The numerical solution of crack problems in plane elasticity in the case of loading discontinuities

The direct quadrature method of numerical solution of Cauchy type singular integral equations encountered in plane elasticity crack problems is applied to the case where the loading distribution along the crack edges presents jump discontinuities. This is made by using a well-known modification of the quadrature method which is free of undesirable errors due to the loading discontinuities. Hence, the method is ideal to treat the aforementioned class of crack problems and, particularly, crack problems where the Dugdale-Barenblatt elastic-perfectly plastic model is adopted. Finally, a numerical application of the method to the problem of a periodic array of cracks with a loading distribution presenting a jump discontinuity is made. The numerical results obtained in this problem compare favorably with the corresponding theoretical results available in this special problem.     

Friction Contact of the Edges of an Interface Crack under the Conditions of Tension and Shear

We study the stressed state of two rigidly fastened elastic half planes made of different materials and containing an interface crack. The friction contact of the crack edges is taken into account. By using the Wiener–Hopf method, the solution of the integral equation of the problem is obtained in the closed form. The lengths of the regions of contact of the crack edges and the distributions of stresses in these regions and over the boundary of the half planes (outside the crack) are found explicitly.     

Non-symmetric extension of a plane crack due to plane SH-waves in a prestressed infinite elastic medium

In an infinite isotropic elastic medium initially in a state of uniform anti-plane shear, the problem of non-symmetric extension of an infinitesimal flaw into a plane shear crack due to two identical linearly varying plane SH-waves with non-parallel wave fronts has been analyzed. Fracture is assumed to initiate at a point a finite time after the waves intersect there and the crack is assumed to extend non-symmetrically along the trace of the wave intersection. Following Cherepanov [10], Cherepanov and Afanas'ev [11] the general solution of the problem has been derived in terms of analytic function of complex variable. Numerical results have been presented to illustrate the nature of variation of the stress intensity factors and the rate of energy flux into the crack edges with the speed of the crack tips and also with the time after fracture initiation.     

Stress intensity factors in two bonded elastic layers containing cracks perpendicular to and on the interface—I. Analysis

In this paper the basic crack problem which is essential for the study of subcritical crack propagation and fracture of layered structural materials is considered. Because of the apparent analytical difficulties, the problem is idealized as one of plane strain or plane stress. An additional simplifying assumption is made by restricting the formulation of the problem to crack geometries and loading conditions which have a plane of symmetry perpendicular to the interface. The general problem is formulated in terms of a coupled system of four integral equations. For each relevant crack configuration of practical interest the singular behavior of the solution near and at the ends and points of intersection of the cracks is investigated and the related characteristic equations are obtained. The edge crack terminating at and crossing the interface, the T-shaped crack consisting of a broken layer and a delamination crack, the cross-shaped crack which consists of delamination crack intersecting a crack which is perpendicular to the interface and a delamination crack initiating from a stress-free boundary of the bonded layers are some of the practical crack geometries considered as examples. The formulation of the problem is given in Part I of the paper. Part II deals with the solution of the integral equations and presentation of the results.     

On the crack propagation with variable velocity

The plane problem of propagation of a straight crack in an elastic medium under arbitrary variable loading is considered. The locations of the edges of the crack are specified as arbitrary smooth functions of time under the only restriction that crack speed at any instant of time is less than the velocity of Rayleigh wave. Solution for the distribution of plane stress components near the crack tip is obtained. In particular, expressions for stress intensity factors at the crack are given, which thus makes it possible to deduce the crack motion under given loading conditions.     

The linear thermoelastic problem of uniform heat flow disturbed by a two-dimensional crack in a strip

Under the assumption of plane strain, a solution for a thermoelastic problem concerning a strip is obtained by the method of dual integral equations. It is assumed that the crack is parallel to the edges of the strip. The variation with the strip width of stress-intensity factor is shown graphically.     

Impact response of a finite crack in an orthotropic strip

Summary The elastodynamic response of a finite crack in an infinite orthotropic strip under normal impact is investigated in this study. The crack is situated symmetrically and oriented in a direction normal to the edges of the strip. Laplace and Hankel transforms are used to reduce the transient problem to the solution of a pair of dual integral equations in the Laplace transform plane. The solution to the dual integral equations is then expressed in terms of a Fredholm integral equation of the second kind. Numerical values on the dynamic stress intensity factor for some fiber-reinforced composite materials are obtained and the results are graphed to display the influence of the material orthotropy.     

General case of multiple crack problems in an infinite plate

General case of multiple crack problems in an infinite plate is a case that the tractions applied on two edges of each crack are arbitrary, generally, are not in equilibrium. Two elementary solutions are present to solve the proposed problem. The first (second) elementary solution is defined as a solution that two pairs of normal and tangential concentrated forces are applied at a point of both edges of a single crack in an infinite isotropic elastic medium, with same magnitude and opposite direction (with same magnitude and same direction). Using the two elementary solutions and the principle of superposition, we found the proposed problem can be converted into a system of Fredholm integral equations. Finally, the system is solved numerically and SIF values at the crack tips can be easily calculated. In order to explain our study, one numerical example is given in this paper.     

Surface and internal cracks in a residually stressed plate

The problem of a surface or an internal crack in a plate which contains residual stresses is examined. The line spring model, which reduces a three-dimensional elasticity problem into a two-dimensional problem in plate theory, is used to model the crack. The Reissner plate theory, which takes into account transverse shear deformations, is used to model the plate. The formulation is based on Fourier Transforms which lead to a pair of singular integral equations that are solved numerically. The line spring method requires the plane strain solution to both the edge and internally cracked strip with crack surface loads representative of tension, bending, and the given residual stress distribution. For general use, plane strain solutions are presented for polynomial loading through the thickness up to the fifth order. Comparisons are made between the results given by the line spring model for the Reissner plate theory and the finite element method.     

Vertical crack inside a semi-infinite plane

In this paper we consider the problem of determining the stress-intensity factors and the crack energy in a semi-infinite plane containing an inside crack perpendicular to the straight boundary of the plane. By the use of Mellin transform, we reduce the problem to solving a single singular integral equation. Approximate solution of the integral equation is obtained as a series of Chebyshev polynomials of the first kind. The coefficients $\text{B}{}_{\mathrm{n}}$ of the series are determined from a system of linear algebraic equations. Expressions for the stress-intensity factors at the edges of the crack, the shape of the crack and the crack energy are derived in terms of the coefficients $\text{B}{}_{\mathrm{n}}$. The numerical values of these quantities have been displayed graphically for three particular cases.     

Closed-form solution for an eccentric anti-plane shear crack normal to the edges of a magnetoelectroelastic strip

Summary The problem of an anti-plane shear crack embedded in a magnetoelectroelastic strip is investigated. The crack is assumed to be normal to the strip edges. By using the finite Fourier transform, the associated mixed boundary-value problem is reduced to triple series equations, then to singular integral equations. Solving the resulting equations analytically, the field intensity factors and energy release rates at the crack tips can be determined in explicit form. The influences of applied electric and magnetic loadings on the normalized energy release rate and mechanical strain energy release rate are presented graphically. Obtained results reveal that applied electric and magnetic loadings affect crack growth, depending on their directions and adopted fracture criteria. The derived solution is applicable to other cases including two collinear cracks distributed symmetrically in a magnetoelectroelastic strip, and a periodic array of collinear cracks in a magnetoelectroelastic plane.     

含任意直线型裂纹的直角域中圆柱夹杂对SH波的散射

摘 要: 采用Green函数及复变函数方法研究了SH波作用下直角域内任意直线型裂纹对圆柱弹性夹杂的影响。首先,取含有圆柱弹性夹杂的直角域内任意一点承受时间谐和的出平面线源荷载作用时的位移函数基本解作为适合本文的Green函数;其次,利用裂纹切割技术构造裂纹,并写出圆柱夹杂与裂纹同时存在时的位移场和应力场;最后,给出圆柱弹性夹杂动应力集中系数的算例和结果,并讨论了裂纹的存在对动应力集中系数的影响。     

On the problem of two non-coplanar parallel cracks in a strip

The problem of two non-coplanar parallel Griffith cracks, located symmetrically in a strip is discussed under the assumption of plane strain condition. The crack surfaces are normal to the edges of the strip and are subjected to the same pressure distribution. When the edges are stress free, solution is assumed in the form of a series of complex eigenfunctions. By the use of certain generalized orthonormality relationships and the calculus of residues, two simultaneous Fredholm integral equations are obtained. Similar equations are obtained when the edges of the strip are constrained by smooth rigid planes. Numerical results are given for the case of constant applied pressure.     

Investigation of an interface crack with a contact zone in a piezoelectric bimaterial under limited permeable electric boundary conditions

Summary The plane strain problem for an interface crack between two bonded piezoelectric semi-infinite planes under remote electromechanical loading is considered. Mechanically frictionless and electrically permeable contact zones are assumed at the crack tips and the remaining part of the crack is considered as electrically limited permeable with a certain permeability of the crack medium. Patron’s way of modelling limited permeable conditions is used. By means of integral transforms the problem is reduced to a nonlinear system of singular integral equations. An iterative scheme together with discretization and utilization of Gauss-Chebishev quadrature rule is applied for the solution of this system. Distributions of the electric displacement along the crack region as well as the stress and electric intensity factors and the energy release rate are found for different electromechanical loads and crack permeabilities. Calculations are performed for an artificial contact zone length, however the way of an easier determination of the associated values for the real contact zone length is shown. As a particular case of the obtained solution the crack in a homogeneous piezoelectric media is considered. The results of the calculations are compared to the corresponding results obtained earlier by means of Hao and Shen’s way of modelling the crack permeability. Even though the electric displacements obtained in the respective framework of these models differ essentially, it appears that the fracture mechanical parameters are in good agreement with each other.     

Approximate solution of singular integro-differential equations in elastic contact problems

A method for the numerical solution of singular integro-differential equations is proposed. The approximate solution is sought in the form of the sum of a power series with unknown coefficients multiplied by a special term which controls the appropriate solution behaviour near and at the edges of the interval. The coefficients are to be determined from a system of linear algebraic equations. The method is applied to the solution of a contact problem of a disk inserted in an infinite elastic plane. Exact analytical solution is obtained for the particular case when the disk is of the same material as the plane. Comparison is made between the exact and the approximate solutions as well as with the solutions previously available in literature. The stability and the accuracy of the present method is investigated under variation of the parameters involved. The applicability of the method to the case when the boundary conditions for the unknown function are nonzero is discussed along with an illustrative example. A FORTRAN subroutine for the numerical solution of singular integro-differential equations is also provided.     

On crack twinning

The general formulation of the crack branching problem is given in terms of perturbation theory. The Wiener-Hopf method has been applied to derive an exact analytical solution of a singular plane problem of elasticity theory for a semi-infinite straight crack with two symmetric rectilinear cuts of finite length issuing from the crack tip. The solution of this problem can be applied to construct a theory for corrosion crack twinning, as well as for cracks having similar geometries, but other physical mechanisms. However, this problem is of considerable interest in studying the retardation in growth of a main crack crossed by a small slit.     

Contact problem for flat crack under two normally incident tension–compression waves with wave mode-shifting

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

Cruciform crack in an orthotropic elastic plane

The problems of determining the stress and displacement fields in an infinite orthotropic plane containing a cruciform crack 387-1, y=0 and 387-2, x=0 when (I) the shape of the crack is prescribed and (II) the cracks are opened by given normal pressures, are reduced to mixed boundary value problems for the quarter plane. Using integral transform techniques, a closed form solution is obtained for problem I, whereas the solution of problem II has been reduced to solving a Fredholm integral equation of second kind with non-singular kernel. Numerical calculation of the stress intensity factor and crack energy in the case of a linear loading function for various crack lengths are presented for problem II, using the values of material constants for a Boron-Epoxy composite.     

Stress intensity factors in a cracked infinite elastic wedge loaded by a rigid punch

The paper considers splitting a plane elastic wedge-shaped solid through the application of a rigid punch. It is assumed that the coefficient of friction on the contact area is constant, the problem has a plane of symmetry with respect to loading and geometry, and the crack lies in the plane of symmetry. The problem is formulated in terms of a system of integral equations with the contact stress and the derivative of the crack surface displacement as the unknown functions. The solution is obtained for an internal crack and for an edge crack. The results include primarily the stress intensity factors at the crack tips, and the measure of the stress singularity at the wedge apex, and at the end points of the contact area.     

Influence of deformation anisotropy on the size of the plastic zone at the tip of an opening mode crack

The form and dimensions of the plastic zone at the tip of an opening mode crack in a plate made of a material with deformation anisotropy were investigated within the limits of the elastic solution. The anisotropy was caused by strengthening during plastic deformation until formation of cracks by loading in a straight trajectory located in the plane of the plate. It was shown that in the case of anisotropy caused by loading in a trajectory which is oriented on a normal to the crack edges the size of the plastic zone decreases and its boundaries are rotated in the direction opposite to the crack growth. Loading in a trajectory in the direction of crack growth leads to broadening of the plastic zone in the transverse direction.Translated from Problemy Prochnosti, No. 1, pp. 73–76, January, 1990.