A Study of Crack Formation and its Effective Internal Surface A Study of Crack Formation and its Effective Internal Surface Area using Area using

X-ray Micro-Tomography (XMT) X-ray Micro-Tomography (XMT)

University of Johannesburg

Energy Postgraduate Conference 2013

Introduction X-ray Microtomography (also referred to as micro CT) is a

radiographic imaging technique that can produce 3D images of

a material's internal structure at a spatial resolution < 1μm

(Landis et al. 2010).

Fractal techniques have been used extensively to characterize

crack formation via details of the crack surface roughness.

•They represent defects in solid materials

• To find convenient scale invariant ways for their characterisation

• Phase reactions occur at the fluid-material interface

• The larger the effective area of interaction the faster the reaction

rate

Why cracks are important to

study

Intuitive definition of dimensionConsider a geometric object F, with measure (F)

F (F) (F)

Line segment Length 21 = 2

Rectangle Area 22 = 4

Parallelepiped Volume 23 = 8

s dim F = (F) r= 1/s resolution

FsF loglogdim

sFF loglogdim

rFF 1loglogdim

If F is scaled by a factor of 2 say, how does the measure scale?

Von Koch Curve (Self-similar Construct)

1 D line segmentDivide into 3 units (s=1/3 unit )

New patterns generated, N= 4

Repeat the recipe for each new straight line segmentRepeat on new smaller scale

We have a theoretical algorithm that creates a self-similar structure – – the Von Koch curve. . It has fractal geometry - a

mixture of 1 D and 2D charracters.

Dimension of the Von Koch

CurveScale (compass setting) Length

nS 31 nL 341/3(1/3)2

(1/3)3

4/3(4/3)2

(4/3)3

sn 13

ns 1log3 34loglog 33 nL

sL 1log34loglog 333 2619.0

3log

34log

3

4log

10

103

Compass dimension (Dcom )

Dfrac = 1 + Dcom =1.2619

Determining the Df of Fracture

Cracks

Crack surface

Section to surface

Fractal character

At large and small distance scales relationship may break down

S = scale

L = Length

SL

Acquisition and

Reconstruction

Tomographic Tomographic reconstructionreconstruction

PenumbraNo penumbra, sharp focus with magnification, M, increasing as df increases:

Microfocus X-ray Geometry

Micro-focus X-ray technology uses micrometer sized source to produce sharp image projections at micrometer scale resolution

ds df s

f

d

dM 1

Source Sample Detector

Preliminary results

Compass dimension(Dcom ) Dcom=0.45

Dfrac = 1 + Dcom

= 1.45

Log/Log plot for the crack of the coal sample

Conclusions

Inherent crack surface geometry will be revealed over many orders of

magnitude and analysed for fractal nature

CT measures with fundamental fractal nature will be identified

Best presentation and interpretation of data forthcoming from fractal

measures will be sought