Reproduced from Soil Science Society of America Journal. Published by Soil Science Society of America. All copyrights reserved. Brittle Fracture of Soil Aggregates: Weibull Models and Methods of Parameter Estimation L. Munkholm* and E. Perfect ABSTRACT experimental data. To our knowledge the three-parameter model has not been previously applied to characterize Brittle fracture of soil aggregates is usually analyzed with the Wei- brittle fracture of soil aggregates. bull “weakest-link” model. Failure is expressed in terms of a probabil- Double logarithmic transformation of an aggregate ity distribution function (pdf) of aggregate strengths. Traditionally a two-parameter Weibull model is fitted to double log-transformed strength pdf combined with linear regression (LIN) is data with the Weibull parameters ( and ) estimated using linear the standard parameter estimation method in soil stud- regression. The main objective of this study was to compare the ies (Braunack et al., 1979; Dexter and Watts, 2000). goodness-of-fit for a three-parameter versus a two-parameter Weibull Lack of a straight line fit is a common problem in soil model. In addition, we compared three common methods of parameter studies (Perfect et al., 1998). This indicates problems estimation: linear regression, nonlinear regression, and maximum like- with model and/or fitting method. For other applica- lihood. The different models and methods of estimation were evalu- tions, the LIN method has been questioned and the ated using previously published and unpublished aggregate rupture maximum likelihood method (ML) has been shown to energy data from three contrasting soil types (Bygholm sandy loam, Maury silt loam, and Karnak silty clay). Overall, the goodness-of-fit provide more accurate parameter estimates for the two- was not markedly improved by using a three-parameter as compared parameter model (Trustrum and Jayatilaka, 1979; Kha- with a two-parameter Weibull model. The choice of model had a lili and Kromp, 1991; Seguro and Lambert, 2000; Clarke, significant effect on the parameter estimates. The three-parameter 2003). In a few studies, Weibull parameters have been model produced lower estimates of  than the two-parameter model. estimated using nonlinear regression (NLIN) (Perfect The data were always best fitted using nonlinear regression. Nonlinear and Kay, 1994; Perfect et al., 1998). The NLIN method regression also resulted in a greater power of distinction between has also been applied to the three-parameter model management treatments and aggregate sizes for  on the Maury soil. (Ferreira et al., 2003). A number of other methods of We recommend fitting aggregate rupture data to a two-parameter parameter estimation have been proposed. The method Weibull model and estimating the model parameters using nonlinear regression. of moments is probably the most well known of the alternative approaches. However, previous studies have shown that this method does not provide more accurate B rittle fracture of soil aggregates is often analyzed estimates than the more common LIN and ML methods probabilistically to investigate soil, management, (Trustrum and Jayatilaka, 1979; Mahdi and Ashkar, 2004). and/or size effects. Several different probabilistic mod- The main objective of this study was to evaluate the els have been proposed for the brittle fracture of hetero- applicability of a three-parameter versus a two-parame- geneous materials (e.g., Srolovitz and Beale, 1988; Herr- ter Weibull model to fit to soil-aggregate brittle fracture mann and Roux, 1990; Frantziskonis, 1995). Of these, the data. An additional objective was to compare three com- Weibull “weakest-link” model (Weibull, 1952; Freuden- mon methods of parameter estimation: LIN, NLIN, and thal, 1968) is the most widely accepted. Failure in this ML. The different models and methods of estimation model is expressed in terms of a pdf of aggregate strengths. were evaluated using measured aggregate rupture en- The Weibull distribution may be expressed as a two- ergy data from three contrasting soil types: Bygholm parameter model, with scale () and shape () parame- sandy loam, Maury silt loam, and Karnak silty clay. The ters, or as a three-parameter model that also includes Bygholm and Maury data sets have been reported in a location parameter (E 0 ). The two-parameter Weibull previous papers (Perfect et al., 1998; Munkholm and model implies that the probability of failure is zero only Kay, 2002; Munkholm and Schjønning, 2004). The Kar- when the rupture energy (E ) is zero, that is, there is nak data have not been previously published. We evalu- always a probability to fail no matter how small the ated the goodness-of-fit, effects on parameter estimates, rupture energy. Shih (1980) suggested using a three- and power of discrimination between management treat- parameter model to characterize the pdf of strengths for brittle materials to obtain a better goodness-of-fit to Abbreviations: AIC, Akaike’s information criterion; ANOVA, analysis of variance; cdf, cumulative probability density function; D, Kolmogorov- Smirnov statistic; E, rupture energy; E 0 , the location parameter—the L. Munkholm, Dep. of Agroecology, Danish Institute of Agricultural value of E where the probability of failure is estimated to be zero; Sciences, Research Centre Foulum, P.O. Box 50, DK-8830 Tjele, Den- F, Fisher’s F statistic; L, likelihood function; LIN, linear regression; mark; E. Perfect, Dep. of Earth and Planetary Sciences, Univ. of LIN-2 = linear regression, two-parameter Weibull model; ML, maxi- Tennessee, 1412 Circle Dr., Knoxville, TN 37996. Received 30 Aug. mum likelihood; ML-2, maximum likelihood, two-parameter Weibull 2004. *Corresponding author (lars.munkholm@agrsci.dk). model; ML-3, maximum likelihood, three-parameter Weibull model; n, number of fits or samples; NLIN, nonlinear regression; NLIN-2, Published in Soil Sci. Soc. Am. J. 69:1565–1571 (2005). Soil Physics nonlinear regression, two-parameter Weibull model; NLIN-3, nonlin- ear regression, three-parameter Weibull model; P, probability; pdf, doi:10.2136/sssaj2004.0290  Soil Science Society of America probability density function; R 2 , coefficient of determination; RSS, residual sums of squares. 677 S. Segoe Rd., Madison, WI 53711 USA 1565 Published online August 25, 2005