Numerical computation of crack growth of Low Cycle Fatigue in the 304L austenitic stainless steel

Engineering Fracture Mechanics 120 (2014) 67–81

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Engineering Fracture Mechanics

journal homepage: www.elsevier .com/locate /engfracmech

Numerical computation of crack growth of Low Cycle Fatiguein the 304L austenitic stainless steel

http://dx.doi.org/10.1016/j.engfracmech.2014.03.0130013-7944/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +216 22789722.E-mail address: [email protected] (O. Ammar).

Ons Ammar a,⇑, Nader Haddar a, Luc Remy b

a Laboratoire de Génie des Matériaux et Environnement, ENIS, B.P.W. 1173-3038, Université de Sfax, Sfax, Tunisiab Centre des matériaux, Ecole Nationale Supérieure des Mines de Paris, B.P. 87, 91003 Evry Cedex, France

a r t i c l e i n f o

Article history:Received 10 May 2013Received in revised form 12 March 2014Accepted 18 March 2014

Keywords:Finite element analysisConstitutive modellingJ-integralCrack growth

a b s t r a c t

This work aims to study the crack behavior of 304L under large scale yielding. An elasto-viscoplastic Chaboche model with a nonlinear kinematic hardening has been chosen. Theidentification of the law parameters was performed by using an experimental databasesissue from uniaxial tests. In addition, the chosen model is validated with TMF tests. Then,a 2D FEA of a tubular specimen is performed with ABAQUS� in order to study the crackgrowth by using the node release technique. This crack growth rate is estimated by usingan energy criteria based on the calculation of the J-integral at different crack lengths.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Various industrial components are subjected to damage caused by thermal fatigue loading, such as hot-forming tool[1–8], engine cylinder heads and blocks in automotive industry [9], gas-turbine blades in aircraft engines [10] and nuclearreactor components like Pressurized Water Reactors (PWR) [11,12]. In the case of PWRs, a leak occurred in the reactor heatremoval system (RHRS) of the Civaux 1 plant, in May 1998. The major root cause of cracking was identified as high cyclethermal fatigue [13,14]. The 304L stainless steel is the major constituent of residual heat removal circuits of PWR.

Indeed, in order to get an accurate description of fatigue material behavior and its properties, it is necessary to choose ordevelop a suitable constitutive model that will accurately describe the materials stress–strain behavior. The constitutivemodelling allows to predict the possible failures in highly loaded engineering components and consequently the optimiza-tion of their design [15]. Many constitutive models were developed to meet the needs for accurate description of materialbehavior and calculation of its lifetime [16–21]. These models consist of a number of parameters which significantly influ-ence the material model behavior. In this paper we consider mainly the inelastic part of the constitutive model, hereaftercalled the inelastic constitutive model. An inelastic constitutive model is called unified, if time dependent and time indepen-dent phenomena are described by a single set of equations [22]. Numerous typical examples of unified inelastic constitutivemodels for isotropic materials were set up [23–26].

Under cyclic loading, the structural materials show complicated mechanical responses involved with the plastic deforma-tion at isothermal and anisothermal conditions.

Moreover, the analysis of fatigue crack growth is very important to ensure the structures reliability under cyclic loadingconditions. Thus, a nonlinear fracture mechanics theory can be applied. Indeed, the J-integral [27] has been established as a

Nomenclature

r�; e� stress and strain tensorl� identity tensorE Young modulusm Poisson ratio_p plastic multipliera� hardening stress tensorA, n parameters of Norton lawJ2 second invariant of stress deviatoric tensorrj0 initial yield stressdevða

�Þ deviatoric stress tensor

Q1 and b material parameters describing the isotropic hardeningC and c material parameters describing the kinematic hardening

Table 1Chemical composition (wt%) of the studied 304L austenitic stainless steel [14].

C Mn Si S Ni Cr Cu Mo P N2

0.029 1.86 0.37 0.004 10.00 18.00 0.02 0.04 0.029 0.056

Fig. 1. Tensile curve of 304L stainless steel at different temperatures.

Fig. 2. LCF specimen used.

68 O. Ammar et al. / Engineering Fracture Mechanics 120 (2014) 67–81

Fig. 3. Tubular specimen with notch used for LCF (and TMF).

Fig. 4. (a) Fatigue crack propagated from a notch on a tubular specimen and (b) schematic notch.

Fig. 5. Plasticity highlighted on the crack tip (em = ±0.14%, T = 165 �C).

O. Ammar et al. / Engineering Fracture Mechanics 120 (2014) 67–81 69

failure criterion for stable or unstable crack growth. This idea has been extended to the analysis of fatigue crack growth rates[28,29]. According to the J-integral approach, the fatigue crack growth rate, da

dN ; is put in relation with the cyclic J-integral(Jcyclic), as follows: da

dN ¼ CðJcyclicÞm [30–34].

Table 2Governing equations of the Chaboche model [34].

Strain decomposition e�¼ e�e þ e

�p þ e�th (1)

Thermal strain e�th ¼ aT:ðT � Tref Þ � l

�(2)

Elastic strain e�e ¼ 1þm

EðTÞr� �m

EðTÞ traceðr�Þ � l�

(3)

Flow rule _e�p¼ _p 3

2

devðr�Þ�devða

�Þ

J2 r��a�Þ

(4)

Flow function _p ¼ AnðTÞðTÞhJ2ðr� XÞ � r0 � rj0inðTÞ (5)

Isotropic hardening r0 ¼ rj0 þ Q1ð1� e�be�

p

Þ (6)

Nonlinear kinematichardening

_a�¼ C 1

rj0ðr�¼ a�Þ _ep ¼ ca

�_ep (7)

Table 3Material parameters at 90 �C and 165 �C [14].

T (�C) Elastic modulus (MPa) Poisson ratio (m) Coefficients of thermal expansion (K�1)

90 195,000 0.3 1.2 � 10�6

165 184,792 0.3 1.6 � 10�6

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-400

-300

-200

-100

0

100

200

300

400

Strain (%)

Stre

ss (M

Pa)

(a)

....... experiment numerical computation

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-400

-300

-200

-100

0

100

200

300

400

Strain (%)

Stre

ss (M

Pa)

(b)

....... experiment numerical computation

Fig. 6. Comparison of numerical computation-experiment (a) T = 90 �C and (b) T = 165 �C, f = 1 Hz, for different strain range.

70 O. Ammar et al. / Engineering Fracture Mechanics 120 (2014) 67–81

O. Ammar et al. / Engineering Fracture Mechanics 120 (2014) 67–81 71

The present paper is organized in two main sections. In the first section, a visco-plastic model able to describe the plasticresponse of the 304L stainless steel under cyclic loading conditions is proposed. The emphasis is put on identifying the com-plete material parameters. The calibration of parameters is carried out, under strain-controlled cyclic loading, by using theavailable experimental stress–strain curves obtained from [14]. In the second section, a computation of crack growth byusing the node release technique is done in order to estimate the evolution of fatigue crack growth rate. da

dN is estimatedby using an energy criteria based on the calculation of the J-integral at different crack lengths.

2. Material and specimens

The material investigated in this study is an AISI 304L, type austenitic stainless steel. Its chemical composition is given inTable 1. In order to homogenize the structure and avoid chromium carbide precipitation Cr23C6, an hyper-quenching heattreatment was performed on specimens with a soaking temperature between 1050–1150 �C. Tensile properties as functionof temperature are shown in Fig. 1.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

50

100

150

200

250

300

Strain (%)

Stre

ss (M

Pa)

(a)

....... experiment numerical computation

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

50

100

150

200

250

Strain (%)

Stre

ss (M

Pa)

(b)

....... experiment numerical computation

Fig. 7. Comparison of numerical computation-experiment (a) T = 50 �C and (b) T = 150 �C for different strain amplitudes.

Table 4Material parameters identified for 304L stainless steel at 90 �C and 165 �C.

T (�C) rj0(MPa) C (MPa) c A (MPa�n s�1) n m f

90 90 37,500 370 8.78 � 10�22 9 0 0.3165 88.9 37,000 371 4.6 � 10�18 8.1 0 0.25

72 O. Ammar et al. / Engineering Fracture Mechanics 120 (2014) 67–81

Cyclic hardening tests were applied on element of volume specimen in Isothermal Fatigue (IF) and in Thermal MechanicalFatigue (TMF) tests for different strain amplitude. IF tests were carried out on cylindrical specimens (Fig. 2). A single periodof 1 s per cycle (1 Hz) used throughout the different levels of mechanical strain per specimen. Furthermore, TMF tests wereconducted on tubular specimens (1 mm thickness and 25 mm gauge length) (Figs. 3 and 4) and the frequency were about0.033 Hz.

-0.1 -0.05 0 0.05 0.1 0.15-200

-150

-100

-50

0

50

100

150

200

Strain (%)

Stre

ss (M

Pa)

(a)

....... experiment numerical computation

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-400

-300

-200

-100

0

100

200

300

400

Strain (%)

Stre

ss (M

Pa)

(b)

....... experiment numerical computation

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-400

-300

-200

-100

0

100

200

300

400

Strain (%)

Stre

ss (M

Pa)

(c)

....... experiment numerical computation

Fig. 8. Comparison of numerical computation-experiment for TMF test,type Out of Phase, for different mechanical strain amplitudes (a) ±0.1%, (b) 0.46% and(c) ±0.8%.

O. Ammar et al. / Engineering Fracture Mechanics 120 (2014) 67–81 73

Crack propagation under large scale yielding was studied on notched tubular specimen (Fig. 3). Tests performed in thiscase highlighted a strong plastic deformation characterized by intrusion/extrusion bands (Fig. 5). This finding led us toconsider in our subsequent analysis the approach of large scale instead of small scale plasticity.

3. Material constitutive behavior

The constitutive equation of the material is a fundamental step of any structural calculation. It provides the indispensablerelation between the strains and the stresses, which is a linear relation in the case of elastic analyses (Hooke’s law) and amuch more complex nonlinear relation in inelastic analyses, involving time and additional internal variables [21].

A unified visco-plastic Chaboche model has been used with a conventional power-law visco-plastic flow and two internalvariables to describe isotropic and nonlinear kinematic hardening. Hardening and softening are not taken into account. Theexpressions used are detailed in Table 2.

In order to better simulate the 304L stainless steel behavior and take into account the viscous component, a Norton law isadded.

_e ¼ Arn ð8Þ

3.1. Material behavior computation

3.1.1. Identification procedureThe present section is focused on the optimization of the constitutive law of 304L austenitic stainless steel by using an

elasto-viscoplastic behavior. Material’s identification was carried out by using the commercial finite element code ABAQUS.The elastic material properties and the coefficients of thermal expansion, used in the present numerical computation, aresummarized in Table 3.

In order to identify the material parameters, in terms of the proposed model, we adopted the following procedure. A firstset of parameters was imposed. It allowed to bring closer, even with a qualitative way, numerical computation to experimen-tal results. Computation were carried out on a volume element (1 mm � 1 mm) under strain control. The comparison weredone by using the experimental curves of cyclic hardening and relaxation tests at different strain levels.

Isothermal tests were carried out via a triangular strain signal under a strain ratio Re = �1 (Re = emin/emax), at 90 and165 �C and at a frequency of 1 Hz.

Plastic parameters rj0; C and c were defined starting from the experimental stress–strain hysteresis loops obtained at halflifetime [14] for isothermal loading at different strain range (ranged from De/2 = 0.1 to 0.8%). In addition, the calibration ofviscous parameters A, n, m and f, is done by using stress relaxation curves at 50 �C and 150 �C.

Thermal–mechanical fatigue (TMF) tests were conducted under a strain ratio Re = �1, a temperature range between 90 �Cand 165 �C and a single period of 30 s per cycle (0.033 Hz). Thermal–mechanical loading is type Out of Phase (OP). Differentstrain levels were tested.

3.1.2. Isothermal loadingFig. 6 illustrates a comparison between the experimental hysteresis loop (dotted lines) with the stabilized stress/strain

hysteresis loop (solid lines) at the 4th cycle. As it can be noticed, the chosen law can well reproduce the cyclic responseof 304L stainless steel for both 90 and 165 �C. The comparison highlights that computation’s results with the identifiedmaterial parameters predict well the experimental stabilized hysteresis loops for all strain amplitudes.

Fig. 9. Fracture surface under large yield scale T = 90 �C, Dem/2 = 0.09%, Re = 0.

74 O. Ammar et al. / Engineering Fracture Mechanics 120 (2014) 67–81

Norton’s parameters are then calibrated by using isothermal relaxation curves (Fig. 7). It should be noted that therelaxation behavior is well simulated for low-levels strain, while the visco-plastic law adopted is not able to simulate thebehavior for high strain levels.

The determined material parameters are summarized in Table 4.

3.1.3. Thermal–mechanical loadingStabilized stress/strain hysteresis loops r = f(e) were plotted with experimental curves at the same strain amplitude

(Fig. 8). It is worth to note that there is a good agreement between experimental (dotted lines) and computed cycle (solidlines). The law chosen simulates correctly the material behavior both for low and high strain levels under thermal–mechan-ical loading.

4. Cracking generalized plasticity

Fatigue crack analyses were performed by using ABAQUS FE code. The definition of geometry is one of the decisive stepsfor beginning a finite element computation. As we mentioned, this test is studied on notched tubular specimen (Fig. 3). Itsinternal diameter and wall thickness are equal to 9 and 1 mm respectively. The dimensions of the notch are 500 lm and100 lm as width and height respectively, and extends through the thickness of the specimens as shown in Fig. 9.

Fig. 10. A 2D finite element computation: boundary conditions and mesh.

O. Ammar et al. / Engineering Fracture Mechanics 120 (2014) 67–81 75

Taking advantage of symmetries, only a quarter of the specimen is modeled in 2D with an initial half crack length of250 lm. Boundary conditions and mesh chosen are illustrated by Fig. 10. The meshing was composed of 2970 four-nodedelements (3100 nodes). A refined meshing was used for the crack tip region (an element size of 10 lm) in order to correctlysimulate the high stress and strain gradients near the crack tip.

Fig. 11 shows the distribution of maximum stress S22 in the case of plane strain modeling. A high stress concentrationaround the crack tip is highlighted. The maximum stress reaches a value of 297 MPa. It is obviously to note that the stressconcentration create a local plastification, therefore, the size of the plastic zone is no longer negligible front crack length andthe hypothesis of generalized plasticity is confirmed.

The calculations of J-integral have been performed with the finite element code ABAQUS, by using *CONTOUR INTEGRALcommand. This integral is theoretically path independent [35]. However, it is widely accepted that the closer contour maynot provide good results because of numerical singularities. Therefore, the J-integral have been obtained from the secondcontour or even further from the crack tip, in order to get a convergent value. In this work, the computation is done withseven contours (Fig. 10). Fig. 12 shows a stabilization of the value of Jmax from the third contour. This result confirms thepath independence of J-integral.

Fig. 11. Stress concentration in the crack tip of cyclic loading, (a) tensile state and (b) compression state (2D, Plan strain assumption).

1 2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

1

1.2

1.4

N°path

Jmax

(N/m

m)

± 0.14%± 0.18%

Fig. 12. Path independence, 2D computation, T = 90 �C.

Fig. 13. Load cycles and node release scheme.

Fig. 14. Modification of mesh to evaluate J-integral, (a) old configuration and (b) new configuration.

Fig. 15. Propagation using release node technique with release node (Plane strain assumption, T = 90 �C).

76 O. Ammar et al. / Engineering Fracture Mechanics 120 (2014) 67–81

O. Ammar et al. / Engineering Fracture Mechanics 120 (2014) 67–81 77

5. Crack growth computation

5.1. Crack growth methodology

Crack growth was simulated by ABAQUS under 2D modelling, via a node release technique <DEBOND> [36]. A crack prop-agation criterion should be chosen. Several criterions are available in ABAQUS. In this paper, the crack length versus timecriterion is used. In order to specify the crack propagation explicitly as a function of total time, a crack length versus timerelationship and a reference point from which the crack length is measured are applied [36]. The load cycle at which noderelease should be made is not entirely clear. Solanki et al. [37] proposed the release of node at minimum load in order toavoid numerical problems caused by sudden change in crack displacement. However, R.C. McClung and H. Sehitoglu et al.[38,39] proposed the maximum load to release the node to express that crack progresses when it is entirely open. Recentwork has shown that good results can be obtained independently of the node release, provided that a suitable mesh refine-ment is used [40].

In this paper, node release was specified at minimum load. This is computationally easier, since there is no sudden changein crack displacement. The increment of crack growth was equal to the element size (10 lm). Fig. 13 shows the node releasescheme adopted in the present work. The crack progresses after each cycle, through the release of successive nodes with asimultaneous changing in boundary conditions (YSYMM).

It was noted that when the crack progresses, the contours chosen for the evaluation of J- integral lengthen more in the ‘ydirection’ and the size of elements along ‘x’ becomes very small compared to the ‘y’ one. Taking into account that the mesh

10-1 100 10110-8

10-7

10-6

ΔJ (kJ/m²)

d/dN

(m/c

yle)

± 0.14% -exp±0.18% -exp0-0.18% -exp± 0.14% -num±0.18% -num0-0.18% -num

Fig. 16. Comparison of numerical computation (-num)-experiment(-exp) of the fatigue crack growth rate da/dN versus DJ, at 90 �C.

10-1 100 101 10210-9

10-8

10-7

10-6

10-5

ΔJ (kJ/m²)

da/d

N (m

/cyc

le)

±0.14% - exp±0.18% - exp±0.3% - exp0-0.18% - exp±0.14% - num±0.18% - num±0.3% - num0-0.18% - num

Fig. 17. Comparison of numerical computation (-num)-experiment(-exp) of the fatigue crack growth rate da/dN versus DJ, at 165 �C.

78 O. Ammar et al. / Engineering Fracture Mechanics 120 (2014) 67–81

must be adapted to the geometry’s change of crack during propagation and in order to ensure a good quality of the evalu-ation of J-integral, an expansion of the mesh’s area including the crack up to the final size is applied (Fig. 14).

Fig. 15 shows various stages of crack propagation with a regular mesh throughout the area of its progress.

5.2. Analysis of the evolution of fatigue crack growth rate for isothermal conditions

Several philosophies describing crack growth in generalized plasticity exist and the ways of evaluating the J-integral arevarious. Some authors have considered that in fatigue, all the energy is recoverable (in tensile such as in compression) andhave admitted that crack opens as soon as reloading. In this case, the term Jcyclic is replaced with DJ on expressions whichevaluate J-integral. Therefore, other authors have considered that crack is closed in compression cycle and will not open onlywhen stress is positive. So, only the positive part of load (r > 0) is used and Jcyclic is defined as Jmax.

The evolution of fatigue crack growth rate as function of DJ at 90 and 165 �C are plotted on Figs. 16 and 17, respectively. Itis pertinent to note that da

dN and the value of DJ increase with the rise of the load level. An acceleration of crack propagation isobserved when DJ increases, according to the following law: da

dN ¼ CðDJÞm. The coefficients C and m depend on material andtest temperature. Many studies have considered the DJ-integral range as a parameter to correlate crack growth rate [41,42].Moreover, an acceptable agreement between numerical and experimental values is shown.

In the case of large scale yielding, the crack is considered open as soon as the deformation becomes irreversible. Therefore,opening stress and the variation of effective stress is:

Dreff ¼ rmax � ropen

In addition, the variation of Jeff [44,45] is written as follows:

Jeff ¼ Jmax � Jopen

In order to numerically define the moment when the crack opens, the evolution of displacement U2 and J (the 4th path) ver-sus time is plotted on Fig. 18. The evolution of da

dN in terms of effective DJ at 90 and 165 �C (Fig. 19) shows a difference in slopebetween low and high strain levels. It is also noted that the test at ±0.3% is not aligned with the rest of the tests, which maysuggest that the crack closes under compression cycle and the opening stress in this case corresponds to the minimum one.

The evolution of crack growth rate in terms of DJeff follows a power law: dadN ¼ CðDJeff Þ

m whose parameters C and m areregrouped on Table 5. As we note, crack growth rate evolves according to effective DJ. In addition, increasing the testtemperature from 90 to 165 �C increases the coefficient ‘m’ from 1.5309 to 1.7589.

815 820 825 830 8350

1

2

3

4

5

6

Time (s)

J (N

/mm

)

816 818 820 822 824 826 828 830 832 834 836

0

1

2

3

4

5

x 10-3

U2

(mm

)

U2

Jouv

Jmax

Δ Jeff

J

Fig. 18. Evaluation of DJ effective.

Table 5Crack growth law’s coefficients.

T (�C) m C

90 1.5309 2 � 10�8

165 1.7589 1 � 10�8

10-1 100 10110-9

10-8

10-7

10-6

ΔJeff (kJ/m²)

da/d

N (m

/cyc

le)

± 0.14%± 0.18%0-0.18%

(a)

10-1 100 101 10210-9

10-8

10-7

10-6

10-5

ΔJeff (kJ/m²)

da/d

N (m

/ccl

e)

± 0.14%± 0.18%± 0.3%0-0.18%

(b)

Fig. 19. Fatigue crack growth rate da/dN as function of DJeffective (a) 90 �C and (b) 165 �C.

10-1 10010-9

10-8

10-7

a (mm)

da/d

N (m

/cyc

le)

TMF-OP0-0.18% 90°C0-0.18% 165°C

Fig. 20. Fatigue crack growth rate da/dN versus crack length.

O. Ammar et al. / Engineering Fracture Mechanics 120 (2014) 67–81 79

10-1 100 10110-9

10-8

10-7

10-6

10-5

ΔJ (kJ/m²)

da/d

N (m

/cyc

le)

TMF-OP Propagation law at 90°C Propagation law at 165°C

Fig. 21. Fatigue crack growth rate da/dN as function of DJ.

80 O. Ammar et al. / Engineering Fracture Mechanics 120 (2014) 67–81

5.3. Analysis of the evolution of fatigue crack growth rate of anisothermal conditions

dadN is plotted vs. crack size in a log–log presentation as showed in Fig. 20. In the case of isothermal loading, an acceleration

of crack propagation is observed with the increasing crack size. However, we notice that dadN evaluated from TMF tests don’t

vary with crack length. We notice that in generalized plasticity, thermal mechanical fatigue type OP exhibits a behavior clo-ser to isothermal conditions at 165 �C.

Fig. 21 is a plot of dadN evolution vs. DJ for isothermal and TMF tests type OP. It can be noted that the crack growth rate

decreases with the rise of temperature, this result is due to the stress relaxation. Thus, the lower stress level is accompaniedwith strain accommodation which is expressed with a low plastic size on crack tip. Moreover, the kinetic of propagation inanisothermal loading is different from the ones obtained at 90 and 165 �C. The stress concentration phenomenon is atten-uated with the variation of temperature. Thus, for a given crack length, the variation of the J integral is lower in the case ofanisothermal loading than for isothermal one. The same result was highlighted by Haddar et al. [43], in the case of F17TNbstainless steel.

6. Conclusion

The use of continuum mechanics constitutive models into engineering application encounters the difficulties to find ref-erences about the material parameters obtained by experimental data.

Numerical computations are carried out on stationary cracks by using an elasto-viscoplastic behavior law with doublehardening (isotropic and kinematics). The computed responses agree reasonably well with the experimental results. Crackpropagation is achieved with node release technique and followed by the evaluation of energy criterion. It can be concludedthat:

� the evolution of da/dN vs. criterion energy follows a power law on isothermal conditions;� the crack propagation is accelerated when DJ reaches high values;� the part of the compression cycle plays a significant role in the crack propagation;� the kinetic of propagation in anisothermal loading is different from that obtained at isothermal loading.

Appendix A. Supplementary material

Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.engfracmech.2014.03.013.

References

[1] Da Silva CRS, Boccalini M. Thermal cracking of multicomponent white cast iron. Mater Sci Technol 2005;21:565–73.[2] Dai WS, Ma M, Chen JH. The thermal fatigue behavior and cracking characteristics of hot-rolling material. Mater Sci Engng A 2007;448:25–32.[3] Persson A, Hogmark S, Bergström J. Thermal fatigue cracking of surface engineered hot work tool steels. Surf Coat Technol 2005;191:216–27.[4] Srivastava A, Joshi V, Shivpuri R. Computer modeling and prediction of thermal fatigue cracking in die-casting tooling. Wear 2004;256:38–43.

O. Ammar et al. / Engineering Fracture Mechanics 120 (2014) 67–81 81

[5] Jean S, Miquel B, Le Roux S, Rézaï-Aria F. An investigation on heat checking of hot work tool steels. In: Proceedings of the 5th international toolingconference. Tool steels of the next century. Leoben (Austria): Institut für Metallkunde und Werkstoffprüfung; 1999. p. 185–9.

[6] Medjedoub F, Dour G, Rézaï-Aria F, Hairy P. Damage to die casting dies through crazing by thermal fatigue: origins mechanisms and approaches. FondFondeur d’Aujourd’hui 2005;244:22–37.

[7] Fazarinc M, Muhic T, Kugler G, Tercelj M. Thermal fatigue properties of differently constructed functionally graded materials aimed for refurbishing ofpressure-die-casting dies. Engng Fail Anal 2012;25:238–49.

[8] Klobcar D, Tušek J. Thermal stresses in aluminum alloy die casting dies. Comput Mater Sci 2008;43:1147–54.[9] Engler-Pinto Jr CC, Su X, Lasecki JV, Chen X, Allison JE. Thermo-mechanical fatigue analysis of cast aluminum cylinder heads. In: Proceedings of

materials science & technology conference, Detroit (USA); 2007. p. 1546–57.[10] Mazur Z, Luna-Ramírez A, Juárez-Islas JA, Campos-Amezcua A. Failure analysis of a gas turbine blade made of Inconel 738LC alloy. Engng Fail Anal

2005;12:474–86.[11] Kamaya M, Taheri S. A study on the evolution of crack networks under thermal fatigue loading. Nucl Engng Des 2008;238:2147–54.[12] Fissolo A, Amiable S, Ancelet O, Mermaz F, Stelmaszyk JM, Constantinescu A, et al. Crack initiation under thermal fatigue: an overview of CEA

experience. Part I: Thermal fatigue appears to be more damaging than uniaxial isothermal fatigue. Int J Fatigue 2009;31:587–600.[13] Cipière MF, Le Duff JA. Thermal fatigue experience in French piping. International Institute of Welding. Document no. XIII-1891-01, Lubyana; 2001.[14] Haddar N. Thermal fatigue of a 304L type austenitic stainless steel: simulation on initiation and short crack growth in isothermal and anisothermal

fatigue. PhD Thesis. Ecole des Mines de Paris, France; 2003 (in French).[15] Franulovic M, Basan R, Prebil I. Genetic algorithm in material model parameters’ identification for low-cycle fatigue. Comput Mater Sci

2009;45:505–10.[16] Ottosen N, Ristinmaa M. The mechanics of constitutive modeling. Elsevier; 2005.[17] Lemaitre J. A course on damage mechanics. Springer; 1996.[18] Lemaitre J, Chaboche JL. Mechanics of solid materials. Cambridge University Press; 1990.[19] Bari S, Hassan T. Anatomy of coupled constitutive models for racheting simulation. Int J Plast 2000;16:381–409.[20] Chaboche JL. Continuum damage mechanics: Part I – General concepts. J Appl Mech 1988;55:59–64.[21] Chaboche JL. A review of some plasticity and viscoplasticity constitutive theories. Int J Plast 2008.[22] Lehn J, Harth T, Schwan S, Kollmann FG. Identification of material parameters for inelastic constitutive models: statistical analysis and design of

experiments. Int J Plast 2004;20:1403–40.[23] Bodner SR, Partom Y. Constitutive equations for elastic-viscoplastic strain hardening materials. J Appl Mech 1975;42:385–9.[24] Chaboche JL. Constitutive equations for cyclic plasticity and cyclic viscoplasticity. Int J Plast 1989;5:283–95.[25] Krempl E et al. Viscoplasticity based on overstress with a differential growth law for the equilibrium stress. Mech Mater 1986;5:35–48.[26] Steck E. A stochastic model for the high-temperature plasticity of metals. Int J Plast 1985;1:243–58.[27] Rice JR. Mathematical analysis in the mechanics of fracture. In: Liebowitz H, editor. Treatise on fracture, vol. 2. New York: Academic Press; 1968. p.

191–312.[28] Dowling NE, Begley JA. Fatigue crack growth during gross plasticity and the J-integral. In: Mechanics of crack growth, ASTM STP 590. Philadelphia,

PA: ASTM; 1976. p. 82–103.[29] Chow CL, Lu TJ. On the cyclic J-integral applied to fatigue cracking. Int J Fract 1989;40:53–9.[30] Koizumi T, Okazaki M. Crack growth and prediction of endurance in thermal mechanical fatigue of 12Cr–Mo–V–W steel. Fatigue Fract Engng Mater

Struct 1979;1:509–20.[31] Taira S, Ohtani R, Komatsu T. Application of J-integral to high temperature crack propagation Part II – Fatigue crack propagation. J Eng Mater Technol

Trans ASME 1979;101:162–7.[32] Ohtani R, Kitamura T, Nitta A, Kuwabara K. High-temperature low cycle fatigue crack propagation and life laws of smooth specimens derived from the

crack propagation laws. In: Solomon HD, Halford GR, Kaisand LR, Leis BN, editors. Low cycle fatigue ASTM STP 942. Philadelphia: American Society forTesting and Materials; 1988. p. 1163–80.

[33] Ohtani R, Kitamura T. Creep-fatigue interaction under high-temperature conditions. In: Carpinteri Andrea, editor. Handbook of fatigue crackpropagation in metallic structures. Elsevier Science; 1994. p. 1347–83.

[34] Renner E, Vehoff H, Neumann P. Life prediction for creep-fatigue based on the growth of short cracks. Fatigue Fract Engng Mater Struct1989;12:569–84.

[35] Courtin S, Gardin C, Bézine G, Ben Hadj Hamouda H. Advantages of the J-integral approach for calculating stress intensity factors when using thecommercial finite element software ABAQUS. Engng Fract Mech 2005;72:2174–85.

[36] ABAQUS user’s manual, version 6.10, volume2.[37] Solanki K, Daniewicz SR, Newman Jr JC. Finite element modelling of plasticity induced crack closure with emphasis on geometry and mesh refinement

effects. Engng Fract Mech 2003;70:1475–89.[38] McClung RC, Sehitoglu H. On the finite element analysis of fatigue crack closure – 1. Basic modelling issues. Engng Fract Mech 1989;33(2):237–52.[39] McClung RC, Sehitoglu H. On the finite element analysis of fatigue crack closure – 2. Numerical results. Engng Fract Mech 1989;33(2):253–72.[40] Solanki KN. Two and three-dimensional finite element analysis of plasticity-induced fatigue crack closure – a comprehensive parametric study. M.sc.,

Mississsipi State University; 2002.[41] Srivastava YP, Garg SBL. Study on modified J-integral range and its correlation with fatigue crack growth. Engng Fract Mech 1988;30(2):119–33.[42] Leslie Banks-Sills, Yehuda Volpert. Application of the cyclic J-integral to fatigue crack propagation of Al 2024-T351. Engng Fract Mech

1991;40(2):355–70.[43] Haddar N, Kchaou Y, Köster A. Numerical computation of the energetic criterion at isothermal and thermal–mechanical cyclic tests under generalized

plasticity of the F17TNb stainless steel. Engng Fract Mech 2013;107:1–13.[44] Ellyin F, Ozah F. 3D modeling of plasticity induced fatigue crack closure-effect of material constitutive relations. Engng Fract Mech 2010;77:1693–707.[45] Wu J, Ellyin F. A study of fatigue crack closure by elastic–plastic finite element analysis for constant-amplitude loading. Int J Fract 1996;82:43–65.