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**0**

**19**

**Statistical Tests Based on the Geometry of**

Viktor Beneš1, Lev Klebanov1, Radka Lechnerová2 and Peter Sláma3 <sup>1</sup>*Charles University in Prague, Faculty of Mathematics and Physics, Department of*

The actual trends in new material development impose the necessity of a thorough knowledge of the relationships between properties and microstructure. In order to meet the requirements on the performance of some materials in their applications, that are often very sophisticated, a very fine tuning of the manufacturing process and its parameters is needed. Therefore, it is indispensable to be able to distinguish between small differences in microstructures produced by the different variations of processing parameters. Another important field of material characterization is the description of microstructure heterogeneities. Such heterogeneities are often related to risks of premature damage nucleation and preferential defects (void, cracks,

One of the most important elements of the microstructure of metallic materials is the set of second phase particles. Particle size and shape distributions and the type of spatial dispersion (homogeneous, long-range or short range ordered, clustered, etc.) are often the major attributes of a particular microstructure (Humphreys & Hatherly, 2004; Polmear, 2006). Thin foils made from aluminium-manganese based alloys, such as AA3003, are the material most frequently used as fins in automotive heat exchangers (Hirsch, 2006). This application imposes very strict requirements on properties and related foil microstructures. The development of an appropriate production technology is contingent on the perfect knowledge of the impact of processing parameters on microstructure transformation, including the

In statistical setting, we deal with microstructures containing random objects in a space or plane, which may be second phase particles, pores, grains and their sections or projections. The question frequently asked is whether two microstructures come from a material with the same geometrical characteristics of microstructure. This statement forms a null hypothesis *H*<sup>0</sup> and the aim is to develop a statistical two-sample test of *H*<sup>0</sup> against an alternative hypothesis that the geometrical characteristics are different. In the literature, parametric models of microstructures as random sets are mostly used (Derr & Ji, 2000; Ohser & Mücklich, 2000) and the authors recommend Monte Carlo testing which is based on the possibility of simulating a random set under the null hypothesis. The evaluation of the test is based on a comparison of the test statistics (describing some characteristics (Tewari & Gokhale, 2006a;b) of the random

**1. Introduction**

corrosion, etc.) occurrence and propagation.

changes of the set of particles (Hirsch, 2006; Slámová et al., 2006).

**Second Phase Particles**

*Probability and Mathematical Statistics*

*Czech Republic*

<sup>2</sup>*Private College of Economic Studies, Ltd., Prague* <sup>3</sup>*COMTES FHT a.s., Metallography, Dobˇrany*

