Thermo-mechanical processing of metallic materials involves a sequence of shaping and heating operations to achieve a component state with target material strength and shape for a specific application. During processing, microstructural features such as dislocation density, grain size, and texture evolve as consequences of plasticity, recovery, and recrystallization [1][2][3]. Using modeling tools to simulate the processes and underlying microstructure evolution reduces the need for experimentation, saving time and cost involved. The models can also be used to explore the influence of adjusting processing parameters and optimize them for final material properties and geometry.
The long-established elasto-plastic self-consistent (EPSC) model is one of the widely used crystal plasticity models for modeling mechanical response and texture evolution during plastic deformation of polycrystals [4,5]. Thermal effects have been considered in EPSC but were limited to influencing material strength and thermal expansion [6,7] and not to predict the effects of recrystallization. Each grain in EPSC is an ellipsoidal inclusion in a homogenous-effective-medium (HEM), which has the mean properties averaged over the constituent grains. These mean properties of the HEM are obtained via the self-consistent (SC) homogenization scheme relying on one value of stress and one value of strain per inclusion, i.e. the first moments [8]. Such standard EPSC formulation provides a favorable balance between accuracy and computational efficiency relative to the more computationally demanding but more accurate full-field models [9][10][11][12][13][14]. Owing to the computational efficiency, the EPSC model has been coupled with finite elements (FE) to simulate larger scale deformation processes involving complex geometries [15][16][17]. Moreover, EPSC has been advanced to incorporate a wide range of sub-models including dislocation density-based hardening [18][19][20], twinning and detwinning [21][22][23], phenomenological backstress [24][25][26], latent hardening [27], and martensitic phase transformation induced plasticity [28][29][30][31].
Deformation textures at very large plastic strains in rolling or tension/compression are predicted sharper than measured using common mean field models such as EPSC, because grains develop high intragranular orientation spreads at large plastic strains such that the consideration of mean orientation per grain becomes insufficient [32]. Such intragranular orientation gradients can cause grain fragmentation, which is another phenomenon typically not considered in mean-field models. Advancing the EPSC formulation to calculate intragranular misorientation spreads and grain fragmentations based on higher order micromechanical fields is the merit of the present paper. Recrystallization of grains in deformed polycrystalline metals is observed to nucleate at inhomogeneities such as shear bands, grain boundaries, transition bands, and precipitate particles [1]. Enabling EPSC to predict the evolution of microstructure during recrystallization via transition-bands and grain boundary nucleation mechanisms along with grain growth driven by granular stored energy are the added objectives of the present paper.
The development of higher order formulations started from the works reported in the areas of composites [33,34] and then adaptations of such formulations in polycrystalline material modeling proceeded [35][36][37]. These homogenization models for the macroscopic response accounted for the second moments of stress fields. Based on the second moment of stress, the evaluations of higher order statistics on updating the lattice rotation rate spreads in grains and resulting intragranular misorientations trends were first accomplished in a viscoplastic self-consistent (VPSC) model [38,39]. Relying on the intragranular misorientations trends, a grain fragmentation (GF)-VPSC [40] and a VPSC capable of modeling recrystallization texture evolution [41] were developed. The recrystallization model considered transition bands and grain boundary nucleation mechanisms [42][43][44], while recrystallized grain growth was driven via the difference in stored energies between the given grain and HEM [45][46][47][48][49][50]. These developments were unified in a comprehensive FF-VPSC model in [51]. The development of FF-VPSC laid down the groundwork for formulating FF-EPSC, since both involve the self-consistent (SC) homogenization. Nevertheless, the fundamental consideration of elasticity and the absence of the viscoplastic power law in the activation of slip systems pertaining to EPSC required amendments to the formulation. The consideration of elasticity is the greatest advantage of EPSC over VPSC. For completeness of literature review, we reflect on other models that handle grain fragmentation [52,53] and recrystallization [45,54,55]. However, these models were based on average values of grain properties and HEM, rather than fluctuations of micromechanical fields.
This paper aims at predicting texture evolution throughout deformation and recrystallization of cubic polycrystals via the novel formulation of FF-EPSC. The novel additions to a standard version of EPSC are the implementations of the second moments of stress rates, lattice rotation rates, and misorientation spreads. The second moments quantities enable the implementation of grain fragmentation and recrystallization models. The consideration of the former model is aimed at improving the predictions of deformation texture evolution, while the latter model is aimed at facilitating the predictions of recrystallization texture evolution via transition-bands and grain boundary nucleation mechanisms, while grain growth would be governed by the stored mechanical energy. The kinetics of nucleation and growth in the model are conceived to be pre-determined/initialized by predicting the deformed state of the metal. The recrystallization formulation in the present work therefore circumvents a common drawback of several other recrystallization models available in the literature pertaining to an arbitrary nuclei orientation selection [56][57][58][59].
The FF-EPSC was first validated by comparing the predicted and measured intragranular misorientation spreads and texture evolution during deformation and recrystallization of a face centered cubic (FCC) aluminum alloy (AA) 5182-O and during deformation and recrystallization of a body centered cubic (BCC) interstitial-free (IF) steel. In doing so, the tradeoffs between transition-bands and grain boundary nucleation mechanisms were elucidated while predicting recrystallization textures. The FF-EPSC model was then integrated in implicit finite elements (FE) as a user material subroutine in Abaqus to facilitate predicting geometrical shape changes under more complex boundary conditions, where every integration point embedded the FF-EPSC constitutive law considering the effects of texture evolution and directionality of deformation mechanisms at the single crystal level. Given that the FF-EPSC incorporated a dislocation density-based hardening law for the evolution of slip resistance and a backstress law to influence the activation of slip systems, the model was applied to simulate 60 % reduction rolling process capturing the through-thickness texture gradients of AA6022-T4, recrystallization of the carried over texture gradients, and finally deep-drawing of a cylindrical cup from a sheet initialized with the texture gradients after recrystallization. The linked simulation processing sequence was aimed to demonstrate the versatility of the simulation framework developed in this paper in predicting texture evolution and phenomena pertaining to behavior of materials and also geometrical changes important for the optimization of metal forming processes.
In this paper, the symbols ⋅ and ⊗ denote a dot product and a tensor product, respectively. Tensors are bold letters. Tensor components and scalars are italic and not bold.
In the following subsections, we first describe the overall EPSC modeling framework including the single crystal constitutive laws in Section 2.1. Then, we provide the partial derivatives of stress rate and shear strain rate in Section 2.2. The partial derivatives are used in the second moment field fluctuations formulation, described in Section 2.3. The section describes the intragranular spreads of stress, spin, and misorientation in the otherwise mean-field model. Finally, we describe the grain fragmentation and recrystallization models enabled with the implementation of the second moments. Table 1 summarizes the key model variables.
The EPSC framework was originally a small strain formulation by Hill [60], implemented by Turner and Tomé [4], and extended to approximate large strain deformation by Neil, Wollmershauser, Clausen, Tomé and Agnew [5]. Under the small strain formulation, the stress rate of a single crystal is:
where σc the Cauchy stress rate for a grain, c, C c is the elastic stiffness, D c is the deformation rate tensor, which is the symmetric part of the corresponding velocity gradient. D pl,c is the plastic stretching rate:
where γs,c is the shear strain rate for a slip system, s, m s,c is the instantaneous symmetric Schmid tensor calculated with the slip plane normal unit vector, n s,c , and Burger's unit vector, b s,c .
The single crystal constitutive equation can also be expressed as:
where L c is the instantaneous elasto-plastic stiffness tensor given by Neil, Wollmershauser, Clausen, Tomé and Agnew [5]:
where I is a second rank identity tensor, and X ss ʹ is a square matrix of size corresponding to the number of active systems as:
Where h ss ʹ = ∂τ s c ∂γ s ʹ is the hardening matrix as a function of shear strain.
The partial derivatives of the critically resolved shear stresses (CRSS), τ s c , can be taken for a known hardening law, such as the dislocation densitybased hardening law adopted in FF-EPSC originally described by Beyerlein and Tomé, 2008 [18,19].
A similar relationship stands for the polycrystalline aggregate:
Where σ is the polycrystal Cauchy stress rate, D is the polycrystal deformation rate tensor, and L is the polycrystal instantaneous elastoplastic stiffness.
In large strain formulation, EPSC considers the rotation of single crystals with respect to the aggregate and the resulting texture evolution:
involving the applied spin, W c,app , which consists of a superimposed macroscopic spin over a polycrystal under an applied deformation, W, and an additional spin that arises from the Eshelby solution for an embedded inhomogeneity inside the HEM subject to applied boundary conditions, Π c = P c (S c ) -1 (D c -D) [61], where S c and P c are the symmetric and antisymmetric Eshelby tensors, respectively. Finally, the plastic spin, W pl,c , is defined using W pl,c = ∑ s γs,c q s,c where q
) is the anti-symmetric Schmid tensor. Now, we can approximate the large strain deformations using the Jaumann rate of Cauchy stress, σc , to account for crystal lattice rotations. The constitutive Eq. ( 1) and Eq. (3) become:
Similarly, for Eq. ( 6) of the polycrystal:
And then we can relate Jaumann and Cauchy stress rates on the single crystal and polycrystal levels in a common frame, considering rotations:
We use the bars on top to indicate a volume average. The volume average is obtained by summing the products of the quantities with w c , the relative weight of the grain. The total weight of the polycrystal is unity.
The interaction equation solves the equilibriums between the single crystal and polycrystal stress and strain rates:
Where L c * is the interaction tensor:
that defines a localization tensor:
which gives the polycrystal instantaneous elastoplastic stiffness tensor:
Since the right-hand side of Eq. ( 14) is also a function of L via Eq. ( 13), these equations must be solved iteratively until convergence with a specified tolerance of 0.001. The tolerance can be adjusted as an input parameter. The chosen value provides a good balance between accuracy and computational efficiency.
The corresponding interaction and localization quantities in terms of compliance are given by Lebensohn, Tomé and Castaneda [62]:
where M c * is the compliance interaction tensor, M is the polycrystalline compliance, B c is the compliance localization tensor, and M c is the single crystal compliance. As EPSC typically uses and calculates stiffness tensors instead of compliance, we will calculate the polycrystalline and single crystal compliances from the inverses of stiffness tensors [63], which then can be used to calculate M c * and B c :
Lastly, slip activities are activated if the resolved shear stress fulfills the following conditions: where τ s,c c is the resistance to slip defining crystal's yield surface, and the right-hand side of the above equations is the driving force or resolved shear stress. τ s bs is the contribution from the phenomenological backstress law, summarized in Appendix A.2. Eq. (17a) indicates a slip system is active when its resolved stress state is on the yield surface. Eq. (17b) then ensures the stress state remains on the yield surface during plastic deformation. The CRSS evolves with plastic deformation via:
The shear strain rate is defined as [15]:
Equations pertaining to the dislocation density-based hardening law and the backstress law implemented in EPSC are provided in Appendix A.1 and A.2. The EPSC is coupled with an implicit FE framework named FE-EPSC, and a brief summary of the implementation is also provided in Appendix A.3.
Before describing the second moments formulation, we first define several partial derivatives specific to the EPSC formulation. These derivatives are also different from those in VPSC as EPSC uses the dislocation density-based hardening law while VPSC employed a simple Voce-type hardening law. These partial derivatives are: stress rate with respect to misorientation vector, ∂ σ ∂δr , shear strain rate with respect to Schmid tensor, ∂γ s ∂m s , and shear strain rate with respect to stress rate, δγ s ∂ σ . These derivatives are used in the second moments calculations described in Section 2.3. For better clarity, the equations in this section are written in the indicial notation.
From Eqs. (17b) and (18), we can write the expression relating shear strain with stress rate and Schmid tensor. We express the second rank 3 × 3 tensors as 6 × 1 vectors via Voigt notation:
Taking derivative with respect to a misorientation vector, δr m , yields:
where δR kl is the rotation matrix representation corresponding to the misorientation vector, δr m , and the partial derivatives, ∂m s ʹ j ∂δR kl and ∂δR kl ∂δrm , are described in Section 2.3 as they are part of the second moment implementation.
The partial derivatives of shear strain rate with respect to Schmid tensor and stress rate can both be derived from Eq. (19). Taking the partial derivative with respect to Schmid tensor gives:
Finally, the partial derivative with respect to stress rate is:
In this section, expressions for the second moments of stress rate, lattice rotation rate, and misorientation are described following the developments in VPSC, which were presented in several recent papers [38][39][40]51]. Since the expressions were presented in detail in these VPSC papers, we provide a summary of the most important expressions sufficient to follow the development of FF-EPSC. The expressions for calculating several additional partial derivatives used to determine the second moments giving rise to the intragranular field fluctuations specific to EPSC are provided in Section 2.2. Given the incremental nature of the EPSC formulation, we define the second moments of stress rate, different from those in VPSC, which uses the second moments of total stress. VPSC solves for total stress. The angle brackets 〈〉 are used to indicate the second moment quantities.
To derive the expressions for the second moments of stress rate, we first consider the following energy rate equation [62][63][64]:
The polycrystalline energy rate is a weighted sum of the individual single crystal energy rates calculated as dot products between crystal stresses and strain rates. The weighted sum is essentially the homogenization procedure to obtain the overall polycrystalline quantity from the single crystal quantities. The overall energy rate can also be expressed using the polycrystalline stress and strain rate. Then, substituting strain for stress using the constitutive equations, applying Eqs. (16a) and (16b), and taking the time rates, we obtain:
Next, we take the partial derivative of Eq. ( 25) with respect to M c . Since the single crystal compliances of each grain are independent of each other, the derivative per grain simplifies to the following:
The expression constitutes the second moments of stress rate per grain, c. Details of the derivation of the partial derivative ∂M/∂M c are provided in Appendix B. To conceptualize the intragranular stress rate distribution, we consider the stress rates at material points, x, inside a grain at a time, t [40]:
where σt is the mean stress rate determined from the standard EPSC formulation. Note that without the fluctuations, all material points would have the same stress rate as the average. Spatial variation of grain properties is the first source that induces fluctuations in stress rate, σt,q
As stress is a function of orientation and thus misorientation, the misorientation field is the second source that induces stress rate fluctuations, σt,δr . In considering the effects of intragranular spreads, the second moment of stress rate from Eq. ( 26) is expanded to the complete form [40]:
where:
is the first moment of mean stress rate,
• 〈 σt,q ⊗ σt,q 〉 c is the second moment of stress rate fluctuations due to mean orientation, q, and mean stress rate of the given grain,
• 〈 σt,δr ⊗ σt,δr 〉 c is the second moment of stress rate fluctuations due to intragranular spread of misorientations, δr, and
terms are the transpose of each other, describing the cross-covariance between stress rate fluctuations from the contributing sources of mean grain properties and misorientations.
The fluctuation due to mean orientation is calculated readily at each time step using:
Using Cholesky decomposition, we can decompose the second moment from Eq. ( 29) into lower-triangular matrices:
where T t-Δt and T t are the lower triangular matrices from the decomposition of the second moment of stress rate fluctuation at a previous time step, t -Δt, and the current time step, t. Then, we can calculate the linear map, Z c , of the stress rate fluctuations as:
Assuming the stress rate fluctuations between the steps are approximately linear, we can propagate the fluctuations as:
Then, we can update the cross-covariance term using the linear map [40]:
where Y δr,c and Y σ ,c are defined as:
where
ij , is the dual vector of the spin tensor from the left hand side of Eq. ( 7), W c , and R inc is the mean increment in rotation represented using the rotational matrix. The partial derivatives of the spin vector with respect to stress rate and misorientation vector are:
where t s k = -1 2 ε ijk q s ij is the dual vector of the anti-symmetric Schmid tensor, q s . The partial derivatives ∂ σ ∂δr , ∂γ s ∂m s , and δγ s ∂ σ for a single crystal are described in Section 2.2, and the partial derivatives ∂m s ∂δR and ∂δR ∂δr are calculated as follows:
Having the second moments of stress rate and the necessary partial derivatives, we can continue to calculate the second moments of lattice rotation rate following a similar form at the current time step, t:
where:
• w t,c is the dual vector of the spin tensor, W c , • w t, σ (x) and w t,δr (x) are the fluctuations in the rotation rate due to fluctuations in stress rate and misorientation, respectively influencing the mean value of w t ,
• 〈w t, σ ⊗ w t, σ 〉 c and 〈w t,δr ⊗ w t,δr 〉 c are the second moments of lattice rotation rate terms caused by intragranular stress rate fluctuations and misorientation fluctuations, respectively producing w t, σ and w t,δr , and
• 〈w t, σ ⊗ w t,δr 〉 c and 〈w t,δr ⊗ w t, σ 〉 c are the cross-covariance terms and are the transpose of each other.
The second moment of lattice rotation rate terms are calculated using:
Finally, we summarize the expression for propagating the second moment of misorientation from the current time step, t, to the next time step, τ:
where δr t inc is the misorientation increment and δr t,rot is the rotated misorientation. The second moment terms are:
) T Δt 2 (40c)
The grain fragmentation model stems from the second moments of misorientation spreads describing the intragranular distribution of local orientations per grain deviating from the grain's average orientation. To formulate the grain fragmentation model, the eigenvalues and eigenvectors of the second moments of misorientation, 〈δr t ⊗ δr t 〉 c , are calculated and denoted as
. The eigen quantities describe the three principal distribution directions. The values are conventionally sorted in the descending order such that the first components, λ 1 and v 1 , correspond to the direction with the largest variation. The misorientation angle in each grain is then [41,51]:
When the misorientation angle in any grain reaches a critical value, set to 20 • in this work, grain fragmentation occurs, and a given grain is split into two grains, labeled as f 1 and f 2 . A transition from low-angle grain boundaries to high-angle boundaries as interfaces between two grains in polycrystalline materials is usually taken as 15 • -20 • misorientation angle. The threshold angle is treated as a fitting parameter, which can be adjusted to match experimental data. The fragments are assumed to each have half of the original grain's weight. The equal weight of the fragments is an assumption. The mean misorientations of the fragments from the mean orientation of the fragmenting grain are obtained using [66,67]:
Then, we construct the fragments' initial misorientation vectors using:
Finally, the mean orientations of the fragments are calculated using the misorientations of the fragments and the mean orientation of the grain:
The fragments also obtain the volume fraction (half of the parent's weight) and diameter/shape. Moreover, the fragments inherit the second moments and other state variables from the fragmenting grain. The fragments are then treated as regular grains in the same phase that evolve their own properties and can fragment further if the criteria for fragmentation are fulfilled.
The recrystallization process involves nucleation of new grains in a deformed microstructure and growth of newly created grains [44,46]. Intragranular orientation spreads and stored energy in the microstructure drive the recrystallization processes [46,68]. Therefore, accurate modeling of recrystallization requires accurate modeling of plastic deformation. The nucleation is known to occur at grain boundaries and transition bands in deformed structures of cubic polycrystalline metals, while the growth of the nucleus occurs by migration of grain boundaries driven by the difference in stored energies of the two sides of the moving boundary [46,68]. A recrystallization model handling both nucleation and growth kinetics was developed based on the second moment quantities in FF-VPSC [41,51]. In this model, the difference in stored energies is approximated by the stored energy of the grain and that of the HEM. The same model is adapted here to FF-EPSC. The formulation is briefly summarized below.
The model simulates both grain boundary and transition band nucleation of recrystallized grains [1]. The subsequent growth of recrystallized grains is driven by the difference of strain energy between each grain and the volume averaged strain energy for the HEM [68]. Fragmented grains that recrystallize via grain boundary nucleation exhibit a "unimodal" distribution of rotation [69] while those recrystallize via transition band nucleation exhibit a "bimodal" distribution [70]. These distributions are visualized in Section 3. Both nucleation mechanisms are governed by probability functions:
where P c gb and P c tb are the probabilities that a recrystallized grain nucleus is formed on grain boundary and transition band in a grain, c, respectively. The probability is calculated for each sub-grain volumes with weight of dw in a grain with total weight w c . dw is an input parameter set to 1 × 10 -10 for all cases in the present work. Numerically, a nucleation event occurs when a randomly generated number is less than the probability at a time step with step size Δt, subdivided into dt steps, set to 1 in this study. A gb , B gb , A tb , and B tb are fitting parameters.
The strain energy density per grain, E c , is calculated as [45]:
where ρ s,c tot is the evolved total dislocation density of a slip system (Appendix A). The remaining two variables are shear modulus μ and the magnitude of its Burgers vector b s,c . As the recrystallized grain grows in each time step, volume from the deformed grains transfers to the recrystallized grain. The grain weight due to its grain boundary migration is updated as [41]:
where M is a fitting parameter representing the grain boundary mobility rate, and E avg = E c is a weighted average of the strain energy of the HEM.
As the recrystallization process is incremental, it is possible to simulate partial recrystallization by limiting the number of recrystallization time steps. In this study, we set the recrystallization time steps for all studied alloys to a large number to simulate complete/full recrystallization. The mobility rate can be adjusted to achieve a balance between nucleation of recrystallized grains and growth. A lower mobility rate would result in a larger number of recrystallized grains and vice versa. However, other parameters and the operating mechanism of nucleation also influence the number of recrystallized grains and resulting grain size.
To clarify the definitions, in this work, we define the original grains in the input texture file as "original-grain", grains generated from the fragmentation process as "frag-grain", and grains generated during recrystallization as "rex-grain".
The rex-grains form from the original-grains and not fragments. To this end, the statistics of all the frag-grains belonging to each originalgrain are "merged" and the potential nuclei are examined inside the merged original-grains using the merged properties. In a recrystallization simulation step, provided the merged properties satisfy the recrystallization nucleation conditions, a rex-grain nucleates and develops an initial weight, which is subtracted from the volume of the merged original-grain weight. In every subsequent recrystallization simulation step, the merged statistics are recalculated based on current volumes, and every merged grain is continuously checked against the nucleation conditions. If the conditions are satisfied, an original-grain that previously nucleated a rex-grain nucleates another rex-grain. In the simulations we performed for the present work, there are instances where up to three rex-grains are nucleated from an original-grain. The number of rex-grains per parent grain is also influenced by adjusting the mobility rate, M, so that the rex-grains grow faster with fewer nucleation events. After a rex-grain is nucleated, its growth is based on Eq. ( 47) in each step. We ensure that the combined volume of all grains is constant and equal to the undeformed initial volume. When the volume of rexgrains alone reaches the undeformed initial volume, the material has fully recrystallized. Note that upon recrystallization, the slip resistances and dislocation densities are reset to initial values in the newly created rex-grains.
The onset of recrystallization process is additionally controlled via four additional input parameters: E gb th and E tb th are the thresholds for strain energy and δθ gb th and δθ tb th are the thresholds for average misorientation angle of local neighboring orientations in each grain for both types of nucleation mechanisms. The latter thresholds are calculated using:
To control grain boundary nucleation of a new grain in grains that developed unimodal misorientation distributions, the given grain needs to have a critical mean misorientation angle, δθ gb th , and a critical stored energy, E gb th . To control transition bands nucleation of a new grain in grains that developed bi-modal misorientation distributions, the given grain must have sufficient strain energy E tb th and mean misorientation angle greater than the threshold value δθ tb th . In our simulations, we set these quantities to zero or small numbers meaning that even very minimal quantities are sufficient for nucleation. The actual values used in the simulations will be provided.
Modelling of recrystallization phenomena is a challenging task because of its multi-scale full-field nature involving the effects of dislocation motion, impurities, precipitation, movement of grain boundaries and is also not completely deterministic [71]. Therefore, a complete modeling of recrystallization requires non-local neighboring information. Sections 3 and 4 will show that we can holistically compare the predictions of the mean-field FF-EPSC model obtained in a computationally efficient manner with experiments reasonably well. The predictions that follow will be primarily influenced by adjusting the probability equations, Eqs. (45a) and (45b), to limit the number of nucleated grains. The implementation also facilitates simulating partial/incomplete recrystallization in addition to full recrystallization. The partial recrystallization can be achieved by limiting the number of recrystallization steps to not give rex-grains sufficient time to fully consume the original grains. For partial recrystallization, the volume of rex-grains and remaining not recrystallized grains and fragments equals the volume of original initial grains.
Pole figures showing measured texture in the initial AA5182-O are provided in Fig. 1. The initial texture is a cube-oriented showing evidence of rolling and recrystallization prior processing of the alloy. The texture is represented using 400 crystal orientations. The initial texture used in the simulations of other alloys (AA6022-T4 and IF steel) was assumed random texture represented with 400 uniquely oriented grains generated using the texture compaction algorithms based on generalized spherical harmonics by Marki and Knezevic [72]. The modeling parameters for AA5182-O and IF steel are calibrated with simple tension and plane strain compression (PSC) stress-strain data [73][74][75]. In order to capture these monotonic responses per material, the following parameters associated with the hardening laws are fit: initial slip resistance, τ α 0 , trapping rate coefficient, k α 1 , activation barrier for de-pinning, g α , drag stress, D α , and the rate coefficient q α . The fitting procedure starts with varying τ α 0 to fit yield stress. Next, k α 1 , is adjusted such that the initial hardening slope is reproduced. Next, g α and D α are adjusted to match the hardening behavior. Finally, q α is adjusted to capture the later stage of the hardening behavior. The following back-stress law parameters are fit: saturation value for back-stresses τ sat bs , asymmetry factor, A, and parameters ν and γ b . Here, both τ sat bs and A are adjusted simultaneously to obtain the asymmetric yield and non-linear unloading upon load reversal. Once this is achieved, tuning ν and γ b simply provides more accurate fits. The backstress parameters were calibrated only for AA6022-T4 with cyclic stress-strain data, as described in detail in [76]. The parameters for AA6022 are the same as those used in Feng, Yoon, Choi, Barrett, Zecevic, Barlat and Knezevic [76]. The calibrated curves are shown in Fig. 1, and the FF-EPSC model parameters are provided in Table 2, Table 3, and Table 4. The direct inheritance of EPSC hardening parameters from [76] for AA6022-T4 was possible because the present FF-EPSC model is a more advanced version of the EPSC model used in [76] having the same dislocation density-based hardening law.
Recrystallization model parameters established for the studied materials are summarized in Table 5. These calibration parameters control the grains that give rise to rex-grain nuclei. The parameters were established considering prior works from the literature initially [51] and fine-tuned as necessary to predict the recrystallization texture evolutions.
The fragmentation process begins in a grain after the misorientation angle reaches 20 • obtained using Eq. (41). While each frag-grain is treated physically as a new independent grain in continuous deformation, it is also recorded as a sub-grain inside the parent original-grain. At fragmentation, frag-grains inherit the properties of the original-grains such as dislocation density and evolved CRSS as their initial state. When frag-grains fragment further, those frag-grains act as a next generation of original-grains. The prior generation of fragments determines the orientations and initial properties of the next frag-grains.
At any point during the deformation process, the distribution of properties in the original-grains can be statistically described using the second moment properties of the frag-grains that originated from each original-grain. An original-grain that nucleates a recrystallized rex-grain at a grain boundary exhibits a "unimodal" type of misorientation distribution considering its fragments, while the original-grain that nucleates at a transition band exhibits a "bimodal" distribution. Example predicted distributions of two grains of each type from a simple tension simulation to 0.45 strain are shown in Fig. 2. The figure shows a discrete spread of 500 randomly sampled misorientation vectors, δr, corresponding to each distribution. The procedure to obtain the 500 randomly sampled points and identify the misorientations colored in red is summarized in Appendix C. The type of distribution for the frag-grains are governed by the original-grain's initial orientation, frag-grains' properties, and deformation mechanics. Therefore, both types of distribution can occur simultaneously in different grains during simulations. If a grain has developed unimodal type misorientation distribution, the spread of misorientation vectors is clustered around a point as shown in Fig. 2a. The set of vectors satisfying the grain boundary nucleation criterion are colored red, and on average have larger magnitudes than other misorientation vectors colored in blue. If a grain has developed bimodal type misorientation distribution, the spread has two modes as shown in Fig. 2b The vectors belonging to the transition band region are colored red and fall between the two modes. During recrystallization, a randomly selected red point is taken as a nucleus. For both nucleation mechanisms, multiple nuclei could be created within an original-grain during recrystallization time steps. Therefore, the number of predicted recrystallized grains is not necessarily equal to the number of original-grains.
The misorientation vectors can be projected along an axis e.g. along the dominant rotation axis, v 1 . The axis is not significant for unimodal distribution as shown in Fig. 3a. However, for bimodal distribution of Fig. 3b, the v 1 axis describes the direction of the bimodal distribution, and the origin region corresponds to the misorientations in the transition band region. Nucleated rex-grains from the transition band region give rise to the nearly cube orientation in recrystallized textures [40].
In the following sections, we first show that the FF-EPSC second moments implementation predicts more accurate texture evolution than standard EPSC by simulating the rolling of AA6022-T4 to a strain of 1.2 strain (~70 % reduction). Then, we validate FF-EPSC by modeling the deformation and subsequent static recrystallization processes for AA5182-O and IF steel. For AA5182-O, we compare the prediction of misorientation spreads and texture after uniaxial tension to 0.45 true
Dislocation density-based hardening law parameters for the simulated alloys. The parameters for AA6022-T4 are taken from the previous study reported in Feng, Yoon, Choi, Barrett, Zecevic, Barlat and Knezevic [76].
Slip mode
2.0 × 10 8 0.028 120 IF-Steel {110}〈111〉 40 1.0 × 10 8 0.016 300 {112}〈111〉 50 1.0 × 10 8 0.02 300 AA6022 {111}〈110〉 56 0.5 × 10 8 0.025 100
Backstress law parameters for AA6022-T4. The parameters are taken from the previous study reported in Feng, Yoon, Choi, Barrett, Zecevic, Barlat and Knezevic [76]. The equations associated with the parameters are given in Appendix A.
[MPa] v γ b A 12 560 0.001 0.01
Recrystallization model fitting parameters for the 3 studied alloys.
strain and texture after static recrystallization against experimental data [51,74,79,80]. Lastly, for IF-steel, we compare the rolled and recrystallized textures [81,82]. Both AA5182-O and IF steel alloys are reported to recrystallize via grain boundary nucleation following these deformations [42,43].
Fig. 4 shows one of the beneficial effects of enabling grain fragmentation during the plasticity to large plastic strains by comparing the deformed textures via pole figures between standard EPSC and FF-EPSC with the grain fragmentations enabled. The simulated textures shown are for a AA6022-T4 sheet rolled to 70 % rolling reduction. As shown, the intensities of the pole figure without grain fragmentation are higher meaning that the predicted texture is sharper i.e. the intensities are overpredicted. The FF-EPSC modeling results including the grain fragmentation match much better with the experimental results reported in [40,83] as far as the intensities than the standalone EPSC modeling results with no grain fragmentation considered. Therefore, the intragranular misorientation-based creation of grain fragments is the key to more accurate texture predictions via the mean-field modeling.
Grain average misorientation (GAM) distribution statistics for the AA5182-O alloy were presented in [51]. The statistics were extracted from inverse pole figure (IPF) maps collected using electron backscattered diffraction (EBSD) of the deformed AA5182-O sample after continuous bending under tension (CBT) to 0.45 strain. The measured statistics are shown in Fig. 5a. To further verify the grain fragmentation model, the corresponding predictions using FF-EPSC are shown in Fig. 5b The figure shows the calculated average intragranular misorientation spreads in grains using Eq. (48). Note that all the sub grains are considered separately. We obtained nearly identical results if we would select a set of discrete orientations from the misorientation spreads per grain to calculate an average misorientation between all orientations in the spread. The FF-EPSC predictions show a similar Gaussian-like GAM distribution to the measured with an average of around 3.3 • The spread predicted by FF-EPSC model matches better the experiment in comparison to the FF-VPSC prediction presented in [51]. The larger spread predicted by FF-EPSC is attributed to the influence of additional elasticity terms in the second moment formulations not available in FF-VPSC.
For the tension simulation of AA5182-O to a strain of 0.45, Fig. 6 compares the measured deformed and recrystallized textures taken from [51] with the FF-EPSC predictions. The tensile deformation to 0.45 strain was achieved using the CBT apparatus [84] and approximated as a uniaxial tension in the FF-EPSC simulation [85]. The deformation texture evolved such that the {111} fiber arises in the pulling direction, i.e. the rolling direction (RD). The fully recrystallized texture was achieved via the grain boundary nucleation mechanism, and the predicted pole figures match reasonably well, supporting the experimental evidence showing the nucleation of rex-grains in aluminum alloys occurs at grain boundaries after tension [79,86]. Essentially, the deformation texture weakened during recrystallization, as expected.
Next, we performed a 60 % rolling reduction simulation of the BCC IF steel in Abaqus. Fig. 7a shows the initial setup for the rolling simulation, based on an earlier work reported in Zecevic, McCabe and Knezevic [87]. The simulation is a single rolling pass resulting in a 60 % reduction in thickness (from 2 to 0.8 in arbitrary units but can be taken as mm). The rolled sheet is uniform in the X-Z plane because the process is simulated in two-dimensions (2D). The initial dimensions of the modeled sheet are 5 mm in length and 2 mm in height. The half-model is used because of the presence of the orthotropic specimen symmetry using the symmetry boundary conditions. Plastic deformation of the half-model is applied via the rotation of the top roll with a radius of R = 8.3 mm by imposing an angular velocity of ω = 1/s. The friction coefficient between the roll and the sheet was assumed to be μ = 0.4. The value was estimated based on the geometry of bite angle of α = 20 • from the roll and the height of the sheet: μ > tan (α) = 0.36. The roller and the
sheet have hard contact. The sheet is meshed with 396 CPE4 bilinear plane strain elements. The full material information pertaining to the grains of the initial texture and associated material states is embedded in each integration point in the FE model that calls local FF-EPSC as a user material (UMAT) subroutine.
The partially rolled sheet is shown in Fig. 7b The figure shows the distribution of equivalent plastic strain during the rolling simulation. Upon entering the rolls, the elements near the surface undergo shearing because of the friction between the contacting rolls and the sheet surfaces. As a result, the polycrystal points near the surface experience different deformation history than the polycrystal points at the center of the sheet. These deformation histories determine the evolution of the texture. The predicted texture at the center resembles a typical FCC texture obtained under plane strain conditions [88], while the predicted texture near the surface resembles a combination of the typical texture components obtained under simple shear and plain strain conditions [89]. The labeled element on the bottom of the model is in the center of the sheet due to the symmetry. During the FEA simulation, grain fragmentation is active in all the grains embeded in the integration points during rolling. Recrystallization is performed for the selected surface and center elements after they exit the rolls, as indicated in the figure. These two recrystallization textures will be sufficient to initialize the cup drawing simulation, as will be presented in the following section.
The resulting deformed and recrystallized textures are shown in Fig. 8. The rolling process involved large and heterogeneous deformation. Large heterogeneous plastic strains caused texture gradients across the sheet. The deformed texture at the center matches a typical plain strain compression texture for BCC metals, while texture at the surface experiences some shear deformation owing to the interactions between the rolls and the sheet. IF steel is known to recrystallize via the grain boundary nucleation mechanism [90,91] and the resulting recrystallized texture preserves some of the deformed texture's features but is greatly weakened. In addition to the pole figures, Fig. 8 compares the measured orientation distribution function (ODF) section plots for the IF steel from literature [81,82] with the FF-EPSC predictions. The FE-FF-EPSC is evidently predicting well the rolled texture features i.e. the formation of γ-fiber and α-fiber [90,92,93]. During the recrystallization process, the intensity of α-fiber weakens, while the intensity of γ-fiber strengthens. The FF-EPSC predicts well these shifts in the recrystallization texture evolution. Therefore, the predictive characteristics of FF-EPSC extend well to the BCC-structured IF steel meaning that the FF-EPSC can be applied to a broad variety of cubic polycrystalline metals. Interestingly, owing to the grain fragmentation predictions by FF-EPSC, the texture of the top element is slightly weaker than the center because the top element predicted more grain fragmentations.
To further illustrate the capabilities of the developed FF-EPSC, we simulate the spatial texture variations during rolling and recrystallization of AA6022-T4 alloy like above for the IF steel, using the same rolling simulation setup. The objective is also to apply the predicted gradient in texture evolution in the through thickness direction of the sheet as the initial texture of the blank for drawing of a cylindrical cup. The cup drawing simulation of AA6022-T4 with EPSC was previously studied by [76], but the initial texture in the sheet was a uniform texture from EBSD measurements. The same cup drawing simulation is showing measured (e, f) and simulated (g, h) textures comparing the rolled (e, g) and recrystallized (f, h) texture evolution. After rolling, a total 1799 fragmented grains are simulated in the top element, while 1607 fragmented grains are simulated in the center element. After recrystallization, the top element has 1076 grains, and the center element has 840 grains. Fig. 9. Pole figures depicting predicted texture evolution for AA6022-T4 using FF-EPSC after: (a, c) 60 % rolling reduction at the two selected polycrystalline material points in Fig. 7 and (b, d) recrystallization of the rolled textures. Pole figures in (a, b) depict texture evolution near the surface of the rolled sheet, while those in (c, d) depict texture evolution at the center of the rolled sheet. After rolling, a total 2071 fragmented grains are simulated in the top element, while 1850 fragmented grains are simulated in the center element. After recrystallization, top element has 880 grains, and center element has 816 grains. performed here. Such linked processing sequence of rolling, followed by recrystallization, and finally deep drawing to predict texture evolution from an initially random texture to all the way after deep drawing is simulated for the first time in the present work.
The rolled and recrystallized textures are visualized in Fig. 9 via pole figures. Since the rolled sheet experienced shearing on the surface, a texture gradient formed in the through-thickness, i.e. Y-direction. The shearing is reflected as a rotation of the pole figures around the TD axis in comparison with the deformed texture at the center. To obtain the cube texture after full recrystallization, the alloy was set to recrystallize via transition-band nucleation, resulting in distinctly different texture resembling that of a cube, unlike rex-grains nucleated from grain boundaries that preserve the deformed texture pole figure features but weakened as shown in the previous section [44]. The predicted recrystallized texture resembles closely the measured initial texture of the as received AA6022-T4 used in [94]. The as received alloy was processed via rolling followed by recrystallization. Additionally, the shearing on the top element induced additional grain fragmentation and FF-EPSC simulated 2071 fragments compared with 1850 fragments in the center. Like results shown in Fig. 7, with more fragmentation activities, the pole figure intensity in the top element is weaker than at the center.
Next, we use the recrystallized textures to initialize an identical cup drawing simulation as presented in [76], a drawing of which is given in Fig. 10. Note that the grain fragmentation model was disabled during the cup drawing simulation for computational efficiency and also to isolate any improvement in the predicted earing profile owing to solely texture gradients. The punch is colored green, die is colored blue, and blank holder is colored red. The blank for a quarter cup is between the die and the die holder, with the center of the blank positioned under the punch. The simulation moves the punch downwards, forming the cylindrical cup. The punch and blank have a soft contact behavior with a coefficient of friction of 0.1. The blank consists of 14,564 C3D8R elements in 4 layers. The recrystallized texture from the rolled center element is assigned as the initial textures to the elements in the 2 center layers, while the texture from the rolled top element is assigned as initial textures to the elements in the top and bottom layers. After the cup is drawn, the punch contact is removed in a second process. Upon removal of the load, the cup exhibits spring back behavior, influenced by the calibrated backstress law, where stress is relaxed and the differences between peaks and valleys on the cup earing profile are slightly reduced.
Since the accuracy of predicting the earing profile depends on the anisotropic hardening and r-ratio, Fig. 11 shows comparisons of measured and predicted r-ratios. R-ratio is defined as the plastic widthto-thickness strain ratio and is known to be a strong function of texture [94,95]. The directional evolution of the r-ratio with plastic strain and the directional values at the 20 MPa plastic work for the rolled and recrystallized texture including both Fig. 9b and d textures are shown in Fig. 11. The experimental data to facilitate the comparisons are taken from Tian, Brownell, Baral and Korkolis [96] for the same alloy in an as-received rolled and recrystallized condition. The same data was used in our earlier modeling works of AA6022 involving the simulations of cup drawing [76,94]. The reasonable agreement between the predicted and measured data further confirms that the predicted rolled and recrystallized texture (Fig. 9) resembles closely the actual measured texture of the alloy demonstrating the capability of the FF-EPSC simulation framework to predict realistic rolling and recrystallization processes.
The simulated cup earing profile after spring back is shown in Fig. 12. Since only a quarter cup is simulated, the FF-EPSC predicted cup heights from 0 • to 90 • are mirrored across the 360 • range. With the gradient texture from a simulated rolling-recrystallization process, the cup height peaks at 0 • and 90 • from rolling direction match the measured values better than the previous study [76], however it preserved the valley at 45 • from the rolling direction, leading to a larger relative cup height profile. A photograph of the experimentally formed cylindrical cup is compared with the corresponding FE mesh after the simulation in the figure. The experimental data is taken from [96]. The initial texture and the latent hardening parameters in addition to the advantages of the novel FF-EPSC model are the differences between the current cup drawing simulation and that from [76]. The texture affects anisotropy more than latent hardening parameters, suggesting that the recrystallized texture gradient is a more accurate representation of the rolled sheet before drawing and therefore the predictions have improved.
We extracted the texture evolution history from select elements at the surface and in the middle of the drawn blank. These deformed textures are shown in Fig. 13, along with the equivalent strain contours over the deformed cup model. All pole figures are plotted in the global frame shown in the figure . The pole figures at the bottom of the cup (locations 3 and 6) are not deformed and not rotated, other pole figures are rotated into the same frame. The top set of pole figures at each location shows the texture at the surface of the cup while the bottom set shows the texture in the center. The cup is most deformed on the top at locations 1 and 4 where the cup experiences shear and circumferential compression and the deformed textures are the sharpest. While the pole figures at locations 2 and 5 are weaker than those at locations 1 and 4, they are more intense than the undeformed textures at locations 3 and 6. The gradient of textures remains during the cup drawing process.
In closing, we reflect on the computational time involved in the rolling and cup drawing simulations. The mean-field EPSC formulation is known to provide a favorable balance between accuracy and computational efficiency relative to the more computationally demanding but more accurate full-field models. The present work advanced the EPSC polycrystal plasticity model into the second moments field fluctuation FF-EPSC formulation incorporating the grain fragmentation and recrystallization models while preserving the computational efficiency. The calculations of the second moments slow down the standard EPSC for about 20 %. As the computational time scales linearly with the number of grains/inclusions, the increased in the number of grains because of the grain fragmentation is the major contributor to computational time. The AA6022-T4 rolling was initialized with 400 grains per integration point and ended with variable number of grains per integration point but on an average about 2000. The simulation took 3.89 h using 16 parallel processes on a workstation with 2.10 GHz Intel Xeon(R) Gold 6130 CPUs. The cup drawing simulation was initialized with 880 grains per integration point of surface elements and with 816 grains per integration point of center elements. The simulation took 344.45 h to complete on the same workstation using 16 CPUs.
This work advanced the EPSC polycrystal plasticity model into the second moments field fluctuation FF-EPSC formulation incorporating grain fragmentation and recrystallization models while preserving the computational efficiency. Specifically, the novel FF-EPSC formulation determines the second moments of lattice rotation rates and resulting intragranular misorientation spreads based on the second moments of stress rate fields inside grains. The calculated intragranular misorientation spreads enabled the grain fragmentation and recrystallization models within FF-EPSC. The former model improves the predictions of the deformation texture evolution, while the latter enables the modeling of recrystallization texture evolution. The recrystallization model considers the transition-bands and grain boundary nucleation mechanisms, while grain growth was governed by stored energy. Furthermore, the FF-EPSC model is physical because it incorporates a dislocation density-based law for slip system hardening and a backstress law to obstruct the dislocation motion. FF-EPSC model is further integrated in an implicit FE framework as a user material subroutine in Abaqus to facilitate predicting geometrical shape changes under complex boundary conditions. In FE-FF-EPSC, every integration point embeds the FF-EPSC constitutive law considering texture evolution and the directionality of deformation mechanisms operating at the single crystal level.
The deformation, grain fragmentation, and recrystallization models of FF-EPSC were validated by predicting tension of an FCC alloy, AA5182-O, and rolling of a BCC alloy, IF steel, and comparing the predictions with the experimental measurements available in literature. Predicted texture evolution during both deformation and recrystallization was found to be in good agreement with the corresponding measurements. Weakening of the texture intensities was successfully predicted after the recrystallization for both alloys via grain boundary nucleation. Moreover, the predicted intragranular misorientation spreads after tension of AA5182-O agreed well with the corresponding measurements. Next, the FF-EPSC model was applied to predict texture gradients formed during rolling and subsequent recrystallization of AA6022-T4. We demonstrated that FE-FF-EPSC can predict the typical rolling texture for FCC metals and the signature cube texture developed during recrystallization. The cube texture measured in AA6022 after rolling and recrystallization was predicted via enabling transition bands nucleation. Although the simulations of each alloy followed strictly one type of recrystallization nucleation, the FF-EPSC model can balance between the transition-bands and grain boundary nucleation mechanisms to achieve a texture with mixed types of nucleation. The nucleation mechanisms were classified by the type of misorientation spreads in the grains: the grains nucleate at transition-bands have bimodal spreads, while the grains that nucleate at grain boundaries have unimodal spreads. Lastly, the FE-FF-EPSC predicted gradient recrystallized textures were used as the initial gradient texture in the AA6022-T4 sheet used in a cup drawing process. The results showed improved drawn cup geometries relative to earlier works that considered a uniform texture. The deformed cup earing profile reflected the anisotropy of AA6022-T4 due to texture. Good predictions in benchmarking and applying the FE-FF-EPSC model demonstrated that the consideration of intragranular misorientation fluctuations and grain fragmentation is essential to more accurately modeling of texture evolution during deformation and permits modeling of recrystallization.
The simulated processing sequence of rolling, recrystallization, and deep-drawing of a cylindrical cup demonstrated the versatility of the developed FE-FF-EPSC simulation framework in predicting texture evolution and phenomena pertaining to behavior of materials and also geometrical changes important for the optimization of metal forming processes. Future works will focus on advancing the FF-EPSC further into a strain gradient SG-EPSC formulation, calculating geometrically necessary dislocations from the misorientation spreads to improve hardening and underlying backstress fields influencing slip system activation in an even more physical sense in a future FE-SG-EPSC.
Zhangxi Feng: Writingoriginal draft, Validation, Software, Methodology, Investigation, Formal analysis. Marko Knezevic: Writing review & editing, Supervision, Software, Resources, Project administration, Methodology, Investigation, Funding acquisition, Conceptualization.
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.
The initial conditions are:
The derivative of Eq. (A1) gives the hardening matrix used in Eqs. ( 5) and ( 18):
Where the components are:
In the above equations, k α 1 is one of the fitting parameters controlling the rate of generation of dislocations, k α 2 is a temperature (T) and the deformation rate tensor ( ε) sensitive parameter driving dynamic recovery, p is a reversibility parameter taken as 0.2, g ss ʹ is another interaction matrix set to g ss = 1 and g ss ʹ = 1 [100,103,104], m is a constant governing the rate of dislocation recombination set to 0.5 [105], and ρ s 0 is the total dislocation density at the point of reversal [102].
The coefficient k α 2 is:
where, k B is the Boltzmann constant, ε0 = 10 7 is a reference strain-rate, g α is another fitting parameter representing effective activation enthalpy, and D α is yet another fitting parameter representing drag stress. The debris/substructure dislocation density evolves using:
where q α is a last fitting parameter set to 4.0 determining the number of dislocations that become debris/substructure. The debris dislocation density evolves from a very small value of 0.1 m -2
When backstress calculations are enabled in the model, Eqs. (17a) and (17b) are modified as:
where τ s bs is the slip system backstress influencing the slip system activation [106,107]. τs bs is evolved similar to τs c :
where h ss ʹ bs = ∂τ s bs ∂γ s is similarly a backstress hardening matrix consisting of partial derivatives. The backstress accumulates during deformation and affects the overall material strength particularly upon load reversals, therefore, when the slip system is active during forward deformation, dγ s + > 0, we distinguish between the backstress accumulated in the positive and negative slip directions:
where τ sat bs is a saturation value, and A and ν are fitting parameters. During load reversal, backstress helps the driving force required to activate the slip in the opposite direction, s -, while the oposit backstress accumulated using:
where τ s + bs,0 is the accumulated backstress at the end of the previous deformation process and γ b is a fitting parameter. Additional details of the phenomenological backstress model can be found in [27,107].
The EPSC model was previously implemented as an UMAT subroutine in the implicit FE framework to simulate complex deformation processes and named finite elements (FE) EPSC [6,7,15,108,109]. The new FF-EPSC formulation went directly into the FE-EPSC framework making FE-FF-EPSC. In this section, the notation FE denotes quantities passed between FE solver and EPSC. For elements with multiple integration points per element, EPSC is called for each integration point independently, and the entire texture data is embedded in each integration point. FE passes boundary condition as strain increment, D FE Δt, and time step, Δt, the updated strain is simply:
Then ESPC returns both a homogenized converged stress from EPSC subroutines, σ t+Δt FE , and calculates a Jacobian matrix, ∂ΔσFE ∂DFEΔt , that estimates a trial displacement field for the FE solver's convergence test:
The Jacobian is therefore the incremental polycrystalline effective stiffness relating Cauchy stress rate and strain rate [109]. More details can be found in the referenced paper.
This appendix presented the derivation of ∂M/∂M c . The section is written in indicial notation for better clarity and the fourth rank symmetric 3 × 3 × 3 × 3 tensors are expressed as second rank 6 × 6 tensors via Voigt notation for simplicity. We start by taking the partial derivative of the effective polycrystalline compliance with respect to the grain-level compliance: , α is the unit vector associated with a point in the Fourier space used as part of the Green functions solutions for the Eshelby inclusion problem, and:
where A is a 4 × 4 matrix defined as:
Taking the derivative of Eq. (B5) with respect to the compliance gives:
where the partial derivatives follow:
The partial derivative ∂L klmn ∂Mpqrs is simply the derivative of the stiffness with respect to the compliance. To calculate the derivative, we write both fourth rank 3 × 3 × 3 × 3 tensors as second rank 6 × 6 tensors using Voigt notation:
Finally, we combine the above together and write the partial derivative of the Eshelby tensor with respect to compliance as:
This appendix describes the process of sampling orientations from misorientation spreads. The process summarized here is originally described in the appendix of Zecevic, Lebensohn, McCabe and Knezevic [41].
After deformation and fragmentation, a discrete statistic of misorientations in each original grain can be sampled based on the second moments of misorientations from the average grain orientation. The algorithm generates random vectors belonging to a Gaussian distribution of zero mean and unit variance based on the gasdev subroutine provided in Press, Flannery, Teukolsky, Vetterling and Kramer [110]. Then, a set of randomly generated vectors from gasdev are mapped to the second moment of misorientation distribution by calculating the Cholesky decomposition of the second moment of misorientation spread:
where L is the lower triangular matrix and transforms random vectors, v, with zero mean and unit variance to random vectors, δr c,rand , with zero mean and second moment 〈δr ⊗ δr〉 c variance, and the inverse L -1 transforms from δr c,rand back to x:
To generate the data used to create Figs. 2 and 3, we specify an arbitrary number of vectors (500) to sample from the misorientation distribution of the original grain. We plot these 500 misorientation vectors as blue points. Some of these discrete misorientation vectors can gave rise to the rex-grains during recrystallization. These are plotted as red points. The criteria for selecting the vectors for grain boundary or transition band nucleation are described below. Note that the actual orientation of a rex-grain needs to be obtained from the selected misorientation vector.
For grain boundary misorientations, if the length of x is larger than an adjustable input constant, c, then the length of δr c,rand is larger than the condition c × SD ( δr c,rand / ⃒ ⃒ δr c,rand ⃒ ⃒ ) , and such a vector is accepted to exist on the grain boundary. SD ( δr c,rand / ⃒ ⃒ δr c,rand ⃒ ⃒ ) is the standard deviation along the direction of the vector δr c,rand . If the length of x is smaller than c, then it is rejected, and a new random vector is drawn. The constant c represents the magnitude of misorientation of grain boundary orientation with respect to the mean orientation of the grain. It is set to a value of 2.0 to ensure the sampled vector is sufficiently large. The first point satisfying the criterion is taken as a nucleus. However, for obtaining Figs. 2 and 3 the algorithm was checked against all 500 points. These were presented as red points.
For transition band misorientations, the random vector δr c,rand is instead projected to the dominant rotation axis, v 1 , and compared to the projection of the modes of the probability density function described by the bimodal grain's 〈δr ⊗ δr〉 c . A similar rejection check ensures the random vector falls between the modes from where a random red point is selected to nucleate a rex-grain.