On a new model for inhomogeneous volume growth of elastic bodies

Abstract

In general, growth characterises the process by which a material increases in size by the addition of mass. In dependence on the prevailing boundary conditions growth occurs in different, often complex ways. However, in this paper we aim to develop a model for biological systems growing in an inhomogeneous manner thereby generating residual stresses even when growth rates and material properties are homogeneous. Consequently, a descriptive example could be a body featuring homogeneous, isotropic material characteristics that grows against a barrier. At the moment when it contacts the barrier inhomogeneous growth takes place. If thereupon the barrier is removed, some types of bodies keep the new shape mainly fixed. As a key idea of the proposed phenomenological approach, we effort the theory of finite plasticity applied to the isochoric part of the Kirchhoff stress tensor as well as an additional condition allowing for plastic changes in the new grown material, only. This allows us to describe elastic bodies with a fluid-like growth characteristic. Prominent examples are tumours where the characteristic macro mechanical growth behaviour can be explained based on cellular arguments. Finally, the proposed framework is embedded into the finite element context which allows us to close this study with representative numerical examples.

Keywords: Biology; Continuum mechanics; Finite plasticity theory; Inhomogeneous growth.