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Elucidation of fatigue crack closure behaviour in surface crack by 3-D finite element analysis
International Journalof Fatigue
International Journal of Fatigue 29 (2007) 168–180
www.elsevier.com/locate/ijfatigue
Elucidation of fatigue crack closure behaviour in surface crack by 3-D finite element analysis Joo-Sung Kim *, Jae Youn Kang, Ji-Ho Song Department of Automation and Design Engineering, Korea Advanced Institute of Science and Technology, Seoul 130-012, Republic of Korea Received 8 January 2005; received in revised form 30 May 2005; accepted 21 January 2006 Available online 3 April 2006
Abstract A 3-D elastic–plastic finite element analysis of fatigue crack closure for semi-circular crack is performed. The stabilization behaviour of crack opening level, the effect of mesh size, the effect of initial crack length, the opening process of crack surface and the shortcoming and sensitivity of experimental measurement methods are investigated. The crack opening load tends to increase with the decrease of mesh size. Irrespective of an initial crack length, all the first nodes behind the crack tip along the front of surface crack are the last to open. The plane strain-opening load at the deepest point measured by extensometer at the midpoint of crack mouth may be estimated to be higher than the actual. The crack opening load experimentally measured by strain gauge boned in the vicinity of the surface intersection point can successfully represent the plane stress crack opening of the free surface. 2006 Elsevier Ltd. All rights reserved. Keywords: Fatigue crack closure; 3D elastic–plastic finite element; Stabilization behaviour; Mesh size effect; Surface crack
1. Introduction Many researchers have performed elastic–plastic finite element analyses to study the plastic induced fatigue crack closure and the several remarkable papers have been reported which reviewed the state of art and current issues of the finite element (FE) crack closure. However, most of studies on the FE analysis of the plastic induced fatigue crack closure have been related to the through thickness crack under two-dimensional plane stress or plane strain conditions. The 2-D FE closure modeling issues have been relatively well established including mesh refinement, stabilization, crack advance schemes and comparisons with experimental results. Ohji et al. [1–3], Newman [4,5], Nakagaki and Atluri [6], Nakamura et al. [7], Blom and Holm [8], and Nicho* Corresponding author. Present address: CAE team, Ssangyong Motor Company, 150-3 Chilgoi-dong, Pyungtaek-si, Gyeonggi-do 459-711, Republic of Korea. Tel.: +82 31 610 3425; fax: +82 31 610 3770. E-mail address: [email protected] (J.-S. Kim).
0142-1123/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2006.01.016
las et al. [9] performed 2-D FE closure analyses for plane stress conditions and the studies of Blom and Holm [8], Lalor and Sehitoglu [10], Fleck [11,12] have been reported for plane strain conditions. Their works have been reported focusing on the qualitative investigations on fatigue crack closure conditions, the relations between the mesh size and the plastic zone size at crack tip, the comparisons of crack opening loads between plane stress and plane strain conditions and the effects of stress ratio. In addition, Ohji et al. [1–3], Newman [4], Wu and Ellyin [13], Dougherty et al. [14], Park et al. [15] investigated the mesh size effects on crack opening level. Fleck [11,12] and McClung and Sehitoglu [16,17] reported the well established works on the effects of mesh refinement, a crack advance scheme, material properties, a constitutive model, stress state, an initial crack length and a maximum applied stress on the opening and closing behaviour of cracks, and the stabilization behaviour of fatigue crack closure level. A relatively few works [18–23] on 3-D plastic induced fatigue crack closure have been reported because of taking
J.-S. Kim et al. / International Journal of Fatigue 29 (2007) 168–180
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very much computation time to do the detailed 3-D FE analysis. Particularly, Skinner and Daniewicz [22], and Hou [23] performed 3-D FE analysis of part-through cracks with very refined mesh through the aid of the rapid progress of computers and FE analysis programs. Skinner and Daniewicz investigated the issues of mesh refinement and stabilization, and compared numerical results with experimental results. Hou studied 3-D fine meshing issues, the contact of the cracked surface, stress distributions around surface crack tip and the variations of opening loads along crack front. In this study, 3-D FE analyses on semi-circular crack were carried out focusing on stabilization behaviour, the mesh size effect, the effect of initial crack length, the opening process of semi-circular crack area and the shortcoming and sensitivity of crack opening levels determined relying on remote displacement in experiments. 2. Finite element model A 3-D FE mesh model was created with the commercial software PATRAN. The commercial software, ABAQUS [24] is used for finite element analysis in present study. The theory of incremental rate independent classical plasticity and von Mises yield criteria are used. To simulate the Bauschinger effect associated with reversed yielding, a nonlinear kinematic hardening rule of Prager–Ziegler implemented in ABAQUS is employed. The part-through crack geometry used in this study is shown in Fig. 1, which consists with the specimen used by Kim and Song [25]. Fig. 2 shows the two typical fine mesh configurations employed in this study. Fig. 2(a) shows the model (Model I) used in order to form the plastic wake behind the crack tip by advancing the semi-circular crack from the initial crack length of 0.15 mm considering that an actual part-through crack progresses from the small flaw of surface. Fig. 2(b) shows the model (Model II) used for the most part in this study. Only one quarter of the geometry is meshed utilizing two planes of symmetry. Contact elements (IRS13) implemented in ABAQUS [24] are used at the crack face so that the reaction force is generated during being closed. Model I is composed of 13,559 eight-node solid elements (C3D8), 235 six-node solid elements (C3D6), 1276 contact elements (IRS13). The total number of elements and nodes of the model I are 15,070 and 15,505, respectively. Model II is composed of 7455 eight-node solid elements (C3D8), 105 six-node solid elements (C3D6) and 705 contact elements (IRS13). The total number of elements and nodes of the model II are 8265 and 8536, respectively. All the meshes of the surface crack area of Model I are used with elemental length, Da = 0.035 mm except that the elemental lengths of 0.0015–0.01 mm are used to investigate the mesh size effect using Model II. Material properties used for the analysis are as follows: Young’s modulus E = 71 GPa, Poisson’s ratio n = 0.33, yield stress ry = 555 MPa and bi-linear strain hardening
Fig. 1. Schematic representation of surface crack specimen.
r=dep ¼ 0:01E ¼ 710 MPa. These propparameter H 0 ¼ d erties are typical of 7075-T6 aluminum alloys. The crack advance scheme, or crack-tip node release scheme has been relatively well discussed [26]. Particularly referring to the work of Park et al. [15], the crack-tip node is released at the minimum load of every cycle. In this study, all the nodes immediately behind the crack tip along the front of semi-circular crack are always the first to open and the last to close. Therefore, the crack opening loads are defined as the moment that the nodes immediately behind the crack tip are opened. 3. Results and discussion 3.1. The effect of material model The effect of the variation of stress–strain curve on crack opening loads is studied. FE analyses are performed by advancing one elemental crack, Da = 0.035 mm per cycle from an initial crack length a0 = 2.21 mm to a final crack length a = 2.56 mm with stress ratio R = 0, rmax = 200 MPa. The crack opening loads calculated using four different stress–strain curves are listed in Table 1. Fig. 3 shows the relation of stress–strain experimentally measured [25]. Comparing the results using multi-linear hardening parameters with the other, as described in Table
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Fig. 2. Finite element models for surface crack, (a) Model I, (b) Model II.
Table 1 Effect of plastic stress–strain curve on crack opening load
AL 7075-T6 BL (H=0.1E)
rop,surface/rmax
rop,depth/rmax
0.405 0.402 0.397 0.359
0.197 0.188 0.185 0.163
1, the opening loads decrease along with the increase of strain hardening and the effect of strain hardening on opening load is significant at the deepest point than at the free surface. If a element size of Da = 0.035 mm is used with a initial crack length of a0 = 2.21 mm and the maximum stress of rmax = 200 MPa, the ratios of Da/xp calculated by Eqs. (1) and (2) are 0.174 at the surface and 0.686 at the deepest point, respectively. It cannot satisfy Da/xp 6 0.1 being necessary for reliable crack closure results that McClung and Sehitoglu [16,17] proposed. However, the main purpose of this section is not to determine the precise crack opening
800 Multi-Linear(Test Data)
Stress (MPa)
Multi-linear Perfectly plastic (Et = 0) Bi-linear (Et = 0.01E) Bi-linear (Et = 0.1E)
1000
BL (H=0.01E)
600
Perfectly Plastic 400
200
0 0.00
Strain
0.05
Fig. 3. Stress–strain curve for AL 7075-T6.
value but to investigate the relative effect of the stress– strain curves on the crack opening value. Therefore, we used rather a large mesh (0.035 mm) because of the huge
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amount of computation time and cost required for obtaining the stabilized crack opening stresses by 3-D FE analysis of semi-circular crack. For bi-linear (H = 0.1E), the normalized opening stresses (rop/rmax) at the free surface and the deepest point were 0.36 and 0.16 in this work while Skinner and Daniewicz [22] reported 0.35 and 0.25, respectively. The discrepancy is due to using the higher ratios of Da/xp than the work of Skinner and Daniewicz. 3.2. Stabilization behaviour Because the stabilization of crack opening level is very important in relation to the computational time and determining the opening load in 3-D FE analyses of plasticity induced crack closure, the variations of crack opening loads both at the depth point and at the free surface were investigated with advancing the surface crack. The monotonic plastic zone sizes, namely, xp,surface of the free surface under plane stress condition and xp,depth of the depth point under plane strain condition are defined, respectively, as follows: 2 1 K max xp;surface ¼ ð1Þ p ry 2 1 K max ð2Þ xp;depth ¼ 3p ry where ry is the yield stress and Kmax is the maximum stress intensity factor calculated by Newman and Raju equation [27]. The previous works acquired by 2-D plane stress FE analyses reported that the crack opening levels were stabilized after the crack advanced the monotonic plastic zone for plane stress state [15,26]. In order to obtain the stabilized crack closure for plane strain condition, Parry et al. [28] and Lee and Song [29] suggested that the crack should be allowed to grow through about four times the monotonic plastic zone sizes while Solanki et al. [30] proposed that crack should advance more than the monotonic plastic zone. For 3-D FE analysis of part-through crack, Skinner and Daniewicz [22] reported that after completion of 10 growth cycles, the opening levels were stabilized at the deepest point of penetration while none of the free surface opening levels were observed to be stabilized because the crack did not pass the original plastic zone of the free surface. Fig. 4 shows the process that the opening levels calculated in this study become stabilized at the deepest point and the free surface. As shown in Fig. 4, the variations of crack opening levels with crack growth for four different initial crack lengths, a0 = 1.16 mm, 1.51 mm, 1.86 mm and 2.21 mm were monitored. The mesh size Da (in the fine mesh zone), or the crack advance increment per cycle, was 0.035 mm, the stress ratio was 0, and the maximum stress rmax was 300 MPa. In the work of Skinner and Daniewicz [22], fatigue crack growth was
171
modeled for 10 cycles. However, in this work, as can be seen in Fig. 4, fatigue crack growth for four initial crack lengths was modeled as follows: • • • •
From From From From
a0 = 1.160 mm a0 = 1.510 mm a0 = 1.860 mm a0 = 2.210 mm
to to to to
a = 2.210 mm (31 cycles). a = 2.875 mm (40 cycles). a = 2. 875 mm (30 cycles). a = 2. 875 mm (20 cycles).
For the plane stress at the free surface, the crack opening level increases monotonically with crack growth and then stabilizes after the crack advances through the plane stress monotonic plastic zone xp,surface regardless of an initial crack length. For the plane strain at the deepest point, the crack opening level increases initially to reach a maximum and then becomes stable after the crack advances eight times the plane strain plastic zone. In 2-D plane strain FE analyses, the similar stabilization behaviour can be found in the works of Ashbaugh et al. [31], Pommier and Bompard [32], Wei and James [33], and Lee and Song [29]. When the crack incremental lengths for the stabilization in 3-D FE analysis is compared with that of the previous 2-D analysis results [15,29], the case of the free surface is similar to the 2-D plane stress results [15] being stabilized after advancing the monotonic plastic zone. However, the opening level of the deepest point is stabilized by growing 8xp,depth which is twice the size required for stabilization in the 2-D plane strain analysis [29]. In the current analysis, the opening level becomes stable earlier at the free surface than at the deepest point. As already noted, this result is very different from the work of Skinner and Daniewicz [22] that the opening level of the free surface would be stabilized later than that of the deepest point. If the small variation of crack opening values at the deepest point during 10 cycle growths did not take into account as in the work of Skinner and Daniewicz, it is possible also in this work to say that the crack opening stresses stabilize more slowly under plane stress conditions than under plain strain conditions. However, because we defined the stabilization location as the point at which crack opening stresses are equal irrespective of initial crack lengths, the crack opening stresses become stable earlier at the free surface than at the deepest point. For four different initial crack lengths, the numbers of elements in the free surface initial forward plastic zones, xp,surface are 5.5 at a0 = 1.16 mm, 6.0 at a0 = 1.51 mm, 8.5 at a0 = 1.86 mm and 11 at a0 = 2.21 mm and the numbers of elements in the deepest point initial forward plastic, xp,depth are 1.5 at a0 = 1.16 mm, 2.0 at a0 = 1.51 mm, 2.3 at a0 = 1.86 mm and 2.8 at a0 = 2.21 mm. However, as can be seen in Fig. 4, the increase of the number of the elements in the initial forward plastic zone hardly influences the trend of stabilization behaviour. For the stabilization of the crack opening loads both at the deepest point and at the free surface, the crack must
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J.-S. Kim et al. / International Journal of Fatigue 29 (2007) 168–180 1.0 0.9
R=0.0, σmax=300MPa
c0 = 1.16 mm c0 = 1.51 mm c0 = 1.86 mm c0 = 2.21 mm
σop at surface /σmax
0.8 0.7 0.6
1ωp,s
1ωp,s
1ωp,s
1ωp,s 0.5 0.4 0.3 0.2 0.1 0.0 1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
c (mm), crack surface length 1.0 0.9
a0 = 1.16 mm a0 = 1.51 mm a0 = 1.86 mm a0 = 2.21 mm
R=0.0, σmax=300MPa
0.8
σop at depth /σmax
0.7 0.6 0.5 4ωp,d
0.4 4ωp,d6ωp,d 8ωp,d
6ωp,d
8ωp,d
0.3 0.2 4ωp,d 6ωp,d 8ωp,d
4ωp,d
6ωp,d
2.6
2.8
0.1 0.0 1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
3.0
a (mm), crack depth Fig. 4. Variation of crack opening load as a function of crack length.
be advanced through the element length, 8xp,depth, however, this takes excessively much computation time. Therefore, in this study, we propose the method that the computation time can be reduced by the process presented in Fig. 5. This method estimates the stabilized value using the crack opening load at the deepest point obtained at xp,surface where the crack opening load at the free surface is stabilized. As shown in Fig. 5, advancing the crack through xp,surface, irrespective of an initial crack length, the crack opening load at the free surface becomes stabilized, but the crack opening load at the deepest point is always higher than the stabilized value at the deepest point. The ratios between two values at the deepest point are fixedly 1.055 irrespective of initial crack lengths (Fig. 6).
Therefore, the stabilized crack opening load at the deepest point can be estimated by multiplying the value obtained after advancing the crack through xp,surface by 1/1.055. Hereafter, the crack opening load at the deepest point is represented by the stabilized one obtained by this method, unless otherwise stated. 3.3. Crack opening behaviour The FE analyses to observe the opening behaviour of surface crack were performed with the two models described in the preceding section. Model I in Fig. 2(a) is used to form the plastic wake behind the crack tip by advancing the surface crack from the initial crack length of 0.15 mm. For model II in Fig. 2(b), an initial crack
J.-S. Kim et al. / International Journal of Fatigue 29 (2007) 168–180 180
surface crack opening load (MPa)
160
1(ωp,s)a =2.21
1(ωp,s)a =1.86
1(ωp,s)a =1.51
173
0
0
0
1.(σop)surface
140 120 2.(σop)depth,prediction
100
A
A
A
=1.055 x σop)depth,stabilized
80 60
B
B
B
3.(σop)depth,stabilized = σop)depth,prediction / 1.055
a0 = 1.16 mm
c0 = 1.16 mm
40
a0 = 1.51 mm
c0 = 1.51 mm
20
a0 = 1.86 mm
c0 = 1.86 mm
a0 = 2.21 mm
0
c0 = 2.21 mm
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
a, c (mm), surface crack length
0.012
1.10
R=0.0, σmax=200MPa a=2.56mm Δa=0.035mm (σop)surface=81MPa
load=200MPa
0.010 1.08 0.008
z (mm)
σop,depth at A of Fig. 5 / σop,depth at B of Fig. 5
Fig. 5. Estimation of the stabilized crack opening load at the deepest point.
1.06 1.055
0.006
150MPa
a0 = 0.15 mm a0 = 2.21 mm
100MPa
0.004
1.04
50MPa
0.002 20MPa
1.02 0.000
0MPa
1.00 1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
0
3.0
1
crack length, a (mm)
3
4
(a) free surface (X-coordinate)
Fig. 6. Ratio of crack opening values at the depth point when the crack is advanced through 1xp,surface.
0.012
R=0.0, σmax=200MPa a=2.56mm Δa=0.035mm (σop)depth=39MPa
load=200MPa
0.010 0.008
z (mm)
length a0 is 2.21 mm. FE analyses are performed with rmax = 200 MPa, rmin = 0 MPa, and element length, Da = 0.035 mm. Fig. 7(a) shows the crack opening profiles on the free surface (X-coordinate) after advancing from an initial crack length a0 = 0.15 mm or 2.21 mm to a final crack length 2.56 mm and Fig. 7(b) shows the profiles along the depth direction (Y-coordinate). The crack opening displacements are considerably different according to an initial crack length before crack length of 2.21 mm, but are identical regardless of an initial crack length after crack length of 2.21 mm. It is due to the overshoot of the opening displacement generated by the large z-direction plastic deformation produced at an initial crack length a0 = 2.21 mm during the first loading.
2
x coordinate (mm)
0.006
150MPa
a0 = 0.15 mm a0 = 2.21 mm
100MPa
0.004 50MPa
0.002 20MPa
0.000
0MPa
0
1
2
3
y coordinate (mm) (b) depth direction (Y-coordinate) Fig. 7. Crack opening profiles at R = 0, rmax = 200 MPa.
4
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Fig. 8 shows that the large initial plastic deformation occurs at an initial crack length a0 = 2.21 mm because there is no compressive residual stress near the crack tip during the 1st loading. When advancing the surface crack from an initial crack length a0 = 0.15 mm, the overshoot of the opening displacement was not observed. In the present study, the large z-direction plastic deformation occurred if an initial surface crack length was more than 0.43 mm at rmax = 200 MPa. However, irrespective of an initial crack length, all the first nodes behind the crack tip along the front of the surface crack are the last to open. Fig. 9 shows the process that the contacted regions of the surface crack become opened as the applied remote stress increases when advancing the crack from an initial crack length a0 = 0.15 mm. The dark areas are the crack surfaces in contact. As seen in Fig. 9(a), at r = 0 MPa, all the area near the free surface is closed, but the inner area from the free surface and the narrow area slightly behind the crack tip are still open. The above results are similar to those examined through 0.0012
R=0.0, σmax=200MPa a0=2.21mm, a=2.56mm
load=20MPa 0.0010
z (mm)
0.0008 0.0006
Δa=0.035mm (σop)surface=81MPa
15MPa
10MPa
0.0004 5MPa 0.0002 0.0000
0MPa a0=2.21mm
0
1
2
3
x coordinate (mm) (a) free surface 0.0012
R=0.0, σ max=200MPa a0=2.21mm, a=2.56mm
load=20MPa 0.0010
z (mm)
0.0008 0.0006
Δa=0.035mm (σop)depth=39MPa
15MPa
10MPa
0.0004 5MPa 0.0002 0.0000
0MPa a0=2.21mm
0
1
2
3
y coordinate (mm) (b) depth direction Fig. 8. Crack surface deformation during the loading range of 0–20 MPa (a0 = 2.21 mm, a = 2.56 mm, Da = 0.035 mm).
the experimental method by Troha et al. [34,35]. At r = 5 MPa, except for the region around the crack tip and the free surface, all the inner nodes become open. As the external load increases, the nodes behind the surface crack tip are opened sequentially from the deepest point to the free surface (Fig. 9(d)–(f)). Fig. 10 shows the state that the crack surface is opened at each load level when the surface crack grows from an initial crack length a0 = 2.21 mm. At r = 0 MPa, most region of the crack surface is closed but the partial area behind the crack tip in the range of 15–90 inward from the free surface is opened (Fig. 10(a)). As the external load increases, the partially opened area is expanded (Fig. 10(b) and (c)) and finally the nodes behind the surface crack tip are opened in order from the deepest point under the plane strain to the free surface under the plane stress (Fig. 10(d)–(f)). This result is similar to that of an initial crack length a0 = 0.15 mm. The opening levels of the deepest point as well as the free surface at the same crack length are not changed depending on an initial crack length. Figs. 9(e) and 10(e) clearly illustrate that the crack tip near the free surface of a part-through crack often lags in the actual fatigue crack growth (Fig. 11) because of more crack closure around the free surface. As shown in Fig. 12, the free surface did not maintain a plane during the loading and unloading cycle and the maximum forward plastic zone occurred approximately 5 inward from the free surface, which was well explained by Daniewicz and Aveline [21]. These are the interesting results that are very difficult to be observed in experimental methods. Fig. 13 shows the forward plastic zone sizes computed by FE analysis with a crack length a = 2.56 mm, rmax = 200 MPa, stress ratio R = 0 and mesh size Da = 0.035 mm using von Mises yield criterion. The lengths of the plastics zones at the free surface and the deepest point by FE analysis are 0.105 mm and 0.045 mm, respectively, and the length at the free surface is 2.33 times longer than at the deepest point. The values estimated by Eqs. (1) and (2) are 0.201 mm and 0.051 mm, respectively, and the length at the free surface is 3.94 times longer than at the deepest point. The ratios of the result of FE analysis to one of Irwin’s equation are 52.2% at the free surface and 88.2% at the deepest point. The length of the maximum forward plastic zone at about 5 inward from the free surface is 0.175 mm, which is 1.67 times longer than at the free surface and 3.89 times longer than at the deepest point. The value is 87.1% of the length calculated by Eq. (1). Fig. 14 shows the cyclic stress–strain curve at the crack tip node for 9th and 10th cycles. Even if the remote applied stress is zero, there is still cyclic plasticity in the material near the crack tip. It results in the plasticity induced crack closure. Crack opening stress is higher at the free surface than at the deepest point because the compressive residual
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175
Fig. 9. Closed regions of semi-circular crack surface advanced from a initial crack length a0 = 0.15 mm.
stress is higher at the free surface than at the deepest point as shown in Fig. 14. 3.4. Mesh refinement The effect of mesh size was examined for the element length ranging from 0.0015 mm to 0.01 mm by performing the FE analysis with a crack length a = 2.0 mm, rmax = 150 MPa and stress ratio R = 0. For each element size, a crack was advanced through the forward plastic zone size xp,surface for the crack opening value to be stabilized at the free surface. The stabilized opening value at the deepest point was obtained by the method given in Fig. 5. We examined the effect of element size in the fine mesh zone within the range of 0.0015–0.01 mm because of the
huge amount of computation time and cost required for obtaining the stabilized crack opening stresses by 3-D FE analysis of semi-circular crack. In addition, as pointed out by Jiang et al. [36], it should be noted that an element size smaller than 0.001 mm is usually not practical. Fig. 15 shows the effects of mesh size on the crack opening load. As shown in the figure, the more refined mesh increases the crack opening loads at both the deepest point and the free surface, but the incremental rates of the crack opening loads decrease. Within the range evaluated, it is not conclusive whether or not the opening load will be minimal or stabilized when the element size is smaller than a certain value. If the converged crack opening results could be obtained only by reducing the mesh size, it would be very
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Fig. 10. Closed regions of semi-circular crack surface advanced from a initial crack length a0 = 2.21 mm.
simple problem to be solved. However, according to the recent work of Jiang et al. [36], the converged crack opening results cannot be obtained only by reducing the mesh size and the complex stress–strain responses for the material near the crack tip cannot be properly described due to inability to mimic the strain ratcheting and stress relaxation when the kinematic hardening rule of Prager–Ziegler is used. It is worthy of note that the crack opening stresses for the CT specimen under plain stress obtained by using the JS cyclic plasticity model proposed by Jiang et al. [36] are not a strong function of the element size. 3.5. Comparison with experimental measurement methods Kim and Song [25] measured the plane strain crack opening at the deepest point by an extensometer of
5.7 mm gauge length spanning the crack at the midpoint of the surface crack mouth and the plane stress crack opening at the free surface by strain gauges boned in the vicinity of the surface intersection point. They determined the crack opening loads on the load–differential displacement curves or the load–differential strain curves with the naked eye. On the viewpoint of measurement sensitivity, the drawbacks of experimental methods were investigated with the load–differential displacement curves or the load–differential strain curves at the identical locations obtained by FE analysis. Fig. 16 shows the load–differential displacements calculated by FE-analysis varying the distance of the z-direction corresponding to the gage length of the extensometer used in experiment [25]. In this case, FE-analysis was performed with an initial crack length a0 = 0.15 mm, stress ratio R = 0, rmax = 200 MPa, final crack length
J.-S. Kim et al. / International Journal of Fatigue 29 (2007) 168–180
177
14
surface (0 degree)
13
z ( mesh ), mesh size=0.035mm
12 11 10 9 8 7
depth (90 degree)
6 5 4
z
3 2 1
5 degree from surface
0
X
-1 -6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
x ( mesh ), mesh size=0.035mm
Fig. 11. Effect of more crack closure at the material surface on crack front shape.
plastic zone sizes from FEM : ω p,surface,FEM = 0.105 mm (ref. theory : 0.201 mm) ω p,5 degre,FEM = 0.175 mm ω p,depth,FEM = 0.045 mm (ref. theory : 0.051 mm)
a = 2.56 mm and the element length Da = 0.035 mm. Crack opening stresses obtained by FE-analysis were determined by monitoring all nodal displacements and nodal reaction forces along the crack. In order to compare the crack opening stresses obtained by FE-analysis with those determined on the load–differential displacement curves with the naked eye, the results obtained by FE-analysis were marked on the load–differential displacement curves. As shown in the figure, the slope change of the load– differential displacement diminishes with the increase of
ω p,surface,FEM / ω p,depth,FEM = 0.105 mm / 0.045 mm = 2.33 / 1 Fig. 13. Plastic zone sizes at crack surface and crack depth calculated by FEM.
the distance of the z-direction. Therefore, in order to increase the accuracy in experimentally measuring the opening load, the span of an extensometer should be as short as possible. As shown in Fig. 17(a), if the opening load at the deepest point were visually determined on the load–differential
Fig. 12. von Mises stress contours and deformations of free surface at rmax = 200 MPa.
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J.-S. Kim et al. / International Journal of Fatigue 29 (2007) 168–180 1000
800 9th loading
σzz (MPa)
600
400 9th unloading 10th loading
200
0
9th,10th cycle R=0, σmax=200MPa
-200
crack tip node at the free surface crack tip node at the deepest depth
-400 0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
0.016
0.018
εzz Fig. 14. Stress–strain response near the crack tip at the free surface and the deepest point (a0 = 2.21 mm, a = 2.56 mm).
250
0.6
z = 0 mm z = 1.225 mm z = 2.432 mm z = 3.675 mm z = 5.520 mm
(σop)surface (σop)depth,corrected
σop / σmax
0.5
R=0.0, σmax=150MPa a=2.0mm
200
R=0.0, σmax=200MPa (σop)surface = 81 MPa
0.4
(σop)depth = 39 MPa a0 = 0.15 mm a = 2.56 mm a = 0.035 mm
150
0.2 0.000
0.002
0.004
0.006
0.008
0.010
0.012
load (MPa)
0.3
100
(σop)surface
Δa, mesh size (mm) Fig. 15. Effect of mesh size on crack opening load.
displace curve calculated at the z-direction distance of 2.5 mm similar to the span of the extensometer used in the experiment [25], the value is considered to be about 44 MPa while the opening load obtained by FE analysis immediately behind the deepest point of the surface crack tip is 39 MPa. This indicates that the plane strain opening load at the deepest point measured experimentally may be estimated to be about 13% higher than the actual. Therefore, the opening loads of the deepest points measured experimentally are in need of being corrected. Fig. 17(b) shows the load–differential strain curve calculated by FE-analysis at the distance of 0.21 mm (1.04xp,surface) forward the crack tip of the free surface. As shown in this figure, the plane stress crack opening load
50
(σop)depth
0
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
differential displacement (μm) Fig. 16. Variation of load–differential displacement curve.
measured visually on the load–differential strain curve is nearly equal to the numerical value obtained immediately behind the crack tip at the free surface. Therefore, it could be found that the crack opening load measured experimentally by strain gauge boned in the vicinity of the surface
J.-S. Kim et al. / International Journal of Fatigue 29 (2007) 168–180 250
250
measurement location x = 2.77 mm (1.04 ω p)
measurement location z = 2.5 mm R = 0.0, σmax = 200 MPa
R = 0.0, σmax = 200 MPa
(σop)surface = 81 MPa
200
(σop)surface = 81 MPa
200
(σop)depth = 39 MPa a = 2.56 mm a = 0.035 mm
(σop)depth = 39 MPa a = 2.56 mm a = 0.035 mm
150
load (MPa)
150
load (MPa)
179
100 (σop)surface
100 (σop)surface visual measurement of σop)surface
50
visual measurement of σop)depth
50 (σop)depth
(σop)depth
0
0
-0.10 -0.05
0.00
0.05
0.10
0.15
0.20
-0.10 -0.05
differential displacement (μm)
0.00
0.05
0.10
0.15
0.20
differential strain (b) at the free surface
(a) at the deepest point
Fig. 17. Comparison between the numerical value and visual measurement one for crack opening load.
intersection point can successfully represent the plane stress crack opening of the free surface. 4. Conclusions A 3-D elastic–plastic FE analysis is performed for semicircular crack, giving special attention to stabilization behaviour, the mesh size effect, the effect of initial crack length, the opening process of crack surface and the comparison with experimental methods. The conclusions obtained are summarized as follows. 1. The opening loads decrease along with the increase of strain hardening and the effect of strain hardening on opening load is significant at the deepest point than at the free surface. 2. For the plane stress at the free surface, the crack opening level stabilizes after the crack advances through the plane stress monotonic plastic zone, xp,surface. For the crack opening loads at the deepest point to be stabilized, the crack must be advanced through 8xp,depth. However, the stabilized crack opening load at the deepest point can be estimated by multiplying the value obtained at the time that the crack is advanced through xp,surface by a correction factor. 3. The crack opening displacements are considerably different according to an initial crack length. However, irrespective of an initial crack length, all the first nodes behind the crack tip along the front of the surface crack are the last to open.
4. The more refined mesh increases the crack opening loads both at the deepest point and at the free surface, but the incremental rates of the crack opening loads decrease. It suggests that the crack opening loads may be converged if using the sufficiently refined mesh. 5. The plane strain crack opening load at the deepest point measured in experiment by an extensometer at the midpoint of crack mouth may be estimated to be higher than the actual. The crack opening load measured experimentally by strain gauge boned in the vicinity of the surface intersection point can successfully represent the plane stress crack opening of the free surface. References [1] Ohji K, Ogura K, Ohkubo Y. On the closure of fatigue cracks under cyclic tensile loading. Int J Fract 1974;10:123–4. [2] Ohji K, Ogura K, Ohkubo Y. Cyclic analysis of a propagating crack and its correlation with fatigue crack growth. Eng Fract Mech 1975;7:457–64. [3] Ogura K, Ohji K. FEM analysis of crack closure and delay effect in fatigue crack growth under variable amplitude loading. Eng Fract Mech 1977;9:471–80. [4] Newman Jr JC. A finite element analysis of fatigue crack closure. ASTM STP 590 1976:280–301. [5] Newman Jr JC. Finite element analysis of crack growth under monotonic and cyclic loading. ASTM STP 637 1977:56–80. [6] Nakagaki M, Atluri SN. Elastic–plastic analysis of fatigue crack closure in mode I and II. AIAA J 1980;18:1110–7. [7] Nakamura H, Kobayashi H, Yanase S, Nakazawa H. Finite element analysis of fatigue crack closure in compact specimen. In: Proceeding of ICM 4; 1983. p. 817–23.
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[8] Blom AF, Holm DK. An experimental and numerical study of crack closure. Eng Fract Mech 1985;22:997–1011. [9] Nicholas T, Palazotto A, Bednarz E. An analytical investigation of plasticity induced closure involving short cracks. ASTM STP 982 1988:361–79. [10] Lalor PL, Sehitoglu H. Fatigue crack closure outside a small scale yielding regime. ASTM STP 982 1988:342–60. [11] Fleck NA. Finite element analysis of plasticity-induced crack closure under plane strain conditions. Eng Fract Mech 1986;25:441–9. [12] Fleck NA, Newman Jr JC. Analysis of crack closure under plane strain conditions. ASTM STP 982 1988:319–41. [13] Wu J, Ellyin F. A study of fatigue crack closure by elastic–plastic finite element analysis for constant-amplitude loading. Int J Fract 1996;82:43–65. [14] Dougherty JD, Padovan J, Srivatsan TS. Fatigue crack propagation and closure behaviour of modified 1070 steel: finite element study. Eng Fract Mech 1997;56(2):189–212. [15] Park SJ, Earmme YY, Song JH. Determination of the most appropriate mesh size for a 2-D finite element analysis of fatigue crack closure behaviour. Fatigue Fract Eng Mater Struct 1997;20(4):533–45. [16] McClung RC, Sehitoglu H. On the finite element analysis of fatigue crack closure – 1. Basic modeling issues. Eng Fract Mech 1989;33:237–52. [17] McClung RC, Sehitoglu H. On the finite element analysis of fatigue crack closure – 2. Numerical results. Eng Fract Mech 1989;33:253–72. [18] Chermahini RG, Shivakumar KN, Newman Jr JC. Three-dimensional elastic–plastic finite element analysis of fatigue crack growth and closure. ASTM STP 982 1988:398–413. [19] Chermahini RG, Palmberg B, Blom AF. Fatigue crack growth and closure behaviour of semi-circular and semi-elliptical surface flaws. Int J Fatigue 1993;15(4):259–63. [20] Zhang JZ, Bowen P. On the finite element simulation of threedimensional semi-circular fatigue crack growth and closure. Eng Fract Mech 1998;60(3):341–60. [21] Daniewicz SR, Aveline CR. Strip-yield and finite element analysis of part-through surface flaws. Eng Fract Mech 2000;67:21–39. [22] Skinner JD, Daniewicz SR. Simulation of plasticity-induced fatigue crack closure in part-through cracked geometries using finite element analysis. Eng Fract Mech 2002;69:1–11.
[23] Hou CY. Three-dimensional finite element analysis of fatigue crack closure behaviour in surface flaws. Int J Fatigue 2004;26:1225–39. [24] ABAQUS user’s manual, vol. 1, Ver. 5.8, Hibbit, Karlsson & Sorensen, Inc.; 1998. [25] Kim JH, Song JH. Crack growth and closure behaviour of surface cracks under axial loading. Fatigue Fract Eng Mater Struct 1992;15(5):477–89. [26] McClung RC. Finite element analysis of fatigue crack closure: a historical and critical review. In: Proceeding of fatigue 99; vol. 1; 1999. p. 495–502. [27] Newman JC, Raju IS. Stress intensity factor equations for cracks in three-dimensional finite bodies. ASTM STP 791 1983:238–65. [28] Parry MR, Syngellakis S, Sinclair I. Numerical modeling of combined roughness and plasticity induced crack closure effects in fatigue. Mater Sci Eng A 2000;291:224–34. [29] Lee HJ, Song JH. Finite element analysis of fatigue crack closure under plane strain conditions: stabilization behaviour and mesh size effect. Fatigue Fract Eng Mater Struct 2005;28:333–42. [30] Solanki K, Daniewicz S, Newman Jr JC. Finite element modeling of plasticity-induced crack closure with emphasis on geometry and mesh refinement effects. Eng Fract Mech 2003;70:1475–89. [31] Ashbaugh NE, Dattaguru B, Khobaib M, Nicholas T, Prakash RV, Ramamurthy TS, et al. Experimental and analytical estimates of fatigue crack closure in an aluminium–copper alloy part II: a finite element analysis. Fatigue Fract Eng Mater Struct 1997;20(7):963–74. [32] Pommier S, Bompard Ph. Bauschinger effect of alloys and plasticityinduced crack closure: a finite element analysis. Fatigue Fract Eng Mater Struct 2000;23:129–39. [33] Wei LW, James MN. A study of fatigue crack closure in polycarbonate CT specimens. Eng Fract Mech 2000;66:223–42. [34] Troha WA, Nicholas T, Grandt Jr AF. Three-dimensional aspects of fatigue crack closure in surface flaws in polymethylmethacrylate material. ASTM STP 982 1988:598–616. [35] Troha WA, Nicholas T, Grandt Jr AF. Observations of three dimensional surface flaw geometries during fatigue crack growth in PMMA. ASTM STP 1060 1990:260–86. [36] Jiang Y, Feng M, Ding F. A reexamination of plasticity-induced crack closure in fatigue crack propagation. Int J Plasticity 2005;21:1720–40.
International Journal of Fatigue 29 (2007) 168–180
www.elsevier.com/locate/ijfatigue
Elucidation of fatigue crack closure behaviour in surface crack by 3-D finite element analysis Joo-Sung Kim *, Jae Youn Kang, Ji-Ho Song Department of Automation and Design Engineering, Korea Advanced Institute of Science and Technology, Seoul 130-012, Republic of Korea Received 8 January 2005; received in revised form 30 May 2005; accepted 21 January 2006 Available online 3 April 2006
Abstract A 3-D elastic–plastic finite element analysis of fatigue crack closure for semi-circular crack is performed. The stabilization behaviour of crack opening level, the effect of mesh size, the effect of initial crack length, the opening process of crack surface and the shortcoming and sensitivity of experimental measurement methods are investigated. The crack opening load tends to increase with the decrease of mesh size. Irrespective of an initial crack length, all the first nodes behind the crack tip along the front of surface crack are the last to open. The plane strain-opening load at the deepest point measured by extensometer at the midpoint of crack mouth may be estimated to be higher than the actual. The crack opening load experimentally measured by strain gauge boned in the vicinity of the surface intersection point can successfully represent the plane stress crack opening of the free surface. 2006 Elsevier Ltd. All rights reserved. Keywords: Fatigue crack closure; 3D elastic–plastic finite element; Stabilization behaviour; Mesh size effect; Surface crack
1. Introduction Many researchers have performed elastic–plastic finite element analyses to study the plastic induced fatigue crack closure and the several remarkable papers have been reported which reviewed the state of art and current issues of the finite element (FE) crack closure. However, most of studies on the FE analysis of the plastic induced fatigue crack closure have been related to the through thickness crack under two-dimensional plane stress or plane strain conditions. The 2-D FE closure modeling issues have been relatively well established including mesh refinement, stabilization, crack advance schemes and comparisons with experimental results. Ohji et al. [1–3], Newman [4,5], Nakagaki and Atluri [6], Nakamura et al. [7], Blom and Holm [8], and Nicho* Corresponding author. Present address: CAE team, Ssangyong Motor Company, 150-3 Chilgoi-dong, Pyungtaek-si, Gyeonggi-do 459-711, Republic of Korea. Tel.: +82 31 610 3425; fax: +82 31 610 3770. E-mail address: [email protected] (J.-S. Kim).
0142-1123/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2006.01.016
las et al. [9] performed 2-D FE closure analyses for plane stress conditions and the studies of Blom and Holm [8], Lalor and Sehitoglu [10], Fleck [11,12] have been reported for plane strain conditions. Their works have been reported focusing on the qualitative investigations on fatigue crack closure conditions, the relations between the mesh size and the plastic zone size at crack tip, the comparisons of crack opening loads between plane stress and plane strain conditions and the effects of stress ratio. In addition, Ohji et al. [1–3], Newman [4], Wu and Ellyin [13], Dougherty et al. [14], Park et al. [15] investigated the mesh size effects on crack opening level. Fleck [11,12] and McClung and Sehitoglu [16,17] reported the well established works on the effects of mesh refinement, a crack advance scheme, material properties, a constitutive model, stress state, an initial crack length and a maximum applied stress on the opening and closing behaviour of cracks, and the stabilization behaviour of fatigue crack closure level. A relatively few works [18–23] on 3-D plastic induced fatigue crack closure have been reported because of taking
J.-S. Kim et al. / International Journal of Fatigue 29 (2007) 168–180
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very much computation time to do the detailed 3-D FE analysis. Particularly, Skinner and Daniewicz [22], and Hou [23] performed 3-D FE analysis of part-through cracks with very refined mesh through the aid of the rapid progress of computers and FE analysis programs. Skinner and Daniewicz investigated the issues of mesh refinement and stabilization, and compared numerical results with experimental results. Hou studied 3-D fine meshing issues, the contact of the cracked surface, stress distributions around surface crack tip and the variations of opening loads along crack front. In this study, 3-D FE analyses on semi-circular crack were carried out focusing on stabilization behaviour, the mesh size effect, the effect of initial crack length, the opening process of semi-circular crack area and the shortcoming and sensitivity of crack opening levels determined relying on remote displacement in experiments. 2. Finite element model A 3-D FE mesh model was created with the commercial software PATRAN. The commercial software, ABAQUS [24] is used for finite element analysis in present study. The theory of incremental rate independent classical plasticity and von Mises yield criteria are used. To simulate the Bauschinger effect associated with reversed yielding, a nonlinear kinematic hardening rule of Prager–Ziegler implemented in ABAQUS is employed. The part-through crack geometry used in this study is shown in Fig. 1, which consists with the specimen used by Kim and Song [25]. Fig. 2 shows the two typical fine mesh configurations employed in this study. Fig. 2(a) shows the model (Model I) used in order to form the plastic wake behind the crack tip by advancing the semi-circular crack from the initial crack length of 0.15 mm considering that an actual part-through crack progresses from the small flaw of surface. Fig. 2(b) shows the model (Model II) used for the most part in this study. Only one quarter of the geometry is meshed utilizing two planes of symmetry. Contact elements (IRS13) implemented in ABAQUS [24] are used at the crack face so that the reaction force is generated during being closed. Model I is composed of 13,559 eight-node solid elements (C3D8), 235 six-node solid elements (C3D6), 1276 contact elements (IRS13). The total number of elements and nodes of the model I are 15,070 and 15,505, respectively. Model II is composed of 7455 eight-node solid elements (C3D8), 105 six-node solid elements (C3D6) and 705 contact elements (IRS13). The total number of elements and nodes of the model II are 8265 and 8536, respectively. All the meshes of the surface crack area of Model I are used with elemental length, Da = 0.035 mm except that the elemental lengths of 0.0015–0.01 mm are used to investigate the mesh size effect using Model II. Material properties used for the analysis are as follows: Young’s modulus E = 71 GPa, Poisson’s ratio n = 0.33, yield stress ry = 555 MPa and bi-linear strain hardening
Fig. 1. Schematic representation of surface crack specimen.
r=dep ¼ 0:01E ¼ 710 MPa. These propparameter H 0 ¼ d erties are typical of 7075-T6 aluminum alloys. The crack advance scheme, or crack-tip node release scheme has been relatively well discussed [26]. Particularly referring to the work of Park et al. [15], the crack-tip node is released at the minimum load of every cycle. In this study, all the nodes immediately behind the crack tip along the front of semi-circular crack are always the first to open and the last to close. Therefore, the crack opening loads are defined as the moment that the nodes immediately behind the crack tip are opened. 3. Results and discussion 3.1. The effect of material model The effect of the variation of stress–strain curve on crack opening loads is studied. FE analyses are performed by advancing one elemental crack, Da = 0.035 mm per cycle from an initial crack length a0 = 2.21 mm to a final crack length a = 2.56 mm with stress ratio R = 0, rmax = 200 MPa. The crack opening loads calculated using four different stress–strain curves are listed in Table 1. Fig. 3 shows the relation of stress–strain experimentally measured [25]. Comparing the results using multi-linear hardening parameters with the other, as described in Table
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Fig. 2. Finite element models for surface crack, (a) Model I, (b) Model II.
Table 1 Effect of plastic stress–strain curve on crack opening load
AL 7075-T6 BL (H=0.1E)
rop,surface/rmax
rop,depth/rmax
0.405 0.402 0.397 0.359
0.197 0.188 0.185 0.163
1, the opening loads decrease along with the increase of strain hardening and the effect of strain hardening on opening load is significant at the deepest point than at the free surface. If a element size of Da = 0.035 mm is used with a initial crack length of a0 = 2.21 mm and the maximum stress of rmax = 200 MPa, the ratios of Da/xp calculated by Eqs. (1) and (2) are 0.174 at the surface and 0.686 at the deepest point, respectively. It cannot satisfy Da/xp 6 0.1 being necessary for reliable crack closure results that McClung and Sehitoglu [16,17] proposed. However, the main purpose of this section is not to determine the precise crack opening
800 Multi-Linear(Test Data)
Stress (MPa)
Multi-linear Perfectly plastic (Et = 0) Bi-linear (Et = 0.01E) Bi-linear (Et = 0.1E)
1000
BL (H=0.01E)
600
Perfectly Plastic 400
200
0 0.00
Strain
0.05
Fig. 3. Stress–strain curve for AL 7075-T6.
value but to investigate the relative effect of the stress– strain curves on the crack opening value. Therefore, we used rather a large mesh (0.035 mm) because of the huge
J.-S. Kim et al. / International Journal of Fatigue 29 (2007) 168–180
amount of computation time and cost required for obtaining the stabilized crack opening stresses by 3-D FE analysis of semi-circular crack. For bi-linear (H = 0.1E), the normalized opening stresses (rop/rmax) at the free surface and the deepest point were 0.36 and 0.16 in this work while Skinner and Daniewicz [22] reported 0.35 and 0.25, respectively. The discrepancy is due to using the higher ratios of Da/xp than the work of Skinner and Daniewicz. 3.2. Stabilization behaviour Because the stabilization of crack opening level is very important in relation to the computational time and determining the opening load in 3-D FE analyses of plasticity induced crack closure, the variations of crack opening loads both at the depth point and at the free surface were investigated with advancing the surface crack. The monotonic plastic zone sizes, namely, xp,surface of the free surface under plane stress condition and xp,depth of the depth point under plane strain condition are defined, respectively, as follows: 2 1 K max xp;surface ¼ ð1Þ p ry 2 1 K max ð2Þ xp;depth ¼ 3p ry where ry is the yield stress and Kmax is the maximum stress intensity factor calculated by Newman and Raju equation [27]. The previous works acquired by 2-D plane stress FE analyses reported that the crack opening levels were stabilized after the crack advanced the monotonic plastic zone for plane stress state [15,26]. In order to obtain the stabilized crack closure for plane strain condition, Parry et al. [28] and Lee and Song [29] suggested that the crack should be allowed to grow through about four times the monotonic plastic zone sizes while Solanki et al. [30] proposed that crack should advance more than the monotonic plastic zone. For 3-D FE analysis of part-through crack, Skinner and Daniewicz [22] reported that after completion of 10 growth cycles, the opening levels were stabilized at the deepest point of penetration while none of the free surface opening levels were observed to be stabilized because the crack did not pass the original plastic zone of the free surface. Fig. 4 shows the process that the opening levels calculated in this study become stabilized at the deepest point and the free surface. As shown in Fig. 4, the variations of crack opening levels with crack growth for four different initial crack lengths, a0 = 1.16 mm, 1.51 mm, 1.86 mm and 2.21 mm were monitored. The mesh size Da (in the fine mesh zone), or the crack advance increment per cycle, was 0.035 mm, the stress ratio was 0, and the maximum stress rmax was 300 MPa. In the work of Skinner and Daniewicz [22], fatigue crack growth was
171
modeled for 10 cycles. However, in this work, as can be seen in Fig. 4, fatigue crack growth for four initial crack lengths was modeled as follows: • • • •
From From From From
a0 = 1.160 mm a0 = 1.510 mm a0 = 1.860 mm a0 = 2.210 mm
to to to to
a = 2.210 mm (31 cycles). a = 2.875 mm (40 cycles). a = 2. 875 mm (30 cycles). a = 2. 875 mm (20 cycles).
For the plane stress at the free surface, the crack opening level increases monotonically with crack growth and then stabilizes after the crack advances through the plane stress monotonic plastic zone xp,surface regardless of an initial crack length. For the plane strain at the deepest point, the crack opening level increases initially to reach a maximum and then becomes stable after the crack advances eight times the plane strain plastic zone. In 2-D plane strain FE analyses, the similar stabilization behaviour can be found in the works of Ashbaugh et al. [31], Pommier and Bompard [32], Wei and James [33], and Lee and Song [29]. When the crack incremental lengths for the stabilization in 3-D FE analysis is compared with that of the previous 2-D analysis results [15,29], the case of the free surface is similar to the 2-D plane stress results [15] being stabilized after advancing the monotonic plastic zone. However, the opening level of the deepest point is stabilized by growing 8xp,depth which is twice the size required for stabilization in the 2-D plane strain analysis [29]. In the current analysis, the opening level becomes stable earlier at the free surface than at the deepest point. As already noted, this result is very different from the work of Skinner and Daniewicz [22] that the opening level of the free surface would be stabilized later than that of the deepest point. If the small variation of crack opening values at the deepest point during 10 cycle growths did not take into account as in the work of Skinner and Daniewicz, it is possible also in this work to say that the crack opening stresses stabilize more slowly under plane stress conditions than under plain strain conditions. However, because we defined the stabilization location as the point at which crack opening stresses are equal irrespective of initial crack lengths, the crack opening stresses become stable earlier at the free surface than at the deepest point. For four different initial crack lengths, the numbers of elements in the free surface initial forward plastic zones, xp,surface are 5.5 at a0 = 1.16 mm, 6.0 at a0 = 1.51 mm, 8.5 at a0 = 1.86 mm and 11 at a0 = 2.21 mm and the numbers of elements in the deepest point initial forward plastic, xp,depth are 1.5 at a0 = 1.16 mm, 2.0 at a0 = 1.51 mm, 2.3 at a0 = 1.86 mm and 2.8 at a0 = 2.21 mm. However, as can be seen in Fig. 4, the increase of the number of the elements in the initial forward plastic zone hardly influences the trend of stabilization behaviour. For the stabilization of the crack opening loads both at the deepest point and at the free surface, the crack must
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J.-S. Kim et al. / International Journal of Fatigue 29 (2007) 168–180 1.0 0.9
R=0.0, σmax=300MPa
c0 = 1.16 mm c0 = 1.51 mm c0 = 1.86 mm c0 = 2.21 mm
σop at surface /σmax
0.8 0.7 0.6
1ωp,s
1ωp,s
1ωp,s
1ωp,s 0.5 0.4 0.3 0.2 0.1 0.0 1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
c (mm), crack surface length 1.0 0.9
a0 = 1.16 mm a0 = 1.51 mm a0 = 1.86 mm a0 = 2.21 mm
R=0.0, σmax=300MPa
0.8
σop at depth /σmax
0.7 0.6 0.5 4ωp,d
0.4 4ωp,d6ωp,d 8ωp,d
6ωp,d
8ωp,d
0.3 0.2 4ωp,d 6ωp,d 8ωp,d
4ωp,d
6ωp,d
2.6
2.8
0.1 0.0 1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
3.0
a (mm), crack depth Fig. 4. Variation of crack opening load as a function of crack length.
be advanced through the element length, 8xp,depth, however, this takes excessively much computation time. Therefore, in this study, we propose the method that the computation time can be reduced by the process presented in Fig. 5. This method estimates the stabilized value using the crack opening load at the deepest point obtained at xp,surface where the crack opening load at the free surface is stabilized. As shown in Fig. 5, advancing the crack through xp,surface, irrespective of an initial crack length, the crack opening load at the free surface becomes stabilized, but the crack opening load at the deepest point is always higher than the stabilized value at the deepest point. The ratios between two values at the deepest point are fixedly 1.055 irrespective of initial crack lengths (Fig. 6).
Therefore, the stabilized crack opening load at the deepest point can be estimated by multiplying the value obtained after advancing the crack through xp,surface by 1/1.055. Hereafter, the crack opening load at the deepest point is represented by the stabilized one obtained by this method, unless otherwise stated. 3.3. Crack opening behaviour The FE analyses to observe the opening behaviour of surface crack were performed with the two models described in the preceding section. Model I in Fig. 2(a) is used to form the plastic wake behind the crack tip by advancing the surface crack from the initial crack length of 0.15 mm. For model II in Fig. 2(b), an initial crack
J.-S. Kim et al. / International Journal of Fatigue 29 (2007) 168–180 180
surface crack opening load (MPa)
160
1(ωp,s)a =2.21
1(ωp,s)a =1.86
1(ωp,s)a =1.51
173
0
0
0
1.(σop)surface
140 120 2.(σop)depth,prediction
100
A
A
A
=1.055 x σop)depth,stabilized
80 60
B
B
B
3.(σop)depth,stabilized = σop)depth,prediction / 1.055
a0 = 1.16 mm
c0 = 1.16 mm
40
a0 = 1.51 mm
c0 = 1.51 mm
20
a0 = 1.86 mm
c0 = 1.86 mm
a0 = 2.21 mm
0
c0 = 2.21 mm
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
a, c (mm), surface crack length
0.012
1.10
R=0.0, σmax=200MPa a=2.56mm Δa=0.035mm (σop)surface=81MPa
load=200MPa
0.010 1.08 0.008
z (mm)
σop,depth at A of Fig. 5 / σop,depth at B of Fig. 5
Fig. 5. Estimation of the stabilized crack opening load at the deepest point.
1.06 1.055
0.006
150MPa
a0 = 0.15 mm a0 = 2.21 mm
100MPa
0.004
1.04
50MPa
0.002 20MPa
1.02 0.000
0MPa
1.00 1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
0
3.0
1
crack length, a (mm)
3
4
(a) free surface (X-coordinate)
Fig. 6. Ratio of crack opening values at the depth point when the crack is advanced through 1xp,surface.
0.012
R=0.0, σmax=200MPa a=2.56mm Δa=0.035mm (σop)depth=39MPa
load=200MPa
0.010 0.008
z (mm)
length a0 is 2.21 mm. FE analyses are performed with rmax = 200 MPa, rmin = 0 MPa, and element length, Da = 0.035 mm. Fig. 7(a) shows the crack opening profiles on the free surface (X-coordinate) after advancing from an initial crack length a0 = 0.15 mm or 2.21 mm to a final crack length 2.56 mm and Fig. 7(b) shows the profiles along the depth direction (Y-coordinate). The crack opening displacements are considerably different according to an initial crack length before crack length of 2.21 mm, but are identical regardless of an initial crack length after crack length of 2.21 mm. It is due to the overshoot of the opening displacement generated by the large z-direction plastic deformation produced at an initial crack length a0 = 2.21 mm during the first loading.
2
x coordinate (mm)
0.006
150MPa
a0 = 0.15 mm a0 = 2.21 mm
100MPa
0.004 50MPa
0.002 20MPa
0.000
0MPa
0
1
2
3
y coordinate (mm) (b) depth direction (Y-coordinate) Fig. 7. Crack opening profiles at R = 0, rmax = 200 MPa.
4
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Fig. 8 shows that the large initial plastic deformation occurs at an initial crack length a0 = 2.21 mm because there is no compressive residual stress near the crack tip during the 1st loading. When advancing the surface crack from an initial crack length a0 = 0.15 mm, the overshoot of the opening displacement was not observed. In the present study, the large z-direction plastic deformation occurred if an initial surface crack length was more than 0.43 mm at rmax = 200 MPa. However, irrespective of an initial crack length, all the first nodes behind the crack tip along the front of the surface crack are the last to open. Fig. 9 shows the process that the contacted regions of the surface crack become opened as the applied remote stress increases when advancing the crack from an initial crack length a0 = 0.15 mm. The dark areas are the crack surfaces in contact. As seen in Fig. 9(a), at r = 0 MPa, all the area near the free surface is closed, but the inner area from the free surface and the narrow area slightly behind the crack tip are still open. The above results are similar to those examined through 0.0012
R=0.0, σmax=200MPa a0=2.21mm, a=2.56mm
load=20MPa 0.0010
z (mm)
0.0008 0.0006
Δa=0.035mm (σop)surface=81MPa
15MPa
10MPa
0.0004 5MPa 0.0002 0.0000
0MPa a0=2.21mm
0
1
2
3
x coordinate (mm) (a) free surface 0.0012
R=0.0, σ max=200MPa a0=2.21mm, a=2.56mm
load=20MPa 0.0010
z (mm)
0.0008 0.0006
Δa=0.035mm (σop)depth=39MPa
15MPa
10MPa
0.0004 5MPa 0.0002 0.0000
0MPa a0=2.21mm
0
1
2
3
y coordinate (mm) (b) depth direction Fig. 8. Crack surface deformation during the loading range of 0–20 MPa (a0 = 2.21 mm, a = 2.56 mm, Da = 0.035 mm).
the experimental method by Troha et al. [34,35]. At r = 5 MPa, except for the region around the crack tip and the free surface, all the inner nodes become open. As the external load increases, the nodes behind the surface crack tip are opened sequentially from the deepest point to the free surface (Fig. 9(d)–(f)). Fig. 10 shows the state that the crack surface is opened at each load level when the surface crack grows from an initial crack length a0 = 2.21 mm. At r = 0 MPa, most region of the crack surface is closed but the partial area behind the crack tip in the range of 15–90 inward from the free surface is opened (Fig. 10(a)). As the external load increases, the partially opened area is expanded (Fig. 10(b) and (c)) and finally the nodes behind the surface crack tip are opened in order from the deepest point under the plane strain to the free surface under the plane stress (Fig. 10(d)–(f)). This result is similar to that of an initial crack length a0 = 0.15 mm. The opening levels of the deepest point as well as the free surface at the same crack length are not changed depending on an initial crack length. Figs. 9(e) and 10(e) clearly illustrate that the crack tip near the free surface of a part-through crack often lags in the actual fatigue crack growth (Fig. 11) because of more crack closure around the free surface. As shown in Fig. 12, the free surface did not maintain a plane during the loading and unloading cycle and the maximum forward plastic zone occurred approximately 5 inward from the free surface, which was well explained by Daniewicz and Aveline [21]. These are the interesting results that are very difficult to be observed in experimental methods. Fig. 13 shows the forward plastic zone sizes computed by FE analysis with a crack length a = 2.56 mm, rmax = 200 MPa, stress ratio R = 0 and mesh size Da = 0.035 mm using von Mises yield criterion. The lengths of the plastics zones at the free surface and the deepest point by FE analysis are 0.105 mm and 0.045 mm, respectively, and the length at the free surface is 2.33 times longer than at the deepest point. The values estimated by Eqs. (1) and (2) are 0.201 mm and 0.051 mm, respectively, and the length at the free surface is 3.94 times longer than at the deepest point. The ratios of the result of FE analysis to one of Irwin’s equation are 52.2% at the free surface and 88.2% at the deepest point. The length of the maximum forward plastic zone at about 5 inward from the free surface is 0.175 mm, which is 1.67 times longer than at the free surface and 3.89 times longer than at the deepest point. The value is 87.1% of the length calculated by Eq. (1). Fig. 14 shows the cyclic stress–strain curve at the crack tip node for 9th and 10th cycles. Even if the remote applied stress is zero, there is still cyclic plasticity in the material near the crack tip. It results in the plasticity induced crack closure. Crack opening stress is higher at the free surface than at the deepest point because the compressive residual
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175
Fig. 9. Closed regions of semi-circular crack surface advanced from a initial crack length a0 = 0.15 mm.
stress is higher at the free surface than at the deepest point as shown in Fig. 14. 3.4. Mesh refinement The effect of mesh size was examined for the element length ranging from 0.0015 mm to 0.01 mm by performing the FE analysis with a crack length a = 2.0 mm, rmax = 150 MPa and stress ratio R = 0. For each element size, a crack was advanced through the forward plastic zone size xp,surface for the crack opening value to be stabilized at the free surface. The stabilized opening value at the deepest point was obtained by the method given in Fig. 5. We examined the effect of element size in the fine mesh zone within the range of 0.0015–0.01 mm because of the
huge amount of computation time and cost required for obtaining the stabilized crack opening stresses by 3-D FE analysis of semi-circular crack. In addition, as pointed out by Jiang et al. [36], it should be noted that an element size smaller than 0.001 mm is usually not practical. Fig. 15 shows the effects of mesh size on the crack opening load. As shown in the figure, the more refined mesh increases the crack opening loads at both the deepest point and the free surface, but the incremental rates of the crack opening loads decrease. Within the range evaluated, it is not conclusive whether or not the opening load will be minimal or stabilized when the element size is smaller than a certain value. If the converged crack opening results could be obtained only by reducing the mesh size, it would be very
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Fig. 10. Closed regions of semi-circular crack surface advanced from a initial crack length a0 = 2.21 mm.
simple problem to be solved. However, according to the recent work of Jiang et al. [36], the converged crack opening results cannot be obtained only by reducing the mesh size and the complex stress–strain responses for the material near the crack tip cannot be properly described due to inability to mimic the strain ratcheting and stress relaxation when the kinematic hardening rule of Prager–Ziegler is used. It is worthy of note that the crack opening stresses for the CT specimen under plain stress obtained by using the JS cyclic plasticity model proposed by Jiang et al. [36] are not a strong function of the element size. 3.5. Comparison with experimental measurement methods Kim and Song [25] measured the plane strain crack opening at the deepest point by an extensometer of
5.7 mm gauge length spanning the crack at the midpoint of the surface crack mouth and the plane stress crack opening at the free surface by strain gauges boned in the vicinity of the surface intersection point. They determined the crack opening loads on the load–differential displacement curves or the load–differential strain curves with the naked eye. On the viewpoint of measurement sensitivity, the drawbacks of experimental methods were investigated with the load–differential displacement curves or the load–differential strain curves at the identical locations obtained by FE analysis. Fig. 16 shows the load–differential displacements calculated by FE-analysis varying the distance of the z-direction corresponding to the gage length of the extensometer used in experiment [25]. In this case, FE-analysis was performed with an initial crack length a0 = 0.15 mm, stress ratio R = 0, rmax = 200 MPa, final crack length
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14
surface (0 degree)
13
z ( mesh ), mesh size=0.035mm
12 11 10 9 8 7
depth (90 degree)
6 5 4
z
3 2 1
5 degree from surface
0
X
-1 -6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
x ( mesh ), mesh size=0.035mm
Fig. 11. Effect of more crack closure at the material surface on crack front shape.
plastic zone sizes from FEM : ω p,surface,FEM = 0.105 mm (ref. theory : 0.201 mm) ω p,5 degre,FEM = 0.175 mm ω p,depth,FEM = 0.045 mm (ref. theory : 0.051 mm)
a = 2.56 mm and the element length Da = 0.035 mm. Crack opening stresses obtained by FE-analysis were determined by monitoring all nodal displacements and nodal reaction forces along the crack. In order to compare the crack opening stresses obtained by FE-analysis with those determined on the load–differential displacement curves with the naked eye, the results obtained by FE-analysis were marked on the load–differential displacement curves. As shown in the figure, the slope change of the load– differential displacement diminishes with the increase of
ω p,surface,FEM / ω p,depth,FEM = 0.105 mm / 0.045 mm = 2.33 / 1 Fig. 13. Plastic zone sizes at crack surface and crack depth calculated by FEM.
the distance of the z-direction. Therefore, in order to increase the accuracy in experimentally measuring the opening load, the span of an extensometer should be as short as possible. As shown in Fig. 17(a), if the opening load at the deepest point were visually determined on the load–differential
Fig. 12. von Mises stress contours and deformations of free surface at rmax = 200 MPa.
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800 9th loading
σzz (MPa)
600
400 9th unloading 10th loading
200
0
9th,10th cycle R=0, σmax=200MPa
-200
crack tip node at the free surface crack tip node at the deepest depth
-400 0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
0.016
0.018
εzz Fig. 14. Stress–strain response near the crack tip at the free surface and the deepest point (a0 = 2.21 mm, a = 2.56 mm).
250
0.6
z = 0 mm z = 1.225 mm z = 2.432 mm z = 3.675 mm z = 5.520 mm
(σop)surface (σop)depth,corrected
σop / σmax
0.5
R=0.0, σmax=150MPa a=2.0mm
200
R=0.0, σmax=200MPa (σop)surface = 81 MPa
0.4
(σop)depth = 39 MPa a0 = 0.15 mm a = 2.56 mm a = 0.035 mm
150
0.2 0.000
0.002
0.004
0.006
0.008
0.010
0.012
load (MPa)
0.3
100
(σop)surface
Δa, mesh size (mm) Fig. 15. Effect of mesh size on crack opening load.
displace curve calculated at the z-direction distance of 2.5 mm similar to the span of the extensometer used in the experiment [25], the value is considered to be about 44 MPa while the opening load obtained by FE analysis immediately behind the deepest point of the surface crack tip is 39 MPa. This indicates that the plane strain opening load at the deepest point measured experimentally may be estimated to be about 13% higher than the actual. Therefore, the opening loads of the deepest points measured experimentally are in need of being corrected. Fig. 17(b) shows the load–differential strain curve calculated by FE-analysis at the distance of 0.21 mm (1.04xp,surface) forward the crack tip of the free surface. As shown in this figure, the plane stress crack opening load
50
(σop)depth
0
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
differential displacement (μm) Fig. 16. Variation of load–differential displacement curve.
measured visually on the load–differential strain curve is nearly equal to the numerical value obtained immediately behind the crack tip at the free surface. Therefore, it could be found that the crack opening load measured experimentally by strain gauge boned in the vicinity of the surface
J.-S. Kim et al. / International Journal of Fatigue 29 (2007) 168–180 250
250
measurement location x = 2.77 mm (1.04 ω p)
measurement location z = 2.5 mm R = 0.0, σmax = 200 MPa
R = 0.0, σmax = 200 MPa
(σop)surface = 81 MPa
200
(σop)surface = 81 MPa
200
(σop)depth = 39 MPa a = 2.56 mm a = 0.035 mm
(σop)depth = 39 MPa a = 2.56 mm a = 0.035 mm
150
load (MPa)
150
load (MPa)
179
100 (σop)surface
100 (σop)surface visual measurement of σop)surface
50
visual measurement of σop)depth
50 (σop)depth
(σop)depth
0
0
-0.10 -0.05
0.00
0.05
0.10
0.15
0.20
-0.10 -0.05
differential displacement (μm)
0.00
0.05
0.10
0.15
0.20
differential strain (b) at the free surface
(a) at the deepest point
Fig. 17. Comparison between the numerical value and visual measurement one for crack opening load.
intersection point can successfully represent the plane stress crack opening of the free surface. 4. Conclusions A 3-D elastic–plastic FE analysis is performed for semicircular crack, giving special attention to stabilization behaviour, the mesh size effect, the effect of initial crack length, the opening process of crack surface and the comparison with experimental methods. The conclusions obtained are summarized as follows. 1. The opening loads decrease along with the increase of strain hardening and the effect of strain hardening on opening load is significant at the deepest point than at the free surface. 2. For the plane stress at the free surface, the crack opening level stabilizes after the crack advances through the plane stress monotonic plastic zone, xp,surface. For the crack opening loads at the deepest point to be stabilized, the crack must be advanced through 8xp,depth. However, the stabilized crack opening load at the deepest point can be estimated by multiplying the value obtained at the time that the crack is advanced through xp,surface by a correction factor. 3. The crack opening displacements are considerably different according to an initial crack length. However, irrespective of an initial crack length, all the first nodes behind the crack tip along the front of the surface crack are the last to open.
4. The more refined mesh increases the crack opening loads both at the deepest point and at the free surface, but the incremental rates of the crack opening loads decrease. It suggests that the crack opening loads may be converged if using the sufficiently refined mesh. 5. The plane strain crack opening load at the deepest point measured in experiment by an extensometer at the midpoint of crack mouth may be estimated to be higher than the actual. The crack opening load measured experimentally by strain gauge boned in the vicinity of the surface intersection point can successfully represent the plane stress crack opening of the free surface. References [1] Ohji K, Ogura K, Ohkubo Y. On the closure of fatigue cracks under cyclic tensile loading. Int J Fract 1974;10:123–4. [2] Ohji K, Ogura K, Ohkubo Y. Cyclic analysis of a propagating crack and its correlation with fatigue crack growth. Eng Fract Mech 1975;7:457–64. [3] Ogura K, Ohji K. FEM analysis of crack closure and delay effect in fatigue crack growth under variable amplitude loading. Eng Fract Mech 1977;9:471–80. [4] Newman Jr JC. A finite element analysis of fatigue crack closure. ASTM STP 590 1976:280–301. [5] Newman Jr JC. Finite element analysis of crack growth under monotonic and cyclic loading. ASTM STP 637 1977:56–80. [6] Nakagaki M, Atluri SN. Elastic–plastic analysis of fatigue crack closure in mode I and II. AIAA J 1980;18:1110–7. [7] Nakamura H, Kobayashi H, Yanase S, Nakazawa H. Finite element analysis of fatigue crack closure in compact specimen. In: Proceeding of ICM 4; 1983. p. 817–23.
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