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Investigation of high harmonics initiation in surface temperature field under fatigue loading
O. Plekhov, T. Palin-Luc1, and N. Saintier1
Institute of Continuous Media Mechanics RAS, Perm, 614013, Russia 1 ENSAM-LAMEFIP (EA 2727), Talence, F-33405, France
The peculiarities of surface temperature time evolution measured by infrared detector are investigated analytically. Based on the Kelvin equation it is shown that initiation of high harmonics in temperature field can be caused by both energy-determined sources (damage, viscous friction and so on) and artifact sources (motion of specimen surface, painting defect, external heating). The spatial distribution of high harmonics caused by artifact sources for different materials and experimental conditions is determined. The way depending on the investigated material properties to decrease the influence of artifact sources on temperature evolution recorded by infrared camera is proposed.
Notation
Ta — mean temperature level;
Tp — maximum temperature of surface; Pt — tensor of thermal expansion; p — mass unit;
cv — specific heat at constant volume; 5e — elastic strain tensor;
<5 a — stress amplitude;
E — Young’s modulus.
1. Introduction
The current progress in the development of infrared detectors makes the infrared thermography as a powerful, nondestructive, non-contact technique for investigation of heat dissipation during different deformation processes. Infrared thermography had been successfully used for fast determination of fatigue limit and shape of Wohler curve [1-3] and
for determination of value of stress intensity factor [4, 5].
The fatigue loading of a great number of modern structural materials are accompanied by development of small defects with cannot be detected until the end structure lifetime. This evolution simultaneously involves a great number of strong non-linear interacting defects at different scales. This leads to the specific changing of the macroscopic material response. This situation leads to the necessity of finding of new techniques for damage monitoring and forecast of
endurance and durability. The infrared thermography is considered as one of promising techniques for these purposes. This technique was successfully used for monitoring of current position of fatigue crack [6] and for determination of small fatigue crack initiation (with characteristic size about 1 mm) [7]. For good estimation of the rest durability requests the monitoring of initiation and development of defects with very small size (about several microns). In this case, the main role plays the way of data treatment.
It is evident that under cyclic loading the energy in the specimen oscillates with double loading frequency (E ~ a(t)e(t) ~ roE_1ctA sin(2rot)). If we propose some mechanical model connecting energy oscillation, damage evolution and heat dissipation, we can expect then damage development should be accompanied by initiation of at least second harmonics in temperature signal. The monitoring of high (second and third) harmonics spatial distribution for detection of place of damage initiation was used for damage detection in composite structures (wind turbine blades) [8] and currently use as dissipation detection tool in some commercial software for infrared cameras. According to our experience the results of this techniques is adequate on the last part of lifetime just before crack initiation and during crack propagation but is unstable in the middle part of test.
The present work is devoted to the investigation under middle-cycle fatigue (around 105 cycles) of the temperature evolution at the surface of smooth specimens made in 35CD4 steel, loaded in fully reversed 4 point plane bending. Temperature evolution has been recorded with an infrared camera (Jade III by CEDIP, thermal resolution: 0.1 mK and maximum framing rate: 500 Hz). The aim of this paper is to show that temperature signals recorded by infrared camera is quite sensitive to the experimental condition and some
© O. Plekhov, T. Palin-Luc, and N. Saintier, 2004
time the experimental conditions (for instance respective motion of the specimen surface and camera lens) can lead to initiation of high harmonics in temperature evolution.
2. Experimental procedure
The material under investigation is 35CD4 steel. The material properties and the chemical composition presented in detail in [7]. The specimens were machined from round heat treated bars. The central portion of the specimen was 13x15x30 mm. The specimens have stress intensity factor K t about 1.05. The specimen was loaded in fully reversed sinusoidal plane bending with a resonant electrodynamic fatigue testing machine (loading frequency ~56 Hz). The tests series were carried out with nominal stress amplitude in the range from 200 to 650 MPa.
During these tests, an infrared camera (CEDIP Jade III) was used to record the evolution of the temperature field of the specimen surface. The spectral camera range is 3-5 jim. The maximum picture size is 320 x240 pixels. The minimum size of pixel is 10-4 m. The maximum framing rate is 500 Hz. The thermal sensitivity is < 25 mK at 300 K.
During experiments, we have observed the initiation of hot spots on the specimen surface and generation of second
harmonics in the recorded temperature signal. Figure 1 presents the infrared image of specimen surface and wavelet analysis of small part of the picture. The wavelet decomposition in time was made for each spatial points based on the Morle function.
It is evident that second harmonic is damage sensitive tool but to quantitative estimate of damage degree, we have to develop the adequate thermo-mechanical model of a solid and take into account influence of experiment conditions. The adequate theoretical model for cold work accumulation in a solid still issue but in next section we have proposed the way to reduce the influence of artifact second harmonics sources caused by relative motion of the specimen and camera lens.
3. Mathematical statement of problem
Let us to assume the possibility of hot spot initiation with temperature Tp. This spot can be caused by damage developing process or plastic deformation localization. In thermography tests, it could be not only real temperature anomaly. The same results can be obtained due to painting defect, small surface scratch and so on.
Let us consider one-dimensional problem. In this case, we can present the hot spot as follow temperature distribution:
T (x) = Ta+ Tbexp(-(x - x„)2), (1)
where x = %r , x0 = %r0 is the center of thermal hot spot, r are the spatial coordinates, x is a constant determining the size hot spot.
Local temperature evolution in the “pure” elastic case can be described by the well-known Kelvin relation:
pcvT = -Тв: E: ee.
(2)
Fig. 1. The infrared image of specimen surface (top). Black square indicate an area for wavelet transform. The instantaneous map of spatial distribution of second harmonics (bottom) at the instant corresponding to that of the top image time
If the external stress depends from time t as 5 = = 5A sin(rot), we obtained the following relation for temperature evolution
T = C0 exp(-p/(pcv) 5A sin(rot)). (3)
The substitution of (1) as an initial temperature distribution in Eq. (3) lead to
T = (Ta + Tp exp(-(x - x0)2)) x
xexp(-Pt/(Pcv) 5A sin(rot))• (4)
Equation (4) describes the thermal evolution in the point x.
The main problem in fatigue experiments is small movement of the specimen surface during experiments relatively the camera lens. In some experiments, the special control system is developed to compensate this movement but in this case a small non-agreement in displacement can be appear.
The main idea of this work is to study the influence of this small displacement on temperature field evolution. Let us assume that camera has only one detector and this detector
Table 1
Material PT • 10-б p, kg/m3 cv, J/(kg K) k = Рт/ Pcv
Aluminum 23.8 2б99 900 9 • 1012
Steel 12.0 7872 481 3 • 1012
Copper (annealed) 1б.4 89б0 385 4.7 • 1012
Ceramics (Al2O3) 7.4 39б0 0.8 2 • 109
Quartz 0.7 2180 750 4 • 1013
has harmonic movement with frequency w relatively the camera lens. The movement of the camera detector can be described as
x = xd — Y sin( wt), (5)
where xd is the position of the detector respectively the unloaded specimen, Y is the displacement of the specimen surface in the point xd (this value can be easily calculated from the specimen displacement).
Finally, we can write the follow equation for thermal response of the movement detector
T = (Ta+ Tp exp[—(r — Ysin(wt))2]) x
X exp( — pT /(pcv ) oA sin(wt)), (6)
where r = xd — x0.
4. Results
Even from the analysis of equation (4) we can conclude that the temperature evolution has more than one harmonics but second harmonics is proportional to k2. From the analysis of Table 1 we can be neglected these components.
Equation (6) can be rewritten as follow
T = (Ta + Tp exp(—r2)exp(2rYsin(wt)))x
xexp(—Y2 sin2(wt))(1 — koA sin(wt)). (7)
It is easy to see that equation (7) has more than one basic harmonics and the value of this harmonics is proportional to the specimen displacement and temperature Tp. In the case r =0 (the infrared detector located in the middle of hot spot) equation (7) can be easily analyzed
T = (Ta + Tp (1 — y2 sin2 (wt) + y4 sin4 (wt))) x
x(1 — koA sin(wt)). (8)
Generally in fatigue experiments, the displacement of specimen surface is small. In our test, the maximum displacement was about 0.01 mm. In this situation equation (7) can be written as
T = (Ta + Tp exp(—r2)exp(1 + 2rY sin(wt) +
+ 4r2y2 sin2(wt)))x (1 — koA sin(wt)). (9)
From equation (9), we can find the main part of the amplitude of the basic harmonics is (Tp + Ta)koA and the main part of amplitude of the second one is
^2 = (Ta + Tp )(a1(ko A)2 + a 2(ko A)4 + k) —
— TpY 2 (a0 + a1(ko A)2 + a 2 (ko A )4 + k) +
+ TpY4(a1 + a3(koa)2 + a4(koa)4 +...) + ... . (10)
It means that we can detect the initiation of second harmonics in the top of hot spot and this harmonics can be caused by an initial surface defect or local heating of the specimen due to any external or internal sources. Figure 2 presents the value of the second harmonics versus specimen surface displacement. We can conclude that for each material we can find the optimal displacement for decreasing of the influence of non-damage sources.
In general case, the analysis of Eq. (7) shows the generation of the current harmonics by all higher one. Figure 3 present the value of the second harmonics versus the surface displacement Y and distance between center of the hot spot and infrared detector position, respectively. From the analysis of Fig. 3 we can conclude that the spatial distribution of the second harmonics is non homogeneous into the hot spot.
5. Discussion and conclusions
Infrared thermography is powerful technique for investigation of damage evolution under fatigue experiments. This technique can be successfully used for monitoring of initiation and propagation of small fatigue cracks. The
A
0.0004 0.0003 0.0002 0.0001
0.01 0.02 0.03 у
A
4-Ю'6 3-Ю'6 2-Ю'6 1-Ю'6
0.001 0.002 0.003 y
Fig. 2. The dependence of the second (top) and third (bottom) amplitude harmonics for koA = 0.01 versus the specimen surface displacement
Fig. 3. The dependence of the second harmonics amplitude versus the surface displacement y and the distance between the hot spot center and infrared detector position (koA = 0.01)
question about the applicability of this technique for monitoring of damage initiation requests additional experimental investigation of heat dissipation in a solid coupled with the microstructure investigation of the specimen during the test. Simultaneously, it is requests the development of an adequate technique for data processing. One of the promising techniques is the monitoring of the second harmonics of temperature evolution initiation and the relation of the amplitude of this harmonics with the probability of fracture in this place.
As results of our investigation, it has been shown that the initiation of high harmonics in the temperature field can be caused by both damage-caused sources and artifact sour-
ces (motion of specimen surface, painting defect, external heating). We have found the spatial distribution of high harmonics caused by artifact sources for different materials and experimental conditions. We have proposed the way depending on the investigated material properties to decrease the influence of artifact sources on temperature evolution recorded by the infrared camera.
Acknowledgement
Dr. Plekhov acknowledges the French Ministry of Research for financial support of his stay in France where this work was started. Dr. Plekhov also acknowledges the Russian Science Support Foundation (grant for talented young researches) for financial support.
The work was partial supported by grant RFBR 04-0196009.
References
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[3] G. Fargione, A. Geraci, G. La Rosa, and A. Risitano, Rapid determination of the fatigue curve by the thermographic method, Int. J. of Fatigue, 24 (2002) 11.
[4] P. Stanley and W.K. Chan, in Proc. ImechE, C262 (1986) 105.
[5] R.A. Tomlinson, A.D. Nurse, and E.A. Patterson, Fatigue Fract. Ingng Struct., 20 (1997) 217.
[6] F.A. Diaz, E.A. Patterson, and J.R. Yates, in Proc. Int. Conf. Fatigue Crack Patch, Parma, Italy, (2002).
[7] O.A. Plekhov, S.V. Uvarov, T. Palin-Luc, and O.B. Naimark, in Proc. Int. Conf. Fatigue Crack Paths, Parma, Italy, (2002).
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