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j. Cosmet. Sci., 53, 249-261 (September/October 2002) Dynamic hairspray analysis. III. Theoretical considerations j. JACHOWICZ, International Specialty Products, Wayne, NJ 07470. Accepted for publication March 15, 2002. Synopsis Model mechanical calculations were carried out to simulate the properties of hair fiber assemblies operating in the bending mode. It was assumed that the fibers were in omega-loop configuration (as in experimental work), and the theory of deformation of thin rings provided a fundamental relationship between stress and strain. A dependence on the stiffness of multi-fiber assembly was derived and verified empirically. Theo- retical bending stiffness for fixative-treated hair was calculated for various model fiber distributions by calculating their area moment of inertia according to the parallel axis theorem. General equations for stiffness of fiber assemblies were derived for cross sections placed on cubic and hexagonal lattices. The results suggest a good agreement between the theoretical and experimental data. INTRODUCTION Quantitative analysis of various parameters related to the bending deformation of hair fibers as well as the effect of fixatives on bending has been recently presented in the literature (1-4). New sensitive instrumentation such as dynamic mechanical analysis and texture analysis has enabled the measurement of the mechanical properties of hair with great precision both as individual single fibers as well as fiber assemblies. This work is important from a theoretical as well as a practical point of view. Theoretical description of the bending behavior of single fibers as well as of multiple fiber assemblies is relevant to the behavior of hair on the scalp, to its properties after styling, and to the effect of styling products on hair properties. In commercial practice, there is a great interest in parameters such as hair stiffness or flexibility after modification with a styling resin, the effect of high humidity on the ability of hair to maintain style (hold), curl memory, curl snap, etc. These quantities define the performance of a product or its key ingredients and are frequently employed in product descriptions or in advertising. In this paper we are mostly concerned with a theoretical definition and interpretation of the parameters obtained in a technique referred to as dynamic hairspray analysis, which employs hair shaped into "omega-loops" to quantify the mechanical properties of un- treated and polymer-modified hair. The most important of those is so-called hair stiff- ness, which was previously quantified by a parameter referred to as the stiffness ratio (4,5). It was defined as a ratio of the measured maximum force at 6.25% deformation (within the elastic region of force vs deformation dependence) for treated and untreated hair. The stiffness ratio was shown to increase 10-20-fold after treatment with styling 249