Proceedings Vol. 16 (2010)
ENGINEERING MECHANICS 2010
May 10 – 13, 2010, Svratka, Czech Republic
Copyright © 2010 Institute of Thermomechanics, Academy of Sciences of the Czech Republic, v.v.i., Prague
ISSN 1805-8248 (printed)
ISSN 1805-8256 (electronic)
list of papers scientific commitee
pages 29 - +11p., full text
The paper discusses the logarithmic strain in one dimension (1D) from the geometrical point of view to highlight the nature of problems when generalizing it to more dimensions (3D or 2D). Starting from geometry of positive real numbers R+, author advocates the geometrical approach via the Riemannian geometry of the space of symmetric positive-deﬁnite n×n matrices (n stands for dimension) of real numbers Sym+ (n, R) ∼ GL+ (n, R)/SO(n, R), which reduces to R+ in the case = of 1D. Based on previous papers, he demonstrates that only such an approach can guarantee consistent and well-deﬁned manipulation with the logarithmic strain in more dimensions. Even though the geometry itself is rather unusual and nonintuitive due to its non-euclidean nature, its proﬁt for the theory of ﬁnite deformations is noticeable and has already been demonstrated formerly – the natural and unambiguous linearization for an incremental approach within ﬁnite deformations, based on covariant derivative instead of on one of many objective time derivatives.
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