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**Consistent tangent operators for rate-independent elastoplasticity.**
*(English)*
Zbl 0535.73025

It is shown that for problems involving rate constitutive equations, such as rate-independent elastoplasticity, the notion of consistency between the tangent (stiffness) operator and the integration algorithm employed in the solution of the incremental problem, plays a crucial role in preserving the quadratic rate of asymptotic convergence of iterative solution schemes based upon Newton’s method. Within the framework of closest-point-projection algorithms, a methodology is presented whereby tangent operators consistent with this class of algorithms may be systematically developed. To wit, associative \(J_ 2\) flow rules with general nonlinear kinematic and isotropic hardening rules, as well as a simple class of non-associative flow rule are considered. The resulting iterative solution scheme preserves the asymptotic quadratic convergence characteristic of Newton’s method, whereas use of the so-called elasto-plastic tangent in conjunction with a radial return integration algorithm, a procedure often employed, results in Newton type of algorithms with suboptimal rate of convergence. Application is made to a set of numerical examples which include saturation hardening laws of exponential type.

### MSC:

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

74S99 | Numerical and other methods in solid mechanics |

74C99 | Plastic materials, materials of stress-rate and internal-variable type |

49M15 | Newton-type methods |

### Keywords:

consistent tangent operators; rate constitutive equations; rate-independent elastoplasticity; closest-point-projection algorithms; associative \(J_ 2\) flow rules; general nonlinear kinematic and isotropic hardening rules; non-associative flow rule; iterative solution scheme; asymptotic quadratic convergence characteristic of Newton’s method; elasto-plastic tangent; radial return integration algorithm; suboptimal rate of convergence; numerical examples; saturation hardening laws of exponential type
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\textit{J. C. Simo} and \textit{R. L. Taylor}, Comput. Methods Appl. Mech. Eng. 48, 101--118 (1985; Zbl 0535.73025)

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### References:

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