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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.

This paper improved Chaboche viscoplastic constitutive model in order to simulate the mechanical behavior of engineering components during service. The anisotropic tensor and Kachanov damage evolution equation were added to the constitutive equation to simulate the creep behavior of anisotropic material. Based on that, the Levenberg-Marquardt nonlinear optimization algorithm was used to obtain material parameters and the constitutive model was compiled as the subprogram of finite element software ABAQUS for its simulation application. Then the improved constitutive model was used to simulate and analyse the uniaxial tensile and creep behavior of specimens made of anisotropic material DZ125 at different loads, and the accuracy and feasibility of material parameters and constitutive model were verified by corresponding test results. Finally, the improved constitutive model was used to simulate the creep behavior of a hollow turbine blade, achieving its engineering application. And the calculated results have great significance for structure design and life prediction of turbine blade.

A constitutive material law for linear thermo-viscoelasticity in the time domain is presented. The time-dependent relaxation formulation is given for full anisotropy, i.e., both the elastic and the viscous properties are anisotropic. Thereby, each element of the relaxation tensor is described by its own and independent Prony series expansion. Exceeding common viscoelasticity, time-dependent thermal expansion relaxation/creep is treated as inherent material behavior. The pertinent equations are derived and an incremental, implicit time integration scheme is presented. The developments are implemented into an implicit FEM software for orthotropic material symmetry under plane stress assumption. Even if this is a reduced problem, all essential features are present and allow for the entire verification and validation of the approach. Various simulations on isotropic and orthotropic problems are carried out to demonstrate the material behavior under investigation.

A three-dimensional visco-hyperelastic constitutive model is developed to describe the finite deformation mechanical behavior of polyurea materials at different strain rates. The constitutive model of finite strain visco-hyperelasticity is founded on the basis of the multiplicative decomposition of the deformation gradient tensor into hyperelastic and viscoelastic parts. The hyperelastic part uses the strain energy function to characterize the equilibrium response, and the viscoelastic part capturing the rate sensitivity uses the time partial derivative of strain energy function to characterize the time-dependent response. The nonlinear mechanical responses of the materials under several common loading conditions are calculated by the new proposed constitutive model. In order to validate the effectiveness of the constitutive model, the nonlinear stress-strain behavior of polyurea under uniaxial tension and compression are carried out in this paper. The experimental verification and the error analysis show that the model is capable of accurately representing the finite deformation stress-strain behavior of polyurea over a strain-rate range of 10 −3 –10 4 /s. In order to further broaden the application of the constitutive model, a method is presented which is available to predict the experimental results of polyurea at quasi-one dimensional strain state.

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.

This paper presents a new thermomechanically coupled constitutive model for polycrystalline shape memory alloys (SMAs) undergoing finite deformation. Three important characteristics of SMA behavior are considered in the development of the model, namely the effect of coexistence between austenite and two martensite variants, the variation of hysteresis size with temperature, and the smooth material response at initiation and completion of phase transformation. The formulation of the model is based on a multi-tier decomposition of the deformation kinematics comprising, a multiplicative decomposition of the deformation gradient into thermal, elastic and transformation parts, an additive decomposition of the Hencky strain into spherical and deviatoric parts, and an additive decomposition of the transformation stretching tensor into phase transformation and martensite reorientation parts. A thermodynamically consistent framework is developed, and a Helmholtz free energy function consisting of elastic, thermal, interaction and constraint components is introduced. Constitutive and heat equations are then derived from this energy in compliance with thermodynamic principles. Time-discrete formulations of the constitutive equations and a Hencky-strain return-mapping integration algorithm are presented. The algorithm is then implemented in Abaqus/Explicit by means of a user-defined material subroutine (VUMAT). Numerical results are validated against experimental data obtained under various thermomechanical loading conditions. The robustness and efficiency of the proposed framework are illustrated by simulating a SMA helical spring actuator.

In the paper, a hyperelastic material with limited tensile and compressive strength is considered. In addition, the maximum value of the tangential component of the stress that can be attained on each plane depends on the intensity of the normal component. Previously formulated for the plane case, the constitutive equation of this material is now extended to three-dimensional bodies, constituting henceforth a generalization of the well-known model of masonry-like materials. The proposed constitutive law has been implemented in the finite element code MADY and has been applied to the study of masonry panels under different load conditions.

An incremental format of the age-dependent constitutive equation was derived by expanding the total form of the constitutive equation by using the first-order Taylor series to describe the persistent change in the creep-causing stress state as well as drying shrinkage and the development of the elastic modulus. The resulting incremental constitutive equation was defined by three basic equations for basic creep, drying shrinkage, and the development of the elastic modulus. Three types of laboratory experiments were carried out to validate the performance of the presented age-dependent constitutive equation; these included cylindrical concrete specimen tests with and without axial reinforcements and reinforced beam specimen tests. The performance of the creep model was compared with those calculated by the other age-dependent constitutive equations.

In this work, we develop a coupled thermo-mechanical, isotropic-plasticity, finite-deformation-based constitutive model for high-temperature shape memory alloys (HTSMAs). This constitutive model is capable of describing austenite-martensite phase transformations, rate-dependent plasticity (creep) in the austenitic phase,and also transformation-induced plasticity (TRIP) due to phase transformations between the austenitic and martensitic phase. The constitutive model was also implemented into a commercially-available finite-element program through a user-material subroutine interface. By using suitably valued material parameters in the constitutive model, we show that the output obtained from our finite-element simulations are able to accurately match the experimental strain-temperature cycling data for a ternary high-temperature shape memory alloy (Ti-Ni-Pd) under various applied stresses. We also show through finite-element simulations that for particular boundary-value problems of practical/engineering significance, the usage of a finite-strain-based constitutive theory gives vastly different results when compared to using a small-strain-based constitutive theory.

Constitutive models are necessary to design shape memory alloy (SMA) components at nano- and micro-scales in NEMS and MEMS. The behavior of small-scale SMA structures deviates from that of the bulk material. Unfortunately, this response cannot be modeled using conventional constitutive models which lack an intrinsic length scale. At small scales, size effects are often observed along with large gradients in the stress or strain. Therefore, a gradient-based thermodynamically consistent constitutive framework is established. Generalized surface and body forces are assumed to contribute to the free energy as work conjugates to the martensite volume fraction, transformation strain tensor, and their spatial gradients. The rates of evolution of these variables are obtained by invoking the principal of maximum dissipation after assuming a transformation surface, which is a differential equation in space. This approach is compared to the theories that use a configurational force (microforce) balance law. The developed constitutive model includes energetic and dissipative length scales that can be calibrated experimentally. Boundary value problems, including pure bending of SMA beams and simple torsion of SMA cylindrical bars, are solved to demonstrate the capabilities of this model. These problems contain the differential equation for the transformation surface as well as the equilibrium equation and are solved analytically and numerically. The simplest version of the model, containing only the additional gradient of martensite volume fraction, predicts a response with greater transformation hardening for smaller structures.

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.

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.

The present work deals with the modeling of multi-defected solids under the action of large deformation. A micromechanics constitutive model, formulated in terms of the compressible anisotropic NeoHookean strain energy density function, is presented to characterize the corresponding nonlinear effective elastic behavior. By employing a scalar energy parameter, a correspondence relation between the effective hyperelastic model and this energy parameter is established. The corresponding effective material coefficients are then evaluated through combined use of the “direct difference approach” and the extended “modified compliance contribution tensor” method. The proposed material constitutive model can be further used to estimate the effective mechanical properties for engineering structures with complicated geometry and mechanics and appears to be an efficient computational homogenization tool in practice.

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.

In this study, developments of an efficient visco-hyperelastic constitutive equation for describing the time dependent material behavior accurately in dynamic and impact loading and finding related materials constants are considered. Based on proposed constitutive model, behaviour of a hollow cylinder elastomer bushing under different dynamic and impact loading conditions is studied. By implementing the developed visco-hyperelastic constitutive equation to LS-DYNA explicit dynamic finite element software a three dimensional model of the bushing is developed and dynamic behaviour of that in axial and torsional dynamic deformation modes are studied. Dynamic response and induced stress under different impact loadings which is rarely studied in previous researches have been also investigated. Effects of hyperelastic and visco-hyperelastic parameters on deformation and induced stresses as well as strain rate are considered.

In many existing theories for incompressible polymeric liquids the Cauchy stress is decomposed as T = − p 1 + S v + S e , where p is an arbitrary pressure, S v = 2 μ s D a deviatoric viscous stress with μs a viscosity and D the deviatoric stretching tensor, and Se is a deviatoric elastic stress which is introduced to account for stiffening arising from the alignment of long-chain polymer molecules during flow. A constitutive equation for Se needs to be prescribed and there are a large number of different proposals in the literature, with most proposals involving a hypoelastic rate constitutive equation for Se given in terms of a suitable frame-indifferent rate, which is usually taken as the Oldroyd or upper-convected rate. As is well-known, a hypoelastic equation for the stress is not thermodynamically consistent, in the sense that the constitutive equation for Se is not derived form a free energy function. The purpose of this paper is to present an alternative — thermodynamically-consistent and frame-indifferent — continuum theory for incompressible viscoelastic liquids. The theory is based on a Kroner-type multiplicative decomposition of the deformation gradient F of the form F = FeFp. In this theory the elastic stress Se is derived from a free-energy function which is prescribed in terms of a suitable measure based on the unimodular elastic distortion tensor Fe. This relation is supplemented by an evolution equation for the unimodular plastic distortion tensor Fp — the plastic flow rule. We study the response of the constitutive theory in steady simple shearing and steady extensional flows. We show: (i) that the theory qualitatively reproduces the experimentally-observed transient shear-thinning and normal stress effects during shearing flows of a polymer melt; and (ii) that it also reproduces the transient extensional response of a polymer melt.

The non-uniqueness of the atomistic stress tensor is a well-known issue when defining continuum fields for atomistic systems. In this paper, we study the non-uniqueness of the atomistic stress tensor stemming from the non-uniqueness of the potential energy representation. In particular, we show using rigidity theory that the distribution associated with the potential part of the atomistic stress tensor can be decomposed into an irrotational part that is independent of the potential energy representation, and a traction-free solenoidal part. Therefore, we have identified for the atomistic stress tensor a discrete analog of the continuum generalized Beltrami representation (a version of the vector Helmholtz decomposition for symmetric tensors). We demonstrate the validity of these analogies using a numerical test. A program for performing the decomposition of the atomistic stress tensor called MDStressLab is available online at http://mdstresslab.org .

Acoustic wave propagation from surrounding medium into a soft material can generate acoustic radiation stress due to acoustic momentum transfer inside the medium and material, as well as at the interface between the two. To analyze acoustic-induced deformation of soft materials, we establish an acoustomechanical constitutive theory by combining the acoustic radiation stress theory and the nonlinear elasticity theory for soft materials. The acoustic radiation stress tensor is formulated by time averaging the momentum equation of particle motion, which is then introduced into the nonlinear elasticity constitutive relation to construct the acoustomechanical constitutive theory for soft materials. Considering a specified case of soft material sheet subjected to two counter-propagating acoustic waves, we demonstrate the nonlinear large deformation of the soft material and analyze the interaction between acoustic waves and material deformation under the conditions of total reflection, acoustic transparency, and acoustic mismatch.

Functionally graded cemented carbides (FGCCs) have an excellent combination of high hardness and high toughness. But the appearance of residual stresses, resulting from cobalt gradient and mismatch between mechanical properties and material constituents, influences the lifetime of FGCCs. To get the thermal stress field distribution, a constitutive equation for WC-Co FGCCs was developed by redefinition of elastic constraint factor and introduction of plastic constraint factor. The constitutive model was applied to thermal stress analysis of WC-Co composites. The distribution of thermal stresses in WC-Co specimen was obtained by the finite element numerical method. Simultaneously, the surface compressive stress of FGCCs alloy was measured by X-ray diffraction (sinψ)2. Numerical results show the thermal stresses of WC-Co composites mainly concentrate in the cobalt gradient zone and the maximum value of principal compressive stress is 380 MPa in the surface zone. This result is in good agreement with X-ray diffraction measurement.

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.

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.

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.

A heterogeneous Cauchy elastic material may display micromechanical effects that can be modeled in a homogeneous equivalent material through the introduction of higher-order elastic continua. Asymptotic homogenization techniques provide an elegant and rigorous route to the evaluation of equivalent higher-order materials, but are often of difficult and awkward practical implementation. On the other hand, identification techniques, though relying on simplifying assumptions, are of straightforward use. A novel strategy for the identification of equivalent second-gradient Mindlin solids is proposed in an attempt to combine the accuracy of asymptotic techniques with the simplicity of identification approaches. Following the asymptotic homogenization scheme, the overall behaviour is defined via perturbation functions, which (differently from the asymptotic scheme) are evaluated on a finite domain obtained as the periodic repetition of cells and subject to quadratic displacement boundary conditions. As a consequence, the periodicity of the perturbation function is satisfied only in an approximate sense, nevertheless results from the proposed identification algorithm are shown to be reasonably accurate.

A second-order rotation gradient theory for non-classical elastic continua is developed. This theory accounts for the higher-order deformation of the material structure where the material particle inside the elastic domain is idealized as a microvolume having three degrees of freedom, namely, a translation, a micro-rotation, and a higher-order micro-deformation. The associated strain energy density is a function of the infinitesimal strain tensor and the first and second gradients of the rotation tensor. It is demonstrated that for materials in nano-scale applications and because of some defects at the material structure level, a higher-order deformation measure may be needed. The second-strain gradient theory has the merit to account for the higher-order deformation of the material particle. However, this theory has limited applications because it depends on 16 additional material constants for isotropic elastic continua. By discussing some physical concepts relevant to the natures of material structures, crystallinity, and amorphousness, the second-strain gradient theory is reduced to the second-rotation gradient theory for certain types of materials. For isotropic materials, the developed second-rotation gradient theory only depends on three additional material constants instead of 16. A continuum model equipped with an atomic lattice model is then proposed to examine the applicability of the available non-classical theories and the applicability of the proposed theory for different types of materials.

The antiplane strain Green's functions for an applied concentrated force and moment are obtained for Cosserat elastic solids with extreme anisotropy, which can be tailored to bring the material in a state close to an instability threshold such as failure of ellipticity. It is shown that the wave propagation condition (and not ellipticity) governs the behaviour of the antiplane strain Green's functions. These Green's functions are used as perturbing agents to demonstrate in an extreme material the emergence of localized (single and cross) stress channelling and the emergence of antiplane localized folding (or creasing, or weak elastostatic shock) and faulting (or elastostatic shock) of a Cosserat continuum, phenomena which remain excluded for a Cauchy elastic material. During folding some components of the displacement gradient suffer a finite jump, whereas during faulting the displacement itself displays a finite discontinuity.

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 consideration of materials capable of undergoing significant plastic changes in volume, an alternative finite strain hyper-elastoplastic constitutive framework is proposed in terms of the Eshelby stress. Taking a phenomenological point of view, a thermodynamically-consistent approach to developing the constitutive equations is presented and discussed. Various Eshelby-like stresses are defined and shown to be energy-conjugate to the plastic velocity gradient, and a general framework is formulated in the stress-free/plastically-deformed intermediate configuration associated with the multiplicative split of the deformation gradient, as well as the current configuration. A novel Eshelby-like stress measure is proposed, which is scaled by the elastic Jacobian, and is shown to be energy-conjugate to the plastic velocity gradient in the spatial representation. Modified Cam–Clay and Drucker–Prager cap plasticity constitutive equations are introduced, and large strain isotropic compression simulations are performed and compared with experimental measurements. The model results are compared with standard approaches formulated in terms of the Mandel and Kirchhoff stresses, which are shown to require the assumption of isochoric plasticity to satisfy the Clausius Planck inequality (Mandel) and preserve that the intermediate configuration remains stress-free (Kirchhoff). The simulations show that both the material and spatial Eshelby-like stress measures presented here produce the same mean Cauchy stress results; whereas, standard formulations, which make use of isochoric plasticity assumptions, diverge from each other at significant plastic volume strains. Standard formulations are further shown to violate the second law of thermodynamics under certain loading conditions. Calibration of model parameters to high pressure isotropic compression of Boulder clay is used to compare the various models.

For the case of metals with large viscoplastic strains, it is necessary to define appropriate constitutive models in order to obtain reliable results from the simulations. In this paper, two large strain viscoplastic Perzyna type models are considered. The first constitutive model has been proposed by Ponthot, and the elastic response is based on hypoelasticity. In this case, the kinematics of the constitutive model is based on the additive decomposition of the rate deformation tensor. The second constitutive model has been proposed by Garcia Garino et al., and the elastic response is based on hyperelasticity. In this case, the kinematics of the constitutive model is based on the multiplicative decomposition of the deformation gradient tensor. In both cases, the resultant numerical models have been implemented in updated Lagrangian formulation. In this work, global and local numerical results of the mechanical response of both constitutive models are analyzed and discussed. To this end, numerical experiments are performed and different parameters of the constitutive models are tested in order to study the sensitivity of the resultant algorithms. In particular, the evolution of the reaction forces, the effective plastic strain, the deformed shapes and the sensitivity of the numerical results to the finite element mesh discretization have been compared and analyzed. The obtained results show that both models have a very good agreement and represent very well the characteristic of the viscoplastic constitutive model.

In the paper a constitutive law formulation for bovine cortical bone tissue is presented. The formulation is based on experimental studies performed on bovine cortical bone samples. Bone tissue is regarded as a non-linear viscoelastic material. The constitutive law is derived from the postulated strain energy function. The model captures typical viscoelastic effects, i.e. hysteresis, stress relaxation and rate-dependence. The elastic and rheological constants were identified on the basis of experimental tests, i.e. relaxation tests and monotonic uniaxial tests at three different strain rates, i.e. λ ˙ = 0.1 min − 1 , λ ˙ = 0.5 min − 1 and λ ˙ = 1.0 min − 1 . In order to numerically validate the constitutive model the fourth-order stiffness tensor was analytically derived and introduced to Abaqus ® finite element (FE) software by means of UMAT subroutine. The model was experimentally validated. The validation results show that the derived constitutive law is adequate to model stress–strain behaviour of the considered bone tissue. The constitutive model, although formulated in the strain rate range λ ˙ = 0.1 − 1.0 min − 1 , is also valid for the strain rate values slightly higher than λ ˙ = 1.0 min − 1 . The work presented in the paper proves that the formulated constitutive model is very useful in modelling compressive behaviour of bone under various ranges of load.

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.

Many rocks exhibit nonlinear behavior under the effect of internal and external factors; thus, stress and initiation models based on linear elastic theory are not applicable for these rocks. In accordance with the deformation theory of plastic mechanics, a nonlinear constitutive model is developed in this study based on the power-hardening equation using a piecewise approximation method. A new model for the elastic–plastic stress field around the wellbore is then proposed considering the in-situ stress anisotropy. Finally, a new elastic–plastic hydraulic fracturing initiation model is developed, coupled with the maximum tensile strength and Mohr–Coulomb criteria. Calculations and analyses reveal that the nonlinearity of the constitutive relation has significant effects on the stress distribution, initiation mode, and pressure. The plastic yield has little effect on the radial stress but a significant effect on the circumferential stress. When rock yielding occurs, the stress concentration around the wellbore is reduced, and the circumferential stress decreases or cannot be tensile. In this case, the initiation pressure is much higher than that of the linear elastic model, and the initiation mode includes tensile and shear failure. The initiation mode is comprehensively controlled by the in-situ stress, cohesion, tensile strength, power-hardening index, yield stress, and internal friction angle. The initiation orientation of both initiation modes is along the maximal horizontal principal stress direction; however, there is a failure angle for the shear failure. It is more accurate to predict the initiation mode and pressure using the piecewise power-hardening constitutive equation than the Hubbert and Willis model.

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.

An anisotropic extension of the Hosford–Coulomb localization criterion is obtained through the linear transformation of the stress tensor argument. Unlike for isotropic materials where the stress state is characterized through the stress triaxiality and Lode parameter, the normalized Cauchy stress tensor is used to describe the stress state in an anisotropic solid. Based on experiments on extruded aluminum 6260-T6 covering stress states from pure shear to equi-biaxial tension for different material orientations, it is shown that this phenomenological and uncoupled model is capable to provide reasonable engineering approximations of the strains and displacements to fracture for thirteen different loading conditions.

In this paper, we study the consequences of the constraint of inextensibility with regard to a class of constitutive relations, where the strain is given as a function of the stress. Such constitutive equations belong to a wider class of implicit constitutive relations, which have been proposed recently in the literature.

The applicability of constitutive equations that incorporate the third deviatoric stress invariant to the description of the nonlinear deformation of coarse-grained metal is analyzed. Published data of tests on thin-wall tubular specimens made of coarse-grained metal are used. It is shown that these equations are in good agreement with data on tension (compression) and torsion of tubular specimens made of gray iron

A comparative study was carried out on the appropriateness of hyperbolic sine, power, and exponential descriptions of Zener–Hollomon parameter (Z) in prediction of high-temperature flow stress by consideration of the effect of strain. It was shown that the main problem of the conventional strain compensation approach is the implementation of the constitutive equations to find the strain-dependent material constants, especially the hot deformation activation energy (Q), at constant strain values, which arises from the change in the microstructure of the material at a given strain for different deformation conditions (different Z values). Subsequently, a simplified approach for each constitutive equation, mainly by taking Q from the peak stress analysis, was proposed to solve this issue. This also resulted in significantly better prediction abilities for unseen deformation conditions and effectively simplified the required calculations.

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...

Simple modeling approaches based on the Hollomon equation, the Johnson–Cook equation, and the Arrhenius constitutive equation with strain-dependent material’s constants were used for modeling and prediction of flow stress for the single-peak dynamic recrystallization (DRX) flow curves of a stainless steel alloy. It was shown that the representation of a master normalized stress–normalized strain flow curve by simple constitutive analysis is successful in modeling of high temperature flow curves, in which the coupled effect of temperature and strain rate in the form of the Zener–Hollomon parameter is considered through incorporation of the peak stress and the peak strain into the formula. Moreover, the Johnson–Cook equation failed to appropriately predict the hot flow stress, which was ascribed to its inability in representation of both strain hardening and work softening stages and also to its completely uncoupled nature, i.e. dealing separately with the strain, strain rate, and temperature effects. It was also shown that the change in the microstructure of the material at a given strain for different deformation conditions during high-temperature deformation is responsible for the failure of the conventional strain compensation approach that is based on the Arrhenius equation. Subsequently, a simplified approach was proposed, in which by correct implementation of the hyperbolic sine law, significantly better consistency with the experiments were obtained. Moreover, good prediction abilities were achieved by implementation of a proposed physically-based approach for strain compensation, which accounts for the dependence of Young’s modulus and the self-diffusion coefficient on temperature and sets the theoretical values in Garofalo’s type constitutive equation based on the operating deformation mechanism. It was concluded that for flow stress modeling by the strain compensation techniques, the deformation activation energy should not be considered as a function of strain.

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.

A new constitutive equation of thermoelasticity for crystals is presented based on the interatomic potential and solid mechanics at finite temperature. Using the new constitutive equation, the calculations for crystal copper and graphene are carried out under different loading paths at different temperatures. The calculated results are in good agreement with those of the previous thermoelasticity constitutive equation based on quantum mechanics, which clearly indicates that our new constitutive equation of thermoelasticity is correct. A lot of comparisons also show that the present theory is more concise and efficient than the previous thermal stress theory in the practical application.