Engineering Mechanics

Proceedings Vol. 14 (2008)

ENGINEERING MECHANICS 2008

Copyright © 2008 Institute of Thermomechanics, Academy of Sciences of the Czech Republic, v.v.i., Prague

 

Fiala Z.
pages 22 - +20p., full text

Stepwise integration of a finitely deformed body is based on an incremental kinematics of a medium under initial stress / strain conditions. As usual in continuum mechanics, deformation and stress tensors at a point are considered to form vector (i.e. Euclidean) spaces. The derivatives therefore are the usual ones, and when dealing with time evolution we have to resort to objective time derivatives. There are two classical textbooks dealing with the incremental approach – the book of Green & Zerna and the book of Biot, both approaching the subject from different perspectives. The first as an incremental strain, the second one as an incremental stress. Employing an approach of differential geometry to the description of finite deformations, resulting in a geometrically consistent linearization, we can simply compare these two descriptions. The contribution reviews the theory of finite deformations both from the finite-dimensional Riemannian geometry viewpoint (standard knowledge) and the infinite-dimensional Riemannian geometry viewpoint (non-standard, disputable approach), considering the space of deformation tensors as a constantly non-negatively curved space.

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