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A mechanically-based model of snow slab and
1
weak layer fracture in the Propagation Saw Test
2
L. Benedettia,b,∗, J. Gaumec,d, J.-T. Fischera
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aBFW - Austrian Research Centre for Forests, Department of Natural Hazards, Innsbruck, Austria
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bCIMNE - International Center for Numerical Methods in Engineering, Barcelona, Spain
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cWSL Institute for Snow and Avalanche Research SLF, Davos, Switzerland
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dEPFL - Swiss Federal Institute of Technology, Lausanne, Switzerland
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Abstract
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Dry-snow slab avalanche release is the result of failure initiation in a weak snowpack
9
layer buried below a cohesive snow slab, which is then followed by rapid crack propa-
10
gation. The Propagation Saw Test (PST) is a field experiment which allows to evaluate
11
the critical crack length for the onset of crack propagation and the propagation distance.
12
Although a widely used method, the results from this field test are difficult to interpret
13
in practice because (i) the fracture process in multilayer systems is very complex and
14
only partially explored and (ii) field data is typically insufficient to establish direct
15
causal links between test results and snowpack characteristics. Furthermore, although
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several studies have focused on the critical crack length assuming linear elasticity for
17
the slab, it still remains unclear how the complex interplay between the weak layer
18
failure and slab fracture impacts the outcome of the PST.
19
To address this knowledge gap, an analytical model of the PST was developed,
20
based on the Euler-Bernoulli beam theory, in order to compute both the critical crack
21
length and the propagation distance as a function of snowpack properties and beam
22
geometry (e.g. beam length and slab height). This work aims to create a link between
23
the two main outcomes of the PST, namely full propagation (END) and slab fracture
24
(SF), and the quantitative results of critical crack length and propagation distance.
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Moreover, introducing empirical relationships based on laboratory experiments (Scapozza, 2004; Sigrist, 2006) between the elastic modulus, the tensile strength and slab density, it is possible to describe the onset of slab fracture for a given geometry of the PST using only the slab density. As a result, the model allows to reproduce the
∗Corresponding author. Address: BFW - Austrian Research Centre for Forests, Department of Natural Hazards, Rennweg 1, 6020 Innsbruck, Austria
Email addresses:lbenedetti@cimne.upc.edu(L. Benedetti),johan.gaume@slf.ch(J. Gaume), Preprint submitted to International Journal of Solids and Structures February 8, 2018
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increasing trend of the propagation distance with increasing slab density, as observed in field experiments. For slabs characterized by low density, slab fracture occurs before reaching the critical crack length (SFb); for intermediate density values, slab fracture occurs after the onset of crack propagation in the weak layer (SFa); then, large densities lead to full propagation in the weak layer without slab fracture (END).
Keywords: Propagation Saw Test, Mechanical Model, Snow Mechanics, Fracture
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Propagation
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1. Introduction
1
Dry-snow slab avalanche release is generally caused by failure initiation in a weak
2
layer buried below cohesive snow slabs, which is then followed by rapid crack prop-
3
agation (Schweizer et al., 2003). The Propagation Saw Test (PST) is an experimen-
4
tal in-situ technique that has been introduced to assess crack propagation propensity.
5
The PST, developed simultaneously by Sigrist and Schweizer (2007) and Gauthier and
6
Jamieson (2008), involves isolating a conventional volume of snow in the downslope
7
direction (Figure 1). Once the weak layer has been identified in the stratigraphy (e.g.,
8
from a manual snow profile or a compression test), a saw is used to cut through it
9
progressively. If the crack length from the sawing reaches a critical value, crack prop-
10
agation can occur. Depending on the snowpack properties, the crack within the weak
11
layer can induce slab fracture (SF) or propagate further to the end of the column, which
12
is called full propagation (END).
13
Statistical studies over large sets of field data have confirmed the relevance of the
14
PST, highlighting a relatively high correlation between the test results and the likeli-
15
hood of avalanche release, despite the number of false alarms (Simenhois and Birke-
16
land, 2009; Ross and Jamieson, 2012; Reuter et al., 2015). Recent PST measurements
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have shown that the bending of the slab is a key element for the onset of crack propa-
18
gation (van Herwijnen et al., 2010; van Herwijnen et al., 2016) and the subsequent dy-
19
namic regime. Numerous experiments were performed by van Herwijnen and Jamieson
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(2005) and van Herwijnen and Birkeland (2014), utilizing high speed cameras and par-
21
ticle tracking velocimetry (PTV), which have revealed important details of the intricate
22
relationship between the propagation of the crack in the weak layer and the slab defor-
23
mation field.
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The seminal work of McClung (1979) provided a first modeling of the shear failure
25
observed in the weak layer and it has been improved, at a later time, by McClung
26
(2003), Chiaia et al. (2008) and Gaume et al. (2013). However, observations of remote
27
triggering of avalanches as well as the bending of the snow slab observed in field PSTs
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(van Herwijnen et al., 2010; van Herwijnen et al., 2016) have challenged these past
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theories. Consequently, this led to the development of models focusing on the weak
30
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layer collapse, such as the anticrack model of Heierli et al. (2008) or the dynamic
1
shear collapse model of Gaume et al. (2015, 2017) which accounts for the mixed-mode
2
failure of the weak layer (Reiweger et al., 2015).
3
Frequently, inconsistencies in results can be given by the unclear interrelations
4
among snowpack properties (Gaume et al., 2017) and the difficulty to treat complex
5
crack propagation phenomena in multilayered systems (Hutchinson and Suo, 1991;
6
Habermann et al., 2008). In addition, most of the existing studies have focused solely
7
on the critical crack length for the onset of crack propagation in the weak layer (assum-
8
ing linear elasticity for the slab as in LEFM), but thus disregarding its influence on the
9
fracture propensity of the slab. Hence, on the one hand, this approach does not allow a
10
thorough understanding and a quantitative evaluation of the complex interplay between
11
crack propagation in the weak layer and slab fracture. On the other hand, there is a lack
12
of scientific evidence on how this interdependence can affect the outcome of the PST.
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In order to address this knowledge gap, this work aims to provide a quantitative
14
analysis of the PST, accounting for the most important elements, specifically the in-
15
teraction between the weak layer and slab fracture, as well as the frictional contact
16
occurring during slab bending. The proposed model is based on the Euler-Bernoulli
17
beam theory, which allows to compute the stress state of the snowpack during the PST,
18
given the crack length, geometrical and mechanical properties. With the introduction
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of this approach, it is possible to compute the critical crack length and the propagation
20
distance and define the outcome of the test in terms of full propagation (END) and slab
21
fracture (SF) outcomes.
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The paper is organized as follows. First, a qualitative mechanical analysis of the
23
three possible PST results (full propagation, slab fracture and fracture arrest) was con-
24
ducted. Based on these cases, the mechanical model considering the interaction of the
25
slab and the weak layer was studied. With the Euler-Bernoulli beam theory and the
26
brittle failure assumption, the stress state of the two layers of interest were analyzed
27
separately. The model was applied to a realistic case study in which snowpack prop-
28
erties were interrelated and, then, model predictions were compared with field data. A
29
detailed parametric study was also performed. Finally, the results are discussed with
1
respect to previous studies highlighting advantages and limitations of the approach in
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Figure 1: Set-up of a typical Propagation Saw Test (photo from the authors).
this work.
3
2. Phenomenology of the PST fracture modes
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The set-up of the test includes an isolated volume of snow ofb = 30 cm for the
5
width and at least 1 meter in lengthltotin the downslope direction (Figure 1). The full
6
depth of the snowpack is considered in the experiment. The case study is composed
7
of a snow slab overlying a rigid substratum (of total lengthltot) with a weak layer in
8
between. Figure 2 shows the side view of the idealized model, where the horizontal is
9
parallel to the slope plane with angleψ.
10
Once the saw reaches the critical crack lengthlc, the crack can propagate in the
11
weak layer, which is followed by a possible slab fracture. The PST results are com-
12
monly reported with the following cases (Gauthier and Jamieson, 2008):
1
• Full propagation (END):the crack propagates through the whole weak layer.
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Figure 2: Side view sketch of the modeled Propagation Saw Test.
• Slab fracture (SF): the crack propagation in the weak layer is stopped by a
3
sudden fracture in the slab.
4
• Fracture arrest without slab fracture (ARR): the propagation in the weak
5
layer stops to a point that is difficult to identify.
6
The results of the test consist of the critical crack length and the propagation dis-
7
tance.
8
The first question to address is the sequence of events that could take place in a
9
Propagation Saw Test. The slab represents a gravitational load on the weak layer. At
10
the beginning of the test, increasing the crack lengthlcreates a volume of snow that
11
is supported on one side while hanging freely on the other one and the weak layer has
12
a reduced resisting area (Figure 3(a)). The layer is displaced both vertically and hori-
13
zontally under its own weight and might touch the rigid substratum (Figure 3(b)). The
14
crack lengthlat which this initial contact (IC) is observed is identified withl = LIC.
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Initially, only the only the lower corner of the beam free end (henceforth called tip)
16
rests on the rigid substratum. Therefore, the resulting effect is a hinged restrain, where
17
the beam does not displace vertically, but has freedom to rotate (Figure 3(c)). Then,
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following the increase ofl, the slab bends back due to its own weight and rests with
19
vertical cross section with respect to the rigid substratum. At this point, not only is
20
the vertical movement constrained but the rotation of the beam is also fixed. The crack
21
lengthl required for this condition to occur is called the length of full contact (FC)
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and it is denoted byLFC(Figure 3(d)). If the sawing continues, then the contact zone
2
of the slab and the substratum layer increases. However, the length between the saw
3
and the first touching point remains constant, being equal to the full contact lengthLFC
4
(Figure 3(e)) due to equilibrium requirements. The beam is now considered as a double
5
supported beam, with a fixed length ofLFC, and, consequently, with a linear bending
6
moment and constant shear on the cross sections. Notably, for typical snowpack con-
7
dition, cases (c), (d) and (e) are rarely observed if the crack length (created by the saw)
8
is lower than the critical one. However, the frictional contact is important during the
9
dynamic phase of the crack propagation in the weak layer (Gaume et al., 2015) and,
10
consequently, for slab avalanche release.
11
Then, the possible failure modes that can appear during the test are discussed. We
12
consider the resulting stresses on the slabσsand on the weak layerσw, with respect to
13
their limit valuesσsyandσwy in a generic sense (i.e. either shear, tension or compres-
14
sion). With different values of the ratiosRs = σs/σysandRw =σw/σwy, it is possible
15
to define various limiting cases. Their values are limited between zero, representing
16
a stress free situation, and one, corresponding to failure. The variation ofRs andRw
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influences the way the slab interacts with the weak layer, as well as the the crack prop-
18
agation process that follows.
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2.1. Rs≈1and Rw<<1: low strength of slab
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The first limiting case presents a slab which has not yet developed enough strength.
21
On the contrary, the weak layer is relatively strong with respect to the load. In this
22
case, slab fracture occurs in relation to a very short crack length without creating a
23
cantilever structure and crack propagation in the weak layer is very unlikely. In this
24
case, the critical crack length tends to the total length (lc→ltot), since the saw has to
25
cut the whole specimen in order to release the entire slab.
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2.2. Rs<<1and Rw≈1: low strength of weak layer
27
In this case, the weak layer has a low strength, whereas the slab strength is high
28
and any initial crack created by the saw provokes failure of the weak layer and crack
29
propagation. Since the stresses in the weak layer can only increase during the test,
1
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Figure 3: Stages of the Propagation Saw Test: after initial bending (a), first contact of slab and rigid sub- stratum is reached at crack lengthl=LIC(b). Following the sawing (c), the full contact length is achievedforl=LFC(d), at which point the cross section has rotated back to vertical. Successively, the length of the beam under bending is kept constant atLFCforl>LFC, while the contact zone is increasing as the crack further progresses (e).
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Figure 4: Example of slab fracture outcomes in the PST: (a) crack propagation in the weak layer followed by a slab fracture (SFa); (b) slab fracture due to high bending and tensile stresses before crack propagation in the weak layer (SFb).
the propagation takes place along the whole specimen as soon as the crack is initiated.
2
Consequently, the critical crack length tends to zero (lc→0).
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2.3. Rs≈1and Rw≈1: slab and weak layer close to failure
4
In this case, the slab and weak layer stresses are comparable with their respective
5
strengths leading to a competitive mechanism of failure between the slab and weak
6
layer.
7
If the slab stress ratio is lower than the weak layer ratio (Rs ≤ Rw), then the
8
crack propagation can be observed in the weak layer upon reaching the critical crack
9
length. Within this context, the propagation may stop when stresses in the slab reach
10
its strength causing a fracture and subsequently releasing the gravitational load (Figure
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4(a)). Conversely, if the weak layer stress ratio is lower than the slab ratio (Rs ≥Rw),
12
the slab may fracture prior to any propagation in the weak layer (Figure 4(b)).
13
This set of different phenomena shows the importance of analyzing and predicting
14
the conditions at which each single mechanism occurs. A detailed analysis of the
15
material properties and resulting stress states can help to reveal the failure mechanism
1
which causes the observed result in the PST.
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Figure 5: PST model: the resisting cross sections for the slab and the weak layer are highlighted by the local coordinate systems. In addition, the geometrical dimensions of the specimen are reported.
3. Mechanical model
3
The objective of this section is to describe analytically the stresses and failure con-
4
ditions for the slab and the weak layer under quasi-static conditions.
5
The isolated volume of snow for the test is presented in Figure 5 in 3D. The total
6
length of the specimen isltot, whilelrepresents the crack length and, consequently,
7
lw=ltot−lis the length of the remaining weak layer.
8
The slab is assumed to consist of a single homogeneous material, with constant
9
widthband height h. A coordinate system x,y,zis placed in the center of mass of
10
the orthogonal cross section at the crack tip. There, the stresses distribution arising in
11
the slab is studied in the case of a free end boundary condition similar to a field PST.
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Furthermore, the end of the slab at x =lis subjected to various boundary conditions
1
depending on its interaction with the other layers.
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The weak layer has a thicknesshwand widthb. It is assumed that the weak layer
3
reacts uniformly to the external load at the interface with the slab, where a coordinate
4
system is designated on the geometrical center of this interface, pointing in the upward
5
directionη. Finally, a perfectly rigid substratum is located under the weak layer.
6
For the purposes of the mechanical analysis, the material composing each layer
7
of the snowpack is assumed to be homogeneous and isotropic. The interface between
8
the layers is considered capable of perfectly transferring the load and maintaining the
9
cross section plane upon deformation. This assumption is required in order to study
10
independently the slab and the weak layer with the beam theory. Subsequently, the
11
failure modes observed in the test are introduced. The critical crack length for the onset
12
of crack propagation in the weak layer is defined aslwc whereas the critical crack length
13
for slab fracture islcs. In addition, the signed crack propagation distancelp=lwc−lcs, for
14
which slab fracture occurs after weak layer failure, is also computed. It is important to
15
note that negative values oflprepresent the case of slab fracture appearing before any
16
propagation in the weak layer.
1
3.1. Mechanical model of the slab
2
The slab is studied as a homogeneous beam of lengthl, with widthband heighth.
3
The load due to gravity is uniformly distributed along the entire beam. Each unitary-
4
length cross section provides a constant loadqwith vertical and horizontal components
5
according to the slope angleψ, as depicted in Figure 6.
6
The uniformly distributed load due to the weight of the slab is:
7
q=ρg h b (1)
8
whereρis the density of the slab andgis the gravitational acceleration. The horizontal
9
and vertical components of the load due to gravity are:
10
qh=qsinψ qv=qcosψ (2)
11
The vertical load, orthogonal to the centerline, causes bending, which rotates the cross
12
sections and lowers the tip of the slab (Figure 6(b)). The horizontal load, parallel to
13
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Figure 6: Beam model with distributed load: (a) with respect to the slope angleψ, (b) projected vertically and (c) projected horizontally.
the beam centerline, causes stretching along the coordinatexin the downslope direc-
14
tion (Figure 6(c)). In general, these two effects are not independent, but as long as the
15
displacements remain small, the hypothesis of superposition of effects holds. Hence,
16
the beam is studied separately for each load case and, then, the results are superim-
17
posed. To describe the variation of the internal forces with respect to the load and to
18
the geometric characteristics, the Euler-Bernoulli beam theory is used.
1
A beam that is subjected to the set of forces previously presented usually experi-
2
ences failure due to bending moment, shear, normal tension or a combination of these
3
(Figure 7). It is important to note that for slender beams, shear stress is significantly
4
smaller than the stress given by bending moment and tensile force. For this reason, we
5
disregard any shear effect in the slab.
6
3.1.1. Bending stresses
7
In the Euler-Bernoulli theory, the behavior of an elastic beam under a distributed
8
loadqvis described by the following ordinary differential equation:
9
d2 dx2
"
EId2v(x) dx2
#
+qv=0 (3)
10
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Figure 7: Stress distribution for the slab modeled as (a) beam: (b) tensile and (c) bending stresses are combined to provide the field of horizontal stresses, which are orthogonal to the resisting cross section.
in terms of the vertical displacement functionvand the horizontal coordinatex. Once
11
v(x) is computed, then it is possible to derive the rotationθ, the curvatureχ, the bending
12
momentMand the shear forceT along the beam:
13
θ= dv(x)
dx χ= dθ(x)
dx M=−EIχ T = dM(x) dx
dT(x)
dx +qv(x)=0 (4)
14
whereE is the elastic modulus and I is the second order moment of inertia of the
15
cross section. To solve this ordinary differential equation, four boundary conditions
16
have to be set. Two of them are imposed on the cross section at the crack tip (x =
17
0), where vertical displacement and rotation are assumed to be null. Likewise, the
18
other two conditions are obtained by considering the free end of the beam (x =l). At
19
the beginning of the PST, the beam is free to rotate (bending moment equal to zero)
20
and displace vertically (shear force equal to zero) at its tip. This situation is denoted
21
Bending Scheme I and depicted in Figure 8(a).
22
The solution for the vertical displacement function of the cantilever is:
23
vI(x)= −qv
6l2x2−4lx3+x4
24EsIs (5)
24
The highest moment is found at the crack tip, corresponding with the clamped end
25
x=0:
26
MI = qvl2
2 (6)
1
When the tip displaces vertically to a height equal tohw, it touches the rigid sub-
2
stratum.LIC is the beam length at which this occurs and it is computed as:
3
LIC = 4
s8EsIshw
qv (7)
4
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Figure 8: (a) First bending scheme: Cantilever. (b) Second bending scheme: Cantilever with hinge. (c) Third bending scheme: Fixed-Fixed roller.
Forl≥LICthe slab is in contact with the rigid substratum and the previous bending
5
scheme is not valid anymore. In the Bending Scheme II, the cross section at the crack
6
tip (x = 0) is assumed to be fixed, whereas the vertical motion is restrained at the
7
other endx =l(Figure 8(b)). The small part that is in contact with the substratum is
8
reacting with a vertical forceFt, but it does not apply any bending moment to restrain
1
the rotation of the cross section at the tip.
2
The resulting vertical displacement functionv(x) is expressed as:
3
vII(x)=−3hwx2
2l2 − qvl2x2 16EsIs +hwx3
2l3 + 5qvlx3
48EsIs − qvx4
24EsIs (8)
4
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The maximum value of the bending moment is located at the clamped end:
5
MII(0)= qvl2 8
1+3L4IC l4
(9)
6
The forceFt represents the reaction of the rigid substratum at the contact point with
7
the slab and it reduces the stresses in the slab as well as in the weak layer:
8
Ft=T(l)= 3qvl
8 −3EsIshw
l3 = 3qvl 8
1− L4IC l4
(10)
9
With further increase of lengthl, the free end of the beam is rotating back to zero
10
due to its weight. Consequently, the Bending Scheme II is valid until the end section
11
of the beam is orthogonal to the rigid substratum. This length is defined as the full
12
contact lengthLFC. Subsequently, all slab in excess ofLFC will be supported by the
13
rigid substratum. To find the full contact lengthLFC,θ(l)=0 and therefore:
14
LFC = 4
s72EsIshw
qv = √
3LIC (11)
15
which shows that the crack length difference between the beginning of Bending Scheme
16
II and the beginning of Bending Scheme III isLFC−LIC =√ 3−1
LIC≈0.75LIC.
17
Here, it is important to recall the concepts presented in Section 2. By increasing
18
the crack length beyond the full contact length (l > LFC), both ends of the beam are
19
restrained in rotation, i.e. the beam must be horizontal at the boundaries. At the same
20
time, in this scheme, the contact zone between the slab and the rigid substratum in-
21
creases with increasing crack length as the length of the beam subjected to bending is
22
alwaysLFC (for a crack lengthl > LFC). In fact, the tip of the beam at x = LFC is
23
in contact (v(LFC) = −hw) and parallel (θ(LFC) = 0) to the rigid substratum and the
24
exceeding crack length is supported directly by the rigid substratum. Consequently,
25
this scheme is also characterized by a frictional force exerted by the weight of the slab
26
on the substratum. This configuration is denoted as the Bending Scheme III.
27
In the bending scheme III, the boundary conditions on a beam of constant length
28
LFCare:
1
vIII|x=0=0 vIII|x=LFC =−hw (12)
2 3
v0IIIx=0=0 v0IIIx=LFC=0 (13)
4
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This yields the vertical displacement function:
5
vIII(x)= hw
L3FC
2x3−3LFCx2
− qvx4
24EsIs(x−LFC)2 (14)
6
Once again, the maximum moment is located at the crack tipx=0, and reads:
7
MIII(0)= qvL2FC
6 (15)
8
It is important to highlight that the value of the bending moment at the cross section
9
of the crack tip will not increase for longer crack lengthsl, since, as it was previously
10
discussed, the length of theactivebeam remains constant at the full contact lengthLFC,
11
while the rest of the slab is directly supported by the underlying substratum.
12
3.1.2. Normal stresses
13
Beside the bending moment, which allows to derive the characteristic lengthsLIC
14
andLFC, the slab is subjected to a horizontal forceNexerted in slope direction.
15
To describe it, two traction schemes are introduced (Figure 9):
16
• Simply supported:This case is linked with the Bending Schemes I and II, where
17
the tip of the beam is either free or only touches the rigid substratum at a sin-
18
gle point, not exerting enough contact force to engage friction. It is assumed
19
that the cross section at the crack tip (x =0) is fixed with respect to horizontal
20
displacements. Givenqh=qsin (ψ), the horizontal force is:
21
N=q(l−x) sin (ψ) forl<LFC (16)
22
• Simply supported with additional frictional resistance:This case occurs when
23
the beam is in full contact with the rigid substratum, i.e. l > LFC, the friction
24
between the slab and the rigid substratum is helping to sustain the horizontal
25
load. The friction force depends on the weight of the part in contactwusand on
1
the static friction coefficientkfbetween the slab and the rigid substratum. Since
2
wusis assumed to be zero up to the full contact lengthLFC, the second horizon-
3
tal force scheme represents the fixed-fixed roller mechanism, corresponding to
4
Bending Scheme III. Then, it is possible to compute the horizontal force as:
5
N=q(l−x) sin (ψ)−kfqbh(l−LFC) cosψ forl≥LFC (17)
6
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Figure 9: Force schemes for the beam under horizontal tensile force.
3.1.3. Slab failure criterion
7
In order to determine the crack length that causes fracture in the slab, the stresses
8
in the resisting cross section are computed. The resisting cross section is defined as the
9
area of the beam, orthogonal to the center line, above the crack tip. As it was discussed
10
before, the highest value of bending moment and horizontal force was observed in this
11
zone. In the elastic range, to evaluate the distribution of stresses (normal to the cross
12
section) due to bending momentMand normal tractionN, it is possible to use Navier’s
13
formula:
14
σ(y)= M Isy+ N
As (18)
15
where the second moment of inertiaIsand the area of the sectionAsare
16
Is= bh3
12 As=bh (19)
17
Navier’s formula gives the distribution of stresses along the height of the beam given
18
the bending momentMand the horizontal forceN.
19
In this model, the fracture of the slab is assumed to be brittle and to appear due
20
to tensile stresses. The value σyts represents the tensile strength (i.e. the maximum
21
admissible tensile stress) for the snow in this layer.
22
Hence, failure is reached when the top surface of the slab reaches the stressσsytdue
1
to the bending momentMand the horizontal forceN:
2
σyts = M Is
h 2 + N
As (26)
3
As the tensile strength is reached, i.e.σts=σsyt, detachment of the slab is predicted.
4
Table 1 outlines the applied bending momentMand horizontal force Nfor each
5
force scheme previously presented, as well as the final expression ofσst, the tensile
6
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Table 1: Summary of the forces and the stress functions computed at the resisting section of the slab.
Scheme I:
l≤LIC
External Forces:
M= qvl2
2 N=qhl (20)
Tensile stress:
σst =ρgh
"l
hsin (ψ)+3l2 h2cos (ψ)
#
(21)
Scheme II:
LIC<l≤LFC
External Forces:
M= qvl2 8
1+3L4IC l4
N=qhl (22)
Tensile stress:
σst =ρgh
l
hsin (ψ)+ 3 4
l2 h2cos (ψ)
1+3L4IC l4
(23)
Scheme III:
l>LFC
External Forces:
M= qvL2IC
2 = qvL2FC
6 N=qhl−kfqv(l−LFC) (24) Tensile stress:
σts=ρgh
l
hsin (ψ)−kfl−LFC
h cos (ψ)+L2FC h2 cos (ψ)
(25)
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stress on the top surface of the slab, as a function of the crack length. By enforcing
7
σst = σyts (i.e. tensile failure), it is possible to invert these expressions in order to
8
determine the critical crack length for slab fracturelcs. Moreover, the variation of the
9
stresses as function of the crack length can be studied.
10
3.2. Mechanical model of the weak layer
11
In this study, a homogeneous weak layer of finite thickness is considered. Recent
12
studies have highlighted the mixed-mode shear-compression failure behavior of weak
13
snowpack layers (Reiweger et al., 2015; Chandel et al., 2015). A simplified failure
14
criterion with constant compressive and shear strength, σwyc and τwys respectively, is
15
introduced as the only mechanical characterization of the weak layer (Figure 10).
Figure 10: Weak layer failure modes: (a) crushing due to compression and (b) shear failure due to the combined effect of horizontal and vertical load.
16
The stresses in the weak layer are described for the load schemes presented in
17
Figure 10, which are representative of the three force schemes in the slab. In order
18
to evaluate the stresses in the weak layer as a function of the crack length, the weight
1
of the slab is considered to be concentrated in its center of gravity. This assumption
2
holds if there is a perfect transfer of forces between the two layers and if the slab
3
is approximated as a rigid solid when computing the forces on the weak layer. The
4
distancelM between the center of gravity of the slab and the center of the resisting
5
section in the weak layer (as previously identified in Figure 5) is given by geometric
6
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considerations as:
7
lM = 2l forl<LIC
lM = 2l forLIC≤l<LFC
lM = L2FC forl≥LFC
(27)
8
In addition to the weight of the slab, it is also necessary to recallFt, the vertical reaction
9
force applied at the tip of the beam which is caused by the contact of the slab with the
10
rigid substratum. Whenl<LIC, the vertical forceFtis zero since there is no contact.
11
Instead, when LIC ≤ l < LFC, the force Ft increases with increasing crack length.
12
Finally, in the case ofl ≥ LFC,Ftis constant because the active beam is alwaysLFC
13
long.
14
Given the uniformly distributed gravitational load q, the verticalWv = qvl and
15
horizontalWh = qhlcomponents of the slab weight result in shear, compression and
16
bending moment on the weak layer, which are summarized in Table 2. Schemes II and
17
III include the reduced vertical weight of the slab due to the presence of the forceFt.
18
The stresses in the weak layer can be calculated with Navier’s and approximate
19
shear expressions:
20
σw(η)= Nw
Aw + Mw
Iw ζ τws = Tw
Aw (34)
21
whereζis the coordinate along the longitudinal dimension of the weak layer (Figure
22
5),IwandAware respectively the second order of inertia and the area of the resisting
23
section:
24
Iw= b(ltot−l)3
12 Aw=b(ltot−l) (35)
25
The stress in the weak layer is limited by the maximum compression at the crack
26
tip or by the maximum shear force at the interface between the layers.
1
Table 2 summarizes the computed shear and compressive stresses applied to the
2
weak layer. A dependence is observed on the inverse of the quantityltot−l, which
3
results in a nonlinear increase of stresses, growing to infinity forl→ltot.
4
Enforcing that the compressive stressσwc and shear stressτws in the weak layer have
5
to be equal to their respective strength valuesσwycandτwys, it is possible to compute the
6
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Figure 11: Equilibrium schemes for the computation of stresses in the weak layer: (a) case forl<LIC; (b) case forLIC ≤l≤LFC; (c) case forl>LFC. CG is the center of gravity of the slab not supported by thepart in contact with the rigid substratum,WhandWvare respectively the horizontal and vertical components of the slab weight,lMis the distance of CG from the center of the reacting weak layer andFtis the vertical reaction given by the contact between slab and rigid substratum.
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Table 2: Summary of the forces acting on the weak layer.
Scheme I:
l≤LIC
External forces:
Nw=Wn Tw=Wh
Mw=Wnl 2 +Whh
2
(28)
Compressiveσcand shearτsstress:
σc= ρghcos (ψ)(lltot
tot−l)
h1+3(l+htan(ψ)) (ltot−l)
i
τs= ρghsin (ψ)(lltot
tot−l)
(29)
Scheme II:
LIC <l≤LFC
External forces:
Nw=Wn−Ft
Tw=Wh
Mw=Wnl 2+Whh
2−Ftltot+l 2
(30)
Compressiveσcand shearτsstress:
σc= ρghcos (ψ)(lltot
tot−l)
1− 8l3ltot
1− Ll4IC4 2lltottot−+ll
+3(l+h(ltan(ψ))
tot−l)
τs= ρghsin (ψ)(lltot
tot−l)
(31)
Scheme III:
l>LFC
External forces:
Nw=Wn−qn(l−LFC)−Ft
Tw=Wh−kfqn(l−LFC)
Mw=(Wn−qn(l−LFC))LFC2 +Whh 2−Ft
LFC+ltot2−l (32) Compressiveσcand shearτsstress:
σc= 34ρghcos (ψ)(lLFC
tot−l)
1+ l1
tot−l
h4hLltot
FCtan (ψ)−(ltot−l−2LFC)i τs= ρghsin (ψ)(lltot
tot−l)
1−kf l−LFC
ltottan(ψ)
(33)
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critical crack lengthslwc,c(in compression) andlwc,s(in shear). The critical crack length
7
lwc is the minimum of the latter two values. Note thatlwc is distinct from the critical
8
crack lengthlcsfor the slab tensile failure.
9
4. Model Application
10
In this section, the features of the presented Propagation Saw test model are high-
11
lighted. First, a realistic study case is introduced, which is helpful to show how the
12
model can predict the results of the PST. The main results of a parameteric study are
13
then presented, while the details are discussed further in the Appendix. Finally, the
14
model is compared with experimental data.
15
4.1. Simple study case
16
A simple PST case study is presented to show the influence of the crack length on
17
the stresses in the slab and in the weak layer, as well as to link the quantitative results
18
(critical crack length and propagation length) to the modeled PST outcomes (END or
19
SF). In this case study, the slab has a width ofb = 30 cm and a height of h = 30
20
cm. The total length of the specimen isltot =2 m and the slope angle is 35◦. Finally,
21
the thickness of the weak layer ishw = 1 mm whereas the static friction coefficient
22
between the rigid substratum and the slab iskf =0.5, equivalent to a friction angle of
23
∼27◦(van Herwijnen and Heierli, 2009).
24
The compressive strength in the weak layer is 2.5 kPa, while the shear strength is
25
0.5 kPa, similarly reported by Reiweger et al. (2015). For the snow in the slab, the
26
elastic modulusEis a function of the density, as in Scapozza (2004):
1
E(ρ)=1.873·105exp0.0149ρ (36)
2
and the tensile strength is given by Sigrist (2006):
3
σsyt=2.4×105 ρ ρice
!2.44
(37)
4
whereρice =917 kg/m3. Recalling expressions in Table 1 and Table 2, it is possible
5
to plot the tensile stress in the slab and the compressive and shear stresses in the weak
6
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layer as functions of the crack length, as it increases or when there is crack propagation.
7
Figure 12 depicts the case for a slab density equal toρ=230 kg/m3. With regard to the
8
tensile stresses in the slab, the initial part of the curveσstshows a quadratic increase of
9
stress due to the growing bending moment. Then, the beam tip comes into contact with
10
the substratum and betweenLICandLFCthe stress locally reduces. For lengths greater
11
than the full contact lengthl > LFC, the bending moment is constant at the crack tip
12
cross section but the stress increases linearly: at this point, the tensile force in the direc-
13
tion of the slope drives the experiment. In contrast, in the weak layer, the compressive
14
σwc and shear stressesτws have a non-linear increase along the whole experiment and
15
the effects of the contact of the slab with the substratum are less pronounced.
Figure 12: Comparison between the tensile stressσstin the slab and the compressiveσwcand shearτwsstresses in the weak layer forρ=230kg/m3. The “X” symbols identify the failure stress at which the critical crack length is computed. The dashed lines represent the approximation introduced for Scheme II (LIC≤l≤LFC)
16
In Figure 12, failure, specifically when the stress reaches the maximum admissible
1
stress, is indicated with the symbol “X”. As it was discussed previously, the slab fails
2
under tensile stress (i.e. whenσstfrom Table 1 is equal toσsyt), induced by the combined
3
effect of bending and horizontal force. Similarly, the weak layer fails due to shear, as
4
τws =τwys, or due to compression, asσwc =σwyc, whereτws andσwcare computed following
1
Table 2. However, it is difficult to relate the various stresses distribution at failure
2
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due to the significant difference in the strength of each layer. Then, to have a better
3
understanding of the phenomena in the Propagation Saw Test, the stressesσts, σwc, τws
4
are scaled with respect to their admissible stress valuesσsyt, σwyc, τwysas elaborated in the
5
following paragraph.
6
Figure 13: Model application results. Figure (a), Full propagation (END) case: shear failure in the weak layer appears at 11 centimeters and propagates to the end of the specimen. Figure (b), Slab Fracture after propagation (SFa): the initial shear failure in the weak layer at 45 centimeters is followed by the slab fracture at 166 centimeters. Figure (c), Slab fracture before propagation (SFb): tensile failure in the slab happens at 59 centimeters, before weak layer failure, which would theoretically appear at 79 centimeters. Figure (d), Critical crack length for the slab fracturelscand the weak layer failurelwc. Note thatlwc is the minimum between the critical crack length for shearlwc,sand compressionlwc,cfailure.
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4.2. Full propagation (END) case
7
In this first case, the density of the slab isρ=280kg/m3, indicative of a stiffslab.
8
Firstly, the elastic modulusE(ρ) and the tensile threshold stress are:
9
E
280 kg/m3
=12.15 MPa (38)
10
11
σyts
280 kg/m3
=13.28 kPa (39)
12
Then, the characteristic lengthsLICandLFCare:
13
LIC =75 cm LFC= √
3LIC=131 cm (40)
14
The plot of normalized stresses is presented in Figure 13(a). On the basis of our hy-
15
potheses, the model predicts that the critical shear stress ratio in the weak layer reaches
16
failure (i.e., unit value ofτs/τys), for a critical crack length of 11 centimeters. It is
17
important to emphasize that, assuming a brittle failure criterion, failure due to shear (or
18
compression) is instantaneous and the potential crack propagation not restrained un-
19
less the stress in the weak layer is suddenly reduced (e.g. due to slab fracture). Hence,
20
the model suggests that crack propagation is initiated in the weak layer atl=11 cm,
21
which causes the stress in the slab to further increase. However, the tensile stress does
22
not reach its maximum admissible value. Consequently, this result would represent the
23
full propagation case (END).
24
It is possible to compute the propagation length by subtracting the critical crack
25
length for the onset of crack propagation in the weak layerlwc to the one required for
26
slab fracturelsc. In the present case the crack would propagate to the end of the column
27
and the propagation length would be:
1
lp=lcs−lwc =200−11=189 cm (41)
2
4.3. Slab Fracture after propagation (SFa) case
3
Consider a slab density ofρ = 230 kg/m3. The elastic modulus E(ρ) and the
4
tensile threshold stress are:
5
E
230 kg/m3
=5.77 MPa (42)
6
