THE ANALYSIS OF FRACTURE SURFACES OF POROUS METAL MATERIALS USING AMT AND FRACTAL GEOMETRY METHODS Sergei Kucheryavski Artem Govorov Altai State University Barnaul, Russia

Ideal life Fractured specimen Fracture surface picture Image Processing and Analysis A priory information about deformation behavior Possible cause of Deformation and fracture

Deformation Structure of Porous Metals Deformation stages (Optical microscope) Fracture surfaces (Electronic microscope)

Known methods  Traditional methods Classical statistics methods (i.e. Mean Absolute Deviation) Textural features methods  Alternative methods Fractal analysis The AMT technique

Fractal geometry  Fractals: irregular, fragmented objects self-similar objects  Fractal geometry methods simulation complex objects like trees, clouds and so on measure of self-similarity quantitative description of irregular, complex structure – fractal dimension D f

Housdorf dimension  = 1 – for squares, cubes  =  /4 – for circles  =  /6 – for spheres D – Housdorf dimension

Fractal dimension N – number of cells

Fractal dimension  Advantages: D – can be considered as the measure of roughness, irregularity of surface The results showed the dependencies between fractal dimension of fracture surfaces and their porosity were obtained  Disadvantages: Some time there is no chance to calculate D It works bad with surfaces that have a small D (from 2 to 2.2) It works bad with noised images

AMT – Angle Measure Technique Algorithm: 1.Image is unfolded into 1D digitized line. 2.A number of points – A – are randomly chosen along the line. 3.For all scales S from 1 to N: Find points B and C – points of intersection of circle with radius S and line; For each point A the Angle and Y-Difference are measured; For all measuring the Mean Angle (MA) and Mean Y Difference (MDY) are calculated. 4.The AMT-spectrum (dependencies of MA and MDY on scale S) is plotted.

AMT-Spectra example

AMT Features  AMT transform the 2D image into 1D spectra without losses the structure information  AMT can be used for data compression  AMT is highly sensitive  Using PCA or PLS for AMT-spectra one can analyze and classify the structures

Fractal Analysis vs. AMT 1. Is there any correlations between fractal dimension of surfaces and their AMT-spectra? 2. Is it possible to use AMT for noised images of surfaces? 3. Apply the AMT to analyze the fracture surfaces of porous metals

Software  Fractal software simulation - C++ program (Diamond-Square Algorithm)  Fractal dimension calculations – C++ program (Box-Counting Algorithm)  AMT-analysis – MATLAB macros (Jun Huang, Telemark University College)  PCA-analysis – The Unscrumbler ®

Simulated fractal surfaces D = 2.1 D = 2.4 D = 2.6 D = 2.9

The results of PCA of AMT spectra 225 specimen with D f from 2.1 to 2.9

The results of PCA of AMT spectra Outliers detection and scores w/o outliers

The results of PCA of AMT spectra The result for specimen with D=2.1 and 2.9

Conclusions  PCA-analysis of AMT-spectra of fractal surfaces allow to make a classification depending on fractal dimension  Scores plot shows that the “clouds” of samples with D<2.5 are overlapped  Score plot shows that the samples with greater D are arranged closely than others

Fractal analysis vs. AMT. Noised images  The real fracture surfaces is differ from simulated fractal surfaces first of all with presence of noise – because of imperfection of devices, external influence and so on.  The task is to add the noise to simulated fractal surfaces and to compare fractal analysis and AMT results.

Simulated surfaces with Gauss noise Original D: 2.1 Calculated D: 2.8 Original D:2.3 Calculated D:2.7 Original D:2.5 Calculated D:2.7 Original D:2.9 Calculated D:2.8

AMT results - Noised Images - w/o Noise

AMT-results - Noised Images - w/o Noise

Conclusions  Fractal analysis doesn’t allow to classify noised images – the calculated and initial fractal dimension are in not close agreement  PCA-results of AMT-spectra of noised images show that “clouds” of samples with equal D are more overlapped and stretched along PC1  In further investigations one can use the fractal dimension of surface as an additional variable in PCA analysis