Interaction between an interface crack and subinterface microcracks in metal/piezoelectric bimaterials

The J-integral analysis is presented for the interaction problem between a semi-infinite interface crack and subinterface matrix microcracks in dissimilar anisotropic materials. After deriving the fundamental solutions for an interface crack subjected to different loads and the fundamental solutions for an edge dislocation beneath the interface, the interaction problem is deduced to a system of singular integral equations with the aid of a superimposing technique. The integral equations are then solved numerically and a conservation law among three values of the J-integral is presented, which are induced from the interface crack tip, the microcracks and the remote field, respectively. The conservation law not only provides a necessary condition to confirm the numerical results derived, but also reveals that the microcrack shielding effect in such materials could be considered as a redistribution of the remote J-integral. It is this redistribution that does lead to the phenomenological shielding effect.     

Further investigation of interaction between interface macrocrack and parallel microcracks in bimaterial anisotropic solids

In this paper, with the aid of superimposing technique and the Pseudo Traction Method (PTM), the interaction problem between an interface macrocrack and parallel microcracks in the process zone in bimaterial anisotropic solids is reduced to a system of integral equations. After the integral equations are solved numerically, a conservation law among three kinds ofJ-integrals is obtained which are induced from the interface macrocrack tip, the microcrack and the remote field, respectively. This conservation law reveals that the microcrack shielding effect in such materials could be considered as the redistribution of the remoteJ-integral. The project supported by the National Natural Science Foundation of China, and the Doctorate Foundation of Xi'an Jiaotong University     

FURTHER INVESTIGATION OF THE J_2-INTEGRAL IN BIMATERIAL SOLIDS

The J_2-integral induced from the interface of bimaterial solids(J_2~(interface))is stud-ied by numerical method.First,the effect on the J_2-integral induced from the interface is verysignificant in bimaterial solids,which is inherently related to that induced from the subinterfacecracks.Moreover,it can be concluded that either the first or the second component of the J_k-vector is always equal to zero when the contour encloses both the cracks and the whole interfacein bimaterial solids.Secondly,it can also be concluded that the interface does produce significanteffect on the J_2-integral induced from the subinterface cracks(J_2~(sub))in bimaterial solids.Thiseffect depends on the geometry of the crack arrangement,which is corresponding to the differentinteraction effect among the cracks and the interface.Moreover,the interface effect on the J_2~(sub)can be neglected when the distance from the crack center to the interface is large enough,whichreveals that the bimaterial solids can be regarded as homogenous solids in fracture analysis whenthe subinterface crack is far enough from the interface.Three examples are given in this paper.     

M-integral analysis for microcrack-damaged anisotropic composite materials

Summary  This paper presents an M-integral analysis for the microcracked anisotropic composite materials. By using an elementary solution derived for a single finite crack subjected to a concentrated force on crack faces, the problem of strong interacting, arbitrarily oriented and located microcracks in an anisotropic composite materials is reduced to a system of Fredholm integral equations. The crack-tip fracture parameters, such as the stress intensity factors, are evaluated from a numerical solution of the system of integral equations. Its dependence on the coordinate system, calculation, and physical interpretation of the M-integral are discussed in the interaction problem. Finally, a numerical example of the damage evaluation by the M-integral analysis is given. Received 24 September 1999; accepted for publication 8 February 2000     

Energy release or absorption due to simultaneous expansion of many interacting nanoholes in elastic materials

The energy release or absorption due to simultaneous expansion of many interacting nanoholes in elastic materials under plane strain deformation is studied as influenced by the surface effect along rims of nanoholes. The M-integral classically used in macro mechanics with defects is extended to treat the problem with many interacting nanoholes. After some manipulations, the energy change due to the simultaneous expansion of many nanoholes represented by the M-integral is evaluated. Four different arrays of many nanoholes under a monotonically increasing tensile loading are considered. Attention is focused on the influence induced from the surface tension, the surface Lamé constants, and the interaction among many nanoholes on the M-integral. It is concluded that the surface tension yields significant influence on the M-integral, whereas the surface Lamé constants offer much smaller influence, which could be neglected with some relative errors less than 2%. It is found that, unlike those in macro mechanics with defects, the simultaneous expansion could either release energy (the positive value of the M-integral) or absorb energy (the negative value of the M-integral), depending on the loading levels. There is a neutral loading point, at which the M-integral transforms from a negative value to a positive value in all arrays of nanovoids under consideration. It is also found that the interaction among multiple nanoholes influences the value of the neutral loading point significantly because the mutual influence induced from both the interacting effect and the surface effect yields a quite different feature from those induced from the interacting effect only. That is, the surface effect always inhibits the influence of the interacting effect on the M-integral.     

界面微裂纹屏蔽J_k矢量投影守恒

用理论推导和电算实践证明，尽管两相材料界面裂纹Ｊ积分的显函数表达式与均质材料中不同，尽管界面裂纹尖端固有的 １／２＋ｉε振荡奇异性和张开滑移型固有的耦合造成近尖区应力场分布的复杂性，但作者在均质材料微裂纹屏蔽问题中发现的Ｊ积分再分配关系 Ｊｋ矢量投影守恒定理在两相材料界面微裂纹屏蔽问题中仍然成立．这再次说明，远场Ｊ积分向界面裂纹尖端传递过程中跨越微裂纹群是有损失的．这个损失可用Ｊｋ矢量在界面裂纹延线坐标轴上的投影来定量地评估     

压电材料中的微裂纹屏蔽问题分析

分析当主裂纹与一个微裂纹在远场I型力（KI）和远场电位移（Ke)作用下的相互干涉问题,得出了在微裂纹的位置角和方向角周时独立变化时,微裂纹对主裂纹的屏蔽作用的全局使命主裂纹扩展,通过电算还发现Ortiz在各向同性材料和各向异性材料中得出的“微裂纹群对主裂纹最大屏蔽效应产生在微裂纹方向与最大主应力垂直的方向”在压电材料中不再成立,进而提出除Hutchinson指出微裂纹屏蔽效应两个来源（即：材料有效刚度的降低和残余应力的释放）外的另一个来源,微裂纹对主裂砂电场的扰动,在对主微裂纹J积分分析时发现J2积分与J1积分具有同等重要的地位。     

A semi-infinite interface crack interacting with subinterface matrix cracks in dissimilar anisotropic materials. II. Numerical results and discussion

Based on the investigation performed in Part I of this series, numerical results for the interaction between a semi-infinite interface crack and multiple subinterface matrix microcracks in three kinds of material combinations are given in Part II. The major interaction behaviors are discussed in detail. Special attention is focused on the influences of the different material combinations, the T-stress, the orientation angles, and the location angles of the microcracks on the local stress intensity factor at the interface crack tip. In addition, the variable tendencies of the interaction effect induced from change of the distance between the interface crack tip and the centers of the microcracks are studied. It is concluded that the different material combinations introduced in this paper have little influence on the variable tendencies of the effect, but have significant influence on the effect in magnitude. Detailed comparisons of the results with those in a homogeneous orthotropic material show that the dissimilar materials shift the maximum amplification angle, the maximum shielding angle, the neutral shielding angle, and the neutral T-stress angle, respectively.     

主裂纹与微裂纹相互作用问题的有效数值计算方法

提出了平面弹性介质中主裂纹与微裂纹相互作用问题的有效数值计算 方法. 通过把适于单一裂纹的Bueckner原理扩充到含有多裂纹的一般体系，将原问题分解 为承受远处载荷不含裂纹的均匀问题，和在远处不承受载荷但在裂纹面上承受面力的多裂纹 问题. 于是，以应力强度因子作为参量的问题可以通过考虑后者(多裂纹问题)来解决，而 利用提出的杂交位移不连续法，这种多裂纹问题是容易数值求解的. 列举 Cai和 Faber为评价主裂纹与微裂纹相互作用问题的近似方法而列举的算例，说明 该数值方法对分析平面弹性介质中主裂纹与微裂纹相互作用问题既简单又非常有效.     

干涉问题中T应力与各向异性的作用

采用离散模型（包括半无限主裂纹和近尖微裂纹）研究了各向异性材料主微裂纹干涉问题中T应力对主裂尖参数的影响,并且与相同情况下各向同性材料的结果进行了比较,比较结果列于文中各图。研究结果表明,在各向异性材料和各向同性材料中T应力对主裂尖应力强度因子的影响趋势是相似的,但是由于T应力与材料各向异性性质的共同作用,使两种情况下T应力对主裂尖参数的影响结果存在着明显的偏差。     

带微裂纹物体的有效断裂韧性

按照等效介质的思想，引进有效表面能密度的概念，建立了带微裂纹物体有效断裂韧性的公式．具体计算了微裂纹群分别平行和垂直于宏观裂纹两种情况的减韧比．表明微裂纹群在产生应力屏蔽（或反屏蔽）效应的同时，也降低了材料的有效断裂韧性，减小了对宏观裂纹的扩展阻力．     

Energy release or absorption due to simultaneous rotation of two nano voids in plane elastic materials as influenced by both surface effect and interacting effect

In this paper, the L-integral analysis for two nano-sized voids in plane elasticity under uniaxial loading is present. Three surface parameters are considered including the surface tension and two surface Lamé constants. Attention is focused on the mutual influence on the L-integral from both the surface effect at voids’ rims and the interacting effect between voids. A close-form expression of L-integral for multiple nano voids is obtained. Comparing with those in macro fracture mechanics, the L-integral shows some different features when the surface effect is taken into account. It is concluded that under tensile loading and due to the mutual influence, the L-integral might be either positive or negative, depending on the loading level. The numerical results show that the surface tension is the dominant one in surface parameters on impacting the L-integral. It is also concluded that the surface effect shields the energy release (positive L-integral value) while enhances the energy absorption (negative L-integral value). The two-state L-integral analysis is performed to clarify the way that the surface effect impacts the L-integral. It is concluded that the contribution to L-integral from the voids’ configuration could either be negative or positive, while that from the surface effect is always negative. Besides, the size dependence in the present problem is studied in detail.     

Energy release or absorption due to simultaneous expansion of many interacting nano-holes in elastic materials

The energy release or absorption due to simultaneous expansion of many interacting nano-holes in elastic materials under plane strain deformation is studied as influenced by the surface effect along rims of nano-holes. The M-integral classically used in macro mechanics with defects is extended to treat the problem with many interacting nano-holes. After some manipulations, the energy change due to the simultaneous expansion of many nano-holes represented by the M-integral is evaluated. Four different arrays of many nano-holes under a monotonically increasing tensile loading are considered. Attention is focused on the influence induced from the surface tension, the surface Lamé constants, and the interaction among many nano-holes on the M-integral. It is concluded that the surface tension yields significant influence on the M-integral, whereas the surface Lamé constants offer much smaller influence, which could be neglected with some relative errors less than 2%. It is found that, unlike those in macro mechanics with defects, the simultaneous expansion could either release energy (the positive value of the M-integral) or absorb energy (the negative value of the M-integral), depending on the loading levels. There is a neutral loading point, at which the M-integral transforms from a negative value to a positive value in all arrays of nano-voids under consideration. It is also found that the interaction among multiple nano-holes influences the value of the neutral loading point significantly because the mutual influence induced from both the interacting effect and the surface effect yields a quite different feature from those induced from the interacting effect only. That is, the surface effect always inhibits the influence of the interacting effect on the M-integral.     

A simple method for calculating interaction of numerous microcracks and its applications

The effects of microcrack interaction on the failure behavior of materials present one problem of considerable interest in micromechanics, which has been extensively argued but has not been resolved as yet. In the present paper, a simple and effective method is presented based on the concept of the effective field to analyze the interaction of microcracks of a large number or of a high density. To determine the stress intensity factors of a microcrack embedded in a solid containing numerous or even countless microcracks, the solid is divided into two regions. The interaction of microcracks in a circular or elliptical region around the considered microcrack is calculated directly by using Kachanov’s micromechanics method, while the influence of all other microcracks is reflected by modifying the stress applied in the far field. Both the cases of tensile and compressive loading are considered. This simplified scheme may yield an estimate for stress intensity factors of satisfactory accuracy, and therefore provide a potential tool for elucidating some phenomena of material failure associated with microcracking. As two of its various promising applications, the above scheme is employed to investigate the size effects of material strength due to stochastic distribution of interacting microcracks and to calculate the effective elastic moduli of elastic solids containing distributed microcracks. Some conventional micromechanics methods for estimating the effective moduli of microcracked materials are evaluated by comparing with the numerical results. Only two-dimensional problems have been considered here, though the three-dimensional extension of the present method is of greater interest.     

一种基于材料细观特征量的PBX拉伸强度理论模型

为给塑性黏结炸药(PBX)的力学强度设计提供支撑、探索材料细观特征量与材料强度之间的定量规律,应用微裂纹扩展区理论,将PBX炸药的单轴拉伸过程中力学响应特征的变化归结为扩展裂纹取向角度的增加,将扩展裂纹最大取向角与拉伸强度相关联,构建了基于材料细观特征量的拉伸强度理论模型,并采用不同温度的单轴拉伸实验验证了该理论模型的有效性。研究表明:该拉伸强度理论模型可以实现对PBX炸药拉伸强度与炸药微裂纹密度、颗粒/黏结剂界面性能以及颗粒/黏结剂体系的表观杨氏模量、泊松比等细观特征量之间关系的定量描述。     

Subinterface crack in an anisotropic piezoelectric bimaterial

In this paper, the problem of a subinterface crack in an anisotropic piezoelectric bimaterial is analyzed. A system of singular integral equations is formulated for general anisotropic piezoelectric bimaterial with kernel functions expressed in complex form. For commonly used transversely isotropic piezoelectric materials, the kernel functions are given in real forms. By considering special properties of one of the bimaterial, various real kernel functions for half-plane problems with mechanical traction-free or displacement-fixed boundary conditions combined with different electric boundary conditions are obtained. Investigations of half-plane piezoelectric solids show that, particularly for the mechanical traction-free problem, the evaluations of the mechanical stress intensity factors (electric displacement intensity factor) under mechanical loadings (electric displacement loading) for coupled mechanical and electric problems may be evaluated directly by considering the corresponding decoupled elastic (electric) problem irrespective of what electric boundary condition is applied on the boundary. However, for the piezoelectric bimaterial problem, purely elastic bimaterial analysis or purely electric bimaterial analysis is inadequate for the determination of the generalized stress intensity factors. Instead, both elastic and electric properties of the bimaterial’s constants should be simultaneously taken into account for better accuracy of the generalized stress intensity factors.     

Constitutive relationship of brittle rock subjected to dynamic uniaxial tensile loads with microcrack interaction effects

A micromechanical model is proposed to describe both stable and unstable damage evolution in microcrack-weakened brittle rock material subjected to dynamic uniaxial tensile loads. The basic idea of the present model is to classify the constitution relationship of rock material subjected to dynamic uniaxial tensile loads into four stages including some of the stages of linear elasticity, pre-peak nonlinear hardening, rapid stress drop, and strain softening, and to investigate their corresponding micromechanical damage mechanisms individually. Special attention is paid to the transition from structure rearrangements on microscale to the macroscopic inelastic strain, to the transition from distribution damage to localization of damage and the transition from homogeneous deformation to localization of deformation. The influence of all microcracks with different sizes and orientations are introduced into the constitutive relation by using the statistical average method. Effects of microcrack interaction on the complete stress-strain relation as well as the localization of damage for microcrack-weakened brittle rock material are analyzed by using effective medium method. Each microcrack is assumed to be embedded in an approximate effective medium that is weakened by uniformly distributed microcracks of the statistically-averaged length depending on the actual damage state. The elastic moduli of the approximate effective medium can be determined by using the dilute distribution method. Micromechanical kinetic equations for stable and unstable growth characterizing the ‘process domains’ of active microcracks are taken into account. These ‘process domains’ together with ‘open microcrack domains’ completely determine the integration domains of ensemble averaged constitutive equations relating macro-strain and macro-stress. Theoretical predictions have shown to consistent with the experimental results.     

Damage analysis of thermopiezoelectric properties: Part II. Effective crack model

Having completed the general formulation for temperature, heat flow, displacement, electric potential and displacements and mechanical stresses of a piezoelectric material as presented in part I of this work, part II is concerned with a generalized self-consistent approximate method for determining the thermoelectroelastic properties of piezoelectric materials weakened by microcracks. A representative area element is adopted; it contains a microcrack surrounded by an elliptic matrix in a solid with effective properties. Numerical results are given for a piezoelectric BatiO₃ ceramic. The effective conductivity and effective modulus are found to decrease with increasing crack density.     

Elastic-plastic crack growth on plane strain bimaterial interface

Finite element computation are carried out to simulate plane strain crack growth on a bimaterial interface under the assumption of small scale yielding. The modified Gurson constitutive equation and the element vanish technique introduced by Tvergaard et al. are used to model the final formation of an open crack. It is found from the calculation that the critical fracture toughness for crack growth is much lower in bimaterials than that in homogeneous material. The critical fracture toughness is strongly dependent on material properties of the bimaterial pair and the mixed mode of remote loads. The interface crack grows in the more compliant (lower hardening) material or in the weaker (lower yield strength) material. In Mode-I loading, the crack grows zigzag along the interface. Project supported by Fok Ying-Tung Education Foundation and National Natural Science Foundation of China.