A viscoelastic model for skin via homogenization theory

In this paper, we carry out the homogenization of a fluid–structure interaction problem consisting in the periodic inclusions of a viscous fluid in an elastic body. We get a macrostructure model, where the body behaves as a viscoelastic material with a long-range memory term. Our aim is not only to get this limit problem but also to study its main properties. By using the micro-structure variables, it is simple to check that it satisfies an energy conservation law assuring in particular the existence and uniqueness of solution. The difficulty is to characterize these properties using only the macroscopic system. We prove that the nonlocal term is given through a convolution kernel which exponentially decreases to zero and satisfies some positive conditions which we write in terms of the Laplace transform. These conditions can be used to directly prove the existence and uniqueness of solution. The results apply to modeling the mechanical behavior of skin as an indicator of what kind of models we should use. In a naive interpretation the fluid inclusions represent the cells and the elastic medium the extracellular matrix.