Seismodynamics of extended underground structures and soils: Statement of the problem and self-similar solutions

In the problems of common vibrations of extended underground structures (pipelines and tunnels) and soil, an approach of the one-dimensional deformation of the medium is developed; this approach is based on the assumption that the soil deformation in the direction of seismic wave propagation coinciding with the pipeline axis is prevailing. The analytic solutions are obtained in the cases where the wave velocity in the soil is respectively less or greater than the wave velocity in the pipeline. The parameters influencing the pipeline fracture are revealed and methods for increasing the seismic stability of such structures are given. The possibility of the pipeline fatigue fracture is pointed out. The statements and solutions of parabolic problems modeling the physical phenomena in soils in the case of discontinuous velocity on the boundaries at the initial time are given. The notion of generalized vorticity diffusion is introduced and the cases of self-similarity existence are classified. A detailed analysis is performed for the non-Newtonian polynomial fluid, the medium close in properties to the rigidly ideally plastic body, and the viscoplastic Shvedov—Bingham body. In the case of physically linear medium, new self-similar solutions are obtained which describe the process of unsteady axially symmetric shear in spherical coordinates. The first approximation to the asymptotic solution of the problem of the vortex sheet diffusion is constructed in a medium with small polynomial nonlinearity. The solutions polynomially decreasing to zero as the self-similar variable increases are proposed in the class of two-constant fluids.

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