The structure of formal solutions to Navier’s equilibrium equation. (English) Zbl 1134.74311

Mladenov, Ivaïlo M. (ed.) et al., Proceedings of the 2nd international conference on geometry, integrability and quantization, Varna, Bulgaria, June 7–15, 2000. Sofia: Coral Press Scientific Publishing (ISBN 954-90618-2-5/pbk). 287-293 (2001).
Summary: The local Lie structure of the orientation-reversing involutions on \(\mathbb R^3\) is used to construct a family of orthogonally invariant operators that produce all formal solutions, up to biharmonic equivalence, of Navier’s equation for elastic equilibrium. In the construction the value of Poisson’s ratio associated with each solution is determined by the hyperbolic geometry of \(\text{sl}_2(\mathbb R)\). Empirically feasible values of the ratio are associated with ‘spacelike’ operators, whereas values outside of this range are associated with ‘timelike’ operators.
For the entire collection see [Zbl 0957.00038].


74B05 Classical linear elasticity