Analytical solution on the overestimation of thickness of interfacial transition zonearound grain with arbitrary convex shape by the section analysis method

This paper employs some related theorems as well as the result s f rom the field of geomet rical probabilityto explore the quantitative relationship between statistical mean values of the apparent interfacial t ransition zone( ITZ) thickness and the actual ITZ thickness. In the 2 dimensional case as well as in the 3 dimensional case , thegeneral solutions are derived for an arbit rary shape grain with closed convex curve (2D) or convex surface (3D) .The result s show that : (1) For the 2D grains with the closed convex curve of an arbit rary shape , the ratio of thestatistical mean apparent ITZ thickness to the actual ITZ thickness is less thanπ/ 2. The exact value depends on theshape of the grains. If the actual ITZ thickness is far less than the size of the grain , this ratio is approximately equaltoπ/ 2. (2) For the 3D grains with the closed convex surface of an arbit rary shape , the ratio of the statistical meanapparent ITZ thickness to the actual ITZ thickness is less than 2. The exact value also depends on the shape of thegrains. If the actual ITZ thickness is far less than the size of the grain , the ratio of the statistical mean value of theapparent ITZ thickness to the actual ITZ thickness is approximately equal to 2.