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On successive
differentiations of the rotation tensor: An application to
nonlinear beam elements
Teodoro Merlini and Marco Morandini |
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Vol. 8 (2013), No. 5-7, 305–340
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Abstract |
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
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Milestones
Received: 27 November 2012
Revised: 1 March 2013
Accepted: 1 March 2013
Published: 18 November 2013
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Authors |
| Teodoro Merlini | |
| Dipartimento di Scienze e Tecnologie
Aerospaziali Politecnico di Milano Campus Bovisa via La Masa 34 20156 Milano Italy |
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| Marco Morandini | |
| Dipartimento di Scienze e Tecnologie
Aerospaziali Politecnico di Milano Campus Bovisa via La Masa 34 20156 Milano Italy |
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