Engineering Mechanics

Proceedings Vol. 12 (2006)

ENGINEERING MECHANICS 2006

Copyright © 2006 Institute of Theoretical and Applied Mechanics, Academy of Sciences of the Czech Republic, Prague

 

Z. Fiala
pages 56 - +12p., full text

Solution of finite deformation problems is sought in the space of all deformation tensor fields. Representation of a deformation process here as a trajectory makes us possible to further classify symmetric second-order tensor fields either as points, vectors, or covectors, and, as a consequence, assign them the corresponding time derivatives. However, as the space of all deformation tensor fields has proved non-euclidean, the time derivative of vector, and covector fields along the trajectory should be defined by the covariant derivative. This approach enables us coherently to formulate an incremental principle of virtual work, and propose the corresponding procedure in solving finite deformation problems. The approach will be demonstrated in finite elasticity. T. Frankel: ”Physics and engineering readers would profit greatly if they would form the habit of translating the vectorial and tensorial statements found in their customary reading of physical articles and books into the language of differential geometry, and of using its methods.” [Frankel 1997, p.xxiii] L. Schwartz: ”Engineers and physicists should take note that virtual work is simply the natural duality between cotangent and tangent bundles.” [BAMS 2003, p.411]

Text and facts may be copied and used freely, but credit should be given to these Proceedings.

All papers were reviewed by members of the scientific committee.