SCIENCE CHINA
Technological Sciences
• Article •
December 2015 Vol.58 No.12: 1–10
doi: 10.1007/s11431-015-5953-6
Numerical simulation of the separation between concrete face slabs
and cushion layer of Zipingpu dam during the Wenchuan
earthquake
KONG XianJing1,2, LIU JingMao1,2* & ZOU DeGao1,2
1
State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116024, China;
Institute of Earthquake Engineering, School of Hydraulic Engineering, Dalian University of Technology, Dalian 116024, China
2
Appendix A
In the generalized plasticity model for granular soils, the
elasto-plastic stiffness tensor is expressed as
Dep  De 
De : n g  n : De
H  n : De : n g
(A1)
where De is the elastic stiffness tensor, n is the loading direction vector, ng is the flow direction vector, and H is the
plastic modulus.
The following surfaces are defined as illustrated in Figure
A1: the peak strength bounding surface p = M gg( )
×exp(kp), the surface of phase transformation Md =
Mgg()exp(km), and the surface of maximum stress ratio
max. Mg, kp, and km are model constants,  is a state param-
eter. In this model, the interpolation function g() is expressed as eq. (A1) [24]. The following parameters are defined as illustrated in Figure A1: , max, p,  ,  .
g (θ ) 
r
max
tc
tc
(1  c2 )2  4c(1  c2 )sin 3  (1  c2 )
2(1  c)sin 3
(A2)
Here, c = (3sincr)/(3+sincr), cr is the critical state angle
in triaxial compression. The Lode’s angle is .
The shear and bulk elastic moduli, G and K, are respectively defined as:
(2.97  e)2
G  G0 pa
1 e
 p   2 K J 
2
  + 0

2
 pa  G0 ( pa ) 
0.25
(A3)
K0 
2(1  v)
G0
3(1  2v)
(A4)
K
2(1  v)
G
3(1  2v)
(A5)
where pa is the atmospheric pressure; p' is the mean effective stress; J2 is the second invariant of the deviatoric stress
tensor; e is the current void ratio; and G0, K0, and v are model parameters.
The critical state line considering particle breakage
[25-27] is defined as
Figure A1
Normalized deviatoric plane of stress
 p 
ec  e 0  ec   ln  
 pa 
ec 
*Corresponding author (email: goliu@mail.dlut.edu.cn)
© Science China Press and Springer-Verlag Berlin Heidelberg 2015
Wp
a  bWp
tech.scichina.com
(A6)
(A7)
link.springer.com
2
Kong X J, et al.
Sci China Tech Sci
where Wp is the accumulative plastic work; e0, , a, and b
are model constants.
The dilatancy of granular soil is defined as
 Mg
dg   
M
 g

 ( d   ) exp(c0 /  )

(A8)
where M g is the critical stress ratio, equal to ON in Figure
A1;  is a model constant; c0 is a small constant (e.g.,
0.0001); and  is the stress ratio.
The plastic flow direction vector is defined in triaxial
space as
 d
1
g
n 
,
 1 d2 1 d2
g
g

T
g








On the normalized shear stress space shown in Figure B1,
two stress surfaces were defined: the “virtual” peak surface
p = Mc  k and the maximum stress surface max. A maximum stress surface f is also defined in -n stress space
(shear stress    x2   y2 , normal stress n). Mc and kp are
model constants,  is the state parameter. The following
parameters are defined as illustrated Figure B1: , max, p,
n, x, and y.
The shear and bulk elastic moduli, Ds and Dn, of the interface are respectively defined as
2
2
1  e   n     
     
Ds  Ds0
e  pa   pa  


(A9)
Dn 
The model is assumed to be non-associative [28], and the
loading direction vector is expressed in the triaxial space as
 d
1
f
nT  
,
 1 d2 1 d2
f
f

December (2015) Vol.58 No.12
(A10)

 (  /  d ) mf  d    exp(c0 /  )





4


 1 

p


2
 1    H ur

(A11)
(A12)
H ur
rv

1
dWp 
(a  bWp )2



max
(B3)
c
dWp (B4)
(a  bWp ) 2

where Wp is the accumulative plastic work; and e0, , a, b,
and c are model constants.
The plastic flow direction vector of interface is defined in
3D space as
 
y
x
nTg  
,
,
2
 d 1  d 2 1
g
g

m

 vr
 
 er  e  
ec 
max
with
 p 


1
H i  H 0 pa   exp 

p
e


e




 a
(B2)
 
ec  eτ 0  ec   ln  n 
 pa 
where mf is a model parameter.
The plastic modulus is expressed as
 Mg
H  Hi 
M
 g
Dn0
Ds
Ds0
(B1)
where Ds0 and Dn0 are model constants, pa is the atmospheric pressure, and e is the current void ratio of the interface.
The critical state line of the interface considering particle
breakage is defined as
with
 Mg
df   
 Mg

0.5


2
d g  1 
dg
(B5)
(A13)
max  r )
r ( tc
tc
  max  d
 

   
(A14)
where H0, m, , rv, and rd are model parameters; and er and
vr are the void ratio and volumetric strain at load reversal,
respectively.
The model for rockfills requires 14 material constants for
monotonic loading, plus 2 more constants for cyclic loading.
The detailed calibration procedure can be found in Liu et al.
(2014) [8].
Appendix B
The elasto-plastic stiffness tensor of soil-structure interface
is also expressed as eq. (A1).
Figure B1 Definitions of some basic concepts. (a) Stress and displacement (b) normalized shear stress space; and (c) maximum stress surface in
~n space.
Kong X J, et al.
Sci China Tech Sci

d g  rd  M c  km  max /    exp(c0 /  )
(B6)
where km, , and rd are model constants; c0 is a small constant (e.g., 0.0001); and  is the stress ratio.
The loading direction vector of interface is expressed in
3D space as
 
y
x
n 
,
,
2
 d 1  d2 1
f
f

T
with


df  rd  M f  km  max /    exp(c0 /  )
with

3
December (2015) Vol.58 No.12


d f2  1 
df2
(B7)
(B8)
where Mf is a model parameter.
The plastic modulus of the interface is expressed as
H  H0
1  n 

  1 
1   pa   p

2
 1    f h

(B9)
where H0 and fh are model parameters.
The 3D interface model requires 12 material constants
for monotonic loading, plus 3 more constants for cyclic
loading. The detailed calibration procedure can be found in
Liu and Ling (2008) [29] and Liu et al. (2014) [9].