连续介质力学中变形梯度张量客观性表述的分歧

张量的客观性是连续介质力学中一个重要的概念,但现有文献对张量客观性的定义不一致导致有关变形梯度张量客观性的表述存在分歧.本文主要基于张量的逆及功共轭角度分析了不同客观性定义的差别,旨在加深对张量客观性,特别是对变形梯度等两点张量客观性的认识.     

损伤体有效弹性性质的细观分析和不变性描述—— 一个考虑微缺陷相互作用的一般理论模式

通过引进微缺陷相互作用张量，建立了一个二维情况下考虑微缺陷（微裂纹或微孔洞）间相互作用的损伤固体有效弹性性质的一般理论模式模型中考虑了微缺陷的几何形状、取向分布和空间分布所造成的有效柔度张量的各向异性和材料中微缺陷之间的相互作用所引起的损伤柔度张量的高阶效应针对微椭圆孔、微圆孔和微裂纹问题，求得了相互作用张量的解析形式     

损伤体有效弹性性质的细观分析和不变性描述—— 一个考虑微缺陷相互作用的一般理论模式

通过引进微缺陷相互作用张量，建立了一个二维情况下考虑微缺陷（微裂纹或微孔洞）间相互作用的损伤固体有效弹性性质的一般理论模式模型中考虑了微缺陷的几何形状、取向分布和空间分布所造成的有效柔度张量的各向异性和材料中微缺陷之间的相互作用所引起的损伤柔度张量的高阶效应针对微椭圆孔、微圆孔和微裂纹问题，求得了相互作用张量的解析形式     

非线性横观各向同性弹性材料的本构方程及其势函数

研究了非线性Green弹性材料弹性张量独立分量,归纳推导出横观各向同性Green弹性材料、各向同性非线性弹性材料独立的弹性常数个数.从张量函数出发,用含有高阶弹性张量的张量多项式,推导出四阶非线性横观各向同性,各向同性材料Green弹性材料本构方程及其势函数.并将本构方程及其势甬数用张量不变量,标量不变量表示.证明了方程是完备的,不可约的,满足张量函数表示定理.     

Lode角对应力张量的导数的基本性质

为计及岩土类材料塑性力学行为的中主应力影响或应力路径相关性,通常将应力张量Lode角/Lode数引入屈服函数与塑性势函数。由此在计算塑性应变增量时必然涉及Lode角/Lode数对应力的导数张量（记为 ）。然而,应力张量主值有重根时 的计算存在困难。本文给出了 的主值计算方法及谱分解表达式并详细讨论了张量 的基本性质。     

非线性正交各向异性弹性材料的本构方程及其势函数

研究了非线性Green弹性材料弹性张量独立分量,归纳推导出各向异性Green弹性材料、具有一个对称面Green弹性材料、 正交各向异性非线性弹性材料独立的弹性常数个数.从张量函数出发,用含有高阶弹性张量的张量多项式,推导出三阶非线性正交各向异性Green弹性材料本构方程及其势函数.并将本构方程及其势函数用张量不变量,标量不变量表示.证明了方程是完备的,不可约的,满足张量函数表示定理.详细研究Green弹性材料势函数存在的充分和必要条件,给出并证明了具有普适性的势函数存在定理.     

从坐标系到张量1）--学习和理解张量的一条捷径

学习和掌握张量基本知识是研究连续介质力学的基础,然而,当前对张量的讲述和介绍方式比较复杂,造成理解和运用的困难.本文利用笛卡尔坐标系引入张量概念及其基本运算,阐明张量本质上是坐标变换,熟悉求和约定和指标表示是其关键,从而使张量能体现出数学本身的简单、和谐和美的统一,也能使读者比较顺利地学习、理解并运用张量.     

从坐标系到张量——学习和理解张量的一条捷径

学习和掌握张量基本知识是研究连续介质力学的基础,然而,当前对张量的讲述和介绍方式比较复杂,造成理解和运用的困难.本文利用笛卡尔坐标系引入张量概念及其基本运算,阐明张量本质上是坐标变换,熟悉求和约定和指标表示是其关键,从而使张量能体现出数学本身的简单、和谐和美的统一,也能使读者比较顺利地学习、理解并运用张量.     

坐标变换系数张量观与杂交张量概念分析

本文致力于澄清一个十分基本的问题:坐标变换系数是否为张量?传统观念认为,坐标变换系数不是张量。为了揭示坐标变换系数的本质,本文采用"从一般到特殊"的研究策略,重塑了张量的内涵和外延,引入了杂交张量概念,进而颠覆了坐标变换系数不是张量的传统观念,确切地讲,它就是度量张量的杂交分量。这一结果扩张了张量概念的集合,提升了张量分析学内在的统一性、对称性和不变性,减少了连续介质力学的运算量。     

含微裂纹材料的损伤理论

本文从含微裂纹材料的变形能出发引出了裂纹的方位张量。在考虑裂纹受压闭合与滑动摩擦的基础上,给出了损伤张量、损伤应变及有效弹性常数。文中给出了损伤机构离散化的方法,并对方位密度给出了演化方程。最后给出一个单向拉压的应力应变关系例子,并揭示了裂纹扩展时的应力突跌现象。     

Anisotropic damage–plasticity model for concrete

A material model for concrete is proposed here within the framework of a thermodynamically consistent elasto-plasticity–damage theory. Two anisotropic damage tensors and two damage criteria are adopted to describe the distinctive degradation of the mechanical properties of concrete under tensile and compressive loadings. The total stress tensor is decomposed into tensile and compressive components in order to accommodate the need for the above mentioned damage tensors. The plasticity yield criterion presented in this work accounts for the spectral decomposition of the stress tensor and allows multiple hardening rules to be used. This plastic yield criterion is used simultaneously with the damage criteria to simulate the physical behavior of concrete. Non-associative flow rule for the plastic strains is used to account for the dilatancy of concrete as a frictional material. The thermodynamic Helmholtz free energy concept is used to consistently derive dissipation potentials for damage and plasticity and to allow evolution laws for different hardening parameters. The evolution of the two damage tensors is accounted for through the use of fracture-energy-based continuum damage mechanics. An expression is derived for the damage–elasto-plastic tangent operator. The theoretical framework of the model is described here while the implementation of this model will be discussed in a subsequent paper.     

含正交排列夹杂和缺陷材料的等效弹性模量和损伤

研究含正交排列夹杂和缺陷材料的等效弹性模量和损伤,推导了以Eshelby－Mori－Tanaka方法求解多相各向异性复合材料等效弹性模量的简便计算公式,针对含三相正交椭球状夹杂的正交各向异性材料,得到了由细观参量（夹杂的形状、方位和体积分数）表示的等效弹性模量的解析表达式．在此基础上,提出了一个宏细观结合的正交各向异性损伤模型,从而建立了以细观量为参量的含损伤材料的应力应变关系．最后,对影响材料损伤的细观结构参数进行了分析．     

A theoretical and computational framework for anisotropic continuum damage mechanics at large strains

The main objective of this work is the formulation and algorithmic treatment of anisotropic continuum damage mechanics at large strains. Based on the concept of a fictitious, isotropic, undamaged configuration an additional linear tangent map is introduced which allows the interpretation as a damage deformation gradient. Then, the corresponding Finger tensor – denoted as damage metric – constructs a second order, internal variable. Due to the principle of strain energy equivalence with respect to the fictitious, effective space and the standard reference configuration, the free energy function can be computed via push-forward operations within the nominal setting. Referring to the framework of standard dissipative materials, associated evolution equations are constructed which substantially affect the anisotropic nature of the damage formulation. The numerical integration of these ordinary differential equations is highlighted whereby two different schemes and higher order methods are taken into account. Finally, some numerical examples demonstrate the applicability of the proposed framework.     

On a finite-strain viscoplastic law coupled with anisotropic damage: theoretical formulations and numerical applications

Based on a dissipation inequality at finite strains and the effective stress concept, a Chaboche-type infinitesimal viscoplastic theory is extended to finite-strain cases coupled with anisotropic damage. The anisotropic damage is described by a rank-two symmetric tensor. The constitutive law is formulated in the corotational material coordinate system. Thus, the evolution equations of all internal variables can be expressed in terms of their material time derivatives. The numerical algorithm for implementing the material model in a finite element programme is also formulated, and several numerical examples are shown. Comparing the numerical simulations with experimental observations indicates that the present material model can describe well the primary, secondary and tertiary creep. It can also predict the anisotropic damage modes observed in experiments correctly.     

Décomposition de Kelvin et concept de contraintes effectives multiples pour les matériaux anisotropes

The effective stress concept, now classical in continuum damage mechanics, is generalized to the case of an initial anisotropy. In order to be used for both damage–elasticity and damage–(visco-)plasticity coupling, the effective stress should not depend on the elastic properties. Kelvin decomposition of the elasticity tensor allows to define such a stress for isotropic and cubic symmetries. For other material symmetries, the concept of multiple effective stresses is proposed. To cite this article: R. Desmorat, C. R. Mecanique 337 (2009).     

Computational multiscale methods for granular materials

The fine-scale heterogeneity of granular material is characterized by its polydisperse microstructure with randomness and no periodicity. To predict the mechanical response of the material as the microstructure evolves, it is demonstrated to develop computational multiscale methods using discrete particle assembly-Cosserat continuum modeling in micro- and macro- scales, respectively. The computational homogenization method and the bridge scale method along the concurrent scale linking approach are briefly introduced. Based on the weak form of the Hu-Washizu variational principle, the mixed finite element procedure of gradient Cosserat continuum in the frame of the second-order homogenization scheme is developed. The meso-mechanically informed anisotropic damage of effective Cosserat continuum is characterized and identified and the microscopic mechanisms of macroscopic damage phenomenon are revealed.     

非饱和颗粒材料的多孔连续体有效压力与有效广义Biot应力

多孔连续体理论框架下的非饱和多孔介质广义有效压力定义和Bishop参数的定量表达式长期以来存在争议,这也影响了对与其直接相关联的非饱和多孔介质广义Biot有效应力的正确预测.基于随时间演变的离散固体颗粒-双联液桥-液膜体系描述的Voronoi胞元模型,利用由模型获得的非饱和颗粒材料表征元中水力-力学介观结构和响应信息,文章定义了低饱和度多孔介质局部材料点的有效内状态变量:非饱和多孔连续体的广义Biot有效应力和有效压力,导出了其表达式.所导出的有效压力公式表明,非饱和多孔连续体的有效压力张量为各向异性,它不仅对非饱和多孔连续体广义Biot有效应力张量的静水应力分量的影响呈各向异性,同时也对其剪切应力分量有影响.文章表明,非饱和多孔连续体中提出的广义Biot理论和双变量理论的基本缺陷在于它们均假定反映非混和两相孔隙流体对固相骨架水力-力学效应的有效压力张量为各向同性.此外,为定义各向同性有效压力张量和作为加权系数而引入的Bishop参数并不包含对非饱和多孔连续体中局部材料点水力-力学响应具有十分重要效应的基质吸力.所导出的非饱和多孔介质广义Biot有效应力和有效压力公式(包括反映有效压力...     

The formulation of constitutive equations for fibre-reinforced composites in plane problems: Part II

Summary From the continuum mechanics points of view, most of commercial fibre-reinforced composites (FRCs) can be considered to be anisotropic materials with one of the five material symmetries: transverse isotropy, orthotropy, tetratropy, hexatropy and octotropy, as illustrated in the preceding paper (Part I) [1]. No properly general formulation of constitutive equations for tetratropic, hexatropic and octoctropic types of FRC has been found in the literature. In this paper, the restriction to the admissible deformation of a FRC imposed by the failure strains of the fibres is investigated. The complete and irreducible two-dimensional tensor function representations regarding tetratropy, hexatropy and octotropy derived in Part I are applied to formulate constitutive equations for FRCs in plane problems of elasticity, yielding and failure in the present work, and of heat conduction, continuum damage and asymmetric elasticity in a continued work (Part III, forthcoming).The supports from the Alexander von Humboldt Foundation, Germany and the Research Foundation of Tsinghua University, P. R. China are acknowledged by the first author.     

Fractal effect and anisotropic constitutive model for concrete

An elasto-anisotropic damage constitutive model for concrete is developed in this work. Disregarding the coupling between the isotropic and the anisotropic damage, the isotropic damage variables are defined as functions of the microcrack fractal dimension, and the anisotropic parts are expressed by the lengths of cracks in concrete which various in different directions. The Helmholtz free energy is decomposed into the elastic deforming, damage and irreversible deforming components, with the last component used to replace the plastic deformation. Therefore the damage constitutive formulas for concrete are derived based on continuum damage mechanics. Evolution laws for both isotropic and anisotropic damage variables are derived, in which the anisotropic parts are obtained by modifying an empirical model. The critical fracture stress and the fracture toughness are investigated for materials with a single fractal crack based on the fractal geometry and the Griffith fracture criterion. Numerical computation is conducted for concrete under the uniaxial and the biaxial compression. The results indicate that the material stiffness degradation can be well addressed when the anisotropic damage is incorporated; the irreversible deformation is greatly related to the behavior of the descending branch beyond the peak load. The validation of the presented model is proofed by comparing results with the experimental data. This model provides an approach to link the macro properties of a material with its micro-structure change.