Strain rate and adhesive energy dependent viscoplastic damage modeling for
nanoparticulate composites: Molecular dynamics and micromechanical
simulations
B.J. Yang1, H. Shin2, H. Kim2, H.K. Lee1,a)
1
Department of Civil and Environmental Engineering,
Korea Advanced Institute of Science and Technology (KAIST),
291 Daehak-ro, Yuseong-gu, Daejeon 305-701, Republic of Korea
2
Graduate School of Energy Environment Water Sustainability (EEWS),
Korea Advanced Institute of Science and Technology (KAIST),
291 Daehak-ro, Yuseong-gu, Daejeon 305-701, Republic of Korea
a)
Electronic mail: leeh@kaist.ac.kr
Parameters U1 and U2 in Eq. (2).
By carrying out the lengthy algebra, the parameters U1 and U 2 can be expressed as
U1  2H 21  H1  31  21  ,
U2  2H2 1
(S.1)
where 1 and m1 denote the Lame’s constants of NPs, and the components H1 and H2 are defined as
follows:
H1 
2 2 1   1 1  2 2 
,
1  2 2 1  31  2 2 
H2 
2
1  2 2
(S.2)
 1    2S21  S1  31  21   ,
 2   2 S2 1
(S.3)
M1   2S21  S1  31  21   1 ,
M 2  2 S2 11
(S.4)
in which
1
2
with
with
1 
10   0 1  3S1 1   0  1  0 
3  3S1  2 S 2 1   0    0   2 S2  1  0   0 
1  0
1 
2  2 S 2  1  0   0 
(S.5)
where  q is the bulk modulus of q-phase (q=0, 1), and S1 and S2 are the parameters of
Eshelby’s tensor with the interface effect which are given in Yang et al.8
 
VE
Derivation process and results of the effective viscoelastic constitutive equation C*
E
First, the effective elasticity tensor C* for the NP-reinforced composites yields

C*E  *Eij kl  *E  ik jl   il jk

(S.6)
with
*E  0  31  2 2  33  2 4  1  20  1  3 


1
(S.7)
*E  20   2   4  
2

Here, 0 and 0 denote Lame’s constants of the matrix, and
L 2 q- 1 =
{2b x
q 2 q- 1
}
+ a q (1- 2x2 q ) f q
(2x2q - 1)(3x2q- 1 + 2x2q - 1)
, L 2q =
b qf q
,
(q = 1, 2)
(S.8)
x2 q = 2S2b qf q ,
( q = 1, 2)
(S.9)
1- 2x2 q
in which
{
}
x2 q- 1 = f q 2S2a q + S1 (3a q + 2b q ) ,
where a 1 and b1 are listed in Eq. (S.5), and a 2 and b 2 are given in
2 
S1
,
 3S1  2S2  1 2S2  1
2 
1
4S2  2
(S.10)
Based on the superposition principle,6,15 the effective viscoelastic constitutive equation can
be derived by combining the effective elasticity tensor and the Laplace transformation (LT).
Hence, the effective elasticity tensor in the transformed domain (TD) CTD
reads
*

TD
TD
CTD
ik jl  il jk
*  *  ij kl  *
with

(S.11)



TD
TD
TD
TD
*TD  0TD 31TD  2TD
2  3 3  2 4  1  2 0 1   3



(S.12)
1
TD
*TD  20TD  TD
2  4  
2

with
L
TD
2 q- 1
{2b
=
TD TD
q
2 q- 1
x
}
+ a qTD (1- 2x2TDq ) f q
(2x2TDq - 1)(3x2TDq- 1 + 2x2TDq - 1)
, L
TD
2q
=
b qTDf q
1- 2x2TDq
,
(q = 1, 2)
(S.13)
with
{
}
x2TDq- 1 = f q 2S2a qTD + S1 (3a qTD + 2b qTD ) ,
x2TDq = 2S2b qTDf q ,
(q = 1, 2)
(S.14)
in which

TD
1

10TD   0TD 1  3S1 1   0TD  1  0TD 
3  3S1  2 S2  1   0TD    0TD   2 S2  1  0TD   0TD 
1TD 
1  0TD
2  2S2  1  0TD   0TD 
 2TD 
 S1
,
 3S1  2S2  1 2S2  1
(S.15)
 2TD 
1
4S2  2
TD
TD
where the LT of the L𝑎́ me constants  0 and  0 are defined as
0TD 
0 0 s
,
0  s0
 0TD 
0 0 s
0  s0
(S.16)
where  0 signify the viscosity of the matrix, and s denotes the Laplace constant. By taking the
inverse Laplace transform of CTD
* , the effective viscoelasticity tensor of the nanocomposites
CVE
* can be obtained by taking the direct inverse Laplace transform, or explicitly described as

VE
VE
CVE
ik jl  il jk
*  *  ij kl  *
in which

(S.17)
*VE
 t 
exp   0 
 0   2 S Z S       3S    

2
3
3
1 4
5
6 
 2 1 2 1

4
*VE
(S.18)
 t 
exp   0 0 7
 0 

2tZ1S 2 1  1 
in which


Z1t 0 2
,
 0  Z10  2 S2 1 1  1  

 
1  exp 
 t 0 
 1
 0 
 2  exp 


t 0 0  Z 2  2 S 2 1  1 


3  exp
 0  0 Z 2  2 S 2 1  1  1  3S1  2 S 2 1  1  

 
 4  6t 0 Z1S 2  3S1  2 S 2 1  1 Z 2  2 S 2  2 S 21 
(S.19)
5  1  4  2   2  31  3  2 
and
Z1  1  2S2 1 1 ,
Z2  1  3S1 1  1 ,
Z3  1  2  1
(S.20)
with
 t 0 
  1 ,
 0 
1  exp 
 2   2 1  1    2   1  1 1  2
2

3  4Z3  2 S2 2  3 0  20 1  1 ,
 t 0  
  1   1  1 2
 0  
 4  2  1  2exp 

5   21  2 11  3 2  2   2  21  51  3
2
(S.21)
6   6 1  3  5  1  6  2  2  2  2 1  41  3
7  6  2 0 1  1  3 0  2  2  31 1  1   2
2
8  3 0  2  2 12  2  1  5  ,
and

 2
 t 0 
  2 1  1 1
 0 

9  3exp 




 t 
 2  201  3 0 exp  0   3  1

 0 


 3  9S12 1  1  2011  3  S2  40 2  7  
 1  40 2  3 0  2  2 1  12  2   5  ,
 4  S2  40   2  1 4  9  2  2   8  ,
 6  2  01  23  1 ,
(S.22)
 5  2  23  S2 2  405  3 0 6 1  1 
 7  4 Z3 2 S2 2 1  1  11  2 S2 2
where t signify the arbitrary time, which is defined as t    t  /  . Here,  denotes the strain
rate.
The fourth-rank tensor T in Eq. (10).
The fourth-rank tensor T can be defined as
T = T 1dij dkl + T 2 (dik d jl + dil d jk )
(S.23)
with
2
T 1 = (3P1 + 2 P2 ) T 1 + 2 P1 (3P1 + 4 P2 )T 2 ,
T 2 = 4 P22T 2
(S.24)
where the fourth-rank tensor Pijkl = P1dij dkl + P2 (dik d jl + dil d jk ) are
P1 =
- (c 1 + c 3 )
,
(2c 2 + 2c 4 + 1)(3c 1 + 2c 2 + 3c 3 + 2c 4 + 1)
P2 =
1
4c 2 + 4c 4 + 2
(S.25)
in which
c 2 q- 1 = f q {(1- 3S1 - 2S2 )a q - 2S1b q },
c 2 q = b qf q (1- 2S2 ),
(S.26)
(q=1, 2)
Furthermore, the fourth-rank tensor Tijkl = T1dij dkl + T2 (dik d jl + dil d jk ) are
f1 é 2
24m0 45 J12 P 12 + 60 J12 P 1P 2 - 4P 22 (P 3 - 5 J12 )
5 êë
f
1
T 2 = 1 (144m02 P 22P 3 )+
5
2
T1 =
}ùúû-
{
1
3
(S.27)
in which
P1 =
m0 - m1
k 0 - k1
+
6m0 {m0 + 2 (m1 - m0 )S2 } 9k 0 {k 0 (3S1 + 2S2 - 1)- 2k 1 (3S1 + 2S2 )}
m0 - m1
P2=
,
4m0 {m0 (2S2 - 1)- 2m1S2 }
with
2
1
2
2
2
1 0
P 3 = J - 72 J1 J 2 + 15 J + 4 J n (7n0 - 10)
(S.28)
J1 =
G(1 + n1 )
{
}
3 (k + 2)(1- 2n1 )+ G(1 + n1 )
r
s
- G éê3G(7 + 5n1 )- 4 {k sr (1 + n 0 )+ msr (2n 0 - 1)+ 3}(10n1 - 7)ù
ú
ë
û
J2 =
3h11
(S.29)
where G= m1 / m0 and n q is the Poisson’s ratio of q-phase. k sr , msr , and h11 are interface
parameters, which are given in Yang et al.8