ant and Planas, 1998]. The governing equations were then obtained from the crack compatibility condition.JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 108, NO. B2, 2119, doi:10.1029/2002JB001884, 2003
[32] Hillerborg et al. [1976], Hillerborg [1985], and Petersson [1981] analyzed mode I cohesive fracture by condensing out all the nodes other than those on the crack line and at the load point from the structural stiffness matrix. Thus they obtained the compliance matrix for the crack surface nodes and the load point [see also Baant and Planas, 1998]. The governing equations were then obtained from the crack compatibility condition.
[33] In the present problem of mode II fracture with a finite residual stress, the condition of compatibility of the crack opening with the deformation of the layer of snow may be written in a dimensionless form as [Zi and Baant, 2003]:
which must be coupled with the condition that the stress intensity factor at the cohesive crack tip KII = 0; here 
= x/D is dimensionless coordinate, 
= 
f D/Ewf = D/2 lch is dimensionless size (lch is Irwin's characteristic length), 
= w/wf is dimensionless crack sliding displacement, 
= /f is dimensionless shear stress, 
i = ai/D is dimensionless length of the weak zone (acting as a notch), 
N = CNE/D is dimensionless compliance corresponding to the nominal stress N = /N , i = Ci E/D is dimensionless compliance corresponding to preexisting stress i = i/f on the weak zone, and = E/C is dimensionless compliance for stress in fracture process zone; i.e., at caused by unit stress at 
(CN, Ci, and C are actual compliances). Note that all the variables in equation (15) are dimensionless.
[34] In Hillerborg's [1985] and Petersson's [1981] approach, the crack compatibility condition of the type of equation (15), coupled with the condition KII = 0, is integrated in small loading steps, which means that the entire history of displacement distributions must be followed even though only the peak load is needed. In the case of size effect studies, the entire histories of displacement distributions must be computed for many different sizes.
[35] Li and Liang [1993], Li and Baant [1997] and Li and Baant [1994] (in a discrete form), and Baant and Li [1995] (in a continuous form), developed for size effect studies a more efficient procedure in which the deformation history need not be computed and the peak load is calculated directly [see also Baant and Planas, 1998, section 7.5.4]. In this procedure, the problem is inverted by searching for the size D for which a given relative crack length corresponds to the peak load (or to N). For the present problem, this solution procedure must be adapted from mode I to mode II fracture, which is quite easy. It must also be generalized for nonzero residual stress r. The way to do that [Zi and Baant, 2003] is sketched in Appendix A.
[36] In this approach, the problem of directly calculating N for various D, without solving the history of displacement distribution, is recast as an eigenvalue problem. The size D for which a given corresponds to the peak load is the eigenvalue in the following dimensionless homogeneous Fredholm integral equation:
in which the subscripts preceded by a comma denote partial derivatives. Equation (16) is almost the same as that of Baant and Li [1995, equation (9)] except that 
is used instead of 
(this is more generally applicable to crack propagation problems in which the slope of the energy release rate changes from negative to positive [Zi and Baant, 2003]). The peak load is characterized by the nominal strength calculated from
[37] Choosing a series of values, one solves for each of them the eigenvalue as well as the eigenmode , (approximated as an eigenvector) from a discrete approximation of equation (16). Knowing and ,, one may then simply evaluate N from the discrete approximation of equation (17).
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[38] The snow layer in Figure 1 is considered as an example. The values E = 987.5 kPa, Poisson ratio 
= 0.25, f = 6.7 kPa, r = 5.0 kPa, and wf = 3.5 mm are chosen as the typical values for snow (based on the work by McClung [1977, 1979]). The crack line is subdivided by nodes into many equal intervals (Figure 4). The discrete values of the compliance (,) are computed by condensing out the interior nodes from a two-dimensional finite element analysis in the vertical plane (Figure 4), assuming the plane strain condition. To take into account the infinite length of the snow slab, the so-called “soak” elements (elements taking into account the effect of an infinite layer) are used at the ends of the meshed domain in computing the compliance.
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[39] The nominal strength values computed by the eigenvalue analysis are plotted in Figure 5a for two initial weak zone lengths a = 5D and 10D. As one can see, the trend of log N versus logD reveals a size effect. This trend can be closely fit by equation (8).
[40] To fit the size effect law (8) to the values obtained by the numerical cohesive crack analysis, it is convenient to rearrange equation (8) as a linear plot of -2 versus D (where = N - r, Figure 5b), given by
Choosing various values of r, one can pass a regression line of Y = (N - r)-2 versus X = D. The Y intercept of this line is 0-2 and its slope is (1/02D0), from which the optimum values of cohesive strength of the material, 0, and of the transitional size D0 may be identified (Figure 5b) for each r. The optimum value of r is that which gives the smallest coefficient of variation of errors. Then, using equation (7), one may calculate the fracture energy GII = 0cf02/E and the half length of fracture process zone, cf = 0D0/2. The GII value must, of course, approximately agree with the shaded area in Figure 2, which is GII = 2.97 N/m. The longer the initial weak zone, the better is the agreement of the input and output values of GII.
Citation: Ba
ant, Z. P., G. Zi, and D. McClung, Size effect law and fracture mechanics of the triggering of dry snow slab avalanches, J. Geophys. Res., 108(B2), 2119, doi:10.1029/2002JB001884, 2003.
