Existence results for a morphoelastic model

TL;DR

This paper establishes existence results for a complex three-dimensional morphoelastic model involving growth, deformation, and nutrient interactions, using regularization, time-discrete schemes, and optimal control formulations.

Contribution

It introduces a regularized framework for proving existence of solutions in morphoelasticity, including coupled growth and nutrient dynamics, and develops an optimal control approach.

Findings

Existence of solutions via regularization and limit processes.

Formulation and proof of existence for optimal controls.

Existence of coupled morphoelastic and nutrient solutions.

Abstract

We present some existence results for three-dimensional quasistatic morphoelasticity. The state of the growing body is described by its deformation and the underlying growth tensor and is ruled by the interplay of hyperelastic energy minimization and growth dynamics. By introducing a regularization in the model, we prove that solutions can be obtained as limits of time-discrete solutions, built by means of an exponential-update scheme. By further allowing the dependence of growth dynamics on an additional scalar field, to be interpreted as a nutrient or inhibitor, we formulate an optimal control problem and prove existence of optimal controls and states. Eventually, we tackle the existence of coupled morphoelastic and nutrient solutions, when the latter is allowed to diffuse and interact with the growing body.