EFFECT OF JOINT SURFACE MORPHOLOGY ON P-WAVE TRANSMISSION AND REFLECTION AT AN OBLIQUELY INCIDENT FRACTAL JOINT

This paper derives the normal and shear stresses of transmitted and reflected P- and S-waves. A fractal constitutive relation for joints was introduced to establish a recursive wave equation, and the transmission/reflection coefficients of P- and S-waves were derived by incorporating the fractal characteristics of the joint. The effect of the joint surface morphology, described by the fractal dimension and intercept, on these coefficients was systematically investigated. The study reveals the following key findings: (1) A significant coupling effect exists between the fractal dimension and intercept. As these two parameters vary, the transmission/reflection coefficients of P- and S-waves exhibit oscillatory behavior — unlike the monotonic trends commonly observed in traditional studies, they alternately increase and decrease. However, the overall trend indicates that with increasing dimension and intercept, the reflection coefficients for both P- and S-waves increase, while their transmission coefficients decrease. (2) When the intercept remains constant, the coefficients oscillate as the dimension traverses consecutive intervals ([2.27, 2.32], [2.32, 2.45], [2.45, 2.57], [2.57, 2.66]). This oscillatory pattern is primarily governed by the fractal dimension. In general, increasing dimension leads to higher reflection coefficients but lower transmission coefficients. (3) When the dimension is held constant, the coefficients vary almost linearly with the intercept. As the dimension progressively increases through the specified values [2.25, 2.33, 2.41, 2.50, 2.55], the rate of change of the transmission/reflection coefficients with respect to the intercept under identical intercept conditions gradually accelerates. This demonstrates that increasing dimension significantly enhances the sensitivity of the coefficients to variations in the intercept.