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A detailed description of the Gurson-Tvergaard-Needleman model within a mixed velocity-pressure finite element formulation. (English) Zbl 1352.74051

Summary: In an effort to implement Gurson-type models into a mixed velocity-pressure finite element formulation with the MINI-element \(P1^{+}/P1\), the algorithm proposed by N. Aravas [Int. J. Numer. Methods Eng. 24, 1395–1416 (1987; Zbl 0613.73029)] to integrate the pressure dependent plasticity as well as the formulations of consistent tangent moduli have been analyzed. This work firstly reviews and clarifies the mathematical basis of the formulations used by N. Aravas [Int. J. Numer. Methods Eng. 24, 1395–1416 (1987; Zbl 0613.73029)] and demonstrates the equality of the tangent moduli formulations proposed by R. M. Govindarajan and N. Aravas [Commun. Numer. Methods Eng. 11, No. 4, 339–345 (1995; Zbl 0826.73019)] and Z. L. Zhang [Comput. Methods Appl. Mech. Eng. 121, No. 1–4, 29–44 (1995; Zbl 0851.73007)], which are widely used in the literature. A unified formulation to calculate the tangent moduli is proven, and its accuracy is also investigated by the finite difference method. The implementation of the Gurson-Tvergaard-Needleman model is then detailed for the mixed velocity-pressure finite element formulation, which employs the MINI-element \(P1^{+}/P1\). Due to the particularity of this element, one needs to calculate two tangent moduli instead of one. The formulas for calculating the ‘linear tangent modulus’ and the ‘bubble tangent modulus’ are then detailed. Finally, comparison tests are carried out with ABAQUS in order to validate the present implementation for both homogeneous and heterogeneous deformations.

MSC:

74C05 Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials)
74S05 Finite element methods applied to problems in solid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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References:

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