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A modified Gurson-type plasticity model at finite strains: formulation, numerical analysis and phase-field coupling

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Abstract

The modeling of failure in ductile materials must account for complex phenomena at the micro-scale, such as nucleation, growth and coalescence of micro-voids, as well as the final rupture at the macro-scale, as rooted in the work of Gurson (J Eng Mater Technol 99:2–15, 1977). Within a top–down viewpoint, this can be achieved by the combination of a micro-structure-informed elastic–plastic model for a porous medium with a concept for the modeling of macroscopic crack discontinuities. The modeling of macroscopic cracks can be achieved in a convenient way by recently developed continuum phase field approaches to fracture, which are based on the regularization of sharp crack discontinuities, see Miehe et al. (Comput Methods Appl Mech Eng 294:486–522, 2015). This avoids the use of complex discretization methods for crack discontinuities, and can account for complex crack patterns. In this work, we develop a new theoretical and computational framework for the phase field modeling of ductile fracture in conventional elastic–plastic solids under finite strain deformation. It combines modified structures of Gurson–Tvergaard–Needelman GTN-type plasticity model outlined in Tvergaard and Needleman (Acta Metall 32:157–169, 1984) and Nahshon and Hutchinson (Eur J Mech A Solids 27:1–17, 2008) with a new evolution equation for the crack phase field. An important aspect of this work is the development of a robust Explicit–Implicit numerical integration scheme for the highly nonlinear rate equations of the enhanced GTN model, resulting with a low computational cost strategy. The performance of the formulation is underlined by means of some representative examples, including the development of the experimentally observed cup–cone failure mechanism.

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Notes

  1. For standard von Mises plasticity, the equivalent plastic strain is defined as

    figure a

  2. Regarding to the relation between the fracture length scale parameter l and the finite element size \(h_e\), we refer to the work of Miehe et. al. [42], where \(l\ge 2 h_e\) is required to resolve the regularized crack surface \(\varGamma _l(d)\), such that we have \(\varGamma _l(d) = \varGamma \) in the finite element approximation.

  3. To overcome the pathological mesh dependency behavior, a non-local GTN damage model can be used, that based on gradient plasticity coupled with fracture phase field, in line with Miehe et. al. [41]. In this approach, a micromororphic regularization on the side of gradient plasticity is introduced by considering an extended set of plastic variables, which are linked by penalty term in a modified energetic response function. This formulation is recently extended towards coupled thermomechanical response of gradient plasticity as outlined in Aldakheel [2] for finite deformations in the logarithmic strain space and Aldakheel and Miehe [4] at small strains.

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Acknowledgements

F. A. wants to thank the late Professor Christian Miehe, whose continuous scientific support and great mentorship will always be remembered. Support for this research was provided by the “German Research Foundation” (DFG) within project WR 19/58-1.

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Authors and Affiliations

  1. Institute of Continuum Mechanics, Leibniz Universität Hannover, Appelstrasse 11, 30167, Hannover, Germany

    Fadi Aldakheel & Peter Wriggers

  2. Institute of Applied Mechanics (CE), Chair I, University of Stuttgart, Pfaffenwaldring 7, 70569, Stuttgart, Germany

    Christian Miehe

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  1. Fadi Aldakheel
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  3. Christian Miehe
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Correspondence to Fadi Aldakheel.

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Aldakheel, F., Wriggers, P. & Miehe, C. A modified Gurson-type plasticity model at finite strains: formulation, numerical analysis and phase-field coupling. Comput Mech 62, 815–833 (2018). https://doi.org/10.1007/s00466-017-1530-0

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