Theoretical and Numerical Solutions of Linear and Nonlinear Elastic Waves in a Thin Rod
نویسندگان
چکیده
In this paper, we report theoretical and numerical solutions of linear and nonlinear elastic waves in a thin rod. First, the classical solution of linear elastic wave in a thin rod is adapted such that it is ready to be compared with the numerical solution of a nonlinear formulation. Based on mass and momentum conservation, we then derive several forms of modeling equations for nonlinear elastic waves, including the contraction/expansion effect of the cross section of the thin rod. We then analyze the eigen-system of the nonlinear equations to show that the system is hyperbolic because the eigenvalues of the Jacobian matrices are real. Analytical forms of eigenvectors and Riemann invariants along characteristic lines are also derived. For numerical solutions, we identify a suitable conservative form, which is then solved by the space-time Conservation Element and Solution Element (CESE) method. Comparison between the numerical solution and the analytical solution is reported.
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