NON-ISOTHERMAL STRESS CYCLE PARAMETERS INFLUENCE

SISOM – 2004, BUCHAREST, 20-21 May
NON-ISOTHERMAL STRESS CYCLE PARAMETERS INFLUENCE
ON THE MIDDLE ALLOY STEELS DURABILITY
Adrian Cătălin DRUMEANU, Niculae Napoleon ANTONESCU, Ion NAE, Marius Gabriel PETRESCU
Universitatea “Petrol-Gaze” din Ploiesti, drumeanu@upg-ploiesti.ro
Abstract: The non-isothermal fatigue characterizes the cracking failure of the metallic materials,
which are subjected to thermal cyclic variations. The thermal stresses can be enough strong, so that
they can produce a local plastic strain. The strain amplitude depending on the maximum temperature
of the cycle, the heating/cooling rate, the thermal material characteristics, the mechanical material
characteristics a.s.o.
The paper presents the experimental results concerning the influence of the non-isothermal stress
cycle parameters on the middle alloy steels durability. These results can be used at the design of the
structures made from this kind of steels, which are cyclic non-isothermal stressed.
Key Words: non-isothermal stress cycle, middle alloy steels, durability
1. INTRODUCTION
The cyclic non-isothermal stresses affect many metallic structures, which are used in mechanical
industry. Thus, can be enumerate some examples like: forging equipment, fire guns, chemical equipment,
heavy-duty mechanical brakes.
The metallic structures cyclic non-isothermal stressed are subjected at thermal stresses and strains,
which together with the microstructural transformations imply the cracks appearance, and finally the
structure damage. For a correctly choosing and using of metallic material, it is necessary to determine their
intrinsic characteristics regarding their cyclic non-isothermal stresses durability.
The experimental determination of these characteristics implies a lot of experiments, which are made in
specific conditions, different from those, used for isothermal mechanical fatigue durability determination.
The specific factors that increase the difficulty of this kind of tests are:
- the stress regime is non-isothermal, and is characterized of two temperature limits between which it is
developed this thermal stress;
- the thermal stresses and strains variations reach high values because of the thermal stress takes place
between two temperature limits; the maximum values of these situate the stress in the elastoplastic range;
- the values of the maximum temperature, deformation speed and heating / cooling speeds are generally
high;
- the durability, which are determined for the metallic structures stressed in these conditions are generally
small, this fact place the fatigue phenomenon in the oligocyclic (low number cycles) range;
- for these kind of experiments it is difficult to mention a constant value of the thermal strain/stress
because of the hardness phenomena which appear during the sample test.
Taking into account of the above statements it can be concludes that the main non-isothermal stress
cycle parameter which influence the durability are: the minimum temperature of the cycle, the maximum
temperature of the cycle, the duration of the cycle and the structure rigidity which implies the level of the
total strain and stress variation.

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Non-isothermal Stress Cycle Parameters Influence on the Middle Alloy Steels Durability
The aim of the paper is to evidence the influence of these parameters on the middle alloy steels
durability, which are used at the construction of the structures that are subjected at non-isothermal cyclic
stresses.
2. THEORETICAL CONSIDERATIONS CONCERNING
THE MATERIALS THERMAL FATIGUE
Thermal fatigue phenomenon explanation will be done considering the degradation (failure, cracking)
which appears after a low number of cycles in non-isothermal stress conditions without maximum
temperature cycle keeping, in fact one of the severe cases of a structure loading. In this situation, the gradual
accumulation process of the degradation is done in the presence of the preponderant plastic deformations, the
material destruction taking place because of the cyclic stresses/deformation effects, which are accumulated
in time.
The basic principle of the material behavior model at thermal fatigue, assumes that the both ends test
piece are fastened by enclosure, during the heating process (Figure 1) [1, 2, 3].
Figure 1. The thermal and mechanical model of the thermal fatigue material behavior:
a) element with elastic link; b) the thermal cycle; c) hysteresis cycle of the elastic and plastic deformation.
This model assumes the existence of tree serial parts: the test piece 1, the infinite rigidity element 2
(J 2 =∞), the unloading element 3 (J 3

Adrian Catalin DRUMEANU, Niculae Napoleon ANTONESCU, Ion NAE, Marius Gabriel PETRESCU 258
m t δ
l
c
c
ε = α ∆ −
(2)
where: ε c is the strain which corresponds to the maximum compression stress; ∆t – the temperature
difference, ∆t= t max - t min ; δ c – the body 1 stretching corresponding to t max .
When the body 1 is cooled from t max to t min , in point 2 it takes place the unloading, with the appearance
of the tensile stress, the face A reaches in the position 3 (Figure 1, a) corresponding to the maximum
stretching δ i . On the cyclic strain diagram (Figure 1, c) the point 3 corresponds to maximum tensile stress
values σ i , with the strain:
m t δ
l
1
1
t
t
ε = α ∆ −
(3)
where: ε i is the strain corresponding to the maximum tensile stress; δ i – the body 1 stretching, corresponding
to t min . At the ulterior heating, the element 1 is unloaded and there are appear compression stresses (Figure 1,
c point 5) with the elastoplastic strain smaller than the compression strain of the first cycle. Thus, the
hysteresis cycle (1-2-3-4-5) is characterized by:
- the elastoplastic deformation:
m t δ −δ
l
- the hysteresis cycle width (the plastic deformation):
t c
∆ε = α ∆ − (4)
min
1
t c
σ δ
∆ε p = ∆ε − − (5)
E E
where E min , E max represent the modulus of elasticity at the minimum, respectively maximum
temperature of the stress cycle. The hysteresis cycle size and form depend on a lot of factors like: the form
and the geometry of the heated element, the environment temperature, the coefficient of thermal
conductivity, the thermal coefficient of expansion and the yield limit of the material etc. For the quantitative
calculus, the influence of these factors is gathered in the thermal strain concentration factor K:
K
∆ε
max
m
= (6)
∆ε
t
where: ∆ε m is mechanical elastoplastic strain; ∆ε t – thermal strain.
Considering that element 2 is cyclic heated with a small variation of the temperature, the element 1
mechanical stretching becomes:
l l l
∆ m1 =∆ t1+∆ t2
−δ 3
where: δ 3 is the face A displacement of the element 1 in the sense of the element 3 shortening (negative
stretching):
∆σ ⋅ S
δ 3 = (8)
J
where: ∆σ is the stress variation (Figure 1, c); S – the element 1 area of section; J – the system rigidity; ∆l t1 ,
∆l t2 – the dilatation of the elements 1 and 2. Considering l 1 = l 2 and:
∆ l t2
=η∆l t1
(7)
(9)
where η is a coefficient, it obtains:
K
δ
l
3
= 1+η− 0≤ η ≤ 1 (10)
∆
t1

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Non-isothermal Stress Cycle Parameters Influence on the Middle Alloy Steels Durability
The variant J 3 → ∞ implies K = 1 + η > 1 , respectively the mechanical strain amplification because of
the thermal dilatation of the element 2. The variant J 3 → ∞ and η = 0 defines the particularly case K = 1,
examined by Coffin, and it is done in real conditions only on the test pieces with both ends fastened by
enclosure. In consequence, the elastoplastic strain variation can be determined using the system rigidity J,
changing either the rigidity of the elastic element J 3 , or the rigidity J 2 , respectively calculating the
supplementary strains of the element 2.
3. CONSIDERATIONS CONCERNING THE TESTING METHODOLOGY
The non-isothermal fatigue of the metallic materials characterizes their durability in oligocyclic range
(low durability). These kinds of tests presume the cyclic axial stress in an elastoplastic regime when the
maximum and minimum temperature values are maintained constants. Unlike to isothermal fatigue
experiments, which are developed under constant amplitude, the non-isothermal fatigue tests present the
strain amplitude variation in time. In the first part of the test, the total elastoplastic strain varies until of the
hysteresis cycle stabilization, and finally it decreases because of the material hardening state appearance [4].
In these conditions, for the calculus of the durability it is used the strain amplitude value corresponding to
hysteresis cycle stabilization. The cylindrical test pieces are recommended for tests with the strain amplitude
up to 2 %. Over this value there are recommended the toroidal test pieces.
The concrete conditions of the tests are: the test pieces were tested in non-isothermal regime; the test
pieces were fastened by enclosure and they were stressed to compression at the maximum temperature of the
cycle; the elastoplastic strain variation was limited using three rigidity steps of the test piece fastened system
(12, 28, 55 MN/m); the average heating speeds of the test piece were in the range of 55…80 °C/s; the
average cooling speeds of the test piece were in the range of 5.5…7.0 °C/s; the absence of the maintaining
time at the maximum temperature of the cycle; the using of the full test pieces with 4 mm diameter and 30
mm length of the calibrated part; the criterion of the testing break was the first crack appearance [5].
4. THE EXPERIMENTAL TESTED STEELS
The experimental determinations were done using four sort of middle alloy steel. Their chemical
composition and their mechanical characteristics are shown in Table 1, and Table 2.
Table 1. Chemical composition of the tested steels
Table 2. Mechanical characteristics of the studied steels
Alloying Elements Composition, % wt Mechanical characteristic Values
C 0.32 … 0.42 Tensile strength 540 … 930 MPa
Mn 0.43 … 1.46 Yield limit 295 … 735 MPa
Cr 0.24 … 1.01 Elongation 12 … 16 %
Ni 0.09 … 0.63 Reduction of area 25 … 50 %
Mo 0.04 … 0.21
Si 0.25 … 0.52
6. RESULTS AND DISCUSSIONS
The aim of the experimental determinations was to emphasize the influence of the main non-isothermal
stress cycle parameters on the tested steels durability. The durability was expressed like the inversions
number until the first crack appearance on the test piece external surface. The main stress cycle parameters,
which were considered, were the maximum temperature of the non-isothermal cycle, the system rigidity, the
total relative strain variation and the cycle duration. For all the tests, the minimum temperature value of the
non-isothermal cycle was 100 o C.
In Figures 2, 3 and 4 it is represented the dependence between the durability and the maximum
temperature of the cycle for different values of the system rigidity. For every rigidity value it was determined

Adrian Catalin DRUMEANU, Niculae Napoleon ANTONESCU, Ion NAE, Marius Gabriel PETRESCU 260
the regression equation and the determination coefficient which were shown on each figure. In all the cases it
can be remarked the good correlation between the variables taken into account.
In Figures 5, 6, and 7 it is represented the dependence between the durability and the total relative
strain variation for different maximum temperature values of the cycle and different rigidities of the system.
In addition, on each figure it was shown the regression analysis equation and its determination coefficient,
which has a high value. Similarly, in Figures 8, 9 and 10 it was plotted the dependence between the
durability and the total duration of the cycle for different maximum temperature values. For these cases, the
determination coefficients have lower values than the situations considered in the figures above enumerated.
The durability, 2N
[inversions]
1500
1100
2N = 78683e -0.0066t
R 2 = 0.9636
700 12 MN/m
300
500 600 700 800 900
The maximum temperature of the cycle, t [ o C]
Figure 2. The durability (2N) vs. the maximum temperature of the cycle (t) for 12 MN/m rigidity value
The durability, 2N
[inversions]
1400
1200
1000
800
600
400
200
28 MN/m
2N = 60631e -0.0066t
R 2 = 0.9786
500 600 700 800 900
The maximum temperature of the cycle, t [ o C]
Figure 3. The durability (2N) vs. the maximum temperature of the cycle (t) for 28 MN/m rigidity value
The durability, 2N
[inversions]
1000
800
600
400
200
55 MN/m
2N = 37147e -0.0064t
R 2 = 0.9416
500 600 700 800 900
The maximum temperature of the cycle, t [ o C]
Figure 4. The durability (2N) vs. the maximum temperature of the cycle (t) for 55 MN/m rigidity value

261
Non-isothermal Stress Cycle Parameters Influence on the Middle Alloy Steels Durability
The durability,
2N [inversions]
1800
1600
1400
1200
1000
800
2N= -4735.3De + 2049.4
R 2 = 0.9403
t=600 o C
600
0 0.1 0.2 0.3
The relative total strain variation, De [%]
12 MN/m
28 MN/m
55 MN/m
Figure 5. The durability vs. the relative total strain variation for maximum cycle temperature of 600 o C and different rigidities values.
The durability,
2N [inversions]
900
800
700
600
500
400
300
2N= -979.11De + 939.29
R 2 = 0.9598
t=700 o C
12 MN/m
28 MN/m
55 MN/m
0 0.1 0.2 0.3 0.4 0.5 0.6
The relative total strain variation, De [%]
Figure 6. The durability vs. the relative total strain variation for maximum cycle temperature of 700 o C and different rigidities values.
The durability,
2N [inversions]
500
400
300
200
100
t=800 o C
12 MN/m
28 MN/m
2N=-318.63De+538.69
55 MN/m
R 2 = 0.932
0 0.2 0.4 0.6 0.8 1 1.2
The relative total strain variation, De [%]
Figure 7. The durability vs. the relative total strain variation for maximum cycle temperature of 900 o C and different rigidities values.
Durability, 2N
[inversions]
1600
1400
1200
1000
800
t=600 o C
2N = -78.434 t + 3885.8
R 2 = 0.8978
12 MN/m
28 MN/m
55 MN/m
25 27 29 31 33 35 37 39
The cycle duration, t [s]
Figure 8. The durability vs. the cycle duration for maximum cycle temperature of 600 o C and different rigidities values

Adrian Catalin DRUMEANU, Niculae Napoleon ANTONESCU, Ion NAE, Marius Gabriel PETRESCU 262
Durability, 2N
[inversions]
800
700
600
500
400
300
2N = -17.586 t + 1837.6
R2 = 0.865
t=700 o C
12 MN/m
28 MN/m
55 MN/m
60 70 80 90
The cycle duration, t [s]
Figure 9. The durability vs. the cycle duration for maximum cycle temperature of 700 o C and different rigidities values
Durability, 2N
[inversions]
500
400
300
2N = -5.7139t + 1045.6
R 2 = 0.8853
t=800 o C
12 MN/m
28 MN/m
55 MN/m
200
100 110 120 130 140 150
The cycle duration, t [s]
Figure 10. The durability vs. the cycle duration for maximum cycle temperature of 800 o C and different rigidities values
It can be observed that between the parameters taking into account and the durability it exists a good
correlation which is proved by the high values of the determination coefficient.
This fact imposes to do a global evaluation of the interdependence between the durability and the main nonisothermal
cycle parameters (the maximum temperature of the cycle, the system rigidity and the cycle
duration), using multiple correlation regression. For this case, it is proposed the next equation:
A
2N
= tW τ
a b c
where: 2N is the durability, inversions; t is the maximum temperature of the non-isothermal cycle, o C; W is the
system rigidity, MN/m; τ is the cycle duration, s; A, a, b, c are the equation constants which have the values
A = 323594, a = 0.26, b = 0.182, c = 0.948.
The characteristics of the regression equation (11) are: the maximum temperature value of the nonisothermal
stress cycle t, in the range of 600…800 o C; the system rigidity W, in the range of 12…55 MN/m; the
duration of the cycle τ, in the range of 25…150 s; the durability 2N, in the range of 200…1600 inversions; the
determination coefficient value R 2 = 0.992.
Regarding the equation regression (11), it can be remarked the good correlation between the independent
variables which are considered and the dependent variable. This good correlation signifies the fact that the
parameters of the non-isothermal stress cycle, which were considered, have the major influence on the tested
steels durability.
(11)
7. CONCLUSIONS
The main conclusions, which can be detached from the experimental results above presented, are:

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Non-isothermal Stress Cycle Parameters Influence on the Middle Alloy Steels Durability
• the high degree verisimilar determinations of the non-isothermal fatigue durability are those who are
done when the test piece is fastened by enclosure at both ends and it is compressed at the maximum
temperature of the cycle;
• the choosing of an adequately rigidity for the test piece enclosure system is very important for the
elastoplastic strain determinations and for the diminution of the stress relaxation phenomenon;
• the size of the temperature variation and the system rigidity have the major weight in durability
determination;
• lower values of the determination coefficients were obtained for the cycle duration influence;
• the regression equation (11) can be used for the middle alloy steels sort which is subjected to nonisothermal
cyclic fatigue:
• the above determined experimental results can be used both in designing and exploitation activities.
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