Mechanical modeling of growth considering domain variation—Part II: Volumetric and surface growth involving Eshelby tensors

The growth of biological tissues is here described at the continuum scale of tissue elements. Relying on a previous work in Ganghoffer and Haussy (2005), the rephrasing of the balance laws for tissue elements under growth in terms of suitable Eshelby tensors is done in the present contribution, considering successively volumetric and surface growth. Balance laws for volumetric growth are written in both compatible and incompatible configurations, highlighting the material forces for growth associated to Eshelby tensors. Evolution laws for growth are written from the expression of the local dissipation in terms of a relation linking the growth velocity gradient to a growth-like Eshelby stress, in the spirit of configurational mechanics. Surface growth is next envisaged in terms of phenomena taking place in a varying reference configuration, relying on the setting up of a surface potential depending upon the surface transformation gradient and to the normal to the growing surface. The balance laws resulting from the stationnarity of the potential energy are expressed, involving surface Eshelby tensors associated to growth. Simulations of surface growth in both cases of fixed and moving generating surfaces evidence the interplay between diffusion of nutrients and the mechanical driving forces for growth.