The effect of surface roughness and mixed-mode loading on the stiffness ratio κx/κz for fractures

The characterization of fractures using elastic waves requires a parameter that captures the physical properties of a fracture. Many theoretical and numerical approaches for wave propagation in fractured media use normal and shear fracture specific stiffness to represent the complexity of fracture topology as it deforms under stress. Most effective medium approaches assume that the normal and shear fracture specific stiffness are equal, yielding a shear-to-normal specific stiffness ratio of one. Yet several experimental studies show that this ratio can vary from zero to three. We conducted a series of experiments to determine the stiffness ratio for fractures with different surface roughness subjected to mixed-mode loading conditions. Specimens containing a single fracture were subjected to either normal loading or combined normal and shear loading during ultrasonic measurements of transmitted and reflected P-and S-waves. Theoretical analysis based on the displacement discontinuity theory shows, for P- and S-waves with the same wavelength, that the theoretical stiffness ratio is not equal to one, but depends on the ratio of S- to P-wave velocities. The conventional stiffness ratio limit of unity is determined to be appropriate for very smooth fracture surfaces even under mixed-mode loading conditions. However, rough fracture surfaces result in stiffness ratios that are greater than the theoretical limit and the magnitude of the ratio depended on the relative ratio of shear-to-normal stress. The results from the experiments suggest that the conventional practice of assuming a constant stiffness ratio equal to 1.0 may not be appropriate. Therefore, the ratio of shear-to-normal fracture specific stiffness depends on the roughness of the fracture surface and the loading conditions.