Tensorial Analysis and

Continuum Mechanics

© 2006 Jean-François Gouyet

Jean-François Gouyet

LPMC, Ecole Polytechnique

Palaiseau, France,

19, Clos Désiré I, 91120 Palaiseau, France.

jean-francois.gouyet@tiscali.fr

The package TContinuumMechanics is devoted to the manipulation of tensors in the context of Continuum Mechanics problems. The present version TContinuumMechanics 2.0, which needs Mathematica (version 4.2 or higher) and Tensorial 4.0, has been divided into two parts: the program TContinuumMechanics version 2.0 itself, which is now a shareware as is version 4.0 of Tensorial, and its specific applications which remain freely available. The main application TContinuumMechanics_Fluegge is composed of the twelve chapters of the well known book of Wilhelm Flügge "Tensorial Analysis and Continuum Mechanics", Wilhelm Flügge, Springer, 1972. It shows how TContinuumMechanics 2.0 can be used in continuum mechanics problems. In this application, I closely followed Flügge's book all along the chapters, as it is to my opinion one of the clearest presentation of Continuum Mechanics using tensors extensively. It demonstrates how the Mathematica packages Tensorial and TContinuumMechanics permit to express in a very clear manner the tensorial structure of the equations of Continuum Mechanics.

New in Tensorial 4.0 and ContinuumMechanics 2.0 - January 2006

All the output can now be copied and pasted. See UsingTensorial, Output Format.

The output notation has been improved. Unexpanded partial derivatives now show only a single comma before the first differentiation index. Similarly an unexpanded covariant derivative shows only a single semicolon. (Other differentiation symbols can be used as before.) In addition there is an alternative mode of display for covariant derivatives using the ∇ symbol with subscripts. This can be turned on and off with SetCovariantDisplay. Similarly, SetLieDisplay can be used to display unexpanded Lie derivatives as a £ symbol with subscripts.

The old Dif and Cov wrappers for differentiation indices are out. Unexpanded (to coordinates) partial and covariant derivatives are only broken out by their linear and Liebnizian properties and then left without further evaluation. This means that the old NestedTensor is unnecesssary and it has been removed. To prevent linear and Liebnizian breakout wrap the expression in Tensor instead of NestedTensor. UnnestTensor will still unwrap the expression.

DummySimplify has been removed. There is a new TensorSimplify. It selects terms that have the same pattern, applies any declared symmetries and then applies SimplifyTensorSum on these subsets of terms.

The Curvature section contains new routines (moved from the General Relativity subpackage) that calculate the Riemann tensor, up and down, the Ricci tensor, the scalar curvature and the Einstein tensor.

The new OrthonormalTransformation routine will, given a metric and a signature pattern, calculate a transformation matrix from the coordinated basis to an orthonormal basis.

A number of routines that are more in the class of general expression manipulation rather than tensor routines are gathered together in the Functions & Rules section. Two new routines here are SymbolsToPatterns and LHSSymbolsToPatterns. These are useful in turning derived equations into general rules that can be applied to subsequent expressions.

The $PrePrint for larger font output has been eliminated. Instead the Tensorial style sheet was changed to give a larger Output cell font. This simplifies copying and pasting by eliminating additional box structures. The style sheet was also changed to eliminate the Helvetica font in Headings in favor of Times. Some systems have a difficult time using the Helvetica font.

Also, in the Tensorial style sheet, Output cells are now StandardForm. Derivatives are formated in Leibnizian style but you will now have to use MatrixForm to format arrays.

Many improvements have been made to various functions of the program. I have added as an example a chapter showing how TContinuumMechanics can be used to calculate classical operators like divergence, gradient, curl or laplacian in arbitrary curvilinear coordinates. Practical examples will be progressively added in the future, even if the present notebook shows that TContinuumMechanics (associated to Tensorial ) already allows to cover all the various aspects of continuum mechanics.

Later on I plan to develop the modelization of growing hulls and membranes with tensorial surface tensions, following the course of Jean Garrigues (Ecole Supérieure de Mécanique de Marseille). Some preliminary works have already started in our group, involving liquid crystal and fibered type of structures.

The development of this package led to the creation of operations and rules which are included in part in `TContinuumMechanics`Help`3D package`, in part in Tensorial. It shows how convenient is the symbolic approach which permits now to manipulate the whole basic concepts presented for instance in the standard Flügge's book.

Figures presented in Flügge's book to clarify the theoretical developments, have been introduced here using the Mathematica package `DrawGraphics`. This program may be downloaded on the same web site as TContinuumMechanics and Tensorial.

All the remarks and suggestions concerning this package will be welcome !

Send your comments to jean-francois.gouyet@tiscali.fr

Acknowledgments

I would like to thank Renan Cabrera, the author of Tensorial, and his collaborator David Park, to have accepted my contribution to the improvement of the successive versions of Tensorial, and to have themselves participated to the amendment of the present package.

History

- October 2002 First three Chapters of Flügge's book (Functions : SymmetricStandardOrder, CrossProductExpansion, VolumeSquare, LeviCivitaSimplify, LeviCivitaOrder, TensorComponentsD, RedRule... are included in Tensorial as ToFlavor).

- November 2003 First Version 1, on line:

Chapters 1 to 5 (Functions BasisChange, BasisDerivation, ChristoffeldSymbol, ChristoffeluSymbol, DifToCov, TCurl, TDiv, TGrad, T Laplacian, FullLeviCivitaExpand, FlavorList, TensorNesting, TensorUnNesting, Cov2d, CovariantD2d, MetricRuleD, EvaluateCrossProducts,

- February 2004 Version 1.9 of TContinuumMechanics, includes the nine first chapters of Flügge's book. (Functions CovariantDSimplify, TensorComponentsD, PartialDSimplify, DifToPartialD, StandardDownIndices, AntiSymmetricStandardOrder, ContravariantD, PartialDToDif, Kronecker).

- April 2005, Version 1.12 of TContinuumMechanics. Flügge's book completed (12 Chapters). Addition of the calculation of Div, Curl, Grad, Laplacian, in curvilinear coordinates.

- May 2005, Functional derivatives ( not in Flügge's book). Correction of a bug in TensorUnNesting.

- June 2005, Correction of problems due to some incompatibilities between Mathematica version 4 and version 5, and with the problem of magnification of the results on Unix systems.

- January 2006, Version 2.0 of TContinuumMechanics, which is adapted to Tensorial 4.0.

- February 2006, A new function SymmetricSlots, with the options Symmetric and AntiSymmetric has been introduced in Tensorial 4 which partly replaces SymmetricStandardOrder and AntiSymmetricStandardOrder. However SymmetricStandardOrder and AntiSymmetricStandardOrder give a different manner of ordering the indice (which includes the Voids, see the Help) and also acts on dirivative symbols, SymmetricSlots works only on tensor indices.

- October 2006 to March 2007, Introduction of LowerIndexD, RaiseIndexD, MetricRuleD, MetricSimplifyD, ReleaseHoldD, ToFlavorD, UpDownSwapD, which works on tensorial indices, corresponding (or not) to covariant and contraviant derivatives. Shortcut for the Laplacian notation.

- July 2007, Correction of errors when going from Mathematica version 4.2 to 5.2. So TContinuumMechanics works in both 4 and 5 Mathematica versions.

Copyright

Tensorial and subpackages produced by the developers Renan Cabrera, David Park, Jean-François Gouyet are copyrighted by the developers.

Tensorial and associated subpackages are provided free of charge on the condition that it is acknowledged that we accept no liability for software performance, continued maintenance, or damage to data. The authors retain any and all rights to the the Tensorial software and associated subpackages produced by the developers.

Use of Tensorial and associated subpackages for purposes of commercial enterprise is forbidden without prior arrangement with the authors.

We request that if you make a publication extensively using TContinuumMechanics and Tensorial you reference the packages and principal authors.

Created by Mathematica (November 27, 2007) |