Simulation of bicrystal deformation including grain boundary effects: Atomistic computations and crystal plasticity finite element analysis

https://doi.org/10.1016/j.commatsci.2020.109641Get rights and content

Highlights

Abstract

Grain boundaries play a pivotal role in dictating the deformation behavior of crystalline materials. Modeling their effects within the framework of physically based crystal plasticity approaches demands a multiscale description of the underlying phenomena. This work puts-forth a two-scale atomistic to crystal plasticity approach for determining the deformation behavior of a ductile face-centered cubic material. The approach uses atomistic computations to quantify the activation energies for nucleation of partial dislocations from a grain boundary under tensile loading. To this end, embedded-atom method based atomistic simulations involving nudged elastic band method are utilized to compute stress-dependent activation parameters of the grain boundary. The extracted parameters are then used as input to the flow rule of crystal plasticity at higher length scale, which is based on transition state theory. At this scale, the grain boundaries are explicitly accounted for by assigning a finite thickness and are differentiated from their bulk counterparts by prescribing distinct flow parameters extracted from atomistic simulations. The predictive capabilities of the proposed methodology are then assessed by performing numerical simulations on uniaxial tensile behavior of Ni bicrystal and validating them from the published experiments and computations available in the literature. This research paradigm can be pushed forward by incorporating more complex grain boundary behaviors at higher length scales while preserving the richness provided by atomic scales.

Introduction

Polycrystalline materials- an aggregate of single crystals have their deformation behavior derived from the constituent interfaces, like grain boundaries (GBs) and phase boundaries. This fact has inspired everlasting efforts to investigate the mechanical response of crystalline materials by first probing the behavior of single crystals [1], [2], [3], [4], [5] and then some specific GBs in multicrystals [6], [7], [8], [9], [10]. As a result, the role of GBs in governing the plastic deformation behavior has been established in a variety of contexts [11], [12], [13], [14]. Under the influence of external loading, the GBs can act as important dislocation sources [15], [16], [17] and sinks [12], [18]. The crystallography of GB and the defect content influence the ability of a boundary to absorb/emit dislocations (see, e.g., the extensive reviews by Dillon et al. [19] and Beyerlein et al. [20]). As GBs play a crucial role in dictating the plastic evolution of a real aggregate, the understanding of their structure-property relationship is a key to materials design.

Modeling the effect of GBs on mechanical response of crystalline materials is a challenging problem given the complexity of defect evolution, which involves dislocation annihilation, pile-up, nucleation and multiplication [20], [21]. Moreover, collective behavior of GBs and their interplay with other microstructural features like cracks, voids etc. leads to interactions spanning a range of length and time scales [8], [22], [23], [24], [25]. Mere statistical averaging of such a huge volume of events for use in simulations at higher length scales may be erroneous. This calls for detailed investigation methods that explicitly account for the GB effects on plasticity.

To this end, extensive experimental investigations have been devoted to capture the role of GBs in governing plastic deformation (see the review by Kacher et al. [26]) at the sub-micron scales. In addition, constitutive modeling based on crystal plasticity (CP) finite element method [27], [28] is another important tool that has been employed to study GB effects and connect the related deformation mechanisms to an aggregate polycrystal. Although earlier studies did not take into account the GB effects in their CP formulation [29], [30], it is now well understood that inclusion of GBs in the micromechanical CP based approaches is essential to improve the accuracy of computations. However, accounting for an accurate and physical description of GBs in CP modeling is a major challenge. This is addressed by the present work, which attempts to create a physical basis for the constitutive equations and quantifies the GB activation parameters for input to CP simulations.

One of the pertinent works incorporating the GB effects on plasticity was presented by Shu and Fleck [31], who incorporated additional hardening in the grains due to geometrically necessary dislocations to predict the grain size dependence of yield stress in metallic bicrystals under shear loading. Aifantis and Wills [32] proposed a formulation to explicitly account for interface effects by assigning them a separate interface energy, which was later deployed to interpret size effects in metallic thin films [33]. Beers et al. [34] assigned a representative GB energy in their GB formulation to account for the magnitude and direction of residual defect populations from both the adjoining grains. Similar other GB formulations within different plasticity frameworks are also available in the literature [28], [35].

Most of the aforementioned modeling frameworks treat GBs in a simplified manner with zero thickness, which makes it difficult to assign them different constitutive properties with respect to the adjoining crystals. Although cohesive elements at the interface allow us to define distinct material properties to the GBs [36], [37], [38], [39], the use of zero thickness GBs lack physical interpretation. With the recent advances in numerical modeling techniques and computational power, these simplified GB treatments are no longer a requisite for computational purposes.

In view of this, intensive CP simulations have been directed at modeling the finite thickness GBs in crystalline materials. For instance, Liu and co-workers [40], [41] simulated finite thickness (=7 nm) GBs in a Voronoi tessellation based model of ultrafine grained Ti with average grain size of 150 nm. Finite thickness GBs were also employed by Biglari and Nikbin [42], [43] to simulate the creep and fracture behavior of steel. With grain sizes varying from 50 μm to 150 μm, the authors employed 1 μm thick GBs in their polycrystalline models while adhering to the microstructure observed in electron micrographs. A similar study was performed by Zhao et al. [44] to predict creep-fatigue crack growth in 9Cr-1Mo steel using 1 μm thick GBs in their finite element model. Gottschalk et al. [45] formulated a gradient plasticity GB model, and illustrated the predictive capability of their approach by running numerical examples on finite thickness GBs in FCC crystal structure. The authors employed 10 μm thick GBs in their finite element meshes of bicrystals and polycrystals. Work of Shi and Zikry [46] provides additional motivation, who employed GB width equivalent to 10% of the grain size in Cu bicrystals containing various coincident site lattice boundaries to account for various dislocation-GB interactions in their finite element framework. The authors demonstrated the propensity of such schemes to quantify GB mediated processes of absorption, emission and transmission, which are principal in governing material response.

In all the aforementioned works, the GBs were either assigned artificial damage [42], [43] or slip resistances [45], [44] to emulate a micro-free (zero density of geometrically necessary dislocations at the boundary) or micro–hard (zero plastic strain at the boundary) GB, or were only allowed to differ from the grain interiors by assigning a different dislocation density [40], [41]. Although such treatments enjoy some ease of numerical predictions, they can be reinforced by a more physical basis of the functional form and constitutive parameters for GBs that enter the CP formulation. In this regard, bottom-up molecular dynamics (MD) simulations and coarse-grained approaches can be used to extract parameters and pass-on as input to CP simulations at higher length scales.

Atomistic simulations of crystalline materials have been extensively used to study the isolated events of GB-dislocation interactions or dislocation nucleation from GBs (refer to the excellent reviews by Zhang et al. [47] and Spearot and Sangid [7]). Very recently, we also have investigated dislocation nucleation from a myriad of GBs containing a pre-existing void in FCC Cu and Ni using MD [6]. Simulations have also been utilized to reveal differences in the deformation behavior (plasticity [48], [49], phase transformation [49], shock loading [50], crack growth [51] etc.) of various materials with FCC, BCC and HCP crystal structure. Yet, MD simulations are limited by their spatial and temporal resolutions which deters them to capture the long range stress fields of defects or even some statistical trends in CP. Hence, state-of-the-art computational modeling approaches rely on multi-scale techniques, for example, of hierarchical nature [52], wherein MD, dislocation dynamics and CP are bridged in a sequential order. In this regard, works of Groh et al. [53] and Chandra et al. [54] have already proven their utility by predicting the mechanical response of single crystals and polycrystals, albeit without taking into account the effect of GBs. However, we are still struggling to connect the disparate scales in the case of GBs due to our inability to convincingly extrapolate the critical information on boundaries from the MD simulation data. In order to have a more physical CP simulation, one has to determine the GB properties from lower length scales which eventually leads to predictions of mechanical behavior like plasticity evolution and work hardening at the level of CP.

To improve the versatility of the bottom-up approaches of hierarchical nature, coarse-grained modeling approaches can be deployed to inform higher length scales without sacrificing the richness provided by MD simulations at the nanoscale [52]. One such technique is the nudged elastic band (NEB) method [55] which takes a series of system configurations as input and determines a minimum energy path between them. A series of configurations are created for the system and are relaxed in the presence of physical and spring forces. While the former drives the system into a relaxed (minimum energy) state, fictitious spring forces ensure even spacing across the potential energy surface. These methods have been widely used in the past in conjunction with atomistic simulations to compute activation parameters for various physical phenomena like dislocation cross-slip [56], point defect migration [57] and kink nucleation along a screw dislocation [58].

Herein, we present an efficient predictive modeling methodology to simulate the deformation behavior of metallic bicrystals of face-centered cubic crystal structure. Atomistic simulations and NEB method are exploited to compute the activation parameters for dislocation nucleation from a GB in Ni bicrystal. The computed parameters are passed on to the flow rule of a higher length scale at the level of CP. Simulations at mesoscale are then compared with experimental results for the uniaxial tensile deformation of the Ni bicrystal [59], thereby validating the methodology. The two-scale modeling approach includes the effects of GB properties at nano- and meso-scales, without unduly relying on non-physical parameters or arbitrary constants for GBs. Such a modeling approach can pave way to a completely new level of materials design, wherein, in addition to single crystal properties, GB parameters can also be quantified within the framework of a physically justified hierarchical multiscale modeling scheme.

This work is presented as follows: Section 2 describes the atomistic and CP modeling methodologies. Section 3 presents the results and also investigates the influence of various material parameters of flow and hardening rules of CP model on the numerical results. Section 4 discusses our approach and Section 5 concludes the paper with a few remarks.

Section snippets

Crystal plasticity framework and modeling

The deformation gradient, F, is decomposed multiplicatively into its elastic (Fe) and plastic (Fp) parts as [60], [61]F=FeFpwhere Fe represents elastic part of deformation and Fp represents plastic deformation along discrete crystallographic planes. In CP, the plastic deformation is assumed to take place through glide of dislocations, hence the plastic part of the deformation gradient is expressed in terms of the combination of slip rate, γ̇α, and Schmid tensor asLp=ḞpFp-1=αγ̇αm0αn0αwhere Lp

Interface structure and energy

Local atomic structure of the GB obtained after static relaxation is displayed in Fig. 3, wherein the blue atoms represent near-perfect configuration and green/red color atoms represent partial dislocations and stacking faults (Fig. 3(a)). In order to make the interface structure more discernible, the Dislocation Extraction Algorithm (DXA) [75] is utilized to identify the dislocation content of the interface. The algorithm extracts perfect and partial dislocations and classifies them according

Discussion

Controlled experiments of Ohashi et al. [59] on Ni bicrystals are utilized to assess the predictive capabilities of our methodology, which involves explicit GB modeling within the framework of CP while taking GB constitutive parameters from the atomic scales of inception. To the best of authors’ knowledge, such a modeling attempt is not available in the literature. This provides a direct link between the relevant atomic scale processes (in this case, dislocation nucleation from GB) and its

Concluding remarks

A CP finite element meshing configuration incorporating finite thickness GB in a Ni bicrystal is adopted to predict the uniaxial tensile behavior the material. For the GB elements, the stress dependent activation parameters of flow rule of the CP formulation are calibrated from the NEB method using atomistic simulations. The proposed methodology is in full fidelity with the atomic scale mechanisms of dislocation nucleation from the GB, thereby providing an ideal analysis platform to predict the

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.

CRediT authorship contribution statement

S. Chandra: Data curation, Formal analysis, Investigation, Validation, Visualization, Writing - original draft. M.K. Samal: Conceptualization, Methodology, Investigation, Data curation, Software, Supervision, Writing - review & editing. Rajeev Kapoor: Writing - review & editing. V.M. Chavan: Project administration, Resources, Writing - review & editing.

References (91)

  • E.S. Venkatesh et al.

    The influence of grain boundary ledge density on the flow stress in Nickel

    Mater. Sci. Eng.

    (1978)
  • M.A. Tschopp et al.

    Atomistic simulations of tension-compression asymmetry in dislocation nucleation for copper grain boundaries

    Comput. Mater. Sci.

    (2008)
  • R.F. Zhang et al.

    Manipulating dislocation nucleation and shear resistance of bimetal interfaces by atomic steps

    Acta Mater.

    (2016)
  • R.J. Dikken et al.

    Impingement of edge dislocations on atomically rough contacts

    Comput. Mater. Sci.

    (2017)
  • S.J. Dillon et al.

    The importance of grain boundary complexions in affecting physical properties of polycrystals

    Curr. Opin. Solid State Mater. Sci.

    (2016)
  • I.J. Beyerlein et al.

    Defect-interface interactions

    Prog. Mater Sci.

    (2015)
  • D.W. Adams et al.

    Atomistic survey of grain boundary-dislocation interactions in FCC nickel

    Comput. Mater. Sci.

    (2019)
  • A. Kedharnath et al.

    Molecular dynamics simulation of the interaction of a nano-scale crackwith grain boundaries in α-Fe

    Comput. Mater. Sci.

    (2017)
  • L. Zhang et al.

    Interaction between nano-voids and migrating grain boundary by molecular dynamics simulation

    Acta Mater.

    (2019)
  • S. Rezaei et al.

    Atomistically motivated interface model to account for coupled plasticity and damage at grain boundaries

    J. Mech. Phys. Solids

    (2019)
  • N.C. Admal et al.

    A unified framework for polycrystal plasticity with grain boundary evolution

    Int. J. Plast.

    (2018)
  • J. Kacher et al.

    Dislocation interactions with grain boundaries

    Curr. Opin. Solid State Mater. Sci.

    (2014)
  • F. Roters et al.

    Overview of constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finite-element modeling: theory, experiments, applications

    Acta Mater.

    (2010)
  • A. Alipour et al.

    A grain boundary model for gradient-extended geometrically nonlinear crystal plasticity: Theory and numerics

    Int. J. Plast.

    (2019)
  • V. Hasija et al.

    Deformation and creep modeling in polycrystalline Ti-6Al alloys

    Acta Mater.

    (2003)
  • M. Schurig et al.

    Computation of deformation textures in copper with equilibrium in small grain neighbourhoods

    Comput. Mater. Sci.

    (2002)
  • J.Y. Shu et al.

    Strain gradient crystal plasticity: size-dependent deformation of bicrystals

    J. Mech. Phys. Solids

    (1999)
  • K.E. Aifantis et al.

    The role of interfaces in enhancing the yield strength of composites and polycrystals

    J. Mech. Phys. Solids

    (2005)
  • P.R.M. van Beers et al.

    Grain boundary interface mechanics in strain gradient crystal plasticity

    J. Mech. Phys. Solids

    (2013)
  • P.R.M. van Beers et al.

    Defect redistribution within a continuum grain boundary plasticity model

    J. Mech. Phys. Solids

    (2015)
  • P. Zhang et al.

    A controlled poisson Voronoi tessellation for grain and cohesive boundary generation applied to crystal plasticity analysis

    Comput. Mater. Sci.

    (2012)
  • M. Paggi et al.

    A numerical investigation of the interplay between cohesive cracking and plasticity in polycrystalline materials

    Comput. Mater. Sci.

    (2013)
  • M. Dakshinamurthy et al.

    Crack propagation in TRIP assisted steels modeled by crystal plasticity and cohesive zone method

    Theor. Appl. Fract. Mech.

    (2018)
  • E. Tarleton

    Incorporating hydrogen in mesoscale models

    Comput. Mater. Sci.

    (2019)
  • H. Liu et al.

    Non-equilibrium grain boundaries in titanium nanostructured by severe plastic deformation: Computational study of sources of material strengthening

    Comput. Mater. Sci.

    (2014)
  • H. Liu et al.

    Gradient ultrafine-grained titanium: Computational study of mechanical and damage behavior

    Acta Mater.

    (2014)
  • F. Biglari et al.

    Environmental creep intergranular damage and multisite crack evolution model for engineering alloys

    Comput. Mater. Sci.

    (2014)
  • F. Biglari et al.

    A diffusion driven carburisation combined with a multiaxial continuum creep model to predict random multiple cracking in engineering alloys

    Engg. Fract. Mech.

    (2015)
  • L. Zhao et al.

    Predicting failure modes in creep and creep-fatigue crack growth using a random grain/grain boundary idealised microstructure meshing system

    Mater. Sci. Eng. A

    (2017)
  • D. Gottschalk et al.

    Computational and theoretical aspects of a grain-boundary model that accounts for grain misorientation and grain-boundary orientation

    Comput. Mater. Sci.

    (2016)
  • L. Zhang et al.

    A review on atomistic simulation of grain boundary behaviors in face-centered cubic metals

    Comput. Mater. Sci.

    (2016)
  • S. Rawat et al.

    Molecular dynamics investigation of c-axis deformation of single crystal Ti under uniaxial stress conditions: evolution of compression twinning and dislocations

    Comput. Mater. Sci.

    (2018)
  • N. Amadou et al.

    Effects of orientation, lattice defects and temperature on plasticity and phase transition in ramp-compressed single crystal iron

    Comput. Mater. Sci.

    (2020)
  • D. Singh et al.

    Atomistic simulations to study crack tip behaviour in single crystal of bcc niobium and hcp zirconium

    Curr. Appl. Phys.

    (2019)
  • D.L. McDowell

    A perspective on trends in multiscale plasticity

    Int. J. Plast.

    (2010)
  • Cited by (6)

    • Quantification of α phase strengthening in titanium alloys: Crystal plasticity model incorporating α/β heterointerfaces

      2022, International Journal of Plasticity
      Citation Excerpt :

      The strategy of coupling meso- or nano-scale microstructure features with the CPFE method can be roughly divided into two typical approaches. One is to explicitly model the structures, which have different constitutive behavior from the grain interior or the matrix phase, into the finite element models, such as the grain/phase boundaries with a certain thickness (Chandra et al., 2020; Zheng et al., 2022), and linear elastic nanoscale precipitates (Li et al., 2017). The other typical way is to incorporate the effects of meso- or nano-scale structures into the constitutive model (Benedetti et al., 2016; Mayeur et al., 2015).

    • An atomistic analysis of the effect of grain boundary and the associated deformation mechanisms during plain strain compression of a Cu bicrystal

      2022, Computational Materials Science
      Citation Excerpt :

      Such calculations are rooted within the framework of crystal plasticity (CP) formalisms [15,16] which inherently provide a physical-based representation of the plastic deformation behavior at the mesoscale. The interface mechanics in such formulations is captured by either invoking the concepts of strain gradient plasticity [15,17] or by explicit modeling of GBs using cohesive-zone methods [18] and three-dimensional interface representations utilizing discrete finite elements [19,20]. In any case, to be able to capture the deformation behavior of polycrystalline materials with high fidelity, it is imperative to account for GB phenomena in an efficient way by formulating local rules based on information gleaned from lower scale atomistic simulations.

    • Employing molecular dynamics to shed light on the microstructural origins of the Taylor-Quinney coefficient

      2021, Acta Materialia
      Citation Excerpt :

      More specifically, given that a proper interatomic potential is provided, the evolution of a microstructure is naturally captured in the simulations without any predefined conditions. For instance, MD simulations are employed to measure the mobility of dislocations [39–42], grain boundary migration [43–45], dislocation nucleation from grain boundaries [46], dislocations interaction with grain boundaries [47] and deformation of nanopolycrystalline structures [48–51]. For this reason, MD simulations can provide the bridge to find the relation between the evolution of the microstructure and heat generated/stored during plastic deformation.

    • Harnessing atomistic simulations to quantify activation parameters for dislocation nucleation from a grain boundary in Nickel

      2020, Physics Letters, Section A: General, Atomic and Solid State Physics
      Citation Excerpt :

      Therefore, if one is interested in explicit modeling of GBs at the level of CP, one could resort to a modeling strategy wherein the kinetic activation parameters for any particular (pristine or damaged) GB (∑3 twin, for example) can be extracted from lower length scale MD simulations. Very recently, such motivation urged us to quantify the mechanical response of bicrystal Ni containing a random GB at the level of CP by extracting interface parameters (as done in this work) from MD simulations [50]. Similarly, for a polycrystalline aggregate, one may apply the similar methodology wherein the GB character and defect content can be extracted from EBSD data [31] and activation parameters for that particular GB (pristine or damaged) can be quantified using MD simulations and passed onto the FE level of simulations.

    View full text