Elasto-plastic constitutive relations of unidirectional fiber composites for cyclic loadings

The clasto-plastic constitutive behaviors of continuous fiber reinforced composites under cyclic loadings are studied by the micromechanics method in which the equal-strain model is used in the fiber direction, the equal-stress model in the other directions. It is supposed that fiber is linearly elastic and matrix is clastic-viscoplastic. The constitutive equations of the matrix are described by Bodner-Partom's unified constitutive theory. Boron/Aluminum composite, as an example, is investigated in detail for an understanding of the stress-strain relations and initial yield behaviors of metal matrix composites. Present results are compared with the experimental data.The project was supported by the Chinese National Natural Science Foundation.     

铁电复合材料的力电耦合行为

对铁电复合材料力电耦合行为的研究进行综述．介绍了铁电复合材料的一些基本特点和线性压电有效性能的研究进展．阐述铁电复合材料中由于铁电相的铁电畴变与铁弹畴变以及基体相的弹性、粘弹性和介电弛豫性质所引起的铁电复合材料的非线性力电耦合行为．介绍了基于细观力学方法对铁电复合材料的本构关系进行研究的一些结果．     

Material multipole mechanics of elastic dielectric composites

The overall mechanical and electrical behaviors of elastic dielectric composites are investigated with the aid of the concept of material multipoles. In particular, by introducing a statistical continuum material multipole theory, the effects of the electric-elastic interaction and the microstructure (size, shape, orientation,...) of inhomogeneous particles on the overall behaviors of the composites can be obtained. A basic solution for an ellipsoidal elastic inhomogeneity with electric polarization in an infinite elastic dielectric medium is first given, which shows that classical Eshelby ’s elastic solution is modified by the presence of electric-elastic interaction. The overall macroscopic constitutive relations and their overall macroscopic material parameters accounting for electroelastic interaction effect are then derived for the elastic dielectric composites. Some quantitative calculations on the problems with statistical anisotropy, the shape effect and the electric-elastic interaction are finally given for dilute composites.     

Overall dynamic constitutive relations of layered elastic composites

A method for the homogenization of a layered elastic composite is presented. It allows direct, consistent, and accurate evaluation of the averaged overall frequency-dependent dynamic material constitutive relations without the need for a point-wise solution of the field equations. When the spatial variation of the field variables is restricted by Bloch-form (Floquet-form) periodicity, then these relations together with the overall conservation and kinematical equations accurately yield the displacement or stress mode-shapes and, necessarily, the dispersion relations. The method can also give the point-wise solution of the elastodynamic field equations (to any desired degree of accuracy), which, however, is not required for the calculation of the average overall properties. The resulting overall dynamic constitutive relations are general and need not be restricted by the Bloch-form periodicity.The formulation is based on micromechanical modeling of a representative unit cell of the composite. For waves in periodic layered composites, the overall effective mass-density and compliance (stiffness) are always real-valued whether or not the corresponding unit cell (representative volume element used as a unit cell) is geometrically and/or materially symmetric. The average strain and linear momentum are coupled and the coupling constitutive parameters are always each others' complex conjugates. We separate the overall constitutive relations, which depend only on the composition and structure of the unit cell, from the overall field equations which hold for any elastic composite; i.e., we use only the local field equations and material properties to deduce the overall constitutive relations. Finally, we present solved numerical examples to further clarify the structure of the averaged constitutive relations and to bring out the correspondence of the current method with recently published results.     

On thermoelastostatics of composites with nonlocal properties of constituents II. Estimation of effective material and field parameters

One considers a linear thermoelastic composite medium, which consists of a homogeneous matrix containing a statistically homogeneous random set of ellipsoidal uncoated or coated heterogeneities. It is assumed that the stress–strain constitutive relations of constituents are described by the nonlocal integral operators, whereas the equilibrium and compatibility equations remain unaltered as in classical local elasticity. The general integral equations connecting the stress and strain fields in the point being considered and the surrounding points are obtained. The method is based on a centering procedure of subtraction from both sides of a known initial integral equation their statistical averages obtained without any auxiliary assumptions such as, e.g., effective field hypothesis implicitly exploited in the known centering methods. In a simplified case of using of the effective field hypothesis for analyzing composites with one sort of heterogeneities, one proves that the effective moduli explicitly depend on both the strain and stress concentrator factor for one heterogeneity inside the infinite matrix and does not directly depend on the elastic properties (local or nonlocal) of heterogeneities. In such a case, the Levin’s (1967) formula in micromechanics of composites with locally elastic constituents is generalized to their nonlocal counterpart. A solution of a volume integral equation for one heterogeneity subjected to inhomogeneous remote loading inside an infinite matrix is proposed by the iteration method. The operator representation of this solution is incorporated into the new general integral equation of micromechanics without exploiting of basic hypotheses of classical micromechanics such as both the effective field hypothesis and “ellipsoidal symmetry” assumption. Quantitative estimations of results obtained by the abandonment of the effective field hypothesis are presented.     

Research note on application of a constitutive inequality in nonassociated plasticity

A constitutive inequality proposed by several authors is applied, apparently for the first time, to a nonassociated elastic/plastic constitutive model. The main mathematical difficulty ensues from the presence of a nonsymmetric dyadic matrix in the constitutive equations. A general property of such products is used to derive a simple sufficient condition for the constitutive inequality. This condition involves the hardening modulus and the spectral condition number of a matrix arising from the constitutive equations. Several examples are given.     

Wave propagation in fractured porous media

A theory of wave propagation in fractured porous media is presented based on the double-porosity concept. The macroscopic constitutive relations and mass and momentum balance equations are obtained by volume averaging the microscale balance and constitutive equations and assuming small deformations. In microscale, the grains are assumed to be linearly elastic and the fluids are Newtonian. Momentum transfer terms are expressed in terms of intrinsic and relative permeabilities assuming the validity of Darcy's law in fractured porous media. The macroscopic constitutive relations of elastic porous media saturated by one or two fluids and saturated fractured porous media can be obtained from the constitutive relations developed in the paper. In the simplest case, the final set of governing equations reduce to Biot's equations containing the same parameters as of Biot and Willis.Now at Izmir Institute of Technology, Anafartalar Cad. 904, Basmane 35230, Izmir, Turkey.     

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     

A DAMAGE MECHANICS MODEL FOR FATIGUE LIFE PREDICTION OF FIBER REINFORCED POLYMER COMPOSITE LAMINA

A damage mechanics fatigue life prediction model for the fiber reinforced polymer lamina is established. The stiffness matrix of the lamina is derived by elastic constants of fiber and matrix. Two independent damage degrees of fiber and matrix are introduced to establish constitutive relations with damage. The damage driving forces and damage evolution equations for fiber and matrix are derived respectively. Fatigue tests on 0°and 90°unidirectional laminates are conducted respectively to identify parameters in damage evolution equations of fiber and matrix. The failure criterion of the lamina is presented. Finally, the life prediction model for lamina is proposed.     

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.     

Friction in unconforming grain contacts as a mechanism for tensorial stress-strain hysteresis

Materials composed of consolidated grains and/or containing internal contacts are widespread in everyday life (e.g. rocks, geomaterials, concretes, slates, ceramics, composites, etc.). For any simulation of the elastic behavior of this class of solids, be it in seismology, in NDT, or in the modeling of building constructions, the stress-strain constitutive equations are indispensable. Since the most common loading patterns in nature considerably deviate from simple uniaxial compression, the problem of tensorial stress-strain representation arises. In simple loading cases it may be sufficient to use a phenomenological constitutive model. However, in a more general case, phenomenological approaches encounter serious difficulties due to the high number of unknown parameters and the complexity of the model itself. Simplification of the phenomenology can help only partly, since it may require artificial assumptions. For instance, is it enough just to link the volumetric stress to the volumetric strain, or do we have to include shear components as well, and if yes, in what form? We therefore propose a physical tensorial stress-strain model, based on the consideration of plane cracks with friction. To do this, we combine known relations for normal displacements of crack faces given by contact mechanics, the classical Amonton's law of dry friction for lateral displacements, and the equations of elasticity theory for a collection of non-interacting cracks with given orientation. The major advantages of this model consist in the full tensorial representation, the realistic stress-strain curves for uniaxial stress compression and quantitative comparison with experimental data, and a profound account for hysteretic memory effects.     

纤维加强复合材料的总体粘弹性响应

在某些纤维增强复合材料(FRC)中使用金属或高分子聚合物作为基体材料。在高温等情况下,这类材料具有明显的粘弹性特性。本文采用Riemann—Liouville形式的分数阶导数模型描述基体的粘弹性特性。通过渐近均匀化方法给出了预测FRC整体三维本构关系的解析表达式。给出了应用于基体具有Makris粘弹性关系的具体形式。以圆截面纤维正方形排列的情形为例,给出了等效模量随纤维体积比的变化曲线。结果说明,这类复合材料仍具有粘弹性特性,其整体粘弹性本构关系的弹性部分综合了纤维弹性和基体弹性的贡献,粘性部分来自基体粘性的贡献,复合材料具有和基体相同的粘性系数和分数阶。为分析微结构特征对整体特性的贡献,须求解两类局部问题。可以看出,在整体的等效模量中包含了局部变形的贡献,局部变形增加了复合材料的耦合刚度。     

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.     

颗粒增强复合材料中微观热应力和残余应力分析

运用空间配位体密堆模型和球对称分析单元，分析计算了颗粒增强复合材料经历温度变化后产生的微观热应力和残余应力。分析结果表明，温度升高时界面产生径向张应力，降温时产生压应力。存在一个基体开始发生塑性变形的临界温差ｔ＿ｐ，其值随增强体体积分数Ｖ＿ｐ增加而降低。除组元间热膨胀系数差和弹性常数外，基体材料本构关系和屈服强度对热应力和残余应力均有很大影响。随Ｖ＿ｐ增加，微观应力和水静宏观应力幅值上升。但粒子周围塑性区尺寸近似与Ｖ＿ｐ无关。给出了不同变温条件下残余应力的确定方法。     

无限域分层介质的基本解

本文从三维弹性力学最基本的平衡方程和本构关系出发，推导出状态传递微分方程，在求解状态传递微分方程时，建议了一种对指数矩阵进行分解的方法，避免了直接解法可能导致状态变量的发散的问题，引入了无穷远处的状态为量为有限值的条件，推导出上，下无限层表面的位移与应力关系式，再根据状态传递方程，可得出层状介质任意点的应力和位移的值，此结果可直接退化到无限域经典的Kelvin解。     

Two exact micromechanics-based nonlocal constitutive equations for random linear elastic composite materials

A Hashin-Shtrikman-Willis variational principle is employed to derive two exact micromechanics-based nonlocal constitutive equations relating ensemble averages of stress and strain for two-phase, and also many types of multi-phase, random linear elastic composite materials. By exact is meant that the constitutive equations employ the complete spatially-varying ensemble-average strain field, not gradient approximations to it as were employed in the previous, related work of Drugan and Willis (J. Mech. Phys. Solids 44 (1996) 497) and Drugan (J. Mech. Phys. Solids 48 (2000) 1359) (and in other, more phenomenological works). Thus, the nonlocal constitutive equations obtained here are valid for arbitrary ensemble-average strain fields, not restricted to slowly-varying ones as is the case for gradient-approximate nonlocal constitutive equations. One approach presented shows how to solve the integral equations arising from the variational principle directly and exactly, for a special, physically reasonable choice of the homogeneous comparison material. The resulting nonlocal constitutive equation is applicable to composites of arbitrary anisotropy, and arbitrary phase contrast and volume fraction. One exact nonlocal constitutive equation derived using this approach is valid for two-phase composites having any statistically uniform distribution of phases, accounting for up through two-point statistics and arbitrary phase shape. It is also shown that the same approach can be used to derive exact nonlocal constitutive equations for a large class of composites comprised of more than two phases, still permitting arbitrary elastic anisotropy. The second approach presented employs three-dimensional Fourier transforms, resulting in a nonlocal constitutive equation valid for arbitrary choices of the comparison modulus for isotropic composites. This approach is based on use of the general representation of an isotropic fourth-rank tensor function of a vector variable, and its inverse. The exact nonlocal constitutive equations derived from these two approaches are applied to some example cases, directly rationalizing some recently-obtained numerical simulation results and assessing the accuracy of previous results based on gradient-approximate nonlocal constitutive equations.     

Uniqueness of uniaxial elastic-plastic interface motions

The uniaxial motion of interfaces between regions deforming elastically and regions deforming plastically is considered. The governing constitutive, stress-rate/strain-rate equations in both elastic and plastic regions are taken to be non-linear. Discontinuity relations across such interfaces are established by the repeated differentiation of existing relations. The relations given by previous workers (especially R.J. Clifton, T.C.T. Ting, E.H. Lee and Th. von Kármán) are discussed. The precise situations in which they hold are considered, and it is shown that some of these relations, while apparently derived for different situations, can, in certain circumstances, be shown to be equivalent. It has been shown that six essentially different types of motion can occur, and, when the constitutive equations are linear, each type of motion is unique. This result is extended to the non-linear situation, by means of an established local expansion procedure. For the case of a meeting interaction of stress waves carrying initially linear profiles, the previous (linear) analysis given by L.W. Morland and A.D. Cox fails to distinguish between certain types of motion. This motion is reconsidered and it is shown how non-linearity in the constitutive laws serves to determine uniquely the type of motion that takes place.     

Buckling analysis of fibers in composite materials by wave propagation analogy

The analogy between the governing equations for the analysis of buckling in elastic structures and the elastodynamic equations of motion for wave propagation is presented. By employing this analogy, the exact and approximate buckling stresses of periodic layered materials and continuous fiber composites, respectively, are established. This is performed by utilizing micromechanically based dispersion relations for elastic wave propagating in the composite materials, which provide for a given wave length the corresponding phase velocity. By a specific change of variables in these dispersion relations, the corresponding buckling stresses can be determined. Results are presented and compared with solutions based on the mechanics of materials approach as well as with the well known Rosen’s fiber buckling predictions.     

纤维排列方式对复合材料总体粘弹性常数的影响

对于金属基或高分子聚合物基复合材料，在特定情况下会表现出明显的粘弹性特性。本文采用Riemann—Liouville形式的分数阶导数模型描述基体的粘性特性，通过渐进均匀化方法给出了预测纤维加强复合材料整体本构关系的解析表达式，给出应用于基体具有Makris粘弹性关系的具体形式。最后，考察了圆截面纤维正方形排列和对角排列时的总体粘弹性弹性常数随纤维比的变化曲线。结果表明，这类复合材料仍具有粘弹性特性，其整体粘弹性本构关系的弹性部分综合了纤维弹性和基体弹性的贡献，粘性部分来自基体粘性的贡献，复合材料具有和基体相同的粘性系数和分数阶。为分析微结构特征对整体特性的贡献，须求解两类局部问题。在相同纤维体积比情况下，正方形排列的总体弹性系数大于正方形对角排列，而粘性常数相反。