HAIR BREAKAGE: REPEATED GROOMING EXPERIMENTS 453 fragments are shown in Table IV, while the models obtained from these results are shown in Figure 13. Findings suggest signifi cant systematic trends. The formation of longer fi bers is modeled with a low shape parameter, indicating that the highest incidence of formation takes place early in the combing process. Conversely, formation of the shortest fragments is described by a higher shape factor, showing a greater formation rate with increased grooming cycles. In short, the smallest fi ber fragments are generated in a “wear-out” mechanism, while the longer fragments form by a “premature failure” mechanism. More- over, fi ber sizes between these extremes show intermediate behaviors. To summarize: as anticipated, an increased number of sort fi ber fragments are formed with repeated grooming, presumably due to fl aw propagation in the most-damaged tip area. However, contrary to our thesis, the formation of longer fragments is more abundant earlier in the grooming process. A premature failure mechanism (i.e., a shape parameter less than 1) results from the relatively rapid failure of pre-existing defects, without ap- preciable generation and propagation of new ones. Therefore, one explanation for the long-fi ber results involves a “weeding out” of already damaged fi bers early in the process. An alternative explanation is the hypothesis of Robbins and Kamath, which proposes that longer fragments are the result of tangles that occur during the grooming process, that is, Table IV Weibull Parameters Describing the Formation of Differently Sized Breakage Fragments Fragment size Characteristic lifetime Shape factor ≤ 2 mm 252,000 1.80 2 mm–1 cm 1,940,000 1.13 1 cm–3 cm 34,000,000 0.76 ≥ 3 cm 627,000,000 0.53 Figure 13. Failure prediction curves for different-sized fragments as a function of brushing strokes at 60% RH.

JOURNAL OF COSMETIC SCIENCE 454 a higher incidence of tangles occurs earlier in the grooming process, which are gradually “brushed out” with time. In addition to the size of the broken fi bers, there is also a need to mention the thickness. As has been shown, there is a squared relationship between the radius of a fi ber and the applied stress, and an exponential relationship between the stress and the propensity for breakage. Therefore, theory suggests a considerably higher tendency for breakage of fi ne fi bers within a tress during the performing of such experiments. While we did not spe- cifi cally generate data to test this presumption, general experimental observations do appear to be in agreement. CONCLUSIONS In mechanical testing terms, the grooming of hair represents a fatiguing process wherein individual strands experience repeated exposure to an external stimulus. Therefore, in accordance with fatiguing principles, there is a gradual propagation of fl aws within indi- vidual fi bers until ultimately catastrophic failure (breakage) results. As such, hair fi bers will break upon repeated application of forces considerably lower than those required to induce breakage from a single-force application. Moreover, this process occurs in a pre- dictable manner that can be modeled to provide an understanding of the contributions associated with different variables in hair breakage. The fi rst paper in this series utilized single-fi ber fatigue experiments to generate funda- mental learning on the effects of such variables. Particularly noteworthy was that consid- erably larger differences were observed as a function of hair type and environmental conditions in fatigue experiments as compared to conventional stress-strain testing. Re- sults also showed a strong dependence of breakage on the magnitude of the applied stress, thus demonstrating how lubrication provided by conditioner products lowers grooming stresses and provides considerable protection against breakage. As such, this introduced the idea of using standard fatigue testing approaches to describe and model the data ob- tained from repeated grooming experiments. Failure analysis is often modeled using Weibull statistics, an approach that introduces the novel idea of treating hair breakage as a statistical variable. Therefore, the output from this approach yields predictions as to the likelihood of hair breakage under differing con- ditions. In this instance, a grouped Weibull analysis was used to analyze breakage data from repeated grooming experiments. Perhaps contrary to fi rst impressions, these tests are not as straightforward to perform as may be imagined. Therefore, some time was spent describing the equipment and the importance of certain experimental variables. The Weibull analysis technique is described, and it is shown how generation of the two Weibull parameters (the characteristic lifetime and the shape parameter) provide a means of quantifying these experiments. However, the real strength of the approach involves the ability to generate survival probability plots that provide predictions as to the likelihood of fi ber breakage under different conditions. Therefore, assuming these laboratory exper- iments are a reasonable representation of real-life conditions, it becomes possible to pre- dict breakage rates on actual heads as a function of different habits and practices. Of course, such experiments frequently represent accelerated wear and tear and consequently stray somewhat from real-life practices. However, more accurate predictions may be obtained by performing more lengthy experiments involving more intricate and realistic