Abstract
The branching patterns formed as a result of crack growth in dimethacrylate resins below their glass transition temperatures looked similar to fractal trees. The skeletons of the patterns were analysed numerically for their topological and geometrical properties. The number of branches, N i, mean branch lengths, N i, and branch angles of a particular order, defined according to the Strahler and inverted Weibel schemes, followed exponential scaling behaviour: N i ∼ (R b)−i and Li ∼ (R l)i. Using the relationship for the fractal dimension D=In R B/In R L, a value of D=1.4 was obtained for the fracture pattern. Fractal behaviour was also examined by the box-counting method which indicated a power-law dependence of the mass on the box size with fractal dimension exponent D=1.4 in the case of the fracture pattern. However, the mass-shell method for both the fracture pattern and the fractal trees gave an exponential increase of mass with distance from the origin, rather than the power-law behaviour expected for fractals. This was attributed to the fact that branches of different sizes were distributed in restricted regions of space closer to the periphery, rather than uniformly over the whole pattern.
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References
M. J. Doyle, A. Maranci, E. Orowan Frs and S. T. Stark, Proc. R. Soc. (Lond.) A329 (1972) 137.
E. H. Andrews, “Fracture in Polymers” (American Elsevier, New York, 1968).
A. J. Kinloch and R. J. Young, “Fracture Behaviour of Polymers” (Applied Science, London, New York, 1983).
H. H. Kausch, “Polymer Fracture”, 2nd Edn (Springer, Berlin Heidelberg New York, 1987).
C. S. Smith, Rev. Mod. Phys. (1964) 524.
F. Spaepen and D. Turnbull, Scripta Metall. 8 (1974) 563.
R. E. Horton, Geol. Soc. Am. Bull. 56 (1945) 275.
A. N. Strahler, Trans. Am. Geophys. Union 34 (1953) 345.
Idem, ibid. 38 (1957) 913.
E. R. Weibel, “Morphometry of the human lung” (Springer, Berlin, 1963).
E. L. Hinrichsen, K. J. Maloy, J. Feder and T. Jossang, J. Phys. A 22 (1989) L271.
J. Feder, E. L. Hinrichsen, K. Maloy and T. Jossang, Phys. D 38 (1989) 104.
P. Ossadnik, Phys. Rev. A 45 (1992) 1058.
B. B. Mandelbrot, “The Fractal Geometry of Nature” (Freeman, San Francisco, 1982).
J. Feder, “Fractals” (Plenum, New York, 1988).
T. Vicsek, “Fractal Growth Phenomena” (World Scientific, Singapore, 1989).
P. Meakin, Science 252 (1991) 226.
P. Meakin, G. Li, L. M. Sander, E. Louis and F. Guinea, J. Phys. A 22 (1989) 1393.
T. A. Witten and L. N. Sander, Phys. Rev. Lett. 47 (1981) 1400.
P. Meakin, in “Phase Transitions and Critical Phenomena”, edited by C. Domb and J. L. Lebowitz (Academic Press, New York, 1987) pp. 336–489.
K. J. Maloy, J. Feder and T. Jossang, Phys. Rev. Lett. 55 (1985) 2688.
K. J. Maloy, F. Boger, J. Feder, T. Jossang and P. Meakin, Phys. Rev. A 36 (1987) 318.
H. A. Laroche, J. F. Fernandez, M. Octavio, A. G. Loeser and C. J. Lobb, ibid. 44 (1991) R6185.
E. Ben-Jacob and P. Garik, Phys. D 38 (1989) 16, and references therein.
F. Family and T. Vicsek, “Dynamics of Fractal Surfaces”, (World Scientific, Singapore, 1991).
B. Mandelbrot, Phys. Scripta 32 (1985) 257.
F. Grey and J. K. Kjems, Phys. D 38 (1989) 154.
W. S. Shin, X. F. Li, B. Schwartz, S. Wunder and G. Baran, J. Dent. Mater., 9 (1993) 317.
R. E. Robertson and V. E. Mindroiu, J. Mater. Sci. 20 (1985) 2801.
Idem, Polym. Eng. Sci. 27 (1987) 55.
T. Y. Pan, R. E. Robertson and F. E. Filisko, J. Mater. Sci. 24 (1989) 3635.
J. Vannimenus, in “Universalities in Condensed Matters”, edited by R. Jullien, L. Peliti, R. Rammal and N. Boccara (Springer, Berlin, 1985).
N. MacDonald, “Trees and Networks in Biological Models” (Wiley, New York, 1983).
B. West, “Fractal Physiology and Chaos in Medicine” (World Scientific, Singapore, 1990), and references therein.
F. Caserta, H. E. Stanley, W. D. Eldred, G. Daccord, R. E. Hausman and J. Nittmann, Phys. Rev. Lett. 64 (1990) 95.
F. Family, B. R. Masters and D. E. Platt, Phys. D 38 (1989) 98.
T. Matsuo, R. Okeda, M. Takahashi and M. Funata, Forma 5 19 (1990).
J. Vannimenus and X. G. Viennot, J. Stat. Phys. 54 (1989) 1529.
V. K. Horvath and H. J. Herrmann, Chaos Solitons Fractals 1 (1991) 395.
A. C. Moloney and H. H. Kausch, J. Mater. Sci. Lett. 4 (1985) 289.
K. Horsfield, J. Appl. Physiol. 68 (1990) 457.
D. G. Tarboton, R. L. Bras and I. Rodigueziturbe, Water Resources Res. 24 (1988) 1317.
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Djordjevic, Z.V., Li, X.F., Shin, W.S. et al. Fractal and topological characterization of branching patterns on the fracture surface of cross-linked dimethacrylate resins. JOURNAL OF MATERIALS SCIENCE 30, 2968–2980 (1995). https://doi.org/10.1007/BF00349671
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DOI: https://doi.org/10.1007/BF00349671