Computational Homogenization in Multi-scale Shell Analysis at Large Strains

Abstract

Biological tissues such as those existing in the eye, heart, veins or arteries are heterogeneous on one or another spatial scale and can undergo very large strains in the elastic range (hyperelasticity). Frequently, these tissues are characterized by thin-walled, shell-like structures. The use of computational homogenization schemes [1], [2], [3] together with a formulation of the continua in curvilinear coordinates is a prerequisite for realistic biomechanical simulations of thin-walled soft tissues on different scales. The goal of this contribution is the generalization of the computational homogenization scheme for the formulation of micro-macro transitions in curvilinear convective coordinates. We consider a homogenized macro-continuum with a locally attached representative micro-structure. The deformation and the coordinate system of the micro-structure are assumed to be coupled with the local deformation and the local coordinate system at a corresponding point of the macro-continuum. For a consistent formulation of micro-macro transitions principal material directions are defined on both scales. To formulate the generalized micro-macro transitions in absolute tensor notation the new operations scale-up and scale-down are introduced. The multi-scale analysis discussed in this paper considering arbitrary curvilinear coordinates as well as geometrical and material nonlinearities rests upon a finite-element discretization of the macro-continuum by means of a bilinear finite shell element with a quadratic kinematic assumption in thickness direction [4]. A finite-element discretization of the micro-structure is attached to each integration point of these macro-elements and is modeled with trilinear brick elements.