Comparative Stability Analysis of Stress-and Strain-Driven Growth Laws with Neumann boundary conditions

Biological tissue possesses the capacity to grow and remodel in response to changing boundary conditions, a phenomenon described by a plethora of stress-based and strain-based mathematical models. While previous comparative studies have focused on these models under the assumption of uniform strain, this paper extends previous analyses by relaxing the uniform-strain assumption. We conduct a study on an annulus representing a simple model of arteries or heart chambers, where stress and strain are coupled. We investigate the basic properties of five popular growth laws, assuming an incompressible, hyperelastic material using a multiplicative decomposition of the deformation gradient. Normally, growth is curtailed by Dirichlet or Robin boundary conditions which restrict or penalize deformation (and thus growth) at the boundary. This study aims to determine if these models achieve equilibrium (steady-state growth) and if the resulting stress or strain profiles are biologically realistic under constant Neumann boundary conditions alone. Our results demonstrate that the growth laws either fail to converge to a steady state or do so because they reach a prescribed growth limit. Furthermore, the models produced stress profiles inconsistent with biological expectations such as the uniform stress or strain hypothesis. These findings highlight the difficulty of ascertaining how growth influences stress and strain due to nonlinearities arising in three dimensional geometries.