Diffuseinterface polycrystal plasticity: expressing grain boundaries as geometrically necessary dislocations
 Nikhil Chandra Admal^{1},
 Giacomo Po^{2} and
 Jaime Marian^{1, 2}Email author
DOI: 10.1186/s4131301700060
© The Author(s) 2017
Received: 9 April 2017
Accepted: 20 June 2017
Published: 11 July 2017
Abstract
The standard way of modeling plasticity in polycrystals is by using the crystal plasticity model for single crystals in each grain, and imposing suitable traction and slip boundary conditions across grain boundaries. In this fashion, the system is modeled as a collection of boundaryvalue problems with matching boundary conditions. In this paper, we develop a diffuseinterface crystal plasticity model for polycrystalline materials that results in a single boundaryvalue problem with a single crystal as the reference configuration. Using a multiplicative decomposition of the deformation gradient into lattice and plastic parts, i.e. F(X,t)=F ^{L}(X,t)F ^{P}(X,t), an initial stressfree polycrystal is constructed by imposing F ^{L} to be a piecewise constant rotation field R ^{0}(X), and F ^{P}=R ^{0}(X)^{T}, thereby having F(X,0)=I, and zero elastic strain. This model serves as a precursor to higher order crystal plasticity models with grain boundary energy and evolution.
Keywords
Polycrystal plasticity Multiplicative decomposition Grain texture DislocationsIntroduction
When a polycrystalline material is deformed, its microstructure generally experiences a reorientation of the crystal lattices of each grain towards a preferential distribution of orientations known as crystallographic texture. The study of texture evolution is important because textured metals typically exhibit plastic anisotropy, which plays a significant role on mechanical properties. Predicting the evolution of deformationinduced texture and the accompanying plastic anisotropy is the subject of polycrystal plasticity models (Beaudoin et al. 1993; Sarma and Dawson 1996; Kok et al. 2002; Estrin 2002). These models are typically formulated assuming that the microstructure of the polycrystal is associated with a representation of microscopic crystals whose individual responses, on average, determine the macroscopic response of the polycrystal. At the level of each grain, plastic deformation occurs by the standard mechanism of dislocation slip, and so (i) constitutive equations that relate dislocation motion to crystal deformation must be defined, and (ii) an averaging scheme that relates the response of individual crystals to the macroscopic stressstrain response of the polycrystal must also be defined. For single crystals, a multiplicative kinematic decomposition of the deformation gradient into elastic and plastic parts is typically used. This decomposition adequately describes the distinctly different kinematical mechanisms that operate during the plastic deformation of a crystal. It was formally introduced in continuum plasticity (NematNasser 1979; Simo 1988; Reina and Conti 2014), and then applied to describe the kinematics of single crystals (Asaro 1983; Lubarda 2004; Roters et al. 2010a). A feature of this decomposition is that it introduces an intermediate configuration between the reference and current configurations which is obtained by unloading the crystal to a stressfree state. The elastoviscoplastic constitutive equations are generally written relative to this relaxed configuration.
Many numerical procedures have been proposed to integrate the crystal constitutive equations (Kalidindi et al. 1992; Cuitino and Ortiz 1993; Kuchnicki et al. 2006), generally implicit and semiimplicit procedures which are developed differently by particular selection of the primary variables (stresses (Harewood and McHugh 2007), shear rates (Zikry 1994), plastic deformation gradient (Rice 1971), etc.). Polycrystal plasticity models appear in various levels of sophistication. Along the venerable Sachs and Taylor models –in which the aggregate deformation (Sachs 1928; Kocks 1970a) or stress (Taylor and Quinney 1932; Hutchinson 1964) is computed by averaging from the individual crystal values–, selfconsistent models have been developed and applied that express the global deformation in terms of linearized viscoplastic moduli that must be adjusted selfconsistently (Lebensohn and Tomé 1993; Lebensohn et al. 2007; Acta Materialia 2012; International Journal of Plasticity 2013; Knezevic et al. 2014a). Models that spatially resolve grain boundaries (GB) have started to gain traction recently thanks to a higher efficiency of numerical solvers and a wider availability of computational resources. Roters et al. have provided a comprehensive review of the different variants of such approaches Roters et al. (2010b), which enable the calculation of the fine spatial features of strain and stress fields, including grain shape changes and nonuniform deformation. Some of these advances have also been discussed by Knezevic et al. (2014b).
However, in the above models, grain boundary processes –which are known to be relevant at high stresses and temperatures– cannot be captured by construction. For example, fundamental grain boundary properties such as energies and mobilities are extraneous to spatiallyresolved standard (poly)crystal plasticity models.
The aim of this paper is to present a framework that preserves the ability to model intragrain plasticity, while at the same time enabling a straightforward generalization to include grain boundary processes. To this end, we develop a ‘diffuse’interface crystal plasticity model for polycrystalline materials based on a representation of grain boundaries as a special subclass of geometrically necessary dislocations (in the sense defined by Cermelli and Gurtin (2001, 2002)). In this model, with a single crystal as the reference configuration, a stressfree polycrystal is constructed by imposing a piecewise constant rotation field and its transpose as the lattice and plastic distortions respectively. To make the resulting model numerically tractable, we regularize the piecewise constant rotation field, resulting in a diffuse interface model, that preserves the zerostress character of the grain boundaries. Our main intent here is to introduce the model and its potential, and perform a verification exercise before launching into more ambitious undertakings where grain boundary phenomena can be properly modeled. In the following sections we lay out the essential theoretical developments of our model and provide a verification exercise of the numerical implementation.
Classical crystal plasticity for single crystals
where F ^{L} and F ^{P} are lattice^{1} and plastic components of F respectively, and detF ^{P}=1. In this paper, F ^{P} represents the deformation gradient of an infinitesimal material element, attributed to dislocation slip through its volume. Since such a process renders the lattice invariant, it follows that F ^{P} leaves the lattice undeformed. F ^{L} represents the deformation of the material due to the deformation of its underlying lattice. Note that F ^{L}(X,t) and F ^{P}(X,t)need not be gradients of a deformation map. Instead, since F ^{L} and F ^{P} are invertible, they represent deformation of an infinitesimally small neighborhood of X at time t. In other words, F ^{P}dX represents the deformation of a differential material element d X. The collection of all deformed differential material elements is referred to as lattice configuration. In this sense, F ^{P} maps the reference configuration to the lattice configuration, and F ^{L} maps the lattice configuration to the deformed configuration.

Macroscopic force balance$$\begin{array}{*{20}l} \text{Div}~\boldsymbol{P}(\boldsymbol{X},t) &= \boldsymbol{0}, \quad \boldsymbol{X} \in \mathcal{B}^{0}, t>0, \end{array} $$(7a)$$\begin{array}{*{20}l} \boldsymbol{u} &= \boldsymbol{u}^{0}~\text{on}~\partial \mathcal{B}^{0}, \end{array} $$(7b)where$$\begin{array}{*{20}l} \boldsymbol{P} := \boldsymbol{F}^{\mathrm{L}} \psi,_{{\mathbb{E}}^{\mathrm{L}}} \left(\boldsymbol{F}^{\mathrm{P}}\right)^{\mathrm{T}}, \end{array} $$(8)
is the first Piola–Kirchhoff stress tensor, and \(\psi,_{{\mathbb {E}}^{\mathrm {L}}}\) denotes the derivative of ψ with respect to \({\mathbb {E}}^{\mathrm {L}}\).

Microscopic force balance for each slip system α$$\begin{array}{*{20}l} b^{\alpha} v^{\alpha}(\boldsymbol{X},t) = \psi,_{{\mathbb{E}}^{\mathrm{L}}} \boldsymbol{m}^{\alpha} \cdot \boldsymbol{C}^{\mathrm{L}} \boldsymbol{s}^{\alpha}, \end{array} $$(9)
where b ^{ α }≥0 is the inverse of the mobility associated with the slip v ^{ α }, and C ^{L}=(F ^{L})^{T} F ^{L} is the right CauchyGreen strain tensor.
Note that, the above initial conditions are also used for polycrystals, with the difference that L ^{ P } is evolved in a piecewise way in each grain due to the different orientation of the slip systems, and the free energy density given by \(\psi \left (\boldsymbol {R}^{\mathrm {T}} {\mathbb {E}}^{\mathrm {L}} \boldsymbol {R}\right)\), where R is a piecewise constant rotation field describing the initial orientation of grains.^{2}
In the next section, we first present a diffuseinterface polycrystal plasticity model which operates at a length scale where all grain boundaries are resolved explicitly. In contrast with assumption (10), the proposed framework gives us access to grain boundary dislocation densities, thus enabling us to model grain boundary energies.
Polycrystal plasticity
Consider a sharpinterface polycrystal, i.e. one where the orientation of the lattice is constant in the interior of one grain and has a jump discontinuity along the grain boundary. In this context, crystal plasticity is studied by having the stressfree polycrystal as the reference configuration. Due to the variation in orientation of the grains, the elastic and plastic response of each grain is different. Therefore, the elastic moduli and the slip systems (s ^{ α } and m ^{ α }) are piecewise constant, with jump discontinuities along the grain boundaries. If the polycrystal is stressfree at t=0, then the initial conditions are identical to (10). Thus, within this framework, polycrystal plasticity is identical to single crystal plasticity with the caveat that the elastic moduli, s ^{ α } and m ^{ α } are piecewise constant. While this model is remarkably simple, it is not straightforward to generalize it to model grain boundarymediated deformation, such as shearinduced grain boundary motion, grain shrinkage and rotation, grain boundary sliding, etc. These phenomena can become important during plastic deformation at high stresses and/or temperatures, such as during recovery, recrystallization, and grain growth. In the following section, we present an alternate framework that lays the foundation to model polycrystal plasticity with grain boundary evolution.
Diffuseinterface polycrystal plasticity
The success of single crystal plasticity in describing the materials deformation lies in precisely identifying the independent mechanisms involved, and attributing them appropriately to the evolution of F ^{P}. For example, the rate of change is F ^{P} due to dislocation slip is identified with the slip rate projected on each slip system by way of the Schmid tensor. Similarly, additional mechanisms such as dislocation climb are built into the evolution law for F ^{P} (Weertman 1955; Thomson and Balluffi 1962). In addition to dislocations, a grain boundary sweeping through a material also results in plastic distortion. For example, consider a circular grain with lattice orientation θ _{2} embedded in a larger grain with orientation θ _{1}. The misorientation of θ _{2}−θ _{1} results in a grain boundary energy. In order minimize the internal energy, the circular grain shrinks. As the circular grain boundary sweeps through the material, the lattice in the swept region rotates from an initial configuration of θ _{1} to θ _{2}, while the rest of the lattice remains unchanged. If F ^{P} is equal to identity during this process, then this results in an incompatible F. This conclusively suggests that F ^{P}≢I in the swept area. In other words, grain boundary motion always results in plastic distortion.
Therefore, in the spirit of modeling plasticity due to bulk dislocations, plasticity due to grain boundary motion may thus be modeled by identifying the mechanism for the accompanying plastic distortion, and include it in the evolution law for F ^{P}. Identifying the pertinent GBmediated plastic mechanisms is highly nontrivial. For example, recent atomistic simulations have revealed that for certain misorientations the interior grain not only shrinks but also rotates with no dislocation activity in the bulk. This suggests that unlike dislocation slip, there is no unique fundamental evolution law for F ^{P} that can be attributed to the motion of a grain boundary with a given misorientation. Therefore, we take an alternate approach to modeling plasticity due to grain boundary motion.
The central idea behind this approach is to identify dislocations as the basic defect carriers, and build grain boundaries as continuum aggregates of dislocations. Therefore, any motion of grain boundary is viewed as a collective motion of dislocations that form the boundary. The most important advantage of this approach is plastic distortion due to grain boundary motion emerges from the original flow rule given in (4) without identifying any new mechanisms. This approach can model phenomena such as shearinduced grain boundary motion, grain boundary sliding and grain rotation (Admal and Marian 2017). We next build a framework of polycrystal plasticity based on the idea described above.
We begin by noting that F ^{P}(X,0)=R ^{0}(X)^{T} qualifies to be a plastic distortion due to dislocation slip, since a rotation can always be expressed as a product of three shear deformation tensors (Tanaka et al. 1986; Paeth 1986; Toffoli and Quick 1997).^{3} Interpreting the three resulting shear deformations as latticeinvariant shears obtained due to dislocation slips, the rotation tensor F ^{P}(X,0) may be interpreted as a latticeinvariant deformation. Since an arbitrary rotation rotates the material, it may seem contradictory for it to leave the lattice invariant (except of course when the rotation belongs to the point group of the lattice). The correct mathematical interpretation of a “latticeinvariant” rotation is given using the notion of weakconvergence discussed in Appendix B. In short, weak convergence represents convergence of functions/distributions on the “average”. In Appendix B, we show that, for a sequence of lattice constants a ^{ i }→0 (as i→∞), F ^{P}(X,0) has to be viewed as a weaklimit of a sequence of deformations (F ^{P})^{ i } that leave the a ^{ i }lattice invariant. Therefore, interpreting F ^{P}(X,0)=R ^{T}(X) and F≡I for a discrete lattice in an average sense resolves the apparent contradiction described in the previous paragraph.
An important consequence of the decomposition given in (11) is that the resulting polycrystal is stressfree since the Lagrangian strain, defined in (6), is equal to zero. Therefore, Eq. (11) describes a polycrystalline state which is obtained from a reference single crystal by the right amount of slip in each grain such that grains undergo relative rotation but the polycrystal remains stress free.
where Curl denotes the curl of a tensor field with respect to the material/reference coordinate.^{4} For a given normal n in the lattice configuration, the vector G ^{T} n measures the net Burgers vector of dislocation lines per unit area passing through a plane of normal n.
Using the decomposition of F discussed above, we can now, in principle, study a polycrystal under a single boundaryvalue problem. Numerically, the problem still does not enjoy the nice characteristics of its single crystal counterpart as F ^{L} and F ^{P} are discontinuous. In order to overcome this challenge, we introduce a smoothinterface version of the above sharpinterface model. This can be achieved by constructing a stressfree diffuse interface crystal plasticity at t=0 with F ^{P} a smoothened step function in the space of rotation fields. This alteration ensures that all the resulting fields are smooth.
Numerical implementation
In this section, we discuss a threedimensional numerical implementation of tensile tests of polycrystals of varying textures using the diffuseinterface model introduced in “Diffuseinterface polycrystal plasticity” section. The main aim of this section is to demonstrate the robustness of the diffuseinterface model.
Finite element implementation
where W is the skewsymmetric matrix associated with q, and q and q/q represent the angle and axis of the rotation tensor. Lagrange linear finite element interpolation is then chosen for the variables U ^{P} and q, from which F ^{ P } is locally computed as F ^{ P }=R ^{ P }(q)U ^{ P }. This method guarantees that F ^{ P }(X,0) can describe an exact plastic rotation field without numerical artifacts due to the interpolation method.
respectively. The remaining initial conditions for q, which defines the texture of the polycrystal, is discussed in “Construction of polycrystals with different textures” section.
The system of Eqs. (4), (7), and (16) are evolved in a segregated manner using the MUMPS direct solver, and BDF (Backward Differential Formula) time stepping algorithm implemented in COMSOL5.2. In particular, due to the highly nonlinear nature of (4) expressed in q and U, we enable the “automatic highly nonlinear (Newton)” option to obtain wellbehaved solutions. On the other hand, we rely on the default “constant (Newton)” option for solving (7), and (16). All simulations were performed on a finite element mesh with 99883 elements, and 595620 degrees of freedom.
Construction of polycrystals with different textures
In this section we describe the generation of diffuse interface polycrystals of different textures. The grain orientations are outputted in the form of a smoothened rotation vector field q(X,0) which serves as an initial condition along with those given in (19).
where \(\text {dist}(\boldsymbol {p}^{i},\boldsymbol {\mathcal {P}}^{\beta })\) is the distance between p ^{ i } and \(\boldsymbol {\mathcal {P}}^{\beta }\). Finally, the rotation vector q ^{ α } is associated to the grid point p ^{ i }. The polycrystal is outputted in the form of the rotation vector field on the grid which is then interpolated as a smooth vector field q(X,0) using the nearest neighbor interpolation implemented in COMSOL5.2. Therefore, the texture of the resulting collection of grains depends on the distribution of the initial collection of N random points, and the grain boundary “thickness” is inversely proportional to the resolution of the grid. The pseudocode for the above algorithm is described in Algorithm 1.
Grains with a log normal distribution of sizes are generated by sampling the initial N points from a log normal distribution based on Algorithm 1 described in Appendix A.
Results
Taylor factors for polycrystals of different textures
Texture  Taylor factor 

log normal  3.344 
Flat (axial)  3.471 
Flat (nonaxial)  3.293 
Elongated (axial)  3.315 
Elongated (transverse)  3.273 
Discussion and conclusions
Most metals and alloys in usable form display an internal microstructure characterized by a collection of grains with different lattice orientation separated by grain boundaries. Metals deformation, particularly at high temperatures and stresses, such as during hot working, involves not just intragranular plasticity but also plasticity controlled by grain boundary mechanisms. Standard formulations of crystal plasticity decouple both types of deformation, probably due to our good deal of understanding about low temperature processes, e.g. cold working, which tends to dominate our thinking of plasticity. Indeed, this decoupling has been the governing principle behind the development of new methodologies to study recrystallization in metals (SingerLoginova and Singer 2008; Steinbach 2009; Takaki and Tomita 2010; Abrivard et al. 2012; Kamachali 2013).
However, during dynamic phenomena such as continuous dynamic recrystallization, bulk and grainboundary plastic processes can occur simultaneously, and therefore the underlying plasticity model must be capable of capturing both types of deformations concurrently. This is the motivation behind the present work: to devise a computational model that combines bulk and grain boundary plasticity by design within the same framework. Our purpose at the moment is simply to demonstrate that our formulation is capable of rendering the same response as standard crystal plasticity models for conventional problems in polycrystal plasticity. Only after fulfilling this step can we truly apply our methodology to phenomena involving grain boundary processes. We have undertaken this verification exercise by solving the same problem, standard Taylor hardening in bodycentered cubic Fe, using both methodologies, and comparing the results obtained. To explore the capabilities of our model further, we have considered several different textures and misorientation ranges and have calculated the associated Taylor factors. In all cases, our results agree with those obtained using standard polycrystal plasticity.
In summary, we have developed a diffuseinterface model for polycrystalline materials deformation that expresses grain boundaries as a special class of geometrically necessary dislocations, such that the stressfree nature of the polycrystalline structures obtained is naturally recovered. We have tested the robustness of the method by simulating tensile tests and calculating Taylor factors for polycrystals of varying textures. Our model provides a pathway from which grain boundary energies and mobilities can eventually be obtained directly from dislocation densities, which opens the door to integrated models of intragranular and grain boundarygoverned plasticity such as recrystallization in hot working.
Endnotes
^{1} In the literature, it is more common to refer to the lattice distortion as an elastic distortion using the notation F ^{E}. Since the decomposition given in (2) is purely geometric in nature (as opposed to energetic), we prefer the term “lattice distortion” denoted by F ^{L}, a terminology adopted by Clayton (2010).
^{2} It is important to note that, within the framework of crystal plasticity, a constitutive response function of the form \(\psi (\boldsymbol {R}^{\mathrm {T}} {\mathbb {E}}^{\mathrm {L}} \boldsymbol {R})\) with a nonconstant R does not imply F ^{L}=R and F ^{P}≡I as this would result in an incompatible F.
where n is an arbitrary constant vector, and the curl on the righthandside of the above equation is the curl of a vector field defined as (v)_{ i }=ε _{ ijk } v _{ j,k }, for any vector field v.
where Φ is the cumulative distribution function of the standard normal distribution.
Appendix A: Algorithm to generate polycrystals with different textures
In this section, we describe the algorithm used to generate the different polycrystal textures simulated in this paper. Algorithm 1 is able to generate a polycrystal with a given cumulative distribution function f for grain sizes. In addition, grains of desired aspect ratio can be generated using the scales sx, sy and sz as given in Algorithm 1.
Appendix B: Interpretation of F ^{P}=R ^{T} and F=I using the notion of weak convergence
In this section, we use the notion of weakconvergence to arrive at a physical interpretation of the decomposition given in (11), and depicted in Fig. 1 for a discrete lattice. Recall the apparent contradiction we arrive at by interpreting F ^{P}=R ^{T} in an absolute sense for a discrete lattice. On the one hand, F ^{P}=R ^{T} should be a latticeinvariant deformation, while on the other hand an arbitrary rotation need not preserve the lattice. We will now show that, for a discrete lattice, F ^{P}=R ^{T} and F=I should be viewed in an average sense using the notion of weak convergence Rudin (2006).
Definition 1
for all ϕ in the space of smooth functions with compact support, denoted by \(C_{\mathrm {c}}^{\infty }\).
where we have used the divergence theorem along with ϕ=0 on ∂ Ω to arrive at the first and last equalities, and the uniform convergence of \(\tilde {\boldsymbol {x}}^{i}\) to interchange the limit and the integral signs in the first equality. By the definition of weak convergence, (30) implies (F ^{P})^{ i }→R ^{T} weakly. Assuming F ^{L}=R, it can be similarly shown that the sequence F ^{ i }:=F ^{L}(F ^{P})^{ i } converges weakly to the identity.
Declarations
Acknowledgements
NCA and JM acknowledge support from DOE’s Early Career Research Program, under grant DESC0012774:0001 and the National Science Foundation, Division of Materials Research, award number 611342. GP acknowledges the support of the U.S. Department of Energy, Office of Fusion Energy, through the DOE award number DEFG0203ER54708, the Air Force Office of Scientific Research (AFOSR), through award number FA95501110282, and the National Science Foundation, Division of Civil, Mechanical and Manufacturing Innovation (CMMI), through award number 1563427.
Authors’ contributions
NCA developed the theory, tested the model, ran the simulations, and wrote most of the manuscript. GP assisted with the theoretical developments and with the solution procedure. JM contributed to the theoretical developments and to the writing of the paper. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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