DYNAMIC HAIRSPRAY ANALYSIS 2 5 3 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 1.1 1.2 1.3 1.4 1.5 1.6 1.7 log(R) -m (Linear Fit) Figure 4. Force as a function of diameter for steel rings at a constant deformation. were obtained for rudimentary loops obtained by imparting the circular shape to a wire and then linking its ends by twisting them around each other. Let's consider the deformation of multiple fiber assemblies and how they can be ap- proximated by using equation 1. For example, in order to evaluate the deformation of tresses of fibers characterized by different fiber thickness and the same total volume, the number of fibers in each bundle has to be calculated from the following relationships: V• = V 2 (3) Bundles 1 and 2 are characterized by the same fiber volume (or weight, assuming the same density of fibers in both bundles). nlTrrl 2 1 = n2Trr2 2 I (4) n 2 = n t (r,/r2) 2 (5) where V1, V2, rt, r2, n•, n 2 are volumes, radii, and numbers of fibers in bundles 1 and 2, and 1, is the length of a fiber bundle. A circular (or omega-loop) configuration is applied to both bundles of fibers. At a given deformation, By, the forces in each bundle are given by the following equations: •11•yEI •11•yE 4 P1 = -- (6) 4 4 = = (7) 4 ß rr l

254 JOURNAL OF COSMETIC SCIENCE Thus, the ratio of the force of deformation of fiber bundles with the same fiber volume and different radii, r• and r2, is given by P2 - (8) We have verified equation 8 experimentally by measuring the stiffness of three types of hair with different diameters. The results are presented in Figure 5 and Table I. Figure 5 shows the variation in fiber stiffness for hair tresses with the same volume but differing in fiber diameter. The data demonstrate a decrease in stiffness for thinner fibers. Table I presents a comparison of experimental and theoretically calculated stiffness (according to equation 8) values normalized to the stiffness obtained for Oriental hair, Porient/P. It also includes diameters of the investigated hair measured by using an optical microscope. The values of (Porient/P)th .... and (Pori•nt/P)•xp. are in reasonably good agreement, with experimental ratios being a little smaller than those estimated from the theoretical model. The table also presents the results of Swift's calculations (2), which were per- formed assuming the elliptical nature of Caucasian hair. The effect of fiber ellipticity might explain the lower experimental values of (Potions/P) than those predicted theoret- ically assuming circular cross sections of hair. To analyze the stiffness of hair tresses treated with fixatives we have to consider the total force of deformation of a fiber bundle consisting of n identical fibers with a cross section radius r. Such a force should be a superposition of forces acting on each individual fiber: n•yEI /',, = (9) -- -- For a fiber assembly treated with a fixative, one has to consider the fact that the average thickness of a polymer layer is small as compared to the fiber dimensions as shown in Figure 6. For the amount deposited of 90 mg/g, an increase in radius of 2.58% could 25 20 20.6•1.8 1.4 12.5•1.6 Oriental Caucasian Fine Caucasian Figure 5. Bending stiffness of various types of hair.