Lagrange multiplier based vs micromorphic gradient-enhanced rate-(in)dependent crystal plasticity modelling and simulation

• Published in: Computer methods in applied mechanics and engineering 2020-12, Vol.372, p.113426, Article 113426
• Main Authors: Scherer, Jean-Michel, Phalke, Vikram, Besson, Jacques, Forest, Samuel, Hure, Jérémy, Tanguy, Benoît
• Format: Article
• Language: English
• Subjects: Axial stress; Crystal plasticity; Engineering Sciences; Finite elements; Lagrange multiplier; Lagrange multiplier approach; Mathematical analysis; Mechanics; Mechanics of materials; Micromorphic approach; Plastic properties; Regularization; Saturation; Scalars; Single crystals; Size effects; Strain gradient plasticity; Strain localization
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/j.cma.2020.113426

A reduced strain gradient crystal plasticity theory which involves the gradient of a single scalar field is presented. Rate-dependent and rate-independent crystal plasticity settings are considered. The theory is then reformulated following first the micromorphic approach and second a Lagrange multiplier approach. The finite element implementation of the latter is detailed. Computational efficiency of the Lagrange multiplier approach is highlighted in an example involving regularization of strain localization. The numerical performance improvement is shown to reach up to two orders of magnitude in computation time speedup. Then, size effects predicted by micromorphic and Lagrange multiplier based formulations of strain gradient plasticity are assessed. First of all numerical comparisons are performed on single crystal wires in torsion. Saturation of the size effects induced by the micromorphic approach and absence of saturation with the Lagrange multiplier approach when sample size is decreased are demonstrated. The Lagrange multiplier based formulation is finally applied to characterize size effects predicted for the ductile growth of porous unit-cells at imposed stress triaxiality. Excellent agreement with micromorphic results is obtained. •Micromorphic and Lagrange multiplier based approaches are applied to strain gradient crystal plasticity.•Finite element implementation of Lagrange multiplier non-local method is provided.•A Lagrange multiplier strategy significantly improves computational performances.•Unlike Lagrange multiplier model micromorphic model causes size effects saturation in torsion.•Micromorphic and Lagrange multiplier methods agree on size effects for ductile growth of voids.