Some theoretical and experimental aspects of design with brittle material Public Deposited

http://ir.library.oregonstate.edu/concern/graduate_thesis_or_dissertations/br86b573s

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  • The Weibull theory, an attempt to account for the variability of measured strength in brittle materials, is described in some detail. An important and useful modification is derived, the normalized survival probability distribution, given by the equation [equation] where S is the probability that the specimen will survive the fraction [beta] of the mean strength, and m is ideally a material constant called the Weibull modulus. The symbol [gamma] denotes the gamma function. Some of the theoretical implications were investigated in a series of strength tests made on tubular BeO specimens by pressurizing them to failure. The results of these experiments were in good agreement with the theory. The test apparatus developed for this study appears to offer a useful addition to laboratory methods of measuring the tensile strength of brittle materials. Equation (1) is a simple analytic expression and can be used to deduce many aspects of the strength behavior of brittle materials. For example, the behavior of least values (that is, the strength of the weakest specimen in a sample of size n) may be estimated by the formula for the most probable strength of the weakest specimen. This least strength [beta asterisk] is given by [equation]. Since the behavior of the least values constitutes the engineering limitation to the application of brittle materials, design must be based on an understanding of least values. The possibilities of influencing the least values by proof testing to eliminate the weak elements, or by prestressing (bias), are examined. It appears that the benefits of proof testing may be limited because mechanical damage may be induced even at low stresses. It is suggested that the safety margin be given as the "extreme" safety factor, defined as the ratio of the most probable least strength to the operational stress, which in terms of the requisite failure probability F is given by [equation]. Many of the useful formulas of the Weibull theory are approximated by simple forms which may serve as practical rules of thumb. Some general aspects of design philosophy are considered and a number of design rules are proposed. For purpose of general information, the variance of the Weibull distribution and its least values are computed and plotted, demonstrating the superior reliability of the extreme values and the interesting double-valued nature of standard deviation for certain combinations of sample size n and Weibull modulus m. A table of stress distributions and the corresponding "risks of rupture" is presented and may be used in circumstances where a complex stress distribution is to be approximated by more tractable forms.
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