PSI - Issue 2_A

Pavel Skalny / Procedia Structural Integrity 2 (2016) 3727–3734 Pavel Skalny/ Structural Integrity Procedia 00 (2016) 000 – 000 Where ( ) is the number of covering sets with the diameter not greater then δ . In our implementation the ( ) denotes the number of squares (for other possibilities see Falconer 2014) in the square mesh grid with the edge length equal to . In practical calculation the S is calculated using the least square method approximating the linear dependency between log(1/ ) and log ( ) . As we will denote the box counting dimension in the , direction. The direction corresponds to the direction of crack propagation whether the direction is perpendicular to the direction of the crack propagation. 2.2. Mormal vector characteristics Besides methods of the fractal geometry an alternative approach to identify fracture surface was applied. The fracture surface was covered with the net of triangles. The vertices of triangle correspond to real measurements of the fracture surface. To create the triangle net the Delaunay triangulation was used, see Ceong and Kreveld (2008) The Delaunay triangulation maximizes the lower angle of each triangle, so the final triangulation contents as regular triangles as possible. For every triangle the unit vector perpendicular to the triangle -normal vector was calculated. In a previous research every triangle was evaluated by the greatest angle of its normal vector with normal vectors of neighbouring triangles. Although the angular deviation describes the fracture surface quite well see Skalny and Strnadel (2015), we lose the information about the direction where normal vector change the most. Due to the fact we will add other characteristics to the angular deviation. For every (unit) vector we will use the length of its x and y component. The components describe how much is the surface tilted in the , direction. Furthermore we will consider changes of and components of two vectors with the greatest angle deviation. The changes of two vectors 1 , 2 was computed in following way: After heuristic analysis of roughness based on angular deviations of normal vectors we concern our attention to other characteristics. For given normal vector = ( , , ) related to the centre of gravity = ( , , ) of the triangle ∇ we take into account the pair , (the -th component does not provide valuable information). Values , partially describe inclination of the triangle. ∇ (see Figure ...). For every fixed chose a neighbour ∇ of the triangle ∇ with maximal angular deviation of related normal vectors. Now, as additional input data for cluster the further analysis we compute differences = − – , = − – . (2) Note, the choice of the proper triangle can be realized with respect to the different criterion, e.g. can be done as a neighbour of with minimal distance of the centres of gravity. This approach gives similar results as method described above. Let us summarize, that every triangle with the centre of gravity is evaluated with five values maximal deviation , normal vector components , and differences , . 2.3. Comparison In this section we will present mean values of box counting dimension and normal vector characteristics. All the values were calculated from 30 different DWTT specimens. In table 1 there are presented mean values of studied characteristics calculated for the area with brittle and ductile fracture. The simple paired t-test was applied to test whether the values on the area of brittle and ductile fracture are significantly different. In table 2 there is presented the correlation matrix. The correlation coefficients are presented in the upper triangular matrix. The results of the test of the statistical significance (using the testing statistics = √ 1 − − 2 2 distributed with the student distribution with − 2 degrees of freedom) of correlation coefficients are presented in the lower triangular matrix. Both statistical tests are realized on the significance level 0.05. The alternative hypotheses are formulated in the one tail form. Generally we can conclude, that the box counting dimensions are correlated with the angular deviation and with differences , . Except for vector components all the values are significantly different in the area of the brittle and ductile fracture. 4

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