Weakly Nonlinear Analysis on Two Types of Elastic Waves in a Beam with Geometric Nonlinearity

Abstract

Elastic wave propagation caused by a plane motion of a uniform beam is theoretically investigated. The vibration in a transverse direction of the beam induces a dispersion effect of waves. In this paper, we focus on a geometrical nonlinearity as the finite deformation, the equation of motion based on the special Cosserrat theory is utilized. For simplicity, the shear strain, body force, and damping are ignored. From the approximation up to the third order of approximation of the displacement in a reductive perturbation method, the Korteweg–de Vries (KdV) equation for a long wave and the nonlinear Schrödinger (NLS) equation for a slowly varying envelope wave of the quasi-monochromatic short carrier wave are derived. The nonlinear term is affected by the axial stiffness and the dispersion term is by the rotational inertia and the bending stiffness for the case of the KdV equation. On the other hand, both the dispersion and nonlinear terms are affected by the axial stiffness, rotational inertia, and bending stiffness for the case of the NLS equation.