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Contribution à l'étude des méthodes des équations intégrales et à leur mise en œuvre numérique en élastoplasticité

Abstract : This study deals with Boundary-Domain Element Methods (BDEM) applied to small-strain elastoplasticity. The governing equations are obtained by using the Somigliana displacement identity with initial strains and unlike in Finite Element Methods (FEM), the main unknown of the resulting formulation is the strain or the stress field in the potentially plastic region. In order to solve the problem, two numerical procedures (Collocation and Galerkin methods) are implemented. It is worth noting that the regularity requirements of the unknown fields are not identical in both approaches. An implicit scheme is adopted for constitutive time integration, which leads to the well-known Return Mapping Algorithm ; moreover the Consistent Tangent Operator (CTO) is inserted in the iterative Newton-Raphson method. An unified presentation of the regularisation techniques at regular points of the boundary is given. These concepts are extended to singular boundary points by introducing the "reconstituted displacement gradient". As a result, a new integral representation of the strain field, even valid for singular boundary points, is proposed and implemented in a Collocation method. This evaluation of the boundary strain gives rise to a new Traction Boundary Integral Equation (TBIE), which can be written at boundary nodes. It is also shown that the governing equations, which take into account the Somigliana identity, result from the stationarity conditions of a fully regularised functional. This result gives an energetic meaning to the BDEM formulation. The numerical implementation of the resulting Galerkin scheme is described and a simple quadrature rule is presented to evaluate the singular double integrals. This procedure however needs to be optimised in future work. Some numerical results validate the proposed formulations and algorithms.
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Submitted on : Friday, December 24, 2010 - 2:16:12 PM
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  • HAL Id : tel-00550209, version 1



Bertrand Burgardt. Contribution à l'étude des méthodes des équations intégrales et à leur mise en œuvre numérique en élastoplasticité. Mécanique []. Ecole Centrale de Nantes (ECN), 1999. Français. ⟨tel-00550209⟩



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