QUASI-STATIC TORSIONAL DEFORMATION OF SINGLE HAIR FIBERS 389 Given this apparent dominate contribution of the cuticles to the torsional modulus, it may be useful to use a model to separate out the respective contributions of the cuticles from cortex. To this end, we have used an approach based on the work conducted by Wortmann et al. (23). Applying this core/shell model, the following equation for the torsional storage modulus G′ is defi ned (equation 5), where the polar moment of inertia, I, is the addition of the polar moments of inertia of the cortex and the cuticle, Ico and Icu, respectively: a a +G = co co cu G I Icu Ga I (5) Individual cut icle cells are generally accepted to have an approximate thickness of 0.5 μm (24) and the total number of cuticle cells in unaltered European hair is 6–10. Given that the hair used is not directly taken from the scalp, we have assumed that the hair used has six cuticle cell layers and therefore a thickness of 3 μm. Using this information, the theo- retical polar moment of inertia of the cortex and the cuticle can be deduced by equation (6). Figure 4. Plot of the torsional modulus G as a function of the polar moment of inertia, Ip, of 192 virgin European hair fi bers. Line of best fi t was obtained using the equation for the torsional modulus G′ as described previously. Table I Means for the Experimental Torsional Storage Modulus of Virgin European Hair at 50% RH Based on the Analysis of 192 Fibers where 2 R s is the Residual Variance between the Experimental Data and the Fit Data Based on equation (5) and 2 T s is the Total Variance with a Value of 3.089 × 102 Cuticle layers Thickness (μm) G (GPa) q102 2 sR Gcuticle, (GPa) Gcortex (GPa) r2 4 2 1.923 1.201 6.747 0.655 0.611 6 3 1.923 1.205 5.043 0.592 0.610 8 4 1.923 1.209 4.191 0.523 0.609 10 5 1.923 1.213 3.680 3.680 0.607 The coeffi cient of determination, r2, provides the goodness of fi t based on an assumed number of cuticle layers. Refer to (23) for more details on calculations.

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