Design sensitivity analysis for parameters affecting geometry,elastic–viscoplastic material constant and boundary condition by consistent tangent operator-based boundary element method

This paper presents a design sensitivity analysis method by the consistent tangent operator concept-based boundary element implicit algorithm. The design variables for sensitivity analysis include geometry parameters, elastic–viscoplastic material parameters and boundary condition parameters. Based on small strain theory, Perzyna’s elastic–viscoplastic material constitutive relation with a mixed hardening model and two flow functions is considered in the sensitivity analysis. The related elastic–viscoplastic radial return algorithm and the formula of elastic–viscoplastic consistent tangent operator are derived and discussed. Based on the direct differentiation approach, the incremental boundary integral equations and related algorithms for both geometric and elastic–viscoplastic sensitivity analysis are developed. A 2D boundary element program for geometry sensitivity, elastic–viscoplastic material constant sensitivity and boundary condition sensitivity has been developed. Comparison and discussion with the results of this paper, analytical solution and finite element code ANSYS for four plane strain numerical examples are presented finally.     

金属刚粘塑性变形的能量泛函与动力显式算法有限元分析

设计出刚粘塑性动态变形场的能量泛函，采用罚函数法和Ｌａｇｒａｎｇｉａｎ乘子法取消对运动容许速度场满足体积不可压缩的约束条件，运用虚功原理和广义变分原理，推导出刚粘塑性变形的动力分析显式算法有限元方程，以及速率形式的中心差分求解的时间积分显式，给出了两个金属变形的算例。     

An implicit algorithm for hypoelasto-plastic and hypoelasto-viscoplastic endochronic theory in finite strain isotropic–kinematic-hardening model

This paper is concerned with objective stress update algorithm for elasto-plastic and elasto-viscoplastic endochronic theory within the framework of additive plasticity. The elastic response is stated in terms of hypoelastic model and endochronic constitutive equations are stated in unrotated frame of reference. A trivially incrementally objective integration scheme for rate constitutive equations is established. Algorithmic modulus consistent with numerical integration algorithm of constitutive equations is extracted. The implementation is validated by means of a set of simple deformation paths (simple shear, extension and rotation), two benchmark test in nonlinear mechanics (the necking of a circular bar and expansion of a thick-walled cylinder), a test which demonstrates the capabilities of the proposed model in simulation of cyclic loading and ratcheting in finite strain case (cyclically loaded notched bar) and finally, the analysis of a tensile test, which presents a shear band with a finite thickness independent of the finite element mesh using endochronic viscoplastic constitutive model.     

梁中复合应力波的传播

采用有限差分法讨论了梁中复合应力波的传播．给出了粘塑性悬臂梁当自由端受突加弯矩载荷作用时梁内复合应力波传播的基本图象。指出，在冲击早期响应阶段．截面横向转动惯性效应起着重要作用，是不可忽视的。标志弹塑性边界的塑性效，一开始由自由端向固定端运动，但在反射卸载波的迎面作用下，会出现回退现象。在波动早期阶段，固定端主要处于弹性变形状态。此外，还对弹塑性梁中复合应力波的控制方程进行了必要的讨论。     

Homogenized elastic–viscoplastic behavior of anisotropic open-porous bodies with pore pressure

Constitutive modeling is studied for the homogenized elastic–viscoplastic behavior of pore-pressurized anisotropic open-porous bodies made of metallic base solids at small strains and rotations. For this purpose, by describing micro–macro relations relevant to periodic unit cells of anisotropic open-porous bodies subjected to pore pressure, constitutive features are discussed for the viscoplastic macrostrain rate in steady states. On the basis of the constitutive features found, the viscoplastic macrostrain rate is represented as an anisotropic function of Terzaghi’s effective stress, which is shown using Hill’s macrohomogeneity condition. The resulting viscoplastic equation is used to simulate the homogenized elastic–viscoplastic behavior of an ultrafine plate-fin structure subjected to uniaxial/biaxial loading in addition to pore pressure. The corresponding finite element homogenization analysis is also performed for comparison. It is demonstrated that the developed viscoplastic equation simulates well the anisotropic effect of pore pressure in the viscoplastic range in spite of there being no anisotropic factor and no fitting parameter in Terzaghi’s effective stress itself.     

非连续变形计算力学模型的粘弹性分析方法Ⅰ:基本理论

基于粘弹性广义有限单元和接触力元,发展了适用于多体相互作用系统非连续变形分析的粘弹性数值分析方法,通过虚功原理,给出了其分区参变量最小势能原理,从而阐明了其理论基础。粘弹性广义有限单元的本构关系可由粘弹性退化为弹性或刚性,因此本文所提出的方法可对由刚体、弹性体和粘弹性体所构成的复杂多体系统在外荷载作用下的力学行为进行数值模拟,同时能够比本文精确地直接得到多体之间的接触应力。     

基于显式有限元方法的两相介质弹塑性动力反应计算分析

针对增量形式的流体饱和两相多孔介质弹塑性波动方程组,运用基于显式逐步积分格式的时域显式有限元方法对该波动方程组进行求解,并应用基于SMP破坏准则的弹塑性动力本构模型描述两相介质的动力反应性质,对两相介质在输入地震波作用下的弹塑性动力反应进行计算和分析,将计算结果与相应的弹性动力反应的计算结果进行对比;对本文应用的弹塑性...     

Micromechanical analysis of the finite elastic–viscoplastic response of multiphase composites

Nonlinear thermoelastic–viscoplastic constitutive equations for large deformations with isotropic and directional hardening, are incorporated into a micromechanical finite strain analysis. As a result of this analysis, which is based on the homogenization technique for periodic microstructures, a global thermoinelastic constitutive law is established that governs the overall response of multiphase materials under finite deformations. This constitutive law is expressed in terms of the instantaneous effective mechanical and thermal stress tangent tensors together with the instantaneous global inelastic stress tensor that represents the viscoplastic effects. Results for a thermoinelastic matrix reinforced by a hyperelastic compressible material are given that illustrate the response of fibrous and particulate composites to various types of loading.     

Dynamic neck development in a polymer tube under internal pressure loading

The initiation and growth of necks in polymer tubes subjected to rapidly increasing internal pressure is analyzed numerically. Plane strain conditions are assumed to prevail in the axial direction. The polymer is characterized by a finite strain elastic–viscoplastic constitutive relation and the calculations are carried out using a dynamic finite element program. Numerical results for neck development are illustrated and discussed for tubes of various thicknesses. The sensitivity to the wave number of the thickness imperfections is studied with a focus on comparing a long wave length imperfection and a short wave length imperfection. After some thinning down at the necks, the mode of deformation switches to neck propagation along the circumference of the tube. A case is shown in which the necks have propagated along the entire tube wall, so that network locking in the polymer results in high stiffness against further expansion of the tube. The rate dependence of the necking behavior gives noticeable differences in neck development for slow loading versus fast loading.     

Deformations of an elastic,internally constrained material. Part 1: Homogeneous deformations

The nonlinear elastic response of a class of materials for which the deformation is subject to an internal material constraint described in experiments by James F. Bell on the finite deformation of a variety of metals is investigated. The purely kinematical consequences of the Bell constraint are discussed, and restrictions on the full range of compatible deformations are presented in geometrical terms. Then various forms of the constitutive equation relating the stress and stretch tensors for an isotropic elastic Bell material are presented. Inequalities on the mechanical response functions are introduced. The importance of these in applications is demonstrated in several examples throughout the paper.This paper focuses on homogeneous deformations. In a simple illustration of the theory, a generalized form of Bell's empirical rule for uniaxial loading is derived, and some peculiarities in the response under all-around compressive loading are discussed. General formulae for universal relations possible in an isotropic elastic, Bell constrained material are presented. A simple method for the determination of the left stretch tensor for essentially plane problems is illustrated in the solution of the problem of pure shear of a materially uniform rectangular block. A general formula which includes the empirical rule found in pure shear experiments by Bell is derived as a special case. The whole apparatus is then applied in the solution of the general problem of a homogeneous simple shear superimposed on a uniform triaxial stretch; and the great variety of results possible in an isotropic, elastic Bell material is illustrated. The problem of the finite torsion and extension of a thin-walled cylindrical tube is investigated. The results are shown to be consistent with Bell's data for which the rigid body rotation is found to be quite small compared with the gross deformation of the tube. Several universal formulas relating various kinds of stress components to the deformation independently of the material response functions are derived, including a universal rule relating the axial force to the torque.Constitutive equations for hyperelastic Bell materials are derived. The empirical work function studied by Bell is introduced; and a new constitutive equation is derived, which we name Bell's law. On the basis of this law, we then derive exactly Bell's parabolic laws for uniaxial loading and for pure shear. Also, form Bell's law, a simple constitutive equation relating Bell's deviatoric stress tensor to his finite deviatoric strain tensor is obtained. We thereby derive Bell's invariant parabolic law relating the deviatoric stress intensity to the corresponding strain intensity; and, finally, Bell's fundamental law for the work function expressed in these terms is recovered. This rule is the foundation for all of Bell's own theoretical study of the isotropic materials cataloged in his finite strain experiments on metals, all consistent with the internal material constraint studied here.     

Modeling and validation of the large deformation inelastic response of amorphous polymers over a wide range of temperatures and strain rates

A robust physically consistent three-dimensional constitutive model is developed to describe the finite mechanical response of amorphous polymers over a wide range of temperatures and strain rates, including the rubbery region and for impact loading rates. This thermomechanical model is based on an elastic–viscoplastic rheological approach, wherein the effects of temperature, strain rate, and hydrostatic pressure are accounted for. Intramolecular, as well as intermolecular, interactions under large elastic–inelastic behavior are considered for the mechanisms of deformation and hardening. For a wide range of temperatures and strain rates, our simulated results for poly(methyl methacrylate) (PMMA) and polycarbonate (PC) are in good agreement with experimental observations.     

脆性固体损伤和局部软化的有限元分析

本文研究混凝土、岩石一类工程中常用的应变软化材料的有限元分析方法。在作者以往有关粘塑性损伤本构模型的工作基础上，给出了一组便于有限元计算的本构方程表达式。包括损伤弹性矩阵和局部损伤软化矩阵，分别运用于计算硬化和软化阶段的有限元刚度矩阵；对所提出的本构方程的实验验证计算和一些算例的有限元数值分析，表明文中给出的本构方程是可行的，相应的有限元算法能较好地对损伤固体的局部软化效应进行数值分析，并可成功地追踪应力应变响应的软化曲线     

Constitutive relations for isotropic or kinematic hardening at finite elastic–plastic deformations

The rate-type constitutive relations of rate-independent metals with isotropic or kinematic hardening at finite elastic–plastic deformations were presented through a phenomenological approach. This approach includes the decomposition of finite deformation into elastic and plastic parts, which is different from both the elastic–plastic additive decomposition of deformation rate and Lee’s elastic–plastic multiplicative decomposition of deformation gradient. The objectivity of the constitutive relations was dealt with in integrating the constitutive equations. A new objective derivative of back stress was proposed for kinematic hardening. In addition, the loading criteria were discussed. Finally, the stress for simple shear elastic–plastic deformation was worked out.     

On the response of rubbers at high strain rates—I. Simple waves

In this series of papers, we examine the propagation of waves of finite deformation in rubbers through experiments and analysis; in the present paper, Part I, attention is focused on the propagation of one-dimensional waves in strips of natural, latex and synthetic, nitrile rubber. Tensile wave propagation experiments were conducted at high strain rates by holding one end fixed and displacing the other end at a constant velocity. A high-speed video camera was used to monitor the motion and to determine the evolution of strain and particle velocity in rubber strips. Analysis of the response through the theory of finite waves indicated a need for an appropriate constitutive model for rubber; by quantitative matching between the experimental observations and analytical predictions, an appropriate instantaneous elastic response for the rubbers was obtained. This matching process suggested that a simple power-law constitutive model was capable of representing the high strain-rate response for both rubbers used.     

A three-dimensional nonlinear model for dissipative response of soft tissue

A set of three-dimensional constitutive equations is proposed for modeling the nonlinear dissipative response of soft tissue. These constitutive equations are phenomenological in nature and they model a number of physical features that have been observed in soft tissue. The equations model the tissue as a composite of a purely elastic component and a dissipative component, both of which experience the same total dilatation and distortion. The stress response of the purely elastic component depends on dilatation, distortion and the stretch of material fibers, whereas the stress response of the dissipative component depends on distortional deformation only. The equations are hyperelastic in the sense that the stress is obtained by derivatives of a strain energy function, and they are properly invariant under superposed rigid body motions. In contrast with standard viscoelastic models of tissues, the proposed constitutive model includes the total deformation rate in evolution equations that can reproduce the observed physical feature that the hysteresis loops of most biological soft tissues are nearly independent of strain rate (Biomechanics, Mechanical Properties of Living Tissues, second ed. (1993)). Material constants are determined which produce good agreement with uniaxial stress experiments on superficial musculoaponeurotic system and facial skin.     

材料的弹粘塑性本构关系和应力波的演化

试验表明，大多数工程材料在冲击载荷作用之下的变形一般都同时包含有可恢复的瞬态性弹性变形和不可恢复的粘滞性塑性变形，即其本构关系可以用弹粘塑性模型来描述。本文从内变量理论出发，探讨了时率相关材料的弹粘塑性本构关系的一般特性，建立了增量型的弹粘塑性本构关系的一般理论框架和普适的表达式，并且对两种最常用的本构模型——Bodner-Partom模型和Johnson-Cook模型给出了在一维应变条件下的具体形式。通过计算和讨论一维应变粘塑性靶板中冲击波的衰减机制和应力波的演化规律，特别是考察各种粘塑性本构模型中的材料参数对冲击波的衰减和应力波的演化的影响，得出了一些可以直接应用或具有一定借鉴价值的结果，为研究应力波的其他衰减机制以及在人防工程中智能防护层设计时新材料的选取奠定了基础。     

特征线法在三维弹／粘塑性波数值解析中的应用

根据广义特征理论建立了求解三维非线性应力波传播问题的特征关系式,并采用特征线法对纵向冲击载荷下弹／粘塑性矩形截面棒的三维应力波传播过程进行了数值模拟。     

有限厚度砂床对波浪载荷的响应

本文采用水-土两相介质混合物的连续介质力学理论分析了有限厚度砂床对波浪载荷的响应。土床看作饱和孔隙弹性介质,用Biot理论来描述流动和变形的耦合效应,波浪用线性水波描述。在上述假设下得到了描述土床对波浪载荷响应的基本方程。用解沂-数值方法给出了土床中位移、应力和孔隙水压力的分布并将理沦计算结果与波浪-砂床模型试验的实验资料进行了比较。文章还对影响土床对波浪响应的参量特性进行了分析和讨论。根据参量研究、理论计算与实验资料比较的结果,可以得出结论:本文所提供的理论模型和计算方法能用来估算海床对波浪载荷的影响。     

Finite deformation of a polymer: experiments and modeling

The response of a polymer (polytetrafluoroethylene) to quasi-static and dynamic loading is determined and modeled. The polytetrafluoroethylene is extremely ductile and highly nonlinear in elastic as well as plastic behaviors including elastic unloading. Constitutive model developed earlier by Khan, Huang and Liang (KHL) is extended to include the responses of polymeric materials. The strain rate hardening, creep, and relaxation behaviors of polytetrafluoroethylene were determined through extensive experimental study. Based on the observation that both viscoelastic and viscoplastic deformation of polytetrafluoroethylene are time dependent and nonlinear, a phenomenalogical viscoelasto–plastic constitutive model is presented by a series connection of a viscoelastic deformation module (represented by three elements standard solid spring dashpot model), and a viscoplastic deformation module represented by KHL model. The KHL module is affected only when the stress exceeds the initial yield stress. The comparison between the predictions from the extended model and experimental data for uniaxial static and dynamic compression, creep and relaxation demonstrate that the proposed constitutive model is able to represent the observed time dependent mechanical behavior of polytetrafluoroethylene polytetrafluoroethylene qualitatively and quantitatively.     

From Non-Linear Elasticity to Linearized Theory: Examples Defying Intuition

Material frame indifference implies that the solution in non-linear elasticity theory for a connected body rigidly rotated at its border is a rigid, stress-free, deformation. If the same problem is considered within linear elasticity theory, considered as an approximation to the true elastic situation, one should expect that if the angle of rotation is small, the body still undergoes a rigid deformation while the corresponding stress, though not zero, remains consistently small. Here, we show that this is true, in general, only for homogeneous bodies. Counterexamples of inhomogeneous bodies are presented for which, whatever small the angle of rotation is, the linear elastic solution is by no means a rigid rotation (in a particular case it is an “explosion”) while the stress may even become infinite. If the same examples are re-interpreted as problems in an elasticity theory based upon genuinely linear constitutive relations which retain their validity also for finite deformations, it is shown that they would deliver constraint reaction forces that are not in equilibrium in the actual, deformed, state. This furnishes another characterization of the impossibility of an exact linear constitutive theory for elastic solids with zero residual stress.