An Energy-Deformation Decomposition for Morphoelasticity

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.