On elastic growth modelling of straight hair

Abstract

Certain macroscopic similarities of the nail and hair elongation mechanisms enable the mass-growth activity that takes place within a straight hair follicle to be modelled through a suitable extension of a relevant small-strain pseudo-elasticity model of human nail growth. Basic differences which are taken into consideration are the facts that straight hair (a) resembles the form of a cylindrical rod, rather than a plate, while (b) its material constitution seems microscopically transversely isotropic, rather than isotropic. A complete analytical solution of the obtained governing differential equations is detailed for the case where incompressible mass-growth conditions prevail within the hair matrix. In addition to estimating displacement and stress distributions that develop within the growing matrix, and the resulting hair elongation, that solution enables prediction of a clinically observed zone of hair-fibre hardening that lies between the matrix soft tissue and the hard keratinous hair shaft. It also predicts that the longitudinal dimension of the hair matrix and that of the hair-fibre hardening zone depend on the material properties of the soft tissue of the follicle. Consideration of more advanced micro- or macro-scopic features of the hair follicle, such as layered structure or curved form, can be handled mathematically in a similar manner at the expense of analytical simplicity.