The ground motion of interacting circular lining structure and crack under impact loading

The problems of SH-wave scattering, which is caused by a shallow-embedded circular lining structure and beeline crack of arbitrary length and arbitrary position while bearing impact loading at horizontal interface, are studied in this paper based on the field of linearly elastic dynamic mechanics. This problem can be considered as the problem of the defending of blast. The ground motion is given finally. The methods of Green´s function, complex variables and multi-polar coordinates are used here. Firstly a suitable Green´s function is constructed, which is an essential solution of the displacement field for the elastic half-space possessing circular lining structure under the out-of-plane harmonic line source load at an arbitrary point. Then, using the Green´s function and the method of crack-division, the crack is established: reverse stresses are inflicted along the crack, that is, out-of-plane harmonic line source loads, which are equal in the quantity but opposite in the direction to the stresses produced for the reason of SH-wave scattering by crack or circular lining structure, are loaded at the region where the crack will appear, thus the crack can be made out. The scattering of SH-wave by circular lining structure subjected to the impact loading at horizontal interface is known. Then, the expressions of displacement field and stress field are established when the crack and circular lining structure are both in existent. Thus, according to the boundary condition around the circular lining structure, the solution of the problem can be reduced to a series of algebraic equations and solved numerically by truncating the finite terms of the infinite integral equations. Numerical examples are provided to show the influences of the wave numbers of the incident wave, the distance between the center of the circular lining structure and horizontal surfaces, the distance between the center of the circular lining structure and the tip of crack, the shear modulus ratio- - of the media and the inclusion, the wave number ratio of the inclusion and the media, the angle of crack and the length of crack upon the ground motion.