Constitutive equations for the nonlinear viscoelastic and viscoplastic behavior of thermoplastic elastomers

Observations are reported in uniaxial tensile tests with constant strain rates at moderate finite deformations, as well as in creep and relaxation tests on a thermoplastic elastomer (ethylene–octene copolymer) at room temperature. A constitutive model is developed for the viscoelastic and viscoplastic responses of a polymer at arbitrary three-dimensional deformations with finite strains. A thermoplastic elastomer is treated as an incompressible heterogeneous transient network of strands. Its viscoelastic behavior is associated with separation of active strands from their junctions and merging of dangling strands with the network. The viscoplastic response reflects sliding of junctions between strands with respect to their reference positions. Stress–strain relations are derived by using the laws of thermodynamics. They involve six adjustable parameters that are found by fitting the experimental data. To examine the accuracy of the model predictions, plane-strain compressive tests with constant strain rates and relaxation tests at compression are performed. Good agreement is demonstrated between the observations and the results of numerical simulation.     

Asymptotically approximate model equations for nonlinear dispersive waves in incompressible elastic rods

Summary In literature, nonlinear waves in elastic rods have been studied by many authors. Usually, the Navier-Bernoulli hypothesis (the assumption that plane cross-sections remain planar and normal to the rod axis) is used. Intuitively, one would expect that this would be a good approximation when one is mainly interested in longitudinal waves. However, there are no rigorous theoretical justifications available. Also, a defect of this assumption is that comparing with the exact three-dimensional theory the boundary conditions on the lateral surface can never be satisfied. Recently, three papers have been published to overcome this defect, but they contain some algebraic errors (which implies that the approach adopted there cannot be used to overcome this defect). So, this problem remains open. In this paper, we present our recent research results for this problem, and we have managed to establish asymptotically valid one-dimensional rod equations which are consistent with the lateral boundary conditions. Further, their dispersion relation can match with that of the exact three-dimensional field equations to any asymptotic order in the long-wave limit. For solitary waves in the far field, we derive to the leading order the KdV equation. Comparing its solitary-wave solution with that of the KdV equation obtained through the Navier-Bernoulli hypothesis, we find that the difference is very small. This provides some evidence to the validity of the assumption that plane cross sections remain planar and normal to the rod axis.     

Constitutive modeling of the anisotropic behavior of Mullins softened filled rubbers

Original constitutive modeling is proposed for filled rubber materials in order to capture the anisotropic softened behavior induced by general non-proportional pre-loading histories. The hyperelastic framework is grounded on a thorough analysis of cyclic experimental data. The strain energy density is based on a directional approach. The model leans on the strain amplification factor concept applied over material directions according to the Mullins softening evolution. In order to provide a model versatile that applies for a wide range of materials, the proposed framework does not require to postulate the mathematical forms of the elementary directional strain energy density and of the Mullins softening evolution rule. A computational procedure is defined to build both functions incrementally from experimental data obtained during cyclic uniaxial tensile tests. Successful comparisons between the model and the experiments demonstrate the model abilities. Moreover, the model is shown to accurately predict the non-proportional uniaxial stress-stretch responses for uniaxially and biaxially pre-stretched samples. Finally, the model is efficiently tested on several materials and proves to provide a quantitative estimate of the anisotropy induced by the Mullins softening for a wide range of filled rubbers.     

On the nonlinear elastic response of bodies in the small strain range

The classical linearized approximation to describe the elastic response of solids is the most widely used model in solid mechanics. This approximate model is arrived at by assuming that the norm of the displacement gradient is sufficiently small so that one can neglect the square of the norm in terms of the norm. Recent experimental results on Titanium and Gum metal alloys, among other alloys, indicate with unmistakable clarity a nonlinear relationship between the strain and the stress in the range of strain wherein one would have to use the classical linearized theory of elasticity, namely wherein the square of the norm of the strain can be ignored with regard to the value of the strain, leading to a dilemma concerning the modeling of the response, as the classical nonlinear Cauchy elastic model would collapse to the linearized elastic model in this range. A novel and important generalization of the theory of elastic materials has been suggested by Rajagopal in Appl Math 48: 279–319, 2003 and Zeit Angew Math Phys 58: 309–317, 2007 that allows for an approximation wherein the linearized strain can be a nonlinear function of the stress. In this paper, we show how this new theory can be used to describe the new experiments on Titanium and Gum metal alloys and also clarify several issues concerning the domain of application of the classical linearized theory.     

Constitutive equations for orthotropic nonlinear viscoelastic behaviour using a generalized Maxwell model Application to wood material

This paper presents a nonlinear viscoelastic orthotropic constitutive equation applied to wood material. The proposed model takes into account mechanical and mechanosorptive creep via a 3D stress ratio and moisture change rate for a cylindrical orthotropic material. Orthotropic frame is based on the grain direction (L), radial (R) and hoop (T) directions, which are natural wood directions. Particular attention is taken to ensure the model to fulfill the necessary dissipation conditions. It is based on a rheological generalized Maxwell model with two elements in parallel in addition with a single linear spring taking into account the long term response. The proposed model is implemented in the finite element code ABAQUS/Standard^® via a user subroutine UMAT and simple example is shown to demonstrate the capability of the proposed model. Future works would deal with damage and fracture prediction for wooden structures submitted to climate variations and mechanical loading.     

Constitutive potentials for dilutely voided nonlinear materials

Constitutive relations are derived for an incompressible, isotropic power-law matrix material containing a dilute concentration of spherical voids. The derivation is made for a nonlinearly viscous material used to characterize steady creep. However, the theory applies equally well to small strain nonlinear elasticity (deformation theory), and an extension to a rate-independent flow theory is also discussed. The starting point and key element in the formulation is the potential function for an isolated spherical void in an infinite block of power-law material. Approximate, but accurate, representations for this potential function are given. The overall constitutive relation governing the behavior of the dilutely voided solid is obtained simply and directly using the void potential. An assessment of the range of validity of the dilute concentration results is obtained using numerical solutions to the problem of a spherical void centered in a sphere of finite radius made of the power-law material. The potential function is also given for a dilute concentration of aligned penny-shaped cracks in the same power-law material.     

Constitutive equation for a class of non-simple elastic materials

Summary Initial yield is the upper limit of the purely elastic deformation behaviour of an elasticplastic solid. Thus the choice of the constitutive equation describing the purely elastic deformation behaviour determines the initial yield function. The constitutive equation of a simple elastic material is only compatible with von Mises yield criterion, a conclusion which applies also to the classical infinitesimal theory. A more general form of constitutive equation for an elastic material is formulated by way of the concept of a stress loading function, the proposed constitutive equation being quadratic in the stress. The two loading coefficients associated with the stress loading function are assumed to be deriveable from a generalised isotropic yield criterion which is now assumed to hold over the entire range of deformation, and in this context is referred to as the stress intensity function. The proposed constitutive equation has the same representation in terms of the left Cauchy-Green deformation tensor as that for a simple elastic material. Using the Cayley-Hamilton theorem, this representation is rearranged and expressed in terms of a measure of finite strain which is defined to be one quarter of the difference between the left Cauchy-Green deformation tensor and its inverse. In this way the strain properties of the proposed constitutive equation are formulated by way of the concept of a strain response function. The three response coefficients associated with the strain response function are assumed to be deriveable from a generalised, isotropic, strain intensity function. The predictions of the proposed constitutive equation are considered in the context of the combined stressing of a thin sheet of incompressible material. In this way, it is shown that the proposed constitutive equation is not limited in the same way as the constitutive equation of a simple elastic material.     

Evolution equations for nonlinear waves in a tapered elastic tube filled with a viscous fluid

In this work, employing the reductive perturbation method and treating the arteries as a tapered, thin walled, long and circularly conical prestressed elastic tube, the propagation of weakly nonlinear waves is investigated in such a fluid-filled elastic tube. By considering the blood as an incompressible viscous fluid, depending on the viscosity and perturbation parameters we obtained various evolution equations as the extended Korteweg-de Vries (KdV), extended KdV Burgers and extended perturbed KdV equations. Progressive wave solutions to these evolution equations are obtained and it is observed that the wave speeds increase with the distance for negative tapering while they decrease for positive tapering.     

Constitutive equations in Cosserat elasticity

Summary A new theory for the constitutive equations in Cosserat elasticity is proposed. It is based on the assumption that the rotation vector depending on the displacement vector should be coupled with a rotation vector independent of the displacement vector. This eliminates the indeterminancies in stress and couple-stress encountered earlier.     

Constitutive equations optimized for determining strengths of metallic alloys

We investigate compatibilities of three constitutive equations, the Hollomon, the Swift, and the Voce equations for determination of yield and ultimate tensile strengths based on tensile true stress–strain curves of 27 metal alloys including those with power-law type and linear-type strain-hardening. We analyze each constitutive equation in terms of yield strength determined by the intercept of the linear elastic loading curve and plastic flow curve and ultimate tensile strength evaluated by the concept of instability in tension. We found that the describing plastic flow is very sensitive in determination of the yield strength and tensile strength from parameters of constitutive equation. Voce equation gives estimate yield strength and tensile strength better than Hollomon and Swift equations.     

Constitutive equations of ageing polymeric materials

In a previous paper, a constitutive equation of relaxation behaviour of time-dependent chemically unstable materials has been developed by employing the irreversible thermodynamics of internal variables and Eyring's absolute reaction theory. In that paper, a theoretical expression for the effect of chemical crosslink density,v, on the relaxation rate has been developed. In this paper the creep behaviour of a network polymer undergoing a scission process has been developed. The temperature effect using the WLF equation on the coupled chemomechanical behaviour has also been incorporated into the equation.     

Effect of nonlinear response of concrete on its elastic modulus and strength

The purpose of this paper is to investigate the effect of nonlinear response of concrete on the relationship between modulus of elasticity at static and dynamic loading as well as on its strength. The obtained relationships are based on the thermofluctuation strength theory coupled with a nonlinear stress–strain material model. From the corresponding equations it was found that the ratio of the static to the dynamic modulus of elasticity depends on the strength of concrete, its temperature, rate of loading. Also it was confirmed that the dynamic modulus is greater than static modulus of elasticity. These equations explain the influence of the value of applied stress on the value of the static or dynamic modulus of elasticity. Comparative study shows substantial agreement with existing experimental results and the general equations given in standard BS 8110, Part 2:1985, and ACI documents. Based on the obtained relationships new methods for evaluating the static modulus of concrete and its strength from the results of dynamic tests are described subsequently.     

Constitutive equations for amended non-Gaussian network models of rubber elasticity

New constitutive equations based on an amended form of the Kuhn–Grün probability distribution function due to Jernigan and Flory are derived from the standard James–Guth (JG) 3-chain and Arruda–Boyce (AB) 8-chain non-Gaussian molecular network models. The kinematics describing the stretch of a 1-chain model in an affine deformation shows that the relative stretch of a single molecular chain initially oriented along the diagonal of a cube is determined by the first principal invariant of the Cauchy–Green deformation tensor. The Kuhn–Grün probability distribution for a randomly oriented chain and its more general amended form due to Wang and Guth, are functions of only the relative chain stretch. Hence, any non-Gaussian network model for which the configurational entropy of all chains may be uniform is characterized by an elastic response function that depends on only the first principal invariant of the Cauchy–Green deformation tensor. Both the regular and amended AB 8-chain models are characterized by specific response functions in this class; the regular and amended JG 3-chain models, however, are not. An amended form of the phenomenological composite 3-chain/8-chain model suggested by Wu and van der Giessen is introduced. Analytical relations for several kinds of homogeneous deformations of the standard and amended models are compared with a variety of experimental data by others. It is found that results for the amended 3-chain and 8-chain models do not vary significantly from results for the corresponding regular models. The composite model, on the other hand, shows excellent overall agreement with the diverse data, including equibiaxial deformations for which other models show greater variance; but it offers no improvement in comparison with data for plane strain compression. Some remarks relating the chain parameters of the 3-chain and 8-chain network models, and the limiting chain and continuum stretches for these models are discussed in an appendix.     

Integral results for nonlinear diffusion equations

We show how to construct integral results for the multi-dimensional nonlinear diffusion equation c/t=·(D(c)c) and for some generalisations of this. For appropriate boundary conditions these become integral invariants. An application of these results to determining the large-time behaviour of some radially symmetric problems is indicated.     

Constitutive equation of a non-simple elastic plastic material

Summary The mechanical response predicted by the constitutive equation of a non-simple elastic material is considered in relation to the total strain behaviour of an elastic-plastic solid extensively deformed in the range of plastic strain. Both loading and unloading are considered in relation to the range of total elastic-plastic strain. In the absence of appropriate experimental studies, comparison of the predictions of the proposed constitutive equation of a non-simple elastic material, when applied to the work-hardening behaviour of the material, has been restricted to a study of the characteristic stress-strain behaviour of a strain hardening material. This has centred on the correlation of stress-strain curves characteristic of the mechanical response of a material tested in simple compression, simple torsion and pure shear with the object of obtaining a universal stress-strain curve.With 1 Figure