We analyse the behaviour of thin composite plates whose material properties vary periodically in-plane and possess a high degree of contrast between the individual components. Starting from the equations of three-dimensional linear elasticity that describe soft inclusions embedded in a relatively stiff thin-plate matrix, we derive the corresponding asymptotically equivalent two-dimensional plate equations. Our approach is based on recent results concerning decomposition of deformations with bounded scaled symmetrised gradients. Using an operator-theoretic approach, we calculate the limit resolvent and analyse the associated limit spectrum and effective evolution equations. We obtain our results under various asymptotic relations between the size of the soft inclusions (equivalently, the period) and the plate thickness as well as under various scaling combinations between the contrast, spectrum, and time. In particular, we demonstrate significant qualitative differences between the asymptotic models obtained in different regimes.
Starting from 3D linear elasticity with soft high-contrast inclusions and stiff matrix we derive the limit 2D plate equations. The approach is based on the recent results in the decomposition of the deformations with bounded symmetrized scaled gradients. By calculating limit resolvent and using operator theoretical approach we analyze limit spectrum and limit evolution equations with respect to different relation between size of soft inclusions and thickness of the body, scalings of high-contrast coefficients, spectrum and time. There is a significant qualitative difference in the models obtained in different regimes.
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