Local linear Timoshenko rod

Abstract

In this article we consider a variant of the Simo-Reissner theory for a rod but restrict the study to two-dimensional motion where the rod undergoes flexure, shear and extension but not torsion. Linear elastic behaviour is assumed to formulate constitutive equations; the constitutive equations of the Timoshenko theory adapted for extension and large rotation. We call the model the Local Linear Timoshenko rod model. We show that this model serves as a framework for a class of simpler mathematical models for slender solids in various applications. The advantage is that the more general model can be used to evaluate and compare the simpler models.

Large displacements and rotations of a local linear elastic rod

“…It may also be used to prove existence results, as illustrated in [1] where an existence result is derived for a linear special case of the LLT rod. In addition, in [1] an exact solution is derived for the simple case of a pivoted LLT rod with constant angular velocity. For the general cantilever LLT rod, the solution triple $$\langle u, w, \phi \rangle$$ maps a time interval [0, T ] into the product space V 3 , where V is the closure of the test functions T 1 [0, 1] in the Sobolev space H 1 (0, 1).…”

“…In this case, the equations of motion are given by $$$$\begin{eqnarray} \partial _t^2w&=&\partial _x (S\partial _x w)+\partial _x V + P_2, \end{eqnarray}$$$$ $$$$\begin{eqnarray} \frac{1}{\alpha } \ \partial _t^2\phi &=& V + \partial _x M, \end{eqnarray}$$$$with the constitutive equations for the moment $$M = \partial _x\phi /\beta$$, the shear force $$V = \partial _x w - \phi$$, and the axial load $$S = \partial _x u /\gamma$$. See [1] for details.…”

“…Recently, a model for planar motion of a local linear elastic rod was presented in [1]. In this context, the term rod is used as a collective name for slender elastic objects (e.g., beams, cables, hoses, wires, etc.…”

“…(Clearly $$\phi (x,t)$$ is the angle between $$\bar{e}_n$$ and $$\bar{e}_1$$, or the angle between the cross‐section and $$\bar{e}_2$$.) Remark For more information on the connection between a three‐dimensional model and a one‐dimensional model, as well as the assumption that cross‐sections remain plane, the reader may consult [1]. …”

“…Local linear refers to situations where large deflections and rotations of the rod are possible, but the strains are sufficiently small that linear elasticity theory still applies for the constitutive equations. Timoshenko theory is used for the shear and bending moment and therefore the model is called the Local Linear Timoshenko (LLT) rod in [1], where it is also compared to the models developed in [2] and [3]. It has significant differences with [2], and can be regarded as a linear approximation of the nonlinear model in [3].In [4], Lang and Arnold highlight the importance of rod models in industrial and engineering applications, and the challenge of developing efficient numerical algorithms for solving the model equations.…”