Keywords: solid mechanics, objective time derivative, finite deformations, Riemannian manifold of Riemannian metrics.

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 , for two successive deformations , the following relation holds: . In case of small deformations one neglects all the terms of the second order in magnitude, and so the relation takes the form: . The result is that the corresponding tensor spaces in referential and spatial configurations are identical. In particular, the metric tensors are equal , and the objective time derivative is replaced by the simple material time derivative . Infinitesimal variation around identity mapping at the point (i.e. linearization of mapping in other words) enters the theory of small deformations via infinitesimal variation of the metric through the strain tensors , . Now, for deformation tensors: and , and for the time derivative the following relations hold: and . For plain meaning of the nomenclature, see the paper or [2].

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 of all possible deformation tensors (relative to reference configuration ). It was Rougée [3] who first noticed, and draw significant conclusions from this fact. Having limited to one single material point , he established the space as a six-dimensional Riemannian manifold by introducing a scalar product on its tangent linear space . This way, a naturally defined time derivative via covariant derivative results in the Zaremba-Jaumann objective time derivative.

Provided we eliminate the restriction of the deformation processes to a single material point , we have slightly to modify the above theory (see [2]). Now, the Riemannian metrics are tensor fields of the right Green-Cauchy deformation tensor over the whole reference configuration, and the corresponding manifold of such Riemannian metrics is an infinite dimensional Riemannian manifold. The time derivative thus obtained is a novel objective time derivative.

Based on coordinate approach, the novel objective time derivative has been derived for all the admissible tensor fields over actual configuration . For a 2-covariant symmetric tensor field , it has the form

( 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 , see the paper.

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.

1

Ch-S. Liu, H-K. Hong, "Non-oscillation criteria for hypoelastic models under simple shear deformation", Journal of Elasticity, 57, 201-241, 1999. doi:10.1023/A:1007616117953

2

Z. Fiala, "Time derivative obtained by applying Riemannian manifold of Riemannian metrics to kinematics of continua", C. R. Mecanique, 332(2), 97-105, 2004. doi:10.1016/j.crme.2003.12.001

3

P. Rougée, "Mécanique des grandes transformations", in: Mathématique & Applications 25, Springer-Verlag, 1997.

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