1
Text S1
2
Hu et al. (2009a, b) derived two fundamental elastic formulas to study the
3
preseismic and coseismic stress fields based on the principle of virtual work. On the
4
basis of the elastic formulas, we propose a quasi-static model of fault earthquake
5
defined by domain V with boundary surface S (Figure 1). The domain consists of
6
three regions: (1) the upper crust region, VI , excluding earthquake fault zones, which
7
is isotropic elastic; (2) the earthquake fault zone, VII , in the upper crust, which is
8
transversely elastic; (3) and the lower crust and upper mantle, VIII , which is
9
viscoelastic according to the Maxwell model. The boundary surface is divided into
10
two sub-boundaries, S1 and S 2 , on which the known displacement vector and
11
traction qt are prescribed, respectively.
12
13
Figure 1 The sketch map of the finite element model.
14
15
16
The constitutive relationship of the material in VI is:
σ t  D Iε t ，
(1)
17
where superscript t of σ and ε denote time, σ  [ xx  yy  zz  yz  xz  xy ]T and
18
ε  [ xx  yy  zz  yz  xz  xy ]T are the stress and strain at time t, respectively,
19
20
ux , u y and uz are displacement components in the directions of x, y and z, DI is the
21
elastic material matrixes,


1  
 
1 



1 
 
E
DI 

(1   )(1  2 ) 

0


22
ux 
 
u y  ，
u 
 z








0.5   
0
0.5  
0.5  
The constitutive relationship of the material in VII (Cai and Yin, 1997) in the
23
24

y 


x 


0

T


 x 0 0 0 z




ε t  Lut  0
0
0
y
z





0 0 z y x

material axis system ( x, y , z ) is:
σ  DII ε ,
25
(2)
26
where ε  [ xx  yy  zz  yz  xz  xy ]T , σ  [ xx  yy  zz  yz  xz  xy ]T are the strain
27
and stress, respectively.
 a (1  a 22 )b a 2 (1   1 )b a ( 1  a 22 )b 0

(1   12 )b
a 2 (1   1 )b 0


a (1  a 22 )b 0
DII  
G2


sym.


28
0
0
0
G1
0

0
0
.
0
0

G2 
E1
E2
,b 
, E1 , E2 ,1 , 2 , G2 are five independent parameters, and
E2
(1  1 )(1  1  2 22 a)
29
a
30
G1 
31
0
E1
E2
, G2 
.
2(1  1 )
2(1  2 )
Equation (2) can be transferred into the global coordinate system (x, y, z):
σ t  D IIε t
32
33
where DII  N T DII N and ε  Nε t ， N is the transformation matrix between the
34
material coordinate system ( x, y , z ) and global coordinate system (x, y, z) ，
35
 l12
 2
 l2
 l2
N 3
 2l2l3
 2l l
 31
 2l1l2
36
37
38
39
40
41
42
m12
n12
m1n1
l1n1
2
2
2
3
2
2
2
3


l2 m2 
l3m3 
,
l2 m3  l3m2 
l3m1  l1m3 

l1m2  l2 m1 
l1m1
m
n
m2 n2
l2 n2
m
n
m3n3
l3n3
2m2 m3
2n2 n3
m2 n3  m3n2
l2 n3  l3n2
2m3m1
2n3n1
m3n1  m1n3
l3n1  l1n3
2m1m2
2n1n2
m1n2  m2 n1
l1n2  l2 n1
l, m, n are directional cosines between the two coordinate systems (Table 1).
Table 1 Directional cosine between the two coordinate systems.
Axis
x
y
z
x
l1
m1
n1
y
l2
m2
n2
z
l3
m3
n3
The material in the earthquake fault zones VIII in the lower crust and upper
mantle is taken as Maxwell viscoelastic material. Its constitutive relationship is:
ε  DI 1σ  Q1σ ,
(3)
where Q is viscous material matrixes:
 1
 3

 1
 6

1
Q 1    1
 6





1
6
1
3
1

6

1
6
1

6
1
3

0
1
0
1






.




1

43
After approximating the derivatives of the strain and stress by
εt 
44
 εt  εt t
 σ t  σ t t


, σt 
,
t
t
t
t
Equation (3) is expressed by
45
σ t  D IIIε t  σ 0 ,
46
(4)
47
where D III  (D I1  Q 1t ) 1 , and σ 0  D III D I1σ t t  D IIIε t t . It is found that the
48
viscoelastic constitutive relation can be considered as an elastic one with initial
49
stress σ 0 . Thus, the constitutive relations in the three regions can be expressed as
50
integrate one:
σ t  Dε t  σ 0
51
(5)
52
where D  Di in Vi , i  I, II, III , σ 0  0 in VI and VII if no initial stresses, say, caused
53
by heat, pore fluid pressure and so on.
54
55
The strain and stress fields can be calculated by using ε t  Lut and
σ t  Dε t  σ 0 if ut is a solution of boundary value problem.
56
Two basic equations can be obtained by following Hu et al. (2009b) on the basis
57
of the principle of virtual work. One equation solves for preseismic displacement ut or
58
strain ε t :
 ε Dε dV  
T
59
V
t
V
uTf t dV   uTqt dS   εTσ0dV
S2
V
(6)
60
where u T and ε T are the virtual displacement and virtual strain, respectively,
61
σt  [ xx  yy  zz  yz  xz  xy ]T and εt  [ xx  yy  zz  yz  xz  xy ]T are the stress and
62
strain at time t, respectively, f t and qt are body force and traction force,
63
respectively, and D and σ 0 are the material matrix and initial or previous stresses,
64
respectively.
The other equation solves the co-, and, postseismic displacement, u and
65
66
67
strain ε , which are caused by an earthquake:

V
εT (D  D)εdV  εT D εt dV   εT σ0dV   uT (f )dV   uT (q)dS2 (7)
V
V
V
S2
68
where ε  Lu is the co- or postseismic strain field in the domain V, and
69
D , σ 0 , f , and q are the changes in the material matrix, stress matrix, body force,
70
and traction force, respectively.
71
The material change or weakening, D , is only limited to the fault zone during
72
the coseismic interval. If fault healing is considered in the postseismic process,
73
the D is a function of time. If f and q are much smaller than the contribution of the
74
first term of right hand of Equation (7) during an earthquake sequence or a
75
quasi-earthquake cycle, Equation (7) can be reduced to

76
V
ε T (D  D)εdV   ε T D ε t dV   ε T σ 0 dV .
V
V
(8)
77
After finite element discretization in V, Equations (6) is developed into the
78
finite element formula to solve the preseismic displacement field U t caused by the
79
body force in V and traction on the boundary S2:
KU t  F
80
Ut
s1
(9)
 U s1
81
where U s1 is the displacement vector specified on the boundary S1, and K and F are
82
the preseismic global stiffness matrix and nodal load vector, respectively.
III
83
Ni
K    e B Di BdV ,
T
i  I e 1
84
85
Vi
e
III
Ni
F   (  e [HTf t  BTσ 0 ]dV e   e HTqt dS2e )
i  I e 1
Vi
s2
The summation symbol in the equation means assembling elements. H is the
shape function matrix, B  LH is a connection matrix between the displacement and
86
strain, and Ni is the number of the elements in the region Vi .
87
Equation (7) can be developed into the finite element formula for
88
solving the co- and postseismic displacement field U , induced by material
89
weakening under the control of the preseismic displacement field U t :
K c (D, D)U  F(Ut )  F(f t , qt , σ 0 )
90
U
s1
(10)
 U s1
91
where U s1 is the displacement change vector specified on the boundary S1, and
92
K c    e BT (Di  Di )BdV e , F(Ut )    e BT Di BUt dVi e ,
Ni
III
i  I e 1
III
Vi
i  I e 1
III
93
Ni
Vi
Ni
F(f , q , σ )   (  e [HT f t  BT σ 0 ]dV e   e HT qt dS2e ) .
t
t
0
i  I e 1
Vi
s2
94
References
95
Cai, Y., and Y. Yin (1997), The Finite Element Method and Program Design of
96
97
Thermo-elastic Problem (in Chinese), Beijing, Peking University Press, 348p.
Hu, C., Y. Zhou, and Y. Cai (2009a), A new finite element model in studying
98
earthquake triggering and continuous evolution of stress field, Science in China
99
Series D: Earth Sciences, 52(7), 994-1004, doi:10.1007/S11430-009-0082-3.
100
Hu, C., Y. Zhou, Y. Cai, and C.-Y. Wang (2009b), Study of earthquake triggering
101
in a heterogeneous crust using a new finite element model, Seismological
102
Research Letters, 80(5), 795-803, doi:10.1785/gssrl.80.5.799.