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Modélisation non-locale et stochastique de matériaux à fort gradient de propriétés par développement asymptotique

Abstract : The aim is to propose a macroscopic, deterministic and non-local model, constructed by scale transition for heterogeneous materials with high property gradients and containing a random distribution of inclusions. More precisely, the inclusions are distributed in an elastic matrix according to a stochastic ergodic process. Several non-local models exist in the literature, but they do not allow (or very little) to obtain non-local quantities and/or fields at the macroscopic scale from a scale-transition. Besides, it is often difficult to link the non-local parameters to the microstructure. To this aim, we developed a two-step approach.In the first stage, we combined the method of asymptotic developments with an energetic approach to reveal a second displacement gradient in the strain energy. The advanced model involves three homogenized elasticity tensors functions of the stochastic parameter and of the phase properties. As opposed to the literature, the model involves two characteristic lengths strongly linked to the microstructure. These lengths define two morphological representative elementary volumes on which full field simulations are performed in order to determine the macroscopic strain tensors at orders 0 and 1 involved in the formulation of the model. In order to test this first version of the model, numerical simulations were performed. The estimate of the classical part of the energy, coming from the local part of the fields, has been successfully compared to classical bounds for a composite bar consisting of a random distribution of two homogeneous and isotropic elastic materials. Then, numerical solving of the whole model including the non-local terms has been performed in the three-dimensional case. Two types of microstructures with increasing morphological complexity were used. The first ones are virtual microstructures generated from a given simple pattern randomly distributed throughout the structure and composed of a big inclusion circled by six identical small ones. The second are real microstructures of Ethylène-Propylène-Diène Monomère (EPDM) obtained by tomography and containing clusters of inclusions with complex structures.In order to obtain a macroscopic model that can be used for structure analysis, without any full field intermediate calculations, a second scale transition has been performed using stochastic variational homogenization tools in the ergodic case. More precisely, the Γ-convergence method has been used in order to have a convergence of energy rather than that of mechanical fields, aiming at keeping a strong microstructural content. In fine, the model is macroscopic, non-local, deterministic and strongly connected to the microstructure. Non-local effects are now accounted for by the presence of the second displacement gradient but also by the presence of the virtual (memory) displacement field of the inclusions. The link with microstructure is still manifest through the presence of the stochastic parameter and phase properties, but also by the presence of the asymptotic fractions of the inclusion phase in the material and in each of the morphological volumes defined by the model characteristic lengths. In order to prepare the use of the model for structure calculations, a non-local finite element enriched with Hermit-type interpolations was implemented in FoXtroT, the finite element solver of the Pprime Institute. This element takes into account the virtual (memory) displacement field related to inclusions as well as the gradients of the macroscopic and virtual displacement fields. The first numerical results on this aspect, to our knowledge never discussed in the literature, are promising.
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Submitted on : Monday, June 8, 2020 - 11:40:34 AM
Last modification on : Wednesday, November 3, 2021 - 5:57:11 AM


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  • HAL Id : tel-02860056, version 1



Sami Ben Elhaj Salah. Modélisation non-locale et stochastique de matériaux à fort gradient de propriétés par développement asymptotique. Autre. ISAE-ENSMA Ecole Nationale Supérieure de Mécanique et d'Aérotechique - Poitiers, 2019. Français. ⟨NNT : 2019ESMA0018⟩. ⟨tel-02860056⟩



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