The physical meaning of Grasselli's morphology parameters and its correlations with several other 2D fracture roughness parameters

Grasselli's 2D morphology parameter theta*(max)/(C + 1)(2D) has been widely used in rock engineering. However, the physical meaning of theta*(max)/(C + 1)(2D) is not clear, and the calculation is complex and time-consuming. By introducing a density function to characterize the distribution of the inclination, this study found that the physical meaning of theta*(max)/(C + 1)(2D) is the expected value of the inclination facing the shear direction. Based on this finding, this study found that theta*(max)/(C + 1)(2D) can be simplified as a statistical parameter. Besides, several new 2D roughness metrics expressed in discrete form were proposed. These new roughness metrics have clear physical meanings and can be obtained easily. This study found that two kinds of weight can be employed to calculate the average inclination facing the shear direction. Besides, this study found several generalized equivalent relationships among some 2D roughness metrics.