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Prediction of fatigue crack growth based on low cycle fatigue properties
Accepted Manuscript Prediction of fatigue crack growth based on low cycle fatigue properties K.K. Shi, L.X. Cai, L. Chen, S.C. Wu, C. Bao PII: DOI: Reference:
S0142-1123(13)00322-8 http://dx.doi.org/10.1016/j.ijfatigue.2013.11.007 JIJF 3252
To appear in:
International Journal of Fatigue
Received Date: Revised Date: Accepted Date:
12 May 2013 8 November 2013 11 November 2013
Please cite this article as: Shi, K.K., Cai, L.X., Chen, L., Wu, S.C., Bao, C., Prediction of fatigue crack growth based on low cycle fatigue properties, International Journal of Fatigue (2013), doi: http://dx.doi.org/10.1016/j.ijfatigue. 2013.11.007
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Prediction of fatigue crack growth based on low cycle fatigue properties K.K. Shi1* , L.X. Cai1, L. Chen1, S.C. Wu2, C. Bao1 1. School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu, 610031, China 2. State Key Laboratory of Traction Power, Southwest Jiaotong University, Chengdu, 610031,China
Abstract: Theoretical models of the fatigue crack growth without artifical adjustable parameters were proposed by considering the plastic strain energy and the linear damage accumulation, respectively. The crack was regarded as a sharp notch with a small curvature radius and the process zone was assumed to be the size of cyclic plastic zone. The near crack tip elastic-plastic stress and strain were evaluated in terms of modified Hutchinson, Rice and Rosengren (HRR) formulations. Predicted results from two established models have been soundly compared with open reports for frequently used materials. It is found that experimental results agree well with theoretical solutions. Keywords: Fatigue crack growth; Low cycle fatigue properties; Cyclic plastic zone; Plastic strain energy; Linear damage accumulation
1. Introduction Fatigue crack growth behaviors have been studied for various types of engineering structural materials in the past several decades. In order to correlate the damage evolution with the cyclic deformation characteristics of a material with macroscopic cracks, notched specimens are typically introduced to calibrate the crack growth rate da/dN[1]. It is well-known that fatigue cracks initiate and propagate due to the local cyclic plastic yielding of the materials for exampling Model I cracking problems[2]. A
*Corresponding
author, Tel.: +86-28-87600850; Fax: +86-28-87600797
Email address: [email protected] (K.K. Shi)
1
crack growth law can then be formulated with the help of the stress and strain field ahead of the crack tip together with a suitable failure criterion. The damage-tolerant design method assumes that engineering components contain intrinsic imperfections in the form of macro cracks. Generally, the cracks can propagate only when a certain critical length is approached. Such a local nature of the fatigue phenomenon can be described using a sigmoidal curve in the fatigue crack growth rate of log da/dN vs. log ΔK. The sigmoidal curve is soundly bounded at the extremes by the lower bound of the fatigue threshold ΔKth and by the upper bound of the critical stress intensity factor range ΔKc. In the intermediate range, log da/dN is nearly linearly correlated with log ΔK, as formulated by Paris and Erdogan[3]. Notably, some life relationships between the fatigue crack growth rate and the low cycle fatigue (LCF) properties of the material have also been established. A number of crack growth rate models previously proposed surrounding the crack tip region are individually based on the fatigue ductility[4], the plastic strain energy[5-10] and the weighted value of the local strain[11]. Unfortunately, most of these theoretical models intrinsically contain adjustable material parameters that requires to be determined experimentally or numerically. Although a few equations with material constants have been established to effectively predict the fatigue cracking growth characteristics, the bore some problem is that material constants are especially difficult to be acquired form the viewpoint of cost. The progression of fatigue cracking is generally assumed to be an incremental growth due to a critical energy accumulated at the crack-tip. The material ahead of the crack is necessarily modeled as an assemblage of uniaxial material elements, and thus 2
the crack growth can be regarded as the successive failure process of these elements. Additionally, the width of the material element is assumed to be the length of cycle plastic zone along the progressive crack direction. More importantly, the success of such a theory depends on the specification of an appropriate plastic strain energy criterion and the linear damage accumulation along the crack growth direction inside of the cyclic plastic zone. As the stress and the strain are theoretically singular at the crack tip, the plastic strain and the linear damage are somewhat difficult to define precisely. However, the singularity will disappear through introducing the crack tip blunting in this study. Therefore, finite values of the stress and the strain are the focus of the present study. Two theoretical models of the fatigue crack growth based on the plastic strain energy and linear damage accumulation therefore were developed inside the low cycle plastic zone at the crack tip, respectively. Theoretical models can accommodate both the intermittent growth and the continuum growth currently identified for the majority of engineering materials. Furthermore, these models can be used to analyze the surface-based cracking life of engineering structures such as the nuclear pressure vessels and pipelines, and high pressure storage tanks. In the following sections, two analytical models based on the plastic strain energy (PSE) and the linear damage accumulation (LDA), respectively, will be thoroughly established, which physically remove some adjustable materials constants for the effective predictions fatigue crack growth (FCG) rate. Two theoretical model formulated here are expected to be an important complement to a relatively reliable solution of in-service fatigue life for engineering materials.
2. Fatigue crack growth models
3
An analytical solution of the stress and strain distribution ahead of a crack loaded in antiplane shear (Mode III) during the small-scale yielding has been derived by Schwalbe[12] . Based on the Hutchinson, Rice and Rosengred (HRR) field, a similar analytical solution as for the tensile loading (Mode I) that is the most important solution from an engineering application viewpoint, is available for small scale yielding. Despite the significant plastic deformation in the crack tip zone, the linear elastic fracture mechanics (LFEM) approach is frequently regarded as a well-established theoretical method for the small scale yielding analysis. Consider a material with strain hardening behavior and a defined stress and strain relationship ahead of the crack tip. The following Fig. 1 illustrates the cyclic stress strain relationship. Assuming Masing behaviors are satisfied inside of the cyclic plastic zone at the fatigue crack tip. The increasing branches of hysteresis loops for various loading levels should be coincident, thus the stress-strain hysteresis loops can be approximated. The stress and strain range can thus be written as[13,14] ∆σ ( r ) = 2 K ′
∆ε p
2
2σ yc rc ∆ε p ( r ) = E r
1
n′
(1)
(1+ n′ )
(2)
where σyc=Eεyc is the cyclic tensile yielding stress, E is the elastic modulus, Δεp is the plastic strain range, Δσ is the stress range, n′ is the cyclic strain hardening exponent, K′ is the cyclic strain hardening coefficient, rc is the cyclic plastic zone and r is the distance from the crack tip. The cyclic plastic zone rc can be formulated by[15]
4
∆K 1 rc = 4π (1 + n′) σ yc
2
(3)
in which ΔK is the stress intensity range.
2.1. FCG based on a plastic strain energy (FCG-PSE) In terms of the product of the stress and the strain, Eqs. (1) and (2) can be arranged into the following equation: σ yc ∆σ ⋅ ∆ε p = 4 K ′ E
( n ′+1)
rc . r
(4)
The above equation will exhibit a singularity as r→0. However, such singularity is rather unreasonable due to crack tip blunting from complex material and loading behaviors[16]. With the nature of crack tip blunting, the stress and strain have a finite magnitude in a high strain zone ahead of the crack tip. The cyclic plastic zone is treated as the “process zone” where the majority of the damages occurs. In the process zone, the plastic strain range is much larger than the elastic strain range, thus leading to Δε≈Δεp. Let us now introduce a critical crack blunting radius ρc as
∆K th 1 ρc = 4π (1 + n′) σ yc
2
,
(5)
where ΔKth is the stress intensity threshold.
The critical radius ρc is associated with the threshold stress intensity range. Generally, the fatigue cracks will not propagate when the critical blunting radius is less than the threshold measures from experiments of a material. Thus Eq. (4) can be rewritten as 5
' σ y ∆σ ⋅ ∆ε p = 4 K E
n' +1
rc r + ρc
(6)
From this, it is seen that the product Δσ·Δεp in the cyclic plastic zone can be obtained analytically. By integrating within the range [0, rc-ρc] using above Eq. (6), the plastic strain energy can be determined by the integral as
∫
rc − ρ c
0
σ ∆σ ⋅ ∆ε pdr = 4 K ′ yc E
( n ′+1)
⋅ rc ⋅ ( ln rc − ln ρ c ) ,
(7)
The fatigue resistance of the material ahead of the crack is governed hypothetically by the local state of stress and strain range perpendicularly to the cracking growth examplied for Mode I problems. A fatigue failure criterion can be applied in the cyclic plastic zone, rc. Based on the Manson-Coffin strain/life relationship obtained from smooth specimens, strain/life relationship can be described either in terms of the plastic strain or stress range and the number of cycles to failure 2Nf. Taking the detrimental effect of a mean stress into account since Morrow’s work[15], the superimposition yields the following relationship between the total stain amplitude and the number of cycles to technical failure Nf.
(
' ∆σ = 2 σ f − σ m
( )
' ∆ε p = 2ε f 2 N f
b
) (2 N )
(8)
f
c
(9)
in which εf ' and σf' are the fatigue ductility and strength coefficients, respectively, exponents b and c are the undetermined material properties and the σm is the mean stress. The product of the plastic strain and stress range can be calculated using both Eq. (8) and Eq. (9):
6
(
b+ c
) ( )
' ' ∆σ ⋅ ∆ε p = 4 σ f − σ m ε f 2 N f
(10)
.
By rearranging this Eq. (10) and the processing zone, rc-ρc, we can obtain the following relationship,
(
b+c
) ( )
' ' * ∆σ ⋅ ∆ε p ⋅ ( rc − ρc ) = 4 σ f − σ m ε f 2 N
⋅ ( rc − ρ c ) ,
(11)
in which N* is the number of cycles for the crack to penetrate through rc-ρc. Because Eq. (7) is equal to Eq. (11), the following formulation can be obtained: 1
n' +1 b +c ' σ yc rc K * 1 N = ⋅ ⋅ ⋅ [ ln( rc ) − ln( ρc )] . ' ' 2 σ −σ E rc − ρ c ε m f f
(
)
(12)
Note that in the present model, the mean stress will be equal to zero in the case of the low cycle fatigue tests. Consequently, the crack extension per cycle da/dN can now be finally defined as da rc − ρc = dN N*
(13)
Let us now introduce an effective stress intensity range, ΔKeq, defined as
(
2 2 ∆K eq = ∆K − ∆K th
1/2
)
(14)
Incorporating Eq. (14) into Eq. (13), the following model can be acquired as: ∆Keq2 da = . dN 4π (1 + n′)σ yc2 N *
(15)
The present Eq. (15) actually looks like the correlation between crack tip opening displacement (CTOD) and J integral for small scale yielding and will be used to
7
predict the da/dN-ΔK relationship for six types of engineering materials in the following study. 2.2. FCG based on a linear damage accumulation (FCG-LDA) The linear damage accumulation can be adopted in the model formulated here. In fact, according to numerical simulations such as finite element analyses, the curvature of the crack tip is really non-zero, and then the plastic strain of the crack tip is finite. Let us now introduce a critical crack blunting radius ρc. Therefore, Eq. (2) can be rewritten as 1/(1+ n' )
rc ∆ε p ( r + ρ c ) = E r + ρ c 2σ yc
(16)
.
By combining Eq. (9) and Eq. (16) and a unit damage D=1/Nf, the damage distribution of the nodes along with the crack growth direction in the cyclic plastic zone can be described as follows: σ yc D ( r + ρc ) = 2 Eε ' f
−1/ c
rc r + ρc
−1/(c +cn' ) .
(17)
It is assumed that each step of crack extension equals the cyclic plastic zone size in the growth direction. Then, the sum of accumulated damages that occur during each cycle is expressed as
∫
rc − ρc
0
D ( r + ρc )dr ,
(18)
while a unit average damage is defined as _
D=∫
rc − ρ c
0
D ( r + ρ c )dr / ( rc − ρc ) .
(19)
8
The hypothesis of linear damage accumulation is an important theory for the current concept of in-service life assessment. Consider the Miner linear damage law, in which the crack advances one step when the unit average damage[17] is equal to one unit, that is Ni
∑N i
= 1.0 .
(20)
if
The summands of the damage that are given by Eq. (20) are the quotients of the actual number of cycles at a certain stress amplitude Ni and the respective number of cycles to fracture Nif. However, it should be noted that the Miner rule is often a unsuitable oversimplification. Depending on the load history, the damage sum may be higher or lower than one[18]. By combining Eq. (13) and Eq. (17), a new theoretical model can be expressed as Eε ' r − ρc f = c = 2 * σ yc dN N
da
1/ c c + cn' ⋅r c +cn' +1 c
1 ρ 1+ c c cn' + 1− r c
(21)
where ρc and rc can be calculated using Eq. (5) and Eq. (3), respectively.
3. Validation of the theoretical models In present section, fatigue crack growth rates were calculated for widely used engineering materials by Eq. (15) and Eq. (21), respectively. Calculated results were thoroughly compared with experiments from the literatures for six types of structural materials. The relevant materials data are listed in Table 1. The calculated results are presented from Fig. 2 to Fig. 7 for homogeneous base materials of SAE 1020 steel, 7075-T6 alloys, 4340 steel, A533-B1 steel, API5L X60 steel and E36 steel, respectively.
9
It can be seen that the theoretical model predictions are in good agreement with the experimental data over realistic fatigue crack growth rate ranges. It is surprising that the two theoretical models give good results over the linear range of the da/dN versus ΔK curve, in which fatigue striations are found. As Eq. (21) ignores the elastic
damage accumulation, Fig. 3 shows that the prediction result of Eq. (15) is better than the result from Eq. (21) for a 7075-T6 alloy. This result indicates that the established model used in Eq. (21) is comparatively conservative in region II (the intermediate region of the da/dN vs. ΔK curve).
4. Discussions Two crack growth analytical models developed are compared with the experimental results for widely used engineering materials available in the literatures. Comparison of da/dN vs. ΔK experimental data with model given by Eq. (15) and Eq. (21) is given in Fig. 2 to Fig. 7. The following observations are made: 1. The LCF failure process can be represented by crack initiation, stable growth and catastrophic propagation stages, with the different duration for different material systems. That is to say, multiple cycles within the “process zone” are generally required for the failure of a material. 2. The model prediction results agree well with region II (the intermediate region of the da/dN Versus ΔK plot). Calculated results from Fig. 2 to Fig. 7 show that the proposed models are somewhat conservative when the “process zone” is assumed to be equal to the cyclic plastic zone near the crack tip.
10
3. Comparisons are made on engineering materials with the wide range of strength. For all these ranges the present predictions of life models proposed here are quite satisfactory. 4. Two theoretical models are used in analysis the 7075-T6 Al alloys. From Fig. 3, it is found that energy-based criteria are more suitable than linear damage accumulation near the crack tip. As the plane stress solution of Mode I cracking problems is used in the analysis, it seems to be most appropriate to use two theoretical models described above for the plane stress thin plates. It should be also pointed out that these models are based on the HRR solution relevant to be localized plastic yielding. Besides, the HRR model and stress intensity factor cannot be used effectively if the plastic zone size ahead of the crack tip is larger than the cracked body dimensions.
5. Conclusions To reduce the material constants to be determined, the present work attempts to develop a novel fatigue cracking growth rate model for the life prediction of engineering structures. Based on the plastic strain energy and the linear damage accumulation, two different analytical life models have thus been developed here and validated in details using open published experimental data of low cycle fatigue properties. From the detailed comparisons, the following important conclusions can be drawn as follows: 1.
These models are soundly capable of predicting the FCG relationship independently using fundamental low cycle fatigue properties.
11
2.
The crack growth rate formulas, which are described in Eq. (15) and Eq. (21), do not contain any adjustable parameters.
3.
A fairly good agreement exists between the predicted da/dN-ΔK relationship and the corresponding experimental data. Satisfactory agreement is obtained for low and medium crack growth rate ranges.
It would be of interest to clarify the following issues in the future. First, the present models require to be improved for similar high strength 7075-T6 Al alloys. Second, more experimental data should be used to validate the life models. Third, it is a crucial meaningful for these theoretical models to be applied into fatigue crack growth tests and further verify this approach.
Acknowledgements The authors gratefully acknowledge the financial support for the present work form National Nature Science Foundation of China (Nos.: 11072205 and 51005061).
References 1
Krupp U. Fatigue crack propagation in metals and alloys. Weinheim: Wiley-VCH, 2007.
2
Tomkins B. Micromechanisms of fatigue crack growth at high stress. Metal Sci 1980; 14: 408-17.
3
Paris P, Erdogan F. A critical analysis of crack propagation laws. Transaction of the ASME, J Basic Eng 1963; 85: 528-33.
4
Oh KP. A weakest-link model for the prediction of fatigue crack growth rate. J Eng Mater Technol 1978; 100: 170-74.
5
Kujawski D, Ellyin F. A fatigue crack growth model with load ratio effects, Eng Fract Mech 1987; 28: 367-78.
6
Li D, Nam W, Lee C. An improvement on prediction of fatigue crack growth from low cycle fatigue properties. Eng Fract Mech 1998; 60: 397-406.
12
7
Kujawski D, Ellyin F. A fatigue crack propagation model. Eng Fract Mech 1984; 20: 695-704.
8
Ellyin F. Stochastic modelling of crack growth based on damage accumulation. Theoretical and Applied Fracture Mechanics, 6 (1986) 95-101.
9
Pandey K, Chand S. An energy based fatigue crack growth model, Int J fatigue 2003; 25: 771-78.
10
Pandey K, Chand S. Fatigue crack growth model for constant amplitude loading, Fatigue Fract Eng Mater Struct 2004; 27: 459-72.
11
Glinka G. A cumulative model of fatigue crack growth. Int J Fatigue 1982; 4: 59-67.
12
Schwalbe KH. Comparison of several fatigue crack propagation laws with experimental results. Eng Fract Mech 1974; 6: 325-41.
13
Rice JR. Stresses due to a sharp notch in a work-hardening elastic-plastic material loaded by longitudinal shear. J Appl Mech 1967; 34: 287-98.
14
Kujawski D, Ellyin F. On the size of plastic zone ahead of crack tip, Eng Fract Mech 1986; 25: 229-36.
15
Morrow J. Fatigue properties of metals, Manual Society of Automotive Engineers. ISTC Div, 4 (1964).
16
Shih CF. Relationship between the J-integral and the crack opening displacement for stationary and extending cracks. J Mech Phs Solid 1981; 29: 305-26.
17
Chen L, Cai LX, Yao D. A new method to predict fatigue crack growth rate of materials based on average cyclic plasticity strain damage accumulation. Chin J Aeronaut 2013; 26: 130-35.
18
Castro JTP, Meggiolaro MA, Miranda ACO. Singular and non-singular approaches for predicting fatigue crack growth behavior. Int J Fatigue 2005; 27: 1366-88.
19
Noroozi A, Glinka G, Lambert S. A two parameter driving force for fatigue crack growth analysis, Int J Fatigue 2005; 27: 1277-96.
20
Glinka G. A notch stress-strain analysis approach to fatigue crack growth, Eng Fract Mech 1985; 21: 245-61.
13
Table 1 Mechanical and fatigued properties used in present models
Materials
SAE 1020 steel[18] [19]
7075-T6 alloy [19]
4340 steel
A533-B1 steel
[6]
API5L X60 steel[18] [9,10,20]
E36 steel
σf ’
ΔK th
σyc
K'
/GPa
/MPa
/MPa
205
270
941
0.18
815
-0.114
0.25
-0.54
0.1
11.6
71
469
781
0.088
781
-0.045
0.19
-0.52
0.5
1.98
200
889
1910
0.123
1879
-0.0895
0.64
-0.636
0.7
4.56
200
345
1047
0.1650
869
-0.085
0.32
-0.52
0.1
7.7
200
370
840
0.132
720
-0.076
0.31
-0.53
0.1
8.0
206
350
1255
0.21
1194
-0.124
0.60
-0.570
0
5.0
E
n′
/MPa
b
c
εf'
/MPa·m1/2
∆ε/2 (∆σ/2) Monotonic plastic zone border
∆εtip/2 (∆σtip/2)
Crack tip
o
ρc
Cyclic plastic zone border
Fig. 1 Distribution of stress and strain at the fatigue crack tip
14
R
0.1
Reference data Prediction FCG-LDA Prediction FCG-PSE
da/dN / mm/Cycle
0.01
1E-3
SAE 1020 steel R=0.1 1/2 ∆Kth=11.6 MPa·mm
1E-4
1E-5
1E-6
1E-7 1
10
1/2
100
∆K / MPa·mm
Fig. 2. Fatigue cracking growth rate comparisons between present predictions and experiments for SAE 1020 steel.
0.1
7075-T6 Alloy R=0.5 1/2 ∆Kth=1.98 MPa·mm
da/dN / mm/Cycle
0.01
1E-3
1E-4
1E-5
Reference data Prediction FCG-LDA Prediction FCG-PSE
1E-6
1E-7 1
10
1/2
100
∆K / MPa·mm
Fig. 3. Fatigue cracking growth rate comparisons between present predictions and experiments for 7075-T6 Al alloys.
15
0.1
Reference data Prediction FCG-LDA Prediction FCG-PSE
da/dN / mm/Cycle
0.01
1E-3
1E-4
1E-5
4340 steel R=0.7 1/2 ∆Kth=4.56 MPa·mm
1E-6
1E-7 1
10
1/2
100
∆K / MPa·mm
Fig. 4. Fatigue cracking growth rate comparisons between present predictions and experiments for 4340 steel.
0.1
Reference data Prediction FCG-LDA Prediction FCG-PSE
da/dN / mm/Cycle
0.01
A5333-B1 steel R=0.1 1/2 ∆Kth=7.7 MPa·mm
1E-3
1E-4
1E-5
1E-6
1E-7 1
10
1/2
100
∆K / MPa·mm
Fig. 5. Fatigue cracking growth rate comparisons between present predictions and experiments for A5333-B1 steel.
16
0.1
Reference data Prediction FCG-LDA Prediction FCG-PSE
da/dN / mm/Cycle
0.01
1E-3
1E-4
API5L X60 steel R=0.1 1/2 ∆Kth=8 MPa·mm
1E-5
1E-6
1E-7 1
10
100
1/2
∆K / MPa·mm
Fig. 6. Fatigue cracking growth rate comparisons between present predictions and experiments for API5L X60 steel.
0.1
Reference data Prediction FCG-LDA Prediction FCG-PSE
da/dN / mm/Cycle
0.01
1E-3
1E-4
1E-5
E36 steel R=0 1/2 ∆Kth=5 MPa·mm
1E-6
1E-7 1
10
1/2
∆K / MPa·mm
Fig. 7. Fatigue cracking growth rate comparisons between present predictions and experiments for E36 steel.
17
100
Highlights 1. Models of FCG have been developed considering the plastic strain energy and the linear damage accumulation, respectively. 2. Models are soundly capable of predicting FCG relationship independently using fundamental low cycle fatigue properties. 3. The relationship between FCG and LCF is developed. Crack growth rate formulas do not contain any adjustable parameters.
18
S0142-1123(13)00322-8 http://dx.doi.org/10.1016/j.ijfatigue.2013.11.007 JIJF 3252
To appear in:
International Journal of Fatigue
Received Date: Revised Date: Accepted Date:
12 May 2013 8 November 2013 11 November 2013
Please cite this article as: Shi, K.K., Cai, L.X., Chen, L., Wu, S.C., Bao, C., Prediction of fatigue crack growth based on low cycle fatigue properties, International Journal of Fatigue (2013), doi: http://dx.doi.org/10.1016/j.ijfatigue. 2013.11.007
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Prediction of fatigue crack growth based on low cycle fatigue properties K.K. Shi1* , L.X. Cai1, L. Chen1, S.C. Wu2, C. Bao1 1. School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu, 610031, China 2. State Key Laboratory of Traction Power, Southwest Jiaotong University, Chengdu, 610031,China
Abstract: Theoretical models of the fatigue crack growth without artifical adjustable parameters were proposed by considering the plastic strain energy and the linear damage accumulation, respectively. The crack was regarded as a sharp notch with a small curvature radius and the process zone was assumed to be the size of cyclic plastic zone. The near crack tip elastic-plastic stress and strain were evaluated in terms of modified Hutchinson, Rice and Rosengren (HRR) formulations. Predicted results from two established models have been soundly compared with open reports for frequently used materials. It is found that experimental results agree well with theoretical solutions. Keywords: Fatigue crack growth; Low cycle fatigue properties; Cyclic plastic zone; Plastic strain energy; Linear damage accumulation
1. Introduction Fatigue crack growth behaviors have been studied for various types of engineering structural materials in the past several decades. In order to correlate the damage evolution with the cyclic deformation characteristics of a material with macroscopic cracks, notched specimens are typically introduced to calibrate the crack growth rate da/dN[1]. It is well-known that fatigue cracks initiate and propagate due to the local cyclic plastic yielding of the materials for exampling Model I cracking problems[2]. A
*Corresponding
author, Tel.: +86-28-87600850; Fax: +86-28-87600797
Email address: [email protected] (K.K. Shi)
1
crack growth law can then be formulated with the help of the stress and strain field ahead of the crack tip together with a suitable failure criterion. The damage-tolerant design method assumes that engineering components contain intrinsic imperfections in the form of macro cracks. Generally, the cracks can propagate only when a certain critical length is approached. Such a local nature of the fatigue phenomenon can be described using a sigmoidal curve in the fatigue crack growth rate of log da/dN vs. log ΔK. The sigmoidal curve is soundly bounded at the extremes by the lower bound of the fatigue threshold ΔKth and by the upper bound of the critical stress intensity factor range ΔKc. In the intermediate range, log da/dN is nearly linearly correlated with log ΔK, as formulated by Paris and Erdogan[3]. Notably, some life relationships between the fatigue crack growth rate and the low cycle fatigue (LCF) properties of the material have also been established. A number of crack growth rate models previously proposed surrounding the crack tip region are individually based on the fatigue ductility[4], the plastic strain energy[5-10] and the weighted value of the local strain[11]. Unfortunately, most of these theoretical models intrinsically contain adjustable material parameters that requires to be determined experimentally or numerically. Although a few equations with material constants have been established to effectively predict the fatigue cracking growth characteristics, the bore some problem is that material constants are especially difficult to be acquired form the viewpoint of cost. The progression of fatigue cracking is generally assumed to be an incremental growth due to a critical energy accumulated at the crack-tip. The material ahead of the crack is necessarily modeled as an assemblage of uniaxial material elements, and thus 2
the crack growth can be regarded as the successive failure process of these elements. Additionally, the width of the material element is assumed to be the length of cycle plastic zone along the progressive crack direction. More importantly, the success of such a theory depends on the specification of an appropriate plastic strain energy criterion and the linear damage accumulation along the crack growth direction inside of the cyclic plastic zone. As the stress and the strain are theoretically singular at the crack tip, the plastic strain and the linear damage are somewhat difficult to define precisely. However, the singularity will disappear through introducing the crack tip blunting in this study. Therefore, finite values of the stress and the strain are the focus of the present study. Two theoretical models of the fatigue crack growth based on the plastic strain energy and linear damage accumulation therefore were developed inside the low cycle plastic zone at the crack tip, respectively. Theoretical models can accommodate both the intermittent growth and the continuum growth currently identified for the majority of engineering materials. Furthermore, these models can be used to analyze the surface-based cracking life of engineering structures such as the nuclear pressure vessels and pipelines, and high pressure storage tanks. In the following sections, two analytical models based on the plastic strain energy (PSE) and the linear damage accumulation (LDA), respectively, will be thoroughly established, which physically remove some adjustable materials constants for the effective predictions fatigue crack growth (FCG) rate. Two theoretical model formulated here are expected to be an important complement to a relatively reliable solution of in-service fatigue life for engineering materials.
2. Fatigue crack growth models
3
An analytical solution of the stress and strain distribution ahead of a crack loaded in antiplane shear (Mode III) during the small-scale yielding has been derived by Schwalbe[12] . Based on the Hutchinson, Rice and Rosengred (HRR) field, a similar analytical solution as for the tensile loading (Mode I) that is the most important solution from an engineering application viewpoint, is available for small scale yielding. Despite the significant plastic deformation in the crack tip zone, the linear elastic fracture mechanics (LFEM) approach is frequently regarded as a well-established theoretical method for the small scale yielding analysis. Consider a material with strain hardening behavior and a defined stress and strain relationship ahead of the crack tip. The following Fig. 1 illustrates the cyclic stress strain relationship. Assuming Masing behaviors are satisfied inside of the cyclic plastic zone at the fatigue crack tip. The increasing branches of hysteresis loops for various loading levels should be coincident, thus the stress-strain hysteresis loops can be approximated. The stress and strain range can thus be written as[13,14] ∆σ ( r ) = 2 K ′
∆ε p
2
2σ yc rc ∆ε p ( r ) = E r
1
n′
(1)
(1+ n′ )
(2)
where σyc=Eεyc is the cyclic tensile yielding stress, E is the elastic modulus, Δεp is the plastic strain range, Δσ is the stress range, n′ is the cyclic strain hardening exponent, K′ is the cyclic strain hardening coefficient, rc is the cyclic plastic zone and r is the distance from the crack tip. The cyclic plastic zone rc can be formulated by[15]
4
∆K 1 rc = 4π (1 + n′) σ yc
2
(3)
in which ΔK is the stress intensity range.
2.1. FCG based on a plastic strain energy (FCG-PSE) In terms of the product of the stress and the strain, Eqs. (1) and (2) can be arranged into the following equation: σ yc ∆σ ⋅ ∆ε p = 4 K ′ E
( n ′+1)
rc . r
(4)
The above equation will exhibit a singularity as r→0. However, such singularity is rather unreasonable due to crack tip blunting from complex material and loading behaviors[16]. With the nature of crack tip blunting, the stress and strain have a finite magnitude in a high strain zone ahead of the crack tip. The cyclic plastic zone is treated as the “process zone” where the majority of the damages occurs. In the process zone, the plastic strain range is much larger than the elastic strain range, thus leading to Δε≈Δεp. Let us now introduce a critical crack blunting radius ρc as
∆K th 1 ρc = 4π (1 + n′) σ yc
2
,
(5)
where ΔKth is the stress intensity threshold.
The critical radius ρc is associated with the threshold stress intensity range. Generally, the fatigue cracks will not propagate when the critical blunting radius is less than the threshold measures from experiments of a material. Thus Eq. (4) can be rewritten as 5
' σ y ∆σ ⋅ ∆ε p = 4 K E
n' +1
rc r + ρc
(6)
From this, it is seen that the product Δσ·Δεp in the cyclic plastic zone can be obtained analytically. By integrating within the range [0, rc-ρc] using above Eq. (6), the plastic strain energy can be determined by the integral as
∫
rc − ρ c
0
σ ∆σ ⋅ ∆ε pdr = 4 K ′ yc E
( n ′+1)
⋅ rc ⋅ ( ln rc − ln ρ c ) ,
(7)
The fatigue resistance of the material ahead of the crack is governed hypothetically by the local state of stress and strain range perpendicularly to the cracking growth examplied for Mode I problems. A fatigue failure criterion can be applied in the cyclic plastic zone, rc. Based on the Manson-Coffin strain/life relationship obtained from smooth specimens, strain/life relationship can be described either in terms of the plastic strain or stress range and the number of cycles to failure 2Nf. Taking the detrimental effect of a mean stress into account since Morrow’s work[15], the superimposition yields the following relationship between the total stain amplitude and the number of cycles to technical failure Nf.
(
' ∆σ = 2 σ f − σ m
( )
' ∆ε p = 2ε f 2 N f
b
) (2 N )
(8)
f
c
(9)
in which εf ' and σf' are the fatigue ductility and strength coefficients, respectively, exponents b and c are the undetermined material properties and the σm is the mean stress. The product of the plastic strain and stress range can be calculated using both Eq. (8) and Eq. (9):
6
(
b+ c
) ( )
' ' ∆σ ⋅ ∆ε p = 4 σ f − σ m ε f 2 N f
(10)
.
By rearranging this Eq. (10) and the processing zone, rc-ρc, we can obtain the following relationship,
(
b+c
) ( )
' ' * ∆σ ⋅ ∆ε p ⋅ ( rc − ρc ) = 4 σ f − σ m ε f 2 N
⋅ ( rc − ρ c ) ,
(11)
in which N* is the number of cycles for the crack to penetrate through rc-ρc. Because Eq. (7) is equal to Eq. (11), the following formulation can be obtained: 1
n' +1 b +c ' σ yc rc K * 1 N = ⋅ ⋅ ⋅ [ ln( rc ) − ln( ρc )] . ' ' 2 σ −σ E rc − ρ c ε m f f
(
)
(12)
Note that in the present model, the mean stress will be equal to zero in the case of the low cycle fatigue tests. Consequently, the crack extension per cycle da/dN can now be finally defined as da rc − ρc = dN N*
(13)
Let us now introduce an effective stress intensity range, ΔKeq, defined as
(
2 2 ∆K eq = ∆K − ∆K th
1/2
)
(14)
Incorporating Eq. (14) into Eq. (13), the following model can be acquired as: ∆Keq2 da = . dN 4π (1 + n′)σ yc2 N *
(15)
The present Eq. (15) actually looks like the correlation between crack tip opening displacement (CTOD) and J integral for small scale yielding and will be used to
7
predict the da/dN-ΔK relationship for six types of engineering materials in the following study. 2.2. FCG based on a linear damage accumulation (FCG-LDA) The linear damage accumulation can be adopted in the model formulated here. In fact, according to numerical simulations such as finite element analyses, the curvature of the crack tip is really non-zero, and then the plastic strain of the crack tip is finite. Let us now introduce a critical crack blunting radius ρc. Therefore, Eq. (2) can be rewritten as 1/(1+ n' )
rc ∆ε p ( r + ρ c ) = E r + ρ c 2σ yc
(16)
.
By combining Eq. (9) and Eq. (16) and a unit damage D=1/Nf, the damage distribution of the nodes along with the crack growth direction in the cyclic plastic zone can be described as follows: σ yc D ( r + ρc ) = 2 Eε ' f
−1/ c
rc r + ρc
−1/(c +cn' ) .
(17)
It is assumed that each step of crack extension equals the cyclic plastic zone size in the growth direction. Then, the sum of accumulated damages that occur during each cycle is expressed as
∫
rc − ρc
0
D ( r + ρc )dr ,
(18)
while a unit average damage is defined as _
D=∫
rc − ρ c
0
D ( r + ρ c )dr / ( rc − ρc ) .
(19)
8
The hypothesis of linear damage accumulation is an important theory for the current concept of in-service life assessment. Consider the Miner linear damage law, in which the crack advances one step when the unit average damage[17] is equal to one unit, that is Ni
∑N i
= 1.0 .
(20)
if
The summands of the damage that are given by Eq. (20) are the quotients of the actual number of cycles at a certain stress amplitude Ni and the respective number of cycles to fracture Nif. However, it should be noted that the Miner rule is often a unsuitable oversimplification. Depending on the load history, the damage sum may be higher or lower than one[18]. By combining Eq. (13) and Eq. (17), a new theoretical model can be expressed as Eε ' r − ρc f = c = 2 * σ yc dN N
da
1/ c c + cn' ⋅r c +cn' +1 c
1 ρ 1+ c c cn' + 1− r c
(21)
where ρc and rc can be calculated using Eq. (5) and Eq. (3), respectively.
3. Validation of the theoretical models In present section, fatigue crack growth rates were calculated for widely used engineering materials by Eq. (15) and Eq. (21), respectively. Calculated results were thoroughly compared with experiments from the literatures for six types of structural materials. The relevant materials data are listed in Table 1. The calculated results are presented from Fig. 2 to Fig. 7 for homogeneous base materials of SAE 1020 steel, 7075-T6 alloys, 4340 steel, A533-B1 steel, API5L X60 steel and E36 steel, respectively.
9
It can be seen that the theoretical model predictions are in good agreement with the experimental data over realistic fatigue crack growth rate ranges. It is surprising that the two theoretical models give good results over the linear range of the da/dN versus ΔK curve, in which fatigue striations are found. As Eq. (21) ignores the elastic
damage accumulation, Fig. 3 shows that the prediction result of Eq. (15) is better than the result from Eq. (21) for a 7075-T6 alloy. This result indicates that the established model used in Eq. (21) is comparatively conservative in region II (the intermediate region of the da/dN vs. ΔK curve).
4. Discussions Two crack growth analytical models developed are compared with the experimental results for widely used engineering materials available in the literatures. Comparison of da/dN vs. ΔK experimental data with model given by Eq. (15) and Eq. (21) is given in Fig. 2 to Fig. 7. The following observations are made: 1. The LCF failure process can be represented by crack initiation, stable growth and catastrophic propagation stages, with the different duration for different material systems. That is to say, multiple cycles within the “process zone” are generally required for the failure of a material. 2. The model prediction results agree well with region II (the intermediate region of the da/dN Versus ΔK plot). Calculated results from Fig. 2 to Fig. 7 show that the proposed models are somewhat conservative when the “process zone” is assumed to be equal to the cyclic plastic zone near the crack tip.
10
3. Comparisons are made on engineering materials with the wide range of strength. For all these ranges the present predictions of life models proposed here are quite satisfactory. 4. Two theoretical models are used in analysis the 7075-T6 Al alloys. From Fig. 3, it is found that energy-based criteria are more suitable than linear damage accumulation near the crack tip. As the plane stress solution of Mode I cracking problems is used in the analysis, it seems to be most appropriate to use two theoretical models described above for the plane stress thin plates. It should be also pointed out that these models are based on the HRR solution relevant to be localized plastic yielding. Besides, the HRR model and stress intensity factor cannot be used effectively if the plastic zone size ahead of the crack tip is larger than the cracked body dimensions.
5. Conclusions To reduce the material constants to be determined, the present work attempts to develop a novel fatigue cracking growth rate model for the life prediction of engineering structures. Based on the plastic strain energy and the linear damage accumulation, two different analytical life models have thus been developed here and validated in details using open published experimental data of low cycle fatigue properties. From the detailed comparisons, the following important conclusions can be drawn as follows: 1.
These models are soundly capable of predicting the FCG relationship independently using fundamental low cycle fatigue properties.
11
2.
The crack growth rate formulas, which are described in Eq. (15) and Eq. (21), do not contain any adjustable parameters.
3.
A fairly good agreement exists between the predicted da/dN-ΔK relationship and the corresponding experimental data. Satisfactory agreement is obtained for low and medium crack growth rate ranges.
It would be of interest to clarify the following issues in the future. First, the present models require to be improved for similar high strength 7075-T6 Al alloys. Second, more experimental data should be used to validate the life models. Third, it is a crucial meaningful for these theoretical models to be applied into fatigue crack growth tests and further verify this approach.
Acknowledgements The authors gratefully acknowledge the financial support for the present work form National Nature Science Foundation of China (Nos.: 11072205 and 51005061).
References 1
Krupp U. Fatigue crack propagation in metals and alloys. Weinheim: Wiley-VCH, 2007.
2
Tomkins B. Micromechanisms of fatigue crack growth at high stress. Metal Sci 1980; 14: 408-17.
3
Paris P, Erdogan F. A critical analysis of crack propagation laws. Transaction of the ASME, J Basic Eng 1963; 85: 528-33.
4
Oh KP. A weakest-link model for the prediction of fatigue crack growth rate. J Eng Mater Technol 1978; 100: 170-74.
5
Kujawski D, Ellyin F. A fatigue crack growth model with load ratio effects, Eng Fract Mech 1987; 28: 367-78.
6
Li D, Nam W, Lee C. An improvement on prediction of fatigue crack growth from low cycle fatigue properties. Eng Fract Mech 1998; 60: 397-406.
12
7
Kujawski D, Ellyin F. A fatigue crack propagation model. Eng Fract Mech 1984; 20: 695-704.
8
Ellyin F. Stochastic modelling of crack growth based on damage accumulation. Theoretical and Applied Fracture Mechanics, 6 (1986) 95-101.
9
Pandey K, Chand S. An energy based fatigue crack growth model, Int J fatigue 2003; 25: 771-78.
10
Pandey K, Chand S. Fatigue crack growth model for constant amplitude loading, Fatigue Fract Eng Mater Struct 2004; 27: 459-72.
11
Glinka G. A cumulative model of fatigue crack growth. Int J Fatigue 1982; 4: 59-67.
12
Schwalbe KH. Comparison of several fatigue crack propagation laws with experimental results. Eng Fract Mech 1974; 6: 325-41.
13
Rice JR. Stresses due to a sharp notch in a work-hardening elastic-plastic material loaded by longitudinal shear. J Appl Mech 1967; 34: 287-98.
14
Kujawski D, Ellyin F. On the size of plastic zone ahead of crack tip, Eng Fract Mech 1986; 25: 229-36.
15
Morrow J. Fatigue properties of metals, Manual Society of Automotive Engineers. ISTC Div, 4 (1964).
16
Shih CF. Relationship between the J-integral and the crack opening displacement for stationary and extending cracks. J Mech Phs Solid 1981; 29: 305-26.
17
Chen L, Cai LX, Yao D. A new method to predict fatigue crack growth rate of materials based on average cyclic plasticity strain damage accumulation. Chin J Aeronaut 2013; 26: 130-35.
18
Castro JTP, Meggiolaro MA, Miranda ACO. Singular and non-singular approaches for predicting fatigue crack growth behavior. Int J Fatigue 2005; 27: 1366-88.
19
Noroozi A, Glinka G, Lambert S. A two parameter driving force for fatigue crack growth analysis, Int J Fatigue 2005; 27: 1277-96.
20
Glinka G. A notch stress-strain analysis approach to fatigue crack growth, Eng Fract Mech 1985; 21: 245-61.
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Table 1 Mechanical and fatigued properties used in present models
Materials
SAE 1020 steel[18] [19]
7075-T6 alloy [19]
4340 steel
A533-B1 steel
[6]
API5L X60 steel[18] [9,10,20]
E36 steel
σf ’
ΔK th
σyc
K'
/GPa
/MPa
/MPa
205
270
941
0.18
815
-0.114
0.25
-0.54
0.1
11.6
71
469
781
0.088
781
-0.045
0.19
-0.52
0.5
1.98
200
889
1910
0.123
1879
-0.0895
0.64
-0.636
0.7
4.56
200
345
1047
0.1650
869
-0.085
0.32
-0.52
0.1
7.7
200
370
840
0.132
720
-0.076
0.31
-0.53
0.1
8.0
206
350
1255
0.21
1194
-0.124
0.60
-0.570
0
5.0
E
n′
/MPa
b
c
εf'
/MPa·m1/2
∆ε/2 (∆σ/2) Monotonic plastic zone border
∆εtip/2 (∆σtip/2)
Crack tip
o
ρc
Cyclic plastic zone border
Fig. 1 Distribution of stress and strain at the fatigue crack tip
14
R
0.1
Reference data Prediction FCG-LDA Prediction FCG-PSE
da/dN / mm/Cycle
0.01
1E-3
SAE 1020 steel R=0.1 1/2 ∆Kth=11.6 MPa·mm
1E-4
1E-5
1E-6
1E-7 1
10
1/2
100
∆K / MPa·mm
Fig. 2. Fatigue cracking growth rate comparisons between present predictions and experiments for SAE 1020 steel.
0.1
7075-T6 Alloy R=0.5 1/2 ∆Kth=1.98 MPa·mm
da/dN / mm/Cycle
0.01
1E-3
1E-4
1E-5
Reference data Prediction FCG-LDA Prediction FCG-PSE
1E-6
1E-7 1
10
1/2
100
∆K / MPa·mm
Fig. 3. Fatigue cracking growth rate comparisons between present predictions and experiments for 7075-T6 Al alloys.
15
0.1
Reference data Prediction FCG-LDA Prediction FCG-PSE
da/dN / mm/Cycle
0.01
1E-3
1E-4
1E-5
4340 steel R=0.7 1/2 ∆Kth=4.56 MPa·mm
1E-6
1E-7 1
10
1/2
100
∆K / MPa·mm
Fig. 4. Fatigue cracking growth rate comparisons between present predictions and experiments for 4340 steel.
0.1
Reference data Prediction FCG-LDA Prediction FCG-PSE
da/dN / mm/Cycle
0.01
A5333-B1 steel R=0.1 1/2 ∆Kth=7.7 MPa·mm
1E-3
1E-4
1E-5
1E-6
1E-7 1
10
1/2
100
∆K / MPa·mm
Fig. 5. Fatigue cracking growth rate comparisons between present predictions and experiments for A5333-B1 steel.
16
0.1
Reference data Prediction FCG-LDA Prediction FCG-PSE
da/dN / mm/Cycle
0.01
1E-3
1E-4
API5L X60 steel R=0.1 1/2 ∆Kth=8 MPa·mm
1E-5
1E-6
1E-7 1
10
100
1/2
∆K / MPa·mm
Fig. 6. Fatigue cracking growth rate comparisons between present predictions and experiments for API5L X60 steel.
0.1
Reference data Prediction FCG-LDA Prediction FCG-PSE
da/dN / mm/Cycle
0.01
1E-3
1E-4
1E-5
E36 steel R=0 1/2 ∆Kth=5 MPa·mm
1E-6
1E-7 1
10
1/2
∆K / MPa·mm
Fig. 7. Fatigue cracking growth rate comparisons between present predictions and experiments for E36 steel.
17
100
Highlights 1. Models of FCG have been developed considering the plastic strain energy and the linear damage accumulation, respectively. 2. Models are soundly capable of predicting FCG relationship independently using fundamental low cycle fatigue properties. 3. The relationship between FCG and LCF is developed. Crack growth rate formulas do not contain any adjustable parameters.
18