Interaction of plane harmonic waves with a thin elastic inclusion of zero flexural rigidity

We solve the problem on the interaction of plane elastic harmonic waves with a thin elastic strip-shaped inclusion. The inclusion is contained in an unbounded body (matrix) that is under plane strain conditions. The normal forces applied by the medium to the inclusion side edges are taken into account. Because of the small thickness of the inclusion, we assume that its flexural rigidity is zero and that the shear displacements at any of its points coincide with the displacements of the corresponding points of its midplane. The displacements on the midplane itself can be found from the corresponding equation of the theory of plates. The solution method consists in representing the displacements as discontinuous solutions of the Lamé equations and then determining the unknown jump from a singular integral equation. This equation is solved numerically by the collocation method, and formulas for the approximate calculation of the stress intensity factors near the inclusion ends are obtained.     

Stress Concentration Near an Elliptic Crack in the Interface Between Elastic Bodies under Steady-State Vibrations

The paper addresses the three-dimensional problem on steady-state vibrations of an elastic body consisting of two perfectly joined dissimilar half-spaces with an elliptic mode I crack located in one of the half-spaces normally to the interface. The problem is reduced to a boundary integral equation for the crack opening function. The integration domain of the equation is bounded by the crack domain, and the interaction between the crack and the interface is described by a regular kernel. The equation is solved using the mapping method. Numerical results are obtained for the case where the surfaces of the elliptic crack are subjected to harmonic loading with constant amplitude. The dependences of the stress intensity factors on the wave number are presented for various relationships among the mechanical constants that ensure the absence of near-surface waves     

The scattering of harmonic elastic anti-plane shear waves by a finite crack in infinitely long strip using the non-local theory

In this paper, the scattering of harmonic anti-plane shear waves by a finite crack in infinitely long strip is studied using the non-local theory. The Fourier transform is applied and a mixed boundary value problem is formulated. Then a set of dual integral equations is solved using the Schmidt method instead of the first or the second integral equation method. A one-dimensional non-local kernel is used instead of a two-dimensional one for the anti-plane dynamic problem to obtain the stress occurring at the crack tips. Contraty to the classical elasticity solution, it is found that no stress singularity is present at the crack tip. The non-local dynamic elastic solutions yield a finite hoop stress at the crack tip, thus allowing for a fracture criterion based on the maximum dynamic stress hypothesis. The finite hoop stress at the crack tip depends on the crack length, the width of the strip and the lattice parameter. Supported by the Post Doctoral Science Foundation of Heilongjiang Province, the Natural Science Foundation of Heilongjiang Province and the National Foundation for Excellent Young Investigators.     

Steady-state mode III delamination crack in a periodically layered medium

A non-homogeneous layered space with a semi-infinite interface crack is considered. Isotropic elastic layers having dissimilar thicknesses and shear moduli are arranged periodically and the antiplane stress state is produced by a shear loading applied to the crack faces. Application of the representative cell method based on the discrete Fourier transform allows to reduce the initial problem to the problem for a single bi-layered cell in the transform space and to formulate the Wiener–Hopf equation. The final result is presented in the form of triple quadratures. A parametric study of the stress intensity factor revealed some qualitative differences in its behavior for the cases of equal and non-equal layers thicknesses. The obtained analytical result is employed to derive of the closed form eigensolution corresponding to the traction free crack faces and vanishing remote loading. In addition to the local near tip square root asymptote this solution has also the remote one on a macroscale. The influence of the problem parameters on the ratio of the near to the far stress intensity factor is investigated and expressed by a simple algebraic formula which is confirmed by energy considerations. It is found that when the thin layers are more compliant this ratio is always less than unity, while in the opposite case the local stress singularity may exceed the remote one.     

Vertical vibrations of a rigid edge inclusion under a harmonic load

The paper considers the problem of vibrations of a rigid edge inclusion, which lies in an elastic half-plane and emerges on the surface perpendicular to that half-plane. The vibrations are initiated by a harmonic force acting on the end of the inclusion, which emerges on the surface. The field of translations in the half-plane is shown to be represented by the superposition of two discontinuous solutions with discontinuities at the boundary between the half-plane and the line of the inclusion. The unknown discontinuities are determined from the boundary conditions and the conditions of the inclusion-medium interaction. The problem is thus reduced to one of solving a singular integral equation with an immobile singularity for the jump in shear stresses on the line of the inclusion. The equation obtained is solved numerically by the method of mechanical quadratures. The amplitudes of the inclusion vibrations and the stressed state of the medium near it are studied.Odessa State Marine Academy, Odessa, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 31, No. 7, pp. 46–55, July, 1995.     

双材料中平片裂纹问题的超奇异积分方程解法

利用三维断裂力学的超奇异积分方程方法,对双材料空间中重直于界面的平片裂纹Ⅰ型问题进行了研究。首先根据双材料空间的弹性力学基本解,使用边界积分方程方法,在有限部积分的意义下导出了以裂纹面位罗间断为未知函数的超奇异积分方程,并为其建立了数值法。在此基础上,讨论了用裂纹面位移问题计算应力强度因子的方法。最后用此计算了几个典型的Ⅰ型下片裂纹问题的应力强度因子,其数值结果令人满意。     

On the Torsion of a Class of Nonlinear Fiber Composite Cylinders

This paper presents an analysis of the torsion of a solid or annular circular cylinder consisting of nonlinear material in the form of an elastic matrix with embedded unidirectional elastic fibers parallel to the cylinder axis. The specific class of composite considered is one for which nonlinear fiber-matrix interface slip is captured by uniform cohesive zones of vanishing thickness. Previous work on the effective antiplane shear response of this material leads to a stress–strain relation depending on the interface slip together with an integral equation governing its evolution. Here, we obtain an approximate single mode solution to the integral equation and utilize it to solve the torsion problem. Equations governing the radial distributions of shear stress and interface slip are obtained and formulae for torque–twist rate are presented. The existence of singular surfaces, i.e., surfaces across which the slip and the shear stress experience jump discontinuities are analyzed in detail. Specific results are presented for an interface force law that allows for interface failure in shear.     

用样条边界元与奇异积分方程法联合求解多裂纹柱体的扭转

通过间解的分离，本文将径向多裂纹柱体的导曲函两个调和函数表示，使问题归为解一组混混合型积分方程。针对方程的特点，本文联合使用三次样条边界法与奇异积分方程的数值方法对所得方程建立了数值法，并对裂纹相交情形作了特殊处理。最后对工程中感兴趣的一些典型的多裂纹柱体的扭转作了例题计算，结果表明，本文方法具有收敛快，精度高的特点。     

An Interface Crack for a Functionally Graded Strip Sandwiched Between Two Homogeneous Layers of Finite Thickness

In this paper, the behavior of an interface crack for a functionally graded strip sandwiched between two homogeneous layers of finite thickness subjected to an uniform tension is resolved using a somewhat different approach, named the Schmidt method. The Fourier transform technique is applied and a mixed boundary value problem is reduced to two pairs of dual integral equations in which the unknown variables are the jumps of the displacements across the crack surface. To solve the dual integral equations, the jumps of the displacements across the crack surfaces are expanded in a series of Jacobi polynomials. This process is quite different from those adopted in previous works. Numerical examples are provided to show the effects of the crack length, the thickness of the material layer and the materials constants upon the stress intensity factor of the cracks. It can be obtained that the results of the present paper are the same as ones of the same problem that was solved by the singular integral equation method. As a special case, when the material properties are not continuous through the crack line, an approximate solution of the interface crack problem is also given under the assumption that the effect of the crack surface interference very near the crack tips is negligible. Contrary to the previous solution of the interface crack, it is found that the stress singularities of the present interface crack solution are the same as ones of the ordinary crack in homogenous materials.     

Harmonic Vibrations of a Viscoelastoplastic Sandwich Cylindrical Shell

The free and forced harmonic vibrations of viscoelastoplastic sandwich shells are analyzed. An equation for approximate determination of the amplitudes of near-resonance vibrations is derived. As an example, the problem is solved for a sandwich circular cylindrical shell     

Contact problem for the flat crack under two normally incident shear H-waves with wave mode-shifting

The frictional contact interaction of the finite crack edges in an infinite plane is studied for the case of normal incidence of two harmonic shear H-waves with multiple mode-shifted frequencies. Boundary integral equation method and constrained optimization algorithm are used for the problem solution. The forces of contact interaction and displacement discontinuity are analyzed. Influence of the waves frequencies on the stress intensity factor for different normalized wave numbers and coefficients of friction is considered here.     

功能梯度材料与均质材料交界面上Ⅰ-型裂纹对简谐动载的响应分析

在忽略界面裂尖端裂纹面相互叠入的条件下,对功能梯度材料与均质材料交界面上Ⅰ-型裂纹对简谐动载响应问题进行了分析。利用傅立叶变换,将问题的求解转换为对以裂纹面上位移差为未知函数的对偶积分方程的求解。为了求解对偶积分方程,将裂纹面上的位移差函数展开为雅可毕多项式的级数形式。最终给出了裂纹长度、入射波频率和材料性质对应力强度的影响。结果表明,当界面材料不连续时,获得了具有普通1/2奇异性的近似解。     

Plastic yielding on inclined slip-planes at a crack tip

Plastic yield at crack tips on singular slip-planes, inclined to the crack plane, has been studied under plane-strain conditions for combined tension, hydrostatic stress, and in-plane shear. The singular integral equation, which represents the equilibrium condition of edge dislocations on the slip-planes, is transformed into a Fredholm integral equation in order to avoid difficulties that occur with its numerical solution. Results are presented for the slip-band length, the plastic crack-tip opening displacement, stress fields, and crack-opening contours. A series expansion of the results obtained numerically confirms approximate analytical expressions given by J.R. Rice (1974), up to the third-order in the applied stresses. The results of finite element methods agree with values of the crack-tip opening displacement obtained here to within 10 per cent. Ahead of the crack tip, the principal tensile stresses exceed the principal shear stresses by a factor of 10, approximately.     

Anti-plane fracture of a functionally graded material strip

This paper considers the anti-plane (or mode III) crack problem in a functionally graded material strip. The shear modulus of the strip is considered for a class of functional forms for which the equilibrium equation has an analytical solution. The problem is solved by means of singular integral equation technique. Both a single crack and a series of collinear cracks are studied. The results are tabulated and plotted to show the effect of the material nonhomogeneity and crack location on the stress intensity factors.     

Investigation of the Dynamic Behavior of a Griffith Crack in a Piezoelectric Material Strip Subjected to the Harmonic Elastic Anti-plane Shear Waves by Use of the Non-local Theory

In this paper, the dynamic behavior of a Griffith crack in a piezoelectric material strip subjected to the harmonic anti-plane shear waves is investigated by use of the non-local theory for impermeable crack surface conditions. To overcome the mathematical difficulties, 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 the electric displacement near at the crack tip. By means of the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations. These equations are solved using the Schmidt method. Contrary to the classical solution, it is found that no stress and electric displacement singularity is present near the crack tip. The non-local dynamic elastic solutions yield a finite hoop stress near the crack tip, thus allowing for a fracture criterion based on the maximum dynamic stress hypothesis. The finite hoop stress at the crack tip depends on the crack length, the thickness of the strip, the circular frequency of incident wave and the lattice parameter.     

Interaction of a plane harmonic wave with a thin rigid inclusion of the shape of a cylindrical shell

The paper presents the solution of the problem of determining the stress state in an elastic matrix containing a rigid inclusion of the shape of a thin cylindrical shell. It is assumed that harmonic vibrations occur in the matrix under the conditions of axial symmetry (the symmetry axis is the inclusion axis) and the conditions of full adhesion between the inclusion and the matrix are satisfied. The vibrations are caused by the propagation of a plane wave whose front is perpendicular to the inclusion axis. The solution method is based on representing the displacements in the matrix as discontinuous solutions of the equations of axisymmetric oscillations of an elastic medium with unknown stress jumps on the inclusion surface. The realization of the boundary conditions for these jumps leads to a system of integral equations. Its solution is constructed numerically by the mechanical quadrature method with the use of special quadrature formulas for specific integrals. It is numerically investigated how the ratio of the inclusion geometric dimensions and the propagating wave frequency affect the stress concentration near the inclusion.     

NUMERICAL SOLUTIONS FOR A NEARLY CIRCULAR CRACK WITH DEVELOPING CUSPS UNDER SHEAR LOADING

In this paper, we study the behavior of the solution at the crack edges for a nearly circular crack with developing cusps subject to shear loading. The problem of finding the resulting force can be written in the form of a hypersingular integral equation. The equation is then transformed into a similar equation over a circular region using conformal mapping. The equation is solved numerically for the unknown coeffcients, which will later be used in finding the stress intensity factors. The sliding and tearing mode stress intensity factors are evaluated for cracks and displayed graphically. Our results seem to agree with the existing asymptotic solution.     

裂纹端部细短纤维的应力分析

基于裂纹端部存在与其裂纹面相垂直的二相细短纤维分析模型，采用叠加原理推导了求解纤维表面应力分布函数的积分方程，通过简化得到了该方程的解析表达显式，该积分方程的特征值方程是纤维几何参数，材料常数以及纤维相对于裂纹位置的相关函数，当材料参数不满足特征方程时，积分方程将具有唯一解；并借助数值方法，给出了纤维剪应力分布算例，和纤维对应力强度因子的影响。     

Dynamic penny-shaped cracks in multilayer sandwich composites

A point force method is proposed for obtaining the dynamic elastic response of a multilayer sandwich composite in the presence of a penny-shaped crack under a harmonic loading. The sandwich composite is a multilayered solid whose lower half is the mirror image of the upper half with the center plane as the mirror. The crack is lying on the mirror plane of the composite. The solution of the mode I dynamic crack problem is formulated by integrating the Green’s function of a time-harmonic surface normal point force over the crack surface with an unknown point force distribution. The dual integral equations of the unknown point force distribution are established by considering the boundary conditions, which can be reduced to a Fredholm integral equation of the second kind. A complete solution of the crack problem under consideration can be obtained by solving this Fredholm integral equation. It will be shown that the results obtained by this approach are the same as some existing solutions.     

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.