Asymptotic analysis based modeling of small inhomogeneity perturbation in solids: two computational scenarios

Eduard Marenić 1 D. Brancherie 2 M Bonnet 3
3 POEMS - Propagation des Ondes : Étude Mathématique et Simulation
Inria Saclay - Ile de France, UMA - Unité de Mathématiques Appliquées, CNRS - Centre National de la Recherche Scientifique : UMR7231
Abstract : The presented work is a step towards designing a numerical strategy capable of assessing the nocivity of a small defect in terms of its size and position in the structure with low computational cost, using only a mesh of the defect-free reference structure. We focus here on presenting two computational scenarios allowing to efficiently evaluate flaw criticality. These scenarios are considering either the effect of a fixed flaw for any evalutaion point in solid, or varying flaws on a fixed evaluation point. 1 Motivation, introduction and problem definition The role played by defects in the initiation and development of rupture is crucial and has to be taken into account in order to realistically describe the behavior till complete failure. The difficulties in that context revolve around (i) the fact that the defect length scale is much smaller than the structure length scale, and (ii) the random nature of their position and size. Even in a purely deterministic approach, taking those defects into consideration by standard models imposes to resort to geometrical discretisations at the defect scale, leading to very costly computations and hindering parametric studies in terms of defect location and characteristics. Our current goal is to design an efficient two-scale numerical strategy which can accurately predict the perturbation in terms of stress caused by an inhomogeneity in elastic (back-ground) material. To make it computationally efficient, the analysis uses only a mesh for the defect-free structure, i.e. the mesh size does not depend on the (small) defect scale. We consider a linearly elastic body occupying a smooth bounded domain Ω ⊂ R d (with the spatial dimensionality d = 2 or 3), whose boundary Γ is partitioned as Γ = Γ D ∪Γ N support a prescribed traction ¯ t and a prescribed displacement ¯ u, while a body force density f is applied in Ω. On the basis of this fixed geometrical and loading configuration, we consider two situations, namely (i) a reference solid characterized by a given elasticity tensor C, which defines the background solution u, and (ii) a perturbed solid constituted of the same background material except for a small inhomogeneity whose material is characterized by C , which defines a perturbed solution u a. The aim of this work is to formulate a computational approach allowing to treat case (ii) as a perturbation of the background solution (i), in particular avoiding any meshing at the small inhomogeneity scale. This will be achieved by applying known results on the asymptotic expansion of the displacement
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Eduard Marenić, D. Brancherie, M Bonnet. Asymptotic analysis based modeling of small inhomogeneity perturbation in solids: two computational scenarios. IV ECCOMAS YOUNG INVESTIGATOR CONFERENCE (YIC 2017), Sep 2017, Milan, Italy. ⟨hal-01931121⟩

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