Dynamic stress concentration of an elliptical cavity in a semi-elliptical hill under SH-waves

The method of wave-function expansion in elliptical coordinates, elliptical cosine half-range expansion and Mathieu function were applied to obtain an exact analytical solution of the dynamic stress concentration factor (DSCF) around an elliptical cavity in a shallow, semi-elliptical hill. An infinite system of simultaneous linear equations for solving this problem was established by substituting the wave expression obtained by the Mathieu function including the standing wave expression of elliptical lining given herein into the boundary condition obtained by the region-matching method. The finite equations system with unknown coefficients obtained by truncation were solved numerically, and the results in the case of an ellipse degenerating into a circle were compared with previous results to verify the accuracy of the method. The effects of different aspect ratios, incident wave angles and aperture ratios on the dynamic stress concentration around the elliptical cavity were described. Some numerical results, when the elliptical hill was changed into a circular one, were analyzed and compared in detail. In engineering, this model can be regarded as a semi-cylindrical hill with an elliptical cylindrical unlined tunnel under the action of SH waves, and the results are significant in aseismic design.