A multiscale numerical approach for the finite strains analysis of materials reinforced with helical fibers

Present study provides a multiscale numerical approach based on representative volume elements (RVE) for the finite strain analyses of materials reinforced with helical fibers. An RVE with wavy-like boundaries bioinspired in the microstructure of tendon fascicles is proposed. Due to the unusual geometry of the RVE, a non-periodic mesh mapping will likely occur, precluding the numerical implementation of the periodic boundary condition in a straightforward manner. Moreover, it is verified that the others classical boundary conditions, namely, the linear boundary displacements model and the minimally constrained model, seem not to be suitable choices for the multiscale analyses of this class of RVEs. Motivated by these facts, two mixed boundary conditions allying characteristics of both, linear and minimal models, are suggested. The kinematic constraints on the RVE are enforced via variational principles and Lagrange multipliers. A displacement-controlled triaxial test performed on a numerical specimen larger than the RVE is proposed as a reference solution for the multiscale responses of the RVE. A set of numerical results concerning microscopic strain fields and macroscopic stress-stretch curves points out that one of the proposed mixed models predicts with great accuracy not only the homogenized quantities but also the kinematic fields developed within the specimen. The computational homogenization strategy addressed in this manuscript can be extended to other fiber-reinforced materials composed of different fiber arrangements also including dissipative effects. Moreover, the boundaries surfaces of the RVE are not restricted to any particular layout and the finite element mesh do not require particular mappings nor additional algorithmic handling.