A reexamination of plasticity-induced crack closure in fatigue crack propagation

International Journal of Plasticity 21 (2005) 1720–1740 www.elsevier.com/locate/ijplas

A reexamination of plasticity-induced crack closure in fatigue crack propagation Yanyao Jiang *, Miaolin Feng, Fei Ding Department of Mechanical Engineering (312), University of Nevada, Reno, NV 89557, USA Received in final revised form 8 August 2004 Available online 15 December 2004

Abstract The crack closure concept is often used to consider the R-ratio and overload effects on fatigue crack growth. The presumption is that when the crack is closed, the external load produces negligible fatigue damage in the cracked component. The current investigation provides a reassessment of the frequently used concept with an emphasis on the plasticity-induced crack closure. A center cracked specimen made of 1070 steel was investigated. The specimen was subjected to plane-stress mode I loading. An elastic–plastic stress analysis was conducted for the cracked specimens using the finite element method. By applying the commonly used one-node-per-cycle debonding scheme for the crack closure simulations, it was shown that the predicted crack opening load did not stabilize when the extended crack was less than four times of the plastic zone size. The predicted opening load was strongly influenced by the plasticity model used. When the elastic–perfectly plastic (EPP) stress–strain relationship was used together with the kinematic hardening plasticity theory, the predicted crack opening load was found to be critically dependent on the element size of the finite element mesh model. For R = 0, the predicted crack opening load was greatly reduced when the finite element size became very fine. The kinematic hardening rule with the bilinear (BL) stress–strain relationship predicted crack closure with less dependence on the element size. When a recently developed cyclic plasticity model was used, the element size effect on the predicted crack opening level was insignificant. While crack closure may occur, it was demonstrated that cyclic plasticity persisted in the material near the crack tip. The cyclic plasticity was reduced but not

*

Corresponding author. Tel.: +1 775 784 4510; fax: +1 775 784 1701. E-mail address: [email protected] (Y. Jiang).

0749-6419/$ - see front matter
2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijplas.2004.11.005

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negligible when the crack was closed. The traditional approaches may have overestimated the effect of crack closure in fatigue crack growth predictions.
2004 Elsevier Ltd. All rights reserved. Keywords: Crack closure; Crack tip plasticity; Cyclic plasticity; Elastic–plastic; Finite element

1. Introduction Crack closure explicitly refers to the contacting of the opposing surfaces of a fatigue crack before the minimum load in the loading cycle is reached. In addition, the crack closure concept related to cracking behavior of a material has two implications: (I) It is assumed that fatigue damage to the cracked component is minimal when the crack is closed. (II) A major understanding in using the crack closure concept is the existence of crack closure for loading cases with R (the minimum external load over the maximum external load) ratios being larger than zero. To many researchers in the fatigue research community, crack closure has become an established concept that is useful for assessing crack growth, while to a few others the whole notion is still questionable. The discovery of crack closure came with the experimental observations using a thin center cracked specimen of 2024-T3 aluminum alloy under tension–tension mode I fatigue loading (Elber, 1970). The existence of crack closure was deduced from the nonlinear load versus crack opening displacement curve. The technique for the determination of crack closure was later used by most researchers with little modifications for the crack opening displacement measurement and interpretation of crack closure or opening levels. A great number of technical papers related to crack closure have been published in the last three decades (Allen et al., 1988; McClung, 1999; Sehitoglu et al., 1996; Vasudeven et al., 1994). Noticeably, two volumes of ASTM Special Technical Publications, ASTM STP 982, and ASTM STP 1343, were specifically dedicated to the discussion of crack closure. The crack closure concept has been used to consider the overload and R-ratio effects on crack growth. A number of quantities using the effective stress intensity range were developed based upon the crack closure concept with good reported correlations with the experimental observations (McClung and Sehitoglu, 1989a,b, 1991, 1992; Sehitoglu et al., 1996). In their review paper, Sehitoglu et al. (1996) summarized experimental observations on the influences of a number of factors including the notches, maximum applied stress, crack length, in-plane biaxiality, out-of-plane constraint, and the transient loading. Based on the crack closure concept, all these effects on crack propagation were explained with an effective stress intensity parameter (Sehitoglu et al., 1996).

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ASTM standards were developed to determine the crack close and opening levels, although arguments existed on how accurately the standardized method can provide consistent results (Donald and Phlilips, 1999; Graham et al., 1999). It was found that the results of the crack opening displacement measurement with the compliance technique near the crack tip depended strongly on the position of the displacement gages. The compliance curves of the remote clip gages were often too insensitive to detect the near-crack-tip closure (Pippan et al., 1999). A unique technique was developed to experimentally determine crack closure response near the crack tip at the center of a specimen (Fleck et al., 1983; Fleck and Smith, 1982). The Sop/Smax (Sop = opening nominal applied stress and Smax = maximum nominal applied stress in a loading cycle) ratio is often used as a measure of the degree of crack closure. The experimentally obtained Sop/Smax ratio can be more than 0.9 for near threshold crack growth with R = 0.1 (Davidson, 1988). When R = 0, the Sop/ Smax ratio was reported to range from 0.6 for 7175-T6 to approximately 0.26 for a titanium alloy (Newman, 1999). Noticing the overestimated crack closure effect, partial crack closure concept was developed by introducing a coefficient being less than unit as a multiplier to Sop (Paris et al., 1999; Kujawski, 2002, 2003). On the other hand, Allen et al. (1988) concluded that at a growth rate of the order of 10
5 mm/ cycle and greater, closure did not occur except in the presence of shear lips. Vasudeven et al. (1994) critically reviewed the published crack closure mechanisms for the near-threshold crack growth behavior and concluded that there can be no contribution from plasticity to crack closure. With idealized elastic and elastic–plastic crack scenarios, it was demonstrated that the changes in the slope of the load versus crack opening displacement curves were not necessarily due to crack closure. Crack closure can exist, but its magnitude is either small or negligible (Vasudeven et al., 1994). Despite the reported success, several questions remain. While direct observations can be made on possible contact of cracked surfaces, whether or not such a surface contact results in minimal or no fatigue damage near the crack tip cannot be experimentally confirmed. The remarkably different crack closure behavior in ambient air and in a vacuum environment raises a critical question about the existence of plasticity-induced crack closure. Several problems exist in the finite element (FE) simulations of crack closure. For R P 0 loading cases, crack closure can be predicted only when a propagating crack is used. Generally, an element debonding scheme of one-node-per-loading cycle is used in the FE simulations. Considering the size of the elements often used, the corresponding crack growth rate is usually orders of magnitude higher than the real crack propagation rate. In addition, the influence of the element size in the FE model on the predicted crack closure results is significant when a traditional plasticity model is used. It was shown (Solanki et al., 2003) that for plane-stress problems, there existed a convergence of the crack closure results predicted with respect to the element size used. However, the results that will be shown in this study reveal that such a convergence does not exist when the element size becomes smaller than the range covered by Solanki et al. (2003). The predicted crack closure results are also strongly dependent on the material models and material constants chosen for a particular constitutive model in the FE simulations (McClung and Sehitoglu, 1989b). Furthermore, the definition of the opening/closing

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load is still ambiguous with regard to the contacting position and the contacting area. Does the contact of any single point in the wake of the crack signify crack closure regardless of the position of the contact and the amount of the contact area? More importantly, does the cyclic plasticity of the material near the crack tip become minimal when any point in the wake of the crack is in contact? The current investigation re-examines the crack closure concept by using the finite element simulations. The discussion is limited to plasticity-induced crack closure at the macroscopic material scale where continuum mechanics concepts are valid. Without laboring to argue its validity and general acceptance, an assumption is that the cyclic plasticity of the material near the crack tip is the major driving force for crack advance.

2. Finite element simulations 2.1. Mesh model A center cracked tension (CCT) specimen with the plane-stress condition was considered. The plate had a width of 140 mm (W = 70 mm) and a center crack size of 20 mm (a = 10 mm). As shown in Fig. 1, a two-dimensional FE mesh model was created with the software package HyperMesh (Altair, 2001). Due to the geometric and load symmetry, only one quarter of the specimen was modeled. Plane-stress quadrilateral elements were used. Referring to Fig. 1, on the y = 0 plane the displacements in the y direction of all the nodes except the nodes on the crack surface were constrained. The displacements in the x direction of all the nodes on the x = 0 plane were zero. Uniform pressure load was applied on the elements of the top surface of the plate. Due to the high stress and strain gradients, the elements were the finest near the crack tip.

Fig. 1. Mesh model for center cracked tension (CCT) specimens: (a) overall model, (b) near crack tip.

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The sizes of the smallest elements in the mesh models ranged from 1 lm (0.001 mm) to 240 lm (0.24 mm). The mesh size was identical for the cases simulated in the elastic deformation zone. The number of quadrilateral elements in a model ranged from 9540 to 12,626 depending on the size of the elements used near the crack tip. Surface contact elements were used for the cracked surfaces. The lower symmetric part of the specimen served as the master surface and the upper cracked surface was the slaved surface. For all the FE simulations where a propagating crack was considered, debonding of the crack tip elements occurred when the external load reached its maximum in a loading cycle. The opening load corresponded to the moment that the wake of the crack was fully open. A crack was identified to be closed when any point in the wake of the crack was in contact. The contact of the cracked surfaces was identified by the contact pressure between the two mating elements on the cracked surfaces. The option ‘‘NLGEOM’’ was selected in ABAQUS (1999) to consider the geometric nonlinearity due to possible large deformation in the material near the crack tip. 2.2. Material 1070 steel was used in the investigation. The material has been well studied for its cyclic plasticity deformation properties and fatigue properties (Jiang and Sehitoglu, 1996a,b; Jiang et al., 2002; Jiang and Feng, 2004). More importantly, the material displays short-lived and insignificant cyclic softening at the beginning of cyclic loading and it does not display nonproportional hardening. A stable material facilitates the modeling of the cyclic stress–strain responses. 2.3. Plasticity models Three different plasticity models were used: pure kinematic hardening rule of Prager–Ziegler with the elastic–perfectly plastic (EPP) stress–strain relationship, pure kinematic hardening rule of Prager–Ziegler with the bilinear (BL) stress–strain relationship, and a kinematic hardening model developed by Jiang and Sehitoglu (1996a,b). The mathematical equations used in the Jiang and Sehitoglu (JS) model and the characteristics of the model can be found in earlier publications (Jiang and Sehitoglu, 1996a,b; Jiang et al., 2002; Jiang and Feng, 2004). The pure kinematic hardening model with EPP stress–strain relationship has been frequently used by Newman and co-workers (Fleck and Newman, 1988; Solanki et al., 2003). The plasticity model with the BL stress–strain relationship was used by McClung and Sehitoglu (1989a,b), McClung (1991a,b, 1994), Roychowdhury and Dodds (2003) in the numerical simulations of crack closure. Both the EPP and BL material models can be found in the material model library in ABAQUS (1999). The JS plasticity model was implemented into the generalpurpose FE code ABAQUS through the user-defined subroutine UMAT. Detailed implementation of the model can be found in Jiang et al. (2002). The cyclic plasticity material constants for 1070 steel are listed in Table 1. They were generated from earlier investigations (Jiang and Sehitoglu, 1996b, 1992). The

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Table 1 Material constants for 1070 steel JS cyclic plasticity model

E = 206.8 GPa, l = 0.3, k = 248.3 MPa

c(1) = 204.1, c(2) = 61.7, c(3) = 18.6, c(4) = 5.6, c(5) = 1.6 r(1) = 58.8 MPa r(2) = 94.9 MPa, r(3) = 99.8 MPa, r(4) = 100.2 MPa, r(5) = 128.8 MPa v(1) = v(2) = v(3) = v(4) = v(5) = 5.0

E = Elasticity modulus, l = PoissonÕs ratio, k = yield stress in shear c(1),c(2),c(3),c(4), c(5) = Material constants r(1),r(2), r(3),r(4), r(5) = Material constants v(1),v(2), v(3),v(4), v(5) = Material constants

EPP

E = 206.8 GPa, l = 0.3, k = 248.3 MPa

E = Elasticity modulus, l = PoissonÕs ratio, k = yield stress in shear

BL

E = 206.8 GPa, l = 0.3, k = 248.3 MPa, H = 0.007E

E = Elasticity modulus, l = PoissonÕs ratio, k = yield stress in shear, H = plastic modulus

material and heat treatment were identical to those used by McClung and Sehitoglu (McClung and Sehitoglu, 1989a,b; McClung and Sehitoglu, 1992; McClung, 1991b; Sehitoglu, 1985). To facilitate a comparison, the material constants when using the BL stress–strain relationship were identical to those used by McClung and Sehitoglu (McClung and Sehitoglu, 1989a,b, 1992; McClung, 1991b). Fig. 2 shows the experimentally obtained cyclic stress–strain curve of 1070 steel together with the representations of three different material models. It should be emphasized that the differences among the cyclic plasticity models are more than the modelsÕ capability to describe the stress–strain curves for the fully reversed uniaxial loading. Strain ratcheting and stress relaxation occur for a material under general multiaxial and asymmetric loading. The material near the crack tip generally experiences both strain ratcheting and stress relaxation (Jiang and Feng, 2004).

Stress Amplitude, MPa

1000

1070 Steel

800 600 400 Experiment EPP BL JS

200 0 0.00

0.01

0.02 0.03 Strain Amplitude

0.04

Fig. 2. Cyclic stress–strain curve for 1070 steel.

0.05

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The pure kinematic hardening models of Prager–Ziegler with EPP stress–strain relationship cannot deal with strain ratcheting and stress relaxation. The model with the BL stress–strain relationship does not predict ratcheting and stress relaxation for any loading conditions. The desire to use the JS model was due to its capability of describing the general cyclic material behavior including cyclic strain ratcheting and stress relaxation. The JS model can describe the stress–strain responses under complex loading conditions very well for 1070 steel (Jiang and Sehitoglu, 1996a,b; Jiang and Kurath, 1996a,b). 3. Results and discussions 3.1. Effect of element size A loading case with the load ratio, R, being zero and the maximum nominal applied stress, Smax, of 160 MPa, was used to show the effect of the element size on the predicted crack opening load. For the p CCT ffiffiffiffi specimen, the corresponding stress intensity factor range, DK, was 28:7 MPa m. The cyclic plastic zone size, rp, based on the stress intensity factor range can be determined by using the following equation for plane-stress condition (Bannantine et al., 1990):  2 1 DK rp ¼ ; ð1Þ 8p r0 where DK is the stress intensity factor range and r0 is the yield stress of the material. According to Eq. (1), the cyclic plastic zone size was 0.178 mm when k = 248 MPa was used, where the symbol k denotes the materialÕs yield stress in shear. The FE simulations predicted an average value of the cyclic plastic zone size of 0.35 mm, twice as large as that obtained using the traditional linear elastic based stress intensity factor. It should be noted that in the FE simulations, the cyclic plastic zone boundary was determined using the criterion that no cyclic plastic deformation occurred outside the plastic zone. In the presentation of the results, the plastic zone size of 0.178 mm was used. The variations of the normalized crack opening load with element size are shown in Figs. 3–5 for the results obtained by using the three different material models. In the figures, the symbol r represents the crack extension and rp denotes the plastic zone size (rp = 0.178 mm for the loading case under consideration). Sop is the opening load (nominal stress) and Smax is the maximum nominal applied load. It should be noted that the element size refers to the finest elements near the crack tip. For each simulation, up to 170 loading cycles were simulated for very small element sizes. A propagating crack was simulated and the debonding rate was one node per loading cycle. Again, debonding was set to occur when the load reached its maximum level in a loading cycle. All the simulations presented in Figs. 3–5 were terminated when the crack length, a, reached 10 mm. The crack length, a, refers to the half total crack length in the specimen and the crack extension, r, is the amount of crack growth simulated using the node debonding technique in the FE analysis.

Y. Jiang et al. / International Journal of Plasticity 21 (2005) 1720–1740 0.6

124.8 µm 62.4 µm

0.5 Sop/Smax

1727

31.2 µm

0.4 7.8 µm

0.3 0.2

2.6 µm

0.1

Element Size =1 µm

0.0 0

2

EPP model Smax=160 MPa, R=0, ∆K=28.7MPa m1/2 W=70mm, a=10mm

4

6

8

10

r/rp

Fig. 3. Variations of predicted crack opening load with the crack extension with the influence of element sizes by using the EPP material model.

0.6

Sop/Smax

0.5 0.4 0.3

7.8 µm

31.2 µm

124.8 µm

62.4 µm

Element Size=2.6 µm

0.2

BL Model Smax=160 MPa, R=0, ∆ K=28.7MPa m1/2 W=70mm, a=10mm,

0.1 0.0 0

2

4

6

8

10

r/rp

Fig. 4. Predicted crack opening load with different element sizes by using the BL material model.

0.6

Sop/Smax

0.5 0.4 0.3 0.2 JS model 1/2 Smax=160 MPa, R=0, ∆K=28.7MPa m

0.1

W=70mm, a=10mm

0.0 0

2

4

6

2.6 µm 7.8 µm 31.2 µm 62.4 µm 124.8 µm

8

10

r/rp

Fig. 5. Crack opening load predicted by using the JS material model with the consideration of crack extension and element size.

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The results shown in Figs. 3–5 indicate that the predicted crack opening load continued to increase with the number of loading cycles simulated. For a given element size, the crack extension simulated is proportional to the number of loading cycles. The predicted opening result tended to stabilize. Results obtained for different loading conditions suggest that the stabilization is not only dependent on the material model used in the FE simulation but also on the loading magnitude. It is possible that the extension of crack needed to stabilize the predicted crack opening is also dependent on the material. It is, therefore, difficult to define a certain crack extension, r/rp, after which the predicted opening level can be considered to be stabilized. As compared with the other two models, the EPP model predicts a much greater effect of element size on the predicted crack opening load. In addition, a longer transition period before stabilization is predicted by using the EPP model. By studying the detailed data shown in Figs. 3–5, it was found that the relationship between the opening load, Sop/Smax, and the crack extension simulated, r/rp, can be approximated by using the following exponential function: S op =S max ¼ C 0 þ C 1 expð
C 2 r=rp Þ;

ð2Þ

where C0, C1, and C2 are fitting constants. C0 is the asymptotic value of the function and it can be used to represent the stabilized crack opening level predicted by the FE simulations. Eq. (2) serves two purposes. The asymptotic value can more reasonably represent the stabilized opening load than using a value corresponding to a given crack extension length. Secondly, when a very small element size is used, it generally takes very large number of loading cycles using the one-node-per-cycle debonding scheme in order to simulate a sizable crack length. The equation can be used to extrapolate the results. The effect of the element size on the predicted crack opening load for the three material models is summarized in Fig. 6. The asymptotic value by using Eq. (2) was used in Fig. 6 for a given element size. Significant effect of element size was found when the EPP was used. A maximum crack opening level of 0.57 was

0.6

Sop/Smax

0.5 0.4 0.3 0.2 EPP BL JS

R=0, Smax=160 MPa

0.1

1/2

W=70mm, a=10mm, DK=28.7 MPa m

0.0 2

1

3

2

4 5 6

3

2

4 5 6

10

3

4 5

100

Element Size, µm

Fig. 6. Effect of mesh size on the crack opening load levels predicted by using different material models.

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predicted when the element size was 0.12 mm. When the element size was 1.0 lm (0.001 mm), the predicted opening level was 0.1. As the element size is further decreased, a minimal crack opening load will be predicted by using the EPP model. The influence of the element size on the predicted opening level was less significant when the BL material model was used in the FE simulation. The model predicted a maximum opening level of 0.4 when the element size was 0.12 mm, and the predicted result continued to reduce when the finite element was refined. 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. However, it should be noted that an element size smaller than 1 lm (0.001 mm) is usually not practical for the CCT specimen under consideration. As can be observed in Fig. 5, the crack opening results obtained by using the JS cyclic plasticity model are not a strong function of the element size. When the element size ranges from 0.008 to 0.06 mm, the crack opening level is predicted to be approximately 0.41. The predicted value is reduced when the element size is larger than 0.06 mm or smaller than 0.008 mm. It should be noticed that a logarithmic scale is used for horizontal axis in Fig. 6. Solanki et al. (2003) studied the influence of the FE element size on plasticityinduced crack closure. It was concluded that for plane-stress problem with the CCT specimen a convergence of the crack opening load existed as the element size was decreased. Obviously, the results shown in Fig. 3 from the current investigation contradict to what was found by Solanki et al. (2003). In order to evaluate the consistency of the results, FE simulations were conducted for the CCT specimen with identical specimen dimension, material properties, material model, and loading conditions to those used by Solanki et al. (2003). A CCT specimen with a half width of W = 40 mm and a crack size of a = 4 mm was used. The maximum external applied stress was 69 pMPa and the stress ratio R was 0, corresponding to ffiffiffiffi DK ¼ 7:78 MPa m. Quadrilateral mesh was used in the FE model. The material had an elasticity modulus of 200 GPa and a yield stress of 230 MPa (k = 132.8 MPa). The pure kinematic hardening rule of Prager–Ziegler with the EPP

0.6

Sop/Smax

0.5 0.4 0.3 0.2 0.1 0.0

EPP model E=200 GPa, k=132.8 MPa Plane Stress,R=0, Smax=69 MPa W=40mm, a=4mm

1

2

3

4

5 6 7 8

Solanki et al.(2003) Current Study 2

10 Element Size, µm

3

4

5 6 7 8

100

Fig. 7. Influence of element size on opening load using EPP material model.

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stress–strain relationship was used. The results are shown in Fig. 7. Practically, identical results were reproduced when the mesh sizes were identical to those used by Solanki et al. (2003). However, when the element size was further reduced, it was found that the crack opening load tended to diminish with the same tendency found in Fig. 6 for the EPP model. The pure kimematic hardening rule of Prager–Ziegler with EPP and BL stress– strain relationships cannot properly describe the complex stress–strain responses for the material near the crack tip due to their inability to mimic the strain ratcheting and stress relaxation. The use of the two material models serves two purposes. First, they have been the most widely used models in the FE simulations of crack closure (McClung and Sehitoglu, 1989a,b; Roychowdhury and Dodds, 2003; Solanki et al., 2003; Fleck, 1988). A comparison of the results obtained from different material models reveals the significant influence of the constitutive models used. The second purpose is to show a consistency with those obtained by other investigators. Due to the inherent deficiency associated with these simple material models, the subsequent discussions will be concentrated on the results obtained by using the JS model.

3.2. Element debonding scheme An argument can be made on whether a propagating crack or stationary crack should be used in the FE simulation for crack closure. The use of a stationary crack does not produce crack closure for a loading case with R P 0. It is reasonable to argue that crack extension should be considered in the crack closure simulation. However, when the one-node-per-cycle debonding scheme is used, the mesh size used for the FE simulations of crack closure often corresponds to an unrealistic high crack growth rate. pffiffiffiffiFor example, for 1070 steel with a stress intensity factor range of 28:7 MPa m (a = 10 mm, R = 0, Smax = 160 MPa), the experimentally obtained crack growth rate is approximately 10
4 mm/cycle (Jiang and Feng, 2004). When using a mesh size of 31.2 lm near the crack tip and one-node-per-cycle debonding rate, the simulating crack growth rate is 3.12 · 10
2 mm/cycle which is more than 300 times faster than that of the real crack growth rate. Since whether or not a propagating crack is used in the FE simulation results in either no crack closure or significant crack closure to be predicted for loading cases with R > 0, it might be important to consider a node debonding rate consistent with the real crack growth rate. Two possible methods can be used to keep such a consistency. With the one-node-per-cycle debonding scheme, the element size near the crack tip should be 0.1 lm for the loading case mentioned in the last paragraph. Such a small element is generally unrealistic for the FE simulation since more than1700 loading cycles should be simulated in order to extend the crack for an amount of rp (0.178 mm for the loading case under consideration). As was shown in Fig. 5, the crack extension in the FE simulation has to reach more than one plastic zone size (rp) in order to obtain a near stabilized crack opening result. More importantly, when the element size is very small, the strain obtained from the FE simulation in the crack tip will be very large. It is very difficult to reach a convergence in the

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Sop/Smax,Scl/Smax

FE elastic–plastic stress simulation particularly when a realistic cyclic plasticity model is used. An alternative method to keep a consistent crack growth rate would be the scheme of one node debonding per every 312 loading cycles for the aforementioned loading case (1070 steel, a = 10 mm, R = 0, Smax = 160 MPa). Such a debonding scheme can be closer to the real crack growth since crack growth is generally observed as a discontinuous process at the microscopic scale. An FE simulation was conducted with the one-node-per-cycle debonding scheme for 23 loading cycles in order to establish the crack closure condition. Corresponding the last loading cycle (23rd loading cycle) the Sop/Smax value was 0.41 and Scl/Smax was 0.69, where Scl denotes the crack close load. The subsequent simulation was conducted for 190 loading cycles without crack extension. The variations of the crack opening and closing loads during these 190 loading cycles are shown in Fig. 8. Sop/Smax was reduced from a value of 0.41 to 0.3 after four loading cycles of stationary crack. At the end of the 190 loading cycles, Sop/Smax was 0.25. The crack close load, Scl/Smax, was reduced from 0.69 to 0.24 after four loading cycles of stationary crack. At the end of the 190 loading cycles in the subsequent loading, the Scl/Smax became 0.20. Clearly, if a debonding scheme of one node per a few hundred loading cycles is used, the average crack opening or close level is significantly smaller than that obtained regularly with the one-node-per-cycle debonding scheme. When a smaller loading magnitude is considered, the reduction in the predicted opening level can be more significant when the aforementioned FE simulation scheme is adopted. For example, when the loading amplitude corresponds to a stress intensity factor pffiffiffiffi range of 15 MPa m, the experimentally obtained crack growth rate is approximately 10
5 mm/cycle for 1070 steel (Jiang and Feng, 2004). If a 31.2 lm element size is used, the debonding scheme should be one node per every 3120 loading cycles in order to keep an identical crack growth rate to the real cracking rate. The predicted opening load will be again greatly reduced after a transient period during the loading block of 3120 cycles with a stationary crack. While the predicted crack opening may not necessary reach zero after a great number of loading cycles, the

0.8

R=0, Smax=160 MPa, W=70mm, a=10mm

0.7

∆K=28.7MPa m

1/2

0.6 0.5 Sop/Smax

0.4

Scl/Smax

0.3 0.2 0.1

0

50

100 Number of Cycles

150

200

Fig. 8. Decay of crack opening and close loads with number of loading cycles.

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continuous reduction in the value may create a practical difficulty in using the crack closure concept. It is interesting to note that when the traditional one-node-per-cycle debonding is used in the simulation of crack closure, the crack close load is always predicted to be significantly higher than the crack opening load. This can be seen from the first part of the current simulation where Sop/Smax was 0.41 while Scl/Smax was 0.69 at the end of the 23rd loading cycle. In the second part of the simulation where no crack debonding was considered, Scl/Smax was reduced much faster than that of Sop/Smax with increasing loading cycles. In addition, Scl/Smax was predicted to be smaller than Sop/ Smax during the subsequence loading with stationary crack.

3.3. Cyclic plasticity near crack tip The influence of the contact of cracked surfaces on the fatigue damage of the material near the crack tip can be better evaluated by looking at the cyclic plasticity in the local area near the crack tip. This will provide an assessment for the major implication in using the crack closure concept that crack closure prevents the material near the crack tip from further cyclic plastic deformation. Two examples are used to show that cyclic plasticity of the material near the crack tip is not minimal nor negligible when a crack is ‘‘closed.’’ Consider the CCT specimen with R = 0, Smax = 160 MPa, and a = 10 mm. In the FE simulation, the smallest element size near the crack tip was 7.8 lm. A total of 26 loading cycles were simulated. Since a significant crack extension was needed in order to stabilize the crack closure result, two node debonding rates were used. For the first 22 loading cycles, four nodes were debonded for every loading cycle. In the subsequent four loading cycles, the node debonding rate was one-node-per-cycle. The extended crack was 0.718 mm and the r/rp ratio was 4.0. Sop/Smax was predicted to be 0.39 for the last two loading cycles. This crack opening load is close to the stabilized value shown in Fig. 5. Since significant strain ratcheting is produced for the material near the crack tip due to the crack extension scheme used in the FE simulation when a relatively large element is used, a small element size reduces the strain ratcheting that facilitates the discussion with respect to the crack tip plasticity. The ryy
eyy response shown in Fig. 9 was taken at the last two consecutive loading cycles (25th and 26th cycles) from the Gauss point closest to the crack tip. Due to crack extension used in the FE simulation and the high stress and strain gradients existed for the material near the crack tip, significant strain ratcheting was predicted. Referring to Fig. 9 for the stress–strain hysteresis loop of the 26th loading cycle, crack was predicted to close at A and to open at C. Therefore, A–B–C in Fig. 9 represents the stress–strain response in the y direction for the material point under consideration. Clearly, significant plastic deformation occurred when the crack was closed. Two quantities are often used to measure the cyclic plastic deformation for a material point. They are accumulative equivalent plastic strain and the accumulative plastic strain energy density. In this study, the plastic strain energy density is

Y. Jiang et al. / International Journal of Plasticity 21 (2005) 1720–1740 1500

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R=0, Smax=160MPa, W=70mm, a=10mm 1/2

σyy,MPa

1000

∆K=28.7MPa m

500 A

0

C

-500 Crack Open Crack Close

B

-1000 0.00

0.01

0.02

εyy

0.03

0.04

0.05

Fig. 9. Stress–strain response of a material point near the crack tip.

adopted. The mathematic expression for the accumulative plastic strain energy density in a loading cycle is I rij depij ; ð3Þ DW p ¼ cycle

p

where DW represents the accumulated plastic strain energy density in a loading cycle, rij represent the stress components, and epij are the plastic strain components for a given material point. With the detailed stress and strain results obtained from the FE simulation, DWp can be determined through a numerical integration. By using Eq. (3), the total strain energy density for the case shown in Fig. 9 is 26.3 MJ/m3. If the plastic deformation is ignored when the crack is closed, the strain energy density is 15.9 MJ/m3, 40.0% lower than the real value. For 1070 steel under constant amplitude loading, the crack initiation life can be approximately related to the plastic strain energy density per loading cycle by using the following power law relationship (Jiang and Sehitoglu, 1992): ðDW p Þ

1:75

N f ¼ 32000;

ð4Þ

where Nf is the fatigue life. According to Eq. (4), ignoring the cyclic plasticity during crack closure results in more than a factor-of-two difference in fatigue life prediction for the material point. The estimate of the difference in the fatigue life prediction is approximate since the stress–strain hysteresis loops were not closed due to significant strain ratcheting. It should be also noted that the only purpose in using Eq. (4) is to provide a quantitative assessment of the influence by ignoring the cyclic plasticity when the crack is closed. To better evaluate the cyclic plasticity when a crack is closed, the second example used a CCT specimen made of 1070 steel, and the specimen was subjected a loading condition with the following parameters: R =
2, Smax = 100 MPa, and a p = ffiffiffiffi 10 mm. The resulting maximum stress intensity factor, Kmax, was 17:72 MPa m. The R =
2 case was chosen since crack surface contact can be expected to occur during a significant part of the loading cycle regardless of the definition of crack closure. In

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this example, stationary crack was used in the FE simulation. In the FE model, the finest mesh size near the crack tip was 31.2 lm. The simulation was conducted for 10 loading cycles and the results of the 10th loading cycle were analyzed. Since a stationary crack was used, 10 loading cycles was enough to stabilize the stress–strain responses. The hysteresis loop shown in Fig. 10 was the stress–strain response in the y-direction obtained from the Gauss point closest to the crack tip. Fig. 11 illustrates the crack closure process under R =
2 loading. The crack length shown in Fig. 11 was plotted to the scale. The crack close load was predicted to be
2.6 MPa (Point A in the figures) and the opening load was predicted to be 1.6 MPa (Point D in the figures) for the R =
2 loading case. When the applied load reached its minimum (Point C in the figures), part of the cracked surface closest to the crack tip was still open (Fig. 11). It is noted from Figs. 10 and 11 that the closure of the crack is a progressive process. By using Eq. (3), it was found that the plastic strain energy density per loading cycle, DWp, for the material point under consideration was 15.5 MJ/m3. When ignoring the plastic deformation when the crack was closed, DWp was 11.7 MJ/m3, 25% 1000

Smax =100MPa, R=-2 W=70mm, a=10mm

σyy,MPa

500 Open

0

D A

-500 B

C

Close

-1000 0.000

0.005

0.010

0.015 εyy

0.020

0.025

0.030

Fig. 10. Influence of crack closure on cyclic plasticity for R =
2.

200

a=10mm

A B C

S, MPa

100 0

Crack Tip

A

D B

-100 -200

C

-300 0.0

0.5

1.0

1.5 Time

2.0

2.5

Fig. 11. Crack closure process for R =
2.

3.0

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lower than the real plastic strain energy density. Such a difference in the prediction of the plastic deformation would result in 64% difference in the fatigue life prediction according to Eq. (4). The FE simulation with the consideration of a propagating crack showed that Sop/Smax = 0.23 and Scl/Smax = 0.44 for the R =
2 loading case. When crack extension is used in the FE simulation, the stress–strain hysteresis loop for a material point near the crack tip displays significant ‘‘ratcheting’’ due to the continuous node debonding and high stress and strain gradients near the crack tip. Nevertheless, it can be expected that more significant cyclic plasticity for the material near the crack tip will be predicted when the crack is ‘‘closed’’ according to the traditional definition of crack closure. It would be worthwhile to mention the difference in the crack closure process when propagating crack and stationary crack are used. Fig. 11 shows that when a stationary crack is used in the FE simulation, the central point of the crack starts to contact first. The contacting area extends from the center of the crack gradually to the crack tip as the compressive load increases. A small area near the crack tip will keep open when the external load reaches its minimum. However, when a propagating crack is used in the FE simulation, the very latest debonded node always closes first. The order of crack closure follows the opposite order of node debonding: the last debonded node closes first and the first debonded node comes to contact the last. After all the cracked area created by node debonding is closed, the center point will come to contact. Clearly, the difference in the predicted results between stationary crack and propagating crack is created by the node debonding in the FE simulations. By using the relationship between the applied nominal stress, S, and the crack tip opening displacement, v, Newman (1999) argued that the cyclic plasticity of the crack tip material after crack closure was insignificant. Three points need to be made with regard to such an argument. First, the S–v relationship does not directly and accurately reflect the cyclic plasticity of the crack tip or the material near the crack tip. It has been well established that the S–v relationship is critically dependent on where the displacement is taken near the crack tip. Secondly, the S–v relationship is dependent on the material model used in the numerical simulation. The results shown in Figs. 3–7 for the crack closure predictions clearly indicate that the detailed deformation predicted near the crack tip, thus the crack tip opening displacement, is dependent on the material model used. When the EPP and BL models are used, the S–v relationship is greatly influenced by the element size used in the mesh model. Thirdly, the S–v relationship is also dependent on the loading condition. For example, the enclosed area by the S–v curve shown in Fig. 12 is 98.7 J/m2 for the loading case R =
2, Smax = 100 MPa, and a = 10 mm for the CCT specimen under consideration. The displacement was taken from the two points, A and B, shown in Fig. 12. Ignoring the area when the crack is closed, the result is 57.7 J/m2, a difference of 71% from 98.7 J/m2. The results are different when the displacement of a different location is considered. The two examples illustrated in this section suggest that while the crack surfaces may come to contact during a part of a loading cycle, cyclic plasticity of the material near the crack tip persists during the crack closure period and cannot be ignored.

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Smax=100MPa, R=-2, a=10mm

100

S, MPa

Open

0 Close

0.031mm

-100

A

0.062mm -200

B

0.000

0.001

0.002

Crack tip

0.003

0.004

CrackOpening Displacement, mm

Fig. 12. Load versus displacement curve for R =
2.

4. Further discussions It should be reiterated that the current discussions are based on and limited to the following conditions: (1) macroscopic continuum mechanics, (2) mode I plane-stress, (3) a stable material that does not display any significant cyclic hardening and nonproportional hardening, and, (4) plasticity-induced crack closure. A polycrystalline material consists of grains with random orientations. A grain has a finite size ranging primary from several micrometers to a few millimeters for most engineering structural metallic materials. A grain is an anisotropic material with mechanical properties dependent on the direction of the applied stress with reference to the grain orientation. The continuum mechanics concept with homogeneous material property assumption is limited to a volume of material consisting of a least a number of grains with random orientations. When the element size in an FE simulation is within the range of or smaller than the grain size, the stress and strain results obtained from the numerical analysis do not realistically reflect the real local stress and strain anymore. Therefore, it should be noted that the discussion with very fine element sizes may create a contradiction. The discussion on the element size effect may be viewed as a theoretical evaluation with an idealized material. Many questions remain to be answered. For example, can the models based on the macroscopic continuum mechanics be applicable for the cases where the plastic zone size is smaller than or in the order of the grain size? A proper element size for the FE stress analysis near the crack tip is also subject to further discussion. The existing crack growth models based upon the crack closure concept may have an engineering significance in the sense that it may provide a practical tool to consider some influencing factors on crack growth. However, the equivalence based upon the traditional crack closure concept does not physically reflect the equivalence of cyclic plasticity of the material near the crack tip. Roughness, material phase transformation, and oxidation may cause crack closure, particularly at the near threshold conditions (Blom and Holm, 1985; Ritchie and Suresh, 1982; Vasudeven et al., 1994). However, it should be noted that cyclic plasticity may persist and fatigue damage is not negligible when the crack is closed.

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The crack closure concept is often used to consider the R-ratio and overload effects on crack growth and threshold. These effects can be assessed by considering the cyclic plasticity near the crack tip due to different R-ratios and overload action. Kujawski and Ellyin (1987) examined the R-ratio effect on the cyclic plasticity of the material near the crack tip by using a quantity similar to the strain energy density and successfully modeled the R-ratio effect on crack growth. Recently, by incorporating the critical plane concept in multiaxial fatigue, plastic strain energy, and material memory in cyclic plasticity, a multiaxial fatigue criterion was developed (Jiang, 2000). The fatigue damage model was directly applied to predict crack growth through the elastic–plastic stress analysis of a cracked component (Jiang and Feng, 2004). Crack propagation behavior of a material was obtained without any additional assumptions or fitting. The approach developed is able to quantitatively capture the important fatigue crack propagation behaviors including the overload and the R-ratio effects on crack propagation and threshold. The effect of the R-ratio on the crack growth with R > 0 is attributed to the mean stress effect on the fatigue damage of the material near the crack tip. When R < 0, the gradual contact of the cracked surfaces during a loading cycle reduces the cyclic plasticity of the material near the crack tip. The overload effect is due to the reduced cyclic plasticity after the application of the overload rather than the plasticity-induced crack closure. It would be desirable to evaluate the plasticity-induced crack closure concept through a benchmarked experiment. After a cracked specimen is fatigued to reach the stable crack growth stage, a contact of the crack surfaces can be purposely created. The artificial contact can be designed in such a way that continuous surface contact occurs during the entire period of a loading cycle. This can be done by inserting a pair of small wedges into the wake of the crack near the crack tip when the external load reaches its maximum. According to the crack closure theories, no further crack growth is expected since the crack opening/close load is equal to the maximum applied load. Primary experimental results obtained by the authors show that crack growth rate is reduced for a period of time upon the insert of the small wedges but resumes after a certain period of subsequent loading cycles. Detailed results will be discussed in a future presentation.

5. Conclusions A center cracked specimen subjected to plane-stress mode I fatigue loading was used to investigate the plasticity-induced crack closure using the finite element method. The predicted crack opening load is very sensitive to the material constitutive model used in the numerical stress analysis. When the traditional EPP and BL stress–strain relationships were used together with the kinematic hardening rule, the predicted crack opening load is strongly dependent on the element sizes used for the material near the crack tip. The crack opening load approaches a very small value as the element size is reduced for a loading case with R = 0. When a more realistic cyclic plasticity model is used, the element size does not influence the predicted crack opening results when the element size is within a certain range. The predicted

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crack opening results are dependent on the crack extension simulated when using the one-node-per-cycle debonding scheme in the finite element simulations. Stabilized value can be achieved only after significant crack extension in the simulation. More importantly, while crack may close during a part of a loading cycle, particularly when R < 0, plastic deformation in the material near the crack tip persists and is not negligible during crack closure. The traditional approaches using the crack closure concept may have overestimated the effect of crack closure.

Acknowledgments The authors gratefully acknowledge the financial supports provided by the National Science Foundation (CMS-9984857) and Ford Motor Company. Computer support was obtained with grants from NCSA at the University of Illinois at Urbana-Champaign.

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