Estimates of Nonlinear Elastic Constants and Acoustic Nonlinearity Parameters for Textured Polycrystals

Abstract

In this article, expressions are derived for the Voigt, Reuss, and Hill estimates of the third-order elastic constants for polycrystals with either cubic or hexagonal crystal symmetry and orthorhombic physical symmetry. General forms of the fourth- and sixth-rank elastic stiffness and compliance tensors for crystal and physical symmetries are given. Explicit expressions are reduced from these tensors for the case of polycrystals exhibiting orthorhombic sample symmetry with either cubic or hexagonal crystallites. The estimated third-order elastic constants of the textured polycrystal are obtained in terms of second- and third-order single-crystal elastic constants and orientation distribution coefficients (ODCs), which are used to account for anisotropic physical symmetry. The acoustic nonlinearity parameter, $\bar{\beta}$, is defined through combinations of the second- and third-order Voigt, Reuss, and Hill estimates of the elastic constants for a textured polycrystal. The model predicts that $\bar{\beta}$ is dependent on the type of averaging scheme used and the texture-defining ODCs. The model is quantitatively evaluated for polycrystalline iron, aluminum, and titanium using second- and third-order single-crystal elastic constants and experimentally measured ODCs. The interrelation between $\bar{\beta}$ and polycrystalline anisotropy offers potential for techniques associated with quantitative texture analysis.

Appendices

Appendix A: Relations Between the Single-Crystal Elastic Constants and Compliances

The relations between the second-order single-crystal elastic compliances and single-crystal elastic constants are obtained by solving the set of equations generated from (5). For cubic crystallite symmetry the relations are [26, 28]

$$\begin{aligned} s_{11}&=\frac{c_{11}+c_{12}}{ (c_{11}-c_{12} ) (c_{11}+2c_{12} )},\qquad s_{12}=-\frac{c_{12}}{ (c_{11}-c_{12} ) (c_{11}+2c_{12} )},\qquad s_{44}= \frac{1}{4c_{44}}. \end{aligned}$$

(A.1)

The second-order relations for hexagonal crystallite symmetry are

$$ \begin{aligned} s_{11}&=\frac{c_{11}c_{33} - c_{13}^{2}}{ (c_{12}-c_{11} ) [2c_{13}^{2} - c_{33} (c_{11} + c_{12} ) ]},\quad\quad s_{12}=\frac {c_{13}^{2} - c_{12}c_{33}}{ (c_{12}-c_{11} ) [2c_{13}^{2} - c_{33} (c_{11} + c_{12} ) ]} \\ s_{13}&=\frac{c_{13}}{2c_{13}^{2} - c_{33} (c_{11} + c_{12} )},\qquad s_{33}=\frac{c_{11} + c_{12}}{c_{33} (c_{11} + c_{12} )-2c_{13}^{2}},\qquad s_{44}= \frac{1}{4c_{44}}. \end{aligned} $$

(A.2)

The relations between the third-order single-crystal elastic compliances and single-crystal elastic constants are obtained by solving the set of equations generated from $S_{ijklmn}=-S_{ijpq}S_{klrs}S_{mnuv}C_{pqrsuv}$. The third-order relations for cubic crystallite symmetry are [26, 28]

$$\begin{aligned} \begin{aligned} s_{{111}}&= - \bigl(s_{{11}}^{3}+2s_{{12}}^{3} \bigr)c_{{111}} - 6 \bigl(s_{{11}}^{2}+s_{{11}}s_{{12}}+s_{{12}}^{2} \bigr)s_{{12}}c_{{112}} - 6s_{{11}}s_{{12}}^{2}c_{{123}}, \\ s_{{112}}&= - \bigl(s_{{11}}^{2}+s_{{11}}s_{{12}}+s_{{12}}^{2} \bigr)s_{{12}} (c_{{111}}+2c_{{123}} ) - \bigl(s_{{11}}^{3}+3s_{{11}}^{2}s_{{12}}+9s_{{11}}s_{{12}}^{2}+5s_{{12}}^{3} \bigr)c_{{112}}, \\ s_{{123}}&= -3s_{{11}}s_{{12}}^{2}c_{{111}} - 6 \bigl(s_{{11}}^{2}s_{{12}}+s_{{11}}s_{{12}}^{2}+s_{{12}}^{3} \bigr)c_{{112}} - \bigl(s_{{11}}^{3}+3s_{{11}}s_{{12}}^{2}+2s_{{12}}^{3} \bigr)c_{{123}}, \\ s_{{144}}&= -4s_{{44}}^{2} (s_{{11}}c_{{144}}+2s_{{12}}c_{{155}} ), \qquad s_{{155}}= -4s_{{44}}^{2} \bigl(s_{{12}}c_{{144}}+(s_{{11}}+s_{{12}} )c_{{155}} \bigr), \\ s_{{456}}&= -8s_{{44}}^{3}c_{{456}}. \end{aligned} \end{aligned}$$

(A.3)

The third-order relations for hexagonal crystallite symmetry are

$$\begin{aligned} s_{{111}} = &-c_{{111}}s_{{11}}^{3} - c_{{222}}s_{{12}}^{3} - c_{{333}}s_{{13}}^{3} - 3c_{{112}}s_{{11}}^{2}s_{{12}} - 3c_{{113}}s_{{11}}^{2}s_{{13}} - 3c_{{113}}s_{{12}}^{2}s_{{13}} \\ &{} - 3c_{{133}}s_{{11}}s_{{13}}^{2} - 3c_{{133}}s_{{12}}s_{{13}}^{2} - 3s_{{11}}s_{{12}}^{2} (c_{{111}} + c_{{112}} - c_{{222}} ) - 6c_{{123}}s_{{11}}s_{{12}}s_{{13}}, \\ s_{{112}} = &-s_{{12}}^{3} (c_{{111}} + c_{{112}} - c_{{222}} ) - c_{{112}}s_{{11}}^{3} - c_{{333}}s_{{13}}^{3} - c_{{111}}s_{{11}}^{2}s_{{12}} - 2c_{{112}}s_{{11}}s_{{12}}^{2} \\ &{} - c_{{113}}s_{{11}}^{2}s_{{13}} - c_{{113}}s_{{12}}^{2}s_{{13}} - 2c_{{123}}s_{{11}}^{2}s_{{13}} - 2c_{{123}}s_{{12}}^{2}s_{{13}} - 3c_{{133}}s_{{11}}s_{{13}}^{2} - 3c_{{133}}s_{{12}}s_{{13}}^{2} \\ &{} - c_{{222}}s_{{11}}s_{{12}}^{2}- 2s_{{11}}^{2}s_{{12}} (c_{{111}} + c_{{112}} - c_{{222}} ) - 4c_{{113}}s_{{11}}s_{{12}}s_{{13}} - 2c_{{123}}s_{{11}}s_{{12}}s_{{13}}, \\ s_{{113}} = &-2c_{{133}}s_{{13}}^{3} - c_{{111}}s_{{11}}^{2}s_{{13}} - c_{{112}}s_{{11}}^{2}s_{{13}} - 2c_{{113}}s_{{11}}s_{{13}}^{2} - 2c_{{113}}s_{{12}}s_{{13}}^{2} - 2c_{{123}}s_{{11}}s_{{13}}^{2} \\ &{} - 2c_{{123}}s_{{12}}s_{{13}}^{2} - c_{{113}}s_{{11}}^{2}s_{{33}} - c_{{222}}s_{{12}}^{2}s_{{13}} - c_{{333}}s_{{13}}^{2}s_{{33}} - s_{{12}}^{2}s_{{13}} (c_{{111}} + c_{{112}} - c_{{222}} ) \\ &{} - 2c_{{112}}s_{{11}}s_{{12}}s_{{13}} - 2c_{{123}}s_{{11}}s_{{12}}s_{{33}} - 2c_{{133}}s_{{11}}s_{{13}}s_{{33}} - 2c_{{133}}s_{{12}}s_{{13}}s_{{33}} \\ &{} - 2s_{{11}}s_{{12}}s_{{13}} (c_{{111}} + c_{{112}} - c_{{222}} ), \\ s_{{123}} = &-2c_{{133}}s_{{13}}^{3} - c_{{112}}s_{{11}}^{2}s_{{13}} - c_{{112}}s_{{12}}^{2}s_{{13}} - 2c_{{113}}s_{{11}}s_{{13}}^{2} - 2c_{{113}}s_{{12}}s_{{13}}^{2} - 2c_{{123}}s_{{11}}s_{{13}}^{2} \\ &{} - 2c_{{123}}s_{{12}}s_{{13}}^{2} - c_{{123}}s_{{11}}^{2}s_{{33}} - c_{{123}}s_{{12}}^{2}s_{{33}} - c_{{333}}s_{{13}}^{2}s_{{33}} - s_{{11}}^{2}s_{{13}} (c_{{111}} + c_{{112}} - c_{{222}} ) \\ &{} - s_{{12}}^{2}s_{{13}} (c_{{111}} + c_{{112}} - c_{{222}} ) - c_{{111}}s_{{11}}s_{{12}}s_{{13}} - c_{{112}}s_{{11}}s_{{12}}s_{{13}} - 2c_{{113}}s_{{11}}s_{{12}}s_{{33}} \\ &{} - 2c_{{133}}s_{{11}}s_{{13}}s_{{33}} - 2c_{{133}}s_{{12}}s_{{13}}s_{{33}}- c_{{222}}s_{{11}}s_{{12}}s_{{13}} - s_{{11}}s_{{12}}s_{{13}} (c_{{111}} + c_{{112}} - c_{{222}} ), \\ s_{{133}} = &-2c_{{113}}s_{{13}}^{3} - 2c_{{123}}s_{{13}}^{3} - c_{{111}}s_{{11}}s_{{13}}^{2} -2c_{{112}}s_{{11}}s_{{13}}^{2} - c_{{112}}s_{{12}}s_{{13}}^{2} - c_{{133}}s_{{11}}s_{{33}}^{2} \\ &{} - 4c_{{133}}s_{{13}}^{2}s_{{33}}- c_{{222}}s_{{12}}s_{{13}}^{2} - c_{{333}}s_{{13}}s_{{33}}^{2} - s_{{11}}s_{{13}}^{2} (c_{{111}} + c_{{112}} - c_{{222}} ) \\ &{} - 2s_{{12}}s_{{13}}^{2} (c_{{111}} + c_{{112}} - c_{{222}} )- 2c_{{113}}s_{{11}}s_{{13}}s_{{33}} - 2c_{{113}}s_{{12}}s_{{13}}s_{{33}} - 2c_{{123}}s_{{11}}s_{{13}}s_{{33}} \\ &{} - 2c_{{123}}s_{{12}}s_{{13}}s_{{33}}, \\ s_{{144}} = &-4c_{{144}}s_{{11}}s_{{44}}^{2} - 4c_{{155}}s_{{12}}s_{{44}}^{2} - 4c_{{344}}s_{{13}}s_{{44}}^{2}, \\ s_{{155}} = &-4c_{{144}}s_{{12}}s_{{44}}^{2} - 4c_{{155}}s_{{11}}s_{{44}}^{2} - 4c_{{344}}s_{{13}}s_{{44}}^{2}, \\ s_{{222}} = &-s_{{11}}^{3} (c_{{111}} + c_{{112}} - c_{{222}} ) - c_{{112}}s_{{12}}^{3} - c_{{333}}s_{{13}}^{3} - c_{{111}}s_{{11}}s_{{12}}^{2} - 2c_{{112}}s_{{11}}^{2}s_{{12}} - c_{{113}}s_{{11}}^{2}s_{{13}} \\ &{} - c_{{113}}s_{{12}}^{2}s_{{13}} - 2c_{{123}}s_{{11}}^{2}s_{{13}} - 2c_{{123}}s_{{12}}^{2}s_{{13}} - 3c_{{133}}s_{{11}}s_{{13}}^{2} - 3c_{{133}}s_{{12}}s_{{13}}^{2} - c_{{222}}s_{{11}}^{2}s_{{12}} \\ &{} - 2s_{{11}}s_{{12}}^{2} (c_{{111}} + c_{{112}} - c_{{222}} ) - 4c_{{113}}s_{{11}}s_{{12}}s_{{13}} - 2c_{{123}}s_{{11}}s_{{12}}s_{{13}}, \\ s_{{333}} = &-2c_{{113}}s_{{13}}^{3} - 2c_{{123}}s_{{13}}^{3} - c_{{111}}s_{{11}}s_{{13}}^{2} - 2c_{{112}}s_{{11}}s_{{13}}^{2} - c_{{112}}s_{{12}}s_{{13}}^{2} - c_{{133}}s_{{11}}s_{{33}}^{2} \\ &{} - c_{{133}}s_{{12}}s_{{33}}^{2} - 4c_{{133}}s_{{13}}^{2}s_{{33}} - c_{{222}}s_{{12}}s_{{13}}^{2} - c_{{333}}s_{{13}}s_{{33}}^{2} - s_{{11}}s_{{13}}^{2} (c_{{111}} + c_{{112}} - c_{{222}} ) \\ &{} - 2s_{{12}}s_{{13}}^{2} (c_{{111}} + c_{{112}} - c_{{222}} ) - 2c_{{113}}s_{{11}}s_{{13}}s_{{33}} - 2c_{{113}}s_{{12}}s_{{13}}s_{{33}} - 2c_{{123}}s_{{11}}s_{{13}}s_{{33}} \\ &{} - 2c_{{123}}s_{{12}}s_{{13}}s_{{33}}, \\ s_{{344}} = &-4c_{{144}}s_{{11}}s_{{44}}^{2} - 4c_{{155}}s_{{12}}s_{{44}}^{2} - 4c_{{344}}s_{{13}}s_{{44}}^{2}. \end{aligned}$$

(A.4)

Appendix B: Orientation Distribution Function

The ODF is a probability distribution function that defines the likelihood of crystallites having various orientations. Often, the ODF is expanded as a series of generalized spherical harmonics [51],

$$ w (\zeta,\psi,\phi )=\sum_{l=0}^{\infty}\sum _{m=-l}^{l}\sum_{n=-l}^{l}W_{lmn}Z_{lmn} (\zeta )\exp [-im\psi ]\exp [-in\phi ], $$

(B.1)

where $\zeta=\cos\theta$. $Z_{lmn} (\zeta )$ are the augmented Jacobi polynomials, which are generated by

$$\begin{aligned} Z_{lmn} (\zeta ) =&\sqrt{\frac{2l+1}{2}}\frac{ (-1 )^{l+n}}{2^{l} (l-m )!}\sqrt{ \frac{ (l-m )! (l+n )!}{ (l+m )! (l-n )!}} \\ &{} \times (1-\zeta )^{-\frac{n-m}{2}} (1+\zeta )^{-\frac{n+m}{2}}\frac{d^{l-n}}{d\zeta^{l-n}} \bigl[ (1-\zeta )^{l-m} (1+\zeta )^{l+m} \bigr]. \end{aligned}$$

(B.2)

The Jacobi polynomials $Z_{lmn} (\zeta )$ exhibit the properties [13, 51]:

$$\begin{aligned} \begin{aligned} &\int_{-1}^{1}Z_{lmn} (\zeta)Z_{pmn} (\zeta )\text {d}\zeta=\delta_{lp},\qquad Z_{lmn} (\zeta )=Z_{l\bar{m}\bar{n}} (\zeta ), \\ &Z_{lmn} (\zeta )= (-1 )^{m+n}Z_{l\bar{n}\bar{m}} (\zeta ),\qquad Z_{lm\bar{n}} (\zeta )=Z_{lmn} (-\zeta )=Z_{l\bar{m}n} (\zeta), \end{aligned} \end{aligned}$$

(B.3)

where $\bar{m}=-m$ for any integer $m$. Since $w (\zeta,\psi,\phi )$ is a probability distribution function, it must yield unity when integrated over all possible orientations,

$$ \int_{0}^{2\pi}\int_{0}^{2\pi} \int_{-1}^{1}w (\zeta,\psi,\phi )\mathrm{d}\zeta \mathrm{d}\psi\mathrm{d}\phi=1. $$

(B.4)

From (B.1)–(B.4), the $l=m=n=0$ term in the ODF expansion is always $W_{000}Z_{000}=1/8\pi^{2}$. All terms other than $W_{000}Z_{000}$ are zero if the polycrystal is macroscopically isotropic. Thus, the other nonzero $W_{lmn}$ can be considered the macroscopic anisotropy coefficients of the polycrystal. The number of ODCs are greatly reduced by considering symmetry conditions of the polycrystal and the constituent crystallites. For the case of orthorhombic texture, there exist three orthogonal 2-fold axes of symmetry that are parallel to the axes $x_{1}$, $x_{2}$, and $x_{3}$. For this case, $W_{lmn}=0$ when $m$ is odd, while $W_{lmn}=W_{l\bar {m}n}= (-1 )^{n}W^{\ast}_{lm\bar{n}}$ when $m$ is even where ∗ denotes complex conjugate [51, 53].

The averaged elastic moduli in (17) and (18) and elastic compliances in (24) and (25) contain powers of $\sin\theta$ or $\cos\theta$ up to the rank of the tensor being averaged. The integration over $\zeta$ from $[-1,1]$ for $l$ greater than the rank of the elastic modulus or compliance tensors yields zero. Thus, the index $l$ of the ODCs are bound between $0< l\leq4$ and $0< l\leq6$ for the fourth-rank and sixth-rank Voigt and Reuss averages, respectively. A proof of this identity is found in Man [16]. The nonzero and independent ODCs needed in the ODF expansion to define the Voigt and Reuss averages are $W_{400}$, $W_{420}$, $W_{440}$, $W_{600}$, $W_{620}$, $W_{640}$, and $W_{660}$ for textured polycrystals containing cubic crystallites and $W_{200}$, $W_{220}$, $W_{400}$, $W_{420}$, $W_{440}$, $W_{600}$, $W_{606}$, $W_{620}$, $W_{626}$, $W_{640}$, $W_{646}$, $W_{660}$, and $W_{666}$ for the case of hexagonal crystallites. Bounds on the possible values of $W_{lmn}$ for polycrystals containing cubic and hexagonal crystallites were given by Paroni [67].