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Abstract
This paper presents a method to measure the three-dimensional distribution of uniaxial stress based on Terahertz Time Domain Spectroscopy (THz-TDS). The measurement principle was first established, which combines the computed tomography (CT) method and the photo-elastic effects. A classic filtered back-projection algorithm is adopted to calculate the three-dimensional stress fields from THz-TDS scanning images. Then, in verification experiment, the uniaxial stress distribution in the compressed cylinder and the stretched screw is obtained based on the measurement principle. Finally, the reliability of the proposed method has been verified by comparing the experiment results with the finite element simulation. A reasonable agreement is obtained.
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1. Introduction
Terahertz (THz) radiation is electromagnetic radiation with the frequency in the range of 0.1∼10 THz. Due to the advantages of THz radiation such as strong transmittance to opaque materials, biological harmless and others, it is widely used in non-destructive testing, biomedical imaging, security testing and information technology [1–3]. THz-TDS is a widely used technology for THz spectral analysis because of its high signal-to-noise ratio and high temporal resolution. By analyzing the time domain waveforms of transmitted or reflected THz pulses, rich information about the medium can be obtained, so that THz-TDS is often used for high-precision measurements on thickness, refractive index, birefringence, etc.
Owing to the small energy of THz photons, they are biologically harmless, so THz technology has also obtained good potential in the field of three-dimensional imaging [4], such as THz CT imaging. B. Ferguson et al. first used THz CT to image the three-dimensional structure of turkey bones in 2002 [5]. In 2014, B. Recur et al. optimized Maximum Likelihood for Transmission tomography (ML-TR) algorithm in Gaussian beam model, and reconstructed three-dimensional structure of a nozzle with low noise and artifacts [6]. In 2017, B. Li et al. used a surface-array pyroelectric detector to acquire the projection data, and proposed an angular spectrum diffraction propagation algorithm to suppress the diffraction effect of THz radiation, eventually succeeded to improve the CT imaging speed [7]. In 2021, Peter et al. described THz radiation in the geometric optical system, and established the relationship between the THz-TDS signal and the material distribution of the specimen, then they made a high-resolution imaging of specimen through the analysis and processing of the transmitted signals [8].
Due to the high transmittance of THz radiation in dielectrics, it has also been used to measure stress distributions in optically dielectric materials. As early as 2008, Ebara et al. measured stress birefringence in optically opaque materials using a polarization-sensitive THz-TDS [9]. In 2014, Pfleger et al. overcame the measurement drawbacks of linear polarized light to observe the phenomenon of stress birefringence in elastic polymers, and then performed a quantitative study to obtain refractive index measurements of the material in two principal stress directions under different stress [10]. In 2016, Wang et al. determined the planar stress in the homogeneous material Polytetrafluoroethylene (PTFE) based on the stress birefringence at THz frequency [11]. In 2021, Kang et al. realized a two-dimensional full-field stress distribution measurement based on THz-TDS [10]. But, so far, all works on the stress measurement based on THz-TDS are limited to plane stress, three-dimensional stress measurement is still an untouched issue.
Inspired by the development of three-dimensional CT and the plane stress measurement using THz-TDS, we hope to explore the possibility of conducting three-dimensional stress measurement based on THz-TDS. This paper presents a method of measuring three-dimensional distribution of uniaxial stress based on THz-TDS. The rest of this paper is organized as follows. Section 2 introduces the measurement principle. The device and experimental procedures are introduced in Section 3. In addition, a special square sleeve was added to ensure the measurement effective and an illustrative experiment was introduced to demonstrate its roles. Section 4 presents the experimental results, while some discussions and conclusive remarks on the results are presented in Section 5.
2. Theoretical model
Figure 1(a) shows the schematic diagram of the stress measurement system, which is composed of two parts, a typical polarized THz-TDS system and a mechanical loading device. The coordinate system is also shown in Fig. 1(a). The directions of polarizers and stress are indicated in Fig. 1(b). σ1 and σ2 represent the two principal stresses. φ is the angle between the polarizer II and Y-axis, and φ’ is the angle of the polarizer I. In this system, the two polarizers are orthogonal to each other so that φ’ = φ + π/2. Additionally, θ is the angle between the first principal stress and Y-axis.
Fig. 1. (a) Schematic diagram of optical path composition and the coordinate system. (b) Schematic diagram of the polarization direction of the optical path and the principal stress directions of the device.
Fig. 2. (a) Schematic diagram of the whole experimental system. (b) Schematic diagram of the three-dimensional movement platform.
The received THz signal through the stress-free device is expressed as
The signal through the device can be expressed as
In Eq. (2), Qφ and Qφ’ are the Jones matrixes of the two polarizers, ${J_\mathrm{\theta }}$ is that of the device and R represents the direction of the receiving antenna. Substituting Eqs. (3)∼(6) into Eq. (2), we can obtain
So, the amplitude of E1 in Eq. (7) should be
where δ1$- $δ2 is the phase difference between the polarized components along the two principal stress directions after they transmitted the object. For each differential slice whose thickness is dl, the phase difference between the two transmitted polarized components is where c is the speed of light in vacuum. Δn is the refractive index difference between the principal stress directions. Due to the stress birefringence, Δn can be expressed as where C is the stress-optical coefficient. In this work, the device is made of PTFE, and C was calibrated as -2.4 × 10−10 Pa−1[6].Under this experimental condition, $\Delta \sigma $ is the difference between two principal stresses in the yoz plane which is perpendicular to the direction of the optical path and because of ${\sigma _1} \gg {\sigma _2}$, so
where ${\sigma _{1\; }}\textrm{and\; }{\sigma _2}$ are the two principal stresses in the yoz plane, ${\sigma _z}$ is the Z-directional stress (the vertical stress).Substituting Eq. (11) into Eq. (10), then
Substituting Eq. (12) into Eq. (9) and integrating both sides then
Thus,
In this article, the time domain signal is transformed into a Fourier amplitude spectrum, from which we can determine the frequency f at the center of half-height band. And then the amplitude at the frequency was taken as the measured amplitude, notated as ${A_{\textrm{measure}}}$. Then, the maximum and the minimum measured amplitudes among all ${A_{\textrm{measure}}}$ are selected, and notated as Amax and Amin. A of each point can be attained by the following normalization operation,
In conclusion, we can know the integration of the z-direction stress along the scanning direction by the three-dimensional data field, and obtain the sine diagram corresponding to the stress distribution, and finally reconstruct the three-dimensional distribution of uniaxial stress of the device by a traditional Filtered back-projection (FBP) algorithm.
3. Experiment
As shown in Fig. 2(a), the experimental system consists of two parts including a typical polarized THz-TDS system and a three-dimensional movement platform whose directions of movement are marked in Fig. 2(b).
The system used here is a commercial TDS with a reliable frequency range of 0.2-2.5 THz, a spot diameter of 4 mm, and a signal-to-noise ratio of 491. It uses a pair of photo-conductive antennas to emit and receive polarized THz radiation. In the following experiment, a dark field setup is adopted because of its sensitivity to stresses and low random fluctuation, which is significantly better than a bright field. Dark field represents that the polarization directions of emitting and receiving antennas are orthogonal each other. Furthermore, two polarizers are added after the emitting antenna and before the receiving antenna to obtain better extinction.
The principle of CT is to calculate the distribution of a physical quantity from the captured projection images at several angles. In typical CT system, the optical parts need to be rotated for the data acquisition at every angle. But, in this work, we rotate the device instead of the optical parts. The used three-dimensional movement platform has one rotational degree of freedom and two translational degrees of freedom. The rotational motion is used to rotate the device to a specific projection angle β, while the two-dimensional translational motion is used to scan a projection image. The movement platform was driven by stepper motors. The repeated positioning accuracy of the rotation is 0.01 deg. The scanning range and the repeated positioning accuracy of the two-dimensional platform are 50 mm × 50 mm and 2µm, respectively.
As shown in Fig. 3, The iron sheet is used for fixing the device on the rotary platform. The cylinder and screw are the interested parts, in which the stress distributions will be measured. The contact between the screw and cylinder is close, smooth and frictionless. When the nut is tightened, the cylinder is pressed and the nut is pulled.
Fig. 3. (a) Schematic diagram of the device. (b) Dimensions of the device. (c) Schematic diagram of the section A and B.
It must be noted that there is an important trick used in this work. A square sleeve is set outside the cylinder. This is because there will be a severe signal attenuation when THz radiation incident on the surface of the cylinder as shown in Fig. 3(c). So, a square sleeve is adopted in this work after several attempts to depress signal attenuation, and its effect will be demonstrated in the following illustrative experiment.
Although the following stress distribution was measured under dark field, an illustrative experiment under bright field, as shown in Fig. 4, provides a better illustration of the square sleeve’s role under the same principle. To examine the influence of sample thickness and angle of incidence on the transmitted signal, two experiments were conducted. In the first experiment, THz radiation transmitted a 10mm-thick PTFE plate with an incidence angle of 20°. In the second one, THz radiation perpendicularly transmitted a 30 mm-thick square. The configurations of the two experiments is shown in Fig. 4(a) and Fig. 4(b). There are three signals under bright field were shown in Fig. 4(c). The red one is an air signal, which is used the incidence signal. The blue one is the signal after the transmission like Fig. 4(a) and the green one is signal after the transmission like Fig. 4(b). According to these signals, the transmission rate of perpendicular incidence is 65%, while the transmission rate of 20° incidence is only 10%. So, it is obvious that an angle of incidence is very important to the signal attenuation. As a result, a square sleeve is adopted in the following experiments. The square sleeve only moves with the movement platform but does not rotate with the rotary platform in the experiment, to ensure perpendicular incidences. As shown in Fig. 3(c), the role of the square sleeve is converting incidence with a small angle(red) to a perpendicular one(green). In addition, the square sleeve ensures that the transmission thickness is constant from one scanning point to another, reducing errors due to thickness differences.
Fig. 4. (a) THz radiation transmitted into the surface of the device with a small angle. (b) THz radiation transmitted perpendicularly into the surface of the device. (c) Comparison between different signal under bright field (emitting antenna and receiving antenna in the same direction, no polarizer is added to the optical path).
At the beginning of the experiment, the nut is tightened with a digital torque wrench, and the preload is used to deliver pressure to the cylinder as well as tension to the screw. The displayed data of the torque wrench is used to calculate the pressure loading between the nut and the cylinder, which is also the tension loading on the screw. Because the signal capture of THz-TDS is extremely time-consuming, we only measured the stress at two representative sections (A and B) as shown in Fig. 3(b), instead of fully measuring the three-dimensional stress distribution in the device. Section A is on the middle of the device, so its stress should be homogeneous, while section B is near the nut so its stress will more concentrated at the contact position with the nut. At each section, the device is scanned with the length of 40 mm and the step of 0.5 mm at 60 projection angles. As a result, a 81 × 60 × 2 three-dimensional data fields were obtained for stress analysis.
4. Results
In this part, we first show the experimental results of sections A and B, then verify the effectiveness of the experimental measurement with a FEM simulation and analyzed the reasons. In addition, the role of the sleeve was demonstrated by the contrast of results whether with the square sleeve.
Figure 5 shows the experimental results of section A, including the original projection data, the calculated stress distribution and the simulation results to verify the experimental results. Figure 5(a) presents the sine diagram. In it, the vertical coordinate of each point represents its scanning direction, while the horizontal coordinate of each point represents its position in that scanning direction. The value of every point is “Ameasure” of a specific projection direction, which should be proportional to the integral of the stress according to Eq. (15). To determine the real value of stress in Fig. 5(b), we need to determine the proportional coefficient between the FBP data and stress. Finally, stress distribution in Fig. 5(b) can be attained from Fig. 5(a) by using a classic FBP algorithm and the proportional coefficient.
Fig. 5. The analysis results of stress on the section A. (a) Sine diagram. (b) The stress distribution by FBP algorithm. (c) The stress results from FEM simulation. (d) Comparison between the experimental and simulated results on the diameter (SA- simulated results of section A, EA- experimental results of section A).
According to the experimental parameters, the pixel size of two-dimensional FBP field is 0.5 mm × 0.5 mm, and the number of pixels of Fig. 5(b) should be 56 × 56. The torque coefficient between screw and nut is 0.143. The overall tension of the screw can be calculated as 522 N according to the displayed torque, and the integral should equal to it, so that we can obtain the proportional coefficient between the FBP data and the real stress (in MPa) is 0.761, Fig. 5(b) shows the obtained stress distribution of section A, in which the cylinder is under compressive stress, while the screw is under tensile stress. This is coincident with the loading configuration of the experiment device in Fig. 3(a).
To verify the effectiveness of the experimental measurement, a FEM simulation of the device was conducted. Figure 5(c) shows the simulated stress distribution on section A. Figure 5(d) presents the stress distributions on the diameter from experimental measurement and FEM simulation. A reasonable agreement between simulation and experiment results is obtained. However, the difference between the experimental and simulated results isn’t negligible.
There may be several reasons accounting for these inconsistences. Firstly, the experiments were conducted under dark field configuration and in an open environment instead of a closed low-humidity environment, so that the signal-to-noise ratio of this experimental measurement is not high enough. Secondly, the CT scanning by THz-TDS need a very long time (about 5 hours). Therefore, it is difficult to maintain a stable experimental condition such as temperature and humidity. Thirdly, the plastic deformation of PTFE device is also inevitable. Moreover, there are two interfaces. one is the interface between the cylinder and screw, and another is between cylinder and square sleeve. The contact at these two interfaces is non-ideal.
In addition to the factors in the experiments described above, low intensity of THz wave sources compared to lasers, imperfections of the CT algorithm, large spot diameter of the TDS system all limit the measurement accuracy. Thus, the investigations on the errors caused by these factors and the measures to improve the measurement resolution and accuracy will be conducted in our future work.
The results on section B are presented in Fig. 6. Compared to that of section A, Fig. 6(a) has a significant difference near the interface between the cylinder and screw. By the FBP algorithm, the difference is reflected in the stress distribution of section B in Fig. 6(b). Compared with the homogeneous stress distribution of section A in Fig. 5(b), there is a stress concentration on the section B as shown in Fig. 6(b). In Fig. 6(b), the stresses in the cylinder and in the screw are both not homogeneous. In the cylinder, the compressive stress in the inner area is large, while the tension stress in the outer area of the screw is large. This difference between the stress distributions in Fig. 5(b) and 6(b) originates from our loaded method, when the screw is tightened, the pressure is transferred through the contact surface between nut and cylinder, so the closer to the contact, the more concentrated stresses in the cylinder and screw are.
Fig. 6. The analysis results of stress on the section B. (a) Sine diagram. (b) The stress distribution by FBP algorithm. (c) The stress results from FEM simulation. (d) Comparison between the experimental and simulated results on the diameter (SB- simulated results of section B, EB- experimental results of section B).
Similarly, Fig. 6(c) shows the simulation stress distribution on section B. Figure 6(d) presents the stress distributions on the diameter from the experimental measurement and FEM simulation and obtains a reasonable agreement between them. It is worth noting that there are unexpected large stresses at the center in both two experiments. This should be because that the loss of the THz radiation through the section center is minimal.
In the last section, the square sleeve was introduced to be a method to depress the signal loss. The results in Fig. 6 prove the effects of the square sleeve. Figure 7(a) and (b) shown the measured stress distributions on the two interested sections without using the square sleeve. Compared to the results in Fig. 5(b) and Fig. 6(b), the degeneration in Fig. 6 is obvious. In Fig. 7, the outlines of the two sections are even invisible, let alone the stress distributions on them. Thus, it is suggested to ensure the signal strong enough in the stress CT imaging by different methods, even if the square sleeve used in this work is not a prefer choice.
Fig. 7. (a) Stress distribution of section A without the square sleeve. (b) Stress distribution of section B without the square sleeve.
5. Discussion and conclusion
In this paper, a method is proposed to measure the three-dimensional distribution of uniaxial stress in optically opaque dielectric materials based on THz-TDS. This method combines the THz three-dimensional CT imaging technology and the stress-optic effect. Since the THz radiation transmitted through the device is influenced by its interior stress, the modulation model of three-dimensional uniaxial stress on the transmitted THz radiation was established. The three-dimensional interior stress distribution can be obtained by the traditional FBP algorithm based on the three-dimensional data field from THz-TDS system. Then, a verification experiment was conducted to determine the interior stress in assembled cylinder and screw. Compared to the results of FEM simulation, a reasonable agreement is obtained, and verify the effectiveness of the experimental measurement.
This measurement method has two advantages in measurement of three-dimensional stress distribution. Firstly, in contrast to the necessity of cutting the sample into thin slices in the traditional three-dimensional photoelasticity, the proposed method can measure the internal stress of the object without destroying the structural integrity of the object directly. So, the method can be used to observe some valuable samples that are not suitable for the stress freezing method, while avoiding the possible errors caused by the temperature change of the stress freezing method. On the other hand, the photoelasticity needs to pre-manufactured a model of the object to be researched using epoxy resin. However, the proposed method by us can directly measure three-dimensional distribution of any objects as long as they can be penetrated by THz radiation.
We have noticed that THz cameras are more and more widely used. If terahertz cameras are used instead of the THz-TDS system used in this work, the measurement speed and simplicity will be greatly improved. This will be the direction of our next research.
Funding
National Natural Science Foundation of China (12021002, 12041201, 12072229, 12172251).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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