BENDING OF HAIR AND PERMANENT WAVING 131 YOUNG'S MODULUS AND BENDING STIFFNESS Figure 2 shows typical results for the time dependence of the bending stiffness observed during a treatment sequence. During the initial water treatment, about 15% of the initial stiffness is lost and mechanical equilibrium is approached after approximately 20 min. Throughout the subsequent reducing process, the bending stiffness decreases al- most linearly with time until the first rinse is applied. The rate of this decrease increases with the concentration of TGA as well as with pH. After application of the first rinse, the bending stiffness immediately ceases to fall and remains almost constant during the subsequent treatments, except for some cases in which a tendency for a slight increase was observed during the first rinsing process, especially for strongly reduced fibers. These results are in agreement with the observations made for the analogous extensional experiments (3,4). It is important to mention that the bending relaxation behavior observed during the reducing treatment is markedly different from the extensional behavior under the same conditions (4). First, in contrast to the linear decrease in the bending stiffness, the extensional stress relaxation shows a tendency to level off. Second, the extent of the decrease is significantly larger for the bending stiffness than for the extensional stress at all corresponding conditions. For example, for 0.3 M TGA at pH 9.0, a decrease of about 50-60% occurs for the bending stiffness (see Figure 2) but only about a 40% decrease is observed for the extensional stress (4). Assuming that in an untreated fiber the Young's modulus is homogeneous over the fiber cross-section and equal for extension and compression, the bending stiffness is related to the Young's modulus as B = I X E (14) where E is the Young's modulus of the fiber and I the moment of inertia, which for a circular fiber with radius r is given by I = ,r r4/4 (15) Scott and Robbins (6) verified the applicability of Equation 14 for human hair. In case the Young's modulus, though decreasing during reduction, remains uniform over the cross-section of the fiber, Equation 14 can be inserted into Equation 8 to yield (4): R c = Ere/Ero (16) where Ere and E•o are the Young's moduli under analogous experimental conditions such as Bre and Bro. Though the fiber diameter and hence the moment of inertia change during the treatment due to swelling and deswelling, Ere and Ero both finally act with the same moment of inertia, exhibited by the fiber at the time of release, so that the geometrical effects neutralize each other. It has been shown (4), however, that the calculation based on Young's moduli leads to significantly lower set values compared to the actually observed set. This deviation to lower values is readily explained by the principal difference of the role of the extensional stress and that of the bending stiffness for fiber bending set. As already mentioned, the normalized bending stiffness decreases faster during the re-

132 JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS ducing treatment than the normalized extensional stress, so that the residual of the normalized extensional stress after reduction Ere/E o is larger compared to that of normal- ized bending stiffness Bre/B o. Due to the validity of Equation 14 for the untreated fiber, this result implies the inequality: Bre I• x Ere (17) which on restricting the following considerations to the reduced state of a fiber is generalized as B I X E (18) The discrepancy between the normalized bending stiffness relaxation and the normal- ized extensional stress relaxation supports the hypothesis already put forward in refer- ence 4 that during hair reduction a non-uniform distribution of Young's moduli is generated. The existence of such a distribution becomes plausible from mathematical- mechanical considerations. Assuming that the cross-section of the fiber is circular with radius r and the distribution of the modulus is symmetrical about the center of the fiber, the general formula of the bending stiffness is given by B = E(p)dI = -rr E(p)p3dp (19) where p is a distance from the centre of the fiber and E(p) is the Young's modulus at the distance p. The Young's modulus determined by extensional experiments can be re- garded as the average value over the cross-section of the fiber, so that E = E(p) = E(p)dA/A = 2 E(p)pdp/r 2 (20) where A denotes the area of the cross-section, and a bar over E(p) stands for the aver- aging over the cross-section. Substituting Equations 15, 19, and 20 into Inequality 18 and rearrangement leads to • [E(p)p 3 - 0.5 r2E(p)p] dp 0 (21) Through integration by parts, it can be shown that Inequality 21 is fulfilled if :(r 2 - p2)p2E'(p)dp 0 (22) E'(p) is the first derivative of E(p). Since (r 2 - p2)p2 is always positive or zero (when p = 0 or p = r), the simplest sufficient condition for this relation is E'(p) 0 (23) This result supports the hypothesis that Young's modulus in a reduced fiber decreases continuously from the core towards the surface of the hair fiber. The generation of such a modulus distribution is quite reasonable since during diffusion into a hair fiber, the reductant decreases in concentration through reaction, resulting in a lower degree of reduction of the core parts of the fiber compared to the annular parts. The form of the