1
2
1. Model scaling
Scaling models into their natural prototypes ensures fulfillment of similarity in geometry,
3
kinematics, dynamics and rheology [Ramberg, 1981].
4
It is widely accepted that a continental lithosphere with a normal crustal thickness (30-35km) and
5
stable geotherm, extending at strain-rate magnitude of about 10-15 s-1, can be mechanically
6
described by a four-layer strength profile with alternating high- and low-strength (Figure1b) [Brace
7
and Khoelsted, 1980; Davy and Cobbold, 1991; Brun and Beslier, 1996].
8
Stresses scale down according to:
𝜎 ∗ = 𝜌∗ 𝑔∗ 𝐿∗
9
(1)
10
where the starred symbols are the dimensionless ratios between model and natural prototype for the
11
following parameters: 𝜎* is stress, ρ* is density, g* is gravity, and 𝐿* is the length scale-factor (L*
12
= Lmodel / Lnature = 6.66 x 10-7, that is 15km in nature correspond to 1cm in the model). Time scaling
13
for the viscous material is achieved by
𝑡 ∗ = 𝑡𝑚 ⁄𝑡𝑛 = 𝜂 ∗ ⁄𝜎 ∗
14
(2)
15
where t* is the dimensionless ratio of time, tn and tm the time in nature and in model respectively,
16
and 𝜂* is the dimensionless ratio for viscosities.
17
The values of dimensionless ratios used to scale the lower crust in the models are provided in Table
18
1, while in Table 2 the resultant values for parameters in the models and in nature are listed.
19
Viscous materials have a Newtonian rheology and their strength is provided by [e.g. Ranalli,
20
1995]:
21
τ = 𝛾̇ 𝜂
(3)
22
where 𝛾̇ is the shear-rate (V/h, Velocity/layer thickness).
23
The brittle rheology is simulated by pure quartz sand [Davy and Cobbold, 1991] with average grain-
24
size of 260µm and negligible cohesion. The ductile rheology is simulated by using silicone putty
25
SGM36. In particular, pure silicone putty was mixed with the proper weight of barite powder in
26
order to tune its density and rheology (Table 2). Silicone putty has an almost Newtonian rheology
27
(parameter ‘n’ is 1,053 and 1,067 for the silicone simulating the crust and the mantle, respectively)
28
and is widely used to reproduce the mechanical behavior of ductile layers in normal gravity
29
experiments [e.g. Davy and Cobbold, 1991; Brun and Beslier, 1996; Brun, 1999; Autin et al., 2010].
30
The experimental lithosphere of our models consists of a four-layer multilayer that, from bottom to
31
top, includes a 12mm thick silicone layer simulating ductile mantle material, a 10mm thick sand
32
layer simulating brittle mantle material, a 7mm thick silicone layer simulating the ductile lower
33
crust, and a 13mm thick sand layer simulating the brittle upper crust. This multilayer floats on
34
glucose syrup (𝜌𝐴 =1450 kg/m3) simulating the asthenosphere [i.e. Davy and Cobbold, 1991]. Model
35
velocity (1cm/h) is representative of prototype velocities ranging between 0,2 - 20cm/y according to
36
the lower crust viscosity range (1020 - 1022 Pa.s). To simulate a natural extensional velocity of
37
≈4mm/y we assume a viscosity of 5 x 1021 Pa.s, and a strain-rate of 5,87x10-14 for the prototype
38
lower crust (Table 1). The calculated model to nature ratios of both the “Ramberg number” (Rm* =
39
1) and the “Smoluchowsky number” (Sm* = 1,03), for the ductile and the brittle domains,
40
respectively (Table 1), ensure dynamic scaling of the model (e.g. [Corti et al., 2003]).
41
42
2. Thinning factor calculation
Top and bottom laser-scanned surface pairs underwent a post-acquisition processing to tie them up
43
to the same x-y space matrix. The thickness of the undeformed lithosphere (T0) is calculated as the
44
difference between top (Zt0) and basal (Zb0) surfaces at each point, then averaged over the entire
45
undeformed model surfaces:
46
𝑇0 = 𝑁 ∑𝑛 [𝑍𝑡0 − 𝑍𝑏0 ](𝑛)
47
where N is the number of nodes into the matrix space (N = i x j).
48
The whole lithosphere-stretching factor (β) [McKenzie, 1978] is calculated as the ratio between the
49
mean initial lithosphere thickness (T0) and the lithosphere thickness at the end of experiment (Tf):
50
[𝑇𝑓 ]
51
𝛽(𝑛) =
1
(𝑛)
= [𝑍𝑡𝑓 − 𝑍𝑏𝑓 ]
𝑇0
𝑇𝑓 (𝑛)
(𝑛)
(4)
(5)
(6)
52
Rather than β, it is more convenient to use the thinning factor (TF) [eg. Reston, 2007], which varies
53
between 0 and 1:
54
𝑇𝐹(𝑛) = 1 − 𝛽
55
TF is calculated for the crust (Figure 4a), the lithospheric mantle (Figure 4b), and the whole
56
lithosphere (Figure 4c).
57
1
(𝑛)
(7)