In General Relativity there exists the impressive tool GR-tensor, see Musgrave et al [2]. GRtensor is a MapleV package for the manipulation and calculation of tensor fields in general relativity. For GR-Tensor there is an application called Elasticity [3], which makes it possible to do calculations in nonlinear elasticity. Elasticity is a library to GRtensor.
General Relativity, however, has one space-time only. The idea behind Elasticity is to formally regard the coordinates in the present configuration and the coordinates in the reference configuration as two coordinate systems for the same space. See, e.g., [4]. The mapping from (the coordinates in) the reference configuration to (the coordinates in) the present configuration is thus formally regarded as a coordinate transformation. This way it is possible to calculate many quantities, in particular to set up the equations of equilibrium in the present configuration, see [3].
If one considers the mapping between the two configurations as coordinate transformation, one has two different metric tensor, and thus two ways to raise and lower indices. This becomes particularly troublesome for so-called two-point objects or double tensors, see [6], living with one foot in each of the spaces.
To formulate the equations of equilibrium in the reference configuration one needs the first Piola-Kirchhoff stress tensor rather than the ordinary Cauchy stress tensor. The first Piola-Kirchhoff stress tensor is an important example of a two-point object, a tensor mapping between the configurations. Another object of this kind is the deformation gradient. To be able to handle such tensor field, a setting with two different spaces with a mapping between them is the natural one.
The present package is based on this concept of mapping between two
different spaces. It can handle tensor field in the present configuration
and the reference configuration as well as two-point objects. Much of it
has been inspired by the GR-Tensor.
References
1. Truesdell, C. and Noll, W., The non-linear field theories of mechanics. Handbuch der Physik III/3. Springer-Verlag Berlin 1965.
2. Musgrave, P., Pollney, D. and Lake, K., 1996 www.astro.queensu.ca/~grtensor/GRHome.html
3. Musgrave, P. and Lake, K., Engineering Applications of GRTensorII: Nonlinear Elasticity-Finite Deformations. 130.15.26.62/NewDemo/Eng/frame.html
4. Green, A.E., and Zerna, W., Theoretical Elasticity, 2nd edition. Oxford University Press, Oxford 1968.
5. Söderholm, L.H., Mechanics and thermodynamics of continua, second edition. KTH Department of Mechanics 1997.
6. Ericksen, J.L., Tensor Fields. In Handbuch der Physik III/1. Springer-Verlag,
Berlin 1960.
A pdf version of the user's guide is found here: usguidenew.pdf
A postscript version of the user's guide is found herer usguidenew.ps
Mikael Eriksson Lars Söderholm