Welcome to the cm-package

    This is the official homepage for the cm-package.

    The cm-package -  A MapleV program for continuum mechanics

    The cm-package is a system for calculating tensor objects related to continuum mechanics and for their manipulations. On constructing the package, emphasis has been on creating a simple and easy to understand user's interface. The user who is familiar with continuum mechanics should quickly be able to apply and understand, as well as further expand and customize the package.

    To formulate continuum mechanics in general, one need two spaces, the present configuration (or simply space) and the reference configuration of the body. The basic object in continuum mechanics is the mapping between these two spaces, see [1].

    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].

    In reality, however, the present configuration (or simply space) and the reference configuration are two different spaces, [1, 5]. The reason for this is simply that there are two different distances associated to two neighbouring points (or particles as the nomenclature is in continuum mechanics), the distance in the present configuration and the distance in the reference configuration. To be concrete, in space there is the ordinary Euclidean distance but also the distance of the reference configuration carried over and expressed by the inverse of the left (referential) Cauchy-Green (metric) tensor. Similarly, in the reference configuration, there is the ordinary Euclidean distance, but also the distance of the present configuration carried over, and expressed by the right (referential) Cauchy-Green (metric) tensor.

    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.
     

    The first version of this program has been developed as a master thesis project by Mikael Eriksson under the supervision by Lars Söderholm

     

    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.
     
     

    The cm-package for unix

    A compressed file for the version 1.0 of the cm-package is found here: cm.tar.gz

    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

    Last modified October 13, 1999.