Abstract
An approach for solving of the axisymmetric biharmonic boundary value problems for semi-infinite cylindrical domain was developed in the paper. On the lateral surface of the domain homogeneous Neumann boundary conditions are prescribed. On the remaining part of the domain’s boundary four different biharmonic boundary pieces of data are considered. To solve the formulated biharmonic problems the method of least squares on the boundary combined with the method of homogeneous solutions was used. That enabled reducing the problems to infinite systems of linear algebraic equations which can be solved with the use of reduction method. Convergence of the solution obtained with developed approach was studied numerically on some characteristic examples. The developed approach can be used particularly to solve axisymmetric elasticity problems for cylindrical bodies, the heights of which are equal to or exceed their diameters, when on their lateral surface normal and tangential tractions are prescribed and on the cylinder’s end faces various types of boundary conditions in stresses in displacements or mixed ones are given.
Highlights
Many practically important problems bring the biharmonic equation Δ2w = 0, (1)w = w (x), x ∈ S ⊂ R2 considering in 2D domains S with Lipschitz-continuous boundary ∂S.Here Δ is Laplace differential operator: Δ = ∇ ⋅ ∇, where ∇ stands for the gradient operator in R2 and the dot (⋅) denotes scalar product
In the problem of the first and second kinds the biharmonic functions should be subordinated to boundary conditions (2) and (3) correspondingly [1]: w|x∈∂S = f0 (x)
An idea of the method consists in representing the solution as a series expansion in some complete system of biharmonic functions being solutions of a homogeneous biharmonic problem on infinite strip [11, 14]
Summary
In the mixed problem of the first kind (1) is considered subject to boundary conditions which are weighted combinations of Dirichlet boundary conditions and Neumann boundary conditions (so-called Robin boundary condition). An idea of the method consists in representing the solution as a series expansion in some complete system of biharmonic functions being solutions of a homogeneous biharmonic problem on infinite strip [11, 14]. These functions satisfy homogeneous Neumann-type boundary conditions on the strip’s sides. In this paper we consider the method of least squares on the boundary combined with the method of homogeneous solutions in application to axisymmetric biharmonic problems for a semi-infinite cylindrical domain
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