Nonlinear Mechanics of Arterial Growth

Abstract

In this paper, we formulate a geometric theory of the mechanics of arterial growth. An artery is modeled as a finite-length thick shell that is made of an incompressible nonlinear anisotropic solid. An initial radially-symmetric distribution of finite radial and circumferential eigenstrains is assumed. Bulk growth is assumed to be isotropic. A novel framework is proposed to describe the time evolution of growth, governed by a competition between the elastic energy and a \emph{growth energy}. The governing equations are derived through a two-potential approach and using the Lagrange-d'Alembert principle. An isotropic dissipation potential is considered, which is assumed to be convex in the rate of growth function. Several numerical examples are presented that demonstrate the effectiveness of the proposed model in predicting the evolution of arterial growth and the intricate interplay among eigenstrains, residual stresses, elastic energy, growth energy, and dissipation potential. A distinctive feature of the model is that the growth variable is not constrained by an explicit upper bound; instead, growth naturally approaches a steady-state value as a consequence of the intrinsic energetic competition.