Fatigue Life Prediction Based on Crack Growth Analysis Using an Equivalent Initial Flaw Size Model: Application to a Notched Geometry

Procedia Engineering 114 ( 2015 ) 730 – 737

1877-7058 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of INEGI - Institute of Science and Innovation in Mechanical and Industrial Engineering doi: 10.1016/j.proeng.2015.08.018

ScienceDirect

1st International Conference on Structural Integrity

Fatigue life prediction based on crack growth analysis using an

equivalent initial flaw size model: Application to a notched

geometry

A.S.F. Alves

, L.M.C.M.V. Sampayo

, J.A.F.O. Correia

*, A.M.P. De Jesus

, P.M.G.P.

Moreira

, P.J.S. Tavares

a _{INEGI, University of Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal} b_{Engineering Faculty, University of Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal}

Abstract

Several methods for fatigue life prediction of structural components have been proposed in literature. The fatigue life prediction based on crack growth analysis has been proposed to assess the residual fatigue life of components, which requires the definition of an initial flaw. Alternatively, Fracture Mechanics crack growth-based fatigue predictions may be used to simulate the whole fatigue life of structural components assuming that there are always initial defects on materials, acting as equivalent initial cracks [1,2]. This latter approach is applied to a notched plate made of P355NL1 steel [3]. Fatigue crack growth data of the material is evaluated using CT specimens, covering several stress R-ratios. Also, S-N fatigue data is available for the double notched plate, for a stress R-ratio equal to 0 [3]. An estimate of the equivalent initial flaw size is proposed, using a back-extrapolation calculation [1,2]. The crack propagation model takes into account the elastic–plastic deformations in the crack-tip area within the calculation, based on the cyclic J-integral method. The performances of predictions are analyzed and deviations discussed. © 2015 The Authors. Published by Elsevier Ltd.

Peer-review under responsibility of INEGI - Institute of Science and Innovation in Mechanical and Industrial Engineering.

Keywords: Fatigue; Fracture Mechanics; Crack Growth; Fatigue Life Prediction.

* Corresponding author. Tel.: +351225082151; fax: +351225081584.

E-mail address: [email protected]

© 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Nomenclature

ai initial crack size af final crack size C, m, γ material constants da/dN fatigue crack growth rate E Young’s modulus EIFS equivalent initial flaw size f crack propagation function K’ cyclic hardening coefficient n’ cyclic hardening exponent N number of cycles

Y geometric function Δa discrete crack increment Δε notch strain range

ΔK stress intensity factor range

ΔKeff effective stress intensity factor range ΔKth stress intensity threshold

ΔJ range of the cyclic J-integral

ΔJth threshold value of the cyclic J-integral ΔJeff effective range of the cyclic J-integral ΔJeff,el elastic effective range of the cyclic J-integral ΔJeff,pl plastic effective range of the cyclic J-integral Δσ notch stress range

Δσeff effective range of the gross section nominal stress Δσf fatigue limit

Δσref reference stress range ΔWɛ strain energy density range

]

1. Introduction

Fatigue modelling may be performed based on several alternative approaches such as the global S-N approaches, the local strain-life or stress-life approaches and the fracture mechanics based approaches. The global S-N approaches relate directly a global definition of a stress range with the total number of cycles to failure.

The local approaches are usually associated with local failure modes, such as the macroscopic crack initiation. However, the definition of the macroscopic crack initiation is neither easy nor consensual and usually a crack initiation criteria is postulated, such a crack of 0.25 mm depth.

Fracture Mechanics based fatigue approaches may be used to model fatigue crack propagation from an initial size to final dimensions responsible by the fracture of the component. This approach may be used to complement the local strain-life approaches, modelling the crack propagation from the initial crack to the critical dimensions leading to component collapse.

Fracture Mechanics based fatigue approaches may be further extended to simulate the global fatigue life of components, using the Equivalent Initial Flaw Size (EIFS) concept [1,2] based on cyclic J-integral. The material is assumed to include defects acting like initial flaws. Therefore, the fatigue life of the component is assumed as the number of cycles required to propagate these initial defects until critical dimensions.

This paper proposes the application of the Fracture Mechanics to model the total fatigue life of a rectangular notched plate made of P355NL1 steel. Fatigue crack growth data was evaluated for stress R-ratios, using CT specimens. This data was used to compute the fatigue life of the notched plate, and results are compared with available S-N data. Experimental data was evaluated previously by some of the authors of this paper [3].

2. Fatigue life evaluation based on fracture mechanics

The fatigue life evaluation based on fracture mechanics for notched details is supported by material crack propagation laws. The cyclic J-integral can be used to consider the elastic–plastic deformations in the crack-tip area within the calculation:

da

Δ (1)

where da/dN is the fatigue crack growth rate, ΔJ is the range of the cyclic J-integral, f is a function of the J-integral. The number of cycles to failure may be computed by integrating the crack propagation law between the initial crack size, ai and the final crack size af:

³

In order to allow the computation of the global fatigue life of the component, the initial crack size ai, is assumed a material characteristic and represents the EIFS of the material. The material is assumed to present internal defects acting like initial cracks. In general, the EIFS may be estimated by an inverse (back-extrapolation) analysis. Some authors try do estimate directly the EIFS using the Kitagawa–Takahashi diagram [4,5]:

2 Δ Δ 1 ¸ ¸ ¹ · ¨ ¨ © § Y σ K π a f th i (3)

where ΔKth is the stress intensity threshold, Δσf is the fatigue limit and Y is the geometric function that allows the computation of the stress intensity factor, as follows:

a π Y σ

K (4)

Other key issue on fatigue modelling using the Fracture Mechanics, besides the definition of the EIFS, is the selection of the appropriate fatigue crack propagation laws. The Paris relation was the first one proposed in literature correlating the fatigue crack growth rate with the stress intensity factor range [6]:

m K C dN da Δ (5)

where C and m are material constants. This relation gives a good description of the crack propagation regime II. However, this relation does not model the other propagation regimes and does not account for mean stress effects. Many other propagation models have been proposed in literature, with higher performance and complexity, requiring a significantly high number of constants [7]. In addition, the modification of the Paris relation to account for stress ratio effects is also assumed, as proposed by Walker [8]:

m m γ σ K C R K C dN da Δ ) 1 ( Δ 1 _»» ¼ º « « ¬ ª   (6)

where C, m and γ are constants. Also, an extension of the Paris relation, to account for crack propagation regime I, is presented:

On effect, the crack growth in the propagation regime I is very life consuming, which requires an accurate definition of the crack growth in this regime. The crack propagation threshold, ΔKth is itself influenced by the stress ratio [9].

Paris-type crack growth law based on the cyclic J-integral takes into account the elastoplastic deformations in the crack-tip area within the calculation [2], as follows:

m J C dN da Δ (8)

where C and m are material constants. Based on Rice’s initial introduction of the J-integral [10], the general definition of ΔJ is given by

³

³

The increments ‘‘Δ” of the stress, strain, traction, and displacement quantities designate the changes in these quantities from their respective reference values, whereas ΔJ and ΔWɛ are single-valued functions of their arguments. Γ denotes an arbitrary path from one crack flank to the other. The effective value of ΔJ is calculated taking into account the effective stress–strain state, i.e. the stress–strain values at which the crack opens and closes under cyclic loading [2]. Formulated in effective ranges, this approximation reads:

pl eff el eff eff J J J Δ _, Δ _, Δ  (11)

The elastic part is directly associated with the stress intensity factor, as follows:

' Δ Δ 2 , _E K J_effel eff (12)

where E’=E/(1-υ2_{), assuming plane strain condition at the crack front, and ΔKeff can be calculated by the following} expression: el eff el eff σ π a Y K Δ ˜ ˜ Δ _, (13)

where Δσeff is the effective range of the nominal stress at the gross section. The plastic part of the effective J-integral [2] is approximated by:

1 ' 1 , , ,  ¸ ¸ ¹ · ¨ ¨ © § ' ' ˜ ˜ ' ˜ ' ' n ref eff el eff pl el eff pl eff J Y J J

V

V

]

where,

]

' 1 ' 2 2 2 n K E ¸¹ · ¨ © § '  ' 'H V V ₍₁₅₎

This means that the material is assumed to be in the cyclically stabilized condition and that it obeys to the so-called Masing behaviour, i.e. that the branches of the hysteresis loops can be constructed by doubling the cyclic stress– strain curve with its parameters n’ and K’ [2].

In this paper, an extension of the Paris-type crack growth law, to account for crack propagation regime I, is adopted:

where ΔJth is threshold value of the cyclic J-integral.

A numerical integration of the propagation law based on cyclic J-integral is adopted. The integral of Equation (2) is replaced by the following approximation:

¦

where Δa is a discrete crack increment which is kept in a very small value, bellow the EIFS.

3. Experimental Results

The present research is based on an experimental data derived for the P355NL1 steel supplied with 5 mm thick plates [3]. This steel is intended for pressure vessel applications and is a normalized fine grain low alloy carbon steel. Table 1 summarizes the mechanical properties of this steel.

Table 1: Mechanical properties of the P355NL1 steel [3]. Ultimate tensile strength, VUTS [MPa] 568

Monotonic yield strength, Vy [MPa] 418 Young’s modulus, E [GPa] 205.2 Poisson's ratio, Q 0.275 Cyclic hardening coefficient, K' [MPa] 777 Cyclic hardening exponent, n' 0.1068 Fatigue strength coefficient, V'f [MPa] 840.5 Fatigue strength exponent, b -0.0808 Fatigue ductility coefficient, H'f 0.3034 Fatigue ductility exponent, c -0.6016

Notched plates made of P355NL1 steel were fatigue tested under constant remote stress amplitude. Specimens are double notched rectangles extracted in the longitudinal/lamination direction of the steel sheets. The geometry of the specimens is illustrated in Figure 1. Peterson [12] proposes for this geometry an elastic stress concentration factor, Kt, equal to 2.17. Experimental fatigue tests of notched plates were performed for a stress ratio, R, equal 0, and are presented in the Figure 2.

In addition, fatigue crack propagation tests were performed using Compact Tension specimens with the same thickness as the notched plates and width, W=40. A total of five specimens were tested under constant loading amplitude, covering three distinct stress ratios, namely R=0, R=0.5 and R=0.7. Figure 3 illustrates the resulting fatigue crack propagation data. The analysis of the results shows that they are approximately linear in the log-log presentation, meaning that propagation regime II was covered by the experiments. The propagation regime I, the near threshold propagation regime was not covered by the tests.

Fig. 1. Notched rectangular plate (dimensions in mm).

Fig. 2. S-N fatigue data of the notched plate [3]. Fig. 3. Crack propagation data for the P355NL1 steel [3].

Cycles, N

Exp. data Exp. Mean S-N curve

1.0E4 1.0E5 1.0E6

200 Stress range, ' σ [MPa] 1.0E3 300 400 500 1.0E7 Rσ= 0.0

4. Application and discussion

In the predictions proposed in this paper, an extension of the Paris-type crack growth law (Equation (16)) was used to model the crack propagation on region I. The maximum value of cyclic J-integral from Table 2 was used to define the maximum crack sizes in the simulations. Since the thickness of the CT specimens and the notched plates are the same, these maximum values of cyclic J-integral are a good estimation of the material toughness.

Concerning the estimation of the fatigue crack growth rates in the propagation regime I, the Equation (16) was used. This equation requires the definition of the propagation threshold, which was estimated using the following linear relation [13]:

R

Kth 15290.252

' (N.mm-1.5_{) (18)}

where the value 152 N.mm-1.5 _{corresponds to the propagation threshold for R=0. This value is very consistent with} the threshold value point out in reference [9] for the mild steel with yield stress of 366 MPa. This stress intensity threshold, ΔKth, allows to calculate the threshold value of the cyclic J-integral, ΔJth, using the Equation (12). It was decided to use distinct relations for the propagation regimes I and II, since the use of a single relation (Equation (8)) did not produce satisfactory results. On effect, the single use of Equation (8) leads to a poor correlation of the experimental data in the propagation regime II and also tends to underestimates the propagation threshold. Despite using two distinct crack propagation laws proposed by Equation (16), they gave a continuous representation of the crack propagation data at the crack propagation I-II regimes transition.

Figure 4 illustrates the fatigue propagation laws adopted in this investigation for the stress R-ratio, equal to 0, tested in the structural detail.

The crack was assumed to propagate asymmetrically in one side of the specimens, starting at the notch root, as it was verified in the experimental program. Also, through thickness cracks were assumed. This situation was also observed for many cases in the experimental program.

Finite element models of the detail were used to perform elastoplastic stress analysis for the computation of the range of the cyclic J-integral. Bi-dimensional finite element models of the notched detail were proposed. Figure 5 illustrates a typical finite element mesh of the detail [14]. This mesh exhibits a crack on the left notch. In the practice, cracks started at one side notch root and propagated asymmetrically in the plate. Taking into account the existing symmetry plane, only half of the geometry is modelled. Plane stress quadratic triangular elements were used in the analysis due to the limited specimen thickness. A von Mises yield model, with multilinear kinematic hardening, was used in simulations using ANSYS® 12.0 code. The plasticity model was fitted to the stabilized cyclic curve of the material [3]. The range of the cyclic J-integral were determined based on an elastoplastic finite element analysis using the J-integral method. Figure 6 represents the cyclic J-Integral as a function of the nominal stress range for crack length, a=0.625mm, obtained for the notched plate.

Figure 7, finally presents the estimations of the S-N curves for the notched plate for the tested stress ratio. It is clear that the procedure produced very satisfactory correlation of the experimental data. For this purpose, the EIFS was assumed equal to 7.25 μm. The crack increment used in the numerical integration was 1/10 of the EIFS. Also, the model was able to produce a satisfactory prediction of the fatigue limit regions.

Table 2: Crack propagation constants.

Regime II Regime I R C* _{m ΔK} max [N.mm-1.5] ΔJmax [N/mm] ΔJth [N/mm] 0.0 7.1945E-15 3.4993 1498 10.98 0.11 0.5 6.2806E-15 3.5548 1013 5.00 0.06 0.7 2.0370E-13 3.0031 687 2.30 0.04

da/dN [mm/ cy cle] 'Jel[N/mm] MB02 (R=0.0) MB04 (R=0.0) Paris law - Paris Region

Extended Paris Law - Threshold Region

0.1 1 10 1.0E-6 1.0E-5 1.0E-4 1.0E-2 1.0E-3 1.0E-7 1.0E-8

1.0E-9 Threshold Region Paris Region

݀ܽ ݀ܰ ൌ ͳǤ͹ͷܧ െ ͷ ή ൫οܬΤ ݂݂݁െ οܬݐ݄൯ͳǤ͸Ͳ 0 20 40 60 80 100 200 250 300 350 400 450 500 Δ Jeff [N /mm]

Nominal stress range, Δσ [MPa] Elastoplastic Analysis

Linear-elastic analysis

a=0.625mm

Fig. 4. Fatigue crack growth data.

Fig. 5. Finite element mesh of the rectangular notched plate with a side crack (1/2 geometry).

Fig. 6. Value of cyclic J-Integral as a function of nominal stress range for the notched plate (a=0.625mm).

5. Conclusions

A Fracture Mechanics based fatigue model was assessed in this paper using experimental data available for the P355NL1 steel. The total fatigue life was modelled assuming a crack propagation process, from an equivalent initial flaw size (EIFS). Experimental fatigue crack propagation data was used to calibrate the crack propagation relations. Two relations were used instead of a unique relation to improve the predictions, since a single crack propagation relation tends to reduce the correlation in the crack propagation region II and to underestimates the fatigue threshold. The prediction of the S-N data available for a notched detail made of P355NL1 steel showed a very good agreement between predictions and experimental data. The EIFS method using finite element analysis was able to

Cycles, N

Exp. data EIFS model

1.0E4 1.0E5 1.0E6

200 Stress range, ' σ [MPa] 1.0E3 300 400 500 1.0E7 Rσ= 0.0 EIFS =a0= 7.25μm

account for stress R-ratio, equal 0, and the fatigue limit region was fairly estimated. The EIFS was estimated to be 7.25 μm. The crack propagation thresholds, despite estimated in this paper, it was very consistent with data published in the literature for similar materials.

Fig. 7. S-N curve prediction using a Fracture Mechanics based method and the EIFS.

References

[1] Xiang Y., Lu Z., Liu Y. Crack growth-based fatigue life prediction using an equivalent initial flaw model. Part I: Uniaxial loading. Intenational Journal of Fatigue, Vol. 32, Num. 2, pp. 341-349, 2010.

[2] Savaidis G., Savaidis A., Zerres P., Vormwald M. Mode I fatigue crack growth at notches considering crack closure. Intenational Journal of Fatigue, Vol. 32, Num. 10, pp. 1543-1558, 2010.

[3] H.F.S.G. Pereira, A.M.P. de Jesus, A.S. Ribeiro, A.A. Fernandes, “Fatigue Damage Behavior of a Structural Component Made of P355NL1 Steel under Block Loading”, Journal of Pressure Vessel Technology, Vol. 131, Num. 2, pp. 021407 (9 pages), 2009.

[4] Kitagawa H., Takahashi S. Applicability of fracture mechanics to very small cracks or cracks in the early stage. Proc. of the 2nd Int. Conf on Mech. Behaviour of Materials, ASM, 627–631, 1976.

[5] Kruzic JJ., Ritchie RO. Kitagawa-Takahashi diagrams define the limiting conditions for cyclic fatigue failure in human dentin. Journal of Biomedical Materials Research Part A, 79(3):747-51, 2006.

[6] Paris PC., Erdogan F. A critical analysis of crack propagation laws. Transactions of The ASME. Series E: Journal of Basic Engineering, Vol. 85, pp. 528-534, 1963.

[7] Beden SM., Abdullah S., Ariffin AK. Review of Fatigue Crack Propagation Models for Metallic Components. European Journal of Scientific Research, Vol. 28, pp. 364-397, 2009.

[8]. Dowling NE. Mechanical Behaviour of Materials – Engineering Methods for Deformation, Fracture and Fatigue. Second Edition, Prentice Hall, New Jersey, USA, 1998.

[9] Szata M., Lesiuk G. Algorithms for the estimation of fatigue crack growth using energy method. Archives of Civil and Mechanical Engineering, Vol. IX, Num. 1, pp. 119-134, 2009.

[10] Rice JR. A path independent integral and approximate analysis of strain concentration by notches and cracks. J Appl Mech, 35, pp. 379-386, 1968.

[11] Dankert M., Greuling S., Seeger T. Unified elastic-plastic model for fatigue crack growth at notches including crack closure effects. ASTM Special Technical Publication, (1343), pp. 411-426, 1999.

[12] Peterson RE. Notch Sensitivity. Metal Fatigue, Sines, G. and Waisman, J.L., edition, McGraw-Hill Book Company, Inc., New York, pp. 293-306, 1959.

[13] Szata M., Lesiuk G. Algorithms for the estimation of fatigue crack growth using energy method. Archives of Civil and Mechanical Engineering, Vol. IX, Num. 1, pp. 119-134, 2009.

[14] Correia JAFO., De Jesus AMP., Fernández-Canteli, A. Local unified probabilistic model for fatigue crack initiation and propagation: Application to a notched geometry. Engineering Structures, Volume 52, July 2013, Pages 394-407.