A non-local crystal plasticity constitutive model for hexagonal close-packed polycrystals
Introduction
Understanding the deformation mechanism of hexagonal closed-packed (HCP) polycrystals has been in the center of many studies for decades (Abdolvand et al., 2020; Agnew, 2005; Agnew et al., 2018; Wan and Dunne, 2020; Wen et al., 2018). These polycrystals are being used in various industrial sectors. For example, zirconium and its alloys are used for manufacturing the core components of nuclear reactors due to their low neutron absorption cross section and good corrosion resistance. Magnesium and titanium alloys are used in automotive and aerospace industries due to their low density and good mechanical properties. During fabrication or in service, these materials may undergo macroscopic or localized microscopic plastic deformation. Such localized deformation zones can become susceptible sites for crack nucleation or even accelerated corrosion (Platt et al, 2015, 2016; Wang et al, 2019, 2020). This paper focuses on the development of a strain-gradient, also known as non-local, crystal plasticity finite element (CPFE) model to study the distribution of localized stress or dislocation hotspots that drives accelerated degradation mechanisms while replicating the macroscopic deformation behaviour of HCP polycrystals.
Various numerical approaches have been used to investigate the micromechanics of plastic deformation. For example, discrete dislocation dynamics (DDD) simulations were used to identify the coefficients of latent hardening in pure magnesium (Capolungo, 2011). Such modeling scheme helps understand how the interactions among dislocations of different slip systems affect the macroscopic hardening observed in magnesium alloys (Bertin et al., 2014). Further, Tummala et al. (2018) used DDD modeling to study the stress fields around hydrides in zirconium. It is important to understand how the localized stress fields around zirconium hydrides evolve since hydrogen embrittlement significantly affects the facture toughness of the zirconium alloys used in nuclear industry. Although DDD models take the interaction between individual dislocations into account, they are numerically costly and do not represent the “real” time scale over which a given phenomenon occurs. As a meso-scale modeling scheme, crystal plasticity is used to study heterogeneous deformation of individual or clusters of grains in “real time” by taking into account the effects of plastic slip that occurs on active slip systems (Han et al., 2020; Kasemer and Dawson, 2020; Lu et al., 2019). Crystal plasticity has been implemented in different frameworks, e.g., self-consistent (Mareau and Daymond, 2010), fast Fourier transform (FFT) (Lebensohn and Needleman, 2016; Wang et al., 2013), and finite element (FE) (Abdolvand, 2019; Patra and Mcdowell, 2016). In the self-consistent framework, each grain interacts with a homogenous medium that represents the average properties of the aggregate except for the grain that is investigated. Although it is naturally fast, it is not possible to study the “true” interaction among neighbouring grains in the self-consistent framework. In the FFT or FE frameworks, however, such interactions can be investigated and recently, there has been promising advances in the field (Berbenni et al., 2020; Eghtesad et al., 2020; Liu et al., 2020; Mareau and Daymond, 2016).
It has been shown that the interaction among the neighbouring grains can have significant effects on the evolution of stress within individual grains of HCP polycrystals (Abdolvand et al., 2018a, b). Such interactions can lead to the development of localized stress or dislocation fields that control the nucleation of cracks in materials (Prastiti et al., 2020). Depending on the crystal orientation, slip or twinning are generally active in such localized stress fields. For slip to occur, dislocations should overcome both short-range and long-range obstacles. The primary short-range obstacles are generally assumed to be the other dislocations that intersect the slip plane and can potentially impede the movement of dislocations on the same plane. The long-range obstacles, on the other hand, may include the elastic stress fields due to grain boundaries or far field dislocations and defects. While the short-range obstacles can mainly be overcome by thermal activation, the long-range obstacles are generally independent from temperature and can be overcome by increasing the resolved shear stress (RSS) that acts on the slip system (Nemat-nasser et al., 1998). Many studies have attempted to incorporate the effects of both short- and long-range obstacles in material hardening models. For example, Lu et al. (2019) assumed that the dislocation movement is a function of the resistance induced by both short- and long-range obstacles. It was assumed that the threshold for activating dislocation glide depends on the long-range obstacles which in turn depends on the evolving dislocation density. The resistance due to short-range obstacles was assumed to originate from Peierls barrier, solid solution atoms, or other point defects in the material. Evers et al. (2004) assumed that the resistance induced by the long-range obstacles is a function of the spatial gradient of geometrically necessary dislocations (GND) densities, while that of short-range obstacles was assumed to be dependent on both total GND and statistically stored dislocation (SSD) densities. Therefore, it was assumed that SSDs do not have any long-range effects, while the evolution of both SSDs and GNDs during crystallographic slip equally increase the short-range interactions (Evers et al., 2004; Kocks et al., 1975). In the conventional form of crystal plasticity, no differentiation is made between the two types of obstacles. That is, the critical resolved shear stress (CRSS) required for dislocation movement solely depends on the state of deformation in the current calculation point. However, in the non-local crystal plasticity framework, dislocation densities are calculated based on the gradients of the plastic strain; such gradients depend on the deformation of the neighbouring regions and hence, both short- and long-range effects can be introduced (Arsenlis, 2001; Arsenlis et al., 2004; Arsenlis and Parks, 2002; Ashby, 1970; Gurtin, 2002; Taylor, 1934).
There are two forms of strain gradient plasticity theories, i.e., the lower-order and higher-order theories (Niordson and Hutchinson, 2003). In the lower-order form (Arsenlis and parks, 1999; Bassani, 2001; Busso et al., 2000), the effects of strain gradients are only included in the materials hardening laws, with no subsequent adjustment made to the force-equilibrium equations or boundary conditions. For example, in the crystal plasticity framework, this is mainly done by introducing GND densities to materials hardening equations and correlating GNDs with the gradient of the plastic strain. While it is the simpler form between the two, the drawback of this approach is that it may result in unrealistic higher strain gradients and formation of unusual localized deformation fields (Niordson and Hutchinson, 2003). In the higher-order form (De Borst and Mühlhaus, 1992; Gao et al., 1999; Gurtin, 2000; Yun et al., 2005), both materials hardening law as well as force-equilibrium equations are modified by adding an extra term that represents localized micro-scale stresses. The consideration of micro-scale stresses results in the modification of virtual work equations which in return results in an extra term proportional to the second order gradient of the plastic strain. In order to solve these differential equations, additional boundary conditions are generally required.
Diffraction based experimental techniques are mainly used for measuring internal or localized stress fields as well as the density of dislocations. While the measurement of SSDs is not straightforward, various diffraction techniques are used for measuring GNDs. For example, micro X-ray diffraction is used for measuring the density of dislocation in the vicinity of slip bands or for determining slip activity in HCP polycrystals (Guo et al., 2020; Unga et al., 2007). This method provides a three-dimensional view of the GNDs as well as stress fields, but for a few grains. With high angular resolution electron back scatter diffraction (HR-EBSD) technique it is, however, possible to measure stresses and GND densities in many grains, yet close to the sample surface. In this technique, the Kikuchi diffraction patterns measured within a grain are cross correlated with respect to a pattern collected at a reference point within the same grain. The reference point is normally chosen to be “far” from grain boundaries where the variation of stress and orientation is assumed to be minimum. The displacement gradient can subsequently be calculated and used to extract the “relative” elastic strain and lattice rotation, and by assuming that the stress normal to surface is zero, it is possible to calculate the “relative” stress tensor (Troost et al., 1993; Wilkinson, 1996; Wilkinson et al., 2006). Since the “relative” elastic lattice rotations are known, it is possible to calculate GND densities using Nye tensor (Britton and Wilkinson, 2012). This method has been successfully used for measuring stresses and GND densities at the vicinity of slip bands and twins in HCP polycrystals (Andani et al., 2020; Guo et al, 2014, 2017). For measuring internal strains, as oppose to surface strains, neutron or X-ray diffraction can be used (Long et al., 2016; Pokharel et al., 2019; Zhang et al., 2019). Lattice strains are the elastic strains in the direction of the scattering vector and in the family of grains that can diffract the incident X-ray or neutron beam. The deflections observed in lattice strains correspond to activation of various slip or twinning modes of deformation. This unique property of lattice strains can be used to study deformation mechanism of polycrystals.
In this paper, a non-local CPFE model is developed to study deformation mechanisms of HCP polycrystals. Results from a calibrated conventional CPFE model are used to firstly calibrate the single crystal properties of the non-local model. The non-local model is subsequently validated in two steps; in the first step, the development of internal lattice strains in Zircaloy-2 is simulated and compared to those measured using the neutron diffraction technique. In the second step, the distribution of localized elastic lattice rotation, dislocation density, and stress for individual zirconium crystals from the model are compared to those measured using HR-EBSD. Attention is given to deformation fields measured and simulated in the vicinity of twins. In addition, results from the non-local model are compared to those from the conventional method. The aim of this paper is to understand the benefits of using non-local models over conventional CPFE models and highlight their differences.
Section snippets
Experiments
The numerical results are compared to the previously published data measured by the authors using neutron diffraction and HR-EBSD methods. Details of each experiment are provided in (Abdolvand et al., 2011) and (Abdolvand and Wilkinson, 2016), here just a brief description is provided. For the neutron diffraction experiment, samples were cut from a hot rolled Zircaloy-2 plate with the texture shown in Fig. 1a. Since most of the crystals c-axis are oriented towards the normal direction (ND),
Modeling
The constitutive equations used for both conventional and non-local crystal plasticity models in the finite element framework are described in this section. These equations are implemented into a user material (UMAT) subroutine originally developed by (Abdolvand and Wilkinson, 2016). The equations that govern deformation by slip are firstly described which are followed by the assumptions made to simulate deformation twinning.
Determination of the single crystal parameters
Since the performance of the non-local model will be compared against that of the conventional model, the non-local model is calibrated such that it reproduces identical stress-strain curves to those from the conventional model for zirconium single crystals. The single crystal parameters for the conventional model were previously calibrated using a comprehensive data set for lattice strains measured during an in-situ neutron diffraction experiment conducted on Zircaloy-2 (Abdolvand et al., 2011
Discussion
In previous sections, it was shown that the calculated distribution of stress and elastic lattice rotation fields were similar from the conventional and non-local models. However, the predicted values from the two models are different such that the non-local model predicts higher localized values. The development of such localized deformation fields can significantly be affected by the size of the grain or by the sharpness of the plastic strain gradients. Such sharp gradients should be
Conclusion
A non-local crystal plasticity finite element model is developed, and its performance is critically examined. The results from the non-local model is compared against those from a conventional CPFE model as well as those from two types of diffraction experiments. It is shown that:
- (1)
the evolution of internal lattice strains measured using neutron diffraction is well captured by both non-local and conventional CPFE models. The recorded macroscopic stress-strain curves under two different loading
Author statement
OS updated the CPFE model by including the strain-gradient effects and conducted all simulations. OS prepared the first draft of the manuscript and all figures presented. HA designed and planned the research, supervised, and contributed to the writing of the manuscript as well as the interpretation of the results.
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.
Acknowledgment
This work was supported by a Discovery Grant (#RGPIN/04969–2017) from the Natural Sciences and Engineering Research Council of Canada (NSERC).
References (86)
Progressive modelling and experimentation of hydrogen diffusion and precipitation in anisotropic polycrystals
Int. J. Plast.
(2019)- et al.
Multi-scale modeling and experimental study of twin inception and propagation in hexagonal close-packed materials using a crystal plasticity finite element approach; Part II: local behavior
J. Mech. Phys. Solid.
(2013) - et al.
Incorporation of twinning into a crystal plasticity finite element model: evolution of lattice strains and texture in Zircaloy-2
Int. J. Plast.
(2011) - et al.
On the nucleation of deformation twins at the early stages of plasticity
Acta Mater.
(2020) - et al.
On the effects of reorientation and shear transfer during twin formation: comparison between high resolution electron backscatter diffraction experiments and a crystal plasticity finite element model
Int. J. Plast.
(2016) - et al.
On the state of deformation in a polycrystalline material in threedimension:Elastic strains, lattice rotations, and deformation mechanisms
Int. J. Plast.
(2018) - et al.
Lattice incompatibility and a gradient theory of crystal plasticity
J. Mech. Phys. Solid.
(2000) Plastic anisotropy and the role of non-basal slip in magnesium alloy AZ31B
Int. J. Plast.
(2005)- et al.
In-situ neutron diffraction of a quasicrystal-containing Mg alloy interpreted using a new polycrystal plasticity model of hardening due to { 10 . 2 } tensile twinning
Int. J. Plast.
(2018) - et al.
Measurement and modeling of micro residual stresses in zirconium crystals in three dimension
J. Mech. Phys. Solid.
(2020)
Crystallographic aspects of geometrically necessary and statistically stored dislocation density
Acta Mater.
Modeling the evolution of crystallographic dislocation density in crystal plasticity
J. Mech. Phys. Solid.
On the evolution of crystallographic dislocation density in non-homogeneously deforming crystals
J. Mech. Phys. Solid.
Overview No. 42 Texture development and strain hardening in rate dependant polycrystals
Acta Metall.
Incompatibility and a simple gradient theory of plasticity
J. Mech. Phys. Solid.
Influence of critical resolved shear stress ratios on the response of a commercially pure titanium oligocrystal: crystal plasticity simulations and experiment
Int. J. Plast.
On the strength of dislocation interactions and their effect on latent hardening in pure Magnesium
Int. J. Plast.
High resolution electron backscatter diffraction measurements of elastic strain variations in the presence of larger lattice rotations
Ultramicroscopy
Gradient-dependent deformation of two-phase single crystals
J. Mech. Phys. Solid.
Dislocation junction formation and strength in magnesium
Acta Mater.
On the characterization of geometrically necessary dislocations in finite plasticity
J. Mech. Phys. Solid.
A crystal plasticity FE model for deformation with twin nucleation in magnesium alloys
Int. J. Plast.
A study of microstructural length scale effects on the behaviour of FCC polycrystals using strain gradient concepts
Int. J. Plast.
Lengthscale-dependent, elastically anisotropic, physically-based hcp crystal plasticity: application to cold-dwell fatigue in Ti alloys
Int. J. Plast.
A multi-GPU implementation of a full-field crystal plasticity solver for efficient modeling of high-resolution microstructures
Comput. Phys. Commun.
Non-local crystal plasticity model with intrinsic SSD and GND effects
J. Mech. Phys. Solid.
Mechanism-based strain gradient plasticity - I
Theory. J. Mech. Phys. Solids
Growth of { 1122 } twins in titanium : a combined experimental and modelling investigation of the local state of deformation
Acta Mater.
Slip band – grain boundary interactions in commercial-purity titanium
Acta Mater.
Dislocation density distribution at slip band-grain boundary intersections
Acta Mater.
A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations
J. Mech. Phys. Solid.
On the plasticity of single crystals: free energy, microforces, plastic-strain gradients
J. Mech. Phys. Solid.
Microstructure-based multiscale modeling of large strain plastic deformation by coupling a full-field crystal plasticity-spectral solver with an implicit finite element solver
Int. J. Plast.
A finite element methodology to incorporate kinematic activation of discrete deformation twins in a crystal plasticity framework
Comput. Methods Appl. Mech. Eng.
Physics and phenomenology of strain hardening : the FCC case
Prog. Mater. Sci.
Hydrogen-assisted failure in a twinning-induced plasticity steel studied under in situ hydrogen charging by electron channeling contrast imaging
Acta Mater.
Numerical implementation of non-local polycrystal plasticity using fast Fourier transforms
J. Mech. Phys. Solid.
GND accumulation in bi-crystal deformation: crystal plasticity analysis and comparison with experiments
Int. J. Mech. Sci.
Grain-scale experimental validation of crystal plasticity finite element simulations of tantalum oligocrystals
Int. J. Plast.
Integration of phase-field model and crystal plasticity for the prediction of process-structure-property relation of additively manufactured metallic materials
Int. J. Plast.
Effect of neutron irradiation on deformation mechanisms operating during tensile testing of Zr-2.5Nb
Acta Mater.
Dislocation mechanism based size-dependent crystal plasticity modeling and simulation of gradient nano-grained copper
Int. J. Plast.
A dislocation density based constitutive model for crystal plasticity FEM including geometrically necessary dislocations
Acta Mater.
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