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Implicit Numerical Integration of Nonsmooth Multisurface Yield Criteria in the Principal Stress Space

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Abstract

Plasticity models employing multiple yield surfaces are frequently met in plasticity theory and engineering practice. The multiple yield surfaces may or may not intersect in a smooth manner, with the latter case being a superset of the former, typically encountered more often in engineering problems and covered within this work. Prominent members of this class of plasticity models can be found in a wide range of applications in Soil Mechanics, Rock Mechanics, Damage Mechanics, metal plasticity, concrete modeling or in the modeling of brittle or cohesive/frictional materials.

While the importance of these models is widely acknowledged, difficulties arising from the singularities induced by the nonsmooth intersections of the yield surfaces, introduce severe algorithmic and numerical complexities that usually require specific treatment of each model. Three main problems can be identified, originating mainly from the application of the normality hypothesis, namely that (a) the elastic domain is subdifferential with respect to the stress vector at the intersections of the yield surfaces, (b) severe numerical errors are present in the vicinity of intersections since the derivatives of the yield surfaces are not always defined in these areas and (c) the set of the yield surfaces considered to be ultimately active is not known a priori.

The objectives of this article are (a) to provide a comprehensive review of the related literature and an extensive overview of the solution techniques proposed by different researchers, (b) to present the formulation and propose the algorithmic treatment for the problem of nonsmooth multisurface plasticity models and finally (c) to give implementation details for some of the most widely used nonsmooth multisurface plasticity models.

The proposed algorithm is based on a spectral representation of stresses and strains for the case of infinitesimal deformation plasticity and the reformulation of the return mapping scheme in principal stress directions. The determination of the set of the yield surfaces that will remain ultimately active is identified by involving a systematic enforcement of the Karush–Kuhn–Tucker conditions, providing in this way a surface agnostic implementation.

Four representative examples of nonsmooth, multisurface plasticity models, are extensively presented and examined within the framework of the proposed algorithm. These are the Tresca, the Mohr–Coulomb, the Hoek–Brown and the Drucker–Prager (in the case that it is accompanied with a tension cut–off type surface) yield criteria, all of which are well established in the related literature and engineering practice.

The efficiency, robustness and accuracy of the proposed algorithm is demonstrated through a series of numerical examples with excellent results.

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Acknowledgements

The present work benefited form the input of Prof. Theodoros Chatzigogos, Department of Civil Engineering, Aristotle of University of Thessaloniki, who provided valuable ideas, comments and assistance to the undertaking of the research summarised here.

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  1. Department of Civil Engineering, Aristotle University of Thessaloniki, P.O. Box 424, 54124, Thessaloniki, Greece

    Fotios E. Karaoulanis

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The author gratefully acknowledge financial support from the Greek State Institute of Scholarships (I.K.Y.); contract/grant number: 4506/05.

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Karaoulanis, F.E. Implicit Numerical Integration of Nonsmooth Multisurface Yield Criteria in the Principal Stress Space. Arch Computat Methods Eng 20, 263–308 (2013). https://doi.org/10.1007/s11831-013-9087-3

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