SAMS Subject Classification Number: 3, 30

In this presentation, numerical experiments resulting from finite element calculations will be used to investigate the properties of a semi-linear Timoshenko rod model with axial force suggested by M.H. Sapir and E.L. Reiss [Dynamic Buckling of a nonlinear Timoshenko beam, SIAM J. Appl. Math. 37 (1979), 290-301]. These properties include the possibility of a non-trivial equilibrium as well as dynamic buckling. In order to aid in the investigations, results from the spectral theory of the related linear Timoshenko rod model will also be considered.

The nonlinear system of partial differential equations considered in the model is \[\begin{aligned} \partial^2_t w &=& \partial_x \left(\partial_x w - \phi \right) + \left( \frac{D}{\gamma} +\frac{1}{2\gamma}\int_0^1(\partial_x w)^2\right) \partial^2_x w , \\ \frac{1}{\alpha}\partial^2_t \phi&=& \partial_x w - \phi + \partial_x \left(\frac{1}{\beta} \partial_x \phi \right). \end{aligned}\qquad{(1)}\] This mechanical system is a special case of a problem of the form \[u''=Au+f(u) \label{Form of Differential eqn}\qquad{(2)}\] on some Hilbert space, where \(A\) is a linear operator and \(f\) a nonlinear mapping.