AIAA-2002-1655
A Computational Framework for a Multiscale Continuum-AtomisticHomogenization Method
Peter W. Chung and Raju R. Namburu
Computational and Information Sciences DirectorateU.S. Army Research Laboratory
AMSRL-CI-HAAberdeen Proving Grounds, Maryland 21005-5067 USA
Abstract
Computational approaches for problems spanning multiple length scales continue to attract attentionfrom many di�erent disciplines and researchers. The present investigation focuses on the asymptoticexpansion homogenization (AEH) approach: a weakly convergent formulation based on asymptoticsfor treating heterogeneous continuum media possessing distinct but disparate micro and macro lengthscales. While prior e�orts have exclusively considered length scales separated by one or two ordersof magnitude, a homogenization-based procedure enables the consideration of much larger length scaledi�erences. In this work, atoms are used at the local scale and are represented energetically using aclassical potential. At the global scale, the phenomenological model of hyperelasticity is employed. Thetwo-scale homogenization method establishes coupled self-consistent variational equations in which theinformation at the atomistic scale, formulated in terms of the Lagrangian sti�ness tensor, feeds thematerial information to the quadrature points of the hyperelastic �nite element equations. At the localscale, the atomistic equations are solved using lattice statics. To demonstrate the method, analyticalresults for one-dimensional problems are shown.
Introduction
A large amount of interest has recently focused onthe multiscale problem involving atoms and con-tinua. This stems from two reasons: the widely ac-cepted view that many e�ects on the continuum ger-minate at the atomic level and the ever-improvingcapabilities of computers.
Methodologies for linking a continuum to anatomistic domain can be found in the literature asearly 1971 [1]. An excellent review of some of thesemethods can be found in Cleri et al. [2]. The generalidea common to many of these investigations startswith kinematically linking an atomistic domain witha continuum or �nite element domain.
A seminal development in �nite element-basedcomputational methods is the so-called quasicon-tinuum method [3], which draws upon the compu-tational convenience of the �nite element methodthrough mesh adaption and numerical integration,with the needed atomic resolution provided by theembedded atom method [4]. A similar approach wasdeveloped in [5] for semiconducting materials. Butrecognizing that classical potentials only account forionic degrees of freedom, they added another scalea step lower for the electronic degrees of freedomthrough the so-called tight-binding approximation.
Multiscale methods such as these have tradition-ally been focused on studying localized regions of in-terest. For example, the applications to which thesemethods have been applied involve small sets of dis-locations or cracks and limited analyses of their mu-tual interactions. The localized regions on whichthese simulations are run typically span, at most,several microns because of the bottleneck imposedby a direct interface between the continuum regionand atomistic region. To ensure compatibility, kine-matic constraints are used to tie together the equa-tions and disparate length scales across this inter-face. Implied in this is that the size of the continuum�nite elements must be driven down to lengths com-parable with atom spacings. This intrinsically re-stricts the size of the continuum and leads to smalleroverall dimensions of the problem which can only beovercome by larger use of computer resources whendealing with problems with larger dimensions.
The asymptotic expansion homogenizationmethod has been widely studied by applied math-ematicians for many years. Numerous texts onthe basic theory can be found in the literature,for instance, by Bensoussan et al. [6], Sanchez-Palencia [7], and Bakhvalov and Panasenko [8]. Itsbasic idea is the use of asymptotics to computesolutions to problems in which the microstructure
This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.
1American Institute of Aeronautics and Astronautics
43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Con22-25 April 2002, Denver, Colorado
AIAA 2002-1655
This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.
exerts some physical in uence on the e�ective re-sponse of the macroscale but is \visually" unresolv-able compared to macroscale dimensions. As such,it circumvents the need to de�ne an interface and,instead, is primarily suited for problems in whichatomistic phenomena are periodically distributedbut asymptotically small.
In this work, a computational method for homog-enization of the atomistic problem is presented. Us-ing two concurrent domains, one for the macroscalecontinuum domain and one for the periodic atomicscale domain, self-consistent sets of equations are de-rived. As a direct consequence of the use of the ho-mogenization method, atoms in arbitrary con�gura-tions and structures of unlimited size are permitted.
In the following sections, the methodology of theapproach is described. First, the continuum for-mulations are shown. Then, through the de�ni-tion of the assumptions inherent to the homogeniza-tion method, coupled equations for macro and mi-croscales are derived. The emphasis of this work isin the solution of the microscale equation and thecorresponding details on how this is done are pre-sented. Finally, simple one-dimensional illustrativeanalytical results are shown.
Methodology
This section describes the kinematics, stress de�ni-tions, and linear momentum conservation laws forthe continuum. It forms the basis to which the ho-mogenization method is applied.
The deformation gradient which maps point Xto point x is de�ned by
F =@�
@X=
@x
@X= r0x; (1)
where r0 signi�es the gradient taken with respect tothe undeformed body V . The right Cauchy-Greenstrain tensor is de�ned by
C = FTF; (2)
and the Green strain tensor is de�ned by
E =1
2(C� I): (3)
The material representation for the conservation oflinear momentum is de�ned by
r0 �P+ f0 = 0; (4)
where P is the �rst Piola-Kircho� stress tensor andf0 is the body force per unit of undeformed volume.In rate form, it is given by
r0 � _P+ _f0 = 0: (5)
Then the principle of virtual work states that
ZV
�r0 � _P
��udV +
ZV
_f0 � �udV = 0; 8�u; (6)
where �u is the virtual displacement. Then, inte-grating by parts in the conventional way gives
ZV
_P : r0�udV =
Z@V
_t0 � �udA+
ZV
_f0 � �udV:(7)
From the classical de�nition of hyperelasticity [9]the �rst Piola-Kircho� stress is de�ned as
P =@W@F
; (8)
and the �rst Lagrangian elasticity tensor [10] as
C =@2W@F@F
=@P
@F: (9)
Note that C exhibits major symmetry but lacks mi-nor symmetries. A relationship is needed betweenstress and strain. From equation (9), one can seethat in hyperelastic materials, P is related to F
through
_P = C _F; (10)
where
_F =@ _u
@X=
@v
@X; (11)
and where _u = v denotes the velocity.
Substituting equation (10) into (7) and using(11) yields
ZV
r0�u : C : r0vdV =
Z@V
_t0 � �udA+
ZV
_f0 � �udV; 8�u;(12)
This is the virtual work expression standard in con-tinuum mechanics. The two-scale approach is de-scribed next. It is devised so that traditional �niteelement continuum equation from (12) can be sep-arated into two coupled equations representative ofthe respective coarse and �ne scale.
HomogenizationThe homogenization framework enables the weakcoupling of the continuum to the atoms. By takingthe limit of the time-independent asymptotic expan-sion parameter " ! 0, the weak convergence prop-erties of the scheme can be exploited in order to
2American Institute of Aeronautics and Astronautics
decouple the length scales. At the �ne scale, the do-main contains only atoms with periodic conditionsprescribed on the boundary, and all atom displace-ments are measured relative to a �xed point in thelocal frame of reference. From classical examplesof continuum mechanics of composite materials [11],this enables the method to account for mutual inter-actions of heterogeneities or, in this case, atomisticdefects.
The homogenization method is based on the as-sumption that two scales exist - a coarse scale and a�ne scale. Coordinates in the coarse material scaleare X = (X1; X2; X3), and those in the �ne mate-rial scale are Y = (Y1; Y2; Y3). Likewise, the spatialcoordinates are the lowercase analogues. The twoscales are related by the scale parameter
Y =X
": (13)
Therefore, the ratio of scales is assumed to remainthe same before and after deformation. The aimis to obtain two sets of coupled equations. Theasymptotic series assumption decomposes the dis-placements as
u(X) = u[0](X) + u[1](X) (14)
= u[0](X) + "u[1](Y); (15)
where u[0] represents the displacement at the coarsescale and u[1] represents the perturbed displace-ments due to inhomogeneity at the �ne scale. Squarebrackets denote the order of the term in the asymp-totic series. The representation of the total displace-ment at the �ne scale is given by Takano et al. [12]as
1
"u(X) = umicro(Y)
= F(u[0](X))Y + u[1](Y):
(16)
The variable X in equation (16) is a �xed value withrespect to Y. That is, the deformation gradient ofa point in the coarse scale gets mapped onto a �nescale grid. This point is typically a quadrature pointin a �nite element sense.
The time derivatives are analogous to equations(15) and (16). They are given as
_u(X) = v(X)
= v[0](X) + "v[1](Y); (17)
_umicro(Y) = vmicro(X)
= F(v[0](X))Y + v[1](Y): (18)
Substituting equations (15) and (17) into (12)yields
ZV
rX
��u[0](X) + "�u[1](Y)
�
: C : rX
�v[0](X) +"v[1](Y)
�dV
=
Z@V
��u[0](X) + "�u[1](Y)
�� _t0dA
+
ZV
��u[0](X) + "�u[1](Y)
�� _f0dV;
8�u[0]; �u[1]:(19)
Note that by use of the chain rule and equation (13),
rX�(X;Y) = rX�+@Y
@XrY �
= rX�+1
"rY �:
(20)
Therefore,
rX
�u[0](X) + "u[1](Y)
�
= rXu[0](X) +rY u
[1](Y):(21)
Using equation (21) in (19) and taking the averageover Y gives
ZV
1
jY jZY
�rX�u
[0](X) +rY �u[1](Y)
�
: C :�rXv
[0](X) +rY v[1](Y)
�dY dV
=
Z@V
��u[0](X) + "�u[1](Y)
�� _t0dA
+
ZV
��u[0](X) + "�u[1](Y)
�� _f0dV;
8�u[0]; �u[1]:(22)
Then, in the limit as " ! 0, equation (22) is trueonly if the following two equations are satis�ed,
1
jY jZV
ZY
rX�u[0](X)
: C :�rXv
[0](X) +rY v[1](Y)
�dY dV
=
Z@V
�u[0](X) � _t0dA+
ZV
�u[0](X) � _f0dV;
8 �u[0](23)
3American Institute of Aeronautics and Astronautics
1
jY jZV
ZY
rY �u[1](Y)
: C :�rXv
[0](X) +rY v[1](Y)
�dY dV = 0;
8 �u[1]:(24)
The solution of equation (23) can be determined us-ing conventional computational techniques such asthe �nite element method. However, this is onlytrue if C and v[1] are known. In the homogenizationmethod, the determination of C and v[1] is pred-icated on the successful solution of equation (24).It therefore stands by reasoning that equations (23)and (24) are coupled and must be solved concur-rently due to their mutual dependence on v[1] andv[0]. The focus of the present work is to formu-late equation (24) in terms of atomistic variablesso that its solution, v[1], will feed the appropriateatomistic information into the continuum in equa-tion (23). This will be described in detail in thenext section.
But to complete the general AEH derivation, as-sume v[1] is known. Then the remaining task is toreformulate the global scale boundary value prob-lem so that it accepts this term in the appropriateway. The key distinction between this investigationand conventional continuum formulations, such ashyperelasticity, is the conspicuous presence of v[1],a �ne-scale/atomistic quantity. In its present form,equation (23) does not recognize v[1] as an atom-istic quantity. Therefore some modi�cation must bemade to ensure that the global equation accommo-dates the atomistic input from v[1].
Assuming again that v[1] is now known, substi-tute the de�nition for the �rst Lagrangian elasticitytensor from equation (9) into equation (23). Thisgives
1
jY jZV
ZY
rX�u[0](X) :
@2W@F@F
:�rXv
[0](X) +rY v[1](Y)
�dY dV
=
Z@V
�u[0](X) � _t0dA
+
ZV
�u[0](X) � _f0dV; 8 �u[0]:(25)
Then, using the de�nition of F in equation (1) andassuming a �rst-order Taylor series representationfor the time derivative gives the following identity
@2W@F@F
: rY v[1] =
@2W@F@q
� v[1]: (26)
Substituting into equation (25) �nally gives,
1
jY jZV
ZY
rX�u[0](X) :
@2W@F@F
: rXv[0](X)dY dV
=
Z@V
�u[0](X) � _t0dA+
ZV
�u[0](X) � _f0dV
� 1
jY jZV
ZY
rX�u[0](X) :
@2W@F@q
� v[1](Y)dY dV;
8 �u[0];(27)
where there is now a double contraction on rX�u[0]
and a single contraction on v[1] in the last expres-sion of equation (27). The solution to this equa-tion yields v[0] that takes into consideration the ef-fect of the atoms from v[1]. It is noteworthy thatthe last term is zero when the energy distributionover Y is constant, i.e., when the atom arrangementforms a perfect lattice. This reduces the problem toa classical harmonic approximation where the �rstLagrangian elasticity tensor is assumed to model thematerial behavior. In equation (27), the last termserves as a corrective force in regions of highly en-ergetic atoms, i.e., nonlocal regions, to account fordefects and lattice inhomogeneities.
Atomistic Considerations
Distinct and distinguishable atoms are assumed toreside in the local level cell. By the Cauchy-Bornrule [13], at a point X, F(u[0]) is assumed to givethe energy minimizing con�guration of the atoms.For the sake of simplicity in the derivation and with-out loss of generality, the atoms are assumed to bearranged in a perfect lattice. The positions of theatoms Y are given from the lattice coordinates mby
Y(m) =mei : m 2 L;L = Z3;Z� N; (28)
where ei are the primitive translation vectors and Nis the integer multiple of atoms contained in the unitcell. To avoid confusion in notation, atom labels arenoted in parentheses henceforth and are not subjectto the conventional summation rules associated withindicial notation. The displacement of the atoms arede�ned by
q(m) : m 2 L: (29)
Upon deformation, the new positions of the atomsare given by
y(m) = Y(m) + q(m): (30)
4American Institute of Aeronautics and Astronautics
The deformation gradient is de�ned by
F =@y
@Y: (31)
The vector separating two atoms i and j in the ref-erence con�guration is given by
R(ij) = Y(j) �Y(i); (32)
where Y(j) denotes the position of atom j and Y(i)
the position of atom i. The vector separating twoatoms in the deformed con�guration is given by
r(ij) = y(j) � y(i): (33)
Then the Cauchy-Born rule states that for a systemfree of external loads and at an equilibrium con�g-uration R(ij), upon a deformation F, the con�gura-tion of atoms which minimizes the energy is givenby
r(ij) = FY(j) �FY(i)
= FR(ij):(34)
For the energy associated with the deformationof the atoms, we use the so-called Potential II param-eterization of the Terso�-Brenner potential [14, 15].It takes the form
W =1
N[Eb(Y + q)�Eb(Y)] ; (35)
where W is the energy density of the frozen system,N is the number of atoms, and Eb is the bindingenergy given for a pure carbon system by
Eb(r) =Xi
Xj(>i)
�VR�r(ij)
�� �BVA�r(ij)
��;(36)
�B =1
2
�B(ij) +B(ji)
�; (37)
VR(r) =f(ij)(r)D
(e)
(S � 1)e�p2S�(r�R(e)); (38)
VA(r) =f(ij)(r)D
(e)S
(S � 1)e�p
2S�(r�R(e)); (39)
f(ij)(r) =
8>>><>>>:
1; r < R(1)
12
n1 + cos
h�(r�R(1))(r�R(2))
io;
R(1) < r < R(2)
0; r > R(2)
(40)
B(ij) =
241 + X
k(6=i;j)G(�(ijk))f(ik)
�r(ik)
�35��
;(41)
G(�) = ao
�1 +
c2od2o� c2od2o + (1 + cos �)2
�;(42)
with the constants given in Table 1.
Given that the energy can be written as a func-tion of the atom displacements, equation (24) can beexpressed in a form conducive to atom representa-tions. Equate v[1] to the rate of atom displacementand attempt to solve the equivalent form
@
@YjCijkl @v
[1]k
@Yl= �@Cijkl
@Yj
@v[0]k
@Xl
(43)
under periodic boundary conditions. The solutionto equation (43) is found as the zero of @R in theequation
@R = Kv[1] �D � r0v[0](x); (44)
where K is the N �N Hessian and is given by
K =@2W@q@q
; (45)
where q is the vector of atom displacements of size3N (in three dimensions) and D is a third orderunsymmetric tensor that is obtained from the �rstderivative of the Euler-Lagrange equation with re-spect to the local deformation gradient given by
D = � @2W@q@F
: (46)
The size of D depends on the dimensionality of theproblem. In three dimensions, it can be expressed asan N�9 matrix, where 9 corresponds to the numberof independent components of F. Note that equa-tion (46) is the transpose of the expression found inthe last term of equation (27). So, whereas D mapsthe macroscale gradients into the microscale solu-tion of v[1] in equation (44), DT in the last term ofequation (27) maps the microscale information intothe macroscale solution of v[0].
By virtue of the Cauchy-Born hypothesis and thezero temperature assumption, equation (44) is a lin-ear equation. That is, neither K nor D depend onv[1]. However, if necessary, a mixed strategy can beused to incorporate molecular dynamics to computethe true energy minimizing atom con�guration atthe �ne scale.
From an implementation point of view, the peri-odic boundary condition is being applied to discreteatoms. Therefore, the �rst derivatives of displace-ments have no clear meaning. An instructive dis-cussion of boundary conditions applied to atomisticproblems is available by Allen and Tildesley [16]. Byvirtue of the one-dimensional nature of the problemshere, no signi�cant sophistication is needed in apply-ing the boundary conditions. However, in multidi-mensional problems more careful considerations for
5American Institute of Aeronautics and Astronautics
the unit cell de�nition are warranted. These prob-lems are of a related investigation currently under-way.
Additional Details
In this section, additional details regarding the cal-culation of equations (45) and (46) are shown. Theapproach shown originally by Tadmor et al. [3] fromthe so-called quasicontinuum method is adoptedhere. By extensive use of the chain rule from calcu-lus, the appropriate derivatives of the classical po-tential can be expressed with respect to either thecontinuum deformation gradient or the atomic dis-placements. These are summarized as
K =@2W
@q(n)@q(m)
=1
N
@2Eb
@q(n)@q(m)
=1
N
Xa
Xb
@2Eb
@r(a)@r(b):
�@r(a)
@q(n) @r(b)
@q(m)
�
(47)
where a; b = fij; ik; jkg. Likewise,@2W
@q(m)@F=
1
N
Xa
Xb
@2Eb
@r(a)@r(b):
�@r(a)
@F @r(b)
@q(m)
�
(48)
where it is implied that the fundamental variablesin the Terso�-Brenner potential are the vectors con-necting atoms i, j and k. The bond order term canbe constructed from the normalized dot product ofany two of these vectors. In addition, extensive usewas made of the following identities:
@r(ij)@r(ij)
=@��r(ij)��@r(ij)
=r(ij)
r(ij); (49)
@r(ij)
@q(m)=
8<:�I m = iI m = j0 m 6= (i; j)
; (50)
where I is the 3�3 identity tensor for a 3-D system.
Example Problems
To illustrate the method, 1-D analytical examplesare presented. The Terso�-Brenner potential isagain used to represent the energetics of a 1-D single-species molecular wire composed of carbon atoms.The objective here is to solve equations (43)-(44)for v[1] and demonstrate the method for two cases.In the �rst case, the wire is assumed to be defect-free
and the solution is expected to return zero pertur-bations. This solution is a special case of the secondexample in which a defect is inserted into the wire.The behavior of the perturbation variable v[1] is thenstudied as a function of the defect size and density.
Example 1: 1-D Molecular Wire without DefectOne atom is assumed to comprise the periodic unitcell, but to account for the e�ects of triples, two \�c-titious" atoms are assumed to extend beyond theboundaries of the cell on each side as illustratedin Figure 1. Periodic conditions apply at the cellboundaries. The equilibrium lattice constant for thewire was found to be ro = 1:86868�A using a simplebisection method to �nd the minimum of the energy.
The following two conditions stem from the 1-Dassumption and the equilibrium lattice constant,
� = �;
R(1) < r < R(2):(51)
This simpli�es the expressions shown earlier. Theresulting Hessian for three arbitrary collinear atoms(i; j; k) is obtained as
hK(ijk)
i=
264K(ijk)11 K(ijk)
12 K(ijk)13
K(ijk)21 K(ijk)
22 K(ijk)23
K(ijk)31 K(ijk)
32 K(ijk)33
375 ; (52)
where K(ijk)mn = K(ijk)
nm and the terms are de�ned by
K(ijk)11 =V
00
R � �BV00
A + ao�V0
AB1+ 1
�
(ij) f0
(ik)
+�
2(� + 1)VAB
1+ 2�
(ij)
�aof
0
(ik)
�2(53)
+ao�
2VAB
1+ 1�
(ij) f00
(ik);
K(ijk)12 =� V
00
R + �BV00
A � ao�
2V
0
AB1+ 1
�
(ij) f0
(ik); (54)
K(ijk)13 =� ao�
2V
0
AB1+ 1
�
(ij) f0
(ik)
� �
2(� + 1)VAB
1+ 2�
(ij)
�aof
0
(ik)
�2(55)
� ao�
2VAB
1+ 1�
(ij) f00
(ik);
K(ijk)22 =V
00
R � �BV00
A ; (56)
K(ijk)23 =
ao�
2V
0
AB1+ 1
�
(ij) f0
(ik); (57)
K(ijk)33 =
�
2(� + 1)VAB
1+ 2�
(ij)
�aof
0
(ik)
�2
+ao�
2VAB
1+ 1�
(ij) f00
(ik): (58)
6American Institute of Aeronautics and Astronautics
Upon assembly of the two unique pairs (31,12) andtheir associated triples (123,132) in Figure 1, the �-nal assembled Hessian of the global system is givenby the matrix,
[K] =24 K11 K12 K13
K21 K22 K23
K31 K32 K33
35 ; (59)
which is assembled through the operation,
[K] =(i;j;k)G(m)
(i;j;k)G(n)
hK(ijk)
i= Kmn; (60)
whereFis the addition operator over all unique pair
and triple combinations of (i; j; k) and (m) and (n)are displacement degrees of freedom for each atom.In equation (60), [K] is symmetric once again, andits components are obtained in detail for the prob-lem shown in Figure 1 as follows:
K11 = K(123)11 +K(315)
22 +K(241)33 ; (61)
K12 = K(123)12 +K(241)
13 ; (62)
K13 = K(123)13 +K(315)
12 ; (63)
K22 = K(123)22 +K(241)
11 ; (64)
K23 = K(123)23 ; (65)
K33 = K(123)33 +K(315)
11 : (66)
This constitutes the sti�ness matrix K in equation(44).
The next step is to calculate the right-hand sideof equation (43), which is equivalent to calculatingD and multiplying by the global rate of the defor-mation gradient. Using equations (46) and (48), theright-hand side for three collinear atoms (i; j; k) isobtained as
nD(ijk)
o=
8><>:D(ijk)1
D(ijk)2
D(ijk)3
9>=>; ; (67)
where the components are de�ned by
D(ijk)1 =R(ij)
�V
00
R � �BV00
A
�+�R(ik) � R(ij)
�
��ao�
2V
0
AB1+ 1
�
(ij) f0
(ik)
�
+R(ik)
��
2(� + 1)VAB
1+ 2�
(ij)
�aof
0
(ik)
�2
+ao�
2VAB
1+ 1�
(ij) f00
(ik)
�; (68)
D(ijk)2 =�R(ij)
�V
00
R � �BV00
A
�
�R(ik)
�ao�
2V
0
AB1+ 1
�
(ij) f0
(ik)
�; (69)
D(ijk)3 =�R(ij)
�ao�
2V
0
AB1+ 1
�
(ij) f0
(ik)
�
�R(ik)
��
2(� + 1)VAB
1+ 2�
(ij)
�aof
0
(ik)
�2
+ao�
2VAB
1+ 1�
(ij) f00
(ik)
�: (70)
The assembly operation for the right hand side vec-tor is
fDg =(i;j;k)G(m)
nD(ijk)
o= Dm; (71)
yields the right-hand side of the global system givenby,
fDg =8<:
D1
D2
D3
9=; ; (72)
where the components are
D1 = D(123)1 +D(315)
2 +D(241)3 ; (73)
D2 = D(123)2 +D(241)
1 ; (74)
D3 = D(123)3 +D(315)
1 : (75)
Under the assumption of a 1-D perfect lattice, wehave R(ij) = R(ik) and, consequently, D1 = 0. Then,we can satisfy the periodicity condition and the rigid
body constraint by setting v[1](2) = v
[1](3) = 0. The so-
lution is therefore
v[1](1) = v
[1](2) = v
[1](3) = 0: (76)
In light of equation (76), the last term in equation(27) is zero, which shows that the material prop-erties are obtained simply from equation (9). Thisresult shows that in a defect-free lattice, the homog-enization method correctly returns the degeneratesolution for a homogeneous material. The next sec-tion shows an example in which a defect causes aninhomogeneous energy distribution, leading to a sit-uation where homogenization return a non-trivial re-sult.
Example 2: 1-D Molecular Wire With DefectConsider the problem shown in Figure 2, where thecenter atom is now displaced by a distance L from itsoriginal energy minimizing con�guration. This dis-placement of the center atom constitutes a defect.
7American Institute of Aeronautics and Astronautics
With this change, the key sti�ness matrix termin equation (61) is now
K11 =V00
R(12)� �B(12)V
00
A(12)+ ao�V
0
A(12)B
1+ 1�
(12) f0
(13)
+�
2(� + 1)VA(12)
B1+ 2
�
(12)
�aof
0
(13)
�2
+ao�
2VA(12)
B1+ 1
�
(12) f00
(13) + V00
R(13)
� �B(13)V00
A(13)+ ao�V
0
A(13)B
1+ 1�
(13) f0
(12)
+�
2(� + 1)VA(13)
B1+ 2
�
(13)
�aof
0
(12)
�2
+ao�
2VA(13)
B1+ 1
�
(13) f00
(12);
(77)
and likewise, equation (73) is now
D1 =R(12)
�V
00
R(12)� �B(12)V
00
A(12)
�
+�R(13) �R(12)
��ao�2V
0
A(12)B
1+ 1�
(12) f0
(13)
�
+R(13)
��
2(� + 1)VA(12)
B1+ 2
�
(12)
�aof
0
(13)
�2
+ao�
2VA(12)
B1+ 1
�
(12) f00
(13)
�
�R(13)
�V
00
R(13)� �B(13)V
00
A(13)
�
� �R(12) �R(13)
��ao�2V
0
A(13)B
1+ 1�
(13) f0
(12)
�
�R(12)
��
2(� + 1)VA(13)
B1+ 2
�
(13)
�aof
0
(12)
�2
+ao�
2VA(13)
B1+ 1
�
(13) f00
(12)
�;
(78)
where R(12) = ro � L and R(13) = ro + L. Then,solving equation (44) under periodic boundary con-ditions using the same approach as in Example 1gives
v[1]1 =
D1
K1
@v[0]
@x; v
[1]2 = v
[1]3 = 0: (79)
The v[1]=r0v[0] solution as a function of L=ro is
shown in Figure 3. As expected, the solution hassymmetry about the origin and grows asymptoti-cally larger as the size of the defect (L) grows closerto the cut-o� radius of the potential. Larger defectsizes are avoided intentionally due to the noncon-vex structure of the energy well associated with theTerso�-Brenner potential. Studies of larger defectsare expected to lead to unphysical discontinuities inthe perturbation velocity (v[1]) as a function of L due
to discontinuous second derivatives of the atomisticenergy. This is attributable to the construction ofthe empirical potential in equations (36)-(42), whichis suited, by design, for systems where nearest neigh-bor atoms, even in defect regions, are within thecut-o� radius R(2).
It is also noteworthy that arbitrary defect den-sities can be treated by appropriate modi�cation ofthe unit cell. In most cases, one can tailor the de-sired density by increasing the size of the unit celland performing the summations and the assemblyof the atomistic discrete equations over more atoms.Figure 4 illustrates this idea for the 1-D carbonmolecular wire.
Numerical experiments show that as the size ofthe unit cell increases, the perturbative displacementhas a sharp discontinuity at the defect. Figure 5shows this nonlocal behavior as the number of atomsincreases. The problem is of a single defect in molec-ular wires of increasing size. The defect magnitude isheld �xed at L=ro = 0:01. The nonlocal discontinu-ity of the perturbative velocity qualitatively agreeswith traditional displacement jumps that occur atdislocation cores. The discontinuity indicates that
the material property at the defect ( @2W
@F@F) is mod-
i�ed by the last term in equation (27), an amountproportional to v[1] that serves as a correcting forcefor the nonlocality.
Although the primary details of the methodhave been demonstrated in these two examples, themethod can be extended to consider the multiscaleproblem shown in equation (27) for more generalcases involving self-consistent solutions with equa-tion (44). This paper is presently restricted to an-alytical 1-D examples where the results, as shownin the previous examples, can be reported indepen-dent of the macroscale solution v[0] and reserve morecomplex cases in higher dimensions for a separatenumerical investigation.
Concluding Remarks
Linking atomic scale physics with continuum scalephenomena is of keen interest in the study of fail-ure, fracture, and reliability of engineering struc-tures. The e�ects that dominate the failure at thecontinuum scale typically initiate and evolve fromthe atomic scale. Despite numerous promising meth-ods in the literature that are capable of linking scalesup to the micron level, structures at the meter scaleand beyond have only begun to be studied. To thisend, we have attempted to address this issue by ex-ploiting the weak convergence properties of homog-enization to devise a scheme which passes atomisticinformation to very large continuum scales.
We have applied the Cauchy Born rule to theatom scale by assuming that the con�guration of
8American Institute of Aeronautics and Astronautics
atoms used to solve for the perturbation displace-ment is indeed the minimizing con�guration of theatomistic energy. We have not considered themethod in conjunction with a molecular dynam-ics routine, i.e., various strategies of minimizingthe atomistic energy by quenching through arti�cialtemperature decrease or solving Newton's equationsto minimize the interatom forces.
For this work, the speci�c case of the Terso�-Brenner Type II potential was considered. But theprinciples and the general equations can be extendedto any potential, provided the appropriate deriva-tives can be obtained. Typically for classical sys-tems, onerous tensor algebra and calculus are re-quired. It is common practice, however, to usemore computationally e�cient procedures to obtainderivatives numerically [16].
The aim of this paper was to develop an approachby which atomistic physics can be embedded into acontinuum formulation for large scale systems. Thisgoal has been achieved by formulating a consistentset of equations involving a classical atomistic po-tential at the �ne scale and general �nite strain anddeformation elasticity at the coarse scale. Simple1-D analytical results were shown to illustrate theapproach and its features. More realistic multiax-ial problems in two and three dimensions for moredetailed validation are the subjects of ongoing work.
Acknowledgments
Partial support by the U. S. Army Research Labo-ratory Director's Research Initiative (DRI) Programunder award number FY01-CIS-27 and the NationalResearch Council Resident Research AssociateshipProgram are gratefully acknowledged.
References
[1] J. Sinclair. Improved atomistic model of a bccdislocation core. Journal of Applied Physics,42:5321{5329, 1971.
[2] F. Cleri, S. R. Phillpot, D. Wolf, and S. Yip.Atomistic simulations of materials fracture andthe link between atomic and continuum lengthscales. Journal of the American Ceramic Soci-ety, 81:501{516, 1998.
[3] E. B. Tadmor, M. Ortiz, and R. Phillips. Qua-sicontinuum analysis of defects in solids. Philo-sophical Magazine A, 73(6):1529{1563, 1996.
[4] M. S. Daw and M. I. Baskes. Semiempiri-cal, quantum-mechanical calculation of hydro-gen embrittlement in metals. Physical ReviewLetters, 50(17):1285{1288, 1983.
[5] J. Q. Broughton, F. F. Abraham, N. Bernstein,and E. Kaxiras. Concurrent coupling of lengthscales: Methodology and application. PhysicalReview B, 60(4):2391{2403, 15 July 1999.
[6] A. Bensoussan, J. L. Lions, and G. Papanico-laou. Asymptotic Analysis for Periodic Struc-tures. North-Holland, 1978.
[7] E. Sanchez-Palencia. Non-homogeneous Me-dia and Vibration Theory: Lecture Notes inPhysics, volume 127. Springer-Verlag, 1980.
[8] N. Bakhvalov and G. Panasenko. Homogeniza-tion: Averaging Processes in Periodic Media.Kluwer, 1989.
[9] Y. C. Fung. Foundations of Solid Mechanics.Prentice-Hall, Inc., Englewood Cli�s, New Jer-sey, 1965.
[10] J. E. Marsden and T. J. R. Hughes. Mathemat-ical Foundations of Elasticity. Dover Publica-tions, Inc., New York, 1983.
[11] T. Mura. Micromechanics of Defects in Solids.Martinus Nijho� Publishers, Dordrecht, theNetherlands, 1987.
[12] N. Takano, Y. Ohnishi, M. Zako, andK. Nishiyabu. The formulation of homoge-nization method applied to large deformationproblem for composite materials. InternationalJournal of Solids and Structures, 37:6517{6535,2000.
[13] J. L. Ericksen. Phase Transformations and Ma-terial Instabilities in Solids, pages 61{77. Aca-demic Press, Inc., 1984.
[14] J. Terso�. Empirical interatomic potential forcarbon, with applications to amorphous carbon.Physical Review Letters, 61(25):2879{2882, 19December 1988.
[15] D. W. Brenner. Empirical potential for hydro-carbons for use in simulating the chemical vapordeposition of diamond �lms. Physical Review B,42(15):9458{9471, 15 November 1990.
[16] M. P. Allen and D. J. Tildesley. Computer Sim-ulation of Liquids. Clarendon Press, Oxford,1987.
9American Institute of Aeronautics and Astronautics
Table 1: Parameters for Terso�-Brenner potential.R(e) 1.39 �A
D(e) 6.0 eVS 1.22� 2.1 �A� 0.5
R(1) 1.7 �AR(2) 2.0 �Aao 0.00020813c2o 3302
d2o 3:52
3
Cell boundaries
1 45
Fictitious atoms Fictitious atoms
2
Figure 1: Unit cell of the 1-D carbon molecular wire. The atoms are labeled by identifying numbers.
3
Cell boundaries
2 45
Fictitious atomsFictitious atoms
1
L
Figure 2: Unit cell of one-dimensional carbon molecular wire with periodic defect.
xfig (L/ro)
xfig
(v1 /d
v0)
-0.05 0 0.05-15
-10
-5
0
5
10
15
[0]
[1]
0
L / r
∆
o
/v
v
Figure 3: Distribution of v[1]=r0v[0] solution as a function of the defect size L.
10American Institute of Aeronautics and Astronautics
Defect
Figure 4: Larger wires of atoms in perfect arrangement around the defect region decreases the defect density.
0 0.25 0.5 0.75 1-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
[1]
[0]
0v
x / L
31 atoms21 atoms
11 atoms
UC
51 atoms
∆ v/
1001 atoms
101 atoms
Figure 5: Distribution of v[1]=r0v[0] along unit cell length for varying number of atoms (L=ro = 0:01).
11American Institute of Aeronautics and Astronautics
