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．     

An effective boundary element method for analysis of crack problems in a plane elastic plate

A simple and effective boundary element method for stress intensity factor calculation for crack problems in a plane elastic plate is presented. The boundary element method consists of the constant displacement discontinuity element presented by Crouch and Starfield and the crack-tip displacement discontinuity elements proposed by YAN Xiangqiao. In the boundary element implementation the left or the right crack-tip displacement discontinuity element was placed locally at the corresponding left or right each crack tip on top of the constant displacement discontinuity elements that cover the entire crack surface and the other boundaries. Test examples (i. e. , a center crack in an infinite plate under tension, a circular hole and a crack in an infinite plate under tension) are included to illustrate that the numerical approach is very simple and accurate for stress intensity factor calculation of plane elasticity crack problems. In addition, specifically, the stress intensity factors of branching cracks emanating from a square hole in a rectangular plate under biaxial loads were analysed. These numerical results indicate the present numerical approach is very effective for calculating stress intensity factors of complex cracks in a 2-D finite body, and are used to reveal the effect of the biaxial loads and the cracked body geometry on stress intensity factors.     

TENSION OF A CLAMPED RECTANGULAR PLATE CONTAINING A CENTRAL CRACK

Using the fundamental solution of a single crack and the Fourier transform solution of an infinite strip, the tension problem of a clamped rectangular plate containing a central crack is reduced to solve a system of singular integral equations. Then, the normal stress on clamped side and the stress intensity factars of the central cruck are carried out by means of Gauss-Jacobi integration formulas. The comparison of numerical results is shown in the "table of stress intensity factars".     

Tension of a clamped rectangular plate containing a central crack

Using the fundamental solution of a single crack and the Fourier transform solution of an infinite strip, the tension problem of a clamped rectangular plate containing a central crack is reduced to solve a system of singular integral equations. Then, the normal stress on clamped side and the stress intensity factors of the central crack are carried out by means of Gauss-Jacobi integration formulas. The comparison of numerical results is shown in the “table of stress intensity factors”. This work was supported by the Science Fund of the Academy of Sciences of China.     

3D dynamic stress intensity factors at three rectangular cracks in an infinite elastic medium subjected to a time-harmonic stress wave

Summary Dynamic stresses around three coplanar cracks in an infinite elastic medium are determined in the paper. Two of the cracks are equal, rectangular and symmetrically situated on either side of the centrally located rectangular crack. Time-harmonic normal traction acts on each surface of the three cracks. To solve the problem, two kind of solutions are superposed: one is a solution for a rectangular crack in an infinite elastic medium, and the other one is that for two rectangular cracks in an infinite elastic medium. The unknown coefficients in the combined solution are determined by applying the boundary conditions at the surfaces of the cracks. Finally, stress intensity factors are calculated numerically for several crack configurations. Received 14 July 1998; accepted for publication 2 December 1998     

Fatigue growth modeling of cracks emanating from a circular hole in infinite plate

This paper presents a numerical approach of fatigue growth analysis of cracks emanating from a hole in infinite elastic plate subjected to remote loads. It involves a generation of Bueckner’s principle and a hybrid displacement discontinuity method (a boundary element method) proposed recently by the senior author of the paper. Because of an intrinsic feature of the boundary element method, a general crack growth problem can be solved in a single region formulation. In the numerical simulation, for each increment of crack extension, remeshing of existing boundaries is not necessary. Crack extension is modeled conveniently by adding new boundary elements on the incremental crack extension to the previous crack boundaries. As an example, fatigue growth process of an inclined crack in an infinite plate under uniaxial cycle load is modeled to illustrate the effectiveness of the numerical approach. In addition, fatigue growth of cracks emanating from a circular hole in infinite elastic plate subjected to remote loads is investigated by using the numerical approach. Many numerical results are given     

INTERACTION OF THREE PARALLEL CRACKS IN RECTANGULAR PLATE UNDER CYCLIC LOADS

This paper presents a numerical approach for modeling the interaction between multiple cracks in a rectangular plate under cyclic loads. It involves the formulation of fatigue growth of multiple crack tips under ruixed-mode loading and an extension of a hybrid displacement discontinuity method （a boundary element method） to fatigue crack growth analyses. Because of an intrinsic feature of the boundary element method, a general growth problem of multiple cracks can be solved in a single-region formulation. In the numerical simulation, remeshing of existing boundaries is not necessary for each increment of crack extension. Crack extension is conveniently modeled by adding new boundary elements on the incremental crack extension to the previous crack boundaries. As an example, the numerical approach is used to analyze the fatigue growth of three parallel cracks in a rectangular plate. The numerical results illustrate the validation of the numerical approach and can reveal the effect of the geometry of the cracked plate on the fatigue growth.     

Rectangular tensile sheet with a center notch root crack

This paper deals with the rectangular tensile sheet with a center notch crack. Such a crack problem is called a center notch crack problem for short. By using a hybrid displacement discontinuity method (a boundary element method) proposed recently by Yan, two center notch models are analyzed in detail. By changing the geometrical forms and parameters of the center notch, and by comparing the SIFs of the center notch crack problem with those of the center cracked plate tension specimen (CCT), which is a model frequently used in fracture mechanics, the effect of the geometrical forms and parameters of the center notch on the stress intensity factors (SIFs) of the center cracked plate tension specimen, is revealed. Some geometric characterestic parameters are introduced here, which are used to formulate the notch length and the branch crack length, which are to be determined in mechanical machining of the center cracked plate tension specimen. So we can say that the geometric characterestic parameters and the formulae used to determine the notch length and the branch crack length presented in this paper perhaps have some guidance role for mechanical machining of the center cracked plate tension specimen. In addition, the numerical investigation proves that the conventional angular notched specimen is much less sensitive to the size of notch than is the circular notched specimen.     

Study of the plastic zone near a crack in an anisotropic body

The plastic zone at a crack tip in a finite anisotropic body is studied. A boundary-value problem is formulated in terms of the components of the covariant displacement vector for small strains. Particular attention is given to the case of plain strain. In this case, a numerical solution is found for a long rectangular body with a central crack under tension. As a result, conditions for the occurrence and development of a plastic zone at the crack tip are established. A plastic zone on the lateral surface of the body is discovered. How both zones extend and coalesce is elucidated. The effect of anisotropy on the occurrence of a plastic zone is evaluated __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 7, pp. 29–44, July 2006.     

A numerical analysis of cracks emanating from an elliptical hole in a 2-D elasticity plate

This paper is concerned with the stress intensity factors (SIFs) of cracks emanating from an elliptical hole in an infinite or a finite plate under biaxial loads by using a boundary element method, which consists of the non-singular displacement discontinuity element presented by Crouch and Starfield and the crack-tip displacement discontinuity elements due to the author. In the boundary element implementation the left or the right crack-tip element is placed locally at the corresponding left or right crack tip on top of the constant displacement discontinuity elements that cover the entire crack surface and the other boundaries. A few numerical examples are included to show that the present approach is very efficient and accurate for the calculating the SIFs of crack problems in an infinite or a finite plate. The present numerical results of cracks emanating from an elliptical hole under biaxial loads can reveal the effect of the elliptical aspect ratio and the transverse load on the SIFs.     

Generalized stress intensity factors in the interaction within a rectangular array of rectangular inclusions

Summary To evaluate the mechanical strength of fiber-reinforced composites, it is necessary to consider singular stresses at the end of fibers because they cause crack initiation, propagation, and final failure. A square array of rectangular inclusions under longitudinal tension is considered in this paper. The body-force method is applied to a unit cell region. Then, the problem is formulated as a system of singular integral equations, where the unknown functions are the densities of body forces distributed in infinite plates having the same elastic constants as those of the matrix and inclusions. The unknown functions are expressed as piecewise-smooth functions using power series and two types of fundamental densities which express singular stresses. Generalized stress intensity factors at the corners of inclusions are systematically calculated with varying the shape and spacing of a square array of square and rectangular inclusions.     

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.     

纳米硬度实验中的多位错生成分析

本文利用边界单元法及基于Peierls-Nabarro模型的位错理论，分析了理想纳米触头下多位错的生成，得到了滑移面上多位错的构形以及表面位移与载荷的关系曲线。根据得到的计算结果分析了薄膜厚度的影响，得到了与已有实验结果相同的结论。通过把表面看成一个包含在无穷大弹性体中的无穷大弹性体中无穷大裂纹来考虑表面的影响，从而可以直接采用已有的边界积分方程。该方法在连续介质力学的计算中引入了包含原子信息的层间势能函数，为分析多尺度的力学问题提供了有效的方法。     

Analysis of two-dimensional finite solids withmicrocracks

A general method is presented for solving the plane elasticity problem of finite plateswith multiple microcracks. The method directly accounts for the interactions between differentmicrocracks and the effect of outer boundary of a finite plate. Analysis is based on a superpositionscheme and series expansions of the complex potentials. By using the traction-free conditions oneach crack surface and resultant forces relations along outer boundary, a set of governingequations is formulated. The governing equations are solved numerically on the basis of aboundary collocation procedure. The effective Youngs moduli for randomly oriented cracks andparallel cracks are evaluated for rectangular plates with microcracks. The numerical results arecompared with those from various micromechanics models and experimental data. These resultsshow that the present method provides a direct and efficient approach to deal with finite solidscontaining multiple microcracks.     

Variation of the stress intensity factor along the front of a 3-D rectangular crack subjected to mixed-mode load

Summary  The singular integral equation method is applied to the calculation of the stress intensity factor at the front of a rectangular crack subjected to mixed-mode load. The stress field induced by a body force doublet is used as a fundamental solution. The problem is formulated as a system of integral equations with r ⁻³-singularities. In solving the integral equations, unknown functions of body-force densities are approximated by the product of polynomial and fundamental densities. The fundamental densities are chosen to express two-dimensional cracks in an infinite body for the limiting cases of the aspect ratio of the rectangle. The present method yields rapidly converging numerical results and satisfies boundary conditions all over the crack boundary. A smooth distribution of the stress intensity factor along the crack front is presented for various crack shapes and different Poisson's ratio. Received 5 March 2002; accepted for publication 2 July 2002     

A three-dimensional rectangular crack subjected to shear loading

In this paper a solution is derived to treat the three-dimensional elastostatic problem of a narrow rectangular crack embedded in an infinite elastic medium and subjected to equal and opposite shear stress distribution across its faces. Employing two-dimensional integral transforms and assuming a plane-strain solution across the width of the crack, the stress field ahead of the crack length is reduced to the solution of an integral equation of Fredholm type. A numerical solution of the integral equation and the corresponding mode II stress-intensity factor is obtained for several crack dimensions and Poisson's ratios of the material.     

On stress analysis for a penny-shaped crack interacting with inclusions and voids

An analytical approach to calculate the stress of an arbitrary located penny-shaped crack interacting with inclusions and voids is presented. First, the interaction between a penny-shaped crack and two spherical inclusions is analyzed by considering the three-dimensional problem of an infinite solid, composed of an elastic matrix, a penny-shaped crack and two spherical inclusions, under tension. Based on Eshelby’s equivalent inclusion method, superposition theory of elasticity and an approximation according to the Saint–Venant principle, the interaction between the crack and the inclusions is systematically analyzed. The stress intensity factor for the crack is evaluated to investigate the effect of the existence of inclusions and the crack–inclusions interaction on the crack propagation. To validate the current framework, the present predictions are compared with a noninteracting solution, an interacting solution for one spherical inclusion, and other theoretical approximations. Finally, the proposed analytical approach is extended to study the interaction of a crack with two voids and the interaction of a crack with an inclusion and a void.     

Anti-plane shear of rectangular region with two cracks

An approach for obtaining the solution of multiple cracks in a rectangular region is presented. A pair of concentrated shear force applied to the crack surface is used in conjunction with the method of superposition for generating the results for cracks whose surfaces are free from tractions. A system of Fredholm integral equations can always be obtained to yield numerical results.     

Surface cracks in a plate of finite width under extension or bending

In this paper the problem of a finite plate containing collinear surface cracks is considered. The problem is solved by using the line spring model with plane elasticity and Reissner's plate theory. The main purpose of the study is to investigate the effect of interaction between two cracks or between cracks and stress-free plate boundaries on the stress intensity factors and to provide extensive numerical results which may be useful in applications. First, some sample results are obtained and are compared with the existing finite element results. Then the problem is solved for a single (internal) crack, two collinear cracks and two corner cracks for wide range of relative dimensions. Particularly in corner cracks the agreement with the finite element solution is surprisingly very good. The results are obtained for semielliptic and rectangular crack profiles which may, in practice, correspond to two limiting cases of the actual profile of a subcritically growing surface crack.     

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.