Abstract—
Differential equations for potentials are considered that ensure the fulfillment of the basic vector differential equation of the linear theory of elasticity in the case of a harmonic dependence of the displacement field on time. An alternative scheme for splitting the vector differential equation of the linear theory of elasticity into unrelated equations is developed. It is based on the concept of a gamma vector that satisfies a screw equation. As a result, the problem of finding the vortex component of the displacement field is reduced to the sequential solution of unrelated screw first-order partial differential equations. A theorem on the completeness of the representation of the displacement field using two screw vortex vector fields is formulated and proved.
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Notes
We are talking about two differential equations of the first order (3.8); as indicated in the introductory part of the article, these equations have a simple geometric meaning associated with the orientation of the characteristic directions of the vector field: the vortex vector of the field must be collinear with the direction of the field itself.
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Funding
The work was carried out on the topic of a state assignment (state registration AAAA-A20-120011690132-4) and with partial financial support from the Russian Foundation for Basic Research (project no. 18-01-00844 Modeling of thermomechanical processes in complex media using the principle of thermomechanical orthogonality).
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Translated by I. K. Katuev
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Radaev, Y.N. REPRESENTATION OF DISPLACEMENTS IN A SPATIAL HARMONIC PROBLEM OF THE THEORY OF ELASTICITY USING TWO SCREW VECTORS. Mech. Solids 56, 263–270 (2021). https://doi.org/10.3103/S0025654421020114
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DOI: https://doi.org/10.3103/S0025654421020114