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Hypo-Elasticity Model Based upon the Logarithmic Stress Rate

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Abstract

Recently these authors have proved [46, 47] that a smooth spin tensor Ωlog can be found such that the stretching tensor D can be exactly written as an objective corotational rate of the Eulerian logarithmic strain measure ln V defined by this spin tensor, and furthermore that in all strain tensor measures only ln V enjoys this favourable property. This spin tensor is called the logarithmic spin and the objective corotational rate of an Eulerian tensor defined by it is called the logarithmic tensor-rate. In this paper, we propose and investigate a hypo-elasticity model based upon the objective corotational rate of the Kirchhoff stress defined by the spin Ωlog, i.e. the logarithmic stress rate. By virtue of the proposed model, we show that the simplest relationship between hypo-elasticity and elasticity can be established, and accordingly that Bernstein's integrability theorem relating hypo-elasticity to elasticity can be substantially simplified. In particular, we show that the simplest form of the proposed model, i.e. the hypo-elasticity model of grade zero, turns out to be integrable to deliver a linear isotropic relation between the Kirchhoff stress and the Eulerian logarithmic strain ln V, and moreover that this simplest model predicts the phenomenon of the known hypo-elastic yield at simple shear deformation.

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  1. Institute of Mechanics, Ruhr-University Boclum, D-44780, Bochum, Germany

    H. Xiao, O. T. Bruhns & A. Meyers

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  1. H. Xiao
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  2. O. T. Bruhns
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  3. A. Meyers
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Xiao, H., Bruhns, O.T. & Meyers, A. Hypo-Elasticity Model Based upon the Logarithmic Stress Rate. Journal of Elasticity 47, 51–68 (1997). https://doi.org/10.1023/A:1007356925912

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