A model of fiber reorientation in fiber-reinforced biological materials combining statistical and configurational mechanics

We present a mathematical model for the reorientation of fibers in a soft, fiber-reinforced, fluid-saturated porous medium describing a hypothetical biological tissue. We consider two types of remodeling that, at different scales, concur in determining the structural reorganization of the solid phase of the medium: one pertains to the development of plastic-like distortions, which are introduced as the macroscopic manifestation of processes occurring at lower scales, but not resolved explicitly; the other one, originating at the mesoscale, concerns the capability of the fibers of reorienting in the extracellular matrix. This latter form of remodeling is studied as a Langevin-like process in which two main agencies are recognized: a drift term, which is given by the effects that the deformation of the extracellular matrix exerts on the fibers, and a noise term, which accounts for the interactions among nearby fibers. We employ the framework of the Principle of Virtual Power to present our model, in which we “free” the kinematics of the system from the constraints of isochoricity of remodeling and incompressibility of the mixture by appending the Chetaev forms of these constraints to the Principle of Virtual Power. Then, we specialize our model to articular cartilage. We do this for comparing our results with some experimental curves describing the distribution of collagen fibers in a sample of articular cartilage, and for studying, through the simulation of a uniaxial compression test, the interplay between fluid flow and the two aforementioned forms of remodeling. Our main result is the establishment of a framework that captures effectively the mechanical coupling between the fiber distribution specific to a given medium and the mechanical stimuli exerted on such distribution. Quite differently from other works on this subject, our framework is capable of accounting for the stochastic effects of the fibers on the overall evolution of the considered tissue.