Numerical simulation of crack deflection and penetration at an interface in a bi-material under dynamic loading by time-domain boundary element method

The hybrid time-domain boundary element method (BEM), together with the multi-region technique, is applied to simulate the dynamic process of crack deflection/ penetration at an interface in a bi-material. The whole bi-material is divided into two regions along the interface. The traditional displacement boundary integral equations (BIEs) are employed with respect to the exterior boundaries; meanwhile, the non-hypersingular traction BIEs are used with respect to the part of the crack in the matrix. Crack propagation along the interface is numerically modelled by releasing the nodes in the front of the moving crack tip and crack propagation in the matrix is modeled by adding new elements of constant length to the moving crack tip. The dynamic behaviours of the crack deflection/penetration at an interface, propagation in the matrix or along the interface and kinking out off the interface, are controlled by criteria developed from the quasi-static ones. The numerical results of the crack growth trajectory for different inclined interface and bonded strength are computed and compared with the corresponding experimental results. Agreement between numerical and experimental results implies that the present time-domain BEM can provide a simulation for the dynamic propagation and deflection of a crack in a bi-material.     

Dynamic analysis of interfacial crack problems in anisotropic bi-materials by a time-domain BEM

A time-domain boundary element method (BEM) together with the sub-domain technique is applied to study dynamic interfacial crack problems in two-dimensional (2D), piecewise homogeneous, anisotropic and linear elastic bi-materials. The bi-material system is divided into two homogeneous sub-domains along the interface and the traditional displacement boundary integral equations (BIEs) are applied on the boundary of each sub-domain. The present time-domain BEM uses a quadrature formula for the temporal discretization to approximate the convolution integrals and a collocation method for the spatial discretization. Quadratic quarter-point elements are implemented at the tips of the interface cracks. A displacement extrapolation technique is used to determine the complex dynamic stress intensity factors (SIFs). Numerical examples for computing the complex dynamic SIFs are presented and discussed to demonstrate the accuracy and the efficiency of the present time-domain BEM.     

Dynamic behavior of a bi-material interface-cracked plate

The hyper-singular traction boundary integral equation method (BIEM) has been developed to analyze the dynamic behavior of two-dimensional finite bi-material plates with one or more interface cracks under uniform time-harmonic tension. The multi-region BEM technique is employed. Fracture parameters and scattered wave field far from the crack-tips are computed. The numerical results show the dependence of these dynamic characteristics on the frequency of the applied load, on the Dundurs bi-material constants, on the crack length and on the existence of other neighboring cracks.     

Analysis of dynamic interaction between an inclusion and a nearby moving crack by BEM

In this paper, the dynamic interaction between an inclusion and a nearby moving crack embedded in an elastic medium is studied by the boundary element method (BEM). To deal with this problem, the multi-region technique and two kinds of time-domain boundary integral equations (BIEs) are introduced. The system is divided into two parts along the interface between the inclusion and the matrix medium. Each part is linear, elastic, homogeneous and isotropic. The non-hypersingular traction boundary integral equation is applied on the crack surfaces; while the traditional displacement boundary integral equation is used on the interface and external boundaries. In the numerical solution procedure, square root shape functions are adopted as to describe the proper asymptotic behavior in the vicinity of the crack-tips. The crack growth is modeled by adding new elements of constant length to the moving crack tip, which is controlled by the fracture criterion based on the maximum circumferential stress. In each time step, the direction and the speed of the crack advance are evaluated. The numerical results of the crack growth path, speed, dynamic stress intensity factors (DSIFs) and dynamic interface tractions for various material combinations and geometries are presented. The effect of the inclusion on the moving crack is discussed.     

基于小波内聚力模型的界面裂纹扩展

利用区间B样条小波良好的局部化性能,将内聚力模型（CZM）引入小波有限元法（WFEM）数值分析中,以区间B样条小波尺度函数作为插值函数,构造小波内聚力界面单元,推导了小波内聚力界面单元刚度矩阵,基于虚拟裂纹闭合技术（VCCT）计算界面裂纹应变能释放率（SERR）,采用β-Κ断裂准则,实现界面裂纹扩展准静态分析。将WFEM和传统有限元法（CFEM） 的SERR数值分析结果与理论解进行比较,结果表明:采用WFEM和CFEM计算的SERR分别为96.60 J/m2 和 101.43 J/m2,2种方法的SERR数值解与理论解相对误差分别为1.85%和3.06%,这明确表明WFEM在计算界面裂纹扩展方面能用较少单元和节点数获得较高的计算精度和效率。在此基础上,探讨了界面裂纹初始长度和双材料弹性模量比对界面裂纹扩展的影响,分析结果表明:界面裂纹尖端等效应力随界面裂纹初始长度的增加而增加;双材料弹性模量比相差越大,界面裂纹越易于扩展,且裂纹扩展长度也越大,因此可通过调节双材料弹性模量比来延缓界面裂纹扩展。      

Crack transient torsional wave interaction in an elastic bi-material

An infinitely long cylindrical elastic solid of finite radius (fibre or inner medium) is imbedded in another infinite elastic cylindrical solid (matrix or outer medium) to form an elastic bi-material composite perfectly bonded at their bi-material interface. The early-time response of the composite to a sudden twist applied axisymmetrically over a stationary penny-shaped crack which appears over a cross-section of the fibre and terminates orthogonally at the bi-material interface, is investigated. The magnitude of the applied twist is assumed to be less than that for which debonding or crack propagation can take place. For a bi-material composite the square of whose matrix shear-wave speed is less than twice the square of whose fibre shear-wave speed, uniform asymptotic analytic results, whose accuracy can be improved by taking higher order terms in the solution, are obtained for the time-dependent surface displacement gradient normal to the plane of the crack and outside the crack region, the dynamic stress intensity factor, and also the stress singularity strength eigenvalue-all as functions of the fibre-matrix material disparities.     

A finite element analysis of plane strain dynamic crack growth at a ductile-brittle interface

In this work, steady, dynamic crack growth under plane strain, small-scale yielding conditions along a ductile-brittle interface is analysed using a finite element procedure. The ductile solid is taken to obey the J ₂ flow theory of plasticity with linear isotropic strain hardening, while the substrate is assumed to exhibit linear elastic behaviour. The objectives of this work are to establish the validity of an asymptotic solution for this problem which has been derived recently [12], and to examine the effect of changing the remote (elastic) mode-mixity on the near-tip fields. Also, the influence of crack speed on the stress fields and crack opening profiles near the propagating interface crack tip is assessed for various bi-material combinations. Finally, theoretical predictions are made for the variation of the dynamic fracture toughness with crack speed for crack growth under a predominantly tensile mode along ductile-brittle interfaces. Attention is focused on the effect of mismatch in stiffness and density of the constituent phases on the above aspects.     

Stress intensity factors of square crack inclined to interface of transversely isotropic bi-material

This paper analyzes a square crack in a transversely isotropic bi-material solid by using dual boundary element method. The square crack is inclined to the interface of the bi-material. The fundamental solution for the bi-material solid occupying an infinite region is incorporated into the dual boundary integral equations. The square crack can have an arbitrary angle with respect to the plane of isotropy of the bi-material occupying either finite or infinite regions. The stress intensity factor (SIF) values of the modes I, II, and III associated with the square crack are calculated from the crack opening displacements. Numerical results show that the properties of the anisotropic bi-material have evident influences on the values of the three SIFs. The values of the three SIFs are further examined by taking into account the effect of the external boundary of the internally cracked bi-material.     

A BEM analysis of a penny-shaped interface crack using the open and the frictionless contact models: Range of validity of various asymptotic solutions

The global elastic solution for the problem of a pressurized penny-shaped crack at the interface of two dissimilar half spaces has been numerically obtained employing the boundary element method (BEM). Using the Williams’ open model (for the whole range of feasible bi-material combinations), the comparison of the global BEM solution with an existing analytical asymptotic solution has shown: (i) that the extent of the zone in which the first term is dominant is always larger than the extent of the zone in which the interpenetrations take place and (ii) that, in the former zone, a recently proposed relation between the components of the complex stress intensity factor (SIF) and the components of the energy release rate (ERR) always yields accurate results. Consequently, the appearance of negative values of the normal contribution to the ERR in certain cases has been confirmed by the BEM solution, thus questioning the significance of the asymptotic results obtained from the open model in those cases. If the Comninou's frictionless contact model is employed, the ability of the BEM formulation employed to obtain accurate elastic solutions is shown by comparisons with an existing semi-analytical solution (for a particular bi-material combination).     

垂直界面裂纹应力强度因子的加料有限元分析

为求解裂尖位于界面上的垂直双材料界面裂纹应力强度因子,发展了一种加料有限元方法。该方法应用Williams本征函数展开和线性变换方法求解裂尖渐进位移场,将该位移场加入常规单元位移模式中,得到加料垂直界面裂纹单元和过渡单元的位移模式,给出加料有限元方程。建立了典型垂直界面裂纹平面问题的加料有限元模型,求解加料有限元方程直接得到应力强度因子,与文献结果对比表明该方法具有较高的精度,可方便地推广应用于垂直界面裂纹的计算分析。     

Finite element study of a penny-shaped crack along the interface in a bi-material cylinder

A contemporary approach to the analysis of interface cracks in bi-material cylinders using finite elements is presented. From results obtained with a commercial finite element code using regular and singular isoparametric elements, three fracture mechanics techniques are considered to study the interface crack problem and are presented in a fundamental manner. These are the stress intensity factor evaluation by the crack opening displacement method, the strain energy release rate evaluation using the modified crack closure integral method, and the J-integral evaluation using the virtual crack extension technique. Only the finite element results in the vicinity of the crack are then needed. The accuracy of the proposed approach is assessed by solving standard test problems with known solutions. In particular, the mode I problem of a penny-shaped crack in a homogeneous isotropic cylinder under remote tension loading is used as a standard test case. Finally, the mixed-mode (I and II) problem of a penny-shaped crack along the interface in a bi-material cylinder under three loading conditions is studied in detail. Numerical results are presented to quantify the combined effects of geometry and material discontinuities on the strain energy release rate.     

The effect of material combinations and relative crack size to the stress intensity factors at the crack tip of a bi-material bonded strip

Several types of singular stress fields may appear at the corner where an interface between two bonded materials intersects a traction-free edge depending on the material combinations. Since the failure of the multi-layer systems often originates at the free-edge corner, the analysis of the edge interface crack is the most fundamental to simulate crack extension. In this study, the stress intensity factors for an edge interfacial crack in a bi-material bonded strip subjected to longitudinal tensile stress are evaluated for various combinations of materials using the finite element method. Then, the stress intensity factors are calculated systematically with varying the relative crack sizes from shallow to very deep cracks. Finally, the variations of stress intensity factors of a bi-material bonded strip are discussed with varying the relative crack size and material combinations. This investigation may contribute to a better understanding of the initiation and propagation of the interfacial cracks.     

Transient response of an insulating crack near to the interface between two piezoelectric half-planes under electromechanical impacts by BEM

A time-domain boundary element method (BEM) together with the sub-domain technique is applied to study transient response of an insulating crack near to the interface between two anisotropic piezoelectric half-planes under electromechanical impacts. The present time-domain BEM uses a quadrature formula for the temporal discretization to approximate the convolution integrals and a collocation method for the spatial discretization. Quadratic quarter-point elements are implemented at the crack tip. A displacement extrapolation technique is used to determine the dynamic stress intensity factors (DSIFs) and the dynamic electrical displacement intensity factor. Numerical examples are presented to show the effects of load combination, geometric configuration and material combination on dynamic intensity factors and dynamic energy release rate.     

Fracture parameters of a penny-shaped crack at the interface of a piezoelectric bi-material system

A penny-shaped crack at the interface of a piezoelectric bi-material system is considered. Analytical general solutions based on Hankel integral transforms are used to formulate the mixed-boundary value problem corresponding to an interfacial crack and the problem is reduced to a system of singular integral equations. The integral equations are further reduced to two systems of algebraic equations with the aid of Jacobi polynomials and Chebyshev polynomials. Thereafter, the exact expressions for the stress intensity factors and the electric displacement intensity factor at the tip of a crack are obtained. Selected numerical results are presented for various bi-material systems to portray the significant features of crack tip fracture parameters and their dependence on material properties, poling orientation and electric loading.     

Increase of stress intensity near interface edge of elastic-creep Bi-material under a sustained load

The transition from small-scale creep to large-scale creep ahead of a crack tip or an interface edge with strong elastic stress singularity at the loading instant causes stress relaxation and the decrease of stress intensity in general. However, this study shows that the stress near the interface edge of bi-material with no or weak elastic stress singularity increases after the loading instant and brings about the stress concentration during the transition. In addition, the creep strain distribution of this bi-material after the loading instant is different from that occurred in the transition of an interface edge with strong elastic stress singularity or a crack tip (notch root). The criterion for the increase or decrease of stress intensity near the interface edge proved by the finite element method is proposed in this study. The stress intensity near the interface edge increases when the elastic stress singularity is lower than the creep stress singularity (λ^el < λ^cr) and vice versa.     

An edge dislocation interacting with a slightly wavy interface

The undulant interface of a bi-material, which is due to manufacturing or instability under high stress, may provide certain preferred places where microdefects accumulate. In the present paper, an edge dislocation near a slightly wavy interface is studied via a regular perturbation scheme. Stress investigation for the interaction between the edge dislocation and the wavy bi-material interface has been carried out. The Peach-Koehler force on the dislocation is calculated from the first perturbed solution of the stress fields. The influence of several key parameters, such as the Dundurs bi-material interface constants α, β, the distance of the dislocation to the interface, the wavy extent of the interface, etc., on the behavior of the dislocation has been analyzed. The numerical examples indicate that there are certain “equilibrium” points around which the dislocations are expected to be accumulated. In turn, microdefects, such as microcracks and microvoids, may be initiated near these sites. Results obtained can be applied to study crack initiation or delamination along bonded bi-material interfaces.     

Asymptotic and finite element analyses of mode III dynamic crack growth at a ductile-brittle interface

In this work, dynamic crack growth along a ductile-brittle interface under anti-plane strain conditions is studied. The ductile solid is taken to obey the J ₂ flow theory of plasticity with linear isotropic strain hardening, while the substrate is assumed to exhibit linear elastic behavior. Firstly, the asymptotic near-tip stress and velocity fields are derived. These fields are assumed to be variable-separable with a power singularity in the radial coordinate centered at the crack tip. The effects of crack speed, strain hardening of the ductile phase and mismatch in elastic moduli of the two phases on the singularity exponent and the angular functions are studied. Secondly, full-field finite element analyses of the problem under small-scale yielding conditions are performed. The validity of the asymptotic fields and their range of dominance are determined by comparing them with the results of the full-field finite element analyses. Finally, theoretical predictions are made of the variations of the dynamic fracture toughness with crack velocity. The influence of the bi-material parameters on the above variation is investigated.     

基于分子动力学模拟结果的界面破坏准则

结合材料的破坏通常都是从界面或其附近发生的,但界面破坏的机理及其评价准则尚未十分清楚。采用分子动力学模拟方法,可以对结合材料的界面起裂过程进行模拟,从而获得结合材料界面应力和界面破坏之间的关系。虽然在分子动力学模拟中采用了高度简化的界面模型,但对界面起裂过程的模拟,仍可以帮助人们获得结合材料界面破坏过程的规律性认识。通过在界面附近引入初始裂纹,导致界面上应力集中,从而引起界面起裂。从分子动力学模拟结果出发,提出了一个结合材料界面起裂,即界面破坏的准则。     

Stress intensity factors for penny-shaped cracks perpendicular to graded interfacial zone of bonded bi-materials

This paper examines the stress intensity factors that are associated with a penny-shaped crack perpendicular to the interface of a bi-material bonded with a graded interfacial zone. Elastic modulus of the graded interfacial zone is assumed to be an exponential function of the depth. The stress intensity factors are calculated numerically using a so-called generalized Kelvin solution based boundary element method. Three cases of normal or shear tractions acting on the crack surfaces are examined. Values of the stress intensity factors are examined by taking into account the effects of the following four parameters: (a) the crack front position; (b) the non-homogeneity parameter of the graded interfacial zone; (c) the crack distance to the graded interfacial zone; and (d) the graded interfacial zone thickness. The numerical results are compared well with existing solutions under some degenerated conditions. These results are useful to furthering our knowledge on fracture behavior of bi-material systems with or without a graded interfacial zone.     

Dynamic crack propagation in matrix involving inclusions by a time-domain BEM

The dynamic crack propagation in composites with inclusions is analyzed by a time-domain boundary element method (BEM) in conjunction with the sub-region technique. The crack-growth direction and the crack-tip instantaneous velocity are determined by the maximum circumferential stress criterion. The instantaneous velocity is well smoothed by a bisection technique. New crack-tip elements of inconstant length are added to the active crack-tip to simulate the fast crack growth. Square root shape functions are adopted as to describe the proper asymptotic behavior in the vicinity of the crack-tips. The computation time for the dynamic fracture problems in composites with multiple inclusions is reduced by a numerical method. The influences of the inclusions on the shielding ratio, the crack growing path and the crack-tip instantaneous speed are well investigated.