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
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