Theoretical and Applied Fracture Mechanics 50 (2008) 243–247

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Theoretical and Applied Fracture Mechanics

journal homepage: www.elsevier .com/ locate/ tafmec

Modeling and simulation of a single crystal based on a multiscale field theory

J.D. Lee a,*, Y.P. Chen b

a Department of Mechanical and Aerospace Engineering, The George Washington University, Washington, DC 20052, USAb Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL 32611, USA

a r t i c l e i n f o a b s t r a c t

Article history:Available online 24 August 2008

Keywords:Multiscale field theoryFinite element analysisNano/micro scaleAtomistic systemSingle crystalLarge strainMagnesium oxide

0167-8442/$ - see front matter � 2008 Elsevier Ltd. Adoi:10.1016/j.tafmec.2008.07.016

* Corresponding author.E-mail addresses: jdlee@gwu.edu (J.D. Lee), ypchen

This work presents a multiscale field theory and its applications in modeling and simulation of atomisticsystems. The theoretical construction of the multiscale field theory is briefly introduced. A single crystalis discretized into finite element mesh as if it is a continuous medium. However, each node is a represen-tative unit cell, which contains a specified number of discrete and distinctive atoms. Material behaviors ofa given atomistic system at nano/micro scale, subject to the combination of mechanical loadings, electro-magnetic field and temperature field, can be obtained through numerical simulations. Sample problem ofa single crystal (magnesium oxide) subject to large strain tension/compression has been solved to dem-onstrate the advantage and applicability of this multiscale field theory.

� 2008 Elsevier Ltd. All rights reserved.

1. Introduction a crystal lattice. Essentially the regularity displayed by a crystal lat-

A multiscale field theory has been advanced [1–5] for the con-current atomic-continuum modeling of materials/systems. Theterm ‘‘multiscale” refers to theory and simulation of material prop-erties and behavior across length scales from the atomistic to mac-roscopic. Continuous local densities of fundamental physicalquantities in atomistic systems are derived. By decomposing atom-ic motion/deformation into homogeneous lattice motion/deforma-tion and inhomogeneous internal atomic motion/deformation, andalso decomposing momentum flux and heat flux into homoge-neous and inhomogeneous parts, field description of conservationlaws at atomic scale has been formulated. As a result of the formu-lation, a field representation of atomic many-body dynamics is ob-tained, and time-interval averaged quantities can be solved. Sincethe conservation equations obtained in [1,2,4] are valid at theatomic scale, the field theory can reproduce time averaged atomictrajectories and can be used to investigate phenomena and proper-ties that originated at atomic scale. Since it is a field theory formu-lated in terms of time interval averaged quantities, it is expected tobe computationally more efficient than atomic-level moleculardynamics simulation (MD), and can be applied to simulate phe-nomena at larger length and time scales.

2. Multiscale field theory

Crystalline solids are distinguished from other states of matterby a periodic arrangement of the atoms; such a structure is called

ll rights reserved.

2@ufl.edu (Y.P. Chen).

tice is that of a three-dimensional mesh which divides space intoidentical parallelepipeds. Imagine a number of identical atomsplaced at the intersections of such a mesh; then we have what isknown as a simple lattice (or Bravais lattice).

Microscopic dynamic quantities are functions of phase-spacecoordinates (r,p), i.e., the positions and momenta of atoms. Formulti-element systems, there is more than one atom in the unitcell. Thus, there results

r ¼ fRka ¼ Rk þ Drkaj k ¼ 1;2;3; . . . n; a ¼ 1;2;3; . . . ; mg;p ¼ fmaVka ¼ maVk þmaDvkaj k ¼ 1;2;3 . . . n; a ¼ 1;2;3; . . . mg;

ð1Þwhere the superscript ka refers to the ath atom in the kth unit cell;ma is the mass of the ath atom; Rka and Vka are the position andvelocity vector of the ka atom, respectively; Rk and Vk are the posi-tion and velocity of the mass center of the kth unit cell, respectively;Drka and Dvka are the atomic position and velocity of the ath atomrelative to the mass center of the kth unit cell, respectively. The lo-cal density of any measurable phase-space function a(r, p) can gen-erally be defined as

Aðx; ya; tÞ ¼Xn

k¼1

Xm

n¼1

afrðtÞ;pðtÞgdðRk � xÞ~dðDrkn � yaÞ

� Aaðx; tÞ: ð2Þ

The first delta function in Eq. (2) is a localization function that pro-vides the link between phase space and physical space descriptions.It can be a Dirac d-function [6], or a distribution function [7], such as

dðRk � xÞ ¼ p�3=2l�3e�jRk�xj=l2 : ð3Þ

244 J.D. Lee, Y.P. Chen / Theoretical and Applied Fracture Mechanics 50 (2008) 243–247

The field descriptions of the conservation equations and the constitu-tive relations (the interrelations between field quantities) are foundto be independent of the choices of the localization function[1,2,4,7]. The second delta function in Eq. (2) identifies ya to Drka. Itcan be easily proved that the following normalization condition holdsZ

dðRk � xÞ~dðDrka � yaÞd3x ¼ 1 ðk ¼ 1;2;3; . . . ;nÞ

ða ¼ 1;2; . . . ; mÞ: ð4Þ

Also, it is obvious that the distribution function satisfies the follow-ing identity as the Dirac delta function does

odðRka � xÞoRka ¼ � odðRka � xÞ

ox: ð5Þ

Most current MD applications involve systems those are either inequilibrium or in some time-independent stationary state; whereindividual results are subjected to fluctuation; it is the well-definedaverages over sufficiently long time intervals that are of interest. Tosmooth out the results and to obtain results close to experiments,measurements of physical quantities are necessary to be collectedand averaged over finite time duration. Therefore, in deriving thefield description of atomic quantities and balance equations, it isthe time-interval averaged quantities that are used, and the time-interval averaged (at time t in the interval [t, t + Dt]) local densityfunction takes the form

Aaðx; tÞ ¼ hAai � 1Dt

Z Dt

0Aaðx; t þ sÞds

¼ 1Dt

Z Dt

0

Xn

k¼1

aðrðt þ sÞ;pðt þ sÞÞdðRk � xÞ~dðDrka � yaÞds:

ð6Þ

The mathematical representation of conservation equations for mass,linear momentum and energy at atomic scale has been analyticallyobtained in terms of averaged field quantities [1,2,4]. In this work,we are only concerned with ‘one-way coupling’ with temperatureand electromagnetic fields, i.e., the temperature and electromagneticfields are given as functions of space and time. Then the relevant gov-erning equations are just the balance laws for linear momentum:

o

ot½�qað�vþ D�vaÞ� ¼ rx � ½�ta � �qa�v� ð�vþ D�vaÞ�

þ rya � ½�sa � �qaD�va � ð�vþ D�vaÞ� þ �ua; ð7Þ

where the time interval averaged mass density �qa, linear momen-tum �qað�v þ D�vaÞ, homogeneous atomic stresses �ta

ðkinÞ þ �taðpotÞ and

inhomogeneous atomic stresses �saðkinÞ þ �sa

ðpotÞ, and body force densityare defined as

�qaðx; tÞ �Xn

k¼1

madðRk � xÞ~dðDrka � yaÞ* +

; ð8Þ

�qað�v þ D�vaÞ �Xn

k¼1

maðVk þ DvkaÞdðRk � xÞ~dðDrka � yaÞ* +

; ð9Þ

�taðkinÞ � �

Xn

k¼1

ma eV k � eV kadðRk � xÞ~dðDrka � yaÞ* +

; ð10Þ

�saðkinÞ � �

Xn

k¼1

maD~vka � eV kadðRk � xÞ~dðDrka � yaÞ* +

; ð11Þ

�taðpotÞ � �

12

Xn

k;l¼1

Xm

n;g¼1

ðRk � RlÞ � FknBðk; n; l;g; x; yaÞ* +

; ð12Þ

�saðpotÞ � �

12

Xn

k;l¼1

Xm

n;g¼1

ðDrkn � DrlgÞ � FknBðk; n; l;g; x; yaÞ* +

; ð13Þ

�ua �Xn

k¼1

ukadðRk � xÞ~dðDrka � yaÞ* +

; ð14Þ

where Fkn is the interatomic force acting on the kn atom; uka is thebody force acting the ka atom;eV ka � Vka � �v � D�va; eV k � Vk � �v; D~vka � Dvka � D�va; ð15Þ

Bðk; n; l;g; x; yaÞ �Z 1

0d½Rkkþ Rlð1� kÞ � x�

� ~d½Drknkþ Drlgð1� kÞ � ya�dk: ð16Þ

It is worthwhile to note that, with the atomistic definitions of inter-atomic force and the potential parts of the atomic stresses, one has

rx � �tapot þrya � �sa

pot ¼ �f a; ð17Þ

where �f a is the interatomic force density acting on the ath atom inthe unit cell located at x.

3. Finite element formulation

From now on, we work with time interval averaged quantities.For simplicity, we drop the bars on top of the quantities if it doesnot cause ambiguity. The governing equations, Eq. (7), can nowbe rewritten as

qa €ua ¼ rx � ta þrya � sa þ fa=V� þ ua=V�; ð18Þ

where ua is the displacement vector of the ath atom; V* is the vol-ume of unit cell; and from now on ta and sa are the kinetic part ofthe homogeneous and inhomogeneous atomic stresses associatedwith the ath atom, respectively. For a single-element atomic sys-tem, Cheung and Yip [8] and Haile [9] gave the following definitionsfor kinetic stresses t̂ij and temperature bT :

t̂ij ¼ �XN

l¼1

m~vli~v

lj

* +,bV ; ð19Þ

3NkBbT ¼ XN

l¼1

m~vli~v

li

* +; ð20Þ

where kB is the Boltzmann constant; bV is the volume that the Natoms occupy. In consistent with [8,9], for multi-element atomicsystem, it follows that

taij ¼ �hma~við~vj þ D~vaj Þi=V�; sa

ij ¼ �hmaD~vai ð~vj þ D~va

j Þi=V�; ð21Þ

3kBTa ¼ hmað~vi þ D~vai Þð~vi þ D~va

i Þi: ð22Þ

At high temperature and within harmonic approximation, all modeshave the same energy [10]. Following this thinking, we make thefollowing assumption relating the energies of acoustic modes andoptic modes as [11]:

ðn� 1ÞhM~vi~vii ¼Xt

a¼1

maD~vai D~va

i

* +; ð23Þ

where M �Pt

a¼1ma is the total mass of a unit cell and t is the num-ber of atoms per unit cell. We further assume that the temperaturewithin a unit cell is uniform. This implies Ta = T(x, t) andrya � sa ¼ 0.Now the governing equations, Eq. (18), can be rewritten as

ma€ua ¼ V�r � ta þ fa þ ua; ð24Þ

with taij ¼ �cakBTdij=V�; saij ¼ �ð1� caÞkBTdij=V�, and ca �ma/M.

J.D. Lee, Y.P. Chen / Theoretical and Applied Fracture Mechanics 50 (2008) 243–247 245

In the following we study the cases of which the temperaturefield is constant in space, including T = 0 as a special case. To dem-onstrate the derivation of the finite element formulation in detail,we rewrite Eq. (24) as

ma _vðk;aÞ ¼ fðk;aÞ þ uðk;aÞ; ð25Þ

where f(k,a) is all the interatomic force acting on the ath in the kthunit cell; u(k,a) is all the body force due to external field acting onthe ath in the kth unit cell. Therefore, for a system with pair poten-tial, one may rewrite the governing equation, Eq. (25), as

ma _vðk;aÞ ¼Xn

l¼1

Xt

b¼1

fðk;a; l;bÞ þ uðk;aÞ

ðk ¼ 1;2;3; . . . ; n; a ¼ 1;2;3; . . . ; tÞ; ð26Þ

where f(k,a; l,b) is the interatomic force acting on the ath atom ofthe kth unit cell due to the interaction with the bth atom of the lthunit cell. The inner product of Eq. (26) with virtual displacementdu(k,a) leads to

ma _vðk;aÞ � duðk;aÞ ¼Xn

l¼1

Xt

b¼1

fðk;a; l;bÞ � duðk;aÞ

þ uðk;aÞ � duðk;aÞ: ð27Þ

Sum over all a and k, we obtain the weak form asXn

k¼1

Xt

a¼1

ma _vðk;aÞ � duðk;aÞ ¼Xn

k¼1

Xn

l¼1

Xt

a¼1

Xt

b¼1

fðk;a; l;bÞ � duðk;aÞ

þXn

k¼1

Xt

a¼1

uðk;aÞ � duðk;aÞ:

ð28Þ

It is notice that Eq. (28) can also be expressed asXn

k¼1

Xt

a¼1

ma _vðk;aÞ � duðk;aÞ ¼Xn

k¼1

Xn

l¼1

Xt

a¼1

Xt

b¼1

fðl; b; k;aÞ � duðl;bÞ

þXn

k¼1

Xt

a¼1

uðk;aÞ � duðk;aÞ:

ð29Þ

Because, for systems with pair potential

fðk;a; l;bÞ ¼ �fðl; b; k;aÞ: ð30Þ

Eq. (29) can be re-written asXn

k¼1

Xt

a¼1

ma _vðk;aÞ � duðk;aÞ ¼Xn

k¼1

Xt

a¼1

uðk;aÞ � duðk;aÞ

þ 12

Xn

k¼1

Xn

l¼1

Xt

a¼1

Xt

b¼1

fðk;a; l;bÞ

� ½duðk;aÞ � duðl;bÞ�: ð31Þ

Suppose we have Ne finite elements, each with 8 Gauss points, andwe approximate Eq. (31) asXNe

Ie¼1

X8

g¼1

Xt

a¼1

JðIe;gÞV�

fma _vðIe;g;aÞ � duðIe;g;aÞ�uðIe;g;aÞ

� duðIe;g;aÞ �12

Xn

l¼1

Xt

b¼1

fðIe;g;a; l;bÞ � ½duðIe;g;aÞ � duðl;bÞ�g ¼ 0;

ð32Þ

where J(Ie,g) is the Jacobian of the gth Gauss point of the Ieth ele-ment; f(Ie,g,a; l,b) is the force acting on the ath atom in the unit celllocated at the gth Gauss point of the Ieth element due to the inter-action with the bth atom of the lth unit cell. Notice that J(Ie,g)/V* is a

dimensionless quantity representing the number of unit cells asso-ciated with the gth Gauss point of the Ieth element; and it is a con-stant in time even in the case of large deformation. Actually, J(Ie,g)/V* being a constant in time is the law of conservation of mass at theatomic level. Through the shape functions, one may write

duðIe; g;aÞ ¼X8

k¼1

NkðgÞdUaðIe; kÞ; duðl;bÞ ¼X8

k¼1

NkðlÞdUbðl; kÞ; ð33Þ

where Nk(g) and Nk(l) are the values of the kth shape function eval-uated at the gth Gauss point of the Ieth element and the center ofthe lth unit cell, respectively; dUa(Ie,k) and dUa(l,k) are the virtualdisplacements of the ath atom of the kth node of the Ieth elementand the element where the lth unit cell is located, respectively.Now Eq. (32) can be expressed asXNe

Ie¼1

X8

g¼1

Xt

a¼1

X8

k¼1

JðIe; gÞV�

(NkðgÞ½ma _vðIe; g;aÞ � uðIe; g;aÞ� � dUaðIe; kÞ

�12

Xn

l¼1

Xt

b¼1

fðIe; g;a; l; bÞ � ½NkðgÞdUaðIe; kÞ � NkðlÞdUbðl; kÞ�)¼ 0:

ð34Þ

Let the connectivity of the finite element model be denoted as

IpðkÞ � IJKðk; IeÞ; JpðkÞ � IJKðk; lÞ; ð35Þ

where IJK(k, Ie) and IJK(k, l) are the global nodal numbers of the kthnode of the Ieth element and the element that the lth unit cell is lo-cated, respectively. Eq. (34) can then be written asXNe

Ie¼1

X8

g¼1

Xt

a¼1

X8

k¼1

JðIe; gÞV�

(NkðgÞ½ma _vðIe; g;aÞ � uðIe; g;aÞ� � dUIpðkÞ;a

�12

Xn

l¼1

Xt

b¼1

fðIe; g;a; l; bÞ � ½NkðgÞdUIpðkÞ;a � NkðlÞdUJpðkÞ;b�)¼ 0:

ð36Þ

Going through the process of assembly properly, one may rewriteEq. (36) as

XNe

Ie¼1

X8

g¼1

Xt

a¼1

X8

k¼1

X8

l¼1

JðIe; gÞV�

NkðgÞNlðgÞma _VIpðlÞ;a � dUIpðkÞ;a

¼XNp

Ip¼1

Xt

a¼1

fFIp ;a þ uIp ;ag � dUIp ;a; ð37Þ

where FIp ;a and uIp ;a are the interatomic force and body force due toexternal field, respectively, acting on the ath atom of the Ipth node;dUIp ;a is the virtual displacement of the ath atom of the Ipth node.

It is noticed that the corresponding mass matrix obtained fromEq. (37) is symmetric but not diagonal. This is referred to as thedistributed mass system. For the sake of dramatically reducingthe computational effort, one would like to have a diagonal massmatrix, which is referred to as the lumped mass system. Let theeight Gauss points move toward their corresponding nodal points.Then

NkðgÞ ! dkg ; NlðgÞ ! dlg ; ð38Þ

and Eq. (37) is reduced to

XNp

Ip¼1

Xt

a¼1

VðIpÞV�

ma _VIp ;a � FIp ;a þ uIp ;a� �

� dUIp ;a ¼ 0; ð39Þ

where V(Ip) is the volume associated with the Ipth node. Obtained isa diagonal mass matrix, which is corresponding to the lumped masssystem. It can be verified that the lumped mass of the ath atom ofthe Ipth node is simply maV(Ip)/V*. It is required that Eq. (39)

246 J.D. Lee, Y.P. Chen / Theoretical and Applied Fracture Mechanics 50 (2008) 243–247

remains valid for any arbitrary virtual displacements. The require-ment leads to

ma VðIpÞV�

_VIp ;a ¼ FIp ;a þ uIp ;a ðIp ¼ 1;2;3; . . . ;Np; a ¼ 1;2;3; . . . ; tÞ:

ð40Þ

It is seen that Eq. (40) is a set of 3tIp second order ordinary differ-ential equations, which can be readily solved by the central differ-ence method. Developed is a computer software, named as CMM(Continuum Multiscale Modeling), which is based on the multiscalefield theory and the above-mentioned finite element formulation.

4. Numerical results

Consider a rocksalt-type single crystal (magnesium oxide) sub-ject to large strain tension/compression to demonstrate the advan-tage and applicability of this multiscale field theory. The advantageof the multiscale field theory may be represented by a ratio R de-fined as

R � nNp

; ð41Þ

which is the ratio between the number of unit cells and the numberof nodes in the finite element model. Atomic units are used suchthat

mass ¼ rest mass of electron : me ¼ 9:10938188� 10�31 kg;

length ¼ Bohr radius : ao ¼ 0:529177249� 10�10 m;

electron charge : e ¼ 1:602176462� 10�19 C;

time : so ¼ mea2o=�h ¼ 2:418884326555� 10�17 s;

energy ¼ Hartree : Eh ¼ �h2=mea2

o ¼ 4:3597482� 10�18 J:

Consider a single crystal, MgO, which is a rocksalt-type lattice. Eachunit cell has 8 atoms: 4 Magnesium and 4 Oxygen. The Coulomb–

Fig. 1. The distribution of displacement uz(Bohr) on deformed shape (a) t = 1.5 � 105s(f) t = 10.0 � 105so.

Buckingham potential between two atoms, atom n and atom g,can be expressed as

Vng ¼ eneg

rng þ Ange�rng=Bng � Cng=ðrngÞ6; ð42Þ

where Ang, Bng, and Cng are material constants;rng � krngk � krn � rgk. The interatomic forces can be obtained as

fðn;gÞ ¼ � oVng

orn¼ � oVng

orng

rng

rng

¼ eneg

ðrngÞ3þ Ang

Bngrnge�rng=Bng � 6

Cng

ðrngÞ8

( )rng; ð43Þ

fðg; nÞ ¼ � oVng

org ¼ �fðn;gÞ:

The material constants used in this work for MgO are

lattice constant a ¼ 7:93684912 Bohr;

mass ðMgÞ ¼ 4:57636� 104 me; mass ðOÞ ¼ 3:01251� 104 me;

eMg ¼ 2e; eO ¼ �2e;

AMg—Mg ¼ CMg—Mg ¼ 0;

AMg—O ¼ 47:2 Hartree; BMg—O ¼ 0:56635 Bohr; CMg—O ¼ 0;

AO—O ¼ 350:88 Hartree; BO—O ¼ 0:41415 Bohr;

CO—O ¼ 53:554 Hartree Bohr6:

The atomic interaction is disregarded when rng > 90.7 Bohr.The finite element model of the specimen has 2 � 2 � 4 eight-

node elements and 45 nodes; each element is a cube with volume(95.2421894 Bohr = 12a)3; the total number of unit cell is 27648.The ratio of advantage is R = 614.4. The specimen occupies a region

o; (b) t = 3.5 � 105so; (c) t = 6.0 � 105so; (d) t = 8.5 � 105so; (e) t = 9.5 � 105so and

Fig. 2. The distribution of displacement uz(Bohr) on deformed shape (a) t = 1.6 � 105so; (b) t = 3.2 � 105so; (c) t = 4.8 � 105so; (d) t = 6.4 � 105so; (e) t = 7.2 � 105so and(f) t = 8.0 � 105so.

J.D. Lee, Y.P. Chen / Theoretical and Applied Fracture Mechanics 50 (2008) 243–247 247

0 6 x 6 24a, 0 6 y 6 24a, 0 6 z 6 48a � H. In this analysis, the mag-nitude of the time step is set to be D t = 50so and the system is as-sumed to be initially at rest with the following boundaryconditions:

uax ðx; y; 0; tÞ ¼ ua

yðx; y; 0; tÞ ¼ 0;

uaz ðx; y; 0; tÞ ¼

�0:5eHt=tr if t 6 tr ;

�0:5eH if t > tr;

� ð44Þ

uax ðx; y;H; tÞ ¼ ua

yðx; y;H; tÞ ¼ 0;

uaz ðx; y;H; tÞ ¼

0:5eHt=tr if t 6 tr;

0:5eH if t > tr :

� ð45Þ

In the case of tensile loading, the rise time and the elongation arechosen to be tr = 8.0 � 105so and e = 0.3, respectively. The historyof the distribution of displacement uz on the deformed shape isplotted in Fig. 1. Because there is no damping incorporated intothe system, vibration will never stop. But still, the phenomenon ofnecking is clearly seen. It is also observed that there is no mirrorsymmetry at the center of the specimen with respect to the x–yplane, y–z plane, or z–x plane.

In the case of compressive loading, the rise time and the short-ening are chosen to be tr = 6.4 � 105so and e = �0.25, respectively.The history of the distribution of displacement uz on the deformedshape is plotted in Fig. 2. The phenomenon of bulging is clearlyseen. The results of tension/compression also indicate the capabil-ity of the theory and the computer software to handle large-strainproblems.

5. Discussion

The multiscale field theory is the field representation of atomicN-body dynamics. It is on the same foundation as moleculardynamics (MD) simulation. In our computer software, the forcesacting on each Gauss point is calculated through a huge number,usually in the order of ten thousands, of interatomic potentialsamong various types of atoms. It may work as an alternative toMD for statistical and finite temperature properties of materials,but with significant advantages over MD in studying simulta-neously large length and time scale properties. This has been dem-

onstrated through the solving of sample problems by finiteelement methods with a large ratio of advantage. Also, it is noticedthat the theory and the computer software can be used to analyzedynamic, nonlocal, materially and geometrically nonlinear prob-lems. It is expected to be well suited for nonequilibrium phenom-ena, and especially high temperature, high pressure and highvoltage properties of real materials and devices. As long as thereis an availability of interatomic potentials, this multiscale field the-ory with the computer software works equally well for metallicmaterials and non-metallic materials such as dielectric, piezoelec-tric and ferroelectric materials, and semiconductor materials, etc.In addition to the application of material research, it may also serveto analyze the applicability of various newly developed micro-con-tinuum theories as well as classical continuum theory, and serve asa natural bridge linking atomistic and continuum views.

Acknowledgement

The authors acknowledge the support by National ScienceFoundation under Award Number CMMI-0646674.

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