Asymptotic Analysis of a Linear Isotropic Elastic Composite Reinforced by a Thin Layer of Periodically Distributed Isotropic Parallel Stiff Fibres

Abstract

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Derivation and Application of Effective Interface Conditions for Continuum Mechanical Models of Cell Invasion through Thin Membranes

“…Nonlinear partial differential equations describing reaction-diffusion processes and transport phenomena in spatial domains that comprise different parts separated by thin layers (i.e. films or membranes) arise in the mathematical modelling of various chemical, physical and biological systems [1,2,4,8,13,14,15,16,20,21,27,34,36,43,49,57,60,64,65,68,71,72]. Due to the analytical and numerical challenges posed by the presence of such layers [12], it is often convenient to approximate the original problem by an equivalent transmission problem whereby each thin layer is replaced by an effective interface.…”

Improved impedance conditions for a thin layer problem in a nonsmooth domain

“…The subject still generates active research activities. Let us mention Abdelkader and Moussa, Bellieud et al, and Bonnet et al for mechanical applications; Perrussel and Poignard, Bendali and Poirier, and Schmidt and Chernov for electromagnetic problems; Zhao and Yao for the Stokes system; Li et al for the heat equation; and Cai and Xu and Diaz and Péron for general purposes. On the other hand, the case of a random thickness has been investigated; see Basson and Gérard‐Varet in the context of rough surfaces, and Dambrine et al for a practical application of approximate boundary conditions to compute moments of solutions of boundary value problems inside random domains.…”

“…21 The subject still generates active research activities. Let us mention Abdelkader and Moussa, 22 Bellieud et al, 23 and Bonnet et al 24 for mechanical applications; Perrussel and Poignard, 6 Bendali and Poirier, 25 32 for a practical application of approximate boundary conditions to compute moments of solutions of boundary value problems inside random domains. Let us mention the works [33][34][35] on polarization tensor for thin inclusions of rough layers.…”

Homogenization of periodic graph-based elastic structures

“…However very few explicit examples have been given in which such a phenomenon appears. In [47], [12], [21], [14], [13], [11] the homogenized material becomes a second gradient one: the elastic energy depends on the second gradient of the displacement instead of the first one only. However all these results fall under the framework of couple-stress theory, [54], [55], [40], [42]: the dependence with respect to the second gradient of the displacement is limited to dependence on the gradient of the skew-symmetric part of the gradient of the displacement only.…”