Abstract:
We find the simplest representation of the general solution to the system of static Lamé equations of linear isotropic elasticity in the form of a linear combination of the first derivatives of three functions that satisfy three independent harmonic equations. The representation depends on 12 free parameters choosing which it is possible to obtain various representations of the general solution and simplify the boundary value conditions for the solution of boundary value problems as well as the representation of the general solution for dynamic Lamé equations. The system of Lamé equations diagonalizes, i.e., is reduced to the solution of three independent harmonic equations. The representation implies three conservation laws and a formula for producing new solutions making it possible, given a solution, to find new solutions to the Lamé static equations by derivations. In the two-dimensional case of a plane deformation, the so-found solution immediately implies the Kolosov–Muskhelishvili representation for shifts by means of two analytic functions of complex variable. Two examples are given of applications of the proposed method of diagonalization of two-dimensional elliptic systems.
Keywords:
linear elasticity, isotropic material, static Lame equation, general solution, diagonalization of an elliptic system, symmetry operators, conservation laws.
Citation:
N. I. Ostrosablin, “Diagonalization of the system of Lamé static equations of linear isotropic elasticity”, Sib. Zh. Ind. Mat., 15:3 (2012), 87–98; J. Appl. Industr. Math., 7:1 (2013), 89–99
\Bibitem{Ost12}
\by N.~I.~Ostrosablin
\paper Diagonalization of the system of Lam\'e static equations of linear isotropic elasticity
\jour Sib. Zh. Ind. Mat.
\yr 2012
\vol 15
\issue 3
\pages 87--98
\mathnet{http://mi.mathnet.ru/sjim742}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3098811}
\transl
\jour J. Appl. Industr. Math.
\yr 2013
\vol 7
\issue 1
\pages 89--99
\crossref{https://doi.org/10.1134/S1990478913010092}
Linking options:
https://www.mathnet.ru/eng/sjim742
https://www.mathnet.ru/eng/sjim/v15/i3/p87
This publication is cited in the following 3 articles:
S. I. Senashov, I. L. Savostyanova, “Conservation laws and solutions of the first boundary value problem for the equations of two- and three-dimensional elasticity”, J. Appl. Industr. Math., 18:2 (2024), 333–343
B. D. Annin, N. I. Ostrosablin, “Presentation of the general solution of three-dimensional dynamic equations of a transversely isotropic thermoelastic medium”, J. Appl. Mech. Tech. Phys., 60:2 (2019), 224–233
N. I. Ostrosablin, “General solution for two-dimensional system of static Lame's equations with an asymmetric elasticity matrix”, J. Appl. Industr. Math., 12:1 (2018), 126–135
Citation in format AMSBIB
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