Thermoelectroelastic Green's function and its application for bimaterial of piezoelectric materials

Summary For a two-dimensional piezoelectric plate, the thermoelectroelastic Green's functions for bimaterials subjected to a temperature discontinuity are presented by way of Stroh formalism. The study shows that the thermoelectroelastic Green's functions for bimaterials are composed of a particular solution and a corrective solution. All the solutions have their singularities, located at the point applied by the dislocation, as well as some image singularities, located at both the lower and the upper half-plane. Using the proposed thermoelectroelastic Green's functions, the problem of a crack of arbitrary orientation near a bimaterial interface between dissimilar thermopiezoelectric material is analysed, and a system of singular integral equations for the unknown temperature discontinuity, defined on the crack faces, is obtained. The stress and electric displacement (SED) intensity factors and strain energy density factor can be, then, evaluated by a numerical solution at the singular integral equations. As a consequence, the direction of crack growth can be estimated by way of strain energy density theory. Numerical results for the fracture angle are obtained to illustrate the application of the proposed formulation. Received 10 November 1997; accepted for publication 3 February 1998     

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.     

热压电介质中的渗透型周期裂纹问题

利用Stroh公式,研究了含共线周期裂纹热的压电介质的广义二维问题。该工作有两个特征：一是裂纹被建模为具有渗透表面的缝隙,并假设为跨越上下表面时,电场的切向分量和电位移的法向分量是连续的;另一个特征是,机－电载荷和热载荷被假设作用在无限远处,而不是在裂纹表面。基于这两个假设,我们获得了有关场强因子,以及裂纹内电场的相当简洁的表达式。结果表明：①在裂纹内电场是线性变化的,②电位移的奇异性总是取决于应力的奇异性.③所有场的奇异性与所加的电载荷无关。     

Theoretical investigation of an elliptical crack in thermopiezoelectric material. Part II: Crack propagation

Propagation behavior of an elliptical crack in thermopiezoelectric material subjected to a uniform temperature is investigated in this paper. The three-dimensional strain energy density formulation is used to determine the direction of crack propagation and the shape of the initial fracture increment. It is found that the elliptical crack grows coplanarly under this particular load case but not normal to the crack front. The elliptical crack tends to become a circular one when thermal loading is applied.     

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.     

Crack initiation angles for systems of two interacting cracks in media with rectilinear preferred directions

The direction of initial crack growth of two interacting cracks in solids with rectilinear preferred directions is predicted by using the strain energy density theory. The minimum local strain energy density factors are presented; they correspond to the directions along which the material elements attain maximum dilatation and hence are assumed to coincide with the direction of crack initiation. Numerical results are obtained for different crack systems and presented in graphical forms.     

A new solution for thermopiezoelectric solid with an insulated elliptic hole

A new solution is obtained for thermal analysis of insulated elliptic hole embedded in an infinite thermopiezoelectric plate. In contrast to our previous results, the present formulation is based on the use of exact electric boundary conditions at the rim of the hole, thus avoiding the common assumption of electric impermeability. Using Lekhnitskii's formulation and conformal mapping, the elastic and electric fields can be expressed in a closed form in terms of complex potentials. The solutions for the crack problem are obtained by setting the minor axis of the ellipse approach to zero. As a consequence, the stress and electric displacement (SED) intensity factors and strain energy release rate can be derived analytically. One numerical example is considered to illustrate the application of the proposed formulation and compare with those obtained from impermeable model. The work was performed with the support of Australian Research Council Foundation.     

Triaxial compressive behavior of rock with mesoscopic heterogenous behavior: Strain energy density factor approach

The strain energy density factor approach is used in conjunction with a micromechanics model to investigate the condition and direction of shear failure for brittle rock subjected to triaxial compression. Moderate confinement in addition to localized deformation and damage are considered. Quantified are the effects of the various geometric and load parameters that involve the interaction of microcrack, friction and the confining pressure such that the path of the wing crack is taken into account. The influence of all microcracks with different orientations are introduced into the constitutive relation. The closed-form solution for the complete stress–strain relation of rock containing microcracks is obtained. It is shown that the complete stress–strain relationship includes linear, nonlinear hardening, rapid stress drop and strain softening effects. The theoretical results show that deviation of the direction of wing cracks from the line of the pre-existing crack decreases with increasing confinement pressure and friction coefficient. Theoretical predictions and experimental results show good agreement.     

Initiation and growth characterization of corner cracks near circular hole

The finite element analysis of crack problems often incorporates the asymptotic character of the local solution into the formulation. Embedment of stress or strain singularities can impose serious restrictions on the outcome and inconsistencies in predicting crack and/or growth. These restrictions are discussed in connection with the problem of two diametrically opposite corner cracks near a circular hole subjected to remote uniform tension. Enforced in the numerical treatment is the 1/r character of the strain energy density function local to the corner crack border where r is the radial distance measured from the crack front. The tendency for the corner crack to become a through crack is predicted by assuming that each point of the crack border extends by an amount proportional to the strain energy density factor. The path would correspond to the loci of minimum strain energy density function. Numerical results are displayed graphically and discussed in connection with crack initiation and non-self-similar crack growth.     

Inclined semi-elliptical crack for predicting crack growth direction based on apparent stress intensity factors

Linear elastic criterion of the inclined semi-elliptical crack growth direction is elaborated on the basis of the strain energy density theory. Stress and displacement fields are presented for higher order terms asymptotic expansion. Solutions for elastic stress intensity factors are accounting for the function describing of the crack tip fields near the free surface of plate. The mixed mode behavior of crack growth direction angle along the semi-elliptical crack front for different combination of biaxial loading, inclination crack angle and surface flaw geometry is determined.     

2D dynamic analysis of cracks and interface cracks in piezoelectric composites using the SBFEM

The dynamic stress and electric displacement intensity factors of impermeable cracks in homogeneous piezoelectric materials and interface cracks in piezoelectric bimaterials are evaluated by extending the scaled boundary finite element method (SBFEM). In this method, a piezoelectric plate is divided into polygons. Each polygon is treated as a scaled boundary finite element subdomain. Only the boundaries of the subdomains need to be discretized with line elements. The dynamic properties of a subdomain are represented by the high order stiffness and mass matrices obtained from a continued fraction solution, which is able to represent the high frequency response with only 3–4 terms per wavelength. The semi-analytical solutions model singular stress and electric displacement fields in the vicinity of crack tips accurately and efficiently. The dynamic stress and electric displacement intensity factors are evaluated directly from the scaled boundary finite element solutions. No asymptotic solution, local mesh refinement or other special treatments around a crack tip are required. Numerical examples are presented to verify the proposed technique with the analytical solutions and the results from the literature. The present results highlight the accuracy, simplicity and efficiency of the proposed technique.     

Crack problems of piezoelectric materials

This paper presents an analysis of crack problems in homogeneous piezoelectrics or on the interfaces between two dissimilar piezoelectric materials based on the continuity of normal electric displacement and electric potential across the crack faces. The explicit analytic solutions are obtained for a single crack in piezoelectrics or on the interfaces of piezoelectric bimaterials. A class of boundary problems involving many cracks is also solved. For homogeneous materials it is found that the normal electric displacementD ₂ induced by the crack is constant along the crack faces which depends only on the applied remote stress field. Within the crack slit, the electric fields induced by the crack are also constant and not affected by the applied electric field. For the bimaterials with realH, the normal electric displacementD ₂ is constant along the crack faces and electric fieldE ₂ has the singularity ahead of the crack tip and a jump across the interface. The project is supported by the National Natural Science Foundation of China(No. 19704100) and the Natural Science Foundation of Chinese Academy of Sciences(No. KJ951-1-201).     

压电材料中心裂纹问题

以电位移法向分量及电势连通过裂纹面为边界条件,对均匀电材料的裂纹问题及两种不同压材料界面裂纹问题进行了系统分析,得到了含中心裂纹无限大体封闭形的全场解。证实了裂纹引起的非均匀扰动场只信赖于外加场而外加电场无关。     

Crack bifurcation predicted for dynamic anti-plane collinear cracks in piezoelectric materials using a non-local theory

A non-local theory of elasticity is applied to obtain the dynamic interaction between two collinear cracks in the piezoelectric materials plane under anti-plane shear waves for the permeable crack surface boundary conditions. Unlike the classical elasticity solution, a lattice parameter enters into the problem that make the stresses and the electric displacements finite at the crack tip. A one-dimensional non-local kernel is used instead of a two-dimensional one for the anti-plane dynamic problem to obtain the stress and electric displacement near the crack tips. By means of the Fourier transform, the problem can be solved with the help of two pairs of triple integral equations in which the unknown variable is the jump of the displacement across the crack surface. The solutions are obtained by means of the Schmidt method. Crack bifurcation is predicted using the strain energy density criterion. Minimum values of the strain energy density functions are assumed to coincide with the possible locations of fracture initiation. Bifurcation angles of ±5° and ±175° are found. The result of possible crack bifurcation was not expected before hand.     

An exact and explicit treatment of an elliptic hole problem in thermopiezoelectric media

This paper presents an exact solution for the problem of an elliptic hole or a crack in a thermopiezoelectric solid. First, based on the extended version of Eshelby–Stroh's formulation, the generalized 2D problems of an elliptical hole in a thermopiezoelectric medium subject to uniform heat flow and mechanical–electrical loads at infinity are studied according to exact boundary conditions at the rim of the hole. The complex potentials in the medium and the electric field inside the hole are obtained in closed form, respectively. Then, when the hole degenerates into a crack, the explicit solutions for the field intensity factors near the crack tip and the electric field inside the crack are presented. It is shown that the singularities of all the field are dependent on the material constants, the applied heat load and mechanical loads at infinity, but not on the applied electric loads. It is also found that the electric field inside the crack is linearly variable, which is different from the result based on the impermeable crack model.     

An embedded crack in a functionally graded orthotropic coating bonded to a homogeneous substrate under a frictional Hertzian contact

In this paper, we consider the elasto-static problem of an embedded crack in a graded orthotropic coating bonded to a homogeneous substrate subject to statically applied normal and tangential surface loading. The crack direction is parallel to the free surface. The coating is graded in the thickness direction and is orthogonal to the crack direction. This coating is modelled as a non-homogeneous medium with an orthotropic stress–strain law. The equivalent crack surface stresses are first obtained and substituted in the plane elasticity equations. Using integral transforms, the governing equations are converted into singular integral equations which are solved numerically to yield the displacement field as well as the crack-tip stress intensity factors. This study presents a complete theoretical formulation for the problem in the static case. A numerical predictive capability for solving the singular integral equations and computing the crack-tip stress intensity factors is proposed. Since the loading is compressive, a previously developed crack-closure algorithm is applied to avoid interpenetration of the crack faces. The main objective of the paper is to investigate the effects of the material orthotropy and non-homogeneity of the graded coating on the crack-tip stress intensity factors, with and without using the crack-closure algorithm, for the purpose of gaining better understanding on the behavior and design of graded coatings.     

Upper and lower bounds for constitutive relation of crack-weakened rock masses under dynamic compressive loads

A micro-mechanics-based model is proposed to investigate the rate-dependent constitutive relation for crack-weakened rock masses subjected to dynamic compressive loads. The present micro-mechanical model reveals that the nucleation, growth and coalescence of sliding cracks dominate the failure and macroscopic properties of crack-weakened rock masses subjected to dynamic compressive loads. The interactions among multiple parallel sliding cracks in crack-weakened rock masses subjected to dynamic compressive loads are examined asymptotically in an explicit and quantitative manner in order to reveal fully their so-called shielding and magnification effects on the stress–strain relation. Based on the micro-mechanical framework and the asymptotic analysis, analytical upper and lower bounds are proposed for the rate-relation for rock masses containing multiple rows of echelon cracks subjected to dynamic compressive loads. The factors that affect the rate-dependent properties of crack-weakened rock masses have been analyzed. The strain energy density factor approach, which is related to crack growth velocity and dynamic fracture toughness of rock material, is employed in the analysis. The rate-dependent constitutive relation of crack-weakened rock masses is derived from micro-mechanical framework and the asymptotic analysis. The closed-form explicit expression for the rate-dependent constitutive relation of rock masses containing echelon cracks subjected to dynamic compressive loads is obtained. Finally, the present model is used to analyze the complete stress–strain relation and strength for jointed rock masses at shiplock slope of the Three Gorges Dam.     

Analysis method of planar interface cracks of arbitrary shape in three-dimensional transversely isotropic magnetoelectroelastic bimaterials

Using the fundamental solutions for three-dimensional transversely isotropic magnetoelectroelastic bimaterials, the extended displacements at any point for an internal crack parallel to the interface in a magnetoelectroelastic bimaterial are expressed in terms of the extended displacement discontinuities across the crack surfaces. The hyper-singular boundary integral–differential equations of the extended displacement discontinuities are obtained for planar interface cracks of arbitrary shape under impermeable and permeable boundary conditions in three-dimensional transversely isotropic magnetoelectroelastic bimaterials. An analysis method is proposed based on the analogy between the obtained boundary integral–differential equations and those for interface cracks in purely elastic media. The singular indexes and the singular behaviors of near crack-tip fields are studied. Three new extended stress intensity factors at crack tip related to the extended stresses are defined for interface cracks in three-dimensional transversely isotropic magnetoelectroelastic bimaterials. A penny-shaped interface crack in magnetoelectroelastic bimaterials is studied by using the proposed method.The results show that the extended stresses near the border of an impermeable interface crack possess the well-known oscillating singularity r^?1/2±iε or the non-oscillating singularity r^?1/2±κ. Three-dimensional transversely isotropic magnetoelectroelastic bimaterials are categorized into two groups, i.e., ε-group with non-zero value of ε and κ-group with non-zero value of κ. The two indexes ε and κ do not coexist for one bimaterial. However, the extended stresses near the border of a permeable interface crack have only oscillating singularity and depend only on the mechanical loadings.     

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.     

Dynamic curving cracks in functionally graded materials under thermo-mechanical loading

Mixed-mode dynamic crack growth along an arbitrarily smoothly varying path in functionally graded materials (FGMs) under thermo-mechanical loading is studied. The property gradation in FGMs is considered by varying shear-modulus, mass density, thermal conductivity and coefficient of thermal expansion exponentially along the gradation direction. Asymptotic analysis in conjunction with displacement potentials is used to develop the stress fields around propagating cracks in FGMs. Asymptotic temperature fields are developed first for the exponential variation of thermal conductivity and later these temperature fields are used to derive thermo-mechanical stress fields for a curving crack in FGMs. Using these thermo-mechanical stress fields, various components of the stresses are developed and the effect of curvature parameters, temperature and gradation on these stresses are discussed. Finally, using the minimum strain energy density criterion, the effect of curvature parameters, crack-tip speeds, non-homogeneity values and temperature gradients on crack growth directions are determined and discussed.