Symmetric-Galerkin boundary element analysis of the dynamic T-stress for the interaction of a crack with an auxetic inclusion

In this work, the symmetric-Galerkin boundary element method (SGBEM) for 2-D elastodynamics in the Laplace-space frequency domain (Laplace domain) is employed to study the dynamic stress intensity factors (DSIFs) and the dynamic T-stress (DTS) during the interaction between a crack and an auxetic inclusion under impact loading conditions. It is found that, while the auxeticity has virtually no effect on the DSIFs, its influence on the DTS is noticeable. This finding is particularly important as it implies the imperative need of fracture criteria based on both the DSIFs and DTS for predicting crack growth in composite materials with auxetic phases.     

Dynamic investigation of a functionally graded layered structure with a crack crossing the interface

The dynamic response of a functionally graded layered structure with a crack crossing the interface is analyzed. The in-plane impact loading condition is considered. By using the Laplace and Fourier integral transforms, singular integral equation method and residue theory, the present problem is reduced to a singular integral equation in the Laplace transform domain. The influences of Young’s modulus ratio, thickness ratio, and crack length and location on the dynamic stress intensity factors (DSIFs) are investigated. Particularly, the DSIFs corresponding to different crack locations are shown in the case when the crack center moves from one layer to another layer through the interface. The peak and static values and overshoot characteristics of the DSIFs are analyzed. It is found that these values typically exhibit kinking behavior when the crack tips arrive at the interface. This study is different from previous other investigations in the following respects: (1) the dynamic response of a crack crossing the interface of a functionally graded structure is studied analytically, which has hardly been done in the past and (2) the present model can be reduced to some important problems, such as a functionally graded coating-substrate structure with a crack in the graded coating or homogeneous substrate or one intersecting the interface.     

Precracking and interfacial delamination in a bi-material structure: Static and dynamic loadings

The behavior of a precracked bi-material structure interface under given static and dynamic axial loading is an interest object in the present paper.Firstly,it is shown that the shear-lag model is a proper tool to analyze a delamination process in a precracked bi-material structure undergoing static loading.Secondly,the"shear-lag model"is applied to the structure under dynamic loading.To solve the problem for an interface delamination of the structure and to determine the debond length along the interface,our own 2D boundary element method(BEM)code is proposed in the case of static loading,and the shear-lag model together with the Laplace transforms and half-analytical calculations are used in the case of dynamic loading.The interface layer is assumed as a very thin plate compared with the other two.The parametric(geometric and elastic)analysis of the debond length and interface shear stress is done. The results from the 2D BEM code proved the validity of analytical solutions to the shear-lag model.In the dynamic case,the influence of loading characteristics,i.e.,frequencies and amplitude fluctuations on the shear stress and the value of debond length for an interval of time,is discussed. The analysis of the obtained results is illustrated by an example of the modern ceramic-metal composite,namely cermet, and depicted in figures.     

动载下裂纹应力强度因子计算的改进型扩展有限元法

相较于常规扩展有限元法(extended finite element method, XFEM), 改进型扩展有限元法(improved XFEM) 解决了现有方法线性相关与总体刚度矩阵高度病态问题, 在数量级上提升了总体方程的求解效率, 克服了现有方法在动力学问题中的能量正确传递、动态应力强度因子数值震荡、精度低下问题. 本文基于改进型XFEM, 采用Newmark 隐式时间积分算法, 重点研究了动载荷作用下扩展裂纹尖端应力强度因子的求解方法, 与静力学方法相比, 增加了裂纹扩展速度项与惯性项的贡献. 通过数值算例研究了网格单元尺寸、质量矩阵、时间步长、裂尖加强区域、惯性项、扩展速度项及相互作用积分区域J-domain的网格与单元尺寸对动态应力强度因子求解精度的影响, 验证了改进型XFEM计算动态裂纹应力强度因子方法的有效性. 针对文献中具有挑战性的 "I 型半无限长裂纹先稳定后扩展"问题, 改进型XFEM给出目前为止精度最好的动态应力强度因子数值解.      

The transient fracture behavior for a functionally graded layered structure subjected to an in-plane impact load

The transient fracture behavior of a functionally graded layered structure subjected to an in-plane impact load is investigated. The studied structure is composed of two homogeneous layers and a functionally graded interlayer with a crack perpendicular to the boundaries. The impact load is applied on the face of the crack. Fourier transform and Laplace transform methods are used to formulate the present problem in terms of a singular integral equation in Laplace transform domain. Considering variations of parameters such as the nonhomogeneity constant, the thickness ratio and the crack length, the dynamic stress intensity factors (DSIFs) in time domain are studied and some meaningful conclusions are obtained.The project supported by the National Science Foundation for Excellent Young Investigators (10325208), the National Natural Science Foundation of China (10432030) and the China Postdoctoral Science Foundation (2004036018)The English text was polished by Ron Marshall.     

Transient response of an interface crack between dissimilar piezoelectric layers under mechanical impacts

Using the integral transform and the Cauchy singular integral equation methods, the problem of an interface crack between two dissimilar piezoelectric layers under mechanical impacts is investigated under the permeable electrical boundary condition on the crack surface. The dynamic stress intensity factors (DSIFs) of both mode-I and II are determined. The effects of the crack configuration and the combinations of the constitutive parameters of the piezoelectric materials on the dynamic response are examined. The numerical calculation of the mode-I plane problem indicates that the DSIFs may be retarded or accelerated by specifying different combinations of material parameters. In addition, the parameters of the crack configuration, including the ratio of the crack length to the layer width and the ratio between the widths of two layers, exert a considerable influence on the DSIFs. The results seem useful for design of the piezoelectric structures and devices of high performance.     

Thermal weight functions for bi-material interface crack system using energy principles

In this study, Betti’s reciprocal theorem and the principle of superposition are used to obtain weight functions in a two-dimensional bi-material interface crack system for any loading, in general and thermal loading, in particular. It is shown that the general expression of weight functions for bi-materials interface crack problems is of the same type as that found in a homogeneous mixed mode loading case. Furthermore, a computational approach has been developed for calculation of thermal stress intensity factors for bi-material interface cracks subjected to thermal loading under quasi-static uncoupled thermo-elasticity assumption. The thermal weight function (TWF) expression and computational scheme have been validated using three examples given in the available literature.     

Interaction of a conductive crack and of an electrode at a piezoelectric bimaterial interface

The interaction of a conductive crack and an electrode at a piezoelectric bi-material interface is studied. The bimaterial is subjected to an in-plane electrical field parallel to the interface and an anti-plane mechanical loading. The problem is formulated and reduced, via the application of sectionally analytic vector functions, to a combined Dirichlet–Riemann boundary value problem. Simple analytical expressions for the stress, the electric field, and their intensity factors as well as for the crack faces' displacement jump are derived. Our numerical results illustrate the proposed approach and permit to draw some conclusions on the crack–electrode interaction.     

Dynamic behaviors of mode III interfacial crack under a constant loading rate

In an earlier study on intersonic crack propagation, Gao et al. (J. Mech. Phys. Solids 49: 2113–2132, 2001) described molecular dynamics simulations and continuum analysis of the dynamic behaviors of a mode II dominated crack moving along a weak plane under a constant loading rate. The crack was observed to initiate its motion at a critical time after the onset of loading, at which it is rapidly accelerated to the Rayleigh wave speed and propagates at this speed for a finite time interval until an intersonic daughter crack is nucleated at a peak stress at a finite distance ahead of the original crack tip. The present article aims to analyze this behavior for a mode III crack moving along a bi-material interface subject to a constant loading rate. We begin with a crack in an initially stress-free bi-material subject to a steadily increasing stress. The crack initiates its motion at a critical time governed by the Griffith criterion. After crack initiation, two scenarios of crack propagation are investigated: the first one is that the crack moves at a constant subsonic velocity; the second one is that the crack moves at the lower shear wave speed of the two materials. In the first scenario, the shear stress ahead of the crack tip is singular with exponent ?1/2, as expected; in the second scenario, the stress singularity vanishes but a peak stress is found to emerge at a distance ahead of the moving crack tip. In the latter case, a daughter crack supersonic with respect to the softer medium can be expected to emerge ahead of the initial crack once the peak stress reaches the cohesive strength of the interface.     

An accurate and fast analysis for strongly interacting multiple crack configurations including kinked (V) and branched (Y) cracks

In this study, multiple interacting cracks in an infinite plate are analyzed to determine the overall stress field as well as stress intensity factors for crack tips and singular wedges at crack kinks. The problem is formulated using integral equations expressed in terms of unknown edge dislocation distributions along crack lines. These distributions derive from an accurate representation of the crack opening displacements using power series basis terms obtained through wedge eigenvalue analysis, which leads to both polynomial and non-polynomial power series. The process is to choose terms of the series and their exponents such that the tractions on the crack faces are virtually zero compared to the far field loading. Applying the method leads to a set of linear algebraic equations to solve for the unknown weighting coefficients for the power series basis terms. Since no numerical integration is required unlike in other methods, in most cases, solution takes just a few seconds on a PC. The accuracy and efficiency of the method are first demonstrated with a simple example of three aligned cracks with small ligaments between their tips under tensile loading. The results are compared to exact results as well as to those of other numerical methods, including recent FIE, FEM and BEM approaches said to have fast computation times. Thereafter, some new and challenging crack interaction problems including branched Y-cracks, two kinked V-cracks are solved. From a parametric study of the various crack configurations, stress intensity factors are graphed and tabulated to demonstrate subtleties in the magnitudes of the crack interactions.     

The simulation of dynamic crack propagation using the cohesive segments method

The cohesive segments method is a finite element framework that allows for the simulation of the nucleation, growth and coalescence of multiple cracks in solids. In this framework, cracks are introduced as jumps in the displacement field by employing the partition of unity property of finite element shape functions. The magnitude of these jumps are governed by cohesive constitutive relations. In this paper, the cohesive segments method is extended for the simulation of fast crack propagation in brittle solids. The performance of the method is demonstrated in several examples involving crack growth in linear elastic solids under plane stress conditions: tensile loading of a block; shear loading of a block and crack growth along and near a bi-material interface.     

Dynamic stress intensity factors for homogeneous and smoothly heterogeneous materials using the interaction integral method

Dynamic stress intensity factors (DSIFs) are important fracture parameters in understanding and predicting dynamic fracture behavior of a cracked body. To evaluate DSIFs for both homogeneous and non-homogeneous materials, the interaction integral (conservation integral) originally proposed to evaluate SIFs for a static homogeneous medium is extended to incorporate dynamic effects and material non-homogeneity, and is implemented in conjunction with the finite element method (FEM). The technique is implemented and verified using benchmark problems. Then, various homogeneous and non-homogeneous cracked bodies under dynamic loading are employed to investigate dynamic fracture behavior such as the variation of DSIFs for different material property profiles, the relation between initiation time and the domain size (for integral evaluation), and the contribution of each distinct term in the interaction integral.     

Transient response of an insulating crack between dissimilar piezoelectric layers under mechanical and electrical impacts

Summary  The dynamic response of an interface crack between two dissimilar piezoelectric layers subjected to mechanical and electrical impacts is investigated under the boundary condition of electrical insulation on the crack surface by using the integral transform and the Cauchy singular integral equation methods. The dynamic stress intensity factors, the dynamic electrical displacement intensity factor, and the dynamic energy release rate (DERR) are determined. The numerical calculation of the mode-I plane problem indicates that the DERR is more liable to be the token of the crack growth when an electrical load is applied. The dynamic response shows a significant dependence on the loading mode, the material combination parameters as well as the crack configuration. Under a given loading mode and a specified crack configuration, the DERR of an interface crack between piezoelectric media may be decreased or increased by adjusting the material combination parameters. It is also found that the intrinsic mechanical-electrical coupling plays a more significant role in the dynamic fracture response of in-plane problems than that in anti-plane problems. Received 4 September 2001; accepted for publication 23 July 2002 The work was supported by the National Natural Science Foundation under Grant Number 19891180, the Fundamental Research Foundation of Tsinghua University, and the Education Ministry of China.     

Effects of electric/magnetic impact on the transient fracture of interface crack in piezoelectric-piezomagnetic sandwich structure: anti-plane case

Due to the incompatibility of the interlaminar deformations,the interface debonding or cracking usually happens in a layered magnetoelectric(ME)structure under an applied load.In this paper,the transient responses of the anti-plane interface cracks in piezoelectric(PE)-piezomagnetic(PM)sandwich structures are studied by the standard methods of the integral transform and singular integral equation.Discussion on the numerical examples indicates that the PE-PM-PE structure under electric impact is more likely to fracture than the PM-PE-PM structure under a magnetic impact.The dynamic stress intensity factors(DSIFs)are more sensitive to the variation of the active layer thickness.The effects of the material constants on the DSIFs are dependent on the roles played by PE and PM media during the deformation process.     

Micromechanics analyses of interface crack

A new mechanics model based on Peierls concept is presented in this paper, which can clearly characterize the intrinsic features near a tip of an interfacial crack. The stress and displacement fields are calculated under general combined tensile and shear loadings. The near tip stress fields show some oscillatory behaviors but without any singularity and the crack faces open completely without any overlapping when remote tensile loading is comparable with remote shear loading. A fracture criterion for predicting interface toughness has been also proposed, which takes into account for the shielding effects of emitted dislocations. The theoretical toughness curve gives excellent prediction, as compared with the existing experiment data. The project supported by the National Natural Science Foundation of China     

Analysis of bi-material interface cracks with complex weighting functions and non-standard quadrature

A boundary element formulation is developed to determine the complex stress intensity factors associated with cracks on the interface between dissimilar materials. This represents an extension of the methodology developed previously by the authors for determination of free-edge generalized stress intensity factors on bi-material interfaces, which employs displacements and weighted tractions as primary variables. However, in the present work, the characteristic oscillating stress singularity is addressed through the introduction of complex weighting functions for both displacements and tractions, along with corresponding non-standard numerical quadrature formulas. As a result, this boundary-only approach provides extremely accurate mesh-insensitive solutions for a range of two-dimensional interface crack problems. A number of computational examples are considered to assess the performance of the method in comparison with analytical solutions and previous work on the subject. As a final application, the method is applied to study the scaling behavior of epoxy–metal butt joints.     

SH波入射时半空间界面裂纹与圆形衬砌的相互作用

利用复变函数和Green函数法研究了双相介质半空间界面裂纹及界面附近圆形衬砌对SH 波的 散射与动应力集中问题。首先,采用映像思想构造满足自由边界条件的散射波表达式,进而求解所需的 Green函数;其次,采用裂纹切割技术构造裂纹,并根据连续性条件建立了求解该问题的无穷代数方程组; 最后,给出了不同入射波数时界面裂纹与衬砌的相互作用。结果表明,裂纹的存在显著放大了衬砌界面的动 应力集中。     

Out-of-plane interface cracks in dissimilar piezoelectric materials

Summary This paper investigates the problem of an anti-plane interfacial crack between two dissimilar piezoelectric material layers. A single crack is first considered. The effect of interaction of two collinear cracks in the medium on the field intensity factors is investigated. The solutions of several particular cases, including an infinite piezoelectric bi-material and a piezoelectric material bonded to an elastic medium, are given. The bi-material constants governing the behavior of the crack tip fields are identified. By considering the crack as a notch of finite thickness, it is shown that the thickness of the notch has a pronounced influence on the crack tip field. The results for the assumption of a permeable crack represent the limit case where the notch thickness is reduced to zero.BLW would like to thank the National Science Foundation of China (#10102004) and the City University of Hong Kong (DAG #7100219) for the support of this work. YGS also thanks the Multidiscipline Scientific Research Foundation Project (HIT. MD 2001. 39) of the Harbin Institute of Technology and the SRF for ROCS, SEM.accepted for publication 3 April 2003     

Influence of plasticity on interface toughness in a layered solid with residual stresses

For crack growth along an interface between two adjacent elastic–plastic materials in a layered solid, the resistance curve behaviour is analysed by approximating the behaviour in terms of a bi-material interface under small scale yielding conditions. Thus, it is assumed that the layers are thick enough so that the extent of the plastic regions around the crack tip are much smaller than the thickness of the nearest layers. The focus is on the effect of initial residual stresses in the layered material, or on T-stress components induced during loading. The fracture process is represented in terms of a cohesive zone model. It is found that the value of the T-stress component in the softer material adjacent to the interface crack plays a dominant role, such that a negative value of this T-stress gives a significant increase of the interface fracture toughness, while a positive value gives a reduction of the fracture toughness.     

Path Independent Integral for an Elliptical Hole in a Plate under Tension for Plane Stress Deformation Theory

An exact expression is derived for a path independent integral surrounding an elliptical hole in an infinite plate with tensile tractions at infinity for plane stress loading conditions. The plate is composed of a non-work-hardening material satisfying the Tresca yield condition under proportional loading and small strain assumptions. This problem may serve as a simple classroom example for the derivation of a relationship between crack opening displacement and path independent integral for a nonlinear elastic material satisfying the Tresca yield condition.