General Functions Method in Transport Boundary Value Problems of Elasticity Theory

Abstract

The Lame system describing the dynamics of an isotropic elastic medium affected by a steady transport load moving at subsonic, transonic, and supersonic speed is considered. Its fundamental and generalized solutions in a moving frame of reference tied to the transport load are analyzed. Shock waves arising in the medium at supersonic speeds are studied. Conditions on the jump in the stress, displacement rate, and energy across the shock front are obtained using distribution theory. Transport boundary value problem for an elastic medium bounded by a cylindrical surface of arbitrary cross section and subjected to transport loads is considered in the subsonic and supersonic case with regard to shock waves. To solve problems, the generalized functions method is developed. In the space of generalized functions, generalized solutions are constructed and their regular integral presentations are obtained. Singular boundary equations solving the boundary value problems are presented.

The first boundary value problem of the dynamics of an anisotropic elastic half-space under the action of subsonic transport loads

“…В [2] построены и исследованы фундаментальные и обобщенные решения уравнений Ламе, которые описывают динамику изотропной упругой среды при движении сосредоточенной и распределенной транспортной нагрузки во всем диапазоне скоростей (дозвуковой, звуковой, трансзвуковые и сверхзвуковые). На этой основе в [3][4][5][6] разработан метод граничных интегральных уравнений и метод обобщенных функций для решения транспортных краевых задач в изотропной упругой среде с цилиндрической формой границ. Этот класс задач очень важен для приложений в области динамики подземных сооружений, транспортных тоннелей и подземных выработок.…”

Relativistic Formulae for the Biquaternionic Model of Electro-Gravimagnetic Charges and Currents

“…For this the Hamilton form of Maxwell's equations has been used [5] and its quaternionic record [6]. Based on the hypothesis of a magnetic charge, it's carried out field complexification with the introduction of the gravimagnetic density to the Maxwell equations.…”