# Reproducing rheology of Morency and Doin, 2004#

*This section was contributed by Jonathan Perry-Houts*

Modeling interactions between the upper mantle and the lithosphere can be difficult because of the dynamic range of temperatures and pressures involved. Many simple material models will assign very high viscosities at low temperature thermal boundary layers. The pseudo-brittle rheology described in Morency and Doin [2004] was developed to limit the strength of lithosphere at low temperature. The effective viscosity can be described as the harmonic mean of two non-Newtonian rheologies:

where

where \(B\) is a scaling constant; \(\dot{\epsilon}\) is defined as the quadratic sum of the second invariant of the strain rate tensor and a minimum strain rate, \(\dot{\epsilon}_0\); \(\dot{\epsilon}_\text{ref}\) is a reference strain rate; \(n_v\), and \(n_p\) are stress exponents; \(E_a\) is the activation energy; \(V_a\) is the activation volume; \(\rho_m\) is the mantle density; \(R\) is the gas constant; \(T\) is temperature; \(\tau_0\) is the cohesive strength of rocks at the surface; \(\gamma\) is a coefficient of yield stress increase with depth; and \(z\) is depth.

By limiting the strength of the lithosphere at low temperature, this rheology allows one to more realistically model processes like lithospheric delamination and foundering in the presence of weak crustal layers. A similar model setup to the one described in Morency and Doin [2004] can be reproduced with the files in the directory cookbooks/morency_doin_2004. In particular, the following sections of the input file are important to reproduce the setup:

Note

Morency and Doin [2004] defines the second invariant of the strain rate in a nonstandard way. The formulation in the paper is given as \(\epsilon_{II} = \sqrt{\frac{1}{2} (\epsilon_{11}^2 + \epsilon_{12}^2)}\), where \(\epsilon\) is the strain rate tensor. For consistency, that is also the formulation implemented in ASPECT. Because of this irregularity it is inadvisable to use this material model for purposes beyond reproducing published results.

Note

The viscosity profile in Figure 1 of Morency and Doin [2004] appears to be wrong. The published parameters do not reproduce these viscosities; it is unclear why. The values used here get very close. See Fig. 97 for an approximate reproduction of the original figure.

```
subsection Geometry model
set Model name = box
subsection Box
set X extent = 3000e3
set Y extent = 750e3
set X repetitions = 4
end
end
subsection Compositional fields
set Number of fields = 2
set Names of fields = upper_crust, lower_crust
end
subsection Initial composition model
set Model name = function
subsection Function
set Variable names = x,y
set Function expression = if(y>=725e3,1,0);if((y<725e3&y>700e3),1,0)
end
end
subsection Initial temperature model
set Model name = function
subsection Function
set Variable names = x,y
set Function constants = h=750e3, w=3000e3, mantleT=1350 # deg C
set Function expression = \
if( y < 100e3, \
(100e3-y)/100e3*(1600-mantleT)+mantleT+293, \
if(y>650e3, \
(h-y)/(100e3)*mantleT+293, \
mantleT+293))
end
end
subsection Material model
set Model name = Morency and Doin
subsection Morency and Doin
set Densities = 3300,2920,2920
set Activation energies = 500,320,320
set Coefficient of yield stress increase with depth = 0.25
set Thermal expansivities = 3.5e-5
set Stress exponents for viscous rheology = 3
set Stress exponents for plastic rheology = 30
set Thermal diffusivity = 0.8e-6
set Heat capacity = 1.25e3
set Activation volume = 6.4e-6
set Reference strain rate = 6.4e-16
set Preexponential constant for viscous rheology law = 7e11 ## Value used in paper is 1.24e14
set Cohesive strength of rocks at the surface = 117
set Reference temperature = 293
set Minimum strain rate = 5e-19 ## Value used in paper is 1.4e-20
end
end
```