Disordering of a dynamic planar crack front in a model elastic medium of randomly variable toughness
This paper is the development of the work of \textit{J. R. Rice, Y. Ben- Zion} and \textit{K.-S. Kim} [J. Mech. Phys. Solids 42, No. 5, 813-843 (1994; Zbl 0806.73056)] concerning strictly linearized analysis of the propagation of a planar crack with a nominally straight front in a model elastic solid with a single displacement component. The aim of this paper is to consider a crack propagating in a region of randomly variable toughness and ask how much the crack front deviates, statistically, from straightness. It is supposed that the material rupture is ruled by the local energy release rate criterion. Using the above cited results, the corresponding first-order expression is given for the deviation of a crack front from straightness as a direct integral expression for the deviation of the material toughness from uniformity in the crack plane. Then this expression is used to analyse the autocorrelation of the crack front position when the toughness deviations are random. The authors introduce two autocorrelation functions, namely one is defined by statistical properties of the random toughness, \(\tau(z,t)\), and other is the deviation from straightness of the crack front, \(A(z,t)\), (\(z\) is the coordinate along the crack front, \(t\) is the time). Applying time and space Fourier transform, the power spectra of \(\tau\) and \(A\) are related. From hypothesis that \(A(z,t)\) is a stationary random variable, \(A\) is expressed as a linear functional of \(\tau(z,t)\), and then a statistical analysis of the deviation from straightness \(A\) is performed. It is shown that while the straight shape is indeed recovered for propagation through a zone of precise uniform value of the critical local energy release rate, \(G_{\text{crit}}\), the straight crack front becomes increasingly more disordered as it propagates through a region of arbitrary small but sustained random variation in \(G_{\text{crit}}\). Then the authors conclude that the straight crack front is configurationally unstable. An expression is obtained for autocorrelation function \(A\) which is similar to well-known equation for an antiplane frictional fault. The results obtained indicate that, according to the strictly linearized perturbation analysis, the front of a crack that can run forever in a random unbounded medium grows overwhelmingly wavily. The root mean square deviation in position diverges logarithmically with travel distance across the random toughness region as do the variances of the propagation velocity and slope of the crack front.
scientific article; zbMATH DE number 618142