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Civil-Comp Proceedings
ISSN 1759-3433 CCP: 79
PROCEEDINGS OF THE SEVENTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: B.H.V. Topping and C.A. Mota Soares
Paper 69
Objective Time Derivative Defined as Covariant Derivative Z. Fiala
Institute of Theoretical and Applied Mechanics, The Academy of Sciences of the Czech Republic, Prague, Czech Republic Full Bibliographic Reference for this paper
Z. Fiala, "Objective Time Derivative Defined as Covariant Derivative", in B.H.V. Topping, C.A. Mota Soares, (Editors), "Proceedings of the Seventh International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 69, 2004. doi:10.4203/ccp.79.69
Keywords: solid mechanics, objective time derivative, finite deformations, Riemannian manifold of Riemannian metrics.
Summary
Choice of an objective time derivative is still an open problem, even in elasticity [1].
A geometrically based approach, defining the time derivative as a covariant derivative in an appropriate nonlinear
space - the infinite dimensional Riemannian manifold of deformation tensor fields - will be employed, and,
via coordinate approach, the covariant derivative and its corresponding time derivative of an arbitrary tensor field will be
explicitly expressed. Based on these geometrical grounds, a modification of the Zaremba-Jaumann time derivative will be established,
and its limitation clarified.
There is a fundamental difference between description of kinematics of small and finite deformations: The small deformations are
described in terms of fields, whereas proper setting for finite deformations are diffeomorphisms. In fact, provided we split
the deformation
On the other hand, in case of finite deformations the deformation process no longer keeps moving inside a linear space,
as in the case of small deformations, and the finite difference between initial and terminal deformation tensors loses all the
intermediate information about the deformation. Consequently, the deformation process should be described not by time dependent
strains, but by a trajectory in the nonlinear space
Provided we eliminate the restriction of the deformation processes to a single material point
Based on coordinate approach, the novel objective time derivative has been derived for all the admissible tensor fields over
actual configuration
( which in Cartesian coordinates means
where ZJ stands for the Zaremba-Jaumann time derivative.
For a 2-contravariant symmetric tensor field
( which in Cartesian coordinates means
For general tensor fields Acknowledgements: The research was conducted in the framework of research plan AV0Z2071913. The support of GA CR through the grant GACR103/03/0581, and of AS CR through the project K1010104 is gratefully acknowledged. References
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