Geodesic
Diagonalization of the system of Lam\'e static equations of linear isotropic elasticity
Sibirskij žurnal industrialʹnoj matematiki, Tome 15 (2012) no. 3, pp. 87-98

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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. 
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     author = {N. I. Ostrosablin},
     title = {Diagonalization of the system of {Lam\'e} static equations of linear isotropic elasticity},
     journal = {Sibirskij \v{z}urnal industrialʹnoj matematiki},
     pages = {87--98},
     publisher = {mathdoc},
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     number = {3},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJIM_2012_15_3_a8/}
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N. I. Ostrosablin. Diagonalization of the system of Lam\'e static equations of linear isotropic elasticity. Sibirskij žurnal industrialʹnoj matematiki, Tome 15 (2012) no. 3, pp. 87-98. http://geodesic.mathdoc.fr/item/SJIM_2012_15_3_a8/