Influence of variable stress ratio during train operation on residual fatigue lifetime of railway axles

ScienceDirect Available online at www.sciencedirect.com Available online at www.sciencedirect.com ScienceDirect Structural Integrity Procedia 00 (2016) 000–000 www.elsevier.com/locate/procedia 2452-3216 © 2016 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of PCF 2016. XV Portuguese Conference on Fracture, PCF 2016, 10-12 February 2016, Paço de Arcos, Portugal Thermo-mechanical modeling of a high pressure turbine blade of an airplane gas turbine engine P. Brandãoa, V. Infanteb, A.M. Deusc* aDepartment of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, Portugal bIDMEC, Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, Portugal cCeFEMA, Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, Portugal Abstract During their operation, modern aircraft engine components are subjected to increasingly demanding operating conditions, especially the high pressure turbine (HPT) blades. Such conditions cause these parts to undergo different types of time-dependent degradation, one of which is creep. A model using the finite element method (FEM) was developed, in order to be able to predict the creep behaviour of HPT blades. Flight data records (FDR) for a specific aircraft, provided by a commercial aviation company, were used to obtain thermal and mechanical data for three different flight cycles. In order to create the 3D model needed for the FEM analysis, a HPT blade scrap was scanned, and its chemical composition and material properties were obtained. The data that was gathered was fed into the FEM model and different simulations were run, first with a simplified 3D rectangular block shape, in order to better establish the model, and then with the real 3D mesh obtained from the blade scrap. The overall expected behaviour in terms of displacement was observed, in particular at the trailing edge of the blade. Therefore such a model can be useful in the goal of predicting turbine blade life, given a set of FDR data. © 2016 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of PCF 2016. Keywords: High Pressure Turbine Blade; Creep; Finite Element Method; 3D Model; Simulation. * Corresponding author. Tel.: +351 218419991. E-mail address: [email protected] Copyright © 2016 The Authors. Published by Elsevier B.V. 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 the Scientific Committee of ECF21. 10.1016/j.prostr.2016.06.447 Procedia Structural Integrity 2 (2016) 3585–3592 10.1016/j.prostr.2016.06.447 Available online at www.sciencedirect.com ScienceDirect Structural Integrity Procedia 00 (2016) 000–000 w ww.elsevier.com/locate/procedia 2452-3216 © 2016 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of ECF21. 21st European Conference on Fracture, ECF21, 20-24 June 2016, Catania, Italy Influence of variable stress ratio during train operation on residual fatigue lifetime of railway axles Pavel Pokornýa,b,*, Luboš Náhlíka, Pavel Hutařa adep. CEITEC IPM, Institute of Physics of Materials, Academy of Sciences of the Czech Republic, v. v. i., Žižkova 22, 616 62 Brno, Czech Republic bFaculty of Mechanical Engineering, Brno University of Technology, Technická 2896/2, 616 69 Brno, Czech Republic Abstract Railway axles are subjected to cyclic loading, which could lead to fatigue failure. Therefore, it is desired to know residual fatigue lifetime of railway axles to ensure safe operation of trains. Because of detection of relatively small fatigue cracks is not guaranteed an estimation of residual fatigue lifetime is based on damage tolerance approach. The acting stress ratio is variable due to variable amplitude loading and load caused by existence of press-fitted wheel in the vicinity of assumed crack. The contribution is focused on influence of variable stress ratio in EA4T steel on residual fatigue lifetime of railway axles. The influence of stress intensity factor on fatigue crack propagation rate was experimentally evaluated for three different stress ratios, which correspond to operation conditions. Two different expressions of fatigue crack propagation rate were used and mutually compared to show influence of the stress ratio on residual fatigue lifetime of structure made of EA4T steel. The first expression considers stress intensity factor range (respecting stress ratio R) and the second one uses maximal value of the stress intensity factor. The paper shows ability of both expressions to describe experimental data obtained under different stress ratios and their influence on estimated residual fatigue lifetime values. The results obtained contribute to the better estimation of residual fatigue lifetime of railway axles and generally to the safer rail transportation. © 2016 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of ECF21. Keywords: railway axle; stress ratio, EA4T, fatigue crack 1. Introduction It is a demand of customers and interest of producers to ensure reliability and safety of railway axles. Increasing demands require to involve damage tolerance approach to axle design process. This approach admits possibility that Available online at www.sciencedirect.com ScienceDirect Structural Integrity Procedia 00 (2016) 000–000 w ww.elsevier.com/locate/procedia 2452-3216 © 2016 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of ECF21. 21st European Conference on Fracture, ECF21, 20-24 June 2016, Catania, Italy Influence of variable stress ratio during train operation on residual fatigue lifetime of railway axles Pavel Pokornýa,b,*, Luboš Náhlíka, Pavel Hutařa adep. CEITEC IPM, Institute of Physics of Materials, Academy of Sciences of the Czech Republic, v. v. i., Žižkova 22, 616 62 Brno, Czech Republic bFaculty of Mechanical Engineering, Brno University of Technology, Technická 2896/2, 616 69 Brno, Czech Republic Abstract Railway axles are subjected to cyclic loading, which could lead to fatigue failure. Therefore, it is desired to know residual fatigue lifetime of railway axles to ensure safe operation of trains. Because of detection of relatively small fatigue cracks is not guaranteed an estimation of residual fatigue lifetime is based on damage tolerance approach. The acting stress ratio is variable due to variable amplitude loading and load caused by existence of press-fitted wheel in the vicinity of assumed crack. The contribution is focused on influence of variable stress ratio in EA4T steel on residual fatigue lifetime of railway axles. The influence of stress intensity factor on fatigue crack propagation rate was experimentally evaluated for three different stress ratios, which correspond to operation conditions. Two different expressions of fatigue crack propagation rate were used and mutually compared to show influence of the stress ratio on residual fatigue lifetime of structure made of EA4T steel. The first expression considers stress intensity factor range (respecting stress ratio R) and the second one uses maximal value of the stress intensity factor. The paper shows ability of both expressions to describe experimental data obtained under different stress ratios and their influence on estimated residual fatigue lifetime values. The results obtained contribute to the better estimation of residual fatigue lifetime of railway axles and generally to the safer rail transportation. © 2016 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of ECF21. Keywords: railway axle; stress ratio, EA4T, fatigue crack 1. Introduction It is a demand of customers and interest of producers to ensure reliability and safety of railway axles. Increasing demands require to involve damage tolerance approach to axle design process. This approach admits possibility that Copyright © 2016 The Authors. Published by Elsevier B.V. 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 the Scientific Committee of ECF21. 3586 Pavel Pokorný et al. / Procedia Structural Integrity 2 (2016) 3585–3592 2 Author name / Structural Integrity Procedia 00 (2016) 000–000 railway axle can include some defects incurred during manufacturing process or developed during train operation. The cyclic loading of railway axle (dominant is rotary bending due to weight and movement of the train) can lead to fatigue crack growth and consequently to fatigue failure of the railway axle. The paper is focused on description of procedure for estimation of residual fatigue lifetime of railway axle, i.e. on determination of number of load cycles (or load blocks) for fatigue crack growth from initial defect (crack) to the critical one which leads to axle failure. The three dimensional numerical models were developed for determination of fracture parameters, see e.g. Pokorný et al. (2016). The fatigue crack was assumed and modelled at location where severity of loading is the highest. Such place is often at notch close to press-fitted wheel, see e.g. work of Zerbst et al. (2012). The notch represents a stress concentrator and existence of press-fitted wheel in its vicinity increases local stress concentration. The residual stress caused by press-fitted wheel represents mean stress in load cycle. The variability of axle load is described by representative load spectrum, see e.g. reference Traupe et al. (2004) or Pokorný et al. (2014). Fig. 1a shows typical load spectrum of railway axle with sorted load amplitudes into 36 load levels. The most of load amplitudes in this spectrum are concentrated near basic load level corresponding to rotary bending due to weight of the train. However, there are also rare load cycles with amplitudes circa 3 times higher than basic load. The reliable procedure for residual fatigue lifetime estimation should contain the influence of residual stresses caused by existence of press-fitted wheel or developed during manufacturing process (e.g. hardening or surface deep rolling), see e.g. Hassani-Gangaraj et al. (2015). The total load is given by superposition of (bending) load amplitudes given by load spectrum and by residual stress component. The load stress ratio is generally different in each cycle, therefore suitable procedure should be used for residual fatigue lifetime estimation. The fatigue crack was presumed at the notch close to press fitted wheel, see Fig. 1b. The fatigue crack front was considered as semi-elliptical with perpendicular orientation to railway axle axis. According to Schijve (2008) this assumption is based on fact that fatigue crack grows perpendicular to maximum principal stress and nearly perpendicular crack growth was observed during failure of real railway axles as well. Fig. 1. (a) load spectrum - histogram expression; (b) scheme of wheelset with considered position of fatigue crack. Author name / Structural Integrity Procedia 00 (2016) 000–000 3 2. Effect of variable stress ratio R The stress ratio R can be expressed as: )( )( max, min, aK aK R I I  , (1) where KI,min is minimal and KI,max maximal value of the stress intensity factor during load cycle. Growing crack changes geometry of its crack front during propagation. Corresponding values of stress intensity factor can be determined analytically or numerically, however procedure of their determination is not aim of the paper, see e.g. Pokorný et al. (2016) for details. Fig. 2a shows stress intensity factor values of assessed axle as a function of fatigue crack length a. Both considered loads (bending and press-fit) contribute to opening of fatigue crack. The fatigue crack growth from 1 mm up to 55 mm was assumed in the presented study. The lower limit (1-2 mm) is frequently chosen due to detection limits of non-destructive testing methods. Fig. 2b shows typical dependence of fatigue crack length on number of applied cycles. It is evident that crack growth from 55 mm up to real critical crack length (with consequent failure) does not have significant contribution to total residual fatigue lifetime. Fig. 2. (a) stress intensity factor values for bending load (caused by weight of train) and load caused by press-fitted wheel; (b) typical evolution of fatigue crack; (c) M(T) specimen used for determination of v-K curves. The function belonging to bending loading shown in Fig. 2a corresponds to ride on straight track where dynamic effects are not present. This function describes stress intensity factors belonging to basic level of bending load. Nevertheless, the train also frequently goes to curve track, over switches, crossovers, etc. Mentioned events cause variable amplitude loading and enlarge “bending” load of railway axle. Load spectrum including these events is shown in Fig. 1a. The maximal value of the stress intensity factor in general cycle can be expressed as: )()()( ,,max, akKaKaK BIPFII  , (2) and minimal value of the stress intensity factor in general cycle as: )()()( ,,min, akKaKaK BIPFII  , (3) where KI,PF is stress intensity factor caused by press-fitted wheel, KI,B is stress intensity factor caused by weight of the train (basic level of bending loading), k is the dynamic coefficient (representing multiple of basic “static” load) describing load spectrum according to Fig. 1a. The stress ratio can be then expressed as: )()( )()( ,, ,, akKaK akKaK R BIPFI BIPFI    . (4) Pavel Pokorný et al. / Procedia Structural Integrity 2 (2016) 3585–3592 3587 2 Author name / Structural Integrity Procedia 00 (2016) 000–000 railway axle can include some defects incurred during manufacturing process or developed during train operation. The cyclic loading of railway axle (dominant is rotary bending due to weight and movement of the train) can lead to fatigue crack growth and consequently to fatigue failure of the railway axle. The paper is focused on description of procedure for estimation of residual fatigue lifetime of railway axle, i.e. on determination of number of load cycles (or load blocks) for fatigue crack growth from initial defect (crack) to the critical one which leads to axle failure. The three dimensional numerical models were developed for determination of fracture parameters, see e.g. Pokorný et al. (2016). The fatigue crack was assumed and modelled at location where severity of loading is the highest. Such place is often at notch close to press-fitted wheel, see e.g. work of Zerbst et al. (2012). The notch represents a stress concentrator and existence of press-fitted wheel in its vicinity increases local stress concentration. The residual stress caused by press-fitted wheel represents mean stress in load cycle. The variability of axle load is described by representative load spectrum, see e.g. reference Traupe et al. (2004) or Pokorný et al. (2014). Fig. 1a shows typical load spectrum of railway axle with sorted load amplitudes into 36 load levels. The most of load amplitudes in this spectrum are concentrated near basic load level corresponding to rotary bending due to weight of the train. However, there are also rare load cycles with amplitudes circa 3 times higher than basic load. The reliable procedure for residual fatigue lifetime estimation should contain the influence of residual stresses caused by existence of press-fitted wheel or developed during manufacturing process (e.g. hardening or surface deep rolling), see e.g. Hassani-Gangaraj et al. (2015). The total load is given by superposition of (bending) load amplitudes given by load spectrum and by residual stress component. The load stress ratio is generally different in each cycle, therefore suitable procedure should be used for residual fatigue lifetime estimation. The fatigue crack was presumed at the notch close to press fitted wheel, see Fig. 1b. The fatigue crack front was considered as semi-elliptical with perpendicular orientation to railway axle axis. According to Schijve (2008) this assumption is based on fact that fatigue crack grows perpendicular to maximum principal stress and nearly perpendicular crack growth was observed during failure of real railway axles as well. Fig. 1. (a) load spectrum - histogram expression; (b) scheme of wheelset with considered position of fatigue crack. Author name / Structural Integrity Procedia 00 (2016) 000–000 3 2. Effect of variable stress ratio R The stress ratio R can be expressed as: )( )( max, min, aK aK R I I  , (1) where KI,min is minimal and KI,max maximal value of the stress intensity factor during load cycle. Growing crack changes geometry of its crack front during propagation. Corresponding values of stress intensity factor can be determined analytically or numerically, however procedure of their determination is not aim of the paper, see e.g. Pokorný et al. (2016) for details. Fig. 2a shows stress intensity factor values of assessed axle as a function of fatigue crack length a. Both considered loads (bending and press-fit) contribute to opening of fatigue crack. The fatigue crack growth from 1 mm up to 55 mm was assumed in the presented study. The lower limit (1-2 mm) is frequently chosen due to detection limits of non-destructive testing methods. Fig. 2b shows typical dependence of fatigue crack length on number of applied cycles. It is evident that crack growth from 55 mm up to real critical crack length (with consequent failure) does not have significant contribution to total residual fatigue lifetime. Fig. 2. (a) stress intensity factor values for bending load (caused by weight of train) and load caused by press-fitted wheel; (b) typical evolution of fatigue crack; (c) M(T) specimen used for determination of v-K curves. The function belonging to bending loading shown in Fig. 2a corresponds to ride on straight track where dynamic effects are not present. This function describes stress intensity factors belonging to basic level of bending load. Nevertheless, the train also frequently goes to curve track, over switches, crossovers, etc. Mentioned events cause variable amplitude loading and enlarge “bending” load of railway axle. Load spectrum including these events is shown in Fig. 1a. The maximal value of the stress intensity factor in general cycle can be expressed as: )()()( ,,max, akKaKaK BIPFII  , (2) and minimal value of the stress intensity factor in general cycle as: )()()( ,,min, akKaKaK BIPFII  , (3) where KI,PF is stress intensity factor caused by press-fitted wheel, KI,B is stress intensity factor caused by weight of the train (basic level of bending loading), k is the dynamic coefficient (representing multiple of basic “static” load) describing load spectrum according to Fig. 1a. The stress ratio can be then expressed as: )()( )()( ,, ,, akKaK akKaK R BIPFI BIPFI    . (4) 3588 Pavel Pokorný et al. / Procedia Structural Integrity 2 (2016) 3585–3592 4 Author name / Structural Integrity Procedia 00 (2016) 000–000 Values of stress ratio R are calculated for all considered crack lengths according to relationship Eq. (4) and all considered load amplitudes from the load spectrum. Table 1 shows values of stress ratios R determined for crack length range from 1 to 55 mm and three dynamics coefficients (k = 2.9 - maximum value from the load spectrum, k = 1 - basic load level, k = 0.9 - minimum value from the load spectrum). It is evident from Table 1 that the stress ratio of railway axle varies during ride in the range from R = 0.2 to R = -1. The residual fatigue lifetime is influenced mainly by the highest load amplitudes in load spectrum, which can induce stress intensity factor values higher than the threshold one, see reference Pokorný et al. (2015). The stress ratio of cycles with the highest stress amplitude (damaging amplitudes) is approximately R = -0.5 in the case of short initial cracks (1-2 mm). The effect of press-fit disappears in the case of longer cracks and the acting stress ratio is close to value -1. Table 1. The various stress ratios R caused by variable amplitude loading and effect of press-fitted wheel. crack length a [mm] dynamic coefficient k max. (k = 2.9) mode (k = 1) min. (k = 0.9) 1 -0.49 0.00 0.18 2 -0.55 -0.09 0.10 3 -0.58 -0.13 0.05 5 -0.61 -0.18 0.00 10 -0.69 -0.31 -0.14 20 -0.83 -0.57 -0.43 30 -0.90 -0.74 -0.64 55 -0.99 -0.97 -0.96 2.1. Experimental measurement of fatigue crack propagation rates It is necessary for determination of residual fatigue lifetime to know dependency between the stress intensity factor K and fatigue crack propagation rate v =da/dN. Results in Table 1 were used during an arrangement of experimental measurements of v-K curves of EA4T steel. Three stress ratios (R = -1, -0.5 and 0.1) were chosen to determine the v-K dependence in the range of acting stress ratio of railway axle. For conservative estimation of residual fatigue lifetime it is worth to take into account v-K data, which were measured on specimen with lower in-plane constraint factor than the railway axle exhibits. This requirement is fulfilled by use of M(T) specimen, see Fig. 2c. The tests were performed in the frame of research infrastructure IPMinfra using resonant machine SCHENCK with mean force range  30 kN and maximal load amplitude 30 kN. The testing frequency was in range from 60 Hz to 40 Hz in dependence on crack length. The fatigue cracks were initiated from sharp notches (two sides), which were manufactured by electro discharge method. Fatigue crack propagation rates were measured according to ASTM E647 after crack initiation. The crack length increments were measured optically on both sides of M(T) specimen by using of CCD cameras uEye UI-2280SE-M-GL with lenses Lensagon CMFA2520ND. The positions of cameras were tied with digital indicators Sylvac CO MFPM 25 with accuracy 0.01 mm. Fourteen specimens were used (five specimens for stress ratio R = -1, five for stress ratio R = -0.5 and four for ratio R = 0.1) for determination of v-K data. Obtained experimental data are shown in Fig. 3a. The threshold value of stress intensity factor is of crucial importance in considered kind of application, therefore this value was evaluated from experimental data and compared with formerly published data. The Fig. 3b shows measured threshold values  Kth in dependence on stress ratio R, where data published by Regazzi et al. (2014) and Varfolomeev et al. (2011) are plotted for comparison. Fig. 3c shows comparison of maximal values Kth,max with respect to stress ratio calculated from following relationship: R K Kth th   1 max, . (5) Author name / Structural Integrity Procedia 00 (2016) 000–000 5 Fig. 3. (a) Measured v-K data (R ratios were considered: -1; -0.5; -1); (b) The dependence of threshold value  K th on stress ratio R; (c) The dependence of threshold value K th,max on stress ratio R. 2.2. Fatigue crack propagation rate description Very common approach NASGRO [NASGRO manual (2002)] was chosen for the interpolation of experimental data: q c p th n K K K K K R f C N a dN da                                    max 1 1 1 1, (6) where C, n are material constants and p, q empirical constants describing the curvatures that occur near the threshold and near the instability region of the crack growth curve, respectively and f is crack opening function. The residual fatigue lifetime is dominantly given by propagation rate of relatively small crack in the studied case, see Fig. 2b, therefore the denominator (1-K max /K c ) q (corresponding to rapid propagation of longer cracks) can be neglected. Then the NASGRO can be expressed in the form: p th n K K K R f C N a dN da                             1 1 1 . (7) Note that Eq. (7) was used for fit of material data in Fig. 3a. The stress intensity factor range is function of the crack length a, therefore: )(2)()()( ,min,max, akKaKaKaK BIII  . (8) The effective stress intensity factor range considers different crack closure under different R ratios: K R f Keff           1 1. (9) The opening function f is defined by Newman empirical description: RAAf 10  , -2  R<0 (10) ),max( 3 3 2 210 RARARAARf  , R  0 (11) Pavel Pokorný et al. / Procedia Structural Integrity 2 (2016) 3585–3592 3589 4 Author name / Structural Integrity Procedia 00 (2016) 000–000 Values of stress ratio R are calculated for all considered crack lengths according to relationship Eq. (4) and all considered load amplitudes from the load spectrum. Table 1 shows values of stress ratios R determined for crack length range from 1 to 55 mm and three dynamics coefficients (k = 2.9 - maximum value from the load spectrum, k = 1 - basic load level, k = 0.9 - minimum value from the load spectrum). It is evident from Table 1 that the stress ratio of railway axle varies during ride in the range from R = 0.2 to R = -1. The residual fatigue lifetime is influenced mainly by the highest load amplitudes in load spectrum, which can induce stress intensity factor values higher than the threshold one, see reference Pokorný et al. (2015). The stress ratio of cycles with the highest stress amplitude (damaging amplitudes) is approximately R = -0.5 in the case of short initial cracks (1-2 mm). The effect of press-fit disappears in the case of longer cracks and the acting stress ratio is close to value -1. Table 1. The various stress ratios R caused by variable amplitude loading and effect of press-fitted wheel. crack length a [mm] dynamic coefficient k max. (k = 2.9) mode (k = 1) min. (k = 0.9) 1 -0.49 0.00 0.18 2 -0.55 -0.09 0.10 3 -0.58 -0.13 0.05 5 -0.61 -0.18 0.00 10 -0.69 -0.31 -0.14 20 -0.83 -0.57 -0.43 30 -0.90 -0.74 -0.64 55 -0.99 -0.97 -0.96 2.1. Experimental measurement of fatigue crack propagation rates It is necessary for determination of residual fatigue lifetime to know dependency between the stress intensity factor K and fatigue crack propagation rate v =da/dN. Results in Table 1 were used during an arrangement of experimental measurements of v-K curves of EA4T steel. Three stress ratios (R = -1, -0.5 and 0.1) were chosen to determine the v-K dependence in the range of acting stress ratio of railway axle. For conservative estimation of residual fatigue lifetime it is worth to take into account v-K data, which were measured on specimen with lower in-plane constraint factor than the railway axle exhibits. This requirement is fulfilled by use of M(T) specimen, see Fig. 2c. The tests were performed in the frame of research infrastructure IPMinfra using resonant machine SCHENCK with mean force range  30 kN and maximal load amplitude 30 kN. The testing frequency was in range from 60 Hz to 40 Hz in dependence on crack length. The fatigue cracks were initiated from sharp notches (two sides), which were manufactured by electro discharge method. Fatigue crack propagation rates were measured according to ASTM E647 after crack initiation. The crack length increments were measured optically on both sides of M(T) specimen by using of CCD cameras uEye UI-2280SE-M-GL with lenses Lensagon CMFA2520ND. The positions of cameras were tied with digital indicators Sylvac CO MFPM 25 with accuracy 0.01 mm. Fourteen specimens were used (five specimens for stress ratio R = -1, five for stress ratio R = -0.5 and four for ratio R = 0.1) for determination of v-K data. Obtained experimental data are shown in Fig. 3a. The threshold value of stress intensity factor is of crucial importance in considered kind of application, therefore this value was evaluated from experimental data and compared with formerly published data. The Fig. 3b shows measured threshold values  Kth in dependence on stress ratio R, where data published by Regazzi et al. (2014) and Varfolomeev et al. (2011) are plotted for comparison. Fig. 3c shows comparison of maximal values Kth,max with respect to stress ratio calculated from following relationship: R K Kth th   1 max, . (5) Author name / Structural Integrity Procedia 00 (2016) 000–000 5 Fig. 3. (a) Measured v-K data (R ratios were considered: -1; -0.5; -1); (b) The dependence of threshold value  K th on stress ratio R; (c) The dependence of threshold value K th,max on stress ratio R. 2.2. Fatigue crack propagation rate description Very common approach NASGRO [NASGRO manual (2002)] was chosen for the interpolation of experimental data: q c p th n K K K K K R f C N a dN da                                    max 1 1 1 1 , (6) where C, n are material constants and p, q empirical constants describing the curvatures that occur near the threshold and near the instability region of the crack growth curve, respectively and f is crack opening function. The residual fatigue lifetime is dominantly given by propagation rate of relatively small crack in the studied case, see Fig. 2b, therefore the denominator (1-K max /K c ) q (corresponding to rapid propagation of longer cracks) can be neglected. Then the NASGRO can be expressed in the form: p th n K K K R f C N a dN da                             1 1 1 . (7) Note that Eq. (7) was used for fit of material data in Fig. 3a. The stress intensity factor range is function of the crack length a, therefore: )(2)()()( ,min,max, akKaKaKaK BIII  . (8) The effective stress intensity factor range considers different crack closure under different R ratios: K R f Keff           1 1 . (9) The opening function f is defined by Newman empirical description: RAAf 10  , -2  R<0 (10) ),max( 3 3 2 210 RARARAARf  , R  0 (11) 3590 Pavel Pokorný et al. / Procedia Structural Integrity 2 (2016) 3585–3592 6 Author name / Structural Integrity Procedia 00 (2016) 000–000 where A 0 , A 1 , A 2 and A 3 are polynomial coefficients of Newman’s crack opening function:     1 0max 2 0/ 2 cos)05.034.0825.0(            A , (12) 0max1 /)071,0415,0(   A , (13) 12 103  AAA , (14) 3102 1AAAA  , (15) where α is constraint factor. It is considered in calculations that plane strain prevails and α = 3. The parameter  max is maximum applied stress and  0 is flow stress. The flow stress was taken as average value of yield and ultimate strength according to reference NASGRO manual (2002). Herein considered  0 = 600 MPa. Fig. 4. The dependence of crack propagation rate v = da/dN on (a) effective stress intensity factor range, (b) maximal value of stress intensity factor The Fig. 4a shows measured data where horizontal axis is transformed into effective stress intensity factor range by considering Eq. (9). This transformation leads to effect that v-  K eff curves obtained under different stress ratios fit quite well each other. However, the slightly difference between data obtained for different R in threshold area occurs. The NASGRO fit of threshold data is determined according to procedure described in NASGRO manual (2002). The result of this threshold value fit is evident in Fig. 3b. Based on measured data the empirical fit of threshold values is given by parameters C thp = 2.056, C thm = 0 and  K 1 = 2.94 MPam 0.5 , see reference NASGRO manual (2002) for detail description of these parameters. The Fig. 4b shows dependence of fatigue crack propagation rate v = da/dN on maximal value of stress intensity factor K max . The measured data fit each other pretty well in v-K max expression for different stress ratios R in considered range from -1 to 0.1. It means that fatigue crack propagation rate for particular maximal value of the stress intensity factor K max is almost the same for stress ratio R = -1 and R = 0.1. It follows that for the description of fatigue crack propagation in EA4T steel and operation stress ratio range (from R = -1 to R = 0.1) just one v-K max dependence is sufficient. The measured data exhibits slightly higher propagation rates in linear region of v-K max dependence for stress ratio R = -1, see Fig. 4b. Therefore, data determined for stress ratio R = -1 are used in Eq. (16) for conservative determination of residual fatigue lifetime of railway axle. The description of v-K max dependence is provided by modification of NASGRO model according to Eq. (7):  * max, max, * max, *1 p I th n IK K KC N a dN da            . (16) Author name / Structural Integrity Procedia 00 (2016) 000–000 7 As was mentioned formerly, such description is slightly conservative for R = -1 and simpler than original NASGRO relationship. The material constants C*, n* and p* have the same meaning as C, n and p, but their values are different in general. The threshold value Kmax,th was taken as a constant in relationship (16) according to results depicted in Fig. 3c. Note that Kmax approach for crack propagation rate description under negative R was formerly used e.g. in Vasudevan and Sadananda (1999). 2.3. Estimation of residual fatigue lifetime The load spectrum is described by load block in the procedure of estimation of residual fatigue lifetime. The load block represents specific amount of kilometers during axle operation. The fatigue crack increments are calculated for all load levels contained in load spectrum. Calculation runs in a loop until length of fatigue crack does not overcome the considered critical length ac = 55 mm. The crack increments  a were determined by using of discretized forms of relationships (7) and (16): N K K K R f Ca p th n                          1 1 1 and  N K K KCa p I th n I         * max, max, * max, *1respectively. (17) The crack increment  a equals to zero until the stress intensity factor range or maximal value of stress intensity factor does not overcome the threshold value  Kth or Kth,max respectively. 2.4. Results and discussion Two above mentioned procedures were used for lifetime estimations. The relationship (7) considers as inputs applied stress intensity factor range, stress ratio and also analytically determined function of crack closure. The approach expressed by Eq. (16) uses maximal values of the stress intensity factor for lifetime estimation. It should be emphasized that both approaches consider effect of press-fitted wheel. The press-fit effect in Eq. (16) is included according to Eq. (2), i.e. by increase of stress intensity factor values. Table 2. Estimated residual fatigue lifetimes of railway axle (in load blocks) for different initial crack lengths. initial crack length a0 [mm] 1 1.5 2 3 5 according to v-K (Eq. (7)) 12 344 420 192 121 84 according to v-Kmax (Eq. (16)) 16 001 380 159 100 69 The Table 2 shows estimated residual fatigue lifetimes obtained by both above described approaches. The results are very sensitive to threshold value, because only peak amplitudes from the applied load spectrum exceed the threshold value. This is valid mainly for short cracks. Both approaches provide similar results. It is possible to declare that on the base of measured data the NASGRO approach expressed by Eq. (7) slightly better describes the crack propagation rate for different R in the mid-region of v-K curve. It is due to better description of plasticity induced crack closure in this region. The approach based on Eq. (16) is slightly conservative from this point of view. However, it is difficult to decide what description is better in the threshold region, where description by analytical interpolation function contains its limitations as well as the general validity of the fact that threshold values are constant for negative R, even though some works proved that idea, see e.g. Vasudevan and Sadananda (1999). It is necessary to notice that Kmax approach is suitable only for stress ratio range negative or slightly positive (close to zero). This approach is unsuitable for application with important positive R, where the crack closure plays different role than in the case of low or negative R. The advantage of Kmax approach for application on railway axle can be found in decrease of necessary experimental measurements (v-K curve only for the lowest applied R is needed) and simpler procedure of determination of residual fatigue lifetime. The results obtained are valid for considered EA4T steel. Its generality will be validated in future for other steels of the same class. Pavel Pokorný et al. / Procedia Structural Integrity 2 (2016) 3585–3592 3591 6 Author name / Structural Integrity Procedia 00 (2016) 000–000 where A 0 , A 1 , A 2 and A 3 are polynomial coefficients of Newman’s crack opening function:     1 0max 2 0/ 2 cos)05.034.0825.0(            A , (12) 0max1 /)071,0415,0(   A, (13) 12 103  AAA , (14) 3102 1AAAA  , (15) where α is constraint factor. It is considered in calculations that plane strain prevails and α = 3. The parameter  max is maximum applied stress and  0 is flow stress. The flow stress was taken as average value of yield and ultimate strength according to reference NASGRO manual (2002). Herein considered  0 = 600 MPa. Fig. 4. The dependence of crack propagation rate v = da/dN on (a) effective stress intensity factor range, (b) maximal value of stress intensity factor The Fig. 4a shows measured data where horizontal axis is transformed into effective stress intensity factor range by considering Eq. (9). This transformation leads to effect that v-  K eff curves obtained under different stress ratios fit quite well each other. However, the slightly difference between data obtained for different R in threshold area occurs. The NASGRO fit of threshold data is determined according to procedure described in NASGRO manual (2002). The result of this threshold value fit is evident in Fig. 3b. Based on measured data the empirical fit of threshold values is given by parameters C thp = 2.056, C thm = 0 and  K 1 = 2.94 MPam 0.5 , see reference NASGRO manual (2002) for detail description of these parameters. The Fig. 4b shows dependence of fatigue crack propagation rate v = da/dN on maximal value of stress intensity factor K max . The measured data fit each other pretty well in v-K max expression for different stress ratios R in considered range from -1 to 0.1. It means that fatigue crack propagation rate for particular maximal value of the stress intensity factor K max is almost the same for stress ratio R = -1 and R = 0.1. It follows that for the description of fatigue crack propagation in EA4T steel and operation stress ratio range (from R = -1 to R = 0.1) just one v-K max dependence is sufficient. The measured data exhibits slightly higher propagation rates in linear region of v-K max dependence for stress ratio R = -1, see Fig. 4b. Therefore, data determined for stress ratio R = -1 are used in Eq. (16) for conservative determination of residual fatigue lifetime of railway axle. The description of v-K max dependence is provided by modification of NASGRO model according to Eq. (7):  * max, max, * max, *1 p I th n IK K KC N a dN da            . (16) Author name / Structural Integrity Procedia 00 (2016) 000–000 7 As was mentioned formerly, such description is slightly conservative for R = -1 and simpler than original NASGRO relationship. The material constants C*, n* and p* have the same meaning as C, n and p, but their values are different in general. The threshold value Kmax,th was taken as a constant in relationship (16) according to results depicted in Fig. 3c. Note that Kmax approach for crack propagation rate description under negative R was formerly used e.g. in Vasudevan and Sadananda (1999). 2.3. Estimation of residual fatigue lifetime The load spectrum is described by load block in the procedure of estimation of residual fatigue lifetime. The load block represents specific amount of kilometers during axle operation. The fatigue crack increments are calculated for all load levels contained in load spectrum. Calculation runs in a loop until length of fatigue crack does not overcome the considered critical length ac = 55 mm. The crack increments  a were determined by using of discretized forms of relationships (7) and (16): N K K K R f Ca p th n                          1 1 1 and  N K K KCa p I th n I         * max, max, * max, *1respectively. (17) The crack increment  a equals to zero until the stress intensity factor range or maximal value of stress intensity factor does not overcome the threshold value  Kth or Kth,max respectively. 2.4. Results and discussion Two above mentioned procedures were used for lifetime estimations. The relationship (7) considers as inputs applied stress intensity factor range, stress ratio and also analytically determined function of crack closure. The approach expressed by Eq. (16) uses maximal values of the stress intensity factor for lifetime estimation. It should be emphasized that both approaches consider effect of press-fitted wheel. The press-fit effect in Eq. (16) is included according to Eq. (2), i.e. by increase of stress intensity factor values. Table 2. Estimated residual fatigue lifetimes of railway axle (in load blocks) for different initial crack lengths. initial crack length a0 [mm] 1 1.5 2 3 5 according to v-K (Eq. (7)) 12 344 420 192 121 84 according to v-Kmax (Eq. (16)) 16 001 380 159 100 69 The Table 2 shows estimated residual fatigue lifetimes obtained by both above described approaches. The results are very sensitive to threshold value, because only peak amplitudes from the applied load spectrum exceed the threshold value. This is valid mainly for short cracks. Both approaches provide similar results. It is possible to declare that on the base of measured data the NASGRO approach expressed by Eq. (7) slightly better describes the crack propagation rate for different R in the mid-region of v-K curve. It is due to better description of plasticity induced crack closure in this region. The approach based on Eq. (16) is slightly conservative from this point of view. However, it is difficult to decide what description is better in the threshold region, where description by analytical interpolation function contains its limitations as well as the general validity of the fact that threshold values are constant for negative R, even though some works proved that idea, see e.g. Vasudevan and Sadananda (1999). It is necessary to notice that Kmax approach is suitable only for stress ratio range negative or slightly positive (close to zero). This approach is unsuitable for application with important positive R, where the crack closure plays different role than in the case of low or negative R. The advantage of Kmax approach for application on railway axle can be found in decrease of necessary experimental measurements (v-K curve only for the lowest applied R is needed) and simpler procedure of determination of residual fatigue lifetime. The results obtained are valid for considered EA4T steel. Its generality will be validated in future for other steels of the same class. 3592 Pavel Pokorný et al. / Procedia Structural Integrity 2 (2016) 3585–3592 8 Author name / Structural Integrity Procedia 00 (2016) 000–000 3. Conclusion For reliable estimation of residual fatigue lifetime of railway axles the effect of the variable stress ratio R should be determined. The EA4T steel was chosen as etalon of railway axle material for this investigation and its v-K dependence was experimentally measured for different R values. Two different approaches were used for evaluation of measured data. First was based on NASGRO approach published by Newman and the second one was based on use of only positive part of load cycle (Kmax). It was shown that for given application and material both approaches provides comparable results. It is evident that sophisticated NASGRO approach is more general and can be used in many application for determination of crack growth under different conditions, but the time and costs for determination of all necessary parameters can be quite high in this case (NASGRO requires at least two v-K dependence measured on different stress ratio to evaluate crack closure effects). Presented Kmax approach is valid only for low or negative R, however only v-K curve for the lowest applied stress ratio (R = -1 was used in studied case) is necessary for its use. The future work will be focused on proof of generality of this conclusion at least for the materials suitable for railway axles. The proof of validity of this approach can lead to the faster use of new or modified axle materials or can decrease the costs of experimental determination of material characteristics. Acknowledgements This research was carried out under the project CEITEC 2020 (LQ1601) with financial support from the Ministry of Education, Youth and Sports of the Czech Republic under the National Sustainability Programme II. References Zerbst, U., et al., 2012. Safe life and damage tolerance aspects of railway axles – A review. Engineering Fracture Mechanics 98, 214-271. Traupe, M., Meinen, H., Zenner, H., Sichere und wirtschaftliche Auslegung von Eisenbahnfahrwerken (in German), TU Clausthal, 2004. Pokorný,P., Náhlík, L., Hutař, P., 2014, Comparison of Different Load Spectra on Residual Fatigue Lifetime of Railway Axle, Procedia Engineering 74, 313-316. 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Hutař, P., Seitl, S., Knésl, Z., 2006, Effect of constraint on fatigue crack propagation near threshold in medium carbon steel, Computational Materials Science 37(1–2), 51-57. Regazzi,D., Beretta, S., Carboni, M., 2014, An investigation about the influence of deep rolling on fatigue crack growth in railway axles made of a medium strength steel, Engineering Fracture Mechanics 131, 587-601. ASTM E 647-08, 2009, Standard test method for measurement of fatigue crack growth rates. American Society for Testing and Materials, West Conshohocken, ASTM International. Vasudevan, A.K., Sadananda, K., 1999, Application of unified fatigue damage approach to compression-tension region, International Journal of Fatigue 21, 263-273.