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On the existence of Riemann invariants in the one-dimensional equations of the nonlinear theory of elasticity. (English. Russian original) Zbl 0945.74009
J. Appl. Math. Mech. 63, No. 4, 627-632 (1999); translation from Prikl. Mat. Mekh. 63, No. 4, 655-661 (1999).
For the system of one-dimensional equations of elasticity theory \[ \frac{\partial v^i}{\partial t} = \frac{\partial }{\partial x} \frac{\partial\Phi}{\partial u^i}, \quad \frac{\partial u^i}{\partial t} = \frac{\partial v^i}{\partial x}\quad \bigg(v^i = \frac{\partial w^i}{\partial t},\quad u^i= \frac{\partial w^i}{\partial x}\bigg),\quad i=1,2,3, \] where \(w^i\) are the components of the displacement vector and \( \Phi=\Phi(u^1,u^2,u^3) \) is the elastic potential, the author derives conditions for the existence of Riemann invariants. The author determines the type of the elastic potential for which the system possesses six Riemann invariants, and describes a procedure for their calculation.
74B20 Nonlinear elasticity
35Q72 Other PDE from mechanics (MSC2000)
[1] Kulikovskii, A. G.; Sveshnikova, Ye.I.: The self-similar problem of the effect of a sudden load on the boundary of an elastic half-space. Prikl mat. Mekh. 49, 284-291 (1985)
[2] Kuljkovskii, A. G.: Equations which describe the distribution of non-linear quasi-transverse waves in a slightly non-isotropic elastic solid. Prikl. mat. Mekh. 50, 597-604 (1986)
[3] Chugainova, A. P.: On forming a self-similar solution in the problem of non-linear waves in an elastic half-space. Prikl. mat mekh. 52, No. 4, 62-697 (1988)
[4] Haantjes, J.: On xm ,forming sets of eigenvectors. Indagationes math. 17, 158-162 (1955) · Zbl 0068.14903
[5] Nijenhuis, A.: Xm-forming sets of eigenvectors. Indagationes math. 13, 200-212 (1951) · Zbl 0042.16001
[6] Mokhov, O. I.: Simplectic and Poisson structures in spaces of loops of smooth manifolds and integrable systems. Doctoral dissertation (1996)
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