Weibull Analysis of Mechanical Data for Castings II: Weibull Mixtures and Their Interpretation
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theory of brittle fracture requires a preexisting crack to initiate failure.[1] Similarly, ductile fracture requires a population of pores or cracks.[2,3] Consequently, statistical models for fracture in materials are all built on the concept that the difference between the ideal and actual performances can be attributed to the presence of defects (flaws) raising local stresses, and consequently, resulting in premature failure. Hence the statistical distributions of fracture-related mechanical properties, such as fracture stress, elongation, impact, fracture toughness, fatigue life, etc., can all be linked to the underlying defect size distribution. Statistically, this implies that the worst (largest) defect leading to the highest stress concentration is the one that determines the fracture-related mechanical properties.[4] Hence this defect constitutes the “weakest link”, based on the theory developed by Pierce.[5] The “weakest link” theory applies in situations that are analogous to the failure of a chain when its weakest link has failed.[6] Based on the “weakest link” theory, Weibull[7] introduced an empirical distribution, for which the cumulative probability function is expressed as: r rT m P ¼ 1 exp ; ½1 r0 where P is the probability of failure at a given stress (strain, fatigue life, etc.), σ, or lower. The threshold MURAT TIRYAKIOG˘LU, Director, is with the School of Engineering, University of North Florida, 1 UNF Drive, Jacksonville, FL 32224. Contact e-mail: [email protected] Manuscript submitted June 26, 2014. Article published online October 21, 2014 270—VOLUME 46A, JANUARY 2015
value, σT, is the value below which no specimen is expected to fail. The term, σ0, is the scale parameter, and m is the shape parameter, alternatively referred to as the Weibull modulus. The probability density function, f, for any continuous distribution is found by: f¼
dP dr
For the Weibull distribution, f is expressed as: m r rT m1 r rT m f¼ exp r0 r0 r0
½2a
½2b
One of the most commonly used methods of presenting the Weibull fits to data is the Weibull probability plot. After rearranging, Eq. [1] can be written as ln½ lnð1 PÞ ¼ m lnðr rT Þ m lnðr0 Þ
½3
Note that Eq. [3] has a linear form when the left-hand side of the equation is plotted vs ln(σ − σT) with a slope of m and an intercept of −m ln(σ0). Alternatively, the Weibull probability plot can be obtained when the left-hand side of the equation is plotted vs ln(σ). This method of presentation gives a straight line relationship only when σT = 0. This is demonstrated in Figure 1, in which Weibull distributions for three datasets for tensile strength (ST) of sand cast A356[8,9] are plotted. Probability was assigned to each data point by using the following plotting position formula: P¼
i 0:5 ; n
½4
where i is the rank in ascending order and n is the sample size. Note that the trend of the curve at low values of ln(σ) is METALLURGICAL AND MATERIALS TRANSACTIONS A
2.0
Boom-Filled & Sr-Modified σT = 235.1 MPa σ0 = 43.9
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