Nonlinear Mechanics of Remodeling

Abstract: In this paper we present a large-deformation formulation of the mechanics of remodeling. Remodeling is anelasticity with an internal constraint -- material evolutions that are mass and volume preserving. In this special class of material evolutions the explicit time dependence of the energy function is via a remodeling tensor (or a set of remodeling tensors) that is (are) the internal variable(s) of the theory. The governing equations of remodeling solids are derived using a two-potential approach and the Lagrange-d'Alembert principle. We consider both isotropic and anisotropic solids and derive their corresponding remodeling equations. We study a particular remodeling of fiber-reinforced solids in which the fiber orientation is time dependent in the reference configuration -- SO(3)-remodeling. We consider an additional remodeling energy, which is motivated by the energy spent in living systems to remodel to enhance stiffness in the direction of loading. We consider the examples of a solid reinforced with either one or two families of reorienting fibers and derive their remodeling equations. This is a generalization of some of the proposed remodeling equations in the literature. We study three examples of material remodeling, namely finite extensions and torsion of solid circular cylinders, which are universal deformation for incompressible isotropic solids and certain anisotropic solids. We consider both displacement and force-control loadings. Detailed parametric studies are provided for the effects of various material and loading parameters on the fiber remodeling.