PREDICTION OF FATIGUE CRACK GROWTH DIAGRAMS BY METHODS OF MACHINE LEARNING UNDER CONSTANT AMPLITUDE LOADING

Important structural elements are often under the action of constant amplitude loading. Increasing their lifetime is an actual task and of great economic importance. To evaluate the lifetime of structural elements, it is necessary to be able to predict the fatigue crack growth rate (FCG). This task can be effectively solved by methods of machine learning, in particular by neural networks, boosted trees, support-vector machines, and k -nearest neighbors. The aim of the present work was to build the fatigue crack growth diagrams of steel 0.45% C subjected to constant amplitude loading at stress ratios R = 0, and R = –1 by the methods of machine learning. The obtained results are in good agreement with the experimental data.


INTRODUCTION
Methods of strength and durability evaluation of the responsible structural elements often need the complicated calculations. Therefore, it is important to learn how to solve the problems of fracture mechanics by methods of machine learning, in particular, neural networks (NN), support-vector machines (SVM), k -nearest neighbors and boosted trees, which allow to achieve high accuracy of solutions [1  4] . The structural elements are often fractured by fatigue, gradually accumulating damage. It is possible to observe a small crack which grows under loading. The fatigue crack is formed mainly at the stress concentrator, that is, the place of damage, which weakens the cross-section of the material. The crack grows as long as the material is able to withstand the loading. Therefore, the basic factors that influence the strength of structural elements are the surface defects of the parts, temperature and the environment during operation, the nature of loading and loading conditions [5]. It is known that, the basic parameters characterizing the fatigue crack growth (FCG) rate da / dN are the stress intensity factor K (SIF) and the stress ratio R [6  9]. The fatigue crack growth diagram is usually built in double logarithmic coordinates lg da/dNlg ΔK. It has the form of an S-shaped curve limited on the left by the threshold SIF range ΔKth, and on the right by the critical SIF ΔKfc (cyclic fracture toughness). The threshold SIF ΔKth is determined experimentally. It is an important characteristic of material resistance to fatigue fracture. The diagram consists of three regions: region I corresponds approximately to the rate da / dN ≈10 -10 ...10 -8 m/cycle, in which the rate of the FCG increases significantly with a slight change of ΔK. Region II has the form of a straight line. The rate in this region is in the range of 10 -8 ...10 -6 m/cycle. In particular, it is considered that here the crack grows evenly for each loading cycle. Region III is characterized by accelerated FCG and corresponds the values of da/dN >10 -6 m/cycle [5]. At high SIF values, the rate of crack growth is extremely high. P. Paris and F. Erdogan found out that the FCG rate for metallic materials can be determined by the SIF [10]. In particular, the formula obtained by them describes only the second region of the fatigue fracture diagram and does not take into account the influence of the stress ratio R on the FCG rate [11]. It is known that, with increasing R the FCG rate increases [12][13]. Therefore, the Walker's equation [2,14] is used to describe the FCG rate taking into account the stress ratio R. However, these models don't take into account the variable regions of FCG. The Forman's equation [15][16]

Machine learning: Background and Modeling
Progress of modern technology, in particular, high demands for accuracy and efficiency, have led to the creation of methods that solve a number of important tasks. Therefore, neural networks, support-vector machines, k -nearest neighbors, boosted trees are powerful algorithms of supervised learning, which can be used to predict FCG. NN consist of a very large, though the finite, number of items that form the input layer, one or more hidden layers of computational neurons, and one output layer. The input signal is transmitted over the network in the direction from layer to layer [18]. Such networks are usually called multilayer perceptrons, which quite accurately solve different tasks. NN determines the relationship coefficients between neurons, whereas the computational power of a multilayer perceptron is in its ability to learn on its own experience and the backpropagation algorithm. The idea of this algorithm is based on an error correction. The basic parameters of NN are its topology, algorithm of training and the functions of the neurons activation. In the current study, the sum of squares error function (SOS) was chosen and the training method was Broyden-Fletcher-Goldfarb-Shanno (BFGS) [19][20][21][22]. The stop parameter of learning network was number of epochs, which in this study was equal to 1000. The boosted trees algorithm reflects the natural thinking of human processes while making a decision [23]. The data obtained by building and using boosted trees are logical and easy for visualizing and interpreting. The algorithm of building the boosted trees structure consists of the creating and cancellation stages of trees. In creating trees, one chooses the criteria of splitting and termination of learning, whereas in the course of trees cancellation, some branches are removed. The boosted trees method is used when the results of one decision influence the next, in particular, for making consistent decisions. The ideas of the methods of support-vector machines and k-nearest neighbors are the simplest [24]. In the first method, the data are presented as points in the space. The training data are split into two categories. The training algorithm creates a model attributing new data to a certain category. Geometrically, it looks as if we are trying to draw a straight line centrally between two sets. The nearest to this straight line points are the support vectors. The support-vector machines method, as any method of machine learning, has many parameters. In this case, the basis objects are the regularization parameter, the loss function, which treats as errors only predicted values deviating from the actual values by a distance greater than ε, and the kernel parameter γ. As the kernel function, the radial basis function (RBF) is used. The method of k-nearest neighbor assigns a new object to the class that is the most common over k-nearest neighbors of the training sample. The distance between k-nearest neighbors is usually chosen as Euclidean. The aim of the learning process is to minimize the loss function, which should decrease. In the current study, the loss function was chosen as the mean squared error (MSE) [

Experimental material(s) and methods
During the operation, a railway axle undergoes static and cyclic loading, including random loading, bending, as well as corrosive action of environment and climate temperatures. The FCG rate in axle steel was predicted by methods of machine learning according to the experimental data of FCG obtained for 0.45% C steel and stress ratios R = 0, -1 [25]. The sample consisted of 200 elements, 70% of which were chosen randomly for the training sample and 30% were left for estimating the quality of predictions. The input parameters were the SIF range K and the stress ratio R. The FCG rate da/dN under regular loading for the stress ratios R = 0, -1 was chosen as the output parameter. The input and output parameters were normalized using the decimal log function to decrease the prediction error.

RESULTS AND DISCUSSION
The dependences of the experimental FCG rates da/dNexp on the predicted FCG values da/dNpred for R = 0, -1 are shown in Fig. 1  There were built the experimental and predicted dependences of the FCG rate da /dN on the SIF range K for R = 0, 1 using the methods of machine learning (Fig. 2).  The parameters of the constructed neural networks, support-vector machines, knearest neighbors and boosted trees are summarized in Tables 1−3.   The error of the NN method for the test sample is 4.5%, the support-vector machines is 5.5%, the k -nearest neighbors is 5.5% and the boosted trees is 6.7%.

CONCLUSION
The predicted FCG rate data are in good agreement with the experimental ones.
In the present study, the NN prediction accuracy is 95.5%, which is the best among the applied methods. Support-vector machines, k -nearest neighbors, and boosted trees also show good results in terms of accuracy. The methods of machine learning are powerful and efficient tools which allow evaluating the FCG behavior.