UNCLASSIFIED Defense Technical Information Center Compilation Part Notice ADPO10905 TITLE: A Fractional Brownian Motion Model of Cracking DISTRIBUTION: Approved for public release, distribution unlimited This paper is part of the following report: ITLE: Paradigms of Complexity. Fractals and Structures in the Sciences To order the complete compilation report, use: ADA392358 The component part is provided here to allow users access to individually authored sections f proceedings, annals, symposia, ect. However, the component should be considered within he context of the overall compilation report and not as a stand-alone technical report. The following component part numbers comprise the compilation report: ADP010895 thru ADP010929 UNCLASSIFIED 117 A FRACTIONAL BROWNIAN MOTION MODEL OF CRACKING P.S. ADDISON, L.T. DOUGAN, A.S. NDUMU, W. M.C. MACKENZIE Civil Engineering Group, School of the Built Environment, Napier University, Merchiston Campus, 10 Colinton Road, Edinburgh,E HJO 5DT, Scotland, UK. An attempt is made to find the fractal cutoff of crack profiles on the tension face of concrete beams subjected to uni-axial bending. Previous work by the authors has shown that such cracking can be interpreted as a non-Fickian diffusive phenomenon resulting from a self-affine random fractal process: specifically fractional Brownian motion (IBm). In addition, a spatial description of the cracking geometry can be found from experimental data using both a (Hurst) scaling exponent and a diffusion-type coefficient. Herein the authors find that the fractal description of the crack profiles extends down to less than 0.75pnm. The use of a scanning electron microscope to probe the crack profile (and surface) at smaller scales is discussed and the synthesis of crack surfaces using fBm is described briefly. 1 Introduction An understanding of the behaviour of cracking in structural elements is of great importance in the analysis and subsequent safe design of engineering structures. As yet, however, there is no definitive theoretical framework for the propagation of cracks and resulting fracture energy. It has recently been found that the irregular geometries of both crack surfaces and crack profiles in a variety of materials may be described (and subsequently modelled) using fractal geometry. The use of fractal geometry to describe cracking phenomena is now widespread (e.g. see referencesl"). Previous work by the authors5'6 has shown that crack profiles on concrete beams in tension can be modelled as iBm trace functions which require a Hurst exponent and a spatial diffusion coefficient to completely describe the spatial distribution of the crack. In addition, the authors have linked the fBm description of the cracking phenomena to an effective Fokker-Planck equation which described the diffusive nature of the cracking phenomena through space. In this paper the cracks are investigated at higher resolutions in an attempt to determine whether a fractal cut-off scale exists. The value of such a Euclidean threshold is of significant importance in the determination of the energy of fracture. 2 The Diffusive Nature of fBm Fractional Brownian Motion (mBm) is a generalisation of Brownian motion suggested by Mandelbrot7 which has found a variety of uses in the natural sciences (see for example Addison and Ndumu8 and the references contained therein). Fractional Brownian motion is defined as: y(x) (l I { L(X X') H 2 (.x')H ] dW(x') + fX (x -x') H 2 dW(x1) r(H + Y21) 118 where dW(x) is a Gaussian random function with zero mean and unit standard deviation, H is the Hurst exponent9, and F is the gamma function. When H=0.5 Eq. (1) models classical Brownian motion which produces normal, or Fickian, diffusion. From Eq. (1) it may be seen that the immpr ocess is correlated over all length scales, i.e., it has an infinite memory associated with it. 800 700 600 500 A y(x) 400 A 300 200 100 -100 ... I.. . . .. 1.., . 0 50 100 150 200 250 300 350 400 450 500 x Figure 1: Diffusive scaling of an fBm (H=O.7 5 Kf=lO) An example of a superdiffusive fBm (i.e. one with H > 0.5) is shown in Fig. 1. The diffusive scaling of the fBm process shown in Fig. 1 may be defined as y= sH (2) where cy is the standard deviation of the y-excursions (Ay) on the trace for a window length s; Kf is a fractal diffusion coefficient. Eq. (2) is in fact the standard deviation of the probability density function P(y,x) =4rKx 2M ep{ 4Kxf.2. (3) which is a non-Fickian scaling of a Gaussian probability density function through space. (If H = ½then Eq. (3) reduces to the solution of a Fickian based diffusion from a point 2 source.) Furthermore, it has been shown by Wang and Lung'° that Eq. (3) is the solution to the effective Fokker-Planck equation: 9P(y,x) _ 2H Kf x2H-1 2 p(Y'X) (4) which describes the probability of occurrence ofy(x) at spatial location x. Eq. (4) is in fact a generalisation of the classical Fickian diffusion equation with a spatial diffusion coefficient. The equation reduces to the classical equation for H=0.5. 119 It can be seen from the above that, over a large number of realisations, fBm approximates a non-Fickian diffusive process described by Eq. (4). The authors have previously shown that persistent fBm (H > 0.5) is a suitable model for cracking on the tension face of a concrete beam in bending5.In addition, both H and Kf are required for a complete geometric description of the cracking phenomena. It was shown by the authors how these parameters can be found from experiment. The mean values of Kf and H were found for a series of flexure cracks in concrete beams to be 0.084 and 0.77, respectively (i.e. superdiffusive surfaces with fractal dimensions between 1 and 1.5). This gives the standard deviation of the cracking across the beam as ary = %x .084 s077 where both s and expressed in millimetres. Thus, for the 40mm wide specimens used in the study the Sare expected standard deviation of the crack displacement across the beam is 7.02mm. These experimentally derived parameters can be used to synthesise crack patterns using fBms. An example of this is shown in Fig. 2 using the nmgme neration method described by Addison et al., 11.In the figure a crack with measured values of H and Kf of 0.75 and 0.133 respectively is shown together with a synthesised crack with the same parameter values. The similarity between the two traces is evident from a visual inspection of the plot. 1 T- 0 Actual crack trace from digitised image = -1 Distance -2 (H0"75,Kf=0"133) Along -3 . Beam -4 / (mm) -5 (-6 " - Synthesised trace - - (H=O. 75, Kf=O. 133) -8 1 . . . i . . i 0 5 10 15 20 25 30 35 40 Distance Across Beam (mm) Figure 2: Comparison of synthesised and experimental cracks 3 Crack Profile Analysis Natural fractals tend to exhibit fractal characteristics over a limited range of scales12. Below a cut-off level, the natural fractal object tends to revert to a Euclidean form. The authors have recently pursued the search for the cut-off length scale in the crack patterns as it has implications for the measurement of the true areas of crack surfaces and hence energy dissipation across the surface. Fig. 3 shows one of the cracks studied by the authors at a magnification level of 6x. Seven boxes are placed on the crack profile indicating regions where a closer inspection was taken of the crack profile at the higher magnification of 40x. In addition, two smaller boxes indicate locations where the crack was studied at 50x and 100x magnification, respectively. Fig. 4 contains a log-log plot of ay against s. From such a plot it is possible to calculate both H and Kj. A line of slope H=1 is also given in the plot, corresponding to a dimension of unity, i.e. a smooth Euclidean curve. It would be expected that the plotted curves tend to this slope at the Euclidean cut-off. 120 0 -1 -2 Distance ABeloanmg --43 -E D 5x (mm) -5 D G -6 -7 50x magnification -8 . . I ., . . . I ,. , ., . I . . . ., I ,. . . . I . . , . I .-. . .i 0 5 10 15 20 25 30 35 40 Distance Across Beam Figure 3: Crack profile showing analysed regions 0 .. ....... i ........... ............. --- --..-..-........... ....! ...... ... ......... --- --..--.-..... .......... ..... ... .... -2 -6 ---- ------ - 4----------- 2----0-1-2.............-..-. . ..... ........ r .. In(s)) ------ ------I- --- --+- - + ----- ..... -- ------ - -2 .-.......... --------.- I--(cid:127) ---- - ................... ........ 3--6 -------..- -.-. 1 -i. . ...;.. . . ..---. . ..---,. .... .-.-.. ,-----,--.-.. .---- O RIG-NA - F-ur 4: Loai-h6 i:lt -4 faesssfrteca-k2s oni --f-i u--r- -- 3-. T-h- -g--r3-p - --shwe4tnig ot bottom left) two sections representing 50x and 100× magnifications. Table 1 contains the H and Kf values for the various regions of the crack in Fig. 3 measured from Fig. 4. It can be seen from the table that measured H values for the whole crack and the average values from the seven boxes A to G are in good agreement. However, the Kf values between the two are significantly different. In fact the KJ values vary over a large range (0.004-0.118) across the selected boxes. The reason for this variability is as yet unclear. Zooming in at 50× and l00x again produces persistent values of the Hurst exponent. It can be seen from the plot that the fractal description of the curves extends down to the lower limits of the 100×x magnification: this relates to a resolution of 7.5× l0"mm of crack per pixel. 121 Tablel: The Measured H and Kf Values for the Crack in Figure 3. H Kf Whole Crack (6x) 0.74 0.069 A (40x) 0.87 0.093 B (40x) 0.64 0.004 C (40x) 0.68 0.008 D (40x) 0.61 0.018 E (40x) 0.72 0.015 F (40x) 0.82 0.118 G (40x) 0.83 0.091 Average (A-G) 0.74 0.050 50x 0.71 0.025 10OX 0.74 0.030 4 Concluding Remarks From Fig. 4, the lower limit to the fractal behaviour of the cracking patterns (if it exists!) appears to be below 0.7511m. This is significantly less than the value between 10 and 20 Ium (which is the approximate size of calcium silicate hydrate) suggested by Souma and Barton4.The authors have recently initiated research to search for a fractal cut-off scale at higher resolutions at the crack edge using a scanning electron microscope (e.g. Fig. 5). This work has so far proved inconclusive due to the difficulty in finding reasonable vertical sections through the crack edge at these higher scales. It is hoped that an improvement in the experimental techniques will lead to a better understanding of the crack geometry at these smaller scales. Figure 5: An electron microscope image of the crack profile at 2000x magnification. Note the difficulty in defining the edge of the crack. 122 The determination of the true fractal cut-off scale has implications for the synthesis of prefractal fBm crack profiles and surfaces and hence the calculation of crack energy across the crack surface. If the crack surface is in fact fBm, then the crack profile may be treated as the result of a vertical cut through the surface"3.If this proves to be the case, then it is known that the fractal dimension of the surface DsuOce is equal to Dprof'ie+l. It is relatively simple to generate such a surface using a variety of methods. It should be possible therefore to synthesise the crack surface using the Kf and H values found from experiment. Fig. 6 shows an fBm surface generated using the turning bands method' 4.The authors intend to pursue the measurement and synthesis of crack profiles and surfaces in order to define the fracture energy of cracking in terms of a fractal geometric framework based on fBms. VI N 73ý~- \0,3 Figure 6: An iBm surface generated using the turning bands method (H=0.8) References 1. Carpinteri A. (1994a). 'Scaling laws and renormalization groups for strength and toughness of disordered materials.' Int. J. Solids Struct., 31(3), 291-302. 2. Carpinteri A. (1994b). 'Fractal nature of material microstructure and size effects on apparent mechanical properties.' Mechanics ofM aterials, 18, 89-101. 3. Borodich F.M. (1997). 'Some fractal models of fracture.' J. Mech. Phys. Solids, 45(2), 239-259. 4. Saouma V.E., Barton C.C. and Gamaleldin N.A. (1990). 'Fractal characterisation of fracture surfaces in concrete.' Engng. Fract.M ech., 35, 47-53. 123 5. 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