Basics of Continuum Plasticity by Kwansoo Chung & Myoung-Gyu Lee

Author:Kwansoo Chung & Myoung-Gyu Lee
Language: eng
Format: epub
Publisher: Springer Singapore, Singapore

As for the integration of the rate of deformation tensor itself, comparing Eqs. (2.​1) and (11.39) confirms that D is the three-dimensional version of the true strain rate; therefore, the natural strain increment is defined as

(11.43)

and , which might be the true strain (or natural strain) as a three-dimensional version of Eq. (2.​2) (note that here and in Eq. (10.​20) are not related to each other). However, even though this integration is performed following a material element, it still has an Eulerian nature so that the resulting is not the true strain and is not so useful. In fact, it is so important to note that Eqs. (2.​1) and (2.​2) share the same principal material lines (meaning that their principal directions correspond to the same material lines, which are aligned in the tensile direction and its normal directions). Therefore, the true strain tensor should be Lagrangian in nature and, furthermore, its principal directions are supposed to be materially fixed.

Now, from Eqs. (11.18) and (11.31),

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