Logarithmic strain in 1 versus 3(2) dimensions

Abstract eng:
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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