SofteningHealing MohrCoulomb Joint Model In 3DEC
The MohrCoulomb joint model has the following deficiencies when used to model seismicity.
The stress drop is instantaneous when slipping starts (shear strength drops instantaneously from a peak to a residual value). This means that all slip events are seismic (see Figure 1).
Once slip has occurred, there is no way for the strength on the fault to recover back to the peak value. This means that stickslip behavior is not possible.
Formulation
The Softening Healing MohrCoulomb model is based on the standard i MohrCoulomb model, with two additions.
When the joint starts slipping, the shear strength evolves from the peak value to the residual value over a userspecified critical slip distance \(D_c\).
When slipping stops, the shear strength recovers instantaneously to the peak value.
There is no softening or healing for the tensile forces. The tensile strength drops instantaneously when the strength is exceeded (no softening) and the tensile strength remains at the residual value (no healing).
For a nonslipping joint, the value of cohesion and friction are the peak values. When the strength is exceeded, the strength is a function of the slip distance. When the slip distance exceeds the critical slip distance, the residual values of cohesion and friction are used. Two formulations for the evolution of shear strength are provided.
Linear
For linear softening, the shear strength is calculated by:
where \(c\) is the cohesion, \(\sigma_n\) is the normal stress, \(\phi\) is the friction angle, \(c_{resid}\) is the residual cohesion, \(\phi_{resid}\) is the residual friction angle, \(D_c\) is the critical slip distance, and \(u_s^p\) is the plastic shear displacement (slip) since the start of slipping. When slipping stops, the shear strength returns instantaneously to the peak value and \(u_s^p\) is set back to 0.
An example of a subcontact with no cohesion is shown in Figure 2.
NonLinear
For nonlinear softening, the shear strength is calculated by:
where \(\alpha\) is an exponent (> 1) that dictates the severity of the strength drop (see Figure 3).
Dilation (\(\psi\)) behaves in the same way as cohesion or friction. If a nonzero residual dilation is specified, the dilation varies according to the following equation for \(\alpha\) = 0 (linear):
For the nonlinear case, the dilation is given by:
If a value of \(\psi_{resid}\) is not provided, it defaults to \(\psi_{peak}\) .
Subcontact failure states are the same as for i MohrCoulomb Joint Model In 3DEC.
Summary of SofteningHealing MohrCoulomb Parameters
The model parameters associated with the sofdteninghealing mohrcoulomb model are summarized in Table 1. The model is accessed in 3DEC with the block contact jmodel assign
softeningmohr
command.
Parameter 
Description 
Keyword 

\(k_n\) 
joint normal stiffness (STRESS/LENGTH) 

\(k_s\) 
joint shear stiffness (STRESS/LENGTH) 

\(\phi\) 
friction angle (DEGREES) 

\(\phi_{res}\) 
residual friction angle (DEGREES). Defaults to phi. 

\(c\) 
cohesion (STRESS) 

\(c_{res}\) 
residual cohesion (STRESS). Defaults to 0. 

\(\psi\) 
dilation angle (DEGREES) 

\(u_{cs}\) 
shear displacement at which dilation stops (LENGTH). Default is infinity 

\(T_f\) 
tensile strength (STRESS) 

\(T_f^{res}\) 
residual tensile strength (STRESS). Default is 0. 

\(D_c\) 
critical slip distance (LENGTH) 

\(\alpha\) 
exponent that dictates the severity of nonlinear stress drop 

boolean to indicate if healing is on or off (default is on) 


\(\phi_{current}\) 
current friction angle (DEGREES). Read only. 

\(u_s^p\) 
plastic slip (LENGTH). Read only. 

Example
To view this example in 3DEC, use the menu command . The data files used are shown at the end of this example.
A springslider problem is simulated with three blocks as shown in Figure 4. A normal stress of 1 MPa is applied to the top surface of the slider block. A velocity of 1 mm/s is applied to the right side of the system. The model is then run until one or more slip events occurs.
The test is a simulation of a direct shear test, which consists of a single horizontal joint that is first subjected to a normal confining stress, and then to a unidirectional shear displacement. Figure 5 shows the model; the joint is defined by one contact that is composed of 10 subcontacts.
The following strength properties are assigned to the sliding joint.
Peak friction 
\(\phi\) 
25 degrees 
Residual friction 
\(\phi_{resid}\) 
20 degrees 
Critical slip distance 
\(D_c\) 
1 × 10^{4}, 1 × 10^{5} 
Alpha exponent 
\(\alpha\) 
0 (linear) 
Plots of friction on the joint, and shear velocity versus shear displacement are shown for the different values of \(D_c\) in Figure 5 to Figure 8. Velocities are plotted with the same maximum in each plot to facilitate comparison. The effect of the softening is clear, and the higher velocities observed in the model with the lower value of \(D_c\) suggest a possible criterion for differentiating seismic and aseismic slip events.
To test changing dilation, the test was run three more times: with a constant dilation of 10 degrees, with a constant dilation of 2 degrees, and with dilation evolving from 10 to 2 over a critical slip distance of \(D_c\) of 1e4. Graphs of normal displacement versus shear displacement for the three cases are shown in Figure 9.
Data Files
springslider.dat
model new
model random 10000
model configure energy
model largestrain off
; slider brick
block create brick 1 1 1 1 1 1
; spring brick
block create brick 1 3 1 1 1 1
; base brick
block create brick 1.5 3.5 1.5 1.5 1.5 1
block zone generate edgelength 0.5
block zone cmodel assign elastic
block zone property density 2000 bulk 1e11 shear 0.6e11
; spring
block contact jmodel assign elastic
block contact property stiffnessnormal 1e9 stiffnessshear 1e9
;sliderbase contact  mohrcoulomb with 0 friction
block hide range positionx 1 1 positionz 1 1
block contact jmodel assign mohr
block contact property stiffnessnormal 1e11 stiffnessshear 1e11
block hide off
; sliding contacts
block hide range positionx 1 3 positionz 1 1
block contact jmodel assign softeningmohr
; joint properties
block contact property stiffnessnormal 1e11 stiffnessshear 1e11 ...
friction 25 frictionresidual 20 tension 1e12 dc 1e4
block hide off
block insitu stress 0 0 1e6 0 0 0
block face apply stresszz 1e6 range positionz 1
block fix range positionz 1.5 1
model solve
model save 'blocks'
=====================================
block gridpoint initialize displacement 0 0 0
block contact reset displacement
; ==== FISH forigin histories =====
[cxi0 = block.subcontact.near(0,0,0)]
fish define sh_mu
sh_mu = block.subcontact.prop(cxi0,'frictioncurrent')
sh_slip = block.subcontact.prop(cxi0,'slip')
end
block gridpoint apply velocityx 1.0e3 range positionx 2.9 3.1
block contact history stressnormal position 0 0 0
block contact history stressshear position 0 0 0
block contact history displacementshear position 0 0 0
; get velocity
block hide range positionz 1.5 1
block history velocityx position 0 0 0
block hide off
fish history sh_mu
fish history sh_slip
block history fob
model save 'initial'
================================
; dc = 1e4
model restore 'initial'
history interval 100
model cycle 30000
history interval 10
model cycle 30000
model save 'springsliderdc1e4'
===========================================
; dc 1e5
model restore 'initial'
block hide range positionx 1 3 positionz 1 1
block contact property dc 1e5
block hide off
history interval 100
model cycle 30000
history interval 10
model cycle 25000
model save 'springsliderdc1e5'
========================================
dilation.dat
; File to test changing dilation angle with slip
model restore 'blocks'
block gridpoint initialize displacement 0 0 0
block contact reset displacement
block gridpoint apply velocityx 1.0e3 range positionx 2.9 3.1
block contact history stressnormal position 0 0 0
block contact history stressshear position 0 0 0
block contact history displacementshear position 0 0 0
block contact history displacementnormal position 0 0 0
; function to compute dilation
fish define get_dilation
local cx0 = block.subcontact.near(0,0,0)
local ndis = block.subcontact.dis.norm(cx0)
local sdis = math.mag(block.subcontact.dis.shear(cx0))
global dilation = math.atan(ndis/sdis)/math.degrad
local status = io.out('Dilation = '+string(dilation))
end
model save 'initial'
; ========= dilation constant 10 =============
; set joint properties
block hide range positionx 1 3 positionz 1 1
block contact property dilation 10
block hide off
history interval 100
model cycle 30000
history interval 10
model cycle 15000
[get_dilation]
history export '4' vs '3' table 'dilation10'
table 'dilation10' export 'dilation10.tab'
; ========= dilation constant 2 =============
model restore 'initial'
; set joint properties
block hide range positionx 1 3 positionz 1 1
block contact property dilation 2
block hide off
history interval 100
model cycle 30000
history interval 10
model cycle 15000
[get_dilation]
history export '4' vs '3' table 'dilation2'
table 'dilation2' export 'dilation2.tab'
; ========= dilation changing from 10 to 2 =============
model restore 'initial'
; set joint properties
block hide range positionx 1 3 positionz 1 1
block contact property dilation 10 dilationresidual 2
block hide off
history interval 100
model cycle 30000
history interval 10
model cycle 15000
[get_dilation]
history export '4' vs '3' table 'dilation_10_2'
table 'dilation10' import 'dilation10.tab'
table 'dilation2' import 'dilation2.tab'
Properties
 alpha f
An exponent (>= 1) that dictates the severity of friction drop. If
alpha
= 0, the linear friction reduction relationship is used.
 cohesion f
Joint cohesion.
 cohesionresidual f
Residual joint cohesion.
 dc f
Slip distance over which friction reduces from peak to residual friction.
 dilation f
Joint dilation angle in degrees.
 dilationzero f
Shear displacement at which joint will cease to dilate.
 friction f
Peak friction angle in degrees.
 frictioncurrent f
Current friction angle in degrees (read only).
 frictionresidual f
Residual joint friction angle in degrees.
 healing b
If set to true, friction recovers to the peak value when slipping stops. Default is true.
 slip
Plastic shear slip (read only).
 stiffnessnormal f
Joint normal stiffness (stress / distance).
 stiffnessshear f
Joint shear stiffness (stress / distance).
 tension f
Tensile strength.
 tensionresidual f
Residual joint tensile tensile.
References
Rice, J., 1983. “Constitutive relations for fault slip and earthquake instabilities,” Pure Appl. Geoph., 116, 790806, 1983.
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