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The perturbation series method based on the logarithm equation for fatigue crack growth prediction
Accepted Manuscript The Perturbation Series Method Based on the Logarithm Equation for Fatigue Crack Growth Prediction Zhiping Qiu, Jingjing Zhu PII: DOI: Article Number: Reference:
S0167-8442(18)30532-9 https://doi.org/10.1016/j.tafmec.2019.102239 102239 TAFMEC 102239
To appear in:
Theoretical and Applied Fracture Mechanics
Received Date: Revised Date: Accepted Date:
12 November 2018 18 March 2019 29 April 2019
Please cite this article as: Z. Qiu, J. Zhu, The Perturbation Series Method Based on the Logarithm Equation for Fatigue Crack Growth Prediction, Theoretical and Applied Fracture Mechanics (2019), doi: https://doi.org/10.1016/ j.tafmec.2019.102239
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The Perturbation Series Method Based on the Logarithm Equation for Fatigue Crack Growth Prediction
Zhiping Qiu*, Jingjing Zhu Institute of Solid Mechanics, School of Aeronautic Science and Engineering, Beihang University, Beijing 100083, P. R. China *Corresponding author:
Zhiping Qiu Institute of Solid Mechanics School of Aeronautic Science and Engineering Beihang University Beijing 100083, P.R. China Fax: 86-010-82339628 Email: [email protected]
The Perturbation Series Method Based on the Logarithm Equation for Fatigue Crack Growth Prediction
Abstract This paper gives the perturbation series method for the fatigue crack growth prediction accounting for the error in initial crack length. The proposed method utilizes the logarithm form of the crack growth equation and the solution of it is expressed in the form of asymptotic series by introducing a small parameter. The initial crack length is considered as a sum of the nominal value and the perturbation which mainly originates from manufacturing and measurement errors. Based on the Taylor series expansion and perturbation series method, a series of perturbation equations to predict the fatigue crack growth history are obtained. Then these equations can be solved by the successive approximation approach and the a-N curve which describes the relationship between crack length and number of load cycles can be obtained. Two examples with respect to fatigue crack growth experiments are given to verify the accuracy and validation of the presented method. The results show that the perturbation series method is closer to the experimental data compared with the original solution of crack growth equation and it can provide a more accurate prediction for the fatigue crack growth process.
Keywords: Crack growth; Logarithm equation; Initial crack length; Taylor series expansion; Perturbation series method
1. Introduction Fatigue failure is a process in which damage accumulates under cyclic loading that may be well below the yield point. Fatigue fracture is particularly serious for aircrafts, ships, bridges and other engineering structures under dynamic loads, and this problem continues to prevail in spite of the exhaustive amount of past research. Through the 1
investigation of the fatigue crack growth process, information is obtained which can be utilized to predict the crack length history and estimate the fatigue life of structures. Therefore, the research on fatigue crack growth problem has attracted continual attention [1-4]. From a fracture mechanics point of view, the fatigue damage of a structural component subjected to dynamic loads can be measured by the size of the dominant crack, and failure will occur when the crack reaches a critical magnitude. In order to describe the process of fatigue fracture, several crack-growth laws taking different forms have been presented in the literatures. Paris et al. [5] suggested an approach that related the crack growth rate ( da / dN ) to the maximum stress intensity factor K max in 1961. Liu [6] subsequently implied that the crack growth rate was a function of the cyclic stress range S . In 1963, Paris and Erdogan [1] proposed the fatigue crack growth rate equation based on experiment, which described the relationship between the crack growth rate and the stress intensity factor range under a fatigue stress regime. This is the celebrated Paris law, which is the most popular fatigue crack growth model applied in materials science and fracture mechanics. Over the years, the Paris formula has continued to be modified to account for variety of observations, including the effect of stress ratio and the maximum stress intensity factor [7-8], the crack closure phenomenon [9] and the threshold stress intensity factor amplitude [10]. Generally, the typical fatigue growth behavior of cracks could be described by the curve relating fatigue crack growth rate ( da / dN ) to stress intensity factor range K , as shown in Fig.1. The process of fatigue crack growth can be divided into three stages, i.e. crack initiation (stage I), subcritical crack growth (stage II) and rapid crack growth (stage III). Thereinto, the stage II follows Paris law and accounts for the main part of fatigue life.
2
Fig.1. A schematic of the typical fatigue growth behavior of cracks. Nowadays, the influence of parameters fluctuation on the safety of structures has aroused more and more attention [11-15]. In practical engineering application, fatigue crack growth in engineering materials and structural components is generally a complex process, in which the wide scattering exists even for identical specimens. This may be caused by the uncertainties of loadings, environmental conditions, material characteristics, geometry of the structures and crack initiation, which contribute to the non-deterministic behavior of fatigue crack growth. Many stochastic fatigue crack growth approaches have been proposed to consider uncertainties for accurate fatigue life predictions [16-18]. The fatigue crack growth models based on fracture mechanics, which describe how the crack length increases under cyc lic loading, are widely used to predict the fatigue life and estimate the reliability of structural components. In order to use the fatigue crack growth models, the prerequisite is to determine the initial crack length. The initial crack length has direct effect on the crack growth process. Initial crack length of practical engineering structures can be obtained by several ways. One practice is to use an empirically assumed crack length, such as 0.25–1 mm for metals [19]. An alternative way is to use the results from nondestructive inspection (NDI) [20]. Due to the limit of detection level, measurement results usually have errors. In the experimental investigation of the fatigue crack growth process, the initial crack size of specimen with precast crack also consists of manufacturing errors. Tremendous dispersions on fatigue life of 3
structures could be caused by these errors under realistic working conditions and the life prediction will be unreliable. Taking the error in the initial crack size into consideration, this paper will present a perturbation series method for the fatigue crack growth prediction. Perturbation series method is a powerful tool to analyze the effect of a small perturbation of the parameters on the response of structures. The perturbation method, which is based on the pioneer works of Bueckneer [21] and Rice [22], has been widely used in mechanics problems, such as structural eigenvalue problems [23], structural response analysis [24] and model updating [25]. For fatigue fracture problem, the perturbation technique also has some applications. Stress intensity factors are analyzed for a non-collinear crack by using a second order perturbation method [26]. Stepanova and Igonin [27-28] utilized the perturbation technique to study the growth of fatigue near-crack-tip fields in a damaged material. Mousa and Reza [29] analyzed the free vibration of simply supported beam with breathing crack by using perturbation method. However, there has been rare research about the effect on the crack length evolution caused by the error in initial crack length. Up to now, a few researches have studied the effect of the uncertainties including material properties, applied loadings, geometry and initial conditions on fatigue crack growth process in the case of inadequate data. Wang et al. [12, 30] investigated the reliability estimation of fatigue crack growth prediction based on the interval mathematics. Long et al. [31] did good works in the interval analysis method for fatigue crack growth life prediction considering uncertainties in propagation rate constants, applied loadings, fracture toughness as well as initial crack length. The above works quantified the uncertainties of the system in the framework of non-probabilistic analysis. Based on the fatigue crack growth model, the upper and lower bounds of fatigue life are evaluated by employing the interval analysis methods. This paper focuses on the perturbation series method for the fatigue crack growth prediction with considering the error in initial crack length (mainly owing to measurement errors). The perturbation in the initial crack length is quantified by introducing a small correction parameter and the modified crack growth equations are 4
derived based on the polynomial theorem. It provides a modified model to solve the deterministic fatigue crack growth problem, which aims to realize a more reliable and precise prediction of fatigue crack growth evolution. In the perturbation method, the error in initial crack length is quantified as a perturbation by introducing a small parameter and the crack length is expanded as asymptotic series with respect to . The fatigue crack growth equation is established in logarithm form and a series of perturbation equations are obtained by Taylor series expansion and perturbation series method. The results show that the proposed method can better predict the crack growth process and estimate the fatigue life of structure component with cracks more accurately. The purpose of this paper is to develop a modified method for the prediction of fatigue crack growth process with considering the error in initial crack length. The paper is organized as follows. In Section 2, the fatigue crack growth rate equation based on fracture mechanics is described. In Section 3, based on the logarithm form of crack growth rate equation, the Taylor series expansion and perturbation series method are used to derived the perturbation equations. Then the perturbation series solution can be obtained by the successive approximation approach. In Section 4, two examples are given to demonstrate the accuracy and effectiveness of the proposed method. In the final section, we draw the conclusions of the current research.
2. Basic fatigue crack growth equation In practical engineering applications, the stage II of the fatigue crack growth process is taken as the main part of the fatigue crack growth life. The crack growth rate in stage II can be described by the Paris law, which relates the stress intensity factor range to subcritical crack growth under a fatigue stress regime. The basic formula reads da m C K dt
(1)
where da / dt is the crack growth rate, t stands for the time or the stress cycle number, K is the stress intensity factor range, C and m are material constants which relate to environment, frequency, temperature, stress ratio and can be 5
determined experimentally. The stress intensity factor range K is defined as
K Kmax Kmin
(2)
where K max and K min are the maximum and minimum stress intensity factors, respectively. In terms of linear elastic fracture mechanics, the stress intensity factor can be calculated by the following formula [32]
K f a,W ,
a
(3)
where is the cyclic stress applied on the structure, a is the crack length,
f a, W ,
is the geometry form factor which is generally a function of the
geometrical dimensions of structure (e.g. the width of the plate W) and the crack length a. By virtue of Eq. (3), K can be rewritten as
K f a,W ,
max min a f a,W , a
(4)
where max and min are the maximum cyclic stress and minimum cyclic stress, respectively, and max min is the cyclic stress amplitude. Substituting Eq. (4) into Eq. (1), we get da m C K C f a,W , dt
a
m
(5)
Due to the limits of the Paris law, researchers have proposed some modified forms of the crack growth rate equation. Considering the effect of the stress ratio R and the peak load K max , Walker [8] presented the following modified formula n da m C 1 R K max dt
(6)
where the material constants C, m, n are obtained from the experimental data at different stress ratios, the stress ratio R max / min . Based on the relationship between R and K , Eq. (6) can be rewritten as da n m CK max K dt
(7)
Considering the fatigue crack closure phenomenon existing in the crack growth test, the fatigue crack closure theory proposed by W. Elber [9] can be expressed as 6
da C Keff dt
n
U mC K
m
(8)
where U max top / is the crack closure parameter and top is the opening stress. The aforementioned crack growth rate equations can be simplified to the general form as follows da Q a,W , , top , R, K max , dt
a
b
(9)
where Q is a parameter related to material, geometry, loads and environment, and b is a constant. In engineering practice, the size of fatigue crack is generally very small relative to the size of structure. The fatigue crack growth is only affected by the stress state, material and environment around the crack. Therefore, for the complex structures possessing arbitrary geometries, the region containing crack can be regarded as a plate model with crack. Based on the stress analysis of the structure, the applied loading on the plate model can be acquired by extracting the stress field around the crack from the stress results of whole structure. For an infinite plate with center crack, the stress intensity factor can be calculated by
K a
(10)
For the finite plate with crack, the stress intensity factor can be obtained by introducing a geometry correction factor Y [33]
K Y a
(11)
Hence, the crack can be assumed as a center or a single edge crack in a plate model. Under this assumption, the parameter Q can be treated as a constant, and the crack growth rate equation can be rewritten as da Qa b dt
(12)
where Q and b are treated as constants for specific structure and load condition [34,35]. In general, the values of these two parameters are measured by experiments. By measuring the data of crack length a and corresponding time or stress cycle number t, the parameters Q and b can be obtained by curve fitting. 7
When the initial crack length is given, the crack length evolution process can be predicted by the following initial value problem
da t b Q a t dt a t a 0 0
(13)
where a0 is the initial crack length.
3. The perturbation series method for the fatigue crack growth equation 3.1 The pe rturbation series expansion of the fatigue crack growth equation Traditionally, the fatigue crack growth process, i.e. the history of crack length versus time, can be predicted by solving Eq. (13) and the solution can be expressed as
a0eQt , a t 1 1b 1 a0 Q 1 b t b ,
b 1 b 1
(14)
Eq. (14) is defined as the original solution in this paper. However, as we discussed above, the initial crack length is generally affected by some disturbance mainly originated from the measurement and manufacturing errors. As the disturbance is usually a small quantity, the perturbation series method can be applied to obtain the approximate solution of Eq. (12), and the initial condition can be expressed by introducing a small parameter
a t0 a0 a0
(15)
where is a scalar quantity much less than unity, a0 is the nominal value of initial crack length, and a0 is the perturbation which stands for the error in initial crack length. Therefore, Eq. (13) can be rewritten as
da t b Q a t dt a t a a 0 0 0
(16)
By resorting to the perturbation series method (see in Appendix A), the crack length, 8
i.e. the solution of fatigue crack growth rate equation, can be expressed as the form of asymptotic series with a t a0 t a1 t a2 t 2
where the bracket
a t a t a t 0
1
2
2
ak t k
(17)
k 0
is a series in and is made
convergent by choosing small enough. Similarly, the initial crack length can be written as
ak t0 k
a t0 a0 t0 a1 t0 a2 t0 2
(18)
k 0
By comparing Eq. (15) and Eq. (18), we have
a0 t0 a0 , a1 t0 a0 a2 t0 2 Considering that the series
,
ak t0 k 2
, k ,
,
(19)
0
(20)
is linearly independent, the
coefficient of each term in Eq. (20) equals to zero, i.e.
a2 t0 0, a3 t0 0,
, ak t0 0,
(21)
Then we have
a0 t0 a0 , a1 t0 a0 , a2 t0 a3 t0
0
(22)
These above equations are the initial values of the coefficients in Eq. (17). By using the logarithm operation, Eq. (12) can be rewritten as
ln
da t ln Q b ln a t dt
(23)
Based on Eq. (17), the derivative of a t with respect to t can be written as
da t dak t k dt dt k 0
(24)
In order to expand the above equation by Taylor series, the equation can be rewritten as
9
1 da0 t da0 t dak t k ln ln 1 dt dt k 1 dt ln Q b ln a0 t b ln 1 a01 t ak t k k 1
(25)
n
a t
By virtue of perturbation series method,
k
k
k 0
approaches to the exact
solution when n . Generally, in practical applications, remaining finite terms could be accurate enough. Ignoring the higher order terms, a t and
da t can be dt
approximately expressed as the sum of the first m+1 terms m
a t ak t k
(26)
da t m dak t k dt dt k 0
(27)
k 0
By the Taylor series expansion and remaining the first n terms, we have 1 1 j 1 m da0 t dak t k n 1 m da0 t dak t k ln 1 dt j k 1 dt dt k 1 dt j 1 n
j 1
1
da0 t dak t k dt k 1 dt
j 1
m 1 k a0 t ak t k 1
j
j 1
a0 j t ak t k k 1
j
j
m n 1 ln 1 a01 t ak t k j k 1 j 1
1 j j 1 n
For the convenience of writing, ak t and
j
j 1
dak t dt
m
m
k 0,1, 2,
j
(28) j
(29)
, m can be
abbreviated as ak , ak , respectively. According to the polynomial theorem (the detailed proof is given in Appendix B), we have
10
j
m k 2 m j ak t a1 t a2 t am t k 1 2 j! ! ! ! a1 1 a2 2 1 2 m j 1 2 m
j! a11 a22 m ! m j 1 ! 2 !
1 2
a
m m
m
amm 11 22
(30)
m m
j
m k 2 m j ak t a1 t a2 t am t k 1 2 j! ! ! ! a1 1 a2 2 1 2 m j 1 2 m
1 2
where k k 1, 2,
j! a11 a22 m ! m j 1 ! 2 !
a
m m
m
amm 11 22
(31)
m m
, m are integers and 0 k j .
Substituting Eqs. (30)-(31) into Eqs. (28)-(29) respectively, we get
m ln 1 a01ak k k 1 m n 1 1 j 1 11 2 2 m m j
m 1
1 2
m 1 !
1 ! 2 ! m !
a0
1 2 m
a
1
1
a22
amm j
m n
h j j j 1
(32)
m ln 1 a01ak k k 1 m n 1 1 j 1 11 2 2 m m j
m 1
1 2
1 ! 2 ! m !
m 1 !
a0
1 2 m
a
1
1
a22
amm j
m n
p j j j 1
(33) where
11
hj
pj
11 2
1
11 2
2
1 2
m 1!
1 ! 2 ! m !
m m j
2
1 m 1
1
1 m 1
1 2
m 1!
1 ! 2 ! m !
m m j
Substituting Eqs. (32)-(33) into Eq.
a0
a
1
a22
amm
a0
a
a22
amm
1 2 m
1 2 m
1
1
1
(25), one obtains
mn
mn
j 1
j 1
ln a0 h j j ln Q b ln a0 b p j j
(34)
By regrouping terms containing , we can get a set of perturbation equations
da0 t Qa0b t dt
(35)
da1 t Qba0b 1 t a1 t dt
(36)
da2 t b b 1 b 2 Q ba0b1 t a2 t a0 t a12 t dt 2
(37)
0 :
1 : 2 :
m :
dam t b b 1 (b 1 m 1) b 1 2 Q a0 dt 1 ! 2 ! m ! 11 2 2 m m m a1 t a2 t 1
m
t
(38)
am t m
2
As we all know, integral equation can reduce the dimension and computing time and relax the restrictions on the property of unknown function compared with differential equation. Therefore, the differential Eqs. (35)-(38) are transformed into integral equations as follows
0 : a0 t a0 t0 Q a0b d
(39)
1 : a1 t a1 t0 Q ba0b1 a1 d
(40)
t
t0
t
t0
2 : a2 t a2 t0 Q ba0b1 a2 + t t
0
12
b b 1 b2 a0 a12 d 2
(41)
t
0
11 2
m : am t am t0 Q t
b b 1
2
m m m
(b 1 m 1) b 1 2 a0 1 ! 2 ! m !
a11 a22
m
amm d
(42) By
solving
ak t k 0,1, 2,
the
above
equations,
we
can
get
the
coefficients
, m in Eq. (26) and accordingly obtain the solution of Eq. (16),
i.e. a t . 3.2 The successive approximation approach for the perturbation equations The above integral equations belong to non- linear second Volterra equations and the solutions of these equations can be obtained by employing the successive approximation approach. The general form of the non- linear second Volterra equation can be expressed as
u x y x Y x, t , u t dt a x
(43)
where y x is the free term, Y x, t , u t is the known functional with respect to
u t , is a parameter, and a is a constant. The solution by using successive approximation approach can be obtained by the following iterative formulas
u0 x y x ui x y x Y x, t , ui 1 t dt , i 1, 2, a x
(44) (45)
Then we can get the solution
u x lim ui x i
(46)
Generally, finite iteration can convergent to the solution
u x uq x
(47)
where q is a positive integer. Based on Eqs. (44)-(45), the iterative formula for integral Eq. (39) can be calculated 13
a0 0 t a0
a0i t a0 Q a0i 1 d , i 1, 2, t
t0
(48) (49)
where the integral on the right side of Eq. (49) can be solved by numerical integration methods. Similarly, the solutions of Eqs. (40)-(42) can be derived by the same way. By substituting these solutions into Eq. (26), the solution of the fatigue crack growth Eq. (16), i.e. the history of crack length versus time, can be obtained. The error in the initial crack length is considered as a modified parameter by the perturbation series method. The perturbation series solution is investigated as modification of the original solution obtained by deterministic method. The effectiveness and feasibility of the proposed method are illustrated in Section 4 with two numerical examples.
4. Examples As a demonstration of the method described in the previous section, two numerical examples are given in this section. In practical engineering, due to the influence of measurement and manufacturing errors, there generally exits a perturbation in the initial crack length, i.e. a0 . The value of the perturbation in the initial crack length can be evaluated by the results of multiple sets of measurements or the accuracy of measuring instruments. 4.1 Example 1 In this example, the perturbation series method is used to predict the fatigue crack growth process in a 7075-T761 aluminum alloy MT specimen, which is investigated by Song et al. [36]. As shown in Fig. 2, the specimen is 75mm wide, 2.4mm thick. A circular hole is located at the center of the wide specimen which initiates the fatigue crack as a stress raiser. The experiments ran under different stress ratios and temperatures and the experimental data correspond to those in [36]. The precast crack length is 2a0 8mm. However, the initial crack length in each specimen usually has deviation. Let’s consider the specimen D1 and the constants Q and b in fatigue crack 14
growth rate equation (Eq. (16)) are listed in Table 1. The initial crack length with perturbation can be expressed as
a0 4mm, a0 0.06mm
(50)
where a0 is the nominal value and a0 is the perturbation mainly originated from manufacturing error. Table 1 Parameters of the fatigue crack growth rate equation (Eq. (16)) Q
b
3.95 108
3.38
The crack growth process is predicted by the original solution (Eq. (14)), and the perturbation series solution (Eq. (26)), respectively. We take different numbers of perturbation terms in perturbation series method to calculate the crack length at a specific number of cycles N 317622 and the results are plotted in Fig. 3. We can see that the crack length change slightly when the number of perturbation terms
k 3 . So retaining the first three perturbation terms is enough to guarantee the precision of the result. The crack length a and the corresponding number of cycles N calculated by these two methods are listed in Table 2. The results based on original solution are obtained with the assumption that the initial crack length equals to the nominal value a0 . The error in initial crack length is considered by the perturbation series method. The experimental data is used as validation for the proposed method. Moreover, the relative errors of the original solution and perturbation series solution with respect to experimental data are also given in Table 2. The a-N curves obtained by the experimental data, original solution and perturbation series solution are plotted in Fig. 3.
15
Fig. 2 Schematic diagram of MT specimen of 7075-T761 aluminium alloy
Fig. 3. Effect of number of perturbation terms on crack length at N=317622 (7075-T761 aluminium alloy)
16
Fig. 4. a-N curves based on experiment, original solution and perturbation series solution (7075-T761 aluminium alloy)
Table 2 Crack growth history obtained by experiment, original solution and perturbation series solution for 7075-T761 aluminium alloy Crack length
Experiment
Original solution
Perturbation series solution
a(mm)
N(cycles)
N(cycles)
Error(%)
N(cycles)
Error(%)
4.2767
171359
176473
2.98
162810
-4.99
4.8301
246022
260621
5.93
246970
0.39
5.3835
299878
317622
5.92
303968
1.36
5.9078
336597
356025
5.77
342385
1.72
6.3447
359853
380252
5.67
366625
1.88
7.0728
388005
410079
5.69
396479
2.18
7.7427
406365
429667
5.73
416104
2.40
8.4126
421053
444274
5.52
430757
2.30
9.0534
433293
454999
5.01
441539
1.90
9.8398
443084
465102
4.97
451724
1.95
From Fig. 4, the predicted crack growth process is mainly in accordance with the experimental data. The deviation between the original solution and the perturbation 17
series solution indicates that the error in initial crack length has an effect on the crack length evolution which cannot be ignored. The results of original solution and perturbation series solution show little difference in the early stage of crack growth. As crack grows, the perturbation series solution match with the experimental data more precisely compared with the original solution. The data in Table 2 show that the relative errors of the perturbation series solution are mainly smaller than those of original solution. It means that the perturbation series method can predict the crack growth process more accurately. 4.2 Example 2 In this example, we utilize the proposed perturbation series method to predict the crack growth process in an AA7075-T651 aluminium alloy specimen [37]. The experiment used center cracked tension (CCT) fatigue crack growth specimen as shown in Fig. 5. The fatigue crack growth experiment was conducted under uniaxial tensile loading condition using servo hydraulic fatigue testing machine under constant amplitude loading, and the cyclic stress amplitude =75MPa . The constants Q and b in fatigue crack growth rate equation (Eq. (16)) are listed in Table 3. The nominal value of initial crack length is a0 3mm . The data were measured by a traveling microscope with an accuracy of 0.01mm. So we take 0.01mm as the perturbation of the initial crack length, i.e.
a0 3mm, a0 0.01mm
(51)
where a0 is the nominal value and a0 is the perturbation mainly originated from measurement error.
Fig. 5 Dimensions of CCT specimen of AA7075-T651 aluminium alloy
18
Table 3 Parameters of the fatigue crack growth rate equation (Eq. (16)) Q
b
8.46 108
3.09
In perturbation series method, the crack lengths calculated by taking different numbers of perturbation terms at N 939590 are shown in Fig. 6, which shows that the precision of the solution ignoring higher order perturbation terms ( n 3 ) is satisfactory. Therefore we retain the first three perturbation terms of the perturbation series solution for the following calculations. The a-N curves based on experimental data, original solution and perturbation series solution are plotted in Fig. 7. From Fig. 7, the a-N curve based on the perturbation series solution is closer to the experimental data for the most period of crack growth. The crack length a and the corresponding number of cycles N calculated by these two methods and the experimental data are listed in Table 4. Generally, the perturbation series solution gives smaller error in the number of cycles at a certain crack length as a comparison with original solution. As Fig. 7 shows, though the results based on original solution and perturbation series solution are very close, the perturbation series method indeed gives better prediction for crack growth process. That is to say, the error of the initial crack length should be considered to achieve high-precision fatigue crack growth prediction.
Fig. 6 Effect of number of perturbation terms on crack length at N=935950 19
(AA7075- T651 aluminium alloy)
Fig. 7 a-N curves based on experiment, original solution and perturbation series solution (AA7075-T651 aluminium alloy)
Table 4 Crack growth history obtained by experiment, original solution and perturbation series solution for AA7075-T651 aluminium alloy Crack length
Experiment
Original solution
Perturbation series solution
a(mm)
N(cycles)
N(cycles)
Error(%)
N(cycles)
Error(%)
4.47619
603581
600485
-0.51
606835
0.54
4.95238
757034
687931
-9.13
695597
-8.12
5.49205
787724
760187
-3.50
767800
-2.53
5.96825
859335
807953
-5.98
815376
-5.12
6.4762
920716
847447
-7.96
854860
-7.15
6.98415
930947
878421
-5.64
885987
-4.83
7.49205
941177
903146
-4.04
910556
-3.25
7.96825
961637
922053
-4.12
929460
-3.35
In each of the above examples, the feasibility and effectiveness of the proposed method are validated by comparing the fatigue crack growth process obtained by the proposed method with the experimental results for a typical test specimen. The fatigue 20
crack growth of the specimen is illustrated by an a-N curve as depicted in Figs. 4 and 7. The prediction of the a-N curve is carried out by the proposed method for this specimen, which is shown to provide a more accurate solution closer to the experimental results. Certainly, for the experimental tests involving multiple samples, due to the uncertainties existing in material, loading, initial crack length and geometry, the a-N curve would be an interval containing upper and lower bounds. For this problem, the analysis for fatigue crack growth can be combined with the interval analysis methods to predict the interval of a-N curve.
5. Conclusions In this paper, a perturbation series method is proposed to predict the fatigue crack growth process of engineering materials and structural components. The error in initial crack length, as one of the main factors influencing the crack growth process, is investigated in this paper. For the sake of errors of measurement and manufacturing, the actual initial crack length is expressed as the sum of nominal value and perturbation by introducing a small parameter. The solution of the fatigue crack growth equation, i.e. crack length evolution, can be expressed in the form of asymptotic series with the small parameter. The logarithmic form of the fatigue crack growth rate equation is exploited. Based on the Taylor series expansion and the perturbation series method, a series of perturbation equations which serve as the modified fatigue crack growth equations are obtained and the equations can be solved by the successive approximation approach. The perturbation series solution can be used to predict the fatigue crack growth process and the fatigue life of the structural component when the critical crack length is given. Two examples, related to two fatigue crack growth experiments respectively, are given to investigate the accuracy and effectiveness of the proposed method. The results show that the a-N curves obtained by the perturbation series method basically tallies with the experiment data. Compared with the original solution of fatigue crack growth equation, the perturbation series solution can predict the crack growth process more reliably and accurately. The presented perturbation series method is a correction 21
method for crack growth problem with perturbation parameters in initial crack length and it is very significant for the design and use of engineering structures. The proposed method in this paper serves as a modified numerical model that can be used to more accurately evaluate the fatigue crack growth process by properly accounting for the error in initial crack length. Besides, the fatigue crack growth problem with large uncertainty existing in structures, which is not considered in this paper, is also a topic worthy to be studied. Combined with uncertainty analysis methods, the perturbation series theory can be further extended to the fatigue crack growth problem with uncertainty. For the problems with large uncertainty, the high order perturbation method should be implemented to improve accuracy, in which the high order terms in perturbation series expansion need to be calculated by recurrence. It is worth to mention that it is not necessary to calculate the high order sensitivities in the high order perturbation method. The extensions of the proposed method in uncertainty analysis will be investigated in our future research.
Acknowledgements This work was supported by the Defense Industrial Technology Development Program (No. JCKY2016204B101, No. JCKY2017601B001); National Nature Science Foundation of the P.R. China (No. 11772026, No. 11872089).
Appendix A. Perturbation series theory Perturbation method provides the most versatile tool available in nonlinear analysis of engineering problems, and they are constantly being developed and applied to ever more complex problems. In the perturbation method for deterministic structure, the governing equation for physical problem is expressed as an equatio n with a small parameter
L u, x, 0 22
(A1)
where L is the general linear operator, u u x, is the solution of the problem, x is the independent variable, and is a small scalar quantity, which is inherent in Eq. (A1) or introduced factitiously. Since u is the function of the independent x and the small parameter , the solution u can be represented as an asymptotic series in
u x, u0 x u1 x
un x n
(A2)
where the coefficient ui x is independent of . Substituting Eq. (A2) into Eq. (A1) and regrouping terms containing , we get
L0u0 h L0u1 L1u0 L0u2 L1u1 L2u0 2 where L0 , L1 , L2 ,
0
(A3)
are linear operators in space U and h is a real function of x,
which can be determined by specific problems. Since Eq. (A3) are valid for all values of and the sequence , 2 , 3 ,
are
linearly independent, all the coefficients of are equal to zero. Then a set of recursive equations about ui x are obtained L0u0 h
L0u1 L1u0 L0u2 L1u1 L2u0
(A4)
By solving the above equations, we can get the coefficients ui x i 0,1, 2,
.
According to Eq. (A2), an approximate solution of Eq. (A1) can be obtained.
Appendix B. Proof of the polynomial theorem Here, we give the detailed proof of the polynomial theorem (Ladd, 1878). The polynomial theorem: For the real numbers xi i 1, 2,
, k and the positive
integer n, we have
x1 x2
xk n
1
2
n! x11 x22 1 ! 2 ! k ! k n
1
xkk
(B1)
where 1 , 2 ,
, k are nonnegative integers.
Proof: Let us write down the sum of k quantities n times:
x1 x2 x3 x4 x1 x2 x3 x4 x1 x2 x3 x4
(B2)
It is obvious that the terms of the product will contain every possible combination of one or more of the quantities with different exponents in such a manner that the sum of the exponents is equal to n. The coefficient of each term stands for the number of ways in which that term can be formed. x1n , x2n and x3n can be formed in the only one way, hence their coefficient is 1. The term x1n 1 x2 is formed by taking x2 of any row and multiplying it by x1 of every other row. But since any one of the n x2 ’s can be taken, there should be n terms x1n 1 x2 , or the coefficient of x1n 1 x2 is n; and the coefficient should be the same for all terms of the same form x2n 1 x4 , x1n1 x3 ,
. To
obtain the coefficient of x1n 2 x22 , the number of ways in which two x2 ’s can be selected out of n x2 ’s is
x3 ’s in n2
1 n n 1 . For the term x1n5 x23 x32 , we can select the two 2
1 n n 1 ways. After taking a certain combination of 2 x3 ’s, we have 2
x2 ’s
out
of
n 2 n 3 n 2 3 1 3!
which
to
select
3
x2 ’s.
There
could
be
ways for the selection, and only one combination of
the n 5 x1 ’s remains. Therefore, the coefficient of x1n5 x23 x32 is
n n 1 n 2 n 3 n 2 3 1 2! 3!
(B3)
Generally, for the coefficient of x1n2 3 4 x22 x33 x44 , we have the product of the combinations of n things 4 at a time by the combinations of n 4 things 3 at 2
a time by the combinations of n 4 3 things 2 at a time, that is n n 1
n 4 1 n 4 n 4 1 n 3 4 1
4 !
3 !
(B4)
n 4 3 n 4 3 1 n 3 4 2 1 2 !
or
n n 1
n 2 3 4 1
(B5)
2 ! 3 ! 4 !
Let n 2 3 4 1 and multiply denominator and numerator of Eq. (B5) by
1 ! and we get n n 1
1 1 1 1 1 1 2
1
(B6)
1 ! 2 ! 3 ! 4 ! or n! 1 ! 2 ! 3 ! 4 !
(B7)
for the coefficient of x11 x22 x33 x44 , in which 1 2 3 4 n . From the manner in which this coefficient is obtained it is evident that all the terms of the form x11 x32 x43 x24 , x21 x32 x43 x54 , and so on, have the same coefficient. Applying the same way for all terms, then we may write
x1 x2
n! n xk x11 x22 1 ! 2 ! k !
xkk
(B8)
where the within the brackets stands the sum of the terms formed by all possible combinations of the k quantities with a given set of exponents, and the without the brackets stands for the sum of all the sets of terms obtained by giving to the exponents all possible values such that
1 2
k n
m Then Eq. (B8) can be written as Eq. (B1). Based on Eq. (B1), ak t k k 1 3
(B9) j
and
m k ak t k 1
j
can take the form of polynomial, see Eq. (30) and Eq. (31).
References [1] P.C. Paris, F. Erdogan, A critical analysis of crack propagation laws, J. Basic Eng. 85 (1963) 528-533. [2] N. Pugno, M. Ciavarella, P. Cornetti, A. Carpinteri, A generalized Paris’ law for fatigue crack growth, J. Mech. Phys. Solids 54 (2006) 1333-1349. [3] Y.C. Lam, K.S. Lian, The effect of residual stress and its redistribution of fatigue crack growth, Theor. Appl. Fract. Mec. 12 (1989) 59-66. [4] J.H. Kim, S.B. Lee, S.G. Hong, Fatigue crack growth behavior of Al7050-T7451 attachment lugs under flight spectrum variation, Theor. Appl. Fract. Mec. 40 (2003) 135-144. [5] P.C. Paris, M. P. Gomez, W.E. Anderson, A rational analytic theory of fatigue, The Trend in Engineering 13 (1961) 9-14. [7] R.G. Forman, V.E. Kearney, R.M. Engle, Numerical analysis of crack propagation in cyclic-loaded structures, J. Basic Eng. 89 (1967) 459-463. [8] E.K. Walker, The effect of stress ratio during crack propagation and fatigue for 2024-t3 and 7075-t6 aluminum, ASTM International (1970). [9] W. Elber, The significance of fatigue crack closure. In: Damage tolerance in aircraft structures, ASTM Spec. Techn. Publ. No. 486 (1971) 230–242. [10] R.J. Donahue, H.M. Clark, P. Atanmo, R. Kumble, A.J. Mcevily, Crack opening displacement and the rate of fatigue crack growth, Int. J. Fracture 8 (1972) 209-219. [11] P.H. Wirsching, Fatigue reliability for offshore structures, Struct. Eng-ASCE 110 (1984) 2340-2356. [12] L. Wang, J. X. Liang, Y.W. Yang, Y.N. Zheng, Time-dependent reliability assessment of fatigue crack growth modeling based on perturbation series expansions and interval mathematics, Theor. Appl. Fract. Mec. 95 (2018) 4
104-118. [13] L. Wang, X.J. Wang, H. Su, G.P. Lin, Reliability estimation of fatigue crack growth prediction via limited measured data, Int. J. Mech. Sci. 121 (2017) 44-57. [14] Z. Lv, Z.P. Qiu, An iteration method for predicting static response of nonlinear structural systems with non-deterministic parameters, Appl. Math. Model. 68 (2019) 48-65. [15] L. Wang, C. Xiong, X.J. Wang, M.H. Xu, Y.L. Li, A dimension-wise method and its improvement for multidisciplinary interval uncertainty analysis, Appl. Math. Model. 59 (2018) 680-695 [16]. Y. Liu and S. Mahadevan, Probabilistic fatigue life prediction using an equivalent initial flaw size distribution, Int. J. Fatigue 31 (2009) 476-487. [17]. J. Dominguez, J. Zapatero, J. Pascual, Effect of load histories on scatter of fatigue crack growth in aluminum alloy 2024-T351, Eng. Fract. Mech. 56 (1997) 65-76. [18]. W.A. Grell, P.J. Laz, Probabilistic fatigue life prediction using AFGROW and accounting for material variability, Int. J. Fatigue 32 (2010) 1042-1049. [19] A. Merati, G. Eastaugh, Determination of fatigue related discontinuity state of 7000 series of aerospace aluminum alloys, Eng. Fail. Anal. 14 (2007) 673-685. [20] B.R. Krasnowski, K.M. Rotenberger, W.W. Spence, A damage tolerance method for helicopter dynamic components, J. Am. Helicopter Soc. 36 (1991) 52-60. [21] H.F. Bueckner, A novel principle for the computation of stress intensity factors, Z. augew. math. Mech. 50 (1970) 529-546. [22] J.R. Rice, First-order variation in elastic fields due to variation in location of a planar crack front, J Appl. Mech. 52 (1985) 571-579. [23] Z.P.
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[25] H.H. Khodaparast, J.E. Mottershead, M.I. Friswell, Perturbation methods for the estimation of parameter variability in stochastic model updating, Mech. Syst. Signal Process. 22 (2008) 1751-1773. [26] Y. Sumi, A second order perturbation solution of a non-collinear crack and its application to crack path prediction of brittle fracture in weldme nt, Nav. Arch. Ocean 28 (1990) 143-156. [27] L.V. Stepanova, S.A. Igonin, Asymptotics of the near-crack-tip stress field of a growing fatigue crack in damaged materials: Numerical experiment and analytical solution, Numer. Anal. Appl. 8 (2015) 168-181. [28] L. Stepanova, S. Igonin, Higher-order asymptotic solution for the fatigue crack growth problem based on continuum damage mechanics, Procedia Mater. Sci. 3 (2014) 421-427. [29] M. Rezaee, R. Hassannejad, Free vibration analysis of simply supported beam with breathing crack using perturbation method, Acta Mech. Solida Sin. 23 (2010) 459-470. [30] L. Wang, X.J. Wang, H. Su, G.P. Lin, Reliability estimation of fatigue crack growth prediction via limited measured data, Int. J. Mech. Sci. 121 (2017) 44-57. [31] X.Y. Long, C. Jiang, K. Liu, X. Han, W. Gao, B.C. Li, An interval analysis method for fatigue crack growth life prediction with uncertainty, Comput. Struct. 210 (2018) 1–11. [32] G.R. Irwin, Fracture, In: Handbuch der Physik, Bd. IV, Springer-Verlag, Berlin, (1958). [33] Y.D. Li, Theoretical and applied fracture mechanics, Beijing: The Publishing House of Ordnance Industry (2005) 71-79. [34] Y. Xiang, Z. Lu, Y. Liu, Crack growth-based fatigue life prediction using an equivalent initial flaw model. Part I: Uniaxial loading, Int. J. Fatigue 32 (2010) 341-349 [35] V. Balasubramanian, V Ravisankar, G.M. Reddy, Influences of pulsed current welding and post weld aging treatment on fatigue crack growth behaviour of AA7075 aluminum alloy joints, Int. J. Fatigue 30 (2008) 341-349. 6
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7
S0167-8442(18)30532-9 https://doi.org/10.1016/j.tafmec.2019.102239 102239 TAFMEC 102239
To appear in:
Theoretical and Applied Fracture Mechanics
Received Date: Revised Date: Accepted Date:
12 November 2018 18 March 2019 29 April 2019
Please cite this article as: Z. Qiu, J. Zhu, The Perturbation Series Method Based on the Logarithm Equation for Fatigue Crack Growth Prediction, Theoretical and Applied Fracture Mechanics (2019), doi: https://doi.org/10.1016/ j.tafmec.2019.102239
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The Perturbation Series Method Based on the Logarithm Equation for Fatigue Crack Growth Prediction
Zhiping Qiu*, Jingjing Zhu Institute of Solid Mechanics, School of Aeronautic Science and Engineering, Beihang University, Beijing 100083, P. R. China *Corresponding author:
Zhiping Qiu Institute of Solid Mechanics School of Aeronautic Science and Engineering Beihang University Beijing 100083, P.R. China Fax: 86-010-82339628 Email: [email protected]
The Perturbation Series Method Based on the Logarithm Equation for Fatigue Crack Growth Prediction
Abstract This paper gives the perturbation series method for the fatigue crack growth prediction accounting for the error in initial crack length. The proposed method utilizes the logarithm form of the crack growth equation and the solution of it is expressed in the form of asymptotic series by introducing a small parameter. The initial crack length is considered as a sum of the nominal value and the perturbation which mainly originates from manufacturing and measurement errors. Based on the Taylor series expansion and perturbation series method, a series of perturbation equations to predict the fatigue crack growth history are obtained. Then these equations can be solved by the successive approximation approach and the a-N curve which describes the relationship between crack length and number of load cycles can be obtained. Two examples with respect to fatigue crack growth experiments are given to verify the accuracy and validation of the presented method. The results show that the perturbation series method is closer to the experimental data compared with the original solution of crack growth equation and it can provide a more accurate prediction for the fatigue crack growth process.
Keywords: Crack growth; Logarithm equation; Initial crack length; Taylor series expansion; Perturbation series method
1. Introduction Fatigue failure is a process in which damage accumulates under cyclic loading that may be well below the yield point. Fatigue fracture is particularly serious for aircrafts, ships, bridges and other engineering structures under dynamic loads, and this problem continues to prevail in spite of the exhaustive amount of past research. Through the 1
investigation of the fatigue crack growth process, information is obtained which can be utilized to predict the crack length history and estimate the fatigue life of structures. Therefore, the research on fatigue crack growth problem has attracted continual attention [1-4]. From a fracture mechanics point of view, the fatigue damage of a structural component subjected to dynamic loads can be measured by the size of the dominant crack, and failure will occur when the crack reaches a critical magnitude. In order to describe the process of fatigue fracture, several crack-growth laws taking different forms have been presented in the literatures. Paris et al. [5] suggested an approach that related the crack growth rate ( da / dN ) to the maximum stress intensity factor K max in 1961. Liu [6] subsequently implied that the crack growth rate was a function of the cyclic stress range S . In 1963, Paris and Erdogan [1] proposed the fatigue crack growth rate equation based on experiment, which described the relationship between the crack growth rate and the stress intensity factor range under a fatigue stress regime. This is the celebrated Paris law, which is the most popular fatigue crack growth model applied in materials science and fracture mechanics. Over the years, the Paris formula has continued to be modified to account for variety of observations, including the effect of stress ratio and the maximum stress intensity factor [7-8], the crack closure phenomenon [9] and the threshold stress intensity factor amplitude [10]. Generally, the typical fatigue growth behavior of cracks could be described by the curve relating fatigue crack growth rate ( da / dN ) to stress intensity factor range K , as shown in Fig.1. The process of fatigue crack growth can be divided into three stages, i.e. crack initiation (stage I), subcritical crack growth (stage II) and rapid crack growth (stage III). Thereinto, the stage II follows Paris law and accounts for the main part of fatigue life.
2
Fig.1. A schematic of the typical fatigue growth behavior of cracks. Nowadays, the influence of parameters fluctuation on the safety of structures has aroused more and more attention [11-15]. In practical engineering application, fatigue crack growth in engineering materials and structural components is generally a complex process, in which the wide scattering exists even for identical specimens. This may be caused by the uncertainties of loadings, environmental conditions, material characteristics, geometry of the structures and crack initiation, which contribute to the non-deterministic behavior of fatigue crack growth. Many stochastic fatigue crack growth approaches have been proposed to consider uncertainties for accurate fatigue life predictions [16-18]. The fatigue crack growth models based on fracture mechanics, which describe how the crack length increases under cyc lic loading, are widely used to predict the fatigue life and estimate the reliability of structural components. In order to use the fatigue crack growth models, the prerequisite is to determine the initial crack length. The initial crack length has direct effect on the crack growth process. Initial crack length of practical engineering structures can be obtained by several ways. One practice is to use an empirically assumed crack length, such as 0.25–1 mm for metals [19]. An alternative way is to use the results from nondestructive inspection (NDI) [20]. Due to the limit of detection level, measurement results usually have errors. In the experimental investigation of the fatigue crack growth process, the initial crack size of specimen with precast crack also consists of manufacturing errors. Tremendous dispersions on fatigue life of 3
structures could be caused by these errors under realistic working conditions and the life prediction will be unreliable. Taking the error in the initial crack size into consideration, this paper will present a perturbation series method for the fatigue crack growth prediction. Perturbation series method is a powerful tool to analyze the effect of a small perturbation of the parameters on the response of structures. The perturbation method, which is based on the pioneer works of Bueckneer [21] and Rice [22], has been widely used in mechanics problems, such as structural eigenvalue problems [23], structural response analysis [24] and model updating [25]. For fatigue fracture problem, the perturbation technique also has some applications. Stress intensity factors are analyzed for a non-collinear crack by using a second order perturbation method [26]. Stepanova and Igonin [27-28] utilized the perturbation technique to study the growth of fatigue near-crack-tip fields in a damaged material. Mousa and Reza [29] analyzed the free vibration of simply supported beam with breathing crack by using perturbation method. However, there has been rare research about the effect on the crack length evolution caused by the error in initial crack length. Up to now, a few researches have studied the effect of the uncertainties including material properties, applied loadings, geometry and initial conditions on fatigue crack growth process in the case of inadequate data. Wang et al. [12, 30] investigated the reliability estimation of fatigue crack growth prediction based on the interval mathematics. Long et al. [31] did good works in the interval analysis method for fatigue crack growth life prediction considering uncertainties in propagation rate constants, applied loadings, fracture toughness as well as initial crack length. The above works quantified the uncertainties of the system in the framework of non-probabilistic analysis. Based on the fatigue crack growth model, the upper and lower bounds of fatigue life are evaluated by employing the interval analysis methods. This paper focuses on the perturbation series method for the fatigue crack growth prediction with considering the error in initial crack length (mainly owing to measurement errors). The perturbation in the initial crack length is quantified by introducing a small correction parameter and the modified crack growth equations are 4
derived based on the polynomial theorem. It provides a modified model to solve the deterministic fatigue crack growth problem, which aims to realize a more reliable and precise prediction of fatigue crack growth evolution. In the perturbation method, the error in initial crack length is quantified as a perturbation by introducing a small parameter and the crack length is expanded as asymptotic series with respect to . The fatigue crack growth equation is established in logarithm form and a series of perturbation equations are obtained by Taylor series expansion and perturbation series method. The results show that the proposed method can better predict the crack growth process and estimate the fatigue life of structure component with cracks more accurately. The purpose of this paper is to develop a modified method for the prediction of fatigue crack growth process with considering the error in initial crack length. The paper is organized as follows. In Section 2, the fatigue crack growth rate equation based on fracture mechanics is described. In Section 3, based on the logarithm form of crack growth rate equation, the Taylor series expansion and perturbation series method are used to derived the perturbation equations. Then the perturbation series solution can be obtained by the successive approximation approach. In Section 4, two examples are given to demonstrate the accuracy and effectiveness of the proposed method. In the final section, we draw the conclusions of the current research.
2. Basic fatigue crack growth equation In practical engineering applications, the stage II of the fatigue crack growth process is taken as the main part of the fatigue crack growth life. The crack growth rate in stage II can be described by the Paris law, which relates the stress intensity factor range to subcritical crack growth under a fatigue stress regime. The basic formula reads da m C K dt
(1)
where da / dt is the crack growth rate, t stands for the time or the stress cycle number, K is the stress intensity factor range, C and m are material constants which relate to environment, frequency, temperature, stress ratio and can be 5
determined experimentally. The stress intensity factor range K is defined as
K Kmax Kmin
(2)
where K max and K min are the maximum and minimum stress intensity factors, respectively. In terms of linear elastic fracture mechanics, the stress intensity factor can be calculated by the following formula [32]
K f a,W ,
a
(3)
where is the cyclic stress applied on the structure, a is the crack length,
f a, W ,
is the geometry form factor which is generally a function of the
geometrical dimensions of structure (e.g. the width of the plate W) and the crack length a. By virtue of Eq. (3), K can be rewritten as
K f a,W ,
max min a f a,W , a
(4)
where max and min are the maximum cyclic stress and minimum cyclic stress, respectively, and max min is the cyclic stress amplitude. Substituting Eq. (4) into Eq. (1), we get da m C K C f a,W , dt
a
m
(5)
Due to the limits of the Paris law, researchers have proposed some modified forms of the crack growth rate equation. Considering the effect of the stress ratio R and the peak load K max , Walker [8] presented the following modified formula n da m C 1 R K max dt
(6)
where the material constants C, m, n are obtained from the experimental data at different stress ratios, the stress ratio R max / min . Based on the relationship between R and K , Eq. (6) can be rewritten as da n m CK max K dt
(7)
Considering the fatigue crack closure phenomenon existing in the crack growth test, the fatigue crack closure theory proposed by W. Elber [9] can be expressed as 6
da C Keff dt
n
U mC K
m
(8)
where U max top / is the crack closure parameter and top is the opening stress. The aforementioned crack growth rate equations can be simplified to the general form as follows da Q a,W , , top , R, K max , dt
a
b
(9)
where Q is a parameter related to material, geometry, loads and environment, and b is a constant. In engineering practice, the size of fatigue crack is generally very small relative to the size of structure. The fatigue crack growth is only affected by the stress state, material and environment around the crack. Therefore, for the complex structures possessing arbitrary geometries, the region containing crack can be regarded as a plate model with crack. Based on the stress analysis of the structure, the applied loading on the plate model can be acquired by extracting the stress field around the crack from the stress results of whole structure. For an infinite plate with center crack, the stress intensity factor can be calculated by
K a
(10)
For the finite plate with crack, the stress intensity factor can be obtained by introducing a geometry correction factor Y [33]
K Y a
(11)
Hence, the crack can be assumed as a center or a single edge crack in a plate model. Under this assumption, the parameter Q can be treated as a constant, and the crack growth rate equation can be rewritten as da Qa b dt
(12)
where Q and b are treated as constants for specific structure and load condition [34,35]. In general, the values of these two parameters are measured by experiments. By measuring the data of crack length a and corresponding time or stress cycle number t, the parameters Q and b can be obtained by curve fitting. 7
When the initial crack length is given, the crack length evolution process can be predicted by the following initial value problem
da t b Q a t dt a t a 0 0
(13)
where a0 is the initial crack length.
3. The perturbation series method for the fatigue crack growth equation 3.1 The pe rturbation series expansion of the fatigue crack growth equation Traditionally, the fatigue crack growth process, i.e. the history of crack length versus time, can be predicted by solving Eq. (13) and the solution can be expressed as
a0eQt , a t 1 1b 1 a0 Q 1 b t b ,
b 1 b 1
(14)
Eq. (14) is defined as the original solution in this paper. However, as we discussed above, the initial crack length is generally affected by some disturbance mainly originated from the measurement and manufacturing errors. As the disturbance is usually a small quantity, the perturbation series method can be applied to obtain the approximate solution of Eq. (12), and the initial condition can be expressed by introducing a small parameter
a t0 a0 a0
(15)
where is a scalar quantity much less than unity, a0 is the nominal value of initial crack length, and a0 is the perturbation which stands for the error in initial crack length. Therefore, Eq. (13) can be rewritten as
da t b Q a t dt a t a a 0 0 0
(16)
By resorting to the perturbation series method (see in Appendix A), the crack length, 8
i.e. the solution of fatigue crack growth rate equation, can be expressed as the form of asymptotic series with a t a0 t a1 t a2 t 2
where the bracket
a t a t a t 0
1
2
2
ak t k
(17)
k 0
is a series in and is made
convergent by choosing small enough. Similarly, the initial crack length can be written as
ak t0 k
a t0 a0 t0 a1 t0 a2 t0 2
(18)
k 0
By comparing Eq. (15) and Eq. (18), we have
a0 t0 a0 , a1 t0 a0 a2 t0 2 Considering that the series
,
ak t0 k 2
, k ,
,
(19)
0
(20)
is linearly independent, the
coefficient of each term in Eq. (20) equals to zero, i.e.
a2 t0 0, a3 t0 0,
, ak t0 0,
(21)
Then we have
a0 t0 a0 , a1 t0 a0 , a2 t0 a3 t0
0
(22)
These above equations are the initial values of the coefficients in Eq. (17). By using the logarithm operation, Eq. (12) can be rewritten as
ln
da t ln Q b ln a t dt
(23)
Based on Eq. (17), the derivative of a t with respect to t can be written as
da t dak t k dt dt k 0
(24)
In order to expand the above equation by Taylor series, the equation can be rewritten as
9
1 da0 t da0 t dak t k ln ln 1 dt dt k 1 dt ln Q b ln a0 t b ln 1 a01 t ak t k k 1
(25)
n
a t
By virtue of perturbation series method,
k
k
k 0
approaches to the exact
solution when n . Generally, in practical applications, remaining finite terms could be accurate enough. Ignoring the higher order terms, a t and
da t can be dt
approximately expressed as the sum of the first m+1 terms m
a t ak t k
(26)
da t m dak t k dt dt k 0
(27)
k 0
By the Taylor series expansion and remaining the first n terms, we have 1 1 j 1 m da0 t dak t k n 1 m da0 t dak t k ln 1 dt j k 1 dt dt k 1 dt j 1 n
j 1
1
da0 t dak t k dt k 1 dt
j 1
m 1 k a0 t ak t k 1
j
j 1
a0 j t ak t k k 1
j
j
m n 1 ln 1 a01 t ak t k j k 1 j 1
1 j j 1 n
For the convenience of writing, ak t and
j
j 1
dak t dt
m
m
k 0,1, 2,
j
(28) j
(29)
, m can be
abbreviated as ak , ak , respectively. According to the polynomial theorem (the detailed proof is given in Appendix B), we have
10
j
m k 2 m j ak t a1 t a2 t am t k 1 2 j! ! ! ! a1 1 a2 2 1 2 m j 1 2 m
j! a11 a22 m ! m j 1 ! 2 !
1 2
a
m m
m
amm 11 22
(30)
m m
j
m k 2 m j ak t a1 t a2 t am t k 1 2 j! ! ! ! a1 1 a2 2 1 2 m j 1 2 m
1 2
where k k 1, 2,
j! a11 a22 m ! m j 1 ! 2 !
a
m m
m
amm 11 22
(31)
m m
, m are integers and 0 k j .
Substituting Eqs. (30)-(31) into Eqs. (28)-(29) respectively, we get
m ln 1 a01ak k k 1 m n 1 1 j 1 11 2 2 m m j
m 1
1 2
m 1 !
1 ! 2 ! m !
a0
1 2 m
a
1
1
a22
amm j
m n
h j j j 1
(32)
m ln 1 a01ak k k 1 m n 1 1 j 1 11 2 2 m m j
m 1
1 2
1 ! 2 ! m !
m 1 !
a0
1 2 m
a
1
1
a22
amm j
m n
p j j j 1
(33) where
11
hj
pj
11 2
1
11 2
2
1 2
m 1!
1 ! 2 ! m !
m m j
2
1 m 1
1
1 m 1
1 2
m 1!
1 ! 2 ! m !
m m j
Substituting Eqs. (32)-(33) into Eq.
a0
a
1
a22
amm
a0
a
a22
amm
1 2 m
1 2 m
1
1
1
(25), one obtains
mn
mn
j 1
j 1
ln a0 h j j ln Q b ln a0 b p j j
(34)
By regrouping terms containing , we can get a set of perturbation equations
da0 t Qa0b t dt
(35)
da1 t Qba0b 1 t a1 t dt
(36)
da2 t b b 1 b 2 Q ba0b1 t a2 t a0 t a12 t dt 2
(37)
0 :
1 : 2 :
m :
dam t b b 1 (b 1 m 1) b 1 2 Q a0 dt 1 ! 2 ! m ! 11 2 2 m m m a1 t a2 t 1
m
t
(38)
am t m
2
As we all know, integral equation can reduce the dimension and computing time and relax the restrictions on the property of unknown function compared with differential equation. Therefore, the differential Eqs. (35)-(38) are transformed into integral equations as follows
0 : a0 t a0 t0 Q a0b d
(39)
1 : a1 t a1 t0 Q ba0b1 a1 d
(40)
t
t0
t
t0
2 : a2 t a2 t0 Q ba0b1 a2 + t t
0
12
b b 1 b2 a0 a12 d 2
(41)
t
0
11 2
m : am t am t0 Q t
b b 1
2
m m m
(b 1 m 1) b 1 2 a0 1 ! 2 ! m !
a11 a22
m
amm d
(42) By
solving
ak t k 0,1, 2,
the
above
equations,
we
can
get
the
coefficients
, m in Eq. (26) and accordingly obtain the solution of Eq. (16),
i.e. a t . 3.2 The successive approximation approach for the perturbation equations The above integral equations belong to non- linear second Volterra equations and the solutions of these equations can be obtained by employing the successive approximation approach. The general form of the non- linear second Volterra equation can be expressed as
u x y x Y x, t , u t dt a x
(43)
where y x is the free term, Y x, t , u t is the known functional with respect to
u t , is a parameter, and a is a constant. The solution by using successive approximation approach can be obtained by the following iterative formulas
u0 x y x ui x y x Y x, t , ui 1 t dt , i 1, 2, a x
(44) (45)
Then we can get the solution
u x lim ui x i
(46)
Generally, finite iteration can convergent to the solution
u x uq x
(47)
where q is a positive integer. Based on Eqs. (44)-(45), the iterative formula for integral Eq. (39) can be calculated 13
a0 0 t a0
a0i t a0 Q a0i 1 d , i 1, 2, t
t0
(48) (49)
where the integral on the right side of Eq. (49) can be solved by numerical integration methods. Similarly, the solutions of Eqs. (40)-(42) can be derived by the same way. By substituting these solutions into Eq. (26), the solution of the fatigue crack growth Eq. (16), i.e. the history of crack length versus time, can be obtained. The error in the initial crack length is considered as a modified parameter by the perturbation series method. The perturbation series solution is investigated as modification of the original solution obtained by deterministic method. The effectiveness and feasibility of the proposed method are illustrated in Section 4 with two numerical examples.
4. Examples As a demonstration of the method described in the previous section, two numerical examples are given in this section. In practical engineering, due to the influence of measurement and manufacturing errors, there generally exits a perturbation in the initial crack length, i.e. a0 . The value of the perturbation in the initial crack length can be evaluated by the results of multiple sets of measurements or the accuracy of measuring instruments. 4.1 Example 1 In this example, the perturbation series method is used to predict the fatigue crack growth process in a 7075-T761 aluminum alloy MT specimen, which is investigated by Song et al. [36]. As shown in Fig. 2, the specimen is 75mm wide, 2.4mm thick. A circular hole is located at the center of the wide specimen which initiates the fatigue crack as a stress raiser. The experiments ran under different stress ratios and temperatures and the experimental data correspond to those in [36]. The precast crack length is 2a0 8mm. However, the initial crack length in each specimen usually has deviation. Let’s consider the specimen D1 and the constants Q and b in fatigue crack 14
growth rate equation (Eq. (16)) are listed in Table 1. The initial crack length with perturbation can be expressed as
a0 4mm, a0 0.06mm
(50)
where a0 is the nominal value and a0 is the perturbation mainly originated from manufacturing error. Table 1 Parameters of the fatigue crack growth rate equation (Eq. (16)) Q
b
3.95 108
3.38
The crack growth process is predicted by the original solution (Eq. (14)), and the perturbation series solution (Eq. (26)), respectively. We take different numbers of perturbation terms in perturbation series method to calculate the crack length at a specific number of cycles N 317622 and the results are plotted in Fig. 3. We can see that the crack length change slightly when the number of perturbation terms
k 3 . So retaining the first three perturbation terms is enough to guarantee the precision of the result. The crack length a and the corresponding number of cycles N calculated by these two methods are listed in Table 2. The results based on original solution are obtained with the assumption that the initial crack length equals to the nominal value a0 . The error in initial crack length is considered by the perturbation series method. The experimental data is used as validation for the proposed method. Moreover, the relative errors of the original solution and perturbation series solution with respect to experimental data are also given in Table 2. The a-N curves obtained by the experimental data, original solution and perturbation series solution are plotted in Fig. 3.
15
Fig. 2 Schematic diagram of MT specimen of 7075-T761 aluminium alloy
Fig. 3. Effect of number of perturbation terms on crack length at N=317622 (7075-T761 aluminium alloy)
16
Fig. 4. a-N curves based on experiment, original solution and perturbation series solution (7075-T761 aluminium alloy)
Table 2 Crack growth history obtained by experiment, original solution and perturbation series solution for 7075-T761 aluminium alloy Crack length
Experiment
Original solution
Perturbation series solution
a(mm)
N(cycles)
N(cycles)
Error(%)
N(cycles)
Error(%)
4.2767
171359
176473
2.98
162810
-4.99
4.8301
246022
260621
5.93
246970
0.39
5.3835
299878
317622
5.92
303968
1.36
5.9078
336597
356025
5.77
342385
1.72
6.3447
359853
380252
5.67
366625
1.88
7.0728
388005
410079
5.69
396479
2.18
7.7427
406365
429667
5.73
416104
2.40
8.4126
421053
444274
5.52
430757
2.30
9.0534
433293
454999
5.01
441539
1.90
9.8398
443084
465102
4.97
451724
1.95
From Fig. 4, the predicted crack growth process is mainly in accordance with the experimental data. The deviation between the original solution and the perturbation 17
series solution indicates that the error in initial crack length has an effect on the crack length evolution which cannot be ignored. The results of original solution and perturbation series solution show little difference in the early stage of crack growth. As crack grows, the perturbation series solution match with the experimental data more precisely compared with the original solution. The data in Table 2 show that the relative errors of the perturbation series solution are mainly smaller than those of original solution. It means that the perturbation series method can predict the crack growth process more accurately. 4.2 Example 2 In this example, we utilize the proposed perturbation series method to predict the crack growth process in an AA7075-T651 aluminium alloy specimen [37]. The experiment used center cracked tension (CCT) fatigue crack growth specimen as shown in Fig. 5. The fatigue crack growth experiment was conducted under uniaxial tensile loading condition using servo hydraulic fatigue testing machine under constant amplitude loading, and the cyclic stress amplitude =75MPa . The constants Q and b in fatigue crack growth rate equation (Eq. (16)) are listed in Table 3. The nominal value of initial crack length is a0 3mm . The data were measured by a traveling microscope with an accuracy of 0.01mm. So we take 0.01mm as the perturbation of the initial crack length, i.e.
a0 3mm, a0 0.01mm
(51)
where a0 is the nominal value and a0 is the perturbation mainly originated from measurement error.
Fig. 5 Dimensions of CCT specimen of AA7075-T651 aluminium alloy
18
Table 3 Parameters of the fatigue crack growth rate equation (Eq. (16)) Q
b
8.46 108
3.09
In perturbation series method, the crack lengths calculated by taking different numbers of perturbation terms at N 939590 are shown in Fig. 6, which shows that the precision of the solution ignoring higher order perturbation terms ( n 3 ) is satisfactory. Therefore we retain the first three perturbation terms of the perturbation series solution for the following calculations. The a-N curves based on experimental data, original solution and perturbation series solution are plotted in Fig. 7. From Fig. 7, the a-N curve based on the perturbation series solution is closer to the experimental data for the most period of crack growth. The crack length a and the corresponding number of cycles N calculated by these two methods and the experimental data are listed in Table 4. Generally, the perturbation series solution gives smaller error in the number of cycles at a certain crack length as a comparison with original solution. As Fig. 7 shows, though the results based on original solution and perturbation series solution are very close, the perturbation series method indeed gives better prediction for crack growth process. That is to say, the error of the initial crack length should be considered to achieve high-precision fatigue crack growth prediction.
Fig. 6 Effect of number of perturbation terms on crack length at N=935950 19
(AA7075- T651 aluminium alloy)
Fig. 7 a-N curves based on experiment, original solution and perturbation series solution (AA7075-T651 aluminium alloy)
Table 4 Crack growth history obtained by experiment, original solution and perturbation series solution for AA7075-T651 aluminium alloy Crack length
Experiment
Original solution
Perturbation series solution
a(mm)
N(cycles)
N(cycles)
Error(%)
N(cycles)
Error(%)
4.47619
603581
600485
-0.51
606835
0.54
4.95238
757034
687931
-9.13
695597
-8.12
5.49205
787724
760187
-3.50
767800
-2.53
5.96825
859335
807953
-5.98
815376
-5.12
6.4762
920716
847447
-7.96
854860
-7.15
6.98415
930947
878421
-5.64
885987
-4.83
7.49205
941177
903146
-4.04
910556
-3.25
7.96825
961637
922053
-4.12
929460
-3.35
In each of the above examples, the feasibility and effectiveness of the proposed method are validated by comparing the fatigue crack growth process obtained by the proposed method with the experimental results for a typical test specimen. The fatigue 20
crack growth of the specimen is illustrated by an a-N curve as depicted in Figs. 4 and 7. The prediction of the a-N curve is carried out by the proposed method for this specimen, which is shown to provide a more accurate solution closer to the experimental results. Certainly, for the experimental tests involving multiple samples, due to the uncertainties existing in material, loading, initial crack length and geometry, the a-N curve would be an interval containing upper and lower bounds. For this problem, the analysis for fatigue crack growth can be combined with the interval analysis methods to predict the interval of a-N curve.
5. Conclusions In this paper, a perturbation series method is proposed to predict the fatigue crack growth process of engineering materials and structural components. The error in initial crack length, as one of the main factors influencing the crack growth process, is investigated in this paper. For the sake of errors of measurement and manufacturing, the actual initial crack length is expressed as the sum of nominal value and perturbation by introducing a small parameter. The solution of the fatigue crack growth equation, i.e. crack length evolution, can be expressed in the form of asymptotic series with the small parameter. The logarithmic form of the fatigue crack growth rate equation is exploited. Based on the Taylor series expansion and the perturbation series method, a series of perturbation equations which serve as the modified fatigue crack growth equations are obtained and the equations can be solved by the successive approximation approach. The perturbation series solution can be used to predict the fatigue crack growth process and the fatigue life of the structural component when the critical crack length is given. Two examples, related to two fatigue crack growth experiments respectively, are given to investigate the accuracy and effectiveness of the proposed method. The results show that the a-N curves obtained by the perturbation series method basically tallies with the experiment data. Compared with the original solution of fatigue crack growth equation, the perturbation series solution can predict the crack growth process more reliably and accurately. The presented perturbation series method is a correction 21
method for crack growth problem with perturbation parameters in initial crack length and it is very significant for the design and use of engineering structures. The proposed method in this paper serves as a modified numerical model that can be used to more accurately evaluate the fatigue crack growth process by properly accounting for the error in initial crack length. Besides, the fatigue crack growth problem with large uncertainty existing in structures, which is not considered in this paper, is also a topic worthy to be studied. Combined with uncertainty analysis methods, the perturbation series theory can be further extended to the fatigue crack growth problem with uncertainty. For the problems with large uncertainty, the high order perturbation method should be implemented to improve accuracy, in which the high order terms in perturbation series expansion need to be calculated by recurrence. It is worth to mention that it is not necessary to calculate the high order sensitivities in the high order perturbation method. The extensions of the proposed method in uncertainty analysis will be investigated in our future research.
Acknowledgements This work was supported by the Defense Industrial Technology Development Program (No. JCKY2016204B101, No. JCKY2017601B001); National Nature Science Foundation of the P.R. China (No. 11772026, No. 11872089).
Appendix A. Perturbation series theory Perturbation method provides the most versatile tool available in nonlinear analysis of engineering problems, and they are constantly being developed and applied to ever more complex problems. In the perturbation method for deterministic structure, the governing equation for physical problem is expressed as an equatio n with a small parameter
L u, x, 0 22
(A1)
where L is the general linear operator, u u x, is the solution of the problem, x is the independent variable, and is a small scalar quantity, which is inherent in Eq. (A1) or introduced factitiously. Since u is the function of the independent x and the small parameter , the solution u can be represented as an asymptotic series in
u x, u0 x u1 x
un x n
(A2)
where the coefficient ui x is independent of . Substituting Eq. (A2) into Eq. (A1) and regrouping terms containing , we get
L0u0 h L0u1 L1u0 L0u2 L1u1 L2u0 2 where L0 , L1 , L2 ,
0
(A3)
are linear operators in space U and h is a real function of x,
which can be determined by specific problems. Since Eq. (A3) are valid for all values of and the sequence , 2 , 3 ,
are
linearly independent, all the coefficients of are equal to zero. Then a set of recursive equations about ui x are obtained L0u0 h
L0u1 L1u0 L0u2 L1u1 L2u0
(A4)
By solving the above equations, we can get the coefficients ui x i 0,1, 2,
.
According to Eq. (A2), an approximate solution of Eq. (A1) can be obtained.
Appendix B. Proof of the polynomial theorem Here, we give the detailed proof of the polynomial theorem (Ladd, 1878). The polynomial theorem: For the real numbers xi i 1, 2,
, k and the positive
integer n, we have
x1 x2
xk n
1
2
n! x11 x22 1 ! 2 ! k ! k n
1
xkk
(B1)
where 1 , 2 ,
, k are nonnegative integers.
Proof: Let us write down the sum of k quantities n times:
x1 x2 x3 x4 x1 x2 x3 x4 x1 x2 x3 x4
(B2)
It is obvious that the terms of the product will contain every possible combination of one or more of the quantities with different exponents in such a manner that the sum of the exponents is equal to n. The coefficient of each term stands for the number of ways in which that term can be formed. x1n , x2n and x3n can be formed in the only one way, hence their coefficient is 1. The term x1n 1 x2 is formed by taking x2 of any row and multiplying it by x1 of every other row. But since any one of the n x2 ’s can be taken, there should be n terms x1n 1 x2 , or the coefficient of x1n 1 x2 is n; and the coefficient should be the same for all terms of the same form x2n 1 x4 , x1n1 x3 ,
. To
obtain the coefficient of x1n 2 x22 , the number of ways in which two x2 ’s can be selected out of n x2 ’s is
x3 ’s in n2
1 n n 1 . For the term x1n5 x23 x32 , we can select the two 2
1 n n 1 ways. After taking a certain combination of 2 x3 ’s, we have 2
x2 ’s
out
of
n 2 n 3 n 2 3 1 3!
which
to
select
3
x2 ’s.
There
could
be
ways for the selection, and only one combination of
the n 5 x1 ’s remains. Therefore, the coefficient of x1n5 x23 x32 is
n n 1 n 2 n 3 n 2 3 1 2! 3!
(B3)
Generally, for the coefficient of x1n2 3 4 x22 x33 x44 , we have the product of the combinations of n things 4 at a time by the combinations of n 4 things 3 at 2
a time by the combinations of n 4 3 things 2 at a time, that is n n 1
n 4 1 n 4 n 4 1 n 3 4 1
4 !
3 !
(B4)
n 4 3 n 4 3 1 n 3 4 2 1 2 !
or
n n 1
n 2 3 4 1
(B5)
2 ! 3 ! 4 !
Let n 2 3 4 1 and multiply denominator and numerator of Eq. (B5) by
1 ! and we get n n 1
1 1 1 1 1 1 2
1
(B6)
1 ! 2 ! 3 ! 4 ! or n! 1 ! 2 ! 3 ! 4 !
(B7)
for the coefficient of x11 x22 x33 x44 , in which 1 2 3 4 n . From the manner in which this coefficient is obtained it is evident that all the terms of the form x11 x32 x43 x24 , x21 x32 x43 x54 , and so on, have the same coefficient. Applying the same way for all terms, then we may write
x1 x2
n! n xk x11 x22 1 ! 2 ! k !
xkk
(B8)
where the within the brackets stands the sum of the terms formed by all possible combinations of the k quantities with a given set of exponents, and the without the brackets stands for the sum of all the sets of terms obtained by giving to the exponents all possible values such that
1 2
k n
m Then Eq. (B8) can be written as Eq. (B1). Based on Eq. (B1), ak t k k 1 3
(B9) j
and
m k ak t k 1
j
can take the form of polynomial, see Eq. (30) and Eq. (31).
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7