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Extrapolation and C e ‐based implicit integration of anisotropic constitutive behavior
Author(s) -
Areias Pedro,
Rabczuk Timon,
Ambrósio Jorge
Publication year - 2021
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.6661
Subject(s) - mathematics , hyperelastic material , cauchy stress tensor , mathematical analysis , constitutive equation , finite strain theory , nonlinear system , finite element method , physics , quantum mechanics , thermodynamics
For finite strain plasticity with both anisotropic yield functions and anisotropic hyperelasticity, we use the Kröner‐Lee decomposition of the deformation gradient to obtain a differential‐algebraic system (DAE) in the semi‐implicit form and solve it by an implicit Richardson‐extrapolated method based on intermediate substeps. The source is here the right Cauchy‐Green tensor and the consistent Jacobian of the second Piola‐Kirchhoff stress is determined with respect to this source. The system is composed by a smooth nonlinear first‐order differential equation and a non‐smooth algebraic equation. The development of a Richardson‐extrapolated implicit integrator for any hyperelastic case and any yield function is the goal of this work. The integration makes use of a backward‐Euler method for the flow law complemented by the solution of a yield constraint. The resulting system is solved by the Newton‐Raphson method to obtain the plastic multiplier and the elastic right Cauchy‐Green tensorC e. To ensure power consistency, we make use of the elastic Mandel stress construction. Iso‐error maps for three yield functions and three numerical examples are presented.