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The Reduction of Orthotropic Bodies Equations to Solving the Partial Differential Equation of the Sixth Order and Its Investigation

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Abstract

The linear statement of the theory of elasticity of a three-dimensional body was used. Issues related to the elimination of unnecessary functions from the known solutions of the theory of elasticity for isotropic and orthotropic materials were examined. A system of three second order partial differential equations was written down. Linear operators have been introduced, and the original system is transformed so each equation includes only two elastic displacements. It is proved that the solution of the resulting system of equations can be reduced to the integration of one sixth-order partial differential equation. Conditions for the introduced operators were found under the general solution of the system of three equations can be expressed in terms of one displacement function. The equation contains nine coefficients that depend on nine independent elastic constants describing the elastic state of an orthotropic material. It is shown that this equation depends on three variables and therefore, in the general case, cannot be decomposed into three factors. The method of separation of variables has been used and a technique has been developed for computing the solution of a sixth-order equation in an orthotropic prism. The characteristic equation was written down. Different variants of the values of its roots were investigated: complex and real. The analytical expression of displacements, strains and stresses have been obtained through the introduced displacement function.

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Authors and Affiliations

  1. Pidstryhach Institute for Applied Problems of Mechanics and Mathematics NASU, 79060, Lviv, Ukraine

    V. P. Revenko

  2. Institute of the Applied Mechanics, Russian Academy of Sciences, 125040, Moscow, Russia

    V. N. Bakulin

Authors
  1. V. P. Revenko
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  2. V. N. Bakulin
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Correspondence to V. P. Revenko or V. N. Bakulin.

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(Submitted by D. A. Gubaidullin)

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Revenko, V.P., Bakulin, V.N. The Reduction of Orthotropic Bodies Equations to Solving the Partial Differential Equation of the Sixth Order and Its Investigation. Lobachevskii J Math 42, 2030–2036 (2021). https://doi.org/10.1134/S1995080221080266

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  • DOI: https://doi.org/10.1134/S1995080221080266

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