Toward a physical model for strain hardening in fcc crystals
Introduction
As noted by Cottrell in 1953 [1] and, nearly 50 years later, in 2002 [2], work hardening is the most challenging problem in dislocation theory. Progress in multiscale modeling, now provides means for incorporating more physics into current strain hardening models. For instance, dislocation core mechanisms like cross-slip can be investigated using atomistic simulations (see e.g. [3]), whereas dislocation dynamics (DD) simulations [4], [5] provide integrated values for elastic processes like junction formation and destruction, as well as an insight into collective dislocation behavior [6].
Substantial progress has also been achieved regarding the connection between mesoscale studies and the continuum framework. Codes for crystal plasticity appeared some years ago [7], [8]. Recent codes for the polycrystal treat the local nature and number of active slip systems inside the grains, as well as their evolution. The conditions for compatible grain deformation are also computed, like in any finite element code, but it is no longer necessary to make use of the traditional approximations or homogenization procedures of continuum mechanics. The current trend is, then, to develop dislocation-based constitutive formulations at the scale of slip systems [9], [10], [11], [12]. Such models involve, however, extensive parameter fitting and are, paradoxically, totally unable to treat the complex behavior of single crystals.
The objective of the present work is to show that the predictive ability of models for strain hardening can be considerably improved by making use of the available information arising from theory, experiment and simulation. For this purpose, the monotonic deformation of fcc single crystals is taken as a benchmark test. In what follows, we discuss a model deriving from the storage–recovery frame developed in scalar form by Kocks and Mecking (see [13] for a review) and further expanded into tensor form by Teodosiu et al. [9]. The storage–recovery framework presents the advantage of incorporating the scaling laws for linear hardening (stage II) and dynamic recovery (stage III).
In the present model, like in several other ones, the microstructural variable is a stored dislocation density per slip plane, which is uniform in space and evolves with strain. To obtain a hardening matrix, in which each coefficient represents the continuously evolving strain hardening rate in the slip systems, one first expresses the critical stress for the onset of glide in a slip system as a function of the stored dislocation density (Section 2). A second set of equations is then needed for describing the evolution with strain of these stored densities (Section 3). A third set of equations, which is not discussed here, accounts for the small temperature and strain rate dependencies of the critical stresses [14]. It involves two parameters, of which the values are not critical and are now estimated from a simplified model for jog hardening.
All model parameters related to elastic dislocation properties are numerically estimated from DD simulations, in order to reduce as much as possible the number of free parameters. Preliminary results obtained by implementing such a constitutive framework in a plasticity code for the single crystal are shown in Section 4 and the limits of the model are discussed in Section 5. For the sake of brevity, only a few major critical points are addressed here.
Section snippets
The Taylor relation
The Taylor relation is traditionally written in the form τ = αμbρ1/2, where all symbols have their usual meaning and ρ is the total dislocation density, from which one often singles out the density of forest dislocations, that is of dislocations interacting in non-parallel slip planes. The pseudo-constant α reflects the average strength of dislocation interactions and includes both long-range elastic interactions and short-range reactions between dislocations. When line tension effects are
The basic model
The stored density is defined as the density of dislocations that are immobile at a given moment. Beyond the yield stress, this density is always large in comparison to the mobile density and it can be approximated by the total density ρ. Whereas storage is known to result in a linear and athermal deformation stage (stage II), dynamic recovery describes a temperature and strain rate dependent process (stage III), during which the strain hardening rate continuously decreases due to
Preliminary results
Preliminary results are presented below for the model outlined in the previous sections. As already mentioned, the main unknowns reside in the evaluation of the mechanisms related to the cross-slip process. On the other hand, all athermal mechanisms follow scaling laws in terms of the (isotropic) elastic constants, and of the magnitude of the Burgers vector. The constitutive formulation is inserted into a crystal plasticity code [11], [14], which computes the tensile deformation of a single
Concluding discussion
Most models for strain hardening in monotonic deformation are based on the use of spatially averaged dislocation densities. There is no real justification to this assumption other than empirical. For instance, the Taylor relation holds rather well from very low stresses up to rather high stresses, well into stage III [16]. In the same way, it was shown in Section 4 that reasonably realistic predictions for the stress–strain behavior are obtained in monotonic deformation. Thus, everything is as
References (33)
- et al.
Mater. Sci. Eng. A
(2001) - et al.
Acta Mater.
(2004) - et al.
Acta Metall.
(1983) - et al.
Acta Mater.
(2001) - et al.
Acta Mater.
(2001) - et al.
J. Mech. Phys. Solids
(2006) - et al.
Prog. Mater. Sci.
(2003) - et al.
Acta Metall.
(1980) - et al.
Scripta Mater.
(2006)
Prog. Mater. Sci.
Scripta Metall.
Acta Metall.
Mater. Sci. Eng. A
Acta Mater.
Cited by (64)
Eutectic structure induced by ordering of solute atoms leading to strengthening effects
2024, Materials Science and Engineering: ARate controlling deformation mechanisms in SS316L stainless steel manufactured using laser powder bed fusion technique
2023, International Journal of PlasticityThe effect of small orientation deviation from [001] to [011] on high-temperature creep properties of nickel-based single crystal
2023, International Journal of PlasticityComposition design of a novel Ti-6Mo-3.5Cr-1Zr alloy with high-strength and ultrahigh-ductility
2022, Journal of Materials Science and TechnologyIdentification of dislocation reaction kinetics in complex dislocation networks for continuum modelling using data-driven methods
2022, Journal of the Mechanics and Physics of SolidsCitation Excerpt :Taking into account the microstructure evolution, different continuum approaches have been developed to represent the hardening behaviour. The models introduced include physical based formulations using the intrinsic material behaviour and phenomenological motivated formulations using extrinsic properties and observations, e.g. Kubin et al. (2008b), Andreoni and Yip (2020), Roters et al. (2019). One well-known crystal plasticity approach represents the coupling of the plastic slip-controlled dislocation multiplication of a Kocks–Mecking formulation (Kocks and Mecking, 2003) with a Taylor- or Franciosi-like yield stress (Taylor, 1934b,a; Franciosi et al., 1980) to consider the strain hardening, e.g. Ma et al. (2006a), Kubin et al. (2008a, b), Alankar et al. (2012a), Reuber et al. (2014), Roters et al. (2019).