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Local stresses and material damping in very high cycle fatigue
International Journal of Fatigue 32 (2010) 1669–1674
Contents lists available at ScienceDirect
International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue
Local stresses and material damping in very high cycle fatigue V. Kazymyrovych *, J. Bergström, F. Thuvander Department of Materials Engineering, Karlstad University, SE-651 88, Sweden
a r t i c l e
i n f o
Article history: Received 3 November 2009 Received in revised form 2 March 2010 Accepted 22 March 2010 Available online 3 April 2010 Keywords: Ultrasonic fatigue testing Stress gradient Damping
a b s t r a c t The increased interest in the very high cycle fatigue (VHCF) properties of materials has resulted in a broadened use of ultrasonic fatigue testing. The accuracy of the obtained fatigue data depends to a large extent on the estimate of the actual stresses acting in the ultrasonic test specimen. Finite element modeling (FEM) is used in this paper to study local stresses at the fatigue initiating defects during ultrasonic fatigue testing of an AISI H11 tool steel. Material damping was included in the calculations, which was found to reduce the magnitude of the stresses during 20 kHz ultrasonic fatigue testing. By combining local stresses with the material damping effect, the actual stresses operating in the region of the fatigue initiating defect were established. In the very long life regime microstructural defects, such as slag inclusions, often serve as fatigue crack origins. Locally amplified stresses enable fatigue crack formation and growth. Therefore, in this work the relationship between actual stresses, the defect geometry and fatigue life is examined. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Ultrasonic fatigue testing has received considerable attention in the last two decades. This is mainly due to the increasing use of engineering materials in applications with service lives reaching up to 1010 load cycles. To match different fatigue conditions met in practice, a variety of ultrasonic equipment modifications has evolved. These include fatigue testing in tension, torsion, bending or fretting, with variable or constant amplitude loading, at different temperatures and in a variety of environments [1]. Tool steels are widely used in fatigue applications. Their high strength and good toughness contribute to excellent fatigue properties. It has been shown [2–6] that the most frequent VHCF initiation sites in high performance steels are at slag inclusions. Consequently, the fatigue strength generally increases with improved cleanness of the steel in question. Other fatigue initiation sites are carbides or their segregations. VHCF failures of steels are characterized by the formation of a structure called ‘‘fish-eye”, which could be described as a circular pattern that develops on the fracture surfaces during internally initiated fatigue failures [1,3,5,6]. One of the most frequently used VHCF testing techniques employs uni-axial tensile loading of hour-glass shaped specimens. The nominal stresses during testing are established with respect to the smallest cross-section where stresses are the highest. However, within this cross-section due to specimen geometry there is a stress gradient from the surface towards the centre. Moreover, the majority of fatigue failures take place at some distance from the
plane of the smallest cross-section, which effectively reduces the actual fatigue stresses and, therefore, the fatigue strength is overestimated. How much local stresses acting at the fatigue initiating defects differ from nominal stresses is discussed in this paper. Materials’ damping properties also influence the magnitude of actual stresses during ultrasonic fatigue testing. To the author’s knowledge, this subject has not been discussed in the literature where ultrasonic stresses are calculated, like in [1]. Therefore, in this work an attempt is made to illustrate the dependence of fatigue stresses on damping. It is considered in terms of mass proportional Rayleigh damping factor a, stiffness proportional Rayleigh damping factor b, as well as testing frequency. When local stresses, defined by the location of the fatigue initiating defect, are corrected by the effect of material damping, the estimate of an actual fatigue stress is obtained. This stress in combination with the defect size and shape would define the stress state at the interface between the microstructural defect and the matrix during fatigue crack initiation stage. Taking into account that in VHCF most of the fatigue life is consumed during initial crack growth [7,8], there should be a traceable relationship between actual stress and defect geometry on the one side and VHCF life on the other. In the paper this relationship is presented through the stress intensity parameter.
2. Experimental 2.1. VHCF testing
* Corresponding author. E-mail address: [email protected] (V. Kazymyrovych). 0142-1123/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2010.03.007
Very high cycle fatigue testing of an AISI H11 high strength steel, Table 1, was performed at 20 kHz frequency using ultrasonic
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piezo-electric equipment. The main feature of the equipment is a resonance system, which receives high frequency mechanical vibrations from a set of piezo-electrics. These vibrations are amplified by a special horn and transmitted to a specimen where the magnitude of the achievable fatigue stress is defined by the electrical voltage applied to the piezo-electrics. For full description of the equipment readers are referred to [1]. The hour-glass shaped specimens with the diameter of the smallest cross-section of 6 mm and hour-glass radius of 31 mm were extracted from the billet in the transverse direction, which represents the worst condition for the fatigue properties. The choice of specimens with the smallest diameter of 6 mm and not 3 mm, as is often found in literature, was made in order to maximize the material volume subjected to approximately the same stress levels. This should enhance finding the largest defects in the material. Austenitizing, quenching and tempering was made to achieve a martensitic microstructure and a hardness of 480 HV corresponding to a tensile strength of 1430 MPa and a 0.2% proof stress of 1260 MPa. Elastic modulus and density were 210 GPa and 7800 kg/m3, respectively. Testing was conducted at minimum to maximum cyclic load ratio R = 0.1. Air cooling was applied in order to maintain the temperature on the specimens’ surface close to room temperature. The staircase test method [9] was used to estimate the material’s fatigue strength at 109 load cycles.
2.2. FEM modeling of VHCF testing An axi-symmetric ABAQUS finite element model was created to simulate ultrasonic fatigue testing. The application of fatigue stresses was accomplished in two consecutive steps: (1) pre-stressing of the specimen in the tensile direction in order to achieve the load ratio R = 0.1; (2) introduction of sinusoidal pulsating stresses by means of assigning micron-scale displacement at the specimen’s ends. The displacement was applied as a sinusoidal function with the period representing test frequency, and with the amplitude of displacement evaluated from measurements on the free end of the ultrasonic amplification horn. The effect of material damping on the magnitude of stresses during fatigue testing was evaluated by incorporating Rayleigh damping into the FEM model. Four-node bilinear axi-symmetric quadrilateral elements with reduced integration were used in the model. 2.2.1. Local stresses at the fatigue initiation sites The nominal stresses were calculated analytically using Eq. (1) as described in [1]. Here specimen geometry, elastic modulus, density and applied displacement at the specimen ends were used as input for the calculation of resonance stresses. The stress obtained for the smallest specimen’s cross-section was then used as nominal during fatigue testing.
rðxÞ ¼ Ed A0 u Table 1 Chemical composition of AISI H11 steel, wt.%. C
Si
Mn
Cr
Mo
V
0.38
1.10
0.40
5.00
1.30
0.40
½b coshðbxÞ coshðaxÞ a sinhðbxÞ sinhðaxÞ 2
cosh ðaxÞ
ð1Þ
where Ed is the dynamic elastic modulus, A0 the displacement amplitude at the end of the specimen, u, b, a the parameters dependent on the specimen geometry and material properties, and x is the distance along the ‘‘hour-glass” section of the specimen.
Fig. 1. An example of FEM calculated stress state in an ‘‘hour-glass” specimen with the 6 mm diameter of the smallest cross-section. Effect of damping is neglected: (a) schematic representation of a specimen’s quarter; (b) stress gradient in radial direction; (c) stress gradient in longitudinal direction obtained by FEM and analytically.
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Due to the ‘‘hour-glass” geometry there is a stress gradient in the middle section of the specimen, with the maximum stress at the surface of the specimen in its smallest cross-section, Fig. 1. The FEM results show that at the waist, in radial direction the difference in stress between the centre of the specimen, point O, and the most stressed region at the surface, point R, is 10%, Fig. 1b, while in the longitudinal direction the stress in point O, Fig. 1c, is about four times as high as in point L. Thus, the actual stresses acting at the fatigue initiating sites differ from the nominal stress levels depending on the location of the initiation sites within the specimen. The severity of the stress gradient would be enhanced if specimens with smaller ‘‘hour-glass” radius were used. In order to obtain a more realistic S–N representation of the test results, it is necessary to consider local stresses at the fatigue initiating defects instead of nominal. It should be clarified that under ‘‘local stress” in this paper is meant the variation of stress within the specimen caused exclusively by specimen geometry. This is not to be confused with the resulting stress at fatigue initiating defects due to their stress raising capacity. The probability of a fatigue crack initiating in the most stressed region is high, but the crack could also evolve in less stressed regions of a specimen if a sufficiently large defect with enough potential for crack initiation is present. A possible way to estimate this potential is proposed later in this paper. 2.2.2. Effect of damping The equation of motion for a finite element structure with consideration of damping could be presented as in [10]:
€ þ ½CU_ þ ½KU ¼ F ½MU
ð2Þ
€ the acceleration, [C] the damping where [M] is the mass matrix, U matrix, U_ the speed, [K] the stiffness matrix, U the spatial displacement, and F is the external force (corresponds to the applied resonance oscillations). Material damping can be described by the Rayleigh damping parameter. The Rayleigh damping is defined by specifying two damping factors: a – for mass proportional damping and b – for stiffness proportional damping [11]. For a given angular velocity x the damping ratio n can be expressed in terms of the damping factors a and b as:
n¼
a 2x
þ
bx 2
ð3Þ
The equation implies that the mass proportional Rayleigh damping factor a is important at low frequencies and the stiffness proportional factor b damps the higher frequencies. The b factor introduces damping proportional to the strain rate, which can be considered as damping associated with the mechanical properties [11].
Normalised stress amplitude
V. Kazymyrovych et al. / International Journal of Fatigue 32 (2010) 1669–1674
β =1E-06
0.9 β =1E-05
0.8 β =1E-04
0.7 0.6 0.5
0
5
10
15
20
25
30
35
40
Frequency, kHz Fig. 2. The effect of test frequency and stiffness proportional damping factor b on the fatigue stress amplitude. Mass proportional damping factor a = 0.01.
frequency a difference of about 20% in fatigue stress amplitude could be observed depending on the damping characteristics of the investigated material. This is of significance as 20% difference in stress amplitude is expected to greatly influence the eventual fatigue life. For the tested steel grade the stiffness proportional damping factor b was estimated by using inverse modeling. This was accomplished by measuring the fatigue strain amplitude in the specimen’s smallest cross-section using micro-strain-gages. The calculated value of 253 MPa was then matched with the results of FEM calculations at 20 kHz, Fig. 3, and the damping ratio n was estimated to be 0.1, which corresponds to the factor b in the order of 105. 3.1. Very high cycle fatigue testing The data obtained from the ultrasonic fatigue testing is presented in Fig. 4, which describes the relationship between the nominal applied stress and fatigue life for each specimen. There seems to be a large scatter in fatigue life for specimens tested at the same nominal stress. This could be attributed to the difference in size of the fatigue initiating defects. As shown later in the paper the size of the defect has a major influence on the VHCF life. In addition, the local stresses that are determined by the defect location in the specimen are not accounted for in Fig. 4. Furthermore, as in all testing, there is inherent variation in test conditions and specimen properties. The fractographic analysis of the fracture surfaces has shown that all the fatigue failures initiated from slag inclusions of a stringer type. Moreover, all fatigue initiations were internal apart from in one specimen (sp.3) where one end of the inclusion stringer reached the specimen’s surface. For this specimen early fatigue
300
Stress amplitude, MPa
3. Results and discussion Fig. 2 presents results obtained by the FEM calculations. The graph shows a variation of the stress amplitude in the middle section of a fatigue specimen as a function of b and test frequency. For frequencies in the range of 20 kHz a change in mass proportional damping factor a at different values of b produces negligible changes in stress amplitude. However, the effect of stiffness proportional damping factor b on the stress amplitude increases with increasing test frequency, Fig. 2. The stress amplitude increase with higher test frequency is most pronounced for low damping (low b). Damping ratio n has been calculated for different values of b at 20 kHz frequency and then plotted against the corresponding stress amplitudes, Fig. 3. The figure suggests that at 20 kHz testing
1
280 260 240 220 200 0.01
0.1
1
10
100
Damping ratio ξ Fig. 3. Damping ratio n versus fatigue stress amplitude during testing at 20 kHz. Mass proportional damping a = 0.01.
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sp.3
Fig. 4. Nominal stress–number of cycles (S–N) plot for AISI H11 tool steel. In sp.3 fatigue crack initiated in the presence of air.
crack development took place in the presence of air, contrary to vacuum for other specimens. Therefore, in the following discussion sp.3 is ignored. The width and length of the stringers vary from 19 to 79 lm and from 79 to 600 lm, respectively, Table 2. The influence of defect width to length ratio W/L on fatigue life is later examined in Fig. 9. The test results show that the AISI H11 steel does not have a fatigue limit at 107 load cycles as the failures continue to occur at much longer lives. The fatigue strength of 260 ± 34 MPa at 109 load cycles was estimated using staircase method described in [9]. It was assumed that the stress in point R, Fig. 1, represents the nominal stress. Then the geometrical location of each fatigue initiating defect was identified and using the FEM model the corresponding local stress has been calculated as a fraction of the nominal. The S–N plot in Fig. 4 is then transformed into the one as in Fig. 5 in which the local stresses in the fatigue initiating region are accounted for. In this particular case the difference between nominal and local stress is not significant for most specimens, but for some it constitutes to over 30 MPa in stress amplitude, Fig. 5. One of the aims of this paper is to raise the awareness that fatigue failures might initiate in locations where stresses are much lower than nominal if sufficiently large defects are present. Therefore, for research purposes where accurate stress estimation at fatigue initiating defects is vital, the local and not nominal stress concept should be applied. It should be noted that residual stresses in the specimen will influence the stress state and location of fatigue crack initiation. Following the tempering treatment of the specimen one does not expect any residual stresses of importance. However, final preparation of the waist surface by grinding will introduce a shallow layer of compressive stresses of the order – 600 MPa. This would contribute to prevent surface initiation.
Fig. 5. The S–N plot for AISI H11 tool steel with consideration of the local stresses at the fatigue initiating defects.
The difference in stress amplitude achievable at 20 kHz with 0.1 and 0.01 damping is 40 MPa, Fig. 3, representing 14% lower stress. When the nominal stresses are adjusted with respect to local stresses and damping (local stress amplitude is lowered by 14% due to 0.1 damping) then Fig. 4 is transformed into Fig. 6. As shown, when testing at ultrasonic frequencies the material damping reduces stresses in the specimens. This effect is material dependent and consistent for all the tested specimens. Therefore, it would not introduce any relative difference between the specimens. Another important parameter that influences fatigue life to failure is the size of the fatigue initiating defect. From the fracture surfaces it could be observed that a fatigue crack tends to initiate in the widest section of an inclusion stringer and then grow as a circular crack, Fig. 7. For an elliptical crack the stress intensity has a peak at the ends of the short axis [12], which is W in Fig. 8. From experimental observation it seems that the crack during early growth propagates in the outward direction from the short axis W, thus in the direction of highest stress intensities. If the local stress at the initiating defect is then scaled in relation to the respective defect width, W, a trend of an increasing fatigue life with the decreasing value of Drlocal(damped) W is observed, Fig. 9. Moreover, the length of the fatigue initiating defect has also a detrimental effect on the resulting fatigue life. This is observed from the fact that the data points positioned below the trend line in Fig. 9 have generally lower width to length ratio than the ones above the trend line. This behaviour supports the assumption that with respect to fatigue the inclusions could be viewed as pre-existing cracks of the same dimensions. By considering the fatigue initiating defects as circular cracks with the diameter equal to the defect width, W, the stress state at the tip of such cracks q could ffiffiffiffiffiffi then be described by the stress intensity factor DK I ¼ p2 Dr p2W , where Dr is the local stress range.
Table 2 Fatigue initiating defects in failed specimens. Specimen
Defect width, W (lm)
Defect length, L (lm)
Aspect ratio, W/L
Nominal stress amplitude (MPa)
Fatigue life (load cycles)
1 3 6 11 12 14 15 16 17 18 19 21
69 19 30 41 79 28 30 55 41 21 23 49
600 79 293 130 128 147 210 142 136 148 196 93
0.12 0.24 0.10 0.32 0.62 0.19 0.14 0.39 0.30 0.14 0.12 0.53
250 250 260 290 280 280 270 280 270 260 250 250
5.97E+07 1.52E+08 1.54E+08 2.75E+08 9.44E+07 1.38E+08 1.02E+09 7.58E+07 3.79E+08 3.62E+08 4.20E+08 3.93E+08
Fig. 6. The S–N plot with consideration of local stresses and damping.
V. Kazymyrovych et al. / International Journal of Fatigue 32 (2010) 1669–1674
Fig. 7. Fatigue crack initiation from a slag inclusion stringer.
Inclusion stringer
Fatigue crack front
W
L Fig. 8. Schematic illustration of a VHCF crack initiating from an inclusion stringer.
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This expression is similar to Drlocal(damped) W and, therefore, the same trend as in Fig. 9 is sustained, Fig. 10. The stress intensity can be viewed as a factor describing a very local highly stressed region that experiences large enough stresses to accumulate fatigue damage, which eventually will lead to crack propagation. Depending on the stress intensity for the life controlling fatigue initiation stage [7,8], higher or slower rates of the damage accumulation are expected, resulting in shorter or longer fatigue lives to failure, respectively, Fig. 10. The same trend of increasing fatigue life with lower stress intensities for fatigue initiating defects was observed by Hui in [13], however, only fatigue lives up to 107 load cycles were evaluated. The long crack threshold stress intensity DKth for steels is comp monly known to vary between 3 and 4 MPa m [14]. Consequently, the results in Fig. 10 confirm the idea of below threshold short crack growth for very long lives. DKI is a parameter that considers both local stresses and defect sizes. Fig. 10 indicates that a high stress intensity during fatigue initiation results in shorter VHCF live. From Figs. 9 and 10 it is seen that the VHCF life is governed by stress at the fatigue initiating defect and the size of the defect (its width in case of stringer type defects). The modified stress parameter Drlocal(damped) enhances the accuracy of the results even though the trends in Figs. 9 and 10 could be sustained if nominal stresses were used. On the other hand, the width, W, of the fatigue initiating defect varies with a factor 4 from 19 to 79 lm, meaning that it would play a major role as to the respective fatigue life. It lies beyond the scope of this paper to investigate the mechanisms of fatigue crack development. However, it should be mentioned that most of the VHCF life is consumed during initial crack growth. In this respect the local stress in combination with the defect geometry are two main contributing factors to the early crack formation. 4. Conclusions The following main conclusions have been drawn from the results of VHCF experiments and from their finite element modeling:
sp.3
Fig. 9. The effect of applied stress and defect size on fatigue life. The values at data points represent the defects’ width to length ratio. In sp.3 fatigue crack initiated in the presence of air.
AISI H11 tool steel does not have a fatigue limit at 107 load cycle. Instead, the fatigue failures have occurred at as high as 109 load cycles. During ultrasonic fatigue testing the difference of 10% or more between the nominal test stress and the actual local stress at the fatigue initiating defect could be expected, due to the specimen’s ‘‘hour-glass” geometry. Consideration of damping properties of investigated material reduces the actual stress amplitude during VHCF testing by as much as 20%. The VHCF life is inversely proportional to the dimensions of the fatigue initiating inclusion stringer, while being most sensitive to its width. References
sp.3
Fig. 10. The relationship between specimens’ fatigue life and the stress intensity at the tip of a circular crack that represent the fatigue initiating defect. In sp.3 fatigue crack initiated in the presence of air.
[1] Bathias C, Paris PC. Gigacycle fatigue in mechanical practice. NY (USA): Marcel Dekker; 2005. [2] Marines I, Bin X, Bathias C. An understanding of very high cycle fatigue of metals. Int J Fatigue 2003;9:1101–7. [3] Bathias C, Drouillac L, Le Francois P. How and why the fatigue S–N curve does not approach a horizontal asymptote. Int J Fatigue 2001:143–51. [4] Wang QY et al. High-cycle fatigue crack initiation and propagation behaviour of high-strength spring steel wires. Fatigue Fract Eng Mater Struct 1999;8:673–7. [5] Sohar CR et al. Gigacycle fatigue behavior of a high chromium alloyed cold work tool steel. Int J Fatigue 2008;7:1137–49. [6] Sakai T, Sato Y, Oguma N. Characteristic S–N properties of high-carbon– chromium-bearing steel under loading in long-life fatigue. Fatigue Fract Eng Mater Struct 2002;8–9:765–73.
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[7] Paris PC. The relationship of effective stress intensity, elastic modulus and Burgers-vector on fatigue crack growth. In: Proceedings of the international conference on very high cycle fatigue III, Kyoto, Japan; September 2004. [8] Paris PC. Fatigue crack growth from small to large cracks in gigacycle fatigue with fish-eye failures. In: 9th International fatigue congress, Atlanta, USA; May 2006. [9] Yung-Li L. Fatigue testing and analysis. Theory and practice. Oxford: Elsevier; 2005. [10] Cook R, Malkus D, Plesha M. Concepts and applications of finite element analysis. 3rd ed. New York: John Wiley and Sons Inc.; 1989.
[11] Abaqus 6.5 documentation. Material damping. In: Analysis user’s manual volume III: materials. USA: Abaqus Inc.; 2004 [section 12.1.1]. [12] Makino T. The effect of inclusion geometry according to forging ratio and metal flow direction on very high-cycle fatigue properties of steel bars. Int J Fatigue 2008;8:1409–18. [13] Hui W. High cycle fatigue fracture behaviour of ultrahigh strength steels. J Mater Sci Technol 2008;5:787. [14] Randelius M. Influence of microstructure on fatigue and ductility properties of tool steels. Stockholm: Royal Institute of Technology; 2008.
Contents lists available at ScienceDirect
International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue
Local stresses and material damping in very high cycle fatigue V. Kazymyrovych *, J. Bergström, F. Thuvander Department of Materials Engineering, Karlstad University, SE-651 88, Sweden
a r t i c l e
i n f o
Article history: Received 3 November 2009 Received in revised form 2 March 2010 Accepted 22 March 2010 Available online 3 April 2010 Keywords: Ultrasonic fatigue testing Stress gradient Damping
a b s t r a c t The increased interest in the very high cycle fatigue (VHCF) properties of materials has resulted in a broadened use of ultrasonic fatigue testing. The accuracy of the obtained fatigue data depends to a large extent on the estimate of the actual stresses acting in the ultrasonic test specimen. Finite element modeling (FEM) is used in this paper to study local stresses at the fatigue initiating defects during ultrasonic fatigue testing of an AISI H11 tool steel. Material damping was included in the calculations, which was found to reduce the magnitude of the stresses during 20 kHz ultrasonic fatigue testing. By combining local stresses with the material damping effect, the actual stresses operating in the region of the fatigue initiating defect were established. In the very long life regime microstructural defects, such as slag inclusions, often serve as fatigue crack origins. Locally amplified stresses enable fatigue crack formation and growth. Therefore, in this work the relationship between actual stresses, the defect geometry and fatigue life is examined. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Ultrasonic fatigue testing has received considerable attention in the last two decades. This is mainly due to the increasing use of engineering materials in applications with service lives reaching up to 1010 load cycles. To match different fatigue conditions met in practice, a variety of ultrasonic equipment modifications has evolved. These include fatigue testing in tension, torsion, bending or fretting, with variable or constant amplitude loading, at different temperatures and in a variety of environments [1]. Tool steels are widely used in fatigue applications. Their high strength and good toughness contribute to excellent fatigue properties. It has been shown [2–6] that the most frequent VHCF initiation sites in high performance steels are at slag inclusions. Consequently, the fatigue strength generally increases with improved cleanness of the steel in question. Other fatigue initiation sites are carbides or their segregations. VHCF failures of steels are characterized by the formation of a structure called ‘‘fish-eye”, which could be described as a circular pattern that develops on the fracture surfaces during internally initiated fatigue failures [1,3,5,6]. One of the most frequently used VHCF testing techniques employs uni-axial tensile loading of hour-glass shaped specimens. The nominal stresses during testing are established with respect to the smallest cross-section where stresses are the highest. However, within this cross-section due to specimen geometry there is a stress gradient from the surface towards the centre. Moreover, the majority of fatigue failures take place at some distance from the
plane of the smallest cross-section, which effectively reduces the actual fatigue stresses and, therefore, the fatigue strength is overestimated. How much local stresses acting at the fatigue initiating defects differ from nominal stresses is discussed in this paper. Materials’ damping properties also influence the magnitude of actual stresses during ultrasonic fatigue testing. To the author’s knowledge, this subject has not been discussed in the literature where ultrasonic stresses are calculated, like in [1]. Therefore, in this work an attempt is made to illustrate the dependence of fatigue stresses on damping. It is considered in terms of mass proportional Rayleigh damping factor a, stiffness proportional Rayleigh damping factor b, as well as testing frequency. When local stresses, defined by the location of the fatigue initiating defect, are corrected by the effect of material damping, the estimate of an actual fatigue stress is obtained. This stress in combination with the defect size and shape would define the stress state at the interface between the microstructural defect and the matrix during fatigue crack initiation stage. Taking into account that in VHCF most of the fatigue life is consumed during initial crack growth [7,8], there should be a traceable relationship between actual stress and defect geometry on the one side and VHCF life on the other. In the paper this relationship is presented through the stress intensity parameter.
2. Experimental 2.1. VHCF testing
* Corresponding author. E-mail address: [email protected] (V. Kazymyrovych). 0142-1123/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2010.03.007
Very high cycle fatigue testing of an AISI H11 high strength steel, Table 1, was performed at 20 kHz frequency using ultrasonic
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V. Kazymyrovych et al. / International Journal of Fatigue 32 (2010) 1669–1674
piezo-electric equipment. The main feature of the equipment is a resonance system, which receives high frequency mechanical vibrations from a set of piezo-electrics. These vibrations are amplified by a special horn and transmitted to a specimen where the magnitude of the achievable fatigue stress is defined by the electrical voltage applied to the piezo-electrics. For full description of the equipment readers are referred to [1]. The hour-glass shaped specimens with the diameter of the smallest cross-section of 6 mm and hour-glass radius of 31 mm were extracted from the billet in the transverse direction, which represents the worst condition for the fatigue properties. The choice of specimens with the smallest diameter of 6 mm and not 3 mm, as is often found in literature, was made in order to maximize the material volume subjected to approximately the same stress levels. This should enhance finding the largest defects in the material. Austenitizing, quenching and tempering was made to achieve a martensitic microstructure and a hardness of 480 HV corresponding to a tensile strength of 1430 MPa and a 0.2% proof stress of 1260 MPa. Elastic modulus and density were 210 GPa and 7800 kg/m3, respectively. Testing was conducted at minimum to maximum cyclic load ratio R = 0.1. Air cooling was applied in order to maintain the temperature on the specimens’ surface close to room temperature. The staircase test method [9] was used to estimate the material’s fatigue strength at 109 load cycles.
2.2. FEM modeling of VHCF testing An axi-symmetric ABAQUS finite element model was created to simulate ultrasonic fatigue testing. The application of fatigue stresses was accomplished in two consecutive steps: (1) pre-stressing of the specimen in the tensile direction in order to achieve the load ratio R = 0.1; (2) introduction of sinusoidal pulsating stresses by means of assigning micron-scale displacement at the specimen’s ends. The displacement was applied as a sinusoidal function with the period representing test frequency, and with the amplitude of displacement evaluated from measurements on the free end of the ultrasonic amplification horn. The effect of material damping on the magnitude of stresses during fatigue testing was evaluated by incorporating Rayleigh damping into the FEM model. Four-node bilinear axi-symmetric quadrilateral elements with reduced integration were used in the model. 2.2.1. Local stresses at the fatigue initiation sites The nominal stresses were calculated analytically using Eq. (1) as described in [1]. Here specimen geometry, elastic modulus, density and applied displacement at the specimen ends were used as input for the calculation of resonance stresses. The stress obtained for the smallest specimen’s cross-section was then used as nominal during fatigue testing.
rðxÞ ¼ Ed A0 u Table 1 Chemical composition of AISI H11 steel, wt.%. C
Si
Mn
Cr
Mo
V
0.38
1.10
0.40
5.00
1.30
0.40
½b coshðbxÞ coshðaxÞ a sinhðbxÞ sinhðaxÞ 2
cosh ðaxÞ
ð1Þ
where Ed is the dynamic elastic modulus, A0 the displacement amplitude at the end of the specimen, u, b, a the parameters dependent on the specimen geometry and material properties, and x is the distance along the ‘‘hour-glass” section of the specimen.
Fig. 1. An example of FEM calculated stress state in an ‘‘hour-glass” specimen with the 6 mm diameter of the smallest cross-section. Effect of damping is neglected: (a) schematic representation of a specimen’s quarter; (b) stress gradient in radial direction; (c) stress gradient in longitudinal direction obtained by FEM and analytically.
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Due to the ‘‘hour-glass” geometry there is a stress gradient in the middle section of the specimen, with the maximum stress at the surface of the specimen in its smallest cross-section, Fig. 1. The FEM results show that at the waist, in radial direction the difference in stress between the centre of the specimen, point O, and the most stressed region at the surface, point R, is 10%, Fig. 1b, while in the longitudinal direction the stress in point O, Fig. 1c, is about four times as high as in point L. Thus, the actual stresses acting at the fatigue initiating sites differ from the nominal stress levels depending on the location of the initiation sites within the specimen. The severity of the stress gradient would be enhanced if specimens with smaller ‘‘hour-glass” radius were used. In order to obtain a more realistic S–N representation of the test results, it is necessary to consider local stresses at the fatigue initiating defects instead of nominal. It should be clarified that under ‘‘local stress” in this paper is meant the variation of stress within the specimen caused exclusively by specimen geometry. This is not to be confused with the resulting stress at fatigue initiating defects due to their stress raising capacity. The probability of a fatigue crack initiating in the most stressed region is high, but the crack could also evolve in less stressed regions of a specimen if a sufficiently large defect with enough potential for crack initiation is present. A possible way to estimate this potential is proposed later in this paper. 2.2.2. Effect of damping The equation of motion for a finite element structure with consideration of damping could be presented as in [10]:
€ þ ½CU_ þ ½KU ¼ F ½MU
ð2Þ
€ the acceleration, [C] the damping where [M] is the mass matrix, U matrix, U_ the speed, [K] the stiffness matrix, U the spatial displacement, and F is the external force (corresponds to the applied resonance oscillations). Material damping can be described by the Rayleigh damping parameter. The Rayleigh damping is defined by specifying two damping factors: a – for mass proportional damping and b – for stiffness proportional damping [11]. For a given angular velocity x the damping ratio n can be expressed in terms of the damping factors a and b as:
n¼
a 2x
þ
bx 2
ð3Þ
The equation implies that the mass proportional Rayleigh damping factor a is important at low frequencies and the stiffness proportional factor b damps the higher frequencies. The b factor introduces damping proportional to the strain rate, which can be considered as damping associated with the mechanical properties [11].
Normalised stress amplitude
V. Kazymyrovych et al. / International Journal of Fatigue 32 (2010) 1669–1674
β =1E-06
0.9 β =1E-05
0.8 β =1E-04
0.7 0.6 0.5
0
5
10
15
20
25
30
35
40
Frequency, kHz Fig. 2. The effect of test frequency and stiffness proportional damping factor b on the fatigue stress amplitude. Mass proportional damping factor a = 0.01.
frequency a difference of about 20% in fatigue stress amplitude could be observed depending on the damping characteristics of the investigated material. This is of significance as 20% difference in stress amplitude is expected to greatly influence the eventual fatigue life. For the tested steel grade the stiffness proportional damping factor b was estimated by using inverse modeling. This was accomplished by measuring the fatigue strain amplitude in the specimen’s smallest cross-section using micro-strain-gages. The calculated value of 253 MPa was then matched with the results of FEM calculations at 20 kHz, Fig. 3, and the damping ratio n was estimated to be 0.1, which corresponds to the factor b in the order of 105. 3.1. Very high cycle fatigue testing The data obtained from the ultrasonic fatigue testing is presented in Fig. 4, which describes the relationship between the nominal applied stress and fatigue life for each specimen. There seems to be a large scatter in fatigue life for specimens tested at the same nominal stress. This could be attributed to the difference in size of the fatigue initiating defects. As shown later in the paper the size of the defect has a major influence on the VHCF life. In addition, the local stresses that are determined by the defect location in the specimen are not accounted for in Fig. 4. Furthermore, as in all testing, there is inherent variation in test conditions and specimen properties. The fractographic analysis of the fracture surfaces has shown that all the fatigue failures initiated from slag inclusions of a stringer type. Moreover, all fatigue initiations were internal apart from in one specimen (sp.3) where one end of the inclusion stringer reached the specimen’s surface. For this specimen early fatigue
300
Stress amplitude, MPa
3. Results and discussion Fig. 2 presents results obtained by the FEM calculations. The graph shows a variation of the stress amplitude in the middle section of a fatigue specimen as a function of b and test frequency. For frequencies in the range of 20 kHz a change in mass proportional damping factor a at different values of b produces negligible changes in stress amplitude. However, the effect of stiffness proportional damping factor b on the stress amplitude increases with increasing test frequency, Fig. 2. The stress amplitude increase with higher test frequency is most pronounced for low damping (low b). Damping ratio n has been calculated for different values of b at 20 kHz frequency and then plotted against the corresponding stress amplitudes, Fig. 3. The figure suggests that at 20 kHz testing
1
280 260 240 220 200 0.01
0.1
1
10
100
Damping ratio ξ Fig. 3. Damping ratio n versus fatigue stress amplitude during testing at 20 kHz. Mass proportional damping a = 0.01.
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sp.3
Fig. 4. Nominal stress–number of cycles (S–N) plot for AISI H11 tool steel. In sp.3 fatigue crack initiated in the presence of air.
crack development took place in the presence of air, contrary to vacuum for other specimens. Therefore, in the following discussion sp.3 is ignored. The width and length of the stringers vary from 19 to 79 lm and from 79 to 600 lm, respectively, Table 2. The influence of defect width to length ratio W/L on fatigue life is later examined in Fig. 9. The test results show that the AISI H11 steel does not have a fatigue limit at 107 load cycles as the failures continue to occur at much longer lives. The fatigue strength of 260 ± 34 MPa at 109 load cycles was estimated using staircase method described in [9]. It was assumed that the stress in point R, Fig. 1, represents the nominal stress. Then the geometrical location of each fatigue initiating defect was identified and using the FEM model the corresponding local stress has been calculated as a fraction of the nominal. The S–N plot in Fig. 4 is then transformed into the one as in Fig. 5 in which the local stresses in the fatigue initiating region are accounted for. In this particular case the difference between nominal and local stress is not significant for most specimens, but for some it constitutes to over 30 MPa in stress amplitude, Fig. 5. One of the aims of this paper is to raise the awareness that fatigue failures might initiate in locations where stresses are much lower than nominal if sufficiently large defects are present. Therefore, for research purposes where accurate stress estimation at fatigue initiating defects is vital, the local and not nominal stress concept should be applied. It should be noted that residual stresses in the specimen will influence the stress state and location of fatigue crack initiation. Following the tempering treatment of the specimen one does not expect any residual stresses of importance. However, final preparation of the waist surface by grinding will introduce a shallow layer of compressive stresses of the order – 600 MPa. This would contribute to prevent surface initiation.
Fig. 5. The S–N plot for AISI H11 tool steel with consideration of the local stresses at the fatigue initiating defects.
The difference in stress amplitude achievable at 20 kHz with 0.1 and 0.01 damping is 40 MPa, Fig. 3, representing 14% lower stress. When the nominal stresses are adjusted with respect to local stresses and damping (local stress amplitude is lowered by 14% due to 0.1 damping) then Fig. 4 is transformed into Fig. 6. As shown, when testing at ultrasonic frequencies the material damping reduces stresses in the specimens. This effect is material dependent and consistent for all the tested specimens. Therefore, it would not introduce any relative difference between the specimens. Another important parameter that influences fatigue life to failure is the size of the fatigue initiating defect. From the fracture surfaces it could be observed that a fatigue crack tends to initiate in the widest section of an inclusion stringer and then grow as a circular crack, Fig. 7. For an elliptical crack the stress intensity has a peak at the ends of the short axis [12], which is W in Fig. 8. From experimental observation it seems that the crack during early growth propagates in the outward direction from the short axis W, thus in the direction of highest stress intensities. If the local stress at the initiating defect is then scaled in relation to the respective defect width, W, a trend of an increasing fatigue life with the decreasing value of Drlocal(damped) W is observed, Fig. 9. Moreover, the length of the fatigue initiating defect has also a detrimental effect on the resulting fatigue life. This is observed from the fact that the data points positioned below the trend line in Fig. 9 have generally lower width to length ratio than the ones above the trend line. This behaviour supports the assumption that with respect to fatigue the inclusions could be viewed as pre-existing cracks of the same dimensions. By considering the fatigue initiating defects as circular cracks with the diameter equal to the defect width, W, the stress state at the tip of such cracks q could ffiffiffiffiffiffi then be described by the stress intensity factor DK I ¼ p2 Dr p2W , where Dr is the local stress range.
Table 2 Fatigue initiating defects in failed specimens. Specimen
Defect width, W (lm)
Defect length, L (lm)
Aspect ratio, W/L
Nominal stress amplitude (MPa)
Fatigue life (load cycles)
1 3 6 11 12 14 15 16 17 18 19 21
69 19 30 41 79 28 30 55 41 21 23 49
600 79 293 130 128 147 210 142 136 148 196 93
0.12 0.24 0.10 0.32 0.62 0.19 0.14 0.39 0.30 0.14 0.12 0.53
250 250 260 290 280 280 270 280 270 260 250 250
5.97E+07 1.52E+08 1.54E+08 2.75E+08 9.44E+07 1.38E+08 1.02E+09 7.58E+07 3.79E+08 3.62E+08 4.20E+08 3.93E+08
Fig. 6. The S–N plot with consideration of local stresses and damping.
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Fig. 7. Fatigue crack initiation from a slag inclusion stringer.
Inclusion stringer
Fatigue crack front
W
L Fig. 8. Schematic illustration of a VHCF crack initiating from an inclusion stringer.
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This expression is similar to Drlocal(damped) W and, therefore, the same trend as in Fig. 9 is sustained, Fig. 10. The stress intensity can be viewed as a factor describing a very local highly stressed region that experiences large enough stresses to accumulate fatigue damage, which eventually will lead to crack propagation. Depending on the stress intensity for the life controlling fatigue initiation stage [7,8], higher or slower rates of the damage accumulation are expected, resulting in shorter or longer fatigue lives to failure, respectively, Fig. 10. The same trend of increasing fatigue life with lower stress intensities for fatigue initiating defects was observed by Hui in [13], however, only fatigue lives up to 107 load cycles were evaluated. The long crack threshold stress intensity DKth for steels is comp monly known to vary between 3 and 4 MPa m [14]. Consequently, the results in Fig. 10 confirm the idea of below threshold short crack growth for very long lives. DKI is a parameter that considers both local stresses and defect sizes. Fig. 10 indicates that a high stress intensity during fatigue initiation results in shorter VHCF live. From Figs. 9 and 10 it is seen that the VHCF life is governed by stress at the fatigue initiating defect and the size of the defect (its width in case of stringer type defects). The modified stress parameter Drlocal(damped) enhances the accuracy of the results even though the trends in Figs. 9 and 10 could be sustained if nominal stresses were used. On the other hand, the width, W, of the fatigue initiating defect varies with a factor 4 from 19 to 79 lm, meaning that it would play a major role as to the respective fatigue life. It lies beyond the scope of this paper to investigate the mechanisms of fatigue crack development. However, it should be mentioned that most of the VHCF life is consumed during initial crack growth. In this respect the local stress in combination with the defect geometry are two main contributing factors to the early crack formation. 4. Conclusions The following main conclusions have been drawn from the results of VHCF experiments and from their finite element modeling:
sp.3
Fig. 9. The effect of applied stress and defect size on fatigue life. The values at data points represent the defects’ width to length ratio. In sp.3 fatigue crack initiated in the presence of air.
AISI H11 tool steel does not have a fatigue limit at 107 load cycle. Instead, the fatigue failures have occurred at as high as 109 load cycles. During ultrasonic fatigue testing the difference of 10% or more between the nominal test stress and the actual local stress at the fatigue initiating defect could be expected, due to the specimen’s ‘‘hour-glass” geometry. Consideration of damping properties of investigated material reduces the actual stress amplitude during VHCF testing by as much as 20%. The VHCF life is inversely proportional to the dimensions of the fatigue initiating inclusion stringer, while being most sensitive to its width. References
sp.3
Fig. 10. The relationship between specimens’ fatigue life and the stress intensity at the tip of a circular crack that represent the fatigue initiating defect. In sp.3 fatigue crack initiated in the presence of air.
[1] Bathias C, Paris PC. Gigacycle fatigue in mechanical practice. NY (USA): Marcel Dekker; 2005. [2] Marines I, Bin X, Bathias C. An understanding of very high cycle fatigue of metals. Int J Fatigue 2003;9:1101–7. [3] Bathias C, Drouillac L, Le Francois P. How and why the fatigue S–N curve does not approach a horizontal asymptote. Int J Fatigue 2001:143–51. [4] Wang QY et al. High-cycle fatigue crack initiation and propagation behaviour of high-strength spring steel wires. Fatigue Fract Eng Mater Struct 1999;8:673–7. [5] Sohar CR et al. Gigacycle fatigue behavior of a high chromium alloyed cold work tool steel. Int J Fatigue 2008;7:1137–49. [6] Sakai T, Sato Y, Oguma N. Characteristic S–N properties of high-carbon– chromium-bearing steel under loading in long-life fatigue. Fatigue Fract Eng Mater Struct 2002;8–9:765–73.
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[7] Paris PC. The relationship of effective stress intensity, elastic modulus and Burgers-vector on fatigue crack growth. In: Proceedings of the international conference on very high cycle fatigue III, Kyoto, Japan; September 2004. [8] Paris PC. Fatigue crack growth from small to large cracks in gigacycle fatigue with fish-eye failures. In: 9th International fatigue congress, Atlanta, USA; May 2006. [9] Yung-Li L. Fatigue testing and analysis. Theory and practice. Oxford: Elsevier; 2005. [10] Cook R, Malkus D, Plesha M. Concepts and applications of finite element analysis. 3rd ed. New York: John Wiley and Sons Inc.; 1989.
[11] Abaqus 6.5 documentation. Material damping. In: Analysis user’s manual volume III: materials. USA: Abaqus Inc.; 2004 [section 12.1.1]. [12] Makino T. The effect of inclusion geometry according to forging ratio and metal flow direction on very high-cycle fatigue properties of steel bars. Int J Fatigue 2008;8:1409–18. [13] Hui W. High cycle fatigue fracture behaviour of ultrahigh strength steels. J Mater Sci Technol 2008;5:787. [14] Randelius M. Influence of microstructure on fatigue and ductility properties of tool steels. Stockholm: Royal Institute of Technology; 2008.