Multiple cracks in thermoelectroelastic bimaterials

The formulation for thermal stress and electric displacement in an infinite thermopiezoelectric plate with an interface and multiple cracks is presented. Using Green's function approach and the principle of superposition, a system of singular integral equations for the unknown temperature discontinuity defined on each crack face is developed and solved numerically. The formulation can then be used to calculate some fracture parameters such as the stress–electric displacement and strain energy density factor. The direction of crack growth for many cracks in thermopiezoelectric bimaterials is predicted by way of the strain energy density theory. Numerical results for stress–electric displacement factors and crack growth direction at a particular crack tip in two crack system of bimaterials are presented to illustrate the application of the proposed formulation.     

Crack growth prediction of an inclined crack in a half-plane thermopiezoelectric solid

The strain energy density theory is used to determine the direction of crack propagation in the case of an inclined crack in a half-plane thermopiezoelectric sheet subjected to uniform heat flow. Based on the Stroh's formulation and the thermoelectroelastic Green's function, a system of singular integral equations for the unknown thermal analog of dislocation density defined on the crack faces is presented and used to calculate the corresponding strain energy density. The direction of crack extension is then determined by way of the minimum strain energy density criterion. The study shows that the criterion is easy-to-use for thermal fracture problems.     

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.     

Arbitrarily oriented crack near interface in piezoelectric bimaterials

Arbitrarily oriented crack near interface in piezoelectric bimaterials is considered. After deriving the fundamental solution for an edge dislocation near the interface, the present problem can be expressed as a system of singular integral equations by modeling the crack as continuously distributed edge dislocations. In the paper, the dislocations are described by a density function defined on the crack line. By solving the singular integral equations numerically, the dislocation density function is determined. Then, the stress intensity factors (SIFs) and the electric displacement intensity factor (EDIF) at the crack tips are evaluated. Subsequently, the influences of the interface on crack tip SIFs, EDIF, and the mechanical strain energy release rate (MSERR) are investigated. The J-integral analysis in piezoelectric bimaterals is also performed. It is found that the path-independent of J₁-integral and the path-dependent of J₂-integral found in no-piezoelectric bimaterials are still valid in piezoelectric bimaterials.     

Green's function for thermopiezoelectric plates with holes of various shapes

Summary Thermoelectroelastic problems for holes of various shapes embedded in an infinite matrix are considered in this paper. Based on Lekhnitskii's formalism, the technique of conformal mapping and the exact electric boundary conditions on the hole boundary, the thermoelectroelastic Green's function has been obtained analytically in terms of a complex potential. As an application of the proposed function, the problem of an infinite plate containing a crack and a hole is analysed. A system of singular integral equations for the unknown temperature discontinuity and the discontinuity of elastic displacement and electric potential (EDEP) defined on crack faces is developed and solved numerically. Numerical results for stress and electric displacement (SED) intensity factors of the crack-hole system are presented to illustrate the application of the proposed formulation. Received 7 October 1998; accepted for publication 26 January 1999     

Hypersingular integral equation method for a three-dimensional crack in anisotropic electro-magneto-elastic bimaterials

Using Green’s functions, the extended general displacement solutions of a three-dimensional crack problem in anisotropic electro-magneto-elastic (EME) bimaterials under extended loads are analyzed by the boundary element method. Then, the crack problem is reduced to solving a set of hypersingular integral equations (HIE) coupled with boundary integral equations. The singularity of the extended displacement discontinuities around the crack front terminating at the interface is analyzed by the main-part analysis method of HIE, and the exact analytical solutions of the extended singular stresses and extended stress intensity factors (SIFs) near the crack front in anisotropic EME bimaterials are given. Also, the numerical method of the HIE for a rectangular crack subjected to extended loads is put forward with the extended crack opening dislocation approximated by the product of basic density functions and polynomials. At last, numerical solutions of the extended SIFs of some examples are obtained.     

同震断层三维裂纹扩展波动时域超奇异积分法

应用波动时域超奇异积分法将P波、S波和磁电热弹多场耦合作用下同震断层任意形状三维裂纹扩展问题转化为求解以广义位移间断率为未知函数的超奇异积分方程组问题;定义了广义应力强度因子,得到裂纹前沿广义奇异应力增量解析表达式;应用波动时域有限部积分概念及体积力法,为超奇异积分方程组建立了数值求解方法,编制了FORTRAN程序,以三维矩形裂纹扩展问题为例,通过典型算例,研究了广义应力强度因子随裂纹位置变化规律;分析了同震断层裂纹扩展中力、磁、电场辐射规律.      

Crack kinking in piezoelectric materials

A solution is presented for a class of two-dimensional electroelastic branched crack problems. Explicit Green's function for an interface crack subject to an edge dislocation is developed using the extended Stroh formulation allowing the branched crack problem to be expressed in terms of coupled singular integral equations. The integral equations are obtained by the method that models a kink as a continuous distribution of edge dislocations, and the dislocation density function is defined on the line of the branch crack only. Competition between crack extension along the interface and kinking into the substrate is investigated using the integral equations and the maximum energy release rate criterion. Numerical results are presented to show the effect of electric field on the path of crack extension. The work was supported by the Australian Research Council through a Queen Elizabeth II fellowship and by the Australian Academy of Science through the J.G. Russell Award.     

压电体中孔边Ⅲ型界面裂纹的动应力强度因子

采用Green函数法研究界面上含圆孔边界径向有限长度裂纹的两半无限压电材料对SH波的散射和裂纹尖端动应力强度因子问题.首先构造出具有半圆型凹陷半空间的位移Green函数和电场Green函数,然后采用裂纹"切割"方法构造孔边裂纹,并根据契合思想和界面上的连接条件建立起求解问题的定解积分方程.最后作为算例,给出了孔边界面裂纹尖端动应力强度因子的计算结果图并进行了讨论.     

Two-dimensional Green's functions in anisotropic multiferroic bimaterials with a viscous interface

We derive, by virtue of the unified Stroh formalism, the extremely concise and elegant solutions for two-dimensional and (quasi-static) time-dependent Green's functions in anisotropic magnetoelectroelastic multiferroic bimaterials with a viscous interface subjected to an extended line force and an extended line dislocation located in the upper half-plane. It is found for the first time that, in the multiferroic bimaterial Green's functions, there are 25 static image singularities and 50 moving image singularities in the form of the extended line force and extended line dislocation in the upper or lower half-plane. It is further observed that, as time evolves, the moving image singularities, which originate from the locations of the static image singularities, will move further away from the viscous interface with explicit time-dependent locations. Moreover, explicit expression of the time-dependent image force on the extended line dislocation due to its interaction with the viscous interface is derived, which is also valid for mathematically degenerate materials. Several special cases are discussed in detail for the image force expression to illustrate the influence of the viscous interface on the mobility of the extended line dislocation, and various interesting features are observed. These Green's functions can not only be directly applied to the study of dislocation mobility in the novel multiferroic bimaterials, they can also be utilized as kernel functions in a boundary integral formulation to investigate more complicated boundary value problems where multiferroic materials/composites are involved.     

Controlling parameter of the stress intensity factors for a planar interfacial crack in three-dimensional bimaterials

Numerical solutions of singular integral equations are discussed in the analysis of a planar rectangular interfacial crack in three-dimensional bimaterials subjected to tension. The problem is formulated as a system of singular integral equations on the basis of the body force method. In the numerical analysis, unknown body force densities are approximated by the products of the fundamental density functions and power series, where the fundamental density functions are chosen to express singular behavior along the crack front of the interface crack exactly. The calculation shows that the present method gives smooth variations of stress intensity factors along the crack front for various aspect ratios. The present method gives rapidly converging numerical results and highly satisfied boundary conditions throughout the crack boundary. The stress intensity factors are given with varying the material combination and aspect ratio of the crack. It is found that the stress intensity factors K_I and K_II are determined by the bimaterial constant ε alone, independent of elastic modulus ratio and Poisson’s ratio.     

Thermal response of a partially insulated interface crack in a graded coating-substrate structure under thermo-mechanical disturbance: Energy release and density

Consider the thermal fracture problem of a functionally graded coating-substrate structure of finite thickness with a partially insulated interface crack subjected to thermal-mechanical supply. A new model is proposed that the heat conduction through the crack region occurs and the temperature drop across the crack surfaces is the result of the thermal resistance. For the first time, real fundamental solutions are derived for the fracture analysis of functionally graded materials. The complicated mixed boundary problems of equations of heat conduction and elasticity are converted analytically into singular integral equations, which are solved numerically. The asymptotic expressions with higher order terms for the singular integral kernels are considered to improve the accuracy and efficiency of the numerical integration. Explicit expressions of various failure modes including stress intensity factors, energy release rate and strain energy density, are provided. Numerical results are presented to illustrate the effects of non-homogeneity parameters and the dimensionless thermal resistance on the temperature distribution along the crack surfaces and extended crack line, the thermal stress intensity factors and minimum strain energy density.     

Displacement discontinuity method for modeling axisymmetric cracks in an elastic half-space

This paper describes a displacement discontinuity method for modeling axisymmetric cracks in an elastic half-space or full space. The formulation is based on hypersingular integral equations that relate displacement jumps and tractions along the crack. The integral kernels, which represent stress influence functions for ring dislocation dipoles, are derived from available axisymmetric dislocation solutions. The crack is discretized into constant-strength displacement discontinuity elements, where each element represents a slice of a cone. The influence integrals are evaluated using a combination of numerical integration and a recursive procedure that allows for explicit integration of hyper- and Cauchy singularities. The accuracy of the solution at the crack tip is ensured by adding corrective stresses across the tip element. The method is validated by a comparison with analytical and numerical reference solutions.     

Electroelastic interaction between a piezoelectric screw dislocation and a confocal elliptic blunt crack with elliptical inhomogeneity

The electroelastic interaction between a piezoelectric screw dislocation and an elliptical inhomogeneity containing a confocal blunt crack under infinite longitudinal shear and in-plane electric field is investigated. Using the sectionally holomorphic function theory, Cauchy singular integral, singularity analysis of complex functions and theory of Rieman boundary problem, the explicit series solution of stress field is obtained when the screw dislocation is located in inhomogeneity. The intervention law of the interaction between blunt crack and screw dislocation in inhomogeneity is discussed. The analytical expressions of generalized stress and strain field of inhomogeneity are calculated, while the image force, field intensity factors of blunt crack are also presented. Moreover, a new matrix expression of the energy release rate and generalized strain energy density (SED) are deduced. With the size variation of blunt crack, the results can be reduced to the case of the interaction between a piezoelectric screw dislocation and a line crack in inhomogeneity. Numerical analysis are then conducted to reveal the effects of the dislocation location, the size of inhomogeneity and blunt crack and the applied load on the image force, energy release rate and strain energy density. The influence of dislocation on energy release rate and strain energy density is also revealed.     

Singular fields of a mode III interface crack in a power-law hardening bimaterial

A high order of asymptotic solution of the singular fields near the tip of a mode III interface crack for pure power-law hardening bimaterials is obtained by using the hodograph transformation. It is found that the zero order of the asymptotic solution corresponds to the assumption of a rigid substrate at the interface, and the first order of it is deduced in order to satisfy completely two continuity conditions of the stress and displacement across the interface in the asymptotic sense. The singularities of stress and strain of the zeroth order asymptotic solutions are −1/(n ₁+1) and −n/(n ₁+1) respectively. (n=n ₁,n ₂ is the hardening exponent of the bimaterials.) The applicability conditions of the asymptotic solutions are determined for both zeroth and first orders. It is proved that the Guo-Keer solution^[10] is limited in some conditions. The angular functions of the singular fields for this interface crack problem are first expressed by closed form. The project supported by National Natural Science Foundation of China     

Interaction of general plane P wave and cylindrical inclusion partially debonded from its viscoelastic matrix

The interaction of a general plane P wave and an elastic cylindrical inclusion of infinite length partially debonded from its surrounding viscoelastic matrix of infinite extension is investigated. The debonded region is modeled as an arc-shaped interface crack between inclusion and matrix with non-contacting faces. With wave functions expansion and singular integral equation technique, the interaction problem is reduced to a set of simultaneous singular integral equations of crack dislocation density function. By analysis of the fundamental solution of the singular integral equation, it is found that dynamic stress field at the crack tip is oscillatory singular, which is related to the frequency of incident wave. The singular integral equations are solved numerically, and the crack open displacement and dynamic stress intensity factor are evaluated for various incident angles and frequencies. The project supported by the National Natural Science Foundation of China (19872002) and Climbing Foundation of Northern Jiaotong University     

Magnetoelastic formulation of soft ferromagnetic elastic problems with collinear cracks: energy density fracture criterion

Magnetoelasticity is applied to solve collinear crack problems for soft ferromagnetic materials in two dimension. Complex functions are used for reducing the problem to the solution of a system of singular integral equations. The energy density factors are derived for determining how an off-axis magnetic field without mechanical load would influence the direction of crack initiation. The critical conditions are also determined for the case when both magnetic and mechanical load are present.     

Interface crack extension at any constant speed in orthotropic or transversely isotropic bimaterials––I. General exact solutions

A semi-infinite crack along the interface of two dissimilar half-spaces extends under in-plane loading. Each half-space belongs to a class of orthotropic or transversely isotropic elastic materials, the crack can extend at any constant speed, and all six possible relations between the four body wave speeds are considered. A steady dynamic situation is treated, and exact full displacement fields derived. A key step is a factorization that produces, despite anisotropy, simple solution forms and compact crack speed-dependent functions that exhibit the Rayleigh and Stoneley speeds as roots. These roots are calculated for various representative bimaterials.Closed-form crack opening displacement gradient and interface stress fields are also derived from a general set of coupled singular integral equations. The equation eigenvalues can, depending on crack speed, be complex/imaginary conjugates, purely real, or zero. This suggests possibilities observed in other studies: oscillations and square-root singular behavior at the crack edge, non-singular behavior, singular behavior not of square-root order, and the radiation of displacement gradient discontinuities at crack speeds beyond the purely sub-sonic range.These possibilities are explored further in terms of two important special cases in Part II of this study [Int. J. Solids Struct., 39, 1183–1198].     

Anti-plane stress intensity, energy release and energy density at crack tips in a functionally graded strip with linearly varying properties

The fracture problem of a crack in a functionally graded strip with its properties varying in a linear form along the strip thickness under an anti-plane load is considered. The embedded anti-plane crack is located in the middle of strip half way through the thickness. The third mode stress intensity factor is derived using two different methods. In the first method, by employing Fourier integral transforms, the governing equation is converted to a singular integral equation, which is subsequently solved numerically by the collocation method based on Chebyshev polynomials. Then, the problem is solved by means of finite element method in which quadrilateral 8-node singular elements around each crack tip are used. After inspecting the validity of the solution technique, effects of crack geometry and non-homogeneous material parameter on the stress intensity, energy release and energy density are studied and the results of analytical and FEM solutions are compared.     

Dynamic stress intensity factors of multiple cracks in a functionally graded orthotropic half-plane

In the present paper dynamic stress intensity factor and strain energy density factor of multiple cracks in the functionally graded orthotropic half-plane under time-harmonic loading are investigated. By utilizing the Fourier transformation technique the stress fields are obtained for a functionally graded orthotropic half-plane containing a Volterra screw dislocation. The variations of the material properties are assumed to be exponential forms which the equilibrium has an analytical solution. The dislocation solution is utilized to formulate integral equation for the half-plane weakened by multiple smooth cracks under anti-plane deformation. The integral equations are of Cauchy singular type at the location of dislocation which are solved numerically to obtain the dislocation density on the faces of the cracks. The dislocation densities are employed to determined stress intensity factor and strain energy density factors (SEDFs) for multiple smooth cracks under anti-plane deformation. Numerical examples are provided to show the effects of material properties and the crack configuration on the dynamic stress intensity factors and SEDFs of the functionally graded orthotropic half-plane with multiple curved cracks.