Photoelasticity

Last updated September 17, 2024

In materials science, 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.

History

The photoelastic phenomenon was first discovered by the Scottish physicist David Brewster, who immediately recognized it as stress-induced birefringence.^[1]^[2] That diagnosis was confirmed in a direct refraction experiment by Augustin-Jean Fresnel.^[3] 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. Max M. Frocht published the classic two volume work, Photoelasticity, in the field.^[4] 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,^[5] however this was proved inadequate almost a century later by Nelson & Lax ^[6] 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 finite elements or boundary elements.^[7] 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^[8] and polymer.^[9] Dentistry utilizes photoelasticity to analyze strain in denture materials.^[10]

Photoelasticity can successfully be used to investigate the highly localized stress state within masonry^[11]^[12]^[13] or in proximity of a rigid line inclusion (stiffener) embedded in an elastic medium.^[14] 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.^[15] Another important application of the photoelasticity experiments is to study the stress field around bi-material notches.^[16] Bi-material notches exist in many engineering application like welded or adhesively bonded structures.^{[ citation needed ]}

For example, some elements of Gothic cathedrals previously thought decorative were first proved essential for structural support by photoelastic methods.^[17]

Formal definition

For a linear dielectric material the change in the inverse permittivity tensor $\Delta (\varepsilon ^{-1})_{ij}$ with respect to the deformation (the gradient of the displacement $\partial _{\ell }u_{k}$ ) is described by ^[18]

\Delta (\varepsilon ^{-1})_{ij}=P_{ijk\ell }\partial _{k}u_{\ell }

where $P_{ijk\ell }$ is the fourth-rank photoelasticity tensor, $u_{\ell }$ is the linear displacement from equilibrium, and $\partial _{l}$ denotes differentiation with respect to the Cartesian coordinate $x_{l}$ . For isotropic materials, this definition simplifies to ^[19]

\Delta (\varepsilon ^{-1})_{ij}=p_{ijk\ell }s_{k\ell }

where $p_{ijk\ell }$ is the symmetric part of the photoelastic tensor (the photoelastic strain tensor), and $s_{k\ell }$ is the linear strain. The antisymmetric part of $P_{ijk\ell }$ 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:^[20]

\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

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.^{[ citation needed ]}

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.^[21]

Two-dimensional photoelasticity

Photoelastic experiment showing the internal stress distribution inside the cover of a Jewel case Photoelasticimetry1.JPG — Photoelastic experiment showing the internal stress distribution inside the cover of a Jewel case

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.^{[ citation needed ]} 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.^{[ citation needed ]}

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.^[22] 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.^{[ citation needed ]}

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.^{[ citation needed ]}

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.^{[ citation needed ]}

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.^{[ citation needed ]}

Related Research Articles

The Fresnel equations describe the reflection and transmission of light when incident on an interface between different optical media. They were deduced by French engineer and physicist Augustin-Jean Fresnel who was the first to understand that light is a transverse wave, when no one realized that the waves were electric and magnetic fields. For the first time, polarization could be understood quantitatively, as Fresnel's equations correctly predicted the differing behaviour of waves of the s and p polarizations incident upon a material interface.

$Refractive index Property in optics$

In optics, the refractive index of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium.

In physics, total internal reflection (TIR) is the phenomenon in which waves arriving at the interface (boundary) from one medium to another are not refracted into the second ("external") medium, but completely reflected back into the first ("internal") medium. It occurs when the second medium has a higher wave speed than the first, and the waves are incident at a sufficiently oblique angle on the interface. For example, the water-to-air surface in a typical fish tank, when viewed obliquely from below, reflects the underwater scene like a mirror with no loss of brightness (Fig. 1).

Optical rotation, also known as polarization rotation or circular birefringence, 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. Circular birefringence and circular dichroism are the manifestations of optical activity. 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.

<span class="mw-page-title-main">Polarization (waves)</span> Property of waves that can oscillate with more than one orientation

Polarization is a property of transverse waves which 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.

<span class="mw-page-title-main">Waveplate</span> Optical polarization device

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 rotates the polarization direction of linearly polarized light, and the quarter-wave plate, which converts between different elliptical polarizations

$Birefringence Refractive property of materials$

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 described as birefringent or birefractive. 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.

Stress–strain analysis is an engineering discipline that uses many methods to determine the stresses and strains in materials and structures subjected to forces. In continuum mechanics, stress is a physical quantity that expresses the internal forces that neighboring particles of a continuous material exert on each other, while strain is the measure of the deformation of the material.

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 for the Kerr effect 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 Scottish physicist John Kerr.

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.

<span class="mw-page-title-main">Polarimetry</span> Measurement and interpretation of the polarization of transverse waves

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.

$Delamination Mode of failure for which a material fractures into layers$

Delamination is a mode of failure where a material fractures into layers. A variety of materials, including laminate composites and concrete, can fail by delamination. Processing can create layers in materials, such as steel formed by rolling and plastics and metals from 3D printing which can fail from layer separation. Also, surface coatings, such as paints and films, can delaminate from the coated substrate.

<span class="mw-page-title-main">Magneto-optic Kerr effect</span> Changes to light reflected from a magnetized surface

In physics the magneto-optic Kerr effect (MOKE) or the surface magneto-optic Kerr effect (SMOKE) is one of the magneto-optic effects. It describes the changes to light reflected from a magnetized surface. It is used in materials science research in devices such as the Kerr microscope, to investigate the magnetization structure of materials.

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.

A Fresnel rhomb is an optical prism that introduces a 90° phase difference between two perpendicular components of polarization, by means of two total internal reflections. If the incident beam is linearly polarized at 45° to the plane of incidence and reflection, the emerging beam is circularly polarized, and vice versa. If the incident beam is linearly polarized at some other inclination, the emerging beam is elliptically polarized with one principal axis in the plane of reflection, and vice versa.

<span class="mw-page-title-main">Acousto-optics</span> The study of sound and light interaction

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.

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.

Geometrically necessary dislocations are like-signed dislocations needed to accommodate for plastic bending in a crystalline material. They are present when a material's plastic deformation is accompanied by internal plastic strain gradients. They are in contrast to statistically stored dislocations, with statistics of equal positive and negative signs, which arise during plastic flow from multiplication processes like the Frank-Read source.

$Huygens principle of double refraction Optical principle$

Huygens principle of double refraction, named after Dutch physicist Christiaan Huygens, explains the phenomenon of double refraction observed in uniaxial anisotropic material such as calcite. When unpolarized light propagates in such materials, it splits into two different rays, known as ordinary and extraordinary rays. The principle states that every point on the wavefront of birefringent material produces two types of wavefronts or wavelets: spherical wavefronts and ellipsoidal wavefronts. These secondary wavelets, originating from different points, interact and interfere with each other. As a result, the new wavefront is formed by the superposition of these wavelets.

References

↑ Brewster, David (1815). "Experiments on the depolarisation of light as exhibited by various mineral, animal, and vegetable bodies, with a reference of the phenomena to the general principles of polarisation". Philosophical Transactions of the Royal Society of London. 105: 29–53. doi: 10.1098/rstl.1815.0004 .
↑ Brewster, David (1816). "On the communication of the structure of doubly refracting crystals to glass, muriate of soda, fluor spar, and other substances, by mechanical compression and dilatation". Philosophical Transactions of the Royal Society of London. 106: 156–78. doi:10.1098/rstl.1816.0011. S2CID 108782967.
↑
Fresnel, Augustin (1822). "Note sur la double réfraction du verre comprimé". Annales de Chimie et de Physique. Série 2. 20: 376–83.
- Reprinted in H. de Senarmont, E. Verdet, and L. Fresnel (eds.), Oeuvres complètes d'Augustin Fresnel, vol. 1 (1866), pp. 713–18
- Translated as Fresnel, Augustin-Jean (2021). "Note on the double refraction of compressed glass". Translated by Putland, Gavin Richard. doi:10.5281/zenodo.4706835.{{cite journal}}: Cite journal requires |journal= (help)
↑ Leven, M.M.; Frocht, M.M., eds. (1969). "Vita Max Mark Frocht". Photoelasticity. Pergamon. pp. xi–xii. doi:10.1016/B978-0-08-012998-3.50005-7. ISBN 978-0-08-012998-3.
↑ Pockels, F. (1890). "Ueber die durch einseitigen Druck hervorgerufene Doppelbrechung regulärer Krystalle, speciell von Steinsalz und Sylvin". Annalen der Physik und Chemie. 275 (3): 440–69. Bibcode:1890AnP...275..440P. doi:10.1002/andp.18902750313.
↑ Nelson, D. F.; Lax, M. (1970). "New Symmetry for Acousto-Optic Scattering". Physical Review Letters. 24 (8): 379–80. Bibcode:1970PhRvL..24..379N. doi:10.1103/PhysRevLett.24.379.
↑ Frocht, M.M., Photoelasticity. J. Wiley and Sons, London, 1965
↑ Ajovalasit, A.; Petrucci, G.; Scafidi, M. (2012). "RGB photoelasticity applied to the analysis of membrane residual stress in glass". Measurement Science and Technology. 23 (2): 025601. Bibcode:2012MeScT..23b5601A. doi:10.1088/0957-0233/23/2/025601. hdl: 10447/61842 . S2CID 53600215.
↑ Kramer, Sharlotte; Beiermann, Brett; Davis, Douglas; Sottos, Nancy; White, Scott; Moore, Jeffrey (2013). "Characterization of Mechanochemically Active Polymers Using Combined Photoelasticity and Fluorescence Measurements". Application of Imaging Techniques to Mechanics of Materials and Structures, Volume 4. Conference Proceedings of the Society for Experimental Mechanics Series. pp. 167–78. doi:10.1007/978-1-4419-9796-8_21. ISBN 978-1-4419-9528-5.
↑ Fernandes, Cláudio P.; Glantz, Per-Olof J.; Svensson, Stig A.; Bergmark, Anders (2003). "Reflection photoelasticity: A new method for studies of clinical mechanics in prosthetic dentistry". Dental Materials. 19 (2): 106–17. doi:10.1016/s0109-5641(02)00019-2. PMID 12543116.
↑ Bigoni, Davide; Noselli, Giovanni (2010). "Localized stress percolation through dry masonry walls. Part I – Experiments". European Journal of Mechanics - A/Solids. 29 (3): 291–98. Bibcode:2010EuJMA..29..291B. doi:10.1016/j.euromechsol.2009.10.009.
↑ Bigoni, Davide; Noselli, Giovanni (2010). "Localized stress percolation through dry masonry walls. Part II – Modelling". European Journal of Mechanics – A/Solids. 29 (3): 299–307. Bibcode:2010EuJMA..29..299B. doi:10.1016/j.euromechsol.2009.10.013.
↑ Bigoni, D. (2012). Nonlinear Solid Mechanics: Bifurcation Theory and Material Instability. Cambridge University Press. ISBN 9781107025417.
↑ Noselli, G.; Dal Corso, F.; Bigoni, D. (2010). "The stress intensity near a stiffener disclosed by photoelasticity". International Journal of Fracture. 166 (1–2): 91–103. doi:10.1007/s10704-010-9502-9. S2CID 56221414.
↑ Shukla, A. (2001). "High-speed fracture studies on bimaterial interfaces using photoelasticity—a review". The Journal of Strain Analysis for Engineering Design. 36 (2): 119–42. doi:10.1243/0309324011512658. S2CID 137504535.
↑ Ayatollahi, M.R.; Mirsayar, M.M.; Dehghany, M. (2011). "Experimental determination of stress field parameters in bi-material notches using photoelasticity". Materials & Design. 32 (10): 4901–08. doi:10.1016/j.matdes.2011.06.002.
↑ Lewis, Michael (3 September 2024). "The Canary: Michael Lewis on Chris Mark of the Department of Labor". The Washington Post .
↑ J.F. Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices, Oxford University Press, 1957. ^{[ ISBN missing ]}^{[ page needed ]}
↑ R.E. Newnham, Properties of Materials: Anisotropy, Symmetry, Structure, Oxford University Press, 2005. ^{[ ISBN missing ]}^{[ page needed ]}
↑ Dally, J.W. and Riley, W.F., Experimental Stress Analysis, 3rd ed., McGraw-Hill Inc., 1991 ^{[ ISBN missing ]}^{[ page needed ]}
↑ Ramesh, K., Digital Photoelasticity, Springer, 2000 ^{[ ISBN missing ]}^{[ page needed ]}
↑ Solaguren-Beascoa Fernández, M.; Alegre Calderón, J.M.; Bravo Díez, P.M.; Cuesta Segura, I.I. (2010). "Stress-separation techniques in photoelasticity: A review". The Journal of Strain Analysis for Engineering Design. 45: 1–17. doi:10.1243/03093247JSA583. S2CID 208518298.

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[1] Brewster, David (1815). "Experiments on the depolarisation of light as exhibited by various mineral, animal, and vegetable bodies, with a reference of the phenomena to the general principles of polarisation". Philosophical Transactions of the Royal Society of London. 105: 29–53. doi: 10.1098/rstl.1815.0004 .

[2] Brewster, David (1816). "On the communication of the structure of doubly refracting crystals to glass, muriate of soda, fluor spar, and other substances, by mechanical compression and dilatation". Philosophical Transactions of the Royal Society of London. 106: 156–78. doi:10.1098/rstl.1816.0011. S2CID 108782967.

[fresnel-1822s-3] Fresnel, Augustin (1822). "Note sur la double réfraction du verre comprimé". Annales de Chimie et de Physique. Série 2. 20: 376–83.
Reprinted in H. de Senarmont, E. Verdet, and L. Fresnel (eds.), Oeuvres complètes d'Augustin Fresnel, vol. 1 (1866), pp. 713–18
Translated as Fresnel, Augustin-Jean (2021). "Note on the double refraction of compressed glass". Translated by Putland, Gavin Richard. doi:10.5281/zenodo.4706835.{{cite journal}}: Cite journal requires |journal= (help)

[mw9Q] Reprinted in H. de Senarmont, E. Verdet, and L. Fresnel (eds.), Oeuvres complètes d'Augustin Fresnel, vol. 1 (1866), pp. 713–18

[mw_Q] Translated as Fresnel, Augustin-Jean (2021). "Note on the double refraction of compressed glass". Translated by Putland, Gavin Richard. doi:10.5281/zenodo.4706835.{{cite journal}}: Cite journal requires |journal= (help)

[4] Leven, M.M.; Frocht, M.M., eds. (1969). "Vita Max Mark Frocht". Photoelasticity. Pergamon. pp. xi–xii. doi:10.1016/B978-0-08-012998-3.50005-7. ISBN 978-0-08-012998-3.

[5] Pockels, F. (1890). "Ueber die durch einseitigen Druck hervorgerufene Doppelbrechung regulärer Krystalle, speciell von Steinsalz und Sylvin". Annalen der Physik und Chemie. 275 (3): 440–69. Bibcode:1890AnP...275..440P. doi:10.1002/andp.18902750313.

[6] Nelson, D. F.; Lax, M. (1970). "New Symmetry for Acousto-Optic Scattering". Physical Review Letters. 24 (8): 379–80. Bibcode:1970PhRvL..24..379N. doi:10.1103/PhysRevLett.24.379.

[7] Frocht, M.M., Photoelasticity. J. Wiley and Sons, London, 1965

[8] Ajovalasit, A.; Petrucci, G.; Scafidi, M. (2012). "RGB photoelasticity applied to the analysis of membrane residual stress in glass". Measurement Science and Technology. 23 (2): 025601. Bibcode:2012MeScT..23b5601A. doi:10.1088/0957-0233/23/2/025601. hdl: 10447/61842 . S2CID 53600215.

[9] Kramer, Sharlotte; Beiermann, Brett; Davis, Douglas; Sottos, Nancy; White, Scott; Moore, Jeffrey (2013). "Characterization of Mechanochemically Active Polymers Using Combined Photoelasticity and Fluorescence Measurements". Application of Imaging Techniques to Mechanics of Materials and Structures, Volume 4. Conference Proceedings of the Society for Experimental Mechanics Series. pp. 167–78. doi:10.1007/978-1-4419-9796-8_21. ISBN 978-1-4419-9528-5.

[10] Fernandes, Cláudio P.; Glantz, Per-Olof J.; Svensson, Stig A.; Bergmark, Anders (2003). "Reflection photoelasticity: A new method for studies of clinical mechanics in prosthetic dentistry". Dental Materials. 19 (2): 106–17. doi:10.1016/s0109-5641(02)00019-2. PMID 12543116.

[11] Bigoni, Davide; Noselli, Giovanni (2010). "Localized stress percolation through dry masonry walls. Part I – Experiments". European Journal of Mechanics - A/Solids. 29 (3): 291–98. Bibcode:2010EuJMA..29..291B. doi:10.1016/j.euromechsol.2009.10.009.

[12] Bigoni, Davide; Noselli, Giovanni (2010). "Localized stress percolation through dry masonry walls. Part II – Modelling". European Journal of Mechanics – A/Solids. 29 (3): 299–307. Bibcode:2010EuJMA..29..299B. doi:10.1016/j.euromechsol.2009.10.013.

[13] Bigoni, D. (2012). Nonlinear Solid Mechanics: Bifurcation Theory and Material Instability. Cambridge University Press. ISBN 9781107025417.

[14] Noselli, G.; Dal Corso, F.; Bigoni, D. (2010). "The stress intensity near a stiffener disclosed by photoelasticity". International Journal of Fracture. 166 (1–2): 91–103. doi:10.1007/s10704-010-9502-9. S2CID 56221414.

[15] Shukla, A. (2001). "High-speed fracture studies on bimaterial interfaces using photoelasticity—a review". The Journal of Strain Analysis for Engineering Design. 36 (2): 119–42. doi:10.1243/0309324011512658. S2CID 137504535.

[16] Ayatollahi, M.R.; Mirsayar, M.M.; Dehghany, M. (2011). "Experimental determination of stress field parameters in bi-material notches using photoelasticity". Materials & Design. 32 (10): 4901–08. doi:10.1016/j.matdes.2011.06.002.

[17] Lewis, Michael (3 September 2024). "The Canary: Michael Lewis on Chris Mark of the Department of Labor". The Washington Post .

[18] J.F. Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices, Oxford University Press, 1957. ^{[ ISBN missing ]}^{[ page needed ]}

[19] R.E. Newnham, Properties of Materials: Anisotropy, Symmetry, Structure, Oxford University Press, 2005. ^{[ ISBN missing ]}^{[ page needed ]}

[20] Dally, J.W. and Riley, W.F., Experimental Stress Analysis, 3rd ed., McGraw-Hill Inc., 1991 ^{[ ISBN missing ]}^{[ page needed ]}

[21] Ramesh, K., Digital Photoelasticity, Springer, 2000 ^{[ ISBN missing ]}^{[ page needed ]}

[22] Solaguren-Beascoa Fernández, M.; Alegre Calderón, J.M.; Bravo Díez, P.M.; Cuesta Segura, I.I. (2010). "Stress-separation techniques in photoelasticity: A review". The Journal of Strain Analysis for Engineering Design. 45: 1–17. doi:10.1243/03093247JSA583. S2CID 208518298.

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[4]

[5]

[6]

[7]

[8]

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[13]

[14]

[15]

[16]

[17]

[18]

[19]

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[21]

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