THE DEVELOPMENT AND APPLICATION OF MAGNETOPHOTOELASTICITY
A. R. Conway1, R. A. Tomlinson1 and G. W. Jewell2
1
Department of Mechanical Engineering
2
Department of Electronic and Electrical Engineering
University of Sheffield, Sheffield, UK
ABSTRACT
A pulsed magnetopolariscope has been developed to measure the internal stress distribution of components that exhibit
temporary birefringence. The instrument utilises pulsed magnetic fields. The benefits of this are two-fold; a high magnetic
field increases the Faraday rotation through a given model and also allows intensity data to be collected at a large number of
discrete magnetic field strengths simultaneously. The present work shows how the equivalence theorem, expressed in Mueller
calculus, can be used to solve for an unknown stress distribution by using data collected at different field strength values.
Results are presented that demonstrate how the technique has been applied to measure the distribution of residual stresses in
toughened glass.
Introduction
The added complexity which differentiates all forms of three dimensional photoelasticity from the two dimensional method is
the rotation of the light vector through the thickness of the birefringent material. This can be a consequence of the principal
stress axes rotating through the material, or, as in the case of magnetophotoelasticity, because of an additional induced
rotation. It follows that for any given layer in a three dimensional material, the state of polarisation is dependent upon the
polarising properties of all the preceding layers of the material through which the light has passed. In this way, the polarising
properties of all the layers are coupled and are ultimately dependent upon the orientation of the incident polarised light wave.
The challenge that integrated photoelasticity presents is an inverse one, that is, to use the observable emergent light to
determine the stress distribution along the light path which caused such a light pattern to emerge.
Magnetophotoelasticity was conceived as a viable method of solving the problem of integrated three-dimensional
photoelasticity [1]. In particular, the technique allows for the solution of stress distributions that vary along the light path in a
manner which produces an integrated optical effect of zero. By placing a three dimensional birefringent material in a magnetic
field which is parallel to the direction of the incident polarised light, the light experiences a rotation in part due to the
birefringence of the model and in part due to an induced Faraday rotation. As a consequence, it becomes possible to
measure an optical effect and describe the stress distribution along the light path.
To date, application of the technique has relied upon determining the optical parameters measured in integrated
photoelasticity in both the presence and absence of the magnetic field. These parameters, collectively known as the
characteristic parameters, together with an assumption regarding the form of the stress field, have enabled the stress
distribution to be described [1,2,3,4].
Perhaps the greatest drawback of the technique is the relatively small Faraday rotation that is achievable under a static
magnetic field. Therefore, previous work has dealt with increasing the Faraday rotation in order to reduce the unusually high
experimental accuracy which is otherwise needed to obtain the characteristic parameters. Methods have included the use of
modeling materials with a high Verdet constant [1], the use of optically active modeling materials in addition to the use of a
magnetic field [5], the use of multiple reflections of light through the model [6] and the use of pulsed magnetic fields [3,4].
In the present work the latter technique was adopted to produce a sufficiently high magnetic field to measure the characteristic
parameters over a circular, 80 mm diameter area. In order to measure the characteristic parameters which define the
integrated optical pattern, light intensity images were collected with a high speed video camera triggered by the pulse of the
magnetic field. These images were then processed using the Fourier polarimetry technique, originally developed for integrated
photoelasticity by Berezhna et al [7] and later adapted for magnetophotoelasticity by Gibson [8].
This method of data collection lends itself well to a ‘multiple field’ solution of the inverse problem which removes the need for
an assumption regarding the form of the stress distribution. This widens the applicability of magnetophotoelasticity to
completely arbitrary stress distributions. Despite having been proposed in earlier work, this solution method has not yet been
used in practice [9]. Results will also be presented that show how the solution technique has been applied to experimental
data.
Theory
Consider a flat birefringent plate orientated normally to a monochromatic incident light source of wavelength, λ, in the presence
of a magnetic field, parallel to the direction of light, of field strength Ho. The plate is orientated in an x y z Cartesian coordinate
system so that the face of the plate is parallel to the x y plane and the origin exists on the incident surface of the plate. This
arrangement is perhaps best described by Figure 1.
Figure 1. Orientation of the birefringent plate
By introducing a magnetic field the plane of polarisation rotates as it passes through the plate. This Faraday rotation is defined
as follows:
ψ = VH ot
where
(1)
V = Verdet constant - an optical material constant dependent upon wavelength
Ho = Strength of the applied magnetic field
t = Thickness of the plate
For such an arrangement the components of the electric light vector, B, were derived by Aben as follows [1]:
dB '1
1
d (ϕ − ψ )
B' 2
= − iC ' (σ 1 − σ 2 ) B '1 +
dz
2
dz
dB ' 2
d (ϕ − ψ )
1
=−
B '1 + iC ' (σ 1 − σ 2 ) B ' 2
dz
dz
2
(2)
where, B1 and B2 represent components of the light vector at the orientation of the principal stresses at a given point along the
z axis, σ1 and σ2 are the first and second principal stresses, φ is the rotation of the principal stress axes from the origin to the
point of interest, ψ is the induced Faraday rotation from the origin to the point of interest and i is the imaginary unit. C’ is a
lumped parameter which includes the stress optic coefficient C, a measure of the material’s stress induced birefringence, and
the wavelength of the incident light:
C' =
360C
λ
(3)
Equation (2) can be shown to be analogous to the conventional two dimensional photoelasticity equation by removing the
‘magnetic’ term i.e. dψ/dz = 0 and by removing the rotation of the principal stresses through the third dimension, i.e. dφ/dz = 0, a
prerequisite for two dimensional photoelasticity. The solution for this simplified set of differential equations is as follows:
B '1 = B '01 e
1
− iC '(σ 1 −σ 2 ) z
2
B ' 2 = B '02 e
(4)
1
iC '(σ 1 −σ 2 ) z
2
In Equations (4), B’01 and B’02 are the components of the incident light vector in the principal stress directions. The phase
difference between these two components, effectively the optical retardation over the entire plate, can be expressed as:
t
∆* = C ' ∫ (σ 1 − σ 2 )dz
(5)
0
where
∆* = Retardation along the entire light path
Equation (5) is commonly referred to as the generalised Wertheim Law. A consequence of this law is that if the integral of the
principal stress difference over the entire plate is zero, then there is no retardation of the light and no optical effect can be
recorded. However, if the integral is non zero an optical effect is produced. For a three dimensional stress distribution with a
zero or near zero principal stress integral, e.g. the parabolic form of stress in toughened glass or the antisymmetric form of
stress in a plate in bending, then the magnetophotoelastic effect can be used to determine the characteristic parameters.
The solution proposed by Aben was to take the general magnetophotoelasticity equations and to simplify them by considering
the case of constant principal stress difference as well as fixed principal axes through a given thickness of the plate i.e. σ1-σ2 is
constant, dφ/dz = 0. A further assumption was that the magnetic field was constant over the thickness i.e. dψ/dz is constant. It
follows that if the entire plate is considered to be made up of constant stress discrete layers then by cumulatively multiplying
the layers together a final form of the set of equations can be reached with the relative contribution of each layer included.
It is not possible to solve such a set of equations for the stresses in each layer of the model without some additional
information. In previous work this has taken the form of an assumption regarding the stress distribution in the material.
Having made such an assumption a solution can still only be achieved indirectly. Aben’s approach was to use a nomogram, a
two dimensional map relating the principal stress difference to the total Faraday rotation, ψt and the characteristic quantities, α
and ∆*. This approach was later automated by Clarke [2] and in recent years Gibson developed an automatic search routine
based upon a genetic algorithm to reach a solution [3].
The main limitations of this method are that an assumption regarding the form of the stress distribution must be made and that
the solution is only applicable to stress distributions where the principal axes do not rotate. For the type of stress distributions
for which the technique was originally conceived this may be acceptable. However, in order to make the technique applicable
to an arbitrary stress distribution two things need to be altered. Firstly, a solution method that allows for rotation of the
principal stress axes through the entire medium needs to be used. Secondly, in order to do this, more information regarding
the integrated stress distribution needs to be obtained.
Mueller Calculus
Using Mueller calculus a two dimensional birefringent plate that imposes a retardation, δ, at a given orientation, θ, can be
described as:
0
1
0 cos 2 2θ + sin 2 2θ cos δ
M θ (δ ) = 
0 (1 − cos δ ) sin 2θ cos 2θ

sin 2θ sin δ
0
0
(1 − cos δ ) sin 2θ cos 2θ
sin 2 2θ + cos 2 2θ cos δ
− cos 2θ sin δ
0

− sin 2θ sin δ 
cos 2θ sin δ 

cos δ

(6)
A three dimensional birefringent medium can be represented by a combination of such matrices. Each matrix therefore
represents a layer in the three dimensional medium through which the principal axes do not rotate and the principal stresses
do not vary. If a sufficiently large number of layers are used to model a medium then by multiplying the matrices for all the
layers together, a final matrix that describes the optical properties of the entire medium can be calculated:
M θ (δ )total = M θ 1 (δ1 ) M θ 2 (δ 2 )...M n (δ n )
(7)
To take this representation further, a three dimensional medium in a magnetic field can be represented by a chain of such
matrices, alternating with rotator matrices that impart the Faraday rotation for that particular layer. A rotator matrix for one
layer, which induces a rotation of ψ, is given by Equation (8) and the full multiplication process is given by Equation (9).
0
1
0 cos 2ψ
ψi = 
0 sin 2ψ

0
0
0
− sin 2ψ
cos 2ψ
0
0
0
0

1 i
(8)
M TF = M θ 1 (δ )1ψ 1 M θ 2 (δ ) 2ψ 2 ...M θ n (δ ) nψ n
(9)
Poincaré’s Equivalence Theorem
In addition to the layered model, a three dimensional birefringent medium can also be optically defined by the characteristic
parameters. In optical terms, polarised light entering a medium is phase shifted at a particular orientation, which is determined
by the orientation of the principal axes within the incident layer of the medium. As light continues to pass through the medium
this orientation changes due to the changing orientation of the principal stresses. Therefore, the medium can be described as
a birefringent plate in combination with a rotator and as such can be represented in Mueller calculus as follows:
0
1
0 cos 2α
Rα M ϕ ( ∆*) = 
0 sin 2α

0
0
0
− sin 2α
cos 2α
0
0  1
0
0
0



2
2
0  0 cos 2ϕ + sin 2ϕ cos ∆ * (1 − cos ∆*) sin 2ϕ cos 2ϕ − sin 2ϕ sin ∆ * (10)
×
0  0 (1 − cos ∆*) sin 2ϕ cos 2ϕ sin 2 2ϕ + cos 2 2ϕ cos ∆ * cos 2ϕ sin ∆ * 
 

1  0
sin 2ϕ sin ∆ *
− cos 2ϕ sin ∆ *
cos ∆ *

φ = Primary Characteristic Direction
α = Characteristic Rotation
∆* = Characteristic Retardation
where
It is the optical equivalence of the two representations, known as Poincaré’s equivalence theorem. Mathematically this can be
described by the following:
Rα M ϕ (∆*) = M θ 1 (δ )1ψ 1 M θ 2 (δ ) 2ψ 2 ...M θ n (δ ) nψ n
(11)
Equation (11) is a more general form of previously published solutions. By setting the rotation of the principal axes to zero, the
two equations are equivalent i.e. θ1, θ2,…,θn are constant. In the absence of such simplifying assumptions, a solution can only
be obtained by generating different sets of characteristic parameters at different magnetic field strengths. This was proposed
mathematically, in terms of the Jones calculus, by Theocaris [9]. In terms of Mueller calculus, the ‘multiple field’ technique can
be described mathematically as follows:
{R
{R
α
M ϕ (∆*)} = M θ 1 (δ )1 {ψ 1 } M θ 2 (δ ) 2 {ψ 2 } ...M θ n (δ ) n {ψ n }
α
M ϕ (∆*)} = M θ 1 (δ )1 {ψ 1 } M θ 2 (δ ) 2 {ψ 2 } ...M θ n (δ ) n {ψ n }
1
2
1
1
2
1
2
2
(12)
M
{R
α
M ϕ (∆*)} = M θ 1 (δ )1 {ψ 1 } M θ 2 (δ ) 2 {ψ 2 } ...M θ n (δ ) n {ψ n }
N
N
N
N
Once again, this system of equations can’t be solved directly and an indirect approach to the solution has not been proposed
previously. In the present work the set of nonlinear equations was solved using a least squares optimisation routine based
upon the Levenberg-Marquardt method [10, 11].
Simulation
An advantage of the Mueller calculus is that the representation lends itself very well to computational simulation. In fact, by
using known stress data for a birefringent material, the manner in which light would behave in the material can be predicted.
By optically defining the material, it is possible to determine the intensity of the emergent light at given orientations of the
polariser and analyser. The Fourier polarimetry technique developed by Yang et al [12] can then be applied to determine the
characteristic parameters. Finally, the solution method can be applied to the calculated characteristic parameters for a number
of magnetic field strengths. A comparison between the known input stress field and the calculated output field can then be
made. A flowchart showing this process is given in Figure 2 and the processes common to both simulated and experimental
data are highlighted.
Figure 2. A flowchart showing the order of data processing for both simulated and experimental data. The common processes
are surrounded by a dashed line.
In this way the accuracy of the solution can be quantified in the absence of experimental error. This is a key element of
successfully using a multiple field methodology, because although using multiple field strength values will enable a greater
number of layers to be solved for, a point will ultimately be reached where the simultaneous equations become badly
conditioned and accuracy in the solution is lost.
A single point measurement was considered. A general parabolic stress distribution was used to define the stress at discrete
intervals through the thickness of the model. Each discrete interval represented a constant two dimensional layer. The
Mueller matrix for each layer was determined from the expressions for retardation and principal stress orientation in two
dimensions:
δ i = Cσi (σ 1i − σ 2 i )t i
tan 2θ i =
where
2τ xy
i
(13)
σ xi − σ yi
i = Layer number,
δ = Retardation in the layer
θ = Orientation of the principal axes
t = Thickness of a layer.
σx, σy and τxy = x, y and shear stresses respectively.
Having established these values for each data point, a corresponding Mueller matrix was calculated. Equation (9) was then
used to calculate a final model matrix, MTF, for each field strength value used.
This optical model was then used to determine the emergent light intensity in a plane polariscope using the Mueller
representation of the polarising optics and the Stokes vector of unpolarised, incident light. The Stokes vector of unpolarised
light, Sin, is given by:
1
0 
S in =  
0 
 
0 
(14)
A linear polariser that subtends an angle β to the vertical axis, y, is represented by:
 1

1 cos 2 β
Pβ = 
2  sin 2 β

 0
cos 2 β
sin 2 β
2
cos 2 β
cos 2 β sin 2 β
cos 2 β sin 2 β
sin 2 2 β
0
0
0
0
0

0
(15)
The output Stokes vector for a plane polariscope in the presence of a magnetic field can be obtained from:
S out = Pβ M TF Pα S in
(16)
In Equation (16), the polariser, orientated at an angle, β, is represented by Pβ and the analyser orientated at an angle, α, is
represented by Pα. The output Stokes vector, Sout, is a 4 element column vector, the first element of which represents the
intensity of the emergent light.
x 10 1
0.12
4
0.1
2
0.08
Principal Axes Orientation (rad)
1st Principal Stress Difference (N/mm2)
6
0
-2
-4
-6
0.04
0.02
0
Input distribution
3 Layers
6 Layers
10 Layers
15 Layers
20 Layers
-8
-10
0.06
0
0.005
0.01
0.015
Distance Z direction (m)
0.02
-0.02
0.025
-0.04
0
0.005
0.01
0.015
0.02
0.025
Distance Z direction (m)
Figure 3. Simulation results for a single point parabolic stress distribution. The solid blue line indicates the input distribution
and the discrete data points indicate the output distribution for increasing numbers of layers.
The emergent light intensity was calculated at the orientations of the polariser and analyser dictated by the Fourier Polarimetry
technique. The characteristic parameters were then calculated over a range of Faraday rotations up to 12°. The characteristic
parameters were used to calculate the equivalent Mueller matrices from Equation (9). This information was passed to a
function which implemented the equivalence theorem. By minimising the output of the function the unknown retardation and
principal stress orientation could be determined for each layer of the model using the Levenberg-Marquardt algorithm.
The results are shown in Figure 3 for an evenly distributed range of Faraday rotations. For the three highest order solutions,
10, 15 and 20 layers, the data points are connected simply to aid visualisation. Figure 3 shows that as the number of layers in
the solution increases, the solution deviates from the input distribution. This is particularly clear when 20 field strength values
2
are used, where the maximum principal stress deviation is 54.2 N/mm and the maximum orientation deviation is 5.73°. This
deviation is a result of the large set of simultaneous equations becoming badly conditioned. However, for lower order solutions
the output correlates very well with the input distribution.
This process can be readily expanded to generate an optical model for a three dimensional medium. Rather than using
arbitrary single point stress data, stress data can be generated for any material, geometry and loading condition using the finite
element method. The stress data for each node of the finite element model can then be used to calculate the retardation and
principal stress orientation using Equation (13). A corresponding Mueller matrix for each node of the model can then be
generated. This data can be stored conveniently using a cellular array in a programming language such as Matlab®. The
Mueller matrices for each of the nodes can then be multiplied through the thickness, that is, the z direction, to complete the
optical model. This method has potential benefits in general photoelasticity because a predicted light field can be generated
directly from a finite element simulation for any combination and orientation of polarising or phase changing optics.
Experimental Apparatus
The pulsed magnetopolariscope consists of a plane polariscope arrangement with two polarising optics mounted on individual
rotation stages. A collimated, highly stable light source was required and for this purpose a 457nm wavelength Argon laser
was used. By using appropriate optical hardware the laser beam was filtered, expanded and collimated to provide an 80 mm
diameter beam. In order to apply the pulsed field, a 10 kJ magnetiser, capable of discharging this stored energy over a 6 ms
time period was connected to an 80 mm diameter continuous bore solenoid resulting in a peak field of 3.7 Tesla. Finally,
intensity data was captured using a high speed video camera capable of recording 12,000 frames per second (fps) at a 512 ×
384 pixel resolution with a 9.8µs exposure time. The apparatus is shown in schematically in Figure 4.
Figure 4. The Pulsed Magneto Polariscope (PMP) apparatus
Preliminary Test Results
A toughened float glass sample was used as a test specimen. The specimen’s x, y, and z dimensions were 20.2mm × 50.7mm
× 10.2mm respectively. As a result of the large residual stresses in the specimen, a high fringe order was visible around the
machined edges of the specimen. For the analysis only the area within half a fringe order, essentially the centre of the
specimen, was considered to avoid data wrapping.
The magnetic pulse characteristic and trigger signal were recorded with a digital oscilloscope. These data were then used to
determine the magnetic field strength for every frame of the recording. For an average pulse of 5.9ms, 88 frames were
recorded. Because the pulse characteristic was symmetrical, only the rising half was used in the solution, therefore 44 frames
were available on average. Potentially this allowed for a 44 layer solution to be processed.
To maximise the accuracy of the characteristic parameters, the Fourier Polarimetry technique [12] was applied using a sample
of 72 images. The characteristic parameters are shown over the surface of the specimen in Figure 5. A three layer solution
was generated using data below the wrapping period of the characteristic parameters. The processed area corresponded to a
295 × 78 pixel or a 24.73 × 6.54 mm area. Each layer of the solution represented an effective thickness of 3.4 mm. A stress
optic coefficient of 2.72 Brewsters was used to calculate the principal stress difference in each layer, using Equation (13). The
principal stress differences and principal stress orientations are shown in Figure 6. The average values for the principal stress
2
difference over the examined area were calculated to be -5.35, 32.93 and 21.92 N/mm for the first, second and third layers of
the solution respectively. The principal stresses are, on average, orientated at -25.82°, -3.20° and 16.80° to the vertical y axis.
Characteristic
Retardation
Characteristic
Rotation
Primary Characteristic
Direction
3
1
1
0
0
-1
-1
2
1
Figure 5. Characteristic parameters (radians)
Layer 1
Layer 2
Layer 3
Layer 1
Layer 2
Layer 3
60
3
40
2
20
1
0
0
-20
-1
-40
-2
-60
-3
1st Principal Stress Difference (N/mm2)
Principal Stress Orientation (rad)
Figure 6. Calculated stress difference in 3 layers of the specimen
Discussion of Results
The principal stress difference measurements shown in Figure 6 display a predictably high stress difference for the central
layer and lower stress difference values for the two outer layers. A quadratic fit to the average values is shown in Figure 7.
2
Using the fitted curve, the predicted stress difference values at the incident and emergent surfaces are -43 and -2 N/mm
respectively. The fact that both predicted stresses are compressive conforms to the expected shape of the residual stress
distribution in toughened glass. By taking the roots of the equation the transition between compressive and tensile stress can
be determined. For the incident surface this transition occurs at 1.99 mm from the surface and for the emergent surface this
transition occurs at 0.119 mm from the surface. If this latter result is correct, the specimen would fracture if the glass were to
be penetrated to this depth. This remains to be validated experimentally but sand blasting techniques employed to remove
very fine layers from the glass surface could be used to this effect. However, the concept of a multiple field solution is that an
assumption regarding the stress distribution is unnecessary and although curve fitting in this way provides an analogy to
previous work in the area, it contravenes this idea.
The orientations of the principal stresses show a clear rotation through the three layers. Again, this is difficult to verify
independently, but it does indicate that the stress orientation is certainly not constant through the thickness of the specimen
and therefore goes some way to justifying the more general theoretical approach used in this work.
1st Principal Stress Difference (N/mm2)
30
20
10
0
-10
-20
-30
-40
-50
0
1
2
3
4
5
6
7
8
9
10
Distance in z (mm)
Figure 7. The average principal stress difference data with a fitted parabola
To date, problems encountered in the experimental process have been primarily due to mechanical vibration. In the first
instance this was due to the pulsed magnetic field. Because of the high repellent forces generated during the discharge pulse,
vibration was initiated in the optical components. The spatial filter assembly was particularly vulnerable to this vibration,
because movements in the order of tenths of millimetres can cause the laser beam to become unfocused. Because the
mechanical vibration is initiated towards the end of magnetic pulse, this doesn’t in itself affect the results for a given pulse.
However, the repeatability of results is severely affected as the spatial filter assembly has to be realigned after each pulse
measurement. For this reason, the results presented here were collected without the use of a filtering pinhole. Although the
expanded light is not Gaussian in this arrangement, the Fourier polarimetry technique is not adversely affected by background
inhomogeneity, provided that the inhomogeneity is consistent from frame to frame. However, the fact that the optical light
pattern is visible in the results as variations in hue demonstrates that the light source did move laterally during the testing.
One limitation of the solution method is wrapping of the characteristic parameters due to the periodic nature of the equations
used in the Fourier Polarimetry method [12]. Conventionally, this is dealt with by detecting sudden shifts in the output data and
correcting the phase accordingly. However, in the case of three dimensional photoelasticity, wrapping can also occur along
the z axis. If, for example, the combined rotation of the principal axis and the induced Faraday rotation exceeds the wrapping
period, the solution will be compromised. Because the principal stress rotation is unknown, this represents a limit to how the
technique can be applied to stress fields where no a prior knowledge of the stress distribution exists.
Conclusion
A theoretical approach has been presented which has the potential to generalise the technique of magnetophotoelasticity and
ultimately provide more accurate measurement of three dimensional stress fields. This approach has been implemented on
both simulated and experimental data. A solution method has been shown to work well on single point simulated data.
A prototype pulsed magnetopolariscope has been developed. Fourier polarimetry has been applied to capture the
characteristic parameters successfully. Experimental data has been processed using the same methodology which was
applied to the simulated data and a three layer solution for a specimen of toughened glass has been derived.
Acknowledgements
The authors would like to extend their thanks to both the Pilkington Group Ltd. and the Engineering and Physical Sciences
Research Council for their financial backing, without whom, this research would not be possible.
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2.