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1 Journal of Applied Mathematics and Computational Mechanics

1 2024

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The method of developing the complex stress tensor by basic states for the construction solutions of spatial boundary problems of the linear elasticity theory

Pabyrivskyi Viktor, Pukach Petro, Pabyrivskyi Volodymyr

Journal of Applied Mathematics and Computational Mechanics

2024

Vol. 23, nr 2

66--78

The apparatus of holomorphic functions of many complex variables is applied to solving spatial boundary value problems of the linear theory of elasticity. The construction of the solution of the boundary value problem is based on the representation of the displacement vector in the form of J. Dougall through spatial harmonic potentials. The transition from spatial harmonic potentials to holomorphic functions of two complex variables z1, z2 was carried out and a boundary value problem for the above functions was formulated. By presenting these holomorphic functions in the form of homogeneous polynomials of order k relative to complex variables z1 , z2 , solutions were constructed by the method of development of the complex tensor of stresses by basic states. The application of this technique is illustrated in the examples of marginal problems, the real components of solutions that correspond to the solutions of Grashof’s problem for an elastic beam. Imaginary components of exact analytical solutions are obtained and corresponding structures of external load vectors for elastic beams of complex cross-section are constructed.