An Energy-Deformation Decomposition for Morphoelasticity

TL;DR

This paper introduces an energy-deformation decomposition method for modeling biological growth, addressing limitations of traditional kinematic approaches by focusing on mechanical energy, with proofs and computational validation for crystalline and network tissues.

Contribution

It proposes a novel energy-based decomposition for growth modeling, overcoming fundamental incompatibilities of the traditional multiplicative approach with shear-resistance.

Findings

Traditional multiplicative decomposition is incompatible with shear-resistance.

Energy-deformation decomposition accurately captures growth effects on mechanical energy.

Results apply broadly to various tissue structures and growth models.

Abstract

Mathematical models of biological growth commonly attempt to distinguish deformation due to growth from that due to mechanical stresses through a hypothesised multiplicative decomposition of the deformation gradient. Here we demonstrate that this hypothesis is fundamentally incompatible with shear-resistance and thus cannot accurately describe growing solids. Shifting the focus away from the kinematics of growth to the mechanical energy of the growing object enables us to propose an "energy-deformation decomposition" which accurately captures the influence of growth on mechanical energy. We provide a proof and computational verification of this for tissues with crystalline structure. Our arguments also apply to tissues with network structure. Due to the general nature of these results they apply to a wide range of models for growing systems.