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Shape and Topology Sensitivity Analysis for Elastic Bodies with Rigid Inclusions and Cracks

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Computer Methods in Mechanics

Part of the book series: Advanced Structured Materials ((STRUCTMAT,volume 1))

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Abstract

In the keynote lecture we describe how the asymptotic analysis in singularly perturbed domains can be employed to determine effectively the influence of nucleation of small voids on some shape functionals. To this end the classical shape sensitivity analysis is combined with asymptotic expansions in order to determine the singular limits of shape derivatives which are called topological derivatives of shape functionals. The topological derivatives are determined for elastic bodies weakened by cracks on boundaries of rigid inclusions. On the crack faces the nonpenetration conditions are prescribed, such conditions are non linear and assure that the displacements of the crack lips or surfaces cannot penetrate each other. Small voids are located on the finite distance from the crack, so there is no interaction between the crack and the voids.

In such a way nucleations of small voids can be implemented in the numerical procedures of optimum design or solution of inverse problems. A nonlinear model in the framework of damage theory is presented in details for modeling and sensitivity analysis. The example of an elastic body with a rigid inclusion and a crack located at the boundary of the inclusion is considered. The asymptotic analysis which leads to topological derivatives is performed in two and three spatial dimensions. The derived formulas can be used in numerical methods of shape and topology optimization.

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Author information

Authors and Affiliations

  1. Laboratoire de Mathématiques, Institut Elie Cartan, UMR 7502 Nancy-Université-CNRS-INRIA, Université Henri Poincaré Nancy 1, B.P. 239, 54506, Vandoeuvre lés Nancy Cedex, France

    Jan Sokołowski

  2. Systems Research Institute of the Polish Academy of Sciences, ul. Newelska 6, 01-447, Warszawa, Poland

    Antoni Żochowski

Authors
  1. Jan Sokołowski
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  2. Antoni Żochowski
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Editors and Affiliations

  1. Inst. Building Engineering, University of Zielona Góra, ul. Z. Szafrana 1, 65-516, Zielona Góra, Poland

    Mieczysław Kuczma

  2. Rue Diderot 4, 13469, Berlin, Germany

    Krzysztof Wilmanski

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Sokołowski, J., Żochowski, A. (2010). Shape and Topology Sensitivity Analysis for Elastic Bodies with Rigid Inclusions and Cracks. In: Kuczma, M., Wilmanski, K. (eds) Computer Methods in Mechanics. Advanced Structured Materials, vol 1. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-05241-5_5

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