- Email: [email protected]
Low-cycle fatigue of welded joints: coupled initiation propagation model
Nuclear Engineering and Design 228 (2004) 179–194
Low-cycle fatigue of welded joints: coupled initiation propagation model Yazid Madi a,∗ , Naman Recho b,1 , Philippe Matheron c,2 a
c
Centre des Matériaux, Ecoles des Mines de Paris, B.P. Evry Cedex 91003, Paris, France b Lermes—Université, Blaise Pascal, B.P. 206, 63174 Aubierre, France DM2S/SEMT/LISN, Commissariat à l’Energie Atomique, Centre d’Etudes de Saclay, 91191 Gif-Sur-Yvette, France Received 19 February 2003; received in revised form 3 June 2003; accepted 23 June 2003
Abstract This paper deals with the low-cycle fatigue (LCF) design of welded structures, the aim being the critical analysis of the rule used in the RCC-MR [Design and construction rules for mechanical components of FBR nuclear islands, AFCEN, 1993], for the design and construction of fast breeder reactors. The study takes into account the evolution of the material behavior laws and damage accumulation during the fatigue loading. The adopted model consists of analyzing separately the behavior and the damage evolutions. It allows us to determine the damage ratio corresponding to initiation and propagation of a significant crack in order to determine the life duration. This model suggests the existence of a threshold level of loading, above which micro-cracks initiate. The initiation fatigue life can then be neglected below the threshold level. This work shows also that the RCC-MR rules are valid below this threshold load level. © 2003 Elsevier B.V. All rights reserved.
1. Introduction As a result of their size, fast breeder reactor vessels are not made of a single material but of many parts welded together. The welds frequently represent a weak point, the fatigue resistance of welded joints depends on the surrounding geometry, on the loading to which they are subjected and on the presence of two materials with different mechanical properties. The lower fatigue resistance of welded structures is provided for in the French RCC-MR code by applying the reduction factor Jf to the Manson–Coffin fatigue ∗ Corresponding author. Tel.: +33-160-76-30-12; fax: +33-160-76-31-50. E-mail addresses: [email protected] (Y. Madi), [email protected] (N. Recho), [email protected] (P. Matheron). 1 Tel.: +33-470-02-20-28; fax: +33-470-02-20-78. 2 Tel.: +33-169-08-28-47; fax: +33-169-08-87-84.
curve of the parent metal. For 316L(N), Jf is equal to 1.25 (DCRC, 1991). In order to better assess this factor, using previous work on large plates and part of a welded vessel, Berton (1996) and Le Ber et al. (1998) showed that the mechanical behavior of a welded joint is influenced by the geometry of the weld and by the interaction of the different cyclic plastic behavior of the two materials: base metal (BM) and weld metal (WM). The Jf value is a function of the structures. In order to improve this method of design, it would be necessary to establish experimental procedures allowing its characterization. With this intention, a new campaign (named FFAST) was performed on welded joint specimens extracted from butt-welded pipe connections (uniaxial tensile-compressive load). The purpose of this study is to experimentally and numerically analyze the behavior of specimens taken from a butt-welded annular joint to determine coefficient Jf .
0029-5493/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.nucengdes.2003.06.017
180
Y. Madi et al. / Nuclear Engineering and Design 228 (2004) 179–194
A new experimental approach was adopted in order to analyze the local behavior of the welded joint (see companion paper). Non-linear calculations integrating the consolidation of two materials were done in order to deduce the fatigue life to crack initiation (Madi, 2001). The approach developed in this paper allows us to analyze the design procedure used in RCC-MR rules, which is based on the reduction factor concept Jf . Finally, a new procedure is proposed to study the fatigue life of welded structures.
2. Position of the problem Many results in technical literature show that the curve of Manson–Coffin, usually used to predict crack initiation, represents actually the two stages of the fatigue process, namely: crack initiation and crack propagation. The results of Levaillant (1984) on steel VIRGO 316 L bring quantitative elements to this observation. Within the framework of this study, the fatigue life will be made up of two parts relating, respectively, to the initiation and the propagation of crack. To evaluate each part of the fatigue life, two approaches are associated: one is based on the fracture mechanics and the other on the damage mechanics. The fracture mechanics approach is applied when a crack of significant size (higher or equal to 0.5 mm) exists within the structure. Contrarily, the approach based on damage mechanics deals with the micro-defects. In the field of the cracks known as “short” (=0.1 mm), some studies seem to be necessary. It is to be noted that the elastic–plastic fracture mechanics makes it possible to describe the evolution of the micro-defects in this field. The aim of this work is to propose a specific approach which will take into account the initiation and the propagation parts of the micro-cracks and the evaluation of the fatigue life observed experimentally. First, we describe the approach in order to evaluate the fatigue life on smooth test specimens in the case of a numerical cycle by cycle description of loading. Then, we apply this approach to interpret the Bi-materials tests under 600 ◦ C with regard to the fracture pictures observed in experiments according to the level of loading. Finally, we manage to justify the use of the fatigue life reduction factor Jf in the field of the low levels of loading.
3. Initiation of a “significant” crack The fatigue design based on the critical plane approach is that which comes closest to the physical reality of fatigue damage. During low-cycle fatigue, the material is submitted to a plastic strain field at each applied cycle. This deformation is related to the movement of dislocations in the slip surfaces, movement induced by shear. This ultimate analysis considers the shear strain amplitude as a significant parameter. This step, initiated by Brown and Miller (1979), was followed by many researchers. These authors consider that the critical plane corresponds to the maximum shear plane. This plane, which controls the fatigue crack initiation, is defined like a combination of maximum amplitude of the shear strain γ max and the amplitude of the normal strain εn in the same plane submitted to γ max . A recent bibliographical study (You and Lee, 1996) shows that multiaxial fatigue is well predicted by using the linear combination of the modified Brown and Miller’s parameters γ max and εn proposed by Socie and Shield (1984): σn0 γmax,p + εn,p + (1) = γR (NR )co E where γ max,p and εn,p are the plastic values of the Brown and Miller’s parameters, σ n0 is the average stress applied on the plane of maximum shear stress, co and γR are two parameters which depend on the material, E is the Young’s modulus, and NR is the number of cycles to failure. As for the particular case of our study, the load applied on the Bi-materials specimens and on the welded tube is an uniaxial type.3 This formulation can be used because the strains are lower than the limits of the domain of the tests validity ( ε < 2%) (Socie and Shield, 1984; Socie et al., 1985). Finite element calculations show that the presence of the weld does not influence significantly the degree of triaxiality of the stress field in the neighborhood of the weld. One can thus reasonably regard the overall stress field as uniaxial. By noting ε1 , ε2 , ε3 the principal strains (ε1 > ε2 = ε), ε1 being the applied strain, the Brown and Miller’s parameters are written as: 3 In this paper, the effect on damage of biaxial loading (as internal pressure in the tube) has not been studied. The application of the model to multiaxial loading is assumed to be covered by Eq. (1).
Y. Madi et al. / Nuclear Engineering and Design 228 (2004) 179–194
181
ε1 − ε3 ε1 + ε3 and εn = (2) 2 2 In the case of alternative loading and taking into account the relation (2), Eq. (1) becomes:
we determine an approximate value of this one with the relation established by Levaillant (1984):
ε1,p = γR (NR )co
Let us note that this relation is in agreement with other results such as those of Manson (1966) which proposes a simple relation. The difference between the relation of Levaillant and that of Manson comes from the definition of the initial crack size a0 obtained at the end of the initiation period. Manson estimates the value a0 at 50 m while Levaillant selects 20 m.
γmax =
(3)
One finds consequently the Manson–Coffin approach usually used in the nuclear design. One can thus use the criterion in terms of strain associated with the Manson–Coffin approach within the framework of this study. However, the definite concept of a number of cycles to initiation calculated as being the number of cycles to failure NR of a smooth specimen will not be adopted. One will use the definition rather approaching the physics recommended by Levaillant (1984) and Jacquelin et al. (1983), which states that NR of the Eq. (3) represents the initiation fatigue life defined as the number of cycles necessary to initiate the microscopic crack. The length of this microscopic crack corresponds to the size of the grain (estimated at a0 = 20 m). The Eq. (3) becomes then: ε1,p = γR (Ni )co
(4)
3.1. Determination of the number of cycles to crack initiation The strain variation ε being given, it makes it possible to determine the number of cycles to failure NR by the relation of Manson–Coffin. Being difficult to estimate the number of cycles to crack initiation Ni ,
Ni = NR − 12NR0.62 + 0.226NR0.90 + 74
(5)
4. Analysis of crack propagation 4.1. Observations 4.1.1. Location and fracture mode The observation of the fracture topographies shows the existence of two types of failure. At a low level loading ( εbm < 0.6%), one notes that the crack initiation occurs on the surface (type I, Fig. 1). At high level loading ( εbm < 0.6%), the crack initiation does not occur on the surface but in the interface between weld and parent metal. The presence of a transverse defect comparable with a crossing centered crack implies a different type of failure (type II, Fig. 1) directed from an internal defect to the external surface of the specimen. We also point out that for the loading level εbm < 0.6%, the two types of failure are observed. The rupture of specimen B573 is done according to
Fig. 1. Fracture mode of the Bi-materials tests.
182
Y. Madi et al. / Nuclear Engineering and Design 228 (2004) 179–194
type I failure in 6670 cycles while that of specimen B576 is done according to the type II failure in 1305 cycles (Madi, 2001). 4.1.2. Striation spacing measurement Several investigators (Jacquelin et al., 1983; Wareing and Vaughan, 1977) have shown that, within the strain range explored in our study and for this type of steel, a relationship between striations and fatigue cycles can be assumed. From direct observation of the fatigue fracture surface, a good estimate of the number of cycles spent to propagate the cracks can be obtained by plotting striation, i, versus crack depth, a, and by integrating along the fitting curve. This procedure has been used only on B571 specimens tested at room temperature (see Fig. 2 and companion paper) where fracture occurs according type II mode. The life part corresponding to crack initiation was determined (Ni = 19 cycles). Compared to the fatigue life (NR = 794 cycles), the result shows that the crack initiation represents a negligible part of the type II fracture mode.
4.2. Determination of the number of cycles during crack propagation In the light of these observations, one can suppose the presence of a threshold loading level, having the value εbm < 0.6%. in the case of the welded junction in 316L/16-8-2 of our study, beyond whose the micro-defects present in the welded junction are activated more quickly close to the interface. That led to a very fast crack initiation period. It will be noted that for the weld having an X shape and for the alternating tensile loading, the critical points associated with the maximum variations of strain, obtained by finite elements calculations, are situated at the weld toe. It is thus normal that the micro-defects present in this zone are responsible for crack initiation. In addition, when the loading level εbm is higher than the threshold level ( εbm = 0.6%), the number of cycles during crack propagation NP is given while following the following step: By supposing that the crack growth rate is connected to the variation of J-Integral J, Usami et al.
Fig. 2. Striation spacing measurements of type II fracture mode.
Y. Madi et al. / Nuclear Engineering and Design 228 (2004) 179–194
183
10
Crack depth a (mm)
Np_t = 0.51mJ Np_c
1
Transverse defect semicircular defect
0.1
0.01 0
500
1000
1500
2000
2500
Num ber of cycles N Fig. 3. Evolution of crack propagation according to the micro-defects.
(1983) proposed an approach which is summarized in Fig. 3 and described below. The ratio of the number of cycles to crack propagation of a transverse defect Np t with those of a semicircular defect Np c can be evaluated with 0.51mJ , where mJ is the exponent of the following propagation law: da = CJ ( J)mJ dN
(6)
J and NP are given, respectively, by the following equations: J ∼ = (2π We + f(n) Wp )Qa Q−mJ (a1−mJ − a01−mJ ) (1 − mJ )g(n) R 1−n f(n) ∼ = (n + 1) 3.85 √ n + πn
NP =
We =
( σ)2 , 2E
Wp =
(7) (8)
σ εp 1+n
Where, E is the Young’s modulus, n is the cyclic hardening exponent in the relation determined by the cyclic stabilized behavior εp /2 = ( σ/2k)1/n , Q is fixed
at 0.51 for a semicircular defect and 1 for a transverse defect. For 304 STEEL, one showed that the ratio Np c /Np t (Np c , a number of cycles for a semicircular defect; Np t , a number of cycles for a transverse defect) is about 2.64 for a temperature of 550 ◦ C. At 600 ◦ C, without the crack propagation law for 316L STEEL, one will base oneself on these bibliographical data to estimate this coefficient. One knows that for the 316L, the exponent m of the Paris law da/dN = C( K)m which is a function of the temperature always lies between 2 and 4. One has more precise values with the temperature of 650 ◦ C, near to the temperature of our study. Curtit (1999) and Polvora (1998) obtain relatively coherent respective values with m = 2.34 and 2.46. We adopt for our study the average value of m = 2.4. In confined plasticity, the relation binding the stress intensity factor to the J-Integral is the following one: J =
K2 E
(9)
As a result, the exponent mJ of Eq. (5) is related to the exponent m of Paris law by: m = 2mJ . The value of mJ is then equal to 1.2.
184
Y. Madi et al. / Nuclear Engineering and Design 228 (2004) 179–194
Consequently the ratio Np c /Np t is equal to 2.24. Finally, the number of cycles to crack propagation is calculated as follows: • For a semi-circular defect ( εbm < 0.6%): Ni = f(NR ) = NR − 12NR0.62 + 0.226NR0.90 + 74 Np
c
= NR − Ni (10)
• For a transversal defect ( εbm > 0.6%): Ni = 0 Np t = 0.51mJ Np
(11) c
5. Evaluation of fatigue damage The crack initiation and the crack propagation periods are taken into account in the fatigue damage law. For that, the formulation used is that developed by Savalle and Cailletaud (1982). We adopt this formulation by using the Manson–Coffin approach. Thus, we suppose that a damage variable DiF exists and equal to zero for non-damaged material. It is equal to 1 when a micro-crack initiation occurs. Start from the moment that develops “the true” fatigue damage DF which is measurable at the macroscopic level. The crack initiation and the crack propagation periods are interpreted by the following relations: dN dDiF = (12) Ni ( ε) H(DiF − 1)dN dDF = NP ( ε)
(13)
Where H is a Heavyside function, H(DiF − 1). It is defined as follow: 0 if DiF < 1 H(DiF − 1) = (14) 1 if DiF ≥ 1 The integration of the first relation gives the relationship between DiF and the variation of applied strain. At the moment when DiF = 1, the number of cycles carried out is Ni (a number of cycles to crack initiation) and the fatigue damage variable DF , which remained null until there, starts to move. The failure is obtained when DF = 1, by carrying out NP additional cycles in the crack propagation period, in other words when: NR = Ni + NP
(15)
In the case of a smooth specimen, the fatigue damage variable DF is calculated by the use of the Manson–Coffin’s curves. The use of Manson–Coffin curves is thus a simple matter to deal with the problems of low-cycle fatigue since it makes it possible to determine in a simple practical way, the failure of a smooth specimen (including the periods of micro-crack initiation and micro-crack propagation) corresponds to the initiation of a crack of about 1 mm in depth in a structural component. The sequence effect is taken into account by this formulation provided that the evolution curve of the damage according to the fatigue life is independent of the loading. Also, one finds by this method the technique suggested by Manson and Halford (1981) based on the “Double linear Damage Rule.”
6. Analysis of Bi-material tests under 600 ◦ C The analysis of the fatigue life of Bi-material tests is carried out while following the step discussed in the paragraphs above. This method consists in determining the numbers of cycles NiWJ to crack initiation and NPWJ to crack propagation of the welded joint by integration of the relations (12) and (13). The level of total applied cyclic loading εbm being known, the calculation of the rates of use of crack initiation and crack propagation is carried out as follows: 1. When the level of applied loading εbm is lower than the threshold of 0.6%, the numbers of cycles to crack initiation Ni and to crack propagation NP of the relations (12) and (13) are determined by the relation (9). Thus, crack initiation is supposed to initiate on the surface of the welded joint. 2. When the level of applied loading εbm is higher than the threshold of 0.6%, the number of cycles to crack initiation NiWJ is neglected (NiWJ = 0). The damage variable to initiation DiF is thus equal to 1. Only the damage of propagation DF is considered in calculation. The number of cycles to crack propagation NP of the relation (12) is determined by the relation (10). Let us note that the phase of propagation is accelerated in order to take into account the shape of the initiated transverse defect. 3. When the level of applied loading εbm is equal to the threshold of 0.6%, one determines the number
Y. Madi et al. / Nuclear Engineering and Design 228 (2004) 179–194
of cycles to failure according to the two earlier alternatives 1 and 2. The cycle by cycle finite element calculations described below, are carried out by taking into account two materials in order to model the test specimen. To show the results of calculations, we will use two specific terms: • Damage fraction per cycle: damage for one cycle of loading • Damage accumulation: sum of the damage fraction per cycle up to the considered cycle From a behavioral point of view, calculations (Madi, 2001) showed that the strain varies cyclicly in the
185
weld. In fact, whereas the strain variation is constant in the parent metal, the strain variation in the weld increases with cycling. So the maximum damage fraction per cycle also varies with cycling. 6.1. Finite element calculation 6.1.1. Simplified finite element calculation 3D finite element calculations were carried out using the reduced cyclic curves attained at half-lifetime as the response law. The material model is an isotropic elastoplastic one of the Ramberg–Osgood type. The calculations indicate the points where the strain concentration is the highest (Fig. 4). These agree with the experimental observations. In this approach, the
Fig. 4. Location of maximum equivalent strain in the welded joint, σ/2 = 270 MPa.
186
Y. Madi et al. / Nuclear Engineering and Design 228 (2004) 179–194
load history is not taken into account because the time-related aspect of cyclic consolidation is not modeled.
The Hill plastic flow rule is: σ − X1 − X2 3 dεp = dλ 2 J2 (σ − X1 − X2 )
6.1.2. Cycle by cycle (CPC) finite element calculation In order to get a better understanding of the phenomena involved in the cyclic behavior of the welded joint components, a cyclic non-unified law, called DDI model, Contesti and Cailletaud (1989), has been identified. This model has been improved in order to take into account the specific cyclic behavior of the weld metal characterized by cyclic hardening and a subsequent softening. So, the plastic equations of this model can be described as: The scale yielding equation is: J2 (σ − X1 − X2 ) − Rp = 0
(17)
where dεp is the plastic strain rate, dλ is the plastic multiplier and σ is the deviatoric part of σ The non-linear kinematic and isotropic hardening functions are: dX1 = 23 cp1 dεp − dp1 X1 dp
(18)
dX2 = 23 ϕC (p)cp2 dεp − ϕD (p)dp2 X2 dp
(19)
ϕC (p) = ϕCS + (1 − ϕCS )exp(−ωp) ϕD (p) = ϕDS + (1 − ϕDS )exp(−ωp)
(16)
where σ is the stress tensor, X1 and X2 the two kinematic hardening tensors, Rp the isotropic hardening function.
Rp = Rp0 + Qp (1 − exp(−bp p)) + Qpe (1 − exp(−bpe p))
(20)
9.E-05
1
8.E-05
0.9
Damage fraction per cycle
Damage fract ion PMA
6.E-05
0.7 0.6
Accumulated Damage PMA
5.E-05
0.5 4.E-05 0.4
Damage Fraction PMB
3.E-05
0.3 2.E-05
0.2 Accumulated Damage PMB
1.E-05
0.1
0.E+00
0 0
2000
4000
6000
8000
10000
12000
14000
Number of cycles Fig. 5. Evolution of maximum damage, εbm = 0.5%, basic approach.
16000
Accumulated damage
0.8
7.E-05
Y. Madi et al. / Nuclear Engineering and Design 228 (2004) 179–194
where p is the accumulated plastic strain, cpi , dpi , Rp0 , ω, ϕCS , ϕCD , Qp , bp , Qpe , and bpe are model’s coefficients. The 12 parameters have been identified for both base and weld metals, on the hysteresis loops of the characterization tests for different strain variations. 3D finite element calculations were carried out using this model coupled with the proposal damage rule in order to predict the life duration of the welded joint thereafter. All details of this finite element calculation are described by Madi (2001). 6.2. CPC calculation of the damage The developed analyses consider that fatigue proceeds in a uniaxial global load, by using equivalent strain. For the CPC calculation, the equivalent strain of the welded junction are calculated at every moment and in each node of the finite element mesh, while using, for example, a model of the type DDI. For each couple of time t and t of the cycle, it is de-
fined: εii (t, t ) = εii (t)−εii (t ). The variation of equivalent strain, according to the definition of the RCC-MR (1993), is then given, by the following relation: εeq = max(t,t ),i =j × (εii (t, t ) − εjj (t, t ))2 + 6(εij (t, t ))2 (21) 6.2.1. Approach without distinction of initiation and propagation (basic approach) We present in Figs. 5–6, the evolution of the position of the more damaged point during various cycles for a low loading level of ( εbm = 0.5%) as well as for a higher loading level of ( εbm = 0.8%). Points PMA and PMB represent (in term of the strain values at GAUSS points calculation) the most damaged point, respectively, in the weld and the parent metal. One observes that, for the loading, εbm = 0.5%, the more damaged points are always situated in the weld metal (point PMA). The predicted rupture occurs after 16,500 cycles in the weld metal in
1
8.E-04
0.9
7.E-04 Accumulated Damage PMA
0.8 0.7
5.E-04
Damage fraction PMA
0.6 0.5
4.E-04 Accumulated Damage PMB
0.4
3.E-04
0.3
Damage fraction PMB
2.E-04
0.2 1.E-04
0.1
0.E+00 0
500
1000
1500
2000
Number of cycles Fig. 6. Evolution of maximum damage, εbm = 0.8%, basic approach.
0 2500
Accumulated damage
6.E-04
Damage fraction per cycle
187
188
Y. Madi et al. / Nuclear Engineering and Design 228 (2004) 179–194
accordance with the experimental results. However, for the loading, εbm = 0.8%, the more damaged points during the first 500 cycles are located in the parent metal (point PMB), then, the point moves directly in the weld metal (point PMA) out of external fiber and close to the interface. The damage accumulated in the weld at the point PMA then exceeds the damage calculated at point PMB after approximately 1000 cycles. The predicted failure occurs finally after 2460 cycles in the weld metal. This is not in conformity with the experimental observations since the failure is observed in the parent metal and close to the interface after 610 cycles. At high loading level, one shows obviously that the fatigue analysis with a simple damage rule (using only Manson–Coffin’s fatigue curves) does not reproduce experimental reality. The experimental observations
show, for these levels of loading, that micro-crack initiation appears rather quickly close to the weld metal and to the parent metal, more stressed in terms of applied strain variation. The presence of such a defect then involves the propagation in the weakened zone, responsible probably for the effective failure of the specimen. 6.2.2. Proposed approach with distinction of initiation and propagation phases We adopt, now, the prediction method with a threshold associated to a damage fraction per cycle to initiation dDiF and to propagation dDF . One points out that this procedure is described in Fig. 10. According to the applied loading level on the welded structure evaluated in the parent metal, the calculation of the damage variables DiF and DF are
1 Damage DF PMA
Accumulated Damage
0,8 Analysis without threshold and without acceleration
0,6
0,4 Damage Di F PMA
Damage DF PMA
0,2
0 0
300
600
900
1200
1500
1800
2100
Number of cycles N Fig. 7. Evolution of maximum damage, εbm = 0.8%, proposed approach.
2400
2700
Y. Madi et al. / Nuclear Engineering and Design 228 (2004) 179–194
carried out in the following way:
(dDF )wm =
• When the total loading level in the parent metal εeq global is lower than the threshold εeq th (0.6%), the method consists in calculating, initially, a crack initiation damage DiF until it is equal to 1 when the crack initiation occurs. Then, as from this moment, one enters the crack propagation damage DF , equal to 1 when the failure occurs. • When the total loading level in the parent metal εeq global is higher than the threshold εeq th , in the weld, only the crack propagation damage DF is considered in the analysis since the number of cycles to crack initiation NiWJ is regarded as zero. Moreover, to take into account the shape of the transverse defect, this phase is accelerated, using relation (22), by applying the reduction coefficient of 2.24. This last analysis with threshold and acceleration is illustrated in Fig. 7. When the maximum strain in weld metal is close to surface, the shape of the initiated defect is semicircular, there is no acceleration.
189
dN [NPwm ( ε)0.51mJ ]
(22)
One presents, moreover, on this figure, the damage analysis of initiation/propagation without threshold and without acceleration. The failure of the welded joint specimen, for this example, is noted after 610 cycles. The analysis without threshold and without acceleration predicts NRWJ = 2580 cycles (with NiWJ = 1540 cycles and NPWJ = 1040 cycles). The analysis with threshold and with acceleration predicts NRWJ = 630 cycles (with NiWJ = 0 and NPWJ = NRWJ ). The correlation with the overall experimental results is represented in Fig. 8. The adopted step thus proves that a good knowledge of the cyclic behavior of each material composing the welded junction and of the type of rupture, allows, using the proposed approach, prediction of the failure of the specimen in a very satisfactory way. The localization of the failure also is predicted better. At low loading level, the failure systematically appears in the weld metal in contrast to the case at high
100000 type I fracture mode sim ple dam age fatigue approach
B575
Ncalcul
10000
B577 B573 type II fracture mode
1000
B576
B574 B639
NR_FEM Ncal= Nexp 100 100
1000
10000
Nexp Fig. 8. Calculation of the number of cycles.
100000
190
Y. Madi et al. / Nuclear Engineering and Design 228 (2004) 179–194
1
Accumulated Damage
0,8
0,6
Damage DiF PMB
0,4 Dama ge DiF PMA
Damage DF PMB
0,2
Dama ge DF PMA
0 0
1000
2000
3000
4000
5000
6000
Number of cycles N Fig. 9. Evolution of maximum damage, εbm = 0.6%, proposed approach.
loading level, during which the failure is located in the parent metal. The prediction is thus in conformity with the experimental observations. For εbm = 0.6%: With the CPC calculation, the crack initiation is first observed in the parent metal at NiWJ = 3990 cycles. The crack initiation in the weld metal occurs at NiWJ = 4280 cycles. The progressive increase of the strain variation in the weld metal implies that the damage related to the crack propagation is more significant in the weld where the rupture finally appears (Fig. 9). This result accounts perfectly for the observations of the failure mechanism of the test specimen B573: the fracture topography presents two principal sites of initiation, the first one is in the weld metal and the second is in the parent metal (Madi, 2001).
joint coefficients for each loading level by the following relation: Jf =
εR εi
(23)
where, εR corresponds to Fig. 11, a fictitious strain variation deduced from the fatigue curve of parent metal (knowing NRWJ ), and εi is the reference strain variation deduced from the fatigue curve of parent metal knowing the number of cycles to failure of the homogeneous structure (HS) NRSH . When the computed value, for a given loading is lower than that codified by the RCC-MR (Jf = 1.25 in our case), we propose to use directly the codified value. However, when Js is higher than the value of 1.25, in particular when the total loading is higher than the threshold εeq th , the recommended value is that calculated by our approach.
7. Determination of the fatigue life reduction factor J f of welded structures 8. Application to the butt-welded pipe The presented methodology makes it possible to use Jf according to the studied welded structure. Thus, as from the calculated fatigue life, one can deduce the
The simplified suggested method is applied in order to evaluate the fatigue life of the welded tube tested
Y. Madi et al. / Nuclear Engineering and Design 228 (2004) 179–194 BM
Finite Element Calculation
The variations of equivalent strain in the base metal ∆εeq_bm and in the weld metal ∆εeq_wm in each node of the FEM mesh Loop on the cycles Crack initiation damage fraction per cycle in each node
WM
Welded Joint (JW) ∆ε eq_global Base Metal
∆ε eq_bm
Weld Metal
∆ε eq_wm
Cumulated damage DiF
While DiF ≤ 1
191
NRbm NRwm
Si DiF > 1
NRbm = Nibm + NPbm
NRwm = Niwm + NPwm
Crack propagation damage fraction per cycle in each node
Accumulated damage DF
Decomposition using relation (10) of the number of cycles to rupture N R in a number of cycles to crack initiation N i and a number of cycles to crack propagation N P .
If ∆ε eq_global > ∆ε eq_th
If ∆ε eq_global < ∆ε eq_th
Base Metal
Weld Metal ∆εeq_wm close to the surface
D iF =
∆εeq_wm within the weld
1
∑N
cycles
i
D F = H ( D iF − 1) ∑
1 N cycles P
( D iF ) wm = 1 ( D F ) wm =
( D iF ) wm = 1 1
∑ N wm
cycles
P
Fig. 10. CPC calculation procedure.
( D F ) wm =
∑ N wm+0.51m 1
cycles
P
J
192
Y. Madi et al. / Nuclear Engineering and Design 228 (2004) 179–194
∆ε
Base Metal
∆εR
Jf =
Weldment
∆ε i
∆ε R ∆ε i
Jf NR
N RBM
WJ
Numbre of cycles To fracture
Fig. 11. Definition of Jf .
(Madi, 2001), showed that, for this loading, the axial rotations are almost similar to those measured. Also let us recall that the maximum strain variations calculated in the weld are in good agreement with those obtained
within the framework of this study. The total loading is applied to the welded structure so as to obtain, in the useful part of the homogeneous tube, an axial strain level of equal to 0.6% ( εglobal = 0.6%). Calculations Table 1 Life prediction of pipes with CPC approach Tests
CPC approach
Type of test
εglobal (%)
NR cycles
Nabm cycles
NPbm cycles
NRSH cycles
Homogeneous tube (TH) Butt-welded tube (TW)
0.6 0.6
2765 2067
3750 4020
1350 1430
5100
Nawm cycles
NPwm cycles
NRWJ cycles
0
2010
2010
Accumulated Damage 1 0.9 0.8 Damage DF PMA
0.7 0.6 0.5 0.4 0.3
Damage DF PMB
Damage DiF PMB
0.2 0.1 0 0
500
1000
1500
2000
2500
3000
3500
4000
4500 5000
Number of cycles N Fig. 12. Evolution of maximum damage in butt-welded tube, proposed approach.
5500
Y. Madi et al. / Nuclear Engineering and Design 228 (2004) 179–194
on Bi-material tests for the applied loading εbm = 0.6%. 8.1. Life prediction with CPC approach The results of the life prediction of tubes, by applying the calculation procedure illustrated in Fig. 10, are presented in Table 1. For the welded tube, the damage evolution allowing determination of the number of cycles to failure NRWJ is illustrated in Fig. 12. The damage analysis is done in two critical points of the structure. The more damaged point in the parent metal (point PMB) presents the two phases related to crack initiation damage DiF and to crack propagation damage DF . However, the damage at the more damaged point in the weld metal (point PMA) is expressed only in terms of damage fraction per cycle of propagation since the loading is higher than the threshold εeq th . The failure of the welded tube is predicted in the weld metal. 8.2. Fatigue life reduction factors Jf Let us recall that the failure of the homogeneous tube is predicted, in accordance with the tests, in the internal fillet (radius of connection) of the tube. The applied strain εeq bm , responsible for the failure of the tube, is higher than the global reference applied strain εglobal , in the useful part of the tube. Consequently, the number of cycles to failure NR which corresponds to a failure in the uniform part of the tube would be higher than that observed and calculated in the tube. As a result the value of the fatigue life reduction factor Jf = 1.11, established by using the experimental number of cycles, is underestimated. While considering, now, the value of reference, εi , equal to the maximum equivalent strain variation in the useful part of the homogeneous tube, the obtained fatigue life reduction factor is given in Table 2. This coefficient results from the application of our sugTable 2 Fatigue life reduction factor Jf of the tube at 600 ◦ C Experience (Jf -exp)
1.11
CPC approach εi
εR
Jf
0.671
0.950
1.42
193
gested method of prediction. The computed value is lower than that given by RCC-MR, i.e. Jf = 1.25. One thus finds the results obtained on the Bi-material tests. At a high loading level, the method of RCC-MR based on the fatigue life reduction factor is not useful.
9. Conclusions The welded joint tests reveal two types of crack initiation according to the loading level. The first type of failure (type I) is characteristic of the low levels loading. It results in a crack initiation on the surface in the weld metal. The second type of failure (type II) is characterized by a crack initiation in the interior of the specimen close to the interface of weld and parent metal. This type II is observed for the high load levels. A threshold of load level which separates these two types of failure was introduced. It corresponds to a strain variation in the parent metal εbm = 0.6%. Higher than this threshold value, the experimental results show that the crack initiation period then represents a negligible share of the fatigue life. In this paper, we propose a method which consists of approaching the mechanical behavior and the damage behavior in an uncoupled way. It makes it possible to determine the damage rate to crack initiation and to crack propagation of a significant crack and by the same method, one to calculate the number of cycles to failure NR . This method suggests the existence of a threshold loading level above which the micro-defects in the weld propagate directly; thus the crack initiation phase becomes negligible. An acceleration of the crack propagation is introduced to better describe the crack propagation following a crack initiation within the specimen. The application of our method to the tested tube shows that taking into account the threshold loading level εth in the procedure makes it possible to predict the failure in weld metal in accordance with the experimental observations. In this case, the crack initiation is obtained on the surface, the number of cycles to failure NRWJ is given then without the acceleration of the crack propagation phase. Without considering the threshold loading level, the predicted failure would be located in the radius of the junction and in the parent metal. The threshold loading level εth is thus validated by the tests. The developed
194
Y. Madi et al. / Nuclear Engineering and Design 228 (2004) 179–194
method allows us to criticize the RCC-MR design method which consists in using the Fatigue life reduction factor Jf . The values of Jf calculated for the welded joint test are comparable with the coefficients of experimental joints. The value of Jf , codified by the RCC-MR as equal to 1.25, is valid only for the low loading levels, ( εbm ≤ εth ). For the joint with an X-shaped weld, the threshold value εth is established as equal to 0.6%. For another weld shape type of welded joint, Bi-material tests will be necessary in order to determine at the same time the value of the threshold as well as the fatigue life reduction factor Jf .
Acknowledgements The present work is cofinanced by the CEA (Commissariat à l’Energie Atomique, France) and EDF (Electricité De France); it is led by the Laboratory of Integrity of the Structures and Standardization (LISN) of the CEA Saclay in collaboration with the Laboratory of Studies and Research in Structural Mechanics (LERMES) of the university Blaise Pascal in Clermont-Ferrand, France. References Berton, M.N., 1996. CEA Report. Saclay Research Institute, France. Brown, M.W., Miller, K.J., 1979. Initiation and growth of cracks in biaxial fatigue. Fatigue Fracture Eng. Mater. Struct. 1, 231– 246. Contesti, E., Cailletaud, G., 1989. Description of creep-plasticity with non-unified constitutive equations. Nucl. Eng. Des. 116 (3), 265–280. Curtit, F. 1999. Propagation de Fissures Semi-Elliptiques en Fatigue-Fluage à 650 ◦ C dans les Plaques d’Acier 316L(N)
Avec et Sans Joints Soudés. Ph.D. thesis. Ecole des Mines de Paris, France. DCRC, 1991. Concluding Report No. 5: Design Rules for Welds, 1st ed. Jacquelin, B., Hourlier, F., Pineau, A., 1983. Crack initiation under low-cycle multiaxial fatigue in type 316L stainless steel. J. Pressure Vessel Technol. 105, 138–143. Le Ber, L., Sainte Catherine, C., Waeckel, N., Turbat, M., 1998. Low cycle fatigue strength of high temperature welded joints: an efficient method to predict life of austenitic 316L(N) weldments. ECF12, Emas Publishing, Sheffield, UK. Levaillant, C., 1984, Approche métallographique de l’endommagement d’aciers inoxydables austénitiques sollicités en fatigue plastique ou en fluage: description et interprétation physique des interactions fatigue-fluage-oxydation. Ph.D. thesis. Univ. de Technologie de Compiegne, France. Madi, Y., 2001, Comportement et durée de vie d’une jonction tubulaire soudée en 316L à 600 ◦ C. Ph.D. thesis. Université B. Pascal—Clermont II, France. Manson, S.S., 1966. Interfaces between fatigue, creep and fracture. Int. J. Fracture Mech. 2 (1), 327–363. Manson, S.S., Halford, G.R., 1981. Practical implementation of the double linear damage curve approach for treating cumulative fatigue damage. Int. J. Fracture 17 (2), 169–192. Polvora, J.P., 1998. Propagation de Fissure à Haute Température dans un Acier Inoxydable Austénitique. Ph.D. thesis. Ecole des Mines de Paris, France. RCC-MR, 1993. Design and construction rules for mechanical components of FBR nuclear islands. AFCEN. Savalle, S., Cailletaud, G., 1982. Micro-amorçage, micropropagation et endommagement. La Recherche Aérospatiale 6, 395– 411. Socie, D.F., Shield, T.W., 1984. Trans. ASME. J. Eng. Mater. Technol. 106, 227. Socie, D.F., Waill, L.A., Dittmer, D.F., 1985. Biaxial Fatigue of Inconel 718 Including Mean Stress Effects, Multiaxial Fatigue, ASTM STP 853, Philadelphia, pp. 463–481. Usami, S. Fukuda, Y., Shida, S., 1983. Micro-crack initiation, propagation and threshold in elevated temperature inelastic fatigue. In: ASME Paper 83-PVP-97. Wareing, J., Vaughan, H.G., 1977. Metal Sci. 11, 439. You, B.-R., Lee, S.-B., 1996. A critical review on multiaxial fatigue assessments of metals. Int. J. Fatigue 18 (4), 235–244.
Low-cycle fatigue of welded joints: coupled initiation propagation model Yazid Madi a,∗ , Naman Recho b,1 , Philippe Matheron c,2 a
c
Centre des Matériaux, Ecoles des Mines de Paris, B.P. Evry Cedex 91003, Paris, France b Lermes—Université, Blaise Pascal, B.P. 206, 63174 Aubierre, France DM2S/SEMT/LISN, Commissariat à l’Energie Atomique, Centre d’Etudes de Saclay, 91191 Gif-Sur-Yvette, France Received 19 February 2003; received in revised form 3 June 2003; accepted 23 June 2003
Abstract This paper deals with the low-cycle fatigue (LCF) design of welded structures, the aim being the critical analysis of the rule used in the RCC-MR [Design and construction rules for mechanical components of FBR nuclear islands, AFCEN, 1993], for the design and construction of fast breeder reactors. The study takes into account the evolution of the material behavior laws and damage accumulation during the fatigue loading. The adopted model consists of analyzing separately the behavior and the damage evolutions. It allows us to determine the damage ratio corresponding to initiation and propagation of a significant crack in order to determine the life duration. This model suggests the existence of a threshold level of loading, above which micro-cracks initiate. The initiation fatigue life can then be neglected below the threshold level. This work shows also that the RCC-MR rules are valid below this threshold load level. © 2003 Elsevier B.V. All rights reserved.
1. Introduction As a result of their size, fast breeder reactor vessels are not made of a single material but of many parts welded together. The welds frequently represent a weak point, the fatigue resistance of welded joints depends on the surrounding geometry, on the loading to which they are subjected and on the presence of two materials with different mechanical properties. The lower fatigue resistance of welded structures is provided for in the French RCC-MR code by applying the reduction factor Jf to the Manson–Coffin fatigue ∗ Corresponding author. Tel.: +33-160-76-30-12; fax: +33-160-76-31-50. E-mail addresses: [email protected] (Y. Madi), [email protected] (N. Recho), [email protected] (P. Matheron). 1 Tel.: +33-470-02-20-28; fax: +33-470-02-20-78. 2 Tel.: +33-169-08-28-47; fax: +33-169-08-87-84.
curve of the parent metal. For 316L(N), Jf is equal to 1.25 (DCRC, 1991). In order to better assess this factor, using previous work on large plates and part of a welded vessel, Berton (1996) and Le Ber et al. (1998) showed that the mechanical behavior of a welded joint is influenced by the geometry of the weld and by the interaction of the different cyclic plastic behavior of the two materials: base metal (BM) and weld metal (WM). The Jf value is a function of the structures. In order to improve this method of design, it would be necessary to establish experimental procedures allowing its characterization. With this intention, a new campaign (named FFAST) was performed on welded joint specimens extracted from butt-welded pipe connections (uniaxial tensile-compressive load). The purpose of this study is to experimentally and numerically analyze the behavior of specimens taken from a butt-welded annular joint to determine coefficient Jf .
0029-5493/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.nucengdes.2003.06.017
180
Y. Madi et al. / Nuclear Engineering and Design 228 (2004) 179–194
A new experimental approach was adopted in order to analyze the local behavior of the welded joint (see companion paper). Non-linear calculations integrating the consolidation of two materials were done in order to deduce the fatigue life to crack initiation (Madi, 2001). The approach developed in this paper allows us to analyze the design procedure used in RCC-MR rules, which is based on the reduction factor concept Jf . Finally, a new procedure is proposed to study the fatigue life of welded structures.
2. Position of the problem Many results in technical literature show that the curve of Manson–Coffin, usually used to predict crack initiation, represents actually the two stages of the fatigue process, namely: crack initiation and crack propagation. The results of Levaillant (1984) on steel VIRGO 316 L bring quantitative elements to this observation. Within the framework of this study, the fatigue life will be made up of two parts relating, respectively, to the initiation and the propagation of crack. To evaluate each part of the fatigue life, two approaches are associated: one is based on the fracture mechanics and the other on the damage mechanics. The fracture mechanics approach is applied when a crack of significant size (higher or equal to 0.5 mm) exists within the structure. Contrarily, the approach based on damage mechanics deals with the micro-defects. In the field of the cracks known as “short” (=0.1 mm), some studies seem to be necessary. It is to be noted that the elastic–plastic fracture mechanics makes it possible to describe the evolution of the micro-defects in this field. The aim of this work is to propose a specific approach which will take into account the initiation and the propagation parts of the micro-cracks and the evaluation of the fatigue life observed experimentally. First, we describe the approach in order to evaluate the fatigue life on smooth test specimens in the case of a numerical cycle by cycle description of loading. Then, we apply this approach to interpret the Bi-materials tests under 600 ◦ C with regard to the fracture pictures observed in experiments according to the level of loading. Finally, we manage to justify the use of the fatigue life reduction factor Jf in the field of the low levels of loading.
3. Initiation of a “significant” crack The fatigue design based on the critical plane approach is that which comes closest to the physical reality of fatigue damage. During low-cycle fatigue, the material is submitted to a plastic strain field at each applied cycle. This deformation is related to the movement of dislocations in the slip surfaces, movement induced by shear. This ultimate analysis considers the shear strain amplitude as a significant parameter. This step, initiated by Brown and Miller (1979), was followed by many researchers. These authors consider that the critical plane corresponds to the maximum shear plane. This plane, which controls the fatigue crack initiation, is defined like a combination of maximum amplitude of the shear strain γ max and the amplitude of the normal strain εn in the same plane submitted to γ max . A recent bibliographical study (You and Lee, 1996) shows that multiaxial fatigue is well predicted by using the linear combination of the modified Brown and Miller’s parameters γ max and εn proposed by Socie and Shield (1984): σn0 γmax,p + εn,p + (1) = γR (NR )co E where γ max,p and εn,p are the plastic values of the Brown and Miller’s parameters, σ n0 is the average stress applied on the plane of maximum shear stress, co and γR are two parameters which depend on the material, E is the Young’s modulus, and NR is the number of cycles to failure. As for the particular case of our study, the load applied on the Bi-materials specimens and on the welded tube is an uniaxial type.3 This formulation can be used because the strains are lower than the limits of the domain of the tests validity ( ε < 2%) (Socie and Shield, 1984; Socie et al., 1985). Finite element calculations show that the presence of the weld does not influence significantly the degree of triaxiality of the stress field in the neighborhood of the weld. One can thus reasonably regard the overall stress field as uniaxial. By noting ε1 , ε2 , ε3 the principal strains (ε1 > ε2 = ε), ε1 being the applied strain, the Brown and Miller’s parameters are written as: 3 In this paper, the effect on damage of biaxial loading (as internal pressure in the tube) has not been studied. The application of the model to multiaxial loading is assumed to be covered by Eq. (1).
Y. Madi et al. / Nuclear Engineering and Design 228 (2004) 179–194
181
ε1 − ε3 ε1 + ε3 and εn = (2) 2 2 In the case of alternative loading and taking into account the relation (2), Eq. (1) becomes:
we determine an approximate value of this one with the relation established by Levaillant (1984):
ε1,p = γR (NR )co
Let us note that this relation is in agreement with other results such as those of Manson (1966) which proposes a simple relation. The difference between the relation of Levaillant and that of Manson comes from the definition of the initial crack size a0 obtained at the end of the initiation period. Manson estimates the value a0 at 50 m while Levaillant selects 20 m.
γmax =
(3)
One finds consequently the Manson–Coffin approach usually used in the nuclear design. One can thus use the criterion in terms of strain associated with the Manson–Coffin approach within the framework of this study. However, the definite concept of a number of cycles to initiation calculated as being the number of cycles to failure NR of a smooth specimen will not be adopted. One will use the definition rather approaching the physics recommended by Levaillant (1984) and Jacquelin et al. (1983), which states that NR of the Eq. (3) represents the initiation fatigue life defined as the number of cycles necessary to initiate the microscopic crack. The length of this microscopic crack corresponds to the size of the grain (estimated at a0 = 20 m). The Eq. (3) becomes then: ε1,p = γR (Ni )co
(4)
3.1. Determination of the number of cycles to crack initiation The strain variation ε being given, it makes it possible to determine the number of cycles to failure NR by the relation of Manson–Coffin. Being difficult to estimate the number of cycles to crack initiation Ni ,
Ni = NR − 12NR0.62 + 0.226NR0.90 + 74
(5)
4. Analysis of crack propagation 4.1. Observations 4.1.1. Location and fracture mode The observation of the fracture topographies shows the existence of two types of failure. At a low level loading ( εbm < 0.6%), one notes that the crack initiation occurs on the surface (type I, Fig. 1). At high level loading ( εbm < 0.6%), the crack initiation does not occur on the surface but in the interface between weld and parent metal. The presence of a transverse defect comparable with a crossing centered crack implies a different type of failure (type II, Fig. 1) directed from an internal defect to the external surface of the specimen. We also point out that for the loading level εbm < 0.6%, the two types of failure are observed. The rupture of specimen B573 is done according to
Fig. 1. Fracture mode of the Bi-materials tests.
182
Y. Madi et al. / Nuclear Engineering and Design 228 (2004) 179–194
type I failure in 6670 cycles while that of specimen B576 is done according to the type II failure in 1305 cycles (Madi, 2001). 4.1.2. Striation spacing measurement Several investigators (Jacquelin et al., 1983; Wareing and Vaughan, 1977) have shown that, within the strain range explored in our study and for this type of steel, a relationship between striations and fatigue cycles can be assumed. From direct observation of the fatigue fracture surface, a good estimate of the number of cycles spent to propagate the cracks can be obtained by plotting striation, i, versus crack depth, a, and by integrating along the fitting curve. This procedure has been used only on B571 specimens tested at room temperature (see Fig. 2 and companion paper) where fracture occurs according type II mode. The life part corresponding to crack initiation was determined (Ni = 19 cycles). Compared to the fatigue life (NR = 794 cycles), the result shows that the crack initiation represents a negligible part of the type II fracture mode.
4.2. Determination of the number of cycles during crack propagation In the light of these observations, one can suppose the presence of a threshold loading level, having the value εbm < 0.6%. in the case of the welded junction in 316L/16-8-2 of our study, beyond whose the micro-defects present in the welded junction are activated more quickly close to the interface. That led to a very fast crack initiation period. It will be noted that for the weld having an X shape and for the alternating tensile loading, the critical points associated with the maximum variations of strain, obtained by finite elements calculations, are situated at the weld toe. It is thus normal that the micro-defects present in this zone are responsible for crack initiation. In addition, when the loading level εbm is higher than the threshold level ( εbm = 0.6%), the number of cycles during crack propagation NP is given while following the following step: By supposing that the crack growth rate is connected to the variation of J-Integral J, Usami et al.
Fig. 2. Striation spacing measurements of type II fracture mode.
Y. Madi et al. / Nuclear Engineering and Design 228 (2004) 179–194
183
10
Crack depth a (mm)
Np_t = 0.51mJ Np_c
1
Transverse defect semicircular defect
0.1
0.01 0
500
1000
1500
2000
2500
Num ber of cycles N Fig. 3. Evolution of crack propagation according to the micro-defects.
(1983) proposed an approach which is summarized in Fig. 3 and described below. The ratio of the number of cycles to crack propagation of a transverse defect Np t with those of a semicircular defect Np c can be evaluated with 0.51mJ , where mJ is the exponent of the following propagation law: da = CJ ( J)mJ dN
(6)
J and NP are given, respectively, by the following equations: J ∼ = (2π We + f(n) Wp )Qa Q−mJ (a1−mJ − a01−mJ ) (1 − mJ )g(n) R 1−n f(n) ∼ = (n + 1) 3.85 √ n + πn
NP =
We =
( σ)2 , 2E
Wp =
(7) (8)
σ εp 1+n
Where, E is the Young’s modulus, n is the cyclic hardening exponent in the relation determined by the cyclic stabilized behavior εp /2 = ( σ/2k)1/n , Q is fixed
at 0.51 for a semicircular defect and 1 for a transverse defect. For 304 STEEL, one showed that the ratio Np c /Np t (Np c , a number of cycles for a semicircular defect; Np t , a number of cycles for a transverse defect) is about 2.64 for a temperature of 550 ◦ C. At 600 ◦ C, without the crack propagation law for 316L STEEL, one will base oneself on these bibliographical data to estimate this coefficient. One knows that for the 316L, the exponent m of the Paris law da/dN = C( K)m which is a function of the temperature always lies between 2 and 4. One has more precise values with the temperature of 650 ◦ C, near to the temperature of our study. Curtit (1999) and Polvora (1998) obtain relatively coherent respective values with m = 2.34 and 2.46. We adopt for our study the average value of m = 2.4. In confined plasticity, the relation binding the stress intensity factor to the J-Integral is the following one: J =
K2 E
(9)
As a result, the exponent mJ of Eq. (5) is related to the exponent m of Paris law by: m = 2mJ . The value of mJ is then equal to 1.2.
184
Y. Madi et al. / Nuclear Engineering and Design 228 (2004) 179–194
Consequently the ratio Np c /Np t is equal to 2.24. Finally, the number of cycles to crack propagation is calculated as follows: • For a semi-circular defect ( εbm < 0.6%): Ni = f(NR ) = NR − 12NR0.62 + 0.226NR0.90 + 74 Np
c
= NR − Ni (10)
• For a transversal defect ( εbm > 0.6%): Ni = 0 Np t = 0.51mJ Np
(11) c
5. Evaluation of fatigue damage The crack initiation and the crack propagation periods are taken into account in the fatigue damage law. For that, the formulation used is that developed by Savalle and Cailletaud (1982). We adopt this formulation by using the Manson–Coffin approach. Thus, we suppose that a damage variable DiF exists and equal to zero for non-damaged material. It is equal to 1 when a micro-crack initiation occurs. Start from the moment that develops “the true” fatigue damage DF which is measurable at the macroscopic level. The crack initiation and the crack propagation periods are interpreted by the following relations: dN dDiF = (12) Ni ( ε) H(DiF − 1)dN dDF = NP ( ε)
(13)
Where H is a Heavyside function, H(DiF − 1). It is defined as follow: 0 if DiF < 1 H(DiF − 1) = (14) 1 if DiF ≥ 1 The integration of the first relation gives the relationship between DiF and the variation of applied strain. At the moment when DiF = 1, the number of cycles carried out is Ni (a number of cycles to crack initiation) and the fatigue damage variable DF , which remained null until there, starts to move. The failure is obtained when DF = 1, by carrying out NP additional cycles in the crack propagation period, in other words when: NR = Ni + NP
(15)
In the case of a smooth specimen, the fatigue damage variable DF is calculated by the use of the Manson–Coffin’s curves. The use of Manson–Coffin curves is thus a simple matter to deal with the problems of low-cycle fatigue since it makes it possible to determine in a simple practical way, the failure of a smooth specimen (including the periods of micro-crack initiation and micro-crack propagation) corresponds to the initiation of a crack of about 1 mm in depth in a structural component. The sequence effect is taken into account by this formulation provided that the evolution curve of the damage according to the fatigue life is independent of the loading. Also, one finds by this method the technique suggested by Manson and Halford (1981) based on the “Double linear Damage Rule.”
6. Analysis of Bi-material tests under 600 ◦ C The analysis of the fatigue life of Bi-material tests is carried out while following the step discussed in the paragraphs above. This method consists in determining the numbers of cycles NiWJ to crack initiation and NPWJ to crack propagation of the welded joint by integration of the relations (12) and (13). The level of total applied cyclic loading εbm being known, the calculation of the rates of use of crack initiation and crack propagation is carried out as follows: 1. When the level of applied loading εbm is lower than the threshold of 0.6%, the numbers of cycles to crack initiation Ni and to crack propagation NP of the relations (12) and (13) are determined by the relation (9). Thus, crack initiation is supposed to initiate on the surface of the welded joint. 2. When the level of applied loading εbm is higher than the threshold of 0.6%, the number of cycles to crack initiation NiWJ is neglected (NiWJ = 0). The damage variable to initiation DiF is thus equal to 1. Only the damage of propagation DF is considered in calculation. The number of cycles to crack propagation NP of the relation (12) is determined by the relation (10). Let us note that the phase of propagation is accelerated in order to take into account the shape of the initiated transverse defect. 3. When the level of applied loading εbm is equal to the threshold of 0.6%, one determines the number
Y. Madi et al. / Nuclear Engineering and Design 228 (2004) 179–194
of cycles to failure according to the two earlier alternatives 1 and 2. The cycle by cycle finite element calculations described below, are carried out by taking into account two materials in order to model the test specimen. To show the results of calculations, we will use two specific terms: • Damage fraction per cycle: damage for one cycle of loading • Damage accumulation: sum of the damage fraction per cycle up to the considered cycle From a behavioral point of view, calculations (Madi, 2001) showed that the strain varies cyclicly in the
185
weld. In fact, whereas the strain variation is constant in the parent metal, the strain variation in the weld increases with cycling. So the maximum damage fraction per cycle also varies with cycling. 6.1. Finite element calculation 6.1.1. Simplified finite element calculation 3D finite element calculations were carried out using the reduced cyclic curves attained at half-lifetime as the response law. The material model is an isotropic elastoplastic one of the Ramberg–Osgood type. The calculations indicate the points where the strain concentration is the highest (Fig. 4). These agree with the experimental observations. In this approach, the
Fig. 4. Location of maximum equivalent strain in the welded joint, σ/2 = 270 MPa.
186
Y. Madi et al. / Nuclear Engineering and Design 228 (2004) 179–194
load history is not taken into account because the time-related aspect of cyclic consolidation is not modeled.
The Hill plastic flow rule is: σ − X1 − X2 3 dεp = dλ 2 J2 (σ − X1 − X2 )
6.1.2. Cycle by cycle (CPC) finite element calculation In order to get a better understanding of the phenomena involved in the cyclic behavior of the welded joint components, a cyclic non-unified law, called DDI model, Contesti and Cailletaud (1989), has been identified. This model has been improved in order to take into account the specific cyclic behavior of the weld metal characterized by cyclic hardening and a subsequent softening. So, the plastic equations of this model can be described as: The scale yielding equation is: J2 (σ − X1 − X2 ) − Rp = 0
(17)
where dεp is the plastic strain rate, dλ is the plastic multiplier and σ is the deviatoric part of σ The non-linear kinematic and isotropic hardening functions are: dX1 = 23 cp1 dεp − dp1 X1 dp
(18)
dX2 = 23 ϕC (p)cp2 dεp − ϕD (p)dp2 X2 dp
(19)
ϕC (p) = ϕCS + (1 − ϕCS )exp(−ωp) ϕD (p) = ϕDS + (1 − ϕDS )exp(−ωp)
(16)
where σ is the stress tensor, X1 and X2 the two kinematic hardening tensors, Rp the isotropic hardening function.
Rp = Rp0 + Qp (1 − exp(−bp p)) + Qpe (1 − exp(−bpe p))
(20)
9.E-05
1
8.E-05
0.9
Damage fraction per cycle
Damage fract ion PMA
6.E-05
0.7 0.6
Accumulated Damage PMA
5.E-05
0.5 4.E-05 0.4
Damage Fraction PMB
3.E-05
0.3 2.E-05
0.2 Accumulated Damage PMB
1.E-05
0.1
0.E+00
0 0
2000
4000
6000
8000
10000
12000
14000
Number of cycles Fig. 5. Evolution of maximum damage, εbm = 0.5%, basic approach.
16000
Accumulated damage
0.8
7.E-05
Y. Madi et al. / Nuclear Engineering and Design 228 (2004) 179–194
where p is the accumulated plastic strain, cpi , dpi , Rp0 , ω, ϕCS , ϕCD , Qp , bp , Qpe , and bpe are model’s coefficients. The 12 parameters have been identified for both base and weld metals, on the hysteresis loops of the characterization tests for different strain variations. 3D finite element calculations were carried out using this model coupled with the proposal damage rule in order to predict the life duration of the welded joint thereafter. All details of this finite element calculation are described by Madi (2001). 6.2. CPC calculation of the damage The developed analyses consider that fatigue proceeds in a uniaxial global load, by using equivalent strain. For the CPC calculation, the equivalent strain of the welded junction are calculated at every moment and in each node of the finite element mesh, while using, for example, a model of the type DDI. For each couple of time t and t of the cycle, it is de-
fined: εii (t, t ) = εii (t)−εii (t ). The variation of equivalent strain, according to the definition of the RCC-MR (1993), is then given, by the following relation: εeq = max(t,t ),i =j × (εii (t, t ) − εjj (t, t ))2 + 6(εij (t, t ))2 (21) 6.2.1. Approach without distinction of initiation and propagation (basic approach) We present in Figs. 5–6, the evolution of the position of the more damaged point during various cycles for a low loading level of ( εbm = 0.5%) as well as for a higher loading level of ( εbm = 0.8%). Points PMA and PMB represent (in term of the strain values at GAUSS points calculation) the most damaged point, respectively, in the weld and the parent metal. One observes that, for the loading, εbm = 0.5%, the more damaged points are always situated in the weld metal (point PMA). The predicted rupture occurs after 16,500 cycles in the weld metal in
1
8.E-04
0.9
7.E-04 Accumulated Damage PMA
0.8 0.7
5.E-04
Damage fraction PMA
0.6 0.5
4.E-04 Accumulated Damage PMB
0.4
3.E-04
0.3
Damage fraction PMB
2.E-04
0.2 1.E-04
0.1
0.E+00 0
500
1000
1500
2000
Number of cycles Fig. 6. Evolution of maximum damage, εbm = 0.8%, basic approach.
0 2500
Accumulated damage
6.E-04
Damage fraction per cycle
187
188
Y. Madi et al. / Nuclear Engineering and Design 228 (2004) 179–194
accordance with the experimental results. However, for the loading, εbm = 0.8%, the more damaged points during the first 500 cycles are located in the parent metal (point PMB), then, the point moves directly in the weld metal (point PMA) out of external fiber and close to the interface. The damage accumulated in the weld at the point PMA then exceeds the damage calculated at point PMB after approximately 1000 cycles. The predicted failure occurs finally after 2460 cycles in the weld metal. This is not in conformity with the experimental observations since the failure is observed in the parent metal and close to the interface after 610 cycles. At high loading level, one shows obviously that the fatigue analysis with a simple damage rule (using only Manson–Coffin’s fatigue curves) does not reproduce experimental reality. The experimental observations
show, for these levels of loading, that micro-crack initiation appears rather quickly close to the weld metal and to the parent metal, more stressed in terms of applied strain variation. The presence of such a defect then involves the propagation in the weakened zone, responsible probably for the effective failure of the specimen. 6.2.2. Proposed approach with distinction of initiation and propagation phases We adopt, now, the prediction method with a threshold associated to a damage fraction per cycle to initiation dDiF and to propagation dDF . One points out that this procedure is described in Fig. 10. According to the applied loading level on the welded structure evaluated in the parent metal, the calculation of the damage variables DiF and DF are
1 Damage DF PMA
Accumulated Damage
0,8 Analysis without threshold and without acceleration
0,6
0,4 Damage Di F PMA
Damage DF PMA
0,2
0 0
300
600
900
1200
1500
1800
2100
Number of cycles N Fig. 7. Evolution of maximum damage, εbm = 0.8%, proposed approach.
2400
2700
Y. Madi et al. / Nuclear Engineering and Design 228 (2004) 179–194
carried out in the following way:
(dDF )wm =
• When the total loading level in the parent metal εeq global is lower than the threshold εeq th (0.6%), the method consists in calculating, initially, a crack initiation damage DiF until it is equal to 1 when the crack initiation occurs. Then, as from this moment, one enters the crack propagation damage DF , equal to 1 when the failure occurs. • When the total loading level in the parent metal εeq global is higher than the threshold εeq th , in the weld, only the crack propagation damage DF is considered in the analysis since the number of cycles to crack initiation NiWJ is regarded as zero. Moreover, to take into account the shape of the transverse defect, this phase is accelerated, using relation (22), by applying the reduction coefficient of 2.24. This last analysis with threshold and acceleration is illustrated in Fig. 7. When the maximum strain in weld metal is close to surface, the shape of the initiated defect is semicircular, there is no acceleration.
189
dN [NPwm ( ε)0.51mJ ]
(22)
One presents, moreover, on this figure, the damage analysis of initiation/propagation without threshold and without acceleration. The failure of the welded joint specimen, for this example, is noted after 610 cycles. The analysis without threshold and without acceleration predicts NRWJ = 2580 cycles (with NiWJ = 1540 cycles and NPWJ = 1040 cycles). The analysis with threshold and with acceleration predicts NRWJ = 630 cycles (with NiWJ = 0 and NPWJ = NRWJ ). The correlation with the overall experimental results is represented in Fig. 8. The adopted step thus proves that a good knowledge of the cyclic behavior of each material composing the welded junction and of the type of rupture, allows, using the proposed approach, prediction of the failure of the specimen in a very satisfactory way. The localization of the failure also is predicted better. At low loading level, the failure systematically appears in the weld metal in contrast to the case at high
100000 type I fracture mode sim ple dam age fatigue approach
B575
Ncalcul
10000
B577 B573 type II fracture mode
1000
B576
B574 B639
NR_FEM Ncal= Nexp 100 100
1000
10000
Nexp Fig. 8. Calculation of the number of cycles.
100000
190
Y. Madi et al. / Nuclear Engineering and Design 228 (2004) 179–194
1
Accumulated Damage
0,8
0,6
Damage DiF PMB
0,4 Dama ge DiF PMA
Damage DF PMB
0,2
Dama ge DF PMA
0 0
1000
2000
3000
4000
5000
6000
Number of cycles N Fig. 9. Evolution of maximum damage, εbm = 0.6%, proposed approach.
loading level, during which the failure is located in the parent metal. The prediction is thus in conformity with the experimental observations. For εbm = 0.6%: With the CPC calculation, the crack initiation is first observed in the parent metal at NiWJ = 3990 cycles. The crack initiation in the weld metal occurs at NiWJ = 4280 cycles. The progressive increase of the strain variation in the weld metal implies that the damage related to the crack propagation is more significant in the weld where the rupture finally appears (Fig. 9). This result accounts perfectly for the observations of the failure mechanism of the test specimen B573: the fracture topography presents two principal sites of initiation, the first one is in the weld metal and the second is in the parent metal (Madi, 2001).
joint coefficients for each loading level by the following relation: Jf =
εR εi
(23)
where, εR corresponds to Fig. 11, a fictitious strain variation deduced from the fatigue curve of parent metal (knowing NRWJ ), and εi is the reference strain variation deduced from the fatigue curve of parent metal knowing the number of cycles to failure of the homogeneous structure (HS) NRSH . When the computed value, for a given loading is lower than that codified by the RCC-MR (Jf = 1.25 in our case), we propose to use directly the codified value. However, when Js is higher than the value of 1.25, in particular when the total loading is higher than the threshold εeq th , the recommended value is that calculated by our approach.
7. Determination of the fatigue life reduction factor J f of welded structures 8. Application to the butt-welded pipe The presented methodology makes it possible to use Jf according to the studied welded structure. Thus, as from the calculated fatigue life, one can deduce the
The simplified suggested method is applied in order to evaluate the fatigue life of the welded tube tested
Y. Madi et al. / Nuclear Engineering and Design 228 (2004) 179–194 BM
Finite Element Calculation
The variations of equivalent strain in the base metal ∆εeq_bm and in the weld metal ∆εeq_wm in each node of the FEM mesh Loop on the cycles Crack initiation damage fraction per cycle in each node
WM
Welded Joint (JW) ∆ε eq_global Base Metal
∆ε eq_bm
Weld Metal
∆ε eq_wm
Cumulated damage DiF
While DiF ≤ 1
191
NRbm NRwm
Si DiF > 1
NRbm = Nibm + NPbm
NRwm = Niwm + NPwm
Crack propagation damage fraction per cycle in each node
Accumulated damage DF
Decomposition using relation (10) of the number of cycles to rupture N R in a number of cycles to crack initiation N i and a number of cycles to crack propagation N P .
If ∆ε eq_global > ∆ε eq_th
If ∆ε eq_global < ∆ε eq_th
Base Metal
Weld Metal ∆εeq_wm close to the surface
D iF =
∆εeq_wm within the weld
1
∑N
cycles
i
D F = H ( D iF − 1) ∑
1 N cycles P
( D iF ) wm = 1 ( D F ) wm =
( D iF ) wm = 1 1
∑ N wm
cycles
P
Fig. 10. CPC calculation procedure.
( D F ) wm =
∑ N wm+0.51m 1
cycles
P
J
192
Y. Madi et al. / Nuclear Engineering and Design 228 (2004) 179–194
∆ε
Base Metal
∆εR
Jf =
Weldment
∆ε i
∆ε R ∆ε i
Jf NR
N RBM
WJ
Numbre of cycles To fracture
Fig. 11. Definition of Jf .
(Madi, 2001), showed that, for this loading, the axial rotations are almost similar to those measured. Also let us recall that the maximum strain variations calculated in the weld are in good agreement with those obtained
within the framework of this study. The total loading is applied to the welded structure so as to obtain, in the useful part of the homogeneous tube, an axial strain level of equal to 0.6% ( εglobal = 0.6%). Calculations Table 1 Life prediction of pipes with CPC approach Tests
CPC approach
Type of test
εglobal (%)
NR cycles
Nabm cycles
NPbm cycles
NRSH cycles
Homogeneous tube (TH) Butt-welded tube (TW)
0.6 0.6
2765 2067
3750 4020
1350 1430
5100
Nawm cycles
NPwm cycles
NRWJ cycles
0
2010
2010
Accumulated Damage 1 0.9 0.8 Damage DF PMA
0.7 0.6 0.5 0.4 0.3
Damage DF PMB
Damage DiF PMB
0.2 0.1 0 0
500
1000
1500
2000
2500
3000
3500
4000
4500 5000
Number of cycles N Fig. 12. Evolution of maximum damage in butt-welded tube, proposed approach.
5500
Y. Madi et al. / Nuclear Engineering and Design 228 (2004) 179–194
on Bi-material tests for the applied loading εbm = 0.6%. 8.1. Life prediction with CPC approach The results of the life prediction of tubes, by applying the calculation procedure illustrated in Fig. 10, are presented in Table 1. For the welded tube, the damage evolution allowing determination of the number of cycles to failure NRWJ is illustrated in Fig. 12. The damage analysis is done in two critical points of the structure. The more damaged point in the parent metal (point PMB) presents the two phases related to crack initiation damage DiF and to crack propagation damage DF . However, the damage at the more damaged point in the weld metal (point PMA) is expressed only in terms of damage fraction per cycle of propagation since the loading is higher than the threshold εeq th . The failure of the welded tube is predicted in the weld metal. 8.2. Fatigue life reduction factors Jf Let us recall that the failure of the homogeneous tube is predicted, in accordance with the tests, in the internal fillet (radius of connection) of the tube. The applied strain εeq bm , responsible for the failure of the tube, is higher than the global reference applied strain εglobal , in the useful part of the tube. Consequently, the number of cycles to failure NR which corresponds to a failure in the uniform part of the tube would be higher than that observed and calculated in the tube. As a result the value of the fatigue life reduction factor Jf = 1.11, established by using the experimental number of cycles, is underestimated. While considering, now, the value of reference, εi , equal to the maximum equivalent strain variation in the useful part of the homogeneous tube, the obtained fatigue life reduction factor is given in Table 2. This coefficient results from the application of our sugTable 2 Fatigue life reduction factor Jf of the tube at 600 ◦ C Experience (Jf -exp)
1.11
CPC approach εi
εR
Jf
0.671
0.950
1.42
193
gested method of prediction. The computed value is lower than that given by RCC-MR, i.e. Jf = 1.25. One thus finds the results obtained on the Bi-material tests. At a high loading level, the method of RCC-MR based on the fatigue life reduction factor is not useful.
9. Conclusions The welded joint tests reveal two types of crack initiation according to the loading level. The first type of failure (type I) is characteristic of the low levels loading. It results in a crack initiation on the surface in the weld metal. The second type of failure (type II) is characterized by a crack initiation in the interior of the specimen close to the interface of weld and parent metal. This type II is observed for the high load levels. A threshold of load level which separates these two types of failure was introduced. It corresponds to a strain variation in the parent metal εbm = 0.6%. Higher than this threshold value, the experimental results show that the crack initiation period then represents a negligible share of the fatigue life. In this paper, we propose a method which consists of approaching the mechanical behavior and the damage behavior in an uncoupled way. It makes it possible to determine the damage rate to crack initiation and to crack propagation of a significant crack and by the same method, one to calculate the number of cycles to failure NR . This method suggests the existence of a threshold loading level above which the micro-defects in the weld propagate directly; thus the crack initiation phase becomes negligible. An acceleration of the crack propagation is introduced to better describe the crack propagation following a crack initiation within the specimen. The application of our method to the tested tube shows that taking into account the threshold loading level εth in the procedure makes it possible to predict the failure in weld metal in accordance with the experimental observations. In this case, the crack initiation is obtained on the surface, the number of cycles to failure NRWJ is given then without the acceleration of the crack propagation phase. Without considering the threshold loading level, the predicted failure would be located in the radius of the junction and in the parent metal. The threshold loading level εth is thus validated by the tests. The developed
194
Y. Madi et al. / Nuclear Engineering and Design 228 (2004) 179–194
method allows us to criticize the RCC-MR design method which consists in using the Fatigue life reduction factor Jf . The values of Jf calculated for the welded joint test are comparable with the coefficients of experimental joints. The value of Jf , codified by the RCC-MR as equal to 1.25, is valid only for the low loading levels, ( εbm ≤ εth ). For the joint with an X-shaped weld, the threshold value εth is established as equal to 0.6%. For another weld shape type of welded joint, Bi-material tests will be necessary in order to determine at the same time the value of the threshold as well as the fatigue life reduction factor Jf .
Acknowledgements The present work is cofinanced by the CEA (Commissariat à l’Energie Atomique, France) and EDF (Electricité De France); it is led by the Laboratory of Integrity of the Structures and Standardization (LISN) of the CEA Saclay in collaboration with the Laboratory of Studies and Research in Structural Mechanics (LERMES) of the university Blaise Pascal in Clermont-Ferrand, France. References Berton, M.N., 1996. CEA Report. Saclay Research Institute, France. Brown, M.W., Miller, K.J., 1979. Initiation and growth of cracks in biaxial fatigue. Fatigue Fracture Eng. Mater. Struct. 1, 231– 246. Contesti, E., Cailletaud, G., 1989. Description of creep-plasticity with non-unified constitutive equations. Nucl. Eng. Des. 116 (3), 265–280. Curtit, F. 1999. Propagation de Fissures Semi-Elliptiques en Fatigue-Fluage à 650 ◦ C dans les Plaques d’Acier 316L(N)
Avec et Sans Joints Soudés. Ph.D. thesis. Ecole des Mines de Paris, France. DCRC, 1991. Concluding Report No. 5: Design Rules for Welds, 1st ed. Jacquelin, B., Hourlier, F., Pineau, A., 1983. Crack initiation under low-cycle multiaxial fatigue in type 316L stainless steel. J. Pressure Vessel Technol. 105, 138–143. Le Ber, L., Sainte Catherine, C., Waeckel, N., Turbat, M., 1998. Low cycle fatigue strength of high temperature welded joints: an efficient method to predict life of austenitic 316L(N) weldments. ECF12, Emas Publishing, Sheffield, UK. Levaillant, C., 1984, Approche métallographique de l’endommagement d’aciers inoxydables austénitiques sollicités en fatigue plastique ou en fluage: description et interprétation physique des interactions fatigue-fluage-oxydation. Ph.D. thesis. Univ. de Technologie de Compiegne, France. Madi, Y., 2001, Comportement et durée de vie d’une jonction tubulaire soudée en 316L à 600 ◦ C. Ph.D. thesis. Université B. Pascal—Clermont II, France. Manson, S.S., 1966. Interfaces between fatigue, creep and fracture. Int. J. Fracture Mech. 2 (1), 327–363. Manson, S.S., Halford, G.R., 1981. Practical implementation of the double linear damage curve approach for treating cumulative fatigue damage. Int. J. Fracture 17 (2), 169–192. Polvora, J.P., 1998. Propagation de Fissure à Haute Température dans un Acier Inoxydable Austénitique. Ph.D. thesis. Ecole des Mines de Paris, France. RCC-MR, 1993. Design and construction rules for mechanical components of FBR nuclear islands. AFCEN. Savalle, S., Cailletaud, G., 1982. Micro-amorçage, micropropagation et endommagement. La Recherche Aérospatiale 6, 395– 411. Socie, D.F., Shield, T.W., 1984. Trans. ASME. J. Eng. Mater. Technol. 106, 227. Socie, D.F., Waill, L.A., Dittmer, D.F., 1985. Biaxial Fatigue of Inconel 718 Including Mean Stress Effects, Multiaxial Fatigue, ASTM STP 853, Philadelphia, pp. 463–481. Usami, S. Fukuda, Y., Shida, S., 1983. Micro-crack initiation, propagation and threshold in elevated temperature inelastic fatigue. In: ASME Paper 83-PVP-97. Wareing, J., Vaughan, H.G., 1977. Metal Sci. 11, 439. You, B.-R., Lee, S.-B., 1996. A critical review on multiaxial fatigue assessments of metals. Int. J. Fatigue 18 (4), 235–244.