The effect of “creep acceleration” in the theory of viscoelasticity

A particular case of constitutive relations in the Pobedrya nonlinear theory of viscoelasticity is considered. It is shown that these relations can be used in the problem of describing the material softening under pulsed loading.     

Coupled nonlinear constitutive models for rarefied and microscale gas flows: subtle interplay of kinematics and dissipation effects

The constitutive relations of gases in a thermal nonequilibrium (rarefied and microscale) can be derived by applying the moment method to the Boltzmann equation. In this work, a model constitutive relation determined on the basis of the moment method is developed and applied to some challenging problems in which classical hydrodynamic theories including the Navier–Stokes–Fourier theory are shown to predict qualitatively wrong results. Analysis of coupled nonlinear constitutive models enables the fundamentals of gas flows in thermal nonequilibrium to be identified: namely, nonlinear, asymmetric, and coupled relations between stresses and the shear rate; and effect of the bulk viscosity. In addition, the new theory explains the central minimum of the temperature profile in a force-driven Poiseuille gas flow, which is a well-known problem that renders the classical hydrodynamic theory a global failure.     

非线性粘弹理论中的单积分型本构关系

本文综述了非线性粘弹理论中的单积分本构表达,评述了多种有代表性的单积分型非线性粘弹理论,对几种本构方程加以分析比较,以揭示它们的内涵,明了其非线性表述原理。      

含裂纹梁刚塑性动态断裂的非线性线弹簧模型分析

用非线性线弹簧模型分析了带裂纹梁的刚塑性动态断裂问题.在塑性势理论基础上,建立了全塑性状态下的弹簧本构关系,并用此关系导出带裂纹梁刚塑性动态断裂分析的基本方程,计算了在冲击载荷作用下,裂纹梁的动态断裂响应.     

The Debye Theory of Rotary Diffusion: History, Derivation, and Generalizations

Motivated by the Debye theory of rotary diffusion in a dipolar fluid, we systematically develop a continuum mechanical theory of rotary diffusion. This theory generalizes classical kinematics to include continuous rotary degrees of freedom and introduces an additional balance law associated with the rotary degrees of freedom. Various constitutive relations are proposed in accordance with standard procedures of nonlinear continuum mechanics. The resulting set of equations provides a properly invariant and thermodynamically consistent theory that allows for constitutive nonlinearities. In particular, the classical Debye theory along with the Nernst-Einstein relations are shown to follow from a special case of linear constitutive relations and an assumption of ideality in which the free energy consists only of a classical entropic contribution. Within our theory, the notion of osmotic pressure arises naturally as a consequence of accounting for forces that act conjugate to the rotary degrees of freedom and serves as the driving force for rotary diffusion. Accepted November 18, 2000?Published online April 23, 2001     

General Reduced Forms of Constitutive Relations in the Classical Continuum Mechanics

A theory of constitutive relations describing the resistance of bodies to deformation is developed. This theory takes into account the presence of internal body forces and internal kinematic constraints. A number of axioms and a general reduced form of constitutive relations for classical media are proposed. For simple bodies it is proved that the Il'yushin and Noll constitutive relations are equivalent.     

A micromechanical analysis of the nonlinear elastic and viscoelastic constitutive relation of a polymer filled with rigid particles

Micromechanical theory is applied to study the nonlinear elastic and viscoelastic constitutive relations of polymeric matrix filled with high rigidity solid particles. It is shown that Eshelby's method can be extended to the case of nonlinear matrix and Eshelby's tensor still exists provided that Poisson's ratio of the nonlinear matrix assumes constant value in deforming process and the rigidity of elastic filling particles is much higher than that of the matrix. A new method for averaging process is proposed to overcome the difficulty that occured in applying the ordinary equivalent inclusion method or the self-consistant method to nonlinear matrices. A rather simple constitutive equation is obtained finally and the strengthening effect of solid particles to composites is investigated. The work supported by the LNM, Institute of Mechanics, Chinese Academy of Sciences and by the National Natural Science Foundation of China     

Potentiality of isotropic nonlinear tensor functions relating two deviators

In the theory of constitutive relations, isotropic quadratic nonlinear tensor functions modeling media with second-order effects, in particular, with misalignment of the force and kinematic tensors, are considered. It is very interesting to consider tensor functions with a scalar potential relating two symmetric deviators of rank two. In this case, the potentiality conditions are integrated, and it is shown that the first integral contains two arbitrary functions of the quadratic invariant of the tensor argument and one arbitrary function of the cubic invariant. A tensorially nonlinear generalization of the rigid-viscoplasticmodel (a two-contact Binghamsolid) is carried out.     

Paradoxical Bending Behavior of Shearable Nonlinearly Elastic Rods

In nonlinear elasticity the exact geometry of deformation is combined with general constitutive relations. This allows a very sophisticated interaction of deformations in different material directions. Based on the Cosserat theory for planar deformations of nonlinearly elastic rods we demonstrate some paradoxical bending effects caused by a nontrivial interaction of extension, flexure, and shear. The analytical results are illustrated by numerical examples.     

A note on incremental equations for a new class of constitutive relations for elastic bodies

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

A generalized multilength scale nonlinear composite plate theory with delamination

A new type of plate theory for the nonlinear analysis of laminated plates in the presence of delaminations and other history-dependent effects is presented. The formulation is based on a generalized two length scale displacement field obtained from a superposition of global and local displacement effects. The functional forms of global and local displacement fields are arbitrary. The theoretical framework introduces a unique coupling between the length scales and represents a novel two length scale or local-global approach to plate analysis. Appropriate specialization of the displacement field can be used to reduce the theory to any currently available, variationally derived, displacement based (discrete layer, smeared, or zig-zag) plate theory.The theory incorporates delamination and/or nonlinear elastic or inelastic interfacial behavior in a unified fashion through the use of interfacial constitutive (cohesive) relations. Arbitrary interfacial constitutive relations can be incorporated into the theory without the need for reformulation of the governing equations. The theory is sufficiently general that any material constitutive model can be implemented within the theoretical framework. The theory accounts for geometric nonlinearities to allow for the analysis of buckling behavior.The theory represents a comprehensive framework for developing any order and type of displacement based plate theory in the presence of delamination, buckling, and/or nonlinear material behavior as well as the interactions between these effects.The linear form of the theory is validated by comparison with exact solutions for the behavior of perfectly bonded and delaminated laminates in cylindrical bending. The theory shows excellent correlation with the exact solutions for both the inplane and out-of-plane effects and the displacement jumps due to delamination. The theory can accurately predict the through-the-thickness distributions of the transverse stresses without the need to integrate the pointwise equilibrium equations. The use of a low order of the general theory, i.e. use of both global and local displacement fields, reduces the computational expense compared to a purely discrete layer approach to the analysis of laminated plates without loss of accuracy. The increased efficiency, compared to a solely discrete layer theory, is due to the coupling introduced in the theory between the global and local displacement fields.     

Distribution-enhanced homogenization framework and model for heterogeneous elasto-plastic problems

Multi-scale computational models offer tractable means to simulate sufficiently large spatial domains comprised of heterogeneous materials by resolving material behavior at different scales and communicating across these scales. Within the framework of computational multi-scale analyses, hierarchical models enable unidirectional transfer of information from lower to higher scales, usually in the form of effective material properties. Determining explicit forms for the macroscale constitutive relations for complex microstructures and nonlinear processes generally requires numerical homogenization of the microscopic response. Conventional low-order homogenization uses results of simulations of representative microstructural domains to construct appropriate expressions for effective macroscale constitutive parameters written as a function of the microstructural characterization. This paper proposes an alternative novel approach, introduced as the distribution-enhanced homogenization framework or DEHF, in which the macroscale constitutive relations are formulated in a series expansion based on the microscale constitutive relations and moments of arbitrary order of the microscale field variables. The framework does not make any a priori assumption on the macroscale constitutive behavior being represented by a homogeneous effective medium theory. Instead, the evolution of macroscale variables is governed by the moments of microscale distributions of evolving field variables. This approach demonstrates excellent accuracy in representing the microscale fields through their distributions. An approximate characterization of the microscale heterogeneity is accounted for explicitly in the macroscale constitutive behavior. Increasing the order of this approximation results in increased fidelity of the macroscale approximation of the microscale constitutive behavior. By including higher-order moments of the microscale fields in the macroscale problem, micromechanical analyses do not require boundary conditions to ensure satisfaction of the original form of Hill's lemma. A few examples are presented in this paper, in which the macroscale DEHF model is shown to capture the microscale response of the material without re-parametrization of the microscale constitutive relations.     

Large deformations of nonlinear viscoelastic and multi-responsive beams

This study presents analyses of deformations in nonlinear viscoelastic beams that experience large displacements and rotations due to mechanical, thermal, and electrical stimuli. The studied beams are relatively thin so that the effect of the transverse shear deformation is neglected, and the stretch along the transverse axis of the beams is also ignored. It is assumed that the plane that is perpendicular to the longitudinal axis of the undeformed beam remains plane during the deformations. The nonlinear kinematics of the finite strain beam theory presented by Reissner [27] is adopted, and a nonlinear viscoelastic constitutive relation based on a quasi-linear viscoelastic (QLV) model is considered for the beams. Deformation in beams due to mechanical, thermal, and electric field inputs are incorporated through the use of time integral functions, by separating the time-dependent function and nonlinear measures of field variables. The nonlinear measures are formulated by including higher order terms of the field variables, i.e. strain, temperature, and electric field. Responses of beams under mechanical, thermal, and electrical stimuli are illustrated and the effects of nonlinear constitutive relations on the overall deformations of the beams are highlighted.     

填药柱壳在内部爆炸载荷下的变形和破坏

采用一种计及三轴因子的损伤模型,以本构关系的内变量理论为基础得到了热塑性本构关系的普适显武表达式;得到了改进的Johnson-Cook本构模型的增量形式; 考虑温度和损伤对材料参数的影响,计入温度和损伤对材料塑性变形发展的耦合作用,给出完备的计算方程组;用Lagrange显示差分的方法对填药柱壳在内部爆炸载荷下的变形和破坏进行了数值模拟,并对结果进行分析,与实验结果进行了比较.     

Green’s function for a prestressed thermoelastic half-space with an inhomogeneous coating

A mathematical model is developed for an inhomogeneous thermoelastic prestressed half-space consisting of a stack of homogeneous or functionally graded layers rigidly attached to a homogeneous base. Each component of the inhomogeneous medium is subjected to initial mechanical stresses and temperature. Successive linearization of the constitutive relations of the nonlinear mechanics of a thermoelastic medium is performed using the theory of superposition of small deformations on finite deformations with the inhomogeneity of the medium taken into account. Integral formulas are derived to explore dynamic processes in inhomogeneous prestressed thermoelastic media.     

A constitutive theory for shape memory polymers. Part II: A linearized model for small deformations

A constitutive theory is developed for shape memory polymers. It is to describe the thermomechanical properties of such materials under large deformations. The theory is based on the idea, which is developed in the work of Liu et al. [2006. Thermomechanics of shape memory polymers: uniaxial experiments and constitutive modelling. Int. J. Plasticity 22, 279-313], that the coexisting active and frozen phases of the polymer and the transitions between them provide the underlying mechanisms for strain storage and recovery during a shape memory cycle. General constitutive functions for nonlinear thermoelastic materials are used for the active and frozen phases. Also used is an internal state variable which describes the volume fraction of the frozen phase. The material behavior of history dependence in the frozen phase is captured by using the concept of frozen reference configuration. The relation between the overall deformation and the stress is derived by integration of the constitutive equations of the coexisting phases. As a special case of the nonlinear constitutive model, a neo-Hookean type constitutive function for each phase is considered. The material behaviors in a shape memory cycle under uniaxial loading are examined. A linear constitutive model is derived from the nonlinear theory by considering small deformations. The predictions of this model are compared with experimental measurements.     

A constitutive theory for shape memory polymers. Part I: Large deformations

A constitutive theory is developed for shape memory polymers. It is to describe the thermomechanical properties of such materials under large deformations. The theory is based on the idea, which is developed in the work of Liu et al. [2006. Thermomechanics of shape memory polymers: uniaxial experiments and constitutive modeling. Int. J. Plasticity 22, 279-313], that the coexisting active and frozen phases of the polymer and the transitions between them provide the underlying mechanisms for strain storage and recovery during a shape memory cycle. General constitutive functions for nonlinear thermoelastic materials are used for the active and frozen phases. Also used is an internal state variable which describes the volume fraction of the frozen phase. The material behavior of history dependence in the frozen phase is captured by using the concept of frozen reference configuration. The relation between the overall deformation and the stress is derived by integration of the constitutive equations of the coexisting phases. As a special case of the nonlinear constitutive model, a neo-Hookean type constitutive function for each phase is considered. The material behaviors in a shape memory cycle under uniaxial loading are examined. A linear constitutive model is derived from the nonlinear theory by considering small deformations. The predictions of this model are compared with experimental measurements.     

Steady-state response of axially moving viscoelastic beams with pulsating speed: comparison of two nonlinear models

Principal parametric resonance in transverse vibration is investigated for viscoelastic beams moving with axial pulsating speed. A nonlinear partial-differential equation governing the transverse vibration is derived from the dynamical, constitutive, and geometrical relations. Under certain assumption, the partial-differential reduces to an integro-partial-differential equation for transverse vibration of axially accelerating viscoelastic nonlinear beams. The method of multiple scales is applied to two equations to calculate the steady-state response. Closed form solutions for the amplitude of the vibration are derived from the solvability condition of eliminating secular terms. The stability of straight equilibrium and nontrivial steady-state response are analyzed by use of the Lyapunov linearized stability theory. Numerical examples are presented to highlight the effects of speed pulsation, viscoelascity, and nonlinearity and to compare results obtained from two equations.     

Finite element computation of torsional plastic waves in a thin-walled tube

Summary  A finite element technique is presented for the analysis of one-dimensional torsional plastic waves in a thin-walled tube. Three different nonlinear consitutive relations deduced from elementary mechanical models are used to describe the shear stress–strain characteristics of the tube material at high rates of strain. The resulting incremental equations of torsional motion for the tube are solved by applying a direct numerical integration technique in conjunction with the constitutive relations. The finite element solutions for torsional plastic waves in a long copper tube subjected to an imposed angular velocity at one end are given, and a comparison with available experimental results to assess the accuracy of the constitutive relations considered is conducted. It is demonstrated that the strain-rate dependent solutions show a better agreement with the experimental results than the strain-rate independent solutions. The limitations of the constitutive equations are discussed, and some modifications are suggested. Received 9 February 1999; accepted for publication 28 March 2000