General stress-strain relations in non-linear theory of viscoelasticity for large deformations

Conclusion General phenomenoligical stress-strain relations in non-linear theory of visco-elasticity for large deformations have been presented.In the first place, according to V. V. Novozhilov 1 we express the generalized equilibrium equation for large deformations in the Lagrange representation, and we apply the generalized Hamilton's principle to the equation of energy conservation, which denotes that the sum of the elastic energy and the dissipative energy is equal to the work done by the body force and the surface on the substance; so that we obtain the required general stress-strain relations in comparison with the above two equations.On the condition that the elastic potential is a function only of the strain, and the dissipation function is a function of the rate of strain and of strain; such a substance is reduced to the Voigt material necessarily, and the stresses which act on the substance are given by the sum of elastic- and viscous stresses, and the stress-strain relations are reduced to the so-called Lagrangian form.If elongations, shears and angles of rotation are small and also the strains and rates of strain are sufficiently small, the stress-strain relations are expressed by a linear Voigt model constituting a Hookian spring in parallel with a Newtonian dashpot.Non-linearity in the theory is classified into two groups i. e. the geometrical non-linearity and the physical non-linearity. The former is introduced into the theory through the definition of the generalized strain and of the generalized stress and through the equilibrium equation for large deformation, and the latter through the general stress-strain relations.The main result of this paper is that the general stress-strain relations in viscoelasticity are deduced necessarily from the physically appropriate assumptions.     

Variational principles in elasticity with nonlinear stress-strain relation

Since 1979, a series of papers have been published concerning the variational principles and generalized variational principles in elasticity such as [1] (1979), [6] (1980), [2,3] (1983) and [4,5] (1984). All these papers deal with the elastic body with linear stress-strain relations. In 1985, a book was published on generalized variational principles dealing with some nonlinear elastic body, but never going into detailed discussion. This paper discusses particularly variational principles and generalized variational principles for elastic body with nonlinear stress-strain relations. In these discussions, we find many interesting problems worth while to pay some attention. At the same time, these discussions are also instructive for linear elastic problems. When the strain is small, the high order terms may be neglected, the results of this paper may be simplified to the well-known principles in ordinary elasticity problems.     

A variational definition of the strain energy for an elastic string

Using the variational point of view, the constitutive equations of an elastic one-dimensional string are deduced from the stress-strain relations of nonlinear three-dimensional elasticity, by passing to the limit when the other dimensions go to zero. The assumptions made on the three-dimensional model are not very restrictive.     

Inversion of a full-range stress-strain relation for stainless steel alloys

This paper presents an approximate inversion of the stress-strain relation for stainless steel alloys. Using currently available stress-strain relations based on a modified Ramberg-Osgood equation, a new expression for the stress σ as an explicit function of the total strain ε is obtained. The new expression is valid over the full-range of the stress well beyond the 0.2% proof stress σ_0.2, defined as the stress level corresponding to the plastic strain value of 0.2%. The validity of the inverted expression is tested over a wide range of material parameters. The tests show that the new expression results in stress-strain curves which are both qualitatively and quantitatively consistent with the fully iterated numerical solution of the full-range stress-strain relation.     

ON LARGE DEFORMATION UNILATERAL CONTACT PROBLEM WITH FRICTION (I)—INCREMENTAL VARIATIONAL EQUATIONS

Based on the theory and technique of nonlinear geometric field theory of continuum, a more general incremental variational equation for elastic and plastic large deformation in co-moving coordinate is established in this paper. An expression for two and three-dimensional continua is derived, and the incremental variational equation for large deformation of changing boundary contact and the variational inequality in rate form are obtained, which provides the theoretical basis for the computation of elastic-plastic large deformation contact problem with friction.     

Consequences of the dissipative restrictions in finite anisotropic elasto-plasticity

The paper is devoted to the dissipation postulate in anisotropic finite elastoplasticity, properly formulated in terms of the total strain histories, on small cycles only. The equivalence between the dissipation postulate and the existence of the stress potential together with the dissipation inequality is proved. The modified flow rules compatible with the dissipation postulate follow as a necessary condition. The convexity and normality properties can be treated as an equivalent issue of the dissipation postulate only within the framework of Σ models. We identify such classes of Σ-models based on the pre-image theorem. The difficulties arise from the non-injectivity of the Mandel's stress measure, as dependent on the elastic strain. We define the yield stress function and the admissible elastic stress range in Σ-space. The equivalence is achieved only if it possible to construct the elastic range in strain space, having just the topological properties originally assumed as a basis of the dissipation postulate. The normality to the admissible elastic stress range does not mean an associative flow rule. The results are exemplified for transversely isotropic elasto–plastic materials as well as for models with small elastic strains.     

Fundamental equations of certain electromagnetic-acoustic discontinuous fields in variational form

We present, in the first part of the paper, the well-known fundamental electromagnetic-acoustic equations, that is, the coupled Maxwells and Newtons equations for an elastic dielectric continuum in differential form, and we also discuss the uniqueness of their linear solutions. In the second part, from a general principle of physics, we deduce a three-field variational principle that operates on the mechanical displacements, the electric potential, and the electromagnetic vector potential of the dielectric continuum. Then, we extend it through an involutory (or Friedrichss) transformation in deriving a nine-field unified variational principle that operates on the mechanical, electrical, and magnetic continuous linear fields under the infinitesimal strains. This variational principle generates Maxwells and Newtons equations, the coupled linear constitutive relations, and the associated natural boundary conditions for the regular region of the dielectric continuum as its Euler-Lagrange equations. In the third part, we further generalise the unified variational principle so as to incorporate the jump conditions across a surface of discontinuity within the dielectric region. We also show that the integral and differential types of variational principles that apply to the linear motions of the elastic dielectric region with a fixed internal surface of discontinuity are in agreement with and recover, as special cases, some of the earlier variational principles. Further, the variational principles may be directly used in linear electromagnetic and/or acoustic field computations and in consistently establishing the lower order one- or two-dimensional equations of the elastic dielectric continuum.Received: 9 January 2002, Accepted: 26 May 2003, Published online: 5 December 2003PACS: 03.40, 41.10, 77.60 Correspondence to: G.A. Altay     

Elastic dynamics variational principle for nonlinear elastic body

Based on paper, the variational principle and generalized variational principle of elastic dynamics for elastic body with nonlinear stress-strain relations are introduced in this paper. The generalized instantaneous variational principle is also raised for the mixed harmonious displacement element and the mixed harmonious stress element.     

A damage model for crack initiation and growth

A damage accumulation model is presented for the study of the problem of crack initiation and stable growth in an elastic-plastic material. A centre-cracked specimen subjected to a uniform stress perpendicular to the crack plane is considered. A coupled stress and failure analysis is performed by using a finite element computer program based on J₂-plasticity theory in conjunction with the strain energy density theory. After initial yielding, each material element follows a different equivalent uniaxial stress-strain behavior depending on the amount of energy dissipation by permanent deformation. A host of uniaxial stress-strain curves constituting parts of the same stress-strain curve were assigned to material elements for each increment of loading. The path-dependent nature of the onset of crack initiation and growth was revealed. The proposed model predicts faster crack growth rates than those obtained on the basis of a single uniaxial stress-strain curve and is closer to experimental observation.     

Anisotropic Hyperelasticity Based Upon General Strain Measures

This paper presents stress-strain constitutive equations for anisotropic elastic materials. A special attention is given to the logarithmic strain. Assuming a constitutive equation for the specific internal energy the equation governing the Cauchy stress is derived. Mathematical relations presented take a relatively simple form and concern a very wide class of elastic materials. The dependence of third-order elastic constants on the choice of strain measure is shown. The third-order elastic constants measured experimentally in relation to the Green strain are recalculated here for the logarithmic strain. This revised version was published online in July 2006 with corrections to the Cover Date.     

不同应变率下聚乙烯材料的压缩力学性能

为了研究聚乙烯材料在不同应变率下的压缩力学性能,通过准静态实验和动态实验获得聚乙烯材料不同应变率下的应力应变曲线,分析发现:聚乙烯的弹性模量和屈服强度随应变率增大而增大,具有明显的黏弹塑性;聚乙烯材料进入塑性阶段,其应力应变曲线在不同应变率下具有相近的变化趋势,即塑性切向模量近似相同。根据聚乙烯材料的压缩力学性能,建立了弹性区、屈服点和塑性区的分段本构模型。该模型的屈服点和塑性段与实验结果吻合较好,由于弹性段采用线弹性模型,与实验结果存在一定偏差,可近似描述材料的弹性行为。     

一种孔隙介质力学模型及在水泥基材料的应用

基于细观力学和断裂力学的基本理论提出一个新的分析模型, 对孔隙介质的力学性能进行了分析. 依据孔隙介质内部孔隙的几何描述和状态参数,如孔隙率、形状、尺度及分布等,通过等效夹杂理论获得孔隙介质的等效本构方程,其最终变量为应力、应变和孔隙的形态参数. 根据断裂理论中材料承受载荷作用下破坏增长过程中的能量守恒,对孔隙介质变形过程中机械能、弹性应变能和载荷提供的势能进行分析, 根据能量守恒定律建立能量守恒方程,其最终变量也为应力、应变和孔隙的形态参数. 根据等效本构方程和能量守恒方程,获得孔隙介质承受载荷作用下的应力应变关系. 最后将该力学模型应用于水泥基材料,计算水泥基材料的力学性能并与文献中的结果进行对比分析,结果显示模型的计算结果准确有效.      

A micromechanics-based nonlocal constitutive equation incorporating three-point statistics for random linear elastic composite materials

A variational formulation employing the minimum potential and complementary energy principles is used to derive a micromechanics-based nonlocal constitutive equation for random linear elastic composite materials, relating ensemble averages of stress and strain in the most general situation when mean fields vary spatially. All information contained in the energy principles is retained; we employ stress polarization trial fields utilizing one-point statistics so that the resulting nonlocal constitutive equation incorporates up through three-point statistics. The variational structure is developed first for arbitrary heterogeneous linear elastic materials, then for randomly inhomogeneous materials, then for general n-phase composite materials, and finally for two-phase composite materials, in which case explicit variational upper and lower bounds on the nonlocal effective modulus tensor operator are derived. For statistically uniform infinite-body composites, these bounds are determined even more explicitly in Fourier transform space. We evaluate these in detail in an example case: longitudinal shear of an aligned fiber or void composite. We determine the full permissible ranges of the terms involving two- and three-point statistics in these bounds, and thereby exhibit explicit results that encompass arbitrary isotropic in-plane phase distributions; we also develop a nonlocal “Milton parameter”, the variation of whose eigenvalues throughout the interval [0, 1] describes the full permissible range of the three-point term. Example plots of the new bounds show them to provide substantial improvement over the (two-point) Hashin–Shtrikman bounds on the nonlocal operator tensor, for all permissible values of the two- and three-point parameters. We next discuss further applications of the general nonlocal operator bounds: to any three-dimensional scalar transport problem e.g. conductivity, for which explicit results are given encompassing the full permissible ranges of the two- and three-point statistics terms for arbitrary three-dimensional isotropic phase distributions; and to general three-dimensional composites, where explicit results require future research. Finally, we show how the work just summarized, treating elastostatics, can be generalized to elastodynamics, first in general, then explicitly for the longitudinal shear example.     

An adaptation of finite linear viscoelasticity theory for rubber-like viscoelasticity by use of a generalized strain measure

In this paper a simplified three-dimensional constitutive equation for viscoelastic rubber-like solids is derived by employing a generalized strain measure and an asymptotic expansion similar to that used by Coleman and Noll (1961) in their derivation of finite linear viscoelasticity (FLV) theory. The first term of the expansion represents exactly the time and strain separability relaxation behavior exhibited by certain soft polymers in the rubbery state and in the transition zone between the glassy and rubbery states. The relaxation spectra of such polymers are said to be deformation independent. Retention of higher order terms of the asymptotic expansion is recommended for treating deformation dependent spectra.Certain assumptions for the solid theory are relaxed in order to obtain a constitutive equation for uncross-linked liquid materials which exhibit large elastic recovery properties.Apart from the strain energyW(I₁,I ₂), which alternatively characterizes the long-time elastic response of solids or the instantaneous elastic response of elastic liquids, only the linear viscoelastic relaxation modulus is required for the first-order theory. Both types of material functions can be obtained, in theory, from simple laboratory testing procedures. The constitutive equations for solids proposed by Chang, Bloch and Tschoegl (1976) and a special form of K-BKZ theory for elastic liquids are shown to be particular cases of the first-order theory.Previously published experimental data on a cross-linked styrene-butadiene rubber (SBR) and an uncross-linked polyisobutylene (PIB) rubber is used to corroborate the theory.     

The role of dissipation and defect energy in variational formulations of problems in strain-gradient plasticity. Part 1: polycrystalline plasticity

A general set of flow laws and associated variational formulations are constructed for small-deformation rate-independent problems in strain-gradient plasticity. The framework is based on the thermodynamically consistent theory due to Gurtin and Anand (J Mech Phys Solids 53:1624–1649, 2005), and includes as variables a set of microstresses which have both energetic and dissipative components. The flow law is of associative type. It is expressed as a normality law with respect to a convex but otherwise arbitrary yield function, or equivalently in terms of the corresponding dissipation function. Two cases studied are, first, an extension of the classical Hill-Mises or J ₂ flow law and second, a form written as a linear sum of the magnitudes of the plastic strain and strain gradient. This latter form is motivated by work of Evans and Hutchinson (Acta Mater 57:1675–1688, 2009) and Nix and Gao (J Mech Phys Solids 46:411–425, 1998), who show that it leads to superior correspondence with experimental results, at least for particular classes of problems. The corresponding yield function is obtained by a duality argument. The variational problem is based on the flow rule expressed in terms of the dissipation function, and the problem is formulated as a variational inequality in the displacement, plastic strain, and hardening parameter. Dissipative components of the microstresses, which are indeterminate, are absent from the formulation. Existence and uniqueness of solutions are investigated for the generalized Hill-Mises and linear-sum dissipation functions, and for various combinations of defect energy. The conditions for well-posedness of the problem depend critically on the choice of dissipation function, and on the presence or otherwise of a defect energy in the plastic strain or plastic strain gradient, and of internal-variable hardening.     

On a Fluid-Structure Model in Which the Dynamics of the Structure Involves the Shear Stress Due to the Fluid

In this paper we consider a model for fluid-structure interaction. The hybrid system describes the interaction between an incompressible fluid in a three-dimensional container with interior a fixed domain and a thin elastic plate, the interface, which coincides with a flexible flat part of the surface of the vessel containing the fluid. The motion of the fluid is described by the linearized Navier–Stokes equations and the deformation of the plate by the classical plate equations for in-plane motions, modified to include the viscous shear stress which the fluid exerts on the plate as well as damping of Kelvin–Voigt type. We establish the existence of a unique weak solution of the interactive system of partial differential equations by considering an appropriate variational formulation. Uniform stability of the energy associated with the model is shown under the assumption that the potential plate energy is dominated by the dissipation induced by the viscosity of the fluid. The retention of the physical parameters in the problem is an a priori requirement in this physical condition.      

A non-linear uniaxial integral constitutive equation incorporating rate effects,creep and relaxation

A previously proposed first order non-linear differential equation for uniaxial viscoplasticity, which is non-linear in stress and strain but linear in stress and strain rates, is transformed into an equivalent integral equation. The proposed equation employs total strain only and is symmetric with respect to the origin and applies for tension and compression. The limiting behavior for large strains and large times for monotonic, creep and relaxation loading is investigated and appropriate limits are obtained. When the equation is specialized to an overstress model it is qualitatively shown to reproduce key features of viscoplastic behavior. These include: initial linear elastic or linear viscoelastic response: immediate elastic slope for a large instantaneous change in strain rate normal strain rate sensitivity and non-linear spacing of the stress-strain curves obtained at various strain rates; and primary and secondary creep and relaxation such that the creep (relaxation) curves do not cross. Isochronous creep curves are also considered. Other specializations yield wavy stress-strain curves and inverse strain rate sensitivity. For cyclic loading the model must be modified to account for history dependence in the sense of plasticity.     

Anisotropic plastic stress fields at a slowly propagating crack tip

Under the condition that any perfectly plastic stress components at a crack tip are nothing but the functions of 0 only making use of equilibrium equations. Hill anisotropic yield condition and unloading stress-strain relations, in this paper, we derive the general analytical expressions of anisotropic plastic stress fields at the slowly steady propagating tips of plane and anti-plane strain. Applying these general analytical expressions to the concrete cracks, the analytical expressions of anisotropic plastic stress fields at the-slowly steady propagating tips of Mode I and Mode III cracks are obtained. For the isotropic plastic material, the anisotropic plastic stress fields at a slowly propagating crack tip become the perfectly plastic stress fields.     

The molecular theory of viscoelasticity for thermoplastic elastomer SBS (SIS) at large deformations

A microstructure model for SBS and SIS triblock copolymers with hard domains as multifunctional reinforcing fillers is proposed. Based on this model and proposed mechanism of large deformations, the probability distribution function of the end-to-end vector for each constituent chain and the free energy of deformation for the total networks was calculated by the combination of statistical thermodynamics and kinetics. A new molecular theory of non-linear visco-elasticity for SBS and SIS at large deformations is presented. It is successful in relating the viscoelastic state to molecular constitution by three important parameters (C ₁₀₀,C ₀₂₀, andC ₂₀₀) of the networks. The relations of stress to strain for four types of deformation, the elastic modulus and the constitutive equation for the stress relaxation were derived from this theory. It provides a theoretical foundation for studying the relationships of multiphase network structures and mechanical properties at large deformations. An excellent agreement between the theoretical relationships and experimental data from the experiments and the reference was obtained.Project supported by the National Natural Foundation of China