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Photoelasticity: theoretical aspects
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Photoelasticity: theoretical aspects
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We use an experimental technique to retrieve grain-scale stresses. The technique is based on birefringence properties of disks. In a birefringent material, the speed of light, and consequently the index of refraction depends on wave polarization. In other cases, such as glass and polymeric materials, birefringence arises only when the material is subject to anisotropic stress. In other words, the refractive indexes depend on the eigenvalues of local stress tensor. This phenomenon is called photoelasticity and has been utilized in granular experiments for several decade (Howell
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This wiki is dedicated to the use of photoelasticity in a general manner. Before going more into detail about [how to make](https://git-xen.lmgc.univ-montp2.fr/PhotoElasticity/Main/wikis/method-make) photoelastic samples, [how to image](https://git-xen.lmgc.univ-montp2.fr/PhotoElasticity/Main/wikis/method-image) them and even [how to get quantitative information](https://git-xen.lmgc.univ-montp2.fr/PhotoElasticity/Main/wikis/method-analyze) from photoelasticity, it is important to begin by some physical explanations about this phenomenon. The goal is not here to go deep into details. For those who would like more information, we suggest you to read the excellent [wikipedia page] about photoelasticity(https://en.wikipedia.org/wiki/Photoelasticity) or this very nice {lecture by W. Wang](http://depts.washington.edu/mictech/optics/me557/photoelasticity.pdf).
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et al., 1999).
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Using photo-elasticity, we can measure the internal stress. This measurement is
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The photoelastic phenomenon is based on birefringence properties of some transparent materials. In a birefringent material, the speed of light, and consequently the index of refraction depends on wave polarization. In other cases, such as glass and polymeric materials, birefringence arises only when the material is subject to anisotropic stress. In other words, the refractive indexes depend on the eigenvalues of local stress tensor. This phenomenon is called photoelasticity and has been utilized in experimental science for several decade.
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best performed using circularly polarized light, which provides isotropic polarization.
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Circularly polarized light is a composition of two orthogonal linearly polarized waves
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Using photoelasticity, we can measure the internal stress. This measurement is best performed using circularly polarized light, which provides isotropic polarization. Circularly polarized light is a composition of two orthogonal linearly polarized waves with a quarter-wave phase shift. As shown in the figure below, this polarization can be obtained by passing unpolarized light through a linear polarizer and a quarter-wave plate. The quarter-wave plate creates a $`\pi/2`$ phase shift between two orthogonal components of the light polarization. Passing unpolarized light through the combination of linear polarizer and quarter-wave, as shown in the figure, results in circularly polarized light. On the other side, covering the camera, sits another circular polarizer with opposite polarity, which blocks the unperturbed light. If there exists a photoelastic material in between under anisotropic stress, the wave components, polarized in the principles directions of local stress tensor, travel with different speeds. This speed difference results in phase shifts in the components of the wave and changes circularly polarized light to elliptical. Consequently, a part of the wave subject to this change is not completely blocked by the second circular polarizer and is passed through it, recorded by the camera.
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with a quarter-wave phase shift. As shown in Fig. 3.3, this polarization can be
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obtained by passing unpolarized light through a linear polarizer and a quarter-wave
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plate. The quarter-wave plate creates a π{2 phase shift between two orthogonal com-
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ponents of the light polarization. Passing unpolarized light through the combination
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of linear polarizer and quarter-wave, as shown in the figure, results in circularly po-
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larized light. On the other side, covering the camera, sits another circular polarizer
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with opposite polarity, which blocks the unperturbed light. If there exists a photo-
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elastic material in between under anisotropic stress, the wave components, polarized
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in the principles directions of local stress tensor, travel with different speeds. This
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speed difference results in phase shifts in the components of the wave and changes
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circularly polarized light to elliptical. Consequently, a part of the wave subject to
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this change is not completely blocked by the second circular polarizer and is passed
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through it, recorded by the camera.
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Photo-elasticity can be used to quantitatively measure local stress. Assuming
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that the relation between stress and refractive index is linear:
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$`n_1-n_2=C(\sigma_1-\sigma_2)`$
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where $`\sigma_i`$ is an eigenvalue of the local stress tensor, n i is the refractive index for polar-
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ized light in the corresponding direction, and C is called the stress-optical coefficient.
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The relative phase shift of wave components in the eigen-directions of local stress
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tensor can be calculated using:
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$`\alpha=\fract{2\piCd}{\lambda}(\sigma_1-\sigma_2)`$
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where $`\lambda`$ is the wavelength of the light and d is the distance traveled inside the
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material. Given this phase shift, the intensity of the emergent wave is:
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$`I=I_0 sin^2(\frac{\alpha}{2})=sin^2(\fract{\piCd}{\lambda}(\sigma_1-\sigma_2)) `$
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Eq. 3.3 relates internal stress and intensity for a single wavelength. In the
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experiment, both the light and camera channels cover a wide range of wavelengths.
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Stress calculation is less noisy using narrow range of wavelength and single channel
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from the camera. However, it is still possible to retrieve stresses using white light.
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In such as case, stress-optical relation can be evaluated using empirical calibration.
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Here, we investigate how the photo-elastic response can be used to retrieve local
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stress in our experiment.
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*The photoelastic technique: The combination of linear polarizer 1 and quarter-wave plate 1, with 45 degrees difference of principle directions, turns unpolarized white light into circularly polarized light passing through the photoelastic material. The second combination of polarizer and plate creates a circular polarizer with opposite polarity, blocking the unperturbed light. In case of local anisotropic stress in the photoelastic material, there will be different phase shifts for different components of the light polarization. This phase shifts perturbs the polarity of the wave, causing the light to be observed by the camera.*
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Photoelasticity can be used to quantitatively measure local stress. Assuming that the relation between stress and refractive index is linear:
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$`n_1-n_2=C(\sigma_1-\sigma_2)`$
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where $`\sigma_i`$ is an eigenvalue of the local stress tensor, $`n_i`$ is the refractive index for polarized light in the corresponding direction, and $`C`$ is called the stress-optical coefficient. The relative phase shift of wave components in the eigen-directions of local stress tensor can be calculated using:
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$`\alpha=\frac{2\piCd}{\lambda}(\sigma_1-\sigma_2)`$
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where $`\lambda`$ is the wavelength of the light and $`d`$ is the distance traveled inside the material, which corresponds with the sample stiffness. Given this phase shift, the intensity of the emergent wave is:
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$`I=I_0 sin^2(\frac{\alpha}{2})=sin^2(\frac{\piCd}{\lambda}(\sigma_1-\sigma_2)) `$
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This equation relates internal stress and intensity for a single wavelength. In the experiment, both the light and camera channels cover a wide range of wavelengths.
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Right now there is not so much here so we prefer to send you to the excellent [wikipedia page about photoelasticity](https://en.wikipedia.org/wiki/Photoelasticity) or to [this lecture by W. Wang](http://depts.washington.edu/mictech/optics/me557/photoelasticity.pdf).
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Stress calculation is less noisy using narrow range of wavelength and single channel from the camera. However, it is still possible to retrieve stresses using white light. In such as case, stress-optical relation can be evaluated using empirical calibration.
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[<go back to home](https://git-xen.lmgc.univ-montp2.fr/PhotoElasticity/Main/wikis/home) |
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[<go back to home](https://git-xen.lmgc.univ-montp2.fr/PhotoElasticity/Main/wikis/home) |
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