Multiscale characterization of the mechanics of curved fibered structures with application to biological materials

Curved fibered structures are ubiquitous in nature and this organization is found in the majority of biological tissues. Indeed, the mechanical behavior of these materials is of pivotal importance in biomechanics and mechanobiology fields. In this paper, we develop a multiscale formulation to characterize the macroscopic mechanical nonlinear behavior from the microstructure of fibered matrices. From the analysis of the mechanics of a randomly curved single fiber, a fibered matrix model is built to determine the macroscopic behavior following a homogenization approach. The model is tested for tensile, compression and shear loads in a number of applications reminiscent to collagen extracellular matrices. However, any other fibered microstructures can be studied following the proposed formulation. The presented approach naturally recovers instabilities at compression as well as the strain stiffening regime, which are observed experimentally in the mechanical behavior of collagen matrices. Indeed, it was found that the bending energy associated to fiber unrolling, is the most important source of energy developed by fibers for the analyzed cases in tensile and shear in all deformation regions (except the strain stiffening region), whereas bending energy dominates at compression too during buckling. The proposed computational framework can also be used to perform multiscale simulations in the referred applications. As a result, the developed methodology may be an interesting and complementary tool to characterize the nonlinear behavior and evolution of curved fibered structures present in biology and engineered materials.