Thermomechanical Continuum Interpretation of Atomistic …dprl2.me.gatech.edu/.../default/files/PDFpub/tmec_ijmce.pdf · 2015-05-12 · 2. THERMOMECHANICAL EQUIVALENT CONTINUUM 2.1

International Journal for Multiscale Computational Engineering, 3(2)177-197(2005)

Thermomechanical ContinuumInterpretation of Atomistic Deformation

Min Zhou∗

George W. Woodruff School of Mechanical EngineeringGeorgia Institute of Technology

Atlanta, GA 30332-0405

ABSTRACT

This paper describes a framework for obtaining thermomechanical continuum inter-pretations of the results of molecular dynamics calculations. This theory is a furtheradvancement from a pure mechanical equivalent continuum theory developed re-cently. The analysis is based on the decomposition of atomic particle velocity into astructural deformation part and a thermal oscillation part. On one hand, balance ofmomentum at the structural level yields fields of stress, body force, traction, massdensity, and deformation as they appear to a macroscopic observer. The full dynamicequivalence between the discrete system and continuum system includes (i) preser-vation of linear and angular momenta; (ii) conservation of internal, external, andinertial work rates; and (iii) conservation of mass. On the other hand, balance ofmomentum for the thermal motions as it appears to an observer moving at the struc-tural velocity yields the fields of heat flux and temperature. These quantities can becast in a manner as to conform to the continuum phenomenological equation for heatconduction and generation, yielding scale-sensitive characterizations of specific heat,thermal conductivity, and thermal relaxation time. The coupling between the struc-tural deformation and the thermal conduction processes results from the fact that theequations for structural deformation and for heat conduction are two different formsof the same balance of momentum equation at the fully time-resolved atomic level.This coupling occurs through an inertial force term in each of the two equations,induced by the other process. For the structural deformation equation, the inertialforce term induced by thermal oscillations of atoms gives rise to the phenomenolog-ical dependence of deformation on temperature. For the heat equation, the inertialforce term induced by structural deformation takes the phenomenological form of aheat source.

∗Address all correspondence to [email protected]

0731-8898/04/$35.00 c© 2005 by Begell House, Inc. 177

178 ZHOU

1. INTRODUCTION

Although molecular dynamics (MD) theoriesprovide explicit resolution of particle motionsand structures of atomic systems, continuumtheories provide approximate characterizationsof the atomic events and atomic structuralstates in an aggregate sense. Continuum mod-els often have far fewer numbers of degrees offreedom (DOF) than MD models. At the atomiclevel, structural deformation and thermal fluc-tuation make up the total atomic motion. MDmodels explicitly track the total displacement.Continuum theories, however, treat these twoparts of atomic motion separately and, to a de-gree, independently. Specifically, although thestructural deformation part is explicitly mod-eled at the continuum level, the thermal oscil-lation part is only accounted for phenomeno-logically, in the form of heat energy. Recon-ciliation of the differences in the two descrip-tions and, more importantly, the integrationof the two frameworks of analyses for cross-scale characterization are a great challenge inphysics, material science, and mechanics. Sta-tistical mechanics is an important theoreticalapproach in this endeavor. This approach,however, does not maintain strict fidelity tomolecular processes. For example, statisticalmechanics characterizations do not resolve de-formation mechanisms associated with disloca-tions at the nanoscale. They cannot be usedto analyze the generation and reaction of dis-locations and other defects. Since they do notresolve nonlocal interactions and discreteness-and nonlocality-induced length scales, they arenot suitable for analyzing nanoscale processes.Ultimately, in the context of multiscale mod-eling and characterization of material behav-ior “from the ground up” (from ab initio, firstprinciple, MD, to micro-, meso-, and macro-scopic continuum models), the formulation ofcontinuum descriptions from the results of MDsimulations must be carried out with a high

degree of faithfulness. As an important firststep, the continuumization of atomic descrip-tions should maintain time-resolved equiva-lence in momentum, work rates, mass, andkinetic energy. The nonlocality of atomic in-teractions and discreteness- and nonlocality-induced length scales must be described in thecontinuum representation. Also, the thermaloscillation part and the structural deformationpart of the atomic motion must be delineatedas one approaches higher length scales. Otherissues include scaling, reduction of numberof degrees of freedom, and extraction of phe-nomenological quantities.

An equivalent continuum (EC) representa-tion has recently been developed for discreteatomistic systems under conditions of generaldynamic deformation. A polar version of thistheory is given in [1] for particle systems ex-hibiting moment interactions as well as cen-tral force interactions. A nonpolar version ofthis theory is given in [2] for systems with onlycentral force interactions. This new theoreti-cal framework uses an explicit expression ofall atomic degrees of freedom. Not only arework-conjugate continuum stress and deforma-tion fields defined, but also specified are allother work- and momentum-preserving kineticquantities and mass distributions for the equiv-alent continuum. The continuum is equivalentto its corresponding MD system in that, at alltimes, it preserves the linear and angular mo-menta of the particle system, it conserves the in-ternal and external mechanical work rates, andit has an equal amount of kinetic energy andcontains the same amount of mass as the parti-cle system. The EC is a fully dynamic represen-tation of the MD system rather than a less rig-orous, lower-order thermodynamic representa-tion. The equivalence between the discrete sys-tem and EC holds for the entire system and forvolume elements defined by any subset of par-ticles in the system; therefore, averaging andcharacterization across different length scales

International Journal for Multiscale Computational Engineering

THERMOMECHANICAL CONTINUUM INTERPRETATION OF ATOMISTIC DEFORMATION 179

are possible and size-scale effects can be ex-plicitly analyzed. The development of the ECuses the fact that discrete atomistic and con-tinuum theories are based on the same funda-mental laws, including Newton’s laws of mo-tion, conservation of energy, and conservationof mass. This equivalent continuum theory isa nanoscale mechanical theory. Just like in theMD system represented by the EC, the kineticenergy of the EC includes both the thermal partand the structural dynamic part. In this sense,the EC is a fully faithful continuum form repre-sentation of the MD model. By faithful here, wemean time-resolved, explicit equivalence in allwork rates, kinetic energy, mass, and deforma-tion fields. Just like MD models explicitly trackthe absolute particle motion and the total ki-netic energy, the EC model does the same by ex-plicitly following the motion of the particles ina fully time-resolved manner. The EC develop-ment can be and, perhaps, should be regardedas the continuumization of discrete models thatoffers a high degree of fidelity to the discretedescription. Following this initial step, furtherdevelopment could involve scaling in time andspace. In this paper, the EC theory is extendedto account for the thermal and mechanical pro-cesses at continuum scales. The new thermo-mechanical equivalent continuum (TMEC) the-ory described herein allows scale-dependentcontinuum descriptions of material behaviorin the form of coupled thermomechanical pro-cesses of deformation and heat conduction tobe obtained from molecular dynamics simula-tions at the atomic level.

2. THERMOMECHANICAL EQUIVALENTCONTINUUM

2.1 Two Perspectives of Particle SystemDeformation

Consider a dynamically deforming system ofN particles that occupies space V and has an

envelope of surface S, as shown in Fig. 1. Attime t, particle i has position ri, displacementui, and velocity ri = ui. ri = dri/dt de-notes the material time derivative. The inter-particle force applied on particle i by parti-cle j is fij (rij) where rij = |rij | = |rj − ri|is the central distance between particle i andparticle j. Note that Newton’s third law re-quires that fij = −fji. This analysis hereadmits pairwise Lennard-Jones type potentialsand multibody interactions models such as theembedded atom method (EAM) potential [3],the modified EAM potentials [4,5], and otherpotentials [6,7].

In most cases, however, the high-frequency(thermal) part of the atomic velocity cannot beor is not explicitly resolved. Instead, the ther-mal oscillation part is phenomenologically ac-counted for as heat in continuum theories. Thisis both convenient and necessary in most mi-croscopic and macroscopic theories. In order toadmit such an analysis in the EC theory and al-low a transition to higher-scale continuum the-ories, a thermomechanical equivalent contin-uum (TMEC) theory is developed in this sec-tion. To this objective, a decomposition of theatomic displacement ui into a structural defor-mation part ui and a thermal oscillation part

j

i

fji

fij

f o

n f o

n

VS

Ve1

2 3

5

4

6

8

7

Se

j

i

fji

fij

f o

n f o

n

VS

Ve1

2 3

5

4

6

8

7

Se

FIGURE 1. Particle system and equivalent contin-uum at arbitrary size scale

Volume 3, Number 2, 2005

180 ZHOU

ui is considered. Based on this decomposition,a thermomechanical version of the equivalentcontinuum theory is developed. The structural-thermal decomposition can be written as

ui = ui + ui (1)

The fact that ui is the high-frequency part andui is the relatively “slowly” changing part dic-tates that the decomposition be carried out withregard to the time evolution of ui and not besolely accomplished in a spatial manner with-out regard to time history. This decomposi-tion is dependent on both the size scale andtime scale of the analysis pursued. Therefore,there is no unique demarcation between thetwo parts that can be universally applied un-der all scales. However, some basic features ofthe decomposition can be outlined. Specifically,this division can be obtained through a Fourieranalysis in time or in both time and space. Thismay be accomplished, for example, by obtain-ing a spectral representation of ui in the formof

ϑi(ν) =1√2π

∞∫

−∞u(t)eiνtdt

= ϑi(ν) + ϑi(ν); where

ϑi(ν) =

{ϑi(ν), if ν < νcutoff

0, if ν ≥ νcutoff

and

ϑi(ν) =

{0, if ν < νcutoff

ϑi(ν), if ν ≥ νcutoff

(2)

In the above expressions, i =√−1, ν is fre-

quency, and νcutoff is a cutoff frequency whosechoice depends on the time and size scales ofanalysis. Since thermal oscillations are typicallyat frequencies in the range of 0.5–50 THz [8] andstructural deformation occurs at much lowerfrequencies, νcutoff can be unambiguously de-termined. Inverse transforms would yield thestructural and the thermal velocities, i.e.,

˙ui(t) =1√2π

∞∫

−∞ϑi(ν)e−iνtdν and

˙ui(t) =1√2π

∞∫

−∞ϑi(ν)e−iνtdν

(3)

The spectral analysis outlined here is, per-haps, only one out of many possible approachesfor decomposition. Even though the analysis iscarried out for each atom separately in a purelytemporal manner, it can allow the spatial prop-agation of lattice waves at different frequenciesto be revealed. This is simply because the spa-tial distribution of the structural part of atomicmotions is represented in ui for each atom.

We note that thermal oscillations are usuallyassociated with no net momentum at the sys-tem level. Specifically,

N∑

i=1

miui =0,

N∑

i=1

mi˙ui =0, and

N∑

i=1

miüi =0 (4)

Under the decomposition in Eq. (1), the posi-tion of particle i can be written as

ri = ri + ui (5)

This decomposition of atomic positions de-fines the difference between the actual configu-ration (V e and Se) and the macroscopically per-ceived configuration (V e and Se) of an arbitraryvolume element. An illustration of the macro-scopically perceived configuration and the ac-tual configuration is given in Fig. 2. Althoughthe pure mechanical EC theory in Refs. [1] and[2] is formulated over the actual configura-tion, the thermal mechanical equivalent contin-uum (TMEC) here must be formulated over themacroscopically perceived configuration.

The force on particle i due to atoms or agentsthat are external to the system under considera-tion is fo

i . The total force on i is



eSeV

eV

eS Domain of thermal

oscillation

iuir

ir

1e

2e

3e

o

( )en

( )en

~

FIGURE 2. Actual and macroscopically perceivedconfigurations

fi =∑

j

fij + foi = f int

i + f exti (6)

Here, the summation is over those particlesinside the system of N particles that interact di-rectly with particle i. It is worthwhile to pointout that due to nonlocal interactions, externalforce can exist for particles both in the interiorof V and on surface S. Note that the conceptsof internal and external forces are specific tothe particular subvolume V e of V considered.So, in general, f int

i 6= ∑j fij and f ext

i 6= foi ,

except for V e = V . For example, for the vol-ume element V e illustrated in Fig. 1, the forcesbetween atoms 1 and 2 (f12, f21) as well asthose between atoms 3 and 4 (f34, f43) are in-ternal; whereas the forces between atoms 5 and6 (f56, f65) and those between atoms 7 and 8(f78,f87) are external. Note that all these forcesare internal when the system is considered as awhole.

Note that balance of momentum must be sat-isfied by a system as a whole and by any por-tion of the system. For a given portion of a sys-tem, internal forces and external forces are fun-damentally different and play different roles.Therefore, if two systems are to be equivalent

and the balance of momentum is to be satisfiedat any size scale, their internal force work rate,external force work rate, and inertial work rateat all size scales must be equivalent.

To develop a scalable representation of theequivalent continuum, consider a subvolumeelement V e ⊂ V with closed surface Se asso-ciated with a subset of M(≤ N) particles in theensemble. Assume that MS out of the M par-ticles (MS ≤ M) are on surface Se, thereforedefining it. The remaining M − MS particlesare in the interior of V e and are considered asinternal particles for V e.

The equation of motion for an atom is

f inti + f ext

i = mi

(üi + üi

)(7)

There are two distinctive yet related per-spectives toward this equation. These two as-pects can be illustrated using two different ob-servers. An illustration of this issue is givenin Fig. 3. The first observer is stationary andhas a macroscopic resolution for atomic motion.This observer uses a slower-responding oscillo-scope that is insensitive to motions at frequen-cies above νcutoff . Consequently, ui is not ex-plicitly resolved or is “invisible” at the size andtime scales analyzed by this macroscopic ob-server. To understand the motion as it appearsto the macroscopic observer, we rewrite Eq. (7)as

f inti +f ext

i +(−mi

üi

)= f int

i +f exti +

_

f i =mi üi (8)

where_

f i is the high-frequency inertial force−mi

üi. Since üi is not directly measurable

for the macroscopic observer,_

f i is not directlymeasurable. The macroscopically observed tra-jectory is ri(t) = ri0 + ui(t) (see Fig. 3).

However, the observer can “feel” the effect

of_

f i or infer its existence from the deviation of


182 ZHOU

FIGURE 3. Actual, macroscopically perceived, and macroscopically predicted trajectories for an atomicparticle

the average path ri(t) (which he or she actuallymeasures) from what is predicted by

f inti (ri) + f ext

i (ri) = miui (9)

This equation consists of only kinetic and kine-matic quantities perceivable to the observer. Inparticular, f int

i (ri) and f exti (ri) are “slowly-

varying” forces evaluated using the macroscop-ically observed positions ri; in contrast, f int

i (ri)and f ext

i (ri) are actual forces evaluated usingthe actual positions ri. Equation (9) describesthe trajectory if thermal oscillations simply donot occur (ui = 0 at all times). We must notethat, although ri(t) describes the “perceived”deformation and Eq. (9) defines the “macro-scopically predicted” deformation of an atomicsystem at longer time and higher size scales, itis Eq. (8) that governs the actual deformation onwhich the “perception” is based. The perceivedstate ri(t) is important since it defines the vol-ume V e and surface Se that form the configura-

tion required by a macroscopic analysis. Theanalysis of macroscopic deformation (balanceof momentum and balance of energy), however,must use Eq. (8).

The second perspective toward Eq. (7) isbased on the observation of an observer trav-eling with the structural velocity ˙ui of an atom.Assume that this observer uses an oscilloscopethat can resolve motions at all frequencies. Tothis observer, the particle motion then appearsto be governed by

f inti +f ext

i +(−mi üi

)=f int

i +f exti +

^

f i =miüi (10)

where^

f i is the inertial force −mi üi due to thestructural deformation. Compared to

_

f i,^

f i isslow-changing. Since üi is invisible to this mov-ing observer, he can only infer the effect (and,

therefore, the existence) of^

f i by noting the de-viation of ui from what is predicted by



f inti (ri0 + ui) + f ext

i (ri0 + ui) = miui∼(11)

In the above equation, f inti and f ext

i are, re-spectively, the internal and external forces onatom i calculated using the atomic position(ri0 + ui) as perceived by the observer mov-ing with ui . ui∼

represents the solution to the

above equation which in general differs fromthe displacement solutions to Eqs. (7-10). Equa-tion (11) consists of only kinetic and kinematicquantities perceivable to the moving observer.It describes the trajectory of the thermal oscil-lations if the structural deformation simply didnot occur (ui = 0 at all times). It is important topoint out that, just like Eq. (9), Eq. (11) is citedhere for comparison and perspective only. Itis not used to develop any equation of conse-quence in this paper.

It will become clear later that Eqs. (8) and (10)govern, respectively, the macroscopically ob-served deformation and the thermal behaviorof a material system. These two equations yieldthe coupled mechanical and thermal equationsat various scales, respectively. We will also see

that the inertial forces_

f i and^

f i give rise to thecoupling between the mechanical and the ther-mal processes. The fact that Eq. (10) is writ-ten from the perspective of an observer movingwith a particle’s macroscopic velocity ˙ui indi-cates that heat conduction analyzed here is rel-ative to material mass at the macroscopicallyperceived positions ri. Therefore, a materialtime derivative should be used later in the con-tinuum representation of internal energy. Thisalso requires the heat flux to be defined relativeto macroscopically perceived, current, atomicpositions ri (as opposed to ri or ri0).

2.2 Structural Deformation: Balance Lawsand Field Quantities

To analyze the behavior of the system asso-ciated with macroscopically observable struc-

tural deformation ui, we consider a variationof this velocity (δ ˙ui) and invoke the principleof virtual work for an arbitrary element as it ap-pears to the macroscopic observer. Under this con-dition, the equivalence can be stated as

M∑

I=1

f intI · δ ˙uI +

M∑

I=1

(f ext

I +_

fI

)· δ ˙uI

+MS∑

I=1

(1− κI)f extI · δ ˙uI =

M∑

I=1

ςImIüI · δ ˙uI

= −∫

V e

σ(e) : δD(e)dV +∫

V e

(b(e)+

_

b(e)

)· δ ˙u(e)dV

+∫

Se

t(e) · δ ˙u(e)dS =∫

V e

ρ(e) ü(e) · δ ˙u(e)dV

(12)

where V e and Se are the volume and surfaceof the element defined by the structural part ofthe atomic position vectors rI , as illustrated inFig. 3. ςI is defined on V e and Se. ς I is the frac-tion of atom I that is attributed to element V e.For periodic and amorphous structures alike, ςI

can be defined through ςeI = ϕe/

∑qe=1 ϕe, with

ϕe being the solid angle (3D) or angle (2D) sub-tended by an element and q being the numberof elements connected to atom at rI . Note thatthe delineation of f int

I and f extI is relative to V e

with η as

f intI =

M∑

J 6=I

ηIJfIJ ,

f extI =

N∑

j,(J 6=1,2,···M)

fIj + foI

(13)

Here, ηIJ is the fraction of the atomic bondthat is spatially within element V e. It pertainsto the bond between atoms I and J that areboth inside V e. In general, when atoms are ran-domly distributed (as in amorphous materials),ηIJ is determined by the dihedral angle of theelement as a fraction of the sum of such angles


184 ZHOU

(≤ 360o) of all elements associated with the par-ticular bond. Specifically, ηe

IJ = φeIJ/

∑qe=1 φe

IJ ,with φe

IJ being the dihedral angle in element(e) associated with the bond between atoms Iand I , and q being the number of elements con-nected to the bond.

The high-frequency term in the above equa-tion is

_

f I = −ς ImIüI (14)

The association of_

f I with the external forcef ext

I in Eq. (12) rather than f intI may appear to

be somewhat arbitrary at first glance. How-ever, it is strictly required by Newton’s thirdlaw and by the necessity of maintaining the bal-ance of momentum at all size scales. It is alsorequired by the stipulation that full work rateequivalence be maintained between the EC andthe molecular system. Furthermore, it is theonly treatment that allows consistent handlingof internal, external, and inertial work rates be-tween the explicitly resolved particle motion asperceived by the observer and an equivalentcontinuum representation based on the macro-scopically perceived particle motion. In partic-ular, we note that Newton’s third law requiresthat the sum of forces internal to a system or aportion of a system vanish. Otherwise, the sys-tem or the portion of that system would accel-erate by its own effects and create energy andmomentum. Here, the high-frequency inertial

force term (_

f I) cannot be associated with theinternal force because, in general

M∑

I=1

(f int

I +_

f I

)6= 0 and

M∑

I=1

rI×(f int

I +_

f I

)6= 0

(15)

The component equations resulting fromEq. (12) are

−∫

V e

σ(e) : δD(e)dV =M∑

I=1

f intI · δ ˙uI

∫

Se

t(e) · δ ˙u(e)dS =MS∑

I=1

(1− κI)f extI · δ ˙uI

∫

V e

(b(e)+

_

b(e)

)· δ ˙u(e)dV =

M∑

I=1

(κIf

extI +

_

f I

)· δ ˙uI

∫

V e

ρ(e) ü(e) · δ ˙u(e)dV =M∑

I=1

ς ImIüI · δ ˙uI

(16)

where ςI is defined relative to V e. It is impor-tant to note that the thermal velocity term inEqs. (12) and (16) accounts for the effect of theinertial force associated with the thermal fluc-tuations of the atoms on structural motion. Itacts like an “invisible” external force. Clearly,it is not associated with a neighboring (or “in-teracting”) atom, since the neighboring atomcan have totally different thermal velocity andthermal acceleration. The inertial forces of twoatoms are not equal in magnitude and oppo-site in direction. As a matter of fact, no part ofthe thermal velocities of any two neighboringatoms can be regarded as being coupled or outof mutual interactions. The inertial force termcannot be treated together with the internal (in-teractive) forces or stress. Ultimately, this is oneof the reasons why thermal motions of atomsdo not give rise to a contribution in terms ofatomic velocity to stress directly.

The continuum description of the structuraldeformation is obtained via

δ ˙u(e)(x) =M∑

I=1

NI(x)δ ˙uI (17)

where x represents continuum positions insideV (e), just like x represents positions inside V (e).NI(x) are defined using the structural part ofthe atomic positions rI over V (e). The corre-



sponding gradient of the virtual velocity fieldis

∂δ ˙u(e)

∂x=

M∑

I=1

δ ˙uI ⊗ ∂NI

∂x=

M∑

I=1

δ ˙uI ⊗ BI (18)

The equations for the stress over V e are∫

V e

σ(e) · BI dV = −f intI (19)

To obtain the traction over the surface areaSe of V e, consider a surface element ∆Se ⊂ Se

defined by L particles. The continuum virtualvelocity over ∆S is

δ ˙u(e)(x) =L∑

I=1

NI(x)δ ˙uI (20)

where NI(x) is the appropriate shape functionover ∆S. The corresponding traction field is

t(e)(x) =L∑

J=1

NJ(x)λJ (21)

where λJ are vector solutions of the linear sys-tem of equations in the form

L∑

J=1

cIJ λJ = ξI (1− κI) f extI (22)

In the above equations, cIJ =∫∆S NI(x)NJ(x)dS, with I, J = 1, 2, · · ·L.

ξI is the fraction of (1− κI) f extI that can be at-

tributed to ∆S, since ∆S may be only a portionof Se and particle I may be on the boundary of∆S (shared by the rest of Se). ξI can be defined

through ξiI =

(∆S

)i

I/

q∑i=1

(∆S

)i

I, with q being

the number of surface areas connected to atomI .

The equations for the body force density b(e)

resulting from the third equation in Eq. (16) are

b(e)(x) =M∑

J=1

NJ(x)υJ (23)

where υJ are the vector solutions of

M∑

J=1

dIJ υJ = κIfextI (24)

In the above relations, dIJ =∫V e NI(x)NJ(x)dV , with I, J = 1, 2, · · ·M .

Similarly, the thermal oscillation-induced

“body force” density_

b(e)

are

_

b(e)

(x) =M∑

J=1

NJ(x)_υJ (25)

where _υJ are the vector solutions of

M∑

J=1

dIJ

_υJ =

_

f I (26)

The mass density for V e is

ρ(e)(x) =M∑

K=1

NK(x)gK (27)

where gK (K = 1, 2, · · ·M) are solutions of

M∑

K=1

gKχIK = ςImIüI (28)


186 ZHOU

with χIK =∫

V e NI(x)NK(x)üedV and üe =∑MI=1 NI(x)üI . Equation (28) and the require-

ment that ρ(e) be independent of üI yields

∑M

K=1dIK gK = ς ImI , (I = 1, 2, · · ·M)

(29)

The replacement of δ ˙u in Eqs. (12) and (16) withthe actual structural deformation part of the ve-locity ˙u yields

M∑

I=1

f intI · ˙uI +

M∑

I=1

(κIf

extI +

_

f I

)· ˙uI

+MS∑

I=1

(1− κI)f extI · ˙uI =

ddt

M∑

I=1

12 ςI mI

˙uI · ˙uI

= −∫

V e

σ(e) : D(e)dV +∫

V e

(b(e)+

_

b(e)

)· ˙u(e)dV

+∫

Se

t(e) · ˙u(e)dS =ddt

∫

V e

12ρ(e) ˙u(e) · ˙u(e)dV

(30)

Direct term-to-term equivalence is main-tained here, establishing conservation of en-ergy and equivalence of work rates from theperspective of the macroscopic observer. Theterm associated with the thermal oscillations ofatoms is

M∑

I=1

_

f I · ˙uI =∫

V e

_

b(e)· ˙u(e)dV (31)

This term represents the coupling or rate ofenergy exchange between the structural modeof the deformation ( ˙uI) and the thermal modeof the deformation ( ˙uI) as it appears to themacroscopic observer. A positive value of thisterm indicates energy input from the thermalmode into the mechanical mode. Such cases

arise, for example, in thermally driven defor-

mations. Although the thermal force_

f I is “in-visible” to the macroscopic observer, its effecton structural deformation and, perhaps moreimportantly, the energy exchange the aboveterm represents can be clearly measured andplays an important role in microscopic andmacroscopic phenomena. Examples includethermoelastic dissipation, thermoplastic dissi-pation, electromagnetically driven heat gener-ation and deformation, thermal shock, and de-formation driven by temperature changes.

It is important to point out that the “macro-scopic” representation of Eq. (30) differs fromthe regular continuum balance of energy rela-tion in the form of

−∫

V e

σ(e) :D(e)dV +∫

V e

b(e) ·u(e)dV +∫

Se

t(e) ·u(e)dS

=ddt

∫

V e

12 ρ(e) u(e) · u(e)dV

(32)

Although this relation is quite similar in form tothat in Eq. (30), they have somewhat differentmeanings, depending on the scale. The quanti-ties in Eq. (32) (denoted by an underscore) arephenomenological quantities developed for ap-proximate descriptions of the atomic counter-parts in Eq. (30). They may not be based onstrict, full equivalence of momentum, energy,work rates, and mass at the atomic level. Sincethey are higher-scale concepts with much fewernumbers of degrees of freedom (DOF), they donot provide resolution of atomic scale behavioras the quantities in Eq. (30) (denoted by a su-perimposed bar) do. The following correspon-dence can be outlined based on physical mech-anisms, without necessarily implying equality(equality holds if all the atomic DOF are usedin the continuum representation).



∫

V e

σ(e) :D(e)dV ⇔∫

V e

σ(e) :D(e)dV +∫

V e

_

b(e)· ˙u(e)dV

=M∑

I=1

f intI · ˙uI +

M∑

I=1

_

f I · ˙uI

∫

V e

b(e) · u(e)dV ⇔∫

V e

b(e) · ˙u(e)dV

=M∑

I=1

κIfextI · ˙uI

∫

V e

t(e) · u(e)dS ⇔∫

Se

t(e) · ˙u(e)dS

=MS∑

I=1

(1− κI)f extI · ˙uI

∫

V e

12 ρ(e) · u(e) · u(e)dV ⇔

∫

V e

12 ρ(e) ˙u(e) · ˙u(e)dV

=ddt

M∑

I=1

12 ςI mI

˙uI · ˙uI

(33)

Of particular interest is the identification ofthe high-frequency thermal coupling term∫

V e

_

b(e)· ˙u(e)dV =

∑MI=1

_

f·I˙uI with the stress

work rate in the first correspondence (ratherthan with the body force term in the second cor-respondence) in the above table. This identifi-cation reflects the continuum phenomenologi-cal description of thermoelastic dissipation andthermoplastic dissipation through the stresswork, instead of through a “body-force-like”term, such as that in Eq. (30). Although thiscontinuum phenomenological treatment is con-venient and allows material behavior to be de-scribed with various degrees of accuracy at dif-ferent length and time scales, the theoreticalanalysis here shows that it is fundamentally in-compatible with the requirement of full workrate and momentum equivalence between thecontinuum description and the atomic reality.

The form of such a treatment does not allowfull momentum and work equivalence to beobtained, even if the model is assigned thesame number of degrees of freedom as thediscrete atomic system. From the viewpointof multiscale material behavior characteriza-tion from the “ground up” through moleculardynamics simulations, this association of ther-mal dissipation and thermostructural couplingwith stress work rate can be used to formulateproper higher-scale, lower-DOF, phenomeno-logical constitutive laws. For example, it can bedefined that

∫

V e

_

b(e)· ˙u(e)dV =

∫

V e

_σ

(e): D(e)dV (34)

Therefore,

∫

V e

σ(e) :D(e)dV +∫

V e

_

b(e)· ˙u(e)dV=

∫

V e

_

σ(e)

:D(e)dV

(35)

It must be pointed out that Eq. (34) does not

give a unique definition for _σ

(e). More impor-

tantly, it is not possible to choose _σ

(e)such that

_

σ(e)

can be made to satisfy the balance of mo-mentum [Eq. (12) or (19)]. Furthermore, thisphenomenological association of thermal dissi-pation with the stress work rate cannot be con-strued as to imply dependence of stress on ther-mal velocity.

2.3 Thermal Oscillations: Balance ofEnergy and Thermal Fields

The analysis above is carried out from the per-spective of a higher-scale observer with focuson the structural part of the atomic deformation( ˙uI). We now focus our analysis on the thermal


188 ZHOU

part of the deformation ( ˙uI), from the perspec-tive of an observer moving at velocity ˙uI withfull resolution for thermal motions.

Equation (10) and the principle of virtualwork over V e with respect to δ ˙uI yield (afterreplacement of δ ˙uI by ˙uI in the end)

M∑

I=1

f intI · ˙uI +

M∑

I=1

(κIf

extI +

^

f I

)· ˙uI

+MS∑

I=1

(1− κI)f extI · ˙uI =

ddt

M∑

I=1

12 ςImI

˙uI · ˙uI

= −∫

V e

σ(e) : D(e)dV +∫

V e

(b(e) +

^

b(e)

)· ˙u(e)dV

+∫

Se

t(e) · ˙u(e)dS =ddt

∫

V e

12 ρ(e) ˙u(e) · ˙u(e)dV

(36)

In the above relation,

^

f I = −ς ImIüI (37)

Also, term-to-term equality is maintained and

D(e) = 12

[∂ ˙u(e)

∂x +(

∂ ˙u(e)

∂x

)T]

= 12

M∑I=1

(˙uI ⊗ BI + BI ⊗ ˙uI

) (38)

where ˙u(e) is defined over V e through

δ ˙u(e)(x) =M∑

I=1

NI(x)δ ˙uI (39)

Also,^

b(e)

is calculated through Eqs. (25) and

(26) with_

b(e)

, _υJ , and

_

f I being replaced by^

b(e)

,^υJ , and

^

f I , respectively.∫

V e

^

b(e)· ˙u(e)dV =

∑MI=1

^

f I · ˙uI represents work coupling betweenthe thermal mode of deformation and the struc-tural mode of deformation as it appears to anobserver moving with the structural velocity˙uI . Specifically, it is the rate of energy input intothe thermal mode of atomic motion.

Equation (36) represents the balance of en-ergy as it appears to the observer traveling withthe structural velocity ˙uI of an atom in the dis-crete description or to an observer travelingwith velocity ˙u(e) of the (continuumized) masspoint occupying x at time t in the equivalentcontinuum description. This equation is theatomic description of the thermal process in acoupled thermomechanical deformation event.The continuum phenomenological characteri-zation is

∫

V e

h(e)dV +∫

Se

n(e) ·q(e)dS =ddt

∫

V e

ρ(e)(eT +eS)dV

(40)

where h(e) is the rate of phenomenological den-sity of the heat source (thermal energy gener-ated in the continuum per unit volume and perunit time), q(e) is the phenomenological heatflux (thermal energy flowing across unit sur-face area per unit time), eT is the density ofthe thermal energy or the part of the internalenergy associated with the thermal velocity ofatoms, and eS is the density of the part of the in-ternal energy associated with interatomic forceinteractions. Since Eq. (40) is phenomenologi-cal in nature and involves much fewer degreesof freedom than Eq. (36), full and direct equiv-alence between the atomic equation and themacroscopic phenomenological equation maynot be established. The following correspon-dences can be outlined based on physical mech-anisms, without necessarily implying equality(equality holds if all the atomic DOF are usedin the continuum representation).



Heat source :∫

V e

h(e)dV ⇔∫

V e

(b(e) +

^

b(e)

)· ˙u(e)dV

=M∑

I=1

(κIf

extI +

^

f I

)· ˙uI

Heat flow :∫

Se

n(e) · q(e)dS ⇔∫

Se

t(e) · ˙u(e)dS

=MS∑

I=1

(1− κI)f extI · ˙uI

Thermal energy :ddt

∫

V e

ρ(e)eT dV ⇔ ddt

∫

V e

12 ρ(e) ˙u(e) · ˙u(e)dV

=ddt

M∑

I=1

12 ςImI

˙uI · ˙uI

Structural energy :ddt

∫

V e

ρ(e)eSdV ⇔∫

V e

σ(e) : D(e)dV

= −M∑

I=1

f intI · ˙uI

(41)

Note that the Reynolds transport theorem [9]specifies that d

dt

∫V e ρ(e)eSdV =

∫V e ρ(e)eSdV .

Equations (36) and (41) allow phenomeno-logical forms of heat equations at various scalesto be formed based on the results of MD calcu-lations. Specifically, the second relation abovecan be used to obtain characterizations of thethermal behavior of materials. To this end, wedefine the temperature at an atomic positionthrough

12 mi

˙ui · ˙ui = 32 kTi (42)

where k is the Boltzmann constant. Wenote that the standard definition of the av-

erage temperature (T ) for several atoms isonly given in the statistical sense over a vol-ume V (e) of a system with M atoms through∑M

I=112 mI

˙uI · ˙uI =32 k

∑MI=1 TI = 3

2MkT . Theextension of this definition to individual atomshere is consistent with the statistical interpre-tations of temperature for aggregates of atoms.The statistical consistency is in the sense thatthe temperature of an aggregate of atoms isthe average of the temperatures of the individ-ual atoms. Some may argue that thermal en-ergy (therefore, temperature) can only be mea-sured (primarily) at higher scales in a statisti-cally averaged, phenomenological manner, andit is difficult to measure the temperature of indi-vidual atoms and, therefore, the concept of thetemperature of an individual atom is difficultto justify. A new and different perspective canbe offered here. The advancement in scanningtunneling microscopy (STM) and atomic forcemicroscopy (AFM) has brought to reality theability to resolve and manipulate the positionsof individual atoms (see, e.g., Refs. [10,11]). Thethermal motions of individual atoms play a sig-nificant role in such processes and can be ana-lyzed. If the thermal velocity of an atom canbe clearly defined, the kinetic energy (thermalenergy) associated with it can be defined andquantified as well. The atomic temperaturedefined in Eq. (42) has relevance, application,and scientific justification for the nanoscience ofindividual atoms and small clusters of atoms.This extension of the definition of temperatureto an individual atom allows thermal analysesto be carried out at the nanoscale. We also notethat this definition fully lends itself to interpre-tations in a statistical sense for higher-scale sys-tems. Under this condition, the temperaturefield inside V e can be expressed as

T (e)(x) =M∑

I=1

NI(x)TI (43)

It is specifically noted that the concept oftemperature, just like the concept of thermal ve-


190 ZHOU

locity, is inherently scale dependent. The equiv-alence of thermal energy at an arbitrary sizescale of V e allows specific the heat c(e) to be de-fined through

ρ(e)eT = ρ(e)c(e)T (e) = 12 ρ(e) ˙u(e) · ˙u(e) (44)

The governing equation for the heat flux insideV (e) is the local version of Eq. (40) in the formof

∂

∂x· q(e) = ρ(e) (eT + eS)− h(e)

=(ρ(e) ˙u(e) · ü(e)+σ(e) :D(e)

)−

(b(e) +

^

b(e)

)· ˙u(e)

(45)

This equation and the boundary requirementthat

n(e) · q(e) = t(e) · ˙u(e) (46)

over S(e) allows q(e) to be uniquely determined.The determination of q(e) at all times yields q(e).

The thermal fields of temperature, specificheat, and heat flux carry information about theintrinsic thermal constitutive behavior of theatomic system. They can be used to formulatephenomenological continuum constitutive de-scriptions of the thermal behavior of materialsystems using MD calculations. Proper formsof thermal constitutive relations should be cho-sen within the context of specific systems, timescale, size scale, and the conditions involved.To illustrate how this can be achieved, we rec-ognize the wave nature of heat conduction atthe atomic level and consider the commonlyused phenomenological relation between heatflux and temperature gradient in the form of

τ · q(e) + q(e) = −K(e) · ∂T (e)

∂x(47)

where τ is the tensor of relaxation times andK(e) is the thermal conductivity. τ is positive-definite and, for simplicity, can be taken as

isotropic, i.e., τ = τI, with I being the second-order identity tensor. τ is positive, reflectingthe fact that the speeds of thermal waves [12]through atomic structures are finite. K(e) is alsopositive-definite, reflecting the fact that heatflows from high-temperature regions to low-temperature regions. Equation (47) leads toheat equations that are hyperbolic in nature,accounting for the fact that ultimately at theatomic level, the conduction of heat is the prop-agation of high-frequency mechanical (thermaloscillation) waves through the atomic structure.This is especially the case at short time scalesin applications such as heating induced by fastlaser pulses (see, e.g., Refs. [13,14]). At highersize and time scales, the wave nature of thethermal process is smeared out and diffusioncharacteristics dominate, since sufficient time isavailable for relaxation to fully occur. To statethis alternatively, if the time scale of analysis isorders of magnitude longer than τ , a transitionto the limiting case with τ → 0 is justified, re-sulting in a parabolic heat equation commonlyused at higher scales.

The determinations of the relaxation timetensor τ and thermal conductivity K(e) may re-quire two separate calculations. First, the de-termination of K(e) should, preferably, be car-ried out under steady-state thermal conditionsfor which q(e) = 0, obviating the need forknowledge of τ. After the conductivity is de-termined, Eq. (47) can then be used to deter-mine τ, based on the results of at least one sep-arate MD calculation involving transient ther-mal responses. This should involve the choiceof the nine components of τ through, for ex-ample, curve-fitting, for the best description ofthe MD data. Materials with symmetries havefewer numbers of independent components forτ and K.

The specific heat and the thermal conductiv-ity defined here are not thermodynamic quan-tities. They are fully dynamic, instantaneousmeasures for the thermal behavior of atomic



systems at various size scales. These defini-tions do not reflect statistical averaging, whichis an intrinsic thermodynamic concept. Rather,they are quantities describing the thermal be-havior of atomic systems with explicit resolu-tions in time and in structure, yet reckoned inthe forms of standard (statistical) specific heatand thermal conductivity so as to conform tothe continuum heat equation. Naturally, thesedynamic quantities show oscillations and de-pendence on size, atomic structure, and com-position of lattice waves at the atomic level. Itis reasonable to expect that as the size scale ofanalysis is increased and as proper spatial andtemporal averages are taken, these quantitieswill approach the microscopically or macro-scopically measured values. Such analyses con-stitute future development. The quantificationand approach introduced here provides an al-ternative to statistical mechanics for reachingmicroscopic, mesoscopic, and macroscopic con-

tinuum scales from a discrete molecular dy-namics framework.

3. TENSILE DEFORMATION OFA RECTANGULAR LATTICE

Although the solutions of dynamic deforma-tions of atomic systems are, in general, com-plex and require numerical treatment, a sim-ple example can be analyzed here to illustratethe perspectives that the novel thermomechan-ical equivalent continuum theory has broughtabout. Some of the insights obtained here arerevealing in terms of the form of stress, scale de-pendence, and thermal-mechanical coupling.

Consider the uniform tension of a rectangu-lar lattice under external forces f ext in Fig. 4.For simplicity, we assume that the deformationis strictly uniaxial. Thus, all atomic displace-ments and velocities are in the horizontal direc-tion. Furthermore, we assume that the defor-

f32

f13f12

1

2

3

f32 4

q

ext

fext

/b=t f=b

iu

iu

a

f32

f13f12

1

2

3

f32 4

q

ext ext

/b==b const

i

i

a

~

(a) (b)

FIGURE 4. Tensile deformation of a lattice with thermal fluctuations under conditions of uniaxial strain


192 ZHOU

mation is uniform in the vertical direc-tion;therefore, ui(x, t), ui(x, t) and ui(x, t) areonly functions of time and of x(x), and are in-dependent of y(y) and z(z). Although such asituation is idealized and cannot be easilybrought about in the laboratory, it is perfectlyallowed and has all the fundamental attributesof a dynamically deforming atomistic particlesystem, including the satisfaction of Eq. (4). Itshould also be pointed out at the outset of thisanalysis that although a time-resolved analy-sis is carried out at the scale of a unit latticecell here, interpretation of the thermal behav-ior discussed has practical meaning primarilyin the sense that the results be viewed as timeaverages over time. Full consistency of thethermal behavior with higher-scale characteri-zations occurs only when the size scale of anal-ysis is much larger than what is the case hereand when a proper time average is taken. How-ever, this example allows the issue of a contin-uum representation of the MD system and thesolution procedure to be illustrated.

As shown in Fig. 4, the deformed lattice hasdimensions a and b in the horizontal and verti-cal directions, respectively. The material is ho-mogeneous and the mass of each atom is m. Forsimplicity, we assume the cutoff radius Rc issuch that a < Rc < 2a and b < Rc < 2b, therefore,nonlocal interactions do not occur. Under thedecomposition of Eq. (1), the atomic displace-ment has a uniform structural deformation partand a thermal oscillation part. Consequently

a(t) = a(t) + a(t) and b(t) = b(t) + b(t) (48)

Since the deformation is assumed to be strictlyone-dimensional, b(t) = b(t) and b(t) = 0. Thecalculation for the TMEC fields here uses 2Dshape functions for rectangular elements (see,e.g., [15]). For the rectangular element definedby atoms 1, 2, 3, and 4 in Fig. 4, the shapes func-tions are

N1(x) =(

1− x− x1

a

)y − y3

b

N2(x) =(

1− x− x1

a

)(1− y − y3

b

)

N3(x) =x− x1

a

(1− y − y3

b

)

N4(x) =x− x1

a

y − y3

b

N1(x) =(

1− x− x1

a

)y − y3

b

N2(x) =(

1− x− x1

a

)(1− y − y3

b

)

N3(x) =x− x1

a

(1− y − y3

b

)

N4(x) =x− x1

a

y − y3

b

(49)

The interatomic forces vary with time [i.e.,f12 = f12(a,b), f13 = f13(a,b), and f32 =f32(a,b)] while they are uniform in the y direc-tion since the deformation is uniform in that di-rection. All relevant continuum fields are sum-marized in Table 1. Both the results for themacroscopic TMEC representation and for themechanical EC representation with full atomicdisplacement resolution are given. The stress,traction, body force, and deformation fields areconsistent with the macroscopic expectations.It is important to note the difference and rela-tionship between the stress representations atthe macroscopic level and at the fully resolvedatomic level. Both representations have sim-ilar forms. Although the stress (traction andbody force, as well) at the fully resolved atomiclevel is evaluated relative to the instantaneous,true positions rI of the atoms that define V (e),the stress (traction and body force, as well) atthe macroscopic level is evaluated relative tothe macroscopically perceived (structural) po-sitions of the atoms rI that define V (e). Bothevaluations use the full instantaneous forces fI

on the atoms. Specifically, (see Eq. (50)).



TABLE 1. Mechanical and thermal quantities for a rectangular lattice in uniform tension

Macroscopic resolution for ui = ui + ui

(Thermomechanical EC)Full atomic resolution for ui

(Mechanical EC)Stress σ(e) =( (

f32+2f13 sin θ)/b 0

0(f12+2f13 cos θ

)/a

) σ(e) =((f32+2f13 sin θ) /b 00 (f12+2f13 cos θ) /a

)

Rate ofdeformation

D=( ˙a/a 0

0 ˙b/b

)

,

D =

(˙a/a 0

0 ˙b/b

)D =

(a/a 00 b/b

)

Stress workrate

σ(e) : D(e) = 1ab

(f32 ˙a + f12

˙b + 2f13 ˙r)

r =√

a2 + b2

σ(e) : D(e) = 1ab

(f32a + f12b + 2f13r

)

r =√

a2 + b2

Traction t = − (fext1 /b

)ex along side 1− 2

t =(fext3 /b

)ex along side 3− 4

t = ± (f12 + 2f13 cos θ) ey/aalong sides 1− 4 and 2− 3

t = − (fext1 /b) ex along side 1− 2

t = (fext3 /b) ex along side 3− 4

t = ± (f12 + 2f13 cos θ) ey/aalong sides 1− 4 and 2− 3

Body force b(e)(x) = 0 b(e)(x) = 0Massdensity

ρ(e)(x) = mab

ρ(e)(x) = mab

Inertialforces

_

b(e)

(x) = − m2ab

(2ü1 − ü3

) (1− x−x1

a

)

+(2ü3 − ü1

)x−x1

a

^

b(e)

(x) = − m2ab

[ (2ü1 − ü3

) (1− x−x1

a

)+

(2ü3 − ü1

)x−x1

a

]

Temperature T (e)(x) = m3k

[˙u21

(1− x−x1

a

)+ ˙u2

3x−x1

a

]

Specific heat c(e) = 3km

Heat flux

q(e)x (x) =

∫

(ρ(e)˙u(e) ·ü(e) + σ(e) : D(e)

)

−(

b(e) +^

b(e)

)· ˙u(e)

dx

withq(e)x (x)

∣∣∣x=x1

= − 1b

f ext1 · ˙u1, q

(e)x (x)

∣∣∣x=x3

= 1b

f ext3 · ˙u3

ü(e)(x) = ü1

(1− x−x1

a

)+ ü

x−x1a

3

Thermalconductivity

τ q(e)x + q

(e)x = −K

(e)xx

∂T (e)

∂x

(steady-state calculation with q(e)x = 0)

Relaxationtime

τ q(e)x + q

(e)x = −K

(e)xx

∂T (e)

∂x

(transient calculation with q(e)x 6= 0 after

K(e)xx is

determined)

a) The atomic level uniaxial strain assumption implies that u2 = u1 and u4 = u3. x2 = x1 andx4 =x3 are the horizontal coordinates of the atoms in the element analyzed.

b) Here, κI = 0 and ςI = 0.25.


194 ZHOU

σ(e) =(

(f32 + 2f13 sin θ) /b 00 (f12 + 2f13 cos θ) /a

)=

12V (e)

M∑

I

M∑

J(6=I)

rIJ ⊗ (ηIJfIJ)

σ(e) =((

f32 + 2f13 sin θ)/b 0

0(f12 + 2f13 cos θ

)/a

)=

12V (e)

M∑

I

M∑

J(6=I)

rIJ ⊗ (ηIJfIJ) (50)

For the volume element defined by atoms 1,2, 3, and 4 in Fig. 4, ηIJ = ηIJ = 1/2 for f12 andf32 since these forces are shared with neighbor-ing cells. However, ηIJ = ηIJ = 1 for f13 sincethis force is fully within the element. These co-incide with the Cauchy stress expressions σ(e)

and σ(e) in Table 1.The heat flux component in the x direction

for the element analyzed in Table 1 is

q(e)x (x) =

∫

(ρ(e) ˙u(e) · ü(e) + σ(e) : D(e)

)

−(

b(e) +^

b(e)

)· ˙u(e)

dx

(51)

where the arbitrary constant resulting from theintegration must be determined by Eq. (46)with takes the form of

q(e)x (x)

∣∣∣x=x1

= −1b

f ext1 · ˙u1,

q(e)x (x)

∣∣∣x=x3

=1b

f ext3 · ˙u3 (52)

Clearly, q(e)x (x) in Eq. (51) measures the flow

rate of energy in the form of high-frequencymechanical waves through the element V (e).This flow is apparently driven by the rate atwhich the thermal motion of the atoms on theboundary S(e) can extract high-frequency me-chanical work from or impart high-frequencymechanical work to its surroundings. Ofcourse, the heat flow, so driven by the exter-nal excitation on the boundary S(e), are effectedthrough an imbalance inside V (e) between therate of change of the kinetic and potential ener-gies associated with the thermal motions of the

atoms and the rate of work of the body forceand inertial force over the thermal oscillations.To put it differently, heat flow across V (e) is amanifestation, in the form of energy exchange,of the interaction of the high-frequency motionof V (e), itself, with its surroundings.

At the macroscopic scale, a homogenous sin-gle crystal is expected to have a spatially uni-form thermal conductivity K. The mathemat-ical forms of T (e), c(e), and K

(e)xx in Table 1 for

V (e) do not necessarily conform to such macro-scopic continuum phenomenological expecta-tions, since they show spatial variations at thelattice level. The perspective obtained here isthreefold.

First, explicit, time-resolved balance of mo-mentum and conservation of energy is main-tained in the TMEC analysis at all times. Theresult on the heat flux is quite illustrative. Thesignificance of the results here should be moreunderstood in the form of proper averages intime. For example, the temperature, specificheat, thermal conductivity, and relaxation timein Table 1 are more meaningful if interpreted as

⟨T (e)

⟩=

m

3k

[⟨˙u21

⟩(1− x− x1

a

)+

⟨˙u23

⟩ x−x1

a

]

⟨c(e)

⟩=

⟨˙u(e) · ˙u(e)

⟩

2⟨T (e)

⟩ and

τ⟨q(e)x

⟩+

⟨q(e)x

⟩= −

⟨K(e)

xx

⟩⟨∂T (e)

∂x

⟩(53)

where the angular brackets denote proper timeaveraging. As in experiments, the determina-tion of c(e) and K

(e)xx should be carried out when

a steady state has been reached. The determina-



tion of τ , on the other hand, requires a transientimpulse at one end.

Second, the atomic structure is explicitly ac-counted for. Under such conditions, the gov-erning equations for the thermal process arethen cast in the form of the macroscopic heatequation. The results at the nanoscopic scalenaturally reflect the nature of the atomic struc-ture. Consistency with higher length scale con-tinuum expectations should emerge as one ap-proaches higher scales. This entails the useof progressively large volume elements. Inthe process, the lattice-level variations of thesequantities should average out and the long-range order should become apparent and dom-inant.

Third, thermal oscillations and structuraldeformations of atomic structures consist ofwaves of different frequencies (wavelengths).The wave spectrum is another source giv-ing rise to time- and scale-dependence of re-sponses. Macroscopic responses can only beobserved when the analysis is carried out atspatial and temporal scales sufficient to repre-sent all components of the lattice waves. Theunit cell analysis above is at a scale well be-low many significant component wavelengthsin the lattice. While this particular analysis atthis particular scale shown in this example doesnot necessarily represent the higher-scale be-havior, it indeed characterizes, rather faithfullyand accurately, the atomic interactions at thelattice scale, albeit in a continuum form usedat higher scales. The use of such a continuumform at all scales allows comparison, analysisof scaling effects, and transition from one scaleto another.

4. CONCLUDING REMARKS

The thermomechanical equivalent continuum(TMEC) theory described here establishes,within the meaning of classical mechanics, theatomic origin of coupled thermomechanical de-

formation processes. It provides a frameworkfor the formulation of coupled thermomechan-ical continuum descriptions of material behav-ior at any spatial and temporal scale “from theground up,” using molecular dynamics simula-tions. This theory is based on a decompositionof the atomic particle velocity into a relativelyslower-varying structural deformation part anda high-frequency thermal oscillation part. Thecontinuumization of the discrete molecular dy-namics model is carried out with strict adher-ence to balance of momentum, balance of en-ergy, and conservation of mass, in a fully dy-namic and time-resolved manner. The estab-lishment of equivalence between the discretesystem and the continuum system follows thesame approach used for the pure mechanicaltheory. This approach uses the dynamic prin-ciple of virtual work. Since the decomposi-tion of atomic velocity into a structural defor-mation part and a thermal oscillation part isintrinsically dependent on both size and timescales, the development has followed a generalframework of analysis, allowing scale-sensitivecharacterizations of both the structural defor-mation as it appears to a macroscopic observerand the thermal oscillation as it appear to anobserver moving at the structural velocity ofan atom. On one hand, balance of momen-tum at the structural level yields fields of stress,body force, traction, mass density, and defor-mation that maintain full dynamic equivalencebetween the discrete system and continuumsystem. This equivalence includes (i) preserva-tion of linear and angular momenta; (ii) conser-vation of internal, external, and inertial workrates; and (iii) conservation of mass. On theother hand, balance of momentum in terms ofthe thermal motions yields the fields of heatflux and temperature. These quantities havebeen cast in a manner so as to conform tothe continuum phenomenological equation forheat conduction and heat generation, allowingthe specific heat, thermal conductivity, and re-


196 ZHOU

laxation time at different scales to be quanti-fied. The coupling of the thermal equation ofheat conduction and the mechanical equationof structural deformation, as phenomenologi-cally known at higher continuum scales, occursbecause the balance equations at the structurallevel and the thermal level are only two dif-ferent forms of the same balance of momen-tum equation at the fully time-resolved atomiclevel. This coupling occurs through an inertial

force term in each of the two equations [_

b(e)

in Eq. (30) and^

b(e)

in Eq. (36)] induced bythe other process. For the structural deforma-tion equation, the inertial force term induced

by thermal oscillations of atoms (_

b(e)

) gives riseto the phenomenological dependence of defor-mation on temperature. This point can be seenfrom the first correspondence in Eq. (33). Forthe heat equation, the inertial force term in-

duced by structural deformation (^

b(e)

) takesthe phenomenological form of a heat source[see Eq. (41)]. Equations (30) and (36) representconservation of energy as it appears to, respec-tively, the macroscopic observer (who sees ˙ui)and the traveling observer (who sees ˙ui). Takentogether, these two equations specify conserva-tion of energy for the atomic system in the fullytime-resolved sense. In summary, the TMECtheory developed here allows scale-dependentcontinuum descriptions of material behaviorin the form of coupled thermomechanical pro-cesses of deformation and heat flow to be ob-tained from purely mechanical molecular dy-namics simulations at the atomic level.

The framework here fully recognizes the spa-tial and temporal scale dependence of deforma-tion. First, the choice of element V (e) is totallyarbitrary, allowing size dependence of atomicbehavior to be analyzed. Specifically, V (e) canbe as small as the tetrahedron defined by fourneighboring atoms, and as large as the wholesystem V . Length scale effects due to lattice

spacing, thermal and structural wave lengths,and structural features such as voids and grainscan be explicitly analyzed as the size of V (e) isvaried. Such analyses, of course, carry a verysignificant computational cost.

The theoretical approach also fully admitsthe analysis of time scale influences. Specif-ically, the decomposition of atomic motion inEq. (1) is highly dependent on time resolution.At the macroscopic time scale, high-frequencycomponents of atomic oscillations are not ex-plicitly resolvable and are, therefore, includedin ˙ui. As the time resolution is increased, rel-atively lower-frequency components of atomicoscillations become explicitly resolved and areincluded in ˙ui, causing ˙ui to decrease. Inthe limit of an analysis with full resolution foratomic motion, ˙ui = 0 and ui = ˙ui. Conse-quently, the pure mechanical theory of equiv-alent continuum is recovered. Of course, tem-poral and spatial resolutions are often relatedand cannot be fully separated. Temporal res-olutions are higher at small scales. For exam-ple, at the size scale of nanowires or electroniccomponents, structural waves along the axialdirection may need to be analyzed, causing thelong wavelength (of the order of 1–2 nm) com-ponents of atomic motion to be included in ˙ui.Such wave components may be regarded asthermal waves at the macroscopic level.

Finally, it is important to point out that thestarting point of the theoretical development inthis paper is molecular dynamics. The frame-work used here naturally inherits the assump-tions and limitations of molecular dynamics. Inparticular, we note that the effect of free elec-trons on heat conduction in metals and alloysis not explicitly accounted for in standard MDmodels. Reflecting this characteristic of MD,the heat conduction analyzed in the TMEC the-ory developed in this paper concerns solely thecontribution of atomic vibrations (or phonons)to thermal conductivity. In order to accountfor the contributions of free electrons (met-



als, alloys, and semiconductors; see, e.g., [16]),of electron/phonon interactions, and of bipo-lar carriers consisting of electron/hole pairs athigh temperatures (semiconductors; see e.g.,[17]) to thermal conductivity, modifications tothe MD framework are required, if a frame-work within classical mechanics is to be used.This would constitute a significant extensionof molecular dynamics, which is possible and,perhaps, worthwhile.

ACKNOWLEDGMENT

Support from the NSF through CAREER AwardNo. CMS9984289, from the NASA LangleyResearch Center through Grant No. NAG-1-02054, and from an AFOSR MURI at GeorgiaTech is gratefully acknowledged.

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