Basic solutions of multiple parallel symmetric mode-III cracks in functionally graded piezoelectric/piezomagnetic material plane

The Schmidt method is adopted to investigate the fracture problem of multiple parallel symmetric and permeable finite length mode-III cracks in a functionally graded piezoelectric/piezomagnetic material plane. This problem is formulated into dual integral equations, in which the unknown variables are the displacement jumps across the crack surfaces. In order to obtain the dual integral equations, the displacement jumps across the crack surfaces are directly expanded as a series of Jacobi polynomials. The results show that the stress, the electric displacement, and the magnetic flux intensity factors of cracks depend on the crack length, the functionally graded parameter, and the distance among the multiple parallel cracks. The crack shielding effect is also obviously presented in a functionally graded piezoelectric/piezomagnetic material plane with mul- tiple parallel symmetric mode-III cracks.     

功能梯度材料涂层平面裂纹分析

研究粘接于均质基底材料上功能梯度涂层平面裂纹问题. 假设功能梯度材料剪切模量的倒数为坐标的线性函数,而泊松比为常数. 采用Fourier变换和传递矩阵法将该混合边值问题化为奇异积分方程组,通过数值求解获得 应力强度因子. 考察了材料梯度变化形式、结构几何尺寸和材料梯度参数对裂纹应力强度因子的影响,发现 功能梯度材料涂层尺寸、裂纹长度以及材料梯度参数均对应力强度因子有显著影响.     

Elastic wave scattering from a penny-shaped interface crack in coated materials

This paper is concerned with the elastic wave scattering induced by a penny-shaped interface crack in coated materials. Using the integral transform, the problem of wave scattering is reduced to a set of singular integral equations in matrix form. The singular integral equations are solved by the asymptotic analysis and contour integral technique, and the expressions for the stress and displacement as well as the dynamic stress intensity factors (SIFs) are obtained. Using numerical analysis, this approach is verified by the finite element method (FEM), and the numerical results agree well with the theoretical results. For various crack sizes and material combinations, the relations between the SIFs and the incident frequency are analyzed, and the amplitudes of the crack opening displacements (CODs) are plotted versus incident wavenumber. The investigation provides a theoretical basis for the dynamic failure analysis and nondestructive evaluation of coated materials.     

Thermoelastic-numerical hybrid analysis of holes and cracks in composites

An effective thermoelastic-numerical hybrid method is demonstrated for obtaining stress information on, and adjacent to, geometric discontinuities in orthotropic composites. This includes determining stress intensity factors. The technique involves evaluating the stress functions from measured thermoelastic data around the external boundary of a subregion containing the geometric discontinuity of interest. Knowing the stress functions, stresses throughout the subregion surrounding the geometric discontinuity, and stress intensity factors, are available.     

Crack shielding and material deterioration in damaged materials: an antiplane shear fracture problem

Summary A simple damage evolution model is proposed for a quasibrittle material in the framework of continuum damage mechanics. The model is used to obtain a closed form solution for a mode-III stationary crack under small scale damage conditions. It is found that the crack tip stress intensity factor is reduced, i.e., the crack is shielded by the damage. However, this shielding effect is completely offset by the material deterioration caused by the damage. It also holds for steady state crack growth. When the most effective shielding is reached for the stationary crack, the zone dominated by the stress intensity factor shrinks to the crack tip. The results for the antiplane shear problem should shed some light on the in- plane fracture problem. Received 4 August 1997; accepted for publication 7 October 1997     

Stress intensity factors for an oblique edge crack in a coating/substrate system with a graded interfacial zone under antiplane shear

This paper investigates the edge crack problem for a coating/substrate system with a functionally graded interfacial zone under the condition of antiplane deformation. With the interfacial zone being modeled by a nonhomogeneous interlayer having the continuously varying shear modulus between the dissimilar, homogeneous phases of the coated medium, the coating is assumed to contain an edge crack at an arbitrary angle to the interfacial zone. The Fourier integral transform method is used in conjunction with the coordinate transformations of basic field variables. Formulation of the proposed crack problem is then reduced to solving a singular integral equation with a generalized Cauchy kernel. The mode III stress intensity factors are defined and evaluated in terms of the solution to the integral equation. In the numerical results, the values of the stress intensity factors are plotted, illustrating the effects of the crack orientation angle for various material and geometric combinations of the coating/substrate system with the graded interfacial zone.     

Transient dynamic response of a coated piezoelectric strip with a vertical crack

In this study, the dynamic response of a coated piezoelectric strip containing a crack vertical to the interfaces under normal impact load is considered. Based on the superposition principle and the integral transform techniques, the solution in the Laplace transformed plane is obtained in terms of a singular integral equation. The order of stress singularity around the tip of the terminated crack is also obtained. The singular integral equation is solved by using the Gauss–Jacobi integration formula, and the numerical Laplace inversion is then carried out to obtain the resulting dynamic stress and electric displacement intensities. The effects of the material properties and the geometric parameters on the dynamic stress intensity factor and the dynamic energy density factors are shown graphically.     

FRACTURE ANALYSIS OF A FUNCTIONALLY GRADED STRIP UNDER PLANE DEFORMATION

In this paper the plane elasticity problem for a functionally graded strip containing a crack is considered. It is assumed that the reciprocal of the shear modulus is a linear function of the thickness-coordinate, while the Possion's ratio keeps constant. By utilizing the Fourier transformation technique and the transfer matrix method, the mixed boundary problem is reduced to a system of singular integral equations that are solved numerically. The influences of the geometric parameters and the graded parameter on the stress intensity factors and the strain energy release rate are investigated. The numerical results show that the graded parameters, the thickness of the strip and the crack size have significant effects on the stress intensity factors and the strain energy release rate.     

Propagation of a mode-III interfacial crack in a piezoelectric–piezomagnetic bi-material

The transient response of a semi-infinite mode-III interfacial crack propagating between piezoelectric (PE) and piezomagnetic (PM) half spaces is investigated in this paper. The integral transform method together with the Wiener–Hopf and Cagniard–de Hoop techniques is used to solve the mixed boundary value problem under consideration. The existence of generalized Maerfeld–Tournois interfacial wave is discussed and the solutions of the coupled fields are derived for four different cases of bulk shear wave velocity. The dynamic intensity factors of stress, electric displacement and magnetic induction as well as energy release rate (ERR) are obtained in explicit forms. The numerical results of the universal functions and dimensionless ERR for several different material combinations are presented and discussed in details. It is found that the Bleustein–Gulyaev (generalized Maerfeld–Tournois) waves dominate the dynamic characteristics of the interfacial crack propagation in PE–PM bi-material.     

A moving mode-III crack in functionally graded piezoelectric material: permeable problem

In this paper a moving mode-III crack in functionally graded piezoelectric materials (FGPM) is studied. The crack surfaces are assumed to be permeable. The governing equations for FGPM are solved by means of Fourier cosine transform. The mathematical formulation for the permeable crack condition is derived as a set of dual integral equations, which, in turn, are reduced to a Fredholm integral equation of the second kind. The results obtained indicate that the stress intensity factor of moving crack in FGPM depends only on the mechanical loading. The gradient parameter of the FGPM and the moving velocity of the crack do have significant influence on the dynamic stress intensity factor.     

含圆形嵌体弹性平面中径向裂纹问题的超奇异积分方程方法

根据含圆形嵌体平面问题在极坐标下的弹性力学基本解,使用Betti互换定理,在有限部积分意义下将问题归结为两个以裂纹岸位移间断为基本未知量、对于Ⅰ型和Ⅱ型问题相互独立的超奇异积分方程,对含圆形嵌体弹性平面中的径向裂纹问题进行了研究.根据有限部积分原理,建立了问题的数值算法.计算结果表明,嵌体半径、裂纹位置及材料剪切弹性模量等都对裂纹应力强度因子具有较为明显的影响.     

Oblique edge crack in an anisotropic material under antiplane shear

An oblique edge crack in an anisotropic material under antiplane shear loadings is investigated. The antiplane problems are formulated based on a linear transformation method. An anisotropic solid containing an edge crack subjected to concentrated forces is first considered. The stress intensity factor for the edge crack with concentrated forces is obtained from the solution of the transformed edge crack in an isotropic material which is solved by using conformal mapping technique and complex function theory. The solution of the edge crack under concentrated loads is used to construct the stress intensity factor for the oblique edge crack in the anisotropic material subjected to antiplane distributed loads. Some numerical computations are carried out to calculate the stress intensity factors for the edge crack in inclined orthotropic materials subjected to point forces as well as distributed tractions.     

Mixed-mode stress intensity factors of 3D interface crack in fully coupled electromagnetothermoelastic multiphase composites

This contribution presents an extended hypersingular intergro-differential equation (E-HIDE) method for modeling the 3D interface crack problem in fully coupled electromagnetothermoelastic anisotropic multiphase composites under extended electro-magneto-thermo-elastic coupled loads through theoretical analysis and numerical simulations. First, based on the extended boundary element method, the 3D interface crack problem is reduced to solving a set of E-HIDEs coupled with extended boundary integral equations, in which the unknown functions are the extended displacement discontinuities. Then, the behavior of the extended singular stress indices around the interface crack front terminating at the interface is analyzed by the extended main-part analysis. The extended stress intensity factors near the crack front are defined. In addition, a numerical method for a 3D interface crack problem subjected to extended loads is proposed, in which the extended displacement discontinuities are approximated by the product of basic density functions and polynomials. Finally, the radiation distribution of extended stress intensity factors at the interface crack surface are calculated, and the results are presented toward demonstrating the applicability of the proposed method.     

Three-dimensional interfacial fracture analysis of a one-dimensional hexagonal quasicrystal coating

In this paper, the three-dimensional (3D) interfacial fracture is analyzed in a one-dimensional (1D) hexagonal quasicrystal (QC) coating structure under mechanical loading. A planar interface crack with arbitrary shape is studied by a displacement discontinuity method. Fundamental solutions of interfacial concentrated displacement discontinuities are obtained by the Hankel transform technique, and the corresponding boundary integral-differential equations are constructed with the superposition principle. Green’s functions of constant interfacial displacement discontinuities within a rectangular element are derived, and a boundary element method is proposed for numerical simulation. The singularity of stresses near the crack front is investigated, and the stress intensity factors (SIFs) as well as energy release rates (ERRs) are determined. Finally, relevant influencing factors on the fracture behavior are discussed.     

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.     

The behavior of two non-symmetrical permeable cracks emanating from an elliptical hole in a piezoelectric solid

Based on the Stroh-type formalism, we present a concise analytic method to solve the problem of complicated defects in piezoelectric materials. Using this method and the technique of conformal mapping, the problem of two non-symmetrical collinear cracks emanating from an elliptical hole in a piezoelectric solid is investigated under remotely uniform in-plane electric loading and anti-plane mechanical loading. The exact solutions of the field intensity factors and the energy release rate are presented in closed-form under the permeable electric boundary condition. With the variation of the geometrical parameters, the present results can be reduced to the well-known results of a mode-III crack in piezoelectric materials. Moreover, new special models used for simulating more practical defects in a piezoelectric solid are obtained, such as two symmetrical edge cracks and single edge crack emanating from an elliptical hole or circular hole, T-shaped crack, cross-shaped crack, and semi-infinite plane with an edge crack. Numerical results are then presented to reveal the effects of geometrical parameters and the applied mechanical loading on the field intensity factors and the energy release rate.     

三维体内含垂直于双材料界面混合型裂纹的超奇异积分解法

利用双材料位移基本解和Somigliana公式,将三维体内含垂直于双材料界面混合型裂纹问题归结为求解一组超奇异积分方程。使用主部分析法,通过对裂纹前沿应力奇性的分析,得到用裂纹面位移间断表示的应力强度因子的计算公式,进而利用超奇异积分方程未知解的理论分析结果和有限部积分理论,给出了超奇异积分方程的数值求解方法。最后,对典型算例的应力强度因子做了计算,并讨论了应力强度因子数值结果的收敛性及其随各参数变化的规律。     

陶瓷基复合材料基体唇形裂纹失效规律研究

建立横向拉伸载荷下的唇形裂纹数学模型,采用复变函数的方法,通过保角映射,推导了唇形裂纹尖端应力场和位移场的解析解,建立了唇形裂纹的应力强度因子准则和最大能量释放率准则,结合算例分析陶瓷基复合材料基体唇形裂纹的几何参数、外载荷和纤维分布对失效准则的影响规律.结果 表明,(1)裂纹尖端应力场和位移场的解析解与有限元计算结果...     

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．     

Stress intensity factors of a crack in a composedcircular cylinder

A composed circular cylinder, formed by a core circular cylinder, containing a crackand enclosed by a layered hollow circular cylinder, is investigated in regard to the evaluation ofstress intensity factors. Analytic solutions to the problem are provided, with which the upper andlower bounds of stress intensity factors in a cracked circular cylinder, the stress distribution in alayered hollow circular cylinder, and the stress intensity factors for a crack in the composedcircular cylinder can precisely be determined. Numerical materials, demonstrating the discretevalues of the stress intensity factors, as well as the general pattern according to which the stressintensity factors vary with the material and geometric constants, are presented. The solutions aredeveloped based on a simplified and modified solution to the Hilbert problem, and the matrixpresentation and manipulation of functions and variables, used in the circuit theory.