# A Shell Theory for Growth-induced Finite-strain Deformations

Project: Research

## Description

Growth-induced deformations have been an active research topic in recent years. One aim is to understand pattern formations in soft biological tissues. Although the formed morphogenesis may be caused by generic and chemistry factors, mechanical effects may induce instabilities, leading to particular patterns. In this project, we intend to develop a shell theory for studying instabilities and deformations in shell-like structures due to mechanical effects induced by growth. Many biological objects can be viewed as shells, such tree leaves, skins, mucous membranes, cardiac valves and arteries, etc. One feature of shells is the thickness is much smaller than the in-plane length scale and the radius of curvature. For modeling, if they are treated as three-dimensional objects, the computations are very costly for the small aspect ratio. One approach is to do a dimension reduction in order to obtain a two-dimensional shell model, which is then used to study the mechanical behaviors. For growth-induced deformations in plates, some plate theories for small strains based on ad hoc assumptions were presented by other researchers, and recently a consistent finite-strain plate was established by us. However, it appears that a consistent shell theory of growth which combines both stretching and bending effects, although much needed, is still not available. In this project, we shall derive a consistent finite-strain shell theory of growth through a novel procedure. The main difference between a shell and a plate is that the curvature effect has to be taken into account for the former, and as a result some knowledge of differential geometry is needed. To model growth, the deformation gradient will be decomposed into a growth part and an elastic part. It will be supposed that the shell is composed of a general incompressible hyperelastic material. The starting point of our technique is series expansions about the bottom surface. Although the problem is highly nonlinear with constraint, a key observation is that certain linear relations exists among stress, Lagrange multiplier and displacement/position expansion coefficients. Then, it becomes possible to establish recursive relations for displacement coefficients from the three-dimensional governing system. As a result, only three equations for the three components of the first displacement expansion coefficient can be obtained, which are 2D shell equations. Shell boundary conditions will also be considered, together with associated weak formulations. We shall use the derived shell theory to study several problems. One is the post-bifurcation analysis on the growth-induced deformation in an annulus. The linear bifurcation analysis has been done in literature, and here the focus is on the derivation of the amplitude equation, which can shed more light on the pattern formation. Another is on the post-bifurcation analysis on the growth-induced axisymmetric deformation in a circular tube. In particular, the influences of various growth fields on the amplitude will be examined. The third one is on the semi-inverse analytical solutions of the shell theory and their applications. Shape-programming through solving the inverse problem will also be considered. It is hoped that the derived finite-strain shell theory of growth could provide a solid theoretical framework for studying growth-induced deformations in soft-material shell-like structures.## Detail(s)

Project number | 9042858 |
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Status | Not started |

Effective start/end date | 1/01/20 → … |