Dynamic hairspray analysis. III. Theoretical considerations.
Model mechanical calculations were carried out to simulate the properties of hair fiber assemblies operating in the bending mode. It was assumed that the fibers were in omega-loop configuration (as in experimental work), and the theory of deformation of thin rings provided a fundamental relationship between stress and strain. A dependence on the stiffness of multi-fiber assembly was derived and verified empirically. Theoretical bending stiffness for fixative-treated hair was calculated for various model fiber distributions by calculating their area moment of inertia according to the parallel axis theorem. General equations for stiffness of fiber assemblies were derived for cross sections placed on cubic and hexagonal lattices. The results suggest a good agreement between the theoretical and experimental data.
Some simple theoretical considerations on the bending stiffness of human hair
Factors affecting the cantilever bending of human hair have been investigated by theoretical structural mechanics in which the hair was treated as if it was of isotropic structure and internal mechanical property. Despite the simplicity of the model it is a valuable one for assessing the major geometric factors affecting a hair's bending behaviour in relation to perceptions of coarseness and fineness. Simple equations have been used for predicting relative bending stiffness between hairs of different racial types using published fibre cross‐sectional information and these accord with existing perceptions of behaviour. Strikingly it is the hair's minor elliptic diameter which has a dominant effect upon bending resistance and it is shown that the hair's cuticle could be making a very high proportional contribution. It is shown that triangular hairs of the uncombable hair syndrome will be stiffer than normal elliptical hairs of the same cross‐sectional area and that this goes some way to explaining the condition.
Mechanics of deformable directed surfaces
Abstract The paper develops a simple, yet complete and consistent non-linear dynamic thermo-elastic shell theory based on the concepts of a two-dimensional directed continuum. The constitutive equations are postulated in a general form. Stress-strain relations are derived using the equation of energy balance and the two entropy production inequalities, which are rewritten in a special form. The linear theory is discussed in detail. The structure of the stress-strain relations is presented in the most general form and then specialized to concrete examples, in particular, to the classical shell theory, using their invariance properties with respective to certain symmetry groups. The elastic moduli are determined by demanding the coincidence of the frequency dispersion surfaces obtained using the two-dimensional theory with the lowest foils of the corresponding surfaces resulting from the three-dimensional theory of elasticity.