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Effect of overload on crack closure in thick and thin specimens via digital image correlation
International Journal of Fatigue 56 (2013) 17–24
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International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue
Effect of overload on crack closure in thick and thin specimens via digital image correlation F. Yusof a,b,⇑, P. Lopez-Crespo a,c, P.J. Withers a a
School of Materials, University of Manchester, Grosvenor Street, Manchester M13 9PL, UK School of Mechanical Engineering, University of Science Malaysia, Nibong Tebal, 14300 Penang, Malaysia c Department of Civil and Materials Engineering, University of Malaga, C/Dr. Ortiz Ramos s/n, 29071 Malaga, Spain b
a r t i c l e
i n f o
Article history: Received 18 February 2013 Received in revised form 27 June 2013 Accepted 2 July 2013 Available online 18 July 2013 Keywords: Crack closure Overload effects Thickness effects Digital image correlation
a b s t r a c t The displacement field of compact tension (CT) specimens have been mapped by digital image correlation (DIC) local to growing fatigue cracks to study overload effects for plane stress and plane strain. We have extracted crack opening displacement (DCOD) and stress intensity (K) determined by a Muskhelishvili fit to the crack tip displacement field to infer the closure load. In both cases a classical knee was observed upon unloading consistent with closure which disappeared during the accelerated growth following OL, before increasing during retardation. In both cases following OL the crack growth rate is perturbed for a distance similar to the plastic zone. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Fatigue crack closure analysis is central to understanding crack growth, especially when the amplitude of the fatigue cycle varies. Since Elber’s [1] initial discovery of the phenomenon, universal agreement on a number of aspects remain elusive, particularly with respect to the origin of closure [2]. The understanding of crack closure is a subject of argument within the fatigue community. This is largely caused by difficulty in its measurement and the complexity of the crack closure mechanics. Some researchers suggest that crack closure may not occur [3], others believe it may occur only under conditions of plane stress [4]. An overview of the classic compliance and electrical resistance method has been outlined in [5,6]. The compliance method gives more reliable and consistent measurement of the crack closure response than the electrical method. However, conventional crack closure measurements characterize the global response to local crack-tip phenomenon and as such provide little micromechanical information from which the pertinent crack shielding mechanisms can be unambiguously inferred [7]. Recent advances in optical experimental mechanics of fatigue crack growth and fracture problems such as the digital image correlation (DIC) have made it possible to infer the local crack-tip ⇑ Corresponding author at: School of Mechanical Engineering, University of Science Malaysia, Nibong Tebal 14300, Penang, Malaysia. Tel.: +6045996316: Fax: +6045941025. E-mail address: [email protected] (F. Yusof). 0142-1123/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijfatigue.2013.07.002
displacement fields at sufficient accuracy [8–10]. DIC has been used to study fatigue crack growth in thin single edge cracked samples in tension [11] and thickness effects in compact tension (CT) specimens [12], to study development of plastic zone size [13] and to extract the instantaneous stress intensity variation, K, by high speed photography [14]. DIC has also been used to study crack closure [15] and the effect of crack closure due to a single OL perturbation during fatigue cycling of different thickness CT samples [16], based on a near tip crack opening displacement (DCOD) as a function of load. Both approaches involve the collection of a set of images within a loading cycle to study changes in closure. For a complete loading and unloading cycle, it was shown that the DCOD changes with distance behind the crack-tip while the crack closure load within a typical DIC length scale behind the cracktip is similar for a given fatigue crack growth condition [16]. However following an OL event, the crack closure changes according to characteristic of (i) acceleration, (ii) attenuation, and (iii) recovery regimes of crack growth rate [17]. It has been demonstrated that a change of the crack closure in the acceleration and attenuation regimes occurs within small numbers of fatigue cycles (order of tens to hundreds) from the point of OL while the recovery regimes occurs over thousands of cycles [18]. The existing literature has usually focused between ten thousand and a hundred thousand cycles and important information on crack closure development within the different regimes may have been missed due to this. A DIC crack-tip displacement field can give a huge number of displacement vectors to identify a state of crack closure at any
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stage of crack growth. However, the distance at which DCOD should be measured behind the crack-tip to determine crack closure load has never been regularized. In [12,16], a minimum distance behind the crack-tip of 150 lm and increments of 100 lm from the minimum distance for additional measures of DCOD were used. During the acceleration and attenuation regimes, fatigue crack growth rates in the near threshold region for a typical steel specimen can be around a hundredth of a micron over a number of cycles [19]. Based on this, it was hypothesized that the distance of DCOD from the crack-tip must be a function of the growth rate associated with the different regimes for accurate determination of the state of crack closure after an OL. In this paper, the main objective of the experimental study is to elucidate the state of crack closure perturbed by an OL for CT specimens of plane strain (hereafter denoted thick) and plane stress (hereafter denoted thin) by biasing the incremental cyclic intervals for DIC analysis as a function of different regimes of crack growth rate. The second objective is to compare two different methods of inferring the crack closure load from the DIC data namely: (i) the DCOD and (ii) the stress intensity factor inferred by the K-Muskhelishvili fit to the near crack-tip displacement field. The DCOD technique is widely used and it is essentially a local measurement of crack closure with reference to the crack tip however the K-Muskhelishvili fit uses an array of displacement vectors around a crack tip to give a fracture mechanics parameter K independent of the K calibration factors. Finally, based on the experimental observations, the crack growth regimes for the thick and thin CT specimens are to be compared to a fatigue crack growth deformation zone proposed by [20]. This paper is organized as follows. First, the experimental details which include the materials and specimens, fatigue crack growth and the DIC method employed in this work are presented. Next the DIC data are manipulated to give DCOD and the K-Muskhelishvili fit as a function of load. Following this the measured crack closure for the thick and thin specimens based on the DCOD and the K-Muskhelishvili fit are discussed in detail. The change of crack closure loads due to the single OL are mapped over the crack length and number of elapsed cycles using the DCOD and the KMuskhelishvili approaches and this is related to Rice’s fatigue crack growth deformation size. Finally, some conclusions and future work are given based on the current investigation.
2. Experimental details 2.1. Materials and specimens The material used in the study is 316L austenitic stainless steel alloy having a Young’s modulus, E around 195 GPa and yield stress, ro = 304 MPa. Fatigue crack growth specimens were prepared following a standard CT specimen configuration according to [21]. Usually a thick specimen can be specified according to the ratio of the thickness, B to the uncracked ligament length, (W-a), as B/ðW aÞ P 1 and a thin specimen can be specified as B=ðW aÞ 6 0:1. Due to experimental limitations, such as the load range of the universal testing machine and the as supplied specimen materials, the ratio of the thickness to the uncracked ligament length was chosen to be 0.5 for the thick (B = 12 mm) specimen and 0.1 for the thin (B = 3 mm) specimen. For the thick specimen the in-plane ligament length for the CT specimens was W = 50 mm. Prior to the experimental work, to ensure that the deformation field for the selected thickness can represent the plane strain and plane stress field, finite element analysis of the selected CT geometries under a given K-field were compared to a displacement boundary layer formulation as given in [22]:
ux uy
K I r 1=2 ¼ 2ð1 þ mÞ E 2p
(
2
cos ðh=2½j 1 þ 2 sin ðh=2Þ
)
2
sin ðh=2Þ½j þ 1 2 sin ðh=2Þ
ð1Þ
where ux and uy are the in-plane horizontal and vertical displacements and where (r, h) are polar coordinates centered at the crack-tip. j ¼ ð3 4mÞ for plane strain and j ¼ ð3 mÞ=ð1 þ mÞ for a generalized plane stress conditions where m is the Poisson’s ratio. The crack-tip displacement field generated from the boundary layer formulation at identical applied remote K field was compared to the K field inferred from the CT specimen through the DIC. It was found that the deformation fields for the thick and thin CT specimens were identifiable as the plane strain and plane stress field respectively within the linear elastic fracture mechanics limit. Due to space limitations, results are not shown here but the technique adopted has been discussed [14] with good agreement. 2.2. Fatigue crack growth experiments Four specimens for each thickness were used in this experiment. For reference purposes the critical fracture toughness for the thick and thin specimens were determined to be pffiffiffiffiffi pffiffiffiffiffi K c ¼ 35 MPa m and 45 MPa m respectively. The fatigue crack growth experiments were conducted at room temperature on a servo-hydraulic testing machine with a ±10 kN loading range. A schematic of the experiment is shown in Fig. 1. The samples were fatigued at a frequency, f = 30 Hz based on a triangular waveform and the DIC images were captured at various stages of crack growth at a much lower fatigue frequency (1/100 Hz). The triangular waveform allows for a relatively simple identification of any knee in the crack opening as a sharp change from the unloading to loading sequence is otherwise expected within a fatigue cycle. For the thick specimens the loading amplitude was represented by maximum and minimum loads of Pmax = 2.95 kN and Pmin = 0.15 kN giving a stress ratio, R ¼ 0:05 while for the thin specimen the fatigue cycling conditions were set at Pmax = 2.1 kN and Pmin = 0.105 kN giving a similar nominal stress ratio to that for the thick specimen. The OL cycle was applied once the fatigue crack growth rate had become constant some distance from the fatigue pre-cracking zone. To initiate a sharp crack from the machined notch, and to ensure that the final stress intensity factor amplitude, DK, from the pre-cracking does not exceed the initial DK for the crack closure experiment, the fatigue pre-cracking was initiated from the machined notch following a 3 stepped reduction in load amplitude to the baseline DK for the crack closure experiment. Before the OL cycle was introduced, the sharp crack was grown to a distance greater than twice the wake zone size of the fatigue pre-cracking. The OL cycle involved raising the stress intensity
Fig. 1. Schematic representation of the experimental configuration.
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pffiffiffiffiffi pffiffiffiffiffi from a baseline value K max ¼ 17 MPa m to K OL ¼ 34 MPa m for the thick CT specimen (100% OL) and from a baseline of pffiffiffiffiffi pffiffiffiffiffi K max ¼ 21:4 MPa m to K OL ¼ 42:8 MPa m for the thin CT specimen (100% OL). The fatigue crack growth curves for the thick and thin specimens are shown first in terms of crack length, a, with respect to the number of cycles, N, in Fig. 2a and c. The crack growth rate, da=dN, plotted as a function of ratio of crack length to ligament length, a=W, for the thick and thin specimens are shown in Fig. 2b and d. Typical behavior of thick and thin CT specimens can be identified from the figures. In general, both the thick and thin specimens demonstrated the classical acceleration, retardation and recovery regimes of crack growth rate after OL [17]. Fig. 2 shows a much greater acceleration of the crack growth for the thin sample compared to the thick sample immediately after OL. Also in common with previous work, it takes the thin specimen a greater number of cycles (25% more) after OL to recover to the original crack growth rate than the thick specimen.
by 75% giving 150,000 displacement vectors over the viewing area (Fig. 3a). For the COD and K-Muskhelishvili data analysis, it was found that using a fraction of displacement vectors (average of 6 thousands displacement vectors) can give sufficiently accurate results thus reducing the computing time to process the displacement data to get DCOD directly and K fitted from the Fourier fit (Fig. 3b). For the present work, an average of 200 images was correlated per cycle of crack growth.
3. Experimental results 3.1. Crack opening displacement (DCOD) From the correlated images, a post processing routine has been used to identify the change of crack opening displacement, DCOD, at various distances behind the crack-tip by subtracting the vertical displacements (uy direction shown in Fig. 1) of the top flank and bottom flank of the crack mouth: bot DCODðxÞ ¼ utop y uy
ð2Þ
2.3. Crack monitoring by digital image correlation where the DCOD varies as a function of distance behind the crack-tip, x. In the present work, DCODðxÞ has been characterized at different distances, x, behind the crack-tip. Fig. 4a and b show the evolution of the DCOD (100 lm behind the crack-tip) through selected loading cycles. The variation of the DCOD for the thick and thin CT specimens at different fatigue crack growth stages and measured at various distances behind the crack-tip are shown in much more detail in Figs. 5 and 6. Normally the point of closure might be identified by a knee in the crack compliance curve recorded using a back-face strain gauge [12]. Here we see a similar knee as the DCOD remains at zero until a significant load has been applied after which it increases linearly with further loading.
Crack length, a (mm)
37
(a) +125k
35
+25k
33 31
+100
29
+4k
-10k 27 0
50
100
150
Crack growth rate (x1e-8), da/dN
The DIC method described by [14,18] was applied in this work. The experimental setup comprised a macro-lens with a teleconvertor mounted on a 4-mega pixel CCD camera (see Fig. 1). A fiber-optic ring was attached to the periphery of the lens to achieve uniform illumination of the specimen surface that had been finely abraded with SiC paper to obtain a random abrasion pattern giving sufficient contrast for the correlation algorithm (Fig. 3a). An area of 10 mm 10 mm was imaged corresponding to a 2048 2048 pixels from which each pixel represents about 5 lm. DIC was performed using commercially available [23] image correlation software correlating 32 32 pixel patches overlapped
200
16
(b)
12
-10k +125k
8
+25k
4
+100 +4k 0 0.55
0.6
Number of cycles, kN
+160k
25
(c)
+100
23
+90k
0
21
+40k 19
-13k 17 0
100
200
Number of cycles, kN
300
Crack growth rate (x1e-8) da/dN
Crack length, a (mm)
27
0.65
0.7
Crack length, a/W 30
(d) 25
+100 20 15
+160k
10
+90k
5
-13k +40k
0 0.3
0.35
0.4
0.45
Crack length, (a/W)
Fig. 2. Fatigue crack growth for thick (a and b) and thin (c and d) CT specimens.
0.5
0.55
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Fig. 3. (a) An inferred DIC displacement map in the area surrounding the crack-tip superimposed upon one of the reference images and (b) the vertical (uy – crack opening direction) displacement field for 5676 points for maximum load, Pmax, for the thick fatigue crack growth at +125k OL cycles. Both figures demonstrate the crack-tip displacement field used to infer DCOD and the stress intensity, K.
150
25
(a)
(b) 125 0 (OL)
ΔCOD (micron)
ΔCOD (micron)
20 0 (OL)
15 +125k
10
+100 -50k
5
100
+160k -13k
75
+100
50 25
+90k
+45k
0
0 0
1
2
3
4
0
1
Fraction of Pmax
2
3
4
Fraction of Pmax
Fig. 4. The opening load as a function of the normalized load (P/Pmax) measured at a distance of 100 lm behind the crack-tip during overload and selected other cycles for (a) the thick and (b) the thin CT samples.
14
312
(a)
208
12
104
10 8 6 4 2 0
0
0.2
0.4
0.6
0.8
1
1.2
18 16 14 12 10 8 6 4 2 0
312 208 104
0
0.2
(d)
785 681 577 473 421 317 213 109 5
0
0.2
0.4
0.6
0.8
Fraction of Pmax (kN)
0.4
0.6
0.8
1
1.2
18 16 14 12 10 8 6 4 2 0
1
1.2
18 16 14 12 10 8 6 4 2 0
312 208 104
0
0.2
(e)
426 218 104
0.2
0.4
0.6
0.8
Fraction of Pmax (kN)
0.4
0.6
0.8
1
1.2
Fraction of Pmax (kN)
842
0
(c)
416
Fraction of Pmax (kN)
ΔCOD (micron)
ΔCOD (micron)
Fraction of Pmax (kN) 18 16 14 12 10 8 6 4 2 0
(b)
416
ΔCOD (micron)
416
ΔCOD (micron)
16
ΔCOD (micron)
ΔCOD (micron)
18
1
1.2
18 16 14 12 10 8 6 4 2 0
(f)
2787 416 312
208 104
0
0.2
0.4
0.6
0.8
1
1.2
Fraction of Pmax (kN)
Fig. 5. The crack opening displacement for thick CT specimens at different crack growth intervals measured at increasing distances behind the crack tip as shown by the legend in each plot.
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16
12 10 8 6 4
12 10
4
0
0
0.4
0.6
0.8
1
1.2
0
16
0.2
6 4 2 0
0.2
0.4
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0.8
1
0.6
0.8
Fraction of Pmax (kN)
6 4
0
0.2
215
10
100
1
1.2
0.4
0.6
0.8
1
16
8 6 4 2
0.4
8
0
1.2
(e)
330
12
0
(c)
1.2
Fraction of Pmax (kN)
445
14
ΔCOD (micron)
ΔCOD (micron)
8
0
(d)
750 700 600 500 400 300 200 100 50 25
10
10
Fraction of Pmax (kN)
16 12
12
2
Fraction of Pmax (kN)
14
445 330 215 100
14
6 2
0.2
16
(b)
8
2 0
445 330 215 100
14
ΔCOD (micron)
ΔCOD (micron)
14
(a)
ΔCOD (micron)
445 330 215 100
(f)
445 330 215 100
14
ΔCOD (micron)
16
12 10 8 6 4 2
0
0.2
0.4
0.6
0.8
1
1.2
0
0
Fraction of Pmax (kN)
0.2
0.4
0.6
0.8
1
1.2
Fraction of Pmax (kN)
Fig. 6. The crack opening displacement for thin CT specimens at different crack growth intervals measured at increasing distances behind the crack-tip as shown by the legend in each plot.
Accordingly, the opening (closing) load can be estimated as the load at which a drastic change in slope takes place during the loading (unloading) portion of the cycle. In this work, both the opening and closure loads will be used interchangeably because in our case they occur at the same load. Figs. 5 and 6 show the variation in the DCOD for the thick and thin CT specimens as a function of distance from the crack-tip for fatigue cycles before or after the OL event from which a number of trends are evident. 3.1.1. Prior to the OL A clear knee is observed at all positions behind the crack (perhaps indicative of instantaneous full-face contact) at a baseline closure load under steady state fatigue cycling. 3.1.2. During the OL event A large amount of plastic deformation occurs during the OL for both the thick and thin CT specimens significantly increasing the crack opening as a result of which immediately after the OL cycle the crack is left with a significant residual DCOD. This amounts to 8 lm and 95 lm for the thick and thin samples at x = 100 lm. In other words the crack faces are significantly open after the OL, more so for the thin than the thick sample. 3.1.3. Acceleration phase No knee (closure) upon loading and unloading during the acceleration phase immediately after the OL. 3.1.4. During the retardation phase An increasingly prominent knee initiating at much earlier upon unloading than for baseline fatigue (closure load 3 greater) near to the crack-tip. The knee occurs at a non-zero DCOD and at lower and lower loads during unloading with increasing distance from the crack. 3.1.5. Recovered crack growth rate When the crack has progressed sufficiently beyond the OL event there is a return to the baseline closure load.
These results suggest that during baseline crack growth the crack faces close instantaneously, but that the significant plastic opening of the crack faces during OL means that the crack faces do not meet during the acceleration phase. After further crack growth the crack faces close progressively from the crack-tip towards the notch and cease to close further at a load which is lower the greater the distance from the crack-tip, but rather than touch along their length the faces appear to be held open at low loads. A distinct difference between the thick and thin specimen can be seen in that the thick specimen shows a more significant change of the closure load with increasing distance behind the crack-tip. This progressive closing of the crack from the crack-tip along the crack faces in a zipping action is particularly marked for the thick sample. As a result, the DCOD is dependent on the distance measured from behind the crack-tip. In general, the knee signifies some form of crack ‘closure’ has occurred in that the crack ceases to close further although the crack may not actually be closed at that location. 3.2. K-Muskhelishvili method The displacement fields such as those shown in Fig. 3b can also be used to infer the stress intensity factor, K, based on the Muskhelishvili method [24], as reported in [25]. To correct the local cracktip plasticity effect within linear elasticity, Irwin’s K-correction approach; acorr ¼ a þ rp is used. In this instance, the crack-tip displacement field from the images can be represented by the Muskhelishvili elastic solutions. The Muskhelishvili method is based on the determination of the complex potential from a given boundary conditions. A system of real linear equations can then be developed based on a known crack-tip location: N X
Ajk ak þ
k¼N N X k¼N
N X
Bjk bk ¼ 2l ujx
ð3Þ
Djk bk ¼ 2l ujy
ð4Þ
k¼N
C jk ak þ
N X k¼N
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where j = 1, 2, . . ., p; ujx and ujy are the displacements at point j; Ajk ; Bjk ; C jk and Djk are known functions of positions; l is the shear modulus and
ak ¼ ak þ ibk
ð5Þ
The complex stress intensity factor is given according to [26]:
rffiffiffiffi K ¼ K I iK II ¼ 2
N pX
a k¼N
kak
ð6Þ
where a ¼ 2r cc is half-length of a center crack in an infinite plane. Fig. 7 shows the change in stress-intensity, DK, field inferred from the crack-tip displacement field as a function of the fraction of the maximum load, Pmax . For both the thick and thin CT specimens, two characteristic shapes of the curves can be identified: (i) Curved for which there is a knee below which the stress intensity does not increase despite increasing applied load followed by a linear relation until a maximum value is reached. (ii) Proportional response rising linearly from zero stress-intensity at a minimum load, Pmin to a maximum, K max at maximum load, Pmax . For the thin CT specimen (Fig. 7c) a clear knee is evident for the pffiffiffiffiffi baseline fatigue attaining a value of around DK ¼ 23 MPa m at Pmax . Upon OL the curves lose the knee showing a linear depenpffiffiffiffiffi dence of DK with applied load reaching a value of 20 MPa m at Pmax (Fig. 7d), with increasing numbers of cycles as the crack grows through the retardation zone the curve becomes increasingly non pffiffiffiffiffi linear at low loads such that DK attains a level of 18 MPa m at Pmax . Once out of the retardation zone (cycles >+90k) DK is non-linpffiffiffiffiffi ear and reaches a higher value of DK 25 MPa m at Pmax because the crack is now considerably longer (a=W = 0.5) than for the original baseline fatigue (a/W = 0.35).
The response for the thick sample does not show such a clear knee either for the baseline fatigue or in the retardation region possibly because we are examining the surface (plane stress) displacements of a thick (predominantly (plane strain) sample. As for the thick sample the curve is linear in the accelerated growth regime (+4k cycles) but the development of a knee in the retardation regime (>+4k cycles) is slight (Fig. 7b). As a result while the DK pffiffiffiffiffi does decrease for +45k cycles the DK range at (20 MPa m) is greatpffiffiffiffiffi er than for the baseline fatigue case (19 MPa m at) contrary to expectations. However, it should be noted that the DK curves do not appear to be very sensitive to the changes in growth rate occurring as the crack grows through the different regimes.
4. Discussions The current DCOD results suggest that the baseline fatigue cycles show significant crack closure prior to the introduction of a 100% OL cycle. The variation in the load at which a knee is observed upon unloading ðP cl P min Þ=ðP max Pmin Þ indicative of ‘closure’ is plotted in Fig. 8 as a function of crack growth. The baseline fatigue level is about 0.15 for both thick and thin samples. Immediately after the introduction of the OL, the opening load reduces to the minimum load, P min , of the fatigue crack growth cycle. At this point for both thick and thin CT specimens, a significant permanent opening has developed between the crack flanks as shown in Fig. 4. Therefore the crack flanks do not touch each other at this point even at Pmin and no closure effects are observed. This regime continues until the acceleration phase is completed (Fig. 8). In the retardation stage the normalized closure load, (Pcl Pmin)/(Pmax Pmin), increases to a maximum value of about 0.5 similar to that determined by DIC both for the thick and thin specimens (Fig. 8). However, an important observation is made for the thick specimen that the crack closure load is dependent on distance behind the crack-tip. In contrast, the thin specimen
(a)
(c)
(d)
Fig. 7. The stress intensity factor, DK, at different crack length for (a and b) thick CT specimens; and (c and d) thin CT specimens.
23
0
-2
-1
0
1
2
3
+90k
+60k
COD100micron K_6.5mm_dia
+100
0.3
+20k
0.4
0.2 0.1 0
4
COD400micron
+160k
0.1
0.5
-13k OL
K_6.5mm_dia
+125k
0.2
(Pcl-Pmin)/(Pmax-Pmin)
0.3
(b)
COD100micron
OL +100
0.4
0.6
COD400micron
+15k
0.5
+85k
(a)
-20k
(Pcl-Pmin)/(Pmax-Pmin)
0.6
+45k
F. Yusof et al. / International Journal of Fatigue 56 (2013) 17–24
-2
-1
0
Crack length from OL (mm)
1
2
3
4
5
6
7
8
9
10
Crack length from OL (mm)
Fig. 8. The change of closure load with crack length for (a) thick and (b) thin CT specimens referred to the OL cycle based on DCOD and K-Muskhelishvili method.
shows a crack closure load that is largely independent of distance behind the crack-tip. The fact that the DCOD measurements for 100 lm and 400 lm behind the crack in the thick CT specimen give the same response for N < +15k and N > +85k suggest that over this regime the crack faces meet or at least stop moving towards one another approximately instantaneously. This is also true for the whole response for the thin sample. In the intermediate regime of the thick CT specimen (+15k < N < +85k) namely the highly retarded growth regime, only the very near crack-tip region appears to meet. The fact that in this regime, the DCOD is strongly dependent on where it is measured may explain why some researchers claim that crack closure does not occur in thick specimen [3] while numerical analysis [4] suggests it does occur in thick specimens. It should be remembered that the COD measurements are of DCOD and while the faces appear to stop moving towards one another this does not necessarily mean that they meet nor that significant load is transferred across the faces at this distance from the crack. Indeed, there is some evidence that, at least further from the crack-tip, the crack faces are held open and cease to approach one another at loads before they actually touch. This may be because the reverse plastic zone at the crack-tip – or points of contact local to the plastically deformed OL location act to hold the faces apart as the sample is unloaded. Of course it should also be remembered that these measurements while on a sample chiefly in plane strain are measured at the surface [12], suggested that DIC measurements match traditional back face strain gauge measurements only for thin samples because of this. Fig. 8 also demonstrates that the crack closure load returns to the pre-OL condition at different crack lengths for the (x > 3 mm) thick and (x > 7–8 mm) thin CT specimen. Following [20], the forward and reverse plastic zone size ahead of the crack-tip can be represented by:
rp ¼
2 1 K OL ap bro
ð7Þ
where a is 2 for plane stress and 3 for plane strain while b = 1 and 2 for the forward and reverse plastic zones respectively. The plane strain constant values were adopted from [27]. Using Eq. (7), the yield stress value for 316L steel is typically around 300 MPa while significant work hardening can lead to ultimate tensile strength of 600 MPa. This value fits with the synchrotron analysis of [18] that suggested the plasticity for 316L steel was well represented by a work hardened yield stress of around 600 MPa. Taking the maximum values, the calculated plastic zones are 3.4 mm and 0.8 mm for the Kmax and the Kmin for the thick specimen and 8.4 mm and 2.1 mm respectively for the thin specimen. From this it can be inferred that the distance ahead of the OL event over which the crack growth rate is affected, i.e. 2 mm and 6 mm for the thick and
thin samples respectively (in Fig. 8) is in good agreement with the affected plastic zone given by Eq. (7). The opening load inferred from the K-Muskhelishvili fit has also been plotted in Fig. 8. A similar pattern is seen in the closure load derived from the K-Muskhelishvili method. However, the extent of crack growth over which the closure level is affected by the OL is smaller compared to the DCOD method. The reason for this is probably that the K-Muskhelishvili method uses a full displacement field that smoothes out the average displacement field over regions close to and further from the crack plane, therefore providing a less sensitive tool for identifying crack closure. These results suggest that it may be difficult to get an accurate assessment of the opening load in problems of crack closure arising from OL or non-uniform loading as the history of the load changes the displacement field and this impairs the Fourier fit used to infer the stress-intensity, K. The fact that, for the thick sample, the center is under plane strain and the surface plane stress may also compound the issue of the Fourier fit over the full displacement field. In contrast, when the displacement field contains a uniform displacement field, the closure load is comparable to the DCOD method as shown in (Fig. 8a and b) in the distances before the OL and when the crack growth returns to pre-OL crack growth level.
5. Conclusions In this paper, the surface displacement field has been mapped by DIC local to growing fatigue cracks. This has been used to study OL effects on fatigue growth for thick and thin CT specimens. We have used DCOD and a K-Muskhelishvili method to infer the closure load. The effect of closure is known to increase as the thickness of the sample is reduced because of the larger plastic zone associated with plane stress. The current work shows a good correlation between the likely extent of the extended plastic zone introduced by the OL event and the extent of the perturbed fatigue crack growth zone. From the DCOD and the K-Muskhelishvili methods, the classical knee interpreted in terms of crack closure following an OL can be observed. It should be remembered that while the faces appear to stop moving towards one another this does not necessarily mean that the crack faces touch or that significant load is transferred across the faces at this distance from the crack. Indeed, there is some evidence that, at least further from the crack-tip, the crack faces are held open and cease to approach one another at loads before they actually touch. This may be due to the reverse plastic zone at the crack-tip, or points of contact local to the plastically deformed OL location, act to hold the faces apart as the sample is unloaded. We have found for the thick specimen that there is a period during crack retardation where the crack zips and unzips behind the crack-tip during loading.
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The reason for this behavior may be linked to the inherent stress triaxiality that influences the displacement field. Possibly due to variation of the displacement field within the DIC image window or the transition from plane strain in the interior to the plane stress at the surface where the strains are monitored, the K-Muskhelishvili approach to infer the stress intensity factor has not been able to capture as sensitive a picture of the closure load as that obtained by the DCOD observations. Although useful insight to the state of crack closure in thick and thin specimens at the surface has been obtained by DIC measurements, these results need to be linked to bulk analysis of the mid-thickness region for the thick CT and the thin CT specimen so that a complete picture of closure after an OL event can be obtained. This is the aim of further on-going research. Acknowledgements The authors are grateful to Dr. Anton Shterenlikht who led the development of the stress intensity factor algorithms on which the K-Muskhelishvili approach was founded. Thanks are also due to Dr. Joseph Kelleher for providing the AISI 316L steel alloys and Mr. David Mortimer of the Mechanical, Aerospace and Civil Engineering (MACE) Department of the University of Manchester for access to a servo-hydraulic fatigue machine. Acknowledgements are also due to Engineering and Physical Sciences Research Council (EPSRC), UK, through Grants EP/D029201/1 and EP/F028431/1 from which this Project was funded. Lastly, we also thank the two anonymous reviewers for their valuable comments to improve the quality of the paper. References [1] Elber W. Fatigue crack closure under cyclic tension. Eng Fract Mech 1970;2:37–45. [2] James MN. Some unresolved issues with fatigue crack closure – measurement, mechanisms and interpretation problems. In: Karihaloo BL, (editor). Advances in fracture research, proceedings of the ninth international conference on fracture. Sydney, Australia: Pergamon Press; 1996. [3] Sadananda K, Vasudevan AK, Holtz RL, Lee EU. Analysis of overload effects and related phenomenon. Int J Fatigue 1999;21:181–92. [4] Alizadeh H, Hills DA, de Matos PFP, Nowell D, Pavier MJ, Paynter RJ, et al. A comparison of two and three-dimensional analyses of fatigue crack closure. Int J Fatigue 2007;29:222–31. [5] Fleck NA. Compliance methods for measurement of crack length. In: Marsh KJ, Smith RA, Ritchie RO, editors. Fatigue crack measurement: techniques and applications. EMAS Publishing; 1991. [6] Donald JK, Ruschau J. Direct current potential difference fatigue crack measurement techniques. In: Marsh KJ, Smith RA, Ritchie RO, editors.
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Fatigue crack measurement: techniques and applications. EMAS Publishing; 1991. Ritchie RO. Mechanisms of fatigue crack propagation in metals, ceramics and composites: role of crack tip shielding. Mater Sci Eng, A 1988;103:15–28. Withers PJ. Strain measurement by digital image correlation: guest editorial. Strain 2008;44:421–2. Lopez-Crespo P, Shterenlikht A, Yates JR, Patterson EA, Withers PJ. Some experimental observations on crack closure and crack-tip plasticity. Fatigue Fract Eng Mater Struct 2009;32:418–29. Nowell D, Paynter RJH, de Matos PFP. Optical methods for measurement of fatigue crack closure: moiré interferometry and digital image correlation. Fatigue Fract Eng Mater Struct 2010;00:1–13. http://dx.doi.org/10.1111/ j.1460-2695.2010.01447.x. Zhang W, Liu Y. Investigation of incremental fatigue crack growth mechanisms using in situ SEM testing. Int J Fatigue 2012;42:14–23. http://dx.doi.org/ 10.1016/j.ijfatigue.2011.03.004. De Matos PFP, Nowell D. Experimental and numerical investigation of thickness effects in plasticity-induced fatigue crack closure. Int J Fatigue; 2008 [doi:10.1016/j.ijfatigue.2008.12.003]. Zhang W, Liu Y. Plastic zone size estimation under cyclic loadings using in situ optical microscopy fatigue testing. Fatigue Fract Eng Mater Struct 2011;34:717–27. http://dx.doi.org/10.1111/j.1460-2695.2011.01567.x. Yusof F, Withers PJW. Real-time acquisition of fatigue crack images for monitoring crack-tip stress intensity variations within fatigue cycles. J Strain Anal Eng Des 2009;44:149–58. http://dx.doi.org/10.1243/03093247JSA440. Zhang W, Liu Y. In situ SEM testing for crack closure investigation and virtual crack annealing model development. Int J Fatigue doi:10.1016/ j.ijfatigue.2012.04.003. Nowell D, de Matos PFP. Application of digital image correlation to the investigation of crack closure following overload. Fatigue 2010;2:1035–43. Carlson RL, Kardomateas GA, Bates PR. The effects of overloads in fatigue crack growth. Int J Fatigue 1991;13:453–60. Lopez-Crespo P, Withers PJ, Yusof F, Dai H, Steuwer A, Kelleher JF, et al. Overload effects on fatigue crack-tip fields under plane stress conditions: surface and bulk analysis. Fatigue Fract Eng Mater Struct 2012;36:75–84. http://dx.doi.org/10.1111/j.1460-2695.2012.01670.x. Suresh S. Fatigue of materials. 2nd ed. Cambridge: CUP; 1998. Rice JR. The mechanics of crack tip deformation and extension by fatigue. In: Symposium on fatigue crack growth, ASTM STP 415. Philadelphia: American Society for Testing and Materials; 1967. ASTM E647. Standard test method for measurement of fatigue crack growth rates. West Conshohocken, PA, USA: ASTM International; 2000. Rice JR. Mathematical analysis in the mechanics of fracture. In: Liebowitz H., editor. Fracture: an advanced treatise vol. II. Mathematical fundamentals; 1968. p. 191–311. La Vision GmbH, PIV Software Manual: Da Vis. (La Vision GmbH, Gottingen, Germany). www.lavision.de; 1999. Muskhelishvili NI. Some basic problems in the mathematical theory of elasticity. 4th ed., Leyden, the Netherlands: Noordhoff International; 1977. Lopez-Crespo P, Shterenlikht A, Patterson EA, Yates JR, Withers PJ. The stress intensity of mixed mode cracks determined by digital image correlation. J Strain Anal Eng Des 2008;43(8):769–80. http://dx.doi.org/10.1243/ 03093247JSA419. Sih GC, Paris PC, Erdogan F. Crack-tip stress-intensity factors for plane extension and plate bending problems. J Appl Mech 1962;29:306–12. Hutchinson JW. Fundamentals of the phenomenological theory of nonlinear fracture mechanics. J Appl Mech 1983;105:1042–51.
Contents lists available at ScienceDirect
International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue
Effect of overload on crack closure in thick and thin specimens via digital image correlation F. Yusof a,b,⇑, P. Lopez-Crespo a,c, P.J. Withers a a
School of Materials, University of Manchester, Grosvenor Street, Manchester M13 9PL, UK School of Mechanical Engineering, University of Science Malaysia, Nibong Tebal, 14300 Penang, Malaysia c Department of Civil and Materials Engineering, University of Malaga, C/Dr. Ortiz Ramos s/n, 29071 Malaga, Spain b
a r t i c l e
i n f o
Article history: Received 18 February 2013 Received in revised form 27 June 2013 Accepted 2 July 2013 Available online 18 July 2013 Keywords: Crack closure Overload effects Thickness effects Digital image correlation
a b s t r a c t The displacement field of compact tension (CT) specimens have been mapped by digital image correlation (DIC) local to growing fatigue cracks to study overload effects for plane stress and plane strain. We have extracted crack opening displacement (DCOD) and stress intensity (K) determined by a Muskhelishvili fit to the crack tip displacement field to infer the closure load. In both cases a classical knee was observed upon unloading consistent with closure which disappeared during the accelerated growth following OL, before increasing during retardation. In both cases following OL the crack growth rate is perturbed for a distance similar to the plastic zone. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Fatigue crack closure analysis is central to understanding crack growth, especially when the amplitude of the fatigue cycle varies. Since Elber’s [1] initial discovery of the phenomenon, universal agreement on a number of aspects remain elusive, particularly with respect to the origin of closure [2]. The understanding of crack closure is a subject of argument within the fatigue community. This is largely caused by difficulty in its measurement and the complexity of the crack closure mechanics. Some researchers suggest that crack closure may not occur [3], others believe it may occur only under conditions of plane stress [4]. An overview of the classic compliance and electrical resistance method has been outlined in [5,6]. The compliance method gives more reliable and consistent measurement of the crack closure response than the electrical method. However, conventional crack closure measurements characterize the global response to local crack-tip phenomenon and as such provide little micromechanical information from which the pertinent crack shielding mechanisms can be unambiguously inferred [7]. Recent advances in optical experimental mechanics of fatigue crack growth and fracture problems such as the digital image correlation (DIC) have made it possible to infer the local crack-tip ⇑ Corresponding author at: School of Mechanical Engineering, University of Science Malaysia, Nibong Tebal 14300, Penang, Malaysia. Tel.: +6045996316: Fax: +6045941025. E-mail address: [email protected] (F. Yusof). 0142-1123/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijfatigue.2013.07.002
displacement fields at sufficient accuracy [8–10]. DIC has been used to study fatigue crack growth in thin single edge cracked samples in tension [11] and thickness effects in compact tension (CT) specimens [12], to study development of plastic zone size [13] and to extract the instantaneous stress intensity variation, K, by high speed photography [14]. DIC has also been used to study crack closure [15] and the effect of crack closure due to a single OL perturbation during fatigue cycling of different thickness CT samples [16], based on a near tip crack opening displacement (DCOD) as a function of load. Both approaches involve the collection of a set of images within a loading cycle to study changes in closure. For a complete loading and unloading cycle, it was shown that the DCOD changes with distance behind the crack-tip while the crack closure load within a typical DIC length scale behind the cracktip is similar for a given fatigue crack growth condition [16]. However following an OL event, the crack closure changes according to characteristic of (i) acceleration, (ii) attenuation, and (iii) recovery regimes of crack growth rate [17]. It has been demonstrated that a change of the crack closure in the acceleration and attenuation regimes occurs within small numbers of fatigue cycles (order of tens to hundreds) from the point of OL while the recovery regimes occurs over thousands of cycles [18]. The existing literature has usually focused between ten thousand and a hundred thousand cycles and important information on crack closure development within the different regimes may have been missed due to this. A DIC crack-tip displacement field can give a huge number of displacement vectors to identify a state of crack closure at any
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stage of crack growth. However, the distance at which DCOD should be measured behind the crack-tip to determine crack closure load has never been regularized. In [12,16], a minimum distance behind the crack-tip of 150 lm and increments of 100 lm from the minimum distance for additional measures of DCOD were used. During the acceleration and attenuation regimes, fatigue crack growth rates in the near threshold region for a typical steel specimen can be around a hundredth of a micron over a number of cycles [19]. Based on this, it was hypothesized that the distance of DCOD from the crack-tip must be a function of the growth rate associated with the different regimes for accurate determination of the state of crack closure after an OL. In this paper, the main objective of the experimental study is to elucidate the state of crack closure perturbed by an OL for CT specimens of plane strain (hereafter denoted thick) and plane stress (hereafter denoted thin) by biasing the incremental cyclic intervals for DIC analysis as a function of different regimes of crack growth rate. The second objective is to compare two different methods of inferring the crack closure load from the DIC data namely: (i) the DCOD and (ii) the stress intensity factor inferred by the K-Muskhelishvili fit to the near crack-tip displacement field. The DCOD technique is widely used and it is essentially a local measurement of crack closure with reference to the crack tip however the K-Muskhelishvili fit uses an array of displacement vectors around a crack tip to give a fracture mechanics parameter K independent of the K calibration factors. Finally, based on the experimental observations, the crack growth regimes for the thick and thin CT specimens are to be compared to a fatigue crack growth deformation zone proposed by [20]. This paper is organized as follows. First, the experimental details which include the materials and specimens, fatigue crack growth and the DIC method employed in this work are presented. Next the DIC data are manipulated to give DCOD and the K-Muskhelishvili fit as a function of load. Following this the measured crack closure for the thick and thin specimens based on the DCOD and the K-Muskhelishvili fit are discussed in detail. The change of crack closure loads due to the single OL are mapped over the crack length and number of elapsed cycles using the DCOD and the KMuskhelishvili approaches and this is related to Rice’s fatigue crack growth deformation size. Finally, some conclusions and future work are given based on the current investigation.
2. Experimental details 2.1. Materials and specimens The material used in the study is 316L austenitic stainless steel alloy having a Young’s modulus, E around 195 GPa and yield stress, ro = 304 MPa. Fatigue crack growth specimens were prepared following a standard CT specimen configuration according to [21]. Usually a thick specimen can be specified according to the ratio of the thickness, B to the uncracked ligament length, (W-a), as B/ðW aÞ P 1 and a thin specimen can be specified as B=ðW aÞ 6 0:1. Due to experimental limitations, such as the load range of the universal testing machine and the as supplied specimen materials, the ratio of the thickness to the uncracked ligament length was chosen to be 0.5 for the thick (B = 12 mm) specimen and 0.1 for the thin (B = 3 mm) specimen. For the thick specimen the in-plane ligament length for the CT specimens was W = 50 mm. Prior to the experimental work, to ensure that the deformation field for the selected thickness can represent the plane strain and plane stress field, finite element analysis of the selected CT geometries under a given K-field were compared to a displacement boundary layer formulation as given in [22]:
ux uy
K I r 1=2 ¼ 2ð1 þ mÞ E 2p
(
2
cos ðh=2½j 1 þ 2 sin ðh=2Þ
)
2
sin ðh=2Þ½j þ 1 2 sin ðh=2Þ
ð1Þ
where ux and uy are the in-plane horizontal and vertical displacements and where (r, h) are polar coordinates centered at the crack-tip. j ¼ ð3 4mÞ for plane strain and j ¼ ð3 mÞ=ð1 þ mÞ for a generalized plane stress conditions where m is the Poisson’s ratio. The crack-tip displacement field generated from the boundary layer formulation at identical applied remote K field was compared to the K field inferred from the CT specimen through the DIC. It was found that the deformation fields for the thick and thin CT specimens were identifiable as the plane strain and plane stress field respectively within the linear elastic fracture mechanics limit. Due to space limitations, results are not shown here but the technique adopted has been discussed [14] with good agreement. 2.2. Fatigue crack growth experiments Four specimens for each thickness were used in this experiment. For reference purposes the critical fracture toughness for the thick and thin specimens were determined to be pffiffiffiffiffi pffiffiffiffiffi K c ¼ 35 MPa m and 45 MPa m respectively. The fatigue crack growth experiments were conducted at room temperature on a servo-hydraulic testing machine with a ±10 kN loading range. A schematic of the experiment is shown in Fig. 1. The samples were fatigued at a frequency, f = 30 Hz based on a triangular waveform and the DIC images were captured at various stages of crack growth at a much lower fatigue frequency (1/100 Hz). The triangular waveform allows for a relatively simple identification of any knee in the crack opening as a sharp change from the unloading to loading sequence is otherwise expected within a fatigue cycle. For the thick specimens the loading amplitude was represented by maximum and minimum loads of Pmax = 2.95 kN and Pmin = 0.15 kN giving a stress ratio, R ¼ 0:05 while for the thin specimen the fatigue cycling conditions were set at Pmax = 2.1 kN and Pmin = 0.105 kN giving a similar nominal stress ratio to that for the thick specimen. The OL cycle was applied once the fatigue crack growth rate had become constant some distance from the fatigue pre-cracking zone. To initiate a sharp crack from the machined notch, and to ensure that the final stress intensity factor amplitude, DK, from the pre-cracking does not exceed the initial DK for the crack closure experiment, the fatigue pre-cracking was initiated from the machined notch following a 3 stepped reduction in load amplitude to the baseline DK for the crack closure experiment. Before the OL cycle was introduced, the sharp crack was grown to a distance greater than twice the wake zone size of the fatigue pre-cracking. The OL cycle involved raising the stress intensity
Fig. 1. Schematic representation of the experimental configuration.
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pffiffiffiffiffi pffiffiffiffiffi from a baseline value K max ¼ 17 MPa m to K OL ¼ 34 MPa m for the thick CT specimen (100% OL) and from a baseline of pffiffiffiffiffi pffiffiffiffiffi K max ¼ 21:4 MPa m to K OL ¼ 42:8 MPa m for the thin CT specimen (100% OL). The fatigue crack growth curves for the thick and thin specimens are shown first in terms of crack length, a, with respect to the number of cycles, N, in Fig. 2a and c. The crack growth rate, da=dN, plotted as a function of ratio of crack length to ligament length, a=W, for the thick and thin specimens are shown in Fig. 2b and d. Typical behavior of thick and thin CT specimens can be identified from the figures. In general, both the thick and thin specimens demonstrated the classical acceleration, retardation and recovery regimes of crack growth rate after OL [17]. Fig. 2 shows a much greater acceleration of the crack growth for the thin sample compared to the thick sample immediately after OL. Also in common with previous work, it takes the thin specimen a greater number of cycles (25% more) after OL to recover to the original crack growth rate than the thick specimen.
by 75% giving 150,000 displacement vectors over the viewing area (Fig. 3a). For the COD and K-Muskhelishvili data analysis, it was found that using a fraction of displacement vectors (average of 6 thousands displacement vectors) can give sufficiently accurate results thus reducing the computing time to process the displacement data to get DCOD directly and K fitted from the Fourier fit (Fig. 3b). For the present work, an average of 200 images was correlated per cycle of crack growth.
3. Experimental results 3.1. Crack opening displacement (DCOD) From the correlated images, a post processing routine has been used to identify the change of crack opening displacement, DCOD, at various distances behind the crack-tip by subtracting the vertical displacements (uy direction shown in Fig. 1) of the top flank and bottom flank of the crack mouth: bot DCODðxÞ ¼ utop y uy
ð2Þ
2.3. Crack monitoring by digital image correlation where the DCOD varies as a function of distance behind the crack-tip, x. In the present work, DCODðxÞ has been characterized at different distances, x, behind the crack-tip. Fig. 4a and b show the evolution of the DCOD (100 lm behind the crack-tip) through selected loading cycles. The variation of the DCOD for the thick and thin CT specimens at different fatigue crack growth stages and measured at various distances behind the crack-tip are shown in much more detail in Figs. 5 and 6. Normally the point of closure might be identified by a knee in the crack compliance curve recorded using a back-face strain gauge [12]. Here we see a similar knee as the DCOD remains at zero until a significant load has been applied after which it increases linearly with further loading.
Crack length, a (mm)
37
(a) +125k
35
+25k
33 31
+100
29
+4k
-10k 27 0
50
100
150
Crack growth rate (x1e-8), da/dN
The DIC method described by [14,18] was applied in this work. The experimental setup comprised a macro-lens with a teleconvertor mounted on a 4-mega pixel CCD camera (see Fig. 1). A fiber-optic ring was attached to the periphery of the lens to achieve uniform illumination of the specimen surface that had been finely abraded with SiC paper to obtain a random abrasion pattern giving sufficient contrast for the correlation algorithm (Fig. 3a). An area of 10 mm 10 mm was imaged corresponding to a 2048 2048 pixels from which each pixel represents about 5 lm. DIC was performed using commercially available [23] image correlation software correlating 32 32 pixel patches overlapped
200
16
(b)
12
-10k +125k
8
+25k
4
+100 +4k 0 0.55
0.6
Number of cycles, kN
+160k
25
(c)
+100
23
+90k
0
21
+40k 19
-13k 17 0
100
200
Number of cycles, kN
300
Crack growth rate (x1e-8) da/dN
Crack length, a (mm)
27
0.65
0.7
Crack length, a/W 30
(d) 25
+100 20 15
+160k
10
+90k
5
-13k +40k
0 0.3
0.35
0.4
0.45
Crack length, (a/W)
Fig. 2. Fatigue crack growth for thick (a and b) and thin (c and d) CT specimens.
0.5
0.55
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Fig. 3. (a) An inferred DIC displacement map in the area surrounding the crack-tip superimposed upon one of the reference images and (b) the vertical (uy – crack opening direction) displacement field for 5676 points for maximum load, Pmax, for the thick fatigue crack growth at +125k OL cycles. Both figures demonstrate the crack-tip displacement field used to infer DCOD and the stress intensity, K.
150
25
(a)
(b) 125 0 (OL)
ΔCOD (micron)
ΔCOD (micron)
20 0 (OL)
15 +125k
10
+100 -50k
5
100
+160k -13k
75
+100
50 25
+90k
+45k
0
0 0
1
2
3
4
0
1
Fraction of Pmax
2
3
4
Fraction of Pmax
Fig. 4. The opening load as a function of the normalized load (P/Pmax) measured at a distance of 100 lm behind the crack-tip during overload and selected other cycles for (a) the thick and (b) the thin CT samples.
14
312
(a)
208
12
104
10 8 6 4 2 0
0
0.2
0.4
0.6
0.8
1
1.2
18 16 14 12 10 8 6 4 2 0
312 208 104
0
0.2
(d)
785 681 577 473 421 317 213 109 5
0
0.2
0.4
0.6
0.8
Fraction of Pmax (kN)
0.4
0.6
0.8
1
1.2
18 16 14 12 10 8 6 4 2 0
1
1.2
18 16 14 12 10 8 6 4 2 0
312 208 104
0
0.2
(e)
426 218 104
0.2
0.4
0.6
0.8
Fraction of Pmax (kN)
0.4
0.6
0.8
1
1.2
Fraction of Pmax (kN)
842
0
(c)
416
Fraction of Pmax (kN)
ΔCOD (micron)
ΔCOD (micron)
Fraction of Pmax (kN) 18 16 14 12 10 8 6 4 2 0
(b)
416
ΔCOD (micron)
416
ΔCOD (micron)
16
ΔCOD (micron)
ΔCOD (micron)
18
1
1.2
18 16 14 12 10 8 6 4 2 0
(f)
2787 416 312
208 104
0
0.2
0.4
0.6
0.8
1
1.2
Fraction of Pmax (kN)
Fig. 5. The crack opening displacement for thick CT specimens at different crack growth intervals measured at increasing distances behind the crack tip as shown by the legend in each plot.
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16
12 10 8 6 4
12 10
4
0
0
0.4
0.6
0.8
1
1.2
0
16
0.2
6 4 2 0
0.2
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Fraction of Pmax (kN)
6 4
0
0.2
215
10
100
1
1.2
0.4
0.6
0.8
1
16
8 6 4 2
0.4
8
0
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(e)
330
12
0
(c)
1.2
Fraction of Pmax (kN)
445
14
ΔCOD (micron)
ΔCOD (micron)
8
0
(d)
750 700 600 500 400 300 200 100 50 25
10
10
Fraction of Pmax (kN)
16 12
12
2
Fraction of Pmax (kN)
14
445 330 215 100
14
6 2
0.2
16
(b)
8
2 0
445 330 215 100
14
ΔCOD (micron)
ΔCOD (micron)
14
(a)
ΔCOD (micron)
445 330 215 100
(f)
445 330 215 100
14
ΔCOD (micron)
16
12 10 8 6 4 2
0
0.2
0.4
0.6
0.8
1
1.2
0
0
Fraction of Pmax (kN)
0.2
0.4
0.6
0.8
1
1.2
Fraction of Pmax (kN)
Fig. 6. The crack opening displacement for thin CT specimens at different crack growth intervals measured at increasing distances behind the crack-tip as shown by the legend in each plot.
Accordingly, the opening (closing) load can be estimated as the load at which a drastic change in slope takes place during the loading (unloading) portion of the cycle. In this work, both the opening and closure loads will be used interchangeably because in our case they occur at the same load. Figs. 5 and 6 show the variation in the DCOD for the thick and thin CT specimens as a function of distance from the crack-tip for fatigue cycles before or after the OL event from which a number of trends are evident. 3.1.1. Prior to the OL A clear knee is observed at all positions behind the crack (perhaps indicative of instantaneous full-face contact) at a baseline closure load under steady state fatigue cycling. 3.1.2. During the OL event A large amount of plastic deformation occurs during the OL for both the thick and thin CT specimens significantly increasing the crack opening as a result of which immediately after the OL cycle the crack is left with a significant residual DCOD. This amounts to 8 lm and 95 lm for the thick and thin samples at x = 100 lm. In other words the crack faces are significantly open after the OL, more so for the thin than the thick sample. 3.1.3. Acceleration phase No knee (closure) upon loading and unloading during the acceleration phase immediately after the OL. 3.1.4. During the retardation phase An increasingly prominent knee initiating at much earlier upon unloading than for baseline fatigue (closure load 3 greater) near to the crack-tip. The knee occurs at a non-zero DCOD and at lower and lower loads during unloading with increasing distance from the crack. 3.1.5. Recovered crack growth rate When the crack has progressed sufficiently beyond the OL event there is a return to the baseline closure load.
These results suggest that during baseline crack growth the crack faces close instantaneously, but that the significant plastic opening of the crack faces during OL means that the crack faces do not meet during the acceleration phase. After further crack growth the crack faces close progressively from the crack-tip towards the notch and cease to close further at a load which is lower the greater the distance from the crack-tip, but rather than touch along their length the faces appear to be held open at low loads. A distinct difference between the thick and thin specimen can be seen in that the thick specimen shows a more significant change of the closure load with increasing distance behind the crack-tip. This progressive closing of the crack from the crack-tip along the crack faces in a zipping action is particularly marked for the thick sample. As a result, the DCOD is dependent on the distance measured from behind the crack-tip. In general, the knee signifies some form of crack ‘closure’ has occurred in that the crack ceases to close further although the crack may not actually be closed at that location. 3.2. K-Muskhelishvili method The displacement fields such as those shown in Fig. 3b can also be used to infer the stress intensity factor, K, based on the Muskhelishvili method [24], as reported in [25]. To correct the local cracktip plasticity effect within linear elasticity, Irwin’s K-correction approach; acorr ¼ a þ rp is used. In this instance, the crack-tip displacement field from the images can be represented by the Muskhelishvili elastic solutions. The Muskhelishvili method is based on the determination of the complex potential from a given boundary conditions. A system of real linear equations can then be developed based on a known crack-tip location: N X
Ajk ak þ
k¼N N X k¼N
N X
Bjk bk ¼ 2l ujx
ð3Þ
Djk bk ¼ 2l ujy
ð4Þ
k¼N
C jk ak þ
N X k¼N
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F. Yusof et al. / International Journal of Fatigue 56 (2013) 17–24
where j = 1, 2, . . ., p; ujx and ujy are the displacements at point j; Ajk ; Bjk ; C jk and Djk are known functions of positions; l is the shear modulus and
ak ¼ ak þ ibk
ð5Þ
The complex stress intensity factor is given according to [26]:
rffiffiffiffi K ¼ K I iK II ¼ 2
N pX
a k¼N
kak
ð6Þ
where a ¼ 2r cc is half-length of a center crack in an infinite plane. Fig. 7 shows the change in stress-intensity, DK, field inferred from the crack-tip displacement field as a function of the fraction of the maximum load, Pmax . For both the thick and thin CT specimens, two characteristic shapes of the curves can be identified: (i) Curved for which there is a knee below which the stress intensity does not increase despite increasing applied load followed by a linear relation until a maximum value is reached. (ii) Proportional response rising linearly from zero stress-intensity at a minimum load, Pmin to a maximum, K max at maximum load, Pmax . For the thin CT specimen (Fig. 7c) a clear knee is evident for the pffiffiffiffiffi baseline fatigue attaining a value of around DK ¼ 23 MPa m at Pmax . Upon OL the curves lose the knee showing a linear depenpffiffiffiffiffi dence of DK with applied load reaching a value of 20 MPa m at Pmax (Fig. 7d), with increasing numbers of cycles as the crack grows through the retardation zone the curve becomes increasingly non pffiffiffiffiffi linear at low loads such that DK attains a level of 18 MPa m at Pmax . Once out of the retardation zone (cycles >+90k) DK is non-linpffiffiffiffiffi ear and reaches a higher value of DK 25 MPa m at Pmax because the crack is now considerably longer (a=W = 0.5) than for the original baseline fatigue (a/W = 0.35).
The response for the thick sample does not show such a clear knee either for the baseline fatigue or in the retardation region possibly because we are examining the surface (plane stress) displacements of a thick (predominantly (plane strain) sample. As for the thick sample the curve is linear in the accelerated growth regime (+4k cycles) but the development of a knee in the retardation regime (>+4k cycles) is slight (Fig. 7b). As a result while the DK pffiffiffiffiffi does decrease for +45k cycles the DK range at (20 MPa m) is greatpffiffiffiffiffi er than for the baseline fatigue case (19 MPa m at) contrary to expectations. However, it should be noted that the DK curves do not appear to be very sensitive to the changes in growth rate occurring as the crack grows through the different regimes.
4. Discussions The current DCOD results suggest that the baseline fatigue cycles show significant crack closure prior to the introduction of a 100% OL cycle. The variation in the load at which a knee is observed upon unloading ðP cl P min Þ=ðP max Pmin Þ indicative of ‘closure’ is plotted in Fig. 8 as a function of crack growth. The baseline fatigue level is about 0.15 for both thick and thin samples. Immediately after the introduction of the OL, the opening load reduces to the minimum load, P min , of the fatigue crack growth cycle. At this point for both thick and thin CT specimens, a significant permanent opening has developed between the crack flanks as shown in Fig. 4. Therefore the crack flanks do not touch each other at this point even at Pmin and no closure effects are observed. This regime continues until the acceleration phase is completed (Fig. 8). In the retardation stage the normalized closure load, (Pcl Pmin)/(Pmax Pmin), increases to a maximum value of about 0.5 similar to that determined by DIC both for the thick and thin specimens (Fig. 8). However, an important observation is made for the thick specimen that the crack closure load is dependent on distance behind the crack-tip. In contrast, the thin specimen
(a)
(c)
(d)
Fig. 7. The stress intensity factor, DK, at different crack length for (a and b) thick CT specimens; and (c and d) thin CT specimens.
23
0
-2
-1
0
1
2
3
+90k
+60k
COD100micron K_6.5mm_dia
+100
0.3
+20k
0.4
0.2 0.1 0
4
COD400micron
+160k
0.1
0.5
-13k OL
K_6.5mm_dia
+125k
0.2
(Pcl-Pmin)/(Pmax-Pmin)
0.3
(b)
COD100micron
OL +100
0.4
0.6
COD400micron
+15k
0.5
+85k
(a)
-20k
(Pcl-Pmin)/(Pmax-Pmin)
0.6
+45k
F. Yusof et al. / International Journal of Fatigue 56 (2013) 17–24
-2
-1
0
Crack length from OL (mm)
1
2
3
4
5
6
7
8
9
10
Crack length from OL (mm)
Fig. 8. The change of closure load with crack length for (a) thick and (b) thin CT specimens referred to the OL cycle based on DCOD and K-Muskhelishvili method.
shows a crack closure load that is largely independent of distance behind the crack-tip. The fact that the DCOD measurements for 100 lm and 400 lm behind the crack in the thick CT specimen give the same response for N < +15k and N > +85k suggest that over this regime the crack faces meet or at least stop moving towards one another approximately instantaneously. This is also true for the whole response for the thin sample. In the intermediate regime of the thick CT specimen (+15k < N < +85k) namely the highly retarded growth regime, only the very near crack-tip region appears to meet. The fact that in this regime, the DCOD is strongly dependent on where it is measured may explain why some researchers claim that crack closure does not occur in thick specimen [3] while numerical analysis [4] suggests it does occur in thick specimens. It should be remembered that the COD measurements are of DCOD and while the faces appear to stop moving towards one another this does not necessarily mean that they meet nor that significant load is transferred across the faces at this distance from the crack. Indeed, there is some evidence that, at least further from the crack-tip, the crack faces are held open and cease to approach one another at loads before they actually touch. This may be because the reverse plastic zone at the crack-tip – or points of contact local to the plastically deformed OL location act to hold the faces apart as the sample is unloaded. Of course it should also be remembered that these measurements while on a sample chiefly in plane strain are measured at the surface [12], suggested that DIC measurements match traditional back face strain gauge measurements only for thin samples because of this. Fig. 8 also demonstrates that the crack closure load returns to the pre-OL condition at different crack lengths for the (x > 3 mm) thick and (x > 7–8 mm) thin CT specimen. Following [20], the forward and reverse plastic zone size ahead of the crack-tip can be represented by:
rp ¼
2 1 K OL ap bro
ð7Þ
where a is 2 for plane stress and 3 for plane strain while b = 1 and 2 for the forward and reverse plastic zones respectively. The plane strain constant values were adopted from [27]. Using Eq. (7), the yield stress value for 316L steel is typically around 300 MPa while significant work hardening can lead to ultimate tensile strength of 600 MPa. This value fits with the synchrotron analysis of [18] that suggested the plasticity for 316L steel was well represented by a work hardened yield stress of around 600 MPa. Taking the maximum values, the calculated plastic zones are 3.4 mm and 0.8 mm for the Kmax and the Kmin for the thick specimen and 8.4 mm and 2.1 mm respectively for the thin specimen. From this it can be inferred that the distance ahead of the OL event over which the crack growth rate is affected, i.e. 2 mm and 6 mm for the thick and
thin samples respectively (in Fig. 8) is in good agreement with the affected plastic zone given by Eq. (7). The opening load inferred from the K-Muskhelishvili fit has also been plotted in Fig. 8. A similar pattern is seen in the closure load derived from the K-Muskhelishvili method. However, the extent of crack growth over which the closure level is affected by the OL is smaller compared to the DCOD method. The reason for this is probably that the K-Muskhelishvili method uses a full displacement field that smoothes out the average displacement field over regions close to and further from the crack plane, therefore providing a less sensitive tool for identifying crack closure. These results suggest that it may be difficult to get an accurate assessment of the opening load in problems of crack closure arising from OL or non-uniform loading as the history of the load changes the displacement field and this impairs the Fourier fit used to infer the stress-intensity, K. The fact that, for the thick sample, the center is under plane strain and the surface plane stress may also compound the issue of the Fourier fit over the full displacement field. In contrast, when the displacement field contains a uniform displacement field, the closure load is comparable to the DCOD method as shown in (Fig. 8a and b) in the distances before the OL and when the crack growth returns to pre-OL crack growth level.
5. Conclusions In this paper, the surface displacement field has been mapped by DIC local to growing fatigue cracks. This has been used to study OL effects on fatigue growth for thick and thin CT specimens. We have used DCOD and a K-Muskhelishvili method to infer the closure load. The effect of closure is known to increase as the thickness of the sample is reduced because of the larger plastic zone associated with plane stress. The current work shows a good correlation between the likely extent of the extended plastic zone introduced by the OL event and the extent of the perturbed fatigue crack growth zone. From the DCOD and the K-Muskhelishvili methods, the classical knee interpreted in terms of crack closure following an OL can be observed. It should be remembered that while the faces appear to stop moving towards one another this does not necessarily mean that the crack faces touch or that significant load is transferred across the faces at this distance from the crack. Indeed, there is some evidence that, at least further from the crack-tip, the crack faces are held open and cease to approach one another at loads before they actually touch. This may be due to the reverse plastic zone at the crack-tip, or points of contact local to the plastically deformed OL location, act to hold the faces apart as the sample is unloaded. We have found for the thick specimen that there is a period during crack retardation where the crack zips and unzips behind the crack-tip during loading.
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The reason for this behavior may be linked to the inherent stress triaxiality that influences the displacement field. Possibly due to variation of the displacement field within the DIC image window or the transition from plane strain in the interior to the plane stress at the surface where the strains are monitored, the K-Muskhelishvili approach to infer the stress intensity factor has not been able to capture as sensitive a picture of the closure load as that obtained by the DCOD observations. Although useful insight to the state of crack closure in thick and thin specimens at the surface has been obtained by DIC measurements, these results need to be linked to bulk analysis of the mid-thickness region for the thick CT and the thin CT specimen so that a complete picture of closure after an OL event can be obtained. This is the aim of further on-going research. Acknowledgements The authors are grateful to Dr. Anton Shterenlikht who led the development of the stress intensity factor algorithms on which the K-Muskhelishvili approach was founded. Thanks are also due to Dr. Joseph Kelleher for providing the AISI 316L steel alloys and Mr. David Mortimer of the Mechanical, Aerospace and Civil Engineering (MACE) Department of the University of Manchester for access to a servo-hydraulic fatigue machine. Acknowledgements are also due to Engineering and Physical Sciences Research Council (EPSRC), UK, through Grants EP/D029201/1 and EP/F028431/1 from which this Project was funded. Lastly, we also thank the two anonymous reviewers for their valuable comments to improve the quality of the paper. References [1] Elber W. Fatigue crack closure under cyclic tension. Eng Fract Mech 1970;2:37–45. [2] James MN. Some unresolved issues with fatigue crack closure – measurement, mechanisms and interpretation problems. In: Karihaloo BL, (editor). Advances in fracture research, proceedings of the ninth international conference on fracture. Sydney, Australia: Pergamon Press; 1996. [3] Sadananda K, Vasudevan AK, Holtz RL, Lee EU. Analysis of overload effects and related phenomenon. Int J Fatigue 1999;21:181–92. [4] Alizadeh H, Hills DA, de Matos PFP, Nowell D, Pavier MJ, Paynter RJ, et al. A comparison of two and three-dimensional analyses of fatigue crack closure. Int J Fatigue 2007;29:222–31. [5] Fleck NA. Compliance methods for measurement of crack length. In: Marsh KJ, Smith RA, Ritchie RO, editors. Fatigue crack measurement: techniques and applications. EMAS Publishing; 1991. [6] Donald JK, Ruschau J. Direct current potential difference fatigue crack measurement techniques. In: Marsh KJ, Smith RA, Ritchie RO, editors.
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