Determination of the mean relative photoelastic dispersion factor by means of photoelasticity

This work proposed to determine the mean dispersion factor, < α C > , which is related to the relative photoelastic dispersion coefficient of a photoelastic material, subjected to consecutive external stresses, by means of a direct computational method associated with error theory. An important feature of photoelasticity, related to the light wavelength, is the photoelastic dispersion coefficient, which is determined in traditional methods by indirect statistical processes, as it depends on refractive indices, which are difficult to determine in photoelastic materials. Therefore, finding an alternative form to determine this coefficient is necessary and can help in the processes that involve the use of photoelasticity. The mean dispersion factor obtained was < α C > = (8,90 ± 0,42) x 10 -5 , which works as a system calibrator to allow the determination of the photoelastic dispersion coefficient of photoelastic materials by photoelasticity, with greater accuracy.


INTRODUCTION
Several photoelasticity techniques and methods can be used to determine the relative photoelastic dispersion coefficient. In this sense, some published works show the relevance of applications in the areas of Physics, Mechanical Engineering, Materials Engineering and Dentistry DOBRANSZKI et. al, 2010;RAMESH, 2000;SOARES, 1997).
Photoelasticity is a branch of optics that studies the distribution of stresses and strains in photoelastic materials with the aid of polarized light (FERREIRA, 2003;HECHT, 2002;GUENTER, 1990;PRADO et. al, 2020). Photoelastic materials have the property of temporary birefringence and, mainly, due to their transparency and elasticity, they are widely used in industry for indirect determination of the properties of materials such as iron, steel, concrete, etc. (BREWSTER, 1815;PARTHASARATHI, 2018;TORO et. al, 2017).
An important feature of photoelasticity, related to the wavelength of light, is the relative coefficient of photoelastic dispersion, which is determined, in traditional methods, by indirect statistical processes, as it depends on refractive indices, which are difficult to determine in photoelastic materials. It is essential to search for new methods for the treatment of data obtained by the various optical techniques that already exist and are being improved. In this direction, a computational system capable of processing a large amount of data quickly and efficiently will contribute to the advancement of studies on photoelastic materials (FERREIRA, 2003;HECHT, 2002;GUENTHER, 1990).
The method proposed in this paper determined the mean dispersion factor, , related to the relative photoelastic dispersion coefficient of photoelastic samples, through computer programs, from algorithms that performed the partial processes, which automated the method of calculating the mean strains of the evolution of photoelastic fringes, observed by a digital camera, during increasing external stresses produced in the samples, through the linear transmission polariscope technique.
The programs carry out the partial processes from recording the captured video and its separation into frames, through the calculation of mean strains and relating them to the mean strains to the construction of strain versus strain graphs to obtain the mean relative photoelastic dispersion factor.

THEORY
For a photoelastic material that exhibits the effect of temporary birefringence under the action of external stress, the optical stress law relates changes in the refractive index with its plane stress state (TORO et. al, 2017;KUSKE;ROBERTSON, 1974;FERREIRA JUNIOR, 2003). These changes are linearly proportional to the external mechanical stresses on the material and, consequently, these refractive indices are associated with the stresses and strains inside. Figure 1 shows a scheme of these refractive indices in an infinitesimal element of the material. Without external stresses the refractive indices, , are equal in both directions. When there is an external stress, the voltage difference promotes a difference in refractive indices in both directions, characterizing the temporary birefringence of the material. In practice, the birefringence of photoelastic materials observed in a polariscope with polychromatic light, which is the polarizer-based instrument for observing this effect, the result is a set of colored and dark fringes in the plane parallel to the direction of the applied external stress. The equation that represents this idea is: (1) and are the refractive indices in the extraordinary and ordinary directions, respectively, according to the reference system in Figure 1. and are the internal stresses in the extraordinary and ordinary directions, respectively.
is called the relative optical stress coefficient. In specific cases, this coefficient is considered a wavelength-independent material constant, , but in general cases is wavelength dependent and is called birefringence dispersion or photoelastic dispersion.  Assuming that is directly proportional to the mean external stress, , and that is proportional to the difference between the mean relative distances between fringes , less than a constant , mean scatter factor, for a given stress, then: (2) The index represents each stress applied to the sample. Defining , we have: ( The mean relative displacements of the photoelastic fringes , are obtained as follows: (4) e , e are the mean longitudinal and transverse distances, respectively, and are the reference distances.
The experimental values of and with adjustment by the least squares method (VUOLO, 2003), will produce the graph outlined in Figure 3. From the graph, the slope is obtained, such that: is obtained directly from linear regression and from typical values in the literature. The uncertainty of the equation (6) is, by error propagation method, described in Vuolo (2003) and Macwilliams and Sloane (1977) is used as the system calibration coefficient.

METHODOLOGY
Epoxy resin produced by a polymerization process was used to make the photoelastic sample (DA SILVA et. al, 2015). This resin offers excellent adhesion to a large amount of materials and exhibits optical fringes when subjected to external stresses.    The spherical wavefront, produced by a white light source (1), is altered by the neutral density filter and the lens (2), making it approximately flat. Then, a color filter, (3), only allows light of a certain wavelength to pass through, so that only fringes of the same color are observed to facilitate the determination of the distances between the fringes. Two linear polarizers, (4) and (6), with orthogonal polarization states prevent, a priori, the passage of light. With the stresses of the device, the polarization state of the light changes as it passes through the sample (A), allowing the digital camera (7) to observe the isochromatic fringes produced. Quarter wave plates, (5) were added before and after the sample to eliminate most of the isoclinic (dark) fringes and allow for a purer image of the isochromatic fringes (of the chosen color).
With the aid of the loading device in Figure 5, automatic loading sequences were performed on the photoelastic sample in Figure 4 and a video was recorded with the digital camera. M videos were obtained each containing Q frames. The proposed method used computational analysis associated with error theory treatment (VUOLO, 2003;MACWILLIAMS;SLOANE, 1977). frames of produced by a video were chosen from the consecutive compression stresses on the photoelastic sample.
groups, each containing image frames, were separated for data processing. The mean mechanical elasticity modulus was determined by the method presented: "… the external stresses, applied to the photeleastic sample, longitudinal were used. The mean deformations can be calculated from values obtained by lines of pixels along a chosen direction of the static fringes image, outlined in Figure 6, caused by the stresses applied to the photoelastic sample.  The i-th relative intensity is determined from the equation, Where I i represents the i-th intensity, for the i-th pixel, with I = 1, 2, …, n. I max represents the maximum intensity of the selected pixel line. The mean distance, <Dd>, between fringes is determined by the equation, where Dp i represents the difference between the positions of pixels in consecutive valleys or peaks, along the line of pixels. To determine the mean strain, <e>, of each image, the following equation is used, where is the reference distance, chosen accordingly, depending on the selected image. To determine the elasticity modulus, E, of the photoelastic sample, Hooke's Law represented in Equation (2) is used, such that, where is the j-th external mean stress and is the j-th mean strain, on the photoelastic sample, with j = 1, 2, …, m. From Equation (11), a graph similar to the schema in Figure 3 can be obtained…" (PRADO et. al, 2020).
With the mean mechanical elasticity modulus, represented by , Figure  3, and typical values in the literature , for the optical dispersion coefficient, , the dispersion factor, , was determined, through equation (6), as well as its uncertainty, through the error propagation method portrayed in the works of Vuolo (2003) and Macwilliams and Sloane (1977), equation (7). Table 1, was made for the experiment in question. The video obtained by the digital camera produced = 4096 frames of images, from which = 4080 were selected, divided into = 12 groups of = 340 frames of images. Images have been converted to default containing 256 shades of grey, 8-bit. The graph in Figure 8 shows the relationships between mean external stresses versus mean longitudinal strains for the photoelastic sample utilized. From the value and the mean typical value of the mean optical dispersion coefficient obtained in the literature, Da Silva (2017), , the mean dispersion factor was obtained, by equation (6), white your respective uncertainty, , by equation (7), such that: is dimensionless, since the dimension of the modulus of elasticity, [ , is the inverse of the dimension of the optical dispersion coefficient, [ .

CONCLUSIONS AND PERSPECTIVES
The method proved to be efficient in determining the mean dispersion factor, , and, with it in hand, it is possible to use the same methodology to directly determine the optical dispersion coefficient of photoelastic samples. For this, just make samples from the same batch and with similar dimensions, then choose a test sample to determine and use this coefficient to directly determine . Thus, it is possible to characterize the photoelastic samples, determining more accurately their typical values, both for the modulus of elasticity, , and for their Poisson coefficients, . Another possibility is to indirectly determine values of and other materials inserted in photoelastic samples.
The perspectives from this work are: to characterize several photoelastic samples, determining their values of ; characterize several photoelastic samples, determining their values of and ; apply the method presented in data obtained by the reflection photoelasticity technique, to other materials (example: metals) covered with thin layers of photoelastic resins; characterize other materials, determining values of ; characterize other materials, determining their values of and etc.