The purpose of this section is to identify the role of the loading stress tensor \(\mathbf {\Sigma }\) on the behavior of INS distribution and its respective statistical (central) moments in a general isotropic linear-elastic polycrystalline material subjected to \(\mathbf {\Sigma }\). This will allow us later to develop a method to extend the existing INS distributions (calculated at specific \(\mathbf {\Sigma }\)s) to arbitrary loading conditions.
Since a polycrystalline aggregate under loading stress
\(\mathbf {\Sigma }\) is assumed to be isotropic,
3 the INS distributions should be invariant to any aggregate rotation
\({\textbf{R}}\). Equivalently, the INS distributions should be invariant to any loading stress rotation
\({\textbf{R}}\mathbf {\Sigma }{\textbf{R}}^{T}\). Selecting such
\({\textbf{R}}\) that
\({\textbf{R}}\mathbf {\Sigma }{\textbf{R}}^T\) becomes diagonal, we may proceed with the most general loading stress
$$\begin{aligned} \mathbf {\Sigma }= & {} \left( \begin{array}{ccc} \Sigma _1 &{} 0 &{} 0\\ 0 &{} \Sigma _2 &{} 0\\ 0 &{} 0 &{} \Sigma _3 \end{array} \right) =\Sigma _1 \left( \begin{array}{ccc} 1 &{} 0 &{} 0\\ 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 \end{array} \right) + \Sigma _2 \left( \begin{array}{ccc} 0 &{} 0 &{} 0\\ 0 &{} 1 &{} 0\\ 0 &{} 0 &{} 0 \end{array} \right) + \Sigma _3 \left( \begin{array}{ccc} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 1 \end{array} \right) \nonumber \\ {}= & {} \Sigma _1 {\textbf{I}}_1+ \Sigma _2 {\textbf{I}}_2 + \Sigma _3 {\textbf{I}}_3 \end{aligned}$$
(1)
where
\(\Sigma _i\) are loading stress eigenvalues and
\({\textbf{I}}_{i}\) unit loading tensile stresses along the
i direction. If a local INS solution on an arbitrarily selected GB is
\(\sigma _{nn}^{(i)}\) for applied
\({\textbf{I}}_{i}\), then a local INS solution for applied
\(\mathbf {\Sigma }\) is
$$\begin{aligned} \sigma _{nn}=\Sigma _1 \sigma _{nn}^{(1)}+ \Sigma _2 \sigma _{nn}^{(2)}+ \Sigma _3 \sigma _{nn}^{(3)}, \end{aligned}$$
(2)
following the linearity principle (of the assumed generalized (3D) Hooke’s law). Since all three directions
i are equivalent in an isotropic polycrystalline aggregate, the average INS scales with the trace of loading (see also [
15]),
$$\begin{aligned} \left\langle \sigma _{nn}\right\rangle =\sum _{i=1}^3 \Sigma _i \left\langle \sigma _{nn}^{(i)}\right\rangle =C \textrm{tr}(\mathbf {\Sigma }) \end{aligned}$$
(3)
where
\(\left\langle \ldots \right\rangle \) represents the averaging over all GBs or a subset of GBs of specific type
4 in a polycrystalline aggregate and
\(C\equiv \left\langle \sigma _{nn}^{(1)}\right\rangle =\left\langle \sigma _{nn}^{(2)}\right\rangle =\left\langle \sigma _{nn}^{(3)}\right\rangle \) is a constant, which depends on elastic material parameters and correlation strength between grain shapes and grain lattice orientations [
15]. If the grains possess cubic lattice symmetry, then
\(C=1/3\) (see Appendix A). Here we assumed that the grains are composed of the same material and they differ only by their crystallographic orientations.
The
m-th central moment of the INS distribution is defined as
$$\begin{aligned} \mu ^m=\left\langle (\sigma _{nn}-\left\langle \sigma _{nn}\right\rangle )^m\right\rangle \big \vert _{\mathbf {\Sigma }} \end{aligned}$$
(4)
for the applied stress
\(\mathbf {\Sigma }\). It is convenient to decompose
\(\mathbf {\Sigma }\) into hydrostatic (
\(\mathbf {\Sigma }_{\text {hyd}}\)) and deviatoric parts (
\(\mathbf {\Sigma }_{\text {dev}}\)),
\(\mathbf {\Sigma }=\mathbf {\Sigma }_{\text {hyd}}+\mathbf {\Sigma }_{\text {dev}}\), and analyze both contributions to
\(\mu ^m\) separately. Since
\(\mathbf {\Sigma }_{\text {hyd}}\) is fully symmetric, its effect on
\(\mu ^m\) is significantly simplified.
2.1.1 Effect of \(\mathbf {\Sigma }_{\text {dev}}\)
We begin by deriving the central moments under the assumption of purely deviatoric loading. In fact, this loading is the sole contributor to
\(\mu ^m\) when the grains posses cubic lattice symmetry.
5 For non-cubic crystal lattices, however, the effect of hydrostatic loading is non-zero. This aspect is discussed later in Sects.
2.1.2 and
2.1.3.
The evaluation of the central moments can be reduced to
$$\begin{aligned} \mu _{\text { dev}}^m=\left\langle (\sigma _{nn}-\left\langle \sigma _{nn}\right\rangle )^m\right\rangle \big \vert _{\mathbf {\Sigma }_{\text {dev}}}=\left\langle \sigma _{nn}^m\right\rangle \big \vert _{\mathbf {\Sigma }_{\text {dev}}} \end{aligned}$$
(5)
when the applied stress
\(\mathbf {\Sigma }\) is replaced by the deviatoric stress
\(\mathbf {\Sigma }_{\text {dev}}\), which is traceless by definition,
$$\begin{aligned} \mathbf {\Sigma }_{\text{ dev }}= & {} {} \mathbf {\Sigma }-\frac{1}{3}\text {tr}(\mathbf {\Sigma }) \mathbbm {1}_{3\times 3}\nonumber \\ {}= & {} {} \frac{2\Sigma _1-\Sigma _2-\Sigma _3}{3} {{\textbf {I}}}_{1}+ \frac{-\Sigma _1+2\Sigma _2-\Sigma _3}{3} {{\textbf {I}}}_{2}+ \frac{-\Sigma _1-\Sigma _2+2\Sigma _3}{3} {{\textbf {I}}}_{3}\nonumber \\ {}\equiv & {} {} {\tilde{\Sigma }}_1 {{\textbf {I}}}_{1}+ {\tilde{\Sigma }}_2 {{\textbf {I}}}_{2}+ {\tilde{\Sigma }}_3 {{\textbf {I}}}_{3}.\end{aligned}$$
(6)
In this respect, only two out of three deviatoric stress invariants,
\(J_2\equiv \textrm{tr}(\mathbf {\Sigma }_{\text {dev}}^2)/2\) and
\(J_3\equiv \textrm{det}(\mathbf {\Sigma }_{\text {dev}})\), control the shape of the INS distribution. In the case of cubic crystal lattices, any hydrostatic contribution enclosed in the first stress invariant
\(I_1\equiv \textrm{tr}(\mathbf {\Sigma })= \textrm{tr}(\mathbf {\Sigma }_{\text {hyd}})\), manifests only as a shift of the INS distribution.
6
Using Eqs. (
2), (
5) and (
6), the
m-th central moment can be expressed as
$$\begin{aligned} \mu _{\text { dev}}^m=\sum _{i=0}^m\sum _{j=0}^i \left( {\begin{array}{c}m\\ i\end{array}}\right) \left( {\begin{array}{c}i\\ j\end{array}}\right) {\tilde{\Sigma }}_1^j {\tilde{\Sigma }}_2^{i-j} {\tilde{\Sigma }}_3^{m-i} \left\langle \left( \sigma _{nn}^{(1)}\right) ^j \left( \sigma _{nn}^{(2)}\right) ^{i-j} \left( \sigma _{nn}^{(3)}\right) ^{m-i}\right\rangle . \end{aligned}$$
(7)
Applying again the statistical equivalence of three spatial directions, the material average term (i.e., independent of loading) in Eq. (
7) becomes invariant to the permutation of the three indices 1,2 and 3,
$$\begin{aligned} M(i,j,k)\equiv & {} \left\langle \left( \sigma _{nn}^{(1)}\right) ^i \left( \sigma _{nn}^{(2)}\right) ^j \left( \sigma _{nn}^{(3)}\right) ^k\right\rangle \nonumber \\= & {} M(i,k,j)=M(j,i,k)=M(j,k,i)=M(k,i,j)=M(k,j,i). \end{aligned}$$
(8)
Thus, a 3-parameter average
M(
i,
j,
k) (with
\(i,j,k=0,1,2,\ldots \) and
\(i+j+k=m\)) can be reduced to a 2-parameter average (using a commutative property of the multiplication of three indices),
$$\begin{aligned} M(i,j,k)\rightarrow M_m((i+1)(j+1)(k+1)). \end{aligned}$$
(9)
This allows us to rewrite the
m-th central moment in a symmetrized form
$$\begin{aligned} \mu _{\text { dev}}^m=\sum _{i=0}^m\sum _{j=0}^i \left( {\begin{array}{c}m\\ i\end{array}}\right) \left( {\begin{array}{c}i\\ j\end{array}}\right) {\tilde{\Sigma }}_1^j {\tilde{\Sigma }}_2^{i-j} {\tilde{\Sigma }}_3^{m-i} M_m((j+1)(i-j+1)(m-i+1)) \end{aligned}$$
(10)
where the averaging function
\(M_m\) depends solely on the (linear-elastic) properties of the grains for the assumed zero crystallographic and zero morphological texture of the aggregate. Moreover, if
\({\tilde{\Sigma }}_1\),
\({\tilde{\Sigma }}_2\) and
\({\tilde{\Sigma }}_3\) in Eq. (
10) are expressed in terms of the two (commonly chosen) deviatoric stress invariants
\(J_2\) and
\(J_3\), the expressions for
\(\mu _{\text { dev}}^m\) can be further simplified,
$$\begin{aligned}{} & {} \mu _{\text { dev}}^1=0\rightarrow 0\nonumber \\{} & {} \mu _{\text { dev}}^2=2 J_2\left( M_2(3)-M_2(4)\right) \rightarrow J_2 M_{1,0}\nonumber \\{} & {} \mu _{\text { dev}}^3=3 J_3\left( M_3(4)-3M_3(6)+2M_3(8)\right) \rightarrow J_3 M_{0,1}\nonumber \\{} & {} \mu _{\text { dev}}^4=2 J_2^2 \left( M_4(5)-4 M_4(8)+3 M_4(9)\right) \rightarrow J_2^2 M_{2,0}\nonumber \\{} & {} \quad \ \ \vdots \end{aligned}$$
(11)
Here, a new parameter
\(M_{i,j}\) is introduced to collect all the material prefactors
\(M_m(k)\) in front of each
\(J_2^i J_3^j\) loading term (with
\(i,j=0,1,2,\ldots \) and
\(2i+3j=m\)). In this way, the
\(\mu _{\text { dev}}^m\) finally simplifies to
$$\begin{aligned} \mu _{\text { dev}}^m=\sum _{\begin{array}{c} i,j\ge 0\\ 2i+3j=m \end{array}}J_2^i J_3^j M_{i,j}. \end{aligned}$$
(12)
The first 20 central moments are presented in Table
1.
Table 1
The first 20 central moments \(\mu _{\text { dev}}^m\) of the general INS distribution for the assumed deviatoric stress loading \(\mathbf {\Sigma }_{\text {dev}}\), expressed in terms of the two deviatoric stress invariants \(J_2\equiv \textrm{tr}(\mathbf {\Sigma }_{\text {dev}}^2)/2\) and \(J_3\equiv \textrm{det}(\mathbf {\Sigma }_{\text {dev}})\), and material invariants \(M_{i,j}\) with \(2i+3j=m\)
1 | 0 | 11 | \(J_2^4 J_3 M_{4,1}+J_2 J_3^3 M_{1,3}\) |
2 | \(J_2 M_{1,0}\) | 12 | \(J_2^6 M_{6,0}+J_2^3 J_3^2 M_{3,2}+J_3^4 M_{0,4}\) |
3 | \(J_3 M_{0,1}\) | 13 | \(J_2^5 J_3 M_{5,1}+J_2^2 J_3^3 M_{2,3}\) |
4 | \(J_2^2 M_{2,0}\) | 14 | \(J_2^7 M_{7,0}+J_2^4 J_3^2 M_{4,2}+J_2 J_3^4 M_{1,4}\) |
5 | \(J_2 J_3 M_{1,1}\) | 15 | \(J_2^6 J_3 M_{6,1}+J_2^3 J_3^3 M_{3,3}+J_3^5 M_{0,5}\) |
6 | \(J_2^3 M_{3,0}+J_3^2 M_{0,2}\) | 16 | \(J_2^8 M_{8,0}+J_2^5 J_3^2 M_{5,2}+J_2^2 J_3^4 M_{2,4}\) |
7 | \(J_2^2 J_3 M_{2,1}\) | 17 | \(J_2^7 J_3 M_{7,1}+J_2^4 J_3^3 M_{4,3}+J_2 J_3^5 M_{1,5}\) |
8 | \(J_2^4 M_{4,0}+J_2 J_3^2 M_{1,2}\) | 18 | \(J_2^9 M_{9,0}+J_2^6 J_3^2 M_{6,2}+J_2^3 J_3^4 M_{3,4}+J_3^6 M_{0,6}\) |
9 | \(J_2^3 J_3 M_{3,1}+ J_3^3 M_{0,3}\) | 19 | \(J_2^8 J_3 M_{8,1}+J_2^5 J_3^3 M_{5,3}+J_2^2 J_3^5 M_{2,5}\) |
10 | \(J_2^5 M_{5,0}+J_2^2 J_3^2 M_{2,2}\) | 20 | \(J_2^{10} M_{10,0}+J_2^7 J_3^2 M_{7,2}+J_2^4 J_3^4 M_{4,4}+J_2 J_3^6 M_{1,6}\) |
Finding a general expression for \(\mu _{\text { dev}}^m\) under purely deviatoric stress loading (note, however, that \(\mu ^m=\mu _{\text { dev}}^m\) for cubic crystal lattices) by decoupling the loading (\(J_2, J_3\)) and material (M) contributions is the first main result of this study.
It is relevant to note that the simplicity of
\(\mu _{\text { dev}}^m\) in Eq. (
12) results from the following assumed symmetries: (i) the isotropy of the polycrystalline material, (ii) the linearity of the employed Hooke’s law and (iii) the absence of hydrostatic loading (or its zero effect on the shape of INS distribution in the case of cubic lattice symmetry). Moreover, in the expression for
\(\mu _{\text { dev}}^m\), the two deviatoric stress invariants
\(J_2, J_3\) are independent parameters which are, however, bounded
7 by
\(J_2\ge 0\) and
\(|J_3|\le 2 J_2^{3/2}/\sqrt{27}\). Also, since odd central moments
\(\mu _{\text { dev}}^m\) (
m odd) scale with only odd powers of
\(J_3\), the sign inversion
\(J_3\rightarrow -J_3\) implies a reflection of the INS distribution about the
\(\left\langle \sigma _{nn}\right\rangle \) point,
\(\textrm{PDF}(\sigma _{nn}-\left\langle \sigma _{nn}\right\rangle )\rightarrow \textrm{PDF}(-\sigma _{nn}+\left\langle \sigma _{nn}\right\rangle )\), with
\(J_3=0\) denoting a symmetric INS distribution.
Having a relatively simple expression for
\(\mu _{\text { dev}}^m\) for a
general isotropic linear-elastic polycrystalline material subjected to purely deviatoric loading stress
\(\mathbf {\Sigma }_{\text { dev}}\) (i.e.,
\(J_2\) and
\(J_3\)), allows to extract the unknown material parameters
\(M_{i,j}\) from
N existing INS distributions obtained at few specific
\(\mathbf {\Sigma }_{\text { dev}}\). For example, if two (
\(N=2\)) such INS distributions are available (e.g., from FE simulations or literature), along with their central moments, one can identify all
\(M_{i,j}\) of the first 11 (and also 13-th) central moments (see Table
1), assuming that
\(J_2\ne 0\) and
\(J_3\ne 0\) of the
\(N=2\) existing INS distributions.
8 With this, one can calculate the new central moments
\(\mu _{\text { dev}}^m\) for
\(m\le 11\) and arbitrary
\(\mathbf {\Sigma }_{\text { dev}}\) (i.e.,
\(J_2\) and
\(J_3\)). As demonstrated in Sect.
3,
\(m\le 11\) is usually sufficient to accurately (re)construct the INS distributions.
2.1.2 Effect of \(\mathbf {\Sigma }_{\text {hyd}}\)
Grains with lower (non-cubic) crystal lattice symmetries yield non-trivial local responses even under purely hydrostatic (symmetric) loading conditions, \(\mathbf {\Sigma }=\mathbf {\Sigma }_{\text {hyd}}\). This indicates that the corresponding central moments are nonzero, \(\mu ^m\ne 0\big \vert _{\mathbf {\Sigma }_{\text {hyd}}}\). Consequently, in the derivation of \(\mu ^m\), it is essential to consider a complete loading, \(\mathbf {\Sigma }=\mathbf {\Sigma }_{\text {hyd}}+\mathbf {\Sigma }_{\text {dev}}\).
Following the same derivation steps as in Sect.
2.1.1, the final expression for
\(\mu ^m\), defined in Eq. (
4) for arbitrary loading
\(\mathbf {\Sigma }\), simplifies to (
\(m>1\))
$$\begin{aligned} \mu ^m=\sum _{\begin{array}{c} i,j,k\ge 0\\ i+2j+3k=m \end{array}}I_1^i I_2^j I_3^k M_{i,j,k} \end{aligned}$$
(13)
where all three stress invariants,
\(I_1\equiv \textrm{tr}(\mathbf {\Sigma })\),
\(I_2\equiv \left( \textrm{tr}(\mathbf {\Sigma })^2 -\textrm{tr}(\mathbf {\Sigma }^2)\right) /2\) and
\(I_3\equiv \textrm{det}(\mathbf {\Sigma })\), multiply the unknown material prefactors
\(M_{i,j,k}\). Since several combinations (
i,
j,
k) satisfy
\(i+2j+3k=m\) for a given
m-th central moment, many more (
\(N\gg 1\)) existing INS distributions are required to extract
\(M_{i,j,k}\) even for relatively low orders
m,
$$\begin{aligned}{} & {} \mu ^1=0\nonumber \\{} & {} \mu ^2=I_1^2 M_{2,0,0} + I_2 M_{0,1,0}\nonumber \\{} & {} \mu ^3=I_1^3 M_{3,0,0} + I_1 I_2 M_{1,1,0} + I_3 M_{0,0,1}\nonumber \\{} & {} \mu ^4=I_1^4 M_{4,0,0} + I_1^2 I_2 M_{2,1,0} + I_2^2 M_{0,2,0} + I_1 I_3 M_{1,0,1}\nonumber \\{} & {} \quad \ \ \vdots \end{aligned}$$
(14)
For example, if four (
\(N=4\)) existing INS distributions (obtained at different loadings
\(\mathbf {\Sigma }\) with
\(I_i\ne 0\)) are available, it is possible to determine the corresponding
\(M_{i,j,k}\) for only the first four central moments. Clearly, the method outlined in Sect.
2.1.1 for extracting
\(M_{i,j}\) from
\(\mu _{\text { dev}}^m\) proves impractical here for any realistic application.
To circumvent this, we propose an alternative path. The core idea is that on any given GB the two INS contributions,
\(\sigma _{nn}=\sigma _{nn}^{\text {dev}}+\sigma _{nn}^{\text {hyd}}\), denoted as
\(\sigma _{nn}^{\text {dev}}\) for purely deviatoric loading
\(\mathbf {\Sigma }_{\text {dev}}\) and
\(\sigma _{nn}^{\text {hyd}}\) for purely hydrostatic loading
\(\mathbf {\Sigma }_{\text {hyd}}\), can be approximated as being independent of each other. This follows from the notion that
\(\sigma _{nn}^{\text {dev}}\) is strongly dependent on GB normal orientation
\({\textbf{n}}\), while
\(\sigma _{nn}^{\text {hyd}}\) is not (or much less) since there is no preferred direction in a (random) aggregate under purely symmetric (hydrostatic) loading.
9 Under this approximation,
\(\sigma _{nn}^{\text {hyd}}\) can be
modeled as stochastic variable with assumed normal probability distribution
10 centered around the mean value for isotropic (or cubic crystal lattice) grains
\(I_1/3\), and with variance
\(\mu _{\text {hyd}}^2\),
$$\begin{aligned} \sigma _{nn}^{\text {hyd}}\sim {\mathcal {N}}\left( \frac{I_1}{3}, \mu _{\text {hyd}}^2\right) . \end{aligned}$$
(15)
Since the response of grains is linear with respect to (hydrostatic) loading, the variance scales with
\(I_1^2\) so that
\(\mu _{\text {hyd}}^2=I_1^2 M_{2,0,0}\), where
\(M_{2,0,0}\) is unknown material parameter
11 to be extracted from a single existing INS distribution obtained at hydrostatic loading
\(\mathbf {\Sigma }_{\text {hyd}}=\frac{1}{3} I_1 \mathbb {1}_{3\times 3}\). With this, all higher (even) central moments of normal distribution follow straightforwardly,
$$\begin{aligned} \mu _{\text {hyd}}^m=\left\{ \begin{array}{ll} 0;&{} m \text { odd}\\ I_1^m \left( M_{2,0,0}\right) ^{m/2} (m-1)!!;&{} m \text { even}\end{array}\right. . \end{aligned}$$
(16)
2.1.3 Effect of \(\mathbf {\Sigma }_{\text {hyd}}+\mathbf {\Sigma }_{\text {dev}}\)
Since the (exact) approach outlined in Eq. (
13) is not feasible, the final (approximate) expression for the central moment
\(\mu ^m\) is obtained from the probability theory, which states that the probability distribution of the sum of two or more independent random variables is the convolution of their individual distributions. In this respect,
$$\begin{aligned} \mu ^m\approx \sum _{i=0}^m \left( {\begin{array}{c}m\\ i\end{array}}\right) \mu _{\text {dev}}^i \mu _{\text {hyd}}^{m-i} \end{aligned}$$
(17)
where
\(\mu _{\text {dev}}^m\) is the (exact)
m-th central moment of INS distribution evaluated for purely deviatoric loading, Eq. (
12), while
\(\mu _{\text {hyd}}^m\) is the (approximate)
m-th central moment of INS distribution evaluated for purely hydrostatic loading, Eq. (
16). In the limit of vanishing hydrostatic contribution (
i.e., when grains posses cubic lattice symmetry or
\(I_1=0\)), the expression in Eq. (
17) correctly reduces to
\(\mu ^m=\mu _{\text {dev}}^m\).
To summarize, we outline the following methodology for computing central moments for arbitrary crystal lattice symmetry and external loading
\(\mathbf {\Sigma }\), utilizing a limited number (
\(N+1\)) of existing INS distributions evaluated at specific
\(\mathbf {\Sigma }\)s:
1.
determine the unknown material parameters \(M_{i,j}\) in \(\mu _{\text {dev}}^m\) from N existing INS distributions obtained at N specific purely deviatoric loadings \(\mathbf {\Sigma }_{\text { dev}}\) (with \(I_1=0\)),
2.
determine the unknown material parameter \(M_{2,0,0}\) from a single existing INS distribution obtained at purely hydrostatic loading \(\mathbf {\Sigma }_{\text {hyd}}=\frac{1}{3} I_1 \mathbb {1}_{3\times 3}\) (with \(I_1\ne 0\)),
3.
calculate the central moments
\(\mu ^m\) from obtained prefactors
\(M_{i,j}\) and
\(M_{2,0,0}\) and arbitrary loading
\(\mathbf {\Sigma }\) (any
\(I_1\),
\(J_2\),
\(J_3\)) using Eqs. (
12), (
16), (
17) and Table
1.
The proposed method is exact (limited by the accuracy of the inputs) for isotropic linear-elastic polycrystals composed of grains that exhibit (i) cubic lattice symmetry under any external loading
\(\mathbf {\Sigma }\), or (ii) non-cubic lattice symmetry under purely deviatoric external loading
\(\mathbf {\Sigma }_{\text {dev}}\). In both cases,
\(\mu ^m=\mu _{\text {dev}}^m\). As validated in Sect.
3, the method performs very accurately also for most general crystal lattices and loadings.