ABSTRACT
The diffusivity of ultrasound in an untextured aggregate of cubic crystallites is studied theoretically with a view towards nondestructive characterization of microstructures. Multiple scattering formalisms for the mean Green’s dyadic and for the covariance of the Green’s dyadic (and therefore for the energy density) based upon the method of smoothing are presented. The first-order smoothing approximation used is accurate to leading order in the anisotropy of the constituent crystallites. A further, Born, approximation is invoked which limits the validity of the calculation to frequencies below the geometrical optics regime.
从理论上研究了超声波在无纹理立方晶体集合体中的扩散性，以期实现微结构的无损表征。基于平滑方法，提出了 Green 二元平均值和 Green 二元协方差（以及能量密度）的多重散射形式。所使用的一阶平滑近似法精确到了组成晶体各向异性的前导阶。此外，还引用了 Born 近似，将计算的有效性限制在几何光学机制以下的频率。
Known result for the mean field attenuations are recovered. The covariance is found to obey an equation of radiative transfer for which a diffusion limit is taken. The resulting diffusivity is found to vary inversely with the fourth power of frequency in the Rayleigh, long wavelength, regime and inversely with the logarithm of frequency on the short wavelength, stochastic, asymptote. The results are found to fit the experimental data.
复现了平均场衰减的已知结果。发现协方差服从一个辐射传递方程，其中采用了扩散极限。结果发现，在长波的 Rayleigh 机制中，扩散率与频率的四次方成反比变化，而在短波长的随机渐近线上，扩散率与频率的对数成反比变化。结果符合实验数据。
INTRODUCTION
The microstructure of polycrystalline aggregates has been probed by means of ultrasonic scattering for at least 40 years, since the work of MASON and MCSKIMM (1947). That ultrasound with frequencies of the order of $10\text{ MHz}$ and inverse wave numbers of order 50 microns would be sensitive to microstructural variations on the length scales of relevance in typical polycrystals has long been appreciated. The names of PAPADAKIS (1968, 1981), BHATIA (1959), Adler (FITTING and ADLER, 198 1). Goebbels (GOEBBELS and HOLLER, 1978 ; GOEBBELS,1980 ; Guo et al., 1985) and VARY (1978) are amongst those associated with laboratory efforts in the pursuit of the ability to assess microstructure, and potentially therefore, mechanical properties, using nondestructive ultrasound. In spite. though, of considerable effort over many years, the most common technique, that of ultrasonic attenuation, though proven in the laboratory, has not seen much application in the field.
自 MASON 和 MCSKIMM（1947 年）的研究以来，利用超声波散射探测多晶体聚集体的微观结构至少已有 40 年的历史。人们早就意识到，频率为 $10$ 兆赫、反波数为 50 微米的超声波对典型多晶体相关长度尺度上的微观结构变化非常敏感。PAPADAKIS (1968, 1981), BHATIA (1959), Adler (FITTING and ADLER, 1981).Goebbels（GOEBBELS 和 HOLLER，1978 年；GOEBBELS，1980 年；Guo 等人，1985 年）和 VARY（1978 年）等人的名字都与实验室在利用无损超声波评估微观结构和可能的机械性能方面所做的努力有关。尽管多年来做出了大量努力，但最常用的超声衰减技术虽然在实验室中得到了验证，但在现场应用却不多。
That technique calls for the measurement of the exponential rate of spatial decay of an ensemble averaged plane wave. Due to a host of potential systematic errors, the unambiguous measurement of that rate is, however, quite difficult (TRUELL ef al., 1969).
The finite beam width to wavelength ratio of ultrasonic transducers is responsible for geometric, non exponential, decays as the beam transmits from near to far field conditions. Most studies are carried out in slab-like geometries and the wave amplitude monitored as it reflects between opposite faces. Imperfections in the reflection coefficients at these faces can artificially enhance the apparent attenuation.
这种技术要求测量系综平均平面波的空间指数衰减速率。然而，由于存在大量潜在的系统误差，要准确测量该速率相当困难（TRUELL 等人，1969 年）。
超声波换能器的有限波束宽度与波长比是造成波束从近场条件传输到远场条件时出现几何非指数衰减的原因。大多数研究都是在板状几何结构中进行的，波幅在相对面之间反射时受到监测。这些面上反射系数的缺陷会人为地增强表观衰减。
Faces which are insufficiently parallel can distort the apparent pulse amplitudes. Internal friction, or absorption, which is more properly viewed as a temporal decay than a spatial one, contributes in such a way as to be indistinguishable, in these configurations, from scattering based decay. High attenuation rates are immeasurable except over distances so short as to bring the ergodic hypothesis, crucial to comparisons between theory and experiment, into severe question.
不够平行的面会扭曲表观脉冲幅度。内摩擦/吸收与其说是空间衰减，不如说是时间衰减，在这些配置中，内摩擦或吸收与基于散射的衰减无法区分。高衰减率是不可测量的，除非距离极短，以至于理论与实验比较的关键–遍历假说受到严重质疑。
In part due to the above difficulties there has been an increasing effort in recent years to measure microstructure from the energy which is incoherently and singly backscattered from a beam (GOEBBELS and HOLLER, 1978; FAY, 1973; SANIIE and BILGUTAY, 1986; SANE et al., 1988). The potential for systematic error in these configurations is perhaps less severe than in the conventional parallel wall slab configurations.
部分由于上述困难，近年来人们越来越努力地利用光束的非相干和单一反向散射能量来测量微观结构（GOEBBELS 和 HOLLER，1978 年；FAY，1973 年；SANIIE 和 BILGUTAY，1986 年；SANE 等人，1988 年）。与传统的平行壁结构相比，这些构型潜在的系统误差较小。
The technique is also attractive in that it makes fewer demands on specimen geometry and can potentially give information on microstructure as a function of depth.
The chief difficulties appear to be traceable to the need for extensive spatial averaging of signal powers in order to minimize the effect of the inevitable meaningless fluctuations of an incoherent wave field, the need for minimum slab thicknesses in order to resolve backwall echoes and distinguish them from the important incoherent grass and the absence of a complete theory relating backscattered power to microstructure.
该技术的吸引力还在于，它对试样的几何形状要求较低，并有可能提供有关深度函数的微观结构信息。
主要的困难似乎可以归结为：需要对信号功率进行广泛的空间平均，以尽量减少非相干波场不可避免的无意义波动的影响；需要最小的板厚度，以分辨后壁回波并将其与重要的非相干草区分开来；缺乏将背向散射功率与微观结构相关联的完整理论。
It is proposed here that measurement of the evolution of the incoherent and Multiplanscattered, fully diffuse, ultrasonic wave field in a material with a scattering microstructure may provide a more robust measure of that microstructure. The published literature addressed to such wave fields is meagre, and seems to be confined to the paper by Guo et al. (1985) and to two by WEAVER(1989a, b).
WEAVER(1989a) considered sub MHz range ultrasound in a specimen with an artificial centimeter scale microstructure. While not directly applicable to the usual materials of interest in NDE, it did demonstrate, over several orders of magnitude, the applicability of the concept of diffusion to the evolution of these fields. That is, it was shown that the ultrasonic energy density evolved in accordance with a diffusion, or heat, equation, modified by the inclusion of an extra term representing temporal decay.
$$ \frac{\partial E}{\partial t} = D\nabla^{2}E - \xi E + \text{source} $$
本文提出，测量具有散射微观结构的材料中的非相干和多平面散射、完全漫射超声波场的演化情况，可以更可靠地测量该微观结构。关于这种波场的已发表文献很少，似乎仅限于 Guo 等人（1985 年）的论文和 WEAVER（1989a，b）的两篇论文。
WEAVER（1989a）考虑了在具有人工厘米级微结构的试样中使用亚兆赫范围的超声波。虽然该研究并不直接适用于无损检测中常见的相关材料，但它确实在几个数量级上证明了扩散概念对这些(声)场演化的适用性。也就是说，研究表明，超声波能量密度的演变符合扩散方程或热方程，并加入了一个代表时间衰减的额外项。
where $E$ is the elastodynamic spectral energy density at frequency $\omega$; $D(\omega)$ is the frequency dependent diffusivity and $\xi(\omega)$ is the absorption rate, taken to be uninteresting for the purposes of the present communication and henceforth neglected.
WEAVER (1989b) has also shown strong correlations between measured diffusivities in steels and their heat treatments and fracture toughnesses. Guo et al. (1985) after a study of such fields in a range of materials, also report a good fit to the predictions of a diffusion model.
其中，$E$ 是(角)频率为 $\omega$ 时的弹性能谱能量密度；$D(\omega)$ 是与频率相关的扩散率; $\xi(\omega)$ 是吸收率。
WEAVER (1989b) 也表明钢测得的扩散率与其热处理和断裂韧性之间有很强的相关性。Guo 等人（1985 年）在对一系列材料中的此类场进行研究后，也报告了与扩散模型预测的良好拟合。
Guo et al. and Weaver have both emphasized that the diffuse-field technique promises to be able to distinguish between temporal decay, or absorption, and scattering. Each is potentially a nondestructive testing parameter of interest. the former having received little attention in the past due to the difficulty of its in situ measurement, (e.g. TVRDOKHLEBOV,1986).
Guo 等人和 Weaver 都强调，扩散场技术有望区分 时间衰减/吸收 和 散射. 前者由于实地测量困难，过去很少受到关注（如 TVRDOKHLEBOV，1986 年）。
No theory for the diffusivity of an ultrasonic field in a polycrystal is immediately obvious. Guo et al. (1985) suggest, in analogy to a random-walk model, that the diffusivity $D$ should be one-third of the product of a mean free ray path and the wave speed. They then identify that mean free path with the inverse of the attenuation and suggest that the diffusion constant $D$ is simply an inverse measure of attenuation.
超声场在多晶体中的扩散率理论并不显然。Guo 等人（1985 年）类比随机行走模型，认为扩散系数 $D$ 应该是 平均自由射程 与 波速 乘积的三分之一。然后，他们将 平均自由程 与 衰减的倒数 相提并论，认为扩散常数 $D$ 就是 衰减的倒数。
The argument is attractive due to its simplicity, but it must be modified to account for the presence of different wave modes, the transverse ($T$) and longitudinal ($L$), each with its own wave speed $c$ and attenuation $\alpha$. We must further recognize that the appropriate mean free path should be half of the inverse of the attenuation (as it is the energy which is diffusing and the attenuation rate of the coherent energy is twice that, $\alpha$, of coherent amplitude). With these identifications we write, as a conjecture,
$$ D = f_{T}\frac{c_{T}}{6\alpha_{T}} + f_{L}\frac{c_{L}}{6\alpha_{L}} $$
where $f_{T}$ and $f_{L} = 1 - f_{T}$ are the mixing weights for the transverse and longitudinal contributions to the diffusivity. Their values are not obvious. The only a priori likely possibility is
$$ \frac{f_{T}}{f_{L}} = \frac{2c_{L}^{3}}{c_{T}^{3}} $$
corresponding to the equilibrium partition of the energy of a fully developed diffuse wave field (WEAVER,1982).
这一论证因其简洁而颇具吸引力，但必须加以修改，以考虑到不同波模的存在，即横波（$T$）和纵波（$L$），每种波模都有自己的波速 $c$ 和衰减 $\alpha$。我们必须进一步认识到，适当的平均自由程应该是衰减的倒数的一半（因为扩散的是能量，而相干能量的衰减率是相干振幅的两倍，即 $\alpha$）。使用上述记号我们就可以写出这样的猜想：
$$ D = f_{T}\frac{c_{T}}{6\alpha_{T}} + f_{L}\frac{c_{L}}{6\alpha_{L}} \tag{1.2}\label{eq1.2} $$
其中，$f_{T}$ 和 $f_{L} = 1 - f_{T}$ 是横波和纵波对扩散性贡献的混合权重。它们的值并不明显。唯一可能的先验值是
$$ \frac{f_{T}}{f_{L}} = \frac{2c_{L}^{3}}{c_{T}^{3}} $$
与完全发展的散射波场的能量平衡分区相对应（WEAVER，1982 年）。
Equation $\eqref{eq1.2}$ has the virtue of being dependent only on quantities available from standard theory and/or experiment. As Guo et al. indicate it also implies that the diffusivity should, for frequencies below the geometric optics regime, scale inversely with frequency with a power between two and four. This follows from the standard theory (for which the reader is directed to the recent work by STANKEand KINO, 1984 or HIRSEKORN,1982, 1983. 1985, 1986 or KARALand KELLER, 1964) or to the general discussions (PAPADAKIS,1981; BHATIA, 1959 ; FITTING and ADLER, 1981) for the attenuation of mean waves in polycrystals.
等式 $\eqref{eq1.2}$ 的优点是只依赖于标准理论和/或实验中可测的量。正如 Guo 等人所指出的，它还意味着，对于低于几何光学体系的频率，扩散率应与频率幂成反比，指数在 $2$ 到 $4$ 之间。根据标准理论（读者可参阅 STANKE 和 KINO，1984 年或 HIRSEKORN，1982、1983、1985、1986 年或 KARALand KINO，1984 年的最新研究成果），这一点是正确的。1985，1986 或 KARAL 和 KELLER，1964），或多晶体中平均波衰减的一般讨论（PAPADAKIS，1981；BHATIA，1959；FITTING 和 ADLER，1981）。
That theory indicates that attenuation scales with the fourth power of frequency in a domain for which wavelength is much larger than microscale length (the ‘Rayleigh’ domain) and with the square of frequency in the opposite limit (the ‘stochastic asymptote’) but is ultimately bounded by the geomtrical optics limit where $\alpha^{-1}$ is of the order of the microscale length.
该理论表明，在波长远大于微观尺度的领域（“Rayleigh” 区），衰减与频率的四次方成正比，而在相反的极限（“随机渐近”），衰减与频率的平方成正比，但最终受限于几何光学极限，其中 $\alpha^{-1}$ 是微尺度长度的数量级。
The frequency dependence of the diffusivity observed by Guo et al. was however very clearly an inverse first power. In consequence, $\eqref{eq1.2}$ cannot be correct. We will see in this communication that that error is due to the identification of that “mean free ray path” required by the random walk model with $1/2\alpha$. The model requires a mean free path which corresponds to the distance travelled by a typical ray before it is significantly scattered away from its original direction.
然而，Guo 等人观察到的扩散率的频率依赖性很明显是一种逆 一次 幂。因此，$\eqref{eq1.2}$ 不可能是正确的。我们将在这篇论文中看到，这一错误是由于将随机漫步模型所需的 “平均自由射线程” 与 $1/2\alpha$ 相混淆造成的。该模型要求的平均自由程相当于典型射线在明显偏离其原始方向之前所走过的距离。
At low frequencies. in the Rayleigh regime, grain scattering is isotropic; equal energies are scattered in all directions; hence $1/2\alpha$ does correspond to a mean free path. At higher frequencies the scattering is increasingly biased towards the forward direction; typical rays are scattered through small angles; hence the appropriate mean free path is somewhat greater than $1/2\alpha$.
在低频下，即在 Rayleigh 机制中，晶粒散射是各向同性的；在所有方向上散射的能量相同；因此 $1/2\alpha$ 确实对应于 平均自由程。在较高频率下，散射越来越偏向前进方向；典型射线被小角度散射；因此适当的平均自由路径应当略大于 $1/2\alpha$。
The present communication is addressed towards the derivation of a more accurate expression for $D$. The derivation is confined, like the attenuation calculation of STANKE and KINO (1984), to the case of an untextured aggregate of cubic-symmetry crystallites. This is the simplest polycrystal and includes iron as a special case, allowing comparison with the results from one of the specimens studied by Guo el al.
本论文旨在推导出更精确的 $D$ 表达式。与 STANKE 和 KINO（1984 年）的衰减计算一样，推导仅限于立方对称晶体的无纹理集合体的情况。这是最简单的多晶体，包括作为特例的铁，可与 Guo 等人研究样品之一的结果进行比较。
After discussion of the assumed statistics of the polycrystal in Section 2, Section 3 presents, by means of consideration of the ensemble average Green’s function the standard result for the attenuations, explicit expressions for which are given in Section 7. Sections 4 and 5 present the analogous theory for the mean square Green’s functions, and therefore for the energy.
第 2 节讨论了多晶体的假定统计量，第 3 节通过对系综平均 Green 函数的考虑，给出了衰减的标准结果，第 7 节给出了其明确表达式。第 4 节和第 5 节介绍了均方 Green 函数以及能量的类似理论。
Section 6 solves the resulting equations of radiative transfer and concludes with an expression (6.26) for the diffusivity. That diffusivity is found to be given by a function of longitudinal and transverse wavespeeds and attenuations, as anticipated in (1.2), but to depend upon scattering-angle weighted versions of the attenuations as well, as anticipated above. Upon choosing a form for the microstructural spatial correlation function, the diffusivity is evaluated in Section 7.
第 6 节求解了由此得出的辐射传递方程，最后给出了扩散率的表达式 (6.26)。正如 (1.2)$(*)$ 所预计的那样，扩散率是由纵向和横向波速及衰减的函数给出的，但正如上文所预计的那样，扩散率还取决于衰减的散射角加权版本。选择微结构空间相关函数的形式后，将在第 7 节评估扩散率。
It is shown that the frequency dependence of $D$ spans a range from strong to weak. The low frequency dependence is with the inverse fourth power of frequency. At high frequencies but below the geometric optics limit $D^{-1}$ scales only logarithmically with frequency. Section 7 concludes with recommendations for further work.
结果表明，$D$ 的频率依赖性从强到弱不等。低频依赖于频率的倒四次方。在高频率但低于几何光学极限时，$D^{-1}$ 与频率的关系仅为对数。第 7 节最后提出了进一步工作的建议。
MATHEMATICAL PRELIMINARIE
Green’s function (dyadic) for the response of a heterogeneous anisotropic passive elastodynamic medium is given by the causal solution to
$$ [-\delta_{li}\partial_{t}^{2} + \partial_{k}C_{klij}(\mathbf{x})\partial_{j}]G_{i\alpha}(\mathbf{x}, \mathbf{x}^{\prime};t) = \delta_{i\alpha}\delta^{3}(\mathbf{x} - \mathbf{x}^{\prime})\delta(t). $$
In $\eqref{eq2.1}$ units have been used such that the material density is equal to one, and $G_{ij}(\mathbf{x},\mathbf{x}^{\prime})$ is the displacement response in the $i$ direction at position $\mathbf{x}$ due to a unit impulse applied in the $j$ direction at position $\mathbf{x}^{\prime}$ at time zero.
The usual fourth rank stiffness tensor is given by
$$ C_{klrj}(\mathbf{x}) = C_{klrj}^{\circ} + \gamma_{klrj}(\mathbf{x}) $$
with
$$ \langle\gamma(\mathbf{x})\rangle = 0. $$
异质各向异性被动弹性动力学介质响应的 Green 函数（二元）由以下因果解给出
$$ [-\delta_{li}\partial_{t}^{2} + \partial_{k}C_{klij}(\mathbf{x})\partial_{j}]G_{i\alpha}(\mathbf{x}, \mathbf{x}^{\prime};t) = \delta_{i\alpha}\delta^{3}(\mathbf{x} - \mathbf{x}^{\prime})\delta(t). \tag{2.1}\label{eq2.1} $$
在 $\eqref{eq2.1}$ 中使用的单位是材料密度等于 $1$，而 $G_{ij}(\mathbf{x},\mathbf{x}^{\prime})$ 是在时间为零时，在位置 $\mathbf{x}$ 的 $i$ 方向上，由于在位置 $\mathbf{x}$ 的 $j$ 方向上施加单位脉冲而产生的位移响应。
通常的四阶刚度张量由以下给出:
$$ C_{klrj}(\mathbf{x}) = C_{klrj}^{\circ} + \gamma_{klrj}(\mathbf{x}) $$
其中
$$ \langle\gamma(\mathbf{x})\rangle = 0. $$
Angular brackets represent ensemble averages, hence $\mathbf{C}$ has a mean value of $\mathbf{C}^{\circ}$ and $\gamma$ represents the modulus fluctuations. We shall be considering the system $\eqref{eq2.1}$ for the case $\gamma$ small, and discussing mean responses and mean square reponses to leading order in the magnitude of $\gamma$. As such, all relevant statistical information regarding the material heterogeneity is contained in the covariance function
$$ \langle\gamma_{klrj}(\mathbf{x})\gamma_{\alpha\beta\gamma\delta}(\mathbf{x}^{\prime})\rangle = \Xi^{\alpha\beta\gamma\delta}_{klrj}\eta(|\mathbf{x} - \mathbf{x}^{\prime}|). \label{2.4} $$
Any other statistical information about the medium is necessarily of order $\gamma^{n}$ with $n > 2$, and therefore negligible.
角括号代表系综平均值，因此 $\mathbf{C}$ 的平均值为 $\mathbf{C}^{\circ}$, $\gamma$ 代表模量波动。我们将考虑 $\gamma$ 较小情况下的系统 $\eqref{eq2.1}$ ，并讨论平均响应和均方响应，以 $\gamma$ 的大小为前导阶。因此，关于材料异质性的所有相关统计信息都包含在协方差函数中:
$$ \langle\gamma_{klrj}(\mathbf{x})\gamma_{\alpha\beta\gamma\delta}(\mathbf{x}^{\prime})\rangle = \Xi^{\alpha\beta\gamma\delta}_{klrj}\eta(|\mathbf{x} - \mathbf{x}^{\prime}|). \tag{2.4}\label{eq2.4} $$
关于介质的任何其他统计信息必然是 $n > 2$ 的 $\gamma^{n}$ 阶，因此可以忽略不计。
Two assumptions regarding the statistics of the polycrystal are implicit in the form of $\eqref{eq2.4}$.
The first assumption is that the tensorial character of the covariance function decouples from the spatial dependence. This is equivalent to an assertion that there are no orientation correlations between different crystallites. The assumption is a standard one. We have also assumed that the polycrystal’s second order statistics are homogeneous (a function of $\mathbf{x} - \mathbf{x}^{\prime}$ rather than $\mathbf{x}$ and $\mathbf{x}^{\prime}$ separately) and isotropic (independent of the direction of $\mathbf{x} - \mathbf{x}^{\prime}$).
$\eqref{eq2.4}$ 的形式隐含了两个关于多晶体的统计假设。
第一个假设是协方差函数的张量特征与空间依赖性脱钩。这相当于断言不同晶体之间不存在取向相关性。这是一个标准假设。
我们还假设多晶体的二阶统计量是均匀的（是 $\mathbf{x} - \mathbf{x}^{\prime}$ 的函数，而非 $\mathbf{x}$ 和 $\mathbf{x}^{\prime}$ 各自的函数）和各向同性的（与 $\mathbf{x} - \mathbf{x}^{\prime}$ 的方向无关）。
If $\nu$ is taken to be a dimensional measure of $\gamma$ then we define $\eta(r)$ as $\nu^{2}$ times the probability that two points separated by a distance $r$ lie within the same crystallite. Hence the eighth rank tensor $\Xi$ is dimensionless. Like STANKE and KINO (1984). we shall choose an $\eta$ of the form
$$ \eta(r) = \nu^{2}e^{-\beta r}. $$
The present work will not, however, call for any specification of $\eta$ until necessary, in Section 7. We will occasionally cite $\beta^{-1}$, though, as a measure of microscale length.
如果把 $\nu$ 作为 $\gamma$ 的维度度量，那么我们就可以把 $\eta(r)$ 定义为 $\nu^{2}$ 与 相距 $r$ 的两点位于同一晶体内的概率 的乘积。因此，八阶张量 $\Xi$ 是无量纲的。与 STANKE 和 KINO (1984)一样，我们将选择一个 $\eta$:
$$ \eta(r) = \nu^{2}e^{-\beta r}. $$
然而，在第 7 节中，在必要之前，本研究不会要求对 $\eta$ 进行任何说明。不过，我们偶尔会引用 $\beta^{-1}$ 作为微观长度的度量。
With respect to crystal axes, the moduli of a cubic crystallite are
$$ \begin{aligned} C_{ijkl} &= \lambda^{I}\delta_{ij}\delta_{kl} + \mu^{I}(\delta_{ik}\delta_{jl} + \delta_{il}\delta_{jk}) + \nu\delta_{ijkl}\\ &= C_{ijkl}^{I} + \nu\delta_{ijkl} \end{aligned} $$
where the last term in each line is not a tensor, and is defined as vanishing unless all indices are equal, and being unity otherwise.
相对于晶轴，立方晶体的模量为
$$ \begin{aligned} C_{ijkl} &= \lambda^{I}\delta_{ij}\delta_{kl} + \mu^{I}(\delta_{ik}\delta_{jl} + \delta_{il}\delta_{jk}) + \nu\delta_{ijkl}\\ &= C_{ijkl}^{I} + \nu\delta_{ijkl} \end{aligned} $$
其中每行的最后一项不是张量，除非所有指数相等，否则定义为 $0$，否则定义为 单位矩阵。
With respect to laboratory axes the crystallite modulus tensor is given by
$$ \begin{aligned} C_{ijkl} &= C_{ijkl}^{I} + \nu\sum_{n = 1}^{3}a_{i}^{n}a_{j}^{n}a_{k}^{n}a_{l}^{n}\\ &= C_{ijkl}^{I} + t_{ijkl} \end{aligned} $$
where $a_{i}^{n}$ is an element of the rotation matrix representing the transformation between the crystallite axes and the laboratory axes.
If the rotation between crystal and laboratory axes is represented by the three Euler angles $\phi$, $\Theta$ and $\xi$, the rotation matrix elements are given by
$$ \begin{aligned} a_{1}^{1} &= -\cos{\Theta}\sin{\phi}\sin{\xi} + \cos{\phi}\cos{\xi},\quad a_{2}^{1} = \cos{\Theta}\cos{\phi}\sin{\xi} + \sin{\phi}\cos{\xi},\\ a_{3}^{1} &= \sin{\Theta}\sin{\xi}.\\ a_{1}^{2} &= -\cos{\Theta}\sin{\phi}\cos{\xi} - \cos{\phi}\sin{\xi},\quad a_{2}^{2} = \cos{\Theta}\cos{\phi}\cos{\xi} - \sin{\phi}\sin{\xi},\\ a_{3}^{2} &= \sin{\Theta}\cos{\xi}.\\ a_{1}^{3} &= \sin{\Theta}\sin{\phi},\quad a_{2}^{3} = -\sin{\Theta}\cos{\phi},\quad a_{3}^{3} = \cos{\Theta}. \end{aligned} $$
as found in Eq. (2.50) of Bunge’s treatise (BIJNGE,1982).
相对于实验系，晶体模量张量的计算公式为
$$ \begin{aligned} C_{ijkl} &= C_{ijkl}^{I} + \nu\sum_{n = 1}^{3}a_{i}^{n}a_{j}^{n}a_{k}^{n}a_{l}^{n}\\ &= C_{ijkl}^{I} + t_{ijkl} \end{aligned} $$
其中，$a_{i}^{n}$ 是旋转矩阵的元素，代表晶体轴和实验室轴之间的转换。
如果晶体轴和实验室轴之间的旋转用三个欧拉角 $\phi$、$\Theta$ 和 $\xi$ 表示，则旋转矩阵元素为
$$ \begin{aligned} a_{1}^{1} &= -\cos{\Theta}\sin{\phi}\sin{\xi} + \cos{\phi}\cos{\xi},\quad a_{2}^{1} = \cos{\Theta}\cos{\phi}\sin{\xi} + \sin{\phi}\cos{\xi},\\ a_{3}^{1} &= \sin{\Theta}\sin{\xi}.\\ a_{1}^{2} &= -\cos{\Theta}\sin{\phi}\cos{\xi} - \cos{\phi}\sin{\xi},\quad a_{2}^{2} = \cos{\Theta}\cos{\phi}\cos{\xi} - \sin{\phi}\sin{\xi},\\ a_{3}^{2} &= \sin{\Theta}\cos{\xi}.\\ a_{1}^{3} &= \sin{\Theta}\sin{\phi},\quad a_{2}^{3} = -\sin{\Theta}\cos{\phi},\quad a_{3}^{3} = \cos{\Theta}. \end{aligned} $$
见 Bunge 论文（BIJNGE，1982 年）中的公式 (2.50)。
