Thermomechanics of volumetric growth in uniform bodies

Abstract

Abstract A theory of material growth (mass creation and resorption) is presented in which growth is viewed as a local rearrangement of material inhomogeneities described by means of first- and second-order uniformity “transplants”. An essential role is played by the balance of canonical (material) momentum where the flux is none other than the so-called Eshelby material stress tensor. The corresponding irreversible thermodynamics is expanded. If the constitutive theory of basically elastic materials is only first-order in gradients, diffusion of mass growth cannot be accommodated, and volumetric growth then is essentially governed by the inhomogeneity velocity “gradient” (first-order transplant theory) while the driving force of irreversible growth is the Eshelby stress or, more precisely, the “Mandel” stress, although the possible influence of “elastic” strain and its time rate is not ruled out. The application of various invariance requirements leads to a rather simple and reasonable evolution law for the transplant. In the second-order theory which allows for growth diffusion, a second-order inhomogeneity tensor needs to be introduced but a special theory can be devised where the time evolution of the second-order transplant can be entirely dictated by that of the first-order one, thus avoiding insuperable complications. Differential geometric aspects are developed where needed.