materials Article

An Artificial Neural Network-Based Algorithm for Evaluation of Fatigue Crack Propagation Considering Nonlinear Damage Accumulation Wei Zhang *, Zhangmin Bao, Shan Jiang and Jingjing He School of Reliability and Systems Engineering, Beihang University, Haidian District, Beijing 100089, China; [email protected] (Z.B.); [email protected] (S.J.); [email protected] (J.H.) * Correspondence: [email protected]; Tel.: +86-10-8231-4649 Academic Editor: Mark T. Whittaker Received: 5 March 2016; Accepted: 10 June 2016; Published: 17 June 2016

Abstract: In the aerospace and aviation sectors, the damage tolerance concept has been applied widely so that the modeling analysis of fatigue crack growth has become more and more significant. Since the process of crack propagation is highly nonlinear and determined by many factors, such as applied stress, plastic zone in the crack tip, length of the crack, etc., it is difficult to build up a general and flexible explicit function to accurately quantify this complicated relationship. Fortunately, the artificial neural network (ANN) is considered a powerful tool for establishing the nonlinear multivariate projection which shows potential in handling the fatigue crack problem. In this paper, a novel fatigue crack calculation algorithm based on a radial basis function (RBF)-ANN is proposed to study this relationship from the experimental data. In addition, a parameter called the equivalent stress intensity factor is also employed as training data to account for loading interaction effects. The testing data is then placed under constant amplitude loading with different stress ratios or overloads used for model validation. Moreover, the Forman and Wheeler equations are also adopted to compare with our proposed algorithm. The current investigation shows that the ANN-based approach can deliver a better agreement with the experimental data than the other two models, which supports that the RBF-ANN has nontrivial advantages in handling the fatigue crack growth problem. Furthermore, it implies that the proposed algorithm is possibly a sophisticated and promising method to compute fatigue crack growth in terms of loading interaction effects. Keywords: fatigue crack growth; artificial neural network; nonlinear multivariable function; retardation; loading interaction

1. Introduction As the damage tolerance concept is now widely accepted and applied in the aerospace and aviation industries, it has become increasingly important to analyze how a fatigue crack grows. Linear elastic fracture mechanics (LEFM), moreover, is the fundamental theory for establishing the analytical model of fatigue crack propagation. Paris and Erdogan [1] correlate the stress intensity factor (SIF) range with the fatigue crack growth rate, and propose this seminal model as Equation (1). da “ CP p∆KqmP dN

(1)

where ∆K is the SIF range and CP and mP are the fitting parameters. This equation shows a linear relationship between da/dN and ∆K in the log-log coordinate. However, a major limitation of the Paris equation is that CP has to change along with the variation of the stress ratio (R). Additionally, it is only applicable to the linear region without the consideration of the threshold SIF (∆Kth ) and the critical SIF (Kc ). Materials 2016, 9, 483; doi:10.3390/ma9060483

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To perfect the Paris equation, many researchers have attempted to make modifications in order to involve more nonlinear factors. Forman et al. [2] take the effects of R and Kc into consideration and propose a modified model as shown in Equation (2). CF p∆Kqm F da “ p1 ´ Rq Kc ´ ∆K dN

(2)

where CF and mF are the fitting parameters. Furthermore, based on the Paris equation, some researchers develop more general formulas by employing additional parameters to account for nonlinearity, such as the NASGRO formula [3] in Equation (3): ´ „ n 1 ´ da p1 ´ f q “ CNa ∆K ´ dN p1 ´ Rq 1´

∆Kth ∆K Kmax Kc

¯PNa ¯q Na

(3)

where CNa , PNa and qNa are the fitting parameters; and f is Newman’s [4] crack opening function determined by the experimental measurement. The series of models discussed above are put forward to illustrate the nonlinear relationship between the crack growth rate and the SIF range for constant amplitude loadings. Moreover, many researchers have investigated the loading interaction effect, which is complicated and of great significance to the variable amplitude loading. In this paper, the constant amplitude loading with overload, which is the typical and simple variable amplitude loading, is studied. Wheeler proposes a plastic-zone-based model [5] to describe the crack growth retardation caused by overload as shown in Equation (4). $ ´ ¯ da & da dN VA “ γ dN ´ r ¯m (4) % γ “ p,i λ where γ is the retardation factor; rp,i is the size of current plastic zone; λ is the distance between the current crack tip and the edge of the plastic zone caused by overload; and m is the fitting parameter. De Koning [6] also develops a plastic-deformation-based model to deal with the overload effect. Wheeler’s and Koning’s models support that the plastic zone (monotonic plastic zone or reversed plastic zone) is the key parameter to correlate with loading interaction effects in the fatigue crack growth calculation. Most studies focus on accurately quantifying the nonlinear relationship between the crack growth rate and the driving parameters by using an explicit and simple function. To achieve these goals, many studies have been undertaken to introduce more parameters to construct a formula which can fit the experimental data better. However, the current formulas are not flexible enough to positively handle all the situations. Overall, the process of fatigue crack growth is a nonlinear and multivariable problem under both constant and variable amplitude loading. Fortunately, the artificial neural network (ANN) has an excellent ability to fit the nonlinear multivariable relationship, which makes it a sophisticated and promising approach to the fatigue crack growth problem. ANN is a family of algorithms based on the imitation of biological neural networks. It has the strong ability to estimate the tendency of nonlinear and multivariable functions based on a large amount of data [7]. Thanks to these advantages, ANN is widely applied to damage estimation in the material sciences [8–10]. Furthermore, it is used to deal with some fracture problems including creep, fatigue and even corrosion fatigue [11–15]. A novel ANN-based algorithm is proposed in this paper to evaluate the process of fatigue crack growth. In the following sections, the ANN is first established and its training outlined. While establishing the ANN, the equivalent SIF is used to account for the influence of the loading history. Subsequently, the ANN-centered algorithm is developed and validated by using experimental data under the constant

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amplitude Materials 2016,loading 9, 483 with different stress ratios or overloads. Some classical models are also employed 3 of 20 for comparison. In the final section, some conclusions and considerations are given. Subsequently, the ANN-centered algorithm is developed and validated by using experimental data 2. Methodology under the constant amplitude loading with different stress ratios or overloads. Some classical models are also employed for comparison. In the final section, some conclusions and considerations 2.1. Radial Basis Function Artificial Neural Network are given. In the 1980s, ANN technology became popular for dealing with practical problems. As it is 2. Methodology inspired by biological neural networks, it shares some features with the human brain, particularly learning by example. Radial basis F = function (RBF) network is a type of ANN which uses radial 2.1. Radial BasisasFunction Artificialfunction. Neural Network basis function the activation Because of the RBF network’s ability to produce optimal approximate solutions and local learning, it is used in function approximation, system control, etc. In the 1980s, ANN technology became popular for dealing with practical problems. As [16]. it is The RBF ANN structure is displayed in Figure 1, where {x , x . . . x } is the input vector; m is the m0 1 2 0 inspired by biological neural networks, it shares some features with the human brain, particularly dimension of the vector; w , w , . . . , w are the connection weights between the middle layer and n 1 basis 2 learning by example. Radial F = function (RBF) network is a type of ANN which uses radial the output layer; is the number of the radial of basis in the middle layer. As shown in basis function as and the N activation function. Because the functions RBF network’s ability to produce optimal Figure 1, the RBF ANN and consists three layers: input layer, the middle layer and the output layer. approximate solutions localoflearning, it is the used in function approximation, system control, etc. The input layer is composed of m source points which connect the ANN to the external environment. 0 [16]. The RBF ANN structure is displayed in Figure 1.

Figure 1. 1. Schematic of radial radial basis basis function function (RBF)-artificial (RBF)-artificial neural neural network network (ANN). (ANN). Figure Schematic illustration illustration of

where {x1, x2…xm0} is the input vector; m0 is the dimension of the vector; w1, w2, …, wn are the The second layer is the only hidden layer in the RBF network. Its function is to transform the connection weights between the middle layer and the output layer; and N is the number of the radial input space into the hidden space nonlinearly. The hidden layer consists of N cells that can be defined basis functions in the middle layer. As shown in Figure 1, the RBF ANN consists of three layers: the mathematically by the radial function shown in Equation (5). The RBF network is good at local input layer, the middle layer and the output layer. The input layer is composed of m0 source points approximation because the radial function in the hidden layer responds to theinput partially. which connect the ANN to the external environment. ` layer in the ˘ RBF network. Its function is to transform the The second layer is the only hidden ϕ j pxq “ ϕ ||x ´ x j || , j “ 1, 2, ¨ ¨ ¨ ¨ ¨ ¨ , N (5) input space into the hidden space nonlinearly. The hidden layer consists of N cells that can be definedx mathematically by the radial function shown in Equation th (5). The RBF network is good at where j means the center of the radial function defined by the j source point; and x is the signal local directly approximation because the radial function the hidden responds to used the which acts on the input layer. Additionally, the in Gaussian functionlayer is the most widely input partially. radial function, and the cells in the hidden layer can be defined as in Equation (6) shown below. ¸ ( )= ˜ ‖ − (5) , = 1,2,······, ` ˘ 1 2 ϕ pxq “ ϕ x ´ x j “ exp ´ 2 ||x ´ x j || j “ 1, 2, ¨ ¨ ¨ ¨ ¨ ¨ , N (6) where xj means thej center of the radial function 2σjdefined by the jth source point; and x is the signal which directly acts on the input layer. Additionally, the Gaussian function is the most widely used radial function, and the hiddenGaussian layer can function, be defined in Equation below. where σj is the width of cells the jthinxthe xj as is the center of(6) theshown jth basis function, j -centered ||x ´ xj || is the vector norm of x ´ xj which means1the distance between x and xj . Finally, the nodal ) =will generate − = ) = 1,2,······, points in the output(layer theexp(− output data.‖ − (6) 2 RBF ANN is one type of feedforward static neural network. The feedforward network is the simplest network as the information canGaussian only move in onexdirection. Theoforiginal feedforward the width of the jth xj-centered function, j is the center the jth basis function, where σj is network is the a single perceptron based the on other networks consisting multiplethe layers of vector norm of layer x − xjnetwork which means distance between x and xof j. Finally, nodal ‖x − xj‖ is computational units layer such as the RBF network. The data. RBF network is able to fit a continuous nonlinear points in the output will generate the output process a satisfied automatically the weight offeedforward the functionsnetwork in the hidden RBFinANN is oneprecision type of by feedforward staticadjusting neural network. The is the layer. Some studies that the can ANN hasmove advantages dealing The with original nonlinear problems. simplest network as indicate the information only in one in direction. feedforward network is a single perceptron layer network based on other networks consisting of multiple layers of computational units such as the RBF network. The RBF network is able to fit a continuous nonlinear process in a satisfied precision by automatically adjusting the weight of the functions in the hidden layer. Some studies indicate that the ANN has advantages in dealing with nonlinear

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problems. Ghandehari et al. [17] discuss the advantages of RBF over the back propagation (BP) network, which the most widely used and popular feedforward Fathi andnetwork, Aghakouchak Ghandehari et al.is[17] discuss the advantages of RBF over the backnetwork. propagation (BP) which [18], wellwidely as Abdalla andpopular Hawileh [19] applied the RBF network to fatigue crack is theas most used and feedforward network. Fathi and Aghakouchak [18], problems as well as successfully. In this paper, the RBF was chosen due to its multiplesuccessfully. advantages;Ininthis view of Abdalla and Hawileh [19] applied thenetwork RBF network to fatigue crack problems paper, the capacity, is suitable for establishing the in function crack growthit the RBF RBF network’s network was chosenitdue to its multiple advantages; view ofbetween the RBF fatigue network’s capacity, and the driving parameters.the function between fatigue crack growth and the driving parameters. is suitable for establishing 2.2. The The Establishment Establishment and and Training Training of of the the Artificial Artificial Neural Neural Network Network (ANN) 2.2. (ANN) In this this section, section,the theMATLAB MATLAB(©1984–2011 (©1984–2011MathWorks. MathWorks. All Allrights rightsreserved, reserved,MathWorks, MathWorks,Natick, Natick, In MA, USA) USA)software softwareisisused usedtotoestablish establishand andtrain trainthe theRBF-ANN RBF-ANNasasshown shownininFigure Figure First MA, 2. 2. First of of all,all, a a multi-input single-output RBF-ANNisisestablished establishedby byanalyzing analyzingthe thephysical physicalprocess processof offatigue fatiguecrack crack multi-input single-output RBF-ANN growth. The Theraw rawexperimental experimentaldata datathen thenneed needto tobe bepreprocessed preprocessed before before training training the the ANN. ANN. The The data data growth. preprocessing includes two steps: the first step is to take the logarithm of ∆K to reduce reduce the the preprocessing ΔK and da/dN da/dN to influence from the order of the magnitude; the second step is to normalize the data from the first step. influence from the order of the magnitude; the second step is to normalize the data from the first step. After preprocessing, the experimental a number of vectors, which are After experimentaldata datahave havebeen beentransformed transformedinto into a number of vectors, which usedused to train the ANN. ANN can be can trained automatically by using by the using MATLAB During the are to train the ANN. ANN be trained automatically the toolbox. MATLAB toolbox. training, some parameters can be tuned for optimization, including: the mean square error (MSE) During the training, some parameters can be tuned for optimization, including: the mean square goal, (MSE) expansion of RBF, maximum of neurons, For example, the example, MSE goalthe controls error goal,speed expansion speed of RBF,number maximum numberetc. of neurons, etc. For MSE the fitting accuracy. By comparing the output with testing data balancing thebalancing accuracy and goal controls the fitting accuracy. By comparing thethe output with theand testing data and the efficiency, the optimal tuning parameters can be determined. accuracy and efficiency, the optimal tuning parameters can be determined.

Data preprocessing

Establish and train the RBF ANN

Compare with the experimental data

Tuning the parameters and optimize ANN Figure Figure2. 2.Procedure Procedureof of developing developing aa well-trained well-trained ANN. ANN.

2.2.1. 2.2.1. The TheConstant Constant Amplitude Amplitude Loading Loading The The experimental experimental data data [20] [20] are are plotted plotted in in Figure Figure 3. 3. The The x-axis x-axis isis the the SIF SIF range; range; the the y-axis y-axis is is the the crack growth rate; and the different kinds of dots represent the testing data with different stress crack growth rate; and the different kinds of dots represent the testing data with different stress ratios. ratios. It can be seen that the testing data do not follow a perfect linear tendency. It can be seen that the testing data do not follow a perfect linear tendency. With the experimental data in Figure 3, the ANN can be trained following the procedure in With the experimental data in Figure 3, the ANN can be trained following the procedure in Figure 2. For the constant amplitude loading, the plasticity, on behalf of the historical load, is Figure 2. For the constant amplitude loading, the plasticity, on behalf of the historical load, is proportional to the current loading. The SIF and stress ratio are therefore chosen to be the inputs, proportional to the current loading. The SIF and stress ratio are therefore chosen to be the inputs, and and the crack growth rate is the output. The training vectors are preprocessed to make them suitable the crack growth rate is the output. The training vectors are preprocessed to make them suitable for the for the ANN. During training, the ANN can learn deeply from the limited data and establish the ANN. During training, the ANN can learn deeply from the limited data and establish the continuous continuous function between the inputs and the output. function between the inputs and the output. The fitting surface by well-trained ANN and the testing data are shown in Figure 4. The blue The fitting surface by well-trained ANN and the testing data are shown in Figure 4. The blue dots dots represent the training data, and the red crosses are the data for validation. It can be observed represent the training data, and the red crosses are the data for validation. It can be observed that the that the fitting surface can match all the experimental data well, even though they are not perfectly fitting surface can match all the experimental data well, even though they are not perfectly log-linear. log-linear. The nonlinearity of the data can be studied by the ANN so that its prediction has a higher The nonlinearity of the data can be studied by the ANN so that its prediction has a higher accuracy accuracy than the tradition log-linear formulas. Moreover, the ANN can offer a continuous than the tradition log-linear formulas. Moreover, the ANN can offer a continuous predicting surface in predicting surface in the domain of definition based on the limited and discrete training data. This the domain of definition based on the limited and discrete training data. This example shows ANN’s example shows ANN’s advantage in fitting and extrapolating the crack growth rate under constant advantage in fitting and extrapolating the crack growth rate under constant amplitude loading with amplitude loading with different stress ratios. different stress ratios.

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Figure 3. 3. The The experimental experimental data Figure data of of Al7075-T6. Al7075-T6. Figure 3. The experimental data of Al7075-T6.

Figure 4. The experimental data of Al7075-T6 and the corresponding ANN. Figure 4. The experimental data of Al7075-T6 and the corresponding ANN. Figure 4. The experimental data of Al7075-T6 and the corresponding ANN.

2.2.2. Single Overload 2.2.2. Single Overload 2.2.2. Single For theOverload variable amplitude loading, the load interaction effects cannot be ignored, because the For the variable amplitude loading,isthe load interaction cannot be ignored, theas influence of historical loading sequence dependent on the effects current load cycle. Single because overload, For the variable amplitude loading, the load interaction effects cannot be ignored, because the influence of historical loading sequence is dependent the current load cycle. Single overload, the simplest and most typical variable amplitude on loading, is investigated in this paper asto influence of historical loading sequence is dependent on the current load cycle. Single overload, as the the simplesttheand mostinteraction typical variable demonstrate loading effect. amplitude loading, is investigated in this paper to simplest and most typical variable amplitude loading, is investigated in this paper to demonstrate the demonstrate the loading interaction effect. As is well known, an applied overload can lead to fatigue crack growth retardation or even crack loadingAsinteraction effect. is well known, an applied overload can lead to fatigue crack growth retardation or even crack arrest. This phenomenon is caused by the loading interaction effect, and its existence obviously As is well known, an applied can leadinteraction to fatigueeffect, crackand growth retardation or even arrest. This phenomenon is causedoverload by the loading its existence obviously stimulates nonlinear damage accumulation. Wheeler [5], De Koning [6] and many other researchers crack arrest.nonlinear This phenomenon is causedWheeler by the [5], loading interaction and its existence stimulates damage accumulation. De Koning [6] andeffect, many other researchers [21–27] have introduced additional parameters to describe the influence of the historical loading obviously stimulates nonlinear damage accumulation. Wheeler [5], De Koning and many other [21–27] have introduced additional parameters to describe the influence of the [6] historical loading sequence. Wheeler [5] models the retardation by correlating the plastic zone size ahead of the crack sequence. [21–27] Wheelerhave [5] models the retardation correlatingtothe plastic the zoneinfluence size ahead crack researchers introduced additionalby parameters describe of of thethe historical tip with the crack growth rate. Topper and Yu [28] use the plasticity-induced crack closure to explain tip with the crackWheeler growth rate. Topperthe andretardation Yu [28] useby thecorrelating plasticity-induced crack closure to explain loading sequence. [5] models the plastic zone size ahead of the this phenomenon. Above all, the plasticity ahead of the crack tip is a reasonable parameter to this tip phenomenon. Above all, the ahead the crack tip is a reasonablecrack parameter toto crack with the crack growth rate.plasticity Topper and Yu of [28] use the plasticity-induced closure account for the loading interaction effect. In this paper, a concept “equivalent stress intensity factor”, account forphenomenon. the loading interaction In this paper, a concept “equivalent stress intensity factor”,to explain this Above all,effect. the plasticity ahead of the crack tip is a reasonable parameter which is derived from the equivalent plastic zone, is employed as input data to handle the nonlinear which for is derived from interaction the equivalent plastic zone, is employed as “equivalent input data tostress handle the nonlinear account the loading effect. In this paper, a concept intensity factor”, damage accumulation. The details are discussed in Section 3.2.1. damage accumulation. The details are discussed in Section 3.2.1.

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2.3. A Fatigue Life Prediction Method which is derived from the equivalent plastic zone, is employed as input data to handle the nonlinear There are three steps calculating the fatigue length. damage accumulation. Thetodetails are discussed incrack Section 3.2.1.First the crack increment within one load cycle is computed; the the crack length, the geometric factor, and the SIF are subsequently 2.3. A Fatigue Life Prediction updated, thereby preparingMethod the inputs for the next cycle. In repeating this process, the fatigue crack propagation is simulated cycle by cycle. The framework is shown in Equations (7) and (8). There are three steps to calculating the fatigue crack length. First the crack increment within one load cycle is computed; the the crack length, the geometric factor, and the SIF are subsequently updated, thereby preparing the=inputs this process, the fatigue crack + for the next = cycle. + In repeating (∆ , ,···) propagation is simulated cycle by cycle. The framework is shown in Equations (7) and (8).

$ I = (∆ , ř I ,···) ` ˘ ř (7) ’ ’ a “ a ` da “ a ` g ∆K j , R, ¨ ¨ ¨ ’ 0 0 I j ’ ’ ’ j “1 & ∆j“da1 = ∆` ×˘ “ g ∆K , R, ¨¨¨ (7) j dI ’ ? ’ ∆K “ ∆σ πa ˆ q ’ = ( ) j j ’ ´a ¯ ’ ’ % q “ Q wj where aI is crack length in the Ith cycle; a0 is the initial crack length; g(ΔKj, R, …) denotes the general relationship between the crackthgrowth rate and applied load; daj is the increment during the jth cycle; where aI is crack length in the I cycle; a0 is the initial crack length; g(∆Kj , R, . . . ) denotes the general Δσ is the stress amplitude; Q is the geometric factor while w is the width of the specimen. relationship between the crack growth rate and applied load; daj is the increment during the jth cycle; Furthermore, once the failure criterion or the critical crack length is provided, the fatigue life can ∆σ is the stress amplitude; Q is the geometric factor while w is the width of the specimen. Furthermore, be determined. once the failure criterion or the critical crack length is provided, the fatigue life can be determined. In this study, the ANN is used to quantify the relationship between the loading and the crack In this study, the ANN is used to quantify the relationship between the loading and the crack increment per cycle instead of the traditional equation. Equation (7) can therefore be transformed increment per cycle instead of the traditional equation. Equation (7) can therefore be transformed into into Equation (8). Equation (8). I ÿ ` ˘ aI = “ a0 `+ f ANN (∆ ∆K j , R, ¨¨ (8) (8) , ¨,···) j “1

where . . )represents representsaageneral generalANN ANNfunction functiondescribing describingthe therelationship relationship between (ΔK, R, R, .…) between wherefANN fANN (∆K, driving parameters and crack growth rate. Generally, the driving parameters would include SIF range driving parameters and crack growth rate. Generally, the driving parameters would include SIF (∆K), stress zone, etc. Additionally, an ANN-based framework for fatigue crackcrack growth range (ΔK),ratios, stress plastic ratios, plastic zone, etc. Additionally, an ANN-based framework for fatigue calculation can be established. The flow chart is shown in Figure 5. growth calculation can be established. The flow chart is shown in Figure 5.

Figure 5. 5. The Figure The flow flowchart chartof ofthe theANN-based ANN-basedalgorithm. algorithm.

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3. 3. Validation Validationand andComparison Comparison 3.1. Loading with with Different Different Stress Stress Ratios Ratios 3.1. Validation Validation and and Comparison Comparison of of the the Constant Constant Loading 3.1.1. ANN Training As the ANN can quantify the relationship from experimental data, it is significant to select the suitable data data for for the the training. training.In Inthis thissection, section,the thetesting testingdata data[20] [20]ofof7075-T6 7075-T6 aluminum alloy is used aluminum alloy is used to to train ANN globally. information of the experiment is listed in Table train thethe ANN globally. TheThe information of the experiment is listed in Table 1. 1. Table 1. The experimental information of Al7075-T6.

Material Material Specimen type Specimen type Specimen Specimenlength length Specimen Specimenwidth width Specimen thickness Specimen thickness Initial cracklength length Initial crack Loading type Loading type

Al7075-T6 Al7075-T6 Middle cracked tension specimen Middle cracked tension specimen 889 mm 889 mm 305 mm 305 mm 2.28 mm 2.28 mm 2.5 mm 2.5 mm Tension-tension, constant amplitude Tension-tension, constant amplitude

With these between thethe loading andand fatigue crack growth rate these experimental experimentaldata datathe therelationship relationship between loading fatigue crack growth can be by the rate canfitted be fitted byANN. the ANN. experimental data data with with different different stress stress ratios. ratios. At first, the ANN is trained by all five sets of the experimental The fitting curves are plotted with the original data in Figure 6. In this figure, the x-axis is the stress ratios (from 0 to 1); the y-axis is the SIF in logarithmic coordinate; and the z-axis is the crack growth The blue blue cycles cycles represent represent the experimental experimental data; and the dark blue rate in logarithmic coordinate. The is observed observed clearly clearly that that the the curves curves fit fit the the experimental experimental data data well. well. lines are the ANN prediction. ItIt is Additionally, Additionally, the the projections projections of of the the fitting fittingcurves curvesare arealso alsoprovided. provided.

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To observe the fitting accuracy clearly, Figure 7 displays the prediction and the experimental To observe the fitting accuracy clearly, Figure 7 displays the prediction and the experimental data in a 2D plot. From the picture it can be seen that the nonlinear fitting curves by ANN can fit the data in a 2D plot. From the picture it can be seen that the nonlinear fitting curves by ANN can fit the experimental data well. experimental data well.

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In this part part the impact impact from data data size on on the fitting fitting performance is is investigated. This the In this This time time the In this part the the impact from from data size size on the the fitting performance performance is investigated. investigated. This time the training vectors only include thre sets of experimental data with stress ratios 0.02, 0.33 and 0.75; the training vectors vectors only only include include thre thre sets sets of of experimental data with with stress stress ratios ratios 0.02, 0.02, 0.33 0.33 and and 0.75; 0.75; the the training experimental data other used for for validation. The other experimental experimental data data are are used The ANN ANN prediction prediction and and the the experimental experimental data data are are other experimental data are used for validation. validation. The ANN prediction and the experimental data are shown in Figure 8. In this picture only the purple crosses are the experimental data used to train the shown in Figure 8. In this picture only the purple crosses are the experimental data used to train the shown in Figure 8. In this picture only the purple crosses are the experimental data used to train the ANN. It It is obvious obvious that the the fitting accuracy accuracy is still still satisfactory compared compared with Figure Figure 8. ANN. ANN. It is is obvious that that the fitting fitting accuracy is is still satisfactory satisfactory compared with with Figure8. 8.

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Figure 8. The fitting curves of ANN trained with part of data. Figure of data. data. Figure 8. 8. The The fitting fitting curves curves of of ANN ANN trained trained with with part part of

Forman’s equation is also utilized to calculate fatigue crack growth under different stress ratios Forman’s equation is also utilized to calculate fatigue crack growth under different stress ratios to make a comparison. shows to thecalculate calibration indices ofgrowth Equation (9). different The fitting parameters Forman’s equation Table is also2utilized fatigue crack under stress ratios to to make a comparison. Table 2 shows the calibration indices of Equation (9). The fitting parameters are calibrated with the global database. The prediction by Equation (9) (9). is displayed in parameters Figure 9. It can make a comparison. Table 2 shows the calibration indices of Equation The fitting are are calibrated with the global database. The prediction by Equation (9) is displayed in Figure 9. It can be concluded predictions Forman’s equation are linear log-log coordinate calibrated withthat the the global database.by The prediction by Equation (9) isin displayed in Figure 9.while It canthe be be concluded that the predictions by Forman’s equation are linear in log-log coordinate while the ANN prediction curves are nonlinear. concluded that the predictions by Forman’s equation are linear in log-log coordinate while the ANN ANN prediction curves are nonlinear. prediction curves are nonlinear. where r is the coefficient of association; RMSE is the root-mean-square 2. The fitting indices of Al7075-T6. coefficient. error; SSE is the sum of squares Table for error and DC is the determination Table 2. The fitting indices of Al7075-T6. The Fitting Indexes ´9 Number 3.5241 ThedaFitting Indexes Number 2.9838 ˆ 10 ˆ ∆K 0.796 “ r dN r p1 ´ Rq Kc ´ ∆K 0.796 Chi-Square 0.000513 Chi-Square 0.000513−5 RMSE 1.47 × 10−5 RMSE 1.47 × 10−8 SSE 7.52× 10−8 SSE 7.52× 10 DC 0.796 DC 0.796

(9)

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Table 2. The fitting indices of Al7075-T6.

Materials 2016, 9, 483

The Fitting Indexes

9 of 20

Number

where r is2016, the9, 483 coefficient of association; RMSE is the root-mean-square error; SSE is the 9sum Materials of 20 of r squares for error and DC is the determination coefficient. 0.796 Chi-Square 0.000513 where r is the coefficient of association; RMSE is the 1.47 root-mean-square error; SSE is the sum of ´5 . RMSE × ∆ˆ 10´8 2.9838 × 10 squares for error and DC is the determination coefficient. SSE 7.52 ˆ 10 (9) = DC (1

-8

−0.796 ∆ .

×∆ −∆

(9)

experimental data R=0.02 fitting curve R=0.02 experimental data R=0.1 fitting curve R=0.1 experimental data R=0.02 fitting curve R=0.02 experimental data R=0.33 experimental data R=0.1 fitting curve R=0.33 fitting curve R=0.1 experimental data R=0.5 experimental data R=0.33 fitting curve R=0.5 fitting curve R=0.33 experimental data R=0.75 experimental data fitting curve R=0.75R=0.5 fitting curve R=0.5 experimental data R=0.75 fitting curve R=0.75

-10-8

-10 -12

-12 -14

-14 -16

lnKmax

lnKmax

− )

2.9838 × 10 = (1 − )

-16 -18 -18 -20 -20

-22 -22

-24

1

-24

1.5 1

1.5

2

2.5

2

2.5

lnda/dN

3 3

3.5 3.5

4 4

4.5 4.5

lnda/dN

Figure 9. 9. The The fitting fitting by by Forman’s Forman’s model model vs. vs. the the testing testing data data for for Al7075-T6. Al7075-T6. Figure Figure 9. The fitting by Forman’s model vs. the testing data for Al7075-T6.

With the the good good performance performance of of ANN ANN under under the the discrete discrete R R values, values, it it is is reasonable reasonable that that the the ANN ANN With Withathe good performance of ANN under the discrete R values, it is reasonable that the ANN can deliver good fitting surface within the continuous domain as shown in Figure 4. can deliver a good fitting surface within the continuous domain as shown in Figure 4. canFurthermore, deliver a good fitting surface within the continuous domain as Figure shown 10 in Figure additional material is utilized utilized to test test the the ANN. shows 4. the fitting fitting surface surface Furthermore, additional material is to ANN. Figure10 10shows shows the Furthermore, additional material is utilized to test the ANN. Figure the fitting surface by ANN and experimental data of Al2024-T315 [21]. Four sets of experimental data (stress ratios: 0, by by ANN and experimental Four sets setsof ofexperimental experimentaldata data (stress ratios: ANN and experimentaldata dataofofAl2024-T315 Al2024-T315 [21]. [21]. Four (stress ratios: 0, 0, 0.1, 0.33 0.33 and and 0.5) 0.5) are are all all used to to train the the ANN globally. globally. 0.1, 0.1, 0.33 and 0.5) are allused used totrain train theANN ANN globally.

Figure 10. The experimental data of Al2024-T315 and the corresponding ANN. Figure 10. 10. The data of of Al2024-T315 Al2024-T315 and and the the corresponding corresponding ANN. ANN. Figure The experimental experimental data

To observe the fitting accuracy clearly, Figure 11 shows the prediction and the experimental To observe the fitting accuracy clearly, Figure 11 shows the prediction and the experimental data in a 2D plot. From the figure it can be seen that the nonlinear fitting curves by ANN can fit the data in a 2D plot. From the figure it can be seen that the nonlinear fitting curves by ANN can fit the experimental data well.

experimental data well.

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To observe the fitting accuracy clearly, Figure 11 shows the prediction and the experimental data in a 2D plot. From the figure it can be seen that the nonlinear fitting curves by ANN can fit the Materials 2016, 9,data 483 well. 10 of 20 experimental Materials 2016, 9, 483

10 of 20 0

10

0

fitting curve R=0 fitting curve R=0 experimental data R=0 experimental data R=0 fitting curve R=0.1 fitting curve R=0.1 experimental data R=0.1 experimental data R=0.1 fitting curve R=0.33 fitting curve R=0.33 experimental data R=0.33 experimental data R=0.33 fitting curve R=0.5 fitting curve R=0.5 experimental data R=0.5

10 -1

10

-1

10 -2

-2

10

-3

10

lnda (mm)

lnda (mm)

10

-3

experimental data R=0.5

10

-4

10 10-4 -5 -5

10 10 -6

-6

10 10

-7

10 -7

10

1.4

10 1.4

10

1.5

101.5

10

1.6

10 1.6

10

1.7

10

1.8

1.7

10

lnKmax (MPa*m0.5)

lnKmax (MPa*m

10

1.9

1.8

10

10

1.9

10

0.5

)

Figure 11. The fitting by ANN vs. the testing data for Al2024-T315.

Figure11. 11. The The fitting fitting by by ANN ANNvs. vs. the the testing testing data data for for Al2024-T315. Al2024-T315. Figure 3.1.2. Crack Growth Calculation under Constant Amplitude Loading

3.1.2. Crack Crack Growth Growth Calculation Calculation under underConstant ConstantAmplitude AmplitudeLoading Loading 3.1.2. With the well-trained ANN, the algorithm for the crack propagation is programmed by With the well-trained ANN, the algorithm for the crack propagation is programmed MATLAB and the flow chart is shown in Figure 12. At first, a loading spectrum is generated the by With the well-trained ANN, the algorithm for the crack propagation is programmed byand MATLAB parameters are initialized. The input vector is then prepared following the same procedure in the MATLAB and the flow chart is shown in Figure 12. At first, a loading spectrum is generated and and the flow chart is shown in Figure 12. At first, a loading spectrum is generated and the parameters Section 2.2. After that, this vector is entered into the well-trained ANN and the crack increment is in parameters initialized. The input vector following is then prepared theSection same 2.2. procedure are initialized.are The input vector is then prepared the same following procedure in After that, worked With the crack increment in the current cycle, the crack length gets forincrement the next is Section 2.2.isout. After that, is entered ANN andupdated the out. crack this vector entered intothis thevector well-trained ANNinto andthe thewell-trained crack increment is worked With the crack iteration. When the whole loop is repeated until the last cycle, the simulation of the fatigue cracknext worked out. the crack increment inlength the current cycle, thefor crack gets updated increment in With the current cycle, the crack gets updated the length next iteration. Whenfor thethe whole propagation is accomplished. iteration. When until the whole is the repeated until of thethe last cycle,crack the simulation of isthe fatigue crack loop is repeated the lastloop cycle, simulation fatigue propagation accomplished. propagation is accomplished.

Figure 12. Flow chart of programing the predicting algorithm. Figure 12. Flow chart of programing the predicting algorithm.

Figure 12. Flow chart of programing the predicting algorithm.

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To validate the theANN-centered ANN-centeredalgorithm, algorithm,the the experimental data of Al7075-T6 are used To validate experimental data of Al7075-T6 are used [20]. [20]. The The ANN has been trained with the da/dN-∆K data as shown in Figure 4. Some additional data ANN has been trained with the da/dN-ΔK data as shown in Figure 4. Some additional data (a-N (a-N curves) are then utilized to compare the model prediction. In Table 3, the testing information curves) are then utilized to compare withwith the model prediction. In Table 3, the testing information of of these a-N curves are listed. Moreover, Forman’s equation also serves as a comparison. these a-N curves are listed. Moreover, Forman’s equation also serves as a comparison. Table 3. Loading information for the a-N curves. Table 3. Loading information for the a-N curves.

σσmin RR min 0.33 0.33 51.2 51.2MPa MPa MPa 0.50.5 6969MPa 168.7MPa MPa 0.70.7 168.7

σmax σ max 155MPa MPa 155 138 138MPa MPa 241 MPa 241 MPa

Figure13, 13, x-axis iscycle the number, cycle number, the y-axis is of thethe length the crack. The In Figure thethe x-axis is the and theand y-axis is the length crack. of The experimental experimental data and the predictions by the two different models are all visualized in data and the predictions by the two different models are all visualized in different lines. It isdifferent obvious lines.the It is obvious thatofthe performance of ANN is better than Forman’s model. that performance ANN is better than Forman’s model. 0.02

Forman R=0.33 Forman R=0.5 Forman R=0.7 ANN R=0.33 ANN R=0.5 ANN R=0.7 testing data R=0.33 testing data R=0.5 testing data R=0.7

0.018

0.016

crack length/m

0.014

0.012

0.01

0.008

0.006

0.004

0.002

0

0

5000

10000

15000

20000

25000

30000

35000

40000

cycle

Figure Figure 13. 13. The The predictions predictions vs. vs. the the experimental experimental data data for for Al7075-T6 Al7075-T6 aluminum aluminum alloy. alloy.

Once the failure criterion is given, the corresponding fatigue life can be determined. Assuming Once the failure criterion is given, the corresponding fatigue life can be determined. Assuming that the that the critical crack length is 0.008 m, 0.01 m, and 0.012 m, the corresponding errors of the two critical crack length is 0.008 m, 0.01 m, and 0.012 m, the corresponding errors of the two models are shown models are shown in Table 4. It is evident that the accuracy and stability of ANN is much better than in Table 4. It is evident that the accuracy and stability of ANN is much better than Forman’s equation. Forman’s equation. Table 4. The errors of the results by two models for Al7075-T6. Table 4. The errors of the results by two models for Al7075-T6.

ac

ac

R 0.33 0.008 0.008 0.5 0.75 0.010.33 0.01 0.5 0.75 0.012 0.33 0.012 0.5 0.75

R

ANN Algorithm ANN Algorithm 0.33 ´4.72% −4.72% 0.5 ´2.18% −2.18% 0.75 ´1.20% −1.20% 0.33 ´2.59% 0.5 ´6.06% −2.59% 0.75 ´1.61% −6.06% 0.33 ´3.97% −1.61% 0.5 ´10.92% −3.97% 0.75 ´2.03% −10.92% −2.03%

Forman Algorithm

Forman Algorithm ´20.85% −20.85% 4.73% 4.73% ´34.34% −34.34% ´21.38% ´8.48% −21.38% ´37.10% −8.48% ´23.10% −37.10% ´4.76% −23.10% ´38.07% −4.76% −38.07%

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Furthermore, additional D16 aluminum alloy are are used for model validation. The Furthermore, additionaltesting testingdata datainin D16 aluminum alloy used for model validation. Furthermore, additional testing data in D16 aluminum alloy are used for model validation. The information of the is listed in Table 5 [29]. Similarly, The information of experiment the experiment is listed in Table 5 [29]. Similarly,the theANN ANNisistrained trainedwith with the the crack crack information of the experiment is listed in Table 5 [29]. Similarly, the ANN is trained with the crack growth rate data (da/dN-ΔK) under three stress ratios. The 3D fitting surface and the 2D projections growth rate data (da/dN-∆K) under three stress ratios. The 3D fitting surface and the 2D projections of growth rate data (da/dN-ΔK) under three stress ratios. The 3D fitting surface and the 2D projections of the ANN shown in Figures respectively. the ANN areare shown in Figures 14 14 andand 15,15, respectively. of the ANN are shown in Figures 14 and 15, respectively. Table alloy. Table 5. The experimental information of D16 aluminum alloy. Table 5. The experimental information of D16 aluminum alloy.

Specimen Material Specimen Material Specimen Material Crack type Crack type Crack type Specimen length Specimen length Specimen length Specimen width Specimen width Specimen width Specimen thickness Specimen thickness Specimen thickness Initial crack length Initial crack length InitialLoading crack length type Loading type Loading type

D16 Aluminum Alloy D16 Aluminum D16 AluminumAlloy Alloy Middle cracked tension specimen Middle cracked cracked tension Middle tensionspecimen specimen 500 mm 500 mm 500 mm 100mm mm 100 100 mm 0.04mm mm 0.04 0.04 mm 10 mm 10 mm mm amplitude Tension-tension,10constant Tension-tension, constant amplitude Tension-tension, constant amplitude

Figure 14. The experimental data of D16 aluminum and the corresponding ANN. Figure 14. 14. The The experimental experimental data data of of D16 D16 aluminum aluminum and and the the corresponding corresponding ANN. Figure ANN. -11 -11 -12 -12

lnda(mm) lnda(mm)

-13 -13 -14 -14

fitting curve R=0.75 fitting curve R=0.75 experimental data R=0.75 experimental data R=0.75 fitting curve R=0.5 fitting curve R=0.5 experimental data R=0.5 experimental data R=0.5 fitting curve R=0 fitting curve R=0 experimental data R=0 experimental data R=0

-15 -15 -16 -16 -17 -17 -18 -182 2

2.5 2.5

3 3

0.5

3.5 3.5

4 4

lnKmax (MPa*m0.5) lnKmax (MPa*m ) Figure 15. The fitting by ANN vs. the testing data for D16 aluminum alloy. Figure Figure 15. 15. The The fitting fitting by by ANN ANN vs. vs. the the testing testing data data for for D16 D16 aluminum aluminum alloy. alloy.

Forman’s model is still employed for comparison. The calibrated equation is Equation (10) and Forman’s model is still employed for comparison. The calibrated equation is Equation (10) and the fitting indices are listed inemployed Table 6. Figure 16 shows theThe fitting lines of Forman’s Forman’s model is still for comparison. calibrated equation equation. is Equation (10) and the fitting indices are listed in Table 6. Figure 16 shows the fitting lines of Forman’s equation. the fitting indices are listed in Table 6. Figure 16 shows the fitting. lines of Forman’s equation. 1.5903 × 10 × ∆ . (10) = 1.5903 × 10´9 × ∆3.2215 (10) (1 ˆ − 10) ˆ− da= 1.5903 ∆K∆ (1 − ) − ∆ “ (10) dN p1 ´ Rq Kc ´ ∆K

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Table6.6. The The fitting fitting indices Table indices of ofD16. D16. TableIndex 6. The fitting indices of D16. Number Index Number

r Index r Chi-Square r Chi-Square RMSE Chi-Square RMSE SSE SSE RMSE DC DC SSE

-5

10

-5

10

-6

10

-6

da (mm) da (mm)

10

0.984 Number 0.984 1.93 × 10−6 1.93 ˆ0.984 10´6 −8 7.80 ××´8 10 1.93 10−6 7.80 ˆ 10 −13 ´13 9.24 ×× 10 9.24 ˆ 10 7.80 10−8 0.964 0.964 9.24 × 10−13

DC

0.964

fitting curve R=0 experimental data R=0 fitting curve R=0 fitting curve R=0.33 experimental data R=0 experimental data R=0.33 fitting curve R=0.33 fitting curve R=0.75 experimental data R=0.33 experimental data R=0.75 fitting curve R=0.75 experimental data R=0.75

-7

10

-7

10

-8

10

-8

10

-9

10

-9

10

10 10

20 20 0.5 ) Kmax (MPa*m Kmax (MPa*m0.5 )

40

60

40

60

Figure16. 16.The Thefitting fittingby byForman’s Forman’s model model vs. alloy. Figure vs. the thetesting testingdata datafor forD16 D16aluminum aluminum alloy. Figure 16. The fitting by Forman’s model vs. the testing data for D16 aluminum alloy.

Similarly, the experimental data of D16 are used to validate the algorithm. Some additional data Similarly, thethe experimental data are used to validatethe thealgorithm. algorithm. Some additional data Similarly, experimental dataof ofD16 D16 usedprediction. to validate Some additional data (a-N curves) are utilized to compare with theare model In Table 7 the testing information of (a-N(a-N curves) are utilized to compare with the model prediction. In Table 7 the testing information curves) are utilized to compare with prediction. In Table 7 thepredictions testing information of of these a-N curves are listed. In Figure 17the themodel experimental data and the by the two these curves are listed. In Figure the experimental data the the predictions thedifferent two these a-N a-N curves areare listed. In Figure 17 the and predictions thebytwo different models plotted together. It17 isexperimental clear that thedata results ofand the proposed by model match the different models together. are plotted together. It isthe clear that the results of the model proposed model match the models are plotted It is clear that results of the proposed match the testing data testing data better than those by Forman’s model. testing better than thosemodel. by Forman’s model. better thandata those by Forman’s 0.035 0.035

Forman R=0 Forman Forman R=0 R=0.33 Forman R=0.33 R=0.75 Forman Forman R=0.75 ANN R=0 ANN ANN R=0 R=0.33 ANN ANN R=0.33 R=0.75 ANN testingR=0.75 data R=0 testing testing data data R=0 R=0.33 testing testing data data R=0.33 R=0.75 testing data R=0.75

0.03 0.03 cracklength/m length/m crack

0.025 0.025 0.02 0.02

0.015 0.015 0.01 0.01 0.005 0.0050 0

5,0000 5,0000

10,0000 25,0000 30,0000 30,0000 10,0000 15,0000 15,0000 20,0000 20,0000 25,0000 cycle cycle

Figure for D16 D16 aluminum aluminum alloy. alloy. Figure17. 17.The Thepredictions predictionsvs. vs. the the experimental experimental data data for Figure 17. The predictions vs. the experimental data for D16 aluminum alloy.

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Table 7. Loading information for the a-N curves. R

σ min

σ max

0.75 0.33 0

105 MPa 96 MPa 64 MPa

140 MPa 32 MPa 0 MPa

Assuming that the critical crack length is 0.015 m, 0.018 m, and 0.020 m, the relative errors of the two models are compared in Table 8. It is clear that the proposed model has very high accuracy and stability. Table 8. The errors of the results by two models. ac

R

ANN Algorithm

Forman Algorithm

0.015

0 0.33 0.75

´1.40% ´1.32% 2.90%

120.95% 96.60% 15.46%

0.018

0 0.33 0.75

´1.13% ´3.26% 2.16%

111.78% 82.71% 10.82%

0.02

0 0.33 0.75

1.05% ´1.69% 2.46%

105.70% 80.00% 8.61%

3.2. Validation and Comparison of the Constant Loading with a Few Overloads 3.2.1. Equivalent Stress Intensity Factor It is indicated that the plasticity ahead of the crack tip affects fatigue crack growth behavior. The retardation effects due to overload can be correlated with the plastic deformation. The plastic state caused by the previous loads is traced. Subsequently, the equivalent stress intensity factor is calculated, which is based on the equivalent plastic zone concept. The general expression of the equivalent plastic zone can be written as: , $ i i iÿ ´1 & . ÿ ÿ a0 ` da j ` Deq,i “ max a0 ` da j ` di , a0 ` da j ` Deq,i´1 (11) % j “1

j “1

j “1

where Deq.i means the size of equivalent plastic zone in the ith cycle; a0 means the initial crack length; ř da means the crack increment; di means the current plastic zone size in the ith cycle; a0 ` ij“1 da j means the crack length in the ith cycle; i means the current cycle number. A schematic sketch is given to illustrate the equivalent plastic zone concept. The loading sequential process and the corresponding plastic state variation are shown in Figure 18. The dashed zigzag lines represent the loading history. The large plastic zones have been formed at “t1 ,” and the crack tip is “O1 ” at that moment. The monotonic and reverse plastic zones can be expressed as Equation (12) [30]: $ ’ &

´ ¯2 π Kmax 8 σy ´ ¯2 π Kmax ´Kop 8 2σy

dm “

’ % dr “

(12)

where dm is the monotonic plastic zone size; and dr is the reverse plastic zone size. The current load is applied at “t2 ” and the new crack tip is “O2 ”. The large forward and reverse plastic zones, which are the dotted ellipses, form during the largest load cycle in the previous loading history. Before “t2 ”, the following plastic zones do not reach their boundaries respectively even though the crack grows.

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solid ellipses represent the equivalent plastic zones ahead of the crack tip O2. In addition, the actual of 20 plastic zone is butterfly-shaped instead of round; theoretically, however,15 their diameters along the crack direction are identical, as shown in Figure 18. In the current investigation, the equivalent plastic zone is in directly proportional to the circular diametric distance and the The solid ellipses represent the equivalent plastic zones ahead of the crack tip O2 . In addition, the proportionality coefficient is equal to or slightly greater than 1. Equation (12) can be rewritten as: actual contour of the plastic zone is butterfly-shaped instead of round; theoretically, however, their diameters along the crack direction are identical, as shown in Figure 18. In the current investigation, + zone+is in, directly + + the, , circular + + , = maxproportional , distance , the equivalent plastic to diametric and the proportionality coefficient is equal to or slightly greater than 1. Equation (12) can be rewritten as: Materials 483 contour2016, of 9,the

$ , # ∗ř , = i i iř ´1 ’ ř ’ 8 ’ a0 ` da j ` Dm,eq,i “ max a0 ` da j ` dm,i , a0 ` da j ` Dm,eq,i´1 ’ ’ ’ ’ j “1 j “1 j “1 ’ ’ ´ ¯ ’ ’ + + , , = d “ Ψ+˚ π Kmax,i + 2, , + + , , & m,i 8 σ y # i i iř ´1 ’ ř ř ’ ’ a0 ` da j ` Dr,eq,i “ max a0 ` da, j − ` dr,i ,, a0 ` da j ` Dr,eq,i´1 ’ ’ ’ ∗ j “1 j“1 j “1 ’ , = ’ 8 ´ K 2´K ¯2 ’ ’ ’ max,i op,i % dr,i “ Ψ ˚ π8 2σy

(13) (13)

in iithth cycle respectively; whereDDmm,eq,i where and Dr,eq,i arethe theequivalent equivalentmonotonic monotonic and and reverse reverse plastic zone in r ,eq,iare ,eq,i and Ψ is is the the geometry geometrymodification modificationfactor factorof ofplastic plasticzone. zone. Ψ

Figure Figure18. 18. Schematic Schematic illustration illustration of ofthe theequivalent equivalentplastic plasticzone zoneconcept. concept.

Theequivalent equivalentstress stressintensity intensityfactors factorsKKEE can can be be calculated calculated by by solving solving the the following followingequation: equation: The

= bπDD , , m,eq,i

KE “ σS

(14) (14)

where Dm,eq,i means the plastic zone in this cycle and σs means the yield limit. where Dm,eq,i means the plastic zone in this cycle and σs means the yield limit. 3.2.2. Single Overload 3.2.2. Single Overload Unlike the constant amplitude loading case, the algorithm for the single overload needs an Unlike the constant amplitude loading the algorithm for the single needs an additional parameter called equivalent stresscase, intensity factor to account for theoverload nonlinear loading additional parameter called equivalent stress intensity factor to account for the nonlinear loading interaction effect. To obtain this parameter, the equivalent plastic zone has to be calculated. Figure 19 interaction effect. To obtain this parameter, the equivalent plastic zone has to be calculated. Figure 19 shows the procedure to calculate the equivalent plastic zone. shows the procedure to calculate the equivalent plastic zone, where Dm,eq,i means the plastic zone i i 1 which characterizes the influence of the following the flow K history load.i Once Dm,eq,i is estimated d m.i ( max.i ) 2 a0 da j Dm.eq.i max{a0 da j d m.i , a0 da j Dm.eq.i 1} chart, the equivalent SIF can be calculated by using jEquation (14). Then get trained 8 y 1 j 1 the ANN can j 1 by using the training data vectors, in which the equivalent SIF, SIF and stress ratios are inputs and 19.rate The is calculation of theAdditionally, equivalent plastic the corresponding crack Figure growth the output. all zone. the training data have to be preprocessed following the procedure in Section 2.2. At last the fatigue crack growth with retardation can be estimated.

3.2.2. Single Overload Unlike the constant amplitude loading case, the algorithm for the single overload needs an additional parameter called equivalent stress intensity factor to account for the nonlinear loading interaction Materials 2016,effect. 9, 483 To obtain this parameter, the equivalent plastic zone has to be calculated. Figure 16 of19 20 shows the procedure to calculate the equivalent plastic zone. Materials 2016, 483 Materials 2016, 9, 9, 483

16 16 of 20 of 20

K

i

i

i 1

max.i 2 where m,eq,i means the plastic influence of0 the history load. Dm,eq,i which d m.i zone zone (which ) characterizes a0 dathe da Once daOnce j Dinfluence m.eq .i max{aof j d m.i , a0load. j Dm.eq 1} where DD m,eq,i means the plastic the the history D.i m,eq,i 8 y characterizes j 1 j 1 j 1 is estimated following the flow chart, the equivalent SIF can be calculated by using Equation (14). is estimated following the flow chart, the equivalent SIF can be calculated by using Equation (14). Then the ANN can get trained by using the training data vectors, in which the equivalent SIF, SIF Then the ANN can get Figure trained19. byThe using the training data vectors, in which calculation equivalent plastic zone. the equivalent SIF, SIF Figure 19.the Thecorresponding calculation of of the the equivalent plastic zone. and stress ratios are inputs and crack growth rate is the output. Additionally, all and stress ratios are inputs and the corresponding crack growth rate is the output. Additionally, all the training data have to be preprocessed following the procedure in Section 2.2. At last the fatigue the training data have to be preprocessed following the procedure in Section 2.2. At last the fatigue crack growth with retardation can be estimated. Thegrowth experimental data in D16 alloy [29] are employed to validate the model. The basic crack with retardation canaluminum be estimated. The experimental data in D16 aluminum alloy [29] are employed validate model. The information about the experiment canaluminum be seen in Table 5. da/dN-∆K curvetoto serves as the the The experimental data in D16 alloy [29] are employed validate thetraining model. vector, The basic information about the experiment can be seen in Table 5. da/dN-ΔK curve serves as the training and the a-N curve is used to validate the whole prediction algorithm. basic information about the experiment can be seen in Table 5. da/dN-ΔK curve serves as the training vector, and the a-N curve is used to validate the whole prediction algorithm. As shown ina-N Figure 20,isthe x-axis is the equivalent the y-axis is the K ; and the z-axis is the vector, and the curve to validate the whole SIF; prediction algorithm. As shown in Figure 20,used the x-axis is the equivalent SIF; the y-axis is the Kmaxmax ; and the z-axis is the crack growth rate. The red triangles the experimental data;the the blue curve the As growth shown in Figure 20,small the x-axis is theare equivalent SIF; the y-axis the K max; and therepresents z-axis is crack rate. The red small triangles are the experimental data; is blue curve represents thethe well-trained ANN; and three broken curves are its projections. It is obvious that the curve by ANN crack growth rate. The red small triangles are the experimental data; the blue curve represents the well-trained ANN; and three broken curves are its projections. It is obvious that the curve by ANN well-trained ANN; and three brokenofcurves are its projections. It is obvious that the curve by ANN can fit highly nonlinear tendency dataperfectly. perfectly. canthe fit the highly nonlinear tendency ofthe theexperimental experimental data can fit the highly nonlinear tendency of the experimental data perfectly.

-14

-14 -15 lnda(mm) lnda(mm)

-15 -16 -16 -17 -17 -18 -18 -19 -19 -20 80 -20 80

60 40

60

40 KE(MPa*m0.5)

20

20

0

0

KE(MPa*m0.5)

0

0

0.2

0.2

0.6

0.4

0.8

0.6 0.4 lnKmax (MPa*m0.5)

1

0.8

1

0.5

lnKmax (MPa*m alloy. ) Figure Thefitting fittingvs. vs.the the experimental experimental data Figure 20.20. The datafor forD16 D16aluminum aluminum alloy.

Figure 20. The vs. the experimental for D16 as aluminum alloy. Fatigue crack growth withfitting the overload effect is thusdata simulated shown in Figure 21. The extra

Fatigue crack growth with the overload effect is thus simulated as shown in Figure 21. The extra experimental information for a-N curve is listed in Table 9. Fatigueinformation crack growthfor with the overload effect thus 9, simulated as shown in Figure 21. stress The extra experimental a-N curve is listed inisTable where the Sol is the overload level. 0.03 experimental information for a-N curve is given listed for in Table 9. The prediction by Wheeler’s model is also comparison. 0.03 0.025

cracklength/m cracklength/m

0.025 0.02

0.02 0.015

0.015 0.01

0.01 0.005

0.005 0

0

2,0000

4,0000

6,0000

8,0000

10,0000 cycle

12,0000

14,0000

16,0000

18,0000

20,0000

Figure 021. The predictions vs. the experimental data for D16 aluminum alloy. 0

2,0000

4,0000

6,0000

8,0000

10,0000 cycle

12,0000

14,0000

16,0000

18,0000

20,0000

Figure21. 21.The Thepredictions predictions vs. vs. the the experimental experimental data Figure datafor forD16 D16aluminum aluminumalloy. alloy.

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Table 9. Applied loading for D16 specimen. Table 9. Applied loading for D16 specimen. Type of of Loading Loading Type CA+single CA + singleoverload overload

SS min min

MPa 00MPa

∆SΔS

Smax Smax

RR

MPa 6464 MPa 0 0 6464MPa MPa

Sol Sol

128MPa MPa 128

where the Sol is the overload stress level. The prediction by Wheeler’s model is also given for comparison. The figure shows that there is an overload applied when the crack length reaches 0.01 m. After that, The figure shows that there is an overload applied when the crack length reaches 0.01 m. After a conspicuous retardation phenomenon can be seen. The slope of the curve decreases dramatically that, a conspicuous retardation phenomenon can be seen. The slope of the curve decreases until the crack grows out of the retardation effect area after another 60,000 cycles. The prediction by dramatically until the crack grows out of the retardation effect area after another 60,000 cycles. The the ANN-based approach has a very good agreement with the testing data in this figure. However, the prediction by the ANN-based approach has a very good agreement with the testing data in this curve by However, Wheeler’sthe model growingmodel after stops the overload applied. Ribeiro al. [31]Ribeiro indicate figure. curvestops by Wheeler’s growing is after the overload is et applied. that Wheeler’s model has some difficulties in crack growth calculation when the overload is et al. [31] indicate that Wheeler’s model has some difficulties in crack growth calculation when larger the than twice that of the σ . However, the approach proposed in this paper does not have this kind max overload is larger than twice that of the σmax. However, the approach proposed in this paper does not of problem which it more generally have this kindmakes of problem which makesapplicable. it more generally applicable.

3.2.3. Multiply Overloads 3.2.3. Multiply Overloads Other differentmaterials materialsareare employed formodel the model validation [32]. Othertesting testingdata data for for different employed herehere for the validation [32]. The The information of this experiment is listed in Table 10. information of this experiment is listed in Table 10. Table of350 350WT WTsteel. steel. Table10. 10.The Theexperimental experimental information information of

Specimen Material 350WT WT Steel Specimen Material 350 Steel Crack Type Center Cracked Tension Specimen Crack Type Center Cracked Tension Specimen Specimen length 300 mm Specimen length 300 mm Specimen width 100mm mm Specimen width 100 Specimen thickness 5 mm Specimen thickness 5 mm Initial crack length 20 mm Initial crack length 20 mm of loading Tension-tension, Constant amplitude withwith overload TypeType of loading Tension-tension, Constant amplitude overload 11.4 SminSmin 11.4MPa MPa S 114 MPa Smax max 114 MPa Sol 190.95 MPa Sol 190.95 MPa

Similarly, the bythe theexperimental experimental data the result is shown in Figure Similarly, theANN ANNis is trained trained by data andand the result is shown in Figure 22. The22. The small red tangles experimental the curve blue curve the fitting the and ANN; small red tangles areare thethe experimental data;data; the blue is the is fitting by the by ANN; the and threethe three broken curves are its projections. It is seen that the ANN delivers a good fitting. broken curves are its projections. It is seen that the ANN delivers a good fitting.

lnda/dN(mm)

-14

-16

-18

1 0.8

-20 70

0.6 60

0.4 50

40

EK(MPa*m0.5)

0.2 30

20

0

lnKmax(MPa*m0.5)

Figure22. 22.The Thefitting fittingvs. vs. the the experimental experimental data Figure datafor for350 350WT WTsteel. steel.

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The The predictions predictions made made by by the the ANN-based ANN-based approach approach and and Wheeler’s Wheeler’s model model are are visualized visualized in in Figure 23 with the experimental data as well. Both the methods give good agreements with the testing Figure 23 with the experimental data as well. Both the methods give good agreements with the data. The proposed model performs slightly slightly better around the 150,000th cycle. cycle. testing data. The proposed model performs better around the 150,000th 0.035

0.03

crack length/m

0.025

0.02

0.015

0.01 ANN prediction experimental data Wheeler prediction

0.005

0

0

5,0000

10,0000

15,0000

20,0000

25,0000

30,0000

cycle

Figure 23. 23. The The predictions predictions vs. vs. the the experimental experimental data data for for 350 350 WT WT steel. steel. Figure

From the validations above, it can be concluded that the proposed method can deal with the From the validations above, it can be concluded that the proposed method can deal with the nonlinear and multivariable fatigue damage accumulation process successfully. nonlinear and multivariable fatigue damage accumulation process successfully. 4. Conclusions 4. Conclusions and and Future Future Work Work In this this paper, paper,aanovel novelmethod methodto topredict predictthe thefatigue fatigue crack crack growth growth based based on on aaradial radial basis basis function function In (RBF)-artificialneural neuralnetwork network(ANN) (ANN)isisdeveloped. developed.The The ANN-centered algorithm is also validated (RBF)-artificial ANN-centered algorithm is also validated by by comparison with the experimental data under the constant and variable amplitude loading of comparison with the experimental data under the constant and variable amplitude loading of different different materials. and Wheeler’s models are also employed for comparisons. is clear materials. Forman’sForman’s and Wheeler’s models are also employed for comparisons. It is clearItthat the that the proposed model has very high accurate and stable performance in all the examples. proposed model has very high accurate and stable performance in all the examples. All the thevalidations validationsabove above prove advantages the ANN-based algorithm in nonlinear All prove the the advantages of theofANN-based algorithm in nonlinear fatigue fatigue crack growth problems. This method still has some limitations that need further crack growth problems. This method still has some limitations that need further investigation. investigation. One major issue is that the size of the training data has a significant impact on the One major issue is that the size of the training data has a significant impact on the prediction accuracy. prediction accuracy. The other is that the method may be time consuming and computationally The other is that the method may be time consuming and computationally expensive due to its expensive duenature. to its cycle-by-cycle nature. cycle-by-cycle Acknowledgment: This Foundation of of China China Acknowledgments: This research research is is financially financially supported by the National Natural Science Foundation (No. (No. 51405009) 51405009) and and the the Fundamental FundamentalResearch ResearchFunds Fundsfor forthe theCentral CentralUniversities. Universities. Author AuthorContributions: Contributions: Wei WeiZhang Zhangorganized organizedthe theresearch; research;Zhangmin ZhangminBao Baocarried carriedon onthe themodel modelsimulation simulation and and wrote the manuscript; Shan Jiang wrote programs and checked the manuscript; Jingjing He helped write the wrote the manuscript; Shan Jiang wrote programs and checked the manuscript; Jingjing He helped write the manuscript and provide the academic support. manuscript and provide the academic support. Conflicts of Interest: The authors declare no conflict of interest. Conflicts of Interest: The authors declare no conflict of interest.

References Reference 1. 1. 2. 2. 3. 3.

Paris, P.; Erdogan, F. A critical analysis of crack propagation laws. J. Basic Eng. 1963, 85, 528–534. [CrossRef] Paris, P.;R.G.; Erdogan, F. AV.E.; critical analysis of crack propagation Basic Eng. 1963, 85, 528–534.structures. Forman, Kearney, Engle, R.M. Numerical analysis oflaws. crack J.propagation in cyclic-loaded R.G.; Kearney, V.E.; [CrossRef] Engle, R.M. Numerical analysis of crack propagation in cyclic-loaded J.Forman, Fluids Eng. 1967, 89, 459–463. ® J. structures. Fluids Eng. 1967, and 89, 459–463. NASGRO Fracture Mechanics Fatigue Crack Growth Analysis Software; Version 4.02; NASA-JSC and SwRI: ® NASGROTX, Fracture Mechanics and Fatigue Crack Growth Analysis Software, Version 4.02; NASA-JSC and Houston, USA, 2002. SwRI: Houston, TX, USA, 2002.

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Newman, J.C. Prediction of Crack Growth under Variable-Amplitude Loading in Thin-Sheet 2024-T3 Aluminum Alloys; NASA, Langley Research Center: Hampton, VA, USA, 1997. Wheeler, O.E. Spectrum loading and crack growth. J. Fluids Eng. 1972, 94, 181–186. [CrossRef] De Koning, A.U. A simple crack closure model for prediction of fatigue crack growth rates under variable-amplitude loading. ASTM STP 1980, 743, 63–85. Hagan, M.T.; Demuth, H.B.; Beale, M.H. Neural Network Design; PWS Publisher: Boston, MA, USA, 1996. De Albuquerque, V.H.C.; de Alexandria, A.R.; Cortez, P.C.; Tavares, J.M.R.S. Evaluation of multilayer perceptron and self-organizing map neural network topologies applied on microstructure segmentation from metallographic images. NDT E Int. 2009, 42, 644–651. [CrossRef] De Albuquerque, V.H.C.; Tavares, J.M.R.S.; Cortez, P.C. Quantification of the microstructures of hypoeutectic white cast iron using mathematical morphology and an artificial neural network. Int. J. Microstruct. Mater. Prop. 2010, 5, 52–64. [CrossRef] De Albuquerque, V.H.C.; Cortez, P.C.; de Alexandria, A.R.; Tavares, J.M.R.S. A new solution for automatic microstructures analysis from images based on a backpropagation artificial neural network. Nondestruct. Test. Eval. 2008, 23, 273–283. [CrossRef] Kang, J.Y.; Song, J.H. Neural network applications in determining the fatigue crack opening load. Int. J. Fatigue 1998, 20, 57–69. [CrossRef] Venkatesh, V.; Rack, H.J. A neural network approach to elevated temperature creep-fatigue life prediction. Int. J. Fatigue 1999, 21, 225–234. [CrossRef] Artymiak, P.; Bukowski, L.; Feliks, J.; Narberhaus, S.; Zenner, H. Determination of S-N curves with the application of artificial neural networks. Fatigue Fract. Eng. Mater. Struct. 1999, 22, 723–728. [CrossRef] Pleune, T.T.; Chopra, O.K. Using artificial neural networks to predict the fatigue life of carbon and low-alloy steels. Nucl. Eng. Des. 2000, 197, 1–12. [CrossRef] Genel, K. Application of artificial neural network for predicting strain-life fatigue properties of steels on the basis of tensile tests. Int. J. Fatigue 2004, 26, 1027–1035. [CrossRef] Lowe, D.; Broomhead, D. Multivariable functional interpolation and adaptive networks. Complex Syst. 1988, 2, 321–355. Ghandehari, S.; Montazer-Rahmati, M.M.; Asghari, M. Modeling the Flux Decline during Protein Microfiltration: A. Comparison between Feed-Forward Back Propagation and Radial Basis Function Neural Networks. Sep. Sci. Technol. 2013, 48, 1324–1330. [CrossRef] Fathi, A.; Aghakouchak, A.A. Prediction of fatigue crack growth rate in welded tubular joints using neural network. Int. J. Fatigue 2007, 29, 261–275. [CrossRef] Abdalla, J.A.; Hawileh, R. Modeling and simulation of low-cycle fatigue life of steel reinforcing bars using artificial neural network. J. Frankl. Inst. 2011, 348, 1393–1403. [CrossRef] Porter, T.R. Method of analysis and prediction for variable amplitude fatigue crack growth. Eng. Fract. Mech. 1972, 4, 717–736. [CrossRef] Zhan, W.; Lu, N.; Zhang, C. A new approximate model for the R-ratio effect on fatigue crack growth rate. Eng. Fract. Mech. 2014, 119, 85–96. [CrossRef] Hudson, C.M. Effect of Stress Ratio on Fatigue-Crack Growth in 7075-T6 and 2024-T3 Aluminum-Alloy Specimens; National Aeronautics and Space Administration: Washington, DC, USA, 1969. Wolf, E. Fatigue crack closure under cyclic tension. Eng. Fract. Mech. 1970, 2, 37–45. [CrossRef] McEvily, A.J. Phenomenological and microstructural aspects of fatigue. Microstruct. Des. Alloys 1973, W36, 204–225. Mohanty, J.R.; Mahanta, T.K.; Mohanty, A.; Thatoi, D.N. Prediction of constant amplitude fatigue crack growth life of 2024 T3 Al alloy with R-ratio effect by GP. Appl. Soft Comput. 2015, 26, 428–434. [CrossRef] Wöhler, A. Über die Festigkeits-Versuche mit Eisen und Stahl [on Strength Tests of Iron and Steel]; Ernst & Korn: Berlin, Germany, 1870; Volume 20, pp. 73–106. Vecchio, R.S.; Hertzberg, R.W.; Jacard, R. An overload-induced fatigue crack propagation behaviour in aluminium and steel alloy. Fatigue Fract. Eng. Mater. Struct. 1984, 7, 181–194. [CrossRef] Topper, T.H.; Yu, M.T. The effect of overloads on threshold and crack closure. Int. J. Fatigue 1985, 7, 159–164. [CrossRef] Schijve, J.; Skorupa, M.; Skorupa, A.; Machniewicz, T.; Gruszczynski, P. Fatigue crack growth in the aluminium alloy D16 under constant and variable amplitude loading. Int. J. Fatigue 2004, 26, 1–15. [CrossRef]

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An Artificial Neural Network-Based Algorithm for Evaluation of Fatigue Crack Propagation Considering Nonlinear Damage Accumulation Wei Zhang *, Zhangmin Bao, Shan Jiang and Jingjing He School of Reliability and Systems Engineering, Beihang University, Haidian District, Beijing 100089, China; [email protected] (Z.B.); [email protected] (S.J.); [email protected] (J.H.) * Correspondence: [email protected]; Tel.: +86-10-8231-4649 Academic Editor: Mark T. Whittaker Received: 5 March 2016; Accepted: 10 June 2016; Published: 17 June 2016

Abstract: In the aerospace and aviation sectors, the damage tolerance concept has been applied widely so that the modeling analysis of fatigue crack growth has become more and more significant. Since the process of crack propagation is highly nonlinear and determined by many factors, such as applied stress, plastic zone in the crack tip, length of the crack, etc., it is difficult to build up a general and flexible explicit function to accurately quantify this complicated relationship. Fortunately, the artificial neural network (ANN) is considered a powerful tool for establishing the nonlinear multivariate projection which shows potential in handling the fatigue crack problem. In this paper, a novel fatigue crack calculation algorithm based on a radial basis function (RBF)-ANN is proposed to study this relationship from the experimental data. In addition, a parameter called the equivalent stress intensity factor is also employed as training data to account for loading interaction effects. The testing data is then placed under constant amplitude loading with different stress ratios or overloads used for model validation. Moreover, the Forman and Wheeler equations are also adopted to compare with our proposed algorithm. The current investigation shows that the ANN-based approach can deliver a better agreement with the experimental data than the other two models, which supports that the RBF-ANN has nontrivial advantages in handling the fatigue crack growth problem. Furthermore, it implies that the proposed algorithm is possibly a sophisticated and promising method to compute fatigue crack growth in terms of loading interaction effects. Keywords: fatigue crack growth; artificial neural network; nonlinear multivariable function; retardation; loading interaction

1. Introduction As the damage tolerance concept is now widely accepted and applied in the aerospace and aviation industries, it has become increasingly important to analyze how a fatigue crack grows. Linear elastic fracture mechanics (LEFM), moreover, is the fundamental theory for establishing the analytical model of fatigue crack propagation. Paris and Erdogan [1] correlate the stress intensity factor (SIF) range with the fatigue crack growth rate, and propose this seminal model as Equation (1). da “ CP p∆KqmP dN

(1)

where ∆K is the SIF range and CP and mP are the fitting parameters. This equation shows a linear relationship between da/dN and ∆K in the log-log coordinate. However, a major limitation of the Paris equation is that CP has to change along with the variation of the stress ratio (R). Additionally, it is only applicable to the linear region without the consideration of the threshold SIF (∆Kth ) and the critical SIF (Kc ). Materials 2016, 9, 483; doi:10.3390/ma9060483

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To perfect the Paris equation, many researchers have attempted to make modifications in order to involve more nonlinear factors. Forman et al. [2] take the effects of R and Kc into consideration and propose a modified model as shown in Equation (2). CF p∆Kqm F da “ p1 ´ Rq Kc ´ ∆K dN

(2)

where CF and mF are the fitting parameters. Furthermore, based on the Paris equation, some researchers develop more general formulas by employing additional parameters to account for nonlinearity, such as the NASGRO formula [3] in Equation (3): ´ „ n 1 ´ da p1 ´ f q “ CNa ∆K ´ dN p1 ´ Rq 1´

∆Kth ∆K Kmax Kc

¯PNa ¯q Na

(3)

where CNa , PNa and qNa are the fitting parameters; and f is Newman’s [4] crack opening function determined by the experimental measurement. The series of models discussed above are put forward to illustrate the nonlinear relationship between the crack growth rate and the SIF range for constant amplitude loadings. Moreover, many researchers have investigated the loading interaction effect, which is complicated and of great significance to the variable amplitude loading. In this paper, the constant amplitude loading with overload, which is the typical and simple variable amplitude loading, is studied. Wheeler proposes a plastic-zone-based model [5] to describe the crack growth retardation caused by overload as shown in Equation (4). $ ´ ¯ da & da dN VA “ γ dN ´ r ¯m (4) % γ “ p,i λ where γ is the retardation factor; rp,i is the size of current plastic zone; λ is the distance between the current crack tip and the edge of the plastic zone caused by overload; and m is the fitting parameter. De Koning [6] also develops a plastic-deformation-based model to deal with the overload effect. Wheeler’s and Koning’s models support that the plastic zone (monotonic plastic zone or reversed plastic zone) is the key parameter to correlate with loading interaction effects in the fatigue crack growth calculation. Most studies focus on accurately quantifying the nonlinear relationship between the crack growth rate and the driving parameters by using an explicit and simple function. To achieve these goals, many studies have been undertaken to introduce more parameters to construct a formula which can fit the experimental data better. However, the current formulas are not flexible enough to positively handle all the situations. Overall, the process of fatigue crack growth is a nonlinear and multivariable problem under both constant and variable amplitude loading. Fortunately, the artificial neural network (ANN) has an excellent ability to fit the nonlinear multivariable relationship, which makes it a sophisticated and promising approach to the fatigue crack growth problem. ANN is a family of algorithms based on the imitation of biological neural networks. It has the strong ability to estimate the tendency of nonlinear and multivariable functions based on a large amount of data [7]. Thanks to these advantages, ANN is widely applied to damage estimation in the material sciences [8–10]. Furthermore, it is used to deal with some fracture problems including creep, fatigue and even corrosion fatigue [11–15]. A novel ANN-based algorithm is proposed in this paper to evaluate the process of fatigue crack growth. In the following sections, the ANN is first established and its training outlined. While establishing the ANN, the equivalent SIF is used to account for the influence of the loading history. Subsequently, the ANN-centered algorithm is developed and validated by using experimental data under the constant

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amplitude Materials 2016,loading 9, 483 with different stress ratios or overloads. Some classical models are also employed 3 of 20 for comparison. In the final section, some conclusions and considerations are given. Subsequently, the ANN-centered algorithm is developed and validated by using experimental data 2. Methodology under the constant amplitude loading with different stress ratios or overloads. Some classical models are also employed for comparison. In the final section, some conclusions and considerations 2.1. Radial Basis Function Artificial Neural Network are given. In the 1980s, ANN technology became popular for dealing with practical problems. As it is 2. Methodology inspired by biological neural networks, it shares some features with the human brain, particularly learning by example. Radial basis F = function (RBF) network is a type of ANN which uses radial 2.1. Radial BasisasFunction Artificialfunction. Neural Network basis function the activation Because of the RBF network’s ability to produce optimal approximate solutions and local learning, it is used in function approximation, system control, etc. In the 1980s, ANN technology became popular for dealing with practical problems. As [16]. it is The RBF ANN structure is displayed in Figure 1, where {x , x . . . x } is the input vector; m is the m0 1 2 0 inspired by biological neural networks, it shares some features with the human brain, particularly dimension of the vector; w , w , . . . , w are the connection weights between the middle layer and n 1 basis 2 learning by example. Radial F = function (RBF) network is a type of ANN which uses radial the output layer; is the number of the radial of basis in the middle layer. As shown in basis function as and the N activation function. Because the functions RBF network’s ability to produce optimal Figure 1, the RBF ANN and consists three layers: input layer, the middle layer and the output layer. approximate solutions localoflearning, it is the used in function approximation, system control, etc. The input layer is composed of m source points which connect the ANN to the external environment. 0 [16]. The RBF ANN structure is displayed in Figure 1.

Figure 1. 1. Schematic of radial radial basis basis function function (RBF)-artificial (RBF)-artificial neural neural network network (ANN). (ANN). Figure Schematic illustration illustration of

where {x1, x2…xm0} is the input vector; m0 is the dimension of the vector; w1, w2, …, wn are the The second layer is the only hidden layer in the RBF network. Its function is to transform the connection weights between the middle layer and the output layer; and N is the number of the radial input space into the hidden space nonlinearly. The hidden layer consists of N cells that can be defined basis functions in the middle layer. As shown in Figure 1, the RBF ANN consists of three layers: the mathematically by the radial function shown in Equation (5). The RBF network is good at local input layer, the middle layer and the output layer. The input layer is composed of m0 source points approximation because the radial function in the hidden layer responds to theinput partially. which connect the ANN to the external environment. ` layer in the ˘ RBF network. Its function is to transform the The second layer is the only hidden ϕ j pxq “ ϕ ||x ´ x j || , j “ 1, 2, ¨ ¨ ¨ ¨ ¨ ¨ , N (5) input space into the hidden space nonlinearly. The hidden layer consists of N cells that can be definedx mathematically by the radial function shown in Equation th (5). The RBF network is good at where j means the center of the radial function defined by the j source point; and x is the signal local directly approximation because the radial function the hidden responds to used the which acts on the input layer. Additionally, the in Gaussian functionlayer is the most widely input partially. radial function, and the cells in the hidden layer can be defined as in Equation (6) shown below. ¸ ( )= ˜ ‖ − (5) , = 1,2,······, ` ˘ 1 2 ϕ pxq “ ϕ x ´ x j “ exp ´ 2 ||x ´ x j || j “ 1, 2, ¨ ¨ ¨ ¨ ¨ ¨ , N (6) where xj means thej center of the radial function 2σjdefined by the jth source point; and x is the signal which directly acts on the input layer. Additionally, the Gaussian function is the most widely used radial function, and the hiddenGaussian layer can function, be defined in Equation below. where σj is the width of cells the jthinxthe xj as is the center of(6) theshown jth basis function, j -centered ||x ´ xj || is the vector norm of x ´ xj which means1the distance between x and xj . Finally, the nodal ) =will generate − = ) = 1,2,······, points in the output(layer theexp(− output data.‖ − (6) 2 RBF ANN is one type of feedforward static neural network. The feedforward network is the simplest network as the information canGaussian only move in onexdirection. Theoforiginal feedforward the width of the jth xj-centered function, j is the center the jth basis function, where σj is network is the a single perceptron based the on other networks consisting multiplethe layers of vector norm of layer x − xjnetwork which means distance between x and xof j. Finally, nodal ‖x − xj‖ is computational units layer such as the RBF network. The data. RBF network is able to fit a continuous nonlinear points in the output will generate the output process a satisfied automatically the weight offeedforward the functionsnetwork in the hidden RBFinANN is oneprecision type of by feedforward staticadjusting neural network. The is the layer. Some studies that the can ANN hasmove advantages dealing The with original nonlinear problems. simplest network as indicate the information only in one in direction. feedforward network is a single perceptron layer network based on other networks consisting of multiple layers of computational units such as the RBF network. The RBF network is able to fit a continuous nonlinear process in a satisfied precision by automatically adjusting the weight of the functions in the hidden layer. Some studies indicate that the ANN has advantages in dealing with nonlinear

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problems. Ghandehari et al. [17] discuss the advantages of RBF over the back propagation (BP) network, which the most widely used and popular feedforward Fathi andnetwork, Aghakouchak Ghandehari et al.is[17] discuss the advantages of RBF over the backnetwork. propagation (BP) which [18], wellwidely as Abdalla andpopular Hawileh [19] applied the RBF network to fatigue crack is theas most used and feedforward network. Fathi and Aghakouchak [18], problems as well as successfully. In this paper, the RBF was chosen due to its multiplesuccessfully. advantages;Ininthis view of Abdalla and Hawileh [19] applied thenetwork RBF network to fatigue crack problems paper, the capacity, is suitable for establishing the in function crack growthit the RBF RBF network’s network was chosenitdue to its multiple advantages; view ofbetween the RBF fatigue network’s capacity, and the driving parameters.the function between fatigue crack growth and the driving parameters. is suitable for establishing 2.2. The The Establishment Establishment and and Training Training of of the the Artificial Artificial Neural Neural Network Network (ANN) 2.2. (ANN) In this this section, section,the theMATLAB MATLAB(©1984–2011 (©1984–2011MathWorks. MathWorks. All Allrights rightsreserved, reserved,MathWorks, MathWorks,Natick, Natick, In MA, USA) USA)software softwareisisused usedtotoestablish establishand andtrain trainthe theRBF-ANN RBF-ANNasasshown shownininFigure Figure First MA, 2. 2. First of of all,all, a a multi-input single-output RBF-ANNisisestablished establishedby byanalyzing analyzingthe thephysical physicalprocess processof offatigue fatiguecrack crack multi-input single-output RBF-ANN growth. The Theraw rawexperimental experimentaldata datathen thenneed needto tobe bepreprocessed preprocessed before before training training the the ANN. ANN. The The data data growth. preprocessing includes two steps: the first step is to take the logarithm of ∆K to reduce reduce the the preprocessing ΔK and da/dN da/dN to influence from the order of the magnitude; the second step is to normalize the data from the first step. influence from the order of the magnitude; the second step is to normalize the data from the first step. After preprocessing, the experimental a number of vectors, which are After experimentaldata datahave havebeen beentransformed transformedinto into a number of vectors, which usedused to train the ANN. ANN can be can trained automatically by using by the using MATLAB During the are to train the ANN. ANN be trained automatically the toolbox. MATLAB toolbox. training, some parameters can be tuned for optimization, including: the mean square error (MSE) During the training, some parameters can be tuned for optimization, including: the mean square goal, (MSE) expansion of RBF, maximum of neurons, For example, the example, MSE goalthe controls error goal,speed expansion speed of RBF,number maximum numberetc. of neurons, etc. For MSE the fitting accuracy. By comparing the output with testing data balancing thebalancing accuracy and goal controls the fitting accuracy. By comparing thethe output with theand testing data and the efficiency, the optimal tuning parameters can be determined. accuracy and efficiency, the optimal tuning parameters can be determined.

Data preprocessing

Establish and train the RBF ANN

Compare with the experimental data

Tuning the parameters and optimize ANN Figure Figure2. 2.Procedure Procedureof of developing developing aa well-trained well-trained ANN. ANN.

2.2.1. 2.2.1. The TheConstant Constant Amplitude Amplitude Loading Loading The The experimental experimental data data [20] [20] are are plotted plotted in in Figure Figure 3. 3. The The x-axis x-axis isis the the SIF SIF range; range; the the y-axis y-axis is is the the crack growth rate; and the different kinds of dots represent the testing data with different stress crack growth rate; and the different kinds of dots represent the testing data with different stress ratios. ratios. It can be seen that the testing data do not follow a perfect linear tendency. It can be seen that the testing data do not follow a perfect linear tendency. With the experimental data in Figure 3, the ANN can be trained following the procedure in With the experimental data in Figure 3, the ANN can be trained following the procedure in Figure 2. For the constant amplitude loading, the plasticity, on behalf of the historical load, is Figure 2. For the constant amplitude loading, the plasticity, on behalf of the historical load, is proportional to the current loading. The SIF and stress ratio are therefore chosen to be the inputs, proportional to the current loading. The SIF and stress ratio are therefore chosen to be the inputs, and and the crack growth rate is the output. The training vectors are preprocessed to make them suitable the crack growth rate is the output. The training vectors are preprocessed to make them suitable for the for the ANN. During training, the ANN can learn deeply from the limited data and establish the ANN. During training, the ANN can learn deeply from the limited data and establish the continuous continuous function between the inputs and the output. function between the inputs and the output. The fitting surface by well-trained ANN and the testing data are shown in Figure 4. The blue The fitting surface by well-trained ANN and the testing data are shown in Figure 4. The blue dots dots represent the training data, and the red crosses are the data for validation. It can be observed represent the training data, and the red crosses are the data for validation. It can be observed that the that the fitting surface can match all the experimental data well, even though they are not perfectly fitting surface can match all the experimental data well, even though they are not perfectly log-linear. log-linear. The nonlinearity of the data can be studied by the ANN so that its prediction has a higher The nonlinearity of the data can be studied by the ANN so that its prediction has a higher accuracy accuracy than the tradition log-linear formulas. Moreover, the ANN can offer a continuous than the tradition log-linear formulas. Moreover, the ANN can offer a continuous predicting surface in predicting surface in the domain of definition based on the limited and discrete training data. This the domain of definition based on the limited and discrete training data. This example shows ANN’s example shows ANN’s advantage in fitting and extrapolating the crack growth rate under constant advantage in fitting and extrapolating the crack growth rate under constant amplitude loading with amplitude loading with different stress ratios. different stress ratios.

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Figure 3. 3. The The experimental experimental data Figure data of of Al7075-T6. Al7075-T6. Figure 3. The experimental data of Al7075-T6.

Figure 4. The experimental data of Al7075-T6 and the corresponding ANN. Figure 4. The experimental data of Al7075-T6 and the corresponding ANN. Figure 4. The experimental data of Al7075-T6 and the corresponding ANN.

2.2.2. Single Overload 2.2.2. Single Overload 2.2.2. Single For theOverload variable amplitude loading, the load interaction effects cannot be ignored, because the For the variable amplitude loading,isthe load interaction cannot be ignored, theas influence of historical loading sequence dependent on the effects current load cycle. Single because overload, For the variable amplitude loading, the load interaction effects cannot be ignored, because the influence of historical loading sequence is dependent the current load cycle. Single overload, the simplest and most typical variable amplitude on loading, is investigated in this paper asto influence of historical loading sequence is dependent on the current load cycle. Single overload, as the the simplesttheand mostinteraction typical variable demonstrate loading effect. amplitude loading, is investigated in this paper to simplest and most typical variable amplitude loading, is investigated in this paper to demonstrate the demonstrate the loading interaction effect. As is well known, an applied overload can lead to fatigue crack growth retardation or even crack loadingAsinteraction effect. is well known, an applied overload can lead to fatigue crack growth retardation or even crack arrest. This phenomenon is caused by the loading interaction effect, and its existence obviously As is well known, an applied can leadinteraction to fatigueeffect, crackand growth retardation or even arrest. This phenomenon is causedoverload by the loading its existence obviously stimulates nonlinear damage accumulation. Wheeler [5], De Koning [6] and many other researchers crack arrest.nonlinear This phenomenon is causedWheeler by the [5], loading interaction and its existence stimulates damage accumulation. De Koning [6] andeffect, many other researchers [21–27] have introduced additional parameters to describe the influence of the historical loading obviously stimulates nonlinear damage accumulation. Wheeler [5], De Koning and many other [21–27] have introduced additional parameters to describe the influence of the [6] historical loading sequence. Wheeler [5] models the retardation by correlating the plastic zone size ahead of the crack sequence. [21–27] Wheelerhave [5] models the retardation correlatingtothe plastic the zoneinfluence size ahead crack researchers introduced additionalby parameters describe of of thethe historical tip with the crack growth rate. Topper and Yu [28] use the plasticity-induced crack closure to explain tip with the crackWheeler growth rate. Topperthe andretardation Yu [28] useby thecorrelating plasticity-induced crack closure to explain loading sequence. [5] models the plastic zone size ahead of the this phenomenon. Above all, the plasticity ahead of the crack tip is a reasonable parameter to this tip phenomenon. Above all, the ahead the crack tip is a reasonablecrack parameter toto crack with the crack growth rate.plasticity Topper and Yu of [28] use the plasticity-induced closure account for the loading interaction effect. In this paper, a concept “equivalent stress intensity factor”, account forphenomenon. the loading interaction In this paper, a concept “equivalent stress intensity factor”,to explain this Above all,effect. the plasticity ahead of the crack tip is a reasonable parameter which is derived from the equivalent plastic zone, is employed as input data to handle the nonlinear which for is derived from interaction the equivalent plastic zone, is employed as “equivalent input data tostress handle the nonlinear account the loading effect. In this paper, a concept intensity factor”, damage accumulation. The details are discussed in Section 3.2.1. damage accumulation. The details are discussed in Section 3.2.1.

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2.3. A Fatigue Life Prediction Method which is derived from the equivalent plastic zone, is employed as input data to handle the nonlinear There are three steps calculating the fatigue length. damage accumulation. Thetodetails are discussed incrack Section 3.2.1.First the crack increment within one load cycle is computed; the the crack length, the geometric factor, and the SIF are subsequently 2.3. A Fatigue Life Prediction updated, thereby preparingMethod the inputs for the next cycle. In repeating this process, the fatigue crack propagation is simulated cycle by cycle. The framework is shown in Equations (7) and (8). There are three steps to calculating the fatigue crack length. First the crack increment within one load cycle is computed; the the crack length, the geometric factor, and the SIF are subsequently updated, thereby preparing the=inputs this process, the fatigue crack + for the next = cycle. + In repeating (∆ , ,···) propagation is simulated cycle by cycle. The framework is shown in Equations (7) and (8).

$ I = (∆ , ř I ,···) ` ˘ ř (7) ’ ’ a “ a ` da “ a ` g ∆K j , R, ¨ ¨ ¨ ’ 0 0 I j ’ ’ ’ j “1 & ∆j“da1 = ∆` ×˘ “ g ∆K , R, ¨¨¨ (7) j dI ’ ? ’ ∆K “ ∆σ πa ˆ q ’ = ( ) j j ’ ´a ¯ ’ ’ % q “ Q wj where aI is crack length in the Ith cycle; a0 is the initial crack length; g(ΔKj, R, …) denotes the general relationship between the crackthgrowth rate and applied load; daj is the increment during the jth cycle; where aI is crack length in the I cycle; a0 is the initial crack length; g(∆Kj , R, . . . ) denotes the general Δσ is the stress amplitude; Q is the geometric factor while w is the width of the specimen. relationship between the crack growth rate and applied load; daj is the increment during the jth cycle; Furthermore, once the failure criterion or the critical crack length is provided, the fatigue life can ∆σ is the stress amplitude; Q is the geometric factor while w is the width of the specimen. Furthermore, be determined. once the failure criterion or the critical crack length is provided, the fatigue life can be determined. In this study, the ANN is used to quantify the relationship between the loading and the crack In this study, the ANN is used to quantify the relationship between the loading and the crack increment per cycle instead of the traditional equation. Equation (7) can therefore be transformed increment per cycle instead of the traditional equation. Equation (7) can therefore be transformed into into Equation (8). Equation (8). I ÿ ` ˘ aI = “ a0 `+ f ANN (∆ ∆K j , R, ¨¨ (8) (8) , ¨,···) j “1

where . . )represents representsaageneral generalANN ANNfunction functiondescribing describingthe therelationship relationship between (ΔK, R, R, .…) between wherefANN fANN (∆K, driving parameters and crack growth rate. Generally, the driving parameters would include SIF range driving parameters and crack growth rate. Generally, the driving parameters would include SIF (∆K), stress zone, etc. Additionally, an ANN-based framework for fatigue crackcrack growth range (ΔK),ratios, stress plastic ratios, plastic zone, etc. Additionally, an ANN-based framework for fatigue calculation can be established. The flow chart is shown in Figure 5. growth calculation can be established. The flow chart is shown in Figure 5.

Figure 5. 5. The Figure The flow flowchart chartof ofthe theANN-based ANN-basedalgorithm. algorithm.

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3. 3. Validation Validationand andComparison Comparison 3.1. Loading with with Different Different Stress Stress Ratios Ratios 3.1. Validation Validation and and Comparison Comparison of of the the Constant Constant Loading 3.1.1. ANN Training As the ANN can quantify the relationship from experimental data, it is significant to select the suitable data data for for the the training. training.In Inthis thissection, section,the thetesting testingdata data[20] [20]ofof7075-T6 7075-T6 aluminum alloy is used aluminum alloy is used to to train ANN globally. information of the experiment is listed in Table train thethe ANN globally. TheThe information of the experiment is listed in Table 1. 1. Table 1. The experimental information of Al7075-T6.

Material Material Specimen type Specimen type Specimen Specimenlength length Specimen Specimenwidth width Specimen thickness Specimen thickness Initial cracklength length Initial crack Loading type Loading type

Al7075-T6 Al7075-T6 Middle cracked tension specimen Middle cracked tension specimen 889 mm 889 mm 305 mm 305 mm 2.28 mm 2.28 mm 2.5 mm 2.5 mm Tension-tension, constant amplitude Tension-tension, constant amplitude

With these between thethe loading andand fatigue crack growth rate these experimental experimentaldata datathe therelationship relationship between loading fatigue crack growth can be by the rate canfitted be fitted byANN. the ANN. experimental data data with with different different stress stress ratios. ratios. At first, the ANN is trained by all five sets of the experimental The fitting curves are plotted with the original data in Figure 6. In this figure, the x-axis is the stress ratios (from 0 to 1); the y-axis is the SIF in logarithmic coordinate; and the z-axis is the crack growth The blue blue cycles cycles represent represent the experimental experimental data; and the dark blue rate in logarithmic coordinate. The is observed observed clearly clearly that that the the curves curves fit fit the the experimental experimental data data well. well. lines are the ANN prediction. ItIt is Additionally, Additionally, the the projections projections of of the the fitting fittingcurves curvesare arealso alsoprovided. provided.

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Figure data. Figure 6. 6. The The fitting fitting curves curves of of ANN ANN trained trained with with all all data.

To observe the fitting accuracy clearly, Figure 7 displays the prediction and the experimental To observe the fitting accuracy clearly, Figure 7 displays the prediction and the experimental data in a 2D plot. From the picture it can be seen that the nonlinear fitting curves by ANN can fit the data in a 2D plot. From the picture it can be seen that the nonlinear fitting curves by ANN can fit the experimental data well. experimental data well.

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Figure 7. The fitting by ANN vs. the testing data for Al7075-T6. Figure 7. The fitting by ANN vs. the testing data for Al7075-T6.

In this part part the impact impact from data data size on on the fitting fitting performance is is investigated. This the In this This time time the In this part the the impact from from data size size on the the fitting performance performance is investigated. investigated. This time the training vectors only include thre sets of experimental data with stress ratios 0.02, 0.33 and 0.75; the training vectors vectors only only include include thre thre sets sets of of experimental data with with stress stress ratios ratios 0.02, 0.02, 0.33 0.33 and and 0.75; 0.75; the the training experimental data other used for for validation. The other experimental experimental data data are are used The ANN ANN prediction prediction and and the the experimental experimental data data are are other experimental data are used for validation. validation. The ANN prediction and the experimental data are shown in Figure 8. In this picture only the purple crosses are the experimental data used to train the shown in Figure 8. In this picture only the purple crosses are the experimental data used to train the shown in Figure 8. In this picture only the purple crosses are the experimental data used to train the ANN. It It is obvious obvious that the the fitting accuracy accuracy is still still satisfactory compared compared with Figure Figure 8. ANN. ANN. It is is obvious that that the fitting fitting accuracy is is still satisfactory satisfactory compared with with Figure8. 8.

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Figure 8. The fitting curves of ANN trained with part of data. Figure of data. data. Figure 8. 8. The The fitting fitting curves curves of of ANN ANN trained trained with with part part of

Forman’s equation is also utilized to calculate fatigue crack growth under different stress ratios Forman’s equation is also utilized to calculate fatigue crack growth under different stress ratios to make a comparison. shows to thecalculate calibration indices ofgrowth Equation (9). different The fitting parameters Forman’s equation Table is also2utilized fatigue crack under stress ratios to to make a comparison. Table 2 shows the calibration indices of Equation (9). The fitting parameters are calibrated with the global database. The prediction by Equation (9) (9). is displayed in parameters Figure 9. It can make a comparison. Table 2 shows the calibration indices of Equation The fitting are are calibrated with the global database. The prediction by Equation (9) is displayed in Figure 9. It can be concluded predictions Forman’s equation are linear log-log coordinate calibrated withthat the the global database.by The prediction by Equation (9) isin displayed in Figure 9.while It canthe be be concluded that the predictions by Forman’s equation are linear in log-log coordinate while the ANN prediction curves are nonlinear. concluded that the predictions by Forman’s equation are linear in log-log coordinate while the ANN ANN prediction curves are nonlinear. prediction curves are nonlinear. where r is the coefficient of association; RMSE is the root-mean-square 2. The fitting indices of Al7075-T6. coefficient. error; SSE is the sum of squares Table for error and DC is the determination Table 2. The fitting indices of Al7075-T6. The Fitting Indexes ´9 Number 3.5241 ThedaFitting Indexes Number 2.9838 ˆ 10 ˆ ∆K 0.796 “ r dN r p1 ´ Rq Kc ´ ∆K 0.796 Chi-Square 0.000513 Chi-Square 0.000513−5 RMSE 1.47 × 10−5 RMSE 1.47 × 10−8 SSE 7.52× 10−8 SSE 7.52× 10 DC 0.796 DC 0.796

(9)

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Table 2. The fitting indices of Al7075-T6.

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where r is2016, the9, 483 coefficient of association; RMSE is the root-mean-square error; SSE is the 9sum Materials of 20 of r squares for error and DC is the determination coefficient. 0.796 Chi-Square 0.000513 where r is the coefficient of association; RMSE is the 1.47 root-mean-square error; SSE is the sum of ´5 . RMSE × ∆ˆ 10´8 2.9838 × 10 squares for error and DC is the determination coefficient. SSE 7.52 ˆ 10 (9) = DC (1

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×∆ −∆

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Figure 9. 9. The The fitting fitting by by Forman’s Forman’s model model vs. vs. the the testing testing data data for for Al7075-T6. Al7075-T6. Figure Figure 9. The fitting by Forman’s model vs. the testing data for Al7075-T6.

With the the good good performance performance of of ANN ANN under under the the discrete discrete R R values, values, it it is is reasonable reasonable that that the the ANN ANN With Withathe good performance of ANN under the discrete R values, it is reasonable that the ANN can deliver good fitting surface within the continuous domain as shown in Figure 4. can deliver a good fitting surface within the continuous domain as shown in Figure 4. canFurthermore, deliver a good fitting surface within the continuous domain as Figure shown 10 in Figure additional material is utilized utilized to test test the the ANN. shows 4. the fitting fitting surface surface Furthermore, additional material is to ANN. Figure10 10shows shows the Furthermore, additional material is utilized to test the ANN. Figure the fitting surface by ANN and experimental data of Al2024-T315 [21]. Four sets of experimental data (stress ratios: 0, by by ANN and experimental Four sets setsof ofexperimental experimentaldata data (stress ratios: ANN and experimentaldata dataofofAl2024-T315 Al2024-T315 [21]. [21]. Four (stress ratios: 0, 0, 0.1, 0.33 0.33 and and 0.5) 0.5) are are all all used to to train the the ANN globally. globally. 0.1, 0.1, 0.33 and 0.5) are allused used totrain train theANN ANN globally.

Figure 10. The experimental data of Al2024-T315 and the corresponding ANN. Figure 10. 10. The data of of Al2024-T315 Al2024-T315 and and the the corresponding corresponding ANN. ANN. Figure The experimental experimental data

To observe the fitting accuracy clearly, Figure 11 shows the prediction and the experimental To observe the fitting accuracy clearly, Figure 11 shows the prediction and the experimental data in a 2D plot. From the figure it can be seen that the nonlinear fitting curves by ANN can fit the data in a 2D plot. From the figure it can be seen that the nonlinear fitting curves by ANN can fit the experimental data well.

experimental data well.

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To observe the fitting accuracy clearly, Figure 11 shows the prediction and the experimental data in a 2D plot. From the figure it can be seen that the nonlinear fitting curves by ANN can fit the Materials 2016, 9,data 483 well. 10 of 20 experimental Materials 2016, 9, 483

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Figure 11. The fitting by ANN vs. the testing data for Al2024-T315.

Figure11. 11. The The fitting fitting by by ANN ANNvs. vs. the the testing testing data data for for Al2024-T315. Al2024-T315. Figure 3.1.2. Crack Growth Calculation under Constant Amplitude Loading

3.1.2. Crack Crack Growth Growth Calculation Calculation under underConstant ConstantAmplitude AmplitudeLoading Loading 3.1.2. With the well-trained ANN, the algorithm for the crack propagation is programmed by With the well-trained ANN, the algorithm for the crack propagation is programmed MATLAB and the flow chart is shown in Figure 12. At first, a loading spectrum is generated the by With the well-trained ANN, the algorithm for the crack propagation is programmed byand MATLAB parameters are initialized. The input vector is then prepared following the same procedure in the MATLAB and the flow chart is shown in Figure 12. At first, a loading spectrum is generated and and the flow chart is shown in Figure 12. At first, a loading spectrum is generated and the parameters Section 2.2. After that, this vector is entered into the well-trained ANN and the crack increment is in parameters initialized. The input vector following is then prepared theSection same 2.2. procedure are initialized.are The input vector is then prepared the same following procedure in After that, worked With the crack increment in the current cycle, the crack length gets forincrement the next is Section 2.2.isout. After that, is entered ANN andupdated the out. crack this vector entered intothis thevector well-trained ANNinto andthe thewell-trained crack increment is worked With the crack iteration. When the whole loop is repeated until the last cycle, the simulation of the fatigue cracknext worked out. the crack increment inlength the current cycle, thefor crack gets updated increment in With the current cycle, the crack gets updated the length next iteration. Whenfor thethe whole propagation is accomplished. iteration. When until the whole is the repeated until of thethe last cycle,crack the simulation of isthe fatigue crack loop is repeated the lastloop cycle, simulation fatigue propagation accomplished. propagation is accomplished.

Figure 12. Flow chart of programing the predicting algorithm. Figure 12. Flow chart of programing the predicting algorithm.

Figure 12. Flow chart of programing the predicting algorithm.

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To validate the theANN-centered ANN-centeredalgorithm, algorithm,the the experimental data of Al7075-T6 are used To validate experimental data of Al7075-T6 are used [20]. [20]. The The ANN has been trained with the da/dN-∆K data as shown in Figure 4. Some additional data ANN has been trained with the da/dN-ΔK data as shown in Figure 4. Some additional data (a-N (a-N curves) are then utilized to compare the model prediction. In Table 3, the testing information curves) are then utilized to compare withwith the model prediction. In Table 3, the testing information of of these a-N curves are listed. Moreover, Forman’s equation also serves as a comparison. these a-N curves are listed. Moreover, Forman’s equation also serves as a comparison. Table 3. Loading information for the a-N curves. Table 3. Loading information for the a-N curves.

σσmin RR min 0.33 0.33 51.2 51.2MPa MPa MPa 0.50.5 6969MPa 168.7MPa MPa 0.70.7 168.7

σmax σ max 155MPa MPa 155 138 138MPa MPa 241 MPa 241 MPa

Figure13, 13, x-axis iscycle the number, cycle number, the y-axis is of thethe length the crack. The In Figure thethe x-axis is the and theand y-axis is the length crack. of The experimental experimental data and the predictions by the two different models are all visualized in data and the predictions by the two different models are all visualized in different lines. It isdifferent obvious lines.the It is obvious thatofthe performance of ANN is better than Forman’s model. that performance ANN is better than Forman’s model. 0.02

Forman R=0.33 Forman R=0.5 Forman R=0.7 ANN R=0.33 ANN R=0.5 ANN R=0.7 testing data R=0.33 testing data R=0.5 testing data R=0.7

0.018

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0.014

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Figure Figure 13. 13. The The predictions predictions vs. vs. the the experimental experimental data data for for Al7075-T6 Al7075-T6 aluminum aluminum alloy. alloy.

Once the failure criterion is given, the corresponding fatigue life can be determined. Assuming Once the failure criterion is given, the corresponding fatigue life can be determined. Assuming that the that the critical crack length is 0.008 m, 0.01 m, and 0.012 m, the corresponding errors of the two critical crack length is 0.008 m, 0.01 m, and 0.012 m, the corresponding errors of the two models are shown models are shown in Table 4. It is evident that the accuracy and stability of ANN is much better than in Table 4. It is evident that the accuracy and stability of ANN is much better than Forman’s equation. Forman’s equation. Table 4. The errors of the results by two models for Al7075-T6. Table 4. The errors of the results by two models for Al7075-T6.

ac

ac

R 0.33 0.008 0.008 0.5 0.75 0.010.33 0.01 0.5 0.75 0.012 0.33 0.012 0.5 0.75

R

ANN Algorithm ANN Algorithm 0.33 ´4.72% −4.72% 0.5 ´2.18% −2.18% 0.75 ´1.20% −1.20% 0.33 ´2.59% 0.5 ´6.06% −2.59% 0.75 ´1.61% −6.06% 0.33 ´3.97% −1.61% 0.5 ´10.92% −3.97% 0.75 ´2.03% −10.92% −2.03%

Forman Algorithm

Forman Algorithm ´20.85% −20.85% 4.73% 4.73% ´34.34% −34.34% ´21.38% ´8.48% −21.38% ´37.10% −8.48% ´23.10% −37.10% ´4.76% −23.10% ´38.07% −4.76% −38.07%

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Furthermore, additional D16 aluminum alloy are are used for model validation. The Furthermore, additionaltesting testingdata datainin D16 aluminum alloy used for model validation. Furthermore, additional testing data in D16 aluminum alloy are used for model validation. The information of the is listed in Table 5 [29]. Similarly, The information of experiment the experiment is listed in Table 5 [29]. Similarly,the theANN ANNisistrained trainedwith with the the crack crack information of the experiment is listed in Table 5 [29]. Similarly, the ANN is trained with the crack growth rate data (da/dN-ΔK) under three stress ratios. The 3D fitting surface and the 2D projections growth rate data (da/dN-∆K) under three stress ratios. The 3D fitting surface and the 2D projections of growth rate data (da/dN-ΔK) under three stress ratios. The 3D fitting surface and the 2D projections of the ANN shown in Figures respectively. the ANN areare shown in Figures 14 14 andand 15,15, respectively. of the ANN are shown in Figures 14 and 15, respectively. Table alloy. Table 5. The experimental information of D16 aluminum alloy. Table 5. The experimental information of D16 aluminum alloy.

Specimen Material Specimen Material Specimen Material Crack type Crack type Crack type Specimen length Specimen length Specimen length Specimen width Specimen width Specimen width Specimen thickness Specimen thickness Specimen thickness Initial crack length Initial crack length InitialLoading crack length type Loading type Loading type

D16 Aluminum Alloy D16 Aluminum D16 AluminumAlloy Alloy Middle cracked tension specimen Middle cracked cracked tension Middle tensionspecimen specimen 500 mm 500 mm 500 mm 100mm mm 100 100 mm 0.04mm mm 0.04 0.04 mm 10 mm 10 mm mm amplitude Tension-tension,10constant Tension-tension, constant amplitude Tension-tension, constant amplitude

Figure 14. The experimental data of D16 aluminum and the corresponding ANN. Figure 14. 14. The The experimental experimental data data of of D16 D16 aluminum aluminum and and the the corresponding corresponding ANN. Figure ANN. -11 -11 -12 -12

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fitting curve R=0.75 fitting curve R=0.75 experimental data R=0.75 experimental data R=0.75 fitting curve R=0.5 fitting curve R=0.5 experimental data R=0.5 experimental data R=0.5 fitting curve R=0 fitting curve R=0 experimental data R=0 experimental data R=0

-15 -15 -16 -16 -17 -17 -18 -182 2

2.5 2.5

3 3

0.5

3.5 3.5

4 4

lnKmax (MPa*m0.5) lnKmax (MPa*m ) Figure 15. The fitting by ANN vs. the testing data for D16 aluminum alloy. Figure Figure 15. 15. The The fitting fitting by by ANN ANN vs. vs. the the testing testing data data for for D16 D16 aluminum aluminum alloy. alloy.

Forman’s model is still employed for comparison. The calibrated equation is Equation (10) and Forman’s model is still employed for comparison. The calibrated equation is Equation (10) and the fitting indices are listed inemployed Table 6. Figure 16 shows theThe fitting lines of Forman’s Forman’s model is still for comparison. calibrated equation equation. is Equation (10) and the fitting indices are listed in Table 6. Figure 16 shows the fitting lines of Forman’s equation. the fitting indices are listed in Table 6. Figure 16 shows the fitting. lines of Forman’s equation. 1.5903 × 10 × ∆ . (10) = 1.5903 × 10´9 × ∆3.2215 (10) (1 ˆ − 10) ˆ− da= 1.5903 ∆K∆ (1 − ) − ∆ “ (10) dN p1 ´ Rq Kc ´ ∆K

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13 of 20 13 of 20

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Table6.6. The The fitting fitting indices Table indices of ofD16. D16. TableIndex 6. The fitting indices of D16. Number Index Number

r Index r Chi-Square r Chi-Square RMSE Chi-Square RMSE SSE SSE RMSE DC DC SSE

-5

10

-5

10

-6

10

-6

da (mm) da (mm)

10

0.984 Number 0.984 1.93 × 10−6 1.93 ˆ0.984 10´6 −8 7.80 ××´8 10 1.93 10−6 7.80 ˆ 10 −13 ´13 9.24 ×× 10 9.24 ˆ 10 7.80 10−8 0.964 0.964 9.24 × 10−13

DC

0.964

fitting curve R=0 experimental data R=0 fitting curve R=0 fitting curve R=0.33 experimental data R=0 experimental data R=0.33 fitting curve R=0.33 fitting curve R=0.75 experimental data R=0.33 experimental data R=0.75 fitting curve R=0.75 experimental data R=0.75

-7

10

-7

10

-8

10

-8

10

-9

10

-9

10

10 10

20 20 0.5 ) Kmax (MPa*m Kmax (MPa*m0.5 )

40

60

40

60

Figure16. 16.The Thefitting fittingby byForman’s Forman’s model model vs. alloy. Figure vs. the thetesting testingdata datafor forD16 D16aluminum aluminum alloy. Figure 16. The fitting by Forman’s model vs. the testing data for D16 aluminum alloy.

Similarly, the experimental data of D16 are used to validate the algorithm. Some additional data Similarly, thethe experimental data are used to validatethe thealgorithm. algorithm. Some additional data Similarly, experimental dataof ofD16 D16 usedprediction. to validate Some additional data (a-N curves) are utilized to compare with theare model In Table 7 the testing information of (a-N(a-N curves) are utilized to compare with the model prediction. In Table 7 the testing information curves) are utilized to compare with prediction. In Table 7 thepredictions testing information of of these a-N curves are listed. In Figure 17the themodel experimental data and the by the two these curves are listed. In Figure the experimental data the the predictions thedifferent two these a-N a-N curves areare listed. In Figure 17 the and predictions thebytwo different models plotted together. It17 isexperimental clear that thedata results ofand the proposed by model match the different models together. are plotted together. It isthe clear that the results of the model proposed model match the models are plotted It is clear that results of the proposed match the testing data testing data better than those by Forman’s model. testing better than thosemodel. by Forman’s model. better thandata those by Forman’s 0.035 0.035

Forman R=0 Forman Forman R=0 R=0.33 Forman R=0.33 R=0.75 Forman Forman R=0.75 ANN R=0 ANN ANN R=0 R=0.33 ANN ANN R=0.33 R=0.75 ANN testingR=0.75 data R=0 testing testing data data R=0 R=0.33 testing testing data data R=0.33 R=0.75 testing data R=0.75

0.03 0.03 cracklength/m length/m crack

0.025 0.025 0.02 0.02

0.015 0.015 0.01 0.01 0.005 0.0050 0

5,0000 5,0000

10,0000 25,0000 30,0000 30,0000 10,0000 15,0000 15,0000 20,0000 20,0000 25,0000 cycle cycle

Figure for D16 D16 aluminum aluminum alloy. alloy. Figure17. 17.The Thepredictions predictionsvs. vs. the the experimental experimental data data for Figure 17. The predictions vs. the experimental data for D16 aluminum alloy.

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Table 7. Loading information for the a-N curves. R

σ min

σ max

0.75 0.33 0

105 MPa 96 MPa 64 MPa

140 MPa 32 MPa 0 MPa

Assuming that the critical crack length is 0.015 m, 0.018 m, and 0.020 m, the relative errors of the two models are compared in Table 8. It is clear that the proposed model has very high accuracy and stability. Table 8. The errors of the results by two models. ac

R

ANN Algorithm

Forman Algorithm

0.015

0 0.33 0.75

´1.40% ´1.32% 2.90%

120.95% 96.60% 15.46%

0.018

0 0.33 0.75

´1.13% ´3.26% 2.16%

111.78% 82.71% 10.82%

0.02

0 0.33 0.75

1.05% ´1.69% 2.46%

105.70% 80.00% 8.61%

3.2. Validation and Comparison of the Constant Loading with a Few Overloads 3.2.1. Equivalent Stress Intensity Factor It is indicated that the plasticity ahead of the crack tip affects fatigue crack growth behavior. The retardation effects due to overload can be correlated with the plastic deformation. The plastic state caused by the previous loads is traced. Subsequently, the equivalent stress intensity factor is calculated, which is based on the equivalent plastic zone concept. The general expression of the equivalent plastic zone can be written as: , $ i i iÿ ´1 & . ÿ ÿ a0 ` da j ` Deq,i “ max a0 ` da j ` di , a0 ` da j ` Deq,i´1 (11) % j “1

j “1

j “1

where Deq.i means the size of equivalent plastic zone in the ith cycle; a0 means the initial crack length; ř da means the crack increment; di means the current plastic zone size in the ith cycle; a0 ` ij“1 da j means the crack length in the ith cycle; i means the current cycle number. A schematic sketch is given to illustrate the equivalent plastic zone concept. The loading sequential process and the corresponding plastic state variation are shown in Figure 18. The dashed zigzag lines represent the loading history. The large plastic zones have been formed at “t1 ,” and the crack tip is “O1 ” at that moment. The monotonic and reverse plastic zones can be expressed as Equation (12) [30]: $ ’ &

´ ¯2 π Kmax 8 σy ´ ¯2 π Kmax ´Kop 8 2σy

dm “

’ % dr “

(12)

where dm is the monotonic plastic zone size; and dr is the reverse plastic zone size. The current load is applied at “t2 ” and the new crack tip is “O2 ”. The large forward and reverse plastic zones, which are the dotted ellipses, form during the largest load cycle in the previous loading history. Before “t2 ”, the following plastic zones do not reach their boundaries respectively even though the crack grows.

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solid ellipses represent the equivalent plastic zones ahead of the crack tip O2. In addition, the actual of 20 plastic zone is butterfly-shaped instead of round; theoretically, however,15 their diameters along the crack direction are identical, as shown in Figure 18. In the current investigation, the equivalent plastic zone is in directly proportional to the circular diametric distance and the The solid ellipses represent the equivalent plastic zones ahead of the crack tip O2 . In addition, the proportionality coefficient is equal to or slightly greater than 1. Equation (12) can be rewritten as: actual contour of the plastic zone is butterfly-shaped instead of round; theoretically, however, their diameters along the crack direction are identical, as shown in Figure 18. In the current investigation, + zone+is in, directly + + the, , circular + + , = maxproportional , distance , the equivalent plastic to diametric and the proportionality coefficient is equal to or slightly greater than 1. Equation (12) can be rewritten as: Materials 483 contour2016, of 9,the

$ , # ∗ř , = i i iř ´1 ’ ř ’ 8 ’ a0 ` da j ` Dm,eq,i “ max a0 ` da j ` dm,i , a0 ` da j ` Dm,eq,i´1 ’ ’ ’ ’ j “1 j “1 j “1 ’ ’ ´ ¯ ’ ’ + + , , = d “ Ψ+˚ π Kmax,i + 2, , + + , , & m,i 8 σ y # i i iř ´1 ’ ř ř ’ ’ a0 ` da j ` Dr,eq,i “ max a0 ` da, j − ` dr,i ,, a0 ` da j ` Dr,eq,i´1 ’ ’ ’ ∗ j “1 j“1 j “1 ’ , = ’ 8 ´ K 2´K ¯2 ’ ’ ’ max,i op,i % dr,i “ Ψ ˚ π8 2σy

(13) (13)

in iithth cycle respectively; whereDDmm,eq,i where and Dr,eq,i arethe theequivalent equivalentmonotonic monotonic and and reverse reverse plastic zone in r ,eq,iare ,eq,i and Ψ is is the the geometry geometrymodification modificationfactor factorof ofplastic plasticzone. zone. Ψ

Figure Figure18. 18. Schematic Schematic illustration illustration of ofthe theequivalent equivalentplastic plasticzone zoneconcept. concept.

Theequivalent equivalentstress stressintensity intensityfactors factorsKKEE can can be be calculated calculated by by solving solving the the following followingequation: equation: The

= bπDD , , m,eq,i

KE “ σS

(14) (14)

where Dm,eq,i means the plastic zone in this cycle and σs means the yield limit. where Dm,eq,i means the plastic zone in this cycle and σs means the yield limit. 3.2.2. Single Overload 3.2.2. Single Overload Unlike the constant amplitude loading case, the algorithm for the single overload needs an Unlike the constant amplitude loading the algorithm for the single needs an additional parameter called equivalent stresscase, intensity factor to account for theoverload nonlinear loading additional parameter called equivalent stress intensity factor to account for the nonlinear loading interaction effect. To obtain this parameter, the equivalent plastic zone has to be calculated. Figure 19 interaction effect. To obtain this parameter, the equivalent plastic zone has to be calculated. Figure 19 shows the procedure to calculate the equivalent plastic zone. shows the procedure to calculate the equivalent plastic zone, where Dm,eq,i means the plastic zone i i 1 which characterizes the influence of the following the flow K history load.i Once Dm,eq,i is estimated d m.i ( max.i ) 2 a0 da j Dm.eq.i max{a0 da j d m.i , a0 da j Dm.eq.i 1} chart, the equivalent SIF can be calculated by using jEquation (14). Then get trained 8 y 1 j 1 the ANN can j 1 by using the training data vectors, in which the equivalent SIF, SIF and stress ratios are inputs and 19.rate The is calculation of theAdditionally, equivalent plastic the corresponding crack Figure growth the output. all zone. the training data have to be preprocessed following the procedure in Section 2.2. At last the fatigue crack growth with retardation can be estimated.

3.2.2. Single Overload Unlike the constant amplitude loading case, the algorithm for the single overload needs an additional parameter called equivalent stress intensity factor to account for the nonlinear loading interaction Materials 2016,effect. 9, 483 To obtain this parameter, the equivalent plastic zone has to be calculated. Figure 16 of19 20 shows the procedure to calculate the equivalent plastic zone. Materials 2016, 483 Materials 2016, 9, 9, 483

16 16 of 20 of 20

K

i

i

i 1

max.i 2 where m,eq,i means the plastic influence of0 the history load. Dm,eq,i which d m.i zone zone (which ) characterizes a0 dathe da Once daOnce j Dinfluence m.eq .i max{aof j d m.i , a0load. j Dm.eq 1} where DD m,eq,i means the plastic the the history D.i m,eq,i 8 y characterizes j 1 j 1 j 1 is estimated following the flow chart, the equivalent SIF can be calculated by using Equation (14). is estimated following the flow chart, the equivalent SIF can be calculated by using Equation (14). Then the ANN can get trained by using the training data vectors, in which the equivalent SIF, SIF Then the ANN can get Figure trained19. byThe using the training data vectors, in which calculation equivalent plastic zone. the equivalent SIF, SIF Figure 19.the Thecorresponding calculation of of the the equivalent plastic zone. and stress ratios are inputs and crack growth rate is the output. Additionally, all and stress ratios are inputs and the corresponding crack growth rate is the output. Additionally, all the training data have to be preprocessed following the procedure in Section 2.2. At last the fatigue the training data have to be preprocessed following the procedure in Section 2.2. At last the fatigue crack growth with retardation can be estimated. Thegrowth experimental data in D16 alloy [29] are employed to validate the model. The basic crack with retardation canaluminum be estimated. The experimental data in D16 aluminum alloy [29] are employed validate model. The information about the experiment canaluminum be seen in Table 5. da/dN-∆K curvetoto serves as the the The experimental data in D16 alloy [29] are employed validate thetraining model. vector, The basic information about the experiment can be seen in Table 5. da/dN-ΔK curve serves as the training and the a-N curve is used to validate the whole prediction algorithm. basic information about the experiment can be seen in Table 5. da/dN-ΔK curve serves as the training vector, and the a-N curve is used to validate the whole prediction algorithm. As shown ina-N Figure 20,isthe x-axis is the equivalent the y-axis is the K ; and the z-axis is the vector, and the curve to validate the whole SIF; prediction algorithm. As shown in Figure 20,used the x-axis is the equivalent SIF; the y-axis is the Kmaxmax ; and the z-axis is the crack growth rate. The red triangles the experimental data;the the blue curve the As growth shown in Figure 20,small the x-axis is theare equivalent SIF; the y-axis the K max; and therepresents z-axis is crack rate. The red small triangles are the experimental data; is blue curve represents thethe well-trained ANN; and three broken curves are its projections. It is obvious that the curve by ANN crack growth rate. The red small triangles are the experimental data; the blue curve represents the well-trained ANN; and three broken curves are its projections. It is obvious that the curve by ANN well-trained ANN; and three brokenofcurves are its projections. It is obvious that the curve by ANN can fit highly nonlinear tendency dataperfectly. perfectly. canthe fit the highly nonlinear tendency ofthe theexperimental experimental data can fit the highly nonlinear tendency of the experimental data perfectly.

-14

-14 -15 lnda(mm) lnda(mm)

-15 -16 -16 -17 -17 -18 -18 -19 -19 -20 80 -20 80

60 40

60

40 KE(MPa*m0.5)

20

20

0

0

KE(MPa*m0.5)

0

0

0.2

0.2

0.6

0.4

0.8

0.6 0.4 lnKmax (MPa*m0.5)

1

0.8

1

0.5

lnKmax (MPa*m alloy. ) Figure Thefitting fittingvs. vs.the the experimental experimental data Figure 20.20. The datafor forD16 D16aluminum aluminum alloy.

Figure 20. The vs. the experimental for D16 as aluminum alloy. Fatigue crack growth withfitting the overload effect is thusdata simulated shown in Figure 21. The extra

Fatigue crack growth with the overload effect is thus simulated as shown in Figure 21. The extra experimental information for a-N curve is listed in Table 9. Fatigueinformation crack growthfor with the overload effect thus 9, simulated as shown in Figure 21. stress The extra experimental a-N curve is listed inisTable where the Sol is the overload level. 0.03 experimental information for a-N curve is given listed for in Table 9. The prediction by Wheeler’s model is also comparison. 0.03 0.025

cracklength/m cracklength/m

0.025 0.02

0.02 0.015

0.015 0.01

0.01 0.005

0.005 0

0

2,0000

4,0000

6,0000

8,0000

10,0000 cycle

12,0000

14,0000

16,0000

18,0000

20,0000

Figure 021. The predictions vs. the experimental data for D16 aluminum alloy. 0

2,0000

4,0000

6,0000

8,0000

10,0000 cycle

12,0000

14,0000

16,0000

18,0000

20,0000

Figure21. 21.The Thepredictions predictions vs. vs. the the experimental experimental data Figure datafor forD16 D16aluminum aluminumalloy. alloy.

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Table 9. Applied loading for D16 specimen. Table 9. Applied loading for D16 specimen. Type of of Loading Loading Type CA+single CA + singleoverload overload

SS min min

MPa 00MPa

∆SΔS

Smax Smax

RR

MPa 6464 MPa 0 0 6464MPa MPa

Sol Sol

128MPa MPa 128

where the Sol is the overload stress level. The prediction by Wheeler’s model is also given for comparison. The figure shows that there is an overload applied when the crack length reaches 0.01 m. After that, The figure shows that there is an overload applied when the crack length reaches 0.01 m. After a conspicuous retardation phenomenon can be seen. The slope of the curve decreases dramatically that, a conspicuous retardation phenomenon can be seen. The slope of the curve decreases until the crack grows out of the retardation effect area after another 60,000 cycles. The prediction by dramatically until the crack grows out of the retardation effect area after another 60,000 cycles. The the ANN-based approach has a very good agreement with the testing data in this figure. However, the prediction by the ANN-based approach has a very good agreement with the testing data in this curve by However, Wheeler’sthe model growingmodel after stops the overload applied. Ribeiro al. [31]Ribeiro indicate figure. curvestops by Wheeler’s growing is after the overload is et applied. that Wheeler’s model has some difficulties in crack growth calculation when the overload is et al. [31] indicate that Wheeler’s model has some difficulties in crack growth calculation when larger the than twice that of the σ . However, the approach proposed in this paper does not have this kind max overload is larger than twice that of the σmax. However, the approach proposed in this paper does not of problem which it more generally have this kindmakes of problem which makesapplicable. it more generally applicable.

3.2.3. Multiply Overloads 3.2.3. Multiply Overloads Other differentmaterials materialsareare employed formodel the model validation [32]. Othertesting testingdata data for for different employed herehere for the validation [32]. The The information of this experiment is listed in Table 10. information of this experiment is listed in Table 10. Table of350 350WT WTsteel. steel. Table10. 10.The Theexperimental experimental information information of

Specimen Material 350WT WT Steel Specimen Material 350 Steel Crack Type Center Cracked Tension Specimen Crack Type Center Cracked Tension Specimen Specimen length 300 mm Specimen length 300 mm Specimen width 100mm mm Specimen width 100 Specimen thickness 5 mm Specimen thickness 5 mm Initial crack length 20 mm Initial crack length 20 mm of loading Tension-tension, Constant amplitude withwith overload TypeType of loading Tension-tension, Constant amplitude overload 11.4 SminSmin 11.4MPa MPa S 114 MPa Smax max 114 MPa Sol 190.95 MPa Sol 190.95 MPa

Similarly, the bythe theexperimental experimental data the result is shown in Figure Similarly, theANN ANNis is trained trained by data andand the result is shown in Figure 22. The22. The small red tangles experimental the curve blue curve the fitting the and ANN; small red tangles areare thethe experimental data;data; the blue is the is fitting by the by ANN; the and threethe three broken curves are its projections. It is seen that the ANN delivers a good fitting. broken curves are its projections. It is seen that the ANN delivers a good fitting.

lnda/dN(mm)

-14

-16

-18

1 0.8

-20 70

0.6 60

0.4 50

40

EK(MPa*m0.5)

0.2 30

20

0

lnKmax(MPa*m0.5)

Figure22. 22.The Thefitting fittingvs. vs. the the experimental experimental data Figure datafor for350 350WT WTsteel. steel.

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The The predictions predictions made made by by the the ANN-based ANN-based approach approach and and Wheeler’s Wheeler’s model model are are visualized visualized in in Figure 23 with the experimental data as well. Both the methods give good agreements with the testing Figure 23 with the experimental data as well. Both the methods give good agreements with the data. The proposed model performs slightly slightly better around the 150,000th cycle. cycle. testing data. The proposed model performs better around the 150,000th 0.035

0.03

crack length/m

0.025

0.02

0.015

0.01 ANN prediction experimental data Wheeler prediction

0.005

0

0

5,0000

10,0000

15,0000

20,0000

25,0000

30,0000

cycle

Figure 23. 23. The The predictions predictions vs. vs. the the experimental experimental data data for for 350 350 WT WT steel. steel. Figure

From the validations above, it can be concluded that the proposed method can deal with the From the validations above, it can be concluded that the proposed method can deal with the nonlinear and multivariable fatigue damage accumulation process successfully. nonlinear and multivariable fatigue damage accumulation process successfully. 4. Conclusions 4. Conclusions and and Future Future Work Work In this this paper, paper,aanovel novelmethod methodto topredict predictthe thefatigue fatigue crack crack growth growth based based on on aaradial radial basis basis function function In (RBF)-artificialneural neuralnetwork network(ANN) (ANN)isisdeveloped. developed.The The ANN-centered algorithm is also validated (RBF)-artificial ANN-centered algorithm is also validated by by comparison with the experimental data under the constant and variable amplitude loading of comparison with the experimental data under the constant and variable amplitude loading of different different materials. and Wheeler’s models are also employed for comparisons. is clear materials. Forman’sForman’s and Wheeler’s models are also employed for comparisons. It is clearItthat the that the proposed model has very high accurate and stable performance in all the examples. proposed model has very high accurate and stable performance in all the examples. All the thevalidations validationsabove above prove advantages the ANN-based algorithm in nonlinear All prove the the advantages of theofANN-based algorithm in nonlinear fatigue fatigue crack growth problems. This method still has some limitations that need further crack growth problems. This method still has some limitations that need further investigation. investigation. One major issue is that the size of the training data has a significant impact on the One major issue is that the size of the training data has a significant impact on the prediction accuracy. prediction accuracy. The other is that the method may be time consuming and computationally The other is that the method may be time consuming and computationally expensive due to its expensive duenature. to its cycle-by-cycle nature. cycle-by-cycle Acknowledgment: This Foundation of of China China Acknowledgments: This research research is is financially financially supported by the National Natural Science Foundation (No. (No. 51405009) 51405009) and and the the Fundamental FundamentalResearch ResearchFunds Fundsfor forthe theCentral CentralUniversities. Universities. Author AuthorContributions: Contributions: Wei WeiZhang Zhangorganized organizedthe theresearch; research;Zhangmin ZhangminBao Baocarried carriedon onthe themodel modelsimulation simulation and and wrote the manuscript; Shan Jiang wrote programs and checked the manuscript; Jingjing He helped write the wrote the manuscript; Shan Jiang wrote programs and checked the manuscript; Jingjing He helped write the manuscript and provide the academic support. manuscript and provide the academic support. Conflicts of Interest: The authors declare no conflict of interest. Conflicts of Interest: The authors declare no conflict of interest.

References Reference 1. 1. 2. 2. 3. 3.

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