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Full-field stress measuring method based on terahertz time-domain spectroscopy

Kai Kang, Yufeng Du, Shibin Wang, Lin an Li, Zhiyong Wang, and Chuanwei Li

Open Access Open Access

Abstract

Terahertz wave has been used to obtain the internal information of materials due to its penetrability to optically opaque materials. In this paper, we propose a full-field stress measuring method using terahertz time-domain spectroscopy. The theoretical model of the method was established based on the stress-optical law, and the method was then validated experimentally. A four-point bending experiment was conducted to determine the stress-optical coefficient of the specimen material, after which the proposed method was used to measure the stress distribution of a diametrically loaded disk. The experimental results present a reasonable agreement with the theoretical solution.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Terahertz wave has been widely applied in various fields due to its advantages such as high transmissivity in dielectric materials, low photon energy, and fingerprint spectrum of molecular chain structures [1,2]. In recent years, terahertz time-domain spectroscopy (THz-TDS) has been one of the most efficient terahertz technologies.

Terahertz wave has also been used in the measurement of the stress in optically opaque dielectrics due to its high transmissivity in dielectrics. As early as in 2008, Ebara et al. measured the stress birefringence of optically opaque materials using a polarization-sensitive THz-TDS system [3]. In 2011, Takahashi et al. obtained intensity images of transmitted terahertz radiation for a loaded polyethylene specimen, but the stress field was not calculated [4]. Later, Song et al. and Schemmel et al. measured the stress-optical coefficients of polytetrafluoroethylene (PTFE) and yttria stabilized tetragonal zirconia (YTZP), respectively, in terahertz frequency through an uniaxial tensile experiment [5,6]. In 2016, Wang et al. determined the plane stress of PTFE, the homogeneous material, based on stress birefringence in terahertz frequency [7]. Recently, the team further represented an anisotropic method to characterize the stress state of single crystal silicon [8]. However, in these studies, the stress was measured only on a single spot; it was difficult to obtain the information of the full-field stress because of the complex experimental procedures.

In this paper, we propose a full-field stress measuring method. The principle of the method is presented in Section 2, where a theoretical model based on the stress-optical law is established to describe the relationship between the amplitude of transmitted terahertz wave and the stress field. Section 3 reports the experiment for validating the method, in which different types of stress field in PTFE specimens were measured using the proposed method. Finally, some discussion and conclusive remarks are presented in Section 4.

2. Proposed measuring method

2.1 Measuring system

The measuring system used in this work is illustrated in Fig. 1. The system is composed of three parts: a polarization-sensitive THz-TDS system, a custom loading device, and a two-dimensional displacement stage.

 figure: Fig. 1.

Fig. 1. The measuring system in this work. (a) The schematic diagram of the experimental system. (b) The polarization-sensitive THz-TDS system. (c) The loading device and the displacement stage. (d) The directions of polarizers and principal stresses.

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Figure 1(b) shows the polarization-sensitive THz-TDS system used in the experiments. The terahertz wave was generated and detected by two rotatable photoconductive antennas. The polarization directions of the antennas were set to be orthogonal. Two polarizers were used to obtain high extinction rate, and the directions of the polarizers were consistent with those of the emitting and receiving antennas, respectively. The light spot of terahertz wave, with a diameter of 4 mm, was focused on the specimen. The highly reliable frequency range of the system was 0.2∼2.5 THz. A commercial Terahertz time domain spectrometer (Terahertz photonics Co. Ltd. TPF15K) was used. Figure 1(c) shows the loading device and the displacement stage. The loading device could provide different types of loading, including the four-point bending and the uniaxial compression. The value of loading was measured by the force sensor, whose maximum measurable loading and accuracy were 2000 N and 0.6 N, respectively. The two-dimensional displacement stage was driven by two stepper motors, and the maximum scanning range and the repeated positioning accuracy of the displacement stage were 50 mm × 50 mm and 2µm, respectively. Figure 1(d) shows the polarization directions and the principal stress directions.

2.2. Measurement principle

The propagation of terahertz wave is described by geometrical optics. Jones matrix is used to express the modulating effects of the terahertz devices and the loaded specimen on terahertz wave. The electric field signal of the polarized terahertz wave from the emitting antenna can be expressed as

$${E_0} = \left[ {\begin{array}{c} {\cos \varphi }\\ {\sin \varphi } \end{array}} \right] \cdot {e^{i(2\pi ft + {\delta _0})}},$$
where f and t are the frequency and time, φ is the angle between the polarization direction and the horizontal direction, and δ0 is the initial phase of terahertz wave.

The terahertz wave propagates through Polarizer1, the loaded specimen, and then Polarizer 2 in turn. The finally received terahertz electric field signal can be expressed as

$${E_1} = R \cdot {Q_{\varphi ^{\prime}}} \cdot {J_\theta } \cdot {Q_\varphi } \cdot {E_0},$$
where
$${Q_\varphi }\textrm{ = }\left[ {\begin{array}{ccc} {{{\cos }^2}\varphi }&{\sin \varphi \cdot \cos \varphi }\\ {\sin \varphi \cdot \cos \varphi }&{{{\sin }^2}\varphi } \end{array}} \right],$$
$${J_\theta }\textrm{ = }\left[ {\begin{array}{ccc} {\cos \theta }&{ - \sin \theta }\\ {\sin \theta }&{\cos \theta } \end{array}} \right] \cdot \left[ {\begin{array}{cc} {{e^{i{\delta_1}}}}&0\\ 0&{{e^{i{\delta_2}}}} \end{array}} \right] \cdot \left[ {\begin{array}{cc} {\cos \theta }&{\sin \theta }\\ { - \sin \theta }&{\cos \theta } \end{array}} \right],$$
$${Q_{\varphi ^{\prime}}}\textrm{ = }\left[ {\begin{array}{cc} {{{\cos }^2}\varphi^{\prime}}&{\sin \varphi^{\prime} \cdot \cos \varphi^{\prime}}\\ {\sin \varphi^{\prime} \cdot \cos \varphi^{\prime}}&{{{\sin }^2}\varphi^{\prime}} \end{array}} \right]$$
and
$$R\textrm{ = }\left[ {\begin{array}{cc} {\cos \varphi^{\prime}}&{\sin \varphi^{\prime}} \end{array}} \right].$$
Qφ and Qφ’ are the Jones matrixes of the two polarizers, and Jθ is the Jones matrix of the loaded specimen. θ is the first principal stress direction. The vector R represents the direction of the receiving antenna which receives only the polarized component of the electric field along φ’ (φ’ = φ + π/2). Substituting Eqs. (3–6) into Eq. (2), we get
$${E_1}\textrm{ = }\sin 2( \theta - \varphi ) \cdot \sin \frac{{{\delta _1} - {\delta _2}}}{2} \cdot {e^{i\left( {2\pi ft + {\delta_0}\textrm{ + }\frac{\pi }{2}\textrm{ + }\frac{{{\delta_1} + {\delta_2}}}{2}} \right)}}.$$

When penetrating the loaded specimen, the terahertz wave is disassembled into two polarized waves due to the stress birefringence. The different propagation speeds cause phase difference between the two waves. After penetrating through the specimen and then Polarizer 2, the two waves merge into one, leading to the change of amplitude. The amplitude of E1 can be expressed as

$$A = \sin 2( \theta - \varphi ) \cdot \sin \frac{{{\delta _1} - {\delta _2}}}{2},$$
where δ1 - δ2 is the phase difference between the principal stress directions. When the thickness d is assumed to be constant, δ1 - δ2 can be calculated using the following equation:
$${\delta _1}\textrm{ - }{\delta _2}\textrm{ = }\frac{{2\pi fd}}{c}\Delta n,$$
where f is the frequency of the terahertz wave and c is the light velocity. Δn is the refractive index difference between the principal stress directions. Due to the stress birefringence, Δn can be expressed as
$$\Delta n\textrm{ = }C \cdot \Delta \sigma ,$$
where C is the stress-optical coefficient, and Δσ = σ1 - σ2 is the principal stress difference. Putting Eqs. (9) and (10) into Eq. (8), we can see that the amplitude A contains information of the first principal stress direction θ and the principal stress difference Δσ. Therefore, two measuring results of A are needed to solve the two unknowns, θ and Δσ. If we capture A when φ = 0 and φ =π/4, θ can be calculated as
$$\theta { ={-} }\frac{1}{2}\arctan \frac{{A|{_{\varphi = 0}} }}{{A|{_{\varphi = \pi /4}} }},$$
Δσ can also be obtained as
$$\Delta \sigma \textrm{ = }\frac{c}{{\pi fdC}} \cdot \arcsin \frac{{A|{_{\varphi \textrm{ = }0}} }}{{\sin 2\theta }}.$$
$$\Delta \sigma { ={-} }\frac{c}{{\pi fdC}} \cdot \arcsin \frac{{A|{_{\varphi \textrm{ = }\pi /4}} }}{{\cos 2\theta }}.$$
The finally calculated Δσ is the average of the results from Eqs. (12) and (13). Note that the amplitudes A in the Eqs. (11) to (13) are the modified values. The measuring amplitude where the principal stress difference Δσ = 0 or the first principal stress direction θ = φ, is used as the reference amplitude. The modified value is calculated as the difference between the measuring amplitude and the reference amplitude.

3. Experimental procedures and results

The specimen material in this work was PTFE. To determine the stress-optical coefficient C of PTFE, a four-point bending experiment was conducted. Figure 2 illustrates the bended specimen. The dotted line indicates the region of interest (ROI); O is the center of specimen. In this experiment, the size of ROI was 28 mm × 2 mm.

 figure: Fig. 2.

Fig. 2. The bended specimen. (a) The custom loading device the loaded specimen. (b) Experimental parameters.

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Based on elastic mechanics [9], the principal stress difference along the stress loading direction can be calculated as

$$\Delta \sigma (x )\textrm{ = }\frac{{6pl}}{{d{h^3}}}x,$$
where l, d and h are the geometric parameters of the bended specimen, p is the static pressure, and x is the position in the direction of p. The experimental parameters are listed in Table 1. According to Eq. (14), the principal stress difference Δσ is proportional to x. In ROI, the first principal stress direction θ is constantly π/2. According to Eq. (13), the stress would modulate the transmitted terahertz time domain signal. The terahertz time-domain signal was measured at multiple points at an interval of 0.5 mm within ROI. The amplitudes were calculated as the peak-trough differences. Figure 3 highlights the experimental results. As can be seen in Fig. 3(a), the amplitude increased with x (from left to right) due to the increasing Δσ. To visualize the change of amplitude in detail, 6 points along x axis (the stress loading direction) were selected, which were -12.5, -7.5, -2.5, 2.5, 7.5 and 12.5 mm, respectively, and the stress differences were calculated by Eq. (14). The waveforms of the transmitted terahertz waves at these points are shown in Fig. 3(b). The amplitudes of signals were modulated obviously by the stresses.

 figure: Fig. 3.

Fig. 3. The results of the four-point bending experiment. (a) The electric field signal amplitude distribution within ROI. (b) The terahertz time-domain signals under different stresses.

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Tables Icon

Table 1. The parameters of the four-point bending experiment

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The angle φ in the experiment was set as π/4. By putting Eq. (9) into Eq. (8), the refractive index difference Δn is calculated as

$$\Delta n\textrm{ = }\frac{c}{{\pi fd}} \cdot \arcsin A|{_{\varphi \textrm{ = }\pi /4}} ,$$
where f is the center frequency of the terahertz wave, which is 1THz. The experimental measuring results of Δn and Δσ are shown in Fig. 4. The stress at each point, Δσ, within ROI could be calculated using Eq. (14). According to Eq. (10), the stress-optical coefficient C is the ratio of Δn to Δσ. C was achieved through linear fitting. Finally, the value of C of the used PTFE specimen was - 2.4×10−10 Pa-1.

 figure: Fig. 4.

Fig. 4. The linear relation of refractive index difference Δn and principal stress difference Δσ.

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After C was obtained, the stress at each point could be calculated from the amplitude at that point using Eqs. (11) to (13), and the full stress field could thus be obtained. An experiment using the specimen of a diametrically loaded PTFE disk was conducted to prove the validity of the proposed measuring method. Figure 5 shows the specimen. The radius, thickness and loading of the disk are listed in Table 2.

 figure: Fig. 5.

Fig. 5. The diametrically loaded disk. (a) The custom loading device and the loaded specimen. (b) Experimental parameters.

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Table 2. The parameters of the diametrically loading experiment

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Based on the theory of elastic mechanics [9], the analytic solution of the stress field of the diametrically loaded disk can be expressed as

$$\left\{ \begin{array}{l} {\sigma_x} = \frac{{2p}}{{\pi d}}\left\{ {\frac{{({r + y} ){x^2}}}{{{{[{{{({r + y} )}^2} + {x^2}} ]}^2}}} + \frac{{({r - y} ){x^2}}}{{{{[{{{({r - y} )}^2} + {x^2}} ]}^2}}} - \frac{1}{{2r}}} \right\}\\ {\sigma_y} = \frac{{2p}}{{\pi d}}\left\{ {\frac{{{{({r + y} )}^3}}}{{{{[{{{({r + y} )}^2} + {x^2}} ]}^2}}} + \frac{{{{({r - y} )}^3}}}{{{{[{{{({r - y} )}^2} + {x^2}} ]}^2}}} - \frac{1}{{2r}}} \right\}\\ {\tau_{xy}} = \frac{{2p}}{{\pi d}}\left\{ {\frac{{{{({r + y} )}^2}x}}{{{{[{{{({r + y} )}^2} + {x^2}} ]}^2}}} - \frac{{{{({r - y} )}^2}x}}{{{{[{{{({r - y} )}^2} + {x^2}} ]}^2}}}} \right\} \end{array} \right.,$$
where p is the static loading, and r and d are the radius and the thickness of the specimen. The first principal stress direction θ and the principal stress difference Δσ are calculated as
$$\left\{ \begin{array}{l} \Delta \sigma = 2\sqrt {{{\left( {\frac{{{\sigma_x} - {\sigma_y}}}{2}} \right)}^2} + {\tau_{xy}}^2} \\ \theta = \frac{1}{2}\arctan \left( {\frac{{ - 2{\tau_{xy}}}}{{{\sigma_x} - {\sigma_y}}}} \right) \end{array} \right..$$
Figure 6 shows the solutions of the first principal stress direction θ and the principal stress difference Δσ of the disk.

 figure: Fig. 6.

Fig. 6. The analytic solutions. (a) The first principal stress direction θ. (b) The principal stress difference Δσ.

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The amplitude of terahertz wave was measured under different polarization settings. The disk was scanned at φ = 0 and φ = 45, respectively. The loading was static during the scanning process. Figure 7 illustrates the measured amplitude distributions of the disk under different settings. The first principal stress direction θ and the principal stress difference Δσ were calculated according to Eqs. (11) to (13), and the results are shown in Fig. 8. The ROI is smaller than the disk because the stress on the disk edge can’t be calculated from the amplitude. The experimental results (Fig. 8) and the analytic solutions (Fig. 6) are in a reasonable agreement. Figure 9 presents a more direct comparison between the solutions and the results along the x-axis and y-axis. As can be seen in Fig. 9(a), the analytic solutions of θ along x-axis and y-axis are constantly zero, and the experimental results of θ are close to their theoretical counterparts. In Fig. 9(b), the analytic and experimental results of Δσ along the x-axis are highly consistent, while those along the y-axis deviate obviously. Such deviation should be caused by the distinct plastic deformation in the disk [4]. In short, the experimental results of the first principal stress direction θ and the principal stress difference Δσ are in agreement with the analytic solutions within the elastic range.

 figure: Fig. 7.

Fig. 7. The electric field signal amplitude distribution of the diametrically loaded disk as we show in Dataset 1 (Ref. [11]) and Dataset 2 (Ref. [12]). (a) φ = 0. (b) φ = π/4.

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 figure: Fig. 8.

Fig. 8. The results of the diametrically loading experiment. (a) The first principal stress direction θ. (b) The principal stress difference Δσ.

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 figure: Fig. 9.

Fig. 9. The analytic solutions and experimental results of the diametrically loading experiment along the x-axis and y-axis. (a) The first principal stress direction θ. (b) The principal stress difference Δσ.

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To further quantify the measuring accuracy, the differences between the analytic and the experimental values are calculated in the elastic range, and their standard deviation is considered the measuring accuracy. Finally, the accuracy of the first principal stress direction θ is 0.014 rad and that of the principal stress difference Δσ is 0.2 MPa, respectively.

4. Discussion and conclusion

In this work, we propose a method to measure the full-field stress of optically opaque dielectrics using a polarization-sensitive THz-TDS system. The measuring model is established based on the stress-optical law. The proposed method was then validated experimentally. The analytic solutions are in agreement with the experimental results in the elastic range.

The proposed method derives from the traditional photoelasticity method for understanding the complex stress field [10]. Application of photoelasticity has to rely on physical models made of some special materials, such as epoxy resin. The method proposed here, by using the high transmissivity of terahertz wave, expands the application range of the traditional photoelasticity method. The full-field stress within dielectric materials, including ceramic material, fibrous material, foam materials and high polymer material, could be measured directly by this method.

More importantly, the proposed method is based on the terahertz wave amplitude, which is mainly influenced by the stress but hardly or slightly influenced by other factors such as the change of environment or the thickness of specimen. Whereas, these factors may not be ignored in methods that directly measure the refractive index [3,5–8]. For this reason, we can spend long time scanning the specimen to get high-quality terahertz images and keep the data noise low enough. As a result, the method can obtain more reliable results. Further, this method would obtain the full-field stress from the images of terahertz camera which can reduce the imaging time.

In our experiment, the stress at each point within the specimen was measured under two polarization settings. This made the measuring procedure a lot simpler than that of some methods in which the stress at a single point needs to be measured more times [3,5–8].

To sum up, this work is an initial step to using terahertz wave in full-field stress measurement. The work is valuable and inspiring because terahertz wave has great potential in the development of convenient internal stress measurement techniques.

Funding

National Natural Science Foundation of China (11772222, 12041201); National Key Research and Development Program of China (2018YFB0703500).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are available in Dataset 1 (Ref. [11]) and Dataset 2 (Ref. [12]).

References

1. B. Ferguson and X. C. Zhang, “Materials for terahertz science and technology,” Nat. Mater. 1(1), 26–33 (2002). [CrossRef]  

2. M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics 1(2), 97–105 (2007). [CrossRef]  

3. S. Ebara, Y. Hirota, M. Tani, M. Hangyo, and Ieee, Highly sensitive birefringence measurement in THz frequency region and its application to stress measurement, 2007 Joint 32nd International Conference on Infrared and Millimeter Waves and 15th International Conference on Terahertz Electronics, Vols 1 and 2 (Ieee, New York, 2007), pp. 651–652.

4. T. Takahashi, Observation of Cavity Interface and Mechanical Stress in Opaque Material by THz Wave (Behaviour of Electromagnetic Waves in Different Media and Structures, 2011).

5. W. Song, L. A. Li, Z. Wang, S. Wang, M. He, J. Han, L. Cong, and Y. Deng, “Experimental verification of the uniaxial stress-optic law in the terahertz frequency regime,” Optics and Lasers in Engineering 52, 174–177 (2014). [CrossRef]  

6. P. Schemmel, G. Diederich, and A. J. Moore, “Direct stress optic coefficients for YTZP ceramic and PTFE at GHz frequencies,” Opt. Express 24(8), 8110–8119 (2016). [CrossRef]  

7. Z. Wang, K. Kang, S. Wang, L. A. Li, N. Xu, J. Han, M. He, L. Wu, and W. Zhang, “Determination of plane stress state using terahertz time-domain spectroscopy,” Sci. Rep. 6(1), 36308 (2016). [CrossRef]  

8. K. Kang, S. Wang, L. A. Li, Z. Wang, and C. Li, “Terahertz-elasticity for single crystal silicon,” Optics and Lasers in Engineering 137, 106396 (2021). [CrossRef]  

9. A. Bertram and R. Glüge, Foundations of Continuum Mechanics (Springer International Publishing, 2015).

10. K. Ramesh, Photoelasticity (Springer, US, 2008).

11. K. Kang, Y. Du, S. Wang, L. A. Li, Z. Wang, and C. Li, “Full-field Stress Measuring Method Based on Terahertz Time-domain Spectroscopy – Dataset 1,” figshare (2021). https://doi.org/10.6084/m9.figshare.15131370 />

12. K. Kang, Y. Du, S. Wang, L. A. Li, Z. Wang, and C. Li, “Full-field Stress Measuring Method Based on Terahertz Time-domain Spectroscopy – Dataset 2,” figshare (2021). https://doi.org/10.6084/m9.figshare.15131358

Supplementary Material (2)

NameDescription
Dataset 1       The electric field signal amplitude distribution of the diametrically loaded disk.when the direction of Polarizer 1 is 90 Deg..
Dataset 2       The electric field signal amplitude distribution of the diametrically loaded disk.when the direction of polarizer is 45 Deg..

Data availability

Data underlying the results presented in this paper are available in Dataset 1 (Ref. [11]) and Dataset 2 (Ref. [12]).

11. K. Kang, Y. Du, S. Wang, L. A. Li, Z. Wang, and C. Li, “Full-field Stress Measuring Method Based on Terahertz Time-domain Spectroscopy – Dataset 1,” figshare (2021). https://doi.org/10.6084/m9.figshare.15131370 />

12. K. Kang, Y. Du, S. Wang, L. A. Li, Z. Wang, and C. Li, “Full-field Stress Measuring Method Based on Terahertz Time-domain Spectroscopy – Dataset 2,” figshare (2021). https://doi.org/10.6084/m9.figshare.15131358

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Figures (9)
Fig. 1.
Fig. 1. The measuring system in this work. (a) The schematic diagram of the experimental system. (b) The polarization-sensitive THz-TDS system. (c) The loading device and the displacement stage. (d) The directions of polarizers and principal stresses.

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Fig. 2.
Fig. 2. The bended specimen. (a) The custom loading device the loaded specimen. (b) Experimental parameters.

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Fig. 3.
Fig. 3. The results of the four-point bending experiment. (a) The electric field signal amplitude distribution within ROI. (b) The terahertz time-domain signals under different stresses.

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Fig. 4.
Fig. 4. The linear relation of refractive index difference Δn and principal stress difference Δσ.

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Fig. 5.
Fig. 5. The diametrically loaded disk. (a) The custom loading device and the loaded specimen. (b) Experimental parameters.

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Fig. 6.
Fig. 6. The analytic solutions. (a) The first principal stress direction θ. (b) The principal stress difference Δσ.

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Fig. 7.
Fig. 7. The electric field signal amplitude distribution of the diametrically loaded disk as we show in Dataset 1 (Ref. [11]) and Dataset 2 (Ref. [12]). (a) φ = 0. (b) φ = π/4.

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Fig. 8.
Fig. 8. The results of the diametrically loading experiment. (a) The first principal stress direction θ. (b) The principal stress difference Δσ.

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Fig. 9.
Fig. 9. The analytic solutions and experimental results of the diametrically loading experiment along the x-axis and y-axis. (a) The first principal stress direction θ. (b) The principal stress difference Δσ.

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Tables (2)

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Table 1. The parameters of the four-point bending experiment

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Table 2. The parameters of the diametrically loading experiment

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Equations (17)

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E 0 = [ cos ⁡ φ sin ⁡ φ ] ⋅ e i ( 2 π f t + δ 0 ) ,
E 1 = R ⋅ Q φ ′ ⋅ J θ ⋅ Q φ ⋅ E 0 ,
Q φ  =  [ cos 2 φ sin ⁡ φ ⋅ cos ⁡ φ sin ⁡ φ ⋅ cos ⁡ φ sin 2 φ ] ,
J θ  =  [ cos ⁡ θ − sin ⁡ θ sin ⁡ θ cos ⁡ θ ] ⋅ [ e i δ 1 0 0 e i δ 2 ] ⋅ [ cos ⁡ θ sin ⁡ θ − sin ⁡ θ cos ⁡ θ ] ,
Q φ ′  =  [ cos 2 φ ′ sin ⁡ φ ′ ⋅ cos ⁡ φ ′ sin ⁡ φ ′ ⋅ cos ⁡ φ ′ sin 2 φ ′ ]
R  =  [ cos ⁡ φ ′ sin ⁡ φ ′ ] .
E 1  =  sin ⁡ 2 ( θ − φ ) ⋅ sin ⁡ δ 1 − δ 2 2 ⋅ e i ( 2 π f t + δ 0  +  π 2  +  δ 1 + δ 2 2 ) .
A = sin ⁡ 2 ( θ − φ ) ⋅ sin ⁡ δ 1 − δ 2 2 ,
δ 1  -  δ 2  =  2 π f d c Δ n ,
Δ n  =  C ⋅ Δ σ ,
θ = − 1 2 arctan ⁡ A | φ = 0 A | φ = π / 4 ,
Δ σ  =  c π f d C ⋅ arcsin ⁡ A | φ  =  0 sin ⁡ 2 θ .
Δ σ = − c π f d C ⋅ arcsin ⁡ A | φ  =  π / 4 cos ⁡ 2 θ .
Δ σ ( x )  =  6 p l d h 3 x ,
Δ n  =  c π f d ⋅ arcsin ⁡ A | φ  =  π / 4 ,
{ σ x = 2 p π d { ( r + y ) x 2 [ ( r + y ) 2 + x 2 ] 2 + ( r − y ) x 2 [ ( r − y ) 2 + x 2 ] 2 − 1 2 r } σ y = 2 p π d { ( r + y ) 3 [ ( r + y ) 2 + x 2 ] 2 + ( r − y ) 3 [ ( r − y ) 2 + x 2 ] 2 − 1 2 r } τ x y = 2 p π d { ( r + y ) 2 x [ ( r + y ) 2 + x 2 ] 2 − ( r − y ) 2 x [ ( r − y ) 2 + x 2 ] 2 } ,
{ Δ σ = 2 ( σ x − σ y 2 ) 2 + τ x y 2 θ = 1 2 arctan ⁡ ( − 2 τ x y σ x − σ y ) .
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