Weight functions for interface cracks in dissimilar anisotropic materials

Bueckner‘s work conjugate integral customarily adopted for linear elastic materials is established for an interface crack in dissimilar anisotropic materials. The difficulties in separating Stroh‘s six complex arguments involved in the integral for the dissimilar materials are overcome and then the explicit function representations of the integral are given and studied in detail. It is found that the pseudo-orthogonal properties of the eigenfunction expansion form (EEF) for a crack presented previously in isotropic elastic cases, in isotopic bimaterial cases, and in orthotropic cases are also valid in the present dissimilar arbitrary anisotropic cases. The relation between Bueckner‘s work conjugate integral and the J-integral in these cases is obtained by introducing a complementary stressdisplacement state. Finally, some useful path-independent integrals and weight functions are proposed for calculating the crack tip parameters such as the stress intensity factors.     

A New Method for Establishing Pseudo Orthogonal Properties of Eigenfunction Expansion Form in Fracture Mechanics

A new and simple method is developed to establish the pseudo orthogonal properties (POP) of the eigenfunction expansion form (EEF) of crack-tip stress complex potential functions for cracked anisotropic and piezoelectric materials, respectively. Di?erent from previous research, the complex argument separation technique is not required so that cumbersome manipulations are avoided. Moreover, it is shown, di?erent from the previous research too, that the orthogonal …     

压电材料中的Bueckner功共轭积分及和J积分M积分的关系

研究横观各向同性压电材料中裂纹问题,提出了Bueckner功共轭积分在这类材料中的表达式：并通过引出两类辅助的应力-位移-电位移-电势场,证明功共轭积分和这类材料中的J积分和M积分仍然存在简单的两倍关系由此,各类在脆性材料断裂问题中已广泛应用的权函数方法可顺理成章地推广到压电材料的研究中来．这对独立地确定电位移强度因子和经典的I、II型应力强度因子提供了有力的数学上的工具．进而通过计算机械应变能释放率对压电材料中裂纹的稳定做出判断．     

Stress intensity factor analysis of a three-dimensional interfacial corner between anisotropic bimaterials under thermal stress

A numerical method using a path-independent H-integral based on the conservation integral was developed to analyze the singular stress field of a three-dimensional interfacial corner between anisotropic bimaterials under thermal stress. In the present method, the shape of the corner front is smooth. According to the theory of linear elasticity, asymptotic stress near the tip of a sharp interfacial corner is generally singular as a result of a mismatch of the materials’ elastic constants. The eigenvalues and the eigenfunctions are obtained using the Williams eigenfunction method, which depends on the anisotropic materials’ properties and the geometry of an interfacial corner. The order of the singularity related to the eigenvalue is real, complex or power-logarithmic. The amplitudes of the singular stress terms can be calculated using the H-integral. The stress and displacement around an interfacial corner for the H-integral are obtained using finite element analysis. In this study, a proposed definition of the stress intensity factors of an interfacial corner, which includes those of an interfacial crack and a homogeneous crack, is used to evaluate the singular stress fields. Asymptotic solutions of stress and displacement around an interfacial corner front are uniquely obtained using these stress intensity factors. To prove the accuracy of the present method, several different kinds of examples are shown such as interfacial corners or cracks in three-dimensional structures.     

The explicit formulations for theL-integral

In this paper, a simple but inberent relation between theL-integral and the Bueckner work conjugate integral is proved for crack problems in isotropic, anisotropic, and dissimilar materials, respectively. It is found that, in the above-mentioned three cases, theL-integral, from the mathematical point of view as well as in principle, arises from Betti's reciprocal theorem. This means that the Bueckner work conjugate integral is a more general path-independent integral than the others since any other path-independent integrals could be derived by using the Bueckner integral while choosing a different subsidiary stress-displacement field.     

Anti-plane shear crack normal to and terminating at the interface of two bonded piezoelectric ceramics

The electroelastic analysis of two bonded dissimilar piezoelectric ceramics with a crack perpendicular to and terminating at the interface is made. By using Fourier integral transform, the associated boundary value problem is reduced to a singular integral equation with generalized Cauchy kernel, the solution of which is given in closed form. Results are presented for a permeable crack under anti-plane shear loading and in-plane electric loading. Obtained results indicate that the electroelastic field near the crack tip in the homogeneous piezoelectric ceramic is dominated by a traditional inverse square-root singularity, while the electroelastic field near the crack tip at the interface exhibits the singularity of power law r^−α, r being distance from the interface crack tip and α depending on the material constants of a bi-piezoceramic. In particular, electric field has no singularity at the crack tip in a homogeneous solid, whereas it is singular around the interface crack tip. Numerical results are given graphically to show the effects of the material properties on the singularity order and field intensity factors.     

Analysis of a crack in a half-plane piezoelectric solid with traction-induction free boundary

In this paper, the problem of a crack embedded in a half-plane piezoelectric solid with traction-induction free boundary is analyzed. A system of singular integral equations is formulated for the materials with general anisotropic piezoelectric properties and for the crack with arbitrary orientation. The kernel functions developed are in complex form for general anisotropic piezoelectric materials and are then specialized to the case of transversely isotropic piezoelectric materials which are in real form. The obtained coupled mechanical and electric real kernel functions may be reduced to those kernel functions for purely elastic problems when the electric effects disappear. The system of singular integral equations is solved numerically and the coupling effects of the mechanical and electric phenomena are presented by the generalized stress intensity factors for transversely isotropic piezoelectric materials.     

Angle cracks in two bonded functionally graded piezoelectric material under anti-plane shear

Consider two bonded functionally graded piezoelectric material (FGPM) with finite height. Each material contains an arbitrary oriented crack. The material properties are assumed in exponential forms in the direction normal to the interface. The crack surface condition is assumed to be electrically impermeable or permeable. Using the Fourier transform technique, the problem can be reduced to a system of singular integral equations, which are then solved numerically by applying the Gauss-Chebyshev integration formula to obtain the stress intensity factors at the crack tips. Numerical calculations are carried out to obtain the energy density factor S and the energy release rate G. In impermeable case, the energy release rate has been shown to be negative as the electric loads are applied. The positive definite characteristic of the energy density factor makes it possible for predicting the fracture behavior of the cracked structure. The influences of the non-homogeneous parameters and crack orientation on the energy density factors at the crack tips are discussed in detail. The results show that the energy density factor at the crack tip will be increased when the crack tip is located within the softer material.     

Dynamic crack tip fields and dynamic crack propagation characteristics of anisotropic material

Based on mechanics of anisotropic material, the dynamic crack propagation problem of I/II mixed mode crack in an infinite anisotropic body is investigated. Expressions of dynamic stress intensity factors for modes I and II crack are obtained. Components of dynamic stress and dynamic displacements around the crack tip are derived. The strain energy density theory is used to predict the dynamic crack extension angle. The critical strain energy density is determined by the strength parameters of anisotropic materials. The obtained dynamic crack tip fields are unified and applicable to the analysis of the crack tip fields of anisotropic material, orthotropic material and isotropic material under dynamic or static load. The obtained results show Crack propagation characteristics are represented by the mechanical properties of anisotropic material, i.e., crack propagation velocity M and fiber direction α. In particular, the fiber direction α and the crack propagation velocity M give greater influence on the variations of the stress fields and displacement fields. Fracture angle is found to depend not only on the crack propagation but also on the anisotropic character of the material.     

功能梯度压电板条中电绝缘型运动裂纹的电弹性场

基于三维弹性理论和压电理论，对材料系数按指数函数规律分布的功能梯度压电板条中的反平面运动裂纹问题进行了求解。利用Fourier积分变换方法将电绝缘型运动裂纹问题化为对偶积分方程，并进一步归结为易于求解的第二类Fredholm积分方程。通过渐近分析，获得了裂纹尖端应力、应变、电位移和电场的解析解，给出了裂纹尖端场各个变量的角分布函数，并求得了裂纹尖端场的强度因子，分析了压电材料物性梯度参数、几何尺寸及裂纹运动速度对它们的影响。结果表明，对于电绝缘型裂纹，功能梯度压电板条中运动裂纹尖端附近的各个场变量都具有-1／2阶的奇异性；当裂纹运动速度增大时，裂纹扩展的方向会偏离裂纹面。     

Mode III fracture problem of an arbitrarily oriented crack in a FGPM strip bonded to a FGPM half plane

This paper studies the mode III electro-elastic field of a cracked functionally graded piezoelectric strip bonded to a functionally graded piezoelectric half plane. The crack is oriented in arbitrary direction. The material properties along x-axis vary in exponential form. By using the Fourier transform, the problem can be formulated into a system of singular integral equations and solved by applying the Gauss–Chebyshev integration formula. The effects come from the edge, crack orientation and the nonhomogeneous material parameters on intensity factors are discussed graphically.     

Dynamic stress intensity factor K III and dynamic crack propagation characteristics of anisotropic materials

Based on the mechanics of anisotropic materials, the dynamic propagation problem of a mode Ⅲ crack in an infinite anisotropic body is investigated. Stress, strain and displacement around the crack tip are expressed as an analytical complex function, which can be represented in power series. Constant coefficients of series are determined by boundary conditions. Expressions of dynamic stress intensity factors for a mode Ⅲ crack are obtained. Components of dynamic stress, dynamic strain and dynamic displacement around the crack tip are derived. Crack propagation characteristics are represented by the mechanical properties of the anisotropic materials, i.e., crack propagation velocity M and the parameter ~. The faster the crack velocity is, the greater the maximums of stress components and dynamic displacement components around the crack tip are. In particular, the parameter α affects stress and dynamic displacement around the crack tip.     

Fracture Analysis for a Frictional Fastener Hole with Edge Cracks in an Anisotropic Plate∗

Abstract

Stress intensity factors are evaluated for a singly or doubly cracked fastener hole with frictional traction in an anisotropic plate, using a special kernel boundary integral equation (BIE) approach. The integration kernel (Green's function) used in this BIE approach has already taken the presence of the crack (or cracks) into account, thus.avoiding the need for element discretization to model the stress singularity at the crack tip. The Green's function employed is that of an infinite anisotropic plate containing an elliptical hole or crack, and subjected to an arbitrarily positioned point force. Several types of normal and shear traction conditions at the pinhole interface are considered. Numerical results are obtained for various geometrical and loading conditions and are compared with known solutions, where available, for their isotropic counterparts.     

Electroelastic field for an impermeable anti-plane shear crack in a piezoelectric ceramics plate

Electroelastic behavior of a cracked piezoelectric ceramics plate subjected to four cases of combined mechanical-electrical loads is analyzed. The integral transform method is applied to convert the problem involving an impermeable anti-plane crack to dual integral equations. Solving the resulting equations, the explicit analytic expressions for electroelastic field along the crack line and the intensity factors of relevant quantities near the crack tip and the mechanical strain energy release rate are obtained. The known results for an infinite piezoelectric ceramics plane containing an impermeable anti-plane crack are recovered from the present results only if the thickness of the plate h → ∞. Biography: LI Xian-fang (1964-)     

Interaction integrals for fracture analysis of functionally graded piezoelectric materials

This paper presents domain form of the interaction integrals based on three independent formulations for computation of the stress intensity factors and electric displacement intensity factor for cracks in functionally graded piezoelectric materials. Conservation integrals of J-type are derived based on the governing equations for piezoelectric media and the crack tip asymptotic fields of homogeneous piezoelectric medium as auxiliary fields. Each of the formulation differs in the way auxiliary fields are imposed in the evaluation of interaction integral and each of them results in a consistent form of the interaction integral in the sense that extra terms naturally appears in their derivation to compensate for the difference in the chosen crack tip asymptotic fields of homogeneous and functionally graded piezoelectric medium. The additional terms play an important role of ensuring domain independence of the presented interaction integrals. Comparison of the numerically evaluated intensity factors through the three consistent formulations with those obtained using displacement extrapolation method is presented by means of two examples.     

Singular electro-mechanical fields near the apex of a piezoelectric bonded wedge under antiplane shear

This paper presents the explicit forms of singular electro-mechanical field in a piezoelectric bonded wedge subjected to antiplane shear loads. Based on the complex potential function associated with eigenfunction expansion method, the eigenvalue equations are derived analytically. Contrary to the anisotropic elastic bonded wedge, the results of this problem show that the singularity orders are single-root and may be complex. The stress intensity factors of electrical and mechanical fields are dependent. However, when the wedge angles are equal (α=β), the orders become real and double-root. The real stress intensity factors of electrical and mechanical fields are then independent. The angular functions have been validated when they are compared with the results of several degenerated cases in open literatures.     

The role of constraint in the fields of a crack normal to the interface between elastically and plastically mismatched solids

Asymptotic solutions are presented for a stationary crack normal to the boundary between two elastically mismatched solids such that the crack tip is located at the interface. The second-order term in the elastic asymptotic expansion was determined as a function of elastic mismatch for a thin cracked film on a substrate and for a thin cracked lamina between two substrates. Elastic-plastic analysis was performed using both modified boundary layer formulations and full field analyses. Analytic and numerical solutions in small strain yielding identify elastic mismatch and the T-stress as the determinants of crack tip constraint. The effect of constraint on the competition between interface failure and penetration is discussed.     

Numerical Modelling of Steady-State Flow in 2D Cracked Anisotropic Porous Media by Singular Integral Equations Method

The equations governing plane steady-state flow in heterogeneous porous media containing curved-line intersecting cracks (Pouya and Ghabezloo in Transp Porous Media 84:511?C532, 2010) and the potential solution obtained for these equations are considered here. The theoretical results are first completed for the mass balance at crack intersections points. Then, a numerical procedure based on a singular integral equations method is described concretely to derive this solution for cracked materials. Closed-form expressions of elementary integrals for special choice of collocation points lead to a very quick and easy numerical method. It is shown that this method can be applied efficiently to the study of the steady-state flow in cracked materials with anisotropic matrix permeability and a dense distribution of curved-line intersecting cracks. Some applications of this method to the permeability of cracked materials are given.     

Numerical solutions of two-dimensional anisotropic crack problems

A complete set of series form solutions of stress and displacement functions, including all higher order terms, around the crack tip for anisotropic crack problems have been newly derived by eigenfunction expansion approach. The analytical solutions of displacement functions were classified into four cases with respect to different types of complex parameters and different corresponding physical meanings. By employing these displacement functions as global interpolation functions, fractal two-level finite element method (F2LFEM) was applied to evaluate the stress intensity factors (SIFs) for various kinds of anisotropic crack problems. In the method of F2LFEM, the infinite number of nodal displacements was transformed to a small set of generalized coordinates by fractal transformation technique. New element matrices need not be generated and the singular numerical integration was avoided completely. Numerical examples of the four cases were studied and high accurate results of SIFs were obtained.     

Stress analysis for an infinite strip weakned by periodic cracks

Stress analysis for an infinite stripcracks were assumed in a horizontal position,weakened by periodic cracks is studied. The and the strip was applied by tension “p“ in y-direction. The boundary value problem can be reduced into a complex mixed one. It is found that the EEVM ( eigenfunction expansion variational method) is efficient to solve the problem. The stress intensity factor at the crack tip and the T-stress were evaluated. From the deformation response under tension the cracked strip can be equivalent to an orthotropic strip without cracks. The elastic properties in the equivalent orthotropic strip were also investigated. Finally, numerical examples and results were given.