Dynamic stress analysis of circular cavity in density inhomogeneous multi-layer soil structures containing a canyon

This study addresses the dynamic behavior of a circular tunnel in inhomogeneous multi-layer soil structures with complex boundary conditions. First, we consider two types of inhomogeneous density: density varies with the square term of coordinates, and density varies with the exponential term of coordinates. By introducing new variables, the variable coefficient Helmholtz equation is transformed into standard form. Then, SH wave is considered as a seismic wave acting on multi-layer soil structures. On the basis of series expansion method, the analytical expression of scattering wave in each soil layer is conducted, and the analytical expression of standing wave is established, which satisfies the stress-free conditions on the boundaries of V-shaped canyon by the fractional Bessel function expansion method and Graf addition theorem. Finally, large-arc assume method is applied, the multi-layer soil structures are divided into three parts along the horizontal interface, the straight boundaries are converted into curved boundaries, and the expressions of scattering waves caused by curved boundaries are obtained. The integral equations are set up through boundary conditions and solved by applying orthogonal function expansion technique and effective truncation. The calculation results analyzed and discussed the dynamic stress concentration factor of tunnel in different soil layers. Besides, the analytical solutions are compared with the finite element solutions to verify the accuracy of the conclusions in this article.