Toward robust scalar-based gradient plasticity modeling and simulation at finite deformations

Abstract

Strain gradient plasticity has been the subject of extensive research in the past forty years in order to model size effects in metal plasticity, on the one hand, and provide finite width shear bands in the simulation of localization phenomena, on the other hand. However, the use of the emerging models is still limited to academic applications and has not yet been adopted by industry practitioners. The present paper systematically reviews the pros and the cons of gradient plasticity at finite strains based on gradient of scalar plastic variables, in particular gradient of the cumulative plastic strain. It proposes benchmark tests addressing both size effect modeling and plastic strain localization simulation. It includes new analytical solutions for validation of FE implementation. It focuses on the micromorphic approach to gradient plasticity, as a convenient method for implementation in FE codes. New features of the analysis include the comparison of three distinct formulations of rate-independent gradient plasticity at finite deformations, based on the multiplicative decomposition of the deformation gradient and on quadratic potentials with respect to gradient terms. The performance of micromorphic and Lagrange-multiplier based strain gradient plasticity models is evaluated for various monotonic and cyclic loading conditions including confined plasticity in simple glide and tension, bending and torsion at large deformations. Limitations are pointed out in the case of bending and torsion, which can be overcome for instance by the use of the gradient of equivalent plastic strain model.

Strain gradient plasticity based on saturating variables

“…The local form of the entropy principle is now exploited with the assumption that the generalized stresses a 0 and b 0 are non-dissipative stresses, following Forest (2016); Abatour et al (2022). Three finite deformation formulations were proposed in the two latter references, depending on the configuration with respect to which the microdeformation gradient, as an argument of the free energy potential, is considered.…”

“…where f (M ∼ , A I ) is the yield function which is evaluated for the Mandel stress tensor and the thermodynamical forces A I , according to Abatour et al (2021). The Eq.…”

“…Severe limitations of scalar based gradient plasticity models have been pointed out in the last ten years both for applications in hardening and softening plasticity. The possibility of vanishing of the yield radius due to excessive values of the Laplacian term 2 has been discussed in (Poh et al, 2011;Wulfinghoff et al, 2014;Jebahi and Forest, 2021;Abatour et al, 2022). This usually leads to divergence of the computation and significantly undermines the robustness of the approach.…”

“…The fact that the cumulative plastic strain p is always increasing during any plastic loading leads to unbounded enhanced hardening which may well be physically irrelevant. This is particularly the case for cyclic loading where unlimited size-dependent cyclic hardening is predicted, especially for cyclic bending and torsion of bars (Phalke et al, 2021;Abatour et al, 2022). A model based on the gradient of the equivalent plastic strain instead of the cumulative one leads to bounded cyclic hardening, as demonstrated by Jebahi and Forest (2021), but its use is also associated with difficulties for vanishing plastic strain values.…”

“…The finite deformation framework incorporates the multiplicative splitting of the deformation gradient, together with the gradient of the micromorphic variable w.r.t. Lagrange coordinates, thus following (Liebe et al, 2003;Geers, 2004;Forest, 2016;Abatour et al, 2021;Friedlein et al, 2022).…”