FFT-based simulations of slip and kink bands formation in 3D polycrystals: Influence of strain gradient crystal plasticity
Graphical abstract
Introduction
The heterogeneous nature of plastic slip is the subject of a long-standing interest in the material science community, as a key to connect the micro-scale physics of dislocations to the macroscopical behavior of crystalline solids. Essential works have precisely described the typology of the various localization phenomena associated to this process (Jaoul, 1964, Neuhäuser, 1983). Among them, the common single-slip deformation occurring at incipient plasticity can yield two different types of localization bands, schematically represented in Fig. 1. Slip bands (SBs) (Fig. 1(a)) are sharp structures formed by intense dislocation glide on a few neighboring crystallographic planes. They are parallel to the active slip plane and are the main signature of plastic slip in crystals. On the other hand, Kink bands (KBs) (Fig. 1-(b)) form orthogonal to the glide direction. Their kinematics results in a strong lattice rotation within the band with respect to the surrounding crystal. They are bounded by high lattice curvature boundaries formed by geometrically necessary dislocation (GND) walls and have a much larger characteristic width than SB. In addition, as evidenced by the recent observations of di Gioacchino and da Fonseca (2015), their structure is characterized by a dense distribution of short SB along the slip plane normal direction. Less common than SBs, KBs are known since the pioneering work of Orowan (1942). They are reported mostly in strongly plastically anisotropic crystals such as ice (Mansuy et al., 2001, Montagnat et al., 2011, Wilson et al., 1986), Zinc (Gilman, 1954, Hagihara et al., 2016a), Magnesium (Hagihara et al., 2016b) or Titanium (Churchman, 1955). They have also been noticed in iron crystals (Jaoul, 1961) and as crack-tip deformation modes in notched single crystals (Crone and Shield, 2001, Kysar and Briant, 2002, Patil et al., 2009, Flouriot et al., 2003). Hence, they arise to accommodate strong strain incompatibilities when only one slip system can be activated.
SB and KB are material-based localization instabilities triggered by softening mechanisms at the dislocation scale (Estrin and Kubin, 1986, Brechet et al., 1993). In some materials, such as irradiated (Sharp, 1974, Onimus et al., 2004, Byun et al., 2006), hyper-quenched (Mori and Meshii, 1969, Bapna and Meshii, 1974) or hydrogen-charged metals (Aubert et al., 2012), these mechanisms are particularly strong. As a result, they induce the formation of very intense SBs, promoting failure and reducing metals ductility. This concern has motivated numerous modeling efforts, relying on dislocation-based softening classical crystal plasticity (CCP) models to simulate such plastic slip localization (Zhang et al., 2010, Barton et al., 2013, Patra and McDowell, 2016, Xiao et al., 2015, Erinosho and Dunne, 2015, Hure et al., 2016). These models effectively lead to the formation of SBs and KBs within polycrystalline simulations. However, SBs and KBs arise in this framework as two equivalent bifurcation modes of CCP equations. Asaro and Rice (1977) demonstrated this classical result by conducting a bifurcation analysis for an homogeneous crystal undergoing single slip. The validity of this SB/KB equivalence has recently been extended to polycrystalline simulations computed with most CCP models (Marano et al., 2019). It results in a purely structural selection mechanism between SB or KB. That is to say, between the SB and the KB, the one having the best suited geometrical orientation to locally accommodate the imposed strain forms, and no other parameter influences the selection. This leads systematically to comparable amounts of these two localization modes in the simulations, as in most crystal structure slip and kink planes are similarly distributed in the microstructure. Yet, in reality, SB are overwhelmingly predominant in real materials. Moreover, the respective formation processes of SBs and KBs strongly differ from a physical point a view. For instance, a KB formation requires the activation of much more dislocation sources than a SB formation. This strongly suggests to include other physical parameters in the modeling, to account for these differences. Consequently, CCP models cannot accurately predict intra-granular plastic slip localization (i.e. slip and kink banding).
It has been evidenced by means of Cosserat plasticity theory by Forest (1998), that accounting for an energetic contribution of lattice curvature allows to break the equivalence between the two modes. Likewise, strain gradient plasticity (SGP) theories that enrich the classical local formulation by accounting for Nye’s geometrically necessary dislocation (GND) density tensor, should provide a natural framework to address this shortcoming of CCP. Indeed, according to Nye’s formula (Nye, 1953), Nye’s tensor is linked to crystal lattice curvature, providing thus a physical measure allowing to distinguish SBs from KBs. Extensive developments in the past decades have spawned numerous formulations of these theories (Fleck and Hutchinson, 1997, Acharya and Bassani, 2000, Gurtin, 2002, Wulfinghoff et al., 2015, Cordero et al., 2010, Kaiser and Menzel, 2019) but have been mostly focused on modeling size effects associated to plasticity. A few studies have presented simulations of SB/KB formation in softening single crystal structures (Ling et al., 2018, Scherer et al., 2019) but they rely on a simplified theory accounting for the gradient of a cumulative scalar slip variable, designed to efficiently regularize both modes with a single length scale. In doing so, they naturally preserve the non-physical SB/KB equivalence. In fact, the promising potential of SGP models to improve the physical relevance of SBs and KBs formation modeling has not been explored yet, and therefore needs a detailed focus.
This paper is dedicated to this task. Using a simple model based on the SGP theory of Gurtin (2002), accounting for the full GND density tensor, its aim is to evidence, by comparison with CCP, the influence of SGP on the modeling of individual slip and kink bands, as well as on the development of complex localization patterns in 3D polycrystals. High resolution simulations are needed to analyze the influence of the microstructure on their formation mechanisms. To perform such simulations, a previous work conducted by the present authors in the framework of CCP models (Marano et al., 2019) has relied on the efficiency of massively parallel FFT-based solvers. In the present work, we take advantage of their recent extension to Gurtin’s theory by Lebensohn and Needleman (2016), and to the Nye’s tensor based Field Dislocation Mechanics (Berbenni et al., 2014, Brenner et al., 2014, Eloh et al., 2019) to apply the same strategy to SGP. However, these authors did not study the influence of their implementation on the numerical modeling of slip localization modes, especially with respect to the differentiation operators used to compute plastic strain gradients, and higher order interface conditions at grain boundaries. A detailed discussion of this matter is also provided in the present work.
This paper is organized as follows. Section 2 presents the kinematics, balance and constitutive equations of the SGP model and their application to individual SB/KB modeling. Section 3 discusses its implementation within the massively parallel FFT-based solver, AMITEX_FFTP, and associated numerical issues. Thereupon, Section 4, resp. Section 5 present the application of the model to slip localization simulations in single crystals, resp. in polycrystals. Finally, the influence of grain boundary interface conditions, the numerical implementation and the physics of slip/kink bands formation within SGP models are discussed in Section 6.
Section snippets
Notation
Throughout this paper, denote respectively scalars, vectors, second, third and fourth order tensors. denote the classical Cartesian basis of the euclidean space. denotes the time derivative of the variable . The following definition of the operator for second order tensors is adopted here: where is the Levi-Civita third order permutation tensor.
The model
As stated in the introduction, the present work is focused on the development of slip localization at incipient plasticity, which occurs at very low local strains. For this reason, our study is limited to a crystal plasticity theory in the context of small deformations. It relies on a plasticity theory that includes the curl of the plastic deformation tensor as non-local variable, initially proposed by Gurtin (2002) and Svendsen (2002), that we recall here. For the sake of brevity, the
Algorithm
The SGP model presented in Section 2.1 is solved numerically using a spectral homogenization scheme, based on the now well-established Fast Fourier Transform (FFT) method. Simulations are performed using the in-house FFT solver AMITEX_FFTP.3 First, its algorithm is briefly presented in the context of classical FFT-based methods, and details are provided about how the additional equations from the SGP model are integrated within this
Simulations of slip and kink bands in single crystals
In this section, the numerical implementation of the SGP model is applied to the simulation of slip localization in single crystals. These simulations will highlight the influence of the SGP framework on individual SB and KB formation. In order to avoid any confusion with the previous numerical validation case, we specify that hereafter all simulations will be based on the constitutive equations detailed in Section 2.1, in particular Eqs. (24), (25) for the equations governing the evolution of
Simulations of slip localization in 2D and 3D polycrystals
In this section, our massively parallel implementation of the SGP model to conduct high resolution is applied to carry out two and three-dimensional polycrystalline simulations. They are aimed at characterizing how the mechanisms of SB/KB competition and KB formation, observed in Section 4, are affected by the polycrystalline microstructure. Crystal structures of increasing complexity will be considered. The results will be compared to a CCP model to evidence the influence of gradient effects.
Grain boundary higher order interface conditions
Here we discuss the ability of our numerical implementation to account for various boundary conditions and their influence on GB induced hardening, as well as their influence on the simulated slip localization band networks. All the results presented and discussed here are obtained with a 64 grains two-dimensional polycrystalline microstructure similar to the one presented in Section 5.2 (1 in-plane slip system per grain), and with the materials coefficients listed in Table 2.
To begin with, the
Conclusions and future prospects
The purpose of this work was to explore the potential of higher order plasticity theory for the modeling of plastic slip localization in polycrystals. To this end, a simple softening gradient crystal plasticity model based on Gurtin’s theory has been applied to the modeling of single slip localization modes. This theory is based on the GND density tensor, a physical measure discriminating between the two single slip localization modes: as shown, it has a non-zero value inside kink bands, and
CRediT authorship contribution statement
Aldo Marano: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Data curation, Writing - original draft, Writing - review & editing, Visualization. Lionel Gélébart: Conceptualization, Methodology, Software, Resources, Data curation, Writing - review & editing. Samuel Forest: Conceptualization, Methodology, Formal analysis, Writing - review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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