A variational structure underpinning higher-order homogenization

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• Manon Thbaut, Ecole Polytechnique
• Friday 09 May 2025, 14:00-15:00
• Oatley 1 Meeting Room, Department of Engineering.

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From an engineering point of view, it is convenient to describe composite materials using homo- geneous effective properties. When the microstructure is periodic, asymptotic homogenizationis particularly well suited for this aim. Classical homogenization corresponds to the dominant order model and yields an effective standard Cauchy medium. At next orders, we can derive addi- tional corrections that depend on the successive strain gradients. These corrections are typically of interest to capture size-effects appearing for microstructures with contrasted stiffness properties. However, these higher-order models present two major limitations. First, the corrections producedby homogenization can handle size-effects that occur in the bulk region, but are not suited to the analysis of the boundaries. In fact, they miss significant boundary effects which can degrade significantly the quality of the predictions. Secondly, these higher-order models present several mathematical inconsistencies, including non-positive strain-gradient stiffnesses. As a result, the effective energy is not necessarily positive and any equilibrium solution is unstable with respect to short-scale oscillations. To handle these two limitations simultaneously, we elaborate a newhomogenization procedure that includes boundary effects. By contrast with usual approaches, inour procedure the homogenization is carried at the energy level, rather than on the strong formof the equilibrium. Besides, the positivity of the resulting energy is guaranteed by an original truncation method [1]. As an example, we consider a 1D spring network. The resulting effective energy contains a bulk term that is positive, plus a boundary term that accounts for the energy generated by the boundary effects. We show that, by contrast with usual asymptotic homogenization, this higher-order model is able to capture size-effects occurring in the interior domain, as well as near the boundaries.

This talk is part of the Engineering - Mechanics and Materials Seminar Series series.

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