JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 108, NO. B2, 2119, doi:10.1029/2002JB001884, 2003

3. Analysis by LEFM

Figure 1. Geometry of snow slab (2a₀ is initial weak zone, 2a is cohesive crack, 2c_f is the fracture process zone); forces acting on an element of slab (top left); and typical shear stress distribution obtained by the cohesive crack analysis (bottom right).

[15] We assume that a sliding (mode II) crack of length 2a develops in snow near the underlying rigid base and propagates symmetrically at both tips (Figure 1). Consider first that the residual shear stress, tau _r, that develops in this sliding crack after large slip is negligible, tau _r approximately 0. By arguments of symmetry, the longitudinal normal stress, sigma , in the sliding layer must vanish at the point of symmetry, i.e., sigma 0 at x = 0, where x is the longitudinal coordinate measured from the center of the crack. The equilibrium condition of an element dx of the layer requires that ( sigma + d sigma )D - sigma D - rho gDsin phi = 0 or d sigma /dx = rho gsin phi (for tau = 0, Figure 1), which means that

where D is snow thickness and rho is the mass density of snow (both assumed to be uniform and equal to the means calculated from thickness and density data) and g is magnitude of gravity acceleration. Beyond the crack (x > a), sigma = 0, and the weight of the snow slab is transmitted to the base entirely by shear stresses. Longitudinally, the upper half of the sliding layer is in tension and the lower half in compression. Evidently, a tensile break must eventually occur in the upper half, but it seems reasonable to assume that this usually happens only after stability loss and is not what triggers the avalanche [McClung, 1981].

[16] The complementary strain energy of one half of the sliding layer is

Here E prime is the effective Young's modulus of the sliding snow layer, b is the lateral width of this layer, and tau _N is the nominal shear stress, defined as the shear stress that would be needed to support the weight of the sliding layer if there were no crack, i.e.,

The nominal stress, tau _N, is a load parameter and represents the component, in the direction of slope, of the gravity force per unit base area. The energy release rate, inline equation , is

In LEFM, the fracture criterion is inline equation = G_II, where G_II is the mode II fracture energy of snow, i.e., the energy required to form a sliding crack of a unit area, considered as a material constant. Setting = G_II, one gets nominal stress at failure

Here alpha is the relative crack length, which will be discussed later. From now on, tau _N will denote nominal shear strength.

[17] From experience with other size effect problems [Ba z caron ant and Planas, 1998], one may expect alpha to be constant when geometrically similar structures of different sizes D are compared. Under that assumption, the LEFM size effect according to (5) is tau _N proportional to D^-1/2, which must have been expected for more fundamental reasons [Ba z caron ant, 1984, 1993]. The approximate size independence of alpha will be better justified later.

Citation: Ba z caron 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.