Advances in Materials Science and Engineering
Volume 2016 (2016), Article ID 3026745, 11 pages
http://dx.doi.org/10.1155/2016/3026745

Research Article

Crack Closure Effects on Fatigue Crack Propagation Rates: Application of a Proposed Theoretical Model

José A. F. O. Correia,¹ Abílio M. P. De Jesus,1,2 Pedro M. G. P. Moreira,¹ and Paulo J. S. Tavares¹

¹Institute of Science and Innovation in Mechanical and Industrial Engineering, University of Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal
²Department of Mechanical Engineering, Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal

Received 10 December 2015; Accepted 18 February 2016

Academic Editor: Antonio Riveiro

Copyright © 2016 José A. F. O. Correia et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Structural design taking into account fatigue damage requires a thorough knowledge of the behaviour of materials. In addition to the monotonic behaviour of the materials, it is also important to assess their cyclic response and fatigue crack propagation behaviour under constant and variable amplitude loading. Materials whenever subjected to fatigue cracking may exhibit mean stress effects as well as crack closure effects. In this paper, a theoretical model based on the same initial assumptions of the analytical models proposed by Hudak and Davidson and Ellyin is proposed to estimate the influence of the crack closure effects. This proposal based further on Walker’s propagation law was applied to the P355NL1 steel using an inverse analysis (back-extrapolation) of experimental fatigue crack propagation results. Based on this proposed model it is possible to estimate the crack opening stress intensity factor, , the relationship between quantity and the stress intensity factor, the crack length, and the stress ratio. This allows the evaluation of the influence of the crack closure effects for different stress ratio levels, in the fatigue crack propagation rates. Finally, a good agreement is found between the proposed theoretical model and the analytical models presented in the literature.

1. Introduction

In the early seventies Elber [1, 2] introduced the crack closure and opening concepts. Initially Elber [1, 2] suggested that crack closure was associated with the effective stress intensity range, using parameter which was assumed as a linear function of stress ratio, , independent of and . Later studies determined that the quantitative parameter is dependent on . After the proposal developed by Elber [1, 2], other proposals appeared, based on elastoplastic analytical concepts [3–7] or the near-threshold fatigue crack growth regimes [8–12]. All approaches are important contributions of the crack closure and opening effects, which occur in materials when subjected to constant amplitude loading, particularly in the near-threshold regime of the fatigue crack growth. All analytical techniques were supported by measurements of the crack-tip opening load. Other approaches have been implemented based on elastoplastic analysis, using finite element methods [13, 14].

This paper proposes a theoretical model based on Walker’s propagation law [15] to estimate the quantitative parameter, , important for the evaluation of the fatigue crack opening stress intensity factor, , taking into account the stress ratio and crack closure and opening effects. The current study is applied to the P355NL1 steel using experimental fatigue crack propagation test results under constant amplitude loading [16]. The proposed model is applied using an inverse analysis of the experimental results. Various analytical models for crack closure are also compared in this study.

2. Overview of the Crack Closure and Opening Concepts and Existing Models

The crack closure and opening concept has been discussed in the scientific community since the first contributions by Elber [1, 2]. Elber suggested [1, 2] that crack closure and opening effects could be characterized in terms of the effective stress intensity factor range, which is normalized by the applied stress intensity factor range, resulting in the ratio, with the following form:

where is the effective stress intensity factor range and is the applied stress intensity factor range. Elber’s formulation is supported by the following assumptions for the parameter: (i) linear function of stress ratio () and (ii) being relatively independent of maximum stress intensity factor () or stress intensity factor range (). The effective stress intensity factor range is defined as

where is the maximum stress intensity factor and is the crack opening stress intensity factor. The applied stress intensity factor range, , is presented by

with being the minimum stress intensity factor. Figure 1 illustrates the previous parameters definitions.

Figure 1: Definition of the effective and applied stress intensity factor ranges.

According to the approach of Elber [1, 2], the fatigue crack growth rate, , is a function of the effective stress intensity factor range, according to the following general expression:

According to studies conducted by Elber for the 2024-T3 aluminum alloy, using stress ratios, , between −0.1 and 0.7,   depends on . The experimental results led to the following linear relationship:

The quantitative parameter of (5) can be substituted into (1) and using (2) results in the following relation for the ratio:

Subsequent studies using various fatigue loading variables clearly showed that and are related variables. Therefore, Elber [1, 2] approach appears to be inconsistent/incomplete.

An improved function of (5) was proposed by Schijve [9] with a more realistic behaviour to account for the crack closure and opening effects for :

Therefore, it is possible to obtain a relation to the crack opening stress intensity factor as follows:

Another proposal was presented by ASTM [10, 11] together with a set of conditions:

Newman Jr. [3] proposed a general crack opening stress equation for constant amplitude loading, function of stress ratio, , stress level, , and four nondimensional constants:

when . The coefficients , , , and are as follows:

where is the flow stress (average between the uniaxial yield stress and uniaxial ultimate tensile strength of the material). For plane stress conditions, , and for plane strain conditions, . The normalized crack opening stress is obtained using the stress ratio for .

This approach proposed by Newman Jr. [3] may be used to correlate fatigue crack growth rate data for other materials and thicknesses, under constant amplitude loading, once the proper constraint factor has been determined.

The crack closure effects may occur due to the surface roughness of the material in the presence of shear deformation at the crack tip [17, 18]. Microstructural studies [18] indicate that fracture surfaces demonstrate the possibility of crack propagation by the shear mechanism primarily in the near-threshold regime of [17]. In this fatigue regime, the crack closure effects have the greatest influence. Consequently, Hudak and Davidson [8] proposed a model defined by the following expression:

where is a constant related to the pure Mode I fatigue crack growth threshold.

The experimental results from fatigue crack propagation tests can be represented by the following generalized relations (see Figure 2):

where is the limiting value above which no detectable closure occurs and is a parameter that can be obtained using the following expression:

The and constants are not material properties in the strict sense since they depend on measurements location and measurements sensitivity, resulting from the crack propagation tests aiming at evaluating the crack closure/opening effects. Note that in the limiting condition of approaching zero (13) reduces to (12) thus describing the local measurements.

Figure 2: Functional relationship between and , Hudak and Davidson [8].

Ellyin [12] proposed an approach to define the effective stress intensity range, , taking into account the stress ratio, , and the threshold value of the stress intensity factor range, , with constant amplitude loading (see Figure 3). This approach is a modified version of proposal by Hudak and Davidson [8]. Generally, the crack opening stress intensity factor range, , is smaller than the threshold stress intensity factor range, ; that is, . Besides, the crack opening or closure stress intensities factors are not the same; that is, . Ellyin [12] proposed the following approach to obtain (see Figure 3):

where is the mean stress, is the maximum stress, and is the fatigue strength coefficient.

Figure 3: Definition of the effective stress intensity range according with Ellyin [12].

The crack closure and opening effects can be estimated using an elastoplastic analysis based on analytical [3, 19] or numerical [13, 14] approaches. Vormwald [4, 5] and Savaidis et al. [6, 7] estimated the crack closure and opening effects proposing the use of an analytical elastoplastic analysis developed by Neuber [19] and the modified crack closure and opening model suggested by Newman Jr. [3]. The numerical elastoplastic analysis, using the finite element method, is proposed by McClung and Sehitoglu [13] and Nakagaki and Atluri [14]. Based on finite element analysis performed by McClung and Sehitoglu [13], it has been shown that the results obtained for may deviate from the results obtained using the crack closure and opening model by Newman Jr. [3]. Nakagaki and Atluri [14] proposed an elastoplastic finite element procedure to account for arbitrary strain hardening material behaviour. In order to determine the crack closure and opening stresses, Nakagaki and Atluri [14] proposed the following procedure: calculating the displacements of the nodes in the crack axis before the closure (and after opening); determining by extrapolating to zero the restraining force at the respective nodal displacement in the corresponding node just after crack closure (and before the opening) by extrapolation against the load level. In all the cases studied, these two sets of extrapolated values for and were found to correlate excellently.

3. Proposed Theoretical Model

In this paper, a theoretical model to obtain the effective stress intensity range, , that takes into account the effects of the mean stress and the crack closure and opening effects is proposed. The starting point for the proposed theoretical model is the initial assumptions proposed by Elber [1, 2] and defined in (1) and (4). The basic fatigue crack propagation law in regime II was initially proposed by Paris and Erdogan [20], which has the following form:

where and are material constants. However, this model does not include the other propagation regimes and does not take into account the effects of mean stress. Many other fatigue propagation models have been proposed in literature, with wider range of application and complexity, requiring a significant number of constants [21]. Nevertheless, the simple modification of the Paris relation in order to take into account the stress ratio effects is assumed, as proposed by Dowling [15]:

where , , and are constants. Besides, an extension of the Paris relation is adopted to account for crack propagation regime I, having the following configuration:

Based on the foregoing assumptions, the effective stress intensity factor range is given by the following expression:

Thus, the expression that allows obtaining the quantitative parameter, , is given by the following formula:

If the material parameter, γ, is equal to 1, this means that the material is not influenced by the stress ratio. Thus, (21) is simplified and assumes the same form as (12) proposed by Hudak and Davidson [8] and is similar to the one proposed by Ellyin [12].

Equation (21) can be presented in another form:

The threshold value of the stress intensity factor range can be described by a general equation, using the following form:

where is the threshold value of stress intensity range for a given stress ratio and is the threshold value of stress intensity range, for . Klesnil and Lukáš [22] proposed a relation between the threshold stress intensity factor range, , and the stress ratio, , which is well known:

The validity of (24) is for . Replacing this result in (22) results in

Finally, it is possible to display (22) in a more simplified form (see Figure 4):

where is the limiting . For stress ratio, , equal to 0 and material parameter, , equal to 1, (22) reduces to

In summary, a theoretical model based on Walker’s propagation law [15] was proposed, which relates the quantitative parameter and (or ) and the stress ratio. The proposed model is based on the same initial assumptions of the analytical models proposed by Hudak and Davidson [8] and Ellyin [12].

Figure 4: Functional relationship between and for proposed theoretical model.

4. Basic Fatigue Data of the Investigated P355NL1 Steel

The P355NL1 steel is a pressure vessel steel and is a normalized fine grain low alloy carbon steel. Its fatigue behaviour has been investigated [16, 23, 24]. A steel plate 5 mm thick was used to prepare specimens for experimental testing. This section presents the strain-life fatigue data and the fatigue crack propagation data obtained for the P355NL1 steel [16].

4.1. Strain-Life Behaviour

The strain-life behaviour of the P355NL1 steel was evaluated through fatigue tests of smooth specimens, carried out under strain controlled conditions, according to the ASTM E606 standard [25]. The Ramberg and Osgood [26] relation, as shown in the following equation, was fitted to the stabilized cyclic stress-strain data:

where and are the cyclic strain hardening coefficient and exponent, respectively; is plastic strain range; and is the stress range.

Low-cycle fatigue test results are very often represented using the relation between the strain amplitude and the number of reversals to failure, , usually assumed to correspond to the initiation of a macroscopic crack. Morrow [27] suggested a general equation, valid for low- and high-cycle fatigue regimes:

where and are, respectively, the fatigue ductility coefficient and fatigue ductility exponent; is the fatigue strength coefficient, is the fatigue strength exponent, and is the Young modulus.

Alternatively to the Morrow relation, the Smith-Watson-Topper fatigue damage parameter [28] can be used, which shows the following form:

where is the maximum stress of the cycle and SWT is the damage parameter. Both Morrow and Smith-Watson-Topper models are deterministic models and are used to represent the average fatigue behaviour of the bridge materials based on the available experimental data.

Two series of specimens were tested under distinct strain ratios, (19 specimens) and −1 (24 specimens). Figure 5 shows a plot of the experimental fatigue data in the form of strain-life and SWT-life curves, for the two strain ratios. The cyclic Ramberg-Osgood and Morrow strain-life parameters of the P355NL1 steel are summarized in Table 1, for the conjunction of both strain ratios [16, 23, 24]. The cyclic curve is shown in Figure 5(a), including the influence of the experimental results from both test series. This research adopted the values obtained by combining the results of the two test series together since the strain ratio effects are considered negligible (tests performed under strain controlled conditions). This could be explained by the cyclic mean stress relaxation phenomenon and also by a lower sensitivity of the material to the mean stress. In addition, Table 2 presents the elastic and monotonic tensile properties of this pressure vessel steel under investigation. The elastic and monotonic tensile properties are represented, respectively, by the Young modulus, , Poisson ratio, , the ultimate tensile strength, , and upper yield stress, .

Table 1: Cyclic elastoplastic and strain-life properties of the P355NL1 steel,   and  .

Material

MPa
—
—
—
MPa
—

P355NL1
1005.50

0.3678

948.35
0.1533

Table 2: Elastic and tensile properties of the P355NL1.

Material

GPa
—
MPa
MPa

P355NL1
205.20
0.275
568.11
418.06

(a)

(b)

(c)

Figure 5: Fatigue data obtained for the P355NL1 steel, and : (a) cyclic curve; (b) strain-life data; (c) SWT-life data.

4.2. Fatigue Crack Propagation Rates

Fatigue crack growth rates of the P355NL1 pressure vessel steel were evaluated, for several stress ratios, using compact tension (CT) specimens, following the recommendations of the ASTM E647 standard [10]. The CT specimens of P355NL1 steel were manufactured with a width  mm and a thickness  mm [16]. All tests were performed in air, at room temperature, under a sinusoidal waveform at a maximum frequency of 20 Hz. The crack growth was measured on both faces of the specimens by visual inspection, using two travelling microscopes with an accuracy of 0.001 mm.

Figures 6(a) and 6(b) represent the experimental data and the Paris law correlations for each tested stress ratio of the P355NL1 steel, , , and . The material constants of the Paris crack propagation relation are presented in Table 3. The crack propagation rates are only slightly influenced by the stress ratio. Higher stress ratios result in higher crack growth rates. The lines representing the Paris law, for and , are approximately parallel to each other.

Table 3: Material constants concerning the Paris crack propagation relation.

—
—
—
N⋅mm−1.5

0.01

3.4993
1498

0.50

3.5548
1013

0.70

3.0031
687

in mm/cycle and in N⋅mm−1.5.

(a)

(b)

Figure 6: Fatigue crack propagation data obtained for the P355NL1 steel: (a) experimental data; (b) Paris correlations for each stress ratio, .

On the other hand, the line representing the Paris law for converges to the other lines as the stress intensity factor range increases. In general, the stress ratio effects are more noticeable for lower ranges of the stress intensity factors. For higher stress intensity factor ranges, the stress ratio effect tends to vanish.

The stress ratio effects on fatigue crack growth rates may be attributed to the crack closure. Crack closure increases with the reduction in the stress ratio leading to lower fatigue crack growth rates. For higher stress ratios, such as , the crack closure is very likely negligible, which means that applied stress intensity factor range is fully effective. Also, the fatigue crack propagation lines for each stress ratio are not fully parallel to each other which means that crack closure also depends on the stress intensity factor range level.

5. Application and Discussion of a Proposed Theoretical Model

The fatigue crack growth data presented in Figure 6 was correlated using the Walker relation (see (18)). The material constants of the Walker crack propagation relation, for the P355NL1 steel, are presented in Table 4. In addition, the Walker modified relation (see (19)) was used to model the fatigue crack propagation on region I and results are presented in Figure 7. Equation (19) requires the definition of the propagation threshold, which was estimated using the following linear relation:

where is the propagation threshold defined in N·mm−1.5, is the stress ratio, and constant value 152 corresponds to the propagation threshold for (in N·mm−1.5); the other constant (90.252) also shows the same units of the propagation threshold. The propagation threshold for is very consistent with the threshold value for the mild steel with yield stress of 366 MPa. The crack propagation threshold is itself influenced by the stress ratio [29]. In [29], for a similar material to the P355NL1 steel, the relationship between and is given, showing a linear relation between the crack propagation threshold and the stress ratio. Despite using two distinct crack propagation laws, they gave a continuous representation of the crack propagation data at the crack propagation I-II regimes transition. Figure 7 illustrates the fatigue propagation laws adopted in this investigation, for each tested stress ratio.

Table 4: Walker constants for the fatigue crack propagation relation of the P355NL1 steel.

3.4993
0.92

in mm/cycle and in N⋅mm−1.5.

(a)

(b)

(c)

Figure 7: Fatigue crack growth data correlations for regimes I and II, using two relations.

The proposed theoretical model is applied using an inverse analysis of the experimental fatigue crack propagation rates available for the P355NL1 steel. Using the inverse analysis of the proposed theoretical model, it is possible to obtain the relationship between the effective stress ratio, , and the applied stress intensity factor range, . accounts for minimum stress intensity factors higher than crack closure stress intensity factor. In this case, the effective stress intensity factor results from the relationship between the crack opening stress intensity factor, , and the maximum stress intensity factor, . Figure 8 shows that for stress ratios above 0.5 the crack closure and opening effects are not significant. For the stress ratio equal to 0 a significant influence on the effective stress ratio of the applied stress intensity factor range is observed. Such influence is more significant for the initial propagation phase of region II, decreasing significantly for higher applied stress intensity ranges. Thus, it is concluded that the crack closure and opening effects are more significant for the initial crack propagation phase, in region II, for higher levels of the stress intensity factor ranges along the crack length. Figure 9 shows the results obtained for the relationship between and , using the inverse analysis of the proposed theoretical model. This figure helps to understand the influence of the crack closure and opening effects, as a function of stress ratios. The effective stress ratio, , shown in Figure 9, is the average of the values obtained as a function of , according to what is evidenced in Figure 8.

Figure 8: Relationship between effective stress ratio, , and applied stress intensity range, (P355NL1 steel).

Figure 9: Effective stress ratio, , as a function of stress ratio (P355NL1 steel).

The proposed relationship between the quantitative parameter, , and the maximum stress intensity, , is also considered. Figure 10 shows the relationship between and for the studied stress ratios, 0, 0.5, and 0.7. The relationship between and for the studied stress ratios is shown in Figure 11. The proposed theoretical model is able to describe the influence of the crack closure and opening effects on different stress ratios.

Figure 10: Influence of the stress ratio on versus relationship (P355NL1 steel).

Figure 11: Influence of the stress ratio on versus relationship (P355NL1 steel).

Figure 12 shows the results for the relationship between and using various analytical models available in the literature. These model results are presented together with the results of the proposed theoretical model. Based on the analysis of Figure 12, it can be noted that the results obtained by applying the proposed theoretical model are between the results obtained using the Hudak and Davidson model (based on the same assumptions) and Savaidis model (with assumptions supported on plastic deformation of the material). The curves of Hudak and Davidson and Savaidis demonstrate being below the other models. Other models are based on direct polynomial relationships of orders 2 and 3, except Newman model that is based on a polynomial relationship of order 3 and uses properties of the plastic deformation of the material.

Figure 12: Relationship between and stress ratio for several models (P355NL1 steel).

Figure 13 shows the fatigue crack propagation rates as a function of effective intensity range using the proposed method. The results obtained, using this theoretical model, are similar to results presented in the literature [3].

Figure 13: Correlation of fatigue crack growth rates using the crack closure analysis (P355NL1 steel).

6. Conclusions

An interpretation of the crack closure and opening effects is possible, without resorting to an excessive computation time resulting from complex approaches.

Fatigue crack propagation tests allowed the estimation of the relationship between the effective stress ratios, , and the applied stress ratios, , without the need for extensive experimental program.

The evaluation of the influence of the crack closure effects was possible using the inverse identification procedure applied with the proposed theoretical model. In this study, it was observed that, for high applied stress ratios, , , the crack closure effect has little significance. But for applied stress ratios, , , there is a significant influence of the crack closure and opening effects. Note that, for applied stress ratios within , the influence of the crack closure and opening effects decreases with increasing of the crack length or stress intensity factor range.

The results of the proposed theoretical model proved to be consistent when compared with other models proposed in the literature [1–12, 19]. It was possible to estimate the constants and of the modified Paris law ( versus ). These parameters are important for fatigue life prediction studies of structural details based on fracture mechanics.

Other fatigue programs under constant and variable amplitude loading with other materials should be considered, through the use of full field optical techniques to experimentally estimate , in order to confirm the results obtained in the inverse analysis of the proposed theoretical model.

Competing Interests

The authors declare that they have no competing interests regarding the publication of this paper.

Acknowledgments

The authors acknowledge the Portuguese Science Foundation (FCT) for the financial support through the postdoctoral Grant SFRH/BPD/107825/2015. The authors gratefully acknowledge the funding of SciTech: Science and Technology for Competitive and Sustainable Industries, R&D project cofinanced by Programa Operacional Regional do Norte (“NORTE2020”), through Fundo Europeu de Desenvolvimento Regional (FEDER).

References

W. Elber, “Fatigue crack closure under cyclic tension,” Engineering Fracture Mechanics, vol. 2, no. 1, pp. 37–45, 1970.

W. Elber, “The significance of fatigue crack closure. Damage Tolerance in Aircraft Structures,” ASTM STP 486, America Society for Testing and Materials, 1971.

J. C. Newman Jr., “A crack opening stress equation for fatigue crack growth,” International Journal of Fracture, vol. 24, no. 4, pp. R131–R135, 1984.

M. Vormwald and T. Seeger, “The consequences of short crack closure on fatigue crack growth under variable amplitude loading,” Fatigue and Fracture of Engineering Materials and Structures, vol. 14, no. 2-3, pp. 205–225, 1991.

M. Vormwald, “Effect of cyclic plastic strain on fatigue crack growth,” International Journal of Fatigue, vol. 82, part 1, pp. 80–88, 2016.

G. Savaidis, M. Dankert, and T. Seeger, “Analytical procedure for predicting opening loads of cracks at notches,” Fatigue and Fracture of Engineering Materials and Structures, vol. 18, no. 4, pp. 425–442, 1995.

G. Savaidis, A. Savaidis, P. Zerres, and M. Vormwald, “Mode I fatigue crack growth at notches considering crack closure,” International Journal of Fatigue, vol. 32, no. 10, pp. 1543–1558, 2010.

S. J. Hudak and D. L. Davidson, “The dependence of crack closure on fatigue loading variables,” in Mechanics of Closure, ASTM STP 982, J. C. Newman Jr. and W. Elber, Eds., pp. 121–138, America Society for Testing and Materials, Philadelphia, Pa, USA, 1988.

J. Schijve, Fatigue of Structures and Materials, Kluwer Academic Publishers, Burlington, Mass, USA, 2004.

American Society for Testing and Materials (ASTM), “ASTM E647: standard test method for measurement of fatigue crack growth rates,” in Annual Book of ASTM Standards, vol. 03.01, pp. 591–630, American Society for Testing and Materials (ASTM), West Conshohocken, Pa, USA, 1999.

N. Geerlofs, J. Zuidema, and J. Sietsma, “On the relationship between microstructure and fracture toughness in trip-assisted multiphase steels,” in Proceedings of the 15th European Conference of Fracture (ECF15 '04), Stockholm, Sweden, August 2004, http://www.gruppofrattura.it/ocs/index.php/esis/ECF15/paper/view/8758.

F. Ellyin, Fatigue Damage, Crack Growth and Life Prediction, Chapman & Hall, New York, NY, USA, 1997.

R. C. McClung and H. Sehitoglu, “Characterization of fatigue crack growth in intermediate and large scale yielding,” Journal of Engineering Materials and Technology, vol. 113, no. 1, pp. 15–22, 1991.

M. Nakagaki and S. N. Atluri, “Fatigue crack closure and delay effects under mode I spectrum loading: an efficient elastic-plastic analysis procedure,” Fatigue of Engineering Materials and Structures, vol. 1, no. 4, pp. 421–429, 1979.

N. E. Dowling, Mechanical Behaviour of Materials—Engineering Methods for Deformation, Fracture and Fatigue, Prentice-Hall, Upper Saddle River, NJ, USA, 2nd edition, 1998.

H. F. S. G. Pereira, A. M. P. De Jesus, A. S. Ribeiro, and A. A. Fernandes, “Fatigue damage behavior of a structural component made of P355NL1 steel under block loading,” Journal of Pressure Vessel Technology, vol. 131, no. 2, Article ID 021407, 9 pages, 2009.

O. N. Romaniv, G. N. Nikiforchin, and B. N. Andrusiv, “Effect of crack closure and evaluation of the cyclic crack resistance of constructional alloys,” Soviet Materials Science, vol. 19, no. 3, pp. 212–225, 1983.

J. R. Lloyd, The effect of residual stress and crack closure on fatigue crack growth [Ph.D. thesis], Department of Materials Engineering, University of Wollongong, Wollongong, Australia, 1999, http://ro.uow.edu.au/theses/1525.

H. Neuber, “Theory of stress concentration for shear-strained prismatical bodies with arbitrary nonlinear stress-strain law,” Journal of Applied Mechanic, vol. 28, pp. 544–550, 1961.

P. C. Paris and F. Erdogan, “A critical analysis of crack propagation laws,” Journal of Basic Engineering, vol. 85, no. 4, pp. 528–534, 1963.

S. M. Beden, S. Abdullah, and A. K. Ariffin, “Review of fatigue crack propagation models for metallic components,” European Journal of Scientific Research, vol. 28, no. 3, pp. 364–397, 2009.

M. Klesnil and P. Lukáš, Fatigue of Metallic Materials, Elsevier Science, Amsterdam, The Netherlands, 2nd edition, 1992.

A. M. P. De Jesus and J. A. F. O. Correia, “Critical assessment of a local strain-based fatigue crack growth model using experimental data available for the P355NL1 steel,” Journal of Pressure Vessel Technology, vol. 135, no. 1, Article ID 011404, 9 pages, 2013.

J. A. F. O. Correia, A. M. P. De Jesus, and A. Fernández-Canteli, “Local unified probabilistic model for fatigue crack initiation and propagation: application to a notched geometry,” Engineering Structures, vol. 52, pp. 394–407, 2013.

ASTM—American Society for Testing and Materials, “ASTM E606-92: standard practice for strain-controlled fatigue testing,” in Annual Book of ASTM Standards, part 10, pp. 557–571, 1998.

W. Ramberg and W. R. Osgood, “Description of stress-strain curves by three parameters,” NACA Technical Note 902, NACA, 1943, http://ntrs.nasa.gov/search.jsp?R=19930081614.

J. D. Morrow, “Cyclic plastic strain energy and fatigue of metals,” Internal Friction, Damping, and Cyclic Plasticity (ASTM STP 378), ASTM International, West Conshohocken, Pa, USA, 1965.

K. N. Smith, P. Watson, and T. H. Topper, “A stress-strain function for the fatigue of metal,” Journal of Materials, vol. 5, no. 4, pp. 767–778, 1970.

M. Szata and G. Lesiuk, “Algorithms for the estimation of fatigue crack growth using energy method,” Archives of Civil and Mechanical Engineering, vol. 9, no. 1, pp. 119–134, 2009.