Application of dynamic fracture mechanics to dry snow slab avalanche release

Abstract

Field observations and measurements show that dry snow slab avalanches initiate by propagating shear fractures within a relatively thin weak layer sandwiched between a planar, stronger, thicker slab above and stronger material below. After initiation, the weak layer fracture can propagate up and down slope for distances which range from about 10 to 100′s of meters to cause tensile fracture through the body of the slab which results in avalanche release. In this paper, dynamic fracture mechanics is applied to slab tensile fracture after which avalanche release is imminent. Two mechanisms for production of tensile stress are explored employing field measurements of slab properties and lab measurements. The first considers inertial effects related to quasi-brittle fracture near the tip of a propagating weak layer shear fracture. The second is concerned with the tensile stress generation from the stress drop behind the crack front as the fracture propagates in the weak layer. Analysis suggest that both mechanisms can contribute to produce the tensile fracture line which precedes avalanche release. Even though both mechanisms may operate together, they are analyzed separately in this paper.

Appendix: Considerations from the elastic equation of motion

Alpine snow is generally not an elastic material. However, it is useful to explore the results of application of the equation of motion which contains the assumption of elasticity. Equation (5) gives the equation of motion employing the separable form (4). As an example, consider initially the mode II stress intensity factor for a semi-infinite crack in an unbounded elastic solid. For the present case:

$$ K_{II} (l,\;0) = \sqrt {\frac{2}{\pi }} \int\limits_{0}^{l} {\frac{{\tau_{g} - \tau_{r} }}{{\sqrt {l - x} }}} dx = 2(\tau_{g} - \tau_{r} )\sqrt {\frac{2l}{\pi }} . $$

(25)

The equation of motion is then given by (e.g. Freund 1990):

$$ \dot{l} = C_{R} \left( {1 - \frac{{l_{c} }}{l}} \right), $$

(26)

where $l_{c} = {{E^{\prime}\Gamma_{II} \pi } \mathord{\left/ {\vphantom {{E^{\prime}\Gamma_{II} \pi } {8\left( {\tau_{g} - \tau_{r} } \right)}}} \right. \kern-\nulldelimiterspace} {8\left( {\tau_{g} - \tau_{r} } \right)}}^{2}$.

From the field measurements of PST, the median value $l_{c} = 0.30{\text{ m}}$. Equation (26) contains the prediction that the crack speed could reach 99% of the Rayleigh speed in a distance of about 30 m. The distances measured by Hamre et al. (2014) had a range: 12–278 m. with a mean 81 m and median 35 m. Equation (18) predicts infinite tensile stress at the Rayleigh speed (Rice 1980).

These results depend on the form of $K_{II} (l,\;0)$. In general, for a stress intensity factor: $K_{II} \propto l^{n}$, it is required that ${{\partial K_{II} } \mathord{\left/ {\vphantom {{\partial K_{II} } {\partial l > 0}}} \right. \kern-\nulldelimiterspace} {\partial l > 0}}$ (or $n > 0$) to produce a running crack (Rice 1968) and maintain acceleration. The equation of motion is then:

$$ \dot{l} = C_{R} \left[ {1 - \left( {\frac{{l_{c} }}{l}} \right)^{2n} } \right]. $$

(27)

Palmer and Rice (1973) proposed a form: $K_{II} \propto l$ which has been applied to initiation of dry slab avalanches (McClung 1981). In this case (n = 1) with $l_{c} = 0.30\;{\text{m}},$ the speed would reach 99% of the Rayleigh speed within 3 m.

The conclusion from these results is that completely elastic modelling is unrealistic which is as expected for alpine snow. The implied avalanche dimensions and propagation distances are too short compared to field measurements.

From (27) the acceleration is given by:

$$ \ddot{l} = 2nC_{R}^{2} \left( {\frac{{l_{c} }}{l}} \right)^{2n} \frac{1}{l}\left[ {1 - \left( {\frac{{l_{c} }}{l}} \right)^{2n} } \right]. $$

(28)

Equation (28) suggests that the acceleration is proportional to $C_{R}^{2}$. From Table 1, the acceleration for density $300\;{\text{kg}}/{\text{m}}^{3}$ is expected to be 10 times that for 100 kg/m³ for 1 Hz and about 8.5 times higher for 100 Hz.

With $l_{c} = 0.30\;{\text{m}}$, the maximum value of $l_{c} /l = 0.025$ from the field measurements of Hamre et al. (2014) which again suggests, assuming elasticity, that the maximum speeds must be very close to $C_{R}$.

The results in this paper are rate independent except for analysis with the limits of $E$ in (16) and (17). The results from the elastic equation of motion suggest that the acceleration predicted is too high to explain the long propagation distances required to explain avalanche dimensions. Full rate dependence and effects such as viscous damping are beyond the scope of this paper. However, it is easy to show that if the resistance to fracture expansion increases with speed, the acceleration will be reduced. Freund (1990) showed that if the fracture resistance increases linearly with speed: $\Gamma_{II} = \Gamma_{0} (1 + \beta \dot{l}/C_{R} )$ then (26) is replaced by:

$$ \frac{{\dot{l}}}{{C_{R} }} = \frac{{1 - l_{c} /l}}{{\left[ {1 + \beta \frac{{l_{c} }}{l}} \right]}}. $$

(29)

From (29), for $\beta > 0$ at position $l$, the speed is reduced compared to (26) or more formally, ${{\partial \ddot{l}} \mathord{\left/ {\vphantom {{\partial \ddot{l}} {\partial \beta < 0.}}} \right. \kern-\nulldelimiterspace} {\partial \beta < 0.}}$ However, such would strictly hold for linear elastic conditions otherwise rate effects in increasing the effective $C_{R}$ by increasing modulus with rate could enter.