Computational micro‐macro transitions at large strains for curvilinear physical directions

Abstract

Biological tissues such as those involved in the eye, heart, veins or arteries are heterogeneous on one or another spatial scale and can undergo very large elastic strains. Frequently, these tissues are characterized by shell-like structures at the macroscopic scale and the physical material directions follow curvilinear paths. We consider a homogenized macro-continuum formulated in curvilinear convective coordinates with locally attached representative micro-structures. Micro-structures attached to different macroscopic points are assumed to be rotated counterparts according to the curvilinear path of the physical material directions at the macro-scale. The solution of such multi-scale problems according to the computational homogenization scheme [1, 2, 3] would need a different RVE at each macroscopic point. The goal of this paper is to use the same initial RVE at each macroscopic point by generalizing the computational homogenization scheme to a formulation considering different physical spaces at the micro-and macro-scale. The deformation and the reference frame of the micro-structure are assumed to be coupled with the local deformation and the local reference frame at the corresponding point of the macrocontinuum. For a consistent formulation of micro-macro transitions physical reference directions are defined on both scales, where the macroscopic one follows a curvilinear path. To formulate the generalized micro-macro transitions in absolute tensor notation the operations scale-up and scale-down are introduced. Definition of micro-macro transitions at finite strains in curvilinear coordinates Kinematics of micro-and macro-continuum in curvilinear convective coordinates at finite strainsLet B M0 and B m0 be the configurations of a macro-and micro-structure at time t 0 in a 3D Euclidean space R 3 . In shell analyses it is preferable to define the position vector as a function of three independent parameters Θ i representing curvilinear coordinates. Note that for shell-like structures Θ 1 and Θ 2 specify the middle surface of the shell and Θ 3 points in thickness direction. The position vector of a typical point P M0 (P m0 ) of B M0 (B m0 ) is denoted by