On low cycle fatigue life of nickel-based superalloy valve membranes under non-proportional cyclic loading

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International Journalof Fatigue

International Journal of Fatigue 30 (2008) 1160–1168

www.elsevier.com/locate/ijfatigue

On low cycle fatigue life of nickel-based superalloy valve membranes under non-proportional cyclic loading M. Abbadi

a,*

, S. Belouettar a, P. Muzzo b, P. Kremer b, O. Oussouaddi c, A. Zeghloul

c

a

c

LTI, CRP Henri Tudor, 29 Bd John F. Kennedy, L-1855 Luxemburg, Luxembourg b Ceodeux Puretec, 24 route de Diekirch – BP 19, L-7505 Lintgen, Luxembourg Laboratoire de Physique et Me´canique des Mate´riaux, UMR CNRS 7554, Universite´ Paul Verlaine, Metz F-57045, France Received 13 February 2007; received in revised form 10 September 2007; accepted 27 September 2007 Available online 10 October 2007

Abstract A numerical model for low cycle fatigue damage analysis of valve membranes is presented in this paper. The isotropic and non-linear kinematic hardening laws of Lemaıˆtre–Chaboche, which define both the cyclic hardening and the ratchetting phenomenon, are adopted. The model for prediction of the number of cycles to crack initiation is based on a combination between Manson–Coffin relationship and Jiang–Sehitoglu fatigue parameter. The applied cyclic loading consists in a load pressure imposed to the internal face of the membrane and a vertical displacement of its hub section. The experimental results and numerical simulation predictions in terms of number of cycles to failure turned out to be in good concordance. Thus, numerical predictions are confirmed by microscopic observations made on membranes failed during testing.  2007 Elsevier Ltd. All rights reserved. Keywords: Low cycle fatigue; Contact fatigue; Crack initiation; Numerical modelling; Nickel-based superalloys

1. Introduction The development of reliable lifetime components is of extreme importance for structures that may undergo severe ratchetting, which may lead, in turn, to failure due to either excessive deformation or ratchetting and fatigue interaction [1,2]. A concrete example of such structures is the valve membrane (see Fig. 1), which may encounter ratchetting under cyclic loading (opening and closing cycles). For this reason, gas valve manufacturers are striving at developing high tightness membranes in order to achieve higher life duration and to comply with environmental legislation. The shape of the various parts of the valve is conceived so that the interaction between the membrane and the different parts leads to minimum stress raisers. Moreover, the nickel-based superalloy membrane and the *

Corresponding author. Tel.: +352 545580530; fax: +352 545580501. E-mail address: [email protected] (M. Abbadi).

0142-1123/$ - see front matter  2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2007.09.010

steel holders make it possible to ensure a perfect contact as well as a good resistance of the membrane to different kinds of gas. To describe the mechanical behaviour of the valve membrane under in-service loading conditions, a finite element model based on the work of Lemaıˆtre and Chaboche [3], was developed for the present study. Then, the number of cycles to failure was computed from the relationship between the fatigue parameter of Jiang–Sehitoglu [4–6], and Manson–Coffin equation. The membrane is considered as being the most sensitive component of the valve here studied, as shown in Fig. 1. This membrane has a circular shape and its dimensions consist of an inside diameter Di and an outside diameter De, while its thickness is h. It should be stressed that this preliminary study consists in only one membrane to better understand the material behaviour and it is obvious that the number of cycles to crack initiation is lower than that of the industrial configuration. Indeed, the last includes four membranes of different nickel-based materials.

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Nomenclature Di De h F f r r0 a adev S r0 Q1 b0 ep a_

inside diameter of the membrane outside diameter of the membrane membrane thickness yield surface function equivalent Mises stress stress current yield stress back-stress deviatoric part of the back-stress tensor deviatoric stress tensor initial yield stress maximum change in the size of the elastic range rate at which the elastic range develops equivalent plastic strain incremental back-stress

C,c k, b e_ p FP De rmax Dc Ds J E Nf rf ef b c

material parameters material parameters equivalent plastic strain rate fatigue parameter normal strain range maximum normal stress shear strain range shear stress range material constant Young’s modulus fatigue life fatigue strength coefficient fatigue ductility coefficient fatigue strength exponent fatigue ductility exponent

To bear the pressure and ensure a certain level of purity, the valve membrane is made of hastelloy C-276. The latter is used because of its excellent general corrosion resistance and good fabricability [7]. The mechanical characteristics of this material are shown in Table 1. Holder 1

2. Finite element analysis Holder 4 Membrane Holder 2

Holder 3 Fig. 1. Sketch of the valve components.

The hub part of the membrane is welded to the stem of the valve and maintained to the hub membrane holders while the rim part is embedded between two rigid bodies. This pinch makes it possible to ensure a good tightness but should not deteriorate the material in the pinch area, to the risk to create cracks. However, this pinch substantially alters the strain material behaviour, which becomes then plastic in a certain zone around the pinch area. The membrane mounting enables the displacement of the hub part with regard to the fixed part (rim). This displacement is function of the desired motion of the stem and is then regulator of the valve delivery. The real working loading conditions consist in introducing the gas into the vessel until filling it up, and once it has been closed at both ends, a pressure intensifier continues introducing gas into the vessel until the desired pressure is reached. The valve components and namely the membrane remains at this high pressure during a period of 20 seconds after which, the gas is emptied out of the vessel through a valve that is opened in order to eliminate the inside pressure.

2.1. Constitutive model The cyclic hardening characteristic is described by the isotropic hardening law and the ratchetting effect by the non-linear kinematic hardening law [8]. This hardening model is based on the work of Lemaıˆtre and Chaboche [3] and is included in this paper for completeness. The pressure-independent yield surface is defined by the function: F ¼ f ðr  aÞ  r0 ¼ 0

ð1Þ

0

where r is the current yield stress, and f(r  a) is the equivalent Mises stress with respect to the back-stress a, and defined as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 ðS  adev Þ : ðS  adev Þ f ðr  aÞ ¼ ð2Þ 2 where S is the deviatoric stress tensor (defined as S = r + pI, where r is the stress tensor, p is the equivalent pressure stress, and I is the identity tensor) and adev the deviatoric part of the back-stress tensor.

Table 1 Mechanical characteristics of C-276 alloy Tensile strength (MPa)

Yield strength (MPa)

Young’s modulus (MPa)

Elongation (%)

934

483

205,500

49

M. Abbadi et al. / International Journal of Fatigue 30 (2008) 1160–1168

p

r0 ¼ r0 þ Q1 ð1  eb0e Þ

ð3Þ

where ep is the equivalent plastic strain, r0 is the initial yield stress at zero plastic strain, and Q1 and b0 the material parameters. Q1 defines the maximum change in the size of the elastic range and b0 the rate at which the elastic range develops. The non-linear kinematic hardening component describes the ratchetting effect by describing the translation of the yield surface in stress space through the back-stress. This is characterized by unsymmetrical cycles of stress caused by ratchetting effect in the direction of mean stress. This law is defined as an additive combination of a linear term and a relaxation term, which introduces the nonlinearity: a_ ¼

C ðr  aÞe_ p  cae_ p r0

ð4Þ

where e_ p is the equivalent plastic strain rate, and C and c are the material parameters that are normally calibrated from cyclic test data. C is a kind of hardening modulus and c defines the rate at which the kinematic hardening modulus decreases as plastic deformation develops. Integration of the kinematic hardening for monotonic loading in one-dimension yields a¼

C p ð1  ece Þ c

ð5Þ

The above-mentioned constitutive model for metals subject to cyclic loading incorporates some material parameters. These parameters are usually determined from the cyclic torsion of metal bar or tube, which is not applicable for flat sheet metals due to the buckling effect. However, a pure isotropic hardening model and a pure non-linear kinematic hardening model can be determined from conventional tensile test stress–strain data. The total stress in uniaxial tension is the combination of the isotropic hardening component by Eq. (3) and the kinematic hardening component by Eq. (5), and is expressed as below: p

r ¼ r0 þ Q1 ð1  eb0e Þ þ

C p ð1  ece Þ c

ð6Þ

The isotropic hardening parameters Q1 and b0 can be calibrated from uniaxial tensile stress–strain curves when the kinematic hardening component is neglected. In the same way, the kinematic hardening parameters C and c can be determined when Q1 and b0 are assigned zero value. The pure isotropic hardening model can also be defined by the tabulated data of true stress and true plastic strain. The isotropic hardening and the kinematic hardening parameters are determined from the experimental data obtained by Tong et al. [9]. Thus, Q1 and b0 are found

to be 154 MPa and 5.54, respectively, while C and c are equal to 242,800 MPa and 395, respectively. 2.2. Loading conditions A finite element model was performed in ABAQUS CAE and executed in ABAQUS 6.5 Standard. The membrane is assumed to be an axisymmetric body with radial symmetry. The stress analysis enables to obtain the stress distribution and the stress concentration, in the critical area, generated by the loadings (displacement of the hub parts and pressure on the inside face of the membrane). Numerical simulations were conducted for a cyclic displacement of 0.35 mm and two cyclic pressures of 75 and 100 bars, respectively. Fig. 2 illustrates the evolution of both the normalized displacement of the membrane and the normalized pressure imposed to the inside face of the membrane. Note that the displacement and the pressure are normalized by the maximum displacement and operating pressure, respectively. The peculiarity of this loading resides in the fact to be non-proportional, i.e. the applied displacement and load pressure are not maintained in strict proportion throughout the loading cycle. Before loading, a tightening force of 3500 N (respectively, 10,000 N) is exerted on the hub parts of the valve membrane (respectively, rim), as shown in Fig. 3. Then, an initial displacement of 0.55 mm is imposed to the hub part of the membrane. 2.3. Mesh Since the body is axisymmetric, any longitudinal section passing through the longitudinal axis can be analyzed, and the analysis is, therefore, 2D, with three normal stress components: radial stress, axial or longitudinal stress, and tangential stress. Thus, a 2D axisymmetric eight nodes element with reduced integration, CAX8R, for the stress calcula-

2.0

1.2

Normalized displacement Normalized pressure

1.0 1.5 0.8 Cyclic displacement

Initial displacement 1.0

0.6

0.4 0.5

Normalized pressure

The isotropic hardening component of the model defines the equivalent stress as a function of the equivalent plastic strain. This evolution can be modelled with a simple exponential law for materials that either cyclically harden or soften as

Normalized displacement

1162

0.2

0.0 0.0

0.5

1.0

1.5

2.0

2.5

0.0 3.0

Time [s] Fig. 2. Evolution of applied loading in terms of load pressure and displacement.

M. Abbadi et al. / International Journal of Fatigue 30 (2008) 1160–1168

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regard to their capacity to model rachetting material response and found that the Jiang–Setitoglu model best simulates the ratchetting and ratchetting rate, compared with experimental results. In the present study, a combined energy–density based and critical plane approach is adopted for the prediction of fatigue life. Indeed, due to the non-proportional loading, material planes that undergo maximum strain amplitude may also undergo considerable shear strain amplitude and shear stresses. This behaviour obeys the fatigue-damage multiaxial fatigue criterion proposed by Jiang and Sehitoglu [4–6]. This latter is based on the Smith– Watson–Topper (SWT) relation, given in Bannantine et al. [23], and can be defined as

Fig. 3. Finite element model: CAD, loading and boundary conditions.

tion is used [8]. A mesh convergence is performed, which resulted in an element size of approximately 20 · 30 lm2 (cf. zoom Fig. 3). 2.4. Boundary conditions After that firm contact has been established between the membrane and the different membrane holders, the membrane is squeezed between the various rigid bodies. The rim parts are completely constrained in all degrees of freedom while the vertical movement of the hub parts is allowed. The contact is defined such that the friction coefficient is equal to 0.1 for non-constrained parts and zero for constrained ones. 3. Fatigue criterion adopted for life prediction Combined energy–density based and critical plane models are based on the assumption that the energy–density should be computed as a damage parameter on every material plane and for every increment of loading. A damage parameter is expressed in products of the stress and strain range components, as kDrDe + bDsDc, where the influence of each product is weighted by the material and load dependent constants, k and b. The material plane subjected to the maximum value of the fatigue parameter during cycle loading is defined as the crack plane. Moreover, this group of models is physically associated with two loading modes for fatigue initiation and fracture: tension is represented by kDrDe and shear by bDsDc. As a consequence, many failure models have been developed for the life prediction of components subjected to fatigue contact [5,10–12]. These models are based on the assumption that either the plane of maximum shear strain amplitude [13,14] or the plane of maximum normal stress range [15,16] or the combination between both the last quantities [4–6] governs the failure of the material. In previous investigations, Ringsberg et al. [17–19] compared three elastic–plastic approaches [3,5,20–22] with

FP ¼

De hrmax i þ J DcDs 2

ð7Þ

where the symbol Æ æ denotes the MacCauley bracket (i.e. Æxæ = 0.5(jxj + x)), FP is the fatigue parameter, De is the normal strain range, rmax is the maximum normal stress, Dc is the shear strain range, Ds is the shear stress range, and J is a load- and material-dependent constant and is obtained from traction/torsion test data. For indication, J is here assumed to be equal to 0.2, i.e. the contribution of mode II fracture is about 20%. All the stress and strain quantities in Eq. (7) are on the critical plane, which is defined as the plane where the fatigue parameter FP is a maximum. Through a tensor rotation for the stress and strain, the maximum FP and the critical plane are determined by surveying all the possible planes at a material point. When the stress loading cycles are determined, the fatigue analysis for each observed material point could be performed. The methods for fatigue analysis are most frequently based on the relation between deformations, stresses and number of loading cycles and are usually modified to fit the nature of the stress cycle, e.g. repeated or reversed stress cycle [24–26]. The number of stress cycles required for a crack initiation may be determined iteratively with the strain-life method e–N. The classical Manson–Coffin relationship [27] does not consider the effects of mean stress or strain on fatigue life. Thus, Smith et al. [15] used Basquin’s formulation for the maximum stress and multiplied it by the strain-life equation to obtain the SWT parameter [28]. In the same way, the number of cycles to failure can be calculated from the following energy-based approach:   De r2 FP max ¼ hrmax i þ J DcDs ¼ f ð2N f Þ2b þ rf ef ð2N f Þbþc 2 E max ð8Þ where E is the elastic modulus, 2Nf is the number of reversals to failure, and rf, b, ef and c are the fatigue strength coefficient, fatigue strength exponent, fatigue ductility coefficient and exponent, respectively.

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4. Computation methodology

1500

Finite element modelling of the membrane and the inservice loading conditions make it possible to reach the values of strains (De, Dc) and stresses (rmax, Ds). These values become stable from a certain number of cycles. The maximum value of fatigue parameter FPmax is then determined for the most deformed element of the membrane. Basic assumption: the elements where the fatigue parameter is a maximum have the same behaviour as a sample that has the size of the element and subjected to a low cycle fatigue loading. When these elements fail, the crack initiation takes place and the number of cycles necessary to this initiation is determined via Eq. (8) using MATLAB.

1000

Stress range (MPa)

Pressure load = 75 bars

500

0

-500

-1000

-1500 0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

Time (s)

5. Numerical results

Fig. 4. Evolution of the normal stress for a pressure load of 75 bars.

The applied loading to the membrane consists in a cyclic load pressure of 75 and 100 bars, respectively, and a cyclic vertical displacement of 0.35 mm. Then, the results predicted by the numerical model are compared to those obtained experimentally. 5.1. For a pressure load of 75 bars Numerical simulation, based on Lemaıˆtre–Chaboche model, shows a stabilization of the response from the 13th cycle (after about 2 h of CPU time). One should state that this delay in stabilization is due to the ratchetting effect. This stabilization is clearly observed in Fig. 4, which depicts the evolution of the normal stress. Then, the appraisal of all the variables necessary for the membrane fatigue life assessment is done at the end of the 13th cycle. The most deformed sections of the membrane are derived and the most deformed element is deduced from Fig. 5. It is noted that this element is located at a distance of 6.56 mm from the stem-axis. The hysteresis loop for all the cycles is shown in Fig. 6 where the ratchetting phenom-

enon is well observed. The values of the maximum normal stress and the plastic strain amplitude are derived from the variations of the normal stress versus the strain diagram. For clarity, the hysteresis loop is shown only for the last cycle (see Fig. 7). In the same way, the variations of the shear stress versus the shear strain, for the last cycle, is shown in Fig. 8. The values of shear stress and shear strain amplitudes are also mentioned. 5.2. For a pressure load of 100 bars In the case of a pressure load of 100 bars, the response becomes also stabilized from the 13th cycle (Fig. 9) and the most deformed sections are shown in Fig. 10. The most deformed element is found to be at a distance of 6.52 mm from the stem-axis. For clarity, the hysteresis loop representing the variation of both the normal stress versus the strain and the shear stress versus the shear strain is depicted only for the last cycle. These are shown in Figs. 11 and 12, respectively.

Fig. 5. Deformed shape of the membrane at the end of cycling for a pressure of 75 bars.

M. Abbadi et al. / International Journal of Fatigue 30 (2008) 1160–1168 1500

1500

Pressure load = 100 bars

1000

500

500

Stress (MPa)

Stress (MPa)

Pressure load = 75 bars, ratchetting effect 1000

13th cycle

0

-500

-1000

1165

0

-500

-1000

-1500 -0.020

-1500 -0.015

-0.010

-0.005

0.000

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

Time (s)

Strain Fig. 6. Stress–strain hysteresis loop for a pressure load of 75 bars.

Fig. 9. Evolution of the normal stress for a pressure load of 100 bars.

6. Experimental validation 1500

6.1. Validation tests σmax=1109

1000

Stress (MPa)

500 Δεp=0.0039

0

-500

-1000 Δεt=0.0146

-1500 -0.015

-0.010

-0.005

0.000

Strain Fig. 7. Stress–strain hysteresis loop for the last cycle corresponding to a pressure load of 75 bars.

150

Δγt=0.0051

Shear stress (MPa)

100

50 Δγp=0.0014

0

Δτ=267

-50

-100

In parallel with the numerical simulation, five test series were conducted for a cyclic displacement of 0.35 mm and two cyclic internal pressures of 75 and 100 bars, respectively, and a lifetime for the membrane is estimated. In the first case, the tests were stopped after the first 500 cycles and then each 100 cycles to observe the membrane surface. While in the second case, tests were stopped each 100 cycles. When the membrane is damaged, a leak of liquid is observed. The low cycle fatigue life is estimated to lie between 900 and 1000 cycles (respectively, 500 and 600) for a pressure load of 75 bars (respectively, 100 bars). 6.2. Microscopic observations Scanning electron microscope (SEM) observations of the fracture surface of different membranes subjected to a pressure load of 75 and 100 bars, respectively, allowed us to detect the crack location zone. Indeed, the microscopic analysis of a membrane surface that has undergone a pressure load of 75 bars displays no crack outside the area located between 5.09 and 6.58 mm from the stem-axis. On the contrary, several microcracks were observed inside this zone (Fig. 13). The final fracture of the membrane is the result of the coalescence of these microcracks. In the same way, the microscopic observations of a membrane surface subjected to a pressure load of 100 bars show that the fracture zone is located in the area that lies between 5.13 and 6.92 mm from the stem-axis (Fig. 14). 7. Discussion

-150 -0.002

0.000

0.002

Shear strain Fig. 8. Variations of the shear stress versus the shear strain for the last cycle corresponding to a pressure load of 75 bars.

According to the results of Section 5 and those of Section 6, it is well noted that the prediction of the membrane fracture location is in a good agreement with

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Fig. 10. Deformed shape of the membrane at the end of cycling for a pressure of 100 bars.

1500

200

σmax=1305

Δγt=0.0073

150

Shear stress (MPa)

Stress (MPa)

1000

500 Δεp=0.0071

0

-500

100 50 Δγp=0.0031

0

Δτ=316

-50 -100

-1000

-150 Δεt=0.0199

-1500 -0.020

-200 -0.015

-0.010

-0.005

0.000

0.005

Strain

-0.006

-0.004

-0.002

0.000

0.002

Shear strain

Fig. 11. Stress–strain hysteresis loop for the last cycle corresponding to a pressure load of 100 bars.

Fig. 12. Variations of the shear stress versus the shear strain for the last cycle corresponding to a pressure load of 100 bars.

microscopic observations performed on samples that failed during testing. The component here investigated, namely the membrane undergoes non-proportional cyclic loading. From Fig. 6, it

is clearly observed that the stress–strain hysteresis loops are not fully closed and an incremental strain is added during cycling. This is associated with ratchetting phenomenon, which is characterized by additional plastic deformation

Fig. 13. Membrane that has undergone a pressure load of 75 bars. (a) Zone located at 5.09 mm from the stem-axis; (b) zone located at 6.58 mm from the stem-axis.

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Fig. 14. Membrane subjected to a pressure load of 100 bars. (a) Zone located at 5.13 mm from the stem-axis; (b) zone located at 6.92 mm from the stemaxis.

for every load cycle. This additional deformation accelerates the failure of the material. To calculate the life duration of the valve membranes, Manson–Coffin parameters used for the studied alloy were approximately derived from experimental measurements performed by Ye et al. [29] on a similar material. Some adjustments of these parameters were needed with regard to tensile and yield strength. Moreover, this material is found to exhibit two different regimes depending on the plastic strain amplitude. Finally, Manson–Coffin parameters adopted are displayed in Table 2. The values used for the determination of the fatigue parameter for pressure loads of 75 and 100 bars, issued from numerical simulation (Section 5), are recapitulated in Table 3. After introducing the fatigue parameter value in the Manson–Coffin law, the membrane fatigue life is found Table 2 Manson–Coffin parameters Dep 2 Dep 2

6 0:2% P 0:2%

Pressure (bar)

rf (MPa)

ef

b

c

75 100

1100 1100

31.45 0.0207

0.0705 0.0705

1.545 0.246

Table 3 The value of the variables used for the determination of the fatigue parameter for a pressure load of 75 and 100 bars, respectively Pressure (bar)

rmax (MPa)

Dep (%)

Detot (%)

Ds (MPa)

Dcp (%)

Dctot (%)

FPmax (MPa)

75 100

1109 1305

0.39 0.71

1.46 1.99

267 316

0.14 0.31

0.51 0.73

2.2373 4.8287

Table 4 Comparison between predicted and experimental fatigue lives for a pressure load of 75 and 100 bars, respectively Pressure load (bar)

75 100

Fatigue life (Cycles) Predicted

Experimental

898 480

900–1000 500–600

to be 898 cycles (480 cycles, respectively) for a pressure load of 75 bars (100 bars, respectively). Numerical and experimental results with respect to the membrane fatigue life are summarized in Table 4. This table shows clearly that numerical predictions are in accordance with experimental results. Furthermore, the numerical model is found to be conservative. 8. Conclusion The numerical model of fatigue-damage initiation due to non-proportional loading of valve membrane is presented in this paper. The constitutive relationship of Lemaıˆtre– Chaboche enabled us to locate the most deformed sections. Then, the different variables necessary for the calculation of the membrane fatigue life were derived after the stabilization of the response. Based on the relationship between the fatigue parameter of Jiang–Sehitoglu and Manson–Coffin equation, the number of cycles to failure of the membrane is predicted. It turns out that the numerical approach predicts a lifetime very close to the experimental one for both pressure loads, namely 75 and 100 bars. Moreover, the prediction of the membrane fracture location is in good agreement with microscopic observations performed on samples that failed during testing. The numerical model developed for the present investigation could be generalized to different amplitudes and shapes of the applied cyclic loadings, materials, geometries, etc. However, more simulations at different amplitudes should be performed before confirming the validation of this model. Finally, this model helps the companies which aim at developing new valve design at reducing the costs related to waste time (testing) and money (manufacture of prototypes). Acknowledgements The authors gratefully acknowledge financial support for this work, via BFR/05/095, from the Ministry for Cul-

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