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Existence and uniqueness of the integrable-exactly hypoelastic equation\(\mathop \tau \limits^ \circ = \lambda (trD){\rm I} + 2\mu D\) and its significance to finite inelasticity

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Summary

In Eulerian rate type finite inelasticity models postulating the additive decomposition of the stretchingD, such as finite deformation elastoplasticity models, the simple rate equation indicated in the above title is widely used to characterize the elastic response withD replaced by its “elastic” part. In 1984 Simo and Pister (Compt. Meth. Appl. Mech. Engng.46, 201–215) proved that none of such rate equations with several commonly-known stress rates is exactly integrable to deliver an elastic relation, and thus any of them is incompatible with the notion of elasticity. Such incompatibility implies that Eulerian rate type inelasticity theory based on any commonly-known stress rate is self-inconsistent, and thus it is hardly surprising that some aberrant, spurious phenomena such as the so-called shear oscillatory response etc., may be resulted in. Then arises the questions: Whether or not is there a stress rate\(\mathop {\tau ^* }\limits^\bigcirc\)? The answer for these questions is crucial to achieving rational, self-consistent Eulerian rate type formulations of finite inelasticity models. It seems that there has been no complete, natural and convincing treatment for the foregoing questions until now. It is the main goal of this article to prove the fact: among all possible (infinitely many) objective corotational stress rates and other well-known objective stress rates\(\mathop {\tau ^* }\limits^\bigcirc\), there is one and only one such that the hypoelastic equation of grade zero with this stress rate is exactly integrable to define a hyperelastic relation, and this stress rate is just the newly discoveredlogarithmic stress rate by these authors and others. This result, which provides a complete answer for the aforementioned questions, indicates that in Eulerian rate type formulations of inelasticity models, the logarithmic stress rate is the only choice in the sense of compatibility of the hypoelastic equation of grade zero that is used to represent the elastic response with the notion of elasticity.

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  1. H. Xiao, O. T. Bruhns & A. Meyers

    Present address: Institute of Mechanics I, Ruhr-University Bochum, D-44780, Bochum, Germany

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    Xiao, H., Bruhns, O.T. & Meyers, A. Existence and uniqueness of the integrable-exactly hypoelastic equation\(\mathop \tau \limits^ \circ = \lambda (trD){\rm I} + 2\mu D\) and its significance to finite inelasticity. Acta Mechanica 138, 31–50 (1999). https://doi.org/10.1007/BF01179540

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