480 JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS A well known product with hair conditioning claims was applied to fibers in a variety of ways. A decrease in fiber stiffness resulted but the original values were restored follow- ing use of a commercially available shampoo (23). Permanent waving of hair with a personal use product caused fiber stiffness to progressively decrease as time allowed for the reduction step was increased (23). Polymerization within fibers is also generally accomplished with an initial reduction step which weakens fibers. Nevertheless, overall increases in stiffness are achieved by proper selection of monomers and reaction conditions (23). Proximal and distal halves of four long fibers from each of three individuals were com- pared for stiffness in an attempt to detect normal wear and aging effects. Unex- pectedly, distal sections of nine fibers were stiffer and stiffness averaged 2% higher for distal halves of all fibers. Moreover, linear density was greater for distal halves of six fibers. Bleached and untreated fibers from the same individual could not be distinguished when stiffness and linear density results were graphed. Measurement of same fibers before and after bleaching should be more discriminating. Dry fiber stiffness is mainly discussed in the present paper because of its important influence on a person's hair behavior. However other experiments show that wet stiff- ness is generally a more sensitive measure of fiber strength changes caused by hair treatments. CONCLUSIONS The Balanced Fiber method for measuring fiber stiffness offers simplicity in experi- mental setup, avoids need for fiber clamping and allows replicate measurements on single fibers. The measuring instrument may be as simple as a ruler or as complex as a traveling microscope. Fibers are not affected by stiffness measurements and can be measured for other physical properties or for changes caused by fiber treatments. The method appears readily adaptable for other materials in filament or sheet forms. Hair fibers can be routinely compared for stiffness using only the distance measure- ment. However this parameter has theoretical significance which qualifies the method for use in research programs. ACKNOWLEDGEMENT Appreciation is expressed to Messrs. E. J. Gibbons and J. C. Jervert for useful dis- cussions and to Ms. P. Redman for obtaining much of the experimental data. APPENDIX LIST OF SYMBOLS A. Average cross-sectional areaof fiber, cm 2 D. Stiffness index, cm EB. Elastic modulus for bending G. Stiffness coefficient H. Hookean slope for extension of a 5-cm fiber, g/m I. Moment of inertia of the fiber cross-sectional area L. Linear density of fiber,/zg/cm

STIFFNESS OF HUMAN HAIR FIBERS 481 M. Bending moment r. Average radius of a fiber, cm R. Radius of curvature T. Tension applied to each fiber end, dynes W. Diameter of wire used assupport, cm Z. Fiber length, cm e. Tensile strain on fiber p. Bulk density of fiber, g/cm a (r. Tensile stress in fiber FIBER SHAPE Physics and engineering texts (24) commonly show equations derived for cantilever beams having very small beam deflections and such equations are often used as a basis for stiffness measurements on fibers. The restriction to small-beam deflections simplifies the derivation steps but, for the fiber hanging over a wire, infinite beam lengths and deflections must be considered in developing suitable theory. Referring to Figure 1, the right half of the fiber suffices for derivation purposes and fiber weight is assumed negligible compared to the attached weight. At any point on the fiber, the bending or clockwise moment, T(D/2 - x), will oppose and at equilibrium will equal the restoring or counterclockwise moment, EB I/R where EB is the elastic modulus for bending, I is the moment of inertia of the cross-sectional area and R is the radius of curvature of the bent fiber. Replacing R with the differential expression (24) for the radius of curvature of an arc, and collecting terms, we obtain - -x dx (1) (1 + p2)a/2 En I where p = dy/dx. This is a standard form of differential which integrates to give (1 + p2)•/= E. I ' (2) The integration constant is zero since at the hook x and p equal zero. At the weighted end of the fiber x = D/2, p is infinitely large and therefore from eq 2 T 8 EB I D • (3) Substituting 8/D • for T/EB I in eq 2 and dy/dx for p gives 4x'/dx D=/ dy = (4) _ _ 4x'/'l An integrated expression is obtained from eq and integrating "by parts." 4 by substituting cos 0 for 4x/D - 4x2/D 2 D D I+F 1og• I + C (5) 1-F