When Volumetric Growth Selects Surface Growth

Abstract

We investigate the relationship between volumetric and surface growth within a recently proposed optimization-driven framework for linearly elastic solids. In this approach, growth is not prescribed through an evolution law; instead, the growth distribution is determined as the solution of a constrained optimization problem. Focusing on processes driven by the minimization of the work performed by external loads in one-dimensional and axisymmetric settings, we derive explicit analytical solutions for the resulting growth distributions. Although growth is initially formulated as a volumetric process through a distributed growth strain tensor, we show that the optimal growth distributions are singular and concentrate on boundaries or internal interfaces. These results provide a variational mechanism through which, under certain conditions, surface growth is selected as the optimal realization of volumetric growth.