ON SINGULAR PERTURBED EQUATIONS OF THIN BODIES

摘要

The problem of solving systems of equations of static and dynamical boundary value problems of elasticity theory for thin bodies (beams, plates, shells) is considered. Taking into account the specific geometry of such bodies, it is shown that in dimensionless values the system of equations is singularly perturbed by a small geometrical parameter. For solving such systems an asymptotic method is used. The solution is combined by the solutions of the inner problem and the boundary layers. Asymptotic orders of the required values are established and iteration processes for their determination are built. It is shown that asymptotics correctly react to the type of conditions, stated on the face surfaces. The connection of asymptotic solutions with the results on classical theories of Bernoulli-Coulomb-Euler beams, Kirchhoff-Love plates and shells is established. The connection of Saint-Venant's principle with the property of the solution for the boundary layer is revealed. A class of problems for which Saint-Venant's principle is mathematically exactly fulfilled, is selected. It is proved that the applied asymptotic method permits to solve new classes of problems of statics and dynamics of thin bodies.