Finite strain visco-elastic growth driven by nutrient diffusion: theory, FEM implementation and an application to the biofilm growth

In this paper, a thermodynamically consistent visco-elastic growth model driven by nutrient diffusion is presented in the finite deformation framework. Growth phenomena usually occur in biological tissues. Systems involving growth are known to be open systems with a continuous injection of mass into the system which results in volume expansion. Here the growth is driven by the diffusion of a nutrient. It implies that the diffusion equation for the nutrient concentration needs to be solved in conjunction with the conservation equation of mass and momentum. Hence, the problem falls into the multi-physics class. Additionally, a viscous rheological model is introduced to account for stress relaxation. Although the emergence of residual stresses is inherent to the growth process, the viscous behaviour of the material determines to what extend such stresses remain in the body. The numerical implementation is performed using the symbolic tool Ace-Gen while employing a fully implicit and monolithic scheme.