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The anisotropic grain size effect on the mechanical response of polycrystals: The role of columnar grain morphology in additively manufactured metals

Additively manufactured (AM) metals exhibit highly complex microstructures, particularly with respect to grain morphology, which typically contains heterogenous grain size distribution, anomalous and anisotropic grain shapes, and the so-called columnar grains. In general, the conventional morphological descriptors are not suitable to represent the complex and anisotropic grain morphology of AM microstructures. In this paper, we formally introduce the notions of axial grain size and grain size anisotropy as robust morphological descriptors, which can adequately represent highly complex grain morphologies. We instantiated a discrete sample of polycrystalline aggregate as a representative volume element, which has random crystallographic orientation and misorientation distributions. However, the instantiated volume element contains the typical morphological features of AM microstructures including distinctive grain size heterogeneity and anisotropic grain size owing to its pronounced columnar grain morphology. We ensured that any anisotropy that arises in the macroscopic mechanical response of the instantiated sample is mainly associated with its underlying anisotropic grain size. The representative volume element was then used for meso-scale full-field crystal plasticity simulations corresponding to uniaxial deformation along different axes via a spectral solver and a physics-based crystal plasticity constitutive model. Through the numerical analysis, we were able to isolate the contribution of anisotropic grain size to the anisotropy in the mechanical response of polycrystalline aggregates, particularly those with the characteristic complex grain morphology of AM metals. Such a contribution can be compactly described by an exponential decay relationship.


Introduction
Deformation behavior of polycrystalline materials is one of the most complex phenomena in Materials Science. The complexity of polycrystal deformation is attributed to the complex hierarchical polycrystalline microstructure (i.e., multi-scale intrinsic structure of polycrystalline materials) and its complex evolution during elasto-viscoplastic deformation. One of the most complex aspects of polycrystal deformation behavior is the anisotropy in the macroscopic mechanical response, which arises due to the presence of microstructural texture [1,2].
Metal additive manufacturing enables production of metallic polycrystals with extremely high microstructural complexity compared to its conventional counterparts [2]. One of the characteristic complexities of additively manufactured (AM) polycrystalline microstructures is associated with their grain morphology. The typical grain morphology in AM microstructures consists of heterogenous multi-modal grain size distribution, anomalous and anisotropic grain shapes, as well as the so-called columnar grains, which are ascribed to directional solidification and epitaxial grain growth during additive manufacturing [3][4][5][6][7].
The crystalline microstructural texture/polarity can be divided into crystallographic texture and morphological texture [1,2]. Conventionally, the morphological texture is represented through morphological descriptors/measures associated with idealized grain shapes [8][9][10][11] (Appendix A). In that approach, the morphological descriptors are parameters corresponding to the equivalent/best-fit sphere and ellipsoids of the grains of a microstructure: equivalent size, equivalent minimum aspect ratio, equivalent maximum aspect ratio, parameters corresponding to the equivalent axes orientation, as well as the number of contiguous neighboring grains. The conventional morphological representation based on grain shape idealization has two major shortcomings, particularly with respect to the heterogenous microstructures processed by unconventional manufacturing technologies such as metal additive manufacturing processes: • Due to the high number of morphological descriptors, it is highly (computationally) demanding to construct a distribution function that preserves the correlations between them. Moreover, having a multivariate joint probability distribution of grain morphology is not very useful with respect to representation and analysis. Therefore, almost always, the distributions of the descriptors are treated without considering their correlation with the other descriptors. In other words, it is presumed that the correlations of the morphological descriptors are fully random. However, in AM metals, the morphological descriptors often have strong correlations. For instance, the columnar grains typically observed in AM metals are characterized by their large equivalent size and highly elongated shapes (low equivalent minimum aspect ratio) whose equivalent major axes are predominantly oriented with low angle with respect to the build direction (BD). Moreover, the typical multi-modal distributions of the conventional morphological descriptors in AM microstructures are attributed to the aforementioned correlations. • The grain shape simplification is often not suitable to adequately represent the anomalous and complex grain shapes in unconventional microstructures, e.g., those typically observed in the microstructures of AM metals. For instance, ellipsoid simplification cannot sufficiently capture the anisotropy of columnar grains in AM microstructures [2]. As another example, in AM microstructures, there are many grains with substantial concavity, while sphere and ellipsoid are fully convex shapes. Moreover, one can usually find small grains fully enclosed by larger grains (i.e., with only one contiguous neighbor).
Given the significance of the grain size effect on the mechanical response of crystalline materials, it is reasonable to define a single-variable morphological descriptor based on grain boundary spacing to properly represent the state of grain size as well as the morphological texture. The equivalent size of a grain approximates its average grain boundary spacing but it does not give any information regarding the anisotropic shape of the grain and its spatial orientation (Appendix A). Even though the equivalent sphere/circle diameter is becoming the method choice to approximate the (scalar) effective grain size, the mean of the so-called chord length distribution provides a more accurate estimate to the effective grain size, particularly in the microstructures with anisotropic grain shapes [79][80][81]. Moreover, the so-called angularly resolved chord length distribution has been used to simultaneously represent the state of grain size and morphological texture of twodimensional (2D) sections of polycrystalline microstructures [82], the state of anisotropic spacing of dispersed particles in 2D sections of a matrix, as well as the size and morphological texture of pores within a threedimensional (3D) matrix [83]. In this context, the chord, or what we refer to as the axial grain size, is defined as a line segment parallel to a certain axis/direction whose end points lie at grain boundaries while its interior points are all located within a single grain. As such, the definition of the axial grain size or chord is not ambiguous. Nevertheless, given a microstructure and a probe axis, estimation of the underlying axial grain size distribution (or chord length distribution) has not yet been properly conceptualized. The standard approach is that grids of scan lines parallel to some axes/directions are intercepted with the microstructure to estimate the corresponding axial grain size distribution.
The equivalent grain size distribution in a microstructure can be represented using the number or volume/area-weighted distributions (Appendix A). In case of a heterogenous (multi-modal) grain size distribution, such as those one typically observe in the microstructures of welded and AM metals, it is recommended to estimate the effective grain size using the volume-weighted distribution of equivalent grain size instead of the number-weighted distribution [1,[84][85][86][87]. Particularly, in the microstructures where there are small number of large grains with significant contributions to the aggregate's volume, the heterogeneity and bimodal character of the grain size distribution will be lost or highly underestimated in representation by number-weighted equivalent grain size distribution. Hence, the effect of the small number of large grains in the microstructure will not be properly expressed in the estimated effective grain size obtained from the corresponding number-weighted equivalent grain size distribution. Therefore, the most precise approach to simultaneously represent the state of grain size and morphological texture of heterogenous microstructures with anisotropic and anomalous grain shapes is through appropriately weighted axial grain size distribution.
In this paper, first, we present a rigorous continuum theory for the axial grain size, the axial grain size distribution, and the grain size anisotropy. Then, we apply the theory to an instantiated discrete sample of (single-phase) polycrystalline mesostructure which has the typical morphological features of AM metallic microstructures including distinctive size heterogeneity and morphological texture owing to its pronounced columnar grain morphology. Based on the notion of axial grain size, the columnar grains are characterized by their large axial grain size along BD relative to the axes orthogonal to BD. In the present study, we seek the contribution of the morphological texture and anisotropic grain size to the anisotropic mechanical response of polycrystals, particularly those with pronounced columnar grain morphology. Therefore, we isolated the morphological texture of the instantiated mesostructure by assigning crystallographic orientation to its grains in such a way to obtain random crystallographic orientation and misorientation distributions (i.e., crystallographically textureless microstructure). As a result, we assume that any anisotropy in the macroscopic mechanical response must be associated with the underlying anisotropic grain size of the microstructure [1,2]. The strain hardening anisotropy associated with the generated aggregate, which describes the anisotropy in its macroscopic strain hardening response under uniaxial deformation, was evaluated. To that end, the instantiated mesostructure sample was used as a representative volume element (RVE) for meso-scale full-field crystal plasticity simulations corresponding to uniaxial (tensile) deformation along different axes via a spectral solver and a physics-based crystal plasticity constitutive model [1].
Using the uniaxial strain hardening as a measure to evaluate the anisotropic mechanical response has several advantages. Firstly, it does not require to find a distinct border as the point of transition from macroscopic fully elastic deformation to macroscopic elasto-viscoplastic deformation (i.e., macroscopic yield point) as treated in the phenomenological orthotropic yield functions [88][89][90][91][92]. Such a macroscopic transition does not occur at an instant during deformation and is often associated with a notably wide strain range, particularly in the absence of yield point elongation, where the arbitrary notion of "proof stress" is used to represent the macroscopic yield stress. Secondly, the term "anisotropy" indicates variation of a certain feature, property, or response of a material along different axes. Therefore, measuring the macroscopic mechanical response of the material to complex multiaxial macroscopic boundary conditions does not explicitly reveal the state of the anisotropy in the macroscopic mechanical response. Thirdly, at small strains, the anisotropy in the stress response of strongly textured polycrystalline materials might remain hidden [1,93]. However, since the strain hardening response is a derivative quantity (derivative of the stress response with respect to strain), it is extremely sensitive to small details in microstructural texture and thus can unravel the anisotropy underlying the mechanical response.

Theory
To seek the contribution of anisotropic grain size on strain hardening anisotropy, first we need a continuum theory for axial grain size and grain size anisotropy as well as uniaxial strain hardening and strain hardening anisotropy. The grain size and the strain hardening are elements of microstructural and micromechanical states (i.e., micro-states), respectively. Even though the grain size anisotropy and the strain hardening anisotropy depend on those micro-states, they represent aspects of the macro-state (i.e., macroscopic state) of the material. In this section, we present formal definitions and theoretical background related to the aforementioned notions. In the following sections, in pursue of a relationship between the grain size and strain hardening anisotropies, we use this theoretical framework to calculate axial grain size, grain size anisotropy, uniaxial strain hardening, and strain hardening anisotropy for an instantiated discrete polycrystalline microstructure.

Microstructural grains and grain boundaries
The polycrystalline microstructure domain as a continuum medium can be divided into finite volumetric regions (subsets of microstructure domain) known as grains/crystallites. The grain boundary corresponding to a grain in a microstructure can be idealized as a closed topological surface embedded in Euclidean 3-space enclosing the grain interior, where each point on the grain boundary is associated with a relatively sharp transition (jump) of microstructural state (combination of crystallographic orientation and phase) across it beyond a certain threshold. Let the finite microstructure domain at time , Ω ≡ Ω( ) ⊂ ℝ 3 , be discretized to ∈ ℕ grains occupying the regions g 1 , g 2 , … , g , and their grain boundaries g 1 , g 2 , … , g , so that: G ≡ {g ⊆ Ω | ∈ ℕ; ≤ }; ∀g , g ∈ G, g ≠ g : g ⋂g = ∅; G ≡ { g ⊆ Ω | ∈ ℕ; ≤ }; ∀ g , g ∈ G: g ⋂ g { ≠ ∅ : g and g are contiguous = ∅ : otherwise ; where G and G respectively represent the set of all grains and grain boundaries in the microstructure domain Ω, which is simply connected. The elements of G (grains) are pairwise disjoint and form a partition of Ω − G. Moreover, according to definition, two grains are contiguous if and only if the intersection of their boundaries is not the empty set.
(2) Therefore, the direction ̂ is a vector from the origin of the unit sphere (sphere with unit radius) to a particular point on the surface of the unit sphere. Thus, an arbitrary direction ̂ belongs to the 3D direction space (set of all 3D directions), which can be represented by the surface of the unit sphere (i.e., 2-sphere) ( 2 ): where ‖⦁‖ denotes the Euclidean norm of ⦁. Direction ̂ as well as its inverse (−̂) uniquely define an axis. Hence, an axis is represented by ̂ or ±̂; and a line is an axis that passes through certain point(s). Therefore, the surface of a unit hemisphere is a fundamental/asymmetrical region/zone of the axis space, which is the set of all possible 3D axes. The fundamental region of the axis space has the property that there is an injective/oneto-one map between its distinct elements to distinct axes, and thus it corresponds to the fundamental region of direction space with antipodal symmetry, which is conventional to be represented by the surface of the "northern" unit hemisphere ( z+ 2 ):

Grain size anisotropy
The axial grain size distribution corresponding to the axis ̂ ( ( |̂)) is the conditional probability density of measuring the axial grain size (̂, ) at random point in the microstructure given the axis ̂. Intuitively, given a microstructure domain Ω ⊂ ℝ 3 and axis ̂∈ 2 , one can think of ( |̂)d as differential volume fraction of material points corresponding to axial grain size falling in an infinitesimal range around : where is the volume of the microstructure domain Ω; d denotes a differential volume; and d | ,̂ is the notation for differential volume fraction of material points in Ω which are associated with axial grain size where d 3 ≡ d d d denotes a differential volume in the vicinity of point ; and ̅ (̂) is the mean axial grain size associated with the axis ̂ or the mean grain boundary spacing along the axis ̂. Note that the mean axial grain size ( ̅ (̂)) and the axial grain size distribution ( ( |̂)) functions inherit the antipodal symmetry of the axial grain size ( (̂, )): ̅ (̂) ≡ ̅ (−̂) and ( |̂) ≡ ( |−̂).
Furthermore, the effective grain size ( ̿ ) is formally defined as the mean of volume weighted distribution of axial grain sizes corresponding to all axes ̂∈ z+ 2 and material points ∈ Ω: where d 2̂≡ sin d d denotes a differential area on the surface of the unit sphere in the vicinity of direction ̂ (i.e., the invariant measure on 2 ); and ∫ d 2ẑ + 2 = 2 is the size of the fundamental region of axis space (i.e., the area of the unit-hemispherical surface). In other words, the effective grain size ̿ is the expectation value of the probability distribution ( ), which returns the probability density of grain boundary spacing in the given microstructure: where ( ,̂) denotes the joint probability density of random variables and ̂; and (̂) ≡ 1 2 is the probability density of axis ̂ within the axis space. Combining Eqs. (12) and (14) leads to Eq. (13).
Given a microstructure and Eqs. (12) and (13), we define the grain size anisotropy (̂(̂)) as mean axial grain size normalized by the effective grain size: The dimensionless grain size anisotropy function ̂(̂) is a map from the surface of the unit sphere to real numbers (̂: 2 → ℝ) and has the unit of "multiple of random distribution" (mrd) representing the state of grain size anisotropy of the microstructure. According to the definitions (Eqs. (13) and (15)), the average of ̂(̂) over its doamin is unity: . Given an axis ̂, ̂(̂) > 1 indicates that the mean axial grain size associated with ̂ is greater than the effective grain size ( ̅ (̂) > ̿ ); and if ̂(̂) < 1, the mean grain boundary spacing parallel to ̂ is lower than the effective grain size ( ̅ (̂) < ̿ ). The function ̂(̂) is a fundamental morphological descriptor of the microstructure which gives a representation of the state of the morphological texture and morphological orientation distribution. The function ̂(̂) and the scalar ̿ together provide a representation of grain morphology (combined morphological size and orientation). In that sense, the axial grain size provides a more comprehensive representation of microstructural grain morphology compared to the conventional representations of grain morphology using shape idealization (Appendix A).
Moreover, one may apply the preceding definitions to the individual grains of a microstructure. Assuming a spherical grain with diameter , we readily find that ̿ = 2 . It is intuitive that the mean axial grain size of a spherical grain is lower than its diameter ( ̿ < ) as the diameter of sphere is the maximum possible axial grain size which can be measured for the sphere (∀̂∈ 2 : 0 < ̅ (̂) ≤ ). Furthermore, the mean axial grain size of a (hypothetical) spherically symmetric grain is independent from the probe axis and has a constant value for any arbitrary axis ̂ (∀̂∈ 2 : ̅ (̂) = ̿ ). Therefore, for a spherical grain: ∀̂∈ 2 : ̂(̂) = 1, as sphere is the only fully isotropic shape. Since each continuous function ̂(̂) uniquly defines a 3D shape, a microstructure with the grain size anisotropy function ̂(̂) has a unique equivalent grain shape which is represented by ̂(̂) .
The grain size anisotropy function ̂(̂) is a periodic real square-integrable antipodally symmetric function on the surface of the unit sphere ( 2 ). In other words, ̂∈ ℝ± 2 ( 2 ), where ℝ± 2 ( 2 ) denotes the Hilbert space of real square-integrable functions with antipodal symmetry on the surface of the unit sphere (Appendix B). Therefore, the grain size anisotropy function ̂(̂) can be expressed by series expansion of spherical harmonic basis functions (Appendix B) as follows: where : 2 → ℂ denotes the spherical harmonic function (SHF) of degree ∈ ℕ 0 and order ∈ { ∈ ℤ | | | ≤ }; ̃∈ ℂ is the Fourier coefficient of SHF of degree and order associated with the function ̂; and asterisk * indicates complex conjugate. Since the SHFs form a complete set of orthonormal basis functions for the Hilbert space ℝ± 2 ( 2 ), the Fourier coefficient ̃ is the inner product of the SHF with the function ̂∈ ℝ± 2 ( 2 ): Furthermore, given a microstructure, we define the grain size anisotropy index as a dimensionless scalar measure characterizing the average deviation from the fully isotropic counterpart of the microstructure: where denotes the anisotropy index functional; [̂] is the grain size anisotropy index; 〈 〉 ∈ ℝ denotes the 2 -norm of function ∈ 2 ( 2 ) (Appendix B).
Let mixed velocity and traction boundary conditions be imposed on the disjoint parts of Ω ( Ω denotes the boundary of the microstructure domain Ω at time ). The balance of linear momentum for Ω in the absence of considerable body and inertia forces can be expressed by ∀ ∈ Ω: ∇ 0 ⋅ = , where ( , ) denotes the first Piola-Kirchhoff stress tensor, which is the work conjugate of . The deformation rate is related to stress rate through the underlying constitutive relationship ̇≡ :, where the fourth-order tensor ≡ ∂ ∂ denotes the elasto-viscoplastic tangent stiffness modulus, which is a spatiotemporal function ( ≡ ( , )).
The uniaxial strain hardening is associated with the macroscopic mechanical response when the microstructure is subjected to homogenous uniaxial (tensile or compressive) deformation along a specific axis at constant strain rate and temperature. Homogenous macroscopic boundary conditions demand that the macromechanical quantities (macroscopic deformation rate and stress on Ω) are the volume averages of the corresponding micromechanical fields (microscopic deformation rate and stress in Ω): where ̅( ) and ̅ ( ) denote the macroscopic/homogenized rate of deformation gradient and first Piola-Kirchhof stress tensors. The macroscopic mechanical boundary conditions associated with homogenous uniaxial deformation can be imposed by prescribing the components of ̅( ) and ̅ ( ) in a mutually exclusive way (with respect to a single reference frame).
Based on the polar decomposition theorem the non-singular deformation gradient ( ) is uniquely decomposed into the product of the symmetric left stretch tensor ( ) and the orthogonal rigid rotation tensor ( ): The true work-conjugate pairs, the true/Cauchy stress tensor ( ( , )) and the true/logarithmic strain tensor ( ( , )), at point ∈ Ω and time are related to ( , ) and ( , ) as follows: where is the Jacobian of the deformation map; and and denote the eigenvalues and eigenvectors of , respectively. As a result of the homogenous macroscopic boundary conditions: where ̅( ) and ̅( ) are the homogenous macroscopic true strain and stress tensors (at Ω), respectively. The true stress and strain tensors are decomposed to their corresponding deviatoric, and volumetric/hydrostatic parts as follows: where ̅ and ̅ denote the deviatoric and volumetric splits of the macroscopic true strain tensor ̅, respectively; and ̅ and ̅ ℎ stand for the deviatoric and hydrostatic splits of the macroscopic true stress tensor ̅, respectively. The equivalent/von-Mises macroscopic stress and strain (scalars) are defined as: where and represent the equivalent macroscopic true stress and strain, respectively; and ‖ ‖ ≡ √ : denotes the 2-norm of second-order tensor . . Moreover, the equivalent macroscopic strain hardening or tangent modulus ( ( )) is defined as:

Strain hardening anisotropy
The mean uniaxial strain hardening associated with the axis ̂ within the fixed equivalent macroscopic true strain interval [0, ] ( ̅ (̂, )) is defined as follows: where (̂, ) denotes the instantaneous equivalent macroscopic strain hardening response of microstructure Ω subjected to the homogenous boundary conditions corresponding to the uniaxial deformation along the axis ̂ at equivalent macroscopic true strain , constant macroscopic nominal strain rate ̇0, and constant temperature . Given ̅ , one could readily arrive at the instantaneous macroscopic stress response of the microstructure under uniaxial deformation along ̂ at any equivalent macroscopic true strain by using ̅ as linear operator: ≡ ̅ . Therefore, (̂, ) fully represents the macroscopic mechanical response of the microstructure subjected to uniaxial deformation at constant macroscopic nominal strain rate ̇0 and constant temperature . Note that, analogous to the mean axial grain size ( ̅ ), the mean uniaxial strain hardening ̅ is antipodally symmetric: ̅ (̂, ) ≡ ̅ (−̂, ). Furthermore, the effective uniaxial strain hardening ( ̿ ( )) is defined as the volume-weighted average of mean uniaxial strain hardenings corresponding to all axes ̂∈ z+ 2 and the equivalent macroscopic strain interval [0, ]: Now, we can define the dimensionless strain hardening anisotropy (̂(̂, )) as the mean uniaxial strain hardening normalized by the effective uniaxial strain hardening: Given an axis ̂ and equivalent macroscopic true strain > 0, ̂(̂, ) > 1 indicates that the mean uniaxial strain hardening associated with ̂ is greater than the effective uniaxial strain hardening ( ̅ (̂, ) > ̿ ( )); and if ̂(̂, ) < 1, the mean uniaxial strain hardening along the axis ̂ is lower than the effective uniaxial strain hardening ( ̅ (̂, ) < ̿ ( )). Assuming a constant final equivalent macroscopic true strain > 0, the strain hardening anisotropy function ̂(̂) ≡̂(̂, )| = is a map from the surface of the unit sphere to real numbers (̂: 2 → ℝ) and has the unit of mrd representing the anisotropy state in the uniaxial strain hardening of the microstructure. Due to the normalization, the average of ̂(̂) over its doamin is unity: Analogous to the grain size anisotropy function ̂(̂) , the strain hardening anisotropy function ̂(̂) is a periodic real square-integrable antipodally symmetric function on the surface of the unit sphere (̂∈ ℝ± 2 ( 2 )). Therefore, the strain hardening anisotropy function ̂(̂) can be expressed by series expansion of spherical harmonic basis functions (Appendix B) as follows: where ̃∈ ℂ is the Fourier coefficient of SHF of degree and order associated with the function ̂. Since the SHFs form a complete set of orthonormal basis functions for the Hilbert space ℝ± 2 ( 2 ), the Fourier coefficient ̃ is the inner product of the SHF with the function ̂: Furthermore, given a microstructure, we define the strain hardening anisotropy index as a dimensionless scalar measure indicating the mean deviation from the fully isotropic counterpart of the microstructure: where [̂] is the strain hardening anisotropy index. Since ̂ is properly normalized, based on the Cauchy-Schwarz inequality: 〈̂〉 ≥ 4 . For a microstructure with an isotropic strain hardening: [̂] = 0.

Strain hardening anisotropy due to grain size anisotropy
We hypothesize that any macroscopic anisotropy in the initial microstructure Ω 0 contributes to the overall anistropy in the mean (macroscopic) uniaxial strain hardening. The sources of microstructural anisotropy in crystalline materials are crystallographic and morphological texture/polarity [2]. Assuming that in a hypothetical crystalline microstructure the only source of microstructural anisotropy is the morphological texture, which is best represented by the grain size anisotropy function, one can isolate the contribution of the grain size anisotropy in the strain hardening anisotropy. According to Eqs. (28)- (30), to derive the strain hardening anisotropy function ̂(̂) , the initial microstructure Ω 0 must be uniaxially deformed to the final equivalent macroscopic strain = along different axes. For large , the deformed microstructure Ω is appreciably different than its initial state Ω 0 mainly due to plasticity. Nevertheless, the mean uniaxial strain hardening associated with any axis ̂ ( ̅ (̂, )) and thus the strain hardening anisotropy ̂(̂) depend on the grain size anisotropy of the initial microstructure ̂0 (̂) (i.e., the initial grain size anisotropy). Therefore, we seek the relationship between the functions ̂(̂) and ̂0 (̂), which can be expressed via functional ℱ so that ̂≡ ℱ[̂0]. In the following, for brevity we drop the subscript 0 from quantities associated with the axial grain size of the initial microstructure Ω 0 as the axial and anisotropic grain size analysis is always carried out on the initial microstructure. Henceforth, the quantities , ̅ , ̿ , and ̂ all correspond to the initial domain Ω 0 .

Methods
Given an initial microstructure Ω 0 , a point 0 ∈ Ω 0 , and an axis ̂∈ z+ 2 , one can unambiguously measure the axial grain size ( 0 ,̂). However, to obtain the axial grain size distribution ( |̂), a spatial discretization/binning/griding/meshing of the microstructure domain Ω 0 is required. Nonetheless, the result of any experimental microstructure characterization, microstructure evolution simulation, or microstructure instantiation is always a discrete microstructure sample. Therefore, the measured axial grain size distribution is, in fact, an approximation of the underlying probability density function ( |̂). Moreover, to obtain the grain size anisotropy ̂(̂) and strain hardening anisotropy ̂(̂) , one also needs a discretization of the axis space z+ 2 (i.e., binning of the spherical coordinates ( , )). Finer discretization resolution of the microstructure domain Ω 0 and axis space z+ 2 leads to more accurate estimation of ( |̂), ̂(̂) , and ̂(̂) .

Instantiation of discrete mesostructure
In the present study, a (single-phase) polycrystalline aggregate with a uniform grid was instantiated, which has the typical morphological features of AM metallic mesostructures including distinctive size heterogeneity and morphological texture corresponding to a pronounced columnar grain morphology. The morphological texture of the instantiated mesostructure was isolated by assigning crystallographic orientation to its grains so that almost random crystallographic orientation and misorientation distributions (i.e., crystallographically textureless microstructure) were obtained. As a result, we assume that any anisotropy that arises in the macroscopic mechanical response must be due to the underlying anisotropic grain size of the aggregate. The instantiated mesostructure was used as a representative volume element (RVE) for axial grain size measurements as well as meso-scale full-field crystal plasticity simulations corresponding to uniaxial (tensile) deformation along different axes.
A robust mesostructure modeling formalism based on grain shape idealization [8,11,[94][95][96], which is implemented in the DREAM.3D code [97], was used to generate the mesostructure RVE. The RVE generation process assembles a sequence of modules (ellipsoidal grain generator, constrained grain packer, and seed point generator-constrained Voronoi tessellation) and produces a statistically realistic model of the corresponding mesostructure based on the conventional morphological descriptors associated with grain shape idealization. A cubic RVE with the edge length of 200 μm and = 1145 grains was instantiated using the microstructure descriptors corresponding to grain shape idealization (Appendix A), which were obtained from experimental microstructure characterization and analysis of an additively manufactured single-phase (fully austenitic) high-Mn steel [1,2]. The RVE was partitioned to a uniform grid of cubic elements of the same size (i.e., voxels) with grid resolution of = 50 3 voxels, where each voxel belongs to one and only one grain. The generated RVE has periodic boundaries: the grains that are cut by one of its faces appear from the opposite parallel face unless the respective face cuts the corresponding grain at its grain boundary. Crystallographic orientations were assigned to the grains of the RVE in such a way that no intragranular misorientation exists and the overall crystallographic orientation and misorientation distributions are almost random (i.e., absence of crystallographic texture). The crystallographic orientation distribution function (ODF) and grain boundary misorientation distribution function (MDF) in terms of Bunge-Euler angles (Appendix C) are computed using a fast algorithm for generalized spherical harmonics expansion based on kernel density estimation [98,99], which is implemented in MATLAB®-based MTEX toolbox [100,101].
The maximum number of grains embedded in a mesostructure volume element is limited by the extent of morphological and crystallographic textures and heterogeneities as well as the grid resolution (e.g., a grid resolution of 10 3 cannot contain more than 10 3 grains). Moreover, the lower bound for the required number of grains inside an RVE also depends on the overall textures and heterogeneities. For instance, to model a single crystal material an RVE with a grid resolution of one voxel is sufficient; and for modeling a grain with an aspect ratio of 3 to 1, at least three voxels must be allocated. The instantiated RVE was then analyzed using the microstructure descriptors based on grain shape idealization to ensure that the morphological and crystallographic textures and heterogeneity characteristics of interest corresponding to typical AM metallic mesostructures were sufficiently captured.

Discretization of axis space
Given a microstructure, to obtain the axial grain size distribution ( |̂) and the corresponding grain size anisotropy ̂(̂) as well as the strain hardening anisotropy ̂(̂) , one needs a uniform discretization of the axis space. A non-uniform discretization of the axis space would lead to biases: the regions on the surface of the unit sphere (i.e., the axis space) where the density of sampled points/axes is higher would have a higher weight in the corresponding distribution compared to the regions with lower density of sampled points. However, discretization of the surface of unit sphere is not as straightforward as uniform discretization of a cubic RVE embedded in the Euclidean 3-space. We adopted the algorithm proposed by Leopardi [102] to partition the surface of the northern hemisphere of the unit sphere ( z+ 2 ) into = 127 ∈ ℕ regions of equal area. The center point of each region, which is denoted by ̂, represents the i-th axis used to measure the axial grain size and the uniaxial strain hardening. As such, R ≡ {̂≡ ( , ) ∈ z+ 2 | ∈ ℕ; ≤ } represents the finite set of discrete measurement axes. Fig. 3 shows the grid of = 127 equispaced axes ̂, which were uniformly sampled from the surface of the northern hemisphere of the unit sphere ( z+ 2 ).

Grain size anisotropy
Prior to the axial grain size measurements, the instantiated cubic RVE was reconstructed so that the segments of the boundary grains (i.e., the grains cut by one of the faces of the RVE) with periodic counterparts are merged. The measurement of axial grain size was carried out at the center point of each voxel in the (undeformed) RVE. X ≡ { ∈ Ω 0 | ∈ ℕ; ≤ } represents the set of measurement nodes associated with the discrete microstructure domain Ω 0 and its corresponding -voxel discretization, where denotes the position of the center of volume of the j-th voxel. Therefore, the measured axial grain size at the node ∈ X along the axis ̂∈ R, which is denoted by ≡ (̂, ), can be obtained using Eqs. (5) or (9), where underscore indicates that the respective quantity is obtained by means of direct measurement. It should be noted that the grain boundaries in the instantiated discrete mesostructure are comprised of voxel faces. Given Eqs. (12)-(15) and the sets D ≡ D(̂) ≡ { | ∈ ℕ; ≤ } and D ≡ { | , ∈ ℕ; ≤ ; ≤ }, the measured mean axial grain sizes, effective grain size, and the grain size anisotropy associated with the discrete microstructure domain Ω 0 are calculated as follows: where ̂ and ̅ respectively denote the measured grain size anisotropy and the measured mean axial grain size associated with the probe axis ̂; and ̿ is the measured effective grain size.
Given Eq. (16), the measured discrete grain size anisotropies ̂ can be used to estimate the underlying continuous and smooth grain size anisotropy function ̂(̂) through finite series expansion of spherical harmonics as follows: where ̂ is the approximant of ̂ with the bandwidth or cut-off degree . The procedure used for spherical harmonics approximation, which is estimating the spherical Fourier coefficients ̃ using the measured discrete values ̂, is detailed in Appendix B. The bandwidth has a substantial impact on the accuracy of the approximant. By increasing the bandwidth, the approximant approaches to the measurement values: ̂= lim →∞̂(̂) . However, at large bandwidths, the sources of measurement noises due to the applied discretization penetrate to the approximant. Therefore, for a noise-reduced/denoised spherical harmonics approximation, the optimal bandwidth must be found. With an optimal bandwidth , the approximant values ̂(̂) are preferred to the measurement values ̂≡̂(̂) as they contain lower noise. In the following, we drop the superscript and use ̂(̂) and ̂(̂) , interchangeably.

Strain hardening anisotropy
The instantiated RVE (the discrete microstructure domain Ω 0 ) was used for meso-scale full-field crystal plasticity simulations corresponding to uniaxial (tensile) deformation along the axes ̂∈ R. We used a physics-based crystal plasticity constitutive model in the framework of full-field method for computational polycrystal homogenization via a spectral solver based on fast Fourier transform. The physics-based crystal plasticity model for the crystal constitutive relationship ̇≡ :̇ (Section 2.5), which is detailed in Motaman et al. [1], was implemented in the modular crystal plasticity code DAMASK [103]. The RVE was generated based on the microstructure analysis of an additively manufactured austenitic high-Mn steel, in which crystal plasticity mechanisms at room temperature are dislocation slip and twinning. The crystal plasticity constitutive model (incrementally and implicitly) computes the constitutive mechanical response as well as evolution and anisotropic interactions of micro-state variables at the underlying deformation (slip/twin) systems of each integration point of the RVE using physically motivated constitutive formulations. The micro-state variables are scalar unipolar and dipolar dislocation densities at slip systems, and twin fraction at twin systems. The RVE was subjected to homogenous macroscopic boundary conditions in terms of the rate of deformation gradient tensor ̅̇ and the first Piola-Kirchhoff stress tensor ̅ (with complementary components) corresponding to uniaxial tensile deformation at constant (room) temperature = 298 K and constant macroscopic nominal strain rate ̇0 = 10 −3 s −1 along the axes ̂∈ R (Section 2.5), which translate to pure displacement periodic boundary conditions on the RVE. The incrementally resolved fields at integration points are then homogenized over the mesoscopic RVE to render the macroscopic mechanical response in terms of the homogenous macroscopic true strain and stress tensors ̅ and ̅ (Section 2.5).
Each simulation was performed in the equivalent macroscopic true strain interval [0, ], where = 0.35. For each simulation, the equivalent macroscopic true stress-strain response ( ( )) was smoothed through Gaussian kernel density estimation and linearly interpolated at the fixed true strain increment ∆ = 0.001. This led to the discrete sequences of equivalent macroscopic true strain = −1 + ∆ (where ∈ ℕ, 0 = 0, where ̂ and ̅ respectively denote the measured strain hardening anisotropy and the measured mean uniaxial strain hardening associated with the probe axis ̂; and ̿ is the measured effective uniaxial strain hardening. Given Eq. (31), the measured discrete uniaxial strain hardening anisotropies ̂ can be used to estimate the underlying continuous and smooth strain hardening anisotropy function ̂(̂) through finite series expansion of spherical harmonic as follows: where ̂ is the approximant of ̂ with the bandwidth . The procedure used for spherical harmonics approximation, which is estimating the spherical Fourier coefficients ̃ using the measured discrete values ̂, is described in Appendix B. By increasing the bandwidth, the approximant ̂(̂) approaches to the measurement values ̂≡̂(̂) . Nevertheless, at large bandwidths, the sources of measurement noises will be included in the approximant. With an optimal bandwidth , the (denoised) approximant values ̂(̂) are preferred to the measurement values ̂≡̂(̂) as they contain lower noise. Henceforth, we drop the superscript and use ̂(̂) and ̂(̂) , interchangeably.

Instantiated RVE
As mentioned in Section 3.1, a discrete mesostructure was generated as a cubic RVE with periodic boundaries, edge length of 200 μm, 1145 grains, and grid resolution of 50 3 voxels. The RVE was instantiated using the conventional microstructure descriptors corresponding to grain shape idealization, which were derived from the experimental microstructure characterization and analysis of an AM austenitic high-Mn steel [1,2]. The instantiated RVE has the grain size heterogeneity and morphological texture corresponding to pronounced columnar grain morphology typically observed in AM metals. Crystallographic orientations were assigned to the grains of the RVE in a way to obtain almost random crystallographic orientation and misorientation distributions. Fig. 4 shows the grain structure and inverse pole figure (IPF) orientation map of the generated RVE.  3 voxels, which has the typical grain size heterogeneity and columnar grain morphology associated with AM metals.
As shown in Fig. 4, most of the grains in the RVE are elongated along the build direction (BD) and the scan direction (SD) relative to the transverse direction (TD). The distributions of the conventional morphological descriptors associated with the generated RVE are shown in Fig. 5. The means of the distributions of the conventional morphological descriptors corresponding to the instantiated RVE are presented in Table 1.  As shown in Fig. 5a, the instantiated RVE has a heterogenous multi-modal equivalent grain size distribution. This high deviation between the number-and volume-weighted mean equivalent grain sizes ( Table 1) is an indicator of the highly heterogeneous grain size distribution present in the RVE. The extremely low mean equivalent aspect ratios imply prevalence of highly elongated grain shapes. The equivalent shapeaxes orientation distribution reveals that the grains in the RVE are dominantly oriented in such a way that their equivalent major semi-axes (̂) and equivalent intermediate semi-axes (̂) have low angles relative to 3 (BD) and 1 (SD), respectively. Consequently, the generated RVE has a pronounced columnar grain morphology resembling the typical microstructures of AM metallic materials. The norm of the equivalent axes ODF (i.e., morphological orientation texture index) associated with the RVE is 36.3. Fig. 6 shows the crystallographic orientation distribution (in terms of IPFs) and the crystallographic misorientation distribution (in axis-angle convention) associated with the instantiated RVE. As shown in Fig.  6a, the RVE has an almost random crystallographic orientation distribution. In Fig. 6b and c, the (correlated) crystallographic misorientation angle and axis distributions in the RVE are compared to their associated uncorrelated/Mackenzie misorientation angle and axis distributions. The negligible difference between the correlated and uncorrelated misorientation distributions implies an almost random crystallographic misorientation distribution in the RVE. The crystallographic orientation and misorientation texture indices (norms of crystallographic ODF and correlated MDF) associated with the RVE are 1.039 and 1.007, respectively. Hence, while the RVE is morphologically strongly textured, it is crystallographically almost textureless. As a result, any notable anisotropy in the mechanical response of the RVE is due to its grain morphology.

Grain size anisotropy
The axial grain sizes ≡ (̂, ) and the mean axial grain sizes ̅ ≡ ̅ (̂) were measured along the uniformly sampled axes ̂∈ at the center point of each voxel ∈ X in the RVE. The distribution of the measured axial grain sizes (i.e., grain boundary spacings) in the instantiated RVE is shown in Fig. 7. The mean of the distribution of the measured axial grain sizes yields the measured effective grain size ̿ = 19.1 μm. The measured effective grain size was used to calculate the measured grain size anisotropies ̂ corresponding to the sampled axes ̂∈ according to Eq. (34). Subsequently, based on the procedure detailed in Appendix B and according to Eq. (35), the underlying grain size anisotropy function ̂(̂) associated with the generated RVE was approximated by finite series expansion of spherical harmonics with the bandwidth = 10. Fig. 8 shows stereographic projection plots of the measured grain size anisotropies ̂ and the corresponding approximated grain size anisotropy function ̂(̂) associated with the generated RVE. The measurement noises corresponding to the high frequency spherical harmonics of degree > with relatively low amplitude are filtered in the (approximated) grain size anisotropy function. The root-meansquare deviation (RMSD) between the grain size anisotropy function (̂(̂)) and the measured grain size anisotropy corresponding to the axes ̂∈ (̂(̂)) is relatively low: The measurement noise sources are associated with the discretization of the microstructure domain and can be divided into two categories: (i) those related to the resolution of the discretization (i.e., coarser resolution leads to higher noise), and (ii) those associated with the shape anisotropy of the discretization elements (e.g., cubic voxels have a certain shape anisotropy [2]).
The generated RVE has a strongly anisotropic grain size, which is suggested by its grain size anisotropy index [̂] = 1.131. As shown in Fig. 8, the grain size anisotropy of the axis 3 (BD) has the highest value (̂(̂3) ≈ 2.4), while the axes oriented closely to 2 (TD) have relatively low grain size anisotropy (̂(̂2) ≈ 0.6). This indicates that the mean axial grain size corresponding to the axis 3 is about four times higher than that associated with the axis 2 . Moreover, the grain size anisotropy of the axis 1 (SD) is between those of the other principal axes of the reference frame (̂(̂1) ≈ 1): ̂(̂3 ) >̂(̂1) >̂(̂2). This is consistent with the conventional equivalent axes orientation distribution shown in Fig. 5d.
Furthermore, as mentioned in Section 2.4, each continuous anisotropy function ̂(̂) uniquly defines a 3D shape. Therefore, the instantiated RVE with the grain size anisotropy function ̂(̂) has an equivalent grain shape represented by ̂(̂) . Fig. 9 shows the principal orthogonal sections of the equivalent grain shape of the generated RVE. As shown in Fig. 9, the RVE has a near ellipsoidal equivalent grain shape whose major, intermediate, and minor semi-axes are aligned with the principal axes 3 , 1 , and 2 , respectively, and its maximum and minimum aspect ratios are roughly 0.41 and 0.25. This was somewhat expected as the first step in the algorithm used for RVE instantiation is the generation of ellipsoidal grains (Section 3.1).   Fig. 9. Principal orthogonal sections of the equivalent grain shape of the generated RVE. Fig. 10 shows the equivalent macroscopic true stress and strain hardening responses of the RVE under homogenous macroscopic boundary conditions corresponding to uniaxial tension at constant temperature = 298 K and constant macroscopic nominal strain rate ̇0 = 10 −3 s −1 along the uniformly sampled axes ̂∈ R. In the following, for brevity, we drop the term "equivalent macroscopic" as the common adjective of true strain, true stress, and strain hardening. As shown in Fig. 10, there is a significant spread in the macroscopic mechanical response of the RVE associated with different tensile axes, which implies a strongly anisotropic macroscopic mechanical response. Even though the spread of true stress at small true strains is relatively low, the spread of true stress at large strains is significant. At true strain = = 0.35, which is associated with the onset of necking in the corresponding modelled material, the spread of true stress is almost 200 MPa. In contrast, the spread of uniaxial strain hardening responses at small strains is higher than those at large strains. At true strain = 0.1, the spread of uniaxial strain hardening responses is more than 600 MPa. In fact, the relatively high initial spread of uniaxial strain hardening responses is the reason for divergence of true stress as true strain increases. As shown in Fig. 10, the lower bounds to the true stress and strain hardening responses of the sampled axes ̂∈ R corresponds to the tensile load axis 3 , while the axis 2 almost serves as upper bound. The effective true stress and strain hardening responses plotted in Fig. 10 represent the average of those associated with all the sampled axes ̂∈ R. Hence, one may think of the effective mechanical response as the would-be isotropic uniaxial mechanical response of the corresponding polycrystal in the total absence of crystallographic and morphological textures. Given the uniaxial strain hardening responses of the RVE and according to Eq. (36), the measured effective uniaxial strain hardening ( ̿ ≈ 1.91 GPa) as well as the measured mean uniaxial strain hardenings ( ̅ ≡ ̅ (̂)) and the measured strain hardening anisotropies (̂≡̂(̂)) associated with the sampled axes ̂∈ R are calculated. Then, based on the procedure described in Appendix B and Eq. (37), the underlying strain hardening anisotropy function ̂(̂) associated with the generated RVE was approximated by finite series expansion of spherical harmonics with the bandwidth = 4. Fig. 11 shows stereographic projection plots of the measured strain hardening anisotropies ̂ and the corresponding approximated strain hardening anisotropy function ̂(̂) associated with the generated RVE.

Strain hardening anisotropy
To filter the notable noise in the measured strain hardening anisotropy data (Fig. 11a), a relatively low bandwidth ( = 4) was chosen. Accordingly, the measurement noises corresponding to the high frequency spherical harmonics of degree > with low amplitude are filtered in the (approximated) strain hardening anisotropy function. The root-mean-square deviation (RMSD) between the strain hardening anisotropy function (̂(̂)) and the measured strain hardening anisotropy corresponding to the axes ̂∈ (̂(̂)) is = 0.015. Owing to the noise present in the crystallographic orientation and misorientation distributions in the RVE, the measurement noise in the measured strain hardening anisotropies ̂ has a more considerable effect compared to the noise in the measured grain size anisotropies ̂. Therefore, the optimal bandwidth for the spherical harmonics approximation of the strain hardening anisotropy function ̂(̂) is lower than that of the grain size anisotropy function ̂(̂) . As shown in Fig. 6, the crystallographic orientation and misorientation distributions in the RVE are nearly but not perfectly random as there are local fluctuations around the corresponding fully random distributions. These local fluctuations in the crystallographic orientation and misorientation distributions and the associated deviations from the corresponding fully random distributions lead to anisotropy fluctuations in the macroscopic mechanical response of the RVE. Since it is impossible to obtain a fully crystallographically textureless RVE, denoising measured strain hardening anisotropies plays a critical role in isolating the effect of morphological texture on the macroscopic mechanical response of the RVE from that associated with the crystallographic texture. Fig. 11. Stereographic projection plots of (a) the measured strain hardening anisotropy at the sampled axes ̂∈ , and (b) the corresponding approximated strain hardening anisotropy function associated with the generated RVE.
The uniaxial strain hardening response of the generated RVE is anisotropic, which is suggested by its strain hardening anisotropy index [̂] = 0.102. As shown in Fig. 11, the strain hardening anisotropy associated with the axis 3 (BD) has the lowest value (̂(̂3) ≈ 0.93), while the axes oriented closely to 2 (TD) have relatively high strain hardening anisotropy (̂(̂2) ≈ 1.07). Even though these values are lower compared to those associated with the grain size anisotropy in the RVE, they represent the considerable spread in the uniaxial mechanical response of the RVE as shown in Fig. 10. Moreover, the strain hardening anisotropy corresponding to the axis 1 (SD) is between those of the other principal axes of the reference frame (̂(̂1) ≈ 1): ̂(̂3 ) <̂(̂1) <̂(̂2).

Anisotropic strain hardening due to anisotropic grain size
Since the crystallographic orientation and misorientation distributions in the instantiated RVE are almost random (i.e., crystallographically textureless), the main source of the observed strain hardening anisotropy is the grain size anisotropy (i.e., morphological texture). By comparison of Fig. 8 and Fig. 11 we find that the axes exhibiting relatively high grain size anisotropy correspond to relatively low strain hardening anisotropy or vice versa. In other words, relatively large mean axial grain size results in relatively low uniaxial strain hardening and relatively small mean axial grain size is associated with relatively high uniaxial strain hardening.
As shown in Fig. 12, the following exponential decay model, which describes the strain hardening anisotropy due to grain size anisotropy, can be fitted to both the measurement and denoised datasets: where and ℎ are two dimensionless constants. The best fit between the model and the denoised dataset (Fig.  12b) corresponds to = 3.15 and ℎ = 0.93. As shown in Fig. 12, the coefficient of determination ( 2 ) is improved from 0.80 to 0.97 after denoising the measurement dataset. The model predicts monotonically decreasing strain hardening anisotropy by increasing grain size anisotropy and the presence of an asymptotic minimum strain hardening anisotropy, which is represented by the constant ℎ.
(a) Measured grain size and strain hardening anisotropies (b) Denoised grain size and strain hardening anisotropies

Outlooks
This study is based on systematic numerical analyses, which includes instantiation of an RVE of a polycrystalline aggregate, measurement of axial grain sizes associated with the RVE, and meso-scale full-field crystal plasticity simulations. Nevertheless, the RVE was generated based on the microstructure of an additively manufactured single-phase polycrystal, which was carefully characterized in previous studies [1,2]. Moreover, the physics-based crystal plasticity constitutive model was developed and calibrated according to the experimental characterization results of the same model material, which exhibits dislocation slip and twinning under uniaxial deformation at quasi-static strain rates and room temperature.
To uncover the contribution of grain size anisotropy to the mechanical response of polycrystals, the adopted numerical approach is inevitable. The main reason is that it is almost impossible to experimentally isolate the morphological texture from the crystallographic texture. In other words, it is highly unlikely to experimentally obtain a polycrystal with strongly anisotropic grain size (i.e., strong morphological texture), and simultaneously random distributions of crystallographic orientation and misorientation (i.e., crystallographically textureless). In practice, the processing conditions that produce strongly anisotropic grain morphologies (e.g., additive manufacturing), also give rise to strong crystallographic textures. In addition, performing experimental uniaxial deformation tests on (exactly) the same material along many different axes is not free of challenges, if not impossible.
The applied physics-based crystal plasticity constitutive model does not account for the following advanced features: strain gradient, gradients of micro-state variables, dislocation density fluxes, and crystal plasticity (slip and twinning) transmission at grain boundaries as a function of their misorientation-dependent opacity. These advanced constitutive modeling features can be integrated in physics-based constitutive models for full-field crystal plasticity simulations of deformation of polycrystals that only exhibit dislocation slip under deformation [104][105][106]. However, large-strain full-field crystal plasticity simulations using a highly nonlinear physics-based constitutive model, which concurrently accounts for dislocation slip, twinning, and the aforementioned modeling features, are too expensive. Those constitutive modeling features lead to relatively high stress concentrations at grain boundaries, which may amplify the size effect on strain hardening. Therefore, as a future perspective, we suggest conducting a similar study on another additively manufactured single-phase polycrystal as the model material, in which dislocation slip is the sole crystal plasticity mechanism, while adopting a physics-based constitutive model that includes the above-mentioned advanced features. In this way, one can ensure that the effect of stress concentration and strain localization at the vicinity of grain boundaries are more realistically accounted for. It is conceivable that such a study would reveal a stronger anisotropic grain size effect on the strain hardening anisotropy.

Conclusions
In this work, we have formally introduced the notions of axial grain size, axial grain size distribution, grain size anisotropy, and strain hardening anisotropy. A discrete sample of single-phase polycrystalline mesostructure was instantiated as a representative volume element (RVE). The distributions of crystallographic orientation and misorientation in the generated RVE are almost random. However, the RVE includes the typical morphological features of additively manufactured microstructures including distinctive grain size heterogeneity and anisotropic grain size owing to its pronounced columnar grain morphology. Hence, any anisotropy observable in the macroscopic mechanical response of the instantiated sample must be mainly a consequence of its anisotropic grain size. The mechanical responses of the RVE subjected to uniaxial tension along different axes were simulated by meso-scale full-field crystal plasticity simulations using a highfidelity physics-based constitutive model. Then, the grain size and strain hardening anisotropies associated with the RVE were estimated. The main findings of this study are summarized in the following: • The conventional morphological descriptors based on grain shape idealization are not suitable to represent complex and anisotropic grain morphologies in polycrystalline aggregates including those typically observed in additively manufactured metals. Given the significance of the grain size effect on the mechanical response of polycrystals, it is reasonable to represent the state of polycrystalline grain morphology based on a single-variable morphological descriptor associated with grain boundary spacing, i.e., the axial grain size. The axial grain size distribution, the mean axial grain size, and the grain size anisotropy function, which are rigorously defined based on the notion of axial grain size, can adequately and compactly represent complex and anisotropic grain morphology states in polycrystals.
• The uniaxial strain hardening is the most suitable measure to represent the anisotropy in the macroscopic mechanical response polycrystals, particularly that related to grain size anisotropy. The mean uniaxial strain hardening and strain hardening anisotropy, which were defined based on the uniaxial strain hardening response, can effectively describe the anisotropy in the mechanical response of polycrystals.
• The contribution of grain size anisotropy to the strain hardening anisotropy of polycrystalline aggregates can be described by: where ̂ represents an axis; and ℎ are two dimensionless constants; ̂(̂) denotes the grain size anisotropy function, which is defined as the mean axial grain size associated with the axis ̂ normalized by the effective grain size; and ̂(̂) represents the strain hardening anisotropy function, which is defined as the mean uniaxial strain hardening associated with the axis ̂ normalized by the effective uniaxial strain hardening. d eq , eq + 1 2 d eq ].
Conventionally, to represent the grain morphological anisotropy and orientation the ellipsoid idealization is used, in which an equivalent ellipsoid is fitted to an arbitrarily shaped grain (best-fit ellipsoid) based on its zeroth-, first-, and second-order geometrical moments [8][9][10][11]. Each ellipsoid can be described by three semiaxes with the lengths , , and (so that ≥ ≥ ), aligned with the orthonormal basis {̂,̂,̂}. The morphological anisotropy can then be represented by two shape parameters associated with its equivalent/bestfit ellipsoid: equivalent minimum aspect ratio ( ≡ ) and equivalent maximum aspect ratio ( ≡ ), where 0 < ≤ ≤ 1. Therefore, the lower the grain equivalent (ellipsoidal) aspect ratios, the higher the grain elongation and hence the grain morphological anisotropy. In addition, the grain morphological orientation can be represented by the spatial orientation of the grain equivalent (ellipsoidal) axes (GEA) with the (auxiliary) orthonormal basis {̂,̂,̂} with respect to the standard/specimen Cartesian coordinate system 1 2 3 having the orthonormal basis {̂1,̂2,̂3} using three Bunge-Euler angles ≡ ( 1 , , 2 ) (Appendix C).
After fitting equivalent ellipsoids to the grains of the microstructure, the morphological anisotropy distribution using the ellipsoid model can be represented by number-or volume-weighted probability distribution of grain equivalent minimum and maximum aspect ratios. Furthermore, the morphological orientation distribution and thus the morphological texture can be represented by distribution of the orientation of GEA using the grain equivalent axes orientation distribution function (GEA-ODF). Given the microstructure domain Ω, the GEA-ODF ( ) is the probability density of measuring the grain equivalent axes orientation ∈ SO(3) at a random point ∈ Ω. GEA-ODF can be represented using series expansion of symmetrized generalized spherical harmonics [107,108] corresponding to the fundamental region of the orientation space SO(3) associated with grain equivalent axes orientation. Grain equivalent axes orientation has the orthorhombic (local) symmetry. It should be noted that GEA-ODF does not contain any information regarding the distribution of morphological anisotropy and size. Moreover, the GEA-ODF can be plotted as pole figures with respect to the equivalent ellipsoidal semi-axes.

Appendix B. Spherical harmonics representation and approximation
For any two complex functions (̂) and (̂) on the surface of unit sphere ( 2 ), we define the following inner product: ∀ , : 2 → ℂ: ⟨ | ⟩ = ∫ * (̂) (̂)d 22 ; ̂≡ ( , ) ∈ 2 ; d 2̂≡ sin d d ; (39) where ⟨ | ⟩ ≡ ⟨ | ⟩ * denotes the inner product of with ; asterik * indicates complex conjugate; ̂ denotes a point on the surface of unit sphere; and are respectively the polar and azimuthal angles (spherical coordinates) of ̂; and d 2̂ represents the invariant measure in 2 (differential area on the surface of unit sphere in the vicinity of ̂). As a result, functions and belong to an infinite-dimensional Hilbert space having the inner product defined by Eq. (39). Using the inner product definition in Eq. (39), the 2 -norm for this Hilbert space is defined as: where the functional 〈 〉 ∈ ℝ denotes the 2 -norm of . The complex functions on the surface of unit sphere whose norm is finite (i.e., square-integrable functions on 2 ) form a Hilbert space denoted by 2 ( 2 ): where is the Kronecker delta. The SHFs form a complete set of orthonormal basis functions for the Hilbert space of square-integrable functions on the surface of unit sphere ( 2 ( 2 )), and thus any square-integrable functions on the surface of unit sphere can be expanded as linear combination of SHFs: where ̃ is the Fourier coefficient of SHF of degree and order associated with the function . Since the SHFs form a complete set of orthonormal basis functions for the Hilbert space 2 ( 2 ), the Fourier coefficients ̃ are the inner product of the SHF with the function ∈ 2 ( 2 ): The functional of Eq. (45) is known as (continuous) spherical Fourier transform.
Therefore, given Eqs. (44), (46) and (49), for any real square-integrable function with antipodal symmetry on the surface of unit sphere, the spherical harmonics representation reads: As mentioned, any function ∈ 2 ( 2 ) can be represented as a linear combination of spherical harmonic functions (SHFs) as they form a complete set of orthonormal bases for the infinite-dimensional Hilbert space 2 ( 2 ). Nevertheless, since it is not feasible to calculate the infinite sum of Eqs. (44) or (50), function ∈ 2 ( 2 ) can be approximated by finite series expansion up to the SHF of finite degree , known as (harmonic) bandwidth or cut-off degree: ∈ ℕ 0 ; ∈ { ∈ ℕ 0 | ≤ }; where (̂) is the approximant of with the bandwidth so that lim →∞ (̂) = (̂). The truncated/bandlimited series of Eq. (51) has ( + 1) 2 terms, as ∀ > : ̃= 0. Hence, increasing has a significant effect on the cost of computing the summation, as direct algorithms for such computation have the computational complexity of ( 2 ). The bandwidth also plays an important role in the accuracy of the approximation of (̂) by its truncated approximant as well as localization error. Therefore, the optimal choice of the bandwidth intricately depends on the desired degree of approximation and available computational resources.
(̂) ∈ 2 ( 2 ) is often unknown a priori while it is measured (with some uncertainty) at a finite set of measurement/sample nodes. Let be the measurement operator corresponding to function , and ≡ (̂) =1 ∈ ( 2 ) an array of finite discrete measurement nodes; where ̂≡ ( , ) ∈ 2 ; ∈ ℕ, ≤ ; ∈ ℕ denotes the number of nodes; and (̂) =1 ≡ (̂1,̂2, … ,̂) is the notation for N-tuple. Therefore, measuring at the measurement nodes gives the array of measurement values ≡ ( ) ≡ ( ) =1 , where ≡ (̂) denotes the measured value of at ̂. The measurement nodes (̂) =1 are generally nonequispaced. In other words, the geodesic distances of pairs of nodes are not constant (i.e., the nodes are not uniformly distributed over the surface of the unit sphere). The objective is to determine the expansion coefficients ̃ of in Eq. (51) such that the distance between the measurements and the truncated estimates is minimized. This distance is defined as the following norm of the weighted residuals [109]: where ≡ ( ) ≡ ( ) =1 denotes the array of quadrature weights corresponding to nodes array ; ≡ (̂) ≥ 0 denotes the non-negative quadrature weight associated with the measurement node ̂; and ∘ is the elementwise/Hadamard product operator. Note that ( ): ( 2 ) → (ℝ + ) denotes a quadrature rule on 2 . The expansion coefficients ̃ (up to the degree ) that minimize the norm can be sought by the least squares method, which is contingent on the evaluation of the following non-uniform/non-equispaced discrete spherical Fourier transform: The numerical integration of Eq. (53) corresponding to the quadrature rule ( ) is an approximation of the definite integral of Eq. (45). A suitable quadrature rule for numerical integration on the unit sphere is based on spherical Voronoi partition/tessellation. The Voronoi partition on 2 associated with ∈ ℕ scattered nodes making the discrete node set R ≡ {̂∈ 2 | ≤ , ∈ ℕ} reads: where ω(R,̂) denotes the (spherical) Voronoi cell corresponding to the node ̂; ω(R) represents the spherical Voronoi partition associated with the node set R; and 2 (̂,̂) denotes the geodesic distance (i.e., length of the shortest path) between the points ̂ and ̂ on the surface of the unit sphere. Based on the preceding definitions, we define the following Voronoi indictor/characteristic functions and Voronoi weights: where ≡ (R,̂) and χ (̂) denote the Voronoi weight and Voronoi indictor function associated with the node ̂∈ R, respectively. The array of Voronoi weights ≡ ( ) ≡ ( ) =1 defines the quadrature rule used in Eqs. (52) and (53). Direct algorithms to compute the Voronoi partition on 2 and the corresponding array of Voronoi quadrature weights have quadratic complexity in the number of nodes ( ( 2 )). However, the algorithm proposed and implemented by Renka [110] computes the Voronoi cells and their associated weights for nodes on 2 with ( log ) arithmetical operations.
Least squares estimation of expansion coefficients ̃ through minimizing the norm (Eq. (52)) by any iterative algorithm requires evaluation of the summations of Eqs. (51) and (53) for many times. Direct algorithms for calculation of the array of expansion coefficients ̃ by Eq. (53) and the array of band-limited function estimates (̂) both have the complexity of ( 2 ). However, fast algorithms to compute the aforementioned arrays with complexity of ( 2 log 2 + ) [111,112], which are based on the nonequispaced fast Fourier transform (NFFT) algorithm [113,114] and implemented in MATLAB®-based MTEX toolbox [100], can be used in combination with the least squares algorithm to achieve higher computational efficiency.

Appendix C. Orientation representation by Bunge-Euler angles
The relative orientation of two Cartesian reference frame belongs to the 3D rotation group SO (3), which is the group of all rotations about the origin of Euclidean 3-space. Such an orientation can be described by a number of orientation representations, such as Euler angles, orientation matrix, and the so-called neo-Eulerian (axis-angle, Rodrigues-Frank, conformal, homochoric, and unit quaternion) vectors [115][116][117][118]. Each representation has distinct advantages and disadvantages with respect to computational efficiency and data visualization and interpretation. Nonetheless, in the crystalline materials science, the Euler angles representation is the most commonly used orientation representation. Eulerian parameterization is particularly convenient to represent orientation distribution functions (ODFs) as linear combination of the generalized spherical harmonics [107,108], but they do not necessarily convey the most efficient visualization and interpretation relative to the neo-Eulerian representations [43,101]. Recently, efficient algorithms have been developed for fast computation of ODFs in terms of Euler angles based on kernel density estimation [98,99]. Nevertheless, ODF representation as series expansion over the hyperspherical harmonics in terms of quaternions is also a viable alternative to the standard Eulerian approach for ODF [119,120].
In the orientation representation by Euler angles, an arbitrary orientation of an auxiliary frame relative to the standard frame is described by three consecutive elemental rotations about predefined axes. The name Euler angles is reserved for those decompositions for which two of the three axes are equal, as in 3 1 3 or Formally, Euler angles are used to describe a second-order (passive) rotation/transformation matrix from a standard (right-handed) Cartesian frame 1 2 3 with the orthonormal basis {̂1,̂2,̂3} to an auxiliary (right-handed) Cartesian frame 1 ′ 2 ′ 3 ′ having the orthonormal basis {̂1 ′ ,̂2 ′ ,̂3 ′ }, so that ≡̂⋅̂′ being the components of the rotation matrix . This transformation is defined by a non-unique sequence of three elemental rotations. In the Bunge notation, the coordinate transformation using Bunge-Euler angles ( 1 , , 2 ) includes the following successive passive elemental rotations [107,108]: (1) Rotation of the standard basis {̂1,̂2,̂3} about ̂3 by the angle 1 (0 ≤ 1 ≤ 2 ) resulting in