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.     

An ‘extended’ volumetric/deviatoric formulation of anisotropic damage based on a pseudo-log rate

Following a framework of elastic degradation and damage previously proposed by the authors, an ‘extended’ formulation of orthotropic damage in initially isotropic materials, based on volumetric/deviatoric decomposition, is presented. The formulation is founded on the concept of energy equivalence and makes use of second-order symmetric tensor damage variables. It is characterized by fourth-order damage-effect tensors (relating nominal to effective stresses and strains) built from the underlying second-order damage tensors and decomposed in product-form in isotropic and anisotropic parts. The formulation is developed in two steps. First, secant relations are established. In the isotropic case, the model embeds a path parameter allowing to range between pure volumetric to pure deviatoric damage. With the two undamaged material constants this makes a total of three constant parameters plus an evolving scalar damage variable, giving rise to a four-parameter model with two varying isotropic material coefficients. In the anisotropic case, the model is still characterized by the same three material constants plus three evolving variables which are the principal values of a second-order damage tensor. This leads to a six-parameter restricted form of orthotropic damage. In the second step, damage evolution rules are formulated in terms of a pseudo-logarithmic rate of damage. This allows to define meaningful conjugate forces that constitute a feasible space in which loading functions and damage evolution rules can be defined. The present ‘extended’ formulation is closed by the derivation of the tangent stiffness.     

Elasto-plastic buckling analysis of laminated plates including interfacial damage

Elasto-plastic buckling of orthotropic laminated plates, which include interfacial damage, is analyzed in detail. Firstly, a novel mixed hardening yield criterion, as an improvement of Hill’s counterpart, is proposed for the orthotropic materials on the basis of the plastic theory. And differing from Hill’s theory, the present yield criterion is related to the spherical tensor of stress. Then, the incremental elasto-plastic constitutive relations of the mixed hardening orthotropic materials are presented. Secondly, the incremental static equilibrium equations for laminated plates including interfacial damage are established based on Von-Karman type theory and the principle of minimum potential energy. Finally, the elasto-plastic buckling of laminated plates are solved by adopting the Galerkin method and iteration scheme. The numerical results show that buckling of the plate occurs easier due to the existence of interfacial damage, and the critical load trends to constant when the interfacial damage approaches a certain degree. Also, the effect of anisotropy on buckling is obvious and the analysis of elasto-plastic buckling is necessary.     

Anisotropic damage law of evolution

A formulation for anisotropic damage is established in the framework of the principle of strain equivalence. The damage variable is still related to the surface density of microcracks and microvoids and, as its evolution is governed by the plastic strain, it is represented by a second order tensor and is orthotropic. The coupling of damage with elasticity is written through a tensor on the deviatoric part of the energy and through a scalar taken as its trace on the hydrostatic part. The kinetic law of damage evolution is an extension of the isotropic case. Here, the principal components of the damage rate tensor are proportional to the absolute value of principal components of the plastic strain rate tensor and are a nonlinear function of the effective elastic strain energy. The proposed damage evolution law does not introduce any other material parameter. Several series of experiments on metals give a good validation of this theory. The coupling of damage with plasticity and the quasi-unilateral conditions of partial closure of microcracks naturally derive from the concept of effective stress. Finally, a study of strain localization makes it possible to determine the critical value of the damage at mesocrack initiation.     

热动力学理论在粘弹性断裂力学中的应用

本文基于热动力学定理推导了线性粘弹性材料的本构方程，并根据已有文献的思想列出了线性粘弹性材料的热力学平衡方程，由此推得二种全局能量释放率的表达开式，同时根据虚功原理进一步推得裂尖局部能量释放率，也就说明了全局能量释放率和被广泛应用的局部能量释放率之间的等价性，本文最后给出了在正交各向异性和向同性粘弹性介质中裂尖能量释放率的具体表达式。     

A thermodynamic approach with internal variables using Lagrange formalism. Part II. Continuous symmetries in the case of the time–temperature equivalence

In the first part of this contribution, the Lie-symmetries of the principle of least action associated to the constitutive equations of the DNLR formalism of relaxation have been presented. We examine in this second part the continuous symmetries corresponding to the simple case of stress relaxation under isothermal conditions. The well-known principle of time/temperature equivalence is discussed in terms of variational symmetry for the Jacobi’s action functional, and connected to the Onsager’s relation near the thermodynamic equilibrium.     

On a geometrically nonlinear damage model based on a multiplicative decomposition of the deformation gradient and the propagation of microcracks

We aim to derive a damage model for materials damaged by microcracks. The evolution of the cracks shall be governed by the maximum energy release rate, which was recently shown to be a direct consequence of the variational principle of a body with a crack (Arch. Appl. Mech. 69 (5) (1999) 337). From this, we get the path of the growing crack by introducing a series of thermodynamically equivalent straight cracks. The equivalence of the energy dissipated by microcrack growth and the damage dissipation leads to our damage evolution law. This evolution law will be embedded in a finite deformation framework based on a multiplicative decomposition into elastic and damage parts. As a consequence of this, we can present the anisotropic damaged elasticity tensor with the help of push and pull operations. The connection of this approach to other well known damage theories will be shown and the advantages of a finite element framework will be worked out. Numerical examples show the possibilities of the proposed model.     

Thermodynamics modelling of plasticity and damage of argilliteModélisation thermodynamique de la plasticité et de l'endommagement d'une argilite.

The study of the physical and mechanical characteristics of argillites allows us to retain the state variables useful for the thermodynamic modelling of their behaviour. We built a thermodynamic potential which reproduces the nonlinear behaviour. Plasticity-damage coupling is taken into account by using an effective stress coming from strain equivalence principle. To cite this article: N. Conil et al., C. R. Mecanique 332 (2004).     

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．     

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     

The effect of material and geometry on the non-linear vibrations of orthotropic circular cylindrical shells

The extensive use of circular cylindrical shells in modern industrial applications has made their analysis an important research area in applied mechanics. In spite of a large number of papers on cylindrical shells, just a small number of these works is related to the analysis of orthotropic shells. However several modern and natural materials display orthotropic properties and also densely stiffened cylindrical shells can be treated as equivalent uniform orthotropic shells. In this work, the influence of both material properties and geometry on the non-linear vibrations and dynamic instability of an empty simply supported orthotropic circular cylindrical shell subjected to lateral time-dependent load is studied. Donnell׳s non-linear shallow shell theory is used to model the shell and a modal solution with six degrees of freedom is used to describe the lateral displacements of the shell. The Galerkin method is applied to derive the set of coupled non-linear ordinary differential equations of motion which are, in turn, solved by the Runge–Kutta method. The obtained results show that the material properties and geometric relations have a significant influence on the instability loads and resonance curves of the orthotropic shell.     

正交异性扁壳的等价关系式和不变量

提出各向同性扁壳比拟法,分析满足条件D_3=D_(12)=(D_1D_2)~(1/2)的正交异性扁壳大挠度弯曲和超屈曲问题,导出了正交异性扁壳与各向同性扁壳之间,两种不同正交异性扁壳之间坐标变量、扁壳厚度和曲率半径、荷载、挠度、转角、弯矩、扭矩、中面应力的等价关系式,还证明了等价正交异性扁壳的几个等价不变量。     

On the coupling of anisotropic damage and plasticity models for ductile materials

In this contribution various aspects of an anisotropic damage model coupled to plasticity are considered. The model is formulated within the thermodynamic framework and implements a strong coupling between plasticity and damage. The constitutive equations for the damaged material are written according to the principle of strain energy equivalence between the virgin material and the damaged material. The damaged material is modeled using the constitutive laws of the effective undamaged material in which the nominal stresses are replaced by the effective stresses. The model considers different interaction mechanisms between damage and plasticity defects in such a way that two-isotropic and two-kinematic hardening evolution equations are derived, one of each for the plasticity and the other for the damage. An additive decomposition of the total strain into elastic and inelastic parts is adopted in this work. The elastic part is further decomposed into two portions, one is due to the elastic distortion of the material grains and the other is due to the crack closure and void contraction. The inelastic part is also decomposed into two portions, one is due to nucleation and propagation of dislocations and the other is due to the lack of crack closure and void contraction. Uniaxial tension tests with unloadings have been used to investigate the damage growth in high strength steel. A good agreement between the experimental results and the model is obtained.     

纤维增强陶瓷基复合材料的D失效判据

经典唯象强度理论适用于正交各向异性线弹性体。对于非线性纤维增强复合材料,通过加卸载试验和损伤力学的分析方法,可以得到一种虚拟的线性化应力-应变关系;依据损伤等效假设,针对线性损伤和非线性损伤,对基于应力的经典二次失效准则进行变换,建立了一种基于损伤的强度理论,即“D失效判据”,这一强度理论可以作为经典判据的补充和扩展。针对平纹编织C/SiC复合材料的拉/剪组合试验,进行了实例计算,结果表明：利用D失效判据预测的失效包络线比蔡-希尔准则的预测曲线低,而且,失效曲线的形式与材料的损伤演化规律相关。     

Limiting state of an elastoplastic orthotropic plate with a periodic system of collinear cracks

A modified Dugdale model is used to study the fracture of an orthotropic elastoplastic plate with a periodic system of rectilinear cracks. The material of the plate obeys a general yield criterion. The general form of solution is obtained in terms of Kolosov-Muskhelishvili potentials. The size of the plastic zone is expressed in terms of the external load and geometrical parameters. The equations for the determination of the stresses in the plastic zone and the crack opening displacement are derived. The effect of anisotropy on the formation of the plastic zones at the crack tip is examined __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 5, pp. 80–88, May 2007.     

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.     

Stress analysis in adhesive-bonded joints for orthotropic plates

In the current paper we present the tensile force, the edge moment and edge transverse force acting on the single-lap joint and get the moment reduction factork for orthotropic plates. We use the variational method based on the principle of complementary energy. Considering Hooke's law for orthotropic materials, carrying out the variationV=0 and using the Euler-Poisson's equation we establish all the expressions of stresses in the joint for orthotropic plates. All numerical calculations can be made on a computer.     

非线性粘弹体的时间-温度-应力等效原理及其应用

考虑粘弹性材料特征时间的应力相关性,认为应力水平与温度、压力、溶剂浓度、损伤及老化等材料特征时间的影响相似,有其等效性。依自由体积理论,推导了温度－应力移位因子的表达式,提出了时间－温度－应力等 效原理。应用此原理,可以将不同温度和应力水平下的儒变曲线移位成某一参考温度和参考应力水平下的主曲线,从而可以通过较高温度和应力水平下的短期蠕变行为来预测较低温度和应力水平下的较长期的蠕变行为。实例分析了高密度聚乙烯（HDPE）非线性蠕变行为的－应力等效性。