A continuum damage mechanics approach to crack tip shielding in brittle solids

This paper focuses firstly on the development of a comprehensive anisotropic theory of continuum damage mechanics for brittle solids suffering progressive deterioration. The basic concept of damage parameterization is re-examined and a new set of damage variables introduced yielding a new damage effect tensor through which the effective stress and strain tensors are defined. Constitutive equations of the damaged material are established incorporating a new hypothesis on equivalence between damaged and undamaged responses of the material. The model is completed by introduction of a general damage characteristic tensor which accounts for the experimentally observed fact that the rate of damage growth depends nonlinearly on applied external loads. The established damage model is next applied to investigate the crack-tip shielding effect due to anisotropic microcracking. The ratio of near-tip to remote stress intensity factors is obtained in closed form. A moderate but definite effect of anisotropy of microcracking is observed. The case of isotropic damage is found to be the least effective in screening remote external loads and is in accord with the results obtained by other researchers using different approaches.     

Damage Constitutive Equations and its Application to Fiber Reinforced Composites under Transverse Impact

Two continuous field variables, called as continuity tensor and damage variable tensor, are used to describe the anisotropic responses of an elastic-brittle material under transverse impact load. Based on the continuum damage mechanics, anisotropic damage constitutive equations in both full and incremental forms are proposed here. The expressions of effective elastic module tensor, damage variable tensor and damage propagation force tensor are further derived, and the methods for determining the tensors are explained in detail. An example of strain and damage response of a fiber reinforced composite laminated plate under transverse impact load is employed to demonstrate the application of this theory. In the example, the damage variable coupled with geometric large deformation of laminated plate is also considered. The calculating results illustrate the influence of damage on strain field in the impacted laminated plate.     

复合固体推进剂黏弹性微裂纹损伤本构模型

为建立复合固体推进剂的损伤本构模型,在介观尺度上视其为微裂纹损伤,选取微裂纹密度为损伤内变量。在Abdel-Tawab本构方程的基础上,基于微裂纹均匀化理论,推导了损伤映射张量的一般形式。该张量通常具有非完全对称性,其物理意义是将真实应力空间中各向异性材料的多轴加载映射为等效应力空间中各向同性材料的更为复杂的多轴加载。其次,基于黏弹性动态裂纹扩展模型和裂纹扩展阻力曲线的概念,建立了损伤内变量的演化方程。该演化方程仅含4个物理意义明确的细观参数,并且参数的取值规律与宏观应力曲线的变化规律相一致。数值结果表明,建立的模型能够有效反映材料损伤的应变率、温度依赖性及各向异性特征,并且具有一定的蠕变损伤预测能力。     

On stiffness and strength reduction of solids due to crack development

Modification of the effective elastic and plastic constants of initially homogeneous and isotropic material with regularly distributed cracks is considered in the paper. The stress-strain relation for linearly elastic range is formulated as a tensor function with two independent variables: the stress tensor and damage tensor describing the current state of the cracked solid. This equation made it possible to evaluate all the elastic constants and is a starting point in the analysis of the plastic behavior of the damaged material. The appropriate yield criterion is derived in the form of an isotropic scalar function with the same variables as in the elastic range. To choose the most important terms of the general representation of this function, the energy of the elastic strain was calculated for homogenized equivalent material. This was done employing the stress-strain relation of elasticity for damaged solid proposed in the paper. The theoretical considerations were verified experimentally. To this end the material constants determined theoretically in the elastic and plastic ranges were compared with those measured experimentally for the models simulating the damaged material.     

弹脆性损伤理论与破坏过程的计算机模拟

本文提出了一种增量形式的受损弹脆性材料的本构关系,此关系考虑了材料与损伤两者的各向异性.而且,导出了损伤模量、有效弹性模量以及动态弹性模量等张量表达式.根据本文提出的理论和方法,给出了考虑局部损伤的结构有限元方法,并应用于模拟复合材料板的损伤一破坏过程.最后,受拉伸的带中心圆孔的玻璃纤维布/环氧树脂复合材料板损伤过程的计算结果用等损伤线图形显示出来.     

A parametric study of residual strength and stiffness for impact damaged composites

A parametric study of residual strength and stiffness for low-velocity impact damaged composites was performed. Possible compression after impact failure mechanisms, which may be caused by stiffness and strength degradation inside the damage region, were discussed. In order to understand the scaling effects on impact damage residual strength, finite element analysis was performed. With stiffness degradation inside the damaged area, the stress fields of small coupon and larger stiffened panels with the same damage were calculated numerically. The stress redistributions were found to be almost identical between the coupon and panels. This indicates that the stress redistribution is a local phenomenon, and will not be affected much by the existence of the stiffeners. It is believed that the residual strength of impact damaged composite structures is related more to the damage severity than to the load redistribution. The dependency of the stress concentration factor on the anisotropic engineering elastic constants of a laminate was extended from an analytical study by Lekhnitskii (Anisotropic Plates, Gordon and Breach Science Publishers, New York, USA, 1968).     

Asymptotic homogenization algorithm for reinforced metal-matrix elasto-plastic composites

The theory of the two-scale convergence was applied to homogenization of initial flow stresses and hardening constants in some exponential hardening laws for elasto-plastic composites with a periodic microstructure. The theory is based on the fact that both the elastic and the plastic part of the stress field two-scale converge to a limit, which can be factorized by parts, one of which depends only on the macroscopic, and the other one – only on the microscopic characteristics. The first factor is represented in terms of the homogenized stress tensor and the second factor – in terms of stress concentration tensor, that relates to the micro-geometry and elastic or plastic micro-properties of composite components. The theory was applied to a composite, that consists of the metallic elasto-plastic matrix with Ludwik and Hocket–Sherby hardening law and pure elastic silica inclusions. Results were compared with those of averaging based on the self-consistent methods.     

A two scale anisotropic damage model accounting for initial stresses in microcracked materials

In a recent study [15], we proposed a class of isotropic damage models which account for initial stresses. The present paper extends this approach to anisotropic damage due to growth of an arbitrarily penny-shaped microcracks system. The basic principle of the upscaling technique in the presence of initial stress is first recalled. Then, we derive a closed-form expression of the elastic energy potential corresponding to a system of arbitrarily oriented microcracks. It is shown that the coupling between initial stresses and damage is strongly dependent of the microcracks density and orientation. Predictions of the proposed model are illustrated through the investigation of the influence of initial stresses on the material response under non monotonous loading paths. Finally, by considering a particular distribution of microcracks orientation, described by a second order damage tensor, it is shown that the model is a generalization of the macroscopic damage model of Halm and Dragon [9], for which a physically-based interpretation is then proposed.     

基于二阶组构张量的各向异性屈服准则

将四个屈服准则:Tresca准则、Mises准则、Mohr-Coulomb准则以及Drucker-Prager准则归类为剪切屈服准则。Tresca准则和Mohr-Coulomb准则是关于最不利截面的剪切屈服准则,而Mises准则和Drucker-Prager准则是关于各方向截面的剪应力和正应力的某种综合度量的八面体剪应力和八面体正应力的剪切屈服准则。从方向函数(ODF)的概念入手,将各方向截面的剪应力和正应力综合度量直接取为所有方向截面上的剪应力和正应力的平均。对各向同性材料,提出了平均剪切屈服度准则:当平均剪应力和平均正应力的组合达到某一极限值时,材料开始屈服。研究表明,平均剪切屈服准则与Drucker-Prager准则具有相同的形式,当不考虑平均正应力对屈服的影响时,它与Mises准则具有相同的形式。针对由各向异性损伤导致的材料各向异性强度问题,定义截面上的有效正应力和有效剪应力则分别为截面上的法向力和切向力与有效承载面积之比,基于截面上的有效应力提出了各向异性材料的平均剪切屈服准则。各向异性损伤引起的截面上有效应力放大系数为方向函数,可以采用二阶组构张量来近似表示,在任意坐标系中,各向异性屈服准则为应力分量的二次齐次式,导出了其中的系数与二阶组构张量之间的显式关系式。在二阶组构张量的主轴坐标系内,各向异性屈服准则与殷有泉的拓展Hill准则形式完全相同,当不考虑正应力对屈服的影响时,它与Hill准则具有相同的形式。     

CONDITIONS FOR ADAPTATION OF DAMAGED ELASTIC–PLASTIC BODIES TO CYCLIC LOADING

Conditions for adaptation of isotropically damaged elastic–plastic bodies with isotropic strain hardening are investigated in the framework of the energy-based coupled elastic–plastic damage model by Ju. The yield function is assumed to be a homogeneous function of the first order in the stress tensor components. Due to this assumption, the notion of effective yield stress can be introduced. The loading program is supposed to be prescribed. Features of the stress path at the post-adaptation stage are considered, which lead to new necessary shakedown conditions expressed by a set of inequalities, and, in turn, to a problem of mathematical programming whose solution yields lower estimates for the damage and strain-hardening parameters. In the event, if the calculated value of the damage parameter is greater than its critical value, an adaptation to a given loading program is impossible. This condition is also necessary for adaptation in the case if only bounds for applied loads are prescribed. A correction of the constitutive material model is proposed which possibly could be good for ductile damage. The derived shakedown condition is not only necessary, but also sufficient for the plastic adaptation. The developed method is expounded in an example.     

Analysis of fatigue damage evolution and associated anisotropic elastic property degradation in random short-fiber composite

A theoretical analysis of cyclic fatigue damage and associated anisotropic property degradation in a random short-fiber composite is presented. The fatigue damage takes various forms of microcracking, originated from microscopic stress concentrators in the highly heterogeneous material system. A probabilistic treatment of the microcracks is introduced to evaluate the statistical nature of the microscopic fatigue damage. Damage evolution and accumulation are analyzed through the development of probabilistic density functions of microcrack length and orientation during the cyclic loading history. Constitutive equations for the damaged fiber composite are then derived on the basis of a self-consistent mechanics scheme in conjunction with a three-dimensional elliptic crack theory and the microcrack density functions. Cyclic fatigue degradation and associated damage-induced anisotropy of composite material properties are determined and checked against experiments. The tensorial nature of material damage and composite stiffness changes during fatigue are evaluated explicitly. A power-law relationship between the rate of damage growth and the fatigue loading cycle is obtained. The rate of fatigue damage growth is found to decrease exponentially with load cycles—a phenomenon unique to the random short-fiber composite. This study provides a comprehensive analytical treatment of the homogeneous fatigue damage problem for random short-fiber composites. The fundamental mechanics and mechanisms of fatigue damage evolution and associated anisotropic property degradation of the composite are elucidated.     

修正偶应力理论层合薄板稳定性模型及尺度效应

该文将各向同性修正偶应力理论推广到各向异性,提出各向异性的细观尺度的复合材料层合板的本构方程,基于虚功原理建立了各向异性修正偶应力理论并用于建立复合材料层合薄板偶应力理论稳定性模型。该理论的偶应力部分的转角不是独立变量(称为C1理论),对于各单层引入纤维和基体材料的不同的两个材料细观参数,建立了适用于层合板/夹层板的偶应力理论模型。该理论的应变不对称,但是,用于各向同性材料与修正偶应力理论等价。为了便于工程应用,忽略基体材料的细观长度参数,建立了各单层只含一个材料细观参数的偶应力层合薄板理论稳定性模型。算例表明建立的偶应力层合板模型能用于分析层合板稳定性的尺度效应。     

On fibre bundle approach to a damage analysis

In this paper the geometric structure of a damaged state in the medium is investigated based on the fibre bundle technique. The damage (a failure) in some region of the medium under loads is regarded as breakdown of the holonomicity in this region. A measure of reduction of load carrying area elements caused by the development of microcracks or microvoids in the damaged medium is identified with the damage tensor. Multiplicative representations of the damage tensor formulated both on the tangent bundle with an affine connection and a fibre bundle with a non-linear connection over the damaged medium are obtained. In the former case the generalization of the damage tensor to the tangent bundle rests on the technique of prolonging the deformation vector to include not only the independent variables and dependent variables appearing in the damage tensor, but also the derivatives of the dependent variables. The so-called lifting technique is used to express the damage tensor by initial, additional and direct (deformation-induced) damages or by direct and transferred ones. In the latter one the higher-order contact geometry approach identified with the Finslerian geometry is employed to analyze the influences of elastic, inelastic, direct, transferred and initial damages on the total damage tensor. Time-dependent relationships of the damage tensor are then presented.     

A finite element approach to anisotropic damage of ductile materials in large deformations Part I: an anisotropic ductile elastic-plastic-damage model

During past decades, many material models using the continuum damage mechanics (CDM) approach have been proposed successfully in the small deformation regime to describe inelastic behaviors and fracturing phenomena of a material. For ductile materials, large deformation takes place at the level of damage appearance. Damage is anisotropic in nature. In this paper, the ductile damage at finite deformations is modeled as an anisotropic tensor quantity. Then, a fourth-order symmetric stress correction tensor is proposed for computationally efficient and easy implementation in the finite element formulations. Consequently, an explicit form of the fourth-order constitutive equations of the proposed elastic-plastic-damage model is derived. Both isotropic and kinematic hardening effects are included in the formulation. The new constitutive model can predict not only the elastic-plastic behaviors, but also the sequential variations of ductile materials. An evaluation of the constitutive and damage evolution equations is presented. This revised version was published online in August 2006 with corrections to the Cover Date.     

An anisotropic linear elastic boundary element formulation for masonry wall analysis

This work presents an application of a Boundary Element Method (BEM) formulation for anisotropic body analysis using isotropic fundamental solution. The anisotropy is considered by expressing a residual elastic tensor as the difference of the anisotropic and isotropic elastic tensors. Internal variables and cell discretization of the domain are considered. Masonry is a composite material consisting of bricks (masonry units), mortar and the bond between them and it is necessary to take account of anisotropy in this type of structure. The paper presents the formulation, the elastic tensor of the anisotropic medium properties and the algebraic procedure. Two examples are shown to validate the formulation and good agreement was obtained when comparing analytical and numerical results. Two further examples in which masonry walls were simulated, are used to demonstrate that the presented formulation shows close agreement between BE numerical results and different Finite Element (FE) models.     

A Damage-Mode Based Three Dimensional Constitutive Model for Fibre-Reinforced Composites

This article presents a three dimensional constitutive model for anisotropic damage to describe the elastic-brittle behavior of unidirectional fibrereinforced laminated composites. The primary objective of the article focuses on the three dimensional relationship between damage of the material and the effective elastic properties for the purpose of stress analysis of composite structures, in extension to the two dimensional model in Matzenmiller, Lubliner and Taylor (1995). A homogenized continuum is adopted for the constitutive theory of anisotropic damage and elasticity. Damage initiation criteria are based on Puck failure criterion for first ply failure and progressive micro crack propagation is based on the idea of continuum damage evolution. Internal variables are introduced to describe the evolution of the damage state under loading and as a subsequence the degradation of the material stiffness. Emphasis is placed on a suitable coupling among the equations for the rates of the damage variables with respect to the different damage modes.     

Anisotropic Gurson‐Tvergaard‐Needleman plasticity and damage model for finite element analysis of elastic‐plastic problems

Implementation and analysis of the anisotropic version of the Gurson‐Tvergaard‐Needleman (GTN) isotropic damage criterion are performed on the basis of Hill's quadratic anisotropic yield theory with the definition of an effective anisotropic coefficient to represent the elastic‐plastic behavior of ductile metals. This study aims to analyze the extension of the GTN model suitable for anisotropic porous metals and to investigate the GTN model extension. An anisotropic damage model is implemented using the user material subroutine in ABAQUS/standard finite element code. The implementation is verified and applied to simulate a uniaxial tensile test on a commercially produced aluminum sheet material for three‐dimensional and plane stress test cases. Spherical and ellipsoidal micro voids are considered in the matrix material, and their effects on the uniaxial stress‐strain response of the material are analyzed. Hill's quadratic anisotropic yield theory predicts substantially large damage evolution and a low stress‐strain curve compared with those predicted by the isotropic model. An approximate model for anisotropic materials is proposed to avoid increased damage evolution. In this approximate model, Hill's anisotropic constants are replaced with an effective anisotropy coefficient. All model‐generated stress‐strain predictions are compared with the experimental stress‐strain curve of AA6016‐T4 alloy.     

Solution of the inverse problems of heat admittance in order to derive characteristics of anisotropic materials

A solution of the inverse problem in order to determine the principal coefficients and principal axes angles of the heat conduction tensor of an anisotropic material is proposed in this paper. It is based upon a previously found solution of the heat conduction problem in an anisotropic half-space heated with a thermal flow. The technique is based upon expansion of the discrepancy composite function into Taylor series and determination of differential vectors of the target coefficients, which are then used for application of iterative gradient descent algorithms. The results have proved fine convergence even if the initial approximation of the coefficients vector is different from the target by more than one hundred percent. The proposed method can be applied to determination of thermophysical characteristics of composite materials used for thermal protection of hypersonic flying vehicles.     

Multiscale modeling of impact on heterogeneous viscoelastic solids containing evolving microcracks

Multiscale computational techniques play a major role in solving problems related to viscoelastic composites due to the complexities inherent to these materials. In this paper, a numerical procedure for multiscale modeling of impact on heterogeneous viscoelastic solids containing evolving microcracks is proposed in which the (global scale) homogenized viscoelastic incremental constitutive equations have the same form as the local‐scale viscoelastic incremental constitutive equations, but the homogenized tangent constitutive tensor and the homogenized incremental history‐dependent stress tensor at the global scale depend on the amount of damage accumulated at the local scale. Furthermore, the developed technique allows the computation of the full anisotropic incremental constitutive tensor of viscoelastic solids containing evolving cracks (and other kinds of heterogeneities) by solving the micromechanical problem only once at each material point and each time step. The procedure is basically developed by relating the local‐scale displacement field to the global‐scale strain tensor and using first‐order homogenization techniques. The finite element formulation is developed and some example problems are presented in order to verify the approach and demonstrate the model capabilities. Copyright © 2009 John Wiley & Sons, Ltd.     

A comparative study of continuum damage models for crack propagation under gross yielding

In this paper, the problem of fracture initiation in an aluminum alloy thin plate containing a central crack is examined by employing several phenomenological continuum damage mechanics models. These models differ mainly in the selection of the kind of tensorial property the damage variable assumes, the nature of the equivalence postulate between damaged and pseudo undamaged material states, and the way damage evolution laws are formulated. Two formulations of damage effect tensor based on the engineering notation and the normative notation of stress and strain, respectively, are compared. In addition, the hypothesis of strain equivalence is compared to that of stress working equivalence. The error in the assumption of isotropic damage development in the crack tip process zone is also checked against that of anisotropic damage. In the numerical algorithm, both updated Lagrangian formulation and small displacement formulation of material non-linearity only are adopted and compared. The influence of non-proportionality in stress histories present in the crack tip region is accounted for by introducing a dynamic coordinate system of principal damage such that the principal direction of damage rotates in accordance with that of the loading. The calculated fracture initiation loads are finally compared with those determined experimentally.