Vol. 8, No. 5-7, 2013

Download this article
Download this article. For screen
For printing
Recent Issues

Volume 12
Issue 3, 249–351
Issue 2, 147–247
Issue 1, 1–146

Volume 11, 5 issues

Volume 10, 5 issues

Volume 9, 5 issues

Volume 8, 8 issues

Volume 7, 10 issues

Volume 6, 9 issues

Volume 5, 6 issues

Volume 4, 10 issues

Volume 3, 10 issues

Volume 2, 10 issues

Volume 1, 8 issues

The Journal
Cover
Editorial Board
Research Statement
Scientific Advantage
Submission Guidelines
Submission Form
Subscriptions
Author Index
To Appear
 
ISSN: 1559-3959
On successive differentiations of the rotation tensor: An application to nonlinear beam elements

Teodoro Merlini and Marco Morandini

Vol. 8 (2013), No. 5-7, 305–340
Abstract
[an error occurred while processing this directive]

Successive differentiations of the rotation tensor are characterized by successive differential rotation vectors. Useful expressions of the differential rotation vectors for differentiations up to third order are derived. In the context of the exponential parameterization, explicit expressions for the differential maps (the maps providing the differential rotation vectors from the differentials of the parameters chosen) are obtained by resorting to an original infinite family of recursive subexponential maps. Useful properties of the mapping tensors are discussed.

The formulation is appropriate for nonlinear problems of computational solid mechanics, when spatial, incremental, and virtual variations of particle orientations must be dealt with together. As an application, the classical problem of modeling space-curved slender beams by finite elements is considered. The variational formulation and the nonlinear interpolation of the orientations, together with the relevant linearizations, consistently exploit the proposed differentiations and lead to an objective beam element. Two test cases are discussed.

Keywords
finite rotation differentiations, exponential map, rotation differential maps, space-curved slender beams, orientation interpolation, nonlinear beam elements
Milestones
Received: 27 November 2012
Revised: 1 March 2013
Accepted: 1 March 2013
Published: 18 November 2013
Authors
Teodoro Merlini
Dipartimento di Scienze e Tecnologie Aerospaziali
Politecnico di Milano
Campus Bovisa
via La Masa 34
20156 Milano
Italy
Marco Morandini
Dipartimento di Scienze e Tecnologie Aerospaziali
Politecnico di Milano
Campus Bovisa
via La Masa 34
20156 Milano
Italy