Mode II fracture parameters of dry snow slab avalanche weak layers calculated from the cohesive crack model

Abstract

Release of dry snow slab avalanches starts with shear failure within a thin weak layer beneath a stronger, cohesive slab. The propagating fracture is within the weak layer as mode II. The fundamental fracture properties include the weak layer critical mode II fracture energy and stress intensity factors. In this paper, mode II fracture energy was calculated from the cohesive crack model from dry slab avalanche fracture line data and in-situ shear fracture tests. One advantage of the cohesive crack model is that it does not require the effective elastic modulus which is highly rate dependent and largely unknown in avalanche release. The results gave weak layer mode II fracture toughness less than mode I fracture toughness for the cohesive slab and fracture energy about one order of magnitude less than for solid ice. From the fracture energy and the mode II stress intensity factor, Irwin’s relation enables calculation of the effective elastic modulus. Calculations of the effective modulus for mode II initiation were compared with other estimates and they were shown to be consistent.

Appendices

Appendix 1: Viscoelastic theory applied to PST and avalanche release

1.1 Time dependence for PST

Propagation saw tests contain both elastic and viscous deformation. The saw is moved at a speed $(V_x)$ through the weak layer which is typically a few centimeters per second and comparable to the size of the FPZ $(\omega )$. The characteristic straining time was given by Palmer and Rice (1973) as: $\omega /V_x$ or the characteristic strain-rate $(\dot{\varepsilon })$ is about 1/s. Shinojima (1967) measured the viscoelastic relaxation time $(\tau )$ for alpine snow as a function of the strain-rate (/s) and obtained: $\tau =0.06({\dot{\varepsilon }}^{-7/8})(\hbox {s})$ for snow densities in the range $130-350 \,\hbox {kg}/\hbox {m}^{3}$. When applied to the scale of the FPZ, this relation yields, $\tau \approx 0.06\hbox { s}$. This result implies viscoelastic relaxation for the PST since it is much shorter than the time for the saw to traverse the FPZ $(\approx 1\hbox { s})$. This was proven experimentally by Reuter and Schweizer (2012) since they showed that the critical cut length (L) is shorter for higher surface energy input into the snowpack which implies higher temperature. Any experiment on alpine snow which displays temperature dependence over a narrow range of temperatures cannot be considered elastic. Since the elastic properties of alpine snow are essentially temperature independent (McClung 1996), the results of Reuter and Schweizer (2012) confirm some viscous deformation for the PST.

The relaxation time relationship of Shinojima (1967) [$\tau =0.06({\dot{\varepsilon }}^{-7/8})(\hbox {s})$] represents an extrapolation since his data were obtained for strain-rates less than about $10^{-3}$/s at which $\tau \approx 15\hbox { s}$. Camponovo and Schweizer (2001) measured both the dynamic viscosity and storage modulus for a strain-rate of 1/s. For snow densities: 215–255 kg/m$^{3}$, the ratio was about 0.04 s which, from the perspective of viscoelastic modelling, is comparable to the relaxation time. It suggests that the relationship provided by Shinojima (1967) gives a reasonable approximation for the relaxation time (0.06 s) for ${\dot{\varepsilon }}=1$/s. Thus, both the estimates of Shinojima (1967) and Camponovo and Schweizer (2001) imply that viscoelastic effects are important for the PST with the important strain-rate expected as about 1/s in terms of fracture initiation. The analysis and the data show that alpine snow must be deformed extremely rapidly to avoid time dependent (viscous deformation).

There are two important rate regimes for alpine snow divided by the brittle to ductile (or fracture—creep) strain-rate which is about $10^{-4}$/s (Narita 1980). Fracture processes are contained within strain-rates faster than about $10^{-4}$/s while creep processes are for slower rates. Scapozza (2004) measured snow properties for strain-rates 10$^{-4}$–10$^{-6}$/s which imply mostly creep deformation. Rates in the fracture regime $(\gg 10^{-4}/s)$ will be of importance for the PST. Experimental data (Mellor 1975) show that the effective elastic modulus increases with increasing rate while the peak strength decreases with increasing rate (Narita 1980). According to Mellor (1975), the effective elastic modulus does not approach time independence until rates approach about 100 Hz. For a given density, the effective elastic modulus increases by more than a factor of 10 for frequency increase from 1 to 100 Hz.

A 4-parameter viscous fluid (Bader 1962; Shinojima 1967; Mellor 1975) has been used as a simple model to describe viscoelasticity of snow for more than 50 years. It consists of a Maxwell fluid in series with a Kelvin solid. The creep compliance (constant stress) is given by:

$$\begin{aligned} J(t)=\frac{1}{E_0 }+\frac{t}{\eta _0 }+\frac{1}{E_1 }\left( {1-e^{-\frac{t}{\tau }}} \right) \end{aligned}$$

(12)

where subscript (0) applies to the Maxwell unit, (1) to the Kelvin unit and t is time. The terms represent: $1/E_0$ (time independent elastic response); $t/\eta _0$ (long time viscous response). The third term is the delayed elastic effect containing an elastic modulus $(E_1)$ with $\tau =\eta _1 /E_1$. Application of the model for PST for a significant fraction of $t\approx 1 \, \hbox {s}$ for the saw to traverse the FPZ with $\tau \approx 0.05\hbox { s}$ gives $1/J\approx \frac{E_0 }{1+{E_0 }/{E_1}}$ as the reduced effective modulus. For a frequency of about 1 Hz, A1 implies $E_0 /E_1 \approx O(10)$ since the effective modulus $1/J(0)\approx E_0$ is about a factor of 20 higher for 100 Hz than for 1 Hz (Mellor 1975; Sigrist 2006). Shinojima (1967) calculated that $E_0$ and $E_1$ were within 4 % of each other by applying A1. However, his experiments were slow so that creep would intervene to reduce $E_0$ by extraction from a tangent modulus (Schulson and Duval 2009).

For PST, the important strain rate over which stress alterations occur for fracture initiation (1 / s) is expected to be much higher than for the bulk of the material. The nominal maximum tensile strain rate $({\dot{\varepsilon }}_N)$ for the bulk outer fiber of a beam of depth (D) and length (L) bending with applied slope normal bending speed $V_y$ can be estimated using beam theory (e.g Timoshenko 1940).

For an unsupported cantilever beam fixed at one end with uniform normal load $\sigma _N$ applied over the length, the maximum displacement at the end of the beam (x = 0; Fig. 1) is (Timoshenko 1940; McClung and Borstad 2012a):

$$\begin{aligned} \delta _y =\frac{3}{2}\frac{\sigma _N }{E^{\prime }}\left( {\frac{L}{D}} \right) ^{3}L \end{aligned}$$

(13)

The maximum tensile stress at the top fiber of the beam is the maximum moment divided by the section modulus (Timoshenko 1940):

$$\begin{aligned} \sigma _{\max } =3\sigma _N \left( {\frac{L}{D}} \right) ^{2}=E^{\prime }\varepsilon _N \end{aligned}$$

(14)

Tensile fractures are sometimes observed in PST at the top of low density slabs (McClung and Borstad 2012a). For median values in (14) from PST $[\sigma _N =0.46 \, \hbox {kPa}; \hbox {L}/\hbox {D} =0.86]$, equating to the tensile strength relationship of Jamieson and Johnston (1990): $\sigma _{\max } =f_t =(80-150)({\bar{\rho }}/917)^{2.4}(\hbox {kPa})$ yields ${\bar{\rho }}=120-150\,\hbox {kg/m}^{3}$. The calculation implies that typically for ${\bar{\rho }} \le 150 \, \hbox {kg}/\hbox {m}^{3}$ tensile fracture can be expected at the top of the slab which is in good agreement with observations. Schweizer et al. (2014) reported ${\bar{\rho }}$ from 100 individual tests with slab fractures. Except for 4 outliers, ${\bar{\rho }} \le 175 \, \hbox {kg}/\hbox {m}^{3}$. From (13), the analysis also suggests that use of a thick saw implies higher tensile stress since $\delta _y$ would be larger which is the other condition which favors slab fracture in PST (McClung and Borstad 2012a).

Combining (13) and (14) gives: $\varepsilon _N =2(D/L)(\delta _y /L)$. For the PST, using median values $(L=0.30\hbox { m}; \, \hbox {L}/\hbox {D} = 0.86)$ with $\delta _y =0.75\hbox { mm}$ (McClung and Borstad 2012a; Fig. 2) yields $\varepsilon _N =0.6\,\%$. Using (14) with the median values from PST including the median modulus estimate $(E^{\prime }=0.3\hbox { MPa})$ gives $\varepsilon _N =0.3\,\%$. These values compare well with failure strains: 0.3–0.7 % measured by Narita (1980) for uniaxial tensile tests.

From (14), the reduced effective modulus is given by: $E^{\prime }=(3/2)(\sigma _N /\delta _y )(L/D)^{3}L$. With median values for the PST $(\sigma _N =0.46\hbox { kPa}; \, \hbox {L}/\hbox {D} = 0.86; \, \hbox {L} = 0.30 \, \hbox {m}), \delta _y =0.75\hbox { mm}, E^{\prime }=0.2\hbox { MPa}$ which compares with the median value estimated for PST (0.3 MPa). For truly elastic deformation, $\delta _y$ would be much smaller to yield a much higher modulus.

When adopted to the PST, (Fig. 1) yields the approximate maximum bending strain-rate: ${\dot{\varepsilon }}_N =2(D/L)(V_y /L)$ where $V_y$ is the speed of the vertical (slope normal) displacements. From McClung and Borstad (2012a; Fig. 2), $V_y \approx 0.4\hbox { mm}/\hbox {s}$ and with median values $(L=0.30\hbox { m}; \, \hbox {L}/\hbox {D} = 0.86)$ yields ${\dot{\varepsilon }}_N =3 \times 10^{-3}$/s. From Shinojima (1967), the estimated relaxation time is about 10 s which is the same order as the duration of the experiments which are typically 5–10 s. Thus, the bulk of the slab material is expected to display viscoelastic behavior and the long term creep term $(t/\eta _0)$ may intervene since the rate approaches the regime when creep processes begin to dominate $(10^{-4}/s)$. For a beam which is supported (McClung and Borstad 2012a), the bulk strain and strain rates will be less.

Palmer and Rice (1973) and Rice (1973) discussed the viscoelastic case for the models applied in this paper. The propagation condition is given by (Rice 1973):

$$\begin{aligned} K_{II}^{2}J(t)=K_{II}^{2}[1/E^{\prime }(\omega /V_x )]=(\tau _p -\tau _r) {\bar{\delta }}=G_{II} \nonumber \\ \end{aligned}$$

(15)

where the creep compliance in plane-strain tension [$1/\hbox {E}^{\prime }(\omega /\hbox {V}_x )$] is evaluated for the strain rate $V_x /\omega$ which is about 1/s for the case here (PST). Camponovo and Schweizer (2001) measured the storage modulus $(\mu )$ for a strain rate 1/s as a function of snow density which by viscoelastic correspondence allows estimation of the modulus $E^{\prime }(1\,\hbox { Hz})=2\mu (1\,Hz)/(1-\nu )$ with $\nu =0.1$ (McClung and Borstad 2012a).

Bažant and Planas (1998) discussed crack growth in a viscoelastic medium in general for quasi-brittle materials. The key relationship is that between $V_x, \tau$ and $\omega$. If $V_x \ll \omega /\tau$, then the FPZ is under nearly complete relaxation and $K_{II} \rightarrow \sqrt{E^{\prime }(\infty )G_{II}}$ where $E^{\prime }(\infty )=E^{\prime }(0)/[1+E^{\prime }(0)/\hbox {E}_1^{\prime }]$ for the model above, ignoring long term creep. For PST, $\omega =0.02\hbox { m}; \, \tau =0.05\hbox { s}, \omega /\tau =0.4 \, \hbox {m}/\hbox {s}$ which may be compared with the saw speed $(V_x)$ of a few cm/s. For $\omega =0.10\hbox { m},\tau =0.05\hbox { s}, \omega /\tau =2\hbox { m}/\hbox {s}$. Given low volume fraction of ice $({<}40\,\%)$ and high temperatures $({>}0.90\,\hbox {T}_m)$, it would be impossible to avoid viscoelastic effects for the PST. Grain re-arrangement would be expected to be immediate for the rate of PST experiments as shown by Camponovo and Schweizer (2001).

1.2 Time dependence for dry snow slab avalanche release

Haefeli (1967) showed the shear viscosity for alpine snow ($-10\,^{\circ }\mathrm {C}$ and density 200 kg/m$^{3}$) is about $10^{3}$ MPa s. From Sect (6), the median effective modulus for natural avalanche release was about 0.5 MPa. Taking the ratio gives $\tau \approx 2000\hbox { s}$ or on the order of half an hour. Use of Shinojima’s relation with $\tau \approx 2000\hbox {s}$ implies a strain-rate of $7 \times 10^{-6}$/s. McClung (1974) showed the shear strain rate in low density snow near the top of the snowpack is on the order of $10^{-7}$/s. The time scale for avalanche release once storm loading begins is on the order of hours which allows time for stress relaxation and long term creep effects in the bulk of the slab.

For avalanche release, the appropriate model depends on $V_x$ compared to $\omega /\tau$ with $V_x ,\tau$ being unknown for the FPZ. Knauss (1970) emphasized that higher strain-rates at the crack tip (FPZ) compared to within the bulk material (slab) imply that the time scale of interest at the tip may be many orders of magnitude smaller than the time scale of any experiment. For alpine snow, high rates at the crack tip imply short relaxation time with viscoelastic effects being important.

Rarely, post control avalanches release on a time scale of hours after explosive application. The time after load application at which quasi-static crack growth begins in a viscoelastic material is called the incubation time and it is on the order of the relaxation time of the material (Knauss 1970). Gubler (1977) estimated that slab strain-rates associated with explosives are in the range 10$^{-3}$–10$^{-4}$/s which imply relaxation time on the order of a minute. In contrast, the time to reach high crack speeds (i.e. avalanche post control release time) is typically several orders of magnitude larger than the incubation time or relaxation time (Knauss 1970). In the post control case, significant stress relaxation might be expected within the FPZ since the initial strain-rate may be high implying short relaxation time. Stress relaxation implies a reduced effective modulus with increase in $K_{II}^{2}/E^{\prime }$ to promote fracture initiation.

In the most common case, avalanches release, at most, on the order of a few seconds after explosive application. Since expected relaxation time greatly exceeds time to failure, viscoelastic slab relaxation seems unlikely. Thus, the slab would appear elastic from the perspective of fracture initiation. However, relaxation at the crack tip may still be possible given several orders of magnitude faster strain-rates there (Knauss 1970).

It is possible that application of explosive control and fully dynamic fracture propagation are the only justifiable cases in snow mechanics for which the slab may be considered elastic over the time scale of fracture. Mellor (1975) suggests relaxation times in the range $10^{-5}$–$10^{-3}$ s for high frequency vibrations which suggests that even in dynamic crack propagation there may be significant, if not total, relaxation within the FPZ since crack speeds are typically in the range 20–35 m/s. It has been suggested (McClung 2007b) that the slow speeds for dynamic avalanche fracture propagation are related to viscous damping. Skier triggering of slab avalanches is known to be temperature dependent with regard to slab properties so viscoelastic effects are important.

Appendix 2: Verification of the pressure term for PST from field measurements

For a PST (Fig. 1), the critical stress intensity factor is from Eq. (10):

$$\begin{aligned} K_{IIc}^{*}= & {} \mu _0 \sigma _N \sqrt{\frac{D}{2}}\left( {\frac{L}{D}-\frac{2c_f }{D}} \right) \nonumber \\&+\,\frac{1}{2}\left( {\frac{\nu }{1-\nu }} \right) \sigma _N \sqrt{\frac{D}{2}} \end{aligned}$$

(16)

When the weak layer cut is made from the interior of the slab, the second term is absent and (16) is replaced by:

$$\begin{aligned} K_{IIc}^{*}=\mu _0 \sigma _N \sqrt{\frac{D}{2}}\left( {\frac{L_A }{2D}-\frac{2c_f }{D}} \right) \end{aligned}$$

(17)

where $L_A$ represents the critical cut length which is theoretically twice that in an experiment with a free surface [Fig. 1 and Eq. (16)] excluding the pressure term. Application of Eq. (11) gives:

$$\begin{aligned} K_{IIc}^{*} = \sqrt{E^{\prime }(G_{II} -\mu _0 \sigma _N \delta _0 )} \end{aligned}$$

(18)

For sample pairs with cuts from the free surface and the slab interior, the right side of (18) should be the same. Equating (16) and (17) then gives:

$$\begin{aligned} L_A =2L+\left( {\frac{\nu }{1-\nu }} \right) \frac{D}{\mu _0 } \end{aligned}$$

(19)

With the constants used in this paper $(\nu =0.1; \, \mu _0=0.6)$, (19) gives:

$$\begin{aligned} L_A =2L+0.19D \end{aligned}$$

(20)

I conducted 11 pairs of field experiments $(L_A ,L)$ with different weak layers (surface hoar: 7; facets: 1; decomposing /fragmented with crust: 3), densities from 145–232 kg/m$^{3}$, D: 0.27–1.13 m and slope angles from $0^{\circ }$–$30^{\circ }$. The ratio of $L_A$ (measured) to $L_A$ (calculated) from Eq. (20) using L measured for the PST had a median: 1.00 with a range 0.84–1.22. Linear regression gave adjusted $R^{2}=0.89$ by relating $L_A$ (measured) to $L_A$ (calculated).

From (20), the ratio: $(L_A -2L)/D$ (all from measured values) had a median value 0.19 which matches the constant in (20). The excellent agreement at the median suggests only that the chosen model values $(\nu ,\mu _0)$ are reasonable, not that the model is extremely accurate. The model contains many simplifying assumptions and approximations.