Analysis of the dynamic stress path under obliquely incident P-waves and its influencing factors

The dynamic stress path of a rock and soil mass under seismic action has a crucial influence on its catastrophic behavior. In soil dynamics, earthquakes are commonly simplified as vertically incident shear waves and the seismic stresses in soil are estimated based on rigid foundation models. However, the great effect of P-waves should not be overlooked in strong earthquakes, which have happened frequently in recent years. The characteristics of the dynamic stress path under longitudinal waves with significant oblique incidence are still unclear. Analytical formulas for the seismic stresses at any depth of a semi-infinite elastic space under obliquely incident P-waves are derived, which degenerate into the traditional rigid foundation method in soil dynamics when both the incident angle and Poisson's ratio are taken as zero. Here, we reveal the fundamental characteristics of a dynamic stress path under obliquely incident P-waves. The stress path is proved mathematically to be an oblique ellipse in the plane of normal stress difference and horizontal shear stress. We identify factors affecting the stress path, including the incident angle, Poisson's ratio, and depth corresponding to unit wavelength. The possible variation in the range of an oblique elliptic stress path is systematically analyzed, which lays a theoretical foundation for further study of the dynamic response of sites under obliquely incident seismic waves.