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Strain localization analysis of Hill’s orthotropic elastoplasticity: analytical results and numerical verification

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Abstract

In this work the strain localization analysis of Hill’s orthotropic plasticity is addressed. In particular, the localization condition derived from the boundedness of stress rates together with Maxwell’s kinematics is employed. Similarly to isotropic plasticity considered in our previous work, the plastic flow components on the discontinuity surface vanish upon strain localization. The resulting localization angles in orthotropic plastic materials are independent from the elastic constants, but rather, they depend on the material parameters involved in the plastic flow in the material axes. Application of the above localization condition to Hill’s orthotropic plasticity in 2-D plane stress and plane strain conditions yields closed-form solutions of the localization angles. It is found that the two discontinuity lines in plane strain conditions are always perpendicular to each other, and for the states of no shear stresses, the localization angle depends only on the tilt angle of the material axes with respect to the global ones. The analytical results are then validated by independent numerical simulations. The B-bar finite element is employed to deal with the incompressibility due to the purely isochoric plastic flow. For a strip under vertical stretching in plane stress and plane strain as well as Prandtl’s problem of indentation by a flat rigid die in plane strain, numerical results are presented for both isotropic and orthotropic plasticity models with or without tilt angle. The influence of various parameters is studied. In all cases, the critical angles predicted from the localization condition coincide with the numerical results, giving compelling supports to the analytical prognoses.

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Acknowledgements

Financial support from the Spanish Government-MINECO-Proyectos de I + D (Excelencia)-DPI2017-85998-P-ADaMANT-Computational Framework for Additive Manufacturing of Titanium Alloy and the Catalan Government ACCIÓ - Ris3cat Transport and \(\hbox {PRO}^{2}\) Project is gratefully acknowledged. This work is also supported by the National Natural Science Foundation of China (51878294; 51678246), the State Key Laboratory of Subtropical Building Science (2018ZC04) and the Funding for Central Universities (2018PY20) to J.Y. Wu.

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Authors and Affiliations

  1. CIMNE, Technical University of Catalonia, Edificio C1, Campus Norte, Jordi Girona 1-3, 08034, Barcelona, Spain

    Miguel Cervera, Michele Chiumenti & Sungchul Kim

  2. State Key Laboratory of Subtropic Building Science, South China University of Technology, Guangzhou, 510641, China

    Jian-Ying Wu

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  1. Miguel Cervera
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  2. Jian-Ying Wu
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  3. Michele Chiumenti
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  4. Sungchul Kim
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Correspondence to Jian-Ying Wu.

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Cervera, M., Wu, JY., Chiumenti, M. et al. Strain localization analysis of Hill’s orthotropic elastoplasticity: analytical results and numerical verification. Comput Mech 65, 533–554 (2020). https://doi.org/10.1007/s00466-019-01782-4

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