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On the current enhancement at the edge of a crack in a lattice of resistors

Part of: Applications to specific types of physical systems

Published online by Cambridge University Press:  01 July 2016

Howard M. Taylor  and
Dennis E. Sweitzer
Howard M. Taylor*
Affiliation:
University of Delaware
Dennis E. Sweitzer*
Affiliation:
University of Delaware
*
∗ Postal address: Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA.
∗ Postal address: Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA.
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Abstract

Consider a network whose nodes are the integer lattice points and whose arcs are fuses of 1Ω resistance. Remove a horizontal segment of N adjacent vertical arcs, forming a ‘crack’ of length N. Subject the network to a uniform potential gradient of v volts per arc in the north-south (or vertical) direction and measure the current in one of the two vertical arcs at the ends of the crack. We write this current in the form e(N)v, and call e(N) the current enhancement.

We show that the enhancement grows at a rate that is the order of the square root of the crack length. Our method is to identify the enhancement with the mean time to exit an interval for a certain integer valued random walk, and then to use some of the well-known Fourier methods for studying random walk. Our random walk has no mean or higher moments and is in the domain of attraction of the Cauchy law. We provide a good approximation to the enhancement using the explicitly known mean time to exit an interval for a Cauchy process. Weak convergence arguments together with an estimate of a recurrence probability enable us to show that the current in an intact fuse, that is in the interior of a crack of length N, grows p roportionally with N/logN.

Keywords

Current enhancementrandom fuse modelrandom walk

MSC classification

Primary: 82D30: Random media, disordered materials (including liquid crystals and spin glasses)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 30 , Issue 2 , June 1998 , pp. 342 - 364
Copyright
Copyright © Applied Probability Trust 1998 

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References

[1] de Arcangelis, L., Redner, S. and Herrmenn, H. J. (1985). J. de Physique Lett. (Paris) 96, L585.Google Scholar
[2] Doyle, P. G. and Snell, J. L. (1984). Random Walks and Electrical Networks. The Mathematical Association of America, Washington, DC.CrossRefGoogle Scholar
[3] Duffin, R. J. and Shaffer, D. H. (1960). Asymptotic expansion of double Fourier transforms. Duke Math. J. 27, 581–596.Google Scholar
[4] Duxbury, P. M., Beale, P. D. and Leath, P. L. (1986). Size effects of electrical breakdown in quenched random media. Phys. Rev. Lett. 57, 1052–1055.CrossRefGoogle ScholarPubMed
[5] Duxbury, P. M., Leath, P. L. and Beale, P. (1987). Breakdown properties of quenched random systems: the random fuse network. Phys. Rev. B, 36, 367–380.Google Scholar
[6] Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edn. John Wiley and Sons, New York.Google Scholar
[7] Flanders, H. (1971). Infinite networks. I. Resistive networks. 18, 326–331.Google Scholar
[8] Flanders, H. (1972). Infinite networks. II. Resistance in an infinite grid. J. Math. Anal. Appl. 40, 30–35.CrossRefGoogle Scholar
[9] Gikhman, I. I. and Skorohod, A. V. (1969). Introduction to the Theory of Random Processes. W. B. Saunders Company, Philadelphia. (Translation).Google Scholar
[10] Hikami, F. and Chou, T.-W. (1990). Explicit crack problem solutions of unidirectional composites: elastic stress concentrations. AIAA J. American Institute of Aeronautics and Astronautics Journal 28, 499–505.Google Scholar
[11] Hsieh, C. and Thomson, R. (1973). Lattice theory of fracture and crack creep. J. Appl. Phys. 44, 2051–2063.CrossRefGoogle Scholar
[12] Kac, M. and Pollard, H. (1950). The distribution of the maximum of partial sums of independent random variables. Canad. J. Math. 2, 375–384.Google Scholar
[13] Kesten, H. (1961). On a theorem of Spitzer and Stone and random walks with absorbing barriers. Illinois J. Math. 5, 246–266.Google Scholar
[14] Kesten, H. (1961). Random walks with absorbing barriers and Toeplitz forms. Illinois J. Math. 5, 267–290.Google Scholar
[15] Phoenix, S. L. and Smith, R. L. (1989). The strength distribution and size effect in a prototypical model for percolation breakdown in materials. (Unpublished draft, 1989).Google Scholar
[16] Spitzer, F. and Stone, C. (1960). A class of Toeplitz forms and their application to probability theory. Illinois J. Math. 4, 253–277.Google Scholar
[17] Spitzer, F. (1964). Principles of Random Walk. Van Nostrand Company, Inc., Princton, NJ.CrossRefGoogle Scholar
[18] Taylor, H. and Karlin, S. (1984). An Introduction to Stochastic Modeling. Academic Press, Inc., Orlando, FL.Google Scholar
[19] Zemanian, A. H. (1984). A classical puzzle: the driving point resistances of infinite grids. IEEE Circuits and Systems Magazine 6, 7–9.Google Scholar
[20] Zhou, S. J. and Curtain, W. A. (1995). Failure of fiber composites: a lattice Green function model. Acta metall. mater. 43, 3093–3104.Google Scholar