New constitutive relations in the nonlinear theory of viscoelasticity

A generalization of Pobedrya’s viscoelasticity constitutive relations is considered for an isotropic material with an elastic behavior of the material volume. Two model problems are discussed for a specimen under uniform compression and uniaxial tension. The resulting expressions for strains are used to formulate some necessary conditions for a nonrelaxing behavior of the material volume. The case of an arbitrary low loading of the specimen is examined. It is proved that the obtained nonrelaxing conditions are not only necessary but are also sufficient.     

On a class of constitutive relations for nonlinear infinitesimal elasticity

The structure of dual potentials in nonlinear infinitesimal elasticity is investigated under a general requirement that the components of strain increase in fixed ratio when the components of stress do. The contour surfaces of such potentials are self-similar and their respective geometries are linked by mutual polar reciprocity. The resulting class of constitutive relations is reduced to canonical form and some general implications are worked out in detail. Much of the analysis is relevant also to viscous creep and plasticity, when transposed by the well known analogues.     

A constitutive model in finite viscoelasticity

A new constitutive model is suggested for the viscoelastic behavior of rubber-like materials at finite strains. The model treats a viscoelastic medium as a system with a variable number of purely elastic links, which can arise and collapse due to micro-Brownian motion of molecules.Assuming that the processes of birth and death for elastic links are independent of stresses, we obtain operator linear constitutive equations in finite viscoelasticity. According to this model, elastic and viscous effects may be distinguished and described independently of each other by a relaxation measure and a strain energy density.The potential energy of deformations is assumed to depend on the principal invariants of the relative Finger tensor of strains. Unlike the standard approach, we do not suggest any expression for the strain energy densitya priori, but suppose that this function is presented as a sum of two functions of one variable which are found by fitting experimental data.The proposed approach allows results of several experiments (uniaxial tension, biaxial tension, and torsion) for styrene butadiene rubber and butyl rubber to be predicted correctly.     

Three-dimensional Maxwell stress-strain relations in viscoelasticity

Conclusion In this paper three-dimensional Maxwell stress-strain relations were deduced phenomenologically.In the first place we applied the Hamilton's principle to the viscoelastic deformation, and obtained the variational equation with respect to the elastic potential and the dissipation function.Then we assumed that the elastic potential is a function only of the stress, and the dissipation function is a function of stress and rate of stress. By the above variational equation of the virtual stress satisfying the equilibrium equation and the boundary conditions, we obtained the relations to be satisfied by the elastic potential and the dissipation function, and the conditions to be satisfied by the dissipation function.From these relations we obtained the required three-dimensional Maxwell stress-strain relations in viscoelasticity. These relations indicate that the strain is the sum of the internal elastic strain and the internal viscous strain.If a given substance is isotropic with respect to stress, the stress-strain relations are expressed by a linear Maxwell model consisting of Hookian spring in series with a Newtonian dashpot.It is the main result of this paper that the three-dimensional Maxwell stress-strain relations in viscoelasticity are deduced from physically appropriate assumptions.     

Implicit constitutive relations in thermoelasticity

In this paper, after a brief discussion on the implicit constitutive relations used in thermoelasticity, based on Fox’s [1] work, we derive a general implicit relation for the heat flux vector. In the section following that we use the methodology suggested by Rajagopal [2] and [69] whereby we derive a class of implicit constitutive relations for q and we show that by selecting appropriate functions appearing in the formulation, we can obtain as special cases the Fourier heat conduction and the Maxwell-Cattaneo-Fox model. We do not discuss the implications of the second law of thermodynamics.     

Identification of quasi-static displacement for constitutive models of viscoelasticity

I.IntroductionQuasi-staticdisplacementbasedtechniqueofinverseanalysisfortheconstitutlveparametersisbeingwidlyusedintheundergroundengineeringandsomeotherfields.Anumberofprogresshasbeenachievedinthefieldofidentificationfortheviscoelasticconstitutiveparallleters.XueandYang(1986)l'].Liu(1986)l=lemployedintegraltransformationtechniquetoidentifyviscoelasticconstitutiveparameters;a3-DinverseanalysisfortheviscoelasticconstitutiveparameterswasconductedbyYang(199I)l'];YallgandWu(1996)l'Jproposedagen…     

The mirror relations and nonlinear viscoelasticity of polymer melts

An attempt is made to incorporate into a quasilinear viscoelastic constitutive equation of the Boltzmann superposition type the two mirror relations of Gleissle, as well as his relation between the steady-state first normal-stress difference and the shear viscosity curve. It is shown that the three relations can hold separately within this constitutive model, but not simultaneously, because they require a different nonlinear strain measure, namelyS ₁₂ () = – a ( – 1) (a = 0 for 1,a = 1 for 1) for the mirroring of the viscosities,S ₁₂ () = – a (–k ²/) (a = 0 for k, a = 1 for k) for the mirroring of the first normal-stress coefficients, and for the third relation. Here denotes the shear strain and erf the error function. Experimental data on melts of a low-density polyethylene, a high-density polyethylene and a polypropylene show that the mirror relations are passable approximations, but that the third relation meets reality surprisingly close if the right value ofk is used.     

Simple constitutive modelling of nonlinear viscoelasticity under general extension

We study the behaviour of a single integral constitutive equation, capable of providing analytic expressions for the viscoelastic stress in extensional flows of a variety of deformation histories and geometries, ranging from uniaxial to equibiaxial. It is based on the use of a stress damping function, with a power-law dependence on the elongation, λ: h(λ) = 1/λⁿ. The parameter n (0 ≤ n ≤ 2) signifies the nonlinear viscoelastic character of the material and, therefore, is an inverse measure of network connectivity strength of the underlying microstructure. This renders the constitutive approach applicable to incompressible polymers of a variable degree of branching, strain hardening and stress thinning behavior. Methods of connecting n with the macromolecular architecture and the alignment strength of the flow are also explored.     

Method of integro-differential relations in linear elasticity

Boundary-value problems in linear elasticity can be solved by a method based on introducing integral relations between the components of the stress and strain tensors. The original problem is reduced to the minimization problem for a nonnegative functional of the unknown displacement and stress functions under some differential constraints. We state and justify a variational principle that implies the minimum principles for the potential and additional energy under certain boundary conditions and obtain two-sided energy estimates for the exact solutions. We use the proposed approach to develop a numerical analytic algorithm for determining piecewise polynomial approximations to the functions under study. For the problems on the extension of a free plate made of two different materials and bending of a clamped rectangular plate on an elastic support, we carry out numerical simulation and analyze the results obtained by the method of integro-differential relations.     

On material representation and constitutive branching in finite compressible elasticity

The mechanical properties of two porous rubbers of different compressibility have been investigated experimentally and represented by aid of a particular isotropic strain-energy function constructed by means of separable distortional and dilatational terms. It is shown that a reduced form of the adopted strain-energy function offers definite advantages for evaluation of experimental data and reproduces well the behaviour of the investigated materials under three different loadings; uniaxial tension, plane strain tension and equibiaxial tension. The possibility of homogeneous branching from fundamental paths of the associated motions is examined and illustrated in detail for axisymmetric loading employing constitutive properties pertinent to the two materials tested.     

A “zero-parameter” constitutive relation for simple shear viscoelasticity

Based on the Cox-Merz rule and Eyring's expression for the nonlinear shear viscosity, a Wagner-type constitutive relation with no nontrivial adjustable parameters is proposed for simple shear viscoelasticity. The predictions for a number of non-steady shear flows are worked out analytically. It is shown that most features of shear viscoelasticity are reproduced by the model.     

Geotechnical yield criteria and constitutive relations in strain space

Based on Hyushins postulate this paper deals with the necessity and features ofresearching the geotechnical elasto-plastie theory in strain space.In the paper,weestablished the relations between stress in variants and elastic strain invariants.broughtabout the transformation from the stress yield surfaces into the strain yield surfaces,derived and discussed the strain expressions from 12 yield criteria expressed by stress.Bynormality rule.we also derived 12 constitutive relations for ideal plastic materialsassociated with the above expressions.The results presented here can be applied to practiceand are helpful to the study of the plastic theory in strain space.     

Effective constitutive relations for inelastic composites

The first special boundary value problem in the mechanics of deformable solids is considered to derive the effective constitutive relations for a heterogeneous inelastic body. The problem is reduced to a number of auxiliary boundary value problems for functions dependent on the shape of the body and on the form of constitutive relations. In the case of a layer of nonuniform thickness, the problem of finding the effective constitutive relations is reduced to an operator equation whose solution is sought by an iterative method of successive approximations. An approximate analytical formula is proposed to find the effective constitutive relations for a laminated composite on the basis of known inelastic constitutive relations for its components. This approximate formula takes into account the character of structural anisotropy in a laminated composite and, in the elastic case, yields the exact values of the effective elastic modulus.     

Non-linear constitutive relations for magnetostrictive materials

In this paper, non-linear deformation behavior of magnetostrictive materials is studied and three magnetoelastic coupling constitutive models are developed. The standard square (SS) constitutive model is developed by means of truncating the polynomial expansion of the Gibbs free energy. The hyperbolic tangent (HT) constitutive equations, which involve a hyperbolic tangent magnetic-field dependence, are proposed to model the magnetic-field-induced strain saturation of magnetostrictive materials in the region of intense magnetic fields. A new model based on density of domain switching (DDS) is established in terms of the basic truth that magnetic domain switching underlies magnetostrictive deformation. In this model, it is assumed that the relation between density of domain switching, defined by the quantity of magnetic domains switched by per unit magnetic field and magnetic field can be described by a density function with normal distribution. The moduli in these constitutive models can be determined by a material function that is proposed to describe the dependence of the peak piezomagnetic coefficient on the compressive pre-stress for one-dimensional cases based on the experimental results published. The accuracy of the non-linear constitutive relations is evaluated by comparing the theoretical values with experimental results of a Terfenol-D rod operated under both compressive pre-stress and bias magnetic field. Results indicate that the SS constitutive equations can accurately predict the experimental results under a low or moderate magnetic field while the HT model can, to some extent, reflect the trend of saturation of magnetostrictive strain under a high magnetic field. The model based on DDS, which is more effective in simulating the experimental curves, can capture the main characteristics of the mechanism of magnetoelastic coupling deformation of a Terfenol-D rod, such as the notable dependence of magnetoelastic response on external stress and the saturation of magnetostrictive strain under intense magnetic fields. In addition, the SS constitutive relation for a general three-dimensional problem is discussed and an approach to characterize the modulus tensors is proposed.     

Finite gradient elasticity and plasticity: a constitutive thermodynamical framework

In Bertram (Continuum Mech Thermodyn. doi: 10.1007/s00161-014-0387-0, 2015), a mechanical framework for finite gradient elasticity and plasticity has been given. In the present paper, this is extended to thermodynamics. The mechanical theory is only briefly repeated here. A format for a rather general constitutive theory including all thermodynamic fields is given in a Euclidian invariant setting. The plasticity theory is rate-independent and unconstrained. The Clausius–Duhem inequality is exploited to find necessary and sufficient conditions for thermodynamic consistency. The residual dissipation inequality restricts the flow and hardening rules in combination with the yield criterion.