Photoelasticity

Sandro Süß, winter semester 2019/20

Photoelasticity or photoelastic analysis is a method that provides the difference of principal stresses/strains and their orientation at every point in the model domain. It is a leading method for gaining phenomenological understanding by providing quantitative information. ^[1]

Physical and technical fundamental principles

In photoelasticity usually polarized light is used which means that the tip of the light vector follows a definite law. If the constraints define the tip of the light vector to follow the circumference of a circle, it is circularly polarized, if it is parallel to a given direction in the wavefront, it is said to be linearly or plane polarized. ^[1]

Birefringence

Most crystalline media except for cubic crystals and isotropic amorphic substances show birefringence behavior. This means that an incident light beam on these substances gives rise to two refracted rays, an ordinary (o) and an extraordinary (e) one, which are plane polarized with perpendicular directions of oscillation. Furthermore, the extraordinary ray doesn’t follow Snell’s law and changes its propagation speed in the medium depending on the angle of incidence. ^[1,2]

The photoelastic effect

Based on the work of Thomas Johann Seebeck in 1813 and David Brewster in 1816, glass and other homogeneous, optically isotropic materials, notably some polymeric plastics, show the property of birefringence when loaded. Therefore, models from such materials show interference under load in a polariscope. The originally isotropic body becomes a material with crystalline properties and has therefore distinguished directions which are related to the direction of the principal stresses. Assuming a two-dimensional stress state, a linearly polarized beam is divided into two components parallel to the principle stress directions $//<![CDATA[ \begin{array}{l}\sigma_{1}\end{array} //]]>$ and $//<![CDATA[ \begin{array}{l}\sigma_{2}\end{array} //]]>$ at every material point. Both components pass the model with different velocities which results in a phase shift $//<![CDATA[ \begin{array}{l}\delta\end{array} //]]>$. As long as the stresses are elastic, we have a linear relation between stress and resulting birefringence. This artificial birefringence is the result of deformations on the atomic or molecular scale. ^[2]

The relative retardation for a crystalline plate in terms of principal stress difference $//<![CDATA[ \begin{array}{l}\delta\end{array} //]]>$ can be written as

$//<![CDATA[ \begin{array}{l}\delta = \frac{2 \pi h}{\lambda} C (\sigma_{1}-\sigma_{2})\end{array} //]]>$

where h is the plate thickness, $//<![CDATA[ \begin{array}{l}\lambda\end{array} //]]>$ is the wavelength of the light source and C is the relative stress-optic coefficient. C depends on the wavelength and temperature, is usually assumed to be a constant for a certain material and calibrated beforehand. To calibrate materials with a high photoelastic sensitivity a bending test is performed, while for less photoelastically sensitive materials a tensile or compression test with compensation is recommendable. ^[1,2]

Above formula can be rewritten as the stress-optic law which relates the stress information to an optical measurement

$//<![CDATA[ \begin{array}{l}\sigma_{1}-\sigma_{2}=\frac{N F_{\sigma}}{h} ; \sigma_{1}>\sigma_{2}\end{array} //]]>$

where the fringe order N is

$//<![CDATA[ \begin{array}{l}N=\frac{\delta}{2\pi}=h\frac{C}{\lambda}(\sigma_{1}-\sigma_{2})\end{array} //]]>$

and the material stress fringe value $//<![CDATA[ \begin{array}{l}F_{\sigma}\end{array} //]]>$ is

$//<![CDATA[ \begin{array}{l}F_ {\sigma}[\frac{\frac{N}{mm}}{fringe}]=\frac{\lambda}{C}\end{array} //]]>$

The principle stress difference can then be obtained for linear stress levels when the fringe order and the material stress fringe values are known.^[1]

Operating principle

Polariscope

To perform a photoelastic analysis, a polariscope is used which can be set up in different optical arrangements. A polariscope consists of at least a light source and two similar polarizers where the first one next to the light source is called polarizer and the second one analyzer. Depending on the polarization of the light we differentiate plane and circular polariscopes. For the bright-field arrangement the polarizer and analyzer are set up in parallel (see Figure 1) while for the dark-field arrangement they are shifted by 90° (see Figure 2). In the bright-field setup a change of oscillation state results in a darkening behind the analyzer while it results in brightening for the dark-field setup. The light source can be either white or monochromatic light. ^[1,2]

The setup of the optical components in Figure 1 and Figure 2 is known as plane polariscope as the light is plane polarized. Behind the analyzer in a dark-field arrangement, black fringe contours appear which correspond to the points where the intensity of transmitted light is zero. Therefore, the plane of polarization of the emergent light from the model is along the vertical axis as the analyzer is kept at 0°. There are two possibilities, the isochromatics and the isoclinics, that fulfill the condition of unaltered incident plane-polarized light passing through the model.^[1]

For a different setup, the so called circular polariscope (see Figure 3), the light is circularly polarized by adding two $//<![CDATA[ \begin{array}{l}\frac{\lambda}{4}\end{array} //]]>$ waveplates. These waveplates delay the plane polarized light parallel to a specific waveplate axis by a quarter wavelength and can therefore create circularly and elliptically polarized light from plane polarized light and vice versa. Isoclinics disappear for circular polariscopes. ^[2]

Isochromatics

Isochromatics appear both in the plane and the circular polariscope. As the incident light is unaltered in all points where the specimen behaves like a full-wave plate, the principal stress difference $//<![CDATA[ \begin{array}{l}\sigma_{1}-\sigma_{2}\end{array} //]]>$ at these points causes a phase shift of $//<![CDATA[ \begin{array}{l}2m\pi\end{array} //]]>$ (m=0, 1, 2, ...). Therefore, a collection of points forms contours that build a fringe field that fulfills this condition as the stress is continuous. For a white light source contours of constant color become visible which roots in the wavelength dependency of waveplates. A single wavelength is cut off at any point and the white light minus the extinct color over the field is visible. For monochromatic sources, isochromatics appear black. [1]

The Isochromatics deliver directly the edge tensions and the lines of equal principal stress differences which are identical with the lines of equal loads. ^[2]

Isoclinics

Isoclinics appear only in the plane polariscope as they become visible when the polarizer axis coincides with one of the principle stress directions at the point of interest. Observable is a dark fringe both with white and monochromatic sources as the light extinction is not wavelength dependent. When rotating the polarizer and analyzer in the same manner around the neutral axis, the isoclinics change and are therefore labeled with the angles they denote such as 0°, 10°, 15°, etc. ^[1]

The isoclinics deliver the lines of equal directions of principal stress. ^[2]

Ordering of isoclinics and isochromatics

To obtain the principle stress directions, the model is viewed through a dark-field plane polariscope and the analyzer and polarizer are rotated in the same manner until the isoclinic band passes through the point of interest. As already mentioned, the orientations of the two polarizers coincide with the orientations of the principal stress directions at that point and the isoclinic represents either the $//<![CDATA[ \begin{array}{l}\sigma_{1}\end{array} //]]>$ or the $//<![CDATA[ \begin{array}{l}\sigma_{2}\end{array} //]]>$ direction which can be clarified by calibrating the polariscope. The orientation of the polarizer axis with the vertical axis is called isoclinic angle.  ^[1]

Ordering of isochromatics is crucial to obtain quantitative information from the fringe field. For that a good understanding must be developed and extended by experience and intuition. To simplify the process, information about source, sink and saddle points is collected to receive information about the fringe gradient direction and the order of at least one fringe forming the fringe packet. Unfortunately, there is no standard procedure to order fringes, but the literature provides some broad guidelines ^[3,4].

Calibration

To calibrate a photoelastic material, the material stress fringe value  is determined by normally using a circular disk under diametral compression as the closed-form stress field solution is known. The advantage of these circular disks compared to simple tension or a beam under pure bending is the easy handling of the compact, easy to machine and easy to load specimen and the accurate measurement of the fringe order at the center of the disk. The principle stress difference $//<![CDATA[ \begin{array}{l}(\sigma_{1}-\sigma_{2})\end{array} //]]>$ can be expressed at any point as

$//<![CDATA[ \begin{array}{l}(\sigma_{1}-\sigma_{2})=\frac{4PR}{\pi h} \frac{R^{2}-(x^{2}+y^{2})}{(x^{2}+y^{2}+R^{2})^{2}-4y^{2}R^{2}}\end{array} //]]>$

where R is the radius of the circular disk, P the compressive load, h the thickness of the disk and x and y the coordinates of a certain point. Using this and the formula from the photoelastic effect gives the material stress fringe value as

$//<![CDATA[ \begin{array}{l}F_ {\sigma}=\frac{8P}{\pi DN}\end{array} //]]>$

For the conventional approach the load is increased and then decreased incrementally while plotting a graph between the load P and the fringe order N, which is then best-fitted with a linear least-squares algorithm to evaluate the value for $//<![CDATA[ \begin{array}{l}F_ {\sigma}\end{array} //]]>$.^[1]

Applications of Photoelasticity

Transmission photoelasticity, as described before to explain the principle of photoelasticity, is only a small part of several experimental techniques that can be used for different applications. Techniques include three-dimensional photoelasticity, dynamic photoelasticity, reflection photoelasticity, photo-orthotropic elasticity and photoplasticity. These approaches increase the possible applications from only glass and other homogeneous, optically isotropic materials to opaque prototypes and composites. Digital photoelasticity is used to automate the analysis by processing the detected images and evaluating fringe orders. Furthermore, it can be used to directly test prototypes for the fit on their digital twin and thus compare experimental results with numerical methods. ^[2]

Literature

[1] Ramesh K.: Photoelasticity. In: Sharpe W. (eds) Springer Handbook of Experimental Solid Mechanics. Springer Handbooks. Springer, Boston, MA (2008), p. 701-742

[2] Wolf H.: Spannungsoptik. Springer, Berlin/Heidelberg (1976), p. 38-107

[3] Durelli, A.J. & Shukla: Identification of isochromatic fringes. Experimental Mechanics (1983) 23:111, p. 111–119.

[4] K. Ramesh: Digital Photoelasticity: Advanced Techniques and Application. Springer, Berlin/Heidelberg (2000)