Abstract
The temperature distribution in a periodic array of parallel cracks acting as heat sinks is studied for the case of stationary crack motion by use of the Wiener–Hopf method. The problem arises in the investigation of cracks propagating into solid material, where the stresses driving crack motion are caused by heat transfer from the solid through the crack surfaces. The solution is given in terms of Fourier integrals involving infinite products. The heat-flux distribution in the vicinity of the crack tips is computed analytically from the high wavenumber asymptotics. Numerical solutions of the temperature distribution are presented for several values of the Biot and Péclet number, and the effect of varying these parameters is discussed qualitatively.
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References
M. Marder and J. Fineberg, How things break. Physics Today 49(9) (1996) 24-29.
L. B. Freund, Dynamic Fracture Mechanics. Cambridge: Cambridge University Press (1998) 581pp.
A. Yuse and M. Sano, Transition between crack patterns in quenched glass plates. Nature 362 (1993) 329-331.
M. Marder, Instability of a crack in a heated strip. Phys. Rev. E 49(1) (1994) R51-R54.
S. Sasa, K. Sekimoto and H. Nakanishi, Oscillatory instability of crack propagations in quasistatic fracture. Phys. Rev. E 50(3) (1994) R1733-R1736.
M. Adda-Bedia and Y. Pomeau, Crack instabilities of a heated glass strip. Phys. Rev. E 52(4) (1995) 4105-4113.
H.-A. Bahr, A. Gerbatsch, U. Bahr and H.-J. Weiß, Oscillatory instability in thermal cracking: a first order phase-transition phenomenon. Phys. Rev. E 52(1) (1995) 240-243.
B. I. Yakobson, Morphology and rate of fracture in chemical decomposition of solids. Phys. Rev. Lett. 67 (1991) 1590-1593.
T. Boeck, H.-A. Bahr, S. Lampenscherf and U. Bahr, Self-driven propagation of crack arrays: a stationary two-dimensional model. Editorially approved for publication in Phys. Rev. E.
B. Noble, Methods Based on the Wiener-Hopf Technique for the Solution of Partial Differential Equations. Oxford: Pergamon Press (1958) 256pp.
G. F. Carrier, M. Krook and C. E. Pearson, Functions of a Complex Variable. New York: McGraw-Hill (1966) 438pp.
D. G. Duffy, Transform Methods for Solving Partial Differential Equations. Boca Raton: CRC Press (1994) 512pp.
J. P. Benthem and W. T. Koiter, Asymptotic approximations to crack problems. In: G. C. Sih (ed.), Mechanics of Fracture, vol. 1. Leiden: Noordhoff International Publishing (1973) 131-178.
W. H. Press, B. P. Flannery, S. A. Teukolsky and W. T. Vetterling, Numerical Recipes in FORTRAN. Cambridge: Cambridge University Press (1993) 992pp.
Wolfram Research, Inc., Mathematica. Version 2.2. (1995).
M. Abramowitz and I. A. Stegun (eds.), Handbook of Mathematical Functions. Frankfurt: Harri Deutsch (1984) 468pp.
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Boeck, T., Bahr, U. Temperature distribution in an array of moving cracks acting as heat sinks. Journal of Engineering Mathematics 37, 289–303 (2000). https://doi.org/10.1023/A:1004552819700
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DOI: https://doi.org/10.1023/A:1004552819700