A fast solution method for three-dimensional many-particle problems of linear elasticity

A boundary element method for solving three-dimensional linear elasticity problems that involve a large number of particles embedded in a binder is introduced. The proposed method relies on an iterative solution strategy in which matrix–vector multiplication is performed with the fast multipole method. As a result the method is capable of solving problems with N unknowns using only 𝒪(N) memory and 𝒪(N) operations. Results are given for problems with hundreds of particles in which N=𝒪(10⁵). © 1998 John Wiley & Sons, Ltd.     

Method of effective stress field in three-dimensional problems of interaction of plane cracks

We propose a modification of the method of boundary integral equations capable of the efficient solution of the problems of interaction of large numbers of arbitrarily oriented plane cracks on the engineering level. The approach is based on the determination of the effective stress field formed in the vicinity of a fixed crack by neighboring cracks interacting with this crack. The reliability of the results obtained by the method of effective stress field is checked by comparing with the exact solution of the problem of interaction of two plane circular cracks for different mutual orientations of the cracks. The efficiency of the proposed approach is illustrated by an example of interaction of an aperiodic system of six cracks located in different planes.     

Solution of plane elasticity problems for orthotropic bodies with cracks by the displacement discontinuity method

 The force problem of fracture mechanics for orthotropic body is solved by boundary element method. The equations of plane strain state are used. The key point of this paper is obtaining the influence functions for finite segment with constant displacement discontinuities in the main axes of orthotropy. They have been deduced by means of two-dimensional Fourier transform and theory of generalized functions. The straight crack under the action of uniform tension in the infinity is considered by using obtained influence functions. The calculations were carried out for isotropic and two orthotropic materials. Received 6 November 2000     

A remark on the solution of the integral equation of planar cracks in three-dimensional elasticity

It is proposed that the classical integral equation of a planar crack under normal loading in three-dimensional isotropic elasticity is solved after an integration with respect to the Laplace operator. Then this equation coincides with the fundamental equation of contact problems for an isotropic elastic half-space. The problem of an elliptical crack under a constant normal loading is solved by the present approach.     

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.     

Singular boundary elements for three-dimensional elasticity problems

The focus of this paper is a set of semi-discontinuous, traction-singular surface elements introduced to help the rigorous boundary integral analysis of problems in three-dimensional solid mechanics. In contrast to the singular boundary elements developed for linear fracture mechanics where the square-root singularity is of primary interest, traction shape functions featuring the proposed four- and eight-node boundary elements can be used to represent power-type singularities of arbitrary order, such as those arising at non-smooth material boundaries and interfaces. Apart from being capable of rigorously handling traction singularities and discontinuities across the domain boundaries and interfaces, these elements also permit a smooth transition to adjacent regular elements. Complemented with a family of suitable displacement and geometry shape functions, the singular surface elements are incorporated into a regularized boundary integral equation method and shown, through a set of benchmark results, to perform well for both static and dynamic problems.     

Application of finite-part integrals to the singular integral equations of crack problems in plane and three-dimensional elasticity

Summary A modification of the form of the singular integral equation for the problem of a plane crack of arbitrary shape in a three-dimensional isotropic elastic medium is proposed. This modification consists in the incorporation of the Laplace operator into the integrand. The integral must now be interpreted as a finite-part integral. The new singular integral equation is equivalent to the original one, but simpler in form. Moreovet, its form suggests a new approach for its numerical solution, based on quadrature rules for one-dimensional finite part integrals with a singularity of order two. A very simple application to the problem of a penny-shaped crack under constant pressure is also made. Moreover, the case of straight crack problems in plane isotropic elasticity is also considered in detail and the corresponding results for this special case are also derived.With 2 Figures     

Solution of plane elasticity problems by the displacement discontinuity method. I. Infinite body solution

This paper is concerned with the development of a numerical procedure for solving complex boundary value problems in plane elastostatics. This procedure—the displacement discontinuity method—consists simply of placing N displacement discontinuities of unknown magnitude along the boundaries of the region to be analyzed, then setting up and solving a system of algebraic equations to find the discontinuity values that produce prescribed boundary tractions or displacements. The displacement discontinuity method is in some respects similar to integral equation or ‘influence function’ techniques, and contrasts with finite difference and finite element procedures in that approximations are made only on the boundary contours, and not in the field. The method is illustrated by comparing computed results with the analytical solutions of two boundary value problems: a circular disc subjected to diametral compression, and a circular hole in an infinite plate under a uniaxial stress field. In both cases the numerical results are in excellent agreement with the exact solutions.     

Adaptive Poly-FEM for the analysis of plane elasticity problems

In this work, we present an adaptive polygonal finite element method (Poly-FEM) for the analysis of two-dimensional plane elasticity problems. The generation of meshes consisting of n ? sided polygonal finite elements is based on the generation of a centroidal Voronoi tessellation (CVT). An unstructured tessellation of a scattered point set, that minimally covers the proximal space around each point in the point set, is generated whereby the method also includes tessellation of nonconvex domains. In this work, we propose a region by region adaptive polygonal element mesh generation. A patch recovery type of stress smoothing technique that utilizes polygonal element patches for obtaining smooth stresses is proposed for obtaining the smoothed finite element stresses. A recovery type a ? posteriori error estimator that estimates the energy norm of the error from the recovered solution is then adopted for the Poly-FEM. The refinement of the polygonal elements is then made on an region by region basis through a refinement index. For the numerical integration of the Galerkin weak form over polygonal finite element domains, we resort to classical Gaussian quadrature applied to triangular subdomains of each polygonal element. Numerical examples of two-dimensional plane elasticity problems are presented to demonstrate the efficiency of the proposed adaptive Poly-FEM.     

Application of the Trefftz method in plane elasticity problems

Direct and indirect approximations using sets of non-singular, complete Trefftz functions, i.e. the complete systems of solutions, have been successfully applied to harmonic problems.¹ In this paper, the procedure is applied to a more complex situation—plane elasticity problems. The examples show that the present method can avoid the difficulties relating to singular integration and has good accuracy compared with traditional boundary elements.     

A hybrid-element approach to crack problems in plane elasticity

The hybrid-element concept and the complex variable technique have been adopted for constructing a special super-element to be used jointly with conventional finite elements for the analysis of elastic stress intensity factors for plane cracks. The use of the complex variable technique permits the proper consideration of the stress intensity at the crack tip, and it also leads to very efficient programming. The use of such a super-element in the finite element solution has been shown to be highly accurate when only a very coarse element mesh is used near the crack.     

An analysis of three-dimensional planar embedded cracks subjected to uniform tensile stress

In this paper we describe an analytical methodology for calculating Stress Intensity Factors (SIF) on planar embedded cracks with an arbitrarily shaped front. The approach is based on a first order expansion of the celebrated integral of Oore-Burns and the actual shapes of three-dimensional planar flaws are analysed in terms of homotopy transformations of a reference disk.The solution is proposed in terms of Fourier series and the first order approximation of the coefficients is given independently from the homotopy transformations.The comparison with numerical results, taken from scientific literature, indicates that the proposed equation is very accurate when the flaw presents a small deviation from the circular shape. Finally, the closed form solution is used to predict the SIF of many types of convex and non-convex planar flaws present in engineering components such as welded structures or casting components.     

Boundary element discretization of plane elasticity and plate bending problems

The paper deals with the discretization of the integral equations arising in the boundary formulation of plane elasticity and plate bending problems. Particular attention is paid to the efficiency of the interpolation used in approximating the boundary quantities and to the precision and computational convenience in evaluating the boundary integrals. The proposed discretization model is based on the use of a quadratic B-spline approximation to represent the boundary variables and on the results from the analytical integration to compute the boundary coefficients. The advantages are those of accuracy and the saving of computer time. Some numerical results allow an analysis of the performance of the model.     

A three-dimensional elasticity solution for functionally graded rotating disks

A semi-analytical three-dimensional elasticity solution for rotating functionally graded disks for both of hollow and solid disks is presented. The aim is to generalize an available two-dimensional plane-stress solution to a three-dimensional one. Although for the thin disks problems the two-dimensional solution provides appropriate results, for the thick disks, a three-dimensional elasticity solution should be considered to avoid poor results. It is shown that although the plane-stress solution satisfies all the governing three-dimensional equations of motion and boundary conditions, it fails to give a compatible three-dimensional strain field. A valid three-dimensional solution has been introduced by modifying the plane-stress solution.     

Boundary element formulation for plane problems in couple stress elasticity

Couple‐stresses are introduced to account for the microstructure of a material within the framework of continuum mechanics. Linear isotropic versions of such materials possess a characteristic material length l that becomes increasingly important as problem dimensions shrink to that level (e.g., as the radius a of a critical hole reduces to a size comparable to l). Consequently, this size‐dependent elastic theory is essential to understand the behavior at micro‐ and nano‐scales and to bridge the atomistic and classical continuum theories. Here we develop an integral representation for two‐dimensional boundary value problems in the newly established fully determinate theory of isotropic couple stress elastic media. The resulting boundary‐only formulation involves displacements, rotations, force‐tractions and moment‐tractions as primary variables. Details on the corresponding numerical implementation within a boundary element method are then provided, with emphasis on kernel singularities and numerical quadrature. Afterwards the new formulation is applied to several computational examples to validate the approach and to explore the consequences of size‐dependent couple stress elasticity. Copyright © 2011 John Wiley & Sons, Ltd.     

On some numerical methods for the solution of the plane elasticity problem for bodies with cracks by means of singular integral equations

Comparison is given of the accuracy of calculation of stress intensity factors at the crack tips by various methods when solving plane elasticity problems for bodies with cruciform and edge cracks. It is shown that, within the range of quadrature formulas for singular integrals discussed, the type of the formula chosen for the solution of an equation, if used correctly, affects negligibly the accuracy of the stress intensity factor evaluation at the crack tip, and in view of this a method is proposed based on simple relationships.

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Résumé On compare l'exactitude des calculs des facteurs d'intensité de contrainte aux extrémités d'une fissure à l'aide de diverses méthodes à l'occasion de la solution de problèmes d'élasticité plane dans des corps présentant des fissures cruciformes et des fissures de bord. On montre que dans les limites des formules de quadrature des intégrales singulières qui sont discutées, le type de formule choisie pour la solution d'une équation n'a qu'une influence négligeable si elle est utilisée correctement sur l'exactitude du facteur d'intensité de contrainte évalué à l'extrémité de la fissure. Dans cette optique, on propose une méthode basée sur des relations simples.

     

Asymptotic solution for long cracks emanated from a pore in compression

The 2-D problem of two collinear radial cracks emanating from a circular pore towards the applied uniaxial compression has been considered. Long crack asymptotic expressions have been obtained for the stress intensity factor and the area of the crack opening. Comparison with numerical results has shown that this solution can still be used even if the crack length reaches half of the pore radius. On the basis of this solution, an approximate method considering the pair of cracks as a single long crack subjected to the stress field generated separately from the pore has been proved to give an acceptable result. The approximate method has also been extended to the 3-D case.The order of authors is determined alphabetically.