Slipweakening law
DieterichRuina law
S = 19
S = 0.8
a = 0.012
b = 0.016
a = 0.014
b = 0.012
Andrea Bizzarri, Massimo Cocco [email protected]
Istituto Nazionale di Geofisica e Vulcanologia, Italy
A FINITE DIFFERENCE ALGORITHM TO MODEL A FULLY 3D DYNAMIC
RUPTURE GOVERNED BY DIFFERENT CONSTITUTIVE LAWS
References
Andrews D. J. (1994): Dynamic growth of mixedmode shear cracks,
Bull. Seism. Soc. Am., 84, No. 4, pp. 11841198.
Andrews D. J. (1999): Test of two methods for faulting in
finitedifference calculations, Bull. Seism. Soc. Am., 89, No. 4,
pp. 931937.
Bizzarri A., Cocco M., Andrews D. J., Boschi E. (2001): Solving
the dynamic rupture problem with different numerical approaches and
constitutive laws, Geophys. J. Int., 144 , pp. 656678.
Dieterich J. H. (1986): A model for the nucleation of earthquake
slip, Earthquake Source Mechanics, Geophysical Monograph, 37, S.
Das, J. Boatwright and C. H. Scholz eds., AGU, Washington, DC, pp.
3747.
Ida Y. (1972): Cohesive force across the tip of a
longitudinalshear crack and Griffith s specific surface energy, J.
Geophys. Res., 77, No. 20, pp. 37963805.
Press W. H., Teukolsky S. A., Vetterling W. T., Flannery B. P.
(1992): Numerical Recipes in FORTRAN. The art of scientific
computing, Second Edition, Cambridge University Press.
Roy M., Marone C. (1996): Earthquake nucleation on model faults
with rate and statedependent friction: effects of inertia, J.
Geophys. Res., 101, No. B6, pp. 1391913932.
Ruina A. L. (1983): Slip instability and state variable friction
laws, J. Geophys. Res., 88, No. B12, pp. 1035910370.
Conclusions
We have proposed a new Finite Difference numerical method to
model the truly 3D (not mixedmode) fully dynamic and spontaneous
crack propagation on a system of six mutually interacting faults,
embedded in an elastic medium, with homogeneous or heterogeneous
properties. The rheology on fault planes is described by different
friction laws: the linear slipweakening and the rate and state
dependent friction laws.
Our methodology allows for the rake rotation during the rupture
propagation. We have shown that such spatial and temporal
variations occour within the breakdown zone, where the physical
processes of the traction drop and the seismic wave emission take
place, especially when the crack tip bifurcates. We have also shown
that the behavior of the solution is quite different adopting a
linear slipweakening constitutive equation or a DieterichRuina law,
generalizing the results of Bizzarri et al. ( 2001).
We are able to simulate realworld events, accounting for the
spatially heterogeneous initial stress, the free surface effect and
the heterogeneities of the frictional parameters on the fault
planes.
In the last decades large effort has been expended to realize
numerical algorithms for solving the fundamental elastodynamic
equation to model earthquake ruptures. Two class of methods have
been extensively implemented and used: the Boundary Integral
Equation (BIE) and the Finite Difference (FD) approaches. In
Bizzarri et al. (2001) we have compared 2D pure inplane solutions
to discuss the main differences existing between these two
different numerical strategies.
In this work we present a new FD numerical method to solve the
fully dynamic spontaneous problem for a truly 3D rupture on planar
faults. We implement the TractionatSplitNodes of Andrews (1999)
fault boundary condition for a system of faults, either vertical or
oblique. For each fault we can assume different constitutive laws:
we can use a slipweakening law to prescribe the traction evolution
within the breakdown zone or a rate and statedependent friction
law, which involve the choice of a governing relation for the state
variable. We have implemented a 3D version of the Rosembrock Stiff
Integration, generalizing that previously used in our 2D model, and
we have included the free surface effect.
In our numerical procedure the initial shear stress is not
necessarily imposed in only one spatial direction and the two
components of slip, slip velocity and traction can vary during the
dynamic crack propagation. Such components are coupled together in
order to satisfy, in norm, the adopted governing law, and therefore
they allow for dynamically controlled rake variations. We can also
model faults with spatially heterogeneous distributions of
constitutive parameters in order to simulate crack arrest and the
healing of slip.
Abstract
The numerical method
Our method allows for the calculation and interactions between a
system of six faults, two vertical and four dipping faults. A grid
of nodes is introduced by using quadrilateral isoparametric
elements and the solutions of the fundamental elastodynamic
equation is obtained by stepping explicitely the solutions at the
previous time level through time: from displacement components at
nodes ui we find strain tensor components at integration points,
then stress tensor components and then net force components fi(n)
at nodes: v(n) v(n 1) a(n) Dt and u(n) u(n 1) v(n) Dt, where a is
the node acceleration derived from the second law of mechanics:
a(n) f(n)/m, m being the mass matrix element. Such a formulation is
equivalent to the local stiffness matrix. All components of slip
velocity, displacement and force are defined at the same node point
and the number of finite difference expressions that are calculated
is 8 times larger than in a simple staggered grid scheme, in which
velocities and stresses are defined at different points in space.
In the latter way is easy to increase the order of accuracy to
fourth order in space, while keeping it second order in time. The
mass matrix is diagonal and the small displacement approximation is
used: in such a way we can refer to the numerical algorithm either
as a Finite Element (FE) scheme and Finite Difference (FD) scheme.
For fault node point solutions are obtained by introducing the
TractionatSplitNodes (TSN) fault boundary condition which assumes
an explicit displacement discontinuity between the fault
surface.
The fault strength Sfault is specified assuming a constitutive
law (Sfault m(Du, Dv, Y, )sneff). For the linear slipweakening law
(Ida, 1972) we first find a trial value of traction on the fault
plane that gives no differential acceleration (aa ab, or Da 0):
tt(n) t0 (ma fa(n) mb fb(n))/ (A ma + A mb), where A is the slip
node area and ma and mb are the split node masses, above and under
the fault plane, respectively. If the fracture criterion is not
satisfied, i. e. if
Dv(n ) va vb 0
tt(n) < Sfault
then the fault slip Du is not changed and the traction t is not
updated and the fault is not slipping in this case. On the
contrary, if the fracture criterion is satisfied, the fault starts
to slip or continues to slip. In this case we update the actual
total dynamic traction by imposing the collinearity between
traction and slip velocity vectors: t(n) c v*(n) where v*(n) Dv(n )
+ A Dt (1/ma + 1/mb) tt(n). From this equation we determine the
director cosines: gi(n) vi*(n)/v*(n). Therefore the two actual
components of total traction are expressed as: ti (n) gi(n)
Sfault.
For the whole class of rate and statedependent friction laws
(Dieterich, 1986; Ruina, 1983; Roy and Marone, 1996) we solve for
Dvi and Y the ODE system:
a1 = daccdtau * (L1 m(Dv, Y)(Dv1/Dv) sneff)
a2 = daccdtau * (L2 m(Dv, Y)(Dv2/Dv) sneff )
(d/dt)Y = evolution equation
where daccdtau is the differential acceleration for unit stress
drop and Li are traction that will give no change of differential
velocity. We generalize to 3D the Rosembrok stiff integration
(Press et al., 1992), that perform better than RungeKutta
algorithm. From Dvi(n) we calculate the director cosines gi(n)
Dvi(n)/Dv(n) and finally the actual value of shear traction: ti (n)
gi(n) Sfault.
Reference cases
We compare solutions obtained with our 2D pure inplane FD code
and with this new fully 3D FD algorithm, adopting either a linear
slipweakening law and a DieterichRuina law. In this simulation, for
the 3D model the initial stress is non null along the inplane
direction only. Nucleation strategies are the same for two methods
(Bizzarri et al., 2001). Our numerical experiments show that the
bifurcation phenomenon is quite different and the superpositions of
the solutions along the time generalize the results of Tinti et al.
(2003).
Coupling of two modes of propagation
At each time level the two components of slip, slip velocity and
traction are calculated independently, with the requirements that
their modula satisfy the assumed governing law and that the
traction is collinear with the slip velocity. We did numerical
simulations with initial rake of 45, either with the slipweakening
and the DieterichRuina law. We show that during the crack
propagation the rake (i. e. the slip velocity azimuth) can vary,
both in space (especially in the cohesive zone, after the tip
bifurcation) and in time.
As observed by Andrews (1994) for a mixedmode crack, the
variation of rake is lower as the initial value of stress
increases, being the strength parameter S constant. All the
simulations shown in the Figures refer to a single vertical
fault.
Vertical lines represent the time duration of the breakdown
process: the first line indicates the time step at which the
traction is at the upper yield value and the second one marks the
time step at which it is at the kinetic level. Location #1 is along
a line passing through the hypocenter and in the x direction;
Location #2 is along a line at 45 degrees and Location #3 is along
a line in the z direction.
Effects of heterogeneities
Our methods allows for the modeling of heterogeneous
configurations, in terms of initial stress distribution (prestress)
and in terms of constitutive parameters. We present here two
numerical experiments, one with the slipweakening law and a barrier
(a region with a high value of strength parameter S) and one with
DieterichRuina law and a velocity strengthening region, surrounding
a velocity weakening one.
In such cases the coupling of the two modes of propagation is
quite different with respect to the homogeneous configurations and
the rake variation is quite complex, especially in the case of
crack arrest.
Model parameters
Slipweakening law (adimensional set)
= 1.vS = 1.vP = 1.732Dx = Dx = Dx = 0.2
t0 = 1.0tu = 1.8tf = 1.0d0 = 1.3
DieterichRuina law
= 2700 Kg/m3vS = 3000 m/svP = 5196 m/s Dx = Dx = Dx = 0.1 m
a = 0.008b = 0.016L = 10 mmsneff = 100 MPa
vinit = 10 mm/s
Frictional parameters
Initial rake = 45 degrees
Slipweakening law with S = 0.8
Test #26
t0 = 1.0tu = 1.8tf = 0.0d0 = 1.3
Test #30
t0 = 3.0tu = 3.8tf = 2.0d0 = 1.3
Test #34
t0 = 4.0tu = 4.8tf = 3.0d0 = 1.3
Test #35
t0 = 10.0tu = 10.8tf = 9.0d0 = 1.3
DieterichRuina law
Test #11
a = 0.0085 b = 0.016L = 10 mm
NMESD 2003
0.08.016.024.0
17.5
12.5
7.5
2.5
x direction
Time
0.04.08.012.016.020.0
Slip
2 - D
25.033.341.750.0
22.5
17.5
12.5
7.5
2.5
z direction
Time
0.04.08.012.016.020.0
Slip
3 - D
25.433.441.449.4
22.5
17.5
12.5
7.5
2.5
x direction
Time
0.04.08.012.016.020.0
Slip
3 - D
Traction vs. Slip
at x_1 = x_init + 18.0
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.60
1.80
2.00
0.00E+00 5.00E-01 1.00E+00 1.50E+00 2.00E+00
Slip
Traction
2 - D
3 - D
Traction vs. Slip velocity
at x_1 = x_init + 18.0
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.60
1.80
2.00
0.00E+00 2.00E+00 4.00E+00 6.00E+00 8.00E+00
Slip velocity
Traction
2 - D
3 - D
Superposition along Time
at x_1 = x_init + 18.0
0.0
0.2
0.4
0.6
0.8
1.0
1.60E+01 1.65E+01 1.70E+01 1.75E+01 1.80E+01 1.85E+01
1.90E+01
Time
Normalized solutions
Traction
Slip velocity
Slip
Superposition along Time
at x_1 = x_init + 18.0
0.0
0.2
0.4
0.6
0.8
1.0
1.70E+01 1.75E+01 1.80E+01 1.85E+01 1.90E+01 1.95E+01
2.00E+01
Time
Normalized solutions
Traction
Slip velocity
Slip
0.0
1.3
2.5
3.8
5.0
1.0e-003
7.8e-004
5.6e-004
3.4e-004
1.3e-004
x direction ( m )
Time ( s )
0.0e+000
5.0e-004
1.0e-003
1.5e-003
Slip ( m )
2 - D
5.0
6.3
7.5
8.8
10.0
1.0e-003
7.8e-004
5.6e-004
3.4e-004
1.3e-004
z direction ( m )
Time ( s )
0.0e+000
5.0e-004
1.0e-003
1.5e-003
Slip ( m )
3 - D
5.0
6.3
7.5
8.8
10.0
1.0e-003
7.8e-004
5.6e-004
3.4e-004
1.3e-004
x direction ( m )
Time ( s )
0.0e+000
5.0e-004
1.0e-003
1.5e-003
Slip ( m )
3 - D
Traction vs. Slip
at x_1 = x_init + 2.0 m
6.00E+07
6.50E+07
7.00E+07
7.50E+07
8.00E+07
0.00E+00 5.00E-05 1.00E-04 1.50E-04 2.00E-04
Slip ( m )
Traction ( Pa )
2 - D
3 - D
Traction vs. Slip velocity
at x_1 = x_init + 2.0 m
6.00E+07
6.50E+07
7.00E+07
7.50E+07
8.00E+07
0.00E+00 1.00E+00 2.00E+00 3.00E+00 4.00E+00
Slip velocity ( m/s )
Traction ( Pa )
2 - D
3 - D
Superposition along Time
at x_1 = x_init + 2.0 m
0.0
0.2
0.4
0.6
0.8
1.0
2.20E-04 2.45E-04 2.70E-04 2.95E-04 3.20E-04 3.45E-04
Time ( s )
Normalized solutions
Traction
Slip velocity
Slip
Superposition along Time
at x_1 = x_init + 2.0 m
0.0
0.2
0.4
0.6
0.8
1.0
2.20E-04 2.70E-04 3.20E-04 3.70E-04
Time ( s )
Normalized solutions
Traction ( Pa )
Slip velocity ( m/s )
Slip ( m )
101.6
134.4
167.2
200.0
200.0
175.0
150.0
125.0
100.0
x direction
z direction
-45
-30
-15
0
15
30
45
Rake variation ( degrees )
Anti
plane
Component
In
plane
Component
x
3
x
1
Local Crack
Enlargement Direction
Local Slip
Velocity Direction
u
3
u
1
Broken Area
Anti
plane
Component
In
plane
Component
x
3
x
1
Local Crack
Enlargement Direction
Local Slip
Velocity Direction
u
3
u
1
Broken Area
Anti
plane
Component
In
plane
Component
x
3
x
1
Local Crack
Enlargement Direction
Local Slip
Velocity Direction
u
3
u
1
Broken Area
25.4
33.9
42.3
50.8
50.0
43.8
37.5
31.3
25.0
x direction
z direction
-45
-30
-15
0
15
30
Rake variation ( degrees )
5.0
6.7
8.3
10.0
10.0
8.8
7.5
6.3
5.0
x direction ( m )
z direction ( m )
-10
-5
0
5
10
Rake variation ( degrees )
Rake vs. Time
-30.0
-10.0
10.0
30.0
50.0
70.0
90.0
110.0
1.30E+01
1.55E+01
1.80E+01
2.05E+01
2.30E+01
2.55E+01
Time
Rake ( degree )
Location #1
Location #2
Location #3
Rake vs. Time
3.50E+01
4.00E+01
4.50E+01
5.00E+01
5.50E+01
3.00E-04
5.50E-04
8.00E-04
Time ( s )
Rake ( degrees )
Location #1
Location #2
Location #3
Rake vs. Time
dist = r_init + 18.0
20.0
30.0
40.0
50.0
60.0
70.0
2.00E+01
2.10E+01
2.20E+01
2.30E+01
2.40E+01
2.50E+01
Time
Rake ( degree )
Test #26
Test #30
Test #34
Test #35
5.0
6.3
7.5
8.8
10.0
14.0
11.6
9.2
6.8
4.4
2.0
x direction ( m )
Time ( s )
-5
-3
-2
-0
2
3
Rake variation ( degrees )
