Linear and non-linear algorithms of photoelastic tomography
Hillar Aben and Andrei Errapart
Institute of Cybernetics, Tallinn University of Technology, 21 Akadeemia tee, 12618 Tallinn, Estonia;
[email protected]
ABSTRACT
Tomography is a powerful method for the analysis of the internal structure of different objects, from human bodies to parts of
atomic reactors. Classical tomography considers only determination of scalar fields. Since stress is a tensor, photoelastic
tomography is much more complicated than the classical one. Photoelastic tomography is based on the relationships of
integrated photoelasticity. Due to the rotation of the principal stress axes, relationships of integrated photoelasticity between
the measurement data and parameters of the stress field are in general non-linear. These relationships can be linearized if
either birefringence is weak or rotation of the principal stress axes is small. In this paper two algorithms of photoelastic
tomography are described. The first one is based on the linearized version of the equations of integrated photoelasticity. In this
case the problem of tensor field tomography is decomposed to several problems of scalar field tomography for normal stress
components of the stress tensor. The second tomographic algorithm is base on non-linear equations of integrated
photoelasticity. In this case the differential evolution algorithm is used to find a stress field that corresponds best to the
experimental measurement data. Both algorithms are illustrated with practical examples.
1.
Introduction
Most of the methods of experimental stress analysis permit measurement of displacements, deformations and stresses only at
the surface of 3-D objects [1]. Exceptions are two photoelastic measurement technologies, the frozen-stress method and the
scattered light method [2]. Although stereolithography facilitates considerably the fabrication of models for the frozen-stress
method, this method is still very time-consuming. As for the scattered light method, in recent years its classical version has
been mostly used for residual stress measurement in glass plates [3,4]. A more sophisticated version of the scattered light
method, optical cutting [5,6] has find limited application in 3-D stress analysis. Certain attempts have been made also to widen
the method of optical correlation to 3-D stress fields [7,8]. However, an efficient non-destructive method for the analysis of 3-D
stress fields is still missing.
A powerful method for the analysis of the internal structure of various objects is tomography [9]. Generalization of the classical,
scalar field tomography for stress measurement is not trivial since stress is a tensor. In this paper we describe two algorithms
of photoelastic tomography. The first algorithm is base on the linear approximation of the equations of integrated
photoelasticity and the second one can be applied in the general case.
2.
Classical tomography
In tomography [9], some radiation (x rays, protons, acoustic waves, light, etc.) is passed through a section of the object in
many directions, and properties of the radiation after it has passed the object (intensity, phase, deflection, etc.) are measured
on many rays (Fig. 1). Experimental data g(l,θ*) for different values of the angle θ* are called projections.
If f(r,φ) is the function that determines the distribution of a certain parameter of the field, experimental data for a real pair l, θ*
can be expressed by the Radon transform of the field,
∞
g (l ,θ *) =
∫ f [r (l,θ *, z),ϕ (l,θ *, z)]dz .
−∞
(1)
When projections for many values of θ* have been recorded, the function f(r,φ) is determined from the Radon inversion
f (r ,ϕ ) =
1
2π 2
π
∞
∫ ∫
dθ ′
0
∂g (l ,θ *)
dl
.
∂l
r cos(θ * −ϕ ) − l
−∞
(2)
Figure 1. Scheme of tomographic measurements.
Figure 2. Illustration explaining measurement scheme of
photoelastic tomography.
Many numerical algorithms for solving Eq. (2) have been elaborated [9].
The question arises whether it is also possible to determine tomographically the stress fields in 3-D objects. This problem is
not trivial, for the following reason. Classical tomography considers only determination of scalar fields, i.e., every point of the
field is characterized by a single number (the coefficient of attenuation of the x rays, the acoustical or optical index of
refraction, etc.). Since stress is a tensor, in stress field tomography every point of the field is characterized by six numbers.
Thus the problem is much more complicated in principle. Let us mention that while a huge number of publications is devoted to
scalar field tomography, there is only a single book, written by Sharafutdinov [10], devoted to mathematical problems of the
tensor field tomography.
3.
Integrated photoelasticity
In integrated photoelasticity [11], the transparent 3-D specimen is placed in an immersion tank, and a beam of polarized light is
passed through the specimen. The transformation of the polarization of the light is measured on many light rays and for many
azimuths of the light beam. This measurement data are related in a complicated way to the stress field.
Transformation of polarization can be described by matrix U
 E x* 
 E x0 




 Ey  = U  Ey  ,
 *
 0
where
 eiξ cosθ
U =  −iζ
 − e sin θ

eiζ sin θ 
,
e −iξ cosθ 
(3)
E x0 , E y 0 are the components of the incident light vector, Ex*, Ey* describe polarization of the light that emerges from
the specimen and ξ, ζ, and θ are functions of the stress distribution between the points of entrance and emergence of the light.
Analysis of the transformation matrix U has shown that there always exist two perpendicular directions of the polarizer at which
the light emerging from the medium is linearly polarized. These directions of polarization of the incident and emerging light are
called the primary and secondary characteristic directions [11]. They are determined through the angles α0 and α* as follows:
sin(ζ + ξ ) sin 2θ
sin(ζ − ξ ) sin 2θ
tan 2α 0 =
,
tan 2α* =
.
(4)
2
2
sin 2ξ cos θ − sin 2ζ sin θ
sin 2ξ cos 2 θ + sin 2ζ sin 2 θ
Due to their exceptional physical properties, the characteristic directions can be measured experimentally. It is also possible to
measure the characteristic optical retardation ∆* between the secondary characteristic vibrations:
cos ∆ ∗ = cos 2ξ cos 2 θ + cos 2ζ sin 2 θ .
(5)
If the characteristic parameters α0, α*, and ∆* are determined experimentally on a light ray, it is possible to calculate the
parameters ξ, ζ, and θ of the transformation matrix U. Since the parameters ξ, ζ, and θ are determined in a non-linear way by
the stress distribution on the ray, in the general case the inverse problem of integrated photoelasticity is very complicated.
4.
Linear algoritthm of photoelastic tomography
It can be shown that in linear approximation an inhomogeneous birefringent medium can be considered as a birefringent plate.
It is possible to measure the parameter of the isoclinic φ and the integrated optical retardation ∆, which are expressed through
the integrals of the components of the stress tensor [12]:
y
y
∫
∫
V1 = ∆ cos 2ϕ = C (σ x − σ z )dy ,
V2 = ∆ sin 2ϕ = 2C τ xz dy .
y0
(6)
y0
Linear approximation is valid if birefringence is weak (optical retardation is less than about 1/3 of the wavelength) or the
rotation of the principal stress axes is small. If no rotation of the principal stresses is present, Eqs. (6) are valid for arbitrary
birefringence. These formulas have been widely used for measuring stress in axisymmetric glass articles [13].
Since Radon inversion for the tensor field does not exist, the problem of stress field tomography can be solved only if we can
reduce it to a problem of scalar field tomography for a single component of the stress tensor. That can be done in the following
way [12]. Let us assume that photoelastic tomographic measurements have been carried out on two parallel sections, a
distance ∆z apart from each other, rotating the specimen around the z axis (Fig. 2). The values of the functions V1 and V2 in
the auxiliary section we denote V´1 and V´2. Considering the equilibrium of the three-dimensional segment ABC in the direction
of the x’ axis (Fig. 2), we may write
C
B
∫
∆z σ x 'dy ' = Tu − Tl ,
A
B
1
1
Tu =
V '2dx' , Tl =
V2dx' ,
2C
2C
∫
∫
l
l
(7)
where Tu and Tl are shear forces on the upper and lower surfaces of the segment, respectively:
Taking into consideration relationships (7) the first equation (6) reveals
C
∫
A
B
σ z dy ' =
B
V
1
( V ' 2 dx'− V2 dx' ) − 1 .
2C∆z
C
∫
∫
l
l
(8)
Since tomographic photoelastic measurement data can be obtained for all the light rays y’, Eq. (8) expresses the Radon
transform of the field of the stress σz. Thus we have reduced a problem of tensor field tomography to a problem of scalar field
tomography for a single stress component σz. The field of σz can be determined using any of the well-known Radon inversion
techniques [9]. Rotating the specimen by tomographic measurements around the axes x and y, the fields of σx and σy can also
been determined.
Photoelastic measurements were carried out with an automatic polariscope AP-05 SM (Fig. 3). As light sources, light-emitting
diodes of λ = 640 nm were used. The polariscope contains two polarizers and two quarter-wave plates, which all are controlled
by stepper motors. Optical information is recorded by a CCD camera. The polariscope can be used as a dark-field or light-field
circular polariscope, and it permits also photoelastic measurements with the phase-stepping method. The algorithm of
tomographic photoelasticity assumes that optical retardation and the parameter of isoclinic are measured experimentally.
Besides, the direction of the first principal stress σ1 is to be determined. For that a specific phase-stepping algorithm [14] is
used. The AP-05 SM polariscope is controlled by an IBM Thinkpad.
For automatic rotation of specimens of small size, a rotary stage has been constructed (Fig. 4). Rotation is effectuated with a
stepper motor which permits rotation of the specimen with a precision of 0.1 deg. The specimen is fixed to the rotary stage so
that the part of the specimen that is investigated is placed in an immersion tank. For stress measurement in larger test objects
it may be sufficient to fix a protractor to the test object and to turn the object manually.
Figure 3. Computer-controlled AP-05 SM polariscope. In the
middle is the rotary stage with an optical fiber perform.
Figure 4. Rotary stage for tomographic stress
measurement in small glass articles.
Figure 5. Geometry of a high-pressure lamp and normal stress field in section AB of the stem; 180 projections. Average stress
is 0.2 MPa.
As an example we have measured stress in a high-pressure lamp. The bulb of the lamp (Fig. 5, centre) has axisymmetric form
and stresses can be measured with axisymmetric algorithms of integrated photoelasticity [13]. In section AB of the stem, which
is not axisymmetric, the normal stress field was determined with the linear algorithm of photoelastic tomography.
5.
Non-linear photoelastic tomography
5.1. Technology of the Experiment
We assume that stress field is axisymmetric. To avoid refraction of light, the specimen is placed in an immersion tank and
investigated in a transmission polariscope with tangential incidence (Fig. 6). As an example, we consider residual stress
measurement in hollow glassware.
Scanning the wall of the specimen in two parallel sections 1 and 2, the parameter of the isoclinic ϕ im and optical retardation
∆mi are measured on n light rays (i = 1, ..., n) in both sections (index “m” denotes experimentally measured values). In the
axisymmetric case ϕ = α0 = α*. Measurement of ϕ im and ∆mi in both sections in many points is the information on the basis of
which we shall determine the stress field.
5.2. Expressions for Stresses
We present stress components σr, σθ, σz and τrz in cylindrical coordinates in the form of polynomials relative to the radial
coordinate r:
m
σr =
∑
k =0
m
a′2k r 2k ,
σθ =
∑
k =0
m +1
m
b2′ k r 2k ,
σz =
∑
k =0
c2′ k r 2 k ,
τ rz =
∑d′
2 k −1r
2 k −1
,
(9)
k =1
where prime denotes section 1. In section 2 stress components are expressed in the same way, distinguishing the coefficients
with a double prime. Our aim is to determine the coefficients a2′ k , b2′ k , c2′ k , d 2′ k −1, a′2′ k , b2′′k , c′2′ k and d 2′′k −1 on the basis of
experimentally measured ϕ im and ∆mi (i = 1, ... , n). The set of the coefficients in Eqs. (9) is named the stress vector. The
number of unknown coefficients can be reduced using equations of the theory of elasticity, boundary conditions and
macrostatic equilibrium conditions.
Figure 6. The specimen is placed in an immersion tank and investigated by tangential incidence.
5.3. The Differential Evolution Algorithm
Our aim is to find the stress vector, which corresponds best to the measurement data. For that we use the differential evolution
(DE) algorithm. Differential evolution is a parallel direct research method for finding optimum values of the components of a
vector. The initial population of the vectors is chosen randomly if nothing is known about the system. In case a preliminary
solution is available, the initial population can be generated by adding normally distributed random deviations to the nominal
solution. The crucial idea behind DE is a scheme for generating trial vectors. DE generates new vectors by adding a weighted
difference vector between two population members to a third member. If the resulting vector yields a lower objective function
than a predetermined population member, the newly generated vector will replace in the following generation the vector, with
which it was compared. The best vector is evaluated for every generation to keep track of the progress that is made during the
minimization process. Extracting distance and direction information from the population to generate random deviations results
in an adaptive scheme with excellent convergence properties.
For the determination of the stress vector, which corresponds best to the measurement data, the following method was used.
First, the parameter of the isoclinic ϕ im and optical retardation are measured in both sections 1 and 2 on n light rays. For
every stress vector it is possible to calculate for the same light rays i the parameters
ϕ ic and ∆ci . For example, by modelling
the test object on a light ray as a pile of birefringent plates, each of which is described by a Jones’ vector [17]. The objective
(penalty) function F
2
2
n  c
 ∆ i cos 2ϕic − ∆mi cos 2ϕim   ∆ci sin 2ϕic − ∆mi sin 2ϕim  
1






F=
+
(10)
 
 

n
ε
ε




i =1 

∑
characterizes how well the stress vector describes the real stress field. Here ε is the measurement error.
General algorithm of the method is shown in Fig. 7 and the algorithm of DE in Fig. 8.
Figure 7. General algorithm of non-linear photoelastic tomography.
Figure 8. Basic algorithm of differential evolution.
In practical application of the algorithm we generated the initial population of 100 stress vectors by adding normally distributed
random deviations to the solution of linear photoelastic tomography. By random crossing (Fig. 8) we used weights of the
crossing vectors as follows
E = p[ A + 0.8( B − C )] + (1 − p) D,
(11)
where p is probability. Practically we used p = 0.9, which has proved to be efficient in practical applications of DE [16].
5.4. Experiment
As an example, residual stresses near the rim in a rim-tempered drinking glass (Fig. 9) were investigated. In section 1 (z =
28.1 mm) optical retardation is less than 100 nm and therefore Eqs. (6) of the linear approximation are valid.
In section 2 (z = 16.8 mm) optical retardation reached 300 nm. In case of rotation of the principal stress axes that is somewhat
more than allowed in linear photoelastic tomography. Figure 10 shows measured data in section 2.
On the basis of the measurement data in section 2, stresses were calculated using the algorithm of linear photoelastic
tomography (Fig. 11). Using measured stresses, theoretical measurement data were calculated (Fig. 10). Figure 10 shows that
the difference between measured and calculated data is considerable, especially for ∆sin 2φ. That is an indication that in
section 2 the linear approximation is not valid.
Using the DE method, final stress distribution in section 2 was obtained (Fig. 12). In comparison with Fig.11 the change of σθ
th
is remarkable. The decrease of the penalty function F during the DE procedure is shown in Fig. 13. After the 150 generation
the penalty function remains about constant, F ≅ 3.
Figure 14 shows a comparison of the experimentally measured and calculated, on the basis of the final stress distribution, data
in section 2. Coincidence of the measured and calculated data is considerably better than that shown in Fig. 10, especially for
the term ∆ sin 2ϕ.
(a)
(b)
Figure 9. Geometry of a rim-tempered glass (a) and
integrated fringe pattern near the rim (b).
Figure 10. Experimentally measured () and calculated
(– – –) data in section 2.
Figure 11. Stresses σz () and σθ (– – –) in section 2,
determined with the linear algorithm.
Figure 12. Stresses σz () and σθ (– – –) in section 2,
determined with the non-linear algorithm.
Figure 13. Dependence of the penalty function on the
number of generation in the DE process.
Figure 14. Experimentally measured (——) and calculated
(– – –), on the basis of the final stress distribution, data in
section 2.
6.
Conclusions
We have described two algorithms of photoelastic tomography. In case of the linear approximation, the problem of tensor field
tomography is decomposed to several problems of scalar field tomography for different components of the stress tensor. In the
non-linear case the differential evolution algorithm is used to find the stress field that corresponds best to the measurement
data. Both algorithms have been illustrated by experimental examples.
Acknowledgements
Support of the Estonian Science Foundation (grant No. 6881) is gratefully acknowledged. The authors thank Jelena Sanko for
consultations concerning the differential evolution method.
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