Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 56, 4, pp. 1109-1122, Warsaw 2018 DOI: 10.15632/jtam-pl.56.4.1109 LIFE PREDICTION FOR LY12CZ NOTCHED PLATE BASED ON THE CONTINUUM DAMAGE MECHANICS AND THE GENETIC ALGORITHM AND RADIAL BASIS FUNCTION METHOD Jiaying Gao, Peng Li School of Aeronautics Science and Engineering, Beihang University, Beijing, China e-mail: gaojiaying 1988@163.com; lpeng210@sina.com (corresponding author) In this paper, a new method based on the Continuum Damage Mechanics (CDM) and the Genetic Algorithm and Radial Basis Function neural network method (GARBF) is propo- sed to predict fatigue life of LY12CZ notched plate. Firstly, the multiaxial fatigue damage evolution equation is derived, and the fatigue life of the notched specimen is predicted based on theCDMmethod. Secondly, theRBFmethod is introduced tomodify the relative devia- tion between the theoretical result and actual life. According to the drawbacks of the RBF method, theGA is adopted to optimize networkparameters to effectively improve themodel quality and reduce the training error.Then, the verification test indicates that the combined method of CDM and GARBF is able to reduce the average relative error of the results of fatigue life prediction to about 7%, which shows that the newmethod to predict the fatigue life is more reliable. At last, compared with the predicted results of the traditional Back Propagation (BP) neural network, the GARBF model proposed in this paper has a better optimization effect and the result is more stable. This research provides a feasible way to predict the fatigue lives of the notched plate based on the CDM andGARBFmethod. Keywords: life prediction, CDM, LY12CZ, notched plate, GARBF 1. Introduction In practical applications, many engineering structures are subject to cyclic loads, and fatigue damage is a common form of failure (Schijve, 2001; Zhan et al., 2013). Fatigue damage is a pro- gressive and localized material behavior which occurs when a structure component is subjected to cyclic loading. Fatigue cracks are prone to initiate at the positions of notches and geometrical irregularity due to stress concentration. The notch is considered as one of the most important problems in the design of structure components. Different from the uniaxial fatigue problem, the fatigue damage of a notched component involves multiaxial stress and strain states. So, it is important to present a method for fatigue life prediction of notched plates. In the past decades, many researchers have been focused on the fatigue experiment and sta- tistic analysis (Suresh, 1998), which is a generalmethod of fatigue life prediction in engineering. However, thismethod takes toomuch time and sometimes the result is not satisfactory. So, it is necessary to find an easy and reliable method. The critical plane approach (Karolczuk andMa- cha, 2005) and stress invariant approaches (Li et al., 2000) have been widely used in researches and applications. However, these approaches often fail to capture the fatigue damage evolution such as material degradation due to damage. Fatigue accumulation damage theory (Lemaitre and Chaboche, 1990; Zhan et al., 2015b, 2016, 2017c) is considered to be one of the most ef- fective methods for fatigue life prediction. TheContinuumDamageMechanics (CDM) provides an effective method to describe degradation of mechanical properties of materials, which take into account damage evolution using the concept of effective stress and has been widely used in 1110 J. Gao, P. Li recent years (Movaghghar and Lvov, 2012; Upadhyaya and Sridhara, 2012; Zhang et al., 2015a, 2017a,b). While the methods based on the damage law are carried out, such problems also attract some researchers engaged in the machine learning field. They attempt to take the advantages of machine learning theories in complex system modeling to enhance the accuracy of fatigue life prediction. Pujol and Pinto (2011) proposed a novel method based on neural network to predict the fatigue life. Guo et al. (2014) introduced the support vectormachine (SVM)method to predict the life of the packaging EMCmaterial. Liu andXuan (2008) adopted the rough sets theory to analyze important affecting parameters on low cycle fatigue life. The fatigue life pre- diction is a complicated problem. There is a large amount of non-linear and non-smooth factors that rapidly influence the actual material life. Compared to traditional theoretical methods, the artificial neural network (ANN)method can better overcome these difficulties (Gao et al., 2015, 2016; Monteiro et al., 2016; Nagarajan and Jonkman, 2013; Reid etal, 2013). Introducing the ANNmethod into the solution of fatigue life prediction has several advantages: (1) ANN is able to approximate any non-linear and non-smooth functionwith any accuracy; (2) ANNhas strong anti-interference ability and good robustness; (3)TheANNstructure is suitable for parallel data processing, and the computation is efficient and fast; (4) ANN has good generalization ability. In this paper, the Radial Basis Function (RBF) neural network method is adopted to predict the fatigue life for LY12CZ notched plate. Some researches have shown that the RBF neural network has better optimal approximation ability than the traditional Back Propagation neural network (BPNN). Compared to BPNN, the RBFmodel has fewer parameters to be learnt. The RBF neural network is composed of 3 layers: an input layer, a hidden layer and an output layer. After suitable data pre-processing, the data will be imported into the hidden layer. The hidden layer consists of several neurons, and each neuron can be regarded as a cluster center which is able to generate the local response by the inputs. When the distance between the inputs and the cluster center are close enough, the hidden neuron will make a non-zero response. The final output of theRBFnetwork is equal to a linear combination of all hidden layer neuron responses. There are also some defects in the use of the RBF method. The RBF is not good at global searching and is easy to fall into the local minimum. In this paper, theGenetic Algorithm (GA) is adopted to optimize the training process of the RBF model. The GA is developed based on the biological evolution process. Several biological concepts such as reproduction, crossover, mutation, competition are introduced into theGAalgorithm.TheGAwill adjust and improve a series of feasible solutions in theprocess of evolution, andfinallyfind the optimumsolution in the whole multi-dimensional space. TheGA is a global optimization algorithm, so it can effectively overcome the drawbacks of traditional methods (Camacho-Vallejo et al., 2015; Goldberg and Samtani, 2015; Zhan et al., 2015c). In this paper, the continuum damagemechanics (CDM) and an optimization model compo- sed of the Genetic Algorithm and Radial Basis Function neural network method (GARBF) are proposed to conduct the fatigue life prediction for LY12CZ notched plate in the case of cyclic loading using the framework is shown in Fig. 1. Firstly, themultiaxial fatigue damage evolution equation is derived. Secondly, according to the fatigue experimental data, the material para- meters in the damage evolution equation are identified. Then the fatigue life prediction of the notched specimen is conducted. On the basis of the CDM method, the Radial Basis Function (RBF)method is introduced tomodify the relative deviation between the theoretical result and actual life. In addition, the Genetic Algorithm (GA) is adopted to improve the RBF training effect in order to obtain amore reliable optimization model (GARBF). Finally, the verification test indicates that the combined method of CDM and GARBF is able to reduce the average relative error of fatigue life prediction to about 7%, and the life prediction result is more re- liable. Compared to the traditional backpropagation (BP) neural network, the GARBF model proposed in this paper has a better optimization effect and the result is more stable. Life prediction for LY12CZ notched plate based on... 1111 Fig. 1. The framework of the methodology 2. The fatigue damage evolution model In uniaxial cycle loading, based on the remaining life and continuum damage concepts, the fatigue cumulative damage model can be illustrated as (Zhan et al., 2015c) Ḋ = dD dN = [1− (1−D)β+1]α(σmax,σm) [ σmax−σm M(σm)(1−D) ]β (2.1) where D is the damage scalar variable and N is the number of cycles. σmax and σm are, re- spectively, themaximum andmean applied stress. β is a material parameter. The expression of α(σmax,σm) is defined as α(σmax,σm)= 1−a 〈σmax−σf(σm) σu−σmax 〉 (2.2) where σu is the ultimate tensile stress, σl0 is the fatigue limit for fully reversed conditions. a and b1 are material parameters. The expression of M(σm) is defined as M(σm)= M0(1−b2σm) (2.3) where M0 and b2 are material parameters. The number of cycles to failure NF for a constant stress condition is obtained by integrating Eq. (2.1) from D =0 to D =1, leading to NF = 1 1+β 1 aM −β 0 〈σu−σmax〉 〈σmax−σf(σm)〉 ( σa 1− b2σm ) −β (2.4) where σa is the stress amplitude during one loading cycle. In the practical engineering application, the stress and strain are always multiaxial. The damage evolution law in the case of multiaxial loading is given as follows Ḋ = dD dN = [1− (1−D)β+1]α ( AII M0(1−3b2σH,m)(1−D) )β (2.5) where AII is the amplitude of octahedral shear stress and σH,m is the mean hydrostatic stress. The parameter α is defined by α =1−a 〈 AII −A∗II σu−σe,max 〉 (2.6) 1112 J. Gao, P. Li where σe,max is themaximumequivalent stress which is calculated bymaximising the vonMises stress over a loading cycle. The Sines fatigue limit criterion A∗II in this model is formulated by A∗II = σl0(1−3b1σH,m) (2.7) By integrating of Eq. (2.5) from D =0 to D =1 for a constant stress condition, the number of cycles to failure NF is NF = 1 1+β 1 aM −β 0 〈σu−σe,max〉 〈AII −A∗II〉 ( AII 1− b2σH,m ) −β (2.8) 3. The material parameters identification Thestaticproperties ofLY12CZaluminiumalloy and fatigue experiment results canbeconsulted from a handbook (Wu, 1996), and the static mechanics properties are presented in Table 1. In themodified damage evolution law, there are five parameters in the damage evolution equation. The four material parameters (β,M0,b1,b2) can be determined by fatigue experimental data of smooth specimens. For smooth specimens under conditions of uniaxial fatigue loading, an S-N curve has been derived. When fatigue tests are carried out at a fixed stress ratio, the relation between the number of cycles to failure NF and the maximum stress σmax can be obtained. Parameters β and 1/[(1+β)aM −β 0 ] come from stress-controlled (R =−1) fatigue tests stress- -life data.With the least squaremethod, parameters b1 and b2 can be obtained from the fatigue tests data at other different stress ratios. Then the independent parameters β and aM −β 0 will be used in the incremental damage formulation (Zhan et al., 2017d), and a is identified numerically by using the fully reversed fatigue test data for the notched specimens. Finally, the identified damage evolution parameters are listed in Table 2. Table 1. Static properties of LY12CZ aluminium alloy E [GPa] ν [–] σb [MPa] σs [MPa] 73 0.3 466 343 Table 2.Material parameters in the fatigue damage evolution equation a β M0 b1 b2 0.62 1.76 712398 0.0001 0.0002 4. Fatigue life prediction for LY12CZ notched plate Two kinds of the LY12CZ notched specimen are used and geometric profiles of which are shown in Fig. 2. The corresponding stress concentration factors KT are 2 and 4, respectively. The fatigue load, stress ratio and the predicted fatigue lives under different stress levels are shown in Table 3. 5. Introduction of an RBF neural network model The establishment of a theoreticalmodel (CDM)provides an important basis and foundation for the solution of notched specimen fatigue life prediction.As shown inTable 3, there is about 20% Life prediction for LY12CZ notched plate based on... 1113 Fig. 2. The geometric profile of the notched specimen (dimensions in mm): (a) KT =2, (b) KT =4 Table 3.Comparisons between experimental lives and predicted results Stress σmax Experimental Numerical Error ratio [MPa] life results [%] KT =2 −0.2 175 34160 26370 22.80 −0.09 155 114000 90450 20.66 0.02 137 217200 238000 9.58 0.25 112 1094000 1354000 23.77 0.4 100 9490000 8035000 15.33 0.4 300 85390 74300 12.98 0.60 262 226100 288900 27.77 0.68 250 778600 881350 13.20 0.75 240 2037000 1672000 17.92 0.79 235 2445000 1906000 22.04 KT =4 0.02 137 45350 35700 21.28 0.25 112 180900 157500 12.93 0.4 100 646000 803200 24.33 0.55 90 2458000 2079000 15.42 0.60 87 3319000 2885000 13.08 0.60 262 45320 36500 19.46 0.68 250 64200 78300 21.96 0.75 240 190400 174000 8.61 0.83 230 753200 845400 12.24 0.91 220 1483000 1727000 16.45 deviation between the actual fatigue life and the results calculated by CDM. The deviation is mainly due to several factors: (1) some parameters in the theoretical model are not accurate; (2) there are structural defects in the theoretical model; (3) experimental environment and hardware conditions lead to certain system errors. In a word, the deviation is appeared by complicated reasons. It is difficult to develop a theoretical approach to calculate. Therefore, the RBF neural network method is adopted to modify the CDM model to obtain a more reliable fatigue life prediction result. The RBF neural network model is composed of 3 layers: an input layer, a hidden layer and an output layer. Its topological structure is shown in Fig. 3. There are two neurons in the input layer, one stands for the stress ratio R and another represents the ultimate stress σmax.In order to enhance the training efficiency and speed up the convergence rate, the input data will be standardized (σmax,R) before imported into the hidden layer. Each weight from the input layer to the hidden layer is equal to 1. The neuron number in the hidden layer is set as h, and each neuron represents a cluster center which is corresponding to the basis vector. The Gauss function is selected as the radial basis function. The input sample in the training set is denoted as:X= [x1,x2, . . . ,xi, . . . ,xp] T, andeach sample contains 2dimensiondata:xi =(σmax,R).The 1114 J. Gao, P. Li Fig. 3. The topological structure for an RBR neural networkmodel output label∆N= [∆N1,∆N2, . . . ,∆Ni, . . . ,∆Np] T is the relative deviation percent between the actual life NA and the theoretical result NF , expressed as: ∆Ni =(NA−NF)/NF . p is the total sample number in the train set. The response output O1ij from the i-th sample to the j-th neuron in the hidden layer can be calculated by Eq. (5.1), in which σ is the variance of the Gauss function, cj stands for the j-th cluster center, ‖xi−cj‖ represents the Euclidean distance between the input vector xi and cj O1ij =exp ( − 1 2σ2 ‖xi−cj‖2 ) (5.1) The final output N of the RBF model is the relative deviation percent between the actual life and the theoretical result. The output is calculated by a linear combination of all Gaussian function outputs in the hidden layer. wj is the weight from the j-th neuron in the hidden layer to the output. Then, the final output for the i-th sample can be calculated as O2i = h ∑ j=1 wjO1ij = h ∑ j=1 wj exp ( − 1 2σ2 ‖xi−cj‖2 ) (5.2) 6. The learning process for the RBF neural network model The learning process for theRBFmodel is divided into the unsupervised learning stage and the supervised learning stage. 6.1. Unsupervised learning stage In the unsupervised learning stage, the K-means method is used to automatically cluster the sample data and locate the center of each Gaussian kernel function. The iteration step is shown as below. Step 1. Initialization: h points are random selected as the initial cluster center c1(σ1,R1),c2(σ2,R2), . . . ,ci(σi,Ri), . . . ,ch(σh,Rh) Step 2. Clustering: Each point is sorted out into the corresponding cluster by the principle of proximity. Life prediction for LY12CZ notched plate based on... 1115 Step 3. Adjustment of the clustering center: The new position of each clustering center is updated by calculating the average value of sample points. If all the clustering centers are not changed any more, then the current centers will be regarded as the Gaussian kernel function centers in the hidden layer of the RBF model, the iteration ends; Else, back to Step 2. 6.2. Supervised learning stage Onthebasis of theK-means clustering result in theunsupervised learning stage, the variance of the Gaussian kernel function in the hidden layer can be obtained by Eq. (6.1), in which dmax represents the maximum distance between the clustering centers σ = dmax√ 2h (6.1) After thedetermination of the clustering centers c =(c1,c2,c3, . . . ,ci, . . . ,ch) and thevarianceσ, the weight matrix can be calculated by solving the linear system in Eq. (6.2) with the least squares method    w11 w12 · · · w1h w21 w12 · · · w2h w31 w12 · · · w3h          O11 O12 ... O1h       =    y1 y2 y3    (6.2) 7. Optimization of the RBF model based on a GA algorithm The selection of the Gaussian kernel center is the core element of the RBF neural network design. In order to improve the neural network quality and reduce the training error, a Genetic Algorithm is adopted tooptimize the initial positionof eachGaussiankernel center.Chromosome encoding, fitness function selection and definition of the genetic operator are 3 parts of the GA algorithm design. 7.1. Chromosome encoding When a GA is used to solve a numerical optimization problem, the binary code cannot achieve very good results. Therefore, the real number chromosome encoding is adopted, as shown in Fig. 4. Each chromosome is composed of several gene segments, and each segment represents a certain Gaussian kernel center position ci(σi,Ri). Fig. 4. A sample of chromosome encoding 7.2. The fitness function and the roulette selection probability The fitness function is used to evaluate the accuracy of calculated results. According to the previous discussion, the objective function of RBF training is expressed as Eq. (7.1)1, and the definition of the fitness function is shown in Eq.(7.1)2 1116 J. Gao, P. Li E = 1 2 i=p ∑ i=1 (O2i−yi)2 Fitness = 1 E Pk = Fitnessk i=Pop ∑ i=1 Fitnessi (7.1) For the k-th individuals in the population, its probability of being chosen for evolution is calcu- lated by the individual fitness and the total fitness, as shown inEq. (7.1)3. The selection process is carried out by the roulette mode, as shown in Fig. 5. Thewhole disc area represents the total fitness of the population, and the fitness of each individual is corresponding to a certain sector area. The area the pointer finally stays determines which individual is selected. In this way, a higher individual fitness means a greater possibility of being chosen, while a lower fitness also has a little possibility to evolve. Fig. 5. The roulette choosingmode 7.3. Genetic operator: crossover and mutation The purpose of the genetic operator is to preserve good chromosomes into the next gene- ration. Assuming the crossover operator is happening on the k-th gene segment of the parent chromosome chi = {chi1,chi2, . . . ,chik, . . . ,chim} and chj = {ch j 1,ch j 2, . . . ,ch j k , . . . ,chjm}, then the real numbers of the k-th gene segment in the two new chromosome (chis,chjs) are calculated by Eq. (7.2), where ψ is the crossover possibility and set as 80%. chisk = ψch i k +(1−ψ)ch j k ch js k = ψch j k +(1−ψ)chik (7.2) Each gene segment is composed of 2 normalized data: the stress ratio and the ultimate stress. Therefore, the crossoverprocessbetween theparents is equivalent to calculation of two symmetric definiteproportionate insertedpoints ina straight linedefinedby (σik,R i k) and (σ j k ,R j k), as shown in Fig. 6. Fig. 6. The crossover process between 2 parent chromosomes Different from binary coding, the mutation operator of real number encoding is to add a random bias d to the selected gene segment, denoted as: chisk = ch i k+d. Themutation operator ensures diversity of population, and its probability is set as 3%. Life prediction for LY12CZ notched plate based on... 1117 7.4. The training process of the GARBF model TheGA algorithm is adopted to optimize theGaussian kernel center positions in the hidden layer during the RBF training process, and the iteration step is shown as below. Step 1. Randomly initialize the population (random selection of the clustering center position in the hidden layer), and encode the chromosomes. Step 2. Calculate the fitness of each individual in the population, and select the optimal indi- vidual. Step 3. If the number of generation reaches the maximum or the optimal individual satisfies the requirements of training accuracy, go to Step4; Else, the next generation of population will be produced by the 3 genetic operator: mutation, crossover and selection, then go to Step 2. Step 4. The chromosomes are anti-encoding and the initial clustering center positions c1,c2, . . . ,ch in the hidden layer are obtained. Step 5. The parameters of the RBFmodel are trained by the supervised learningmethod, the iteration ends. All the experimental data are divided into two parts, 80% for training and 20% for the verification test. There is no intersection between the training set and the test set. The training error descent curve by the GARBF and RBF method are shown in Fig. 6. When the iteration epoch is less than 30, the optimization effect of the GA is not obvious, just as the red circle region shown in Fig. 7; As the iteration epoch is increasing, it is indicated that the GARBF method is superior to simple use of the RBFmethod in both the convergence rate and training error aspect. After about 430 epochs, the training error is reduced and converged to 0.0017. Fig. 7. The training error descent curve of GARBF andRBFmodels 8. The combined method of CDM and GARBF TheCDMandGARBFmethod are combined to accurately predict the fatigue life for LY12CZ notched plate. On the basis of the establishment of theCDMmodel, the trainedGARBFmodel is adopted to modify the theoretical result and obtain the prediction life. In the verification 1118 J. Gao, P. Li test, NF represents the theoretical result calculated by the CDM model, and NGARBF stands for the result after being modified by the trained GARBF model, NA actual life measured in the experiment. The relative error percent between NF and NA is denoted as eFA, and the relative error percent between NGARBF and NA is denoted as eGARBF. eFA and eGARBF can be calculated by Eq. (8.1). Under the two conditions of KT = 2 and KT = 4, the distributions of eFA and eGARBF are shown in Fig. 8, in which X and Y label respectivelymean the input value of σmax and R. The red circles mean the relative error exceed 20%, while the blue ones mean that the relative error is less than 5%. The distribution of the relative error indicates that the prediction results modified by the trained GARBFwill be closer to the actual life in the whole. The combinedmethod of CDM andGARBF proposed in this paper is effective and reliable eF = NF −NA NF ·100.0% eGARBF = NGARBF −NA NGARBF ·100.0% (8.1) Fig. 8. The distribution of relative error calculated by CDM and CDM+GARBF The combination method proposed in this paper is composed of a theoretical model (CDM) and an optimization model (GARBF). Actually, the selection of optimization model is not uni- que. Inorder to compare theeffects ofdifferentoptimizationmodels,Table4 respectivelypresents the optimization performances by traditional BPNNandGARBFmodels under the same verifi- cation test data.The statistic results indicate the average relative error ofGARBF is 6.85%, and the average relative error calculated byBPNN is 7.2%. The twomethods have a similar optimi- zation effect of the average relative error. However, the GARBFmodel is much better than the BPNN model when comparing the variance of the relative error (DBP =6.0,DGARBF =2.49). Figure 9 shows the relative error optimized after BPNN andGARBF under a fixed value of KT and R. The distribution of the relative error indicates that the optimization effect of GARBF is more stable in the whole, while some calculations of BPNN seem to fluctuate significantly. The GARBFmethod adoptsGaussian kernel as the activation function thatmeans the responsewill close to 0when the inputdata is far fromthe cluster centers. TheBPNNmethoduses the sigmo- id function which has a large output region. Therefore, when the input data is a set of “strange Life prediction for LY12CZ notched plate based on... 1119 data” that has not been trained, the output of the BPNN maybe seriously far from the true one, while theGARBFmodelwill output a relative small value and choose to believe the results calculated by CDM theory. In a word, the optimization effect of GARBF is more reliable and stable. Fig. 9. The error distribution optimized by BPNN andGARBF under a fixed value of KT and R 9. Conclusions remarks In this paper, a newmethodbased on theCDMandGARBFneural networkmethod is proposed to predict the fatigue life of LY12CZnotched plate. Some important conclusions are summarized as follows: • Themultiaxial fatigue damage evolution equation is derived and the theoretical model for fatigue life prediction of LY12CZ notched plate is established. The predicted results based on the CDMmethod tally with the fatigue experimental results. 1120 J. Gao, P. Li Table 4.Optimization effect comparisons between the BPNN andGARBF model Stress σmax Relative error Relative error ratio [MPa] of BPNN [%] of GARBF [%] KT =2 −0.2 175 14.78 9.07 −0.09 155 5.09 5.64 0.02 137 0.84 4.04 0.25 112 14.73 10.69 0.4 100 17.79 8.79 0.4 300 2.44 7.06 0.60 262 2.09 6.27 0.68 250 15.16 7.16 0.75 240 2.14 3.05 0.79 235 1.76 2.88 KT =4 0.02 137 10.85 12.03 0.25 112 16.48 8.48 0.4 100 3.47 7.66 0.55 90 2.85 5.64 0.60 87 6.93 3.95 0.60 262 2.87 5.41 0.68 250 3.98 3.83 0.75 240 15.54 8.54 0.83 230 1.6 7.55 0.91 220 2.65 9.27 Mean relative error [%]: eBP =7.2, eGARBF =6.85 Variance of relative error [%2]: DBP =6.0, DGARBF =2.49 • The RBF method is introduced to modify the relative deviation between the theoretical result and the actual life. 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ZhanZ.,MengQ.,HuW., SunY., ShenF., ZhangY., 2017d,Continuumdamagemechanics based approach to study the effects of the scarf angle, surface friction and clamping force over the fatigue life of scarf bolted joints, International Journal of Fatigue, 102, 59-78 Manuscript received January 14, 2018; accepted for print April 10, 2018