Wave curves for the Riemann problem of plane waves in isotropic elastic solids
(From authors' summary.) A modified Riemann problem in which the initial and boundary conditions are constants is considered for plane waves in a half-space occupied by an elastic solid. The governing quasilinear differential equations form a system of hyperbolic conservation laws which possesses three wave speeds \(c_ 1\geq c_ 2\geq c_ 3\). The system is genuinely nonlinear with respect to \(c_ 1\) and \(c_ 3\) and linearly degenerate with respect to \(c_ 2\). Thus it is sufficient to study a two-wave-speed system with \(c_ 1\) and \(c_ 3\). Wave curves for simple waves and shock waves are used to construct the solution. Second- order hyperelastic materials which contain four material constants are considered and the solution in the form of wave curves is obtained of all possible combinations of initial and boundary conditions. The two-wave-speed system has one umbilic point at which \(c_ 1=c_ 3\) and hence the system is not totally hyperbolic (or not strictly hyperbolic). Several interesting and unexpected results are obtained due to the existence of the umbilic point. In one example, we find that a shock wave satisfies the Lax stability condition for a \(V_ 1\) shock as well as a \(V_ 3\) shock. In another, a shock wave which involves only one stress component does not satisfy the Lax stability condition for either a \(V_ 1\) shock or a \(V_ 3\) shock. However, it satisfies the Lax condition if we consider it under the context of a one-wave-speed system.
scientific article; zbMATH DE number 4039403
scientific article