Reinitiation of a crack reaching an interface

We consider here a bi-material made of two layers bonded together by an interface. The specimen is loaded in tension parallel to the interface and the existence of a mode I crack is assumed. The crack initiated in just one layer reaches the interface normally. We then study the second of the two possible cases: the crack crosses the interface and goes straight into the second layer, in mode I also; or the crack debonds the interface before reinitiating in the second layer at the debond tip.In the present study the conditions of the reinitiation of the crack in the second layer after debonding of the interface are presented. The maximum debond distance is calculated by means of a Shear Lag analysis associated with a damage constitutive equation.Qualitative rules for design are pointed out to make the interface a location of crack arrest or at least of crack growth delay. These rules are mainly: small thickness of the possibly cracked layer, strong interface and tough substrate.     

Crack deflection at an interface of alumina/metal joint: A numerical analysis

Deflection of a crack at the bimaterial interface is the initial mechanism required for obtaining enhanced toughness in bimaterial system. In this paper, a criterion is presented to predict the competition between crack deflection and penetration at the interface, using an energy release rate criterion. The finite element methods are used to calculate the strain energy release rates at the crack tip of alumina–metal bimaterial that either deflect or penetrate at the interface as a function of elastic mismatch and length of the deflected or penetrated crack. The effects of the elastic properties of two bonded materials were highlighted in order to evaluate the conditions for the crack deflection by the interface as well as the distance between the crack tip and the interface.     

On the stress intensity factors associated with cracks interacting with an interface between two elastic media

Closed form expressions are obtained for the stresses at a crack tip when a crack is approaching a welded boundary (or a free surface) and when it has just passed through the interface. The solutions which are obtained in terms of a small parameter, the distance from or through the interface, are given in explicit form for the mode 3 situation and for some mode 1 and 2 cases. The importance of the change of stress singularity when the crack meets the interface is demonstrated.     

Analysis of stress intensity factors for three-dimensional interface crack problems in electronic packages using the virtual crack closure technique

In this study the fracture mechanics parameters, including the strain energy release rate, the stress intensity factors and phase angles, along the curvilinear front of a three-dimensional bimaterial interface crack in electronic packages are considered by using finite element method with the virtual crack closure technique (VCCT). In the numerical procedure normalized complex stress intensity factors and the corresponding phase angles (Rice, J Appl Mech 55:98–103, 1988) are calculated from the crack closure integrals for an opening interface crack tip. Alternative procedures are also described for the cases of crack under inner pressure and crack faces under large-scale contact. Validation for the procedure is performed by comparing numerical results to analytical solutions for the problems of interface crack subjected to either remote tension or mixed loading. The numerical approach is then applied to study interface crack problems in electronic packages. Solutions for semi-circular surface crack and quarter-circular corner crack on the interface of epoxy molding compound and silicon die under uniform temperature excursion are presented. In addition, embedded corner delaminations on the interface of silicon die and underfill in flip-chip package under thermomechanical load are investigated. Based on the distribution of the fracture mechanics parameters along the interface crack front, qualitative predictions on the propensity of interface crack propagation under thermomechanical loads are given.     

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 crack growth across an interface

The growth of a crack first in an elastic solid, then across an interface and into an elastic-viscoplastic solid is analyzed numerically. The analyses are carried out within a framework where the continuum is characterized by two constitutive relations; one that relates stress and strain in the bulk material, the other relates the traction and separation across a specified set of cohesive surfaces. Crack initiation, crack growth and crack arrest emerge naturally as outcomes of the imposed loading, without any ad hoc assumptions concerning crack growth or crack path selection criteria. Full transient analyses are carried out using two characterizations of strain rate hardening for the viscoplastic solid; power law strain rate hardening and a combined power law-exponential relation that gives rise to enhanced strain rate hardening at high strain rates. Results are presented for two values of interface strength. For the higher strength interface the crack grows straight through the interface into the elastic–viscoplastic solid, while for the lower strength interface the crack deflects into the interface. This revised version was published online in July 2006 with corrections to the Cover Date.     

双相介质界面附近裂纹的断裂力学特征

复合材料界面附近的力学性态对于材料的性能和强韧化影响是非常重要的。首先研究和讨论了含裂纹的双相介质的J 积分守恒定律的适用性问题, 采用有限元法证明了当裂纹平行靠近界面时, 其J 积分数值与裂纹位置无关的假设。文中建立了一种双相介质界面附近存在斜裂纹的分析模型, 用有限元和数值拟合相结合的方法, 得到了在远场单轴拉应力作用下, 斜裂纹处在不同介质中, 近界面一端裂尖的é 型能量释放率近似计算公式, 和相应的应力强度因子的计算方法。     

A THREE-DIMENSIONAL FINITE ELEMENT ANALYSIS FOR A CRYSTALLOGRAPHIC CRACK NEAR THE INTERFACE OF AN INCOMPATIBLE BICRYSTAL

On the basis of modelling bicrystal deformation, using three-dimensional, anisotropic finite elements, the problem of a crystallographic crack approaching the interface in a [111] tilt bicrystals of 90? misfit angle under shear loads is solved. The influence of thickness direction material heterogeneity across the interface on the distribution of the stresses, strains and crack sliding displacements along the crack front near the interface has been revealed. As the crack approaches the interface, those mechanical parameters are considerably changed by the heterogeneity across the interface. Remarkable variations in the stresses and strains along the crack front have also been identified and are referred to the different constraint across the thickness. The maximum stress may shift from the crack tip to the interface ahead of it, where, as suggested by numerical results and previous experimental observation, a new fracture process core may be activated. The interface-induced crack shielding or antishielding under mode II and III loading is analyzed and discussed.     

Fracture mechanics of piezoelectric materials

This paper presents an analysis of crack problems in homogeneous piezoelectrics or on the interfaces between two dissimilar piezoelectric materials based on the continuity of normal electric displacement and electric potential across the crack faces. The explicit analytic solutions are obtained for a single crack in an infinite piezoelectric or on the interface of piezoelectric bimaterials. For homogeneous materials it is found that the normal electric displacement D₂, induced by the crack, is constant along the crack faces which depends only on the remote applied stress fields. Within the crack slit, the perturbed electric fields induced by the crack are also constant and not affected by the applied electric displacement fields. For bimaterials, generally speaking, an interface crack exhibits oscillatory behavior and the normal electric displacement D₂ is a complex function along the crack faces. However, for bimaterials, having certain symmetry, in which an interface crack displays no oscillatory behavior, it is observed that the normal electric displacement D₂ is also constant along the crack faces and the electric field E₂ has the singularity ahead of the crack tip and has a jump across the interface. Energy release rates are established for homogeneous materials and bimaterials having certain symmetry. Both the crack front parallel to the poling axis and perpendicular to the poling axis are discussed. It is revealed that the energy release rates are always positive for stable materials and the applied electric displacements have no contribution to the energy release rates. This revised version was published online in July 2006 with corrections to the Cover Date.     

Antiplane shear crack terminating at and going through a bimaterial interface

The antiplane shear problem of two bonded elastic half planes containing a crack perpendicular to the interface is considered. The cases of a semi-infinite crack terminating at the interface, a finite crack away from and terminating at the interface, two cracks one on each side of the interface, and a finite crack crossing the interface are separately investigated. The nature of the stress singularity for the crack terminating at and going through the interface is studied, and it is shown that at the irregular point on the interface, for the former the power of singularity is not -1/2 and for the latter the stresses are bounded. For a material pair of aluminum-epoxy some numerical results giving the stress intensity factors, the density functions, and the crack opening displacements are presented.     

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。     

Boundary element analysis of thermal stress intensity factors for interface griffith and cusp cracks

The thermal stress intensity factors for interface cracks of Griffith and symmetric lip cusp types under vertical uniform heat flow in a finite body are calculated by the boundary element method. The boundary conditions on the crack surfaces are insulated or fixed to constant temperature. The relationship between the stress intensity factors and the displacements on the nodal point of a crack-tip element is derived. The numerical values of the thermal stress intensity factors for an interface Griffith crack in an infinite body are compared with the previous solutions. The thermal stress intensity factors for a symmetric lip cusp interface crack in a finite body are calculated with respect to various effective crack lengths, configuration parameters, material property ratios and the thermal boundary conditions on the crack surfaces. Under the same outer boundary conditions, there are no appreciable differences in the distribution of thermal stress intensity factors with respect to each material property. However, the effect of crack surface thermal boundary conditions on the thermal stress intensity factors is considerable.     

Fracture mechanics parameters for cracks on a slightly undulating interface

Typical bimaterial interfaces are non-planar due to surface facets or roughness. Crack-tip stress fields of an interface crack must be influenced by non-planarity of the interface. Consequently, interface toughness is affected. In this paper, the crack-tip fields of a finite crack on an elastic/rigid interface with periodic undulation are studied. Particular emphasis is given to the fracture mechanics parameters, such as the stress intensity factors, crack-tip energy release rate, and crack-tip mode mixity. When the amplitude of interface undulation is very small relative to the crack length (which is the case for rough interfaces), asymptotic analysis is used to convert the non-planarity effects into distributed dislocations located on the planar interface. Then, the resulting stress fields near the crack tip are obtained by using the Fourier integral transform method. It is found that the stress fields at the crack tip are strongly influenced by non-planarity of the interface. Generally speaking, non-planarity of the interface tends to shield the crack tip by reducing the crack-tip stress concentration.     

坝基界面在非线性水压力驱动下的非线性断裂过程模拟

基于多边形比例边界有限元法和粘聚裂缝模型提出了混凝土坝坝基界面在随缝宽非线性变化的水压力驱动下的非线性断裂数值模型。混凝土和基岩采用多边形比例边界单元模拟,界面裂缝的断裂过程区采用粘性界面单元模拟。因为界面裂缝总是处于复合断裂模态,故同时引入了法向和切向的界面单元,且考虑了裂纹面作用有法向和切向任意荷载时的应力强度因子求解。以裂尖为原点,裂尖附近的位移场和应力场在径向上解析求解,在环向具有有限元精度。因此无需在裂尖附近加密网格或采用富集技术即可求得高精度的解。对于界面断裂,可模拟出与两种材料差异性相关的非1/2奇异性。断裂过程区的水压力随缝面宽度变化,采用指数函数的形式进行表征,通过参数调整可实现不同分布的水压力的模拟。水压力与粘聚力考虑为与裂缝宽度相关的组合函数,便于非线性迭代的实现。结合多边形网格生成和重剖分技术,可方便地模拟界面裂缝在水力驱动下的扩展过程。算例研究表明了该文模型的有效性,从中也可看出考虑缝内水压及其具体分布形式对研究坝的稳定性具有重要影响。     

Boundary integral equation formulation for Interface cracks in anisotropic materials

We present a boundary integral formulation for anisotropic interface crack problems based on an exact Green's function. The fundamental displacement and traction solutions needed for the boundary integral equations are obtained from the Green's function. The traction-free boundary conditions on the crack faces are satisfied exactly with the Green's function so no discretization of the crack surfaces is necessary. The analytic forms of the interface crack displacement and stress fields are contained in the exact Green's function thereby offering advantage over modeling strategies for the crack. The Green's function contains both the inverse square root and oscillatory singularities associated with the elastic, anisotropic interface crack problem. The integral equations for a boundary element analysis are presented and an example problem given for interface cracking in a copper-nickel bimaterial.     

Crack deflection at an interface between dissimilar piezoelectric materials

The problem of crack deflection in bimaterial systems is considered in this paper. The material combinations may be of piezoelectric-piezoelectric, or one is piezoelectric and the other is not. Based on the Stroh formulation for anisotropic material, Green's functions for various bimaterial combinations are presented within the framework of two-dimensional electroelasticity, allowing the crack problem to be expressed in terms of coupled singular integral equations. A crack impinging on an interface joining two dissimilar materials may arrest or may advance be either penetrating the interface or deflecting into the interface. The competition between deflection and penetration is investigated using the maximum energy release rate criterion. Numerical results are presented to study the role of remote electroelastic loads on the path selection of crack extension. Key words: Crack, piezoelectric material, interface, Green's function, singular integral equation.     

The use of interlayers in modeling interface crack propagation

Delaminations are a common mode of failure at interfaces between two material layers which have dissimilar elastic constants. There is a well-known oscillatory nature to the singularity in the stress fields at the crack tips in these bimaterial delaminations, which creates a lack of convergence in the modewise energy release rates. This makes constructing fracture criteria somewhat difficult. An approach used to overcome this is to artificially insert a thin, homogeneous, isotropic layer (the interlayer) at the interface. The crack is positioned in the middle of this homogeneous interlayer, thus modifying the original ‘bare’ interface crack problem into a companion ‘interlayer’ crack problem. Individual modes I and II energy release rates are convergent and calculable for the companion problem and can be used in the construction of a fracture criterion or locus. However, the choices of interlayer elastic and geometric properties are not obvious. Moreover, a sound, consistent, and comprehensive methodology does not exist for utilizing interlayers in the construction and application of mixed-mode fracture criteria in interface fracture mechanics. These issues are addressed here. The role of interlayer elastic modulus and thickness is examined in the context of a standard interface fracture test specimen. With the help of a previously published analytical relation that relates the bare interface crack stress intensity factor to the corresponding interlayer crack stress intensity factor, a suitable thickness and elastic modulus are identified for the interlayer in a bimaterial four-point bend test specimen geometry. Interlayer properties are chosen to make the interlayer fracture problem equivalent to the bare interface fracture problem. A suitable mixed-mode phase angle and a form for the fracture criterion for interlayer-based interface fracture are defined. A scheme is outlined for the use of interlayers for predicting interface fracture in bimaterial systems such as laminated composites. Finally, a simple procedure is presented for converting existing bare interface crack fracture loci/criteria into corresponding interlayer crack fracture loci.     

不同界面SiC/SiC复合材料的断裂行为研究

碳化硅纤维增强碳化硅复合材料(SiC/SiC)是极具前景的高温结构材料。通过先驱体浸渍裂解(PIP)工艺分别制备了PyC界面和CNTs界面SiC/SiC复合材料, 对两种SiC/SiC复合材料的整体力学性能以及界面剪切强度等进行了测试表征, 并对材料中裂纹的产生与扩展进行了原位观测。结果表明, 两种界面SiC/SiC复合材料弯曲强度相近, 但PyC界面SiC/SiC复合材料的断裂韧性约为CNTs界面SiC/SiC复合材料的两倍。在PyC界面SiC/SiC复合材料中, 裂纹沿纤维-基体界面扩展, PyC涂层能够偏转或阻止裂纹, 材料呈现伪塑性断裂特征; 而在CNTs界面SiC/SiC复合材料中, 裂纹在扩展路径上遇到界面并不偏转, 初始裂纹最终发展为主裂纹, 材料呈现脆性断裂模式。     

Variable power singular interface elements for a crack normal to the bimaterial interface

Zero thickness crack tip interface elements for a crack normal to the interface between two materials are presented. The elements are shown to have the desired r^λ−1 (0 < λ < 1) singularity in the stress field at the crack tip and are compatible with other singular elements. The stiffness matrices of the quadratic and cubic interface element are derived. Numerical examples are given to demonstrate the applicability of the proposed interface elements for a crack perpendicular to the bimaterial interface.     

Prediction of arbitrary crack growth from the interface between two dissimilar elastic materials

Based on the universal laws of stress distribution around a crack edge, the general analysis of crack start from the interface is given. The solution for the original crack is assumed to be known. In order to classify different possible cases of crack growth beginning, both a slip-region ahead of the opening zone and a core region near the crack edge are introduced. The former provides non-overlapping of the crack surfaces and removes an undesirable oscillating stress field singularity which is produced mathematically at the assumption of a completely opening crack. The latter means that we consider the damage of an elementary volume (geometrical measure of the microstructure) as a discrete fracture operaton. Two asymptotical cases are studied: the slip-region is much smaller or much larger than the cross-section of the core region. Then the stress-strain state on the core region periphery qualitatively is the same as for a completely opening or shear interface crack, respectively. A characteristic crack opening appears instead of a characteristic length of the slip-region for a blunted crack. The angles of crack departure are predicted according to different known criteria of fracture. In passing, parametric, analysis of stresses and energy density angle distributions are given. Plane and penny-shaped cracks are examined as the illustration.