EXAMINATION OF THE OPTIMAL FREQUENCY OF THE CYCLIC LOAD
IN THE INVERSE ANALYSES OF THE HEAT CONDUCTION
OF INFRARED THERMOGRAPHY
K. Machida, S. Miyagawa and K. Hayafune
Department of Mechanical Engineering
Tokyo University of Science
Chiba, 278-8510, Japan
ABSTRACT
The effect of heat conduction changes by change of the heat conductivity of material, thickness of the specimen and a cyclic load
frequency at the time of obtaining images. The optimum frequency of cyclic load for exact principal stress separation was
examined about several kinds of metals in which heat conductivity differs greatly, and the viscoelastic material in which a
frequency influences measurement greatly. In the case of the analyses of the material with low heat conductivity, even if the effect
of heat conduction is not taken into consideration, a highly precise stress analysis is possible. It is necessary to take the effect of
heat conduction into consideration in the stress analysis of a thick object. The stress analyses by the infrared thermography have a
large error in metals with high heat conductivity at low frequency of a cyclic load. However, the effect of the heat conduction can be
modified by the inverse analysis of non-steady heat conduction. The optimal frequency for exact principal stress separation was
examined changing the frequency with 1 to 25 Hz by the comparison with the 3-D finite element method. In the metallic materials
of steel, an aluminum alloy and a copper alloy, it is required to obtain a temperature image at 15 Hz or more. For acrylic material, it
is required to obtain a temperature image at 6 Hz or more.
Introduction
In recent years, as an experimental method by which a crack problem is evaluated, the approach of evaluating object surface
displacement in full field optically [1-4] and the approach of measuring the information relevant to a fracture mechanics parameter
directly like the caustic method are proposed. Although the thermoelastic stress measuring method of this study is classified into a
full field measuring method, since it is the experimental method that direct stress information can be evaluated unlike an optical
experimental method, it is the very useful experiment method at the point that a stress-concentration part can be recognized with
an image to deal with a crack problem. It carries out imaging of the minute temperature change resulting from the thermoelastic
effect with an infrared thermography camera, and makes it possible to obtain the sum of the principal stresses which is the first
invariant of the stress tensor from the temperature field. Although measurement of the sum of the principal stresses can be
performed non-contact and quantitatively, since each stress component cannot be obtained directly, the technique of principal
stress separation for evaluating individual stress components from the sum of the principal stresses has been studied briskly [5-15].
The study in the latest thermal-elasticity stress measurement is shifting to the image-processing approach for the improvement in
measurement accuracy of the thermoelastic stress measuring method itself from the conventional technique of principal stress
separation. It is thought that the main factors of the error in an infrared image of the sum of the principal stresses are the influence
of the edge effect produced at the time of image superposition and the heat conduction inside a specimen. Then, the infrared
hybrid stress-analysis method which combined the intelligent hybrid method with the thermoelastic stress measuring method was
built. After estimating the nodal forces from the sum of the principal stresses by an inverse analysis, the error included in the
experiment field was modified by the intelligent hybrid method proposed by Nishioka et al. [16, 17]. By this, the error near a crack
edge by the edge effect was minimized, reasonable displacement field could be obtained, and it became possible to evaluate a
fracture mechanics parameter [18, 19]. However, it became clear to have the effect by heat conduction on an infrared stress image
in recent years. Furthermore, about the effect of the heat conduction included in the infrared stress image, it was eliminated by the
nonsteady-heat-conduction inverse-analysis method proposed by Inoue et al. [14, 15]. The effect of heat conduction changes
variously by the heat conductivity of materials, thickness of the specimen, the frequency of a cyclic load at the time of image
acquisition and so on.
The temperature images were acquired by the superposition of the temperature field in about 100 cycles of the fixed amplitude of a
cyclic load. Therefore, it is considered that those were obtained in a steady state. Then, the optimum frequency of cyclic load for
exact principal stress separation was examined about several kinds of metals in which heat conductivity differs greatly, and the
viscoelastic material in which the frequency influences measurement greatly. The analytic accuracy by the newly introduced
nonsteady-heat-conduction inverse analysis was quantitatively estimated by comparing with the stress intensity factor obtained
from the three-dimensional finite element method.
Theoretical Background
Non-steady Heat Conduction Analysis.
Heat-conduction inverse analyses use the method previously proposed by Inoue et al. Usually, the thermoelastic effect used by the
infrared stress measurement method assumes the perfect adiabatic condition. The basic equation is expressed by Eq. 1.
ρC p
∂T ( x , t )
∂s ( x , t )
= −αT ( x, t )
∂t
∂t
(1)
where T is the absolute temperature, s is the sum of the principal stresses, x is the space coordinate, t is time, ρ is the mass
density,
Cp
is the specific heat at constant volume, and α is the coefficient of linear expansion. On the other hand, the basic
equation of the thermolelastic effect considering of heat conduction is expressed as follows:
ρC p
∂s ( x, t )
∂T ( x, t )
= k∇ 2T ( x, t ) − αT ( x, t )
∂t
∂t
(2)
where κ is the coefficient of heat conduction.
First, the amplitude data of the sum of the principal stresses which assumed the adiabatic condition acquired from the actual
experiment, Δs0 is obtained. Next, the non-steady heat-conduction analyses by a finite element method are conducted from the
data and Eq. 2. In practice, although a temperature field produces a phase lag to a heat source, it is disregarded and the
temperature amplitude is evaluated. The amplitude of the sum of the principal stresses is corrected by the following equation by
setting this temperature amplitude to ΔＴ１(x).
Δs1 = Δs0 ( x) −
ρC p
[ΔT0 ( x) − ΔT1 ( x)]
αT0
(3)
The exact sum of the principal stresses is obtained by repeating this procedure and correcting the sum of the principal stresses. In
this way, the amplitude of the corrected sum of the principal stresses is applied to the infrared hybrid method, and a stress intensity
factor is evaluated.
Infrared Hybrid Method
A certain limited domain Γ is divided by eight node isoparametric elements. A unit load is applied to the direction of x and y at the
nodes on the boundary, respectively as shown in Figure 1.
J1 at point i in the domain is expressed as follows:
2N
J1i = ∑ J1*ij F j ,
i = 1, M
j = 1, 2 N
(4)
j =1
where M denotes the number of points at which the sum of the principal stresses are given, N denotes the number of nodes on the
boundary,
*
J 1ij
denotes the matrix of the sum of the principal stresses, and Fj denotes the nodal force on the boundary. The
displacement at each node is given by
2N
ui = ∑ uij* F j ,
i = 1, M
j = 1, 2 N
(5)
i = 1, M
j = 1, 2 N
(6)
j =1
2N
vi = ∑ vij* F j ,
j =1
Figure 1. Infrared hybrid domain
where ui and vi denote the displacements in the x and y directions, respectively, and u ij* and vij* denote the matrices of
*
displacement. If J 1ij is calculated about all the unit force of acting on a boundary by FEM, a boundary value Fj can be obtained by
solving the simultaneous equations of the following equation.
{J } = [J * ]{F }
(7)
If Fj is the appropriate value, the error ei expressed with the following equation is zero.
2N
ei = J1i − ∑ J ij* F j
i = 1, M
j = 1, 2 N
(8)
j =1
We applied the least squares method to determine the unknown value Fj. Then, S which is the sum square error is defined as
follows:
M
S = ∑ ei
2
(9)
i =1
It is considered that Fj considered to be appropriate satisfies the following equation.
∂S ∂F j = 0
(10)
Fj can be obtained by solving 2N simultaneous equations of Eq. (10). The displacements u, v of each node are obtained by the
following equation from obtained Fj.
{u} = [u * ]{F }
(11)
{v} = [v * ]{F }
(12)
In this study, the 2-D intelligent hybrid method proposed by Nishioka et al. [16, 17] was employed to evaluate the stress and the
strain using displacement data. The experimental displacement field contains measurement errors so that one obtains the following
finite element equation.
[K ] {Q mod } = {F }− [K ] {Q exp } = {R}
(13)
Material
SS400
A2017
C1100
Acrylic
material
Table1 Material properties
Thermal
Young's
Poisson's
Conductivity
Modulus
Ratio
(GPa)
（W/m・K）
208.4
0.3
145
70.6
0.33
190
122.6
0.343
390
3.558
0.38
0.19
(a) Metallic materials
(b) Acrylic material
Figure 2. Specimen configuration
Figure 3. Images of the sum of the principal stresses obtained by the infrared stress image system (A2017)
(a) SS400
(b) A2017
(c) C1100
Figure 4. Contour maps of σx and σy of each metal obtained by the infrared stress image system
{
where Q
mod
} and {Q } are the nodal displacement vectors of the modifying and experimental fields in entire hybrid analysis
exp
area, [K ] and {F } are the global stiffness matrix and the global nodal force vectors, respectively. {R} is the restoration force.
The modifying displacement field can be evaluated by solving the following equation.
[K ] {Q mod } = {R}
(14)
{Q } can be obtained from the 2-D FEM by putting R as the nodal load and constraining all nodes on the outer boundary in the
mod
x and y directions. Finally, the true displacement field is obtained by the following equation.
{Q} = {Q exp }+ {Q mod }
(15)
If the appropriate displacement is obtained, stress and strain are evaluated by the following equations.
{ε } = [B ]{δ }e
(16)
{σ } = [ D ]{ε }
(17)
0
⎡1 ν
E ⎢
[D] =
ν 1
0
1 −ν 2 ⎢
⎢⎣0 0 (1 − ν)
⎤
⎥
⎥
2⎥⎦
(plane stress)
⎤
⎡
ν (1 −ν )
0 ⎥
⎢ 1
⎥
[D] = E (1 −ν ) ⎢⎢ν (1 −ν )
1
0 ⎥
(1 +ν )(1 − 2ν ) ⎢
(1 − 2ν )⎥
0
0
⎢⎣
2(1 −ν ) ⎥⎦
(plane strain)
(18)
(19)
where E is the Young’s modulus, ν is the Poisson’s ratio, {ε} is the strain vectors in an element, {σ} is the stress vectors in an
element, {δ} is nodal displacement vectors, and [B] is the strain-displacement matrix.
Experiment
The configuration of the specimen is shown in Figure 2. Thickness and crack length of the specimen for metallic materials (Figure
2(a)) are 1, 3, 6 or 10 mm, and 30 mm, respectively. Those for acrylic material (Figure 2(b)) are 10 mm and 30 mm, respectively.
Material is rolled steel for general structures (SS400), an aluminum alloy (A2017), and a copper alloy (C1100). A Young's modulus,
a Poisson's ratio, and thermal conductivity are shown in Table1, respectively. The specimen was created based on JSME S 0011981. A crack was introduced up to 28 mm by electric discharge machining, after that a fatigue pre-crack of 2 mm length was
introduced with the fatigue testing machine, and final crack length was 30 mm. In the experiment with infrared thermography, in
order to make emissivity regularity, the neighborhood of a crack was lightly ground by sand paper, and the heat-resistant black
coating spray was uniformly applied so that there might be no unevenness of consistency. The frequency at the time of thermal
image acquisition is 1-25 Hz.
Results and Discussions
Accuracy Estimation of the Sum of the Principal Stresses
Figure 3 shows an example of the image of the sum of the principal stresses obtained by the infrared stress image system. The
sum of the principal stresses of each node in the hybrid domain of Figure 1 is obtained from this image. An example of the contour
map of the stress components obtained by the stress separation of the sum of the principal stresses by the intelligent hybrid
method is shown in Figure 4. In each material, it turns out that the reasonable stress field is obtained by the infrared hybrid method.
Moreover, since displacement at the outer boundary of the hybrid domain mesh is constrained and the nodal displacement vector
in the entire domain is computed by the finite element method in the hybrid stress analysis, the stress field on the outer boundary
of the hybrid domain is unreliable. Therefore, the contour map of stress of Figure 4 shows the local domain near the crack tip.
Figure 5 shows the distribution of σy in consideration of the heat conduction with the distance from the clack tip. In the case of steel,
an aluminum alloy and a copper alloy, a relative error is large in the range smaller than x= 0.4 mm, but distribution of each stress
approaches that of FEM at 15 Hz or more. As well as acrylic material, the relative error is large in the range smaller than x= 0.4
mm, but distribution of each stress approaches that of FEM from 6 Hz to 10 Hz. The method modifying the sum of the principal
stresses searches for temperature distribution synchronizing with this period using the periodic steady heat source (based on data
of the sum of the principal stresses) given by the sinusoidal wave, and is the method of correcting it from Eq. (3). At this time, the
period of temperature produces phase retardation to the steady heat source actually. However, it was disregarded and it united
with the period of the steady heat source this time. In an actual experiment, even if the infrared stress image system considers
(a) SS400
(b) A2017
(c) C1100
(d) Acrylic material
Figure 5. Distribution of σy with the distance from the clack tip
having taken the maximum and the minimum of temperature synchronizing with a load period, it is thought that it is reasonable.
Moreover, the time of inverse analysis was united with the time which took for image acquisition at the time of the experiment.
It became possible to modify the sum of the principal stresses of material (C1100) with high heat conductivity which had the very
big measurement error with sufficient accuracy, and to separate the reasonable stress components as the result of modification by
this heat-conduction inverse analysis. However, the modified sum of the principal stresses in the low frequency below 5 Hz is
inaccurate in common about the specimen of each material. In the nonsteady-heat-conduction inverse analysis introduced by this
study, the temperature distribution computed by Eq. (1) from the infrared image of the sum of the principal stresses as initial
temperature distribution is used. So, with the low distribution of the sum of the principal stresses with the descending slope in a low
frequency experiment, it is thought that the temperature distribution itself is smoothed, and exact heat-conduction analysis cannot
be conducted. On the other hand, if nonsteady-heat-conduction inverse analysis is introduced to the image of the sum of the
principal stresses obtained by the cyclic load in the high frequency of 20 Hz or 25 Hz, it is possible to acquire considerably
accurate information of the sum of the principal stresses. The method modifying the sum of the principal stresses used this time is
the inverse problem approach using a finite element method, and is easy to treat compared with other inverse problem analysis
methods. Moreover, in the calculation which computes displacement from subsequent sum of the principle stresses data, and the
intelligent hybrid method, there is an advantage of being easy to apply to the infrared hybrid method from a viewpoint of the unity of
the principle.
Accuracy Estimation of Stress Intensity Factor
To simplify the comparison between stress intensity factors, the non-dimensional stress intensity factor FI was evaluated as
follows:
FI =
KI
P
πa
WB
Here, P is the load, W and B are the width and thickness of the specimen, respectively. a is the crack length.
(20)
(a) SS40
(b) A2017
(c) C1100
Figure 6. Variation of the stress intensity factor with the thickness of the specimen
(a) SS400
(b) A2017
(c) C1100
Figure 7. Variation of the stress intensity factor with the frequency of a cyclic load
Figure 6 shows the variation of the stress intensity factor with the thickness of the specimen. These results were obtained by the
three methods which are the 3-D FEM, the original stress image (HYB), and the non-steady heat-conduction analysis of having
taken into consideration the effect of heat conduction (INV).
With material with high heat conductivity like C1100, the stress intensity factor of the infrared hybrid method of not taking the effect
of heat conduction into consideration has 10% or more of error of that of FEM which was made into the reference value. In the
case of the material with low heat conductivity like SS400, even if it did not conduct the analyses which took heat conduction into
consideration, the error was less than 5%, and the accurate result was obtained. The error when not taking the effect of heat
conduction into consideration about A2017 with middle heat conductivity is less than 8%. In the analyses which do not take heat
conduction into consideration, about C1100 with big heat conductivity, clearly, the error became large and increased to a maximum
of 14% at 10 mm thickness as the thickness of the specimen increased. The same tendency was acquired although it was small
compared with C1100 about other materials. This is considered to be because for three-dimensional heat conduction not to be
taken into consideration.
In the nonsteady-heat-conduction inverse analysis conducted this time, the image of the sum of the principal stresses on the
surface of the specimen is again changed into the temperature distribution image, and heat conduction analysis is conducted. The
temperature distribution inside the specimen is not taken into consideration in that case. Naturally, since heat is conducted in three
dimensions, if the difference of internal temperature and surface temperature becomes large, it will be thought that the heatconduction error becomes large which is included in the image of the sum of the principal stresses obtained at the time of the
experiment. Heat is accumulated inside the specimen by this, and it is thought that it interferes with measurement of exact
temperature distribution.
Next, we consider the influence of a frequency change by the comparison of a stress intensity factor. Figure 7 shows the variation
of the stress intensity factor with the frequency of a cyclic load. The sum of the principal stresses obtained by the thermoelastic
stress measurement at lower frequency is greatly influenced by heat conduction, and analytic accuracy is falling with the decrease
in a frequency, as mentioned above. Since a hybrid domain mesh is arranged in the source of a stress concentration near a crack
tip when dealing with a crack problem, the set-up hybrid domain has a big temperature gradient. Therefore, it is indispensable to
the high precision analyses of the crack problem by the infrared hybrid stress analysis which makes a local hybrid domain an
analytical object to remove the effect of heat conduction by nonsteady-heat-conduction inverse analysis. Although high accuracy
with the usual hybrid analyses which do not take the effect of heat conduction into consideration is acquired in SS400 with low heat
conductivity, in low frequency experiments, such as 1 Hz and 5 Hz, accuracy is falling, and using the frequency of 15Hz or more is
recommended. On the other hand, in the case of C1100 with high heat conductivity, even if the sum of the principal stresses
obtained at the frequency of 15 Hz or more is used and it takes heat conduction into consideration, accuracy changes slightly with
the increase in a frequency.
Conclusions
1.
2.
3.
4.
5.
In the case of the material with low heat conductivity, even if the effect of heat conduction is not taken into consideration, a
highly precise stress analysis is possible.
It is necessary to take the effect of heat conduction into consideration in the stress analysis of a thick analytical object.
The stress analyses obtained by infrared thermography at low frequency of a cyclic load have a large error in metals with high
heat conductivity. However, the effect of the heat conduction included in the infrared stress images obtained at high frequency
of a cyclic load can be modified by the nonsteady-heat-conduction inverse analysis.
In the metallic material of steel, an aluminum alloy and a copper alloy, it is necessary to get a temperature image at a cyclic
load of 15 Hz or more.
As well as acrylic material, it is necessary to get a temperature image from 6 Hz to 10 Hz.
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14.
15.
16.
17.
18.
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