Weibull Statistics

Weibull plots are often used in the design of products fabricated from brittle materials. They are used to estimate the cumulative probability that the given sample will fail under a given load. These plots are based on MOR data obtained on a representative population of samples prepared and, where possible, tested in a manner similar to that which the products will experience during use. The data can be used (with some caveats – low probabilities are calculated from extrapolated data) to specify an applied stress at which a given fraction of samples will fail. Thus, if the application requires that less than 1 sample in 10,000 fail during use, the data can be used to estimate the stress that would cause this fraction of failure. Once the information is known, you may be able to use it to design the product such that no part of the sample reaches the stress (i.e. by choosing sample thickness, length, or some other geometrical factor), or by altering the conditions of use to avoid such stresses.

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Figure 4: Weibull plot of cumulative failure probability vs. applied tensile stress on logarithmic scale for various treatment times.

For products used in a given environment, and which have a specified geometry, improvements in the probability of failure require an increased strength. The figure above provides several examples of Weibull plots for glasses strengthened by ion exchange. Importantly, as mentioned in Process Design, short exchange times increase the average MOR but may not improve the weakest strengths significantly and in some cases can actually weaken the glass. As a result, one may have the same, or even greater, number of failures at low stress for the strengthened materials as for the unstrengthened material. Long exchange times typically reduce the number of low-stress failures relative to short exchange times, but do not significantly increase the maximum failure stress. Effects of post-treatment handling are often overlooked in chemical strengthening. However, since the case depth is much smaller than in thermal tempering, it needs to be considered in designing a process. Generally, scratch resistance and abrasion resistance increase, but if the case is not sufficiently deep, even light abrasion can reduce the strengths to values near, or even below, those of unstrengthened glass. Tests of abraded strengths need to be carried out for products which will experience such handling problems.

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Figure 5: The effect of post-treatment abrasion on the strength
of chemically strengthened glass.

Figure 5 is a Weibull plot which includes the effect of abrasion and illustrates that untreated samples and those having a short exchange treatment with post-treatment abrasion will experience 1% cumulative failure at a stress of about 100 MPa.

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