311 / 2019-09-30 00:28:08

Peculiarities of Construction of Analytical Solutions to Boundary Value Problems of the Theory of Elasticity in Finite Domains with the Angular Points of the Boundary

Papkovich–Fadle eigenfunctions,Biorthogonal functions,Lagrange expansions,Boundary value problems,Exact solutions

计算力学在工程中的应用 > 计算岩土力学与岩土工程

The article presents a formulation of the biharmonic problem of the theory of elasticity and the method developed by the authors for its solution using the example of the first basic boundary value problem for a half-strip (rectangle). The solution is constructed in the form of series in Papkovich–Fadle eigenfunctions. The coefficients of the series are determined in a simple closed form on the basis of the theory of the Borel transform developed by the authors in the class of quasi-entire functions of exponential type. The obtained series are equiconvergent with the corresponding trigonometric series. Despite the fact that the mathematical apparatus is quite complex, the final formulas are simple, reliable and can easily be applied in engineering calculations. The exact solutions obtained describe residual stresses. Therefore, they are especially valuable for solving various engineering problems of rock mechanics, where such stresses almost always exist, particularly in the problems associated with the construction of underground structures (such as the metro), the construction of mathematical models of rock bursts and earthquakes, etc.