ENERGY ANALYSIS OF STEEL SUBJECTED
TO HIGH CYCLE FATIGUE
A. Chrysochoos1, B. Berthel1,2, A. Galtier2, F. Latourte1, S. Pagano1, B. Wattrisse1
( ) Lab. of Mechanics and Civil Engineering, CC 048, Pl. E. Bataillon, 34095 Montpellier, France
(2) Arcelor Research SA, Voie Romaine BP 30320, 57283 Maizières-lès-Metz, France
1
ABSTRACT
This paper presents an experimental protocol to locally estimate the different terms of energy balance associated with a
fatigue test on a DP 600 steel. The method involves two quantitative imaging techniques. On the one hand, Digital Image
Correlation (DIC) provides displacement fields and after derivation, fields of strain and strain-rate. A variational method,
associated with an energy functional, is used both to identify the field of elastic parameters and to determine the stress pattern.
The knowledge of stress and strain fields allows then to construct a pattern of deformation energy rate. On the other hand,
Infrared Thermography (IRT) provides thermal images which are used to separately estimate the distributions of thermoelastic
source amplitude and mean dissipation per cycle. The image processing uses a local form of the heat diffusion equation and a
special set of approximation functions taking into account the frequency spectra of the sought sources.
Introduction
Characterization of fatigue in materials and mechanical components requires time-consuming and expensive statistical
processing of the results of numerous mechanical tests. During these two last decades, alternative experimental approaches
have been developed to rapidly provide reliable fatigue characteristics. Among these, thermal methods, based on an analysis
of self-heating during a stepwise loading fatigue test, should be mentioned [1-6]. Indeed, during the high cycle fatigue of a
steel specimen, the stress states remain traditionally within the mesoscopic elastic domain of the material. However, slight
energy dissipation due to the irreversible evolution of the microstructure is observed during the multiplicity of cycles performed
at important loading frequencies. This dissipation is then superimposed with the classical thermoelastic coupling sources due
to the thermal expansion of the crystalline network. In references mentioned above, several authors claimed that the
remarkable change in the heating regime, observed at a certain stress range, is empirically related to the fatigue limit of the
material. Although realistic estimates of this limit were sometimes obtained, the thermal approach often led to questionable
results. Indeed, the physical meaning of the threshold stress evaluated by these thermal methods is not very clear. In [7] it was
shown that the appearance of the persistent slip bands is associated with the loss of linearity of the temperature vs. stress
range correspondence. It was also shown that the persistent slip band number increases with the stress and appears to be a
relevant fatigue indicator.
In previous works [8], we also underlined that the direct use of temperature as a fatigue indicator is not always reliable
because the temperature variation is not intrinsic to the material behaviour. It actually depends on the diffusion properties but
also on thermal boundary conditions and the heat source distribution. In this work, fatigue phenomena were therefore studied
using a local energy approach: the goal was to experimentally determine the distribution of deformation energy rate and that of
various heat sources accompanying the fatigue test.
Speckle image correlation techniques, involving a digital CCD camera [9], were used to assess surface displacement fields.
These kinematical fields enabled us to identify the distributions of elastic parameters and the stress patterns by minimizing a
given energy functional. Besides, a local form of the heat equation was used to derive heat sources from thermal images
provided by an IRFPA camera. A specific method of thermal image processing was therefore developed to separately estimate
dissipated energy and the thermoelastic coupling sources.
In what follows, we first review the form of the different terms of the energy balance associated with the high cycle fatigue of
steels. We then briefly remind the principles of the both kinematical and thermal image processing and the interest of their
combined use. Very first results obtained on dual-phase steel are finally shown to underline the promising potential of this local
energy approach.
Local Form of the Energy Balance
From a thermomechanical standpoint, fatigue is considered as a dissipative quasi-static process. Within the framework of
Generalized Standard Materials [10], the equilibrium state of each volume material element is then described using a set of N
state variables. The chosen variables are: the absolute temperature T, the linearized strain tensor ε (hypothesis of small
perturbations) and N-2 scalar components α1, α2, …αN-2 of vector α which pools the so-called internal variables. These latter
describe the macroscopic effects of complex, coupled microstructural phenomena.
•
The rate of deformation energy w def
was classically defined by:
•
w def
= σ : εɺ
where σ is the Cauchy stress tensor. The symbol
The local heat diffusion equation was written as:
( )
•
means that the time variation of
(1)
( ) is path-dependent.
ρCTɺ − div(K gradT ) = d1 + sthe + sic + rext
(2)
where ρ is the mass density, C the specific heat, K the conduction tensor. The heat sources are, in turn, the intrinsic
dissipation d1, the thermomechanical coupling sources that pool the thermoelastic source sthe and the other internal coupling
sources sic, and the external volume heat supply rext. With the specific free energy ψ(T , ε, α ) , the volume heat sources d1, sthe
and sic can be rewritten as:
d1 = σ : εɺ − ρ ∂ψ ∂ε .εɺ − ρ ∂ψ ∂α .αɺ

2
sthe = ρT ∂ ψ ∂T ∂ε : εɺ ≈ T ∂σ ∂T : εɺ ≈ −λ thT σɺ

2
sic = ρT ∂ ψ ∂T ∂α .αɺ
(3)
where λth is the linear thermal expansion coefficient. In the framework of our fatigue tests, the temperature variations remained
small and could not modify the internal state of the material. Consequently, we neglected the corresponding heat sources sic.
DIC & IRT Image Processing
Digital image correlation: DIC gives the space-time evolution of various kinematic variables on the sample surface [9]
(displacement, velocity, strain, strain-rate, acceleration…). The camera must be carefully set so that the CCD detector remains
parallel to the sample surface throughout the test. Indeed, each out-of-plane movement (translation or rotation) produces a
parallax error which distorts the images.
Figure 1. Shape of the specimen
The image processing was systematically realized after the test into two steps. First, the displacement field was estimated.
Second, the strains (or the strain-rates) were then derived from the displacements by space (and time) differentiation. Each
computational step used a particular numerical processing founded on local least squares fitting widely presented in [9]. The
2
performances of this image processing were tested both on analytic and experimental cases corresponding to rigid body
motions (translation or rotation), or to homogeneous or heterogeneous straining. The influence of all the computation
parameters on the displacement and strain measurement was also analyzed. Using standard parameters, the accuracy on the
-2
-4
displacement measurement was about 5.10 pixel and the accuracy on the strain calculation was 1.10 .
•
To estimate w def
, we used the in-plane displacement field (U,V) measured by DIC under the plane stress hypothesis. The
load applied at the specimen boundary was recorded. For each step of loading, the variational approach provided a couple (E,
σ) which is then one solution of the identification problem if it satisfies the local equilibrium equations, the constitutive
equations and the global equilibrium.
We associated to this problem the functional F defined by [11]:
F ( τ, B ) =
1
( τ − B : ε(u )) : B -1: ( τ − B : ε(u )) dΩ
2 ∫Ω
(4)
where the stress field τ is statically admissible, and where the compliance tensor B is supposed to be symmetrical, non
negative and definite. The functional F(τ, B) is convex and positive, and null if and only if the couple (τ, B) satisfies the
constitutive equation. Therefore the identification is performed by numerically minimizing the functional F(τ, B). A minimisation
over the first and second variable gives the stress field solution and the field of piecewise constant elastic parameters,
respectively [12]. To check the consistency of results given by the variational approach, plane stress components were also
computed by using local equilibrium equations and strain data as already performed in [13,14].
Infrared Thermography: The left-hand side of Equation (2) is a differential operator applied to T, while its right-hand side
groups all possible heat sources accompanying the deformation process. The regularizing effects of heat diffusion limit the
thermal gradients throughout the (small) thickness of the specimen (Figure 1). A depth-wise averaged heat source distribution
can thus be usually estimated by using an integrated form over the sample thickness of the heat equation and by assuming
that thermal data on the surface remain close to the averaged temperature [15,16].
In the framework of our experiments we supposed the material parameters ρ, C, K are constant. We considered an isotropic
heat diffusion and we neglected the convective terms in the material time derivative of the temperature. We also supposed
that, near thermal equilibrium, the internal coupling sources remain negligible. Besides, we verified that the external heat
supply remained time-independent so that:
−K ∆T0 = rext
(5)
where T0 is the equilibrium temperature field of the sample. For tests performed on thin flat specimen, it was shown in [16] that
the temperature measured on surface of the sample remains very close to the depth-wise averaged temperature. Taking into
account all the hypotheses, Eq. (2) could be markedly simplified into a two-dimensional partial derivative equation:
 ∂θ θ 
ρC 
+
 − k ∆θ = d1 + s the
 ∂t τ th 
(6)
where θ = T − T0 is the difference between T and T0, averaged according to thickness. The symbol τth stands for a time
constant characterizing heat losses perpendicular to the plane of the specimen while the heat conduction in the plane is taken
into account by the two-dimensional Laplacian operator.
Construction of the heat source distribution via Eq. (6) requires the evaluation of partial derivative operators applied to noisy
digital signals. To compute reliable estimates of heat sources, it is then necessary to reduce the noise amplitude without
modifying the spatial and temporal thermal gradients. Among several possible methods, a special local least-squares fitting of
the thermal signal was considered in this work. The approximation functions account for the spectral properties of the
underlying heat sources. Moreover, the linearity of Eq. (6) and that of the respective boundary conditions enabled us to
separately analyze the influence of thermoelastic and dissipative heat sources.
Indeed, within the framework of linear thermoelasticity, it is easy to verify that:
the thermoelastic source has the same frequency spectrum as the stress signal; the variation of the thermoelastic
energy wthe vanishes at the end of each loading cycle of period fL−1 , so we get:
wɶ the = ∫
fL−1
sthe dτ = 0
(7)
3
Regarding the dissipative effects, we considered dissipation averaged over a whole number n of complete cycles (e.g. n ∼
number of cycles per second or per block, …):
dɶ1 = ∫
n fL-1
n -1fL d1 d τ
(8)
The new variable dɶ1 may characterize the slow degradation of the material microstructure due to fatigue phenomena. The
average dissipation per cycle dɶ1 is thus a positive heat source whose spectrum is limited to the very low frequencies.
Henceforth by noting ∆ɶ s the as the range of the thermoelastic source averaged over n cycles, the aim of the image processing
is to assess ∆ɶ s the and dɶ1 , separately.
The local fitting function θfit of the temperature charts is chosen as:
θfit ( x, y , τ ) = p1 ( x, y ) τ + p2 ( x, y ) + p3 ( x, y ) cos ( 2πfL τ ) + p4 ( x, y ) sin ( 2πfL τ )
(9)
where the trigonometric time functions describe the periodic part of the thermoelastic effects while the linear time function
takes transient effects due to heat losses, dissipative heating and possible drifts in the equilibrium temperature into account.
nd
Functions pi ( x, y ) , i = 1,…,4, are 2 order polynomials in x and y. These polynomials enabled us to take into account the
possible spatial heterogeneity of the source patterns.
Experimental Results
In the following, very first results obtained by using both imaging techniques are presented. The material under examination is
DP600 steel produced by Arcelor (Dual Phase Carbon Steel). Thin, flat specimens were used with a gauge part of 10 mm
(length), 10 mm (width) and 2.6 mm (thickness). This material is a hot-rolled steel grade containing ferrite and martensite. It is
composed of 0.074 C, 0.84 Mn, 0.038 P, 0.002 S, 0.217 Si, 0.04 Al, 0.702 Cr, and 0.005 N (in wt.%). The thermophysical
properties are given in Table 1.
-1
-1
-1
-1
λth (10 .°C )
ρ (kg.m )
C (J.kg .°C )
k (W.m .°C )
10-11
7800
460
64
-6
-1
-3
Table 1: Thermophysical properties of DP600
The tests involved loading blocks of Nc=2 400 cycles performed at different ∆σ with Rσ = −1 and fL =30 Hz . Between the 5
and the last five blocks, a block of 100 000 cycles was performed at maximal stress range (Figure 2). Note that the visible
-3
CCD camera data aquisition were performed at fL=5.55.10 Hz to increase the ratio fS/fL, fS denoting the sampling frequency
and then to improve the computation of the energy corresponding to the hysteresis area of the stress-strain curve (Figure 3).
The calorimetric analysis showing that the energy dissipated in a cycle was independent of the loading frequency [17], we
assumed that the deformation energy corresponding to the hysteresis loop of the stress-strain curve was also rateindependent. The loading block series of the fatigue test was schematically presented in Figure 2.
th
600
200 σ (MPa)
∆σ (MPa)
100
400
Rσ = −1
∆σ = 471 MPa
0
200
-100
-200
0
Blocks
Figure 2. Series of blocks
ε
x 10-3
-1 -0.5 0
0.5 1
Figure 3. Example of stress-strain hysteresis loop
Analysis of mechanical effects: Figure 3 gives an example of a mean stress-strain hysteresis loop over the next to last block.
In Figure 4, we chose to show the time evolution of the tensile strain components ε xx (t , x, y = 0) captured along the
4
longitudinal axis during a loading cycle performed at ∆σ = 570 MPa. The white curve represents the sinusoidal evolution of the
load. The analysis of the strain distributions enabled us to put forward the existence of small but systematic strain gradients
which were associated with material heterogeneity.
Ox axis (mm)
ε xx (t, x, y = 0)
x10-3
45
1
40
0.5
35
0
30
25
-0.5
Rσ = −1
-1
20 ∆σ = 570 MPa
15
0
50
100
time (s)
150
Figure 4. Time evolution of the Ox profile of tensile strains over a loading cycle
240
0.35
∆σ (MPa)
ν
230
0.3
220
210
4
6
8
block #
10
block #
0.25
12
Figure 5. Mean values of Young’s modulus according to the
block’s number
4
5
6
7
8
9
10 11
Figure 6. Mean values of Poisson’s ratio according to the
block’s number
Using the variational inverse method, the displacement fields obtained by DIC allowed estimating the elastic parameters of the
studied steel [12]. Figures 5 and 6 show the different mean values per block over the sample gauge part of Young’s moduli
and Poisson’s ratios obtained for the eleventh blocks. These values can be compared with those obtained with the second
method [13]. In this last case, the modulus was identified with the slope of the hysteresis loop. A linear regression was
performed for each block and for each couple (ε xx , σ xx ) of the strain and stress field measurements. The mean contraction
coefficient was also directly derived from the strain field measurements by averaging the local and instantaneous
ν( x, y , t ) = −ε yy / ε xx . Observing results of Figures 5 and 6 we have to acknowledge that the elastic properties of the material
remained approximately constant throughout the 124 000 cycles. The non detected influence of fatigue on the elastic
parameters is at the same time quite surprising and interesting if we think about fatigue mechanisms in terms of damage
coupled with elasticity [18] and/or plasticity [19].
Analysis of energy effects: We used both stress and strain field measurements to compute the mean volume energy rate
corresponding to the mean hysteresis area per block Ah. Under the plane stress hypothesis we computed:
fL Ah ( x, y ) =
fL
Nc
Nc fL-1
∫0
σij ( x, y , τ) εɺ ij ( x, y , τ) dτ
(10)
Dividing fL Ah by the volume heat capacity ρC of the material, we obtained a mean deformation energy rate per cycle
-1
expressed in °C.s . This operation makes it possible to define an equivalent heating speed for each type of energy rate and
facilitates comparison between the different terms of the energy balance.
5
The calorimetric terms were derived from the polynomial coefficients of pi ( x, y ) . By denoting P1, .., P24 as the fitting
parameters, the corresponding expressions of ∆ɶ s the and dɶ1 are respectively:

I2
J2
1
 P1ω g + P5ω g + P21ω + 2D

N
N
τth
x
y

∆ɶ sthe
=2
ρC

 2k  P2
I2
J2
P 
+ 62  
 P2 g + P6 g + P22  −

2
ρ
N
N
C
∆
∆
x
y  
x
y




I2
J2
1
+  −P2ω g − P6 ω g − P22ω + 2D

N
N
τth
x
y

2
(11)

 2k  P1
I
J
P 
+ 52  
 P1 g + P5 g + P21  −

2
N
N
ρ
C
∆
x
∆
y  
x
y



2
2
2
dɶ1  P3 P4  I 2  P7 P8  J 2  P23 P24  2k  P4
P 
=  + 2D  g +  + 2D  g + 
+ 2D  −
+ 82 

2
ρC  ∆τ τth
∆y 
 N x  ∆τ τth  N y  ∆τ τth  ρC  ∆x
(12)
i =Nx
i = Ny
where N xg = 2N x + 1 et N yg = 2N y + 1 are the number of pixels of the fitting window, I 2 = ∑ i =−N i 2 et J 2 = ∑ i =− N j 2 , ∆x and
x
∆y are the space resolutions,
∆τ
y
is the time-step associated with the frame-rate (i.e. the sampling frequency fS), and
ω = fL fS . A detailed presentation of the image processing and its check should be available in a next future.
0.5
9.6
w0
fL Ah(1)
fL Ah(2)
0.3
dɶ1
fL Ah(1)
9.5
fL Ah(2)
0.3
w0 (mm)
(°C/s)
0.4
0.4
(°C/s)
Figure 7. Mean 2D distribution of thermoelastic sources ∆ɶ s the ( x, y ) corresponding to the last loading block
9.4
0.2
d1
Rσ = −1
9.3
0.1
0.2
9.2
20
30
40
x (mm)
Figure 8. Mean profile over the sample width of
fL A(2) and dɶ1 .
h
0
178
50
fL Ah(1)
,
267
∆
356
σ (MPa)
444
533
and dɶ1 over the
Figure 9. Mean values of
gauge part and per block.
fL Ah(1) ,
fL Ah(2)
In Figure 7 we presented a 2D distribution of the mean thermoelastic source amplitudes averaged over the last loading block.
The two main features to be noticed are the quite good homogeneity of the coupling source field and the order of magnitude of
-1
-1
the thermoelastic effects. The mean value of ∆ɶ s the is around 92,5 °C.s and the standard deviation of about 0.3 °C.s . Note
ɶ
that the amplitude ∆sthe is much higher than the dissipation intensity. The energy dissipated in a cycle was besides compared
with the anelastic energy corresponding to the hysteresis loop. In Figure 8, the average longitudinal profile of dissipation
corresponding to the last block can be compared with Ah(1) which is the profile of anelastic energy rate derived from the
6
nd
variational approach. We also plotted the profile Ah(2) constructed using the 2 method. In both cases, we observed that the
anelastic work is higher than the energy dissipated in a cycle. We finally plotted the “initial” width variations w0(x) of the
specimen to allow the reader to detect the limits of the gauge part and to check sample geometry. Here, by “initial”, we mean
prior to the observed block.
In figure 9, the energy balance was performed for all blocks by considering mean values of anelastic energy (using both
methods) and dissipated energy over the sample gauge part. This result deserves some comments as far as the hysteresis
area is often identified with the dissipated energy. Let us consider the local expressions of both principles of Thermodynamics:
ρeɺ = σ : εɺ − divq + rext
 ɺ
ρTs − rext + divq = d1
(13)
where e is the specific internal energy, q = -KgradT the heat influx vector and s the specific entropy. By combining both
equations we get the following expression of the volume deformation energy rate:
σ : εɺ = ρeɺ + d1 − ρTsɺ = ρeɺ + d1 − ρCTɺ + s the + sic
(14)
Integrating Eq.(14) over a mechanical cycle leads to the hysteresis area Ah. If the mechanical cycle is also a thermodynamic
cycle, the hysteresis area can come from intrinsic dissipation and/or from thermomechanical coupling sources [20]. In the
framework of DP600 steel fatigue, we underlined that the variation of the thermoelastic energy wthe vanishes at the end of each
loading cycle. Besides we neglected the internal coupling sources. The hysteresis area is then due to variation of internal
energy and dissipated energy as soon as a periodic evolution of the temperature is reached (generally obtained after some
hundreds of cycles). We then interpreted the difference between the energy of the hysteresis loop and the dissipated energy
as variation of internal energy due to the irreversible fatigue mechanisms.
Concluding Comments
In this paper we presented a combined application of DIC and IRT to the fatigue of DP 600 steel. We used the kinematical and
thermal data to estimate the deformation energy rate, the thermoelastic source amplitude and the mean intrinsic dissipation
per cycle. The deformation energy rate was computed using the strain field measurements, the stresses being estimated using
a variational inverse method. The heat sources were derived from thermal data using a local expression of the heat equation.
The linearity of the diffusion equation and that of the boundary conditions allowed us to estimate the sources separately.
Finally, the very first results obtained on DP 600 steel were presented. These results did not show any noticeable influence of
5
the high cycle fatigue on the elastic properties (up to 10 cycles). Moreover, they showed amounts of deformation energy
corresponding to the stress-strain hysteresis loop systematically greater than the dissipated energy per cycle. A
thermodynamic analysis led us to interpret these differences as internal energy variations. Aware of the consequences of such
a finding on the modeling of the fatigue kinetics, we stress the fact that these first results must be considered, as usual, with
precaution. Complementary checks and tests have to be performed in a near future. Nevertheless, the promising character of
the local energy approach by combining DIC with IRT is already very encouraging.
Acknowledgments
The authors would like to thank Arcelor Research SA and Nippon Steel Corporation for their technical and financial support
during this study.
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