Ex) Article Title, Author, Keywords
Current Optics
and Photonics
Ex) Article Title, Author, Keywords
Curr. Opt. Photon. 2024; 8(3): 225-229
Published online June 25, 2024 https://doi.org/10.3807/COPP.2024.8.3.225
Copyright © Optical Society of Korea.
Zhi Qiang Wang^{1}, Wen Jia Ren^{1}, Gui Ying Zhang^{2} , Zhi Yong Wang^{3}
Corresponding author: ^{*}gyzhang84@163.com, ORCID 0009-0006-5461-1132
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper introduces a rapid measurement technique for the stress-optic coefficient, using terahertz time-domain spectroscopy. First we propose a design combining a four-point bending device with a scanning stage to streamline the loading process. Then we detail the measurement principle and outline the signal-processing algorithm. The experiments are carried out on Al_{2}O_{3}, a representative ceramic material. The experimental data reveal that the refractive index of Al_{2}O_{3} exhibits a linear decrease with increasing stress. This work supplies an efficient method for stress measurement rooted in the stress-optic effect.
Keywords: Ceramics, Stress birefringence, Stress-optic coefficient, Terahertz time domain spectroscopy (THz-TDS)
OCIS codes: (040.2235) Far infrared or terahertz; (070.4790) Spectrum analysis; (170.6795) Terahertz imaging; (200.4740) Optical processing; (240.6490) Spectroscopy, surface
Due to the good penetration of terahertz radiation in dielectric materials [1–5], terahertz time-domain spectroscopy (THz-TDS) is an effective means for analyzing the properties of ceramic materials [6–10]. Ma et al. [8] used THz-TDS to determine the dielectric constants and dielectric loss tangents of several ceramics. By combing THz-TDS with photogrammetric reconstruction, Mikerov et al. [6] achieved a measurement of the local dielectric properties of ceramic on a curved surface. Kamba et al. [9] employed THz-TDS to investigate the magnetodielectric effects of BiFeO_{3} over a wide temperature range. Ye et al. [7] used a machine-learning algorithm to process THz-TDS spectral data for yttria-stabilized zirconia, and some valuable information about the interior microscopic defects was obtained. Pfleger et al. [10] realized an efficient technique to determine the bi-refractive of lithium niobate by a standard THz-TDS system.
Currently, thermal barrier coatings (TBCs) used in the thermal power energy are one type of important functional ceramics. As a multilayer assembly designed to protect its metal substrate (e.g. Ni-Cr-Al-Y or Ni-Co-Cr-Al-Y alloy) from thermal damage, TBCs were initially composed of the top coat (TC) and the bond coat (BC). After some work time, thermal growth oxide (TGO) often would grow between the TC and BC layers. The application of THz-TDS to the characterization of TBCs has also received widespread attention [11, 12]. By analyzing the reflected time-domain waveforms, the thickness and refractive index of TC can be determined [13, 14]. By considering the multireflection of the TBCs, the TGO thickness can also be measured, in agreement with the results from microscopy [15, 16]. Cao et al. [17] improved the optical-reflection model by considering the roughness of surface and interface to make the measurement of thickness of a multilayer coating more accurate.
Stress is a major cause of failure in materials, so THz-TDS is often used to measure the stress and strain in materials [18–24]. Ebara et al. [24] observed the stress-induced birefringence of PTFE. Based on polarized THz-TDS, our own group achieved the measurement of the plane stress at a point [18, 19], the stress fields [21, 22], and the three-dimensional stress distribution [20]. This work demonstrated the potential of THz-TDS for measuring stress in dielectric materials.
It is worth noting that, as of now, the fundamental principle behind measuring stress using THz-TDS is the stress-optic effect. This effect asserts that when stress is exerted on a materials, it induces birefringence. Thus research on how to measure the birefringence quickly and accurately by THz-TDS also supplies us with important inspirations and directions for stress measurement.
Since we conduct the stress measurement based on the stress-optic effect, the stress-optic coefficient should be determined first. In fact, there are already some relevant works to introduce how to determine the stress-optic coefficients of materials in the terahertz range. Song et al. [25] demonstrated the stress-optic effect of polytetrafluoroethylene in the terahertz range and measured its stress-optic coefficient. Schemmel et al. [26, 27] detailed measurements of the stress-optic coefficient for yttria-partially-stabilized zirconia in the frequency range 260–380 GHz using both transmission and reflection methods. Wang et al. [28] presented a method for measuring the stress-optic coefficient of TBCs.
Until now, though, all of the reported works on the measurement of the stress-optic coefficient have been conducted by multiple loadings through a uniaxial loading device. Normally the THz-TDS system is in a semi-sealed chamber to maintain the humidity, so multiple loadings will strongly increase the experiment time. Thus it is beneficial for us to improve the speed of measuring the stress-optic coefficient.
The purpose of this paper is to introduce a rapid and convenient technique to determine the stress-optic coefficient of materials. In this study Al_{2}O_{3}, a typical ceramic material, is selected as the specimen material. By introducing a four-point bending device mounted on a scanning stage, we can determine the stress-optic coefficient with application of just a single load. This design can facilitate the measurement process for the stress-optic coefficient and ensure the consistency of experimental measurements. The rest of this paper is organized as follows: Section 2 details the experimental setup and measurement principle. Section 3 presents the experimental results. Sections 4 and 5 supply some specific discussions and a simplified conclusion.
In this work, we employ a conventional transmissive THz-TDS system. The system uses a femtosecond laser characterized by an 88-MHz repetition rate, 26-fs pulse width, and 800-nm wavelength. This laser serves as the pump source with an average power of 10 mW, and is split into two beams to excite both the terahertz transmitter and receiver. The generation and detection of terahertz radiation occur by shorting the dipolar photoconductive antenna gap. To ensure optimal signal stability, the surrounding humidity is maintained below 2% by introducing dry air into a semisealed chamber during measurements. This THz-TDS system boasts a signal-to-noise ratio (SNR) of 80 dB, and its phase-retardance stability stands at 0.5° at 1 THz. This high phase stability is a key factor in ensuring the precision of the results. For the used system, reliable frequency measurements range approximately between 0.3 and 1.50 THz. The specimen is placed between two wired grid polarizers, which are used to control the polarization state of the terahertz radiation. The time-domain waveforms of the THz radiation through the specimen can therefore be captured under different settings of the polarizers using this system.
Figure 1 illustrates the loading device and specimen utilized in this study. As shown in Fig. 1(a), the loading device is a four-point bending device and is securely mounted on a mechanical scanning stage. The loading is applied by the screw. A force sensor is incorporated to determine the applied loading. Figure 1(b) presents the schematic diagram of the loading device, while Fig. 1(c) shows the specimen. As illustrated in Fig. 1(b), the left loading block is driven by the screw so that two compressive forces can be applied to the specimen through the two contact points. The dimensions of the specimen are 75 mm × 25 mm × 5 mm. According to the specification of the manufacturer, Al_{2}O_{3} had a density of 6.0 g/cm^{3}, a Young’s modulus of 200 Gpa, and a Poisson’s ratio of 0.3.
By means of the four-point bending device, we subjected the specimen to what is commonly referred to as pure bending. According to our knowledge of materials mechanics, along the dotted line in Fig. 1(b), from left to right, the stress experienced by the specimen transitions linearly from maximum compressive stress to maximum tensile stress. The scanning stage is driven by a stepper motor so that the loading device can be moved horizontally. Thus we can locate points with different stresses on the focal point of the THz-TDS system. In this study, seven equidistant points along the central line, i.e. the x-axis in Fig. 1(b), are successively positioned at the focal point of the THz-TDS system. In the x-y coordinate plane of Fig. 1(b), the x coordinates of the seven points of interest are from −6 mm to 6 mm in steps of 2 mm.
In the experiment, the loading P = 500 N. According to our knowledge of material mechanics, the y-direction stress should be
where l = 20 mm, h = 25 mm as illustrated in Fig. 1(b), d = 5 mm is the specimen thickness, and x is the horizontal coordinate of the point of interest. Thus the stresses of the seven points of interest range from −2.25 MPa to 2.25 MPa in steps of 0.75 MPa.
According to the stress-optic law, after being subjected to loading, an originally optically isotropic material will exhibit birefringence and become optically anisotropic. Considering the plane stress described by principal stresses σ_{1} and σ_{2}, the change in refractive index of the loaded material can be described by [26]
where n is the initial refractive index before loading, n_{1} and n_{2} are the changed refractive indices along the directions of σ_{1} and σ_{2}, and c_{1} and c_{2} are the stress-optic coefficients.
In the experimental setup of this study the specimen is subjected to a uniaxial stress, that is σ_{2} = 0. The first equation in Eq. (2) becomes
where σ can be determined by Eq. (1) using the co-ordinates of the points of interest.
Upon loading the specimen, not only does the refractive index change, but the thickness also undergoes alteration. Furthermore, the refractive-index change should correspond to the change of the material’s molecular structure. Thus the change of the transmission time of the terahertz radiation for the stressed material should be expressed as
where μ and E are the Poisson’s ratio and the Young’s modulus respectively, d is the original thickness of the specimen, and C is the light speed in air. If the delay time Δt can be experimentally measured, we can calculate the stress-optic coefficient by
In Fig. 2, we present the captured time-domain waveforms through the seven points of interest. Obviously there are different time delays for differently stressed locations. The specific amount of time delay can be determined by analyzing the time-domain waveforms.
In this study, the time-domain waveforms transmitted through the bent specimen at seven different positions are captured. Figure 2 shows the captured time-domain waveforms when the seven points of interest are positioned at the focal point of the THz-TDS system. The magnified views around the peak are also provided. In Fig. 2, the numbers provided in the legend represent the x-coordinates of the selected positions of interest. Obviously, as the stress transitions from compression to tension, the signal gradually shifts to the right, but the amount of shift is tiny; There is only about 0.05 ps between maximum compression and maximum tension.
To ascertain the time delay precisely, we perform a Fourier transform on the time-domain waveforms to obtain their frequency-phase curves. Each of these curves is then subtracted from the counterpart corresponding to zero stress. The resultant curves are presented in Fig. 3. In Fig. 3, the numbers provided in the legend represent the x-coordinates of the selected positions of interest, the same as in Fig. 2. According to the time-shifting property of the Fourier transform, the time shift of the stressed signal relative to the free signal can be calculated by the slope of the frequency-phase curve in Fig. 3 using
The slope can be obtained by linearly fitting each phase curve in Fig. 3. In Eq. (4), Δt is experimentally measured and n, C, d, μ, and E are known quantities, so the stress-optic coefficient can be determined. Table 1 lists the values of these necessary quantities.
TABLE 1 The material and geometrical parameters of the specimen
n | E (GPa) | μ | d (mm) |
---|---|---|---|
5.81 | 200 | 0.3 | 5 |
Figure 4 shows the experimentally measured changes of the refractive index induced by the different stresses, and their linear fitting. In Fig. 4 the blue points represent the experimental data, while the red line is their linear fit. According to Eq. (1), the stress of the three rightmost points should be negative, since their x coordinates are negative. Negative stress means the stress is compressive. On the other hand, the stress of the three leftmost points is positive, which means the stress is tensile. The slope the linear fitting is the stress-optic coefficient of Eq. (2). As a result, the stress-optic coefficient of Al_{2}O_{3} is measured as 0.48 GPa^{−1}.
In this work, a convenient technique is introduced to rapidly determine the stress-optic coefficient of Al_{2}O_{3}. This method is based on a transmission THz-TDS system. By employing a four-point bending device, all measurements can be conducted by loading the specimen only once. Based on the stress-optic law and elasticity theory, a model is established to calculate the stress-optic coefficient by using the relative time delay of the stressed location. The time delay is accurately determined from the frequency-phase curves of the time-domain waveform. Over the frequency range of 0.2–1.4 THz, the refractive index exhibits a negatively linear relationship with the imposed stress. As the final result, the stress-optic coefficient c_{1} of Al_{2}O_{3} is measured as 0.48 GPa^{−1}.
The Tianjin Education Commission Research Program Project of China [grant number 2022KJ118]; the Research Program Project of Tianjin University of Technology and Education [grant number KYQD1625 and KYQD14014].
The authors declare no conflicts of interest.
Data underlying the results presented in this paper are not publicly available at the time of publication, but may be obtained from the authors upon reasonable request.
Curr. Opt. Photon. 2024; 8(3): 225-229
Published online June 25, 2024 https://doi.org/10.3807/COPP.2024.8.3.225
Copyright © Optical Society of Korea.
Zhi Qiang Wang^{1}, Wen Jia Ren^{1}, Gui Ying Zhang^{2} , Zhi Yong Wang^{3}
^{1}Department of Computer Science and Technology, School of Information Technology and Engineering, Tianjin University of Technology and Education, Tianjin 300222, China
^{2}Department of Microelectronics, School of Electronic Information Engineering, Tianjin University of Technology and Education, Tianjin 300222, China
^{3}Department of Mechanics, School of Mechanical Engineering, Tianjin University, Tianjin 300072, China
Correspondence to:^{*}gyzhang84@163.com, ORCID 0009-0006-5461-1132
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper introduces a rapid measurement technique for the stress-optic coefficient, using terahertz time-domain spectroscopy. First we propose a design combining a four-point bending device with a scanning stage to streamline the loading process. Then we detail the measurement principle and outline the signal-processing algorithm. The experiments are carried out on Al_{2}O_{3}, a representative ceramic material. The experimental data reveal that the refractive index of Al_{2}O_{3} exhibits a linear decrease with increasing stress. This work supplies an efficient method for stress measurement rooted in the stress-optic effect.
Keywords: Ceramics, Stress birefringence, Stress-optic coefficient, Terahertz time domain spectroscopy (THz-TDS)
Due to the good penetration of terahertz radiation in dielectric materials [1–5], terahertz time-domain spectroscopy (THz-TDS) is an effective means for analyzing the properties of ceramic materials [6–10]. Ma et al. [8] used THz-TDS to determine the dielectric constants and dielectric loss tangents of several ceramics. By combing THz-TDS with photogrammetric reconstruction, Mikerov et al. [6] achieved a measurement of the local dielectric properties of ceramic on a curved surface. Kamba et al. [9] employed THz-TDS to investigate the magnetodielectric effects of BiFeO_{3} over a wide temperature range. Ye et al. [7] used a machine-learning algorithm to process THz-TDS spectral data for yttria-stabilized zirconia, and some valuable information about the interior microscopic defects was obtained. Pfleger et al. [10] realized an efficient technique to determine the bi-refractive of lithium niobate by a standard THz-TDS system.
Currently, thermal barrier coatings (TBCs) used in the thermal power energy are one type of important functional ceramics. As a multilayer assembly designed to protect its metal substrate (e.g. Ni-Cr-Al-Y or Ni-Co-Cr-Al-Y alloy) from thermal damage, TBCs were initially composed of the top coat (TC) and the bond coat (BC). After some work time, thermal growth oxide (TGO) often would grow between the TC and BC layers. The application of THz-TDS to the characterization of TBCs has also received widespread attention [11, 12]. By analyzing the reflected time-domain waveforms, the thickness and refractive index of TC can be determined [13, 14]. By considering the multireflection of the TBCs, the TGO thickness can also be measured, in agreement with the results from microscopy [15, 16]. Cao et al. [17] improved the optical-reflection model by considering the roughness of surface and interface to make the measurement of thickness of a multilayer coating more accurate.
Stress is a major cause of failure in materials, so THz-TDS is often used to measure the stress and strain in materials [18–24]. Ebara et al. [24] observed the stress-induced birefringence of PTFE. Based on polarized THz-TDS, our own group achieved the measurement of the plane stress at a point [18, 19], the stress fields [21, 22], and the three-dimensional stress distribution [20]. This work demonstrated the potential of THz-TDS for measuring stress in dielectric materials.
It is worth noting that, as of now, the fundamental principle behind measuring stress using THz-TDS is the stress-optic effect. This effect asserts that when stress is exerted on a materials, it induces birefringence. Thus research on how to measure the birefringence quickly and accurately by THz-TDS also supplies us with important inspirations and directions for stress measurement.
Since we conduct the stress measurement based on the stress-optic effect, the stress-optic coefficient should be determined first. In fact, there are already some relevant works to introduce how to determine the stress-optic coefficients of materials in the terahertz range. Song et al. [25] demonstrated the stress-optic effect of polytetrafluoroethylene in the terahertz range and measured its stress-optic coefficient. Schemmel et al. [26, 27] detailed measurements of the stress-optic coefficient for yttria-partially-stabilized zirconia in the frequency range 260–380 GHz using both transmission and reflection methods. Wang et al. [28] presented a method for measuring the stress-optic coefficient of TBCs.
Until now, though, all of the reported works on the measurement of the stress-optic coefficient have been conducted by multiple loadings through a uniaxial loading device. Normally the THz-TDS system is in a semi-sealed chamber to maintain the humidity, so multiple loadings will strongly increase the experiment time. Thus it is beneficial for us to improve the speed of measuring the stress-optic coefficient.
The purpose of this paper is to introduce a rapid and convenient technique to determine the stress-optic coefficient of materials. In this study Al_{2}O_{3}, a typical ceramic material, is selected as the specimen material. By introducing a four-point bending device mounted on a scanning stage, we can determine the stress-optic coefficient with application of just a single load. This design can facilitate the measurement process for the stress-optic coefficient and ensure the consistency of experimental measurements. The rest of this paper is organized as follows: Section 2 details the experimental setup and measurement principle. Section 3 presents the experimental results. Sections 4 and 5 supply some specific discussions and a simplified conclusion.
In this work, we employ a conventional transmissive THz-TDS system. The system uses a femtosecond laser characterized by an 88-MHz repetition rate, 26-fs pulse width, and 800-nm wavelength. This laser serves as the pump source with an average power of 10 mW, and is split into two beams to excite both the terahertz transmitter and receiver. The generation and detection of terahertz radiation occur by shorting the dipolar photoconductive antenna gap. To ensure optimal signal stability, the surrounding humidity is maintained below 2% by introducing dry air into a semisealed chamber during measurements. This THz-TDS system boasts a signal-to-noise ratio (SNR) of 80 dB, and its phase-retardance stability stands at 0.5° at 1 THz. This high phase stability is a key factor in ensuring the precision of the results. For the used system, reliable frequency measurements range approximately between 0.3 and 1.50 THz. The specimen is placed between two wired grid polarizers, which are used to control the polarization state of the terahertz radiation. The time-domain waveforms of the THz radiation through the specimen can therefore be captured under different settings of the polarizers using this system.
Figure 1 illustrates the loading device and specimen utilized in this study. As shown in Fig. 1(a), the loading device is a four-point bending device and is securely mounted on a mechanical scanning stage. The loading is applied by the screw. A force sensor is incorporated to determine the applied loading. Figure 1(b) presents the schematic diagram of the loading device, while Fig. 1(c) shows the specimen. As illustrated in Fig. 1(b), the left loading block is driven by the screw so that two compressive forces can be applied to the specimen through the two contact points. The dimensions of the specimen are 75 mm × 25 mm × 5 mm. According to the specification of the manufacturer, Al_{2}O_{3} had a density of 6.0 g/cm^{3}, a Young’s modulus of 200 Gpa, and a Poisson’s ratio of 0.3.
By means of the four-point bending device, we subjected the specimen to what is commonly referred to as pure bending. According to our knowledge of materials mechanics, along the dotted line in Fig. 1(b), from left to right, the stress experienced by the specimen transitions linearly from maximum compressive stress to maximum tensile stress. The scanning stage is driven by a stepper motor so that the loading device can be moved horizontally. Thus we can locate points with different stresses on the focal point of the THz-TDS system. In this study, seven equidistant points along the central line, i.e. the x-axis in Fig. 1(b), are successively positioned at the focal point of the THz-TDS system. In the x-y coordinate plane of Fig. 1(b), the x coordinates of the seven points of interest are from −6 mm to 6 mm in steps of 2 mm.
In the experiment, the loading P = 500 N. According to our knowledge of material mechanics, the y-direction stress should be
where l = 20 mm, h = 25 mm as illustrated in Fig. 1(b), d = 5 mm is the specimen thickness, and x is the horizontal coordinate of the point of interest. Thus the stresses of the seven points of interest range from −2.25 MPa to 2.25 MPa in steps of 0.75 MPa.
According to the stress-optic law, after being subjected to loading, an originally optically isotropic material will exhibit birefringence and become optically anisotropic. Considering the plane stress described by principal stresses σ_{1} and σ_{2}, the change in refractive index of the loaded material can be described by [26]
where n is the initial refractive index before loading, n_{1} and n_{2} are the changed refractive indices along the directions of σ_{1} and σ_{2}, and c_{1} and c_{2} are the stress-optic coefficients.
In the experimental setup of this study the specimen is subjected to a uniaxial stress, that is σ_{2} = 0. The first equation in Eq. (2) becomes
where σ can be determined by Eq. (1) using the co-ordinates of the points of interest.
Upon loading the specimen, not only does the refractive index change, but the thickness also undergoes alteration. Furthermore, the refractive-index change should correspond to the change of the material’s molecular structure. Thus the change of the transmission time of the terahertz radiation for the stressed material should be expressed as
where μ and E are the Poisson’s ratio and the Young’s modulus respectively, d is the original thickness of the specimen, and C is the light speed in air. If the delay time Δt can be experimentally measured, we can calculate the stress-optic coefficient by
In Fig. 2, we present the captured time-domain waveforms through the seven points of interest. Obviously there are different time delays for differently stressed locations. The specific amount of time delay can be determined by analyzing the time-domain waveforms.
In this study, the time-domain waveforms transmitted through the bent specimen at seven different positions are captured. Figure 2 shows the captured time-domain waveforms when the seven points of interest are positioned at the focal point of the THz-TDS system. The magnified views around the peak are also provided. In Fig. 2, the numbers provided in the legend represent the x-coordinates of the selected positions of interest. Obviously, as the stress transitions from compression to tension, the signal gradually shifts to the right, but the amount of shift is tiny; There is only about 0.05 ps between maximum compression and maximum tension.
To ascertain the time delay precisely, we perform a Fourier transform on the time-domain waveforms to obtain their frequency-phase curves. Each of these curves is then subtracted from the counterpart corresponding to zero stress. The resultant curves are presented in Fig. 3. In Fig. 3, the numbers provided in the legend represent the x-coordinates of the selected positions of interest, the same as in Fig. 2. According to the time-shifting property of the Fourier transform, the time shift of the stressed signal relative to the free signal can be calculated by the slope of the frequency-phase curve in Fig. 3 using
The slope can be obtained by linearly fitting each phase curve in Fig. 3. In Eq. (4), Δt is experimentally measured and n, C, d, μ, and E are known quantities, so the stress-optic coefficient can be determined. Table 1 lists the values of these necessary quantities.
TABLE 1. The material and geometrical parameters of the specimen.
n | E (GPa) | μ | d (mm) |
---|---|---|---|
5.81 | 200 | 0.3 | 5 |
Figure 4 shows the experimentally measured changes of the refractive index induced by the different stresses, and their linear fitting. In Fig. 4 the blue points represent the experimental data, while the red line is their linear fit. According to Eq. (1), the stress of the three rightmost points should be negative, since their x coordinates are negative. Negative stress means the stress is compressive. On the other hand, the stress of the three leftmost points is positive, which means the stress is tensile. The slope the linear fitting is the stress-optic coefficient of Eq. (2). As a result, the stress-optic coefficient of Al_{2}O_{3} is measured as 0.48 GPa^{−1}.
In this work, a convenient technique is introduced to rapidly determine the stress-optic coefficient of Al_{2}O_{3}. This method is based on a transmission THz-TDS system. By employing a four-point bending device, all measurements can be conducted by loading the specimen only once. Based on the stress-optic law and elasticity theory, a model is established to calculate the stress-optic coefficient by using the relative time delay of the stressed location. The time delay is accurately determined from the frequency-phase curves of the time-domain waveform. Over the frequency range of 0.2–1.4 THz, the refractive index exhibits a negatively linear relationship with the imposed stress. As the final result, the stress-optic coefficient c_{1} of Al_{2}O_{3} is measured as 0.48 GPa^{−1}.
The Tianjin Education Commission Research Program Project of China [grant number 2022KJ118]; the Research Program Project of Tianjin University of Technology and Education [grant number KYQD1625 and KYQD14014].
The authors declare no conflicts of interest.
Data underlying the results presented in this paper are not publicly available at the time of publication, but may be obtained from the authors upon reasonable request.
TABLE 1 The material and geometrical parameters of the specimen
n | E (GPa) | μ | d (mm) |
---|---|---|---|
5.81 | 200 | 0.3 | 5 |