of displacement can be neglected in comparison tothe norm of the gradient of displacement, leads inexorably to the classical linearized elastic model. It is however common, especially in work related to inelastic bodies, to see expressions for the Cauchy stress as a nonlinear function of the linearized strain. Even though such models are outside the purview of purely elastic response, the nonlinear relationship between the stress and the linearized strain is also often assumed to hold in the elastic range also. While the linearized strain being a nonlinear function of the stress has no basis within the context of the classical Cauchy elasticity theory, we show that a proper justification can be provided for such models within the context of the new class of constitutive relations that have been developed to describe the response of elastic bodies by Rajagopal [19], and these models can be generalized to also describe the inelastic response in the small strain regime.

© 2017, Springer-Verlag GmbH Austria. We study the heat conduction with a general form of a constitutive equation containing fractional derivatives of real and complex order. Using the entropy inequality in a weak form, we derive sufficient conditions on the coefficients of a constitutive equation that guarantee that the second law of thermodynamics is satisfied. This equation, in special cases, reduces to known ones. Moreover, we present a solution of a temperature distribution problem in a semi-infinite rod with the proposed constitutive equation.

Explicit thermodynamically consistent constitutive equations are employed.Domain is discretized with serendipity polygonal elements.Lagrange type higher order shape functions are constructed based on pairwise products of barycentric coordinates.A new one point integration scheme is proposed to compute the smoothed (corrected) derivatives.The numerical results with new constitutive equations show stress softening behavior even in small strain regime. Orthopaedic implants made of titanium alloy such as Ti-30Nb-10Ta-5Zr (TNTZ-30) are biocompatible and exhibit nonlinear elastic behavior in the small strain regime (Hao et al., 2005). Conventional material modeling approach based on Cauchy or Green elasticity, upon linearization of the strain, inexorably leads to Hookes law which is incapable of describing the said nonlinear response. Recently, Rajagopal introduced a generalization of the theory of elastic materials (Rajagopal, 2003, 2014), wherein the linearized strain can be expressed as a nonlinear function of stress. Consequently, Devendiran et al. (2016) developed a thermodynamically consistent constitutive equation for the generalized elastic solid, in order to capture the response of materials showing nonlinear behavior in the small strain regime. In this paper, we study the response of a long cylinder made of TNTZ-30 with non-circular cross section subjected to end torsion. An explicit form of the constitutive equation derived in Devendiran et al. (2016) is used to study the response of the cylinder. The cross-section is discretized with quadratic serendipity polygonal elements. A novel one point integration rule is presented to compute the corrected derivatives, which are then used to compute the terms in the stiffness matrix. Unlike the conventional Hookes law, the results computed using the new constitutive equation show stress softening behavior even in the small strain regime.

There are many alloys used in orthopaedic applications that are nonlinear in the elastic regime even when the strains are ‘small’ (see Hao et al., 2005; Saito et al., 2003; Sakaguch et al., 2004). By using conventional theories of elasticity, either Cauchy or Green elasticity, it is impossible to systematically arrive at constitutive equations, which would be applicable in the elastic domain of such metals as such materials exhibit non-linear response for small strains 1 where the classical linearized response is supposed to hold in the sense that the norm of the squares of the displacement gradient are much smaller than the displacement gradient. We delineate a new framework for developing constitutive equations for a new class of elastic materials, termed as implicit elastic materials, which can be used to describe the response of such alloys. In addition to a fully implicit constitutive relation, we discuss a non-linear constitutive relation between the linearized strain and the stress that can be properly justified to describe the response of such alloys. By using the example of a rectangular plate with a hole subject to uniform loading, a classical problem, we illustrate the differences in the stress and strain fields when compared to that predicted by the classical linearized relation.

In the finite deformation range, the numbers of orthotropic 2n order elastic constants are studied on the basis of tensor function and of its representation theorem. On the basis of elastic constant research, the elastic orthotropic constitutive equation is derived by using the tensor method. Based on orthotropic elastic constitutive equations an in-depth study on the constitutive theory of orthotropic nonlinear thermal elasticity is carried out, and by considering the deformation produced by the coupling of temperature and load, nonlinear orthotropic thermoelastic constitutive equation is further derived with representation of the tensor invariant and scalar invariant. The constitutive equations could be used very convenient to the application in reality.

The paper puts forwards principal kinematic relations and constitutive equations, which can be applied in designing numerical methods of study of finite elasto-plastic strains. The medium kinematics is considered under the multiplicative decomposition of the total deformation gradient. The constitutive equations are deduced using the theory of flow and the second law of thermodynamics. As a result, we find the dependence of the stress tensor rate on the free energy function and on the yield function.

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.

Some new classes of constitutive relations for elastic bodies have been proposed in the literature, wherein the stresses and strains are obtained from implicit constitutive relations. A special case of the above relations corresponds to a class of constitutive equations where the linearized strain tensor is given as a nonlinear function of the stresses. For such constitutive equations we consider the problem of decomposing the stresses into two parts: one corresponds to a time-independent solution of the boundary value problem, plus a small (in comparison with the above) time-dependent stress tensor. The effect of this initial time-independent stress in the propagation of a small wave motion is studied for an infinite medium.

A simple method, called anisotropic transformed stress (ATS) method, is proposed to develop failure criteria and constitutive models for anisotropic soils. In this method, stress components in different directions are modified differently in order to reflect the effect of anisotropy. It includes two steps of mapping of stress. First, a modified stress tensor is introduced which is a symmetric multiplication of stress tensor and fabric tensor. In the modified stress space, anisotropic soils can be treated to be isotropic. Second, a transformed stress tensor is derived from the modified stress tensor for the convenience of developing anisotropic constitutive models to account for the effect of intermediate principal stress. By replacing the ordinary stress tensor with the transformed stress tensor directly, the Unified Hardening (UH) model is extended to model the anisotropic deformation of soils. Anisotropic Lade’s criterion is adopted for shear yield and shear failure in the model. The form of the original model formulations remain unchanged and the model parameters are independent of the loading direction. Good agreement between the experimental results and predictions of the anisotropic UH model is observed.

One of the basic assumptions of the micropolar theory is that the stress tensor is not symmetric. In this paper, asymmetry of the stress tensor is studied with discrete element method and averaging method. The change of the skew symmetric part of an asymmetric tensor with the rotation of the coordinate system is shown graphically. Averaging method is used to obtain stress tensor from a DEM simulation of biaxial test. Stress asymmetries at different locations, scales and time steps are studied. The importance of the asymmetric stress for setting up a constitutive model for granular materials is discussed.

This paper presents a Gibbs potential-based granular micromechanics approach capable of modeling materialswith complete anisotropy. The deformation energy of each grain–pair interaction is taken as a function of the inter-granular forces. The overall classical Gibbs potential of a material point is then defined as the volume average of the grain–pair deformation energy. As a first-order theory, the inter-granular forces are related to the Cauchy stress tensor using a modified static constraint that incorporates directional distribution of the grain–pair interactions. Further considering the conjugate relationship of the macroscale strain tensor and the Cauchy stress, a relationship between inter-granular displacement and the strain tensor is derived. To establish the constitutive relation, the inter-granular stiffness coefficients are introduced considering the conjugate relation of inter-granular displacement and forces. Notably, the inter-granular stiffness introduced in this manner is by definition different from that of the isolated grain–pair interactive. The integral form of the constitutive relation is then obtained by defining two directional density distribution functions; one related to the average grain–scale combined mechanical–geometrical properties and the other related to purely geometrical properties. Finally, as the main contribution of this paper, the distribution density function is parameterized using spherical harmonics expansion with carefully selected terms that has the capability of modeling completely anisotropic (triclinic) materials. By systematic modification of this distribution function, different elastic symmetries ranging from isotropic to completely anisotropic (triclinic) materials are modeled. As a comparison, we discuss the results of the present method with those obtained using a kinematic assumption for the case of isotropy and transverse isotropy, wherein it is found that the velocity of surface quasi-shear waves can show different trends for the two methods.

In recent papers, Surana et al. presented internal polar non-classical Continuum theory in which velocity gradient tensor in its entirety was incorporated in the conservation and balance laws. Thus, this theory incorporated symmetric part of the velocity gradient tensor (as done in classical theories) as well as skew symmetric part representing varying internal rotation rates between material points which when resisted by deforming continua result in dissipation (and/or storage) of mechanical work. This physics referred as internal polar physics is neglected in classical continuum theories but can be quite significant for some materials. In another recent paper Surana et al. presented ordered rate constitutive theories for internal polar non-classical fluent continua without memory derived using deviatoric Cauchy stress tensor and conjugate strain rate tensors of up to orders n and Cauchy moment tensor and its conjugate symmetric part of the first convected derivative of the rotation gradient tensor. In this constitutive theory higher order convected derivatives of the symmetric part of the rotation gradient tensor are assumed not to contribute to dissipation. Secondly, the skew symmetric part of the velocity gradient tensor is used as rotation rates to determine rate of rotation gradient tensor. This is an approximation to true convected time derivatives of the rotation gradient tensor. The resulting constitutive theory: (1) is incomplete as it neglects the second and higher order convected time derivatives of the symmetric part of the rotation gradient tensor; (2) first convected derivative of the symmetric part of the rotation gradient tensor as used by Surana et al. is only approximate; (3) has inconsistent treatment of dissipation due to Cauchy moment tensor when compared with the dissipation mechanism due to deviatoric part of symmetric Cauchy stress tensor in which convected time derivatives of up to order n are considered in the theory. The purpose of this paper is to present ordered rate constitutive theories for deviatoric Cauchy strain tensor, moment tensor and heat vector for thermofluids without memory in which convected time derivatives of strain tensors up to order n are conjugate with the Cauchy stress tensor and the convected time derivatives of the symmetric part of the rotation gradient tensor up to orders 1n are conjugate with the moment tensor. Conservation and balance laws are used to determine the choice of dependent variables in the constitutive theories: Helmholtz free energy density Φ, entropy density η, Cauchy stress tensor, moment tensor and heat vector. Stress tensor is decomposed into symmetric and skew symmetric parts and the symmetric part of the stress tensor and the moment tensor are further decomposed into equilibrium and deviatoric tensors. It is established through conjugate pairs in entropy inequality that the constitutive theories only need to be derived for symmetric stress tensor, moment tensor and heat vector. Density in the current configuration, convected time derivatives of the strain tensor up to order n, convected time derivatives of the symmetric part of the rotation gradient tensor up to orders 1n, temperature gradient tensor and temperature are considered as argument tensors of all dependent variables in the constitutive theories based on entropy inequality and principle of equipresence. The constitutive theories are derived in contravariant and covariant bases as well as using Jaumann rates. The nth and 1nth order rate constitutive theories for internal polar non-classical thermofluids without memory are specialized for n = 1 and 1n = 1 to demonstrate fundamental differences in the constitutive theories presented here and those used presently for classical thermofluids without memory and those published by Surana et al. for internal polar non-classical incompressible thermofluids.

In this paper, new boundary-domain integral equations are derived for solving two- and three-dimensional elastoplastic problems. In the derived formulations, domain integrals associated with initial stresses (strains) are avoided to use, and material nonlinearities are implicitly embodied in the integrand kernels associated with the constitutive tensor. As a result, only displacements and tractions are explicitly involved in the ultimate integral equations which are easily solved by employing a mature efficient non-linear equation solver. When materials yield in response to applied forces, the constitutive tensor (slope of the stress–strain curve for a uniaxial stress state) becomes discontinuous between the elastic and plastic states, and the effect of this non-homogeneity of constitutive tensor is embodied by an additional interface integral appearing in the integral equations which include the differences of elastic and plastic constitutive tensors. The domain is discretized into internal cells to evaluate the resulted domain integrals. An incremental variable stiffness iterative algorithm is developed for solving the system of equations. Numerical examples are given to verify the correctness of the proposed BEM formulations.

In this work, we study the transformation properties of the local form of the material (referential) balance of energy equation under the superposition of arbitrary material diffeomorphisms. For this purpose, the tensor analysis on manifolds is utilized. We show that the material balance of energy equation, in general, cannot be invariant; in fact an extra term appears in the transformed balance of energy equation, which is directly related to the work performed by the configurational stresses. By making the fundamental assumption that the body and the ambient space manifolds are always related in the course of deformation and by utilizing the metric concept, we determine this extra term. Building on this, we derive several constitutive equations for the material stress tensor. The compatibility of these constitutive equations with the second law of thermodynamics is evaluated. Finally, we postulate that the material balance of energy equation is covariant, and we study this case in detail, as well.

Recently, there has been interest in the study of a new class of constitutive relation, wherein the linearized strain tensor is assumed to be a function of the stresses. In this communication, some boundary value problems are solved using the finite element method and the solid material being described by such a constitutive relation, where the stresses can be arbitrarily ‘large’, but strains remain small. Three problems are analyzed, namely the traction of a plate with hyperbolic boundaries, a plate with a point load, and the traction of a plate with an elliptic hole. The results for the stresses and strains are compared with the predictions that are obtained by using the constitutive equation of the classical linearized theory of elasticity.

Shear stress growth curves for viscoelastic fluids at low shear rates are analyzed using two linear rheological constitutive equations, an integral constitutive equation and a mixed type constitutive equation. It is shown that some published solutions do not satisfy all of the pertinent boundary conditions. For the low shear rate region, available experimental shear stress curves show a monotonic increase with decreasing slope in the shear stress. Shear stress curves calculated using a mixed type constitutive equation are found to exhibit this type of behavior while curves calculated using an integral constitutive equation do not. For the mixed type constitutive equation, the calculated developing velocity distribution is used to examine its effect on the developing shear stress distribution. For low values of E (the elasticity number), there is a moderate effect, but, for sufficiently large values of E, the developing velocity distribution has a negligible effect. It is also shown that results consistent...

A general solution method to the Cauchy Problem (CP) formulated for incremental elastoplasticity is designed. The method extends previous works of the authors on the solution to Cauchy Problems for linear operators and convex nonlinear elasticity in small strain to the case of generalised standard materials defined by two convex potentials. The CP is transformed into the minimisation of an error between the solutions to two well-posed elastoplastic evolution problems. A one-parameter family of errors in the constitutive equation is derived based on Legendre–Fenchel residuals. The method is illustrated by the simple example of a pressurised thick-spherical reservoir made of elastic, linear strain-hardening plastic material. The identification of inner pressure and plasticity evolution has been carried-out using semi-analytical solutions to the elastoplastic behaviours to build the error functional.

Recently, there has been an interest in the development of implicit constitutive relations between the stress and the deformation gradient, to describe the response of elastic bodies as such constitutive relations are capable of describing physically observed phenomena, in which classical models within the construct of Cauchy elasticity are unable to explain. In this paper, we study the consequences of the constraint of incompressibility in a subclass of such implicit constitutive relations.

Implicit constitutive relations afford a logically consistent framework for formulating strain-limiting, nonlinear elastic constitutive models utilizing the classical linearized strain tensor. This is in marked contrast to traditional Cauchy or Green elastic formulations which give rise to customary linear elasticity in the infinitesimal strain limit. Within this strain-limiting elastic constitutive setting, the present paper investigates the asymptotic behavior of the stress field at the tip of a straight plane-strain fracture. It is shown that within the general class of crack-tip asymptotic expansions considered, the only cases satisfying the required boundary conditions correspond to bounded stresses. This is in agreement with previous results for the corresponding anti-plane shear fracture problem.

The mechanical response of a homogeneous isotropic\ud linearly elastic material can be fully characterized by\ud two physical constants, the Young’s modulus and the\ud Poisson’s ratio, which can be derived by simple tensile\ud experiments. Any other linear elastic parameter can\ud be obtained from these two constants. By contrast, the\ud physical responses of nonlinear elastic materials are\ud generally described by parameters which are scalar\ud functions of the deformation, and their particular\ud choice is not always clear. Here, we review in a unified\ud theoretical framework several nonlinear constitutive\ud parameters, including the stretch modulus, the shear\ud modulus, and the Poisson function, that are defined\ud for homogeneous isotropic hyperelastic materials and\ud are measurable under axial or shear experimental\ud tests. These parameters represent changes in the\ud material properties as the deformation progresses,\ud and can be identified with their linear equivalent\ud when the deformations are small. Universal relations\ud between certain of these parameters are further\ud established, and then used to quantify nonlinear\ud elastic responses in several hyperelastic models for\ud rubber, soft tissue, and foams. The general parameters\ud identified here can also be viewed as a flexible basis\ud for coupling elastic responses in multi-scale processes,\ud where an open challenge is the transfer of meaningful\ud information between scales.

A methodology for obtaining implicit constitutive representations involving the Cauchy stress and the Hencky strain for isotropic materials undergoing a non-dissipative process is developed. Using this methodology, a general constitutive representation for a subclass of implicit models relating the Cauchy stress and the Hencky strain is obtained for an isotropic material with no internal constraints. It is shown that even for this subclass, unlike classical Green elasticity, one has to specify three potentials to relate the Cauchy stress and the Hencky strain. Then, a procedure to obtain implicit constitutive representations for isotropic materials with internal constraints is presented. As an illustration, it is shown that for incompressible materials the Cauchy stress and the Hencky strain could be related through a single potential. Finally, constitutive approximations are obtained when the displacement gradient is small.

International audience; Soft tissues are mainly composed of organised biological media giving them an anisotropic mechanical behavior. Soft tissues also have the ability to undergo large elastic reversible deformations. Many constitutive models were developed to describe these phenomena. In this chapter, we discuss several varying models and their constitutive equations which are defined by means of strain components or strain invariants. The notion of tangent moduli will be plotted for two well-known constitutive equations, and, we will illustrate how to implement explicitly a structural kinematics constraint in a constitutive law to derive the resulting Cauchy stress tensor.

The polyurethane, which was the subject of the constitutive research presented in the paper, was based on oligocarbonate diols Desmophen C2100 produced by Bayer®. The constitutive modelling was performed with a view to applying the material as the inlay of intervertebral disc prostheses. The polyurethane was assumed to be non-linearly viscohyperelastic, isotropic and incompressible. The constitutive equation was derived from the postulated strain energy function. The elastic and rheological constants were identified on the basis of experimental tests, i.e. relaxation tests and monotonic uniaxial tests at two different strain rates, i.e.\(\dot \lambda = 0.1\) min−1 and \(\dot \lambda = 1.0\) min−1. The stiffness tensor was derived and introduced to Abaqus® finite element (FE) software in order to numerically validate the constitutive model. The results of the constants identification and numerical implementation show that the derived constitutive equation is fully adequate to model stress-strain behavior of the polyurethane material.

Evaluation of the tangent modulus is one of the crucial steps in the finite element implementation of a constitutive model. The analytical derivation of the fourth order tangent moduli is in general a tedious task as it involves the derivative of stress tensor with respective to an appropriate strain tensor. A constitutive model independent subroutine which numerically evaluates tangent modulus in the place of material model specific closed form analytical expressions is developed in this study. In this context, the concept of numerically approximated Cauchy integral (NACI) is introduced based on Cauchy integral formula for evaluating derivatives. The performance of this method is demonstrated by evaluating tangent moduli for five hyperelastic models. In addition, efficacy of NACI is compared to the other existing numerical methods generally employed to evaluate tangent moduli. NACI is found to be computationally efficient and numerically robust when compared to the existing procedures.

We consider Cauchy's equation of motion for hyperelastic materials. The solution of this nonlinear initial-boundary value problem is the vector field which discribes the displacement which a particle of this material perceives when exposed to stress and external forces. This equation is of greatest relevance when investigating the behaviour of elastic, anisotropic composites and for the detection of defects in such materials from boundary measurements. This is why results on unique solvability and continuous dependence from the initial values are of large interest in materials research and structural health monitoring. In this article we present such a result, provided that reasonable smoothness assumptions for the displacement field and the boundary of the domain are satisfied, for a certain class of hyperelastic materials where the first Piola-Kirchhoff tensor is written as a conic combination of finitely many given tensors., Comment: 33 pages, no figures

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.

Growth (resp. atrophy) describes the physical processes by which a material of solid body increases (resp. decreases) its size by addition (resp. removal) of mass. In the present contribution, we propose a sound mathematical analysis of growth, relying on the decomposition of the geometric deformation tensor into the product of a growth tensor describing the local addition of material and an elastic tensor, which is characterizing the reorganization of the body. The Blatz-Co hyperelastic constitutive model is adopted for an isotropic body, satisfying convexity conditions (resp. concavity conditions) with respect to the transformation gradient (resp. temperature). The evolution law for the transplant is obtained from the natural assumption that the evolution of the material is independent of the reference frame. It involves a modified Eshelby tensor based on the specific free energy density. The heat flux is dependent upon the transplant. The model consists of the constitutive equation, the energy balance, and the evolution law for the transplant. It is completed by suitable boundary conditions for the displacement, temperature and transplant tensor. The existence of locally unique solutions is obtained, for sufficiently smooth data close to the stable equilibrium. The question of the global existence is examined in the simplified situation of quasistatic isothermal equations of linear elasticity under the assumption of isotropic growth.

This paper presents ordered rate constitutive theories in Lagrangian description for compressible as well as incompressible homogeneous, isotropic thermoviscoelastic solid matter with memory in which the material derivative of order m of the deviatoric stress tensor and heat vector are functions of temperature, temperature gradient, time derivatives of the conjugate strain tensor up to any desired order n, and the material derivatives of up to order m−1 of the stress tensor. The thermoviscoelastic solids described by these theories are called ordered thermoviscoelastic solids with memory due to the fact that the constitutive theories are dependent on orders m and n of the material derivatives of the conjugate stress and strain tensors. The highest orders of the material derivative of the conjugate stress and strain tensors define the order of the thermoviscoelastic solid. The constitutive theories derived here show that the material for which these theories are applicable have fading memory. As is well known, the second law of thermodynamics must form the basis for deriving constitutive theories for all deforming matter (to ensure thermodynamic equilibrium during evolution), since the other conservation and balance laws are independent of the constitution of the matter. The entropy inequality expressed in terms of Helmholtz free energy density \({\Phi}\) does not provide a mechanism to derive a constitutive theory for the stress tensor when its argument tensors are stress and strain rates in addition to others. With the decomposition of the stress tensor into equilibrium and deviatoric stress tensors, the constitutive theory for the equilibrium stress tensor is deterministic from the entropy inequality. However, for the deviatoric stress tensor, the entropy inequality requires a set of inequalities to be satisfied but does not provide a mechanism for deriving a constitutive theory. In the present work, we utilize the theory of generators and invariants to derive rate constitutive theories for thermoviscoelastic solids with memory. This is based on axioms and principles of continuum mechanics. However, we keep in mind that these constitutive theories must satisfy the inequalities resulting from the second law of thermodynamics. The constitutive theories for heat vector q are derived: (i) strictly using conditions resulting from the entropy inequality; (ii) using the theory of generators and invariants with admissible argument tensors that are consistent with the stress tensor as well as the theories in which simplifying assumptions are employed which yield much simplified theories. It is shown that the rate theories presented here describe thermoviscoelastic solids with memory. Mechanisms of dissipation and memory are demonstrated and discussed, and the derivation of memory modulus is presented. It is shown that simplified forms of the general theories presented here result in constitutive models that may resemble currently used constitutive models but are not the same. The work presented here is not to be viewed as extension of the current constitutive models; rather, it is a general framework for rate constitutive theories for thermoviscoelastic solids with memory based on the physics and derivations that are consistent within the framework of continuum mechanics and thermodynamics. The purpose of the simplified theories presented in the paper is to illustrate possible simplest theories within the consistent framework presented here.

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] .

We revisit here the concept of the constitutive relation error for the identification of elastic material parameters based on image correlation. An additional concept, so called constitutive compatibility of stress, is introduced defining a subspace of the classical space of statically admissible stresses. The key idea is to define stresses as compatible with the observed deformation field through the chosen class of constitutive equation. This makes possible the uncoupling of the identification of stress from the identification of the material parameters. As a result, the global cost of the identification is strongly reduced. This uncoupling also leads to parametrized solutions in cases where the solution is non-unique as demonstrated on 2D numerical examples.

This paper presents development of rate constitutive theories for compressible as well as in incompressible ordered thermoviscoelastic fluids, i.e., polymeric fluids in Eulerian description. The polymeric fluids in this paper are considered as ordered thermoviscoelastic fluids in which the stress rate of a desired order, i.e., the convected time derivative of a desired order ‘m’ of the chosen deviatoric Cauchy stress tensor, and the heat vector are functions of density, temperature, temperature gradient, convected time derivatives of the chosen strain tensor up to any desired order ‘n’ and the convected time derivative of up to orders ‘m−1’ of the chosen deviatoric Cauchy stress tensor. The development of the constitutive theories is presented in contravariant and covariant bases, as well as using Jaumann rates. The polymeric fluids described by these constitutive theories will be referred to as ordered thermoviscoelastic fluids due to the fact that the constitutive theories are dependent on the orders ‘m’ and ‘n’ of the convected time derivatives of the deviatoric Cauchy stress and conjugate strain tensors. The highest orders of the convected time derivative of the deviatoric Cauchy stress and strain tensors define the orders of the polymeric fluid. The admissibility requirement necessitates that the constitutive theories for the stress tensor and heat vector satisfy conservation laws, hence, in addition to conservation of mass, balance of momenta, and conservation of energy, the second law of thermodynamics, i.e., Clausius–Duhem inequality must also be satisfied by the constitutive theories or be used in their derivations. If we decompose the total Cauchy stress tensor into equilibrium and deviatoric components, then Clausius–Duhem inequality and Helmholtz free-energy density can be used to determine the equilibrium stress in terms of thermodynamic pressure for compressible fluids and in terms of mechanical pressure for incompressible fluids, but the second law of thermodynamics provides no mechanism for deriving the constitutive theories for the deviatoric Cauchy stress tensor. In the development of the constitutive theories in Eulerian description, the covariant and contravariant convected coordinate systems and Jaumann measures are natural choices. Furthermore, the mathematical models for fluids require Eulerian description in which material point displacements are not measurable. This precludes the use of displacement gradients, i.e., strain measures, in the development of the constitutive theories. It is shown that compatible conjugate pairs of convected time derivatives of the deviatoric Cauchy stress and strain measures in co-, contravariant and Jaumann bases in conjunction with the theory of generators and invariants provide a general mathematical framework for the development of constitutive theories for ordered thermofluids in Eulerian description. This framework has a foundation based on the basic principles and axioms of continuum mechanics, but the resulting constitutive theories for the deviatoric Cauchy stress tensor must satisfy the condition of positive work expanded, a requirement resulting from the entropy inequality. The paper presents a general theory of constitutive equations for ordered thermoviscoelastic fluids which is then specialized to obtain commonly used constitutive equations for Maxwell, Giesekus and Oldroyd-B constitutive models in contra- and covariant bases and using Jaumann rates.

Keywords: strain, stress, material constants, constitutive equation, anisotropy. Abstract. If the material is anisotropic, there are differences in stress distribution under the same boundary conditions when it was simplified as an orthotropic material. We established a simple finite element model for rectangular perforated planar material, in which one side was fixed, the opposite side was loaded with uniform force, and the other sides were set free. Based on this model we studied the difference of distribution of stress between anisotropic material and its simplified form, orthotropic material. The results showed differences in some cases quite large, the maximum relative error of extreme stress can reach 341%. In conclusions, this study does not support that the complex anisotropic materials are simplified to orthotropic materials. If researchers only concern the location of extreme stress, this study does not deny that the complex anisotropic materials can be simplified to orthotropic one.

The constitutive equation of large deformation problem is closely related to geometric description. Nowadays, linear strain tensor is no longer unsuitable to describe large deformation. However, the existing non-linear strain tensors have complicated forms as well as no apparent geometric or physical meaning. While, the increment method is used to solve, however, convergence and efficiency are low sometimes. Thus the idea of visual strain tensor is proposed, with distinct meaning and visual image. Beside, it is likely to be used in engineering measurement, and it can be connected with measured constitutive equation directly. Thus this research provides a new idea and method for solving large-deformation problems in practical engineering.

The consequences of the constraint of incompressibility is studied for a new class of constitutive relation for elastic bodies, for which the left Cauchy–Green tensor is a function of the Cauchy stress tensor. The requirement of incompressibility is imposed directly in the constitutive relation, and it is not necessary to assume a priori that the stress tensor should be divided into two parts, a constraint stress and a constitutively specified part, as in the classical theory of nonlinear elasticity.

The roles of the consistent Jacobian matrix and the material tangent moduli, which are used in nonlinear incremental finite deformation mechanics problems solved using the finite element method, are emphasized in this paper, and demonstrated using the commercial software ABAQUS standard. In doing so, the necessity for correctly employing user material subroutines to solve nonlinear problems involving large deformation and/or large rotation is clarified. Starting with the rate form of the principle of virtual work, the derivations of the material tangent moduli, the consistent Jacobian matrix, the stress/strain measures, and the objective stress rates are discussed and clarified. The difference between the consistent Jacobian matrix (which, in the ABAQUS UMAT user material subroutine is referred to as DDSDDE) and the material tangent moduli (Ce) needed for the stress update is pointed out and emphasized in this paper. While the former is derived based on the Jaumann rate of the Kirchhoff stress, the latter is derived using the Jaumann rate of the Cauchy stress. Understanding the difference between these two objective stress rates is crucial for correctly implementing a constitutive model, especially a rate form constitutive relation, and for ensuring fast convergence. Specifically, the implementation requires the stresses to be updated correctly. For this, the strains must be computed directly from the deformation gradient and corresponding strain measure (for a total form model). Alternatively, the material tangent moduli derived from the corresponding Jaumann rate of the Cauchy stress of the constitutive relation (for a rate form model) should be used. Given that this requirement is satisfied, the consistent Jacobian matrix only influences the rate of convergence. Its derivation should be based on the Jaumann rate of the Kirchhoff stress to ensure fast convergence; however, the use of a different objective stress rate may also be possible. The error associated with energy conservation and work-conjugacy due to the use of the Jaumann objective stress rate in ABAQUS nonlinear incremental analysis is viewed as a consequence of the implementation of a constitutive model that violates these requirements.

For several classes of soft biological tissues, modelling complexity is in part due to the arrangement of the collagen fibres. In general, the arrangement of the fibres can be described by defining, at each point in the tissue, the structure tensor (i.e. the tensor product of the unit vector of the local fibre arrangement by itself) and a probability distribution of orientation. In this approach, assuming that the fibres do not interact with each other, the overall contribution of the collagen fibres to a given mechanical property of the tissue can be estimated by means of an averaging integral of the constitutive function describing the mechanical property at study over the set of all possible directions in space. Except for the particular case of fibre constitutive functions that are polynomial in the transversely isotropic invariants of the deformation, the averaging integral cannot be evaluated directly, in a single calculation because, in general, the integrand depends both on deformation and on fibre orientation in a non-separable way. The problem is thus, in a sense, analogous to that of solving the integral of a function of two variables, which cannot be split up into the product of two functions, each depending only on one of the variables. Although numerical schemes can be used to evaluate the integral at each deformation increment, this is computationally expensive. With the purpose of containing computational costs, this work proposes approximation methods that are based on the direct integrability of polynomial functions and that do not require the step-by-step evaluation of the averaging integrals. Three different methods are proposed: (a) a Taylor expansion of the fibre constitutive function in the transversely isotropic invariants of the deformation; (b) a Taylor expansion of the fibre constitutive function in the structure tensor; (c) for the case of a fibre constitutive function having a polynomial argument, an approximation in which the directional average of the constitutive function is replaced by the constitutive function evaluated at the directional average of the argument. Each of the proposed methods approximates the averaged constitutive function in such a way that it is multiplicatively decomposed into the product of a function of the deformation only and a function of the structure tensors only. In order to assess the accuracy of these methods, we evaluate the constitutive functions of the elastic potential and the Cauchy stress, for a biaxial test, under different conditions, i.e. different fibre distributions and different ratios of the nominal strains in the two directions. The results are then compared against those obtained for an averaging method available in the literature, as well as against the integration made at each increment of deformation.

For describing the properties for micro-polar cellar material, the definitions of representative element as well as the corresponding equivalent physical quantities in cellular are introduced in this paper. It involves the Cauchy stress, couple stress, displacement gradient, strain, torsion tensor, temperature gradient and the heat flux respectively. The general principle and mode of solving the boundary value problem with respect to the construction of the equivalent constitutive equations is investigated, and the thermo-mechanical interaction is modeled for the micro-polar cellar material. In the context, a standard method for formulizing the boundary value problem in accordance with the specified representative element volume (REV) is developed, and thereupon the equivalent constitutive equations are deduced. For honeycomb materials, the analytical formula for the equivalent elastic coefficient tensor, temperature coefficients of equivalent stress, equivalent Fourier coefficients and the temperature gradient coefficients of couple stress of the honeycomb materials would be formed.

In the present study, the constitutive law of the deformation theory of plasticity has been derived. And that develop the two-dimensional and three-dimensional finite element program. The results of finite element and analytic of plasticity are compared to verify the derived the constitutive law of the deformation theory and the FEM program. At plastic stage, the constitutive laws of the deformation theory can be expressed as the linear elastic constitutive laws. But, it must be modified by iteration of the secant modulus and the effective Poisson’s ratio. Make it easier to develop finite element program. Finite element solution and analytic solution of plasticity theory comparison show the answers are the same. It shows the derivation of the constitutive law of the deformation theory of plasticity and finite element analysis program is the accuracy.

The main purpose of this study is to investigate the efficacy and usefulness of a class of recently proposed models that could be reasonable candidates for describing the response of brittle elastic materials. The class of models that are considered allows for a non-linear relationship between the linearized elastic strain and the Cauchy stress, and this allows one to describe situations wherein the stress increases while the strain yet remains small. Thus one would be in a position to model the response of brittle elastic bodies in the neighborhood of the tips of cracks and notches. In this paper we study the behavior of such models in a plate with a V-notch subject to a state of anti-plane stress. This geometrical simplification enables us to characterize the governing equation for the problem by means of the Airy stress function, though the constitutive relation is a non-linear relation between the linearized strain and the stress. We study the problem numerically by appealing to the finite element method. We find that the numerical solutions are stable. We are able to provide some information regarding the nature of the solution near the tip of the V-notch. In particular we find stress concentration in the vicinity of the singularity.

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 paper considers developments of constitutive theories in Eulerian description for compressible as well as incompressible ordered homogeneous and isotropic thermofluids in which the deviatoric Cauchy stress tensor and the heat vector are functions of density, temperature, temperature gradient, and the convected time derivatives of the strain tensors of up to a desired order. The fluids described by these constitutive theories are called ordered thermofluids due to the fact that the constitutive theories for the deviatoric Cauchy stress tensor and heat vector are dependent on the convected time derivatives of the strain tensor up to a desired order, the highest order of the convected time derivative of the strain tensor in the argument tensors defines the ‘order of the fluid’. The admissibility requirement necessitates that the constitutive theories for the stress tensor and heat vector satisfy conservation laws, hence, in addition to conservation of mass, balance of momenta, and conservation of energy, the second law of thermodynamics, that is, Clausius–Duhem inequality must also be satisfied by the constitutive theories or be used in their derivations. If we decompose the total Cauchy stress tensor into equilibrium and deviatoric components, then Clausius–Duhem inequality and Helmholtz free energy density can be used to determine the equilibrium stress in terms of thermodynamic pressure for compressible fluids and in terms of mechanical pressure for incompressible fluids, but the second law of thermodynamics provides no mechanism for deriving the constitutive theories for the deviatoric Cauchy stress tensor. In the development of the constitutive theories in Eulerian description, the covariant and contravariant convected coordinate systems, and Jaumann measures are natural choices. Furthermore, the mathematical models for fluids require Eulerian description in which material point displacements are not measurable. This precludes the use of displacement gradients, that is, strain measures, in the development of the constitutive theories. It is shown that compatible conjugate pairs of convected time derivatives of the deviatoric Cauchy stress and strain measures in co-, contravariant, and Jaumann bases in conjunction with the theory of generators and invariants provide a general mathematical framework for the development of constitutive theories for ordered thermofluids in Eulerian description. This framework has a foundation based on the basic principles and axioms of continuum mechanics but the resulting constitutive theories for the deviatoric Cauchy stress tensor must satisfy the condition of positive work expanded, a requirement resulting from the entropy inequality. The paper presents a general theory of constitutive equations for ordered thermofluids which is then specialized, assuming first-order thermofluids, to obtain the commonly used constitutive theories for compressible and incompressible generalized Newtonian and Newtonian fluids. It is demonstrated that the constitutive theories for ordered thermofluids of all orders are indeed rate constitutive theories. We have intentionally used the term ‘thermofluids’ as opposed to ‘thermoviscous fluids’ due to the fact that the constitutive theories presented here describe a broader group of fluids than Newtonian and generalized Newtonian fluids that are commonly referred as thermoviscous fluids.

When the mathematical models for the deforming solids are constructed using the Eulerian description, the material particle displacements and hence the strain measures are not known. In such cases the constitutive theory must utilize convected time derivatives of the strain measures. The entropy inequality provides a mechanism for determining constitutive equations for the equilibrium stress with the additional requirement that the work expanded due to the deviatoric part of the Cauchy stress tensor be positive, but provides no mechanism for establishing the constitutive theory for it. In the development of the constitutive theory in the Eulerian description for thermoelastic solids, one must consider a coordinate system in the current configuration in which the deformed material lines can be identified. Thus the covariant, contravariant and Jaumann convected coordinate systems are natural choices for the development of the constitutive theory. The compatible conjugate pairs of convected time derivatives of the stress and strain measures in these bases in conjunction with the theory of generators and invariants provide a general mathematical framework for the development of the constitutive theory for thermoelastic solids. This framework has a foundation based on the basic principles and axioms of continuum mechanics but the resulting constitutive theory must satisfy the conditions resulting from the entropy inequality to ensure thermodynamic equilibrium of the deforming matter. This paper presents development of rate constitutive theories for compressible as well as incompressible, homogeneous, isotropic solids. The density, temperature, and temperature gradient in the current configuration and the convected time derivatives of the strain tensor up to any desired order in the chosen basis are considered as the argument tensors of the first convected time derivative of the deviatoric Cauchy stress tensor and heat vector. The thermoelastic solids described by these constitutive theories are termed ordered thermoelastic solids due to the fact that the constitutive theories for the deviatoric Cauchy stress tensor and heat vector are dependent on the convected time derivatives of the strain tensor up to any desired order, the highest order defining the order of the solid.

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.

In-depth study of compressible material constitutive equation, using incompressible condition, the nonlinear incompressible elastic solid’s complete irreducible constitutive equation and strain energy function expressed in invariants are derived in this essay. The elastic constants of rubber material are given by fitting the experiment data that was carried out by Treloar with the equation. Then we got evelen exact value of the elastic constants.