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Magnetophotoelasticity—Photoelasticity in a magnetic field

Investigation shows that, if a magnetic field is applied to a photoelastic model, it enables one to investigate states of stress that do not cause any optical effect in an ordinary transmission polariscope

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Abstract

The basic equations governing the propagation of polarized light in a three-dimensional photoelastic model located in a magnetic field are derived. Optical phenomena in this case can be adequately described by the aid of the theory of characteristic directions developed previously by the author. The case when the principal-stress difference, as well as the magnetic field, is constant is considered in detail. An algorithm is developed that permits the study of the optical phenomena in the case of arbitrary stress distribution along the wave normal. As an example, investigation of the bending of plates is considered; graphs are produced that permit the determination of the stress components on the basis of experimental data. Some considerations of the experimental technique are given.

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Abbreviations

\(\bar B'\) :

emergent-light vector

\(\bar B_0 ^\prime \) :

incident-light vector

B 1,B 2 :

components of electric vector of light in arbitrary coordinate axes after transformation [eq (12)]

B 10,B 20 :

initial values ofB 1 andB 2

B′1,B′2 :

components of electric vector of light in principal directions [eq (21)]

B′10,B′20 :

initial values ofB′1 andB′2

C :

\(\frac{\omega }{{2c\sqrt \in }}\)

C 0,C 1 :

photoelastic constants

C′:

CC 0

c :

velocity of light in vacuum

D 1,D 2 :

components of electric-induction vector of light

E 1,E 2 :

components of electric vector of light

g :

\(\frac{1}{2} \int {C'} (\sigma _1 - \sigma _2 )dz\)

\(\bar H_0\) :

magnetic vector

k :

\(\frac{\omega }{c}\sqrt \in\)

M :

matrix of 2m layers of a gyrotropic photoelastic medium

M j :

matrix of a single layer of a gyrotropic photoelastic medium

p j (m),q j (m),r j (m),s j (m):

parameters of the matrix ofm layers

p,q,r,s :

parameters of the matrix of a gyrotropic photoelastic medium

R :

\(\frac{{2\psi _0 }}{\vartriangle }\)

R j :

\(\frac{{2\psi _j }}{{\vartriangle _j }}\)

S :

\(\sqrt {1 + R^2 }\)

S j :

\(\sqrt {1 + R_j ^2 }\)

t :

thickness of the model

U :

unitary two-by-two matrix

\(\bar u\) :

displacement vector

z :

coordinate in the direction of the wave normal

α:

α* - α*

α*, α* :

angles which determine the primary and secondary characteristic direction

2γ:

characteristic phase retardation

Δ:

phase retardation

\(\vartriangle _*\) :

phase retardation determined by eq (37)

δ ij :

Kronecker tensor

ε:

dielectric constant of the nonstressed medium

ε ij :

tensor of dielectric constant

\( \in _{ij} ^\prime \) :

symmetric tensor of dielectric constant

\( \in _{ij} ^{\prime \prime }\) :

antisymmetric tensor of dielectric constant

η:

1/2SΔ

λ:

wavelength

σ:

\(C'\frac{\pi }{{8\lambda }}(\sigma _1 - \sigma _2 )_0 t\)

σ 1 , σ 1 :

principal stresses

σ ij :

stress tensor

(σ1 - σ2)0 :

principal stress difference on the surface of the bent plate

φ:

angle of rotation of principal axes

ψ:

ψ0 z

ψ0 :

Faraday rotation on a unit length

ψ1,ψ2,ψ3 :

Eulerian angles

ω:

frequency of vibration of light

ω1,ω2,ω3 :

projections of the vector of angular velocity on the steady axes

References

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Authors and Affiliations

  1. Institute of Cybernetics, Tallinn, Estonian SSR, USSR

    Hillar K. Aben (Assistant Director)

Authors
  1. Hillar K. Aben
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Aben, H.K. Magnetophotoelasticity—Photoelasticity in a magnetic field. Experimental Mechanics 10, 97–105 (1970). https://doi.org/10.1007/BF02325113

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