Modeling of Damage Interaction in Fatigue Relaxation for Long-Term Life Prediction. Case of Alloy 800 Grade 2 Study at 550°C
Prediction of fatigue life in creep-fatigue at a
 temperature of 550°C has been made for an
 Inconel alloy with the help of 3 models of damage
 evolution: the Chaboche model, the
 Levaillant model, and a new model of creep fatigue
 relaxation. The main advanta...
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| Date: | 2002 |
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
2002
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Modeling of Damage Interaction in Fatigue Relaxation for Long-Term
 Life Prediction. Case of Alloy 800 Grade 2 Study at 550°C / A.El Gharad, G. Pluvinage, Z. Azari, A. Elamraoui, A. Kifani // Проблемы прочности. — 2002. — № 4. — С. 28-47. — Бібліогр.: 8 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860091243491491840 |
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| author | Gharad, A. El Pluvinage, G. Azari, Z. Elamraoui, A. Kifani, A. |
| author_facet | Gharad, A. El Pluvinage, G. Azari, Z. Elamraoui, A. Kifani, A. |
| citation_txt | Modeling of Damage Interaction in Fatigue Relaxation for Long-Term
 Life Prediction. Case of Alloy 800 Grade 2 Study at 550°C / A.El Gharad, G. Pluvinage, Z. Azari, A. Elamraoui, A. Kifani // Проблемы прочности. — 2002. — № 4. — С. 28-47. — Бібліогр.: 8 назв. — англ. |
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| container_title | Проблемы прочности |
| description | Prediction of fatigue life in creep-fatigue at a
temperature of 550°C has been made for an
Inconel alloy with the help of 3 models of damage
evolution: the Chaboche model, the
Levaillant model, and a new model of creep fatigue
relaxation. The main advantages of the
new model are that it requires only few constants
and provides a high accuracy of the fatigue
life prediction.
Прогнозирование усталостной долговечности в условиях ползучести при температуре 550°С
выполнено для жаропрочного и жаростойкого сплава на никелевой основе (Inconel). Представлены
модели эволюции повреждения Шабоша (Chaboche) и Леваяна (Levaillant), а
также новая модель релаксации при совместном действии ползучести и усталости. Предложенная
модель позволяет осуществлять долгосрочное прогнозирование без использования
большого количества постоянных.
Прогнозування утомної довговічності в умовах повзучості при температурі 550°С виконано для жароміцного і жаростійкого сплаву на нікелевій основі (Inconel). Представлено методи еволюції пошкодження Шабоша (Chaboche) і Леваяна (Levaillant), а також нову модель релаксації при сумісній дії повзучості й утоми. За допомогою запропонованої моделі можна проводити довготермінове прогнозування без використання великої кількості сталих.
|
| first_indexed | 2025-12-07T17:23:24Z |
| format | Article |
| fulltext |
UDC 539.4
Modeling of Damage Interaction in Fatigue Relaxation for Long-Term
Life Prediction. Case of Alloy 800 Grade 2 Study at 550°C
A. El G harad ,a G. Pluvinage,b Z. Azari,b A. E lam raoui,a and A. K ifanic
a Laboratory of Applied Mechanics and Applied Technology, Rabat Institute, Rabat,
Morocco
b L.F.M. Faculty of Sciences, University of Metz, Metz, France
c Laboratory of Mechanics and Physics of Materials, Rabat’s Faculty of Sciences,
University of Rabat, Rabat, Morocco
У Д К 539.4
Моделирование разрушающего взаимодействия при усталостной
релаксации при прогнозировании длительной долговечности для
сплава 800 (тип 2) при 550°C
А. Эль Гарада, Г. П лю винаж 6, 3. Азари6, А. Э льам раиа, А. Киф анив
а Лаборатория прикладной механики и технологии, Институт г. Рабат, Марокко
6 Лаборатория механической надежности, Университет г. Метц, Франция
в Лаборатория механики и физики материалов, Университет г. Рабат, Марокко
Прогнозирование усталостной долговечности в условиях ползучести при температуре 550°С
выполнено для жаропрочного и жаростойкого сплава на никелевой основе (Inconel). Пред
ставлены модели эволюции повреждения Шабоша (Chaboche) и Леваяна (Levaillant), а
также новая модель релаксации при совместном действии ползучести и усталости. Пред
ложенная модель позволяет осуществлять долгосрочное прогнозирование без использования
большого количества постоянных.
К лю чевы е слова : сплав 800, ползучесть, усталость, релаксация, долгосрочное
прогнозирование, малые деформации.
1. Introduction.
1.1. Problem Form ulation. Subject of our study is an alloy with 35%
nickel, 20% chrome and an addition of titan and aluminum, which is further
referred to as alloy 800 at the state of grade 1 [1-3]. When construction
components made of this alloy are assembled by welding, thermal gradients are
established in the nearest zones to the welded joints, while welding modes
causing metallurgical changes produce the material structure which is similar to
the state of grade 2. Fatigue relaxation tests carried out on the alloy 800 grade 2 at
500°C have revealed the following [3]:
- An apparition of intergranular failure and its dominance for a long period
of time.
© A. EL GHARAD, G. PLUVINAGE, Z. AZARI, A. ELAMRAOUI, A. KIFANI, 2002
28 ISSN 0556-171X. Проблемы прочности, 2002, N 4
Modeling o f Damage Interaction in Fatigue Relaxation
- Fatigue life reduction (in numbers of cycles), which is observed as soon as
a hold time is introduced into a creep fatigue cycle. This is related to the failure
mode changeover from transgranular to intergranular one that occur during
creep-fatigue-relaxation tests.
Low cycle fatigue damage is different from that of creep [3]. It seems to be
necessary to identify separately each damage mode, for taking them into
consideration, when they are simultaneously present during fatigue relaxation
tests, in a unique law, in order to predict a long life duration under long-term and
small-scale deformation conditions.
1.2. Methodology. In this study, two models among the most recent ones are
briefly reviewed. These deal with the evolution of creep, fatigue and creep/fatigue
and fatigue/relaxation damage. The results of the long life prediction under
small-scale deformations and long term conditions, which are obtained by the two
models in the case of alloy 800 grade 2 to 550oC, are also presented [3]. Next the
following issues are discussed:
1. The definition of our proper model, treating the evolution of damage in the
fatigue relaxation.
2. The model validation on the basis of the same results as the two previous
models.
3. Prediction of the life duration (in cycles) for long hold time periods and
small-scale deformations (e.g., tmt =1000 hours and Ae t = 0.6%).
4. Comparison of our results with the experimental data and with those
obtained by the two previous models.
2. The Chaboche M odel of Creep-Fatigue Interaction.
2.1. Fatigue Model. Chaboche [3, 4, 6] has proposed to describe the
evolution of fatigue damage by the relation (1). This relation depends on the
damage itself, on the maximal stress in tension, and on the average stress of the
cycle and the number of cycles.
dD F = [1 - (1 - D ) l+P ]a
I
t max av
M (a av )(1- D f )
d N , (1)
where o t max = o a is the stress amplitude of the stabilized cycle; o 1 = o 10 is the
inferior limit of failure in fatigue (endurance limit); o u is the final limit of the
curve (stress of static load failure: a failure in the first quarter of the cycle); y3, b,
and M are constants.
Here
(1 — (l) = ao tmax — o 10 (o av V(o u — o t max ),
o 1( o av ) = o av + o 10(1— o av X
M ( o av ) = M 0(1— bo av )-
Hypothesis: The relation (2) has been used to define an approximate value
for the ultimate stress o u , using the results obtained during the tensile test, by (2).
ou = Rm [1 + A(%)]. (2)
ISSN 0556-171X. npoôëeubi npounocmu, 2002, № 4 29
A. El Gharad, G. Pluvinage, Z. Azari, et al.
Chaboche [3, 4, 6] has then proposed the relation (3) to describe the
evolution of the fatigue damage as a function of the number of cycles:
D fat — 1_
( N
1/(1-a) "
1- I-----
_ \ N R I
l/(l+/3)
(3)
2.2. Creep Model. Chaboche [3, 4, 6] has proposed the relation (4) to
describe the creep damage
D c - (a / A )r ( 1 - D c)~k dt (4)
where o is the stress applied to the element of the volume, k is a constant,
independent on the stress o, r is the slope of the creep curve (log tR - log o),
and A is the material constant.
2.2.1. Calculation o f the Time to Failure. Chaboche has introduced in his
study [3, 4, 6] the relation (5) to calculate the time of breaking tR by integrating
the expression (4), between D = 0 and D = D f aiiure and between t = 0 and tR
(time to failure).
t R —
1 I * \ -r a
1 + k A (5)
where o is a constant stress, which is a characteristic of the creep test and can
have one of the 3 values of the described in [3]: (1) the nominal stress o o, (2) the
stress at the end of loading o c, and the average actual stress o vav.
2.2.2. Creep D amage Curve. Chaboche [3, 4, 6] has proposed the expression
(6), which gives the evolution of damage according to the ratio (t/ tR) integrating
the relation (4) for t = 0 to t = t and D = 0 to D = D c.
1 - ( - \\ tR I
1 + k
(6)
2.3. Creep-Fatigue In teraction Model. Under fatigue relaxation conditions,
the fatigue and creep damages are simultaneously present. A nonlinear interaction
proposed in [4, 6, 7] is represented by the elementary sum of fatigue and creep
damage.
dD — (1 - D )~ k dt + [1 - (1 - D ) l+$ ]a
( A a
\2 M (1 - D )
dN (7)
In order to calculate the stress during relaxation, a viscoplastic relaxation
rule named “law of CEA-M” [3, 4] described by Eq. (8) is used:
30 ISSN 0556-171X. npoôëeubi npounocmu, 2002, N2 4
Modeling o f Damage Interaction in Fatigue Relaxation
a
= [1 + ( n - 1)EA( t )p a (8)max
where o is the current stress during relaxation, E is Young’s modulus (MPa), t is
the time from the beginning of the hold period, and n , A, and p are constants
that are determined by a nonlinear regression, when fitting the relation (8) to the
experimental curves.
The application of this model requires determination of 17 coefficients.
These have been determined within the framework of a previous study [3] in the
case of alloy 800 grade 2 at 550oC. These results are compared with the
experimental data, with results, corresponding to the Levaillant model, and with
those obtained by our model.
3. The Levaillant Model of Creep-Fatigue Interaction. Levaillant [5] has
proposed a model of creep-fatigue interaction where the intergranular cracks
propagate according to preferential directions from a broken grain joint to another
due to creep at the front of the fatigue crack.
3.1. Fatigue. The author [3, 5] postulated that the damage in continuous
fatigue be assimilated at the depth of crack a. The propagation ratio of trans-
granular cracks with fatigue striations per cycle is obtained by measuring the
distance between two successive striations (which is called “interstriation
distance” i). By integrating the crack propagation rules of type (9):
The number of the cycles corresponding to the fatigue crack N p can be
calculated as
where a o = 20 /im and a ̂ = 3 mm.
Levaillant proposed the following relation (11) to calculate the number of
where ^ 1, 2 , a 2 , and «1 are constants to be determined for each particular
material.
Thus, two phases are distinguished. The first one is the crack initiation phase
N a , and the second is the crack propagation phase N p . Then, it can be written as
d a /d N = i = f ( a). (9)
'0
(10)
FC
propagation cycles in continuous fatigue N according to the number of cycles
to failure NR1c .
(11)
N r = N a + N p . (12)
ISSN 0556-171X. npoôëeubi npounocmu, 2002, N2 4 31
A. El Gharad, G. Pluvinage, Z. Azari, et al.
The main assumptions made by Levaillant [3, 5] were the following:
FRCrack propagation phase in fatigue relaxation N p is reduced in relation to
FR FCthe continuous fatigue: N p < < N p .
FRCrack initiation phase in fatigue relaxation N aFR is shorter than in
FR FCcontinuous fatigue N a < < N a . For very long maintain period, Levaillant has
FRset this simplifying hypothesis: N a = 0.
The reduction of the crack propagation phase in fatigue relaxation in relation
FRwith the continuous fatigue is defined by the coefficient R FR :
N fc - N fr r pr = _ p N p
N PC (13)
3.2. In te rg ranu lar Creep Damage M easurem ent. In order to give a value
to the intergranular creep damage, which takes place during fatigue relaxation
tests, the intergranular damage coefficient D m is measured as described in [3, 5]
by Eq. (14)
Dm = L f /L a , (14)
where L f represents the accumulated length of cracked grain joints (except for
the isolated cavities), while L a represents the total length of the measured grain
joints on the same micrographic field to a 200 magnification, a scope of
0.55-0.4 m m 2.
3.3. Intergranular Damage p e r Cycle. The evolution of D m is a sensitive
function of the number of cycles, which has allowed the author [3, 5] to attribute
to every cycle an elementary damage coefficient D c defined by Eq. (15)
D c = Dm I N r . (15)
In the same way that the coefficient D c has been defined, author [5] defined
the reduction coefficient of the propagation phase by cycle R c, which is
expressed as R c = R PR/N R pR .Assuming that N pR = 0 and
N rR = N pR + N p R , (16)
the life duration coefficient R c has been obtained in [5] from Eq. (17):
^ N Fp C - N fr
Rc = P P— , (17)
C N p N p
This relation is very important because it allows to calculate the number of
PCcycles to failure in fatigue relaxation, knowing only N p , which can be
32 ISSN 0556-171X. npoôëeubi npounocmu, 2002, N2 4
Modeling o f Damage Interaction in Fatigue Relaxation
FCcalculated by Eq. (11) if the number of cycles to failure N R is known. The
latter one is calculated by using the Coffin-Manson rule.
4. Experim ent.
4.1. M aterial [1, 2, 3]. Our study has been performed on the alloy 800, at
the state of grade 1 (980oC). This alloy presents a structure, which is entirely
austenitic. Its chemical composition and its mechanical properties at the delivery
state are given in Tables 1 and 2.
T a b l e 1
Material Chemical Composition [2, 3]
C S P Si Mn Ni Cr
0.0З- 0.06 0.015 0.015 0.07 100 З2.0 - З2.5 19.0 - 2З.0
0.0З 0.004 0.008 0.50 0.070 ЗЗ.З0 21.0
Mo Co Cu Ti Al Ti+Al N
- 0.25 0.75 0.З0- 050 0.10- 0.25 0.45- 0.75 0.0З0
0.05 0.02 0.02 041 0.1 З 0.54 0.014
Note. Here and in Table 2: over the line are given the data obtained according specification
Novatome-NIRA* and under the line - EDF analysis.
T a b l e 2
Mechanical Characteristics in As-Recieved State [2, 3]
2O0C 4OO0C
R0.002, RRm • A, % Z,% R0.002, R ,Rm ; A, % Z,%
MPa MPa MPa MPa
210 - З50 520 - 700 З0 - 160 455 - -
187-19З 542 - 544 52 - 5З 74 - 75 108-11З 452 - 456 50 69 - 71
4.2. Experim ental M ethod [3]. The test was carried out to estimate the
interaction fatigue creep have lead to specimen tests treated at state of grade 2,
obtained after a heat treatment at 1100oC during 30 min [3].
The fatigue tests consisted in subjecting specimens heated to 550OC to cyclic
deformation comprised between two limits of deformations up to the failure. In
the case of fatigue relaxation, the deformation level is maintained constant during
an interval of variable relaxation (Fig. 1). All the tests have been conducted using
test machines with electromechanic servocontrol [3].
5. Proposed Model.
5.1. Introduction and Objectives. In order to describe the evolution of
damage in continuous fatigue, a mechanical law proposed by Chaboche [3, 6, 7]
has been considered. To describe the evolution of damage during fatigue-
relaxation tests, the intergranular damage measurement by cycle has been used. It
is characterized by the coefficient D c deduced from metallurgic observations and
obtained according to the Levaillant model described above. The following issues
have to be considered:
1. Changing this metallurgic damage D c into a mechanic damage.
2. Combining the fatigue damage and that of creep so as to have a law
describing the interaction between the fatigue and creep damages, which happens
simultaneously when the fatigue relaxation is present.
ISSN G556-Î7ÎX. Проблемыг прочности, 2GG2, № 4 33
A. El Gharad, G. Pluvinage, Z. Azari, et al.
Fig. 1. Loading modes and cycle type obtained during continuous fatigue (a) and fatigue-relaxation
tests (b).
3. Comparing the results of prediction obtained by the present model to these
obtained by experiment and the two previous models.
5.2. Model Description. Our model is based on a simple idea, which
assuming that there are tow damage mechanisms in fatigue relaxation.
a) The f ir s t mechanism : It characterizes the continuous fatigue damage,
where the material degradation is considered as being directly related to the
applied stress during the continuous fatigue tests as focused in [4, 6, 7]. Here the
model of failure is transgranular and presents more often faces marked by fatigue
striations witnessing that there is crack propagation from one or two sites of the
crack initiation [3, 4].
b) The second mechanism: It characterizes the creep damage and stresses
giving, thus, the influence of the hold time, which corresponds to the presence of
intergranular microcracks in the test tube mass. This damage is generated by the
importance of the viscoplastic distortion £ Vp or the relaxed stress o rt. This
second mechanism is based on metallographic observations of the broken tube
tests [3, 5].
5.2.1. Fatigue Case. The damage in fatigue is considered as a function,
which depends on the stress amplitude o a , the average stress o av and the
damage itself. This function has the following form (18).
d D fat = h(0 a > 0 av > D fat )dN , (18)
34 ISSN 0556-171X. npo6n.eubi npounocmu, 2002, N2 4
Modeling o f Damage Interaction in Fatigue Relaxation
where o a is the stress amplitude of a stabilized cycle, o av is the average stress,
and D f at is the fatigue damage.
5.2.2. Creep during Relaxation. Here, the creep damage is a function which
depends on the relaxed stress o rt (or o t max), the applied stress o vav and of the
damage itself. This function has the form (19).
dD flu g [o rt (or o tmax ), o vav, D c ]dt, (19)
where D f u is creep damage, o rt is relaxation stress, o tmax is the mplitude of
maximum stress, o vav is the actual average stress, and D c is the intergranular
damage coefficient by cycle.
5.2.3. Damage Interaction in Fatigue Relaxation. The interaction of the two
factors is accounted for by adding two elementary damages defined by Eqs. (18)
and (19):
dD t = dD fat dD flu ,
dD t = h( o a , o av , D fat ^d N g [° rt (or o t max), o vav , D c ]dt- (20)
It is notew orthy that the two functions h and g can be determined
independently, respectively either by pure fatigue test for the function h or by
creep test with an imposed loading for the function g .
5.3. Proposed Equation for Description of Damage Evolution in Fatigue
Relaxation. We propose a new damage law in fatigue relaxation. The
particularity of this law is that it allows the interactive coupling, that is, the
multiplying coupling of the damage. The global law is not a simple sum of the
elementary damages. It is assumed that in fatigue relaxation with hold under
tension, the fatigue intervenes accelerating of the main crack propagation. The
proposed equation describing the evolution of fatigue relaxation damage is
obtained by multiplying the coefficient representing the intergranular damage
with the number of cycles N, in power y ( Ae t ), that is a function of the total
imposed strained N y(Aet)D c . The creep intervenes with the term D c ,
introducing the creep damage during fatigue relaxation tests. For the*
homogenisation of this equation, the coefficient D c has been considered as the
_1
ratio between the intergranular damage coefficient per cycle D and N R that
yields:
* D c
D c = N r (21)
In order to characterize the fatigue influence, some parameters have been
taken into account, such as the number of cycles N, the ratio ] / NR 1 and
N
parameter £, which is a correction factor defined as £ = k n ~ ■ The term k is a
constant to be determined.
ISSN 0556-171X. npoôëeubi npounocmu, 2002, N 4 35
A. El Gharad, G. Pluvinage, Z. Azari, et al.
The final rule for the evolution of total damage D totai in fatigue relaxation,
is described by Eq. (22):
D total 1 1— k-
N
N r
N y (Aet) D c
N y—1
R )
(22)
m
where y is a coefficient depending on the material and the imposed strain
amplitude y = f (Ae t ), 2 is a coefficient depending on the material, m is a
coefficient allowing to adjust the concavity of the curves of damage evolution as a
function of life duration, and £ is a correction factor introducing the influence of
the fatigue damage evolution.
The damage criteria applied in this study are:
a) if N = 0, then D = 0;
b) if N = N r , then D = 1.
5.4. Calculation of the N um ber of Cycles to Failure N R . At failure, the
total damage is equal to the unit:
k [ N R (A£t) D *] * = 1,
where
* D c
D c = ---- —r.
n R-1
Equation (14) can be obtained as
n r = 7 t It J . (23)
1 11 ' 1/2
Dc
This relation allows computing of the number of cycles to failure N R in
fatigue relaxation, only knowing the intergranular damage coefficient measured
per cycle D c and the two coefficients k and 2.
5.5. D eterm ination of Coefficients of the Model for the Life Prediction.
In order to identify the parameters 2, k , and m considered to be positives, it is
assumed that the equation giving the evolution of the total damage D total is
positive. The evolution of the intergranular damage D c is studied first.
5.5.1. Evolution o f Intergranular Dam age D c .
Evolution o f D c as a function o f the relaxation stress (o rt). The coefficient
D c given by Eq. (21), is in good correlation with the relaxation stress o rt [3], as
shown in Fig. 2. It is considered that the responsible parameter of the grain joint
damaging is the viscoplastic deformation e vp provoked by the relaxation stress
o rt that is generated while holding under relaxation. The linear regression
performed on the couples (D c , o rt) for Ae t = 1.5% yields
Dc (1.5%) = 18 ■ 10—4 (o rt ) 261. (24)
36 ISSN 0556-171X. npo6n.eubi npounocmu, 2002, № 4
Modeling o f Damage Interaction in Fatigue Relaxation
Relaxation stress art, MPa
Fig. 2. Correlation between the damage coefficient Dc and the relaxation stress ort [3].
Evolution o f D c as a function o f hold time (tmt). Figure 3 shows the
evolution of the intergranular damage per cycle D c as a function of the hold time
under tension tmt for Ae t = 1.5% [3].
The relation found between D c and tmt has the form (25):
D c (15%) = p ( tmt) '“(L5%). (25)
The linear regression allows the identification of the two coefficients:
H = 0.4468 and p = 5.2033-10“ 5.
For long-term predictions with hold time tmt = 60,000 min and Ae t = 1.5%,
Ae t = 0.8%, and Ae t = 0.6%, Eqs. (25)-(27) can be used as extrapolation rules to
determine the coefficient D c.
Hold time tmt, min
Fig. 3. Correlation between the intergranular damage coefficient Dc and hold time tmt [3].
ISSN 0556-171X. Проблемы прочности, 2002, № 4 37
A. El Gharad, G. Pluvinage, Z. Azari, et al.
5.5.2. M ethod o f Determination o f the Coefficient D c fo r Small-Scale
Deformations. The method is based on the use of Eq. (25) that allows computing
of the coefficient D c for long-term hold duration and large-scale deformations,
for example: tmt = 10,080 min and Aet = 1.5%. However, the evolution of D c
has to be known for the other strain amplitudes Ae t = 0.8% and Ae t = 0.6%.
Assum ption: In order to determine the two coefficients D c(0.8%) and
D c(0.6%) for various hold periods, it is assumed that the evolution of these
coefficients as a function of the hold time (tmt), is similar to the one
corresponding to D c(1.5%), as shown in Fig. 3. As a matter of fact, the evolution
of D c(0.8%) and D c(0.6%), as a function of tmt, can be given by Eqs. (17) and
(18).
D c (0.8%) = p 1( tmt) (26)
D c (0.6%) = p 2( tmt) ̂ (0-6%)^M1-5%). (27)
In order to determine the two coefficients p 1 and p 2, measurements of D c are
available for the hold period tmt = 90 min - Ae t = 1.5% and Ae t = 0.8%:
D c = 28-10“ 5,
D c = 8 -10_ 5.
However, measurements concerning D c (0.6%) are not available and they have to
be determined herewith.
5.5.3. Determination o f the Coefficient D c(0.6%). Figure 4 shows the
evolution of the coefficient D c as a function of total strain Ae t (%) for the hold
time of 90 min, and this plot allows to obtain D c (0.6%) for the hold time
of 90 min. For this reason, it is assumed that D c (0.6%) has a linear evolution as a
function of the strain Ae t (%).
Total deformation, %
Fig. 4. Evolution of Dc as a function of Aet (%).
The evolution of D c as a function of total strain amplitude As t (%) is given
by Eq. (19), which allows one to determine D c for any total strain amplitude.
38 ISSN 0556-171X. npoôëeMbi npounocmu, 2002, N 4
Modeling o f Damage Interaction in Fatigue Relaxation
Particularly for Ae t = 0.6%, the estimated result by this equation is D c(0.6%) =
2.2856-10- 5 .
D c = 28.572( Ae t ) - 14.857. (28)
As shown in Fig. 5, the evolution of the coefficient D c(1.5°%) is given as a
function of hold duration (tmt) in logarithmic coordinates. Two parallel lines to
the evolution line of D c(1.5°%) have been drawn from the two points representing
the two couples ofvalues: (90 m in -8 -1 0 - 5 ) and (90 m in -2 .2 8 5 6 -10- 5 ). These
two lines represent the evolution of D c (0.8%) and D c (0.6%).
Hold time tmt, min
Fig. 5. Correlation between the coefficient Dc and hold time tmt for three levels of deformation
and the determination of the coefficients pi and / [3, 8].
We have established that:
1. Consequently to the equality of slopes, //(1.5%) = //(0.8%) = //(0.6%) =
0.4468.
2. The linear regression allows to identify coefficients of Eqs. (17) and (18).
Finally, the following relations are obtained:
D c(1.5%) = 5.2033 • 10“ 5 ( tmt ) 0 4468 , (29)
D c (0.8%) = 1.0713 • 10_5( tmt ) 04468, (30)
D c (0.6%) = 0.3061-10_5( tmt ) 04468. (31)
5.5.4. Determination o f Coefficient y(As t ). The fitting of the relation
N r = (D c )_1/7 to the couples (N R p , D — red) in Fig. 6 for various hold
periods at As t = 1.5% made it possible to obtain values of y(1.5%) and y(0.8%)
for tmt = 90 min (see Table 6).
It is found that values of 7(1.5%) vary between a maximum and a minimum.
It is thought that is the same for the values of y(0.8%) and y(0.6%). It is thought
that there is a maximum and a minimum around the points obtained respectively
for As t = 0.8% and As t = 0.6% with tmt = 90 min. The estimation of the values
ISSN 0556-171X. npoôëeubi npounocmu, 2002, N2 4 39
A. El Gharad, G. Pluvinage, Z. Azari, et al.
of y(0.8%) and y(0.6%) for various hold periods has been obtained using the
relation y (Ae t ) = 1.2118 + 0.1478Ae t (see Table 3). The evolution of y[Ae t (%)]
deduced this way, as a function of the total strain Ae t (%) is given in Fig. 6.
T a b l e 3
Values of k for Various Hold Durations
tmt,min 10 30 300 1440 10080
(NrDc )-1 8.7870 7.9926 7.1537 6.015 5.2247
y = 1.2118+ 0.1478Aet
Total deformation Aet , %
Fig. 6. Determination of y for strain amplitudes 0.8 and 0.6% [8].
5.5.5. M ethod o f Identification o f Coefficients k , X, m, and y.
a) Determination o f the value o f the coefficient k. In order to calculate the
value of k, it should be verified that D totai < 1. For this reason, it is necessary that
the expression (20) be positive.
1 - U -
N
N
( N y ( A£t ) D * ) Я
R
> 0.
Since
N
N R
( N y(A£t )D * ) Я < 1.
(32)
(33)к
Therefore, it is necessary that
к < — ( N y (A£t ) D * ) - .
N c
If the function g ( N ) is considered to be defined by (27):
g ( N ) = NN R ( N y (Д£' ) D *c ) ~ * . ( 3 4 )
40 ISSN 0556-171X. Проблемы прочности, 2002, N2 4
Modeling o f Damage Interaction in Fatigue Relaxation
The function (34) is an increasing function, as its derivative g '( N ) is < 0.
As a matter of fact,
g '( N ) = [ N r (D* ) - ] [ - (yA) - 1]N-(yA)-2. (35)
It is assumed that
k = g ( N r ) = min{g( N ), N > 0}. (36)
This allows to obtain k by the following relation:
k = ( N R (A£‘) D* ) - = ( N r D c ) - . (37)
A value of k is obtained for each test.
—X For D totai < 1 for all the tests, the condition k < inf ( N r D c) is necessary.
Assum ption: If it is assumed that A> 1, then k < in f (N r D c) -1 .
The five values obtained for (N r D c) -1 , for five hold periods, respectively
are given in Table 3.
It is accepted that k = 5.224.
b) Determination o f the value o f the coefficient A. The coefficient A can be
log k
determined from Eq. (14), with A = - ---------------- and k = 5.224. For the five
log( n r D c )
different hold times tmt, respectively: 10, 30, 300, 1440, and 10080 min, the five
following values are obtained (Table 4).
T a b l e 4
Values of A for Various Hold Durations
tmt. min 10 30 300 1440 10080
A 0.7607 0.7954 0.8402 0.9214 0.9999
Choice: A> 1. In our case, the five computed values are < 1. It is accepted
A = 1.
c) Determination o f the value o f the coefficient m. In order to determine the
value of coefficient m, the conditions of increasing and of concavity D totai have
been used.
(i) D totai is increasing as its derivative (D totai )' is positive.
1
(Dtotai )' = — m(1+yA)(N r D c y
N r
n
n
( N r ( Aet )
R /
D c y
m—1
. (38)
(ii) For D total be concave, it is necessary that the second derivative
(D total)'' b e p °sitive-
ISSN 0556-171X. npoôëeubi npounocmu, 2002, № 4 41
A. El Gharad, G. Pluvinage, Z. Azari, et al.
( D total )" = N r m k(1+ yX )( D * ) A N yÀ—1
N u
l - l k -
N
N R
( N 7 (A£t ) D * ) ;
m— 2
(39)
This yields
N 7/1+1 * X
yX — k — — ( D c )A [m(1 + yX) — 1]
N R
> 0. (40)
or
m: 7Xn r
kN Ay+1( D * ) A
+1
Xy + 1 ' (41)
1
As the term (Xy + 1) is positive, it is taken that N = N r :
i - ^
m : 1 +
Xy
k ( N r D c ) '
1
Xy + 1
The five values obtained for the following expression
i
1+
Xy
k (N r D c Y Xy + 1
1
are given in Table 5.
T a b l e 5
Values m for Various Hold Durations
tmt. min 10 30 300 1440 10080
m 1.4045 1.3158 1.2184 1.0885 1.0001
Let’s take: m = 1.0001.
5.5.6. Damage Evolution. The application of Eq. (13) is given in Figs. 7 and 8:
Figure 7 shows the damage evolution D totai as a function of the ratio
N / N R for the fatigue-relaxation tests under Ae t = 1.5% and various hold
intervals. It is found that total damage increases since the first cycles and becomes
preponderant while N / N R tends towards 1.
It is found that the life duration is decreasing while:
- The total damage D totai increases sensitively since the first cycles et tends
towards its maximal value assigning its failure.
- The holding interval increases.
- The coefficient of intergranular damage per cycle D c increases.
Table 6 indicates the results obtained by applying different models.
Figures 9 to 12 show the life duration predictions N R (cycles) given by the
model of El Gharad compared to the experimental results [3, 8].
42 ISSN 0556-171X. npoôëeMbi npounocmu, 2002, N2 4
Modeling o f Damage Interaction in Fatigue Relaxation
T a b l e 6
Results of Life Predictions Obtained by the Model of El Gharad [8], Compared
to the Ones Obtained by the Two Models of Chaboche [3, 8] and Levaillant [3, 8]
A£t ,
%
Holding
times
m ,min
Life
duration
Nr
(cycles)
tests
d 1 -10“ 5 У Dc2) -10“ 5 Life prediction Nr (cycles)
by the models
El Gharad
[8]
Chaboche
[3, 8]
Levaillant
[3, 8]
1.5 10 794 14.333 1.4573 14.5572 13353) 744 634
30 439 28.500 1.4745 23.7824 6713) 408 514
90 360 28.000 1.2515 38.8538 7363) - -
300 198 90.600 1.4467 66.5359 2113) 158 278
1440 133 125.000 1.4048 134.1019 1533) 92 170
10080 60 319.000 1.4056 319.9079 603) 52 85
60000 - - 1.3998 709.8340 274) 37 48
0.8 90 1975 8 1.3311 7.9988 23933) 1229 2048
300 1.3311 13.6977 13974) - -
1440 1.3311 27.6074 6934) 784 1029
10080 1.3311 65.8592 2904) 384 530
60000 1.3311 146.1330 1304) 190 253
0.6 90 1.3124 8.7202 2054) 9976 3849
300 1.3124 17.5755 21954) - -
1440 1.3124 17.5755 10894) 930 3122
10080 1.3124 41.9274 4564) 191 2782
60000 1.3124 93.0316 2054) 39 2386
Notes. ̂ Values that are slightly higher than the ones obtained by tests in [3, 8]. 2) Values are
computed with Eqs. (16)—(18).3) Predictions obtained using D1. 4) Predictions obtained using D2).
0 0.5 1
N /N r
Fig. 7. Damage evolution as a function of N /N r in fatigue relaxation for hold intervals from 10 to
10,080 min, by means of the model [8].
ISSN 0556-171X. Проблемы прочности, 2002, № 4 43
A. El Gharad, G. Pluvinage, Z. Azari, et al.
Number of cycles N , cycles
Fig. 8. Comparison of evolutions of total damage Dtotai as a function of number of cycles N from
various fatigue relaxation tests by the model [8].
Hold time, min
Fig. 9. Life duration prediction by the model of El Gharad and comparison of results to the tests [3, 8].
Hold time, min
Fig. 10. Comparison of life duration prediction obtained by the three fatigue relaxation damage
models for Aet = 1.5% [3, 8].
44 ISSN 0556-171X. Проблемы прочности, 2002, № 4
Modeling o f Damage Interaction in Fatigue Relaxation
1 10 100 1000 10000 100000 1000000
Hold time, min
Fig. 11. Comparison of life duration predictions Nr (cycles) obtained by the three fatigue
relaxation damage for Aet = 0.8% [3, 8].
1 10 100 1000 10000 100000 1000000
Hold time, min
Fig. 12. Comparison of life duration N r (cycles), obtained by the three fatigue relaxation damage
models for Aet = 0.6% [3, 8].
6. Discussion. The difference between the three models is characterized by
the simplicity of use and the number of coefficients required to be determined for
each model.
The three models proposed, respectively, by J. L. Chaboche, C. Levaillant,
and A. El Gharad have been compared with the results of continuous fatigue tests,
fatigue relaxation and creep, obtained on the alloy 800 grade 2 at 550oC.
6.1. The Chaboche Model Specification. In order to use this model in
fatigue relaxation:
(i) It’s necessary to identify 17 coefficients, which is very difficult task to
perform.
(ii) It’s also necessary to know a ( t) during the relaxation cycle.
(iii) For a long term, a t min should be known. It can be determined
preliminarily by knowing the stress a t max and the relaxation rule. Finally, the
iterative calculation gives the number of cycles to failure in fatigue relaxation
N f R .
ISSN 0556-171X. npoôëeubi npounocmu, 2002, N 4 45
A. El Gharad, G. Pluvinage, Z. Azari, et al.
It is noteworthy that the long-term prediction result (for Ae t = 1.5%) is in
good accordance with the experiment o tmax and it is also reasonable for the
extrapolation to 1000 hours. Meanwhile, for Ae t = 0.6 and 0.8%, the calculation
of o rt is obtained when the minimal stress o t min is determined by the
relaxation rules, based on the hypothesis about o tmax.
6.2. The Levaillant M odel Specification. It should be noted that this model
is related to the stress o rt, and insofar as it is based on metallographic
observations, an exhaustive metallographic analysis must be performed in order to
apply this model to the case of fatigue relaxation.
RCIn continuous fatigue: We use the fatigue curve to calculate N R , and Eq. (11)
CF
to calculate N p .
In fa tigue relaxation: It is obligatory to know the relaxed stress o rt for the
total deformation Ae t and the hold time tmt.
Within the framework of this model, it is necessary to interrupt tests to
determine o t min or a relaxation rule to know o rt, from where the calculation ofFR
the number of cycles to failure in fatigue relaxation N R is possible.
It is noticed that the long-term prediction results (for Ae t = 1.5%) are in
good accordance with the experimental data on o tmax. Moreover, the obtained
results are sensitive to the relaxation rule used to extrapolate the relaxed stress
o rt , especially in the case of small-scale deformations ( Ae t = 0.6%).
Conclusions. This study has allowed us to give preference to the new model
which permits to predict the life duration in fatigue relaxation with small-scale
deformations and long-term hold periods. In fact, this model is based on a simple
concept of interaction of various damage mechanisms.
The relaxation process is characterized both by fatigue and creep
mechanisms which occur simultaneously when fatigue relaxation tests are carried
out.
In order to apply the above model, the following steps are to be made:
It has been chosen to use the equation proposed by J. L. Chaboche to
describe the fatigue damage [3, 4, 6, 7].
To describe the creep damage evolution, we strongly recommend to use the
parameter D c that characterizes the measure of the intergranular creep damage
per cycle according to the Levaillant method [3, 5].
The relation describing the creep damage progress is characterised by
parameter y( Ae t ) which is a function of the amplitude of the applied
deformation. It’s application has allowed to notice that the damage progress is a
function of the parameter D c, i.e., of the hold time in relaxation.
For the fatigue relaxation, Eq. (3) has been applied. This shows that the
fatigue damage becomes negligible compared to that of creep, which dominates
especially for the tests, whose maintain time is very long. The identification of
this model requires to know: a) 5 coefficients which are easily determined, and
b) measures of intergranular damage of large-scale deformations.
The life prediction is given by Eq. (15), obtained by applying the damage
criterion.
The results obtained are in good accordance with the experiment for
large-scale deformations and satisfying for the small-scale ones.
46 ISSN 0556-171X. npo6n.eubi npounocmu, 2002, N 4
Modeling o f Damage Interaction in Fatigue Relaxation
The comparison of the results obtained by our model with those obtained by
the model proposed by J. L. Chaboche [3, 4, 6, 7] has allowed us to show the
advantage of this model, especially for small-scale deformations. Insofar as the
estimation of the parameter y ( Ae t ) influences the obtained results, it is necessary
to be careful while identifying the model.
Р е з ю м е
Прогнозування утомної довговічності в умовах повзучості при температурі
550оС виконано для жароміцного і жаростійкого сплаву на нікелевій основі
(Inconel). Представлено методи еволюції пошкодження Шабоша (Chaboche)
і Леваяна (Levaillant), а також нову модель релаксації при сумісній дії
повзучості й утоми. За допомогою запропонованої моделі можна проводити
довготермінове прогнозування без використання великої кількості сталих.
1. Ph. Berge, “Choix de matériaux pour générateur de vapeur des réacteurs
surgénérateurs refroidis au sodium,” in: 2ème Colloque “Les Aciers
Spéciaux et Energie Nucléaire,” Paris (1976).
2. D. Guttmann, S. Licheron, and P. Spiteri, “Etude de l’alliage Fer-Nickel type
Incoloy 800 en vue de son utilisation dans les generateurs de vapeur de
réacteurs nucléaires à neutrons rapides. Troisième partie: Essais sur une barre
de fabrication Ugine,” HT/PV D.437 MAT/T40 (1979).
3. A. El Gharad, G. Pluvinage, and Z. Azari, “High-temperature fatigue:
Example of creep lifetime prediction for grade 2 alloy 800 at 5500C,”
Strength o f M aterials, No. 4, 36-42 (1994).
4. G.I.S. Rupture à Chaud., Rapport No. 5, Fascicule 4, Etude 2-3, (Déc. 1982),
E.D.F. Centre de recherche, Les Renardières, France (1982).
5. C. Levaillant, Approche M étallographique de l'Endom m agem ent d 'Aciers
Inoxydables Austénitiques Sollicités en Fatigue Plastique ou en Fluage:
Description et Interprétation Physique des Interactions Fatigue-Fluage-
Oxydation, Doctorat d’Etat, Université de Compiègne (1983).
6. J. L. Chaboche, M écanique des M atériaux Solides, Ed. Dunod, BORDAS,
Paris (1985).
7. J. L. Chaboche, “Une loi différentielle d’endommagement de fatigue avec
cumulation non linéaire,” Revue Française de M écanique, Nos. 50-51
(1974).
8. A. El Gharad, Etude et M odélisation de l'Endom m agem ent en Fatigue
Oligocyclique, en Fluage et en Fatigue-Relaxation des M ateriaux (Cas de
l'A lliage 800 Grade 2). E t Prévision des Durées de Vie, Doctorat d’Etat
Es-Sciences. Faculté des Sciences de Rabat-Maroc (in print).
Received 09. 10. 2001
ISSN 0556-171X. Проблеми прочности, 2002, № 4 47
|
| id | nasplib_isofts_kiev_ua-123456789-46875 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0556-171X |
| language | English |
| last_indexed | 2025-12-07T17:23:24Z |
| publishDate | 2002 |
| publisher | Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| record_format | dspace |
| spelling | Gharad, A. El Pluvinage, G. Azari, Z. Elamraoui, A. Kifani, A. 2013-07-07T17:15:41Z 2013-07-07T17:15:41Z 2002 Modeling of Damage Interaction in Fatigue Relaxation for Long-Term
 Life Prediction. Case of Alloy 800 Grade 2 Study at 550°C / A.El Gharad, G. Pluvinage, Z. Azari, A. Elamraoui, A. Kifani // Проблемы прочности. — 2002. — № 4. — С. 28-47. — Бібліогр.: 8 назв. — англ. 0556-171X https://nasplib.isofts.kiev.ua/handle/123456789/46875 539.4 Prediction of fatigue life in creep-fatigue at a
 temperature of 550°C has been made for an
 Inconel alloy with the help of 3 models of damage
 evolution: the Chaboche model, the
 Levaillant model, and a new model of creep fatigue
 relaxation. The main advantages of the
 new model are that it requires only few constants
 and provides a high accuracy of the fatigue
 life prediction. Прогнозирование усталостной долговечности в условиях ползучести при температуре 550°С
 выполнено для жаропрочного и жаростойкого сплава на никелевой основе (Inconel). Представлены
 модели эволюции повреждения Шабоша (Chaboche) и Леваяна (Levaillant), а
 также новая модель релаксации при совместном действии ползучести и усталости. Предложенная
 модель позволяет осуществлять долгосрочное прогнозирование без использования
 большого количества постоянных. Прогнозування утомної довговічності в умовах повзучості при температурі 550°С виконано для жароміцного і жаростійкого сплаву на нікелевій основі (Inconel). Представлено методи еволюції пошкодження Шабоша (Chaboche) і Леваяна (Levaillant), а також нову модель релаксації при сумісній дії повзучості й утоми. За допомогою запропонованої моделі можна проводити довготермінове прогнозування без використання великої кількості сталих. en Інститут проблем міцності ім. Г.С. Писаренко НАН України Проблемы прочности Научно-технический раздел Modeling of Damage Interaction in Fatigue Relaxation for Long-Term Life Prediction. Case of Alloy 800 Grade 2 Study at 550°C Моделирование разрушающего взаимодействия при усталостной релаксации при прогнозировании длительной долговечности для сплава 800 (тип 2) при 550°C Article published earlier |
| spellingShingle | Modeling of Damage Interaction in Fatigue Relaxation for Long-Term Life Prediction. Case of Alloy 800 Grade 2 Study at 550°C Gharad, A. El Pluvinage, G. Azari, Z. Elamraoui, A. Kifani, A. Научно-технический раздел |
| title | Modeling of Damage Interaction in Fatigue Relaxation for Long-Term Life Prediction. Case of Alloy 800 Grade 2 Study at 550°C |
| title_alt | Моделирование разрушающего взаимодействия при усталостной релаксации при прогнозировании длительной долговечности для сплава 800 (тип 2) при 550°C |
| title_full | Modeling of Damage Interaction in Fatigue Relaxation for Long-Term Life Prediction. Case of Alloy 800 Grade 2 Study at 550°C |
| title_fullStr | Modeling of Damage Interaction in Fatigue Relaxation for Long-Term Life Prediction. Case of Alloy 800 Grade 2 Study at 550°C |
| title_full_unstemmed | Modeling of Damage Interaction in Fatigue Relaxation for Long-Term Life Prediction. Case of Alloy 800 Grade 2 Study at 550°C |
| title_short | Modeling of Damage Interaction in Fatigue Relaxation for Long-Term Life Prediction. Case of Alloy 800 Grade 2 Study at 550°C |
| title_sort | modeling of damage interaction in fatigue relaxation for long-term life prediction. case of alloy 800 grade 2 study at 550°c |
| topic | Научно-технический раздел |
| topic_facet | Научно-технический раздел |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/46875 |
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