Analytical solution on the overestimation of thickness of interfacial transition zonearound grain with arbitrary convex shape by the section analysis method
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摘要: 借助于几何概率中的一些相关理论, 从理论分析的角度, 对表面为凸形曲线或曲面的二维和三维任意形状粒子建立了表观界面过渡区厚度统计平均值和实际界面过渡区厚度之间的定量关系。结果表明, 对于二维凸形粒子, 截面分析法所得的表观界面过渡区厚度的统计平均值与实际界面过渡区厚度之间的比值小于π/ 2 。精确比值与粒子的形状有关。如果实际界面过渡区厚度与粒子的尺寸相比可以忽略的话, 截面分析法所得的表观界面过渡区厚度的统计平均值与实际界面过渡区厚度之间的比值约等于π/ 2 。对于三维凸形粒子, 截面分析法所得的表观界面过渡区厚度的统计平均值与实际界面过渡区厚度之间的比值小于2 , 比值的精确解与粒子的形状有关。如果实际界面过渡区厚度与粒子的尺寸相比可以忽略的话, 截面分析法所得的表观界面过渡区厚度的统计平均值与实际界面过渡区厚度之间的比值约等于2 。
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关键词:
- 截面分析法 /
- 界面过渡区厚度 /
- 粒子形状 /
- 放大倍数 /
- 几何概率 /
- Cauchy 均值定理 /
- Crof ton 均值定理
Abstract: 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. -
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