Influence of toughness on Weibull modulus of ceramic bending strength
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It is widely suspected that tougher materials are more reliable. Indeed, it is said about ceramics that "High toughness leads toflawtolerance and high Weibull modulus,"1 where Weibull modulus m is a measure of reliability, denned by the scatter in a series of bend strength results on nominally identical samples of ceramic.2'3 Similarly, Jayatilaka4 states that "as m increases, the material becomes less brittle." The purpose of this paper is to demonstrate that the above statements are untrue for a brittle material, such as a ceramic, which obeys the Griffith fracture criterion5'6 and which contains a constant distribution of flaw lengths. A simple theoretical argument shows that fracture toughness per se should have no influence on Weibull modulus. Bending strength experiments have confirmed this theory by measuring the Weibull modulus of a very brittle green body and showing that this modulus remained the same on sintering, although the toughness increased by two orders of magnitude. Arguments are then advanced to explain the mistaken notion that toughness influences Weibull modulus. One reason is that materials can be made tougher by removing large flaws. However, the Weibull modulus rises in this case because the flaw distribution is narrowed, not because of a toughness increase. A more significant reason for the idea that toughness increases reliability is that toughening procedures often cause the material to deviate from Griffith behavior in such a manner that toughness increases with crack length. This form of toughening, whether it be R-curve, Dugdale, or fibrous crack-bridging behavior, is shown to give an increased Weibull modulus, not because the toughness is higher but because the toughness increases with crack length.
being defined as n/{q + 1), where n is the number of samples with strength a or less and q is the total number of samples tested] by the function In ln[ l/( 1 - Pf ) ] = m ln[ {a - au )/a0] + const, (1) where au is the stress at which Pf is zero, a0 is a normalizing stress, and m is the Weibull modulus. Taking au to be zero, since there is always some risk of failure even at low stresses, In In [ l/( 1 - Pf) ] = m In a — m In a0 + const. (2) If the material behaves in a brittle, Griffith fashion, then (3) a = Ku/{^c, where c is the length of the criticalflaw,assumed to be a through crack. The distribution of strength values a stems from the distribution offlawlengths c in the samples. For a tougher material,
a2=Kia,/Jm.
(4)
Thus a2/a = Klc2/Ku. (5) For failure of the tougher material, from Eq. (2),
gradient m—-__/ / lnln_L 1-P f /—high toughness
low toughness /
K IC2
In
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