Morphoelasticity: The mechanics and mathematics of elastic growth

Abstract

Growth plays a key role in many fundamental biological processes. In many cylindrical structures in biology, residual stress fields are created through differential growth. We present a general formulation of growth for a three-dimensional nonlinear elastic body and apply it to specific geometries relevant in many physiological and biological systems. The goal of this work is to study the development of residual stress induced by differential growth of biological cylindrical structures and elucidate its possible mechanical role in modifying material properties.

As a tissue grows, it is not only subject to stresses but it also develops stresses by itself. These stresses play an important role in the evolution and regulation of growth, both in physiological and pathological conditions. We explore the interplay between growth and stress and the time evolution it generates. In particular, we show that in the case of spatially homogeneous growth, a general form of time evolution can be obtained leading to a dynamical system coupling the growth and stresses.

The effect of tissue tension on the stability is studied through an analysis of the buckling properties of residually stressed cylindrical tubes. The general method to study stability is through a perturbation expansion in which an incremental deformation is superimposed on some finite deformation. If a solution is found to the incremental equation with the appropriate boundary conditions, then the possibility of instability exists. This method allows us to understand how residual stresses affect the overall stability of the system in the context of plant stem rigidity and arterial buckling.

Lastly, we study the problem of elastic cavitation, the opening of a void in elastic materials. For a particular class of materials, the existence of a bifurcated solution has been shown in which a sphere supports the trivial spherical solution and a cavitated solution with spherical symmetry whose cavity radius vanishes at some critical external pressure. It naturally leads to interesting questions regarding the opening of cavities in residually stressed systems. We show that residual stresses induced by differential growth can induce cavitation and it can also play an important role in microvoid opening.