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The effect of single overloads in tension and compression on the fatigue crack propagation behaviour of short cracks
International Journal of Fatigue xxx (2016) xxx–xxx
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The effect of single overloads in tension and compression on the fatigue crack propagation behaviour of short cracks Xiang Zhou a,b,⇑, Hans-Peter Gaenser a, Reinhard Pippan b,c a
Materials Center Leoben Forschung GmbH, Roseggerstr. 12, 8700 Leoben, Austria Austrian Academy of Science, Erich Schmid Institute of Materials Science, Jahnstr. 12, 8700 Leoben, Austria c Department of Material Physics, Montanuniversität Leoben, Jahnstr. 12, 8700 Leoben, Austria b
a r t i c l e
i n f o
Article history: Received 29 September 2015 Received in revised form 25 January 2016 Accepted 1 February 2016 Available online xxxx Keywords: Overload Crack closure Residual stress Threshold of stress intensity Effective threshold
a b s t r a c t The crack propagation behaviour in cast quenched and tempered steel after one overload cycle in tension as well as in compression on short cracks is investigated in deep notched specimens. The overload cycle exhibits a significant influence on the fatigue life endurance, due to the formation of an overload plastic zones in front of the crack tip. The crack propagation after overload cycles is investigated by inspection of the fatigue threshold R-curve and fatigue crack propagation rate. Tension overload increased the long crack threshold and reduced the R-curve effect, whereas overloads in compression reduced the crack growth resistance and shifted the threshold value to larger crack extension. A simple FE simulation was also performed to investigate the variation in the contribution of plasticity induced crack closure during crack propagation after the overload. Macroscopic mechanistic and dislocation models are introduced to explain the results obtained. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Fatigue crack propagation under variable load amplitude has been extensively investigated for a long time. A major breakthrough was made in the first half of last century by Miner [1] and Palmgren [2], who proposed a linear damage hypothesis to accumulate the damage under variable load amplitudes. However, research in the last few decades revealed the existence of a strong load interaction effect. A change in load amplitude leads to an alteration of the material fatigue properties, resulting in a load sequence dependency in the lifetime [3–7]. Such mechanisms as: 1. residual stress: overload introduces a residual stress field which interacts with the remote stress and affects the subsequent crack propagation [8], 2. crack closure: overload in tension leaves a plastic deformation which increases the crack closure stress intensity [9,10], 3. geometric variation: overload in tension can induce some geometric variation like plastic blunting [11,12], crack deflection or even change in crack front profile [13]. All of these variations lead to a reduction in the local driving force,
4. stain hardening: strain hardening can also take place within the tensile overload induced plastic zone [14,15], can be followed by overload. In ductile material, a change in the contribution of plasticity-induced crack closure (PICC) is a direct consequence of overload. The crack propagation behaviour thereafter can be very complex especially near the threshold. For example, after a single tensile overload the increased PICC can reduce the fatigue crack growth (FCG) rates back to near the threshold regime and thereby promote oxide-induced and roughnessinduced crack closure [16]. For that reason, prediction methods for crack growth are often of empirical nature. The prediction is often based on experiments with certain load time history patterns like single overloads during cyclic loading with a constant load amplitude, block loads or variable amplitudes which correspond to the real load time history in practice. In order to find out the physical reasons of the load variation effect, the current study focuses on one of the simplest cases; the effect of one cycle of large load amplitude (overload) in tension as well as in compression on short open cracks. The material fatigue properties are determined by the R-curve behaviour of the fatigue threshold of stress intensity and FCG rate by constant load amplitude.
⇑ Corresponding author at: Austrian Academy of Science, Erich Schmid Institute of Materials Science, Jahnstr. 12, 8700 Leoben, Austria. http://dx.doi.org/10.1016/j.ijfatigue.2016.02.001 0142-1123/Ó 2016 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Zhou X et al. The effect of single overloads in tension and compression on the fatigue crack propagation behaviour of short cracks. Int J Fatigue (2016), http://dx.doi.org/10.1016/j.ijfatigue.2016.02.001
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2. Material and experimental method The experiments were performed on single-edge-notched-bend (SENB) specimens (Fig. 1) machined from quenched and tempered steel with a yield stress of 630 MPa and an ultimate tensile strength of 760 MPa. The fatigue properties of this material were already well investigated [17]. The R-curve of the fatigue threshold of stress intensity range and the FCG rates are summarised in Figs. 2 and 3. Pre-cracks are introduced at the root of a deep notch (about 4 mm) using compressive-compressive loading. The razor blade polished notch root radius is between 10 and 20 lm. 10,000 load pffiffiffiffiffi cycles with DK ¼ 18 MPa m were applied to the specimen at a load ratio of R = 10. The nucleated pre-crack exhibits about 30 lm in length. Cracks with such length are regarded as physical short cracks [18]. The pre-crack process was performed on a servo hydraulic testing machine at frequency of 10 Hz. One cycle of large load amplitude was applied on the pre-crack. The load was compffiffiffiffiffi pletely released (to K = 0 MPa m) thereafter. The determination of R-curve behaviour of DK th was performed on a resonance testing machine with a testing frequency of 108 Hz. The crack length is measured by potential drop method. The detection limit of our measure device is about 10 lm. In order to obtain the R-curve, a step-wise increasing constant load method was used [19,20]. All tests were started at the intrinsic threshold values pffiffiffiffiffi (DK ¼ 3 MPa m) of the investigated material [17]. For a DK smaller than the intrinsic threshold value, no crack propagation is possible. If the crack started to grow, the load amplitude was kept constant. If the FCG rate decreased to a rate lower than 108 mm/ cyc, the DK was increased to the next step. At load ratio R = 0.1 and pffiffiffiffiffi R = 1, a load increment of 2 MPa m was applied. At load ratio pffiffiffiffiffi R = 0.7, the applied increment was reduced to 0.2 MPa m due to the high mean stress. After the DK exceeded certain value, crack kept extending in a proper way which is well known as stable crack propagation. The short crack behaviour inverted into long crack behaviour. This step is referred to as long crack threshold value (DK th;lc ). For each loading step where the crack propagation got arrested, the total crack extension length was plotted with the current DK in one diagram. The fit curve of these data is the fatigue threshold R-curve. The investigation of stable crack extension is done by recording the fatigue crack growth (FCG) rate. More details about the experimental procedure can be found in [20]. A schematic representation of the current load time history including the compressive-compressive pre-crack process, single overload and step-wise increasing constant load method at a load ratio R = 1 is presented in Fig. 4. Four different overloads in tension pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi Kov = 10 MPa m, 20 MPa m, 30 MPa m, 45 MPa m and two pffiffiffiffiffi pffiffiffiffiffi overloads in compression Kov = 30 MPa m, 40 MPa m were investigated in the current study. The reason for the reduced variety in overloads in compression is that the short open pre-crack was introduced by compressive-compressive loading. The applied pffiffiffiffiffi Kmin during pre-cracking was 20 MPa m. Compressive overloads pffiffiffiffiffi with an amount smaller than 20 MPa m are not regarded as overloads under such initial conditions. Therefore only two comprespffiffiffiffiffi sive overloads, 30 and 40 MPa m, were investigated in this
Fig. 1. Geometry of single-edge-notched-bend (SENB) specimen.
Fig. 2. The R-curves of fatigue threshold of stress intensity range DK th at load ratio R = 1, R = 0.1 and R = 0.7. The R-curves are determined by a fit curve of the measure data.
Fig. 3. The FCG rates of the investigated quenched and tempered steel at load ratio R = 1, R = 0.1 and R = 0.7.
Fig. 4. Schematic presentation of the load time history in the overload experiment pffiffiffiffiffi Kov = 30 MPa m at a load ratio of R = 1.
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study. These overloads still fulfil the small scale yielding requirement. The determination of the R-curve behaviour and FCG rates were performed at load ratio R = 1, R = 0.1 and R = 0.7.
2.1. Finite element analysis A simple elastic–plastic 2D finite element (FE) analysis was implemented in order to estimate the contribution of plasticity induced crack closure (PICC). A half of an SENB was modelled with CPE4 (four-node plane strain) element considering the symmetry conditions. The basic idea is to use the node release technique to simulate the crack extension and investigate thereby the evolution of crack growth resistance. According to the crack propagation condition postulated by Elber [12], load cycles damage the crack tip only if the crack is fully open. It is assumed that the crack is only fully open when the contact force between the crack flanks disappears. As long as the contact force is present, i.e. the crack is closed, the material resistance prevents crack propagation. The obtained R-curve is the transition load level, where the contact force is completely disappeared. The crack surface is assigned by free elements in contrast to the other elements along the y = 0 line which are defined as rigid body with a constraint of the displacement in y direction. A rigid contact is specified between the crack surface and the y = 0 line to realise the contact of both crack surfaces. The loading step is equally divided into 10 sub-steps. After each loading cycle, the first node in front of crack tip is released and thus crack extension is simulated. Since the size of the monotonic plastic zone and the reversed plastic zone is crucial for the degree of finite element mesh refinement [21–23], it is necessary to estimate the size of both plastic zones at first. The estimation is followed by the approximation for plane strain in Mode I proposed by Irwin: Monotonic plastic zone size:
rp ¼
1 3p
K
2
ry
ð1Þ
Reversed plastic zone size:
r p;r ¼
2 1 DK 3p 2ry
3
3. Results 3.1. Experimental results The experimentally obtained R-curve of DK th at load ratio R = 1 is presented in Fig. 6. The cross symbols indicate the load levels tested at which no crack propagation was observed or the crack started to grow but finally was arrested. The circle symbols indicate the load level where stable crack propagation took place. The crack extension at these levels is an assumed length, only used to approximate the exponential nature of R-curve. These DK levels are the upper limit i.e. maximum of the threshold value denoted as DK th;max in the current study. DK th;max increases with increasing overload (Kov). The overload affected R-curve exhibits very short crack extensions Da of about 0.1 mm at the last measurable data (the last cross symbol) compared to the 0.4 mm without overload (Fig. 2). No crack extension was observed before the maximum of threshold value was reached at load ratio R = 0.1 and R = 0.7. i.e., the investigated material does not exhibit a measurable R curve behaviour at these two load ratios. Since the detection limit of the potential drop method is 10 lm, the increase in crack length at each started level was less 10 lm, however, it is possible that the accumulated crack extension is longer than 10 lm which is detectable for our measure device. For that reason, two overloaded specimens were directly tested at high DK levels (closed to DK th;max ). A detectable crack extension was obtained. The resulted R-curves are presented in Fig. 7. The approximation of R-curve is based on the exponential nature of R-curve again. In the FCG rate diagram, the crack growth rate starts at DK th;max and reaches the FCG rate of specimens without overloads in the Paris regime after a certain crack extension Da (Figs. 8–10). The results of the overload in compression experiments are presented in Fig. 11. The R-curve behaviour is approximated by lines based on the measured data. The R-curves at a load ratio of R = 1 show a reduced crack growth resistance at the beginning, returned to the state without the overload after certain crack extension. The FCG rate (Fig. 12) exhibits an abnormal behaviour. After the DK th;max was reached, the crack propagated with a reduced FCG rate, met a minimum at a crack extension of about 2 mm extension in both cases, re-established thereafter the steady state curve in the Paris regime.
ð2Þ
A fundamental requirement for a correct FEM simulation is that the monotonic plastic zone should be within a fine meshed zone. That is the primary condition to ensure that there is no significant mesh influence on the simulated closure results [22,23]. Another requirement is that the size of elements ahead of the crack tip must be at least in the same magnitude of the size of the reversed plastic zone, which ensures a correct implementation of reversible deformation ahead of the crack tip. Simulations were carried out at a load ratio of R = 0 for a comparison with the experiment. Three overloads pffiffiffiffiffi Kov = 10, 20 and 30 MPa m were applied to the pre-crack. The experimentally obtained maximum threshold value at a load ratio pffiffiffiffiffi of R = 0.1 (DK ¼ 6, 9 and 12 MPa m) were used to investigate the development of crack opening levels. Another simulation without any overload was also carried out as reference. The tested DK was pffiffiffiffiffi 6 MPa m. The size of monotonic plastic zone caused by the highest pffiffiffiffiffi overload Kov = 30 MPa m is 241 lm estimated by Eq. (1). and the pffiffiffiffiffi smallest reversed plastic zone at DK = 6 MPa m is 2.5 lm. Therefore, the fine element size was chosen as 2 lm 2 lm. They were defined in an area 150 lm along the crack growth direction (y = 0) and 60 lm in height (y-direction) in order to track the main plastic deformation (Fig. 5).
3.2. Results of finite element analysis The change of the crack opening stress intensity obtained from the finite element analysis are presented in Fig. 13. The last load step with existing contact force is defined as K op . They are symbolised by circles and approximated by a fit curve. In the simulation of pffiffiffiffiffi K ov ¼ 10 MPa m, the maximum of closure stress intensity factor pffiffiffiffiffi of 3 MPa m is first reached at the crack extension of 10 lm. Then the Kop shows a continuous reduction during further crack extenpffiffiffiffiffi sion. After about 100 lm, the Kop decreases to 1.5 MPa m and approaches at larger crack extension the crack closure value obtained by constant amplitude loading (Fig. 13). This continuous reduction reveals that the overload effect decreases with further crack extension and disappears after about 100 lm crack extension. Another phenomenon observed is that after the maximal resistance (Da ¼ 10 lm), the contact force at K op is concentrated at two positions, one directly at the current crack tip and the other at about 10 lm in front of the origin (position of the pre-crack tip or the overload crack tip). In Fig. 14, the position with contact force indicates where premature contact of the crack surfaces takes place. The contact force at the current crack tip is caused by the plastic deformation of the current crack tip and the contact force in front of pre-crack tip is caused by the plastic deformation of
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Fig. 5. Finite element mesh for the estimation of the crack opening stress intensity factor as a function of crack extension.
Fig. 6. The experimentally obtained R-curve behaviour at load ratio R = 1. The cross symbols indicate the load levels, where the crack was arrested or the extension was below the detection limit of the potential drop method. The circle symbols are the DK th;max levels where stable crack propagation took place. The crack extension at DK th;max is an assumed length which is only used to approximate the exponential R-curve behaviour.
the overload cycle. However, the contact force in front of the precrack tip decreases with further crack extension. In the simulation, only the contribution of the plasticity induced crack closure was pffiffiffiffiffi estimated. Taking the intrinsic resistance 3 MPa m into account, the fit curves are located between the R-curves at load ratios R = 0.1 and R = 1 (Figs. 15–17). For the comparison with the experimental results one should always taken into account that the FEM simulation only considers the effect of PICC. 4. Discussion 4.1. Overload in tension Many studies indicate that a sufficient large tensile overload within constant amplitude fatigue loading leads to a delay retardation phenomenon in ductile material [24–26]. There are different explanations for overload effect as mentioned in the introduction, we will follow here the mostly used explanation by the change of
Fig. 7. The experimentally obtained R-curve behaviour at load ratio R = 0.1. The cross symbols indicate the crack arrested levels. The circle symbols are the DK th;max levels where stable crack propagation took place. The crack extension at DK th;max is a given length which is only used to approximate the exponential R-curve behaviour.
crack closure effect [15,27,28]. The overload can change all closure contributions. Unlike the PICC, the change of the roughness induced crack closure and oxide induced crack closure are not a direct consequence of overload. The simulation reveals that overload left the crack open (Fig. 14). This crack opening leads to a significant reduction in these two crack closure effects. However, the crack opening during the following cycles is significantly reduced, so that both roughness and oxide induced crack closure build up again on the newly created crack surfaces. Unfortunately, this effect is still unpredictable at the moment. In the present investigation, we applied the overload on short pre-cracks generated in cyclic compression in deep sharp notched samples. Due to the cyclic compression, the pre-cracks were even open in unloaded state. Such pre-cracks ensure that no crack closure is able to take place behind the pre-crack tips. The very steep R curves for DK th indicates that closure is built up very fast. The build-up of roughness induced crack closure and oxide induced crack closure usually requires a larger crack extension than PICC. Hence for the discussion of the R-curve for DK th after the overload in the next subsection, only the PICC has to be
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Fig. 8. Comparison of FCG rates at load ratio R = 1.
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Fig. 11. The R-curve after overload in compression are compared with the R-curve without overload at load ratio R = 1. The measure data are approached by fit curves. A reduction in crack growth resistance occurred at the beginning. However, the long crack threshold remained unaffected.
Fig. 9. Comparison of FCG rate at load ratio R = 0.1.
Fig. 12. The crack extended with a reduced rate after overload in compression. The pffiffiffiffiffi FCG rate exhibits a minimum at about DK ¼ 14 MPa m (Da 2 mm) and returned to the rate without overload in compression.
taken into account. The transition behaviour to the standard long crack will be discussed in another subsection. 4.1.1. The development of K th after an overload The results of R-curve after overload show that the R-curve is significantly increased after one cycle of tensile overload. The maximal stress intensity factor (K th;max ) at DK th;max shows a linear dependence with overload stress intensity Kov (Fig. 18) at a constant R ratio. The measured data shows a steeper increase at load ratio R = 0.1 and R = 0. The least-square method was used to evaluate the parameters pffiffiffiffiffi in the linear relation between overload Kov and Kth,max (in MPa m) for R = 0.1:
K th;max ¼ 0:32 K ov þ 3:6
ð3Þ
for R = 0: Fig. 10. Comparison of FCG rate at load ratio R = 0.7
K th;max ¼ 0:33 K ov þ 2:7
ð4Þ
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Fig. 15. Comparison of the measured and calculated R curve for K th;max after pffiffiffiffiffi K ov ¼ 10 MPa m.
Fig. 13. The development of opening stress intensity K op with crack extension. After the maximum of K op was reached, the K op decreases in further crack extension towards the K op without overload.
and for R = 1:
K th;max ¼ 0:21 K ov þ 3:1
ð5Þ
The gradient 0.32 at R = 0.1 and 0.33 R = 0 are identical. Since the threshold values at R = 0 are obtained by the simulations in which only the PICC was taken into account, we can conclude that the increase in R-curve is only contributed by PICC. At R = 1, the contribution of PICC is about 0.21 of K ov . The constant term varies from 2.7 to 3.6 around the contribution of intrinsic threshold of pffiffiffiffiffi 3 MPa m in our material.
The R-curves at load ratio R = 1 show a marked extension Da to the DK th;max (Fig. 6). The corresponding Kth,max is also clearly lower at the same Kov (Fig. 18). The significant difference in Rcurve behaviour at different load ratios reveals that the load ratio exhibits an influence on R-curve for DK th after overload. For the pffiffiffiffiffi overload Kov = 30 MPa m, all three load ratios were tested. Since the specimen was unloaded before stepwise increasing load amplitude was started. Unlike the total reversed stress intensity (from K ov to K min ) at load ratio R = 0.7 and R = 0.1, which equals to the K ov , the total reverse stress intensity at load ratio R = 1 (alternating load) is larger. Fuehring and Seeger [29] have already mentioned that a reversed plastic deformation during unloading affects the closure load. A reversed plastic deformation causes a
√ (a) Contact force along the crack surface at Δa = 28µm, Kop = 3M P a m.
√ (b) Contact force along the crack surface at Δa = 64µm, Kop = 2.7M P a m.
(c) Schematic presentation of the contact profile according to the simulation result. pffiffiffiffiffi Fig. 14. Development of contact force along the crack surface at K op during crack extension after K ov ¼ 10 MPa m. The contact force at 10 lm where the maximal plastic deformation induced by overload occured, decreased with increasing crack length and finally disappeared, whereas the contact force direct behind the current crack tip was permanently present.
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Fig. 16. Comparison of the measured and calculated R curve for K th;max after pffiffiffiffiffi K ov ¼ 20 MPa m.
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pffiffiffiffiffi K ov ¼ 10 MPa m, a large plastic deformation occurred in front of the crack tip. The maximum of crack closure level is observed at Da ¼ 10 lm for R = 0. The crack growth resistance decreases with further crack extension due to the decrease of crack closure effect. Another important phenomenon is also indicated by the simulation. After 10 lm crack extension, two local maxima of contact force along the crack surface can be observed. The first maximum is located at 10 lm in front of the pre-crack, where the maximum of crack closure takes place. The second maximum is located immediately behind the current crack tip which is the commonly well documented PICC [9,30]. The first maximum is caused by the overload induced plastic deformation and acting as closure over a long distance. However, this closure (as contact force in the simulation) at that position decreases with further crack extension continuously and became negligible small after about 0.1 mm crack extension, where K op returned back to the constant amplitude state (Fig. 13). The experimentally obtained FCG rate at load pffiffiffiffiffi ratio R = 0.1 under Kov = 10 MPa m returns to its constant amplitude data after about 0.5 mm (Fig. 19). The longer crack extension with reduced FCG rate seems to be an effect of other crack closure mechanisms which require a certain crack extension to build up. The roughness induced crack closure as well as the oxide induced crack closure might contribute to the additional closure effect, because these mechanisms are pronounced at these lower crack growth rates. 4.2. Overload in compression
Fig. 17. Comparison of the measured and calculated R curve for K th;max after pffiffiffiffiffi K ov ¼ 30 MPa m.
Fig. 18. K max;th as a function of the overload K ov . The Kth,max exhibits a linear relation pffiffiffiffiffi with applied single overload for K ov larger than 10 MPa m.
reduction of the overload induced plastic deformation in the present experiments and reduces the contribution of the PICC thereby. 4.1.2. The development of crack growth resistance after K th;max The FE simulation gives an insight into the evolution of closure and crack contact feature. After the overload cycle
The material resistance against crack propagation after overload in compression shows an inverse behaviour in the R-curve for DK th . The crack growth resistance is reduced after overload. At a load ratio R = 1, the R-curve shows a missing closure at the beginning and returns to the R-curve without overload after the crack extended to a certain length continuously (Fig. 11). This length is approxiamtely about 2 times of the size of the monotonic plastic pffiffiffiffiffi zone induced by the overload (rp ¼ 241 lm for K ov ¼ 30 MPa m pffiffiffiffiffi and rp ¼ 428 lm for K ov ¼ 40 MPa m). Due to the plastic compressive deformation, large tensile stresses were present in front of the crack tip. In the subsequent load cycles, the tensile residual stress overlapped with the remote load, resulted in a higher local tensile stress and caused a faster and longer crack extension distance. After the long crack threshold value was reached, the FCG rates show somewhat abnormal behaviour. The crack propagated at a lower rate and met a local minimum at about 2 mm (Fig. 12). After the FCG came out of the minimum, the rate returned to the steady state FCG rate. The distance with a lower crack propagation rate is much larger than the monotonic plastic zone left by the overload in compression. The long influenced range after compressive loading was also mentioned in [31]. These effects of the compression overload can be explained in convenient way by the dislocation shielding concept. Crack propagation in a ductile material is based on a blunting and resharpening process which can be described by emission and annihilation of dislocations at the crack tip [32]. During compressive loading, anti-shielding dislocations are emitted at the crack tip. The crack tip opening distance can be calculated by relation:
CTOD ¼
K2 E ry
ð6Þ
The released opening by emission of a single dislocation is approximately equal to the Burgers vector (typical 0.3 nm). In order to achieve one half of the CTOD 7.2 lm which remains after pffiffiffiffiffi an overload of K ov ¼ 30 MPa m, about 11,000 dislocations have to be emitted from the crack tip on each side of crack tip. In the pffiffiffiffiffi case of K ov ¼ 40 MPa m, 21,000 geometrically necessary
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pffiffiffiffiffi Fig. 19. The FCG rate after Kov = 10 MPa m at a load ratio of R = 0.1 shows a lower FCG rate till 0.5 mm crack extension. The obtained 0.5 mm is longer than the estimated 0.1 mm in the FE simulation. Other crack closure effects such as roughness induced crack closure and oxide induced crack closure seem to be responsible for the reduced crack growth rate in this transition regime.
dislocations on each side are required to achieve a CTOD of 12.8 lm. The stress field caused by such single dislocation is estimated using an equation which is proposed in [33,32] (Fig. 20):
rffiffiffiffiffi bx G 1 3 by G pffiffiffiffi pffiffiffiffi sinh cos h 2 2ð1 mÞ p 2r 2ð1 mÞ p rffiffiffiffiffi 1 1 3 2cos h þ sinh sin h ; 2r 2 2
K¼
ð7Þ
where G is the shear modulus, r is the distance from crack tip to the dislocation. m is the Poisson’s ratio, h is the angle between the connection line from the crack tip to the position of dislocation and the crack plan. bx is the parallel part to the crack propagation direction
bx ¼ b cosðaÞ
ð8Þ
and by is the perpendicular part to the crack propagation direction
by ¼ b sinðaÞ
ð9Þ
First, only one anti-shielding dislocation on the boundary of each plastic zone (241 lm and 428 lm) is considered. The dislocation emitted by Mode I is assumed to be 70° inclined to the crack propagation direction. The resulted stress field (Fig. 21) shows a strong anti-shielding stress intensity within a length, which is about two times of the monotonic plastic zone, and a weak shielding stress field thereafter. Assuming all anti-shielding dislocations are concentrated on the boundary of the plastic zone after overload pffiffiffiffiffi K min ¼ 30 MPa m (in reality they are distributed in an area inside the monotonic plastic zone but outside the cyclic plastic zone). The
Fig. 21. The stress intensity caused by an anti-shielding dislocation in dependence with crack extension Da. The considered anti-shielding dislocation is located at the boundary of the monotonic plastic zone (r = 241 lm and 428 lm) caused by pffiffiffiffiffi pffiffiffiffiffi overload (Kov = 30 MPa m and 40 MPa m).
stress field is approximately enhanced the number of the emitted pffiffiffiffiffi dislocations. At Da ¼ 0 mm, the residual K equals to 9.4 MPa m. Taking the residual stress into account, at the first tested level pffiffiffiffiffi pffiffiffiffiffi DK ¼ 4 MPa m, the local K min was 7.4 MPa m and the local K max pffiffiffiffiffi was 11.4 MPa m which leads to a local load ratio of 0.65. At such a high load ratio, no crack closure takes place. The experimentally obtained R curve also shows a missing closure effect at the beginning. The calculated anti-shielding stress field diminished rapidly. After 0.5 mm crack extension, the anti-shielding stress disappeared completely. The same behaviour can also be observed after pffiffiffiffiffi K ov ¼ 40 MPa m. The evaluated evolution of the stress field shows a good agreement with the approached R-curve based on experimental data, especially in acting length (Fig. 22). The long range shielding stress field afterwards could be the reason for the obtained reduced the FCG rate in Fig. 12. Nevertheless, the effect of an overload in compression is primarily concentrated in the anti-shielding zone.
4.3. Estimation of tensile overload effect by evaluation of stress field of boundary dislocations Fig. 20. Schematic representation of the geometrical arrangement of a single anti shielding dislocation generated during the compression loading. r pl is the size of monotonic plastic zone.
The same dislocation shielding concept is also applied in order to estimate the tensile overload effect. The result is presented in
Please cite this article in press as: Zhou X et al. The effect of single overloads in tension and compression on the fatigue crack propagation behaviour of short cracks. Int J Fatigue (2016), http://dx.doi.org/10.1016/j.ijfatigue.2016.02.001
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Fig. 23. The tensile overload effect is estimated by shielding dislocation concept. The resulted long crack threshold values (at Da ¼ 0) and the acting length thereafter show a good agreement with the experimental results.
material for short cracks in deep, sharp notches. The following conclusions can be made.
Fig. 22. Comparison of the calculated anti-shielding stress intensity of both overloads in compression enhanced by the corresponding number of dislocations with the experimental R-curve for DK th .
Fig. 23. The minimum of stress intensity is directly located at 0 mm crack extension. Since the R-curve behaviour is a result of interaction of dislocations, which is not considered in our simplified model, no R-curve behaviour is obtained in our evaluation. Taking the intrinsic pffiffiffiffiffi resistance (3 MPa m) into account, the resulted long crack threshold values show a good agreement with the experimental results. Due to the fact that the acting length of overload effect only depends on the boundary dislocations, the simplified model provides a precise total acting length of the overload effect. Shielding dislocation concept is based on the analytical expression which exhibits a significant advantage in calculation time compared to the FE simulations. However, to estimate the R-curve behaviour with shielding dislocation concept, the distribution of dislocations has to be considered in a proper manner which requires a large dislocation system and evaluation of the interaction between the dislocations. In such case, the FE simulation is more recommended.
1. After a tensile overload, the material resistance against crack propagation is increased significantly. The Kth,max reveals a linear relation with the applied overload Kov for sufficient large overloads. The increase of Kth,max is about 0.32 of the applied Kov at R = 0.1 and R = 0 and 0.21 of the applied Kov at R = 1. Furthermore, Kth,max decreases with decreasing load ratio. Larger reversed deformation was induced by a lower load ratio which reduced the contribution of the PICC. 2. An elastic plastic FE simulation was carried out for a tensile pffiffiffiffiffi overload of K ov ¼ 10, 20 and 30 MPa m at load ratio R = 0. The results show a good agreement with experimental results. pffiffiffiffiffi The simulation of K ov ¼ 10 MPa m reveals that the overload induced plastic deformation is concentrated at a position of 10 lm in front of the pre-crack. First, this deformation hindered the crack growth into it. After the crack grew through it, the deformed area still caused premature contact and contributed to closure. After about 0.1 mm, the crack growth resistance returned to its original state. However the experimentally obtained FCG diagram shows a crack extension distance of 0.5 mm with reduced FCG rates. We believe that other shielding effects as roughness induced crack closure and oxide induced crack closure contributed to the additional crack growth resistance. 3. The crack growth resistance after overload in compression shows an inverse behaviour. The long crack threshold remained unaffected although the crack growth resistance is reduced after overload. The same behaviour was also observed in the estimation based on the stress field of emitted dislocations during the overloads. High anti-shielding stress are left in front of the crack tip. The anti-shielding stress decreases with increasing Da and even reverted into a very weak shielding stress at larger Da.
Acknowledgements 5. Conclusion In this study the effect of single overloads in tension and compression was systematically studied on a ductile steel as a model
Financial support by the Austrian Federal Government (in particular from Bundesministerium für Verkehr, Innovation und Technologie and Bundesministerium für Wissenschaft, Forschung und
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Wirtschaft) represented by Österreichische Forschungsförderungs gesellschaft mbH and the Styrian and the Tyrolean Provincial Government, represented by Steirische Wirtschaftsförderungsge sellschaft mbH and Standortagentur Tirol, within the framework of the COMET Funding Programme is gratefully acknowledged. References [1] Miner MA. Cumulative damage in fatigue. J Appl Mech 1945;67:159–64. [2] Palmgren A. Die lebensdauer von kugellagern, vol. 68. Berlin: Verfahrenstechnik; 1924. p. 339–341. [3] Corbly DM, Packman PF. On the influence of single and multiple peak overloads on fatigue crack propagation in 7075-t6511 aluminum. Eng Fract Mech 1973;5(2):479–96. IN51–IN52, 497. [4] Cheng Xiaohua, Okuhara Yuji, Yamada Kentaro, Kondo Akimasa. Fatigue crack growth rate measurement of structural steel under overload conditions. Doboku Gakkai Rombun-Hokokushu/Proc Jpn Soc Civil Eng 1994(489 pt 127):71–8. [5] Shuter DM, Geary W. Some aspects of fatigue crack growth retardation behaviour following tensile overloads in a structural steel. Fatigue Fract Eng Mater 1996;19(2):185–99. [6] Hammouda MMI, Ahmad SSE, Seleem MH, Sallam HEM. Fatigue crack growth due to two successive single overloads. Fatigue Fract Eng Mater 1998;21 (12):1537–47. [7] Lang M, Marci G. Influence of single and multiple overloads on fatigue crack propagation. Fatigue Fract Eng Mater 1999;22(4):257–71. [8] Schijve J. Fatigue crack propagation in light alloy sheet material and structures. NRL Report MP 195, National Aeronautical Research Institute, Amsterdam, Holland; 1960. [9] Elber W. Fatigue crack closure under cyclic tension. Eng Fract Mech 1970;2 (1):37–45. [10] Hertzberg RW, Von Eur EFJ, Roberts R. Stress analyses and growth of cracks. ASTM STP 513, American Society for Testing and Materials; 1973. p. 230–59. [11] Christensen RH. Metal fatigue. New York: McGraw-Hill; 1959. [12] Hudson CM, Hardrath HF. Effects of changing stress amplitude on the rate of fatigue crack propagation of two aluminum alloys. NASA Technical Note D960, National Aeronautics and Space Administration, Washington, DC; 1961. [13] Forsyth PJE. Fatigue 84. In: Beevers CJ, editor, vol. 2. Warley, U.K.: EMAS Ltd.; 1984. p. 637. [14] Jones RE. Fatigue crack growth retardation after single-cycle peak overload in ti-6al-4v titanium alloy. Eng Fract Mech 1973;5(3):585–8. IN13–IN16, 589– 604.
[15] Knott JF, Pickard AC. Effects of overloads on fatigue-crack propagation: aluminium alloys. Met Sci 1977;11(8–9):399–404. [16] Vasudevan AK, Sadananda K, Rajan K. Role of microstructures on the growth of long fatigue cracks. Int J Fatigue 1997;19(Suppl. 1):S151–9. [17] Zhou X, Pippan R, Gaenser HP. Effect of compression precracking on the r curve for the threshold of stress intensity range, in preparation. [18] Ritchie RO. Mechanisms of fatigue crack propagation in metals, ceramics and composites: role of crack tip shielding. Mater Sci Eng 1988;103(1):15–28. [19] Pippan R, Plochl L, Klanner F, Stuwe HP. Use of fatigue specimens precracked in compression for measuring threshold values and crack growth. J Test Eval 1994;22(2):98–103. [20] Tabernig B, Pippan R. Determination of the length dependence of the threshold for the fatigue crack propagation. Eng Fract Mech 2002;69(8):899–907. [21] Dougherty JD, Srivatsan TS, Padovan J. Fatigue crack propagation and closure behavior of modified 1070 steel: experimental results. Eng Fract Mech 1997;56(2):167–87. [22] McClung RC, Sehitoglu H. On the finite element analysis of fatigue crack closure 1. Basic modeling issues. Eng Fract Mech 1989;33(2):237–52. [23] McClung RC, Sehitoglu H. On the finite element analysis of fatigue crack closure 2. Numerical results. Eng Fract Mech 1989;33(2):253–72. [24] Bichler Ch, Pippan R. Effect of single overloads in ductile metals: a reconsideration. Eng Fract Mech 2007;74(8):1344–59. [25] Schijve J. Four lectures on fatigue crack growth. Eng Fract Mech 1979;11 (1):167–8. [26] Skorupa M. Load interaction effects during fatigue crack growth under variable amplitude loading – a literature review. Part ii: Qualitative interpretation. Fatigue Fract Eng Mater 1999;22(10):905–26. [27] Wheeler OE. Spectrum loading and crack growth. J Basic Eng, Trans ASME 1972;94(1):181–6. [28] Suresh S. Micromechanisms of fatigue crack growth retardation following overloads. Eng Fract Mech 1983;18(3):577–93. [29] Fuehring H, Seeger T. Dugdale crack closure analysis of fatigue cracks under constant amplitude loading. Eng Fract Mech 1979;11(1):99–122. [30] Elber W. The significance of fatigue crack closure. ASTM Special Technical Publication; 1971. p. 230–242. [31] Newman Jr JC, Ziegler BM, Shaw JW, Cordes TS, Lingenfelser DJ. Fatigue crack growth rate behavior of a36 steel using astm load-reduction and compression precracking test methods. J ASTM Int 2012;9(4). [32] Riemelmoser FO, Gumbsch P, Pippan R. Dislocation modelling of fatigue cracks: an overview. Mater Trans 2001;42(1):2–13. [33] Weertman J. Dislocation based fracture mechanics. World Scientific; 1996.
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International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue
The effect of single overloads in tension and compression on the fatigue crack propagation behaviour of short cracks Xiang Zhou a,b,⇑, Hans-Peter Gaenser a, Reinhard Pippan b,c a
Materials Center Leoben Forschung GmbH, Roseggerstr. 12, 8700 Leoben, Austria Austrian Academy of Science, Erich Schmid Institute of Materials Science, Jahnstr. 12, 8700 Leoben, Austria c Department of Material Physics, Montanuniversität Leoben, Jahnstr. 12, 8700 Leoben, Austria b
a r t i c l e
i n f o
Article history: Received 29 September 2015 Received in revised form 25 January 2016 Accepted 1 February 2016 Available online xxxx Keywords: Overload Crack closure Residual stress Threshold of stress intensity Effective threshold
a b s t r a c t The crack propagation behaviour in cast quenched and tempered steel after one overload cycle in tension as well as in compression on short cracks is investigated in deep notched specimens. The overload cycle exhibits a significant influence on the fatigue life endurance, due to the formation of an overload plastic zones in front of the crack tip. The crack propagation after overload cycles is investigated by inspection of the fatigue threshold R-curve and fatigue crack propagation rate. Tension overload increased the long crack threshold and reduced the R-curve effect, whereas overloads in compression reduced the crack growth resistance and shifted the threshold value to larger crack extension. A simple FE simulation was also performed to investigate the variation in the contribution of plasticity induced crack closure during crack propagation after the overload. Macroscopic mechanistic and dislocation models are introduced to explain the results obtained. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Fatigue crack propagation under variable load amplitude has been extensively investigated for a long time. A major breakthrough was made in the first half of last century by Miner [1] and Palmgren [2], who proposed a linear damage hypothesis to accumulate the damage under variable load amplitudes. However, research in the last few decades revealed the existence of a strong load interaction effect. A change in load amplitude leads to an alteration of the material fatigue properties, resulting in a load sequence dependency in the lifetime [3–7]. Such mechanisms as: 1. residual stress: overload introduces a residual stress field which interacts with the remote stress and affects the subsequent crack propagation [8], 2. crack closure: overload in tension leaves a plastic deformation which increases the crack closure stress intensity [9,10], 3. geometric variation: overload in tension can induce some geometric variation like plastic blunting [11,12], crack deflection or even change in crack front profile [13]. All of these variations lead to a reduction in the local driving force,
4. stain hardening: strain hardening can also take place within the tensile overload induced plastic zone [14,15], can be followed by overload. In ductile material, a change in the contribution of plasticity-induced crack closure (PICC) is a direct consequence of overload. The crack propagation behaviour thereafter can be very complex especially near the threshold. For example, after a single tensile overload the increased PICC can reduce the fatigue crack growth (FCG) rates back to near the threshold regime and thereby promote oxide-induced and roughnessinduced crack closure [16]. For that reason, prediction methods for crack growth are often of empirical nature. The prediction is often based on experiments with certain load time history patterns like single overloads during cyclic loading with a constant load amplitude, block loads or variable amplitudes which correspond to the real load time history in practice. In order to find out the physical reasons of the load variation effect, the current study focuses on one of the simplest cases; the effect of one cycle of large load amplitude (overload) in tension as well as in compression on short open cracks. The material fatigue properties are determined by the R-curve behaviour of the fatigue threshold of stress intensity and FCG rate by constant load amplitude.
⇑ Corresponding author at: Austrian Academy of Science, Erich Schmid Institute of Materials Science, Jahnstr. 12, 8700 Leoben, Austria. http://dx.doi.org/10.1016/j.ijfatigue.2016.02.001 0142-1123/Ó 2016 Elsevier Ltd. All rights reserved.
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2. Material and experimental method The experiments were performed on single-edge-notched-bend (SENB) specimens (Fig. 1) machined from quenched and tempered steel with a yield stress of 630 MPa and an ultimate tensile strength of 760 MPa. The fatigue properties of this material were already well investigated [17]. The R-curve of the fatigue threshold of stress intensity range and the FCG rates are summarised in Figs. 2 and 3. Pre-cracks are introduced at the root of a deep notch (about 4 mm) using compressive-compressive loading. The razor blade polished notch root radius is between 10 and 20 lm. 10,000 load pffiffiffiffiffi cycles with DK ¼ 18 MPa m were applied to the specimen at a load ratio of R = 10. The nucleated pre-crack exhibits about 30 lm in length. Cracks with such length are regarded as physical short cracks [18]. The pre-crack process was performed on a servo hydraulic testing machine at frequency of 10 Hz. One cycle of large load amplitude was applied on the pre-crack. The load was compffiffiffiffiffi pletely released (to K = 0 MPa m) thereafter. The determination of R-curve behaviour of DK th was performed on a resonance testing machine with a testing frequency of 108 Hz. The crack length is measured by potential drop method. The detection limit of our measure device is about 10 lm. In order to obtain the R-curve, a step-wise increasing constant load method was used [19,20]. All tests were started at the intrinsic threshold values pffiffiffiffiffi (DK ¼ 3 MPa m) of the investigated material [17]. For a DK smaller than the intrinsic threshold value, no crack propagation is possible. If the crack started to grow, the load amplitude was kept constant. If the FCG rate decreased to a rate lower than 108 mm/ cyc, the DK was increased to the next step. At load ratio R = 0.1 and pffiffiffiffiffi R = 1, a load increment of 2 MPa m was applied. At load ratio pffiffiffiffiffi R = 0.7, the applied increment was reduced to 0.2 MPa m due to the high mean stress. After the DK exceeded certain value, crack kept extending in a proper way which is well known as stable crack propagation. The short crack behaviour inverted into long crack behaviour. This step is referred to as long crack threshold value (DK th;lc ). For each loading step where the crack propagation got arrested, the total crack extension length was plotted with the current DK in one diagram. The fit curve of these data is the fatigue threshold R-curve. The investigation of stable crack extension is done by recording the fatigue crack growth (FCG) rate. More details about the experimental procedure can be found in [20]. A schematic representation of the current load time history including the compressive-compressive pre-crack process, single overload and step-wise increasing constant load method at a load ratio R = 1 is presented in Fig. 4. Four different overloads in tension pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi Kov = 10 MPa m, 20 MPa m, 30 MPa m, 45 MPa m and two pffiffiffiffiffi pffiffiffiffiffi overloads in compression Kov = 30 MPa m, 40 MPa m were investigated in the current study. The reason for the reduced variety in overloads in compression is that the short open pre-crack was introduced by compressive-compressive loading. The applied pffiffiffiffiffi Kmin during pre-cracking was 20 MPa m. Compressive overloads pffiffiffiffiffi with an amount smaller than 20 MPa m are not regarded as overloads under such initial conditions. Therefore only two comprespffiffiffiffiffi sive overloads, 30 and 40 MPa m, were investigated in this
Fig. 1. Geometry of single-edge-notched-bend (SENB) specimen.
Fig. 2. The R-curves of fatigue threshold of stress intensity range DK th at load ratio R = 1, R = 0.1 and R = 0.7. The R-curves are determined by a fit curve of the measure data.
Fig. 3. The FCG rates of the investigated quenched and tempered steel at load ratio R = 1, R = 0.1 and R = 0.7.
Fig. 4. Schematic presentation of the load time history in the overload experiment pffiffiffiffiffi Kov = 30 MPa m at a load ratio of R = 1.
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study. These overloads still fulfil the small scale yielding requirement. The determination of the R-curve behaviour and FCG rates were performed at load ratio R = 1, R = 0.1 and R = 0.7.
2.1. Finite element analysis A simple elastic–plastic 2D finite element (FE) analysis was implemented in order to estimate the contribution of plasticity induced crack closure (PICC). A half of an SENB was modelled with CPE4 (four-node plane strain) element considering the symmetry conditions. The basic idea is to use the node release technique to simulate the crack extension and investigate thereby the evolution of crack growth resistance. According to the crack propagation condition postulated by Elber [12], load cycles damage the crack tip only if the crack is fully open. It is assumed that the crack is only fully open when the contact force between the crack flanks disappears. As long as the contact force is present, i.e. the crack is closed, the material resistance prevents crack propagation. The obtained R-curve is the transition load level, where the contact force is completely disappeared. The crack surface is assigned by free elements in contrast to the other elements along the y = 0 line which are defined as rigid body with a constraint of the displacement in y direction. A rigid contact is specified between the crack surface and the y = 0 line to realise the contact of both crack surfaces. The loading step is equally divided into 10 sub-steps. After each loading cycle, the first node in front of crack tip is released and thus crack extension is simulated. Since the size of the monotonic plastic zone and the reversed plastic zone is crucial for the degree of finite element mesh refinement [21–23], it is necessary to estimate the size of both plastic zones at first. The estimation is followed by the approximation for plane strain in Mode I proposed by Irwin: Monotonic plastic zone size:
rp ¼
1 3p
K
2
ry
ð1Þ
Reversed plastic zone size:
r p;r ¼
2 1 DK 3p 2ry
3
3. Results 3.1. Experimental results The experimentally obtained R-curve of DK th at load ratio R = 1 is presented in Fig. 6. The cross symbols indicate the load levels tested at which no crack propagation was observed or the crack started to grow but finally was arrested. The circle symbols indicate the load level where stable crack propagation took place. The crack extension at these levels is an assumed length, only used to approximate the exponential nature of R-curve. These DK levels are the upper limit i.e. maximum of the threshold value denoted as DK th;max in the current study. DK th;max increases with increasing overload (Kov). The overload affected R-curve exhibits very short crack extensions Da of about 0.1 mm at the last measurable data (the last cross symbol) compared to the 0.4 mm without overload (Fig. 2). No crack extension was observed before the maximum of threshold value was reached at load ratio R = 0.1 and R = 0.7. i.e., the investigated material does not exhibit a measurable R curve behaviour at these two load ratios. Since the detection limit of the potential drop method is 10 lm, the increase in crack length at each started level was less 10 lm, however, it is possible that the accumulated crack extension is longer than 10 lm which is detectable for our measure device. For that reason, two overloaded specimens were directly tested at high DK levels (closed to DK th;max ). A detectable crack extension was obtained. The resulted R-curves are presented in Fig. 7. The approximation of R-curve is based on the exponential nature of R-curve again. In the FCG rate diagram, the crack growth rate starts at DK th;max and reaches the FCG rate of specimens without overloads in the Paris regime after a certain crack extension Da (Figs. 8–10). The results of the overload in compression experiments are presented in Fig. 11. The R-curve behaviour is approximated by lines based on the measured data. The R-curves at a load ratio of R = 1 show a reduced crack growth resistance at the beginning, returned to the state without the overload after certain crack extension. The FCG rate (Fig. 12) exhibits an abnormal behaviour. After the DK th;max was reached, the crack propagated with a reduced FCG rate, met a minimum at a crack extension of about 2 mm extension in both cases, re-established thereafter the steady state curve in the Paris regime.
ð2Þ
A fundamental requirement for a correct FEM simulation is that the monotonic plastic zone should be within a fine meshed zone. That is the primary condition to ensure that there is no significant mesh influence on the simulated closure results [22,23]. Another requirement is that the size of elements ahead of the crack tip must be at least in the same magnitude of the size of the reversed plastic zone, which ensures a correct implementation of reversible deformation ahead of the crack tip. Simulations were carried out at a load ratio of R = 0 for a comparison with the experiment. Three overloads pffiffiffiffiffi Kov = 10, 20 and 30 MPa m were applied to the pre-crack. The experimentally obtained maximum threshold value at a load ratio pffiffiffiffiffi of R = 0.1 (DK ¼ 6, 9 and 12 MPa m) were used to investigate the development of crack opening levels. Another simulation without any overload was also carried out as reference. The tested DK was pffiffiffiffiffi 6 MPa m. The size of monotonic plastic zone caused by the highest pffiffiffiffiffi overload Kov = 30 MPa m is 241 lm estimated by Eq. (1). and the pffiffiffiffiffi smallest reversed plastic zone at DK = 6 MPa m is 2.5 lm. Therefore, the fine element size was chosen as 2 lm 2 lm. They were defined in an area 150 lm along the crack growth direction (y = 0) and 60 lm in height (y-direction) in order to track the main plastic deformation (Fig. 5).
3.2. Results of finite element analysis The change of the crack opening stress intensity obtained from the finite element analysis are presented in Fig. 13. The last load step with existing contact force is defined as K op . They are symbolised by circles and approximated by a fit curve. In the simulation of pffiffiffiffiffi K ov ¼ 10 MPa m, the maximum of closure stress intensity factor pffiffiffiffiffi of 3 MPa m is first reached at the crack extension of 10 lm. Then the Kop shows a continuous reduction during further crack extenpffiffiffiffiffi sion. After about 100 lm, the Kop decreases to 1.5 MPa m and approaches at larger crack extension the crack closure value obtained by constant amplitude loading (Fig. 13). This continuous reduction reveals that the overload effect decreases with further crack extension and disappears after about 100 lm crack extension. Another phenomenon observed is that after the maximal resistance (Da ¼ 10 lm), the contact force at K op is concentrated at two positions, one directly at the current crack tip and the other at about 10 lm in front of the origin (position of the pre-crack tip or the overload crack tip). In Fig. 14, the position with contact force indicates where premature contact of the crack surfaces takes place. The contact force at the current crack tip is caused by the plastic deformation of the current crack tip and the contact force in front of pre-crack tip is caused by the plastic deformation of
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Fig. 5. Finite element mesh for the estimation of the crack opening stress intensity factor as a function of crack extension.
Fig. 6. The experimentally obtained R-curve behaviour at load ratio R = 1. The cross symbols indicate the load levels, where the crack was arrested or the extension was below the detection limit of the potential drop method. The circle symbols are the DK th;max levels where stable crack propagation took place. The crack extension at DK th;max is an assumed length which is only used to approximate the exponential R-curve behaviour.
the overload cycle. However, the contact force in front of the precrack tip decreases with further crack extension. In the simulation, only the contribution of the plasticity induced crack closure was pffiffiffiffiffi estimated. Taking the intrinsic resistance 3 MPa m into account, the fit curves are located between the R-curves at load ratios R = 0.1 and R = 1 (Figs. 15–17). For the comparison with the experimental results one should always taken into account that the FEM simulation only considers the effect of PICC. 4. Discussion 4.1. Overload in tension Many studies indicate that a sufficient large tensile overload within constant amplitude fatigue loading leads to a delay retardation phenomenon in ductile material [24–26]. There are different explanations for overload effect as mentioned in the introduction, we will follow here the mostly used explanation by the change of
Fig. 7. The experimentally obtained R-curve behaviour at load ratio R = 0.1. The cross symbols indicate the crack arrested levels. The circle symbols are the DK th;max levels where stable crack propagation took place. The crack extension at DK th;max is a given length which is only used to approximate the exponential R-curve behaviour.
crack closure effect [15,27,28]. The overload can change all closure contributions. Unlike the PICC, the change of the roughness induced crack closure and oxide induced crack closure are not a direct consequence of overload. The simulation reveals that overload left the crack open (Fig. 14). This crack opening leads to a significant reduction in these two crack closure effects. However, the crack opening during the following cycles is significantly reduced, so that both roughness and oxide induced crack closure build up again on the newly created crack surfaces. Unfortunately, this effect is still unpredictable at the moment. In the present investigation, we applied the overload on short pre-cracks generated in cyclic compression in deep sharp notched samples. Due to the cyclic compression, the pre-cracks were even open in unloaded state. Such pre-cracks ensure that no crack closure is able to take place behind the pre-crack tips. The very steep R curves for DK th indicates that closure is built up very fast. The build-up of roughness induced crack closure and oxide induced crack closure usually requires a larger crack extension than PICC. Hence for the discussion of the R-curve for DK th after the overload in the next subsection, only the PICC has to be
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Fig. 8. Comparison of FCG rates at load ratio R = 1.
5
Fig. 11. The R-curve after overload in compression are compared with the R-curve without overload at load ratio R = 1. The measure data are approached by fit curves. A reduction in crack growth resistance occurred at the beginning. However, the long crack threshold remained unaffected.
Fig. 9. Comparison of FCG rate at load ratio R = 0.1.
Fig. 12. The crack extended with a reduced rate after overload in compression. The pffiffiffiffiffi FCG rate exhibits a minimum at about DK ¼ 14 MPa m (Da 2 mm) and returned to the rate without overload in compression.
taken into account. The transition behaviour to the standard long crack will be discussed in another subsection. 4.1.1. The development of K th after an overload The results of R-curve after overload show that the R-curve is significantly increased after one cycle of tensile overload. The maximal stress intensity factor (K th;max ) at DK th;max shows a linear dependence with overload stress intensity Kov (Fig. 18) at a constant R ratio. The measured data shows a steeper increase at load ratio R = 0.1 and R = 0. The least-square method was used to evaluate the parameters pffiffiffiffiffi in the linear relation between overload Kov and Kth,max (in MPa m) for R = 0.1:
K th;max ¼ 0:32 K ov þ 3:6
ð3Þ
for R = 0: Fig. 10. Comparison of FCG rate at load ratio R = 0.7
K th;max ¼ 0:33 K ov þ 2:7
ð4Þ
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Fig. 15. Comparison of the measured and calculated R curve for K th;max after pffiffiffiffiffi K ov ¼ 10 MPa m.
Fig. 13. The development of opening stress intensity K op with crack extension. After the maximum of K op was reached, the K op decreases in further crack extension towards the K op without overload.
and for R = 1:
K th;max ¼ 0:21 K ov þ 3:1
ð5Þ
The gradient 0.32 at R = 0.1 and 0.33 R = 0 are identical. Since the threshold values at R = 0 are obtained by the simulations in which only the PICC was taken into account, we can conclude that the increase in R-curve is only contributed by PICC. At R = 1, the contribution of PICC is about 0.21 of K ov . The constant term varies from 2.7 to 3.6 around the contribution of intrinsic threshold of pffiffiffiffiffi 3 MPa m in our material.
The R-curves at load ratio R = 1 show a marked extension Da to the DK th;max (Fig. 6). The corresponding Kth,max is also clearly lower at the same Kov (Fig. 18). The significant difference in Rcurve behaviour at different load ratios reveals that the load ratio exhibits an influence on R-curve for DK th after overload. For the pffiffiffiffiffi overload Kov = 30 MPa m, all three load ratios were tested. Since the specimen was unloaded before stepwise increasing load amplitude was started. Unlike the total reversed stress intensity (from K ov to K min ) at load ratio R = 0.7 and R = 0.1, which equals to the K ov , the total reverse stress intensity at load ratio R = 1 (alternating load) is larger. Fuehring and Seeger [29] have already mentioned that a reversed plastic deformation during unloading affects the closure load. A reversed plastic deformation causes a
√ (a) Contact force along the crack surface at Δa = 28µm, Kop = 3M P a m.
√ (b) Contact force along the crack surface at Δa = 64µm, Kop = 2.7M P a m.
(c) Schematic presentation of the contact profile according to the simulation result. pffiffiffiffiffi Fig. 14. Development of contact force along the crack surface at K op during crack extension after K ov ¼ 10 MPa m. The contact force at 10 lm where the maximal plastic deformation induced by overload occured, decreased with increasing crack length and finally disappeared, whereas the contact force direct behind the current crack tip was permanently present.
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Fig. 16. Comparison of the measured and calculated R curve for K th;max after pffiffiffiffiffi K ov ¼ 20 MPa m.
7
pffiffiffiffiffi K ov ¼ 10 MPa m, a large plastic deformation occurred in front of the crack tip. The maximum of crack closure level is observed at Da ¼ 10 lm for R = 0. The crack growth resistance decreases with further crack extension due to the decrease of crack closure effect. Another important phenomenon is also indicated by the simulation. After 10 lm crack extension, two local maxima of contact force along the crack surface can be observed. The first maximum is located at 10 lm in front of the pre-crack, where the maximum of crack closure takes place. The second maximum is located immediately behind the current crack tip which is the commonly well documented PICC [9,30]. The first maximum is caused by the overload induced plastic deformation and acting as closure over a long distance. However, this closure (as contact force in the simulation) at that position decreases with further crack extension continuously and became negligible small after about 0.1 mm crack extension, where K op returned back to the constant amplitude state (Fig. 13). The experimentally obtained FCG rate at load pffiffiffiffiffi ratio R = 0.1 under Kov = 10 MPa m returns to its constant amplitude data after about 0.5 mm (Fig. 19). The longer crack extension with reduced FCG rate seems to be an effect of other crack closure mechanisms which require a certain crack extension to build up. The roughness induced crack closure as well as the oxide induced crack closure might contribute to the additional closure effect, because these mechanisms are pronounced at these lower crack growth rates. 4.2. Overload in compression
Fig. 17. Comparison of the measured and calculated R curve for K th;max after pffiffiffiffiffi K ov ¼ 30 MPa m.
Fig. 18. K max;th as a function of the overload K ov . The Kth,max exhibits a linear relation pffiffiffiffiffi with applied single overload for K ov larger than 10 MPa m.
reduction of the overload induced plastic deformation in the present experiments and reduces the contribution of the PICC thereby. 4.1.2. The development of crack growth resistance after K th;max The FE simulation gives an insight into the evolution of closure and crack contact feature. After the overload cycle
The material resistance against crack propagation after overload in compression shows an inverse behaviour in the R-curve for DK th . The crack growth resistance is reduced after overload. At a load ratio R = 1, the R-curve shows a missing closure at the beginning and returns to the R-curve without overload after the crack extended to a certain length continuously (Fig. 11). This length is approxiamtely about 2 times of the size of the monotonic plastic pffiffiffiffiffi zone induced by the overload (rp ¼ 241 lm for K ov ¼ 30 MPa m pffiffiffiffiffi and rp ¼ 428 lm for K ov ¼ 40 MPa m). Due to the plastic compressive deformation, large tensile stresses were present in front of the crack tip. In the subsequent load cycles, the tensile residual stress overlapped with the remote load, resulted in a higher local tensile stress and caused a faster and longer crack extension distance. After the long crack threshold value was reached, the FCG rates show somewhat abnormal behaviour. The crack propagated at a lower rate and met a local minimum at about 2 mm (Fig. 12). After the FCG came out of the minimum, the rate returned to the steady state FCG rate. The distance with a lower crack propagation rate is much larger than the monotonic plastic zone left by the overload in compression. The long influenced range after compressive loading was also mentioned in [31]. These effects of the compression overload can be explained in convenient way by the dislocation shielding concept. Crack propagation in a ductile material is based on a blunting and resharpening process which can be described by emission and annihilation of dislocations at the crack tip [32]. During compressive loading, anti-shielding dislocations are emitted at the crack tip. The crack tip opening distance can be calculated by relation:
CTOD ¼
K2 E ry
ð6Þ
The released opening by emission of a single dislocation is approximately equal to the Burgers vector (typical 0.3 nm). In order to achieve one half of the CTOD 7.2 lm which remains after pffiffiffiffiffi an overload of K ov ¼ 30 MPa m, about 11,000 dislocations have to be emitted from the crack tip on each side of crack tip. In the pffiffiffiffiffi case of K ov ¼ 40 MPa m, 21,000 geometrically necessary
Please cite this article in press as: Zhou X et al. The effect of single overloads in tension and compression on the fatigue crack propagation behaviour of short cracks. Int J Fatigue (2016), http://dx.doi.org/10.1016/j.ijfatigue.2016.02.001
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pffiffiffiffiffi Fig. 19. The FCG rate after Kov = 10 MPa m at a load ratio of R = 0.1 shows a lower FCG rate till 0.5 mm crack extension. The obtained 0.5 mm is longer than the estimated 0.1 mm in the FE simulation. Other crack closure effects such as roughness induced crack closure and oxide induced crack closure seem to be responsible for the reduced crack growth rate in this transition regime.
dislocations on each side are required to achieve a CTOD of 12.8 lm. The stress field caused by such single dislocation is estimated using an equation which is proposed in [33,32] (Fig. 20):
rffiffiffiffiffi bx G 1 3 by G pffiffiffiffi pffiffiffiffi sinh cos h 2 2ð1 mÞ p 2r 2ð1 mÞ p rffiffiffiffiffi 1 1 3 2cos h þ sinh sin h ; 2r 2 2
K¼
ð7Þ
where G is the shear modulus, r is the distance from crack tip to the dislocation. m is the Poisson’s ratio, h is the angle between the connection line from the crack tip to the position of dislocation and the crack plan. bx is the parallel part to the crack propagation direction
bx ¼ b cosðaÞ
ð8Þ
and by is the perpendicular part to the crack propagation direction
by ¼ b sinðaÞ
ð9Þ
First, only one anti-shielding dislocation on the boundary of each plastic zone (241 lm and 428 lm) is considered. The dislocation emitted by Mode I is assumed to be 70° inclined to the crack propagation direction. The resulted stress field (Fig. 21) shows a strong anti-shielding stress intensity within a length, which is about two times of the monotonic plastic zone, and a weak shielding stress field thereafter. Assuming all anti-shielding dislocations are concentrated on the boundary of the plastic zone after overload pffiffiffiffiffi K min ¼ 30 MPa m (in reality they are distributed in an area inside the monotonic plastic zone but outside the cyclic plastic zone). The
Fig. 21. The stress intensity caused by an anti-shielding dislocation in dependence with crack extension Da. The considered anti-shielding dislocation is located at the boundary of the monotonic plastic zone (r = 241 lm and 428 lm) caused by pffiffiffiffiffi pffiffiffiffiffi overload (Kov = 30 MPa m and 40 MPa m).
stress field is approximately enhanced the number of the emitted pffiffiffiffiffi dislocations. At Da ¼ 0 mm, the residual K equals to 9.4 MPa m. Taking the residual stress into account, at the first tested level pffiffiffiffiffi pffiffiffiffiffi DK ¼ 4 MPa m, the local K min was 7.4 MPa m and the local K max pffiffiffiffiffi was 11.4 MPa m which leads to a local load ratio of 0.65. At such a high load ratio, no crack closure takes place. The experimentally obtained R curve also shows a missing closure effect at the beginning. The calculated anti-shielding stress field diminished rapidly. After 0.5 mm crack extension, the anti-shielding stress disappeared completely. The same behaviour can also be observed after pffiffiffiffiffi K ov ¼ 40 MPa m. The evaluated evolution of the stress field shows a good agreement with the approached R-curve based on experimental data, especially in acting length (Fig. 22). The long range shielding stress field afterwards could be the reason for the obtained reduced the FCG rate in Fig. 12. Nevertheless, the effect of an overload in compression is primarily concentrated in the anti-shielding zone.
4.3. Estimation of tensile overload effect by evaluation of stress field of boundary dislocations Fig. 20. Schematic representation of the geometrical arrangement of a single anti shielding dislocation generated during the compression loading. r pl is the size of monotonic plastic zone.
The same dislocation shielding concept is also applied in order to estimate the tensile overload effect. The result is presented in
Please cite this article in press as: Zhou X et al. The effect of single overloads in tension and compression on the fatigue crack propagation behaviour of short cracks. Int J Fatigue (2016), http://dx.doi.org/10.1016/j.ijfatigue.2016.02.001
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Fig. 23. The tensile overload effect is estimated by shielding dislocation concept. The resulted long crack threshold values (at Da ¼ 0) and the acting length thereafter show a good agreement with the experimental results.
material for short cracks in deep, sharp notches. The following conclusions can be made.
Fig. 22. Comparison of the calculated anti-shielding stress intensity of both overloads in compression enhanced by the corresponding number of dislocations with the experimental R-curve for DK th .
Fig. 23. The minimum of stress intensity is directly located at 0 mm crack extension. Since the R-curve behaviour is a result of interaction of dislocations, which is not considered in our simplified model, no R-curve behaviour is obtained in our evaluation. Taking the intrinsic pffiffiffiffiffi resistance (3 MPa m) into account, the resulted long crack threshold values show a good agreement with the experimental results. Due to the fact that the acting length of overload effect only depends on the boundary dislocations, the simplified model provides a precise total acting length of the overload effect. Shielding dislocation concept is based on the analytical expression which exhibits a significant advantage in calculation time compared to the FE simulations. However, to estimate the R-curve behaviour with shielding dislocation concept, the distribution of dislocations has to be considered in a proper manner which requires a large dislocation system and evaluation of the interaction between the dislocations. In such case, the FE simulation is more recommended.
1. After a tensile overload, the material resistance against crack propagation is increased significantly. The Kth,max reveals a linear relation with the applied overload Kov for sufficient large overloads. The increase of Kth,max is about 0.32 of the applied Kov at R = 0.1 and R = 0 and 0.21 of the applied Kov at R = 1. Furthermore, Kth,max decreases with decreasing load ratio. Larger reversed deformation was induced by a lower load ratio which reduced the contribution of the PICC. 2. An elastic plastic FE simulation was carried out for a tensile pffiffiffiffiffi overload of K ov ¼ 10, 20 and 30 MPa m at load ratio R = 0. The results show a good agreement with experimental results. pffiffiffiffiffi The simulation of K ov ¼ 10 MPa m reveals that the overload induced plastic deformation is concentrated at a position of 10 lm in front of the pre-crack. First, this deformation hindered the crack growth into it. After the crack grew through it, the deformed area still caused premature contact and contributed to closure. After about 0.1 mm, the crack growth resistance returned to its original state. However the experimentally obtained FCG diagram shows a crack extension distance of 0.5 mm with reduced FCG rates. We believe that other shielding effects as roughness induced crack closure and oxide induced crack closure contributed to the additional crack growth resistance. 3. The crack growth resistance after overload in compression shows an inverse behaviour. The long crack threshold remained unaffected although the crack growth resistance is reduced after overload. The same behaviour was also observed in the estimation based on the stress field of emitted dislocations during the overloads. High anti-shielding stress are left in front of the crack tip. The anti-shielding stress decreases with increasing Da and even reverted into a very weak shielding stress at larger Da.
Acknowledgements 5. Conclusion In this study the effect of single overloads in tension and compression was systematically studied on a ductile steel as a model
Financial support by the Austrian Federal Government (in particular from Bundesministerium für Verkehr, Innovation und Technologie and Bundesministerium für Wissenschaft, Forschung und
Please cite this article in press as: Zhou X et al. The effect of single overloads in tension and compression on the fatigue crack propagation behaviour of short cracks. Int J Fatigue (2016), http://dx.doi.org/10.1016/j.ijfatigue.2016.02.001
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Please cite this article in press as: Zhou X et al. The effect of single overloads in tension and compression on the fatigue crack propagation behaviour of short cracks. Int J Fatigue (2016), http://dx.doi.org/10.1016/j.ijfatigue.2016.02.001