258 JOURNAL OF COSMETIC SCIENCE cubic lattice hexagonal lattice Figure 8. Model fiber packings in a hair tress. 600 I 500 400 300 200 lOO o 0 2 4 6 8 10 12 14 16 18 20 Number of layers m cubic lattice -•- hexagonal lattice Figure 9. Stiffness ratio calculations for cubic and hexagonal lattices. of 1.4 g/cm 3, the calculated tress thickness is 376 pm, which corresponds to five to six layers of tightly packed average Caucasian hair with a diameter of 70 pm. The theoret- ically calculated stiffness ratios for six layers in a cubic and hexagonal configuration are 48 and 36, respectively. Since experimental stiffness ratios were found to be in the range from 20 to 50, this suggests excellent agreement with theoretical predictions. It should also be noted that the proposed mechanical model can explain higher values of stiffness ratio observed for fixative-treated thinner fibers. This experimental observation could be a consequence of the fact that thinner fibers should produce more layers in a tress than thicker fibers, assuming that the total weight or volume of fibers is the same in both cases. Since the stiffness ratios are dependent on fiber dimensions through n (n•/n 2 = r2/r•) , and the theoretical stiffness ratio increases strongly with n, the model predicts higher stiffness ratios for thinner hair treated with fixatives.

DYNAMIC HAIRSPRAY ANALYSIS 2 5 9 CONCLUSIONS By considering a simple model of fiber bending, with the assumption of a circular cross section of hair and a shape in the form of an "omega-loop," it has been shown that the ratio of hair stiffness of hair bundles with the same fiber volume is equal to the ratio of their respective diameters raised to the second power. The model was confirmed by experimental results obtained by using hair with various thickness such as Caucasian fine hair, Caucasian normal hair, and Chinese hair. For polymer-treated hair, calculations were performed for several model fiber assemblies, and for generalized fiber distributions on cubic and hexagonal lattices by using the parallel axis theorem. It was demonstrated that stiffness depends primarily on the number of layers in a tress. Theoretical predictions were in good agreement with experimental results obtained in dynamic hairspray analysis. APPENDIX The area moments of inertia of circular cross sections were calculated by employing the parallel axis theorem. It states that the area moment of inertia of a body about any axis equals the area moment of inertia about the parallel axis through the center of gravity plus the product of the area of the body and the square of the distance between the two parallel axes (6). For a single circular cross section (6) • x 4 ?r r I•/- 4 P• - 1 X" The distance between x and x' is r the area is q'rr 2. I2/= -•- + ?rr2 r2 + -•- + q'rr2 - • P2 2q'rr 4 4