Photoelasticity: determining stress concentration in plastic objects

Annetta Schartl, Wintersemester 2024/25

Photoelasticity is a leading optical technique for visualising an entire stress field (see Figure 1) and thus provides a phenomenological understanding of the difference in principal stresses and strains as well as their orientation at any point in the model domain. It offers quantitative and qualitative results for transparent plastic materials which exhibit two different indices of refraction under stress. ^[1,2]

Physical Principle

Polarization

From the point of view of wave theory and electromagnetic light theory, light is a transverse wave and is considered to be the periodic change of the light vector, perpendicular to the direction of propagation. In ordinary or natural light, this light vector has no distinct plane of oscillation. It is called linearly polarized light if the light vector oscillates along the direction of propagation in only one plane. Polarized light is created from ordinary, unpolarized light through reflection, refraction and natural or artificial birefringence. ^[3] Also polarized light enables the visualization of birefringence in a material. ^[1]

Birefringence

Photoelasticity is based on the phenomenon of temporary or artificial double refractionor birefringence effect. According to it, unstressed optically isotropic transparent materials become optically anisotropic under stress and behave like birefringent crystals. The difference of the principal stresses is proportional to the birefringence of the material. It is maintained only during application of loads and disappears when the loads are removed. ^[4]

Optically anisotropic crystalline media are characterized by the fact that one ordinary and one extraordinary refracted ray will result from a single incident ray and thus shows birefringence - also known as double refraction. These two rays are linearly polarized with their planes of polarization being perpendicular to each other. One of the rays is called extraordinary in the sense that, under suitable circumstances, it violates Snell's law of refraction and does not have to be limited to the plane of incidence. Moreover, its velocity changes continuously with the angle of incidence. The two refractive indices for the extraordinary and ordinary rays are equal only in the direction of an optic axis. Ordinary light can be transmitted from an isotropic medium, whereas light is always polarized when traveling through a crystal. ^[2]

In case of plane-polarized light ray which incidents normally on a crystalline plate of thickness h and is perpendicular to the optic or polarized axes, the ordinary and extraordinary rays travel with two different velocities v₁ and v₂ in the same direction. As the propagation speeds within the crystal are different for these two rays, each requiring the times h/v₁ and h/v₂ to cross the plate, this leads to a net phase difference of δ upon emergence from the crystal plate which is defined as

where:

The light emerging from this so-called retarder plate is generally elliptically polarised. If the thickness is such that there is a phase difference of π/2 radians, it is referred to as a quarter-wave plate (λ/4). Likewise half-wave plates and full-wave plates exists with corresponding phase differences of π and 2π radians which for the latter one the incident light remains unchanged. Usually, these wave plates are made of mica or quartz and the λ/4 plates have a thickness of one millimetre. Also important to mention is that these retarder plates are wavelength-specific. Wave plates have two polarizing axes which are denoted as the fast axis (F) and the slow axis (S). Planar-polarized light can be generated from a natural source by reflection, scattering, the use of polarising films and by a Nicol’s prism.

In summary wave plates or retarders can be used to convert linearly polarised light into circular or elliptically polarised light. It is said to be right-handedly polarized if the tip of the light vector describes a counterclockwise movement, and left-handedly polarized if the path is traversed in a clockwise direction. ^[2]

Stress-Optic Law

Considering a transparent model made of a high polymer, which is subjected to a plane state of stress characterized at a point by the principal stresses σ₁, σ₂ and their orientation θ with reference to a set of axes and the refractive indices for vibrations n₁ and n₂ which are corresponding to these two directions, the Maxwell’s formulation can be written as follows:

In terms of the fringe order N the Maxwell’s formulation can be rewritten as

which can be reformulated as the stress-optic law

where:

The change in fringe order, which usually can be seen as the change in colour, is used to obtain the sign of the boundary stresses. For a positive boundary stress the fringe order will increase and vice versa. The principal stress difference can be determined if the fringe order and the material stress fringe values are known. Even though F_σ and (σ₁ − σ₂) are linearly related as stated in the stress-optic law, this relation becomes nonlinear at higher stress levels and cannot be used anymore. Also the relative stress-optic coefficient C, which is assumed to be a material constant, should be used with care as various studies have shown a wavelength dependecy of C. ^[2]

Operating Principle

Polariscopes

There exist different types of optical arrangements for observing the photoelastic effect. A plane polariscope is one of the simplest, but also circular polariscopes are used. These systems are characterized by the orientation of the used polarizers and additional wave plates and can be maintained with either a white or a monochromatic light source, which results in colored or dark and light fringe patterns, as can be seen in Figure 1. Usually, a dark field arrangement of the polarizing filters is chosen (shown in Figure 2). ^[4]

Plane Polariscope

A plane polariscope as shown in Figure 3 consists of a pair of plane polarizers, whose optical axes are arranged parallel (bright field) or perpendicular (dark field) to each other, and a monochromatic or white light source. Its name is derived from the application of plane polarized light. The field of the polariscope is defined as the region between the two polarizers, in which the model of the analyzed structure is placed. A polarizer is an optical element that splits an incident ray of light into two mutually perpendicular components and ensures that the component parallel to its axis is transmitted, while the component perpendicular to its axis is absorbed. The polarizer near the light source is called the polarizer, whereas the analyzer is defined as the second polarizer of the polariscope downstream the model. In general, the dark-field arrangement with crossed polarizer and analyzer is used in the plane polariscope. ^[4]

As the plane polarized light passes through the model, which is under compression, a change in state of polarization occurs from point to point. This is depending on the magnitude of the principal stress difference as well as on the principal stress direction. One can obtain information about the stress field by studying the state of polarization of the emergent light, which can be easily achieved by introducing the analyzer – a polarizer at 0° with respect to the x axis. As a result, a fringe pattern appears on the screen. If a monochromatic light source is applied, these fringe contours are black and correspond to the points of zero intensity of the transmitted light. This is only possible for light emerging from the model with its plane of polarization along the vertical axis, as the analyser is kept at 0°. In order to give the fringe contours a physical meaning, it is therefore necessary to determine the conditions under which the incident plane-polarized light remains unchanged as it passes through the model. That is at all those points where the model shows the behavior of a full-wave plate, which occurs when the principal stress difference (σ₁ −σ₂) causes a relative phase difference of 2mπ (with m = 0, 1, 2, ...). Since stress is a continuous quantity, an agglomeration of points, which are forming the contours satisfying that above mentioned condition, can be observed. This corresponding fringe field is referred to as the isochromatics. ^[2]

Circular Polariscope

As the name suggests, a circular polariscope uses circularly polarized light which is generated by a combination of a polarizer and a quarter wave plate whose optical axis forms an angle of 45° with the axis of the polarizer. As already described in section for Birefringence a quarter-wave plate splits the incoming light into two linearly polarized beams along its two perpendicular axes, with the optical path length difference between the rays corresponding to a quarter wave (λ/4). A circular polariscope is composed of the following four optical elements arranged in sequence as it is shown in Figure 4:

1. A linear polariser with its optical axis being 90° with respect to the x-axis (vertical).
2. The first quarter-wave plate, whose fast axis (F) forms an angle of 45° with the x-axis.
3. The second quarter-wave plate, whose fast axis (F) forms an angle of 135° with the x-axis (here in the dark-field polariscope arrangement the axes of the two quarter-wave plates are crossed, but they are parallel in a light-field arrangement).
4. A linear polarizer (analyser) with its optical axis being 0° with respect to the x-axis (horizontal).

By inserting two quarter-wave plates into a plane polariscope, it is converted into a circular polariscope. The analyzed model is located in the optical field between the two quarter-wave plates and is illuminated with circularly polarized light which emerges from the first quarter-wave plate after the polarizer. The two crossed quarter-wave plates have no influence on the incident light. In general, a circular polariscope, where the two quarter-wave plates as well as the polarizer and analyser are crossed, provides a dark field, while it provides a light field when the quarter-wave plates are parallel and polarizer and analyzer crossed. ^[4]

Isoclinics

The name isoclinic is derived from the Greek words ‘iso’ for ‘equal’ and ‘clinic’ for ‘inclination’, i.e. lines with the same inclination of the principal stresses. The condition of light extinction (when the intensity I is zero) behind the analyzer of a plane dark-field polariscope holds for angles of one of the principal stresses Ɵ = 0, π/2, ..., n π/2. It indicates that the points of the model appear dark, where the directions of the principal stresses and the ones of the crossed polarizer or analyzer coincide. These locations of the points belong to the same dark fringe, which are called isoclinics. When analysing the optical patterns of plane and circular polariscopes, it is found that the isoclines only appear in the plane polariscope and disappear in the circular polariscope. ^[4]

The ordering of isoclinics can be used to determine the principal stress direction at a desired point on a stressed model. Therefore, a plane polariscope in dark-field arrangement is used, where polarizer and analyser are rotated in unison until the isocline represented by a dark band passes through the point of interest. Here, the principle stress directions coincide with the orientations of the polarizer and analyzer. The isoclinic angle is referred to as the alignment of the polarizer axis with the vertical. It should be noted that it is not clear whether the isoclinic angle represents the σ₁ or σ₂ direction. For this, a calibration of the polariscope is necessary. ^[2]

Isochromatics

The name isochromatics is derived from the Greek words ‘iso’ for ‘equal’ and ‘chromatic’ for ‘colour’, i.e. lines of the same colour. When white light is used, isochromats appear in the same colour. For δ = 0, 2π,..., n 2π (n=0, 1, 2,...), the light downstream the analyzer of either a dark-field plane polariscope or a dark-field circular polariscope gets extinct, such that the light intensity is zero. Points on the model fulfilling this condition appear dark and belong to the same fringe. These loci of the points belong to the same dark fringe at which the difference of the principal stresses (σ₁ − σ₂) becomes zero or an integral number of wavelengths of retardation (δ = 0, 2π, …, n 2π) is produced. These fringes are known as isochromatics of integer order. However, in a light-field circular or plane polariscope the points of the body appearing dark satisfy the condition δ = (1+2n)π, (n = 0, 1, 2, …). They belong to the same fringe, at which the difference of the principal stresses results in an even number of half-wavelengths of retardation (δ = π, 3π, … (1 + 2n)π, n = 0, 1, 2, …), such that these fringes are called isochromatics of half order. ^[4] It should be noted that the intensity equations for both dark and light fields are independent of θ and therefore the extinction condition is only a function of δ. So only isochromats become visible, what helps separating isoclines from isochromats. ^[2]

One of the crucial steps to determine quantitative information from the fringe field is the ordering of isochromatics. There still does not exist a standard procedure for this, but through experience and intuition numbering of isochromatic fringes becomes possible and still it is not an easy task ordering fringes from a given dark- or light-field black-and-white photograph. Several experimental auxiliary methods have been used in the past for this, for example the use of a white light source, such that the color code can be used for identifying the fringe orders and the direction of the fringe gradient. Unfortunately, this approach is only possible up to the third-order fringe in the field, as the colours merge beyond this point. ^[2]

Variants of Photoelasticity and their Applications

The development of different variants of photoelasticity has enabled its use in different application areas and has offered measuring techniques for the stress field inside the model. Compared to metals the moderate stress wave velocities in photoelastic materials make it interesting for studying dynamic phenomena. The Dynamic photoelasticity as a logical extension of transmission photoelasticity is applied for analysing time-varying problems. Therefore, it uses essentially the same stress-optic law by adjusting it to a dynamic experiment to obtain a modified material stress fringe value, which is in general 10-30% higher than the static value for the model material. A challenging factor here is the proper synchronization of the dynamic event with the recording. Dynamic photoelasticity has contributed significantly to the phenomenological understanding of fractures, the propagation of stress waves, the interaction between cracks and stress waves as well as the load transfer in granular media. ^[2]

Since photoelasticity is a strain-induced phenomenon, it is also used in reflection photoelasticity, which is an engineering tool for solving practically important and complicated problems.  Opaque prototypes made of different materials, at which a thin birefringent coating with a reflective backing at the interface is temporary pasted on, can be studied by reflection photoelasticity. By appropriate loading of the prototype and using a reflection polariscope one can collect the optical information. Based on solid mechanics principles, the stresses of the sample are determined from the optical response of the coating, which is related to the coating stresses. In addition, photoelasticity also finds application in orthotropic photoelasticity to study composite materials and in photoplasticity to study strains beyond the elastic limit. ^[2]

Also three-dimensional problems can be analyzed by using photoelasticity. Because of the difficulty of continuously varying principal stresses and their directions not remaining in one plane compared to the 2D case, three approaches for analyzing 3D-photoelasticity have been established. The first approach is experimentally expensive by mechanically slicing the model. It uses concepts of 2D-photoelasticity to solve a 3D case. The second approach, namely the integrated photoelasticity, is mathematically intensive, when attempting to relate the stress distribution along the light path to the integrated delay pattern. This method has been widely used for evaluating residual stresses in glass components and also has been successful in analyzing axisymmetric problems. The scattered light photoelasticity – the third approach – is a non-destructive method and attempts optical slicing with the use of integrated photoelasticity to interpret the data. ^[2]

Modern photoelasticity has been digitalized to automate procedures which were done either optical or manually in the past. Digital photoelasticity techniques can be broadly categorised into spatial- and frequency-domain methods. While the frequency-domain methods are computationally very intensive as they demand more recorded images, the spatial-domain methods are computationally very fast because of the requirement that fewer images have to be recorded. In summary, digital photoelasticity offers a great opportunity for further developments and improved analysis methods in photoelasticity. ^[2]

Literature

[1] Armstrong, T. W.: Introduction to experimental methods. Taylor & Francis Group, 2023, S.143-150, https://doi.org/10.1201/9781003329237-12

[2] Ramesh K.: Photoelasticity. In: Springer handbook of experimental solid mechanics, 2008 701–742, https://doi.org/10.1007/978-0-387-30877-7_25

[3] Wolf H.: Spannungsoptik, Berlin 1976

[4] Gdoutos E.E.: Experimental Mechanics An Introduction, Springer 2022, https://doi.org/10.1007/978-3-030-89466-5