International Conference Vajont2013 - 8 October

1. „Modeling of runout length of high-speed granular masses‟
Francesco Federico & Chiara Cesali
University of Rome “Tor Vergata” - Rome, Italy
Department of Civil Engineering and Information Engineering

2. INTRODUCTION
Rapid granular flow prediction is linked to the understanding of the flow mechanisms.
Several in situ observations pointed out that the runout length, kinematical characters depend
on the geometry of the slope, the debris volume, the susceptibility of the involved material
(Bozzano, Martino, Prestininzi, 2005).
Advanced lab experiments highligthed the key role played by some micro-mechanical
parameters.
An analytical model to estimate the runout length of granular masses, accounting for both
global and micro-mechanical parameters, is proposed.
2

3. Dependence of the runout length “L” on volume “V” of the debris flow.
Correlations L (V) or H/L(V) are shown.
RICKENMANN D. (1999), „Empirical Relationships
for Debris Flows‟, Natural Hazards 19.
M = material
volume;
He = H =
difference
in elevation;
L = total travel
distance;
Lf = runup;
mapp = H/L
STRAUB S. (1997), „Predictability of long runout landslide motion‟, Springer.
LOCAT P. et al. (2006), „Fragmentation energy in rock avalanches‟, Can. Geotech. J. 43.
3

4. COMPARISON WITH EMPIRICAL RELATIONSHIPS
7000
Rickenmann 1 (1999) - (R1)
Rickenmann 2 (1999) - (R2)
6000
(R1)
5000
L [m]
L = 30(VH)1/4
(R2)
0.8
0.7
4000
r  tan  
3000
Scheidegger (1973) - (S)
Corominas (1996) - (C)
Davies (1982) - (D)
H
L
L = 1.9∙V0.16∙H0.83
2000
0.6
1000
0.5
log10 (H/L) = -0.15666∙log10(V) + 0.62419
0
1
2
3
4
5
6
x 10
r
V [m3]
Empirical Relationships
(S)
0.4
log10 (H/L) = -0.085∙log10(V) + 0.047
(C)
0.3
1/(H/L) = 1/tan(fb) + 0.5∙(9.98∙V0.32)/H
(D)
0.2
0.1
1
2
3
3
V [m ]
The runout length L depends only on the
involved “global” geometrical parameters, „V‟
and „H‟. A great variability of the results is
observed.
4
5
x 10
6
4

5. PROPOSED MODEL
Hypotheses: The granular sliding body is composed by two layers of equal basal area Ω and
length l: within the "shear layer" (thickness ss) random and turbulent fluctuations of particles
displacements at high rate induce a „granular temperature‟ (Zhang & Foda, 1999). The
available energy is dissipated through collisions effects and frictional resistance; inertial
forces act on the overlying body (thickness sb).
The global geometry (Ω, l and H) does not change if erosion or deposition processes are
neglected. Otherwise, the geometry may vary according to simple geometrical rules as a
function of the rate of the sliding mass and of a critical value vcr,e beyond which the erosion
phenomenon occurs.
The sum of the masses of the shear
layer (ms) and the overlying block (mb)
First slope (q):
equals the total sliding mass m:
runout
ms and mb vary along the slopes as a
function of rate as well as of erosion
phenomena.
ms(x(t))=ρs ss(x(t)) Ω
mb(x(t))=ρb sb(x(t))Ω
m = mb(x(t))+ ms(x(t))
Transition zone
The thicknesses sb(x(t)), ss(x(t)) along the path (x(t)) are not a priori known.
Counterslope ():
runup
5

6. PROPOSED MODEL: Energy Balance, constant mass
The involved energies are: Ep, potential energy; Ek, kinetic energy; Ecoll, energy dissipated through
collisions; Egt, energy related to granular temperature; Efr, energy dissipated due to the friction.
The energy-balance equation is:
The corresponding Power Balance is:
Power of the potential energy:
shear layer:
shear layer:
Power of the kinetic energy:
shear layer:
They are written as a function of the traveled distance x(t), the rate
, the thicknesses
,
.
6

7. PROPOSED MODEL (constant mass)
Egt , energy stored as granular temperature
Granular temperature, Tg, measures the degree of agitation of solid grains, which influences the mixture bulk
density and the ability of grains to avoid interlocking and to collide.
Tg is determined by the average grains‟ velocity fluctuations, v‟, respect to their mean velocities
(Campbell ,1990): Tg ~ < v‟ 2>
Granular temperature, Tg , can be generated and maintained only by continuous conversion of bulk volume
translational energy, supplied by the sliding of the moving masses, to grain fluctuation energy; this
conversion occurs as grains shear, rotate, impact along irregular
surfaces, collide each on others.
7

8. PROPOSED MODEL (constant mass)
Egt, , energy stored as granular temperature
OGAWA (1978) observed that the
energy stored in the grain-inertial
regime (Egt) is proportional to the
granular temperature (Tg):
Experimental observations, (Straub, 1997):
thickness of shear zone ss ~ 10 dp
Experimental analyses (Capart et al., 2000; Larcher, 2002; Armanini et al. 2003) allow to define a local
relation between the granular temperature and the measured velocity and concentration profiles.
e being the restitution coefficient of the granular phase ϵ [0,1] (as e → 1 (elastic collisions), Tg → ∞);
u(z), the grain velocity; D, the average grain diameter; Fs , the solid fraction; g0 = g0(Fs ) = (1 - 0.5∙
Fs)/(1- Fs)3.
Therefore, Tg depends on the square of the shear rate (Savage & Jeffrey, 1981) through a multiplier
coefficient that depends, in turn, on the restitution coefficient e and the solid fraction Fs .
8

9. PROPOSED MODEL (constant mass)
By taking into account the previous relationships, the Power of energy stored in granular
temperature may be simply written as:
being:
, and
∙
the ratio between the powers lost (Ecoll) due to collisions and stored
∙
(Egt) through the granular temperature (Federico & Favata, 2011).
By assuming binary collisions and constant average mass of the grains composing the sliding
mass, the parameter b (ϵ [0,2],
) represents the ratio between Dv, the relative velocity
between two colliding grains and dv, average value of the modulus of the velocity fluctuation
vector.
If collisions are neglected, the relative velocities Dv of all grains are null (b = 0);
if, for each collision, two colliding grains, moving along the same direction, assume opposite
velocity vectors, their relative velocity Dv doubles the absolute velocity of each grains (b = 2).
9

10. :
PROPOSED MODEL (constant mass)
Ecoll, energy lost due to repeated grain inelastic collisions
The power of the energy lost in granular inelastic collisions is related to the granular
temperature (Tg) (Jenkins & Savage, 1983):
Recalling that relation
holds, it is possible to define:
.
G(ns) describes the dependence of the power Ecoll on the solid fraction ns; w is the ZhangFoda coefficient (suggested value 0.8).
10

11. PROPOSED MODEL
EQUILIBRIUM ORTHOGONALLY TO THE SLIDING PLANAR SURFACES
(SLOPE (q) AND COUNTERSLOPE ()) (constant mass)
To simulate the coupled and unknown effects of the contact/collision as well as of the shear resistance,
acting along the block-base profile, a splitting function r has been introduced in the equilibrium equation
along the direction normal to the sliding surface.
Wbcosz is balanced by the resultants of the effective and
dispersive pressures acting along the base surface. The
relative role of these pressures depends on the rate v and
parameters h, vcrit, which modulate the „splitting rule‟:
cos z
• Wbcosζ (ζ = θ or α);
Shear layer
•
, effective stresses;
z
• pdisp, dispersive pressures, (BAGNOLD, 1954): pdisp = 0.042ρ (λγdp)2 cosϕ (being g the velocity gradient),
stress acting normally to the boundary of the particles moving at high rate in the grain-inertial regime. If a
linear change of velocity, orthogonally to the sliding surfaces, is assumed, the following expression is
obtained:
The equilibrium equation thus gets:
11

12. THE ROLE OF THE FUNCTIONS r AND
The splitting rule
is defined as follows:
η, being a parameter ϵ [0.005,0.5]; vcrit, critical speed for which the grain displacement realm dominated
by the inertial forces turns towards a condition governed by the collisions. Some experimental results
1
(e.g. Savage, 1981) get:
0.9
0.8
(Nsav = 0.1, b = thickness within which the
rate changes from 0 to the average value of
the „shear layer‟ ).
1- r (v)
r(v)
0.6
r(v), 1-r(v)
The parameter η allows to
model the shape of the function
r. It modulates the local
transition between the inertial
and collisional regimes.
0.7
h
h
h
h
0.5
0.4
0.3
vcr = 20 m/s
0.2
0.1
0
0
10
20
30
40
50
v [m/s]
60
70
80
90
100
12

13. THE ROLE OF THE FUNCTIONS r AND r
1
0.9
0.8
1- r (v)
r(v)
0.7
r(v), 1-r(v)
0.6
vcr = 10 m/s
vcr = 20 m/s
0.5
vcr = 30 m/s
vcr = 40 m/s
0.4
0.3
0.2
h
0.1
0
0
10
20
30
40
50
60
70
80
90
100
v [m/s]
If the critical rate (vcrit) increases, the transition between the two regimes occurs at higher values ​of
the speed.
13

14. PROPOSED MODEL (constant mass)
Energy dissipated due to the frictional resistance (Efr):
Efr depends on the weight W of the sliding mass, the dynamic friction angle fb at the base of the
block. Dispersive and interstitial pressures reduce the energy dissipated due to the friction.
The basal frictional resistance is expressed as follows:
Tfr = W cos z - U – (F (x2))
being
It is simply assumed that the interstitial pressures pw assume a constant value along the
sliding surface (s.s) (Iverson & LaHusen, 1997) and that the isopiezic lines are orthogonal to
the s.s.:
dw = 0, if the mass is saturated; dw = h, if the mass is dry; pw(x) may exceed the hydrostatic
value (pore pressure excesses) if rapid pore volume changes following with the continuos
rearrangement of grains involved in the flow motion occur. To simulate this effect, dw < 0,
must be assigned (Musso, Federico, Troiano, 2004).
The energy dissipated due to the friction along the s.s. is obtained by integration along the
path x . The corresponding power is therefore obtained through derivation vs time t.
14

15. PROPOSED MODEL: VARIABLE MASS
The mass of a debris flow may change due to erosion processes.
Assumptions:
• Erosion of the bed and of the walls of the
crossed
channel may occur;
(i)
• The erosion acts if the speed v ≥ vcr,e (critical speed of
erosion);
• The reduction of the volume of the d.f. (detachment,
splitting) is not considered.
• The geometry of the debris flow varies (constant width
B = B0 of the flow or channel) according to the following
laws:
(ii)
(iii)
15

16. PROPOSED MODEL: VARIABLE MASS
The mass of the debris flow varies according to the law:
The thicknesses of the shear layer (ss1(t)) and of the overlying granular mass (sb1(t)), by
recalling the equilibrium equation, are rewritten as follows:
The Power Balance (for v > vcr,e) becomes:
Power due to the change of the inertial mass following
the increase of the granular material.
16

17. PROPOSED MODEL
PARAMETRICAL ANALYSES
Results depend on the parameters describing the micromechanical behaviour : b (є [0,2]), e (restitution coefficient є
[0,1]), k ( in Egt‟s expression) and dp (grain diameter).
Collisional Energy Ecoll; Energy Egt related to the granular temperature Tg; Rate v and total runout length (L = x)
for different values of parameter k (= ¼ (1-e2)b2)
40
k
k
k
k
35
v [m/s]
Parameters:
q= 38 ;  = 0 ;
dp = 15 cm;
fb = 18 ;
e = 0.2, 0.5,
0.7;
dw = 0 (U≠0);
m = m0;
h = 0.005;
b = 0.5, 1, 1.5
ss(t) ~ 10dp
x 10
k
k
k
k
3
=
=
=
=
0.54 (e=0.2,
0.42 (e=0.5,
0.13 (e=0.7,
0.05 (e=0.5,
Ecoll
Egt
2
gt
E ,E
coll
[J]
Ecoll
1.5
1
Egt
0
20
15
10
5
200
400
600
800
0
200
400
600
800
1000
1200
1400
1600
1800
- The thicknesses of the block (min) and the „shear layer‟ (max) slightly
change if the parameter k increases, except for b < 1: the thickness of shear
layer for some set of parameters, assumes a value close to 10∙dp,
(experimental observations, Straub, 1997).
- Collisional energy Ecoll, set the coefficient of restitution (e = 0.5), increases
if the parameter k (and therefore β) increases, while Egt decreases. Fixed the
coefficient β (= 1.5), if the k decreases (or e increases), Ecoll decreases and Egt
increases.
- Set the coefficient e (= 0.5), if k (and β) increases, the runout length
decreases, while the maximun rate increases.
2.5
0
25
x [m]
b
b
b
b
0.5
0.54 (e=0.2, b
0.42 (e=0.5, b
0.13 (e=0.7, b
0.05 (e=0.5, b
30
0
11
3.5
=
=
=
=
1000
x [m]
1200
1400
1600
1800
2000
17
2000

18. PROPOSED MODEL: PARAMETRICAL ANALYSES
[sb(x(t)), x(t)];[ss(x(t)), x(t)]; )];[H, x(t)]; Collisional Energy Ecoll; Energy Egt related to the granular temperature Tg;
Rate v and total runout length (L = x) for different values of grain diameter d p
70
dp = 0.06 m
dp = 0.10 m
dp = 0.14 m
v(x(t))
60
40
30
30
block
20
v(x(t))
20
10
10
shear zone
Ecoll
4
3.5
Ecoll
3
2
Egt
1.5
Egt
1
Ecoll
Egt
0.5
0
0
500
1000
1500
0
500
1000
1500
2000
2500
3000
0
2500
3000
3500
3
x [m]
dp = 0.06 m
dp = 0.10 m
dp = 0.14 m
Egt
2.5
Egt
e = 0.8
Ecoll
Egt
gt
, E [J]
2
coll
1.5
E
-If the average diameter dp increases, the thickness of
the „shear layer‟ becomes smaller; the distance
traveled and the maximum speed decrease.
-dp, coupled to the restitution coefficient e, plays a
key role in Egt end Ecoll; if dp increases (e is fixed), the
collisional energy Ecoll increases, while the energy
associated with granular temperature Egt decreases;
- fixed dp, to an increase of e, Egt increases because of
the reduction of the energy lost due to collisions; the
runout length increases, if e increases.
2000
x [m]
11
0
e = 0.2
2.5
coll
d.f.
dp = 0.06 m
dp = 0.10 m
dp = 0.14 m
gt
[J]
Parameters:
q = 38 ;  =
0;
k = 0.54;
fb = 22 ;
e = 0.2;
dw = 0 (U≠0);
m = m0;
h = 0.005;
b = 1.5
x 10
,E
40
x 10
4.5
60
50
v(x(t))
5
E
thicknesses [m]
50
12
70
v [m/s]
Ecoll
1
Ecoll
0.5
0
0
500
1000
1500
2000
x [m]
2500
3000
3500
18

19. COMPARISON WITH Coulomb and Voellmy MODELS
Parameters:
q= 30 ;  =
0;
k = 0.28;
fb = 18 ;
e = 0.3;
(dw/h = 0);
m = m0;
h = 25 m ;
b = 1.1;
dp = 0.05 m;
M = 8x104
The motion laws:
Kg/m
M = 0, Coulomb‟s Model
M ≠ 0, Voellmy‟s Model,
- C-M model gets runout and rate values greater than G-M and V-M
models; the runout length doesn‟t depend on the volume of the sling
mass;
- In G-M model, the motion lasts more than the other cases;
- In G-M and V-M, the solutions depend on the granular volume;
- For great values of V, G-M and V-M runout length values tend to the
C-M runout length;
- G-M gets greater runout lengths than V-M model, even if the
19
maximum speed is almost equal.

20. COMPARISON WITH EMPIRICAL RELATIONSHIPS
Influence of micromechanical parameters
INFLUENCE OF INTERSTITIAL PRESSURE
Rick. 1
r  tan  
H
L
Rick. 2
Parameters: q = 38 ;  = 0 ; k = 0.91; fb = 15 ; e = 0.3; dp = 0.1 m; b = 2;
H = 616 m
The interstitial pressure greatly affects the traveled distance: the results obtained by
imposing U = 0 (dw = h) better approximate the relationships by Corominas, while the ones
obtained by imposing U ≠ 0 (dw = -h/3 and dw = 0) approximate the relationships proposed
by Rickenmann ((1) and (2)).
20

21. COMPARISON WITH EMPIRICAL RELATIONSHIPS
INFLUENCE OF RESTITUTION COEFFICIENT e
0.8
4500
0.7
4000
Rickenmann 1
0.6
3500
e = 0.5 (k = 0.42)
0.5
3000
Scheidegger
e = 0.3 (k = 0.51)
L [m]
r = H/L
e = 0.8 (k = 0.2)
0.4
Rickenmann 2
2500
e = 0.3 (k = 0.51)
Corominas
0.3
2000
Davies
0.2
e = 0.8 (k = 0.2)
1500
e = 0.5 (k = 0.42)
0.1
0
0.5
1
1.5
2
2.5
3
V [m ]
3
3.5
4
6
x 10
1000
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
V [m3]
5
x 10
fb = 18°, U ≠ 0 (dw = 0, hydr. cond), dp = 5 cm, h = 0.005, b = 1.5, q = 38°,  = 0°; H = 616 m
An agreement between G-M and empirical relationships may be obtained by varying
parameters such as e and dp. As expected, if the restitution coefficient e increases, the energy
lost due to collisions decreases, the energy dissipated due to the collisional effects becomes
less significant, the granular temperature Tg increases and, for a prefixed volume V, the
runout length increases.
21

22. COMPARISON WITH EMPIRICAL RELATIONSHIPS
INFLUENCE OF GRAIN DIAMETER dp
fb = 15°, U ≠ 0 (dw = 0, hydr. cond), e = 0.5, h = 0.005, b = 1.5 (k = 0.42), q = 30°,  = 0°; H = 616 m
The role of the average grain diameter dp becomes less important if the volume V increases.
Relationships proposed by Scheidegger, Corominas and Davies provide smaller runout values than G-M
and Rickenmann‟s criteria .
22

23. BACK ANALYSES
The event of 12 June 1997 in the „Acquabona Creek‟
Berti, Genevois, Simoni and Tecca („Field observations of a debris flow event in the
Dolomites‟ (1999)) studied the dynamics of d.f. through in situ analyses: three measurement
stations, equipped with pressure sensors, geophones and cameras, provided data on pore
pressures, global thickness of the d.f., flow velocity at the surface.
+
+
No counterslope:
the slope q= +18 reduces
to  = +7
23

24. ACQUABONA CREEK: FIELD OBSERVATIONS
12
Maximum
Rate v
(observed):
9 m/s
10
Pos. x [m]
500
700
750
875
1000
1125
1175
1250
1320
1380
1440
1630
v [m/s]
8
6
4
2
12
0
0
200
400
11
600
10
8
9
800
1000
x [m]
7
6
1200
3
2
5 4
1400
1
1600
1800
Runout (observed): 1630 m
From data measured in situ, the following
input parameters for a back analysis
through the G-M model (variable mass)
are chosen: L = 1630 m; CW = 1.5 m2/m;
Ch = 0 (being the average depth of d.f. h =
H0 = 2 m, case (ii)); B0 = 4 m; l0 = 75 m;
W0 = 300 m2.
24

25. ACQUABONA CREEK (variable mass)
25
12
e = 0.6; dp = 1 cm
e = 0.8; dp = 1.5 cm
dw = -h/4
dw = 0b = 1.5; fb = 21 ;
b = 1.5;
h= 0.04;
G-M: 20.2 m/s
fb = 26 ;
e = 0.3; dp = 1 cm
h= 0.03
dw = 0 b = 0.7;
Pore pressure
fb = 19 ;
G-M: 16.5 m/s
excess at the base
h= 0.035;
U≠0
15
Maximum
Rate v
(observed):
9 m/s
10
10
G-M: 12 m/s
(G-M): 10.7 m/s
(G-M): 10 m/s
e = 0.2;
dp = 1 cm
dw = h;
b = 0.6;
fb = 13 ;
h= 0.035;
Maximum
Rate v
(observed):
9 m/s
8
rate v [m/s]
rate v [m/s]
20
e = 0.3; dp = 1 cm
dw = h b = 0.6;
fb = 12 ;
h= 0.03;
6
(G-M): 8.5 m/s
G-M: 10.5 m /s
4
Runout (G-M):
1643 m
5
2
Hydrostatic
condition
0
0
200
400
e = 0.5; dp = 1 cm
dw = h/2 b = 0.7; fb = 16 ;
h= 0.04;
600
800
1772 m
1691 m
1680 m
1660 m
1000
x [m]
1200
1400
1600
1800
0
0
Runout (observed): 1630 m
partially saturated
observed
200
400
600
1760 m
Runout (G-M):
1636 m
U=0
800
1000
1200
1400
1600
1800
x [m]
Runout (observed): 1630 m
e = 0.1; dp = 1 cm
dw = h; b = 0.5; fb = 13 ; h= 0.04;
- Long runout (~1650-1750 m) are measured, although the initial slope assumes small value (q = +18 ),
due to the positive value (+7 ) of the second slope angle.
-To fit the observed values (max rate = 9 m/s; runout length = 1630 m) through the G-M model, if
interstitial pressures are neglected, very small shear resistance angle fb ϵ [12-13 ] at the base of the d.f.,
coupled to small values of restitution coefficient e, must be assigned.
- If U ≠ 0 and high values of e are imposed, the rate v increases (respect the condition U = 0), although
appreciable values of fb are assigned.
25

26. ACQUABONA CREEK (variable mass)
18
observed
d.f. depth
2
e = 0.3;
dp = 1 cm;
b = 0.6;
fb = 12 .
16
U≠0, h=0.1
14
U=0,
h=0.03
1.5
block
U=0
(dw = h)
U≠0
(dw = -h/4)
rate v [m/s]
thickness [m]
12
U≠0
(dw = 0)
U≠0
(dw = h/2)
1
10
8
e = 0.8; dp = 1.5 cm
b = 1.5; fb = 21 ;
h= 0.04;
dw = 0 (U ≠ 0)
6
shear layer
0.5
Rate v
(observed):
9 m/s
4
2
Runout (observed): 1630 m
0
0
200
400
600
800
1000
x [m]
1200
1400
1600
1800
0
0
500
1000
1500
2000
2500
3000
3500
4000
x [m]
- The thickness of the shear layer, for the different sets of assigned micromechanical parameters, falls
within the range [0.4-0.7 m] and it assumes higher values if U≠0.
- G-M model gets the observed values if h ϵ [0.03-0.04]; higher values of h get runout length
greater than 1630 m.
- High values of the parameter b must be assumed if the interstitial pressures are not neglected,
coupled to high values of e.
26

27. ACQUABONA CREEK (constant mass): comparison between G-M, Voellmy and Coulomb models (V-M, C-M)
60
observed
C-M:
fb = 13 ;
U = 0 (dw = h)
C-M:
fb = 23 ;
U ≠ 0 (dw = 0)
54 m /s
50
42 m /s
V-M:
fb = 23 ;
U ≠ 0 (dw = 0);
x= 21 m/m2
C-M
e = 0.8;
dp = 1.5 cm
b = 1.2;
fb = 23 ;
h= 0.04;
dw = 0 (U ≠ 0)
e = 0.3;
dp = 1 cm
b = 0.5;
fb = 13 ;
h= 0.025;
dw = h (U = 0)
rate v [m/s]
40
V-M
30
27.3 m /s
V-M:
fb = 13 ;
U = 0 (dw = h);
x=8 m/m2
20 m /s
20
Rate v
(observed):
9 m/s
10
9.8 m /s
6.2 m /s
0
1550 m
0
500
1000
1500
x [m]
1767 m
1715 m 1900 m
1760 m 2000
2114 m
2500
Runout (observed): 1630 m
C-M provides higher values of runout and rate v than observed values; V-M runout values approximate the
measured values, although the rate v is too high. A better fitting is obtained through the proposed model if
parameters e = 0.8; dp = 1.5 cm; b = 1.2; fb = 23 ; h = 0.04; dw = 0 (U ≠ 0) are assigned.
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28. CONCLUDING REMARKS
- An analytical (two-layers) model describing the characteristics of high speed granular masses (rate,
runout length), based on energy-balance equations and several simplified assumptions, is proposed.
- The model is based on several experimental results reported in technical literature („granular
temperature‟, dispersive pressure, excess pore water pressure, collisions effects, …) regarding the
micro-mechanical behaviour of granular masses.
- The equations describing the sliding of the granular mass along two planar surfaces (slope,
counterslope) have been numerically solved.
- Parametric analyses and the comparison with results obtained through empirical relationships put
into evidence the role played by the considered geometrical and micro-mechanical parameters.
- Comparisons between the results obtained through the G-M, the C-M and V-M models, as well as
some preliminary back analyses of real cases have been shown. The results obtained through the
proposed model fit the observed values for some specific set of values of the considered micromechanical parameters.
- The limits of the proposed model lie in the oversimplified geometry of the debris body as well as in
the assumptions regarding the laws that correlate the micro-mechanical parameters and the
„coupling‟ of dispersive and effective pressures acting within the shear layer that, during the rapid
sliding, generates between the overlying block and the base surface.
28