SUMMARYStatic discontinuities (i.e. distributions of forces along a line or a surface, implying a jump of tractions across it) and kinematic (displacement) discontinuities are considered simultaneously as sources acting on the unbounded elastic space R, along the boundary r of a homogeneous elastic body R embedded in R,. The auxiliary elastic state thus generated in the body is associated with the actual elastic state by a Betti reciprocity equation. Using suitable discretizations of actual and fictitious boundary variables, a symmetric Galerkin formulation of the direct boundary element method is generated.The foliowing topics are addressed: reciprocity relations among kernels with particular attention to the role of singularities; conditions to be satisfied by the boundary field modelling in order to achieve the symmetry of the coefficient matrix; variational properties of the solution.With reference to two-dimensional problems, a technique based on a complex-variable formalism is proposed to perform the double integrations involved in this apprgach. An implementation of this technique for elastic analysis is described assuming straight elements, with continuous linear displacements and piecewise-constant tractions; all the double integrations are carried out analytically.Comparisons, from the computational standpoint, with the traditional non-symmetric method based on collocation and single integration, demonstrate the effectiveness of the present approach.
The subject of this paper is the formulation and the implementation of the symmetric Galerkin BEM for three-dimensional linear elastic fracture mechanics problems. A regularized version of the displacement and traction equations in weak form is adopted and the integration techniques utilized for the evaluation of the double surface integrals appearing in the discretized equations are detailed. By using quadratic isoparametric quadrilateral and triangular elements, some example crack problems are solved to assess the efficiency and robustness of the method.Keywords Fracture, Boundary element method, Galerkin approach IntroductionIn the numerical modelling of linear elastic fracture mechanics problems, boundary element methods (BEM) have distinct advantages over domain approaches, especially when cracks are directly represented as displacement discontinuity loci and the traction integral equation is employed to enforce static conditions on the crack itself. The displacement discontinuity method [6], the dual BEM [11,13,21] and the symmetric Galerkin BEM [4] share the above features and permit single domain formulations for problems with single or multiple cracks embedded in finite bodies or in the infinite medium. However the SGBEM differs from the others in that it is based on a variational (weak) version of the integral equations, thus entailing double integrations, and leads (through an appropriate discretization scheme) to matrix operators which exhibit symmetry and sign-definiteness. A 1998 review paper on the SGBEM [4] shows that the method has been the subject of extensive investigations since it was first proposed in 1979 [24] and the most recent literature gives evidence of a lively interest towards this methodology (see e.g. [10,17,19]).While for 2D linear elasticity a number of implementations of the SGBEM into computer codes have been presented (see, e.g. [7,25]), some of which also addressing crack problems, only few 3D implementations for elastic problems, with or without cracks, have been described in the literature [15,27]. The evaluation of the double surface integrals in the singular cases, not an easy task especially if general isoparametric elements are employed, represents probably the main obstacle which has hampered the application of the method in the 3D context.The present work addresses the application of the SGBEM in the context of 3D linear elastic fracture mechanics using a ''regularized'' symmetric formulation which is essentially the same as that expounded in [5,8,18]; a first fairly general implementation of this approach is documented in [15] where, however, no details are given on the adopted integration techniques. On the contrary here focus is set on the development of efficient algorithms for the crucial singular double surface integrals, according to the new schemes introduced in [1, 9, 23] and further expanded so as to permit the combined use of quadrilateral and triangular quadratic isoparametric elements. With reference to both cracks in the unbounded space and in finite bo...
In this paper two procedures are developed for the identification of the parameters contained in an orthotropic elastic-plastic-hardening model for free standing foils, particularly of paper and paperboard. The experimental data considered are provided by cruciform tests and digital image correlation. A simplified version of the constitutive model proposed by Xia et al. (Int J Solids Struct 39:4053-4071, 2002) is adopted. The inverse analysis is comparatively performed by the following alternative computational methodologies: (a) mathematical programming by a trust-region algorithm; (b) proper orthogonal decomposition and artificial neural network. The second procedure rests on preparatory once-for-all computations and turns out to be applicable economically and routinely in industrial environments.
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