Experimental Mechanics

, Volume 6, Issue 1, pp 13–22

# Optical phenomena in photoelastic models by the rotation of principal axes

Investigation shows that, using the matrix representation of the solution of the equations of photoelasticity, an adequate description of the complicated optical phenomena in three-dimensional photoelastic models can be obtained
• Hillar K. Aben
Article

DOI: 10.1007/BF02327109

Aben, H.K. Experimental Mechanics (1966) 6: 13. doi:10.1007/BF02327109

## Abstract

The basic equations of three-dimensional photoelasticity are derived in a form which is simpler than that of equations known previously. Using the matrix representation of the solution of these equations, it is also shown that when rotation of principal axes is present there always exist two perpendicular directions of polarizer by which the light emerging from the model is linearly polarized. These polarization directions of the incident and emergent light are named primary and secondary characteristic directions, respectively. The experimental determination of characteristic directions, as well as of the phase retardation, gives three equations on every light path to determine the stress components in a three-dimensional model. A general algorithm of the method of characteristic directions is presented, and its application by determination of stress in shells by normal and tangential incidence is described. A further extension of the method to the general axisymmetric problem has been suggested.

### List of Symbols

A1,A2

components of electric vector of light after transformation [eq (4)]

a1,…,ak,b1,…,be,c1,…,cm,d1,…,dk

constants which determine the distribution of stress in an axisymmetric model

B1,B2

components of electric vector of light after transformation [eq 7]

B10,B20

components of the incident-light vector on arbitrary coordinate axes

B10o,B20o

components of the incident-light vector in primary characteristic directions

B1o,B2o

components of the emergent-light vector in secondary characteristic directions

c

velocity of light in vacuum

C

$$\frac{1}{{2k}}\frac{{\omega ^2 }}{{c^2 }}$$

C0,C1

photoelastic constants

C

CC0

D1,D2

components of electric-induction vector of light

E1,E2

components of electric vector of light

f1,…,f4

functions which determine the distribution of stress in an axisymmetric model

G(γ)

diagnonal unitary matrix [eq (20)]

k

ωN/c

N

index of refraction of the non-stressed medium

R

$$\frac{{2\varphi _0 }}{\Delta }$$

r

outer radius of a cylindrical shell; radial coordinate in an axisymmetric model

S

$$\sqrt {1 + R^2 }$$

Sj)

matrix of rotation [eq (19)]

t

thickness of the model

U

unitary matrix [eq (17)]

U1

unitary matrix which transforms the incident-light vector into the plane of symmetry

U1*

transpose ofU1

u′, v′

primary characteristic directions

u″, v″

secondary characteristic directions

x1,x2

principal directions at the point of entrance of light

x1,x2

principal directions at the point of emergence of light

x1,x2,z

rectangular coordinates

z1

z2/rt

α

angle between conjugate characteristic directions

α1, α2

angles which determine the primary and secondary characteristic directions

β

$$\frac{{3(1 - \mu ^2 )}}{{r^2 t^2 }}$$

characteristic phase retardation which corresponds to the matrixU

2γ1

characteristic phase retardation which corresponds to the matrixU1

Δ

C't1 − σ2)

Δ*

phase retardation determined by [eq (37)]

σij

Kronecker's tensor

ij

tensor of dielectric constant

j

principal components of the tensor of dielectric constant perpendicular to the wave normal

μ

Poisson's ratio

ξ, ζ, θ

parameters of the matrixU

σij

stress tensor

σ1, σ2

principal stresses perpendicular to the wave normal

11 − σ22)n, (σ12)n

membrane stress components in a shell

11 − σ22)b, (σ12)b

maximum bending stresses in a shell

σn

½C′(σ11 − σ22)n

σb

½C′(σ11 − σ22)b

σ2b

longitudinal bending stress in a cylindrical shell

σϑb

circumferential bending stress in a cylindrical shell

σϑn

circumferential membrane stress in a cylindrical shell

σ2n

longitudinal membrane stress in a cylindrical shell

σ2b°

max σ2b

σ

$${\raise0.7ex\hbox{1} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{2}}C'\left( {\frac{t}{r}\sigma \vartheta _n - \sigma _{2b} ^\circ } \right)$$

σr, σϑ, σ22, τ2r

stress components in an axisymmetric model

τn

C′(σ12)n

τ2r

shearing stress in a cylindrical shell

τ1r°

max τ1r

τ

C′τ2r°

ϕ

angle of rotation of principal axes

ϑ0

total angle of rotation of principal axes

ψ

1/2SΔ

ω

frequency of vibration of light