Editing Photoelasticity (section)

==Formal definition==
For a linear [[dielectric|dielectric material]] the change in the inverse permittivity tensor <math>\Delta(\varepsilon^{-1})_{ij}</math> with respect to the deformation (the gradient of the displacement <math>\partial_\ell u_k</math>)  is described by <ref>J.F. Nye, ''Physical Properties of Crystals: Their Representation by Tensors and Matrices'', Oxford University Press, 1957. {{ISBN?}} {{page?|date=September 2024}}</ref>

:<math> \Delta(\varepsilon^{-1})_{ij} = P_{ijk\ell} \partial_k u_\ell </math>

where <math>P_{ijk\ell}</math> is the fourth-rank photoelasticity tensor, <math>u_\ell</math> is the linear displacement from equilibrium, and <math>\partial_l</math> denotes differentiation with respect to the Cartesian coordinate <math>x_l</math>. For isotropic materials, this definition simplifies to <ref>R.E. Newnham, ''Properties of Materials: Anisotropy, Symmetry, Structure'', Oxford University Press, 2005. {{ISBN?}} {{page?|date=September 2024}}</ref>

:<math> \Delta(\varepsilon^{-1})_{ij} = p_{ijk\ell} s_{k\ell} </math>

where <math>p_{ijk\ell}</math> is the symmetric part of the photoelastic tensor (the photoelastic strain tensor), and <math>s_{k\ell}</math> is the [[Infinitesimal strain theory|linear strain]]. The antisymmetric part of <math>P_{ijk\ell}</math> 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 [[Hooke's law|mechanical stress]].