Stress field due to elastic mass growth on spherical and cylindrical substrates

Mass growth on cylindrical and spherical substrates is a phenomenon which can be related to the biochemical creation of an elastic actin gel shell by polymerization and cross linking of actin filaments either in vivo on bacteria cylindrical surfaces as a means for their motility or in vitro on spherical beads as a means for experimentally studying the previous in vivo case. Such mass growth is strongly effected by the developed stress field. The objective of this paper is to accurately determine this stress field assuming elasticity of the growing mass and symmetrical growth. Based on the special kinematics of mass growing on spherical and cylindrical substrates, inwards or outwards from them, and various isotropic constitutive laws for both small and finite elastic strains, it is possible to obtain the solution for the stress field in closed analytical form for all cases considered. This expands very significantly recent findings [Dafalias, Y.F., Pitouras, Z., 2008. Stress field in actin gel growing on spherical substrate. Biomechanics and Modeling in Mechanobiology, doi:10.1007/s10237-007-0113-y.] for some constitutive laws and outwards growth on spherical substrates only. The effect of biomass compressibility is shown to be of cardinal importance for the developed stress field, questioning the validity of the simplifying assumption of a zero value Poisson ratio usually made in the relevant biomechanics literature for simplicity. Few selected graphs of stress variation along the radial direction illustrate the analytical findings. The obtained closed form analytical expressions for stress can be a standard reference tool in this important area of stress-modulated soft tissue growth.