Modelling Volumetric Growth in Soft Solids via Residual Stress

In this paper the nonlinear elasticity theory of volumetric growth based on residual stress that was introduced in previous contribution (Huang et al. in J. Elast. 145:223-241, 2021) is developed further, and is then focused on an applications of the theory with computational examples. The main idea here is to use residual stress in an intact unloaded configuration, or the deformation from a fixed and intact reference configuration (which may itself be residually stressed), as a means to assess the growth in a soft solid, the developing unloaded configuration and the accompanying developing residual stress. The general theory is presented in terms of the free energy per unit mass and the associated energy functions relative to the reference configuration and the unloaded configuration. Growth of a thick-walled spherical shell is examined in order to illustrate the theory using simple prototype energy functions. A general programme for obtaining the developing deformed configuration is outlined and several possible growth laws are discussed for the growth of a spherical shell under internal pressure. This study shows that growth modelling based on the unloaded configurations may provide insights into the development of residual stress and morphology, both of which are, in principle, accessible to experimental observation. For several possible growth laws detailed numerical results are provided to illustrate the evolution of growth and the associated residual stress.