Small Strain Growth and the Human Nail

Abstract

is specialised to describe growth processes in which mass-growth may take place elastically at infinitesimally small strain. Growth-strain is considered infinitesimally small but the resulting simplified constitutive equations are in general still non-linear due to combined action of the rate of massgrowth and the rate of volumetric strain. Nevertheless, mass-growth processes which are either incompressible or slightly/moderately compressible are found consistent with linearly elastic material behaviour, and can accordingly be studied with convenient use of linear elasticity concepts. In a relevant example application, combination of the concept of periodically accumulated small strain with a certain nail-elongation criterion enables development of a relatively simple mathematical model of the human nail apparatus and, hence, an interesting interpretation of the nail elongation process. The efficiency of the model is demonstrated through a relevant analytical solution obtained in the particular case that mass-growth in the living part of the nail takes place under constant mass density and is, therefore, incompressible. Consideration of more advanced nail features and physical concepts can be handled in a similar manner, at the expense of parts of the pursued and desirable analytical simplicity.

On the preservation of fibre direction during axisymmetric hyperelastic mass-growth of a finite fibre-reinforced tube

“…Rather than (4.3), such types of growth make use of the full form of the continuity equation with growing mass (2.10) and may thus be able to influence directly not only the dilatational, but also the shear deformation features of axisymmetric tube mass-growth. The present hyperelasticity mass-growth model can then be found helpful in directing relevant compressible mass-growth developments [22,32], where the influence of the divergence of the velocity vector is still accounted for in the corresponding constitutive equation (4.1). Similarly, the presented analysis may assist future developments in which plastic-like mass-growth and/or influence of some non-zero prestress state are further expected to be accounted for [22,29].…”

“…This parameter is considered independent of time, but not necessarily independent of position and, hence, not necessarily constant; see also [32]. However, the constant value s = 1 represents all cases and classes of incompressible mass-growth (4.3), where the ratio coefficient appearing in the last term (4.1) becomes 1.…”

“…It is then observed that forms ofŴ that satisfy (8.22) yield now 32) and, hence, enable again axisymmetric cross-sectional mass-growth to take place in a stress-free manner. Such forms ofŴ enable further (8.29) to obtain the simplified form…”

On elastic growth modelling of straight hair

“…the work by Powell and Rogers [1]) presents considerable similarities with, but also differences from its human nail counterpart (e.g. the works by Soldatos and Baran et al [2, 3]). Nails and hair grow, for instance, at slow rates but, while geometrical features of a human nail resemble closely those of assembled flat plates [3], a straight hair specimen is more naturally thought of as an assembly of consecutively bonded cylindrical rods.…”

“…The latter is made of hard, keratinous, dead tissue and is unable to grow, but keeps elongating in an unbounded manner due to the aforementioned matrix activity. Those macroscopic similarities of the nail and hair elongation mechanisms suggest that the mass-growth activity that takes place within a straight hair follicle may become understood through a suitable extension of the existing elastic model of nail growth [2].…”