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<p>We construct a convergent recurrence scheme for a solution of the three-dimensional Neumann type boundary value problem of the elasticity theory when on the boundary of a homogenous anisotropic elastic body the stress vector is prescribed. By the potential method, the boundary value problem is reduced to the second kind Fredholm integral equation with zero index generated by the single layer potential. The corresponding boundary integral operator has a six-dimensional null-space. Therefore, the nonhomogeneous integral equation possesses a solution if the right-hand side vector-function meets the necessary and sufficient solvability conditions, i.e., it is orthogonal to the space of rigid displacements, which are solutions to the corresponding homogeneous adjoint integral equation. First, we reduce the boundary integral equation to the equivalent first kind Fredholm integral equation with symmetric non-negative compact operator. Afterwards, we construct a uniquely solvable modified boundary integral equation with symmetric compact positive operator having the following property: if the above mentioned necessary conditions are satisfied then the solution of the modified boundary integral equation is a particular solution of the original integral equation. Further, we construct a recursive sequence of vector-functions which converges to the unique solution of the modified boundary integral equation in appropriate Bessel-potential spaces of functions defined on the boundary. Using these approximations as densities of the single layer potential, we construct a sequence converging to a particular solution (a particular displacement vector) of the Neumann type boundary value problem in the appropriate Sobolev-Slobodetskii spaces of functions defined in the region occupied by the elastic body. The general solution of the Neumann type boundary value problem can be constructed by adding an arbitrary rigid displacement vector to the obtained particular solution. Evidently, the corresponding strain and stress tensors are defined uniquely.</p>

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Iteration Method for the Neumann Type Problem of the Elasticity Theory

  • 期刊论文

  • 发表日期：

    2026年6月23日

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  • David Natroshvili,
  • George Gotchoshvili,
  • Maia Mrevlishvili

摘要

We construct a convergent recurrence scheme for a solution of the three-dimensional Neumann type boundary value problem of the elasticity theory when on the boundary of a homogenous anisotropic elastic body the stress vector is prescribed. By the potential method, the boundary value problem is reduced to the second kind Fredholm integral equation with zero index generated by the single layer potential. The corresponding boundary integral operator has a six-dimensional null-space. Therefore, the nonhomogeneous integral equation possesses a solution if the right-hand side vector-function meets the necessary and sufficient solvability conditions, i.e., it is orthogonal to the space of rigid displacements, which are solutions to the corresponding homogeneous adjoint integral equation. First, we reduce the boundary integral equation to the equivalent first kind Fredholm integral equation with symmetric non-negative compact operator. Afterwards, we construct a uniquely solvable modified boundary integral equation with symmetric compact positive operator having the following property: if the above mentioned necessary conditions are satisfied then the solution of the modified boundary integral equation is a particular solution of the original integral equation. Further, we construct a recursive sequence of vector-functions which converges to the unique solution of the modified boundary integral equation in appropriate Bessel-potential spaces of functions defined on the boundary. Using these approximations as densities of the single layer potential, we construct a sequence converging to a particular solution (a particular displacement vector) of the Neumann type boundary value problem in the appropriate Sobolev-Slobodetskii spaces of functions defined in the region occupied by the elastic body. The general solution of the Neumann type boundary value problem can be constructed by adding an arbitrary rigid displacement vector to the obtained particular solution. Evidently, the corresponding strain and stress tensors are defined uniquely.