  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.
