The practical evaluation of stress intensity factors at semi-infinite crack tips

The solution of crack problems in plane or antiplane elasticity can be reduced to the solution of a singular integral equation along the cracks. In this paper the Radau-Chebyshev method of numerical integration and solution of singular integral equations is modified, through a variable transformation, so as to become applicable to the numerical solution of singular integral equations along semi-infinite intervals, as happens in the case of semi-infinite cracks, and the direct determination of stress intensity factors at the crack tips. This technique presents considerable advantages over the analogous technique based on the Gauss-Hermite numerical integration rule. Finally, the method is applied to the problems of (i) a periodic array of parallel semi-infinite straight cracks in plane elasticity, (ii) a similar array of curvilinear cracks, (iii) a straight semi-infinite crack normal to a bimaterial interface in antiplane elasticity and (iv) a similar crack in plane elasticity; in all four applications appropriate geometry and loading conditions have been assumed. The convergence of the numerical results obtained for the stress intensity factors is seen to be very good.     

Elastodynamic analysis of nonplanar cracks in a half-space

Summary The interaction of time harmonic antiplane shear waves with nonplanar cracks embedded in an elastic half-space is studied. Based on the qualitatively similar features of crack and dislocation, with the aid of image method, the problem can be formulated in terms of a system of singular integral equations for the density functions and phase lags of vibrating screw dislocations. The integral equations, with the dominant singular part of Hadamard's type, can be solved by Galerkin's numerical scheme. Resonance vibrations of the layer between the cracks and the free surface are observed, which substantially give rise to high elevation of local stresses. The calculations show that near-field stresses due to scattering by a single crack and two cracks are quite different. The interaction between two cracks is discussed in detail. Furthermore, by assuming one of the crack tips to be nearly in contact with the free surface, the problem can be regarded as the diffraction of elastic waves by edge cracks. Numerical results are presented for the elastodynamic stress intensity factors as a function of the wave number, the incident angle, and the relative position of the cracks and the free surface.     

Application of the Model of Reverse Yield of a Material in Thin Plastic Strips for the Prediction of Growth of Fatigue Cracks

On the basis of the model of thin plastic strips and the method of singular integral equations, we obtain a closed analytic solution of the problem of stress-strain state of a material near the tip of a fatigue crack taking into account its reverse plasticity in the prefracture zone and the effect of crack closure. This solution is used for the prediction of crack growth under loads with constant and variable amplitudes and the analysis of the behavior of small cracks. It is shown that the obtained results agree with the experimental data.     

Thermoelastic interaction of two offset interfacial cracks in bonded dissimilar half-planes with a functionally graded interlayer

This paper deals with the thermoelasticity problem of bonded dissimilar half-planes with a functionally graded interlayer, weakened by a pair of two offset interfacial cracks. The material nonhomogeneity in the graded interlayer is represented by spatially varying thermoelastic moduli expressed in terms of exponential functions. The cracks are assumed to be thermally insulated disturbing a steady-state uniform heat flow, and the solution is obtained within the framework of linear plane thermoelasticity. The Fourier integral transform method is employed, and the formulation of the current nonisothermal crack problem is reduced to two sets of Cauchy-type singular integral equations for temperature and thermal stress fields in the bonded system. In the numerical results, parametric studies are conducted so that the variations in mixed-mode thermal stress intensity factors are presented as a function of offset crack distance for various geometric and material combinations of the dissimilar homogeneous media bonded through the thermoelastically graded interlayer, elaborating thermally induced singular interaction of the two neighboring interfacial cracks.     

Multiple interfacial cracks in dissimilar piezoelectric layers under time harmonic loadings

In this study, the dislocation‐based model is developed to study the interaction between time‐harmonic elastic waves and multiple interface cracks in 2 bonded dissimilar piezoelectric layers. In this model, cracks are represented by a distribution of so‐called electro‐elastic dislocations whose density is to be determined by satisfying the boundary conditions. Using the Fourier transform, this formulation leads to 2 singular integral equations, which can be solved numerically for the densities of electro‐elastic dislocations on a crack surface. The formulation is used to determine dynamic field intensity factors for multiple interface cracks without limitation of number of cracks. The dynamic field intensity factors are then calculated for both permeable and impermeable crack, and finally, numerical results are presented to illustrate the variation of these quantities with the electromechanical coupling, crack spacing, and the frequency of loading.     

Longitudinal Shear of an Infinite Body with a System of Polygonal Cracks

On the basis of singular integral equations, we propose a new method for the solution of antiplane problems of elasticity theory for bodies with a system of polygonal cracks with regard for the singularities of stresses at the corner points. The modified singular integral equations have continuous regular kernels and right-hand sides, i.e., belong to the same type as in the case of smooth boundary contours. A numerical solution of these equations is found by the method of mechanical quadratures. The values of the stress intensity factors at the corner points and tips of both a three-link polygonal crack and a system of two two-link polygonal cracks in an infinite body are presented. The limiting case of two cracks which form a rhombic hole is also analyzed.     

Interaction between cracks and rigid lines in an infinite plate

Summary Interaction between cracks and rigid lines in an infinite plate is investigated in this paper. The rigid lines are assumed in an equilibrium condition and may have some rotation in the deformation process of the adjacent material. After placing some distributed dislocations along the cracks and some distributed body forces along the rigid lines, a system of singular integral equations is obtained. The obtained system of the singular integral equations is reduced to a system of Fredholm integral equations by appropriate substitution of the unknown functions. The regularized integral equations are solved numerically. Stress intensity factors at the crack tips and stress singularity coefficients are investigated in the numerical examples.     

Thermoelectroelastic response of a functionally graded piezoelectric strip with two parallel axisymmetric cracks

A mixed-mode thermoelectroelastic fracture problem of a functionally graded piezoelectric material strip containing two parallel axisymmetric cracks, such as penny-shaped or annular cracks, is considered in this study. It is assumed that the thermoelectroelastic properties of the strip vary continuously along the thickness of the strip and that the strip is under thermal loading. The crack faces are supposed to be insulated thermally and electrically. Using integral transform techniques, the problem is reduced to that of solving two systems of singular integral equations. Systematic numerical calculations are carried out, and the variations of the stress and electric displacement intensity factors are plotted for various values of dimensionless parameters representing the crack size, the crack location and the material non-homogeneity.     

含曲线裂纹柱体扭转问题的新边界元法

研究了带曲线裂纹柱体的扭转断裂问题,推导出了可以直接应用于任意形状截面含有任意形状曲线裂纹的柱体扭转问题的新的边界积分方程,并建立了带裂纹柱体扭转问题的边界元数值计算方法,提出了裂纹尖端的奇异元和线性元插值模型,给出了抗扭刚度和应力强度因子的边界元计算公式。该文对含有圆弧裂纹、曲线裂纹及直线裂纹的不同截面形状柱体的典型问题进行了数值计算,所得结果证明了边界元方法的正确性和有效性。     

Efficient solution of multiple cracks in great number using eigen COD boundary integral equations with iteration procedure

A newly developed computational approach is proposed in the paper for the analysis of multiple crack problems based on the eigen crack opening displacement (COD) boundary integral equations. The eigen COD particularly refers to a crack in an infinite domain under fictitious traction acting on the crack surface. With the concept of eigen COD, the multiple cracks in great number can be solved by using the conventional displacement discontinuity boundary integral equations in an iterative fashion with a small size of system matrix to determine all the unknown CODs step by step. To deal with the interactions among cracks for multiple crack problems, all cracks in the problem are divided into two groups, namely the adjacent group and the far-field group, according to the distance to the current crack in consideration. The adjacent group contains cracks with relatively small distances but strong effects to the current crack, while the others, the cracks of far-field group are composed of those with relatively large distances. Correspondingly, the eigen COD of the current crack is computed in two parts. The first part is computed by using the fictitious tractions of adjacent cracks via the local Eshelby matrix derived from the traction boundary integral equations in discretized form, while the second part is computed by using those of far-field cracks so that the high computational efficiency can be achieved in the proposed approach. The numerical results of the proposed approach are compared not only with those using the dual boundary integral equations (D-BIE) and the BIE with numerical Green's functions (NGF) but also with those of the analytical solutions in literature. The effectiveness and the efficiency of the proposed approach is verified. Numerical examples are provided for the stress intensity factors of cracks, up to several thousands in number, in both the finite and infinite plates.     

Scattering of antiplane shear wave by a kinked crack

In this paper the scattering of antiplane shear waves by a kinked crack for a linearly elastic medium is considered. In order to solve the proposed problem, at first the broken crack problem is reduced to two coupled single cracks. Fourier integral transform method is employed to calculate the scattered field of a single crack. In order to derive the Cauchy type integral equations of a broken crack and analyze the singular stresses at the breakpoint, the scattered field of a single crack is separated into a singular part and a bounded part. The single crack solution is applied to derive the generalized Cauchy type integral equations of a broken crack. The singular stress and singular stress order are analyzed in the paper and the dynamic stress intensity factor (DSIF) at breakpoint is defined. Numerical solution of the obtained Cauchy type integral equations gives the DSIF at the crack tips and at the breakpoint. Comparison of the present results in some special cases with the known results confirms the proposed method. Some typical numerical results and corresponding analysis are presented at the end of the paper.     

An incremental formulation for the prediction of two-dimensional fatigue crack growth with curved paths

This paper presents a new incremental formulation for predicting the curved growth paths of two-dimensional fatigue cracks. The displacement and traction boundary integral equations (BIEs) are employed to calculate responses of a linear elastic cracked body. The Paris law and the principle of local symmetry are adopted for defining the growth rate and direction of a fatigue crack, respectively. The three governing equations, i.e. the BIEs, the Paris law and the local symmetry condition, are non-linear with respect to the crack growth path and unknowns on the boundary. Iterative forms of three governing equations are derived to solve problems of the fatigue crack growth by the Newton–Raphson method. The incremental crack path is modelled as a parabola defined by the crack-tip position, and the trapezoidal rule is employed to integrate the Paris law. The validity of the proposed method is demonstrated by two numerical examples of plates with an edge crack. Copyright © 2007 John Wiley & Sons, Ltd.     

A plastic fracture mechanics model for characterization of multiaxial low-cycle fatigue

The near-tip stress and strain fields of small cracks in power-law hardening materials are investigated under plane-stress, general yielding, and mixed mode I and II conditions by finite element analyses. The characteristics of the near-tip strain fields suggest that the experimental observations of shallow surface cracks (Case A cracks) propagating in the maximum shear strain direction can be explained by a fracture mechanics crack growth criterion based on the maximum effective strain of the near-tip fields for small cracks under general yielding conditions. The constant effective stress contours representing the intense straining zones near the tip are also presented. The results of the J integral from finite element analyses are used to correlate to a fatigue crack growth criterion for Case A cracks. Based on the concept of the characterization of fatigue crack growth by the cyclic J integral, the trend of constant J contours on the Γ-plane for Case A cracks compares well with those of constant fatigue life and constant crack growth rate obtained from experiments.     

Singular integral equation method for multiple Zener–Stroh crack problems in antiplane elasticity

In this paper, the multiple Zener–Stroh crack problems in anti-plane elasticity are studied. The crack faces are assumed to be traction free, and dislocation distributions on the cracks are chosen as the unknown functions in the solution. The singular integral equations for the problem are obtained. The constraint equations are also derived from the condition of the accumulation of dislocation on the cracks. After solving the integral equations, the stress intensity factors at crack tips can be evaluated immediately. Numerical examples are given. It is found that interactions between the Zener–Stroh cracks are quite different from those for the Griffith cracks, in qualitative and quantitative aspects.     

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.     

The mixed-type integral equations for transient elastodynamic plane crack problems

In the present paper, by use of the boundary integral equation method and the techniques of Green fundamental solution and singularity analysis, the dynamic infinite plane crack problem is investigated. For the first time, the problem is reduced to solving a system of mixed-typed integral equations in Laplace transform domain. The equations consist of ordinary boundary integral equations along the outer boundary and Cauchy singular integral equations along the crack line. The equations obtained are strictly proved to be equivalent with the dual integral equations obtained by Sih in the special case of dynamic Griffith crack problem. The mixed-type integral equations can be solved by combining the numerical method of singular integral equation with the ordinary boundary element method. Further use the numerical method for Laplace transform, several typical examples are calculated and their dynamic stress intensity factors are obtained. The results show that the method proposed is successful and can be used to solve more complicated problems.     

A remark on the direct numerical determination of stress intensity factors at crack tips

A numerical method for the direct determination of stress intensity factors at crack tips from the numerical solution of the corresponding singular integral equations is proposed. This method is based on the Gauss-Chebyshev method for the numerical solution of singular integral equations and is shown to be equivalent to the Lobatto-Chebyshev method for the numerical solution of the same class of equations.     

Frobenius’ method for curved cracks

The distribution of stresses produced by an undulated crack in a plane elastic solid, and in particular, at its tips where stresses approach infinity, requires the solution of two coupled singular integral equations. Except for simple crack geometries such as rectilinear and circular arcs in infinite plates, for which explicit analytic solutions have been obtained, the integral equations require numerical solutions. We propose a treatment of the integral equations by Frobenius’ method, which is particularly suitable for evaluating the stress intensity factors of slightly curved cracks.     

Multiple crack problems of antiplane elasticity in an infinite body

Twe elementary solutions are presented for case of a pair of normal or tangential concentrated unit forces acting at a point of both edges of a single crack in an infinite plane isotropic elastic medium. Using these two elementary solutions and the principle of superposition, we found that the multiple crack problems can be easily converted into a system of Fredholm integral equations. Finally, the system obtained is solved numerically and the values of the stress intensity factors at the crack tips can be easily calculated. Two numerical examples are given in this paper. A system of Fredholm integral equations is complex form is also presented. We found that the system of Fredholm integral equations can be easily reduced from the system of singular integral equations given by Panasyuk[1]     

Dual boundary element analysis for fatigue behavior of missile structures

Abstract

The fatigue behavior of a crack in a missile structure is studied using the dual boundary integral equations developed by Hong and Chen (1988). This method, which incorporates two independent boundary integral equations, uses the displacement equation to model one of the crack boundaries and the traction equation to the other. A single domain approach can be performed efficiently. The stress intensity factors are calculated and the paths of crack growth are predicted. In order to evaluate the results of dual BEM, four examples with FEM results are provided. Two practical examples, cracks in a V‐band and a solid propellant motor are studied and are compared with experimental data. Good agreement is found.