Fractal Dimension (FD): image as a single real number

MAST research project
Fractal dimension. Materials and Structures research group

A point in space has zero dimensions, a line has one dimension, an area has two dimensions, and a volume has three dimensions. A standard formula is N=rD, where D is the exponent and r the magnification factor. In Euclidean geometry, D has to be an integer. Mandelbrot [1] introduced the concept of fractals and self-similarity (e.g. Sierpiński Triangle, or Koch Snowflake), which can be quantified as Fractal Dimension (FD). FD occurs when D is a real number, and can be determined by rearranging the equation for D. FD determination is automated within ImageJ software [2] with the FracLac plugin [3]. For patterns with, or without, self-similarity, either the Richardson Plot (length estimate vs length of scale is linear on a log-log plot) or the Box Counting Method (BCM) are applicable [4-7]. Figure 1 shows the boxes containing a feature, in this case resin-rich volumes (RRV), for a typical composite cross section as used in the determination of FD by BCM.

Figure 1: Representative images showing the stages of fractal data generation for a Brochier twill fabric

Figure 1: Representative images showing the stages of fractal data generation for a Brochier twill fabric

Pearce et al [8-9] measured reinforcement permeabilities and conducted composite tensile, compressive and interlaminar shear tests for three reinforcement fabrics. The satin weave provided the best mechanical properties, but the lowest permeability. The cross-plied Injectex satin weave, containing flow-enhancement tows (FET), exhibited a higher permeability than the standard satin but at the expense of mechanical performance. The twill fabric had significantly higher permeability than both the satin and Injectex fabrics, whilst, with the exception of ILSS, possessing slightly better mechanical properties than the Injectex. Image analysis to cross-sectional images revealed that FD was ranked in the same sequence as the tensile and compressive moduli and strengths. Subsequent analysis [9-11] considered the FD and measured permeabilities of the above, and new concept Carr FET, reinforcement fabrics (Figure 2).

Figure 2:  Permeability plotted against fractal dimension for Brochier and new concept Carr fabrics.

Figure 2:  Permeability plotted against fractal dimension for Brochier and new concept Carr fabrics.

Summerscales et al [12] analysed 150 cross-sectional slices of x-ray computed tomography images of plain weave E-glass fibre-reinforced epoxy resin composites using fractal dimensions. The fractal dimension analysis returned a consistent numerical value for each of the slices in the two similar orthogonal planes.

Mahmood [13, 14] investigated RRV in fabric-reinforced composites. Fibre distribution, quantified from cross-sections as FD, was used as a parameter to characterise the static and fatigue properties of the composites tested in four-point bend. The ultimate flexural strength of the composites showed a clear dependence on the fibre distribution. Interlaminar shear strength (ILSS) also correlated with the fibre distribution albeit with lower significance.

Labrosse et al [15] used FD as a novel procedure for the assessment of the surface finish of in-mould gel-coated fibre-reinforced polymer matrix composites. The technique involves photographing the reflection from a coated surface on an opaque screen and analysing the image using fractal dimensions. The ranking of the quality of surface finish from this technique sensibly aligned with that obtained by human observation and by the commercial Wavescan DOI system. The investment required for the new system was primarily that required for a high-resolution digital camera, and hence is less than 5% of the cost of the commercial instrument.

Piasecki and Summerscales [16] reported a novel system to ensure use of the correct reinforcement fabric during composites manufacture. Three carbon fibre reinforcement fabrics (plain weave, single-tow twill and double-two twill) were characterised using a high-resolution scanner, images converted into binary, and then analysed using ImageJ/FracLac software to determine FD. The three fabrics each had a distinct FD value, in the undeformed condition and when sheared up to 30° (the locking angle). Implemented in a quality management system this analysis could eliminate the risk of an operator using fabric from the wrong roll leading to compromised mechanical properties in the component.

A summary review is also available [17].

References

1. BB Mandelbrot, The Fractal Geometry of Nature, W.H. Freeman, New York, 1982. ISBN 0-7167-1186-9. 
3. A Karperien, Fractal Dimension and Lacunarity, 15 September 2015, accessed 19 February 2020. 
4. K Foroutan-Pour, P Dutilleul and D Smith, Advances in the implementation of the box-counting method of fractal dimension estimation, Applied Mathematics and Computation, Nov-Dec 1999, 105(2-3), 195-210.
5. R Pitchumani and B Ramakrishnan, A fractal geometry model for evaluating permeabilities of porous preforms used in liquid composite molding, International Journal of Heat and Mass Transfer, 1 June 1999, 42(12), 2219-2232.
7. E Damrau, M Normand and M Peleg, Effect of resolution on the apparent fractal dimension of jagged force-displacement relationships and other irregular signatures, Journal of Food Engineering, February 1997, 31(2), 171-184.
9. NRL Pearce, Process-property-fabric architecture relationships in fibre-reinforced composites, PhD thesis, University of Plymouth, Plymouth, 2001.
10. NRL Pearce, J Summerscales and FJ Guild, Improving the resin transfer moulding process for fabric-reinforced composites by modification of the fabric architecture, Composites Part A: Applied Science and Manufacturing, December 2000, 31(12), 1433-1441.
11. J Summerscales, FJ Guild, NRL Pearce and PM Russell, Voronoi cells, fractal dimensions and fibre composites, Journal of Microscopy, February 2001, 201(2) 153-162. 
12. J Summerscales, PM Russell, S Lomov, I Verpoest and RS Parnas, The fractal dimension of X-ray tomographic sections of a woven composite, Advanced Composites Letters, 1 March 2004, 13(2), 113–121.
13. AS Mahmood, MN James and J Summerscales, Process-property-performance relationships in CFRP composites using fractal dimension, IOP Conference Series: Materials Science and Engineering, September 2018, 388, 012013.
14. AS Mahmood, Processing-performance relationships for fibre-reinforced composites, PhD thesis, University of Plymouth, Plymouth, 2016.
15. Q Labrosse, CP Hoppins and J Summerscales, Objective assessment of the surface quality of coated surfaces, Insight, January 2011, 53(1), 16-20.
16. D Piasecki and J Summerscales, Characterisation of reinforcement fabrics by fractal dimension, SAMPE Europe Conference 2018, Society for the Advancement of Materials and Process Engineering, Southampton UK, 11-13 September 2018.
17. J Summerscales, Fibre Distribution and the Process-Property Dilemma, Chapter 11 in P Beaumont, C Soutis and A Hodzic (editors), The Structural Integrity of Carbon Fibre Composites, Springer, 2017, pp. 301-317. ISBN 978-3-319-46118-2.