微纳米尺度下材料性能多尺度模拟方法进展

, u, u˙ a
; rj, r˙j [11].
FEAt MAAD
,
MD/FEM
, Abraham
.
2.
125
1
1
HFE/MD = 2 HFE + 2 HMD
(2)
1
1
HTB/MD = 2 HTB + 2 HMD
(3)
,
,
,
,
KFE
=
3 2 (Natom
−
Nmesh)kBT +
1
(4)
KFE + 2 NmeshkBT
,
,
,
,
7 CADD
[33]
2
:/
“” .
.
“”
( 8).
8 CADD
.
,
[13]
, CADD
.
,
8
,
E
=
(b ⊗ m)sym d
+
(m ⊗
b)(b 2d2
⊗
m)
(18)
,
“sym”
,m
, b Burgers , d
.
,
.
,
. CADD
[35]. , CADD
, .
.
CADD
2.2
,
,
,
,
.
129
Nloc
, nβ = 1.
, Ortiz
Cauchy-Born ,
, ai ,F
ε
ai = F · Ai
(24)
, Ai
.,
F
,
Nnl
M
Etot,h ≈
nβ Eβ uh +
neαΩ0ε (Fe)
β=1
e=1
, Ω0
.
(25)
2
QC
,
:/
, ,
,
“ ”(ghost force).
,
, Ortiz
Nrep
2
:/
binding, TB) (FEM)
(MD) .
,
. MAAD
MD/TB ,
MD
FEM
TB
FEM/MD
Htot = HFE(ua, u˙ a) + HFE/MD(rj , r˙j, ua, u˙ a)+ HMD(rj , r˙j ) + HMD/TB(rj , r˙j)+ HTB(rj , r˙j ) (1)
QC
,
, Phillips
QC
[51].
,
.
, MD
[55]
,
. , ,
,
,
.
,
(atomic-scale finite element method, AFEM).
AFEM ,
,
,
[52].
,
lor ,
Tay-
K ·u = P
, K = ∂Utot
,
∂x∂x x=x(0)
, x(0) F
, .
. (27)
2
.
,
[59]
2.1.4
BSM
Belytschko[28,29]
2004
, Xiao
,
Belytschko [30]
.
(bridging sub-domain, BD)
,
BSM ,
6,
.
BSM
,
.
3
,
.
6 BDM
,
,
[29]
,
3
,
H = HM(x, p) + HC(v, F , u) + HBD
(11)
,
M, C BD
=
A(x1, x2, · · ·
, xN )
i=1
(28)
Diαβ
=
1 mi
∂2Φ ∂xiα∂xiβ
(29)
, xi
i
,N
, Φ0
,k
,
THale Waihona Puke Baidu
,
, Di
i
,
Diαβ .
(28)
,
,
.
MST
,
,
,
.,
(CST). CST
MST
,
,
,
Nc
A = Aα,
α=1
(30)
, Aα
α
.
, Nc ,
,
,
(
3
),
,
,
Nc
E(uk, u˙ k) = 3(N − Nnode)kT +
1 2
(Mjku˙ j · u˙ k + uj · Kjk · uk)
j,k
(20)
,k
,T
,N
, Nnode , Mjk
, u˙ j uj
, Kjk
,
−1
Mjk = m
fjμfμk
μ −1
Kjk =
fj μ Dμ−κ1 fκk
μκ
(21) (22)
2.2.1 CGMD
,
MAAD, BSM, BDM
,
MAAD
Broughton
Rudd 1998
——
(coarse-grained molecular dynamics, CGMD) [36,37]. CGMD
,
.
CGMD
,
,
,
.
,
uj = fjμuμ
μ
, uj
, uμ
.,
.
Gibbs
,
(19) , fjμ
[46,47]
[48].
9 (a) QC
. , local) ; , , ,
(a) ,
, , , . ,
,
, ,
( 9). ,
, (non-
(local) . ,
Nnl
Nloc
Etot,h ≈
nβEβ uh +
nαEα uh
β=1
α=1
(23)
; (b) , uh
(b)
[41]
,
,
, Eβ Eα
, Nnl , nβ nα
Nc
A = Aα =
Φαh + 3NαkT ln
α=1
α=1
= A (x1, x2, · · · , xM )
|Dαh|1/6 kT
(31)
, Φαh α
h
α ,M
h
, Nα
, Dαh
α
, xi
i
. , M < N ,CST
.
CST
,
,
. CST MST
CST MST
,
,
.
CST MST
,
, CST
θm−m ,n−n (t − τ )×
m ,n 0
(q0,m ,n (τ )
−u¯0,m ,n (τ )−
R0,m ,n (τ ))dτ
(13)
(11) , ”,
,
f0im,mp,n(t)
“
,
. f0im,mp,n(t) θm−m ,n−n (t − τ ).
MD FE
, .
R0,m,n(t)
“
”,
,x ,P
(27) , Utot
,
.
[52,53]
[54]
[55,56] ,
.
AFEM,
,
.
132
AFEM [55],
. , AFEM ,
, ,
, .
2.2.4
/
,
,
MD
,
;
, QC
,
.
,
,
[57∼59]
(molecular stastical thermo-
dynamics, MST)
(cluster statisti-
, CGMD
CGMD ,
.
,
. (2)
.
. (4)
2011
41
,
,
.
2.2.2 QC
QC ,
(quasi-continuum, QC)
,
Tadmor, Ortiz, Phillips [38,39]
,
2001
Knap Ortiz[40]
. 1996
.
[41].
QC
,
,
,
. QC
[42,43]
,,
[44,45]
41
2
2011 3 25
ADVANCES IN MECHANICS
Vol. 41 No. 2 Mar. 25, 2011
∗
1,2
1 3
5
3
2 4
1,†
4
5
,
100190
,
100049
,
23002
,
100871
,
100191
. ,
.
. ,
,
,
,
,
,
1
21
,
., [1], .
.
,
,
,
MD
(
),
[3].
106/s
VFE
=
3
2
(Natom 1
−
Nmesh)kB T +
(5)
VFE + 2 NmeshkBT
, kB , Nmesh
Abraham [14]
,T
, Natom
.
MAAD
, MAAD
(1)
,
,
,
,
. (2) 3
(
)
,
,
2 FEAt
MAAD
[13]
, FE/MD
,
,
,
,
“”
.
,
.,
,
, MAAD
“ ” .“ ”
973
(2007CB814803)
† E-mail: why@lnm.imech.ac.cn
, .
(hierar-
124
2011
41
Clementi ,
[4,5]
. , .
.
, ,, . ,
. ,
,
,
,
/ .
. MD
, .
,
,
.
,
“
”
, .
, , .
. ,
.
2.1
, ,,
.
2.1.1 FEAt
MAAD
(bridging scale method, BSM).
—— ,
u(x, t) = u¯(x, t) + u (x, t)
(6)
, u (x, t) u (x, t)
.
,
u = N d, d
,N
. u (x)
, u (x, t)
u (x, t)
.
,
.
q,
u = Nd + q − Pq
(7)
,P
,
.
QC (QC-FNL).
, QC-
FNL
,
,
Cauchy-Born .
,
QC
,
[40].
,
(
)
(
),
“ ”.
QC-
FNL
,
.
131
QC
.
, Phillips
QC
QC
QC
MC
Carlo, QCMC),
.
MD
MD ,
,
.
QC
2.2.3
Cauchy-Born ,
[49,50] 1999
. Shenoy
,
(quasicontinuum Monte
,x
,v
,u
.
,p ,F
,
128
,
HBD = (1 − α)HM + αHC
,α , l(x)
l(x)
,α= , l0
l0
α=1
Ω
C 0
−
Ω0int
α = [0, 1]
Ω0int
α=0
Ω0M − Ω0int.
, ,
(12)
, (13) (14) (15) ,
gI = {giI } =
NJ (XI )uiJ − diI = 0 (16)
cal thermodynamics, CST)
/
(hybrid molecular/cluster stastical thermody-
namics, HMCST)
.
, ,
, .
, ,
. MD , (
,
MST
, ) .
, ,
,
,
,
.
,
,
2011
41
N
A = Φ0 + 3kT ln
|Di|1/6 kT
,
“ ”,
.
,
,
,
, 2.1.2
1994 ,
. (3) ,
–
TB
.
[15,16]
,
,
,
.
,
( 3). ,
,
.
/
,
4∼5
.
.
. ,
126
3
–
.
,
, ,
.
. . , .
. ,
Weertman [17]
,
, ,,
2.1.3
. –
Wagner Liu [18−24]
[16]
. ,
2011
41
BSM u(x, t)
(22) , Dμκ
.
(20)
,
.
, N = Nnode, , CGMD
MD.
,
, CGMD
MD,
.
,
,
“
”
,
“”
. Broughton
CGMD,
,
[37].
130
CGMD ,
, CGMD
,
π/ (Nmaxa)( ,
D
,a ),
(1) ,
,
,
(3)
fjμ
, ,
[36], , Nmax
,
. k=
“ ”.
) , ,
.
, (partitioned-
,
;
.
,
, ,
1
FEAt
[7]
1998 , Abraham Broughton [10,11]
FEAt
.
,
,
/
/
“” ,
, MAAD
(macroscopic, atomistic, ab intio dynamics).
”[12].
,
“
MAAD
(tight-
(coupled atomistic and discrete disloca-
tion plasticity, CADD).
(1)
; (2)
,
, (3)
,
.
CADD
,
,
7
.
Giessen Needleman
[34].
,
(7
I,
σ˜ ε
)
(7
II,
σˆ εˆ
)
.
,
,
,,
,
.
, MAAD
, CADD
,
J
λ,
HLBD = HBD + λTg = HBD + λTI gI
I
(17)
HLBD
.
α
λ,
,
,
,
,
[29].
/
. BSM ,
,
,
.
,
,
,
.
BSM ,
,
,
,
,
.
2011
41
2.1.5 CADD
MAAD, BSM, BDM
,
,
(
),
.
, ) .
[31∼33]
( , , Curtin, Shilkrot, Miller
20 90
, Kohlhoff Gumbsch [7,8]
FEAt(finite element combined with
atomistic modelling)
.
(I )
(IV )
(II
III ),
1.
;
;
“”
,
Kroner [9]
.
,
,
.
2
domain multi-scale method)[6]. (
Φ˜(uh) = Φ(uh) − fαG · uhα
α=1
, fαG
“”
,
α Φ˜
(26) ,
.
QC
,
QC
,
.
QC
,
QC
. , Tadmor
QC
(http://www.qcmethod.com/),
QC
,
QC
.
QC
,
, , QC
; QC
. QC
( ).
,
;
,
QC
.
QC
“”
,
Knap Ortiz
QC
[40],
. ,
,
.
,
, MD
,
MD
,
,
.
,
,
(MD) ,
. MD ,
[2]. ,
/
/
,
/
,
,
109
[3],
.,
chical) [4].
, (concurrent) ,
: 2010-04-29,
: 2011-02-17
∗
(109332011, 10772181, 10732090, 10772012, 10721202),
,
BSM
.
,
Lagrangian
,
MAq¨ = f (u)
(8)
M d¨ = N Tf (u)
(9)
(8)
(9)
N Tf (u)
.
,
,
,
,
.
,
,
,
,
,
4.
4 BSM
,
[20]
2
:/
BSM
,
(8)
–
.
u¨ l,m,n (t) =
MA−1Kl−l ,m−m ,n−n ×
l ,m ,n
ul ,m ,n (t) +MA−1fle,xmt,n (t)
(10)
, MA
, l, m, n ; l ,m ,n
Kl−l ,m−m ,n−n
fle,xmt,n
.
Fourier ,
,
MD
; ; Laplace
FE
MAq¨ (t) = f (t) + f0im,mp,n(t) + R0,m,n(t)
(11)
M d¨ = N Tf (u)
(12)
,
t
f0im,mp,n(t) =
,
.
BSM
, , Piola-Kirchoff
, BSM
.,
[25].
,
(1)
θ(t − τ )
,
θ(t − τ ) ,
(corner effect), BSM θ(t − τ )
,
,
,
BSM ,
. (2) BSM
,
.
BSM
,
127
,
[26]
,
,
“
”
.
,
scle method, HMM), .
[27]
(heterogeneous multi-