各向同性弹性损伤的双标量描述

损伤状态的描述是损伤力学中仍未完善解决的基本问题．我们旨在对此问题就最简单的一种情形——各向同性弹性损伤，进行较为全面的研究．首先，指出了古典各向同性损伤理论中，基于应变等效假设，用单个标量损伤变量描述损伤状态的局限性．然后，建立了一个用两个标量损伤变量描述的各向同性弹性损伤模型．此模型解除了古典理论的局限，能完全描述各向同性弹性损伤，并且得到本文数值实验的验证．最后，将本文模型与现有细观力学结果连接，给出了宏细观损伤变量之间的关系，使得细观量可以通过宏观量来反映，建立了一个用细观损伤材料常数描述细观缺陷特征的损伤本构模型     

First-order gradient damage theory

Taking the strain tensor, the scalar damage variable, and the damage gradient as the state variables of the Helmholtz free energy, the general expressions of the firstorder gradient damage constitutive equations are derived directly from the basic law of irreversible thermodynamics with the constitutive functional expansion method at the natural state. When the damage variable is equal to zero, the expressions can be simplified to the linear elastic constitutive equations. When the damage gradient vanishes, the expressions can be simplified to the classical damage constitutive equations based on the strain equivalence hypothesis. A one-dimensional problem is presented to indicate that the damage field changes from the non-periodic solutions to the spatial periodic-like solutions with stress increment. The peak value region develops a localization band. The onset mechanism of strain localization is proposed. Damage localization emerges after damage occurs for a short time. The width of the localization band is proportional to the internal characteristic length.     

损伤力学中的能量等效假设及其实验验证

大多数损伤模型在构造受损材料本构方程时所采用的应变或应力等效假设在一般情况下并不能满足热力学理论的基本规则.本文提出的能量等效假设,在弹性、塑性和蠕变损伤等普遍情形均证明了其有效性.同时,能量等效假设配合以应变能密度准则,可以很好地描述穿孔铜薄板试件的破断行为.     

考虑温度效应的弹性损伤一般理论

现有的各种损伤理论基本上都是关于等温问题的 ,且在不同程度上依赖于某些经验假设。本文在严格的不可逆热力学理论基础之上 ,建立了考虑温度效应的弹性损伤一般理论。推导出热弹性各向同性与各向异性损伤材料全部本构方程的一般形式 ,其中包括应力 应变关系、熵密度方程、损伤对偶张量表达式、热 固 损伤耦合的热传导方程和损伤演化方程。它们的特殊形式包含了等温各向同性与各向异性弹性损伤的本构方程     

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.     

Elasto-plastic postbuckling of damaged orthotropic plates

Based on the elasto-plastic mechanics and continuum damage theory, a yield criterion related to spherical tensor of stress is proposed to describe the mixed hardening of damaged orthotropic materials. Its dimensionless form is isomorphic with the Mises criterion for isotropic materials. Furthermore, the incremental elasto-plastic damage constitutive equations and damage evolution equations are established. Based on the classical nonlinear plate theory, the incremental nonlinear equilibrium equations of orthotropic thin plates considering damage effect are obtained, and solved with the finite difference and iteration methods. In the numerical examples, the effects of damage evolution and initial deflection on the elasto-plastic postbuckling of orthotropic plates are discussed in detail.     

Finite strain solutions in compressible isotropic elasticity

Three classes of compressible isotropic elastic solids are introduced, for each of which the strain energy, expressed as a function of the three principal invariants of the stretch tensors, is linear in two of its arguments and nonlinear in the third argument. One of these is the class of harmonic materials. Several deformation fields are examined, for which the governing equations, for general compressible isotropic elastic response, reduce to a nonlinear ordinary differential equation. For the three special classes of materials, this differential equation may be solved in closed form, giving a deformation field which is independent of the material response function, or its solution may be reduced to a single quadrature, involving the nonlinear material response function.     

Thermodynamically consistent nonlocal theory of ductile damage

In this paper a thermodynamically consistent, weakly nonlocal theory of ductile damage is presented. The theory is based on the classical dynamical balance laws of forces and couples in the physical space and dynamical balance laws of material forces on evolving defects and on the first and second law of thermodynamics formulated for physical and material space. Assuming general constitutive equations their frame-invariant and thermodynamically admissible form is determined. It is shown that physical and material forces and stresses consist of two parts, a nondissipative part derivable from a free energy potential, and a dissipative part, which can be obtained from a dissipation pseudo-potential, if such a pseudo-potential exists.The theory can be considered as a framework with gradient elastoplasticity, isotropic and anisotropic brittle and ductile gradient damage at finite strain as special cases.     

A general nonlinear theory of elastic shells

This paper presents a general nonlinear theory of elastic shells for large deflections and finite strains in reference to a certain natural state. By expanding the displacement components into power series in the coordinate θ³ normal to the undeformed middle surface of shells, the expansions of the Cauchy-Green strain tensors are expressed in terms of these expanded displacement components. Through the modified Hellinger-Reissner variational principle for a three-dimensional elastic continuum, a set of the fundamental shell equations is derived in terms of the expanded Cauchy-Green strain tensors and Kirchhoff stress resultants. The Love-Kirchhoff hypothesis is not assumed and higher order stretching and bending are taken into consideration. For elastic shells of isotropic materials, assuming the strain-energy to be an analytic function of the strain measures, general nonlinear constitutive equations are then derived. Thus, a complete and consistent two-dimensional shell theory incorporating the geometrical and physical nonlinearities is established. The classical theories of shells are directly derivable from the present results by proper truncations of the series.     

On two approaches to determining the axisymmetric elastoplastic stress-strain state of laminated shells made of isotropic and transversely isotropic bimodulus materials

The paper compares two approaches to determining the axisymmetric elastoplastic stress-strain state of laminated shells made of isotropic and transversely isotropic materials with different elastic moduli in tension and compression. The approaches use different nonlinear constitutive equations describing the elastic deformation of the transversally isotropic bimodulus materials. Both approaches employ the theory of deformation along paths of small curvature and the Kirchhoff-Love hypothesis for the whole laminate to describe the deformation of the isotropic materials. The problem is solved by the method of successive approximations. The solutions of specific problems obtained by the two approaches are analyzed __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 6, pp. 59–69, June 2008.     

Constitutive structure in coupled non-linear electro-elasticity: Invariant descriptions and constitutive restrictions

A constitutive framework for electro-sensitive materials in the context of non-linear elasticity is analyzed. Constitutive equations are given in terms of energy functions that depend on several invariants. The study includes both the analysis of the invariants, which are present in the energy functions, and the analysis of constitutive restrictions that have to be obeyed by the constitutive functions. Isotropic as well as non-isotropic electro-sensitive elastomers are studied. The set of invariants that describe each material model is analyzed under two homogeneous deformations: (i) an uniaxial elongation and (ii) a simple shear deformation. These deformations are chosen since they are relevant to specific experiments, from which one may try to fit constitutive equations. The constitutive restrictions developed are based on classical ones used for isotropic non-linear elastic materials, in particular, are based on the Baker–Ericksen inequality and the ellipticity condition.     

APPLICATION AND THEORETICAL ANALYSIS OF C/SiC COMPOSITES BASED ON CONTINUUM DAMAGE MECHANICS

Based on the thermodynamic theory, an orthotropic damage constitutive model was developed to describe the nonlinear mechanical behavior of C/SiC composites. The different nonlinear kinematic and isotropic hardening functions were adopted to describe accurately the damage evolution processes. The damage variables were defined with the damaged modulus and the initial undamaged modulus on energy equivalence principle. The initial orthotropy and damage coupling were presented in the damage yield function. Tensile and in-plane shear loading and unloading tests were performed, and a good agreement between the model and the experimental results was achieved.     

The effect of rotation and initial stress on the propagation of waves in a transversely isotropic elastic solid

In this paper the equations governing small amplitude motions in a rotating transversely isotropic initially stressed elastic solid are derived, both for compressible and incompressible linearly elastic materials. The equations are first applied to study the effects of initial stress and rotation on the speed of homogeneous plane waves propagating in a configuration with uniform initial stress. The general forms of the constitutive law, stresses and the elasticity tensor are derived within the finite deformation context and then summarized for the considered transversely isotropic material with initial stress in terms of invariants, following which they are specialized for linear elastic response and, for an incompressible material, to the case of plane strain, which involves considerable simplification. The equations for two-dimensional motions in the considered plane are then applied to the study of Rayleigh waves in a rotating half-space with the initial stress parallel to its boundary and the preferred direction of transverse isotropy either parallel to or normal to the boundary within the sagittal plane. The secular equation governing the wave speed is then derived for a general strain–energy function in the plane strain specialization, which involves only two material parameters. The results are illustrated graphically, first by showing how the wave speed depends on the material parameters and the rotation without specifying the constitutive law and, second, for a simple material model to highlight the effects of the rotation and initial stress on the surface wave speed.     

A constitutive theory for creep behavior of initially isotropic materials sustaining unilateral damage

The goal of the present work is to modify structure of the creep constitutive equations existing in the literature, and simultaneously to incorporate both damage induced anisotropy and unilateral damage into the constitutive model. The proposed nonlinear-tensor constitutive equation for creep together with the damage evolution equation take into account the secondary and tertiary creep of the initially isotropic materials. The material parameters of the model are determined using basic experiments. It is shown that the creep model is capable of describing available experimental data for the lateral creep responses under uniaxial compression.     

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.     

STUDY ON NONLINEAR CONSTITUTIVE RELATIONSHIP AND FRACTURE BEHAVIOR OF STITCHED C/SiC COMPOSITES$^{\bf 1)}$

C/SiC复合材料具有高比强度、高比模量和优良的热稳定性能等一系列优点, 广泛应用于航空航天领域中. 裂纹扩展进而引起的脆性断裂是其主要失效形式之一, 因而材料的断裂性能分析对材料的结构设计和应用有重要的指导意义. 本文开展了缝合式C/SiC复合材料简单力学试验和断裂试验, 研究了材料在不同载荷下的力学响应及断裂特征. 基于缝合式C/SiC复合材料简单力学试验, 建立了材料宏观非线性损伤本构方程, 并模拟了缝合式C/SiC复合材料单边切口梁和双悬臂梁的断裂行为. 本构方程采用简单函数描述了材料在复杂应力状态下的非线性应力-应变曲线, 并考虑了反向加载过程中造成的裂纹闭合. 基于商业有限元软件ABAQUS, 通过编写UMAT子程序实现非线性损伤本构方程, 采用单个单元验证了建立的本构方程的有效性. 在此基础上, 采用线弹性损伤本构和非线性损伤本构分别模拟了缝合式C/SiC复合材料单边切口梁和双悬臂梁的断裂行为. 采用非线性损伤本构方程模拟的力-位移曲线结果与试验结果更为吻合, 非线性损伤本构预测的失效载荷与试验失效载荷更为接近, 验证了所建立的非线性损伤本构方程的准确性, 为C/SiC复合材料断裂行为的研究提供了借鉴, 为缝合式C/SiC复合材料结构的设计和应用提供了理论基础.     

准脆性材料的弹塑性各向异性损伤耦合本构模型研究

针对准脆性材料的非线性特征：强度软化和刚度退化、单边效应、侧限强化和拉压软化、不可恢复变形、剪胀及非弹性体胀,在热动力学框架内,建立了准脆性材料的弹塑性与各向异性损伤耦合的本构关系。对准脆性材料的变形机理和损伤诱发的各向异性进行了诠释,并给出了损伤构形和有效构形中各物理量之间的关系。在有效应力空间内,建立了塑性屈服准则、拉压不同的塑性随动强化法则和各向同性强化法则。在损伤构形中,采用应变能释放率,建立了拉压损伤准则、拉压不同的损伤随动强化法则和各向同性强化法则。基于塑性屈服准则和损伤准则,构建了塑性势泛函和损伤势泛函,并由正交性法则,给出了塑性和损伤强化效应内变量的演化规律,同时,联立塑性屈服面和损伤加载面,给出了塑性流动和损伤演化内变量的演化法则。将损伤力学和塑性力学结合起来,建立了应变驱动的应力-应变增量本构关系,给出了本构数值积分的要点。以单轴加载-卸载往复试验识别和校准了本构材料常数,并对单轴单调试验、单轴加载-卸载往复试验、二轴受压、二轴拉压试验和三轴受压试验进行了预测,并与试验结果作了比较,结果表明,所建本构模型对准脆性材料的非线性材料性能有良好的预测能力。     

具确定弱区域正交各向异性薄板的弹塑性屈曲分析

基于弹塑性力学和损伤理论,建立了一个与应力球张量有关的具损伤正交各向异性材料的混合硬化屈服准则,该准则无量纲化后与各向同性材料的Mises准则同构,在此基础上,建立了正交各向异性材料的增量型和全量型弹塑性损伤本构方程,并以具确定弱区域正交各向异性矩形薄板为例,根据屈曲时的能量准则和全量理论,以等效塑性应变为内变量,对其弹塑性屈曲问题进行了分析,讨论了几何参数和弱区域对正交各向异性薄板弹塑性屈曲临界应力的影响.     

A continuum damage mechanics framework for modeling micro-damage healing

A novel continuum damage mechanics-based framework is proposed to model the micro-damage healing phenomenon in the materials that tend to self-heal. This framework extends the well-known Kachanov’s (1958) effective configuration and the concept of the effective stress space to self-healing materials by introducing the healing natural configuration in order to incorporate the micro-damage healing effects. Analytical relations are derived to relate strain tensors and tangent stiffness moduli in the nominal and healing configurations for each postulated transformation hypothesis (i.e. strain, elastic strain energy, and power equivalence hypotheses). The ability of the proposed model to explain micro-damage healing is demonstrated by presenting several examples. Also, a general thermodynamic framework for constitutive modeling of damage and micro-damage healing mechanisms is presented.