A. Zaborski, Photoelasticity – basic ideas
PHOTOELASTICITY
Photoelasticity is a group of optical methods that serve to determine experimentally the state of strains. We will
talk here about the basic method only, which uses an observation of light passing through a transparent material
having property of strain-induced birefringence.
Photoelasticity is commonly used to examine models of plane structures. In order to relate experimental results
to reality, the model has to fulfill some conditions of similarity. However, in many cases, like in plane state of
stress, the stress repartition does not depend on the material constants so the experimental results are valid for
any other anisotropic material.
Due to the above-mentioned facts, the photoelasticity has great teaching advantages, visualizing the stresses in
color images and providing easy verification of assumptions and theoretical results. Therefore, despite of great
progress of numerical calculations in mechanics, photoelasticity remains essential part of laboratory course on
the strength of materials.
Light polarization, strain-induced birefringence
Light is a transverse electromagnetic wave, unambiguously determined by the intensity vectors of electric and
magnetic fields. Because the vectors mutually coupled, in practice an analysis of one of them is sufficient. We
will call it a “vector of light”.
The refractive index or index of refraction of a substance is a measure of the speed of light in that substance. It is
expressed as a ratio of the speed of light in vacuum relative to that in the considered medium. The velocity at
which light travels in vacuum is a physical constant, and the fastest speed at which energy or information can be
transferred. However, light travels slower through any given material, or medium, that is not vacuum.
As light crosses from one medium into another, the same wave will travel at different speeds. This process can
be understood as follows. A light wave enters a transparent material and excites an electron of an atom that
makes up that material. The excited electron now emits a light wave of its own, which in turn excites another
electron of another atom. Between atoms, light is traveling at light speed (c = 3×10 8 m/s). However, the time it
takes to excite and emit contributes to the average time it takes for a light wave to "travel" through the medium.
In glass, the apparent light speed is about 2×10 8 m/s, so 2/3 of c. Thus, the speed of light is different in different
materials.
Fig. 1. Polarization: linear, circular and elliptical
A. Zaborski, Photoelasticity – basic ideas
An “ordinary” light is an electromagnetic wave that passes through transparent material in some direction the
material’s atoms are excited in some direction, perpendicular to the light direction. Because the time interval of
the excitation is about 1e-8 second and each atom is excited in some direction we observe only an average of
huge number of the flashes. We observe non-polarized light. Now, let us consider a material composed from
fibers that can be excited in one direction only. The polarization of light is the orientation of the wave's electric
field at a point in space over one period of the oscillation. When light travels in free space, in most cases it
propagates as a transverse wave — the polarization is perpendicular to the wave's direction of travel. The electric
field may be oriented in a single direction (linear polarization), or it may rotate as the wave travels (circular or
elliptical polarization). In the latter cases, the oscillations can rotate either towards the right or towards the left
in the direction of travel.
There are some special optical filters , called polarizers, that cause linear polarization of the light. The light,
after passing such material will be linearly polarized, it means the vector of the light will oscillate in one
direction.
There are media that have some interesting property of optical anisotropy, called birefringence. These materials
are optically anisotropic because they have the refractive index depending on direction of the light polarization.
In such materials, there is one “fast” axis and one “slow” axis. The light, passing through the material, divides
into two rays; one of them passes quickly through the medium and another slowly. In effect, two rays are phase
shifted. The shift we may observe in particular conditions.
Fig.2. The ray of light passing through medium with birefringence
The times of optical transition are different, depending on material thickness and speed:
t1 = d / v1, t2 = d / v2,
the difference in time is:
Δt = t1 − t2 = d (1 / v1 − 1 / v2).
The shift of ray’s components is proportional to the light speed and – consequently – to the difference of
refractive index, n:
δ = v0 Δt = d (v0 / v1 − v0 / v2) = (n1 − n2) d.
According to Maxwell’s theory, the refractive indexes are linear combination of principal stresses:
n1 = n + C1σ1 + C2σ2,
n2 = n + C2σ1 + C1σ2,
and their difference is:
n1 − n2 = (C1 - C2) (σ1 − σ2) = C (σ1 − σ2).
A. Zaborski, Photoelasticity – basic ideas
The shift of ray’s components is proportional to the difference of principal stresses:
δ = Cd (σ1 - σ2).
The birefringence described above is strain-induced birefringence.
The optical bench
The optical bench is an experimental stand with elements arranged in particular order; cf. fig.2. The
monochrome light source, polarizer and/or ¼ wave plate, specimen, analyzer and/or ¼ wave plate and screen or
human eye.
Fig.3. The optical bench
Decomposition of light vector in linear polarizer
The light passing through polarizer possesses linear polarization. When it falls on analyzer the light extinguishes.
The situation does not change when we add a specimen not loaded. When we load the model, we observe some
light effects on screen.
Fig. 4. Decomposition of light ray in linear polarizer
Assume the light vector vibrate like a simple (harmonic) oscillator
Ap = sinωt.
A. Zaborski, Photoelasticity – basic ideas
In principal directions, its components may be written as:
A1 = sinα sinωt,
A2 = cosα sinωt,
and, after the shift caused by the pass through the model:
A1 = sinα sinωt, A2 = cosα sin(ωt − φ),
Horizontal component behind the analyzer is:
Aa = A1 cosα − A2 sinα = sin2α sin0.5φ cos(ωt - 0.5φ).
So, some part of light will be extinguished if sin2α = 0 or sin0.5φ = 0.
In the first case, the directions of principal stresses are parallel to (mutually perpendicular) directions of
polarization of polarizer and analyzer. The rays in such points extinguish and dark bands remain on the screen.
They are called isoclinics.
Isoclinics are the locus of the points in the specimen along which the principal stresses are in the same direction
(as polarizer and analyzer crossed).
The angle between one principal direction and the reference system is the isoclinic’s parameter. The image of
isoclinics depends on polarizer/analyzer setting.
The second case arises when the phase shift is multiplicity of vibration period:
0.5φ = mπ, and φ = m (2π).
then, the shift is multiplicity of the wave length: δ = mλ. Comparing two formulae, we get the equation of
isochromatics:
σ1 − σ2 = km,
where k is elastooptic constant and m is isochromatics order.
Isochromatics are the locus 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.
Circular polariscope
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 polariser and the specimen and the second
quarter-wave plate is placed between the specimen and the analyser. The effect of adding the quarter-wave plates
is that we get circularly polarised light.
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.
Full-field analysis of image
Determination of nominal stress values, stress gradients and full distribution of stress in elastooptic model is the
aim of full-field analysis of image. The analysis may be carried-out based on good knowledge of properties of
the image arising on the screen. There are some principal rules.
Isoclinics
A. Zaborski, Photoelasticity – basic ideas
Isoclinics are observed in the light with linear polarization. They determine the places where the principal
directions of stress are constant and equal to isoclinics parameter. Isoclinics are black bands, lines, spots and
points.
To determine isoclinics properly it is important to distinguish them from black isochromatics of zero order.
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isoclinics change with the polarizer/analyzer revolution,
during loading (in the case of simple loading process of proportional loading growth) isoclinics don’t change
its position,
directions of principal stresses are the same as directions of optical axis of polarizer/analyzer,
directions of principal stresses in any point may be determined by polarizer/analyzer revolution so that the
isoclinics reach that point,
at the isotropic points, where both principal stresses are equal and the isochromatics order is zero, all
isoclinics pass through these points,
the drawing of isoclinic should correspond with the darkest part of isoclinics,
on the basis of the isoclinics drawings the trajectories of principal stresses can be constructed.
Fig. 5. Isoclinics of parameter α
Isochromatics
The set of isochromatics on the screen of circular polariscope is like geographical contour lines because they
have very similar properties. The sea level is like isochromatics of zero order and hills as isochromatics of higher
orders. .
Each isochromatics order has its own characteristic color that is complementary to the color extinguished,
namely: order 0th – black, order 1st – purple, order 2nd – purple light, order 3rd – red/green, order 4th – pink/green
In particular, there are some properties of isochromatics:
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during loading the isochromatics appear first in the points with the highest stresses, the most often at
boundary of the model,
with loading the higher order isochromatics appear, and the former translate to the less stressed regions,
the isochromatics do not cross nor connect with others,
they do not change with the polarizer/analyzer revolutions,
they appear in the same order as contour lines: the isochromatics of n-th order is between isochromatics of
order n-1 and n+1,
the isochromatics of zero order is always black and is the region where every direction is principal,
higher density of isochromatics means greater stress gradient (stress concentrations),
like on the topographic map, the region with uniform color means constant level of stress,
in uniaxial state of stress , where one principal stress value is zero, the value of the second principal value
can be determined directly from isochromatics equation; this is true also for unloaded edge of model (as
results of static boundary conditions),
the sign of stress at the unloaded edge can be determined by simple loading with pencil: if isochromatics
pass to the side of higher isochromatics the edge is compressed,
A. Zaborski, Photoelasticity – basic ideas
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fractional order of isochromatics are determined by compensation methods: Tardy’s (goniometric) method
and with compensators.
Fig. 6. Isochromatics lines
Method of goniometric compensation (Tardy)
For the light of circular polarization the angle of revolution of analyzer shifts the isochromatics into position of
fractional order by α /180 .
Vishay educational polariscope
Fig. 7. Vishay polariscope system
Optical bench
Fig. 8. Optical bench’s system
A. Zaborski, Photoelasticity – basic ideas
Elastooptic models
Fig 9. Object models
Shear bending – stress trajectories
Fig. 10. The isoclinics image in sheared and bent beam
Pure bending – linear distribution of stress
Fig. 11. Equidistant isochromatics in simply bent beam
Curved beam – hyperbolic stress distribution
Fig. 12. Isochromatics image in curved beam and resulting stress distribution