EVALUATION OF ALGORITHMS FOR RESIDUAL STRESS CALCULATION
FROM RADIAL IN-PLANE DSPI DATA
a
M. R. Viotti , and A. V. Fantin
a
a
, A.
Albertazzi
a
Laboratório de Metrologia e Automatização, Universidade Federal de Santa Catarina, CEP 88040970, Florianópolis, SC, Brazil
ABSTRACT
Residual stress measurements can be performed by using the combination of a radial in-plane digital speckle pattern
interferometer and the hole drilling technique. In practical situations it is very usual that displacements due to relieved residual
stresses produce optical phase distribution corrupted by several noise levels. In consequence, algorithms used to compute
these stresses can be influenced by the level noise introducing errors during the computation. This paper reports on the
evaluation and comparison of four methods: (a) a displacement-least squares approach, (b) a strain-least squares approach A,
(c) strain-least square approach B and (d) a strain-least absolute error minimization one. These methods determine residual
stress states from only one optical phase distribution produced by the hole drilling technique. The former approach makes the
computation from the measured displacement field around the hole. On the other hand, the other approaches use the
computed radial strain field from the measured radial in-plane displacement field. To evaluate the performance of these
methods, phase maps for several levels of residual stresses and different levels of optical noise were simulated.
Keywords: Digital speckle pattern interferometry; Residual stresses; Radial in-plane displacements.
1. Introduction
In the last decades, several optical techniques have been developed having the ability to generate fringe patterns from which
the displacement field can be evaluated. Among them, digital speckle pattern interferometry (DSPI) is the most versatile one
[1]. As it is well known, DSPI is based on the generation of speckle distributions by a laser source. In contrast to holographic
interferometry, which needs a photographic process, DSPI uses speckles sufficiently large to be resolved by a video system.
The application of digital techniques in DSPI allows the automation of the data analysis process, which is usually based on the
extraction of the optical phase distribution encoded by the generated correlation fringes [2]. From these data, in-plane and outof-plane whole-field displacement fields can be measured over the surface of any rough object without making contact with it.
A novel and important application of DSPI is its combination with the hole-drilling technique to measure residual stress fields
[3, 4]. Several steps take place during the measurement. First, a set of speckle interferograms is acquired and next the hole is
drilled to remove stressed material. After this, a second set of interferograms is acquired and the optical phase difference
distribution is calculated. Finally, the displacement field due to the relieved stresses is computed. To perform the drilling
process between both image acquisitions, kinematic mountings are used [5]. They allow acquiring the image sets on the
optical bench and to make the hole in an external milling machine. These repositioning systems present some drawbacks,
namely: (a) they can introduce rigid body displacements, (b) they can be used to evaluate only small parts, and (c) they can
not be applied to perform in-situ measurements.
In order to avoid these problems, Albertazzi et al. [6-8] have developed a portable double illumination DSPI device that
enables measuring of residual stresses outside of the optical bench and the laboratory. This system was designed taking into
account the fulfilment of some requirements such as: (a) high stiffness to keep the relative motion between critical parts of the
interferometer below on an acceptable level; (b) strong clamping system to keep negligible the relative motion between the
device and the specimen to be measured; (c) fast and easy positioning in order to place the measuring device precisely in a
given point of interest, fast processing to obtain the radial in-plane displacement field from the analysis of only one correlation
fringe pattern.
Despite the fact that the good performance of this portable device to measure residual stresses has been firmly established
and its use for in-situ measurements, its compact configuration can introduce optical noise in the measured optical phase
distribution. This noise can be increase by the drilling process because it produces local heating and small local rigid body
displacements which can decrease interferograms correlation consequently increasing the signal to noise ratio. This optical
noise affects the measured optical phase distribution and, depending on their magnitude, they could introduce considerable
errors during the residual stress determination. In consequence, it is important to have a processing method capable to deal
with optical phase distributions corrupted with optical noise in order to compute the residual stress field.
This paper reports on the evaluation and comparison of four methods: (a) a displacement-least squares approach, (b) a strainleast squares approach A, (c) strain-least squares approach B and (d) a strain-least absolute error minimization one. These
methods determine residual stress states from only one optical phase distribution produced by the hole drilling technique. The
former method uses Kirsch’s displacement equations to apply the minimizing technique and to perform the computation of
residual stress magnitudes and rigid body translations. On the other hand, the latter methods use the computed radial strain
field from the measured radial in-plane displacement field. To evaluate the performance of these methods, phase maps for
several levels of residual stresses and different levels of optical noise were simulated.
As in practise, it is very difficult to make a specimen with a reference value of residual stress that allows the availability of a set
of phase maps for different residual stress states. In the present work, simulated ones were used as reference. These phase
maps were corrupted for five levels of random optical noise. These new set maps were processed in order to evaluate the
performance and the error of the methods to quantify the residual stress fields.
2. Residual stress measurement
2.1. Displacement-least squares approach.
The polar radial displacement field measured in a circular region provides sufficient information for characterization of
displacements, stresses or residual stresses that occurs in the interrogated region.
If a uniform in-plane translation is applied on the specimen surface, the following radial displacement field is developed [9]
ur (r , θ ) = ut cos(θ − α )
(1)
where ur ( r , θ ) is the polar in-plane radial displacement component, ut is the amount of uniform translation, α is the angle
that defines the translation direction, r and θ are polar coordinates.
When the hole-drilling technique is used, it is possible to obtain a quantitative value of the residual stress from the
measurement of the displacement field generated around a through hole by using the model developed by Kirsch [10] and
around a blind hole by applying the numerical model developed by Schajer [11]. The Kirsch’s model is based on the elastic
solution for an infinite plate subjected to a uniform stress state, when a cylindrical hole is drilled through the plate thickness.
The radial component of the radial in-plane displacement field developed by the introduction of the hole can be written in polar
coordinates as [10]:
ur (r , θ ) = A(r ) (σ1 + σ2 ) + B (r ) (σ1 − σ2 ) cos(2θ − 2β )
(2)
where the functions A(r ) and B (r ) are given by [10]:
A(r ) =
B(r ) = −
1+ υ a2
2E r
1 + υ 2 ⎛⎜ a 2
4
a ⎜⎜ 3 −
⎜
2E
1+ υ
⎝r
(3)
1 ⎟⎞
⎟⎟
r ⎟⎠
(4)
being ur ( r ,θ ) the radial component of the radial in-plane displacement field, σ1 and σ2 are the principal residual stresses, β is
the principal direction, E is the elastic modulus, υ is the Poisson’s ratio, and a is the radius of the hole. Replacing the values of
A(r ) and B (r ) from Eqs (3) and (4) into Eq. (2):
ur ( r , θ ) =
a
a ⎡
(1 + υ ) ρ (σ1 + σ2 ) +
4ρ -(1 + υ )ρ 3 ⎥⎤ (σ1 − σ2 ) cos(2θ − 2β )
⎦
2E
2 E ⎢⎣
(5)
where ρ = a / r is the ratio of the hole radius a to the polar coordinate r .
The radial component of the radial in-plane displacement field developed by the combination of displacements produced by
rigid body motions and relieve residual stresses can be obtained by adding Eqs. (1) and (5) giving
ur (r , θ ) = ut cos(θ − α ) +
a
a ⎡
(1 + ν ) ρ (σ1 + σ2 ) +
4ρ -(1 + ν )ρ 3 ⎤⎥ (σ1 − σ 2 ) cos(2θ − 2β )
⎦
2E
2 E ⎣⎢
(6)
When a least squares approach is used, a set of experimental data is sampled from the measured displacement field and fitted
to a mathematical model by least squares. No particular sampling strategy is required, but it is a good practice to select
sampling points regularly distributed over all measured region. An appropriate mathematical model can be obtained by adding
and rewriting Eqs (1) and (5):
⎡ 4 a ⎛ a ⎞3 ⎤
⎡ 4 a ⎛ a ⎞3 ⎤
a
ur ( r , θ ) = K u 0 R . + Ku1C cos(θ ) + Ku1S sin(θ ) + K u 2C . ⎢⎢
− ⎜⎜ ⎟⎟ ⎥⎥ cos(2θ ) + Ku 2 S . ⎢⎢
− ⎜⎜ ⎟⎟ ⎥⎥ sin(2θ ) + Ku 0
⎟
⎟
r
⎢⎣ (1 + υ ) r ⎜⎝ r ⎠ ⎦⎥
⎢⎣ (1 + υ ) r ⎝⎜ r ⎠ ⎦⎥
(7)
The terms K u 0 R , K u1C , K u1S , K u 2C and K u 2 S are easily identified by comparison with equations (1) and (5). The term K u 0 is
an additional one used to absorb a constant bias in the displacement field, which can be occasionally caused by a thermal
drift.
At least six measured points are necessary to determine the six coefficients. With DSPI a few tens of thousands measured
points are available being used to compute the coefficients by means of the least squares method. Since the coefficients are
all linear, the least squares can be carried out by a straight forward way using a multi-linear fitting procedure. The
displacement and stresses components can be computed from the fitted coefficients of Eq. (7) by
ut = K u21C + K u21S
⎛K ⎞
α = tan −1 ⎜⎜⎜ u1S ⎟⎟⎟
⎜⎝ K ⎠⎟
u1C
(
(
E
K u 0 R + K u22C + K u22 S
(1 + υ ).a
E
σ2 =
K u 0 R − K u22C + K u22 S
(1 + υ ).a
⎛K ⎞
1
β = tan −1 ⎜⎜⎜ u 2 S ⎟⎟⎟
⎜⎝ K ⎠⎟
2
σ1 =
)
)
(8)
u 2C
In practical situations it is very usual that both residual stresses and rigid body displacements appear mixed up in the same
optical phase difference distribution. They can be measured simultaneous and independently since different orthogonal least
squares terms are involved in their computation.
2.2. Strain-least squares approach A
Taking into account that ρ = a / r , it can be replaced in Eq. (5). Thus, this equation can be written in the following way:
ur ( r , θ ) =
(1 + ν ) a 2
1
(σ1 + σ2 ) +
2E r
2E
4 ⎤
⎡ a2
⎢ 4 -(1 + ν ) a ⎥ (σ1 − σ2 ) cos(2θ − 2β )
⎢
r ⎦⎥
⎣ r
(9)
In order to obtain the radial component of the radial in-plane strain εr , Eq. (8) can be derivate in r
(1 + ν ) ⎜⎛ a ⎞⎟
1
⎜⎜ ⎟⎟ (σ1 + σ2 ) +
⎝
⎠
2E r
2E
2
εr (r , θ ) = −
2
4⎤
⎡
⎢ -4 ⎛⎜ a ⎞⎟⎟ + 3(1 + ν )⎛⎜ a ⎞⎟⎟ ⎥ (σ − σ ) cos(2θ − 2β )
⎜⎜ ⎟ ⎥ 1
2
⎢ ⎝⎜⎜ r ⎠⎟
⎝r⎠ ⎥
⎢⎣
⎦
(10)
As Eq. (1) does not depend on the radial position, the radial in-plane strain produced by rigid body displacements is zero.
Thus, by rearranging the terms and replacing again the relationship between the radius of the hole and the radial position, Eq.
10 can be written in a more adequate form
εr (r , θ ) = −
(1 + ν ) 2
1 ⎡ 2
ρ (σ1 + σ2 ) +
-4ρ + 3(1 + ν )ρ 4 ⎤⎥ (σ1 − σ 2 ) cos(2θ − 2β )
⎦
2E
2 E ⎢⎣
(11)
Since the radial in-plane displacement produced by rigid body translations is independent of the radial coordinate, its derivate
is zero for every point to be analyzed. In consequence, the use of radial strains to compute residual stress from phase maps
where displacements produced by relieved residuals stresses and unwanted rigid body translations are mixed up is a very
attractive method because this method is independent of rigid body motions.
The relation between the displacement ur ( r , θ ) and the measured phase difference φr (r , θ ) is given by [2]
φr (r , θ ) =
4π
ur (r , θ ) sin γ
λ
(12)
where λ is the wavelength of the light and γ is the angle between the direction of illumination and the normal to the
specimen surface. Thus, to compute the residual stress field, the displacement field is obtained from the continuous optical
phase difference distribution and then the related strain field is calculated performing the numerical derivate of these
displacements. As it is known, DSPI techniques allows obtaining phase difference maps whose values are between −π and
+π , the continuous phase difference distribution should be computed using phase unwrapping methods which add or subtract
an adequate multiple of 2π to all pixels to remove phase jumps [14].
As before, an appropriate mathematical model can be obtained by rewriting Eq. (11) as follows
⎡ 4 a
⎡ 4 a
⎛ a ⎞3 ⎤
⎛ a ⎞3 ⎤
a
− 3⎜⎜ ⎟⎟ ⎥⎥ cos(2θ ) + K ε 2 S . ⎢⎢
− 3⎜⎜ ⎟⎟ ⎥⎥ sin(2θ ) + K ε 0
εr (r , θ ) = K ε 0 R . + K ε 2C . ⎢⎢
⎜⎝ r ⎠⎟ ⎥
⎝⎜ r ⎠⎟ ⎥
r
⎢⎣ (1 + υ ) r
⎢⎣ (1 + υ ) r
⎦
⎦
(13)
The stress components can be computed from the fitted coefficients of Eq. (12) by
(
(
E
K ε 0 R + K ε22C + K ε22 S
(1 + υ )
E
σ2 =
K ε 0 R − K ε22C + K ε22 S
(1 + υ )
⎛K ⎞
1
β = tan −1 ⎜⎜⎜ ε 2 S ⎟⎟⎟
⎜⎝ K ⎠⎟
2
σ1 =
)
)
(14)
ε 2C
2.3. Strain-least squares approach B
The previous approach evaluates the residual stress field by a strain approach using the strain field computed from the radial
in-plane displacement fields. Previously, an image processing has to be carried out, which includes the application of phase
unwrapping techniques. Taking into account Eq. (12) and the relationship between strain and displacement fields
ur ( r , θ ) =
∂u (r , θ )
λ
.φr (r , θ ) and εr (r , θ ) = r
4π sin γ
∂r
(15)
it is possible to obtain the following relationship between the optical phase distribution and the radial strain component
εr (r , θ ) =
∂φ ( r , θ )
λ
. r
4π sin γ
∂r
(16)
Consider that
∂ cos φr ( r , θ )
∂φ ( r , θ )
= − sin φr ( r , θ ). r
∂r
∂r
∂ sin φr (r , θ )
∂φr ( r , θ )
= cos φr ( r , θ ).
∂r
∂r
(17)
Squaring and adding member to member
∂φr (r , θ )
=±
∂r
⎛ ∂ cos φr (r , θ ) ⎞⎟2 ⎛ ∂ sin φr (r , θ ) ⎞⎟2
⎜⎜
⎟ + ⎜⎜
⎟
⎜⎝
⎠⎟ ⎝⎜
⎠⎟
∂r
∂r
(18)
Thus, it can be noted that Eq. (18) and (16) allows performing the direct computation of the strain field from the optical phase
distribution being unnecessary the use of phase unwrapping algorithms because Eq. (18) uses sin() and cos() functions which
are continuous functions. The sign of the derivative is determined, for example, comparing the sign of
cos φr (r , θ )
with the
sign of
∂ sin φr (r , θ ) / ∂r . The sign of equation (18) is positive if both signs are the same and negative if otherwise. Once
the strain field is computed, Eqs. (13) and (14) can be used in order to compute the associated residual stress field.
2.4. Strain-least absolute error minimization approach
In this approach, an error function is built and minimized for the complete set of points of the processed phase difference map
by using a least absolute minimization method in order to compute the principal residual stresses and their principal direction.
This error function is shown in Eq. 19.
Npo int s
Fe1( r , θ ) = ∑ ε(r , θ ) −∆u (r , θ )
(19)
n =1
where Fe1( r , θ ) is the error function, ∆u (r , θ ) is the numerical derivate of the measured displacements computed by
equations (16) and (18), and Npoints are the number of valid points on the image.
As approach 2.3, the use of the direct computed strain field presents the same advantage avoiding the implementation of a
phase unwrapping algorithm. The least absolute method has some advantages over the least squares ones, namely it is
insensitive to large differences produced by the presence of outlier phase values into the optical phase distribution. The main
drawback is that it is mathematically less stable and computationally heavier to be implemented.
3. Evaluation procedure and results discussion
As it was previously mentioned, a computer simulation was used in order to comparatively evaluate the performance of the
proposal methods. The computer simulation considered that a perfectly relieved displacement fields, as in equation (2), were
measured by a radial in-plane interferometer with a symmetrical dual-beam illumination having a sensitivity vector in the radial
direction.
(a)
(b)
(d)
(c)
(e)
Figure 1. Simulated residual stress phase difference maps for σ1=150 MPa, σ2= 30 MPa and β = 45º. Optical Gaussian noise
distributions were added to the phase values with: (a) standard deviation of 3 %, (b) standard deviation of 6 %,(c) standard
deviation of 9 %, (d) standard deviation of 12 % and (e) standard deviation of 15 % of one fringe order.
The wavelength used in the simulation was chosen as λ = 785 nm and the angle between the direction of illumination and the
normal to the specimen surface was γ = 60 . The specimen was simulated over a circular region of 10 mm in diameter with a
o
resolution of 640× 480 pixels centred at the hole centre. Steel was selected as the specimen material, with an elasticity
module E = 210 GPa and a Poisson’s ratio υ = 0.3 .
Radial in-plane displacement fields for several residual stress states were simulated using Eq. (2). Then, they were
transformed in phase values using Eq. [12] being finally wrapped in order to obtain discontinuous optical phase distributions.
On the other hand, several sets of images containing optical noise with Gaussian distributions were obtained by using
NORM (µ, δ ) = µ + δ −2 ln( rdn).cos(2π.rdn)
(20)
where the function NORM (µ, δ ) in equation (20) generates a Gaussian (normal) distribution, where µ and δ are the mean
value and the standard deviation of the Gaussian distribution, and rdn is a pseudo-random numbers generator with a
rectangular distribution between 0 and 1. Five levels of Gaussian noise were taking into account by using five magnitudes of
standard deviations, namely, 3, 6, 9, 12, and 15 % of one fringe order added to the images of the phase difference maps. As
these images had 256 grey levels (8 bits) the used standard deviations were 7.7, 15.4, 23.0, 30.8 and 38.4 grey levels. To
accomplish a statistical analysis, a set of 10 images were finally obtained for each level of optical noise and for each residual
stress state simulated.
At last, each combined phase map (residual stress added to optical noise) was processed with the proposal approaches and
the residual stress fields were computed.
As an example, Figs. 1 (a), (b), (c), (d) and (e) show simulated residual stress phase maps for σ1=150 MPa, σ2= 30 MPa and
β = 45º and they correspond to 3, 6, 9, 12, and 15 % of a fringe order of standard deviation of the optical noise respectively.
A summary of the computed results is shown in Tables 1, 2 3 and 4. These tables list the stress mean value and its standard
deviation for each set of 10 images. In these tables, the identification number of every method is corresponded to the
approach presentation order in header 2.
Residual stress field: σ 1 = 150 MPa, σ 2 = - 50 MPa, and β=45º
Noise standard
deviation [%]
σ1 [MPa]
σ1 [MPa]
σ1 [MPa]
3
Approach 2.1
Approach 2.2
Approach 2.3
Approach 2.4
Mean value Standard dev. Mean value Standard dev. Mean value Standard dev. Mean value Standard dev.
149,98
0,10
150,44
0,20
145,23
0,17
145,26
0,23
6
149,95
0,33
150,35
0,43
144,58
0,77
144,69
9
150,1
0,37
150,13
1,53
143,37
0,77
143,66
0,67
σ1 [MPa]
σ1 [MPa]
12
150,21
0,50
150,77
1,73
141,42
0,93
142,12
1,03
15
149,79
0,67
151,14
2,50
138,37
0,87
140,72
0,93
σ2
σ2
σ2
σ2
σ2
3
6
9
12
15
-50
-50,05
-49,89
-49,84
-50,27
0,13
0,33
0,37
0,50
0,63
-46,46
-46,61
-46,52
-46,25
-46,39
0,17
0,40
1,60
1,87
2,20
-49,73
-49,29
-48,79
-49,06
-48,08
0,17
0,47
0,90
0,90
1,33
-49,7
-49,26
-48,93
-48,82
-48,17
0,23
0,60
0,80
0,87
1,40
3
6
9
12
15
45
45
44,99
45
45,01
0,00
0,00
0,00
0,03
0,03
45,01
44,99
44,99
45,01
45,1
0,03
0,07
0,07
0,10
0,23
44,99
44,99
45,02
45,07
45,05
0,03
0,07
0,10
0,07
0,17
44,99
44,99
45
45,01
44,93
0,03
0,07
0,10
0,13
0,13
[MPa]
[MPa]
[MPa]
[MPa]
[MPa]
β
β
β
β
β
[º]
[º]
[º]
[º]
[º]
0,50
Table 1. Computed residual stress fields from simulated phase map with five levels of optical noise and for σ1=150 MPa,
σ2= -50 MPa and β = 45º. All results are expressed in MPa.
According to results list in Tables 1, 2, 3 and 4, it is possible to see that the displacement-least squares approach (in these
tables it is called as Approach 2.1) has managed to correctly compute the residual stress from all the phase maps for each
level of optical noise. In addition, this algorithm presented an absolute error between the simulated residual stress field and the
computed one always inside + 0.4 MPa.
These tables also show that the strain-least squares approach A (called as Approach 2.2) had a good performance. This
approach only failed during the computation of one of the principal residual stress, namely the smaller one σ2. For this stress,
the absolute error ranged from 1 to 3 MPa. This method presented an absolute error in the larger stress σ1 which oscillated
between 0.1 MPa and 1.0 MPa.
The other strain approaches presented absolute errors which increase considerably as the optical noise raises. Only for an
optical noise with a standard deviation up to 7.7 grey levels (3%), they showed a reasonable performance having an absolute
error of about 5 MPa. For the other level noises, they presented a poor performance ranging the absolute error between 5 and
10 MPa.
Residual stress field: σ 1 = 150 MPa, σ 2 = 0 MPa, and β=45º
Noise standard
deviation [%]
σ1 [MPa]
σ1 [MPa]
σ1 [MPa]
3
Approach 2.1
Approach 2.2
Approach 2.3
Approach 2.4
Mean value Standard dev. Mean value Standard dev. Mean value Standard dev. Mean value Standard dev.
149,96
0,23
150,49
0,33
146,62
0,23
146,58
0,30
6
149,98
0,27
150,49
0,63
145,57
0,43
145,98
9
149,69
0,43
150,21
0,57
144,51
0,47
144,61
0,80
σ1 [MPa]
σ1 [MPa]
12
149,98
0,43
150,14
1,43
142,59
0,90
142,76
1,03
15
149,59
0,60
149,58
1,40
139,25
1,03
140,55
1,53
σ2
σ2
σ2
σ2
σ2
3
6
9
12
15
-0,03
0,04
-0,3
-0,03
-0,27
0,23
0,30
0,37
0,43
0,60
3,22
3,27
2,69
2,67
2,38
0,33
0,57
0,83
1,33
2,23
-0,58
-0,69
-0,84
-1,07
-1,79
0,20
0,40
0,60
0,43
0,90
-0,57
-0,32
-0,82
-1,12
-2,35
0,30
0,40
0,57
1,07
1,20
3
6
9
12
15
45
44,99
45
45
45,01
0,00
0,00
0,00
0,03
0,03
44,98
44,99
44,98
45,07
45,05
0,03
0,07
0,13
0,13
0,17
45
44,98
45,02
45,04
45,04
0,03
0,03
0,10
0,20
0,13
45
45,06
45,03
44,93
45,1
0,03
0,07
0,13
0,23
0,23
[MPa]
[MPa]
[MPa]
[MPa]
[MPa]
β
β
β
β
β
[º]
[º]
[º]
[º]
[º]
0,70
Table 2. Computed residual stress fields from simulated phase map with five levels of optical noise and for σ1=150 MPa,
σ2= 0 MPa and β = 45º. All results are expressed in MPa.
Residual stress field: σ 1 = 150 MPa, σ 2 = 30 MPa, and β=45º
Noise standard
deviation [%]
σ1 [MPa]
σ1 [MPa]
σ1 [MPa]
3
Approach 2.1
Approach 2.2
Approach 2.3
Approach 2.4
Mean value Standard dev. Mean value Standard dev. Mean value Standard dev. Mean value Standard dev.
149,99
0,10
151
0,17
147,44
0,20
147,25
0,37
6
149,93
0,23
150,95
0,40
146,74
0,37
146,58
9
149,78
0,27
150,64
0,70
145
0,80
145,74
0,93
σ1 [MPa]
σ1 [MPa]
12
149,86
0,27
150,17
1,37
143,22
0,70
143,67
1,27
15
150,09
0,73
149,85
1,63
140,66
1,23
141,57
1,10
σ2
σ2
σ2
σ2
σ2
3
6
9
12
15
30
29,95
29,79
29,84
30,17
0,10
0,20
0,27
0,27
0,73
32,22
32,24
32,27
31,8
31,05
0,30
0,47
0,80
1,30
1,23
29,37
29,67
29,12
28,4
27,63
0,23
0,33
1,03
0,77
0,70
29,36
29,45
29,46
28,65
27,32
0,43
0,77
0,87
1,00
1,40
3
6
9
12
15
45
45,01
45
45
44,99
0,00
0,00
0,03
0,03
0,03
45
45,01
45,02
45,02
44,92
0,07
0,07
0,13
0,20
0,37
44,99
44,95
45,06
45,04
44,92
0,07
0,13
0,17
0,23
0,23
44,98
45,03
45,01
44,98
45,01
0,07
0,10
0,30
0,17
0,17
[MPa]
[MPa]
[MPa]
[MPa]
[MPa]
β
β
β
β
β
[º]
[º]
[º]
[º]
[º]
0,87
Table 3. Computed residual stress fields from simulated phase map with five levels of optical noise and for σ1=150 MPa,
σ2= 30 MPa and β = 45º. All results are expressed in MPa.
As it was previously shown, strain-least squares approach B (2.3) and strain-least absolute error minimization one (2.4)
dispense the application of phase unwrapping techniques to compute the radial strain field from the phase map, greatly
reducing the numerical effort required. According to previous tables, the introduced optical noise influence over the residual
stress absolute error was considerable. This conclusion highlighted the advantage of the application of a robust phase
unwrapping algorithms that use weighting matrix to mask inconsistent pixels avoiding their influence during the residual stress
computation. In consequence, the displacement-least squares approach (2.1) and the strain-least squares approach A (2.2)
are more robust than the strain-least squares approach B (2.3) and the strain-least absolute error minimization approach (2.4).
The mean values of the residuals stresses are always closer to the reference values for the first two approaches, what also
make them more adequate to processing real hole drilling phase maps.
By comparing the displacement-least squares approach (2.1) and the strain-least squares approach A (2.2) and by analysing
the standard deviation of the obtained results, it is possible to see that the standard deviation of both methods increases as the
optical noise level increase. For the former method the standard deviation was smaller than 1 MPa whatever the noise level.
On the other hand, the latter method showed a standard deviation up to 2 MPa. Thus, the displacement-least squares
approach can be considered as the best of these four methods to compute residuals stresses.
Residual stress field: σ 1 = 150 MPa, σ 2 = 75 MPa, and β=45º
Noise standard
deviation [%]
σ1 [MPa]
σ1 [MPa]
σ1 [MPa]
3
Approach 2.1
Approach 2.2
Approach 2.3
Approach 2.4
Mean value Standard dev. Mean value Standard dev. Mean value Standard dev. Mean value Standard dev.
149,97
0,10
151,09
0,23
148,16
0,23
148,26
0,30
6
149,95
0,27
150,81
0,53
147,18
0,37
147,5
0,57
9
149,99
0,40
150,62
0,93
146,27
0,73
146,63
0,87
σ1 [MPa]
σ1 [MPa]
12
149,99
0,63
150,39
1,23
143,43
0,73
144,73
0,70
15
150,11
0,73
150,45
1,37
141,53
0,83
143,58
1,13
σ2
σ2
σ2
σ2
σ2
3
6
9
12
15
74,97
74,96
74,97
74,9
75,04
0,10
0,23
0,40
0,63
0,77
77,29
77,02
76,72
76,81
76,33
0,20
0,57
1,23
1,10
1,07
74,43
74,01
73,51
71,7
70,35
0,30
0,53
0,67
0,77
0,73
74,65
74,18
74,44
74,64
74,48
0,37
0,77
1,07
1,00
0,77
3
6
9
12
15
45
45,01
44,99
44,98
44,99
0,00
0,03
0,03
0,03
0,03
44,96
44,92
45,04
45,1
44,92
0,07
0,17
0,20
0,20
0,47
44,99
44,96
44,9
45,08
44,95
0,07
0,17
0,20
0,27
0,37
44,98
45,07
44,91
44,94
45,09
0,10
0,20
0,23
0,47
0,43
[MPa]
[MPa]
[MPa]
[MPa]
[MPa]
β
β
β
β
β
[º]
[º]
[º]
[º]
[º]
Table 4. Computed residual stress fields from simulated phase map with five levels of optical noise and for σ1=150 MPa,
σ2= 75 MPa and β = 45º. All results are expressed in MPa.
4. Conclusion
The attractiveness of DSPI to the optical metrology community arises not only form its non contacting nature but also from the
relative speed of the inspection procedure, mainly due to the use of video detection and digital image processing. A novel and
important application of this technique is its combination with the hole-drilling technique to measure residual stress fields.
However, phase maps that are generated during the hole drilling process present a sever data contamination by noise and
also include the edge of the hole and decorrelation due to rigid body displacements of the specimen to be measured. This
optical noise affects the measured optical phase distribution and, depending on their magnitude, they could introduce
considerable errors during the residual stress determination. In consequence, it is obvious the need of robust processing
algorithms which should be capable to deal with these real phase maps.
This paper presents some algorithms to compute residual stress fields from the phase difference maps. According to the
preceding results it is possible to see that the displacement-least squares approach (2.1) has presented the better overall
performance. In addition, it has managed to compute residual stress fields with an absolute error ranging between + 0.4 MPa.
According evaluation results, it can be noted that this algorithm is more robust because it is slightly affected by optical noise
level present in the residual stress simulated phase maps and no significant bias error was detected in any simulated map.
However, a robust phase unwrapping algorithm is required.
Further research will be done in order to evaluate the displacement-least squares approach and the strain-least squares
approach A with phase maps generated by combining displacements due to rigid body motions and residual stress fields. The
results will be presented in future works.
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