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Mechanics of Materials 40 (2008) 961–973

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Mechanics of Materials
journal homepage: www.elsevier.com/locate/mechmat

Modeling of fatigue crack growth of stainless steel 304L
Feifei Fan, Sergiy Kalnaus, Yanyao Jiang *
Department of Mechanical Engineering (312), University of Nevada, Reno, NV 89557, USA

a r t i c l e

i n f o

a b s t r a c t
An effort is made to predict the crack growth of the stainless steel 304L based on a newly developed fatigue approach. The approach consists of two steps: (1) elastic–plastic ?nite element (FE) analysis of the component; and, (2) the application of a multiaxial fatigue criterion for the crack initiation and growth predictions based on the outputted stress–strain response from the FE analysis. The FE analysis is characterized by the implementation of an advanced cyclic plasticity theory that captures the important cyclic plasticity behavior of the material under the general loading conditions. The fatigue approach is based upon the notion that a material point fails when the accumulated fatigue damage reaches a certain value and the rule is applicable for both crack initiation and growth. As a result, one set of material constants is used for both crack initiation and growth predictions. All the material constants are generated by testing smooth specimens. The approach is applied to Mode I crack growth of compact specimens subjected to constant amplitude loading with different R-ratios and two-step high–low sequence loading. The results show that the approach can properly model the experimentally observed crack growth behavior including the notch effect, the R-ratio effect, and the sequence loading effect. In addition, the early crack growth from a notch and the total fatigue life can be simulated with the approach and the predictions agree well with the experimental observations. ? 2008 Elsevier Ltd. All rights reserved.

Article history: Received 7 November 2007 Received in revised form 9 June 2008

Keywords: Damage accumulation Fatigue crack growth Fatigue criterion

1. Introduction Load-bearing engineering components are often subjected to cyclic loading and failure due to fatigue is of a great concern. Generally, fatigue process consists of three stages: initiation and early crack growth, stable crack growth, and ?nal fracture. Traditionally, the crack growth rate (da/dN) is expressed as a function of the stress intensity factor range (DK) on a log–log scale. The stable crack growth results under constant amplitude loading with different R-ratios (the minimum load over the maximum load over a loading cycle) are often represented by the Paris law (Paris and Erdogan, 1963) and its modi?cations (Walker, 1970; Kujawski, 2001). Different materials behave differently under constant amplitude fatigue loading. Some materials display a R-ratio effect: crack growth rate curves are coincided for the same R-ratio, but a higher R-ratio re* Corresponding author. Tel.: +1 775 784 4510; fax: +1 775 784 1701. E-mail address: yjiang@unr.edu (Y. Jiang). 0167-6636/$ - see front matter ? 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmat.2008.06.001

sults in a higher crack growth rate (Kumar and Garg, 1988; Pippan et al., 2005; Wu et al., 1998; Zhao et al., 2008). Other metallic materials do not reveal any R-ratio effect, and the curves for constant amplitude loading overlap in a log–log scale (Crooker and Krause, 1972; Kumar and Pandey, 1990; Wang et al., to appear). The fatigue crack growth behavior under variable amplitude loading is another subject that has been studied for a number of years. The application of an overload (tensile load of high magnitude applied over one cycle preceded and followed by constant amplitude loading) or change in the loading amplitude (so-called high–low sequence loading experiments) can introduce profound effects on the fatigue crack growth. For most metallic materials, the application of the abovementioned loading schemes results in a crack growth rate retardation. Based on the linear elastic fracture mechanics (LEFM) concept, such a transient behavior is often modeled by using the stress intensity factor concept and by introducing correction factors to the Paris law on the stable crack growth

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regime. A model of such a type was introduced by Wheeler (1972) and can be viewed as a practical way of treating the effects of variable amplitude loading. Several modi?cations on Wheeler’s model have been proposed (Kim et al., 2004; Yuen and Taheri, 2006; Zhao et al., 2008) targeting the particular shapes of the crack growth curves for different materials subjected to variable amplitude loading. These models have little or no physical basis and the results of the crack growth experiments are needed in order to obtain a set of ?tting constants to calibrate the models. Since its introduction by Elber (1970), the crack closure concept is often used to explain crack growth behavior. The retardation in crack growth rate generated by a single tensile overload was explained by using the crack closure concept in Elber’s later study (Elber, 1971). The concept of Kop was introduced as a stress intensity factor corresponding to the crack opening load, and the effective stress intensity factor range from Kop to Kmax was considered as a crack driving parameter. As a result, the contribution to crack propagation comes from a part of the total stress intensity factor range corresponding to the part of the cycle when the crack is open. Such an approach is used to explain the R-ratio and variable loading effects. However, the crack closure method has been under criticism based upon experimental observations (Lang and Marci, 1999; Sadananda et al., 1999; Silva, 2004; Feng et al., 2005) and numerical simulations (Jiang et al., 2005; Mercer and Nicholas, 1991; Zhao et al., 2004). Crack-tip blunting has been used to explain the crack advance (Gu and Ritchie, 1999; Tvergaard, 2004). The retardation caused by an overload is attributed mainly to the compressive residual stresses ahead of the crack tip, plasticity induced crack closure behind the crack tip, or the combination of these two. The initial acceleration in the crack growth immediately after the application of an overload was explained as a result of the tensile residual stress due to crack-tip blunting (Makabe et al., 2004). The ?nite element analysis was used to analyze the stress distribution and the crack opening displacement which was related to the variable amplitude loading effects (Zhang et al., 1992; Ellyin and Wu, 1999; Tvergaard, 2006). Generally, a fatigue crack is nucleated at a notch due to the stress concentration. The so-called notch effect on short crack behavior exists and the crack growth rate may be higher or lower than that expected based on the stable growth. Extensive research has been carried out to study the crack initiation and early crack growth behavior from a notch. Around a notch, a transition zone exists and the fatigue crack growth rate may decelerate, accelerate, or non-propagate after the crack initiation under constant amplitude loading. In order to model the short crack growth behavior from a notch, efforts were concentrated on the ‘‘effective stress intensity factor” near the notches (Sadanandam and Vasudevan, 1997; Dong et al., 2003; Teh and Brennan, 2005; Vena et al., 2006), notch tip plasticity (Li, 2003; Hammouda et al., 2004), and the combination of crack tip cyclic plasticity and the contact of the crack surfaces (Ding et al., 2007a). A recent effort by Jiang and co-workers (Ding et al., 2007a,b; Feng et al., 2005; Jiang and Feng, 2004a) attempted to use a multiaxial fatigue criterion to unify the

predictions of both crack initiation and crack growth. The notion is that both crack initiation and the subsequent crack growth are governed by the same fatigue criterion. A material point fails to form a crack once the accumulation of the fatigue damage reaches a certain critical value. The approach has been applied to 1070 steel with success. The predictions of the early crack growth from notches (Ding et al., 2007a; Jiang, Ding and Feng, 2007), the stable crack growth (Feng et al., 2005; Jiang and Feng, 2004a; Jiang, Ding, and Feng, 2007), the overload effect (Jiang and Feng, 2004a; Jiang, Ding, and Feng, 2007), the R-ratio effect (Jiang and Feng, 2004a; Jiang, Ding, and Feng, 2007), and the crack growth under direction-changing loading (Ding et al., 2007b) agreed well with the experimental observations. All the predictions of the crack growth were based on the material constants generated from testing the smooth specimens. In the present investigation, the aforementioned approach is used to simulate the crack growth from the notched specimens made of the AISI 304L austenitic stainless steel. The notch effect on the early crack growth, the Rratio effect, and the in?uence of the loading sequence are modeled. The stress analysis is conducted by using the ?nite element method implementing a robust cyclic plasticity model. The predicted results are compared with the results of the crack growth experiments.

2. Crack growth modeling In the present investigation, the fatigue approach developed by Jiang and co-workers (Jiang and Feng, 2004a; Jiang et al., 2007) is used to model the crack growth of the stainless steel 304L. The approach is based on the assumption that any material point fails if the accumulation of the fatigue damage reaches a critical value on a material plane. A fresh crack surface will form on the material plane at the material point. Essentially, the approach consists of two major computational steps: a) Elastic–plastic ?nite element (FE) stress analysis for the determination of the stress and strain history at any material point of a component, and, b) Application of a multiaxial fatigue criterion utilizing the stress and strain obtained from the previous step for the determination of crack initiation and crack growth. The following sub-sections describe the methods employed in the current study. 2.1. Cyclic plasticity model and multiaxial fatigue criterion Earlier studies indicate that an accurate stress analysis is the most critical part for the fatigue analysis of the material (Jiang and Kurath, 1997a,b; Jiang and Zhang, 2008; Kalnaus and Jiang, 2008; Jiang et al., 2007). If the stresses and strains can be obtained with accuracy, fatigue life can be reasonably predicted by using a multiaxial fatigue criterion. The elastic–plastic stress analysis of a notched or cracked component requires the implementation of a

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cyclic plasticity model into FE software package. The selection of an appropriate cyclic plasticity model is crucial for an accurate stress analysis of a component subjected to cyclic loading. Cyclic plasticity deals with the non-linear stress–strain response of a material under repeated external loading. A cyclic plasticity model developed by Ohno and Wang (1993, 1994) and Jiang and Sehitoglu (1996a,b) is used in the present FE simulations of the stress and strain response in a notched or cracked component. The model is based on the kinematic hardening rule of the Armstrong–Frederick type. Basic mathematical equations constituting the model are listed in Table 1. A detailed description of the plasticity model together with the procedures for the determination of material constants can be found in corresponding references (Jiang and Sehitoglu, 1996a,b). The choice of the cyclic plasticity model was based on its capability to describe the general cyclic material behavior including cyclic strain ratcheting and stress relaxation that occur in the material near the notch or crack tip. The plasticity model listed in Table 1 was implemented into the general purpose FE package ABAQUS (2007) through the user de?ned subroutine UMAT. A backward Euler algorithm is used in an explicit stress update algorithm. The algorithm reduces the plasticity model into a non-linear equation that can be solved by Newton’s method. The corresponding consistent tangent operator is derived for the global equilibrium iteration, which ensures the quadratic convergence of the global Newton–Raphson equilibrium iteration procedure (Jiang et al., 2002). A critical plane multiaxial fatigue criterion developed by Jiang (2000) is used for the assessment of fatigue damage. The criterion can be mathematically expressed as follows,

MacCauley bracket h i ensures that when rmr 6 r0 the fatigue damage is zero. The critical plane is de?ned as the material plane where the fatigue damage accumulation ?rst reaches a critical value, D0. The Jiang multiaxial fatigue criterion has been successfully applied to the fatigue predictions of a variety of materials (Ding et al., 2007a,b; Feng et al., 2005; Gao et al., to appear; Jiang, Ding, and Feng, 2007; Jiang et al., 2007; Zhao and Jiang, 2008). The incremental form of the criterion (Eq. (1)) does not require a separate cycle counting method for general loading conditions. Any fatigue criterion making use of the stress/strain amplitude or range requires the de?nition of a loading cycle or reversal. Therefore, a cycle counting method is needed to deal with the variable amplitude loading. Although the rain-?ow cycle counting method is widely accepted for counting the loading reversals/ cycles, it is not well de?ned for general multiaxial loading. The second feature of the criterion expressed by Eq. (1) is its capability to predict the cracking behavior. The Jiang fatigue criterion is a critical plane approach which is capable of predicting different cracking behavior through the introduction of constant b in Eq. (1). The value of constant b is selected to predict a particular mode of cracking based on the smooth specimen experiments. It has been shown (Jiang et al., 2007; Zhao and Jiang, 2008) that the predictions of the cracking behavior based on the Jiang criterion are generally more accurate than the predictions based on the other multiaxial criteria such as the Fatemi–Socie model (Fatemi and Socie, 1988), the Smith–Waltson–Topper model (Smith et al., 1970) and the short-crack based criterion (D?ring et al., 2006). Table 2 lists the material constants used in the cyclic plasticity model and the fatigue model for stainless steel

 dD ?

rmr ?1 r0

m  1?

r rf

  1?b br dep ? s dcp 2

?1?

Table 2 Material constants for SS304L Cyclic plasticity constants Elasticity modulus E = 200 GPa Poisson’s ratio l = 0.3 k = 115.5 MPa c(1) = 1381.0, c(2) = 507.0, c(3) = 172.0, c(4) = 65.0, c(5) = 4.08 r(1) = 93.0 MPa, r(2) = 130.0 MPa, r(3) = 110.0 MPa, r(4) = 75.0 MPa, r(5) = 200.0 MPa v(1) = v(2) = v(3) = v(4) = v(5) = 8.0

In Eq. (1), D represents the fatigue damage on a material plane and b and m are material constants. r and s are the normal and shear stresses on a material plane, and ep and cp are the plastic strains corresponding to stresses r and s, respectively. r0 and rf are the endurance limit and the true fracture stress of the material, respectively. rmr is a memory stress re?ecting the loading magnitude. For constant amplitude loading, rmr is equal to the maximum equivalent von Mises stress in a loading cycle. The use of
Table 1 Cyclic plasticity model used in the ?nite element simulations Yield function
2 f ? ?e ? a? : ?e ? a? ? 2k ? 0 S ~ S ~

Fatigue constants

r0 = 270 MPa; m = 1.5;b = 0.5;rf = 800 MPa;
D0 = 15000 MJ/m3

e ? deviatoric stress S ~ a ? backstress k = yield stress in shear

Flow law

d~p ? 1 hd~ : nin e S ~ ~ h

~ n ? normal of yield surface h = plastic modulus function ~p ? plastic strain e ~ a?i? ? ith backstress part M = number of backstress parts dp = equivalent plastic strain increment c(i), r(i), v(i) = material constants

Hardening Rule

~ a?

PM

i?1

~ a?i?

~ da?i? ? c?i? r ?i?

!  v?i? ?1 ~ ka?i? k ~ a?i? ~ dp n? ~ r ?i? ka?i? k

(i = 1, 2, 3, . . . M)

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304L. The cyclic plasticity material constants were obtained from the cyclic stress–strain curve which was obtained from the experiments on the smooth specimens under fully reversed tension-compression loading. A complete description of procedure for determination of material constants can be found in corresponding references (Jiang and Sehitoglu, 1996a,b). The fatigue material constants were determined by comparing the fatigue data under fully reversed tension-compression and that under pure torsion (Jiang, 2000). 2.2. Finite element model Round compact specimens with a thickness of 3.8 mm were used in the crack growth experiments. The geometry and the dimensions of the specimen are shown in Fig. 1. The crack growth experiments were conducted in ambient air. The specimens were subjected to constant amplitude loading with different R-ratios (the minimum load over the maximum load in a loading cycle) and high–low sequence loading. All of the experiments started without a pre-crack, except two specimens tested under the following loading conditions: R = 0.85, DP/2 = 0.54 kN and R = ?1, DP/2 = 5.0 kN. More detailed information of the experiments and the experimental results were reported in a separate presentation. Due to the small thickness, plane-stress condition was assumed for the round compact specimen. Four-node plane-stress elements were used in FE mesh model. The FE mesh model shown in Fig. 2 was created by using the FE package HyperMesh (Altair HyperMesh, 2004). Due to the symmetry in geometry and loading, only half of the specimen was modeled. To properly consider the high stress and strain gradients in the vicinity of the notch or crack tip, very ?ne mesh size was used in these regions. The size of the smallest elements in the mesh model was 0.05 mm. There were approximately 3058 to 5067 elements used in the mesh model depending on the crack
Fig. 2. Finite element mesh model.

size. The knife edges for the attachment of the open displacement gage in the specimen (Fig. 1) were not modeled because the free end of the specimen does not affect the stress and strain of the material near the crack tip or notch. Referring to the coordinates system employed in Fig. 2, the tensile external load, P, is applied in the y direction uniformly over nine nodes on the upper surface of the loading hole. To mimic the actual loading condition, the compressive load is applied in the negative y direction uniformly over nine nodes on the lower surface of the loading hole. The displacements in the x direction of the middle nodes on the upper edge of the loading hole are set to be zero. The displacements in the y direction for all the nodes on the plane in front of the crack tip or the root of the notch are set to be zero. In order to consider the possible contact between the upper and lower surfaces of a crack, the FE model incorporates the contact pairs de?ned in ABAQUS (2007). The crack surface of the lower symmetric half of the specimen is considered as a rigid surface which acts as the master surface. The corresponding crack surface of the upper half of the specimen serves as the slave surface. 2.3. Determination of crack growth rate For continuous crack growth under constant amplitude loading with small yielding, a simple formula was derived for the determination of the crack growth rate (Jiang and Feng, 2004a),

da A ; ? dN D0
where,

?2?

A?

Z
0

r0

DD?r? dr;

?3?

r is the distance from the crack tip and r0 is the damaging zone size ahead of the crack tip where the fatigue damage is non-zero. DD(r) is the maximum fatigue damage per loading cycle with respect to all possible material planes at a given material point. DD(r) is determined by integrating Eq. (1) over one loading cycle,

DD ?
Fig. 1. Geometry and dimensions of the round compact specimen (all dimensions in mm).

I
cycle



rmr ?1 r0

m  1?

r rf

  1?b br dep ? s dcp 2 ?4?

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for a given material point once the stress–strain response at the point is known. In Eq. (3), A denotes the damaging area enclosed by the DD(r)–r curve. Fig. 3 shows the distribution of DD(r) along the x-direction for Specimen C01 which was subjected to constant amplitude loading with R = 0.1 and DP/2 = 2.475 kN. According to the fatigue criterion, Eq. (1), a material plane will accumulate fatigue damage when the memory stress is higher than the endurance limit and the material point experiences plastic deformation. For a cracked component, only the material near the crack tip accumulates fatigue damage. The values of DD(r) are determined along all radial directions in a polar coordinate system with its origin being at the crack tip. The direction at which the crack growth rate is a maximum or the value of A is a maximum is the predicted cracking direction. The corresponding crack growth rate is the predicted crack growth rate. 2.4. Crack initiation and early crack growth from notch The approach described in the previous sub-sections assumes that a material point fails to form a fresh crack on the critical plane when the accumulation of the fatigue damage on the critical material plane reaches a critical value, D0. The rule applies to the initiation of a crack and the crack extension after a crack has been formed. Therefore, the approach uni?es both the initiation and the subsequent crack propagation stage. The distribution of the stress-plastic strain ?eld in the vicinity of a notch root, however, in?uences the early crack growth, which should be properly considered. The de?nition of crack initiation used in the current study is different from that of the traditional way. The crack initiation of a fatigue crack is judged by using the fatigue criterion, Eq. (1). Once the fatigue damage on a material plane for the material point at the notch root reaches the critical fatigue damage, D0, the notched member is called to have initiated a fatigue crack. The FE stress analysis is conducted with the notched member for the designated loading condition. For a notched component, the maximum fatigue damage occurs at the notch root. The fatigue damage per loading cycle can be determined and it can be plotted as a distribution along the radial direction from the notch root. Fig. 4 shows an example for Specimen C20 (R = 0.2, DP/2 = 2.0 kN, notched depth an = 7.38 mm, notch radius = 2.0 mm). The distance,
50 40
Damaging Zone Size

0.4

ΔDin

Material: SS304L

ΔDi (MJ/m )

3

0.2
Specimen C20 R=0.2, ΔP/2=2.0kN an=7.38mm Notch Radius=2.0mm

0.0 0.0 Notch Root 0.5 1.0 r, Distance fron Notch Root (mm) 1.5

Fig. 4. Fatigue damage per loading cycle during crack initiation.

r, from the notch root is along the x-axis (refer to Fig. 1). DDi denotes the fatigue damage per loading cycle on the critical plane during crack initiation. DDi is a function of the location of the material point. The maximum fatigue damage occurs at the notch root during crack initiation. The crack initiation life is predicted to be,

Ni ?

D0 ; DDin

?5?

where Ni is the predicted crack initiation life, D0 is a material constant, and DDin is the fatigue damage per loading cycle on the critical plane at the root of the notch. DDin is DDi shown in Fig. 4 when r = 0. During crack initiation, the fatigue damage is also accumulated in the vicinity of the notch root and should be considered in the determination of the crack growth near the notch. The area where the fatigue damage accumulation is non-zero during crack initiation (Fig. 4) is referred to as the notch in?uencing zone (NIZ). For a specimen under a given loading condition, the NIZ can be determined by applying the fatigue criterion, Eq. (1), with the stress and strain histories outputted from the FE analysis. For Specimen C20 shown in Fig. 4, the NIZ size is approximately 0.85 mm ahead of the notch root. For each material plane at any material point, the total fatigue damage at the end of the fatigue crack initiation is NiDDi. It should be reiterated that the discussion is based on the assumption that the material is stable in stress– strain response and the applied loading is constant amplitude. The crack growth rate within the NIZ can be determined by using the following equation with the consideration of pre-existing fatigue damage accumulation (Ding et al., 2007a):

ΔD (MJ/m )

da A ? : dN D0 ? Ni DDi ?r?
Material: SS304L

?6?

3

30 20 10 0 0.0 0.2 A

Specimen C01 R=0.1, ΔP/2=2.475kN an= -1.34mm a=7.34mm

0.4 0.6 0.8 r, Distance from Crack Tip (mm)

1.0

Fig. 3. Distribution of fatigue damage per loading cycle radiated from the crack tip.

where A is the damage area enclosed by the DD(r)–r curve, as explained in Section 2.3. In Eq. (6), Ni and DDi(r) are related to the fatigue damage accumulation during crack initiation in the NIZ. For a given crack size within the NIZ the FE analysis is conducted. The distribution of the fatigue damage per loading cycle, DD(r), can be determined as a function of the distance from the crack tip, as shown in Fig. 3. The enclosed area made by the DD(r)–r curve is A in Eq. (6). For any direction radiated from the crack tip, the direction which has the highest crack growth rate is

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the predicted cracking direction and the corresponding crack growth rate is the predicted crack growth rate. It can be seen that the difference between the crack growth rate determination near the notch (Eq. (6)) and that away from the notch root (Eq. (2)) lies in the consideration of the fatigue damage caused during the crack initiation stage. Generally, the stress–strain response becomes stabilized after a limited number of loading cycles. It was shown (Jiang and Feng, 2004a) that the predicted crack growth results obtained based on the stress–strain response from the 10th loading cycle were very close to those based on the stabilized stress and strain response. Therefore, the FE analysis for a given notch or crack length under a designated loading amplitude is conducted for 10 loading cycles. The stress and strain results at the 10th loading cycle are used for the fatigue analysis. The stress and strain results obtained from analyzing the notched component during crack initiation will determine the fatigue damage per loading cycle for each material plane at each material point. Eq. (5) is used to determine the crack initiation life. FE stress analyses are conducted with different crack lengths for a given loading condition. When the crack tip is within the notch in?uencing zone, Eq. (6) is used for the crack rate determination. DDi(r) in Eq. (6) is the fatigue damage per loading cycle for a given material point during crack initiation. Once the crack grows out of the NIZ, Eq. (2) is used for the crack growth rate determination. In fact, DDi(r) is determined during crack initiation. As a result, Eq. (6) can be used for both situations since DDi(r) is zero for the material points out of the notch in?uencing zone. It should be noticed that the FE simulation is conducted cycle by cycle mimicking the real crack growth procedure. The crack initiation life is determined ?rst. The crack growth rates at several crack lengths are predicted by using the approach. Therefore, the prediction is the relationship between the crack growth rate, da/dN, and the crack length for a given notched component. With the crack initiation life obtained from using Eq. (5), the relationship between the crack length and the number of loading cycles can be established through a numerical integration. Simulations are also conducted for the high–low step loading conditions. In a high–low step loading experiment, an external load with higher loading amplitude is applied until a crack length reaches a certain value. The amplitude of the external load is switched to a lower value in the second loading step. In the simulations for the high–low loading sequence, one special consideration is made. The memory stress, rmr, in Eq. (1) is kept the same before and immediately after the change of the external load from a higher amplitude to a lower amplitude. After an extension of the crack in the second loading step, the memory stress returns to that under the lower constant amplitude loading.

class of metastable steels of 300-series. Austenitic steels display a R-ratio effect when subjected to constant amplitude loading, as has been shown for AISI 304 (Mei and Morris, 1990) and AL6-XN (Kalnaus et al., 2008). The experimental data used in the present investigation was the results of a series of experiments conducted by the authors. Fatigue crack growth experiments were performed using round compact specimens made of stainless steel 304L. The compact specimens were machined from an as-received cold rolled round bar. The bar had a diameter of 41.28 mm. The dimensions of the specimens are shown in Fig. 1. The U-shaped notches were made through EDM (Electric Discharge Machining). The width of the slot in the specimen is 0.2 mm. One side of the specimen was polished to facilitate the observation of the crack growth using an optical microscope with a magni?cation of 40. The loading conditions included constant amplitude loading with R-ratios ranging from ?1 to 0.85 and two-step high–low sequence loading. Detailed description of the experiments and the results were reported in a separated presentation. Fig. 5 shows the experimental results under constant amplitude loading with different R-ratios. Ten specimens were subjected to constant amplitude loading with different loading amplitudes and six R-ratios. Clearly, the R-ratio has an effect on the crack growth of the material. The notch effect is re?ected in the crack growth results presented in Fig. 5. It can be found that, except in the case of the specimen with a relatively large notch radius under R = ?1 loading, the notch effect on the crack growth is not signi?cant. For the R = ?1 case (Specimen C24, notch

10

-2

Material: SS304L Compact Specimens Constant Amplitude Loading

da/dN, Crack Growth Rate, mm/cycle

10

-3

10

-4

10

-5

3. Results and discussion 3.1. Crack growth experiments The material under consideration in the present study is AISI 304L austenitic stainless steel which belongs to the

10

-6 4 5 6 7 1 2 3 4 5 6 7

R=0.1 C01 ΔP/2=2.475kN C06 ΔP/2=1.8kN R=0.2 C20 ΔP/2=2.0kN R=0.5 C02 ΔP/2=1.75kN C17 ΔP/2=1.8kN R=0.75 C03 ΔP/2=1.0kN C18 ΔP/2=0.95kN R=0.85 C16 ΔP/2=0.54kN R=-1 C14 ΔP/2=5.0kN C24 ΔP/2=3.2kN

10 10 1/2 ΔK, Stress Intensity Factor Range, MPa m
Fig. 5. Experimental crack growth results under constant amplitude loading.

2

2

3

4

F. Fan et al. / Mechanics of Materials 40 (2008) 961–973

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3.2. Constant amplitude loading The primary results obtained from using the fatigue approach presented in the current investigation is the crack growth rate as a function of the crack length from the notch root for a given loading condition. However, the traditional way to present the crack growth results in da/dNDK, where DK is the stress intensity factor range, is adopted, considering the general familiarity of the traditional presentation of the crack growth results. Comparisons of the experimental crack growth results with the predictions are shown in Fig. 6 for the constant amplitude cases. The markers in the ?gure denote the experimentally obtained crack growth results and the lines are the predictions. A general good agreement can be found between the predictions and the experimental observations. It should be reiterated that all of the specimens were subjected to constant amplitude loading without a pre-crack except for specimens C16 and C14, to which pre-cracking was applied (refer to Fig. 5 for the specimen numbers). Therefore, the results shown in Fig. 6 include the effect of the notch on early crack growth. The notch effect is dependent on the notch size. A larger effect is expected if the notch size is larger. Since all of the specimens except Specimen C20 (R = 0.2, DP/2 = 2.0 kN, notch radius = 2.0 mm) and Specimen C24 (R = ?1, DP/ 2 = 3.2 kN, notch radius = 1.0 mm) had small notch sizes with the notch radius being 0.1 mm, the notch effect is not signi?cant. Fig. 7 shows the results for Specimen C01 (R = 0.1, DP/2 = 2.475 kN, and the notch depth

da/dN, Crack Growth Rate, (mm/cycle)

radius is 1.0 mm), short-crack like phenomenon was observed.

10

-2

Material: SS304L

10

-3

10

-4

notch influencing zone: 0.4mm Specimen C01 R=0.1, ΔP/2=2.475kN an= -1.34mm Notch Radius=0.1mm

10

-5

10

-6

Experiment Prediction

0

5 10 15 a, Crack Length from Notch Root, (mm)
Fig. 7. Notch effect for specimen C01.

an = ?1.34 mm) in terms of the crack growth rate versus the crack length measured from the notch root. The size of the notch in?uencing zone is predicted to be 0.4 mm. 3.3. High–low sequence loading The results of experiments involving two-step high–low loading sequence are shown in Fig. 8 and are displayed as a function of the crack length measured from the notch root. The markers in the ?gure represent the experimental data and solid lines represent the predicted crack growth rate

10

-2

Material: SS304L Compact Specimens Constant Amplitude Loading

10

-2

Material: SS304L Compact Specimens Constant Amplitude Loading

da/dN, Crack Growth Rate, mm/cycle

10

-3

da/dN, Crack Growth Rate, mm/cycle

10

-3

10

-4

10

-5

10

-6 4 5 6 7 1 2 3 4 5 6 7

R=0.1 ΔP/2=2.475kN Prediction R=0.2 ΔP/2=2.0kN Prediction R=0.5 ΔP/2=1.75kN Prediction R=0.75 ΔP/2=1.0kN Prediction R=-1 ΔP/2=5.0kN Prediction
2 2

10

-4

10

-5

10 10 10 1/2 ΔK, Stress Intensity Factor Range, MPa m

-6 4 5 6 7 1 2 3 4 5 6 7

R=0.1 ΔP/2=1.8kN Prediction R=0.5 ΔP/2=1.8kN Prediction R=0.75 ΔP/2=0.95kN Prediction R=0.85 ΔP/2=0.54kN Prediction R=-1 ΔP/2=3.2kN Prediction
2 2

10 10 1/2 ΔK, Stress Intensity Factor Range, MPa m

Fig. 6. R-ratio effect on crack propagation under constant amplitude loading.

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F. Fan et al. / Mechanics of Materials 40 (2008) 961–973

10 da/dN, Crack Growth Rate, mm/cycle

-3 8 6 4 2

Material: SS304L Specimen C07 High-low Sequence Loading

10

-3 8 6 4

Material: SS304L Specimen C08 High-low Sequence Loading

Crack Growth Rate, mm/cycle

2

10

-4 8 6 4 2

10

-4 8 6 4 2

10

-5 8 6 4 2

step1: Pmax=6.0kN Pmin=0.6kN

step2: Pmax=6.0kN Pmin=2.4kN

10

-5 8 6 4 2

10

-6

Experiment Prediction

step1: Pmax=6.0kN Pmin=0.6kN

step2: Pmax=4.2kN Pmin=0.6kN

10 0 5 10 15 a, Crack Length from the Notch Root, mm

-6

Experiment Prediction

0 5 10 15 Crack Length from the Notch Root, mm

10 da/dN, Crack Growth Rate, mm/cycle

-3 8 6 4 2

Material: SS304L Specimen C19 High-low Sequence Loading

10

-4 8 6 4 2

10

-5 8 6 4 2

step1: Pmax=6.0kN Pmin=0.6kN

step2: Pmax=4.0kN Pmin=0.4kN

10

-6

Experiment Prediction

0 5 10 15 a, Crack Length from the Notch Root, mm
Fig. 8. Two-step high–low loading sequence effect on crack propagation: (a) identical maximum load, (b) identical minimum load, (c) identical R-ratio.

based on the current approach. The experiments were designed in the following way. One of the tests was conducted with the maximum load being the same in both high and low steps (Fig. 8(a)). The second test was done with the minimum load being held the same in both steps (Fig. 8(b)). The third specimen was tested under the condition that the R-ratio was kept constant throughout the experiment (Fig. 8(c)). The results shown in Fig. 8 demonstrate that the crack growth behavior under high–low loading depends on the loading conditions in the lower amplitude loading step. When the experiment is performed with the same maximum load in both steps, the ?rst (higher amplitude) step

has little in?uence on the subsequent crack growth in the second (lower amplitude) step. However, in other two cases (Fig. 8(b) and (c)), a signi?cant crack growth retardation can be observed at the beginning of the low amplitude block of the experiment. It can be seen from Fig. 8 that the predictions of the crack growth for the two-step loading agree well with the experimental observations and capture crack growth behavior in all the three cases described above. 3.4. Life prediction An appropriate approach should be able to predict the relationship between the crack length, a, and the number

F. Fan et al. / Mechanics of Materials 40 (2008) 961–973

969

of loading cycles, N. Fig. 9 shows a comparison of the experimentally obtained a ? N results and the predictions obtained from using the fatigue approach. The predicted fatigue life for a given crack length is obtained using the following equation,

10

7

Material: SS304L

N ? Ni ?

Z
0

a1

da ; f ?a?

?7?

Observed Life, cycles

10

6

where N is the number of loading cycle corresponding a crack length of a, and Ni is the crack initiation life predicted for the specimen by using Eq. (5). f(a) is the crack growth rate as a function of the crack length predicted by using the approach. Eq. (7) is integrated numerically and is applicable after crack initiation. An overall good correlation between the experiments and the predictions can be observed from the results shown in Fig. 9. Another way to evaluate the approach for the capability to predict fatigue life is to compare the crack initiation life and the fatigue failure life of the specimen with those observed experimentally. As was mentioned in an early section, the crack initiation used in the current study is different from the traditionally used concept. The current de?nition of the initiation life is a mathematic concept since it is very dif?cult to experimentally identify the exact moment of crack initiation. However, a practical pseudo crack initiation life can be used to check the capability of the models. The pseudo crack initiation can be de?ned as the fatigue life corresponding to a small crack length. In the current study, a crack length of 0.5 mm from the notch root (refer to Fig. 1) is chosen as the ‘‘pseudo crack initiation.” A crack length of 10 mm measured from the line of action of the externally applied load is used for fatigue failure. In other words, two fatigue lives with an order of difference in the crack length are used to check the capability of the models in predicting the fatigue life. Fig. 10 shows the comparison of the predicted fatigue lives and the experimentally measured fatigue lives when the crack lengths are 0.5 mm (N0.5) and 10 mm (N10.0). The vertical axis is for the observed fatigue life and the hor-

10

5

10

4

10

3 3 4 5 6

N0.5 N10.0
7

10

10

10 10 Predicted Life, cycles

10

Fig. 10. Comparison of experimental fatigue life with prediction.

izontal axis is for the predicted fatigue life. Logarithmic scales are used for both axes. The open circles are for the ‘‘pseudo crack initiation” (when the crack length measured from the notch root is 0.5 mm) and the solid markers are the ‘‘fatigue failure” lives corresponding to a crack length of 10 mm from the line of action of the externally applied force. The thick solid diagonal line signi?es a perfect prediction and the two dashed lines are the factor-of-two boundaries. Almost all the results are within the factorof-two boundaries, signifying a very reasonable prediction.

4. Further discussion The approach used in the current investigation for the crack growth prediction is fundamentally different from the commonly used methods where a stress intensity factor or J-integral is used. There are three major features that distinguish the current approach from the traditional methods. The stress intensity factor was developed to avoid the stress and strain singularity of the material at the crack tip. The stress intensity factor is based on the elastic deformation concept and it is a bulk measure of the stressing severity of the material near the crack tip upon the application of the external loading. It has been well known that while the stress intensity factor can be used to deal with the cases with constant amplitude loading, modi?cations and additional coef?cients have to be added in order to consider such factors as the notch effect, the R-ratio effect, and effects of variable amplitude loading. As a result, many constants are introduced and they are determined by best ?tting the experimentally obtained crack growth data. The methods tend to become a curve ?tting technique instead of predictions. The current approach attempts to use the local stress and strain directly for the fatigue damage assessment. Therefore, no crack growth experimental data is used for the determination of the material constants in the models.

15

Crack Length, mm

10

R=0.1 ΔP/2=2.475kN Experiment Prediction R=0.2 ΔP/2=2.0kN Experiment Prediction R=0.85 ΔP/2=2.0kN Experiment Prediction

Material: SS304L

5

0 10
3

10 10 10 Number of Cycles, cycle

4

5

6

10

7

Fig. 9. Crack length versus number of loading cycle.

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F. Fan et al. / Mechanics of Materials 40 (2008) 961–973

Traditionally, crack initiation is modeled using the continuum mechanics method where stress and strain are used to access fatigue damage. A separate model, often based on the stress intensity factor concept, is needed to deal with the crack growth. In order to use the fracture mechanics approach for the crack growth prediction, an initial crack length or an initiation crack length must be de?ned. Since the predicted crack propagation life is very sensitive to the initial crack length, the de?nition of the crack initiation size is more to ?t the experimental data than to have a physical base, or rather, the initial crack size characterizing crack initiation in the traditional methods is a ?tting constant. Within the fatigue approach discussed in the current investigation, the integrated consideration of the crack initiation and crack growth is employed. One single fatigue criterion is used for both crack initiation and crack propagation. The uni?ed consideration allows for a seamless transition from crack initiation to crack growth without necessity to de?ne a crack initiation size. For the prediction of crack growth, a separate criterion is generally needed for the determination of the crack growth direction. For example, a minimum strain energy density factor theory (Sih and Barthelemy, 1980; Sih and Bowie, 1992; Badaliance, 1980) speci?es that the fatigue crack growth rate is related to the range of the strain energy density but the crack growth direction is determined by the minimum strain energy density factor with respect to the orientation of the material plane in a loading cycle. Within the maximum tangential stress approach, the crack growth rate is assumed to be related to an effective stress intensity factor range while the crack growth direction is determined by using a maximum tangential stress criterion (Tian et al., 1982). By using the critical plane multiaxial fatigue criterion, Eq. (1), the current approach predicts the crack growth rate and the crack growth direction in an integrated manner. The fatigue criterion determines the critical plane where the cracking surface is to be formed and the stress and strain quantities on the critical plane determine the crack growth rate. The approach was used to predict with success the cracking direction under a loading condition involving a change in loading directions (Ding et al., 2007b). While the overall predictions of the crack ‘‘initiation” and crack growth are in general and reasonable agreement with the experimental observations, the predicted results are not as satisfactory as those using 1070 steel (Ding et al., 2007a,b; Feng et al., 2005; Jiang and Feng, 2004a; Jiang, Ding, and Feng, 2007). The major reason is the less accurate description of the cyclic plastic deformation of the SS304L than that for 1070 steel. 1070 steel displays very stable stress–strain response with practically no cyclic hardening/softening or non-proportional hardening. The material under consideration exhibits signi?cant non-proportional hardening and cyclic hardening/softening. The simple version of the cyclic plasticity model listed in Table 1 does not consider cyclic hardening/softening or the non-proportional hardening. It is found that while the material displays signi?cant non-proportional hardening and the stress state is multiaxial in the material near the crack tip, the loading is practically proportional for Mode I crack growth. Fig. 11 shows

1000 500

Material: SS304L Specimen C19 High Step R=0.1, ΔP/2=2.70kN a=4.6mm

σyy σxx

σyy, MPa

0 -500

-1000 -1000

-500

0 σxx, MPa

500

1000

Fig. 11. Stress state in the material near the crack tip.

the response of the two normal stresses at the Gauss point closest to the crack tip for a loading cycle. The loading condition is R = 0.1 and DP/2 = 2.7 kN. For the plane-stress condition, the material point under consideration has a minimal shear stress. It can be seen in Fig. 11 that the two normal stress components are practically proportional. It is further con?rmed that for Mode I loading the results obtained from considering the non-proportional hardening are practically the same as those without considering the non-proportional hardening in the ?nite element analysis. The exclusion of the cyclic hardening/softening in the constitutive modeling of the cyclic plastic deformation of stainless steel 304L contributes to the discrepancy between the experimentally observed fatigue behavior and the predictions for the notched member under consideration. Fig. 12 shows the variation of the stress amplitude with the number of loading cycles for a smooth specimen subjected to strain-controlled constant amplitude loading. From Fig. 12 it can be seen that the material experiences a period of cyclic softening followed by cyclic hardening. In other words, the stress response of the stainless steel 304L never becomes stabilized. Cyclic hardening/softening is very dif?cult to model accurately. In the deformation analysis particularly with the FE analysis, the transient cyclic hardening/softening is always ignored. Another difference between the real cyclic plastic deformation and the model simulation is the ignorance of the

Δσ/2, Axial Stress Amplitude, MPa

700 600 500 400 300 200 10
0

Tension-Compression Δε/2=1.0% 0.6% 0.4% 0.32% 0.28% 0.25%

10

1

10 10 10 N, Number of Cycles

2

3

4

10

5

10

6

Fig. 12. Stress amplitude variation with the loading cycles under straincontrolled constant amplitude loading showing signi?cant cyclic hardening/softening.

F. Fan et al. / Mechanics of Materials 40 (2008) 961–973

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1400 1200 Δσ, Stress Range 1000 800 600 400 200 0

Material: AISI 304L

Δε/2 = 0.4%

0.8%

1.0%

1.5%

0.000

0.005

0.010 0.015 0.020 p Δε , Plastic Strain Range

0.025

Fig. 13. Stabilized stress–plastic strain hysteresis loops with the lower tips tied together.

non-Masing or strain-range effect on the cyclic plasticity in the constitutive model. Most engineering materials display non-Masing behavior, while the constitutive model listed in Table 1 is based on Masing behavior. The stress-plastic strain hysteresis loops shown in Fig. 13 were obtained from the strain-controlled constant amplitude loading experiments using dog-bone shaped smooth cylindrical specimens. The stress-plastic strain loops shown in Fig. 13 were taken at the number of cycles corresponding to half of the fatigue life of the specimen. The loops were tied together at the lower tips. If the material displays Masing behavior, all the upper branches of the hysteresis loops shown in Fig. 13 should coincide. It is evident from Fig. 13 that stainless steel 304L displays non-Masing behavior or strain range effect particularly when the loading amplitude is large. It is possible to include the non-Masing behavior, nonproportional hardening, and cyclic hardening/softening in the constitutive relations for cyclic plasticity. The inclusion of all these considerations results in a very complicated constitutive model. It is also required to implement the model into an FE software package such as the UMAT in ABAQUS. Furthermore, the consideration of all the cyclic plasticity behavior will slow down the already slow elastic–plastic FE stress analysis of a real component. Further work is needed to properly consider the material deformation in the numerical stress analysis. The approach by using the multiaxial fatigue criterion, Eq. (1), does not need to de?ne a loading cycle. However, for all the experiments that were used, a loading cycle can be easily de?ned. The only purpose to use the loading cycle as a unit in fatigue damage accumulation was to facilitate the presentation of the results. When a cracked component is subjected to external loading, the material near the crack tip always experiences elastic–plastic deformation and both the stresses and the strains at the crack tip are theoretically in?nite if the material displays strain hardening. Since a cracked component can withstand a certain external load, the theoretical singular problem is caused by the basic assumption of a continuum for a material. The real stresses and strains in a cracked component should be ?nite when the external load is below a certain level. The FE methods average out the stress and strain in the highly gradient area and it may provide reasonable and realistic results to serve a gi-

ven purpose. It is found that the considerations of the non-linear material deformation and the non-linear geometry near the crack tip do not considerably in?uence the stress–strain results near the crack tip for the crack growth rate below the unstable crack growth region. However, the stress and strain results near the crack tip obtained from the elastic–plastic FE stress analysis of a cracked component is sensitive to the FE size near the crack tip. When the mesh size is extremely small for the material near the crack tip, the stresses and strains at the crack tip obtained from the FE analysis will be unrealistically high, which will result in a very high predicted crack growth rate. The FE size near the crack tip is a ‘‘model” constant. A preliminary study (Jiang and Feng, 2004b) reveals that an appropriate element size is in the order of one to three times the average grain size of the polycrystalline material. Within such an element size range, the fatigue results obtained from using the approach are not very sensitive to the element size used near the crack tip. For the stainless steel 304L under investigation, the average grain size is approximately 20 lm. In the current FE simulations, the smallest element size near the crack tip was 50 lm. Further investigations are needed to explore the in?uence of the element size and the element type on the simulation results for the stresses and strains near the crack tip. 5. Conclusions An approach was applied for the prediction of the fatigue behavior of notched members under constant-amplitude loading and step loading. Elastic–plastic stress analysis was conducted to determine the detailed stress and strain in the notched and cracked component. The application of a multiaxial fatigue criterion using the stress and strain outputted from the numerical stress analysis resulted in the prediction of the fatigue initiation and crack growth rate. With the material constants determined solely from testing the smooth specimens, the crack initiation and the crack growth of a notched member can be properly modeled. Acknowledgements The Of?ce of Naval Research (N000140510777) sponsored this work. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the of?cial policies or endorsements, either expressed or implied, of the Of?ce of Naval Research, or the U.S. government. References
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