Abstract
The curvature of surface topography is quantified in this study using a newly developed directional blanket covering curvature (DBCC) method. The novel method addresses a long-standing problem of a measurement of a local surface curvature at individual scales and directions. The curvature data are obtained using the first and second derivations of the quadratic polynomial fitted to the local surface profile extracted from the surface height image data at individual scale and direction. The range of scales is set between an instrument spatial resolution and 1/10 of the image shortest size. Using the surface curvature data, three parameters, i.e. curvature, peak and valley dimensions, are calculated as the slopes of lines fitted to data point subsets of log–log plots of surface areas (differences between dilated and eroded surface curvature matrices) against scales of calculation. The dimensions quantify directional changes in the overall curvature and the curvature of peaks and valleys at individual scales. The scale corresponds to the centre of the subset. A flat surface criterion based on the surface areas was also proposed. Using the criterion, a flat surface was identified in computer images of dome, corrugated plate and fractal surface. The DBCC method was applied to computer-generated fractal surfaces with increasing curvature complexity, sine waves with decreasing curvature at single scale and microscope images of isotropic (sandblasted) and anisotropic (ground) surfaces of titanium plates. Results showed that the method is accurate in the measurement of surface curvature and the detection of minute changes occurring in the curvature of real surfaces over a wide range of scales.
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Abbreviations
- \(A\) :
-
Surface area
- \(A_{\text{PEAK}} ,A_{\text{VALLEY}}\) :
-
Peak and valley surface areas
- \({\text{abs}}\) :
-
Absolute value function
- \(a_{0} , a_{1} ,a_{2}\) :
-
Coefficients of quadratic function
- \(B_{\varepsilon }\) :
-
Structuring element
- \(b\) :
-
Eroded version of K
- c :
-
Constant
- CD:
-
Curvature dimension
- \({\text{CD}}_{\text{MEAN}}\) :
-
Mean curvature dimension
- \(\text{C}_{\text{Str}}\) :
-
Curvature aspect ratio
- \({\text{FD}}\) :
-
Fractal dimension
- \({\text{FD}}_{\text{t}}\) :
-
Theoretical fractal dimension
- \(f\) :
-
Surface function
- \(f_{\kappa }\) :
-
Curvature function
- \(f_{\text{q}}\) :
-
Quadratic function
- G f :
-
Graph
- I :
-
Image function
- J:
-
Rotated image matrix
- \(L_{x} ,L_{y}\) :
-
Spatial domains
- l :
-
Subset size
- \(l_{x}\) :
-
Line length
- \(l_{y}\) :
-
Distance between peak and line
- \(N_{\varepsilon }\) :
-
Number of scales
- \(N_{\theta }\) :
-
Number of directions
- \(N_{x} ,N_{y}\) :
-
Number of grid points in spatial directions
- \(N_{z}\) :
-
Number of greyscale levels
- \({\text{PD}}\) :
-
Peak dimension
- \({\text{PD}}_{\text{MEAN}}\) :
-
Mean peak dimension
- \(\text{P}_{\text{Str}}\) :
-
Peak aspect ratio
- \(r_{c}\) :
-
Peak radius
- \({\text{Sds}}\) :
-
Density of peaks
- Sku:
-
Kurtosis
- Spc:
-
Arithmetic mean curvature of peaks
- Ssc:
-
Mean curvature of summits
- Sta:
-
Texture major axis
- Str:
-
Texture aspect ratio
- Th:
-
Threshold
- V :
-
Accumulated volume
- VD:
-
Valley dimension
- VDMEAN :
-
Mean valley dimension
- VStr :
-
Valley aspect ratio
- u :
-
Dilated version of K
- X, Y :
-
Spatial domains
- x, y :
-
Spatial coordinates
- z :
-
Surface height
- ε :
-
Scale
- θ :
-
Direction
- K :
-
Surface curvature matrix
- K PEAK :
-
Surface peak matrix
- K VALLEY :
-
Surface valley matrix
- κ :
-
Profile curvature
- \({\text{\copyright}}\) :
-
Curvature operation
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Acknowledgments
The authors wish to thank Curtin University and the Department of Mechanical Engineering for their support during preparation of the manuscript. The study was conducted as part of the Implementing Agreement on Advanced Material for Transportation Applications, Annex IV Integrated Engineered Surface Technology. The Implementing Agreement functions within a framework created by the International Energy Agency (IEA). The views, findings and publications of the AMT IA do not necessarily represent the views or policies of the IEA or of all of its individual member countries.
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Podsiadlo, P., Wolski, M. & Stachowiak, G.W. Novel Directional Blanket Covering Method for Surface Curvature Analysis at Different Scales and Directions. Tribol Lett 65, 2 (2017). https://doi.org/10.1007/s11249-016-0786-4
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DOI: https://doi.org/10.1007/s11249-016-0786-4