Calculating of the True Sizes and the Numbers of Spherical Inclusions in Metal

Abstract

Observing inclusions on a polished plane, researchers do not see these inclusions directly, but only their cross sections. Meanwhile, the given size of a section can be obtained from the inclusions of different sizes. The true sizes of the inclusions and their quantities are hidden from the researchers. This paper proposes a new method for determining the size distribution of inclusions derived from the size distribution of their sections. The method based on mathematical statistics allows evaluating the true sizes and quantities of inclusions. A notable feature of the method is that it enables a researcher to derive the confidence sets for the distribution functions. A special technique for testing the method has been developed. Numerous tests have shown good adequacy of the method.

Appendix

The values of the coefficients θ _ij for Eq 20 for the next partition of inclusion diameter are as follows.

$$ \begin{aligned} d = \left( {\begin{array}{*{20}c} {\left( {0 - 1} \right]} & {\left( {1 - 2} \right]} & {\,\,\,\,\,\,\left( {2 - 3} \right]} & {\,\,\,\,\,\,\,\,\,\,\,\left( {3 - 5} \right]} & {\,\,\,\,\,\,\,\left( {5 - 7} \right]} & {\,\,\,\,\,\,\left( {7 - 10} \right]} & {\,\,\left( {10 - 15} \right]} & {\left( {15 - 20} \right]} \\ \end{array} } \right) \times 10^{ - 6} \hfill \\ \varTheta = \left( {\begin{array}{*{20}l} {400} \hfill & { - 158.882} \hfill & {36.01098} \hfill & { - 7.0613} \hfill & {1.66294} \hfill & { - 0.48301} \hfill & {0.042313} \hfill & { - 0.03341} \hfill \\ 0 \hfill & {186.294} \hfill & { - 111.8921} \hfill & {9.475127} \hfill & { - 4.98322} \hfill & {0.416576} \hfill & { - 0.26237} \hfill & {0.013874} \hfill \\ 0 \hfill & 0 \hfill & {139.93307} \hfill & { - 50.849} \hfill & {11.97536} \hfill & { - 3.19812} \hfill & {0.373078} \hfill & { - 0.20961} \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & {39.55505} \hfill & { - 25.4649} \hfill & {3.010473} \hfill & { - 1.12494} \hfill & {0.164829} \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {31.70203} \hfill & { - 15.5809} \hfill & {1.874784} \hfill & { - 0.90048} \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {14.52936} \hfill & { - 6.42709} \hfill & {1.439377} \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {5.597323} \hfill & { - 3.76784} \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {4.67192} \hfill \\ \end{array} } \right) \times 10^{10} \hfill \\ \end{aligned} $$