Scattering Of Harmonic Plane Waves By A Shallow Circular Tunnel In A Poroelastic Half-space

Great progress in the construction of the tunnels and subways has been made with the rapid developments of the economy in the last decades.The soft saturated ground exists in the eastern coastal area of China,which is located in the circum-Pacific seismic belt,where earthquakes are a common occurrence.The requirements of the seismic design of the structures in the saturated medium are different from the elastic case.The dynamic responses of the underground structures during earthquakes in the saturated medium are completely different from that of the ground structures under seismic loads since there are complicated soil-structure interactions.The previous calculation methods also cannot explain the seismic mechanism of the underground structures in the saturated medium,which are based on the elastic theory.Therefore,it is significant to focus on the seismic analysis of the underground structures in saturated medium both in theory and in engineering application.Based on Biot’s theory,an analytical solution for the scattering of plane waves in a poroelastic half-plane with a shallow unlined,and the lined tunnel is obtained using the wave function expansion method and plane complex theory in elastodynamics.The major contribution of the paper is the treatment of the orthogonality of the boundary conditions of the tunnel and the half surface with different coordinates.Based on the plane complex theory,the present solution deals with the mathematical uncertainty of the huge-arc assumption.In light of the wave expansions,the wave fields of the poroelastic medium and the liner with unknown coefficients are obtained based on the Helmholtz decomposition.Complex-valued expressions of the stresses and the displacements of the poroelastic medium and the liner are expressed by the complex variable method and the conformal translation technique.With the boundary conditions and the continuity of the medium-liner interface,the boundary-valued problem results in a series of algebraic equations.The unknown coefficients in the infinite set of algebraic equations can be solved numerically by truncating the series number.The verification with the existed reference using the indirect boundary method shows that the results of this study are in excellent agreement with the reference.A parametric study for the incident P1 and SV waves is performed to investigate the displacements,dynamic stress concentrations,and the fluid stress of the medium and the liner.Numerical results show that the embedment depth of the tunnel,the porosity of the medium,the incident angle,and the frequency of the excitations have considerable influence on the dynamic responses of the medium and the liner:(1)The displacements at the half surface and the dynamic stress concentration factors of the liner and the medium decrease with the increase of the embedment ratio,but the shapes turn out to be more complicated.In the case of the oblique incidence,the maximum value of the displacement amplitude existed on the surface where the wave front is.A parametric study indicates that it is apparent that the incident angle has significant influences on the dynamic response of the liner and the medium;(2)For the shallow tunnel,the porosity of the medium and the angle of the incident plane wave have a significant influence on the dynamic response by the incident P1 and SV waves.With the increase of the porosity,the dynamic stress concentration factors turn to be complex along the cavity.When the porosity is closed to the critical porosity,the porosity contributes greatly to the response of the medium-liner system;(3)The effects of the incident angle are investigated that it has a significant influence on the DSCFs and the fluid stress along the cavity surface.The shielding effect on the half surface by the tunnel is obvious to the incident wave,thus the displacement at the right side of the half surface turns to be gentle.It is observed that the DSCFs along the cavity are relatively high in the low-frequency circumstance,while the value of the fluid stress reduces,which is worthy of being studied.