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XV Portuguese Conference on Fracture, PCF 2016, 10-12 February 2016, Paço de Arcos, Portugal XV Portuguese Conference on Fracture, PCF 2016, 10-12 February 2016, Paço de Arcos, Portugal

Numerical validation of crack closure concept using non-linear Numerical validation of crack closure concept using non-linear crack PCF tip parameters XV Portuguese Conference on Fracture, 2016, 10-12 February 2016, Paço de Arcos, Portugal crack tip parameters b b FV Antunesa,a,*, Rmodeling Brancoaa, L Correia , AL Ramalho , S Mesquitaaa b b turbine blade Thermo-mechanical of a high pressure of an FV Antunes *, R Branco , L Correia , AL Ramalho , S Mesquita CEMUC, Dep. of Mechanical Engineering, University of Coimbra, Rua Luís Reis Santos, 3030-788 Coimbra, Portugal airplane gas turbine engine CEMUC, Escola Superior Tecnologia do Instituto Politécnico Branco, Av.Reis do Empresário, 6000 -Coimbra, 767 Castelo Branco, Portugal CEMUC, Dep. de of Mechanical Engineering, Universityde ofCastelo Coimbra, Rua Luís Santos, 3030-788 Portugal a

b b

a

CEMUC, Escola Superior de Tecnologia do Instituto Politécnico de Castelo Branco, Av. do Empresário, 6000 - 767 Castelo Branco, Portugal

P. Brandãoa, V. Infanteb, A.M. Deusc*

AbstractaDepartment of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, Portugal Abstract b of Mechanical Instituto Superiorissues Técnico, Universidade Lisboa, Av. Rovisco Pais,some 1, 1049-001 CrackIDMEC, closureDepartment concept has been widelyEngineering, used to explain different of fatigue crackde propagation. However, authorsLisboa, have Portugal Crack closure concept has of been widely used and to explain different issues of fatigue crack The propagation. However, some authors have questioned the relevance crack closure have proposed alternative concepts. main objective here is to check the c CeFEMA, Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, questioned of crack closure and have concepts. The maincrack objective here is to Numerical check the effectivenesstheofrelevance crack closure concept by linking the proposed contact Portugal ofalternative crack flanks with non-linear tip parameters. effectiveness of crack closure concept by linking the contact of acrack non-linear crack tipThe parameters. models with and without contact of crack flanks were built for wideflanks range with of loading parameters. non-linearNumerical crack tip models withstudied and without contact built a wide range ofdisplacement loading parameters. tip parameters were the rangeofofcrack cyclicflanks plasticwere strain, thefor crack tip opening (CTOD) The and non-linear the size of crack reversed parameters were the range of cyclic plastic strain, the crackand tip K opening (CTOD) andofthe sizeflanks of reversed plastic zone.studied Well defined relations between non linear parameters were displacement obtained without contact crack which Abstract plastic zone. defined relations between nonmechanics. linear parameters andCTOD K were contact crack flanks which values are reinforces theWell validity of linear elastic fracture When the withobtained contact without is plotted versusofK eff the the values are reinforces linear elastic fracture mechanics. When thethe CTOD is plotted versus During the their operation, modern aircraft engine components are subjected to contact increasingly operating conditions, coincident withvalidity those of obtained without contact, which shows that crackwith closure concept isdemanding valid and K able to explain the eff especially the those high pressure turbine (HPT) blades. Such conditions cause these parts to undergo different types coincident obtained without which shows that the crack closure concept valid of andcrack ableof totime-dependent explain the influence ofwith mean stress. The analysis of contact, overloads and high-low load blocks demonstrated the is validity closure concept degradation, onestress. of which creep. Aofmodel usingand thehigh-low finite element methoddemonstrated (FEM) was developed, be ableconcept to predict influence of amplitude mean Theisanalysis overloads load blocks the validityinoforder cracktoclosure for variable loading. creepamplitude behaviourloading. of HPT blades. Flight data records (FDR) for a specific aircraft, provided by a commercial aviation forthe variable company, were used to obtain © 2016 The Authors. Published bythermal Elsevierand B.V.mechanical data for three different flight cycles. In order to create the 3D model © 2016, PROSTR (Procedia Structural Integrity) Hosting by Elsevier All rights needed forunder the FEM analysis, aElsevier HPT blade was Ltd. scanned, andreserved. its chemical composition and material properties were © 2016 The Authors. Published by B.V. scrap Peer-review responsibility of the Scientific Committee of PCF 2016. Peer-review under responsibility of the Scientific Committee ofthe PCF 2016. obtained. The data that was gathered was fed into FEM model and different simulations were run, first with a simplified 3D Peer-review under responsibility of the Scientific Committee of PCF 2016. rectangular block shape, in order to better establish the model, and then with the real 3D mesh obtained from the blade scrap. The Keywords: Fatigue crack propagation; plasticity induced crack closure; non-linear crack tip parameters; overloads overall expected behaviour in terms of displacement was observed, in particular at the trailing edge of the blade. Therefore such a Keywords: Fatigue crack propagation; plasticity induced crack closure; non-linear crack tip parameters; overloads model can be useful in the goal of predicting turbine blade life, given a set of FDR data.

1. © Introduction 2016 The Authors. Published by Elsevier B.V. 1. Peer-review Introduction under responsibility of the Scientific Committee of PCF 2016. Engineering analysis of FCG is usually performed by relating da/dN to the stress intensity factor range, K. Engineering analysis of that FCG isCreep; usually performed by relating da/dN to the stressdescribe intensitythe factor K. Keywords: Turbine Blade; Finite Element Method; 3Dcould Model; Simulation. Initially it High was Pressure surprising the linear-elastic parameter also successfully raterange, of plastic Initially it was surprising that the linear-elastic parameter could also successfully describe the rate of plastic * Corresponding author. Tel.: +0-000-000-0000 ; fax: +0-000-000-0000 . E-mail address:author. [email protected] * Corresponding Tel.: +0-000-000-0000 ; fax: +0-000-000-0000 . E-mail address: [email protected] 2452-3216 © 2016 The Authors. Published by Elsevier B.V. Peer-review underThe responsibility of theby Scientific Committee of PCF 2016. 2452-3216 © 2016 Authors. Published Elsevier B.V. Peer-review underauthor. responsibility the Scientific Committee of PCF 2016. * Corresponding Tel.: +351of218419991. E-mail address: [email protected]

2452-3216 © 2016 The Authors. Published by Elsevier B.V.

Peer-review under responsibility of the Scientific Committee of PCF 2016. 2452-3216 © 2016, PROSTR (Procedia Structural Integrity) Hosting by Elsevier Ltd. All rights reserved. Peer review under responsibility of the Scientific Committee of PCF 2016. 10.1016/j.prostr.2016.02.013

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processes at the crack tip. Later, Rice (1967) showed that the small-scale cyclic plasticity at the crack tip is, indeed, controlled by the value of K. According to Paris law (Paris and Erdogan, 1963), da/dN is uniquely determined by one loading parameter, the stress intensity factor range, K. However, other parameters, like mean stress or load history, influence da/dN,. The concept of crack closure was proposed by Christensen (1963), leading to a decrease of stress intensity at the crack tip and to an increase of fatigue life. As the crack propagates due to cyclic loading, a residual plastic wake is formed. The deformed material acts as a wedge behind the crack tip and the contact of fracture surfaces is forced by the elastically deformed material, which is called plasticity induced crack closure (Elber, 1970). This concept established for the first time that the crack growth rate is not only influenced by the conditions ahead the crack tip, but also by the nature of the crack flanks contact behind the crack tip. The effect of specimen geometry on crack closure has been accounted for using the T-stress concept (Tong, 2002). However, several questions have been raised, questioning the crack closure concept. The interpretation of experimental and numerical results in order to obtain the crack closure values is subjective, since different strategies can be followed giving significant scatter (Philips, 1989). Considering the uncertainty associated with experimental and numerical measurements, crack closure values cannot now be seen as absolutes. Therefore, the importance and even the existence of crack closure effect have been questioned by different authors. Some researchers suggest that closure can only occur under plane stress (Alizadeh, 2007), while others believe that it may not occur at all. Since 1993, Sadananda and Vasudevan (1993), Kujawski (2001), Glinka (2005) and Toribio (2013) have advocated that because the closure occurs behind the crack tip, it has a rather limited effect on the damage process, which takes place at the ‘process zone’ in front of the crack. According to these researchers the approaches to fatigue behavior based on crack closure (i.e. on what happens behind the crack tip) should be replaced by approaches based on what happens ahead of the crack tip. They argued that closure effects on FCG behavior have been greatly exaggerated, and suggested that the fatigue crack propagation rate is controlled by a two parameter driving force, which is a function of the maximum stress intensity factor, Kmax, and the total stress intensity factor range, K. Clearly there is no general agreement among researchers regarding the significance of closure concept on fatigue crack behavior. The difficulties encountered in the use of da/dN-K curves may however be overcome with the use of non-linear crack tip parameters. The direct link between crack closure and crack tip fields has not been totally exploited. This might be due to experimental difficulties in measuring quantitative strain/stress fields near a fatigue crack tip under applied loads and observing in situ crack tip deformation and failure phenomena during real-time fatigue experiments (Lee, 2011). The effects of thickness, stress state, specimen geometry and overloads will be naturally accommodated by non linear parameters. The non-linear crack tip parameters effectively control fatigue damage, therefore their analysis is a gate for the understanding of da/dN variations. Different parameters have been used in literature. Pokluda (2013) stated that the crack driving force in fatigue is directly related to the range of cyclic plastic strain. Heung et al. (1996) linked fatigue crack growth with the size of reversed plastic zone ahead of crack tip. Zhang et al. (2005) linked fatigue crack growth with crack tip shear band cracking, and this with the size of reversed plastic zone generated during the unloading part of the previous stress cycle. Other authors suggested that the total plastic dissipation per cycle is a driving force for fatigue crack growth in ductile solids, and can be closely correlated with fatigue crack growth rates (Klingbeil, 2003). Bodner et al. (1983) proposed that fatigue crack growth rate is proportional to the total plastic dissipation per cycle occurring throughout the reversed plastic zone ahead of the crack tip. The crack tip opening displacement (CTOD) is another main crack tip parameter. Pelloux (1970), using microfractography, showed that the concept of COD allowed the prediction of fatigue striations spacing and therefore the crack growth rate. Nicholls (1994) assumed a polynomial relation between crack growth rate and CTOD, while Tvergaard (2004), and Pippan and Grosinger (2013) indicated a linear relation between da/dN and CTOD for very ductile materials. The main objective of this paper is therefore to check the effectiveness of crack closure concept by linking the contact of crack flanks to the non-linear crack tip parameters. A close look to non-linear crack tip parameters, like plastic strain range or dissipated energy, is the key for a deeper understanding of FCG and the establishment of physically based relations. Besides, the fatigue models will not be limited by the validity of small-scale yield condition.

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2. Numerical model In this research, a standard M(T) specimen with width (W) of 60mm and length (L) of 300 mm was considered. A straight crack with initial size (a0) of 5 mm (a0/W=0.083) was modelled. A small thickness (t=0.1 mm) was considered to simulate plane stress state. Due to the symmetries of the sample in terms of geometry and loading conditions, only 1/8 of the specimen was simulated using adequate boundary conditions. The contact of crack flanks was simulated placing a symmetry plane with frictionless contact conditions behind the growing crack front. This symmetry plane was removed in the cases without contact. Constant amplitude loading cycles were applied to the specimen. In order to have a wide range of loading parameters, sets with constant Kmin, Kmax, K and R were studied (Table 1). All the load cases were run twice, with and without contact of crack flanks. Table 1. Load parameters (K, Kmax, Kmin=MPa.m1/2)

Set 1 (Kmin=0) K 2.7 3.6 4.6 6.4 8.2 9.1 10.0

R 0 0 0 0 0 0 0

Set 2 (Kmax=6.4) R K 3.6 0.43 5.5 0.14 7.3 -0.14 9.1 -0.43 10.9 -0.71 12.8 -1.00 14.6 -1.29

Set 3 (K=c.te) K 4.6 4.6 4.6 4.6 4.6 4.6

R -2 -1 -0.5 0 0.25 0.5

Set 4 (K=c.te) K 6.4 6.4 6.4 6.4 6.4 6.4

R -2 -1 -0.5 0 0.25 0.5

Set 5 (R=c.te) K 2.9 3.6 4.4 5.1 5.8 6.6

R 0.2 0.2 0.2 0.2 0.2 0.2

The material considered was the 6016-T4 aluminium alloy. Since PICC is a plastic deformation based phenomenon, the hardening behaviour of the material was carefully modelled. An anisotropic yield criterion was considered. The hardening behavior of this alloy was represented using an isotropic hardening model described by a Voce type equation combined with a non-linear kinematic hardening model described by a saturation law (Chaparro, 2008). The finite element mesh was refined near the crack tip and enlarged at relatively remote positions. Square elements with 88 m2 were defined in the most refined region where the crack propagates, while only one layer of elements was considered along the thickness. The total mesh is composed of 6639 linear isoparametric elements and 13586 nodes. Crack propagation was simulated by successive debonding of nodes at minimum load. Each crack increment corresponded to one finite element and two load cycles were applied between increments. In each cycle, the crack propagates uniformly over the thickness by releasing both current crack front nodes. The opening load, Fop, necessary for the determination of the closure level was determined considering two approaches. The first consisted in evaluating the contact status of the first node behind the current crack tip with the symmetry plane. In order to avoid resolution problems associated with the discrete character of load increase, the opening load was obtained from the linear extrapolation of the applied loads corresponding to two increments after opening. The second approach uses the contact forces at minimum load to calculate the stress intensity factor required to open the crack (Antunes, 2014). The numerical simulations were performed with the Three-Dimensional Elasto-plastic Finite Element program (DD3IMP) that follows a fully implicit time integration scheme (Menezes, 2000). The mechanical model and the numerical methods used in the finite element code DD3IMP, specially developed for the numerical simulation of metal forming processes, take into account the large elastic-plastic strains and rotations that are associated with large deformation processes. To avoid the locking effect a selective reduced integration method is used in DD3IMP (Alves, 2001). The optimum values of the numerical parameters of the DD3IMP implicit algorithm have been well established in previous works, concerning the numerical simulation of sheet metal forming processes (Oliveira, 2004) and PICC (Antunes, 2008).

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The analysis of the effect of contact flanks was developed comparing the crack tip parameters obtained with and without contact. For each load condition, the crack was submitted to 160 crack increments and 320 load cycles, which corresponds to a global crack increment a=1608 m=1.280 mm. This is enough to stabilize the crack opening values. This procedure was done with and without the symmetry plane used to simulate the contact of crack flanks. Three non-linear crack tip parameters were measured at the end of this procedure: the size of cyclic plastic zone (rpc), the crack tip opening displacement (CTOD), and the range of plastic strain. This last quantity was measured at the Gauss point immediately ahead of the last crack tip position, and in the last load cycle applied. The size of cyclic plastic zone was determined from the analysis of equivalent plastic strain ahead of crack tip. The increase of plastic deformation with the decrease of load, down to its minimum value, indicates the occurrence of reversed plasticity. The CTOD was assumed to be the vertical displacement of the node behind crack tip at maximum load. The same approach was used by Ellyin and Wu (1999). 3. Results 3.1. CTOD versus load Figure 1 plots the CTOD versus load plotted in the form /ys, being  the remote stress and ys the material’s yield stress, for the load case CA_0_140. This notation is used to designate a constant amplitude test with maximum and minimum remote loads of 140 and 0 N, respectively. The remote stress is calculated from the load, F, by dividing the area of cross section: =F/A, being A=300.1= 3 mm2. The crack closed at minimum load (A) and only opened when the load reached point B. This is the crack closure phenomenon. After opening the CTOD increases linearly, but after point C there is some deviation from linearity which indicates plastic deformation. The extrapolation of the linear regime to the maximum load, as is represented, shows that the major part of the deformation is elastic. The decrease of load from point D produces a linear decrease of CTOD. The rate of variation of CTOD in regions DE and BC is similar. After point E, reversed plastic deformation starts and the crack closes again at point F. It is interesting to note that the crack opening and crack closure levels are slightly different. Similar trends were obtained by Matos and Nowell (2007).

0.5 D

0.4

CTOD [m]

CTOD

Plastic

0.3 E

0.2

C

Elastic

0.1

0.0

F

A 0

0.1

0.2 B

0.3

0.4

/ys

Fig. 1. CTOD versus load (CA_0_140, plane stress).

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Figure 2 plots the stress-strain curve measured at the Gauss point immediately ahead of crack tip position. The points A to F are indicated and therefore a crossed analysis with Figure 1 can be made. At point A there is a compressive residual stress which reduces linearly with load increase. The CTOD is constant and zero between points A and B, however the analysis of contact forces would show a progressive decrease with the increase of remote loading. The crack opens at point B when the first node behind crack tip starts moving. A compressive stress still exists at the Gauss point when the crack opens behind the crack tip. The linearity of stress-strain curve is kept up to point C. The load increase above this point produces a well defined non-linear behavior. The plastic deformation is much more evident in here than that observed for the CTOD. The decrease of load after point D produces a linear decrease of stress with the same rate of region AC. The occurrence of reversed plastic deformation starts at point E and the crack closes at point F. The range of plastic strain, indicated in figure 2, is another nonlinear parameter used in literature (Pokluda, 2013). Additionally, the area of the stress-strain loop is the energy dissipated at the Gauss point studied. 4 D 3

GP CTOD

/ys

2

C

ep,yy

1 0

E 0

0.005

0.01

0.015 B

-1 -2 -3

A

0.02

0.025

0.03

F A eyy

Fig. 2. Stress-strain curve (CA_0_140, plane stress).

3.2. Master curves Figure 3 plots the variations of two non-linear crack tip parameters with K, obtained without contact of crack flanks. Both the size of cyclic plastic zone, rpc, and the crack tip opening displacement, CTOD, show an increase with K, as could be expected. Additionally, well defined relations are obtained without contact of crack flanks, although a wide range of loading parameters are being considered. This indicates that without contact of crack flanks there is no effect of stress ratio. Besides, these results reinforce the validity of linear elastic fracture mechanics, i.e., indicates that K controls the non-linear crack tip parameters and therefore fatigue crack growth rate. These results of crack tip parameters versus K may be seen as master curves, free of the influence of crack closure. The scatter of r pc is higher than that observed for CTOD, which can be explained by the distance of the measurement relatively to the crack tip. In fact, while the CTOD is measured in the first node behind crack tip, which is placed 8 m behind this, the cyclic plastic zone extends further ahead of crack tip, up to about 250 m as can be seen in the results of figure 3.

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0.25

95

5 Size of cyclic plastic zone

4.5

Crack tip opening displacement

0.2

4

rpc [mm]

0.15

3 2.5

0.1

2

CTOD [m]

3.5

1.5

0.05

1 0.5

0

0

0

5

10

15

K [MPa.m 0.5 ]

Fig. 3. Non-linear crack tip parameters versus K.

3.3. Validation of crack closure concept Figure 4 plots the CTOD against K. The master curve, obtained without contact, is plotted and is used as reference. However, when the CTOD with contact is plotted versus K the values are on the right side of the master curve, i.e., are lower than could be expected for the value of K applied to the specimen. This clearly indicates that crack closure influences the non-linear crack tip parameters and therefore da/dN, and contradicts the authors that deny the importance of crack closure. In addition, there is a great scatter, indicating that K is not the only controlling parameter. As has been widely observed in literature, the mean stress has a significant influence on da/dN, and therefore on CTOD. However, when this parameter is plotted against Keff, the points overlap the master curve. This means that crack closure concept is able to explain the effect of mean stress. 2.5 No contact; versus K Master curves

Contact; versus K

2

CTOD [mm]

Contact; versus K eff No contact; versus K (M16)

1.5

1

0.5

0

0

5

10 K, K eff [MPa.m 0.5 ]

Fig. 4. Effect of Keff on CTOD.

15

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Figure 4 also shows results for a mesh with larger elements (1616 m2). The modification of the mesh affects the crack tip parameters, because the positions of the first node behind crack tip and of the Gauss point ahead of crack tip are changed. Figure 5 plots the range of plastic strain versus K and Keff in the case of overloads and Low-High load blocks. The load cases considered were OL_60_140_180, OL_N20_140_180, OL_N100_140_180, OL_N140_140_180, OL_N140_140_20, OL_N60_140_180, HL_0_200_0_180, HL_0_200_0_140 and HL_0_140_0_120. OL indicates an overload and HL indicates a High-Low load pattern. When the plastic deformation is plotted versus K, there is a great scatter and the values are lower than that indicated by the master curves obtained without contact of crack flanks. However, when the same values are plotted against Keff the points overlap the master curve. This clearly indicates that the concept of crack closure is valid and applicable to variable amplitude loading. 0.08 Overload; K

0.07

Overload; K eff High-Low; K

0.06

Master curve (no contact)

High-Low; K eff

ep

0.05 0.04 0.03 0.02 0.01 0

0

2

4

6

8

10

12

14

K [MPa.m0.5]

Fig. 5. Effect of Keff on CTOD for overloads and Low-High load blocks.

4. Conclusions A numerical analysis of non-linear crack tip parameters was developed here. The main conclusions obtained were:  The analysis of CTOD versus load clearly shows the contact phenomenon. Linear variations are obtained immediately after crack opening. Without contact of crack flanks the variation of CTOD is significantly higher, as could be expected;  Well defined relations between non linear parameters and K were obtained without contact of crack flanks, although the wide range of loading parameters considered. This indicates that without contact of crack flanks there is no effect of stress ratio which reinforces the validity of linear elastic fracture mechanics;  When the CTOD with contact is plotted versus K the values are on the right side of the master curve, which indicates that crack closure influences the non-linear crack tip parameters and therefore da/dN. The comparison of CTOD versus load obtained with and without contact of crack flanks also shows the relevance of contact on crack tip parameters.  When the CTOD obtained with contact is plotted versus Keff, the points overlap the master curve, which validates the crack closure concept. Similar results were obtained for overloads and High-Low load blocks, indicating that the crack closure concept is also valid for variable amplitude loading. Further work is required to study the influence of material and numerical parameters on the master curves, and to link the non-linear crack tip parameters with da/dN.

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Acknowledgements This research is sponsored by FEDER funds through the program COMPETE (under project T44950814400019113) and by national funds through FCT – Fundação para a Ciência e a Tecnologia –, under the project PTDC/EMS-PRO/1356/2014. References Alves, J.L., Menezes, L.F., 2001. Application of tri-linear and tri-quadratic 3-D solid FE in sheet metal forming process simulation. In: K. Mori, editors. NUMIFORM 2001, Japan, 639–644. Antunes, F.V., Rodrigues, D.M., 2008. Numerical simulation of plasticity induced crack closure: Identification and discussion of parameters. Engng Fracture Mechanics 75, 3101–3120. Antunes, F.V., Chegini, A.G., Correia, L., Branco, R., 2014. Numerical study of contact forces for crack closure analysis. International Journal of Solids and Structures 51, 1330–1339. Alizadeh, H., Hills, D.A., de-Matos P.F.P., Nowell, D., Pavier, M.J., Paynter, R.J., Smith, D.J., Simandjuntak, S., 2007. A comparison of two and three-dimensional analyses of fatigue crack closure. Int. J. Fatigue 29, 222–231. Bodner, S.R., Davidson, D.L., Lankford, J., 1983. A description of fatigue crack growth in terms of plastic work. Engineering Fracture Mechanics 17(2), 189–191. Chaparro, B.M., Thuillier, S., Menezes, L.F., Manach, P.Y., Fernandes, J.V., 2008. Material parameters identification: Gradient-based, genetic and hybrid optimization algorithms. Computational Materials Science 44(2), 339-346. Christensen, R.H., 1963. Fatigue crack growth affected by metal fragments wedged between opening-closing crack surfaces. Appl. Mater. Res. 2(4), 207-210. Elber, W., 1970. Fatigue crack closure under cyclic tension, Eng. Fracture Mechanics 2, 37-45. Ellyin, F., Wu, J., 1999. A numerical investigation of the effect of an overload on fatigue crack opening and closure behavior. Fatigue and Fracture of Engineering Materials and Structures 22, 835-847. Heung, B.P., Kyung, M.K., Byong, W.L., 1996. Plastic zone size in fatigue cracking, Int. J. Pres. Ves. Piping 68, 279–285. Klingbeil, N.W., 2003. A total dissipated energy theory of fatigue crack growth in ductile solids. Int Journal of Fatigue 25, 117–128. Kujawski, D., 2001. A new (K+Kmax)0.5 driving force parameter for crack growth in aluminum alloys. Int J Fatigue 23, 733–740. Lee, S.Y., Liaw, P.K., Choo, H., Rogge, R.B., 2011. A study on fatigue crack growth behavior subjected to a single tensile overload Part I. An overload-induced transient crack growth micromechanism. Acta Materialia 59, 485–494. Matos, P.F.P., Nowell, D., 2007. On the accurate assessment of crack opening and closing stresses in plasticity-induced fatigue crack closure problems. Engineering Fracture Mechanics 74, 1579–1601. Menezes, L.F., Teodosiu, C. 2000. Three-Dimensional Numerical Simulation of the Deep-Drawing Process using Solid Finite Elements. Journal of Materials Processing Technology 97, 100-106. Nicholls, D.J., 1994. The relation between crack blunting and fatigue crack growth rates. Fatigue Fract Eng Mater Struct 17(4), 459-467. Noroozi, A.H., Glinka, G., Lambert, S., 2005. A two parameter driving force for fatigue crack growth analysis. Int J Fatigue 27, 1277–1296. Oliveira, M.C., Menezes, L.F. 2004. Automatic correction of the time step in implicit simulations of the stamping process. Finite Elements in Analysis and Design 40, 1995–2010. Paris, P.C., Erdogan J. 1963. Critical analysis of crack growth propagation laws. J Basic Eng 85D, 528–34. Pelloux, R.M., 1970. Crack Extension by alternating shear. Eng Fracture Mechanics 1, 170-174. Philips, E.P., 1989. Results of the round robin on opening-load measurement, NASA Tech. Meno. 101601, 1989 (Langley Research Center, Hampton, VA). Pippan, R., Grosinger, W., 2013. Fatigue crack closure: From LCF to small scale yielding. Int Journal of Fatigue 46, 41–48. Pokluda, J., 2013. Dislocation-based model of plasticity and roughness-induced crack closure. Int Journal of Fatigue 46, 35-40. Rice, J.R., 1967. Mechanics of crack tip deformation and extension by fatigue. In: Fatigue crack propagation. Philadelphia: ASTM STP 415; 1967. p. 256–71. Tong, J., 2002. T-stress and its implications for crack growth, Engng Fracture Mechanics 69, 1325–1337. Toribio, J., Kharin, V., 2013. Simulations of fatigue crack growth by blunting–re-sharpening: Plasticity induced crack closure vs. alternative controlling variables. International Journal of Fatigue 50, 72–82. Tvergaard, V., 2004. On fatigue crack growth in ductile materials by crack–tip blunting. Journal of the Mechanics and Physics of Solids 52, 2149 – 2166. Vasudevan, A.K., Sadananda K., Louat, N. 1993. Two critical stress intensities for threshold crack propagation. Scripta Metal 28, 65–70. Zhang, J., He, X.D., Du, S.Y., 2005. Analyses of the fatigue crack propagation process and stress ratio effects using the two parameter method. Int Journal of Fatigue 27, 1314–1318.