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**A finite element formulation for elastoplasticity based on a three-field variational equation.**
*(English)*
Zbl 0591.73082

A three-field variational equation, which expresses the momentum balance equation, the plastic consistency condition, and the dilatational constitutive equation in a weak form, is proposed as a basis for finite element computations in hardening elastoplasticity. The finite element formulation includes algorithms for the integration of the elastoplastic rate constitutive equations which are similar to members of the ”return mapping” family of algorithms employed in displacement formulations, except that the proposed algorithms are not required to explicitly satisfy the plastic consistency condition at the end of each time step. This condition is imposed globally by the inclusion of a variational equation that suitably constrains the solution.

The plastic incompressibility constraint is also treated in an appropriate variational sense. Solution of the nonlinear finite element equations is obtained by use of Newton’s method and details of the linearization of the variational equation are given. The formulation is developed for an associative von Mises plasticity model with general nonlinear isotropic and kinematic strain hardening. A number of numerical test examples is provided.

The plastic incompressibility constraint is also treated in an appropriate variational sense. Solution of the nonlinear finite element equations is obtained by use of Newton’s method and details of the linearization of the variational equation are given. The formulation is developed for an associative von Mises plasticity model with general nonlinear isotropic and kinematic strain hardening. A number of numerical test examples is provided.

### MSC:

74S05 | Finite element methods applied to problems in solid mechanics |

74S30 | Other numerical methods in solid mechanics (MSC2010) |

### Keywords:

three-field variational equation; momentum balance equation; plastic consistency condition; dilatational constitutive equation; weak form; hardening elastoplasticity; algorithms; integration of the elastoplastic rate constitutive equations; return mapping; plastic incompressibility constraint; nonlinear finite element equations; Newton’s method; linearization; von Mises plasticity model; general nonlinear isotropic and kinematic strain hardening; rate-dependent; three-field mixed variational formulation
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\textit{P. M. Pinsky}, Comput. Methods Appl. Mech. Eng. 61, 41--60 (1987; Zbl 0591.73082)

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### References:

[1] | Wilkins, M.L., Calculation of elastic-plastic flow, () |

[2] | Kreig, R.D.; Key, S.W., Implementation of a time-independent plasticity theory into structural computer programs, (), 125-137 |

[3] | Kreig, R.D.; Kreig, D.B., Accuracies of numerical solution methods for the elastic-perfectly plastic model, ASME J. pressure vessel technol., 99, 510-515, (1977) |

[4] | Schreyer, H.L.; Kulak, R.F.; Kramer, J.M., Accurate numerical solutions for elastoplastic models, ASME J. pressure vessel technol., 101, 226-234, (1979) |

[5] | Ortiz, M.; Pinsky, P.M.; Taylor, R.L., Operator split methods for the numerical solution of the elastoplastic dynamic problem, Comput. meths. appl. mech. engrg., 39, 137-157, (1983) · Zbl 0501.73077 |

[6] | Ortiz, M.; Popov, E.P., Accuracy and stability of integration algorithms for elastoplastic constitutive relations, Internat. J. numer. meths. engrg., 21, 1561-1576, (1985) · Zbl 0585.73057 |

[7] | Nagtegaal, J.C.; Parks, D.M.; Rice, J.R., On numerically accurate finite element solutions in the fully plastic range, Comput. meths. appl. mech. engrg., 4, 153-177, (1974) · Zbl 0284.73048 |

[8] | Key, S.W., A variational principle for incompressible and nearly-incompressible anisotropic elasticity, Internat. J. solids structures, 5, 951-964, (1969) · Zbl 0175.22101 |

[9] | Nyssen, C.; Beckers, P., A unified approach for displacement, equilibrium and hybrid finite element models in elastoplasticity, Comput. meths. appl. mech. engrg., 44, 131-151, (1984) · Zbl 0535.73056 |

[10] | Marsden, J.E.; Hughes, T.J.R., Mathematical foundations of elasticity, (1983), Prentice-Hall Englewood Cliffs, NJ · Zbl 0545.73031 |

[11] | Simo, J.C.; Taylor, R.L., Consistent tangent operators for rate-independent elastoplasticity, Comput. meths. appl. mech. engrg., 48, 101-118, (1985) · Zbl 0535.73025 |

[12] | Prager, W.; Hodge, P.G., Theory of perfectly plastic solids, (1963), Wiley New York |

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