Application of bimaterial interface corner failure mechanics to silicon/glass anodic bonds

Motivated by the existence of a universal singular stress field at bimaterial interface corners, a fair amount of work has been performed to support the use of the corresponding critical stress intensities to correlate fracture initiation. The approach is in the spirit of interface fracture mechanics but applicable to a different class of problems, specifically, when a crack does not previously exist (or cannot be detected, at least economically), and when subsequent crack propagation does not necessarily occur along the interface. Here we further progress toward the development, understanding, and application of the approach, both experimentally and theoretically, for a series of silicon/glass anodically bonded structures. To this end we designed and fabricated two series of silicon/glass anodically bonded bimaterial specimens with different interface corner geometries that commonly arise from different silicon etching technologies. Offset three-point flexure tests were performed that resulted in brittle fracture that initiated at the interface corner. From a rigorous stress analysis at the interface corner, we determined the order of the stress singularities and the angular variation of the stress fields. We computed the corresponding stress intensities via full-field finite element analyses of the silicon/glass specimens loaded in offset three-point flexure. Measured fracture data show that although the failure stress varies significantly with bond size, the corresponding critical stress intensity of the dominant mode is constant, thus providing support for its use as a fracture initiation criterion. In the light of both the stress analysis and the measured fracture data, we discuss the effect of mode mixity (loosely shearing versus opening) and show that it has little influence on the results for the specimens and loading considered in this study. Via an idealized model of a small crack, either interfacial or extending into one of the adherends, we study the effects of geometrical perturbations at the interface corner on the stress state, and discuss implications for fracture analysis and interpretation of fracture data. We also explore the prediction of the crack initiation angle and achieve reasonable success with a simple criterion based on the maximum circumferential stress near the uncracked interface corner.     

各向异性介质中Ⅲ型半无限长裂纹的匀速扩展

本文用Laplace变换、Wiener-Hopf和Cagniard-de Hoop方法求解了各向异性介质中一半无限长Ⅲ型裂纹突然以匀速扩展的问题.给出了瞬态的位移场和运动裂纹尖端的动态应力强度因子.     

Experimental investigation of jointed rock breaking under a disc cutter with different confining stresses

Extensive and detailed investigations have been made to better understand the rock-breaking mechanism of the tunnel boring machine (TBM) disc cutter, but the crack propagation and failure modes induced by the disc cutter when the confining stresses and joint characteristics vary have not been comprehensively investigated. To address this area of research, a triaxial testing machine (TRW-3000) is modified to investigate the effect of different confining stresses (0, 2.5, 5, 7.5 and 10 MPa) on the rock breaking of different joint angles (0°, 30°, 60°, and 90°) induced by the disc cutter. In this series of tests, the crack propagation and failure modes of the intact and jointed rock with different confining stresses are analysed. During the experiments, four different types of failure modes have been observed. The failure mode is affected by the joint orientation at low confining stress. The existence of joints has no obvious effect on the failure mode when the confining stress increases to a certain extent.     

Finite boundary effects in problem of a crack perpendicular to and terminating at a bimaterial interface

In this paper, the problem of a crack perpendicular to and terminating at an interface in bimaterial structure with finite boundaries is investigated. The dislocation simulation method and boundary collocation approach are used to derive and solve the basic equations. Two kinds of loading form are considered when the crack lies in a softer or a stiffer material, one is an ideal loading and the other one fits to the practical experiment loading. Complete solutions of the stress field including the T stress are obtained as well as the stress intensity factors. Influences of T stress on the stress field ahead of the crack tip are studied. Finite boundary effects on the stress intensity factors are emphasized. Comparisons with the problem presented by Chen et al. (Int. J. Solids and Structure, 2003, 40, 2731–2755) are discussed also.The project supported by the National Natural Science Foundation of China (10202023 and 10272103), and the Key Project of CAS (KJCX2-SW-L2).     

Torsional impact of transversely isotropic solid with functionally graded shear moduli and a penny-shaped crack

The torsional impact response of a penny-shaped crack in an unbounded transversely isotropic solid is considered. The shear moduli are assumed to be functionally graded such that the mathematics is tractable. Laplace transform and Hankel transform are used to reduce the problem to solving a Fredholm integral equation. The crack tip stress fields are obtained. Investigated are the influence of material nonhomogeneity and orthotropy on the dynamic stress intensity factor. The peak value of the dynamic stress intensity factor can be suppressed by increasing the shear moduli's gradient and/or increasing the shear modulus in a direction perpendicular to the crack surface.     

Green function and its application for a piezoelectric plate with various openings

Summary For a two-dimensional piezoelectric plate subjected to mechanical and electric load, a Green function satisfying traction free and exact electric boundary conditions along a hole is developed using Lekhnitskii's formalism and the technique of conformal mapping. The critical points for the mapping function used is investigated numerically, and the study indicates that the transformation of a polygonal hole in a piezoelectric plate into a unit circle is nonsingle-valued. A simple approach is presented to treat such a situation. Based on the Green function developed in this paper, a system of singular integral equations for the unknown dislocation defined on crack faces is presented to study the interaction between cracks and holes. Numerical results are presented to elucidate the effects of crack orientation on stress and electric displacement (SED) intensity factors and to illustrate the application of the proposed formulation. Received 4 June 1998, accepted for publication 15 July 1988     

变厚度壁板损伤容限特性研究

从损伤容限设计概念出发，进行了变厚度壁板的裂纹尖端应力强度因子、剩余强度、裂纹扩展寿命分析，充分讨论等重量设计原则下变厚度壁板的可靠性、安全性，提出变厚度壁板损伤容限设计原则。给出的变厚度壁板应力强度因子是由基本构型因子组合而成，由此可以直观地看出各参数地影响，同时为工程应用中获取应力强度因子提供简洁有效的途径。     

12结点三维等参奇异单元的构造和应用

通过改变三维8结点六面体等参单元的结点位置、结点数目和形函数,构造了一种12结点三维等参奇异单元,该单元的应力场具有1/(√r)奇异性,可以模拟裂缝前沿的奇异应力场;该单元的位移模式在其中两个坐标方向是线性变化的,因此,该单元与线性单元连接时不需要过渡单元,仍能保证交界面位移协调,克服了20结点三维等参奇异单元不能与线性单元协调连接的缺陷;文章最后将该奇异单元布置在裂缝前沿,应用有限元法计算了三点弯曲梁预制裂缝前沿的应力强度因子,该结果与规范公式计算值基本一致.     

岩石偏心圆孔单裂纹平台圆盘的动态裂纹扩展与止裂

针对平台圆环构型的优点, 提出偏心圆孔单裂纹平台圆盘(cracked eccentrically holed flattened disc, CEHFD), 该试样具有更长的断裂路径。利用霍普金森压杆加载系统, 径向冲击CEHFD试样, 完成Ⅰ型动态断裂实验。砂岩试样表面粘贴应变片和裂纹扩展计, 用于监测裂纹动态起裂、扩展和止裂的全过程。实验表明, 在整个断裂过程中, 裂纹非匀速扩展, 裂纹扩展速度在裂纹起裂后加速上升, 在裂纹止裂前有明显的减速, 与地震时断层的动态破裂全过程完全吻合。采用实验-数值-解析法得到动态应力强度因子, 其时间历程呈现先增大后减小的趋势。根据断裂过程不同时刻, 得到相应的动态起裂韧度、扩展韧度及止裂韧度。在动态断裂全过程中, 动态扩展韧度为速度的函数, 变化趋势与速度一致, 随着时间先增大后减小; 动态起裂韧度大于动态止裂韧度, 止裂韧度随着裂纹最大扩展速度的增大而降低, 并且有较大的离散性。

     

The Anti-plane Dynamic Fracture Analysis of Functionally Graded Materials with Arbitrary Spatial Variations of Material Properties

Anti-plane dynamic fracture analysis is presented for functionally graded materials (FGM) with arbitrary spatial variations of material properties. The FGM with the material properties varying continuously in an arbitrary manner is modeled as a multi-layered medium with the elastic modulus and mass density varying linearly in each sub-layer and continuous at the interfaces between two adjacent sub-layers. With this linearly inhomogeneous multi-layered model, the problem of a crack in a graded interfacial zone bonded to two homogeneous half-spaces or in a coating bonded to a homogeneous half-space subjected to the anti-plane shear impact load is investigated. Laplace and Fourier transforms and transfer matrix are applied to reduce the associated mixed boundary value problem to a Cauchy singular integral equation which is solved numerically in the Laplace transformed domain. The dynamic stress intensity factors (DSIF) are obtained by using the numerical technique of Laplace inversion.     

冲击载荷下含空孔三点弯曲梁的动态断裂行为

为得到圆孔缺陷对运动裂纹扩展过程的影响规律,采用动态焦散线实验方法进行模型实验,研究了冲击载荷下含空孔三点弯曲梁的动态断裂行为。研究结果表明:空孔对裂纹扩展有极大的阻碍作用,在一定范围内,空孔直径越大,阻碍作用越明显。裂纹扩展到空孔附近时,扩展速度会下降。裂纹在空孔上部再次起裂后,最大扩展速度远远大于裂纹与空孔贯通前的最大扩展速度。裂纹扩展至空孔附近时,裂纹尖端动态应力强度因子KdⅠ和KdⅡ均会下降。裂纹在空孔上部再次起裂后,裂尖的应力强度因子KdⅠ和KdⅡ均大于裂纹与空孔贯通前裂尖的KdⅠ和KdⅡ。在整个扩展过程中,裂纹尖端的动态应力强度因子KdⅡ远小于KdⅠ,说明KdⅠ在裂纹扩展过程中起主要作用。     

Plane strain dynamics of elastic solids with intrinsic boundary elasticity, with application to surface wave propagation

In this paper, in a development of the static theory derived by Steigmann and Ogden (Proc. Roy. Soc. London A 453 (1997) 853), we establish the equations of motion for a non-linearly elastic body in plane strain with an elastic surface coating on part or all of its boundary. The equations of (linearized) incremental motions superposed on a finite static deformation are then obtained and applied to the problem of (time-harmonic) surface wave propagation on a pre-stressed incompressible isotropic elastic half-space with a thin coating on its plane boundary. The secular equation for (dispersive) wave speeds is then obtained in respect of a general form of incompressible isotropic elastic strain-energy function for the bulk material and a general energy function for the coating material. Specialization of the form of strain-energy function enables the secular equation to be cast as a quartic equation and we therefore focus on this for illustrative purposes. An explicit form for the secular equation is thereby obtained. This involves a number of material parameters, including residual stress and moment in the properties of the coating. It is shown how this equation relates to previous work on waves in a half-space with an overlying thin layer set in the classical theory of isotropic elasticity and, in particular, the significant effect of omission of the rotatory inertia term, even at small wave numbers, is emphasized. Corresponding results for a membrane-type coating, for which the bending moment, inertia and residual moment terms are absent, are also obtained. Asymptotic formulas for the wave speed at large wave number (high frequency) are derived and it is shown how these results influence the character of the wave speed throughout the range of wave number values. A bifurcation criterion is obtained from the secular equation by setting the wave speed to zero, thereby generalizing the bifurcation results of Steigmann and Ogden (Proc. Roy. Soc. London A 453 (1997) 853) to the situation in which residual stress and moment are present in the coating. Numerical results which show the dependence of the wave speed on the various material parameters and the finite deformation are then described graphically. In particular, features which differ from those arising in the classical theory are highlighted.     

A unified approach to problems of stress concentration near V-shaped notches with sharp and rounded tip

The paper proposes a unified approach to problems of stress concentration near notches with sharp and rounded tip based on the method of singular integral equations. A solution for an elastic region having a V-shaped notch with rounded tip of large curvature is first found. Then, the stress intensity factor at the tip of a sharp-tipped notch is calculated by passing to the limit. Numerical results are obtained for a slit and a square hole in an elastic plane and an edge notch in a half-plane __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 2, pp. 70–87, February 2007. For the centenary of the birth of G. N. Savin.     

Analysis of a bi-piezoelectric ceramic layer with an interfacial crack subjected to anti-plane shear and in-plane electric loading

The behaviour of a bi-piezoelectric ceramic layer with a centre interfacial crack subjected to anti-plane shear and in-plane electric loading has been studied. The dislocation density functions and the Fourier integral transform method have been employed to eliminate the problem of singular integral equations. The normalized energy release rate, stress and electrical displacement intensity factors, G/G₀,K_III/K_III0 and K_D/K_D0, respectively, were determined for different geometric and property parameters by use of two different crack surface electric boundary conditions, i.e. impermeable and permeable. It has been shown that the effects of the thickness and material constants of the piezoelectric layer on all the three parameters, i.e. G/G₀,K_III/K_III0 and K_D/K_D0 were significant.     

The solution of integral equations with strongly singular kernels applied to the torsion of cracked circular cylinder

In this paper,the functions of warping displacement interruption defined on the crack lines are taken for the fundamental unknown functions.The torsion problem of cracked circular cylinder is reduced to solving a system of integral equations with strongly singular kernels.Using the numerical method of these equations,the torsional rigidities and the stress intensity factors are calculated to solve the torsion problem of circular cylinder with star-type and other different types of cracks.The numerical results are satisfactory.