Elastostatic solutoins for semi-infinite orthotropic cantilevered strips

The elastostatic solutions of semi-infinite orthotropic cantilevered strips with traction free edges and loading at infinity are governed by the differential equation φ,_xyxy+(2+δ₀)φ,_xyzzz+φ,_zzzz with δ₀>-4. Based on the work of [10] for δ₀>0 case, this paper completes the case δ₀=0 for isotropic materials and the case 0>δ₀>-4 for orthotropic materials. The solutions of the above problems have important application in the properly formulated boundary conditions of plate theories for prescribed displacement edge data.     

正交异性平面内的椭圆孔或裂纹系问题

应用Ｆａｂｅｒ级数展开和各向异性体平面问题复应力函数的方法，对于含有任意个椭圆孔或裂纹的正交异性平面，给出了孔周应力场解或孔附近裂纹应力强度因子解，其特例与前人结果一致．     

插值矩阵法分析与复合材料界面相交的平面裂纹奇异应力场

提出了用插值矩阵法分析与各向异性材料界面相交的平面裂纹应力奇异性。基于V形切口尖端附近区域位移场渐近展开,将位移场的渐近展开式的典型项代入线弹性力学基本方程,得到关于平面内与复合材料界面相交的裂纹应力奇异性指数的一组非线性常微分方程的特征值问题,运用插值矩阵法求解,获得了平面内各向异性结合材料中与界面以任意角相交的裂纹尖端的应力奇异性指数随裂纹角的变化规律,数值计算结果与已有结果比较表明,本文方法具有很高的精度和效率。     

Scattering of love waves by an interface crack between a piezoelectric layer and an elastic substrate

The scattering of Love waves by an interface crack between a piezoelectric layer and an elastic substrate is investigated by using the integral transform and singular integral equation techniques. The dynamic stress intensity factors of the left and the right crack tips are determined. It is found from numerical calculation that the dynamic response of the system depends significantly on the crack configuration, the material combination and the propagating direction of the incident wave. It is expected that specifying an appropriate material combination may retard the growth of the crack for a certain crack configuration. Project supported by the National Natural Science Foundation of China (No. 19891180), the Fundamental Research Foundation of Tsinghua University (JZ 2000.007) and the Fund of the Education Ministry of China.     

Analysis of stress intensity factor in orthotropic bi-material mixed interface crack

Adopting the complex function approach, the paper studies the stress intensity factor in orthotropic bi-material interface cracks under mixed loads. With consideration of the boundary conditions, a new stress function is introduced to transform the problem of bi-material interface crack into a boundary value problem of partial differential equations. Two sets of non-homogeneous linear equations with 16 unknowns are constructed. By solving the equations, the expressions for the real bi-material elastic constant εt and the real stress singularity exponents λt are obtained with the bi-material engineering parameters satisfying certain conditions. By the uniqueness theorem of limit,undetermined coefficients are determined, and thus the bi-material stress intensity factor in mixed cracks is obtained. The bi-material stress intensity factor characterizes features of mixed cracks. When orthotropic bi-materials are of the same material, the degenerate solution to the stress intensity factor in mixed bi-material interface cracks is in complete agreement with the present classic conclusion. The relationship between the bi-material stress intensity factor and the ratio of bi-material shear modulus and the relationship between the bi-material stress intensity factor and the ratio of bi-material Young's modulus are given in the numerical analysis.     

Determination of stress intensity factors in half-plane containing several moving cracks

The dynamic stress intensity factors in a half-plane weakened by several finite moving cracks are investigated by employing the Fourier complex transformation. Stress analysis is performed in a half-plane containing a single dislocation and without dislocation. An exact solution in a closed form to the stress fields and displacement is ob- tained. The Galilean transformation is used to transform between coordinates connected to the cracks. The stress components are of the Cauchy singular kind at the location of dislocation and the point of application of the the influence of crack length and crack running force. Numerical examples demonstrate velocity on the stress intensity factor.     

Analysis of multiple interfacial cracks in three-dimensional bimaterials using hypersingular integro-differential equation method

By using the concept of finite-part integral, a set of hypersingular integro-differential equations for multiple interracial cracks in a three-dimensional infinite bimaterial subjected to arbitrary loads is derived. In the numerical analysis, unknown displacement discontinuities are approximated with the products of the fundamental density functions and power series. The fundamental functions are chosen to express a two-dimensional interface crack rigorously. As illustrative examples, the stress intensity factors for two rectangular interface cracks are calculated for various spacing, crack shape and elastic constants. It is shown that the stress intensity factors decrease with the crack spacing.     

Singular integral equations and boundary element method of cracks in thermally stressed planar solids

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Fracture analysis on a layered multiferroic smart structure containing non-periodic interfacial cracks under magnetic/electric and mechanical loadings

The problems of periodic cracks have been widely solved in existing literature of fracture mechanics. Although the assumption of periodicity facilitates the theoretical derivation, practical multiple cracks in most cases are not periodically distributed in materials and structures. The present article performs fracture analysis on a piezoelectric/piezomagnetic/piezoelectric (PE/PM/PE) tri-layered smart structure containing nonperiodic interfacial cracks under magnetic/electric and mechanical loadings. The methods of dislocation simulation, Green's function and Cauchy singular integral equations are employed to solve the problem. Stress intensity factors (SIFs) are numerically calculated, and parametric studies are conducted to demonstrate the variation laws of SIFs versus geometrical parameters. The obtained conclusions may provide references for the optimal design of PE/PM/PE smart structures.     

含界面裂纹Reissner板弯曲问题分析的奇异单元

首先,采用特征函数渐近展开法,推导了Reissner板弯曲界面裂纹尖端附近位移场渐近展开的前两阶显式表达式,并利用所获得的位移场渐近表达式构造了一种可用于Reissner板弯曲界面裂纹分析的奇异单元。然后,将该奇异单元与外部的常规有限单元相结合,开展了含界面裂纹Reissner板弯曲断裂问题的数值分析。奇异单元可以较好地描述裂纹尖端附近的内力场与位移场,其优势是它与常规单元进行连接时不需要使用过渡单元,并且可以直接给出应力强度因子等断裂参数的高精度数值结果。最后,通过两个数值算例验证了本文方法的有效性。     

Singular behaviour and asymptotic stress field of a crack terminating at a bimaterial interface

A crack terminating at an interface of two dissimilar elastic materials is investigated. It is found that the asymptotic stress field near the crack tip is in general composed of two parts with each part being characterized by one singularity. The detailed relation of the two singularities with the bimaterial properties is given for some special cases of the crack.     

层状介质垂直界面有限裂纹问题的积分变换解

层状弹性材料包含垂直于界面有限裂纹时，可运用富里叶变换及引用位错密度函数，导出了反映裂纹尖端奇异性的奇异积分方程组，并使用Ｌｏｂａｔｔｏ－ｃｈｅｂｙｓｈｅｖ方法解此方程组，最后得到裂纹尖端应力强度因子，为检验方法的正确性，对某两层含裂实际结构进行了计算，结果是满意的。     

A crack perpendicular to and terminating at a bimaterial interface

Using dislocation simulation approach, the basic equation for a finite crack perpendicular to and terminating at a bimaterial interface is formulated. A novel expansion method is proposed for solving the problem. The complete solution to the problem, including the explicit formulae for theT stresses ahead of the crack tip and the stress intensity factors are presented. The stress field characteristics are analysed in detail. It is found that normal stresses {ie27-1} and {ie27-2} ahead of the crack tip, are characterised byQ fields if the crack is within a stiff material and the parameters |p _T | and |q _T | are very small, whereQ is a generalised stress intensity factor for a crack normal to and terminating at the interface. If the crack is within a weak material, the normal stresses {ie27-3} and {ie27-4} are dominated by theQ field plusT stress. This work was supported by the Swedish Research Council for Engineering Sciences.     

Investigation of the behavior of a griffith crack at the interface between two dissimilar orthotropic elastic half-planes for the opening crack mode

The behaviors of an interface crack between dissimilar orthotropic elastic halfplanes subjected to uniform tension was reworked by use of the Schmidt method. By use of the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations, of which the unknown variables are the jumps of the displacements across the crack surfaces. Numerical examples are provided for the stress intensity factors of the cracks. Contrary to the previous solution of the interface crack, it is found that the stress singularity of the present interface crack solution is of the same nature as that for the ordinary crack in homogeneous materials. When the materials from the two half planes are the same, an exact solution can be otained.     

Transient response of a piezoelectric ceramic with two coplanar cracks under electromechanical impact

The transient response of two coplanar cracks in a piezoelectric ceramic under antiplane mechanical and inplane electric impacting loads is investigated in the present paper. Laplace and Fourier transforms are used to reduce the mixed boundary value problems to Cauchy-type singular integral equations in Laplace transform domain, which are solved numerically. The dynamic stress and electric displacement factors are obtained as the functions of time and geometry parameters. The present study shows that the presence of the dynamic electric field will impede or enhance the propagation of the crack in piezoelectric ceramics at different stages of the dynamic electromechanical load. Moreover, the electromechanical response is greatly affected by the ratio of the space of the cracks and the crack length.     

和界面接触的刚性线夹杂对SH波的散射

利用积分变换方法,得出了两相材料中单位简谐力的格林函数。根据简谐集中力的格林函数得出了和界面接触的刚性线的散射场。利用无穷积分的性质,把和界面接触刚性线的散射场分解为奇异部分和有界部分。通过分解后的散射场建立了和界面接触剐性线在SH波作用下的Cauchy型奇异积分方程。根据所得奇异积分方程和刚性线的散射场得到了刚性线端点的奇异性阶数及奇性应力。应用刚性线端点的奇性应力定义了刚性线端点的应力奇异因子。对所得Cauchy型奇异积分方程的数值求解,可得刚性线端点的应力奇异因子。     

纤维增强复合材料圆柱型界面裂纹分析

以裂纹面上的位错函数为未知量将圆柱型界面裂纹问题化成一组奇异积分方程的求解问题．应用Ｍｕｓｋｈｅｌｉｓｈｖｉｌｉ的奇异积分方程理论，分析了圆柱型界面裂纹尖端应力场．针对裂纹尖端分别存在和不存在接触区两种情况，确定了裂纹尖端应力场的奇异性．利用数值方法计算了圆柱型界面裂纹尖端接触区尺寸对剪应力强度因子的影响．     

基体裂纹与界面刚性线的弹性干涉

研究两种材料界面上的刚性线与其它任意位置处直线裂纹弹性干涉的反平面问题。基于界面上刚性线与任意位置处螺型位错干涉的基本解,运用连续位错密度模型法将问题转化为奇异积分方程。用半开型积分法求解奇异积分方程,得到位错密度函数的离散值,计算裂纹尖端处的应力强度因子。算例说明该方法可用于工程实际问题。     

The dynamic stress intensity factor analysis of adhesively bonded material interface crack with damage under shear loading

This paper studies the dynamic stress intensity factor (DSIF) at the interface in an adhesive joint under shear loading. Material damage is considered. By introducing the dislocation density function and using the integral transform, the problem is reduced to algebraic equations and can be solved with the collocation dots method in the Laplace domain. Time response of DSIF is calculated with the inverse Laplace integral transform. The results show that the mode Ⅱ DSIF increases with the shear relaxation parameter, shear module and Poisson ratio, while decreases with the swell relaxation parameter. Damage shielding only occurs at the initial stage of crack propagation. The singular index of crack tip is -0.5 and independent on the material parameters, damage conditions of materials, and time. The oscillatory index is controlled by viscoelastic material parameters.     

Scattering of SHPlease define abbreviation SH waves by arc-shaped interface cracks between a cylindrical magneto-electro-elastic inclusion and matrix: near fields

Summary In this paper, the scattering of SH waves by a magneto-electro-elastic cylindrical inclusion partially debonded from its surrounding magneto-electro-elastic material is investigated by using the wavefunction expansion method and a singular integral equation technique. The debonding regions are modeled as multiple arc-shaped interface cracks with non-contacting faces. The magneto-electric impermeable boundary conditions are adopted. By expressing the scattered fields as wavefunction expansions with unknown coefficients, the mixed boundary-value problem is firstly reduced to a set of simultaneous dual-series equations. Then, dislocation density functions are introduced as unknowns to transform these dual-series equations to Cauchy singular integral equations of the first type,which can be numerically solved easily. The solution is valid for arbitrary number and size of the arc-shaped interface cracks. Finally, numerical results of the dynamic stress intensity factors are presented for the cases of one debond. The effects of incident direction, crack configuration and various material parameters on the dynamic stress intensity factors are discussed. The solution of this problem is expected to have applications in the investigation of dynamic fracture properties of magneto-electro-elastic materials with cracks.The work was supported by the National Natural Science Fund of China (Project No. 19772029) and the Research Fund for Doctors of Hebei Province, China (Project No. B2001213).