New Classes of Traveling Waves in a Planar Kirchhoff Beam with Nonlinear Bending Stiffness

Abstract

New traveling wave solutions are presented for motion of an inextensible, unshearable, planar Kirchhoff beam endowed with rotary inertia and a generalized strain energy function for bending which models nonlinear stiffening and softening. It is shown that although sonic waves (i.e., wave traveling at the bar speed in the beam) do not exist for constant bending stiffness, nonlinear bending stiffness supports new classes of solutions that admit nonlinear waves traveling with axial tension or compression. In addition, it is shown that waves traveling at the bar wave speed may be compactons (i.e., solitary waves of finite span). Surprisingly, for a large class of constant and nonlinear bending stiffness, initially circular rods do not support traveling waves.

Appendix: Derivation of the Shear Force

Using ([11], Sects. 5.9, 5.28) it follows that the shear force $V$ is determined by the director momentum equation, such that

$$ V = {\mathbf {d}}_{3} \cdot \biggl[ \frac{{\partial }{\mathbf {m}}^{1}}{{\partial }s} - \biggl(\frac{\rho A^{2}}{12} \biggr) y \frac{{\partial }^{2} {\mathbf {d}}_{1}}{{\partial }t^{2}}\biggr]. $$

(65)

In this expression, ${\mathbf {m}}^{1}$ denotes the director couple which determines the mechanical moment ${\mathbf {m}}$ applied on cross-sections of the rod by the expression

$$ {\mathbf {m}}= {\mathbf {d}}_{1} \times {\mathbf {m}}^{1}. $$

(66)

To specify the constitutive equation for ${\mathbf {m}}^{1}$ it is necessary to introduce the kinematic variables

$$ {\mathbf {F}}= {\mathbf {d}}_{i} \otimes {\mathbf {D}}_{i}, \quad \boldsymbol{\beta }_{1} = {\mathbf {F}}^{-1} \frac{{\partial }{\mathbf {d}}_{1}}{{\partial }s} - \frac{d {\mathbf {D}}_{1}}{d s} = - \frac{\beta }{\sqrt{A}} {\mathbf {D}}_{3}, $$

(67)

where the normalized curvature $\beta$ is defined in (8), the reference directors ${\mathbf {D}}_{i}$ are determined by replacing $\theta (s,t)$ in the expressions for ${\mathbf {d}}_{i}$ in (3) with its value $\varTheta (s)$ in the reference configuration, and ${\mathbf {a}}\otimes {\mathbf {b}}$ denotes the tensor product between two vectors ${\mathbf {a}}$, ${\mathbf {b}}$. Also, using the strain energy $\varSigma$ and the expression for the moment $M$ in (7), the director couple is given by

$$ {\mathbf {m}}^{1} = {\mathbf {F}}^{-T} \rho A \frac{{\partial }\varSigma }{{\partial }\boldsymbol{\beta }_{1}} = - M {\mathbf {d}}_{3}, \quad M = \rho A \sqrt{A} \frac{{\partial }\varSigma }{{\partial }\beta }. $$

(68)

Then, substituting (68) into (65) yields (10).