压电压磁复合材料中裂纹对反平面简谐弹性波的散射问题

分析了压电压磁复合材料中裂纹对反平面简谐弹性波的散射问题。利用傅立叶变换,使问题的求解转换为对一对以裂纹表面上的位移差为未知变量的对偶积分方程的求解。为了求解对偶积分方程,把裂纹面上的位移差展开为雅可比多项式形式,进而得到了裂纹长度、入射波波速及入射波频率对裂纹应力强度因子的影响。从数值结果可以看出,压电压磁复合材料中可导通裂纹的反平面问题的动应力奇异性与一般弹性材料中的反平面断裂问题动应力奇异性相同。     

平行于异材界面的三维椭圆平片裂纹分析

讨论了拉伸载荷作用下平行于两相材料界面的椭圆平片裂纹问题．首先,使用有限部积分概念和两相材料界面完全接合时的点力基本解导出了一组以裂纹表面位移差为未知函数的超奇异积分方程组．该组方程表明,此时三种裂纹模型同时存在;其次,在数值求解该组方程的过程中,未知函数裂纹表面位移差被近似为位移差的基本密度函数与多项式之积．基本密度函数反映了裂纹前沿应力奇性性态;最后,以拉伸载荷为例,讨论了椭圆平片裂纹与界面的距离、裂纹形状比和不同材料组合对应力强度因子的影响,并以图表形式给出。     

功能梯度压电压磁材料中断裂问题分析

分析了功能梯度压电/压磁材料中裂纹在反平面剪切载荷下的断裂问题. 为了便于分析，假设材料性质沿着裂纹的法线方向呈指数变化. 利用Fourier变换，问题可以转化为对未知数是裂纹表面张开位移的一对对偶积分方程的求解，此对偶积分方程采用Schmidt方法求解. 最后分析了裂纹长度及表征功能梯度材料的参数βl对应力，电位移和磁通量强度因子的影响.     

垂直磁压电材料界面三维裂纹的超奇异积分法

应用有限部积分概念和广义位移基本解,垂直于磁压电双材料界面三维复合型裂纹问题被转 化为求解一组以裂纹表面广义位移间断为未知函数的超奇异积分方程问题. 进而,通过主部 分析法精确地求得裂纹尖端光滑点附近的奇性应力场解析表达式. 然后,通过将裂纹表面 位移间断未知函数表达为位移间断基本密度函数与多项式之积,使用有限部积分法对超奇异 积分方程组建立了数值方法. 最后,通过典型算例计算,讨论了广义应力强度因子的变化规 律.     

薄膜涂层材料中币形界面裂纹的弹性波散射

研究了薄膜涂层材料中币形界面裂纹的弹性波散射问题,建立了含有币形界面裂纹的覆层半空间模型,采用Hankel积分变换,将裂纹对弹性波散射的问题转化为求解矩阵形式的奇异积分方程。结合渐近分析和围道积分技术得到积分方程的解,进一步推导了散射波的应力场和位移场,以及动应力强度因子的理论计算公式。在数值算例中,分析了不同材料组合和裂纹尺寸情况下动应力强度因子与入射波频率的关系,并给出了裂纹张开位移的结果。为薄膜涂层材料的动态破坏分析提供了一定的理论基础。     

横观各向同性压电材料中裂纹问题的边界元法

采用Somigiliana公式给出了三维横观各向同性压电材料中的非渗漏裂纹问题的一般解和超奇异积分方程,其中未知函数为裂纹面上的位移间断和电势间断．在此基础上,使用有限部积分和边界元结合的方法,建立了超奇异积分方程的数值求解方法,并给出了一些典型数值算例的应力强度因子和电位移强度因子的数值结果,结果令人满意．     

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．     

纤维增强复合材料的轴对称横向裂纹分析

从弹性力学解出发,借助积分变换将纤维和基体内的位移场和应力表示成以裂纹面上位错函数为未知量的积分形式。由边界条件纤维增强复合材料三维轴对称裂纹问题化成求解一组奇异积分方程的问题。     

功能梯度压电压磁材料中裂纹对SH波的散射

研究无限大功能梯度压电/压磁复合材料中裂纹对SH波的散射问题.为了便于分析,假设材料性质沿着裂纹的法线方向是指数变化.利用Fourier余弦变化,将问题转化为对偶积分方程的求解,此对偶积分方程采用Copson方法求解.然后求得应力强度因子、电位移强度因子、磁通量强度因子的解析表达式,最后数值算例给出了材料参数、入射角及波数对标准动应力强度因子的影响.     

梯度材料中矩形裂纹的对偶边界元方法分析

采用对偶边界元方法分析了梯度材料中的矩形裂纹. 该方法基于层状材料基本解,以非裂纹边界的位移和面力以及裂纹面的间断位移作为未知量. 位移边界积分方程的源点配置在非裂纹边界上,面力边界积分方程的源点配置在裂纹面上. 发展了边界积分方程中不同类型奇异积分的数值方法. 借助层状材料基本解,采用分层方法逼近梯度材料夹层沿厚度方向力学参数的变化. 与均匀介质中矩形裂纹的数值解对比,建议方法可以获得高精度的计算结果. 最后,分析了梯度材料中均匀张应力作用下矩形裂纹的应力强度因子,讨论了梯度材料非均匀参数、夹层厚度和裂纹与夹层之间相对位置对应力强度因子的影响.      

The dynamic behavior of two parallel symmetric cracks in functionally graded piezoelectric/piezomagnetic materials

The dynamic behavior of two parallel symmetric cracks in functionally graded piezoelectric/piezomagnetic materials subjected to harmonic antiplane shear waves is investigated using the Schmidt method. The present problem can be solved using the Fourier transform and the technique of dual integral equations, in which the unknown variables are jumps of displacements across the crack surfaces, not dislocation density functions. To solve the dual integral equations, the jumps of displacements across the crack surfaces are directly expanded as a series of Jacobi polynomials. Finally, the relations among the electric, magnetic flux, and dynamic stress fields near crack tips can be obtained. Numerical examples are provided to show the effect of the functionally graded parameter, the distance between the two parallel cracks, and the circular frequency of the incident waves upon the stress, electric displacement, and magnetic flux intensity factors at crack tips.     

Dynamic behavior of unequal parallel permeable interface multi-cracks in a piezoelectric layer bonded to two piezoelectric materials half planes

This study is concerned with the treatment of the dynamic behavior of interacting cracks in a piezoelectric layer bonded to two dissimilar piezoelectric half planes subjected to harmonic anti-plane shear waves. The permeable electric boundary condition is considered. By use of the Fourier transform technique, the problem can be solved with the help of two pairs of dual integral equations in which the unknown variables are the jumps of the displacements across the crack surfaces. To solve the dual integral equations, the jumps of the displacements across the crack surfaces are expanded in two series of Jacobi polynomials. The electromechanical behavior of two pairs of unequal parallel cracks was determined. Numerical examples are provided to show the effects of the geometry of the cracks, the frequency of the incident waves and materials properties upon the dynamic stress intensity factors (DSIFs) and the electric displacement intensity factors.     

An Interface Crack for a Functionally Graded Strip Sandwiched Between Two Homogeneous Layers of Finite Thickness

In this paper, the behavior of an interface crack for a functionally graded strip sandwiched between two homogeneous layers of finite thickness subjected to an uniform tension is resolved using a somewhat different approach, named the Schmidt method. The Fourier transform technique is applied and a mixed boundary value problem is reduced to two pairs of dual integral equations in which the unknown variables are the jumps of the displacements across the crack surface. To solve the dual integral equations, the jumps of the displacements across the crack surfaces are expanded in a series of Jacobi polynomials. This process is quite different from those adopted in previous works. Numerical examples are provided to show the effects of the crack length, the thickness of the material layer and the materials constants upon the stress intensity factor of the cracks. It can be obtained that the results of the present paper are the same as ones of the same problem that was solved by the singular integral equation method. As a special case, when the material properties are not continuous through the crack line, an approximate solution of the interface crack problem is also given under the assumption that the effect of the crack surface interference very near the crack tips is negligible. Contrary to the previous solution of the interface crack, it is found that the stress singularities of the present interface crack solution are the same as ones of the ordinary crack in homogenous materials.     

DYNAMIC BEHAVIOR OF TWO PARALLEL SYMMETRY CRACKS IN MAGNETO-ELECTRO-ELASTIC COMPOSITES UNDER HARMONIC ANTI-PLANE WAVES

The dynamic behavior of two parallel symmetry cracks in magneto-electro-elastic composites under harmonic anti-plane shear waves is studied by Schmidt method. By using the Fourier transform, the problem can be solved with a pair of dual integral equations in which the unknown variable is the jumps of the displacements across the crack surfaces. To solve the dual integral equations, the jumps of the displacements across the crack surface were expanded in a series of Jacobi polynomials. The relations among the electric filed, the magnetic flux and the stress field were obtained. From the results, it can be obtained that the singular stresses in piezoelectric/piezomagnetic materials carry the same forms as those in a general elastic material for the dynamic anti-plane shear fracture problem. The shielding effect of two parallel cracks was also discussed.     

Basic solution of two parallel non-symmetric permeable cracks in piezoelectric materials

The behavior of two parallel non-symmetric cracks in piezoelectric materials subjected to the anti-plane shear loading was studied by the Schmidt method for the permeable crack electric boundary conditions. Through the Fourier transform, the present problem can be solved with two pairs of dual integral equations ip which the unknown variables are the jumps of displacements across crack surfaces. To solve the dual integral equations, the jumps of displacements across crack surfaces were directly expanded in a series of Jacobi polynomials. Finally, the relations between electric displacement intensity factors and stress intensity factors at crack tips can be obtained. Numerical examples are provided to show the effect of the distance between two cracks upon stress and electric displacement intensity factors at crack tips. Contrary to the impermeable crack surface condition solution, it is found that electric displacement intensity factors for the permeable crack surface conditions are much smaller than those for the impermeable crack surface conditions. At the same time, it can be found that the crack shielding effect is also present in the piezoelectric materials.     

Two parallel limited-permeable mode-I cracks or four parallel limited-permeable mode-I cracks in the piezoelectric materials

In this paper, the basic solutions of two parallel mode-I cracks or four parallel mode-I cracks in the piezoelectric materials were investigated by means of the Schmidt method for the limited-permeable electric boundary conditions. The electric permittivity of air in the crack was considered. Through the Fourier transform, the problems can be solved with the help of two pairs of dual integral equations, in which the unknown variables were the jumps of the displacements across the crack surfaces, not the dislocation density functions. To solve the dual integral equations, the jumps of the displacements across the crack surfaces were directly expanded in a series of Jacobi polynomials. Finally, the effects of the distance between two parallel cracks, the distance between two collinear cracks and the electric boundary conditions on the stress and the electric intensity factors in the piezoelectric materials are analyzed. These results can be used for the strength evaluation of the piezoelectric materials with multi-cracks. The crack shielding effect is also present in the piezoelectric materials.     

Investigation of the behavior of a Griffith crack at the interface of a layer bonded to a half plane using the Schmidt method for the opening crack mode

In this paper, the behavior of a Griffith crack at the interface of a layer boned to a half plane subjected to a uniform tension is investigated by use of the Schmidt method under the assumptions that the effect of the crack surface overlapping very near the crack tips is negligible and also there is a sufficiently large component of mode-I loading so that the crack essentially remains open. By use of the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations in which the unknown variables are the jumps of the displacements across the crack surfaces. To solve the dual integral equations, the jumps of the displacements across the crack surfaces are expanded in a series of Jacobi polynomials. Numerical examples are provided to show the effects of the crack length, the thickness of the material layer and the materials constants upon the stress intensity factor of the crack. As a special case in our solution, we also give the solution of the ordinary crack in homogeneous materials. Contrary to the previous solution of the interface crack problem, it is found that the stress singularities of the present interface crack solution are similar with ones for the ordinary crack in homogeneous materials.     

Non-local theory solution for a Mode I crack in piezoelectric materials

In this paper, the non-local theory of elasticity is applied to obtain the behavior of a Griffith crack in the piezoelectric materials subjected to a uniform tension loading. The permittivity of the air in the crack is considered. By means of the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations, in which the unknown variables are the jumps of the displacements across the crack surfaces. To solve the dual integral equations, the jumps of the displacements across the crack surfaces are expanded in a series of Jacobi polynomials. Numerical examples are provided to show the effects of the crack length, the materials constants, the electric boundary conditions and the lattice parameter on the stress and the electric displacement fields near the crack tips. It can be obtained that the effects of the electric boundary conditions on the electric displacement fields are large. Unlike the classical elasticity solutions, it is found that no stress and electric displacement singularities are present at the crack tips. The non-local elastic solutions yield a finite hoop stress at the crack tips, thus allowing us to use the maximum stress as a fracture criterion.     

The closed form solution of a Mode-I crack in the piezoelectric/piezomagnetic materials

In this paper, the behavior of a Mode-I crack in the piezoelectric/piezomagnetic materials subjected to a uniform tension loading is investigated by the generalized Almansi’s theorem. Through the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations, in which the unknown variables are the jumps of the displacements across the crack surfaces. To solve the dual integral equations, the displacement jumps are directly expanded in a series of Jacobi polynomials. Then the closed form solution of this problem can be obtained.     

The nonlocal solution of two parallel cracks in functionally graded materials subjected to harmonic anti-plane shear waves

In this paper, the dynamic interaction of two parallel cracks in functionally graded materials (FGMs) is investigated by means of the non-local theory. To make the analysis tractable, the shear modulus and the material density are assumed to vary exponentially with the coordinate vertical to the crack. To reduce mathematical difficulties, a one-dimensional non-local kernel is used instead of a two-dimensional one for the dynamic problem to obtain stress fields near the crack tips. By use of the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations, in which the unknown variables are the jumps of displacements across the crack surfaces. To solve the dual integral equations, the jumps of displacements across the crack surfaces are expanded in a series of Jacobi polynomials. Unlike the classical elasticity solutions, it is found that no stress singularity is present at the crack tips. The non-local elastic solutions yield a finite hoop stress at the crack tips. The present result provides theoretical references helpful for evaluating relevant strength and preventing material failure of FGMs with initial cracks. The magnitude of the finite stress field depends on relevant parameters, such as the crack length, the distance between two parallel cracks, the parameter describing the FGMs, the frequency of the incident waves and the lattice parameter of materials. The project supported by the National Natural Science Foundation of China (90405016, 10572044) and the Specialized Research Fund for the Doctoral Program of Higher Education (20040213034). The English text was polished by Yunming Chen.