EXTENDING THE APPLICABILITY OF ...

;9,,QWHUQDWLRQDO&ROORTXLXP 0HFKDQLFDO)DWLJXHRI0HWDOV

±6HSWHPEHU %UQR&]HFK5HSXEOLF

EXTENDING THE APPLICABILITY OF PROBABILISTIC S-N MODELS TO THE LCF REGION USING ENERGETIC DAMAGE PARAMETERS A. Fernández-Canteli1,a, C. Przybilla1,b, J.A.F.O. Correia2,c, A.M.P. de Jesus2,d, E. Castillo3,e 1

2

UCVE-IDMEC-LAETA/ Departamento de Engenharias, Escola de Ciências e Tecnologia da Universidade de Trás-os-Montes e Alto Douro, 5001-801 Vila Real, Portugal

3

a

Department of Construction and Manufacturing Engineering, University of Oviedo, Campus de Viesques, 33203 Gijón, Spain

Department of Applied Mathematics and Computational Sciences, University of Cantabria, Avda. De los Castros s/n, 48007 Santander, Spain b

c

d

e

[email protected], [email protected], [email protected], [email protected], [email protected]

Summary—With the aim of avoiding some limitations of the fatigue regression model of Castillo and Fernández-Canteli, the energy parameter ǻıÂİmax is proposed to meet the fatigue damage progression in a more suitable manner than that by using the classical parameter ǻı. The resulting ǻıÂİmax – N field may be reconverted to the conventional ǻı–N field using the fitted model parameters and the cyclic stress-strain diagram of the material, for instance in the RambergOsgood version. In this way, the sigmoidal shape of the S–N percentile curves over the elastic and elasto-plastic regions can be analytically reproduced in agreement with the experimental data. The suitability of this energy damage parameter is discussed in terms of their adequacy to describe the LCF region in the S–N and İ–N fields. Finally, an example of evaluation of a representative material is presented. Keywords: Energetic damage parameters, S-N field, LCF region 1. INTRODUCTION The probabilistic S-N Weibull regression model proposed by Castillo and Fernández-Canteli [1] represents an advance in the testing strategy implying significant time and costs saving along with enhanced reliability. Nevertheless, it evidences some lacks that, among others, impedes its implementation in the LCF region for the assessment of fatigue results and subsequent use in the engineering practice. The consideration of stress range ∆σ and stress level, as independent parameters without mutual interaction, suggests the necessity of resorting to a more general driving force coupling both in a unique parameter for the assessment of the S–N field from experimental results. Only under certain simple conditions, as those represented by constant stress level under linear elastic stress conditions, the conventional simplistic approach could be justified. The consideration of σmax and σmin as determining variables, as suggested in [2], allows us to derive the analytical expression of the general solution of the S–N field on sound physical and statistical conditions, in contrast to the empirical models currently proposed in the literature. Nevertheless, its validity is restricted to the field of elastic stress ranges so that the low cycle fatigue region remains excluded. The so called Smith-Watson-Topper (SWT) parameter [3] represents an important advance suggesting the consideration of ∆ε ⋅ σ max , as the driving force in the fatigue damage process, thus accounting for the influence of the stress level in the model. In this way, an energy parameter of broader application is suggested encompassing also the more simple case i.e., when the stress level does not need to be additionally considered. Unfortunately, its generality is not satisfactorily exploited, irrespective of its utility and conceptual contribution, unless probabilistic considerations 193



are implemented [4], thus retaining some of the limitations of the conventional S-N models. Further, the plasticity effects seem not to be adequately represented using the SWT parameter due to the small variation of σmax involved in the plastic region of the σ – ε diagram even when large strain ranges are applied. In this work, an alternative proposal for the damage driving force in the fatigue S–N field is suggested aiming at a) trying to reproduce a presumable energy character of the damage induced by fatigue b) confirming that the use of ∆σ as the leading parameter in the fatigue field, without consideration of the strain level, would be only justified for linear elastic conditions, and c) extending the model proposed by Castillo and Fernández-Canteli [1] to elasto-plastic conditions, i.e., to the LCF region by the use of this new parameter. 2. THE PROPOSED ENERGY PARAMETER ǻı Â İmax The probabilistic S-N and ε-N Weibull regression models proposed by Castillo et al. [1,5] show superiority in comparison to other current models as that of Basquin and Coffin-Manson, and represent an noticeable advance in the assessment and testing strategy implying relevant time and costs saving with enhanced reliability in the assessment of fatigue results for subsequent use in the engineering practice. Nevertheless, the S-N model, in particular, evidences some lacks that, among others, impede its implementation in the LCF region. This is due to the fact that the existence field of the percentile curves are unlimited when approaching to the low cycle fatigue region. This means unacceptable big values of ∆σ , implying σmax values far over the ultimate stress of the material, when σmin, σmean or R = σmin/σmax are taken as the reference stress level parameter. In an attempt of avoiding this limitation, an energy parameter is searched as a reference for the damaging fatigue process. The shape of the cyclic σ–ε curve suggests the consideration of ∆σ ⋅ ε max as the new parameter representing the driving force. Instead of being considered a simple opportunistic variant to the SWT parameter, it tries to take allowance of the markedly increasing values for İmax compared to those of ımax, when the elastic limit stress is exceeded, maintaining the energy character of any product of stress and strain in the search of a general fatigue parameter capable to reproduce elastic-plastic effects, precisely, as the own SWT parameter, or those used by Noroozi et al. [6] or those related to the hysteresis cycle and J-integral, as proposed by Vormwald [7] and Anthes [8], see also [9,10]. This new proposal is particularly related to the probabilistic model for assessment of fatigue results under load control already mentioned [1] providing an enhanced parameter estimation. A subsequent reconversion of the parameter ∆σ ⋅ ε max in the elasto-plastic regimen to the simple stress range parameter ǻı in the conventional S–N field is possible thus facilitating a probabilistic definition of the percentile curves also in the LCF region. For doing this, the estimated model parameters, just obtained, and an analytical expression of the stress-strain cyclic diagram of the material, for instance in the Ramberg-Osgood version, are used. In this way, the sigmoidal shape of the S–N percentile curves over the elastic and elasto-plastic regions can be analytically reproduced in agreement with the experimental data. It can be observed that, as long as the tests are conducted in the elastic region, the use of the new parameter leads to a translation and scale change in the log-log plot of the fatigue results compared to the traditional S–N field as a function of ǻı, without having consequences in the lifetime prediction:

194



log(∆σ ⋅ ε max ) = log[σ max (1 − R ) ⋅ σ max E ] = log[(1 − R ) E ] + 2 logσ max

(1)

Contrary, when local plastic strains are involved, the product ∆σ ⋅ ε max implies spreading of the ǻı–N results thus allowing a modified S–N field with quasi-asymptotic trend of the results to be assumed in agreement with the Weibull model proposed. The advantages of such an approach can be extended to other fatigue models, as for instance those in which ǻJ is involved as damage parameter [7,8]. Also, the expected correspondence of the present approach with the model formerly proposed by the authors [11] implying the direct conversion from the ǻİ field to the ǻı field, will deserve attention in the next future. Similarly to the SWT damage parameter, the new combined damage parameter may account for the stress level effects in the medium to high-cycle fatigue regimes. Eq. (1) shows that for elastic regimes, the damage parameter may be formulated as a function of the stress ratio, R. 3. INFLUENCE OF THE CYCLIC ı–İ CURVE AND ITS REPERCUSSION ON THE ǻı Â İmax – N FIELD Different methodologies can be envisaged to derive the cyclic ı – İ curve. For materials stabilizing after a relative short number of cycles, the following methods are suitable to be applied: incremental step test (IST), multiple steps with increasing strain, multiple steps with decreasing strain and random cyclic strain amplitude [12,13]. These methods require one specimen to be fatigued by blocks of increasing and decreasing constant strain amplitudes until the material reaches a cyclically stable state [10]. Nevertheless, for materials showing unstable cyclic behaviour, the above referred methods lead to remarkable discrepancies in the resulting ı – İ curves. This variability arising in the ı – İ curves would be the subject of a future investigation focused on its repercussion on the shape of the LCF region of the resulting S–N field from each methodology applied. More representative cyclic ı – İ curves can be obtained by carrying out various constantamplitude tests, also called single step tests (SST), covering different ǻİ. As the plastic response depends on the loading history and the loading path, different cyclic curves are obtained, whereas the curve obtained by IST is reported to lay below the curve obtained by SST [10]. In the absence of cyclic experimental data, alternative empiric approximations can be followed such as the Uniform Material Law [14]. According to the Uniform Material Law, the cyclic behavior of a material is based on its static properties (Young’s modulus and ultimate strength), which is an approximate method without the need to carry out additional experiments. The cyclic ı – İ curve is frequently described by the equation of the Ramberg-Osgood type: 1 n′

ε a = σ a E + (σ a K ′)

(2)

where E is the Young’s modulus, K’ the cyclic strength coefficient and n’ the cyclic strain-hardening coefficient. Though components may experience significant stress-strain levels at notches, the available experimental cyclic data is usually limited to maximum strain amplitudes well below the ultimate strain of the material, due to difficulties on conducting experimental tests on smooth specimens under extreme strain levels (i.e. higher than 1.5–2%). Therefore, the Ramberg-Osgood equation is generally estimated for a limited stress-strain domain, and extrapolations for higher stress-strain levels may lead to inconsistent predictions. For practical purposes, the cyclic ı – İ curve can be assumed to exhibit the ultimate strength as an asymptote. To obtain an asymptote for 195



the ultimate strength Rm, and extend the validity of the cyclic ı – İ mathematical representation, the authors propose to add a factor in Eq. (2) leading to: 1 n′

ε a = σ a E + (σ a K ′)

[1 (1 − η )]α ,

(3)

with η = σ max Rm and α = 2 , as proposed values to be eventually optimized in a next work. Only an accurate representation of the whole cyclic ı – İ curve, including the ultra-low cycle fatigue regime and the asymptotic behaviour to the monotonic tensile strength, will allow the complete description of the S – N field, including the elasto-plastic region, leading to the well-known sigmoidal representation of the S – N data. This approach can be further improved by considering rather a probabilistic definition of the tensile strength Rm, than a deterministic value as done here.

4. APPLICATION TO A PRACTICAL CASE To ensure the utility of the proposed approach, the proposed driving force parameter was applied to the assessment of fatigue results on the steel alloy 42CrMo4 reported in [15], using the Weibull probabilistic model [1]. The procedure consists of the following steps: 1. Obtaining fatigue data in the form ∆σ versus lifetime N. 2. For a stress ratio R = –1, computing the maximum deformation εmax directly with Eq. (2) or (3), because εmax = εa. (For a different R the mean strain has to be summed leading to εmax = εmean + εa) 3. Computing the product ∆σ ⋅ ε max . 4. Estimating the model parameters B, C, λ, δ and β for data ∆σ ⋅ ε max versus N using the Weibull model from [1]. 5. Computing values of N for ∆σ ⋅ ε max for different percentiles, but plotting those values at positions of ǻı.

700

600

σa [MPa]

500

400

300

200

Experimental data - 42CrMo4 Ramberg-Osgood (Eq. 2) Modified Ramberg-Osgood (Eq. 3) Uniform Material Law Ultimate stress

100

0

0

0.5

1

1.5

2

2.5

εa [%]

Fig. 1. Different cyclic ı – İ curves for 42CrMo4 steel obtained according to the methodology used.

196



2000 1800 1600

1%

∆σ ⋅ εmax [MPa ⋅ %]

1400 50% 1200 1000

99%

800 600 400 200 0 1 10

10

2

10

3

4

10 N [cycles]

10

5

10

6

Fig. 2. ǻı·İmax – N curves for 42CrMo4 steel (using Eq. 2).

1600

1400

1% 50%

1200 ∆ σ [MPa]

99% 1000

800

600

Experimental data ∆ σ-N (from ∆ σ εmax-N fit) ∆ σ-N (from ∆ σ-N fit) Ultimate stress

400 0 10

10

2

10 N [cycles]

4

10

6

Fig. 3. S–N curves obtained with different fitting methods for 42CrMo4 steel (using Eq. 2).

Fig. 1 represents the cyclic stress-strain curves obtained with the Ramberg-Osgood law given by Eq. (2) (K’ = 807 MPa and n’ = 0.087), with the modified Ramberg-Osgood law given by Eq. (3) (K’ = 2485 MPa and n’ = 0.2006) and with the uniform material law [14] setting for low-alloy steels, K’ = 1.65 Rm and n’ = 0.15 in Eq. (2). Fig. 2 represents the fit obtained for the fatigue data using the parameter ǻı İmax. Fig. 3 represents the fatigue data fitting once as usual ∆σ versus N and once ∆σ ⋅ ε max versus N using the procedure explained above using Eq. (2). Note that the curvature is not sigmoidal as expected due to the fact that using the conventional Ramberg-Osgood diagram an infinite strain can be only achieved for an infinite stress. However, using the modified Ramberg197



1600

1400

1% 50%

∆ σ [MPa]

1200

99%

1000

800

600

400 0 10

Experimental data ∆ σ-N (from ∆ σ εmax-N fit) ∆ σ-N (from ∆ σ-N fit) Ultimate stress 10

2

10 N [cycles]

4

10

6

Fig. 4. S–N curves obtained with different fitting methods for 42CrMo4 steel (using Eq. 3).

Osgood law given by Eq.(3), for the same fatigue data an asymptotic behaviour of the Wöhler field for stresses close to the ultimate stress is obtained, as represented in Fig. (4). 5. CONCLUSIONS The principal conclusions derived from this work are the following: • The consideration of the energy variable ǻı·İmax as the damage parameter in fatigue assessment enhances the probabilistic Weibull regression model proposed by Castillo and Fernández-Canteli, allowing a probabilistic definition of the LCF region for the conventional S-N field. • The procedure implies proceeding to an initial model parameter estimation by considering the lifetime as a function of the new energy parameter ǻı·İmax then reconverting the resulting Wöhler field to the original ǻı-N field using the fitted model parameters and the ı–İ cyclic diagram of the material, for instance in the Ramberg-Osgood version. • A modification of the original Ramberg-Osgood proposal has been undertaken in order to assume an asymptotic upper bound of the stress range, without surpassing the ultimate stress as the maximum stress. This correction is required due to limitations in available experimental cyclic stress-strain data which usually do not cover the full range of strains due to testing difficulties. • The influence of the variability of the resulting cyclic ı–İ Ramberg-Osgood curve of the material on the LCF region of the S-N field is to be investigated. • Also, the possibility of the proposed damage parameter to describe the stress level effects needs to be investigated using data for distinct stress levels.

198



ACKNOWLEDGEMENTS The authors acknowledge partial support supplied by the Spanish Ministry of Science and Innovation MICINN (Ref: BIA2010-19920) and by the Portuguese Science and Technology Foundation FCT (Ref: SFRH/BD/66497/2009).

REFERENCES [1] [2]

[3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

[14] [15]

E. Castillo and A. Fernández-Canteli: A general regression model for lifetime evaluation and prediction. Int. J. Fract. 107 (2001) 117–137. R. Koller, M.L. Ruiz-Ripoll, A. García, A. Fernández Canteli and E. Castillo: Experimental validation of a statistical model for the Wöhler field corresponding to any stress level and amplitude. Int. J. Fatigue 31 (2009) 231–241. K.N. Smith, P. Watson and T.H. Topper: A Stress-Strain Function for the Fatigue of Metals. J. Mater. 5 (1970) 767–778. J.A.F.O. Correia, A.M.P. de Jesus and A. Fernández Canteli: A procedure to derive probabilistic fatigue crack propagation data. Int. J. Struct. Integr. 3 (2012) 158–183. E. Castillo, A. Fernández-Canteli, H. Pinto and M. López Aenlle: A general regression model for statistical analysis of strain-life fatigue data. Mater. Lett. 62 (2008) 3639–3642. A.H. Noroozi, G. Glinka and S. Lambert: A two parameter driving force for fatigue crack growth analysis. Int. J Fatigue 27 (2005) 1277–1296. M. Vormwald: Anrisslebensdauervorhersage auf der Basis der Schwingbruch-mechanik für kurze Risse. PhD Technische Hochschule Darmstadt, 1989. R. Anthes: Aein neuartiges Kurzrisskonzept zur Anrisslebensdauervorhersage bei wiederholte Beanspruchung. Institut für Stahlbau und Werkstoffmechanik TH Darmstadt, Heft 56, 1997. J. Menke, M. Sander and H.A. Richard: Crack initiation life if notched specimens. In: Proc. of 17th ECF, Brno, 2–5 September, 2008. F. Ellyin: Fatigue Damage, Crack Growth, and Life Prediction. Chapman & Hall Editions, Edmunds, Suffolk, UK, 1997. A. Fernández-Canteli, E. Castillo, H. Pinto and M. López Aenlle: Estimating the S-N field from strainlifetime curves. Strain 47 (2011) e93–e97. J. Jones and R.C. Hudd: Cyclic stress-strain curves generated from random cyclic strain amplitude tests. Int. J Fatigue 21 (1999) 521–530. A. Neslony, Ch. Dsoki, H. Kaufmann and P. Krug: New method for evaluation of the Manson–Coffin– Basquin and Ramberg–Osgood equations with respect to compatibility. Int. J Fatigue 30 (2008) 19671977. A. Bäumel and T. Seeger: Materials data for cyclic loading, Supplement 1. Elsevier Science Publishers, 1990. D. Eifler: Inhomogene Deformationserscheinungen bei Schwingbeanspruchung eines unterschiedlich wärmebehandelten Stahles des Typs 42 CrMo 4. PhD-Thesis, University of Karlsruhe, 1981.

199

EXTENDING THE APPLICABILITY OF ...

EXTENDING THE APPLICABILITY OF ...

Suggest Documents