Analysis of loading history influence on fatigue and fracture surface parameters using the method of induction trees

In fatigue life testing under various loading conditions, researchers observe the profile, surface and morphology of materials. In this study authors research the fatigue life of material and the surface fracture geometry. Areal field and fractal based characterisation are evaluated for the whole fracture surfaces. Results of this test were correlated to notch geometry and loading conditions. It was confirmed, for notched specimens, that the change from torsion to proportional bending with torsion fatigue life increase significantly, the same as changing loading from bending with torsion to bending. The measurement device was equipped with a motorised nosepiece using five dedicated microscopic objective lenses from 2.5× to 100× magnification. This paper presents the application of the induction tree method for analysis of loading history influence on fatigue and fracture surface parameters. In a decision tree, nodes store tests checking values of example attributes and leaves store categories assigned to them. For each of possible test results, there is one branch coming from a node to a subtree. In this way, it is possible to represent any attributes of the hypothesis admissible for a given set. Analysis of selected parameters will estimate their impact on the surface structure.


Introduction
Fracture surface topography is one of the basic macroscopic investigations aimed at determining the cause of the fatigue damage [1]. It allows to determine what kind of the fatigue loading material was subjected. Several of typical macroscopic patterns of fatigue damage can be distinguished. Among them, their functions of type and magnitude of loading. The surface analysis reveals localisation of initiation and crack path propagation, as well as identifies the areas for further microscopic examination. Fracture mechanics tests are usually concentrated on crack growth under uniaxial and multiaxial loadings [2]. Some articles deals only with crack growth, while other scientist carried out quantitative analysis of fracture surface. The study on relationship between fracture toughness and fracture surface fractal dimension began in the 1980s [3]. Since then, the quantitative approach to the morphology has led to many interesting studies on the interconnection with loading or ambient environment [4].The topography of fracture surfaces, especially in bending and torsion fatigue, was investigated and published in [5,6]. Researchers demonstrated, inter alia, the influence of torsion loading constituent on the surface form. Previous studies shown that there are differences in the surface geometry of individual zones (e.g. initiation, propagation) [7].
The optical method Focus Variation Microscope (FVM) was used to measure the entire fracture surface in connection with the relevant Areal parameters [8].
The classification is an important stage in the analysis of acoustic properties. In this stage, properties characteristic for signals of particular microphones are compared with each other. On the basis of obtained results a decision concerning the classification of the signal properties to a given group is made [9] Among the most often applied methods of recognising acoustic signals are: HMM (Hidden Markov Models), VQ (Vector Quantization), LVQ (Learning Vector Quantization), SOM (Self-Organising Maps), ANN (Artificial Neural Network), GMM (Gaussian Mixture Models) [10,11], SVM (Support Vector Machines). In addition, there are classifiers based on the induction of decision trees, ML-HMM (Maximum Likelihood Hidden Markov Model) [12,13]. In a general case, the classification includes two stages: creating of patterns for recognition and identification. In general terms of optimisation, groups of measurement data are provided by a measurement station. Discretisation of ranges of individual measurement data leads to a coded record of construction and operating parameters [14].
Multivalent logical trees determine the importance of design and operation parameters, playing the role of logical decision variables. A number of methods can be distinguished in discrete optimisation, e.g. heuristic method -based on searching for deviations between data values [15,16], decision trees induction algorithm -based on the entropy increase [17,18] neural networks, evolutionary algorithms [19]. These methods, in particular, are used in fault diagnosis and acoustic analysis of signals [20]. This paper focuses on a relationship between type of loading, like bending, torsion and combination bending with torsion and surface fracture Height Parameters, Functional Parameters and Fractal dimension. The induction tree method helped in the search for the importance of individual parameters. Additionally, based on the files, the neural network was used.

Fatigue tests
The fatigue tests were made in the lab of the Department of Mechanics and Machine Design, Opole University of Technology, Poland [21,22]. This scientific body specialises in developing methods for determining the fatigue life of materials [12], as well as performing its own test stands [23,24,25].
The object of the study was rectangular cross-section specimens of the EN AW-2017A-T4 aluminium alloy, shown in figure 1. Specimens had an external, unilateral, sharp and blunt one-sided notches, which radius was ρ = 0.2 mm, 5 mm, 10 mm and 22.5 mm. Fractures was caused by different kinds of loadings in bending and torsion fatigue. Stress ratios for this research were R = -1, -0.5, 0.

Surface parameters measurements
Topography of the surface was measured and calculated using the focus variation microscope (FVM) Alicona Infinite Focus G4 with the MountainsMap 7.4 software.
Fatigue fracture surface was observed on total area with objective magnification 10×. Surface fracture studies were carried out using Height Parameters, according to ISO 25178. The fatigue loading history was checked with selected Height Parameters, such as Rootmean-square height (Sq), Skewness (Ssk), Kurtosis (Sku), Maximum height (Sz), Arithmetical mean height (Sa) and fractal dimension (Df).
Surface parameters are calculated on the whole studied surface, marked in figure 2.

The classification of validity rank of the parameters with the application of induction trees
A decision tree is a structure which has ordinary properties of trees in the meaning assigned to the tree in the information technology, so it is a structure composed of nodes from which branches come to other nodes or leaves. It is convenient to define tree structures in a recursive way. Assuming that a given branch X on which attributes a1, a2 ,..., an and the set of notions C of the category C are determined: 1. The leaf containing any category label d  C is a decision tree.
2. If t: X Rt is a test made on the values of attributes of examples with a set of possible results Rt=  r1, r2, ... rm  are decision trees, then the node containing the test t, from which m branches come out, given that for i= 1, 2, ..., m branch i corresponds to the result ri and leads to the tree Ti, is a decision tree.
For any node of n decision tree, by tn we mean a test connected with it, and for each of its possible results r  Rt by n[r] node or a child leaf, to which the n branch related to the r result leads from the node. The notation described above is presented in Figure 3.  Tables 1 and 2 present the analysed parameters in the form of a training file.  The classification by means of induction trees was made separately for loading ratio r=τmax/(σmax+τmax) as output attributes (wy).

Application of inductive decision trees for analysis of loading history influence on fatigue and fracture surface parameters
Input attributes (we) are the values of fracture surface parameters: Cycles (we); Moment (we); R (we); notch radius ρ (we); αk (we) ; σmax (we) ; τmax (we); b [ The DeTreex module (Aitech Software) was used in the classification of validity rank of the construction parameters and it made it possible to form decision trees. In the described system, the module forming the decision tree requires an appropriate data preparation. In Figures  4 and 5, the first and second part of the tree of the induction tree are presented.  The induction tree determines the degree of importance of the attribute (Moment-M) from the most important one placed in the root, through classification of any example. The inductive search algorithm is an approximate method. To make the calculations more accurate, the local localisation algorithm was used. For this purpose, the neural network method was used [26,27]. Standards for the network were results based on a comparison of surface parameters with the results of fatigue calculations.

Application of neural network
The training process of the applied network was made on the basis of the supervised learning. One of the main parameters determining the receipt of satisfactory results, from the point of view of recognising acoustic emissions of microphones, was an appropriate choice of a training algorithm by the adopted SSN architecture. The following algorithms were tested for the training procedure: GDA (Gradient Descent with Adaptive Learning Rate Backpropagation) and RPROP (Resilient Backpropagation) [28].
The weight correction process of particular neurones for this algorithm takes place according to the dependence described by the formula below: an n for n n n n n for n n n n for    In the last step, an induction algorithm for the neural network was used. Induction tree was generated for arithmetical mean height Sa, fractal dimension Df and Skewness Ssk (for charts from figure 6) (Fig. 10).

Conclusions
The article contains analyse of selected features of surface of EN AW-2017A-T4 alloy caused by different kinds of loadings in bending and torsion fatigue. Induction trees is a method used to examine the importance of individual parameters relative to fatigue life. The parametric tree classifies parameters: R, τmax, αk , σmax, τa and h0. The parametric tree also determines the importance level. The greatest influence on the topography of the breakthrough area (Sq, Sa, Sz, Sku, Ssk as well as Df) according to the induction tree according to Figure 10 is R, then: τmax, and at the end: αk ,τa. The greatest influence on durability (Nf) has the moment M.
In a decision tree, nodes store tests checking values of example attributes and leaves store categories assigned to them. For each of possible test results, there is one branch coming from a node to a subtree. In this way, it is possible to represent any attributes of the hypothesis admissible for a given set. Trees have such a property that they start in the root from which branches are built. The use of trees in the optimization of machine systems is fully useful in the sphere of the concept, because it allows you to select (change) the arithmetic values of the relevant construction and/or operating parameters of a given machine system and evaluate the operation of the system in new conditions. At any stage of optimisation, we can make a tree by choosing the optimal decisions. Then you can attach vertices to the tree, which represent optimal system responses to changes in arithmetic values of construction parameters, etc. Another approach would be to use, for example, a backtrack algorithm. The search "with backtrack" is based on a systematic search with the transition from the subproblem.