S1 Appendix -

<div><p>This paper theoretically explores the propagation attenuation of normally incident P-waves on a single uncoupled joint exhibiting nonlinear deformation behavior. The stress-deformation model of the single uncoupled joint (g-λ model with λ ≥ 1) is employed to depict the nonlinearity of uncoupled joints, with a greater value of the parameter λ signifying a lower degree of non-linearity in the joint model curve. By making use of the characteristic line approach in conjunction with the discontinuous displacement model, we have obtained the finite difference expressions which precisely represent the particle velocity and energy transmission coefficient of the transmitted wave. The expressions for the stiffness transmission coefficient and the stiffness reflection coefficient, which can effectively reflect the nonlinear variation in joint stiffness, have been accomplished. Parametric investigations are carried out to examine the impacts of nonlinear joint normal deformation on P-wave transmission. The findings suggest that when λ is respectively equal to 4.19, 8.57, 10, and 12.15, the peak particle velocity (PPV) of the transmitted waves is significantly close to the incident wave amplitude. Furthermore, when λ is fixed, the energy transmission coefficient increases with the incident wave amplitude but decreases with the incident wave frequency. The stiffness transmission coefficient rises while the stiffness reflection coefficient drops with the increasing joint closure. And if the value of λ assumes larger values, the distortion in the shape of the transmitted wave is associated with the plastic deformation in the uncoupled rock mass. These results broaden the application scope of the g-λ model and can be employed to identify fault positions in jointed rock masses during the advanced geological prediction of mountain tunnels.</p></div>