Photoelasticity

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Photoelasticity describes changes in the optical properties of a material under mechanical deformation. It is a property of all dielectric media and is often used to experimentally determine the stress distribution in a material, where it gives a picture of stress distributions around discontinuities in materials. Photoelastic experiments (also informally referred to as photoelasticity) are an important tool for determining critical stress points in a material, and are used for determining stress concentration in irregular geometries.

History

The photoelastic phenomenon was first discovered by the Scottish physicist David Brewster. [1] [2] Experimental frameworks were developed at the beginning of the twentieth century with the works of E. G. Coker and L. N. G. Filon of University of London. Their book Treatise on Photoelasticity, published in 1930 by Cambridge Press, became a standard text on the subject. Between 1930 and 1940, many other books appeared on the subject, including books in Russian, German and French. At the same time, much development occurred in the field – great improvements were achieved in technique, and the equipment was simplified. With refinements in the technology, photoelastic experiments were extended to determining three-dimensional states of stress. In parallel to developments in experimental technique, the first phenomenological description of photoelasticity was given in 1890 by Friedrich Pockels, [3] however this was proved inadequate almost a century later by Nelson & Lax [4] as the description by Pockels only considered the effect of mechanical strain on the optical properties of the material.

With the advent of the digital polariscope – made possible by light-emitting diodes – continuous monitoring of structures under load became possible. This led to the development of dynamic photoelasticity, which has contributed greatly to the study of complex phenomena such as fracture of materials.

Applications

Photoelasticity has been used for a variety of stress analyses and even for routine use in design, particularly before the advent of numerical methods, such as for instance finite elements or boundary elements. [5] Digitization of polariscopy enables fast image acquisition and data processing, which allows its industrial applications to control quality of manufacturing process for materials such as glass [6] and polymer. [7] Dentistry utilizes photoelasticity to analyze strain in denture materials. [8]

Photoelasticity can successfully be used to investigate the highly localized stress state within masonry [9] [10] [11] or in proximity of a rigid line inclusion (stiffener) embedded in an elastic medium. [12] In the former case, the problem is nonlinear due to the contacts between bricks, while in the latter case the elastic solution is singular, so that numerical methods may fail to provide correct results. These can be obtained through photoelastic techniques. Dynamic photoelasticity integrated with high-speed photography is utilized to investigate fracture behavior in materials. [13] Another important application of the photoelasticity experiments is to study the stress field around bi-material notches. [14] Bi-material notches exist in many engineering application like welded or adhesively bonded structures

Formal definition

For a linear dielectric material the change in the inverse permittivity tensor ${\displaystyle \Delta (\varepsilon ^{-1})_{ij}}$ with respect to the deformation (the gradient of the displacement ${\displaystyle \partial _{l}u_{k}}$) is described by [15]

${\displaystyle \Delta (\varepsilon ^{-1})_{ij}=P_{ijkl}\partial _{k}u_{l}}$

where ${\displaystyle P_{ijkl}}$ is the fourth-rank photoelasticity tensor, ${\displaystyle u_{l}}$ is the linear displacement from equilibrium, and ${\displaystyle \partial _{l}}$ denotes differentiation with respect to the Cartesian coordinate ${\displaystyle x_{l}}$. For isotropic materials, this definition simplifies to [16]

${\displaystyle \Delta (\varepsilon ^{-1})_{ij}=p_{ijkl}s_{kl}}$

where ${\displaystyle p_{ijkl}}$ is the symmetric part of the photoelastic tensor (the photoelastic strain tensor), and ${\displaystyle s_{kl}}$ is the linear strain. The antisymmetric part of ${\displaystyle P_{ijkl}}$ is known as the roto-optic tensor. From either definition, it is clear that deformations to the body may induce optical anisotropy, which can cause an otherwise optically isotropic material to exhibit birefringence. Although the symmetric photoelastic tensor is most commonly defined with respect to mechanical strain, it is also possible to express photoelasticity in terms of the mechanical stress.

Experimental principles

The experimental procedure relies on the property of birefringence, as exhibited by certain transparent materials. Birefringence is a phenomenon in which a ray of light passing through a given material experiences two refractive indices. The property of birefringence (or double refraction) is observed in many optical crystals. Upon the application of stresses, photoelastic materials exhibit the property of birefringence, and the magnitude of the refractive indices at each point in the material is directly related to the state of stresses at that point. Information such as maximum shear stress and its orientation are available by analyzing the birefringence with an instrument called a polariscope.

When a ray of light passes through a photoelastic material, its electromagnetic wave components are resolved along the two principal stress directions and each component experiences a different refractive index due to the birefringence. The difference in the refractive indices leads to a relative phase retardation between the two components. Assuming a thin specimen made of isotropic materials, where two-dimensional photoelasticity is applicable, the magnitude of the relative retardation is given by the stress-optic law: [17]

${\displaystyle \Delta ={\frac {2\pi t}{\lambda }}C(\sigma _{1}-\sigma _{2})}$

where Δ is the induced retardation, C is the stress-optic coefficient, t is the specimen thickness, λ is the vacuum wavelength, and σ1 and σ2 are the first and second principal stresses, respectively. The retardation changes the polarization of transmitted light. The polariscope combines the different polarization states of light waves before and after passing the specimen. Due to optical interference of the two waves, a fringe pattern is revealed. The number of fringe order N is denoted as

${\displaystyle N={\frac {\Delta }{2\pi }}}$

which depends on relative retardation. By studying the fringe pattern one can determine the state of stress at various points in the material.

For materials that do not show photoelastic behavior, it is still possible to study the stress distribution. The first step is to build a model, using photoelastic materials, which has geometry similar to the real structure under investigation. The loading is then applied in the same way to ensure that the stress distribution in the model is similar to the stress in the real structure.

Isoclinics and isochromatics

Isoclinics are the loci of the points in the specimen along which the principal stresses are in the same direction.

Isochromatics are the loci of the points along which the difference in the first and second principal stress remains the same. Thus they are the lines which join the points with equal maximum shear stress magnitude. [18]

Two-dimensional photoelasticity

Photoelasticity can describe both three-dimensional and two-dimensional states of stress. However, examining photoelasticity in three-dimensional systems is more involved than two-dimensional or plane-stress system. So the present section deals with photoelasticity in a plane stress system. This condition is achieved when the thickness of the prototype is much smaller as compared to dimensions in the plane. Thus one is only concerned with stresses acting parallel to the plane of the model, as other stress components are zero. The experimental setup varies from experiment to experiment. The two basic kinds of setup used are plane polariscope and circular polariscope.

The working principle of a two-dimensional experiment allows the measurement of retardation, which can be converted to the difference between the first and second principal stress and their orientation. To further get values of each stress component, a technique called stress-separation is required. [19] Several theoretical and experimental methods are utilized to provide additional information to solve individual stress components.

Plane polariscope setup

The setup consists of two linear polarizers and a light source. The light source can either emit monochromatic light or white light depending upon the experiment. First the light is passed through the first polarizer which converts the light into plane polarized light. The apparatus is set up in such a way that this plane polarized light then passes through the stressed specimen. This light then follows, at each point of the specimen, the direction of principal stress at that point. The light is then made to pass through the analyzer and we finally get the fringe pattern.

The fringe pattern in a plane polariscope setup consists of both the isochromatics and the isoclinics. The isoclinics change with the orientation of the polariscope while there is no change in the isochromatics.

Circular polariscope setup

In a circular polariscope setup two quarter-wave plates are added to the experimental setup of the plane polariscope. The first quarter-wave plate is placed in between the polarizer and the specimen and the second quarter-wave plate is placed between the specimen and the analyzer. The effect of adding the quarter-wave plate after the source-side polarizer is that we get circularly polarized light passing through the sample. The analyzer-side quarter-wave plate converts the circular polarization state back to linear before the light passes through the analyzer.

The basic advantage of a circular polariscope over a plane polariscope is that in a circular polariscope setup we only get the isochromatics and not the isoclinics. This eliminates the problem of differentiating between the isoclinics and the isochromatics.

Related Research Articles

In optics, the refractive index or index of refraction of a material is a dimensionless number that describes how fast light travels through the material. It is defined as

Optical rotation or optical activity is the rotation of the orientation of the plane of polarization about the optical axis of linearly polarized light as it travels through certain materials. Optical activity occurs only in chiral materials, those lacking microscopic mirror symmetry. Unlike other sources of birefringence which alter a beam's state of polarization, optical activity can be observed in fluids. This can include gases or solutions of chiral molecules such as sugars, molecules with helical secondary structure such as some proteins, and also chiral liquid crystals. It can also be observed in chiral solids such as certain crystals with a rotation between adjacent crystal planes or metamaterials. Rotation of light's plane of polarization may also occur through the Faraday effect which involves a static magnetic field, however this is a distinct phenomenon that is not usually classified under "optical activity."

Polarization is a property applying to transverse waves that specifies the geometrical orientation of the oscillations. In a transverse wave, the direction of the oscillation is perpendicular to the direction of motion of the wave. A simple example of a polarized transverse wave is vibrations traveling along a taut string (see image); for example, in a musical instrument like a guitar string. Depending on how the string is plucked, the vibrations can be in a vertical direction, horizontal direction, or at any angle perpendicular to the string. In contrast, in longitudinal waves, such as sound waves in a liquid or gas, the displacement of the particles in the oscillation is always in the direction of propagation, so these waves do not exhibit polarization. Transverse waves that exhibit polarization include electromagnetic waves such as light and radio waves, gravitational waves, and transverse sound waves in solids.

A waveplate or retarder is an optical device that alters the polarization state of a light wave travelling through it. Two common types of waveplates are the half-wave plate, which shifts the polarization direction of linearly polarized light, and the quarter-wave plate, which converts linearly polarized light into circularly polarized light and vice versa. A quarter-wave plate can be used to produce elliptical polarization as well.

Birefringence is the optical property of a material having a refractive index that depends on the polarization and propagation direction of light. These optically anisotropic materials are said to be birefringent. The birefringence is often quantified as the maximum difference between refractive indices exhibited by the material. Crystals with non-cubic crystal structures are often birefringent, as are plastics under mechanical stress.

Linear elasticity is a mathematical model of how solid objects deform and become internally stressed due to prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mechanics.

The Kerr effect, also called the quadratic electro-optic (QEO) effect, is a change in the refractive index of a material in response to an applied electric field. The Kerr effect is distinct from the Pockels effect in that the induced index change is directly proportional to the square of the electric field instead of varying linearly with it. All materials show a Kerr effect, but certain liquids display it more strongly than others. The Kerr effect was discovered in 1875 by John Kerr, a Scottish physicist.

Ellipsometry is an optical technique for investigating the dielectric properties of thin films. Ellipsometry measures the change of polarization upon reflection or transmission and compares it to a model.

Electrostriction is a property of all electrical non-conductors, or dielectrics, that causes them to change their shape under the application of an electric field.

Polarimetry is the measurement and interpretation of the polarization of transverse waves, most notably electromagnetic waves, such as radio or light waves. Typically polarimetry is done on electromagnetic waves that have traveled through or have been reflected, refracted or diffracted by some material in order to characterize that object.

Elastic energy is the mechanical potential energy stored in the configuration of a material or physical system as it is subjected to elastic deformation by work performed upon it. Elastic energy occurs when objects are impermanently compressed, stretched or generally deformed in any manner. Elasticity theory primarily develops formalisms for the mechanics of solid bodies and materials. The elastic potential energy equation is used in calculations of positions of mechanical equilibrium. The energy is potential as it will be converted into other forms of energy, such as kinetic energy and sound energy, when the object is allowed to return to its original shape (reformation) by its elasticity.

A photoelastic modulator (PEM) is an optical device used to modulate the polarization of a light source. The photoelastic effect is used to change the birefringence of the optical element in the photoelastic modulator.

A shear band is a narrow zone of intense shearing strain, usually of plastic nature, developing during severe deformation of ductile materials. As an example, a soil specimen is shown in Fig. 1, after an axialsymmetric compression test. Initially the sample was cylindrical in shape and, since symmetry was tried to be preserved during the test, the cylindrical shape was maintained for a while during the test and the deformation was homogeneous, but at extreme loading two X-shaped shear bands had formed and the subsequent deformation was strongly localized.

Acousto-optics is a branch of physics that studies the interactions between sound waves and light waves, especially the diffraction of laser light by ultrasound through an ultrasonic grating.

The viscous stress tensor is a tensor used in continuum mechanics to model the part of the stress at a point within some material that can be attributed to the strain rate, the rate at which it is deforming around that point.

The acoustoelastic effect is how the sound velocities of an elastic material change if subjected to an initial static stress field. This is a non-linear effect of the constitutive relation between mechanical stress and finite strain in a material of continuous mass. In classical linear elasticity theory small deformations of most elastic materials can be described by a linear relation between the applied stress and the resulting strain. This relationship is commonly known as the generalised Hooke's law. The linear elastic theory involves second order elastic constants and yields constant longitudinal and shear sound velocities in an elastic material, not affected by an applied stress. The acoustoelastic effect on the other hand include higher order expansion of the constitutive relation between the applied stress and resulting strain, which yields longitudinal and shear sound velocities dependent of the stress state of the material. In the limit of an unstressed material the sound velocities of the linear elastic theory are reproduced.

Crack tip opening displacement (CTOD) or is the distance between the opposite faces of a crack tip at the 90° intercept position. The position behind the crack tip at which the distance is measured is arbitrary but commonly used is the point where two 45° lines starting at the crack tip and intercepting the crack faces. The parameter is used in fracture mechanics to characterise the loading on a crack and can be related to other crack tip loading parameters such as the stress intensity factor and the elastic-plastic J-integral.

In continuum mechanics, objective stress rates are time derivatives of stress that do not depend on the frame of reference. Many constitutive equations are designed in the form of a relation between a stress-rate and a strain-rate. The mechanical response of a material should not depend on the frame of reference. In other words, material constitutive equations should be frame-indifferent (objective). If the stress and strain measures are material quantities then objectivity is automatically satisfied. However, if the quantities are spatial, then the objectivity of the stress-rate is not guaranteed even if the strain-rate is objective.

Brillouin spectroscopy is an empirical spectroscopy technique which allows the determination of elastic moduli of materials. The technique uses inelastic scattering of light when it encounters acoustic phonons in a crystal, a process known as Brillouin scattering, to determine phonon energies and therefore interatomic potentials of a material. The scattering occurs when an electromagnetic wave interacts with a density wave, photon-phonon scattering.

JCMsuite is a finite element analysis software package for the simulation and analysis of electromagnetic waves, elasticity and heat conduction. It also allows a mutual coupling between its optical, heat conduction and continuum mechanics solvers. The software is mainly applied for the analysis and optimization of nanooptical and microoptical systems. Its applications in research and development projects include dimensional metrology systems, photolithographic systems, photonic crystal fibers, VCSELs, Quantum-Dot emitters, light trapping in solar cells, and plasmonic systems. The design tasks can be embedded into the high-level scripting languages MATLAB and Python, enabling a scripting of design setups in order to define parameter dependent problems or to run parameter scans.

References

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3. Pockels, F. Ueber die durch einseitigen Druck hervorgerufene Doppelbrechung regulärer Krystalle, speciell von Steinsalz und Sylvin, Annalen der Physik, 275, 1890, 440.
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