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

Abstract

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.

A strain gradient plasticity model of porous single crystal ductile fracture

“…The model presented in previous sections was discretized using an Euler-backward (implicit) scheme and implemented in the finite element software Z-set (Besson and Foerch, 1998). Details on the finite element implementation of the Lagrange multiplier formulation were described in Scherer et al (2020). In the subsequent finite element simulations, 20-node brick finite elements are used.…”

“…This method is compared to the well established Gurson-Tvergaard-Needleman (Tvergaard and Needleman, 1984) approach for void growth and coalescence which relies on an effective porosity. The present work takes advantage of the strain gradient crystal plasticity model developed (without damage) and compared to the micromorphic approach in Scherer et al (2020). This finite strain formulation of strain gradient plasticity is based on a Lagrange multiplier method already successfully applied by Zhang et al (2018) for isotropic materials in the context of ductile fracture.…”

Crystal Plasticity with Micromorphic Regularization in Assessing Scale Dependent Deformation of Polycrystalline Doped Copper Alloys

“…The theoretical and thermodynamic derivation of the strain gradient extension S χ is provided in Reference [18]. An alternative formulation of a similar kind of strain gradient extension using a so-called augmented Lagrangian method can be found in [22].…”

Modeling size effects in microwire torsion: A comparison between a Lagrange multiplier-based and aCurlFpgradient crystal plasticity model

“…The micromorphic approach has been extended to other types of mechanical variables, including plastic strain and damage variables by Forest (2009Forest ( , 2016. It was used to investigate strain localization phenomena by Dillard et al (2006); Anand et al (2012); Mazière and Forest (2015); Brepols et al (2017) and to predict size effects in crystal plasticity by Cordero et al (2010); Aslan et al (2011); Wulfinghoff et al (2013); Ling et al (2018); Scherer et al (2019Scherer et al ( , 2020; Ryś et al (2020). In contrast to Eringen's original micromorphic theory, the reduced-order micromorphic theory relies on a single scalar-valued additional degree of freedom at each material point, akin to accumulated plastic strain or slip (Wulfinghoff and Böhlke, 2012;Wulfinghoff et al, 2013;Erdle and Böhlke, 2017;Ling et al, 2018;Scherer et al, 2019).…”