A short introduction to morphoelasticity: the mechanics of growing elastic tissues

Abstract

Growth is a key process in the life and development of all biological organisms and depends on a number of genetic, bio-chemical, environmental, and mechanical factors. In particular, growth can be affected by mechanical stresses and, in turn, generate new stresses to modify shape, create patterns, and tune the overall response of the tissue. From a mathematical perspective, the modelling of growth processes and in particular its interplay with mechanics is particularly challenging since, unlike traditional mechanical systems, the reference state where key physical quantities need to be evaluated evolve with time. An extra difficulty comes from the fact that the geometry of the object also evolves in time due to the addition of mass. From a modelling perspective, it is particularly important to isolate these effects as they are generated by different processes. In this short introduction, we first give a general overview of the problem of biological growth. Second, the mathematical problem of growth modelling for biological system is considered and illustrated on a number of examples starting with simple one-dimensional systems. Third, we present the general framework of nonlinear morphoelasticity to describe the mechanical response of growing elastic tissues and the remodelling of material properties. The first few introductory pages of these lecture notes are reproduced from (Goriely and Moulton, 2010)