Analysis of a row of elliptical inclusions in a plate using singular integral equations

This paper deals with the interaction problem of a row of elliptical inclusions under uniaxial tension. The body force method is used to formulate the problem as a system of singular integral equations with Cauchy--type and logarithmic singularities, where the unknowns are densities of body forces distributed in infinite plates that have the same elastic constants as those of the matrix and inclusion. In order to satisfy the boundary conditions along the elliptical boundaries, eight kinds of fundamental density functions, proposed in a previous paper, are applied. In the analysis, the number, shape, and position of inclusions are varied systematically; after which the magnitude and position of the maximum stress are examined. For any fixed shape and size of inclusions, the maximum stress is shown to be linear with the reciprocal of the number of inclusions. The present method is found to yield rapidly converging numerical results for various geometrical conditions of inclusions. This revised version was published online in August 2006 with corrections to the Cover Date.     

Stress analysis of arbitrarily distributed elliptical inclusions under longitudinal shear loading

This paper deals with an interaction problem of arbitrarily distributed elliptical inclusions under longitudinal shear loading. The problem is formulated as a system of singular integral equations with Cauchy-type or logarithmic-type singularities, where unknown functions are the densities of body forces distributed in the longitudinal directions of infinite bodies having the same elastic constants as those of the matrix and inclusions. In order to satisfy the boundary conditions along the elliptical inclusions, four kinds of fundamental density functions are introduced in a similar way of previous papers treating plane stress problems. Then the body force densities are approximated by a linear combination of those fundamental density functions and polynomials. In the analysis, elastic constants of matrix and inclusion are varied systematically; then the magnitude and position of the maximum stress are shown in tables and the stress distributions along the boundary are shown in figures. For any fixed shape, size and elastic constant of inclusions, the relationships between number of inclusions and maximum stress are investigated for several arrangements.     

Stress concentration factors of a general triaxial ellipsoidal cavity

The equivalent inclusion method is applied to solve the stress concentration problem concerning the disturbing effect of a general triaxial ellipsoidal cavity on an otherwise uniform normal stress state. Several useful solutions in simple form for limiting cases are derived and numerical results for general cases are obtained. These findings show the general features of the stress concentration factors around the base equator of the cavity. It is found that (1) when Poisson's ratio of the material is zero, the stress concentration factor does not vary along the equator of the cavity; (2) When the aspect ratio c/b of the cavity is very small, the stress concentration factor is also constant along the equator; (3) In general, the variation of stress concentration factors around the equator is less than 0.17 regardless of Poisson's ratio of the material. Thus, it is concluded that the stress concentration factor may be treated as constant around the equator of an ellipsoidal cavity with only a slight error.     

层内混杂复合材料断裂纤维附近的应力重分布

研究了层内混杂复合材料的高模量纤维断裂引起的邻近各层应力重分布问题。通过建立适当的计算模型,利用二维弹性力学精确解和付氏变换法,建立问题的奇异积分方程组,通过求解方程组,计算高、低模量层和基体层的应力集中因子,计算结果对"混杂效应"作出了理论解释,并与一般采用的Shear-Lag理论计算结果作了比较,本文结果更精确、合理,可应用于层内混杂复合材料的设计。     

Variations of stress intensity factor of a semi-elliptical surface crack subjected to mixed mode loading

Maximum stress intensity factors of a surface crack usually appear at the deepest point of the crack, or a certain point along crack front near the free surface depending on the aspect ratio of the crack. However, generally it has been difficult to obtain smooth distributions of stress intensity factors along the crack front accurately due to the effect of corner point singularity. It is known that the stress singularity at a corner point where the front of 3 D cracks intersect free surface is depend on Poisson's ratio and different from the one of ordinary crack. In this paper, a singular integral equation method is applied to calculate the stress intensity factor along crack front of a 3-D semi-elliptical surface crack in a semi-infinite body under mixed mode loading. The body force method is used to formulate the problem as a system of singular integral equations with singularities of the form r ⁻³ using the stress field induced by a force doublet in a semi-infinite body as fundamental solution. In the numerical calculation, unknown body force densities are approximated by using fundamental density functions and polynomials. The results show that the present method yields smooth variations of mixed modes stress intensity factors along the crack front accurately. Distributions of stress intensity factors are indicated in tables and figures with varying the elliptical shape and Poisson's ratio.     

Interaction between crack and arbitrarily shaped hole with stress and displacement boundaries

The interaction between a crack and an arbitrarily shaped hole under stress and displacement boundaries in an infinite plane subjected to a remote uniform load is studied. The Green's functions of a point dislocation for the problems are derived and are then used to analyze the interaction problem. The superposition principle is employed to reduce the original problem to two subsidiary problems. The complex stress functions of each problem are composed of two parts, in which the second parts are always holomorphic. Using analytical continuation in conjunction with rational mapping function, the stress functions are obtained in closed form. The interaction of a hole or an inclusion with a crack is solved using dislocations to model the crack and solving a system of singular integral equations. Stress intensity factors for crack tips and stress concentration factors for inclusion corner are determined and plotted for various cases. The affecting ranges of hole and inclusion are investigated.     

Variation of mixed modes stress intensity factors of an inclined semi-elliptical surface crack

In this paper, a singular integral equation method is applied to calculate the stress intensity factor along crack front of a 3D inclined semi-elliptical surface crack in a semi-infinite body under tension. The stress field induced by displacement discontinuities in a semi-infinite body is used as the fundamental solution. Then, the problem is formulated as a system of integral equations with singularities of the form r ^–3. In the numerical calculation, the unknown body force doublets are approximated by the product of fundamental density functions and polynomials. The results show that the present method yields smooth variations of mixed modes stress intensity factors along the crack front accurately for various geometrical conditions. The effects of inclination angle, elliptical shape, and Poisson's ratio are considered in the analysis. Crack mouth opening displacements are shown in figures to predict the crack depth and inclination angle. When the inclination angle is 60 degree, the mode I stress intensity factor F _I has negative value in the limited region near free surface. Therefore, the actual crack surface seems to contact each other near the surface.     

Variation of stress intensity factor along the front of a 3D rectangular crack by using a singular integral equation method

In this paper a singular integral equation method is applied to calculate the distribution of stress intensity factor along the crack front of a 3D rectangular crack. The stress field induced by a body force doublet in an infinite body is used as the fundamental solution. Then, the problem is formulated as an integral equation with a singularity of the form of r ^–3. In solving the integral equation, the unknown functions of body force densities are approximated by the product of a polynomial and a fundamental density function, which expresses stress singularity along the crack front in an infinite body. 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.     

Interaction of time-harmonic SH waves with a crack-like inclusion: edge region analysis

Elastodynamic stress concentration near the tips of a crack-like inclusion of finite length generated by the diffraction of high-freqency time-harmonic SH waves is analyzed. It is shown that the stress intensity factors at the tips of inclusion are provided by the fields describing the solution of the static boundary value problem for a semi-infinite strip and edge waves travelling between the two inclusion tips. The solution to the problem is expressed in a closed form that is computationally effective and yields accurate results in the resonance region of dimensionless wave numbers. This revised version was published online in July 2006 with corrections to the Cover Date.     

Stress analysis with arbitrary body force by boundary element method

Linear stress analysis without body force can be easily carried out by means of the boundary element method. Some cases of linear stress analysis with body force can also be solved without the domain integral. However domain integrals are generally necessary to solve the linear stress problems with complicated body forces. This paper shows that the linear stress problems with complicated body forces can be solved approximately without the domain integral. In order to solve these problems, the domain is divided into small areas using contour lines of body force. In these areas, the distributions of body force are assumed approximately to satisfy the Laplace equation.     

Stress concentration formula useful for any dimensions of shoulder fillet in a round bar under tension and bending

In this work stress concentration factors (SCFs), K_t for a round bar with a fillet are considered on the basis of exact solutions, now available for special cases, and accurate numerical results. Then, a convenient K_t formula useful for any dimensions of the fillet is proposed. The conclusions can be summarised as follows: (i) For the limiting cases of deep (d) and shallow (s) fillet, the body force method is used to calculate the K_t values. Then, the formula are obtained as K_td and K_ts. (ii) On the one hand, upon comparison of K_t and K_td, it is found that K_t is nearly equal to K_td if the fillet is deep or blunt. (iii) On the other hand, if the fillet is sharp or shallow, K_t is mainly controlled by K_ts and the fillet depth. (iv) The fillet shape is classified into several groups according to the fillet radius and fillet depth. Then the least squares method is applied for calculation of K_t/K_td and K_t/K_ts. (v) Finally, a convenient formula is proposed that is useful for any dimensions of fillet in a round bar. The formula give SCFs with less than 1% error in most cases for any dimensions of fillet under tension and bending.     

Determination of stress intensity factors for bimaterial interface stationary rigid line inclusions by boundary element method

The stress intensity factors for a rigid line inclusion lying along a bimaterial interface are calculated by the boundary element method with the multiregion and the discontinuous traction singular elements. The relationships between the stress intensity factors and the inclusion surface stresses are derived. The numerically computed stress intensity factors for the bimaterial interface rigid line inclusion in the infinite body are proved to be in good agreement within 3% when compared with the previous exact solutions. In the finite bimaterial models, the stress intensity factors for the center and edge rigid line inclusions at the interface are computed with the variation of the rigid line inclusion length and the shear modulus ratio under the uniaxial and biaxial loading conditions.     

Plane problem of macro-microcrack interaction taking account of crack closure

The fracture stability of a macrocrack under the tensile and shear loading in the presence of a system of microcracks is analysed. Interaction of cracks leads to full or partial closure of the crack edges. The boundary problem is formulated and a solution is obtained by the small parameter method. Domains where microcracks are closed, and regions where microcracks cause full or partial closure of the macrocrack are found. The influence of crack contact on the stress intensity coefficient is analysed under the friction free assumption.     

Stress intensity factor for a Griffith crack interacting with a coated inclusion

Stress investigation for the interaction problem between a coated circular inclusion and a near-by line crack has been carried out. The crack and the coated inclusion (a coated fiber) are embedded in an infinitely extended isotropic matrix, with the crack being along the radial direction of the inclusion. Two loading conditions, namely, the tensile and shear loading ones are considered. During the solution procedure, the crack is treated as a continuous distribution of edge dislocations. By using the solution of an edge dislocation near a coated fiber as the Green's function, the problem is formulated into a set of singular integral equations which are solved by Erdogan and Gupta (1972) method. The expressions for the stress intensity factors of the crack are then obtained in terms of the asymptotic values of the dislocation density functions evaluated from the integral equations. Several numerical examples are given for various material and geometric parameters. The solutions obtained from the integral equations have been checked and confirmed by the finite element analysis results.     

Analysis of a short interfacial crack from the corner of a rectangular inclusion

A useful method is proposed to analyze a short interfacial crack emanating from the corner of a rectangular inclusion. We first analyze the singular stress field (and the corresponding singularity intensity factor H) without the crack in an infinite medium having the rectangular inclusion. The singular stress field (and the corresponding stress intensity factor K) at the tip of the short interfacial crack lying in the interface of the rectangular inclusion is also analyzed, giving the relation between H and K. With this relation, the stress intensity factor K is easily obtained for the case of a short interfacial crack from the corner of a different rectangular inclusion with different external boundary. This method is based on the assumption that the singular K-field is embedded in another singular H-field, which is much smaller than the specimen geometry. To meet the assumption, it is found here that the eigenfunction corresponding to the next smallest eigenvalue of the singular H-field has to be considered. An example is presented to show the usefulness of the present method, where a short interfacial crack from the corner of a rectangular lead frame in epoxy compound used in electronic packaging is analyzed. It is found that the result of the present method is in good agreement with that of the well-known method.     

An Analysis of a Quarter-Infinite Solid with a Corner Crack of Arbitrary Shape

In this paper, a versatile body force method for a quarter-infinite solid with a corner crack of arbitrary shape is proposed under two types of pressure: constant and linear. New numerical results are obtained for different corner crack cases. Fatigue crack growth from a corner crack has been analysed successively with the present method. Moreover, the stress intensity factor of a corner crack is proposed in a simple form for an arbitrary shape.     

Tangential derivative of singular boundary integrals with respect to the position of collocation points

This paper investigates the evaluation of the sensitivity, with respect to tangential perturbations of the singular point, of boundary integrals having either weak or strong singularity. Both scalar potential and elastic problems are considered. A proper definition of the derivative of a strongly singular integral with respect to singular point perturbations should accommodate the concomitant perturbation of the vanishing exclusion neighbourhood involved in the limiting process used in the definition of the integral itself. This is done here by esorting to a shape sensitivity approach, considering a particular class of infinitesimal domain perturbations that ‘move’ individual points, and especially the singular point, but leave the initial domain globally unchanged. This somewhat indirect strategy provides a proper mathematical setting for the analysis. Moreover, the resulting sensitivity expressions apply to arbitrary potential-type integrals with densities only subjected to some regularity requirements at the singular point, and thus are applicable to approximate as well as exact BEM solutions. Quite remarkable is the fact that the analysis is applicable when the singular point is located on an edge and simply continuous elements are used. The hypersingular BIE residual function is found to be equal to the derivative of the strongly singular BIE residual when the same values of the boundary variables are substituted in both SBIE and HBIE formulations, with interesting consequences for some error indicator computation strategies. © 1998 John Wiley & Sons, Ltd.     

On the efficient computation of the stress components near a closed boundary in plane elasticity by using classical complex boundary integral equations

Complex boundary integral equations (Fredholm‐type regular or Cauchy‐type singular or even Hadamard–Mangler‐type hypersingular) have been used for the numerical solution of general plane isotropic elasticity problems. The related Muskhelishvili and, particularly, Lauricella–Sherman equations are famous in the literature, but several more extensions of the Lauricella–Sherman equations have also been proposed. In this paper it is just mentioned that the stress and displacement components can be very accurately computed near either external or internal simple closed boundaries (for anyone of the above equations: regular or singular or hypersingular, but with a prerequisite their actual numerical solution) through the appropriate use of the even more classical elementary Cauchy theorem in complex analysis. This approach has been already used for the accurate numerical computation of analytic functions and their derivatives by Ioakimidis, Papadakis and Perdios (BIT 1991; 31 : 276–285), without applications to elasticity problems, but here the much more complicated case of the elastic complex potentials is studied even when just an appropriate non‐analytic complex density function (such as an edge dislocation/loading distribution density) is numerically available on the boundary. The present results are also directly applicable to inclusion problems, anisotropic elasticity, antiplane elasticity and classical two‐dimensional fluid dynamics, but, unfortunately, not to crack problems in fracture mechanics. Brief numerical results (for the complex potentials), showing the dramatic increase of the computational accuracy, are also displayed and few generalizations proposed. Copyright © 2000 John Wiley & Sons, Ltd.     

Stress concentration formulae useful for any shape of notch in a round test specimen under tension and under bending

In this work stress concentration factors, K_t , for a round bar with a circular-arc or V-shaped notch are considered on the basis of exact solutions for special cases and accurate numerical results. Then, a set of K_t formulae useful for any shape of notch is proposed. The conclusions can be summarized as follows. (i) For the limiting cases of deep (d) and shallow (s) notches, the body force method is used to calculate the K_t values. Then, the formulae are obtained as K_td and K_ts . (ii) On the one hand, upon comparison of K_t and K_td it is found that K_t is nearly equal to K_td if the notch is deep or blunt. (iii) On the other hand, if the notch is sharp or shallow, K_t is mainly controlled by K_ts and the notch depth. (iv) The notch shape is classified into several groups according to the notch radius and notch depth. Then, the least-squares method is applied for the calculation of K_t /K_td and K_t /K_ts . (v) Finally, a set of convenient formulae is proposed that are useful for any shape of notch in a round test specimen. The formulae give SCFs with <1% error for any shape of notch.     

Interaction effect of stress intensity factors for any number of collinear interface cracks

In this study, the stress intensity factors for any number of interface cracks are calculated for various spacings, elastic constants and number of cracks and the interaction effect of interface cracks is discussed. The problem is formulated as a system of singular integral equations on the basis of the body force method. In the numerical analysis, the unknown functions of the body force densities which satisfy the boundary conditions are expressed by the products of fundamental density functions and power series. Here, the fundamental density functions are chosen to express the stress field due to a single interface crack exactly. The accuracy of the present analysis is verified by comparing the present results with the results obtained by other researchers and examining the compliance with boundary conditions. The calculation shows that the present method gives rapidly converging numerical results for those problems as well as ordinary crack problems in homogeneous materials. The interaction effect of interface crack appears in a similar way to ordinary collinear cracks having the same geometrical condition and the maximum stress intensity factor is shown to be linearly related to the reciprocal of number of interface cracks. This revised version was published online in August 2006 with corrections to the Cover Date.