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A numerical method for special Cosserat rods based on Antman's description Antman, 2005 is developed for hyperelastic materials and potential forces. This method preserves the relevant properties of the underlying PDE system, namely, the orthonormality of the directors and the conservation of the energy.

Elastic rods are considered in many fields of science and technology; see, for example, [

Over the past years many different approaches to the numerical simulation of elastic rods have been developed; see, for example, [

In this paper we use the description of the rod as a special one-dimensional Cosserat continuum following the formulation in Antman [

We formulate the kinematic and dynamic equations of the Cosserat rod theory in the so-called director-basis. The rod’s curve

Following Antman [

To rewrite these kinematic equations in an appropriate basis, we decompose an arbitrary vector field

The balance laws for momentum and angular momentum yield the dynamic equations of the Cosserat rod theory, cf. [

The special Cosserat rod theory describes the angular momentum as a linear function of the angular velocity. The choice of the representation of the vector fields in the director-basis leads to the time independent matrix

In this paper we restrict ourselves to hyperelastic materials. That means there exists a potential

The more general class of elastic materials are materials where

The kinematic equations (

The presentation of the energy conserving numerical algorithm in Section

For hyperelastic materials (

The system (

For the spatial discretization we use a simple finite difference scheme. We note that similar finite difference schemes have been developed and their properties, in particular, the conservation of invariants, have been investigated in [

We discretize

Now, we come up with our main point, the (semi)discrete energy conservation of this scheme. The energy density is approximated locally at the mesh points

For the time discretization any energy conserving method can be used. We choose a Gauss method, that also guarantees the preservation of the norm of the quaternions. In the numerical realization, we make use of the second order Gauss method, that is, the midpoint rule, to obtain a temporal order that is consistent with the spatial one, at least at the inner points. For the discretization of space and time, and the use of the midpoint rule for the conservation of certain properties we refer also to the above mentioned papers [

To solve the resulting nonlinear equations a Newton method is used. The strict conservation of energy and orthogonality are the main advantages of the straightforward finite difference scheme presented here.

The scheme described above concentrates on preserving the energy of the rod and the orthonormality of the directors. In the sense of numerical methods for hyperbolic systems the scheme is not able to handle shocks properly. It does not have the usual properties like being a TVD scheme or satisfying the entropy condition.

Higher order discretizations are also possible. For example, we could consider the following fourth order numerical flux function:

In this section we present three numerical examples, restricting ourselves to Timoshenko’s material law for a homogeneous cylinder as discussed at the end of Section

The chosen initial configuration of a straight rod and direction of gravity

Initial situation.

In all simulations we choose the CFL-number equal to

We remark that in all simulations energy is strictly conserved according to the above analysis.

We choose

Torsional oscillation-Frequency

We choose

We note that the solution is very accurate although we have chosen a linear material law for the contact force

Transversal oscillation-frequency

We choose

Rod’s curve at

In this paper we use the description of a hyperelastic rod in the formulation of Antman [

For the material law of Timoshenko [

This work has been supported by Deutsche Forschungsgemeinschaft (DFG), KL 1105/18-1 and WE 2003/3-1 and by Rheinland-Pfalz Excellence Center for Mathematical and Computational Modeling