HAIR BREAKAGE: REPEATED GROOMING EXPERIMENTS 441 (Dia-stron CYC800, Dia-stron, UK), where experiments involve repeated application of a user-defi ned force until breakage. The propensity for breakage was found to depend strongly on the magnitude of the repeating stress (i.e., stress = force/unit area), meaning that the magnitude of the repeating force and the fi ber dimensions are major infl uences. This relationship is shown in Figure 1, in what is often termed an S-N curve. From Figure 1, one observes that an exponential relationship exists between the magni- tude of the repeating stress and the number of cycles to fail. Extrapolating these fi ndings into the world of hair care, one observes how lowering the stresses associated with groom- ing will signifi cantly reduce the likelihood of fi ber breakage, and it explains why many conditioning products produce dramatic effects in repeated grooming experiments. These single-fi ber experiments provide fundamental understanding, but the magnitude of the stresses experienced by individual fi bers during everyday grooming is not readily available. As such, while it is possible to model the propensity for breakage as a function of the applied stress, it is unknown where real-life conditions lie. Repeated grooming experiments represent the opposite scenario—in that real-life stresses and strains are pre- sumably replicated relatively well but the magnitude of these stimuli is unknown. Nev- ertheless, this comparision does introduce the idea of treating repeated grooming results by fatigue testing approaches. WEIBULL ANALYSIS OF FAILURE DATA In a fatigue test, failure is commonly attributed to the propagation of pre-existing fl aws within a material. Accordingly, with the distribution of such fl aws on a hair fi ber being statistical in nature, breakage also needs to be treated as a statistical var- iable. Therefore, modeling and characterization of breakage involves fi tting a statis- tical distribution to the data. A convenient approach involves utilization of the highly fl exible Weibull distribution (9). The fl exibility of this expression arises from the presence of the Weibull shape factor, β. By means of illustration, when β = 3.6, Figure 1. S-N curve showing failure data for virgin Caucasian hair at 60% relative humidity.

JOURNAL OF COSMETIC SCIENCE 442 the Weibull function approximates the normal distribution, whereas, when β = 1, the distribution is equivalent to the exponential distribution. Therefore, this ap- proach can be used to model a wide range of data that is subsequently characterized by evaluation of the two Weibull parameters—the shape factor and the characteristic lifetime. The linear form of the Weibull equation is shown below: [ ] = β β α) ln ln (1/1 F(x)) (ln x) (ln – – – where F(x) is the probability of the fi ber breaking in x cycles, 1-F(x) is the probability of surviving x cycles, a is the characteristic lifetime at which 63.2% of the fi bers have bro- ken, and b is the shape factor. Thus, a plot of the double logarithm of the reciprocal of the survival probability function versus the logarithm of the number of cycles-to-fail yields the shape parameter from the slope and the characteristic lifetime from the intercept. Estimation of the cumulative distribution function, F(x), involved the commonly used median rank method. As outlined in the earlier publication, this approach fi rst involves ordering cycles-to-fail results for all fi bers from lowest to highest and then applying the median rank equation given below: 0.3 Median rank( ) 0.4 i Xi n − = − where i is the sample number and n is the total number of samples. After evaluation of the two Weibull parameters, it becomes possible to reconstruct the best-fi tting distribution from which they were derived, and, in doing so, generate predic- tions for the likelihood for survival (or failure) as a function of the number of cycles. The result is termed a survival probability plot, and this process has been described for single-fi ber experiments in previous papers (8,10,11). By means of illustration, Figure 2 shows a sur- vival probability plot obtained for virgin Caucasian hair fi bers exposed to a repeating 0.010-0.011 g/um2 stress at 60% relative humidity. For reference purposes, this magni- tude is approximately half of the break stress for hair in a conventional stress-strain ex- periment under the same conditions. Therefore, from Figure 2 one observes a 100% likelihood of survival when zero stress cycles are applied, with the survival probability progressively decreasing as a function of additional cycles. Thus, from performing comparable experiments under differing condi- tions, it becomes possible to generate similar plots that allow for breakage predictions as a function of specifi c variables (e.g., the magnitude of the applied stress, the nature of the hair, the environmental conditions, etc.). As will be shown in this report, repeated grooming experiments can be modeled using what is termed a grouped Weibull analysis (12). Such experiments involve generating a cumulative distribution function by obtaining a running total of broken fi bers over regularly repeating time intervals (for example, after 1000, 2000, or 3000 grooming strokes).