Speaker
Description
In this talk, we discuss a quasi-stationary model for stress-modulated growth of elastic materials. Inspired by models for crystal plasticity, the model features a multiplicative decomposition of the total deformation gradient into an elastic part and the growth tensor. The growth tensor is given by the solution to an ordinary differential equation on a suitable Banach space that depends, possibly non-linearly, on the elastic stress which is induced by the subsequent elastic deformation. The elastic deformation is given by the minimizer of a variational problem with a determinant constraint, which in turn depends on the solution to the ordinary differential equation, the growth tensor, by means of a coordinate transformation. Moreover, the growth process is driven by the presence of nutrients. In this model the nutrient concentration is determined by an elliptic reaction-diffusion equation. The ordinary differential equation that determines the growth process depends on the solution to this reaction-diffusion equation and the coefficients of the reaction-diffusion equation depend on the growth tensor and the elastic deformation.
We discuss existence and regularity of global solutions to this model in the case of one spatial dimension. Furthermore, we give an outlook onto the multi-dimensional case. Existence of solutions to the ordinary differential equation is proved by means of Picard-Lindelöf's theorem. Hence, for existence and regularity results, it is crucial to prove that the elastic stress depends Lipschitz-continuously on the growth tensor. While in the one-dimensional case the variational approach yields good results due to the fact that minimizers to the variational problem solve the associated Euler-Lagrange equations despite the determinant constraint, this poses a major obstacle in the analysis of the multi-dimensional case in which analyzing the associated equations seems to be more promising.
