22 July 2016 ZERODUR strength modeling with Weibull statistical distributions
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Abstract
The decisive influence on breakage strength of brittle materials such as the low expansion glass ceramic ZERODUR is the surface condition. For polished or etched surfaces it is essential if micro cracks are present and how deep they are. Ground surfaces have many micro cracks caused by the generation process. Here only the depths of the micro cracks are relevant. In any case presence and depths of micro cracks are statistical by nature.

The Weibull distribution is the model used traditionally for the representation of such data sets. It is based on the weakest link ansatz. The use of the two or three parameter Weibull distribution for data representation and reliability prediction depends on the underlying crack generation mechanisms. Before choosing the model for a specific evaluation, some checks should be done. Is there only one mechanism present or is it to be expected that an additional mechanism might contribute deviating results? For ground surfaces the main mechanism is the diamond grains’ action on the surface. However, grains breaking from their bonding might be moved by the tool across the surface introducing a slightly deeper crack. It is not to be expected that these scratches follow the same statistical distribution as the grinding process. Hence, their description with the same distribution parameters is not adequate. Before including them a dedicated discussion should be performed.

If there is additional information available influencing the selection of the model, for example the existence of a maximum crack depth, this should be taken into account also. Micro cracks introduced by small diamond grains on tools working with limited forces cannot be arbitrarily deep. For data obtained with such surfaces the existence of a threshold breakage stress should be part of the hypothesis. This leads to the use of the three parameter Weibull distribution. A differentiation based on the data set alone without preexisting information is possible but requires a large data set. With only 20 specimens per sample such differentiation is not possible. This requires 100 specimens per set, the more the better.

The validity of the statistical evaluation methods is discussed with several examples. These considerations are of special importance because of their consequences on the prognosis methods and results. Especially the use of the two parameter Weibull distribution for high strength surfaces has led to non-realistic results. Extrapolation down to low acceptable probability of failure covers a wide range without data points existing and is mainly influenced by the slope determined by the high strength specimens. In the past this misconception has prevented the use of brittle materials for stress loads, which they could have endured easily.
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Peter Hartmann, "ZERODUR strength modeling with Weibull statistical distributions", Proc. SPIE 9912, Advances in Optical and Mechanical Technologies for Telescopes and Instrumentation II, 991208 (22 July 2016); doi: 10.1117/12.2231608; http://dx.doi.org/10.1117/12.2231608
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KEYWORDS
Surface finishing

Diamond

Mirrors

Failure analysis

Polishing

Etching

Zerodur

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