Basic Theory of Photoelasticity
The term “Photoelasticity“ [5]*) describes methods of investigation and numerical evaluation of some mechanical quantities as strain, stress, mode of deformation, velocity distribution, etc..., in solid and fluid bodies by analyzing the change of state an
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Chapter 1.
Basic Theory of Photoelasticity. J. I. Introduction. The term ~IPhotoelasticity"
I511f)
des-
cribes methods of investigation and numerical evaluation of some mechanical quantities as strain, stress, mode of deformation, velocity distribution, etc ••• , in solid and fluid bodies by analyzing the change of state and velocity of electromagnetic radiation interacting with these bodies. The major
met~ods
of Photo-
elasticity are based on the analysis of induced and orientational birefringence by measuring the change of polarized radiant power transmitted or reflected from the body under observation. The detector and carrier of information in photoelastic systems is the photon radiation. The electromagnetic theory of radiation, that can be considered ,as a particular case of the more general quantum theory, seems to be adequate for formulation of basic and derived theories on which various photoelas-
*) [ .•. ] References.
V. Brčić, Photoelasticity in Theory and Practice © Springer-Verlag Wien 1970
6
Chap. 1. Basic Theory of Photoelasticity
ticity techniques are founded. That theory is based on the concept of an electromagnetic field, represented by two vectors, the electric vector and the magnetic vector. The interaction between that field and matter is described by a set of vectors : the elec tric current density, the electric displacement and the magnetic induction, and by a set of material constants : the specific conductivity, the dielectric cons tant, and the magnetic permeability. Since the majority of materials used in photoelasticity is macroscopically homogeneous, practically non-conducting, and isotropic magnetically, in the basic relations of photoelasticity the magnetic permeability is represented by a scalar, and the electric
anisotropy is chosen as the most satisfactory
optical parameter that alter with strain and stress.
I. 2. Fundamental Equations. [4] The elastic, isotropic and transparent bodies, which are used in photoelastic investigation and which are submitted to a general three-dimensional state of stresses, behave as birefringent crystals of rhombohedral system, thus, they are submitted to similar mathematical representation. The crystal optics and the theory of electromagnetism are the theoretical
Fundamental Equations
7
background of photoelasticity. The principles of conservation of charge and magnetic flux lead to the following field equations [8J which have the form: d~'\)'
I
at
8Q
=
0
~
aB at
=
0
d~v-B
=
0
+
tV
cu.rt E + tV
N
(1. 1)
where ~
is the density of magnetic flux, or the magnetic induction,
E
'"
I '"
Q
is the electric field (or the electric
vec~
tor), is the current density, is the charge density. These field equations are related to
the existence of electromagnetic potentials such that
B
N
=
cu.d A
8A
E = rv
-~
Q =
d~'l)-
rv
=
'"
- gradV
at
D
ru
cu.rt ruH =
aD at
~
(1. 2)
Chap. 1. Basic Theory of Photo elasticity
8 where
A is the magnetic potential,
N
V is the electric potential,
H is the current
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