Collinear periodic cracks in an anisotropic medium

The problem of collinear periodic cracks in an anisotropic medium is examined in this paper. By means of Stroh formalism and the conformal mapping method, we obtain general periodic solutions for collinear cracks. The corresponding stress intensity factors, crack opening displacements and strain energy release rate are found.     

A Crack Emanating from a Wedge In Dissimilar Anisotropic Materials Under Antiplane Shear

A crack in a composite wedge consisting of two dissimilar anisotropic materials under concentrated antiplane loads is analyzed. The problem of a crack in an isotropic composite wedge is solved first by using the Mellin transform and the Wiener-Hopf method. Using a linear transformation of the original composite wedge into an isotropic composite wedge consisting of dissimilar isotropic materials, the antiplane displacement and stresses for the anisotropic composite wedge are obtained from the solution for the transformed wedge. The stress intensity factor for the crack in the original anisotropic composite wedge is obtained from the solution of the crack in the transformed wedge. Special attention is given to the asymptotic problem of a wedge crack in an anisotropic bimaterial. Numerical computations are carried out to obtain the energy release rate for various apex angles and anisotropic parameters as a function of crack angle.     

压电复合材料的共线周期性裂纹平面问题的电弹性场分析

利用复变函数知识、半逆解法及待定系数法, 研究了压电复合材料的共线周期性裂纹问题, 给出了在电不可渗透边界条件下的应力、电位移、应力强度因子、电位移强度因子和机械应变能释放率的解析解。当裂纹间距趋于无穷时, 共线周期性裂纹退化为一条单裂纹, 得到了压电复合材料一条单裂纹的结果。通过数值算例讨论了共线周期性裂纹的裂纹长度、裂纹间距和机电载荷对机械应变能释放率的影响规律。结果表明, 机械应变能释放率随着共线周期性裂纹的裂纹长度、共线周期性裂纹的裂纹间距、机械载荷和正电场的增大而增大, 随着负电场的增大而减小。     

Transient response of three layered composites with two interface cracks due to a line load

Summary The dynamic response of three layered composites, with two interface cracks, subjected to an antiplane line load is analyzed. The Green's function for the uncreacked medium is used along with the representation theorem to derive the scattered field. Satisfaction of the traction free condition at the crack surfaces gives rise to a system of singular integral equations for determining crack opening displacements (COD), which are then solved by expanding the unknown COD in a complete set of Chebyshev polynomials. Numerical results for two sample problems are presented, for both isotropic and anisotropic materials. Results show how significantly material anisotropy and interaction between two cracks can affect the COD in a three layered plate.With 9 Figures     

Stress intensity factors and energy release rates of delaminations in composite laminates

Due to the oscillatory characteristics of stresses near interface crack tips, the stress intensity factor K_i, i = I, II, III, should be modified and the energy release rate G_i, i = 1, 2, 3, of each fracture mode calculated by the virtual crack closure method may not exist. Based upon a near-tip solution for interface cracks between dissimilar anisotropic media, a proper definition for the stress intensity factors and energy release rates for general anisotropic bimaterial interface cracks is provided in this paper, which is applicable for the delaminated composites. Moreover, this definition can be reduced to the classical definition for a crack tip in homogeneous media when the two materials become the same. A simple quadratic relation between K_i and _Gi is derived, which is further reduced explicitly for orthotropic bimaterials. The influence of fiber orientation and the coupling among opening, shearing and tearing mode fracture are studied numerically. The results show that the classical stress intensity factors and energy release rates are still the dominant stress intensity and energy release rate of the mixed mode condition induced by the interface.     

非均匀复合材料中反平面裂纹的动态断裂力学研究

对于非均匀复合材料中多个裂纹的动态断裂力学问题, 提出了一种分析方法, 假设复合材料为正交各向异性并含有多个垂直于厚度方向的裂纹, 材料参数沿厚度方向为变化的, 沿该方向将复合材料划分为许多单层, 假设单层材料参数为常数, 应用柔度矩阵/刚度矩阵方法及Fourier变换法, 在L aplace 域内推导出了控制问题的奇异积分方程组, 并用虚位移原理求解, 给出了应力强度因子及能量释放率的表达式, 然后利用Laplace 数值反演, 得出了裂纹尖端的动态应力强度因子和能量释放率。作为算例, 研究了带有两个裂纹的功能梯度结构, 分析了材料参数的优化对降低应力强度因子的意义。      

Complex-variable and integral-transform methods for elastodynamic solutions of cracked orthotropic strips

Problems concerning cracked bodies in the form of an infinite strip subjected to antiplane stresses or displacements were solved within the context of the linear and anisotropic theory of elasticity. Two basic approaches were utilized, namely a technique based on the complex-variable theory and another based on the exponential Fourier transform and the Wiener-Hopf method. Two geometric configurations were also considered. In the first case the long strip was internally weakened by a constant-length crack, whereas in the second case by a semi-infinite crack. The problems were analyzed under the assumption of a steady-state elastodynamic crack motion.     

Weight function,stress intensity factor and crack opening displacement solutions to periodic collinear edge hole cracks

Periodic collinear edge hole cracks and arbitrary small cracks emanating from collinear holes, which are two typical multiple site damages occurred in the aircraft structures, are studied by using the weigh function method. An explicit closed form weight function for periodic edge hole cracks in an infinite sheet is obtained and further used to calculate the stress intensity factor and crack opening displacement for various loading cases. Compared to finite element method, the present weight function is accurate and highly efficient. The interactions of the holes and cracks on the stress intensity factor and crack opening displacement are quantitatively determined by using the present weight function. An approximate weight function method is also proposed for arbitrary small cracks emanating from multiple collinear holes. This method is very useful for calculating the stress intensity factor for arbitrary small cracks.     

Weight functions and strip yield solution for two equal-length collinear cracks in an infinite sheet

The problem of two equal-length collinear cracks in an infinite sheet is treated using the weight function method. Exact weight functions for the inner and outer crack tips are derived based on the crack opening displacement solution for a reference load case. These weight functions are used to calculate stress intensity factors for different load cases, plastic zone sizes and crack tip opening displacements of the strip yield model. The approach is validated by the perfect agreement between the present strip yield model solutions and Collins and Cartwright’s analytical results based on the direct complex stress function formulation.     

Orthotropic thermoelasticity problem of symmetrical heat flow disturbed by three coplanar cracks

This article presents a study on the plane thermoelasticity problem of an infinite orthotropic plate split by three coplanar cracks under the action of symmetrical heat flow. Using the technique of Fourier transforms, the related four-part mixed boundary value problems are reduced to two kinds of quadruple integral equations with cosine and sine kernels which are solved by use of finite Hilbert transformation. Closed form solutions to the temperature, thermal displacements and thermal stresses on the crack surfaces, and especially, the thermal stress intensity factors at crack tips are obtained for the case of uniform heat flow. The known solutions to the orthotropic thermoelasticity problem of uniform heat flow disturbed by a pair of coplanar cracks or a central planar crack can be deduced from the above results in a straightforward manner, including the solution of thermal stress intensity factors for the corresponding thermoelasticity problem with two collinear cracks which is another form of the solution, equivalent to that of series expressions obtained by the authors in a previous paper, but much simpler. It is found that extremely large magnitudes of stress singularity may occur as the distance between two adjacent cracks approaches zero.     

Analyses of infinite pairs of surface cracks in elastic plates

Stress intensity factors and crack opening displacements are presented for infinite pairs of surface cracks in plates subjected to remote tension by using the three dimensional weight function method developed in [7,8]. A wide range of configuration parameters is considered. The results compare very well with double edge cracks as crack aspect ratio tends to zero; with collinear cracks as it tends to infinity; with a pair of surface cracks in a wide plate when the ratio of crack length to plate width is small; and with a single surface crack in large plates when both the ratios of crack length to plate width and crack depth to plate thickness are small. Also illustrated is the significant difference between a single surface crack and the surface cracks in pairs when the ratio of crack depth to plate thickness is large.     

Collinear periodic crack in anisotropic bimaterials

The problems of collinear periodic cracks in anisotropic bimaterials are examined in this paper. By means of Stroh formalism, conformal mapping and analytic continuation arguments, explicit full domain solutions for the periodic cracks are presented. Following the procedure used by Wu (1990), we also obtain the corresponding stress intensity factors. The solutions are valid not only to plane problems but also to antiplane problems and those problems whose inplane and antiplane deformations couple each other. This revised version was published online in August 2006 with corrections to the Cover Date.     

A wedge crack in an anisotropic material under antiplane shear

A crack emanating from the apex of an infinite wedge in an anisotropic material under antiplane shear is investigated. An isotropic wedge crack subjected to concentrated forces is first solved by using the conformal mapping technique. The solution of an anisotropic wedge crack is obtained from that of the transformed isotropic wedge crack based on a linear transformation method. Expressions for the stress intensity factor for the anisotropic wedge crack with both concentrated and distributed loads are derived. The stress intensity factors are numerically calculated for generally orthotropic wedge cracks with various crack and wedge angles as well as anisotropic parameters.     

Three Collinear Antiplane Interfacial Cracks in Dissimilar Piezoelectric Materials

Three collinear impermeable interfacial cracks in bonded dissimilar transversely isotropic piezoelectric materials under electromechanical loadings is analyzed. A single antiplane mechanical and inplane electrical loads are applied at a point on crack surface. The problem is formulated by the complex function method, and reduced to the vector Hilbert problem. Solving this problem, a closed form solution for the stress intensity and electric displacement intensity factor is obtained. This solution can be used as a Green??s function for different loading conditions.     

Crack-tip fields in anisotropic shells

Asymptotic crack-tip fields including the effect of transverse shear deformation in an anisotropic shell are presented. The material anisotropy is defined here as a monoclinic material with a plane symmetry at x ₃=0. In general, the shell geometry near the local crack tip region can be considered as a shallow shell. Based on Reissner shallow shell theory, an asymptotic analysis is conducted in this local area. It can be verified that, up to the second order of the crack tip fields in anisotropic shells, the governing equations for bending, transverse shear and membrane deformation are mutually uncoupled. The forms of the solution for the first two terms are identical to those given by respectively the plane stress deformation and the antiplane deformation of anisotropic elasticity. Thus Stroh formalism can be used to characterize the crack tip fields in shells up to the second term and the energy release rate can be expressed in a very compact form in terms of stress intensity factors and Barnett–Lothe tensor L. The first two order terms of the crack-tip stress and displacement fields are derived. Several methods are proposed to determine the stress intensity factors and `T-stresses'. Three numerical examples of two circular cylindrical panels and a circular cylinder under symmetrical loading have demonstrated the validity of the approach.     

The elastic plate containing two collinear transverse edge-cracks

Summary The paper solves the problem, in a closed form, of the stress distribution in an infinite plate containing two collinear and transverse edge-cracks, subjected to an axial loading at infinity. The distribution of stresses and displacements all over the cracked plate was effectively solved by applying the method of isostatics, as it has been developed by the author in 1959. The solution of the problem was formulated as a Boussinesq-type problem, where the ligament between the symmetric edge cracks corresponded to the constant displacement zone of the rigid punch. The stress distribution in the cracked plate was expressed by an analytic function of the complex variable, which represented the field of isostatics (i.e. the principal stress trajectories). The advantage of the method over all other existing solutions is that it gives the components of stresses all over the field with the same degree of accuracy, based on a closed-form type of solution. These stresses are given in graphical form along characteristic sections of the plate. Therefore, the method is proved effective for solving completely the problem of externally cracked plates with the same accuracy, as the closed-form solution of an internal crack. The advantages of this fact may be appreciated when one has to find out crack opening displacements all over the lengths of the cracks and, in general, needs data outside the close neighbourhood of the crack tips.With 11 Figures     

Screw Dislocation Interacting with an Interfacial Edge Crack between Two Bonded Dissimilar Piezoelectric Wedges

This letter is concerned with an interfacial edge crack in a piezoelectric bimaterial wedge interacting with a screw dislocation under antiplane mechanical and in-plane electric loading. In addition to a discontinuous electric potential across the slip plane, the dislocation is subjected to a line-force and a line-charge at the tip. The out-of-plane displacement and electric potentials are obtained in closed-form based on conformal mapping technique and the solution for the screw dislocation in an infinite piezoelectric bimaterial with a semi-infinite interfacial crack. The intensity factors (IFs) and energy release rate (EER) are obtained explicitly. These solutions can be used as a base for constructing solutions for arbitrary coupled antiplane mechanical and in-plane electrical loadings.     

Fracture analysis of magnetoelectroelastic solids by using path independent integrals

A solution scheme based on the fundamental solution for a generalized edge dislocation in an infinite magnetoelectroelastic solid is presented to analyze problems involving single, multiple and slowly growing impermeable cracks. The fundamental solution for a generalized dislocation is obtained by extending the complex potential function formulation used for anisotropic elasticity. The solution for a continuously distributed dislocation is derived by integrating the solution for an edge dislocation. The problem of a system of cracks subjected to remote mechanical, electric and magnetic loading is formulated in terms of set of singular integral equations by applying the principle of superposition and the solution for a continuously distributed dislocation. The singular integral equation system is solved by using a numerical integration technique based on Chebyshev polynomials. The J_i and M-integrals for single crack and multi-cracks problems are derived and their dependence on the coordinate system is investigated. Selected numerical results for the M-integral, total energy release rate and mechanical energy release rate are presented for single, double and multiple crack problems. The case of a slowly growing crack interacting with a stationary crack is also considered. It is found that M-integral presents a reliable and physically acceptable measure for assessment of fracture behaviour and damage of magnetoelectroelastic materials.     

Hygrothermal stresses for a plane crack in a generally anisotropic plate

The solutions are presented for the hygrothermal stress field of a generally anisotropic plate under uniform heat flux and moisture concentration transfer obstructed by a hygrothermally insulated crack. For uncoupled diffusion of temperature and moisture, the solutions of both temperature and moisture are obtained directly from the Hilbert problem approach, and are treated as the particular solutions to a pair of nonhomogeneous partial differential equations for an uncoupled hygrothermoelastic system. The associated homogeneous solutions are expressed in terms of three stress functions based on the complex variable approach of Lekhnitskii. With some identities concerning the eigenvalues and eigenvectors, the general expressions of the stress and displacement fields can then be found in an explicit form. The stress intensity factors, crack opening displacements and energy release rate are expressed in terms of the heat flow, moisture concentration, material geometry, elastic and hygrothermal anisotropy. The simultaneous existence of mode I, II and III fracture is found to be due to material inherent anisotropy. Special cases for isotropic and orthotropic materials are also discussed.     

Crack problems in magnetoelectroelastic solids. Part II: general solution of collinear cracks

In this paper, we study mechanically traction-free and electromagnetically permeable crack problems in infinite magnetoelectroelastic solids with linear coupling between the elastic and electromagnetic fields. Using the Stroh-formalism, we first obtain the general solution for collinear cracks in a magnetoelectroelastic medium subjected to arbitrary loads. Then, we give specific solutions for several examples: finite or infinite number of collinear crack subjected to arbitrary remote loads, and a single crack subjected to a line load at an arbitrary point. It is found that in the most general cases, the singularity of electric-magnetic field is always dependent on that of stress. Especially when the medium is only loaded by the remote uniform field, the intensity factor of stress is the same as that of isotropic materials, and the electric-magnetic field inside any crack is uniform.