Statistical aspects of estimating density of penny-shaped inhomogeneities from their intersections with plane sections and lines
摘要
Quantitative evaluation of 3D density of cracks or plates from the data on their intersections with the 2D plane of a polish or a section may serve as a tool for monitoring changes in the effective elastic and conductive properties of materials due to such defects. In previous works co-authored with Mark Kachanov, relationships were established between the 3D density of flat circular cracks or thin plates and the 2D density of their line traces observed on the surfaces of sections; the latter parameter being defined as the sum of the trace half-lengths squared per unit area. In the present paper, the estimate of the 3D crack density, in the isotropic case, was derived using other parameters — numbers of the defect intersections with a plane section and a line — which are easier to determine experimentally than the crack trace lengths. The mentioned relationships were obtained in terms of mathematical expectations; therefore, estimates of the 3D density of cracks or plates, calculated using observed data are spread out around the actual values. Formulas for the dispersion measures of the crack density estimates were derived and analysed depending on various geometric parameters. In particular, it was shown in which cases the variance increases with an increase in the number of observed crack traces, that may seem counterintuitive.