Analytic solutions to problem of elliptic hole with two straight cracks in one-dimensional hexagonal quasicrystals

By means of the complex variable function method and the technique of conformal mapping, the anti-plane shear problem of an elliptic hole with two straight cracks in one-dimensional hexagonal quasicrystals is investigated. The solution of the stress intensity factor （SIF） for mode III problem has been found. Under the condition of limitation, both the known results and the SIF solution at the crack tip of a circular hole with two straight cracks and cross crack in one-dimensional hexagonal quasicrystals can be obtained.     

Anisotropic thermopiezoelectric solids with an elliptic inclusion or a hole under uniform heat flow

The two-dimensional problem of a thermopiezoelectric material containing an elliptic inclusion or a hole subjected to a remote uniform heat flow is studied. Based on the extended Lekhnitskii formulation for thermopiezoelectricity, conformal mapping and Laurent series expansion, the explicit and closed-form solutions are obtained both inside and outside the inclusion (or hole). For a hole problem, the exact electric boundary conditions on the hole surface are used. The results show that the electroelastic fields inside the inclusion or the electric field inside the hole are linear functions of the coordinates. When the elliptic hole degenerates into a slit crack, the electroelastic fields and the intensity factors are obtained. The effect of the heat flow direction and the dielectric constant of air inside the crack on the thermal electroelastic fields are discussed. Comparison is made with two special cases of which the closed solutions exist and it is shown that our results are valid.     

十次对称二维准晶中的热应力分析

推导了当考虑热效应时十次对称二维准晶体平面应变问题的通解表示．作为应用，采用所获得的通解首先得到了十次对称二维准晶体中的一个点热源所引起的声子场和相位子场，给出了点热源所引起的声子场和相位子场应力分量的解析表达式；接着获得了在均匀热流作用下十次对称二维准晶体中-绝缘椭圆孔洞所引起的热应力问题的弹性解答，给出了沿椭圆边界环向应力分布的解析表达式；当椭圆的短轴趋于零时，则获得了裂纹问题的解答，给出了应力强度因子、裂纹表面张开位移及能量释放率的解析表达式；推导了在任意热载荷作用下裂尖附近的渐近场．     

Stress and electric displacement analyses in piezoelectric media with an elliptic hole and a small crack

In this paper, the interactions between an elliptic hole and an arbitrary distributed small crack in plane piezoelectric medium, which are often happened in engineering problems, are discussed. The Green’s functions in a piezoelectric plate with an elliptic hole for a generalized line dislocation and a generalized line force are presented. The small crack is represented by unknown continuous distributed dislocations. By considering traction free conditions on the surface of the small crack, the problem is then reduced to a group of singular integral equations which are solved by using a special numerical technique. Accuracy of the present method is confirmed by comparing the numerical results with those in literatures for PZT-4 when the elliptic hole is degenerated into a crack. The generalized stress intensity factors of cracks and the generalized stress on the edge of the elliptic hole are shown graphically. It is shown that the small crack may have shielding or amplifying effects on the main elliptic hole or crack, which depends on the location and orientation of the small crack. The hole near a crack can significantly reduce the stress intensity factor of the crack. The direction of the electric field is important to shielding effect.     

SINGULAR SOLUTIONS OF ANISOTROPIC PLATE WITH AN ELLIPTICAL HOLE OR A CRACK

In the present paper, closed form singular solutions for an infinite anisotropic plate with an elliptic hole or crack are derived based on the Stroh-type formalism for the general anisotropic plate. With the solutions, the hoop stresses and hoop moments around the elliptic hole as well as the stress intensity factors at the crack tip under concentrated in-plane stresses and bending moments are obtained. The singular solutions can be used for approximate analysis of an anisotropic plate weakened by a hole or a crack under concentrated forces and moments.They can also be used as fundamental solutions of boundary integral equations in BEM analysis for anisotropic plates with holes or cracks under general force and boundary conditions.     

The Stress State of a Ferromagnetic with an Elliptic Crack in a Homogeneous Magnetic Field

The problem on the stress–strain state of an infinite isotropic body made of a magnetically soft material and containing an elliptic crack is considered. It is assumed that the body is under an external magnetic field perpendicular to the crack plane. The basic characteristics of the stress–strain state and the magnetic field induced are determined and their singularities near the elliptic crack are studied. Formulas are given for the stress intensity factors for the force and magnetic fields near the crack tip     

Exact solutions for the plane problem in piezoelectric materials with an elliptic or a crack

Based on the complex potential approach, the two-dimensional problems in a piezoelectric material containing an elliptic hole subjected to uniform remote loads are studied. The explicit, closed-form solutions satisfying the exact electric boundary condition on the hole surface are given both inside and outside the hole. When the elliptic hole degenerates into a crack, the field intensity factors are obtained. It is shown that the stress intensity factors are the same as that of isotropic material, while the electric displacement intensity factor depends on both the material properties and the mechanical loads, but not on the electric loads. In other words, the uniform electric loads have no influence on the field singularities. It is also shown that the impermeable crack assumption used previously to simply the electric condition is not valid to crack problems in piezoelectric materials.     

On the stress problem of large elliptical cutouts and cracks in circular cylindrical shells

Numerical solutions are presented for stresses around an elliptical hole in a long, thin, circular cylindrical shell subjected to axial tension for both the symmetric orientations of the hole with respect to the shell. The method of analysis involves obtaining a series solution to the governing shell equations in terms of Mathieu functions by the method of separation of variables and satisfying the boundary conditions numerically term by term in a Fourier series formulation. Results are presented in the form of charts from which stress concentration factors can be directly read over a wide range of the two parameters, namely, axis ratio of the ellipse and a curvature parameter defining the hole size with respect to dimensions of the shell.An interesting feature of the investigation is the analysis of limiting cases of circumferential and axial cracks for axial tension and internal pressure loadings respectively. The method developed involves determining the solution completely in elliptic coordinates and then determining the singular stresses by carrying out a transformation to polar coordinates with crack tip as the origin through a Taylor series expansion. Membrane and bending stress intensity factors are computed and plotted over a sufficiently wide range of the curvature parameter extending from small to large sized cracks. As an outcome of the analysis, a “hybrid” technique has been developed by which singularity conditions at the crack tip can be handled effectively in dealing with boundary conditions in crack problems.     

Exact solutions for piezoelectric screw dislocations near two asymmetrical edge cracks emanating from an elliptical hole

The interaction between piezoelectric screw dislocations and two asymmetrical interfacial cracks emanating from an elliptic hole under combined mechanical and electric load at infinity is dealt with. The closed-form solutions are derived for complex potentials and generalized stress fields. In the limiting cases, some well-known results can be obtained from the present solutions. Moreover, some new exact solutions are shown. The stress intensity factor and the energy release rate at the right tip due to a screw dislocation near the right interfacial crack are also calculated. The results show that the shielding effect of dislocation on crack expanding decreases with the increase in dislocation azimuth angle and the distance between the dislocation and the crack tip, and the repulsion acting on the dislocation from the other half plane demotes crack propagation. The increasing of the length of the other crack promotes crack growth, but the increasing of the minor semi-axis demotes it.     

Dynamic photoelastic study of wave fields in elastic plates with stress concentrators

The method of dynamic photoelasticity is used to study the dynamic failure of structural members in the form of plates with a curvilinear (circular or elliptic) hole and an isolated crack under impulsive loading. The time-dependences of the stress intensity factors and the crack tip velocity are investigated for two types of models __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 12, pp. 84–92, December 2005.     

The fundamental solutions for the plane problem in piezoelectric media with an elliptic hole or a crack

1.IntroductionItiswell-knownthatthefundame,ltalsolutionsorGreen'sfunctionsplayanimportantroleilllinearelasticity.Forexample,theycanbeusedtoconstructmanyanalyticalsolutionsofpracticalproblems.Itismoreimportantthattheyareusedasthefundamentalsolutionsintheboundaryelementmethod(BEM)tosolvesomecomplicatedproblem.Withthewidely-increasingapplicationofpiezoelectricmaterialsinengineeringproblems,thestudyregardingtheGreen'sfLlnctionsinpiezoelectricsolidshasreceivedmuchinterest.The3DGreen'sfunctionsi…     

冲击载荷下含空孔三点弯曲梁的动态断裂行为

为得到圆孔缺陷对运动裂纹扩展过程的影响规律,采用动态焦散线实验方法进行模型实验,研究了冲击载荷下含空孔三点弯曲梁的动态断裂行为。研究结果表明:空孔对裂纹扩展有极大的阻碍作用,在一定范围内,空孔直径越大,阻碍作用越明显。裂纹扩展到空孔附近时,扩展速度会下降。裂纹在空孔上部再次起裂后,最大扩展速度远远大于裂纹与空孔贯通前的最大扩展速度。裂纹扩展至空孔附近时,裂纹尖端动态应力强度因子KdⅠ和KdⅡ均会下降。裂纹在空孔上部再次起裂后,裂尖的应力强度因子KdⅠ和KdⅡ均大于裂纹与空孔贯通前裂尖的KdⅠ和KdⅡ。在整个扩展过程中,裂纹尖端的动态应力强度因子KdⅡ远小于KdⅠ,说明KdⅠ在裂纹扩展过程中起主要作用。     

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.     

Boundary element weight function analysis of a strip yield crack in a rotating disk

The boundary element method combined with subtration of Bueckner singular fields are used to obtain weight functions for an internal edge crack in a rotating annular disk. A previously developed, general representation of the weight function is used which leads to integrals that can be evaluated analytically to obtain the stress intensity factor and surface displacements of the crack. The determination of crack tip opening displacements for the strip yield crack is reduced to a non-singular integral which can be evaluated in closed form. The strip yield zone length and crack tip opening displacement are determined for an internal radial crack in a rotating annular disk for a range of crack lengths and rotational speeds.     

Permeable cracks between two dissimilar piezoelectric materials

An interfacial crack with electrically permeable surfaces between two dissimilar piezoelectric ceramics under electromechanical loading is investigated. An exact expression for singular stress and electric fields near the tip of a permeable crack between two dissimilar anisotropic piezoelectric media are obtained. The interfacial crack-tip fields are shown to consist of both an inverse square root singularity and a pair of oscillatory singularities. It is found that the singular fields near the permeable interfacial crack tip are uniquely characterized by the real valued stress intensity factors proposed in this paper. The energy release rate is obtained in terms of the stress intensity factors. The exact solution of stress and electric fields for a finite interfacial crack problem is also derived.     

Elastic fields of line defects in a cracked body

Elastic fields are presented for line forces and dislocations in the vicinity of a crack tip and of a contained, double-ended planar crack. The fields of line force couples are also derived. The corresponding stress intensity factors are listed. The use of these results as two-dimensional Green functions for more general cases is discussed.     

基于权函数法的核主泵飞轮应力强度因子研究

采用权函数法确定了含裂纹飞轮在离心力和接触压力作用下的应力强度因子,计算了在同步转速工况下裂纹尖端应力强度因子的值,并与解析法和有限元法计算结果进行了对比。结果表明:权函数法与解析法的误差在3%以内,与有限元法的误差在1%以内,验证了权函数法计算应力强度因子的准确性高;在结构不变的情况下,权函数法可以快速求解不同载荷条件、不同长度裂纹的应力强度因子。通过控制变量法研究了不同参数对应力强度因子的影响,结果表明,飞轮裂纹尖端总应力强度因子随着裂纹尺寸、旋转转速、飞轮尺寸外径与内径比值的增大而增大。     

Piezothermoelastic analysis of a piezoelectric material with an elliptic cavity under uniform heat flow

Summary The problem of a two-dimensional piezoelectric material with an elliptic cavity under a uniform heat flow is discussed, based on the modified Stroh formalism for the piezothermoelastic problem. The exact electric boundary conditions at the rim of the hole are introduced in the analysis. Expressions for the elastic and electric variables induced within and outside the cavity are derived in closed form. Hoop stress around the hole and electric fields in the hole are obtained. The limit situation when the hole is reduced to a slit crack is discussed, and the intensity factors for the problem are obtained. Received 14 April 1998; accepted for publication 25 June 1998     

Study of a Crack at a Fastener Hole by Digital Image Correlation

In this work the efficacy of using digital image correlation to determine stress intensity factors for a crack emanating from a fastener hole has been investigated. To this end a fatigue crack was grown in pure mode I from a 50 mm diameter hole in an Al 7010 alloy plate test-piece. This crack was then loaded elastically under several combinations of mixed mode (I + II) displacements. In each case, images of the sample surface before and after the deformation were recorded using a high resolution digital camera. The surface preparation consisted only of scratching the surface lightly with silicon carbide abrasive paper. The crack location and resulting displacements were then calculated using digital image correlation. The analytical displacement fields for a traction free crack under arbitrary loading conditions based on the Muskhelishvili’s complex function approach were fitted to the experimentally measured displacement fields and the mixed mode stress intensity factor was determined in each case. Good agreement with the nominal applied values was obtained. The uncertainty of the crack tip position has a major influence on the accuracy of the stress intensity factors and so the Sobel edge finding filter was successfully applied to experimental displacement fields to establish precisely the crack tip location. This paper was presented at the 2007 SEM Congress held in Springfield, Massachusetts, USA     

一维六方准晶中平行螺型位错与有限长裂纹的相互作用

论文采用解析函数理论,利用Muskhelishvili著名的复变函数方法,给出了无限大一维六方准晶中多条平行螺型位错与有限长直裂纹相互作用的解析解,并得到了裂纹尖端的场强度因子和作用在位错上的像力.数值算例讨论了镜像力、场强度因子随位错位置的变化,及位错的位置与分布对广义应力场的影响.