An asymptotically consistent morphoelastic shell model for compressible biological structures with finite-strain deformations

TL;DR

This paper develops an asymptotically consistent shell model for biological tissues that accounts for large, finite deformations and compressibility, combining growth models with shell theory for accurate morphoelastic predictions.

Contribution

It introduces a novel morphoelastic shell model derived from 3D theory, incorporating compressibility and finite strains, and validates it through examples.

Findings

Model accurately predicts large deformations in biological tissues.

Shell energy depends on stretching and bending strains.

The model recovers various existing shell models.

Abstract

We derive an asymptotically consistent morphoelastic shell model to describe the finite deformations of biological tissues using the variational asymptotical method. Biological materials may exhibit remarkable compressibility when under large deformations, and we take this factor into account for accurate predictions of their morphoelastic changes. The morphoelastic shell model combines the growth model of Rodriguez et al. and a novel shell model developed by us. We start from the three-dimensional (3D) morphoelastic model and construct the optimal shell energy based on a series expansion around the middle surface. A two-step variational method is applied that retains the leading-order expansion coefficient while eliminating the higher-order ones. The main outcome is a two-dimensional (2D) shell energy depending on the stretching and bending strains of the middle surface. The derived morphoelastic shell model is asymptotically consistent with three-dimensional morphoelasticity and can recover various shell models in literature. Several examples are shown for the verification and illustration.