Useful Conservation Sums in Molecular Dynamics and Atomistics

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1 Useful Conservaton Sums n Molecular Dynamcs and Atomstcs SAEJA O. KIM Department of Mathematcs, Unversty of Massachusetts Dartmouth, North Dartmouth, MA 02747, USA YOON Y. EARMME Department of Mechancal Engneerng, Korea Advanced Insttute of Scence and Technology, Guseong Dong, Yuseong Gu, Daejeon , South Korea KYUNG-SUK KIM Dvson of Engneerng, Brown Unversty, Provdence, RI 02912, USA (Receved 15 August 2009 accepted 13 November 2009) Abstract: Confguraton forces of an arbtrary molecular dynamcs doman are derved n whch the energetc characterstcs are approxmated as those of a non-local gradent elastc feld. The confguraton forces nclude not only the effects of the stran gradents but also those of mcro-polar energetcs. The useful sums represent approxmate expressons of translatonal, expansonal and rotatonal confguraton forces. Partcular homogenzaton of the feld makes the contnuum lmts of the sums approxmately obey confguratonal transformaton symmetres to satsfy Noether s theorem, and to converge to generalzed J ntegral for statc non-local elastc felds of homogeneous meda however, the non-local M and L ntegrals do not have correspondng conservaton laws. Interpolaton and projecton schemes of a molecular dynamcs or atomstcs system onto a system of contnuum motons and energetcs are also derved for elastc as well as non-elastc cases. Key Words: Noether s theorm, conservaton ntegrals, non-local elastcty, molecular dynamcs, contnuum projecton 1. INTRODUCTION Over the past several decades, confguraton forces on varous contnuum felds have been studed [1 9], and projectons of dscrete molecular-dynamc (MD) materal ponts on a contnuum dynamc elastc feld have been attempted wth varous approxmatons [8, 9]. Once the projectons are made, varous confguraton forces of a MD doman can be derved based on nfntesmal transformatons of the Lagrangan (knetc energy mnus potental energy) densty of the elastc feld. Snce the Lagrangan densty comples wth the Hamlton prncple of dynamcs, f the resoluton of the projecton allows us to use local elastc felds for the projecton, the translatonal confguraton force converges to the classcal J ntegral of Mathematcs and Mechancs of Solds 00: , 2009 DOI: / The Author(s), Reprnts and permssons:

2 2 S.O.KIMETAL. Eshelby [2] and Rce [6]. In addton, the expansonal and rotatonal confguraton forces converge to the classcal M and L ntegrals of Budansky and Rce [1] and Knowles and Sternberg [3], respectvely. The classcal J ntegral of quasstatc nfntesmal deformaton of a non-lnear elastc materal becomes path ndependent obeyng Noether s theorem [10] to have translatonal symmetry and the correspondng conservaton law. The classcal M ntegral of two-dmensonal statc nfntesmal deformaton of a lnear elastc materal s path ndependent followng the Noether s theorem to have expansonal symmetry and the correspondng conservaton law. Smlarly, the classcal L ntegral of statc nfntesmal deformaton of a non-lnear elastc but sotropc materal s path ndependent, havng rotatonal symmetry and the correspondng conservaton law. However, multbody nteractons of MD requre projectons on non-local elastc felds n general. In recent years, confguraton forces of varous non-local elastc felds have been studed [4, 5, 7]. ukobrat and Kuzmanovc [7] nvestgated conservaton laws n non-local elastcty wthn the framework of nternal state varables. Lubarda and Markenskoff [4] derved confguraton forces n couple-stress elastcty, and found that the J ntegral of quasstatc deformaton s conserved whle M ntegral s not path ndependent for lnear couple-stress elastcty. Markus and Helmut [5] derved the Eshelby stress tensor, angular momentum tensor and dlataton flux n gradent elastcty they also confrmed that M ntegral s not path ndependent for statc lnear gradent elastcty. Followng an attempt [9] to make a theory of anharmonc lattce statcs for analyss of defectve crystals, Yavar and Marsden [8] nvestgated the connecton between balance laws and energy balance nvarance for a system of nteractng partcles. In ther nvestgaton, balance of energy for a system of partcles embedded n a Remannan manfold was examned va a generalzed form of parwse nteractons by assumng that the parwse potental energy of partcles explctly depends on ther parwse dstances. We present, n ths paper, explct expressons of varous dynamc confguraton forces for gradent elastcty and reduce the expressons for ther quasstatc counterparts. The quasstatc expressons follow Noether s theorem wth approprate constrants,.e. nfntesmal deformaton for J ntegral n general, and elastc sotropy and local elastcty for L ntegral, respectvely. However, M and L ntegrals are not conserved for the gradent elastcty. Based on the expressons, approxmate confguraton forces are derved by embeddng or projectng the MD partcles nto a Remannan manfold. Throughout the paper, bold characters represent vectors or tensors, whle talc subscrpts stand for Cartesan components, commas between subscrpts ndcate dfferentaton, and summaton conventon s appled for repeated ndces n a sngle term. 2. HAMILTON S PRINCIPLE FOR GRADIENT ELASTICITY A contnuum moton xx t s consdered to be a tme-parameter famly of materal confguratons, whch maps the materal pont X to a space pont x at tme t. The contnuum moton xx t s C 2 wth respect to X and C 1 to t. Introducng the dsplacement ux t xx t xx 0 where xx 0 X, the Lagrangan (knetc energy mnus potental energy of the system) of body, bounded by the boundary surface s defned as

3 USEFUL CONSERATION SUMS IN MOLECULAR DYNAMICS 3 L[u u j u jk u X l ] [u u j u jk u X l ]d X Tm u m mn u nm dsx (1a) where [u u j u jk u X l ] 1 2 0X l u m u m u u j u jk X l (1b) s the nternal Lagrangan densty wth 0 X l the mass densty of the contnuum, u u X l t t and u u j u jk X l the elastc energy densty per unt volume n the ntal confguraton, T represents surface tracton and surface couple. Here, the Lagrangan densty functonal depends not only on the functons, u X t, u j X t, u jk X t and u X t, but also explctly on the poston X. The explct dependence on X means that the materal propertes such as mass densty and elastc modul vary wth the poston of the materal pont X n general. Followng Hamlton s prncple,.e. I 0, where I t 2 Ldt for arbtrary t 2 namely, I t2 t2 t2 t2 t2 t2 [u u j u jk u X l ]d X dt u u u j u j Tm u m mn u nm dsx dt u j j kj jk t t2 u jk u u jk u u N j j kjk T u ds X dt Nk kj j u j ds X dt u d X dt u Tm u m mn u nm dsx dt u d X dt t 2 d X 0 (2a) for arbtrary u and u j, and for gven ntal and fnal condtons of or vanshng n. Here u u j u j j j (2b) s the generalzed local stress tensor, kj u jk the couple tensor, N the surface normal, and u the body force densty. Once we set 0 t 2 and t 2 nfntesmally small,

4 4 S.O.KIMETAL. the symmetrc part of, sym, becomes the Cauchy stress and the ant-symmetrc part of, antsym, the ant-symmetrc mcropolar stress [4]. Then, we have the feld equatons as j u kjk 0 n (3) j t u N j j kjk and j N k kj on (4) T Here, we have the generalzed stress j j kjk, the body force densty f u and the lnear momentum densty 0 u u. 3. CONFIGURATION FORCES OF GRADIENT ELASTICITY Confguraton forces arsng n a volume of a body are defned as total system energy varatons per vrtual confguratonal dsplacement, caused by spatal varatons of the materal propertes durng the vrtual dsplacement of the confguraton. The total system energy varaton of ncludes not only that of nternal elastc energy and knetc energy of, but also the potental energy varaton of the external agency to change the loaded state of for one materal property to another. Eshelby [2] noted that the confguraton force densty turns out to be the negatve explct gradent exp of the nternal Lagrangan densty [u u j u jk u X l ]. Here, we consder three useful dynamc confguraton forces, J D M D and L D nduced by three generalzed confguraton dsplacements, translaton, expanson and rotaton transformatons. Correspondng confguraton forces arsng n are expressed as J D expd X (5a) M D X expd X (5b) L D jk X j k expd X In a homogeneous medum the explct gradent of the Lagrangan, exp, vanshes along wth the confguraton forces. Expressons for the confguraton forces are decomposed nto volume and surface ntegrals when expressed n terms of mechancs varables. In the decomposton, f the volume ntegral dsappears under certan condtons, the remanng surface ntegral becomes zero. Therefore, for a feld generated by a source n a homogeneous medum, the surface ntegral enclosng the source s path ndependent under such condtons. Then, the ntegral s called conservatve. In the followng we derve the expressons explctly n terms of mechancs varables. (5c)

5 USEFUL CONSERATION SUMS IN MOLECULAR DYNAMICS Translatonal confguraton force J The translatonal confguraton force J can be derved by consderng the relatonshp between the absolute and the explct gradents of the Lagrangan densty, snce the volume ntegral of the latter gves the net confguraton force n as noted n (5a). The dfference between the absolute and explct gradents of the Lagrangan densty s derved by chan rules of dfferentaton appled on, and smplfed by (3) as exp u k u k u k j u k j u k u k j j u k j u klj u klj u k u klj u k u klj lj l u k u k jl u kl u k u k u k j u k jk t u l jk l u k jlk u k l (6) j k s ob- Therefore, the th component of the translatonal dynamc confguratonal force, J D taned, usng (4), (5a) and (6), as J D t u k d X N T k u k u lk u kl ds X (7) k For statc equlbrum the volume ntegral n (7) vanshes and the translatonal statc confguratonal force s reduced to N T k u k lk u kl ds X (8) J S Equaton (8) shows that J S whch lk 0on. converges to the conventonal J ntegral of local elastcty [6] for 3.2. Expansonal confguraton force M The expansonal confguraton force M can be derved by consderng the dvergence of the poston weghted Lagrangan densty, X,as X X kk u k jk t u ljkl uk jlk u kl j exp X kk k

6 6 S.O.KIMETAL. t u k X jk u ljkl uk X jlk u kl X j exp X k kk jk ljkl uk j jlk u klj, (9) where kk s the contracton of the Kronecker delta whch s 2 for two dmensons and 3 for three dmensons. Then, the dynamc expansonal confguraton force, M ntegral, s derved from (5b) and (9) as M D X u k kk jk t u ljkl uk j jlk u klj d X k X N T k u k lk u kl dsx. (10) The statc M ntegral s subsequently expressed as M S kk jk ljkl uk j jlk u klj dx X N T k u k lk u kl dsx (11) If, n addton, the medum s lnear elastc, jk u k j jlk u klj 2 and thus the M ntegral s reduced to M SLE X N T k u k lk u kl dsx 2 kk ljkl u k j dx. (12) Equaton (12) s reduced to conventonal statc M ntegral of local elastcty [3] for whch lk 0on and ljkl 0n.Theterm2 kk n the volume ntegral vanshes for twodmensonal problems however, the whole volume ntegral does not vansh n general. Ths ndcates that the two-dmensonal M ntegral s not conservatve for lnear gradent elastcty due to ts ntrnsc length scale of the non-local elastcty consttutve relaton, whch s constant and does not scale wth the self-smlar expanson scalng transformaton of the M ntegral Rotatonal confguraton force L Expresson of the rotatonal confguraton force, or confguraton moment, L can be obtaned by consderng the curl of the poston weghted Lagrangan densty X wth (6), as

7 USEFUL CONSERATION SUMS IN MOLECULAR DYNAMICS 7 jk X k j jk j X k jk X k u m j lm t u nlmn um j lnm u mnj l j exp m jk u m j X k lm t u nlmn um j X k lnm u mnj X k l m j exp X k km nkmn um j knm u mnj (13) The dynamc confguraton moment L D s then obtaned from (5c) and (13) as L D jk X j u mk jm t u njmn umk jnm u mnk d X m jk X j Nk T m u mk nm u mnk dsx jk X j t u mk d X u m jk 2 jm 2 sym njmn mk jm 2njmn ant mk jnm u mnk dx jk X j Nk T m u mk nm u mnk Tj u k dsx (14) where jk s the alternatng tensor sym njmn the symmetrc par njmn 2 nmjn and ant njmn the ant-symmetrc par njmn 2 nmjn of njmn and mk the local stran 1 umk u 2 km and mk the local rotaton 1 umk u 2 km. Then the statc L ntegral becomes jk 2 jm 2 sym njmn mk jm 2 ant njmn mk jnm u mnk d X L S jk X j Nk T m u mk nm u mnk Tj u k dsx. (15) Note that the volume ntegral n (15) does not vansh n general for non-local elastcty. For local elastcty t s further reduced to L Slocal 2 jk jm mk jm mk dx jk X j Nk T m u mk Tj u k dsx (16)

8 8 S.O.KIMETAL. If the medum s non-mcropolar, jm vanshes, and f further the materal s sotropc, jk jm mk also vanshes, and we recover the conventonal surface L ntegral of local elastcty. 4. USEFUL SUMS FOR MOLECULAR DYNAMICS AND ATOMISTICS In MD and atomstcs, mechancs of a medum composed of N partcles s descrbed by a set of dscrete motons, m x t x t 0 t 1 N,wherem x t denote the mass and the current poston of the partcle. These dscrete motons are then vewed as a contnuum moton by a contnuum projecton: x t N m Xx Xx 1 (17a) u j x t N u j t x x 1 (17b) where y s a normalzed projecton functon of C 2 n, wth y d y 1, and x x are nterpolaton functons of C 2 n. The contnuum velocty feld s derved by dfferentatng (17b) as u k x t kj N u j 1 k x x 1 N 1 u j x x u j u l l x x (18) Then, we derve the contnuum Lagrangan densty n two dfferent ways, and equate them as x t N m u k u k Xx Xx 1 2 u k u k x t (19) Here, s the atomc or molecular potental of partcle and x t the contnuum elastc energy densty of the body where x t x t u k u k N 1 m u k u k Xx Xx (20a)

9 USEFUL CONSERATION SUMS IN MOLECULAR DYNAMICS 9 N x t Xx Xx 1 (20b) s the projected elastc energy densty. In (20), the projected elastc energy x t s an explct functon of X n general so that the contnuum projected feld for the confguraton force ntegrals s not conservatve. However, under thermodynamc equlbrum lmts such as a constant temperature state, the projected elastc energy converges to a thermodynamc potental energy, such as the Helmholtz free energy for deformatons at constant temperature, and the projected contnuum feld can be conservatve. In a MD smulaton, f the partcle system s close to an elastc medum, we can obtan an approxmate effectve elastc energy densty functon by best-fttng the dstrbuton of x t n the u j u jk space. Once we have ths effectve elastc energy densty functonal, we can obtan the generalzed stress and couple tensors, respectvely, by takng dervatves wth respect to u j and u jk. If the system s far from elastc, the stresses and the couple tensors cannot be derved from the elastc energy potental however, the volume-average stress and couple tensors for a gven volume resoluton can be derved from the prncple of vrtual work, T u j u j dsx f 0 u u g j u j dx j u j kj u jk dx (21) where f s the external body force densty and g j the external body couple densty. Takng tral dsplacement u Q j x j,forallq j, we can derve the volume average stress n gradent elastcty for whch X approaches x as j 1 T x j j dsx 1 f u x j g j dx (22a) where s the mass densty n the current confguraton. Smlarly, by takng u R jk x j x k, for all R jk, we get the average couple as 1 kj xk T x j j x j k dsx 1 xk f u x j g j j x j g k k d x (22b) For a dscrete atomstc system n molecular dynamcs smulatons, we consder only the volume ntegral part n (22a) and (22b). By expressng the contnuum body forces f j, acceleratons a j and couples g j n terms of concentrated ones denoted wth hats as f N 1 f ext 3D x x (23a)

10 10 S.O. KIM ET AL. a N m a 3D x x 1 (23b) g j N 1 g ext j 3D x x (23c) for x,where 3D denotes the three-dmensonal Drac delta functon, we obtan j 1 N 1 g ext j x j f ext m a (24) Then, usng f nt f ext m a and g nt j g ext j 0 (gnorng the molecular moment of nerta), wth superscrpts (nt) and(ext) denotng nternal and external respectvely, we obtan j 1 N 1 g nt j x j f nt 1 2 N N 1 g nt j x j x j f nt (25) for parwse nteractng partcles. Here, the superscrpt par ndcates the nteracton g nt j or f nt on the partcle exerted by the partcle. Smlarly, takng tral dsplacement u R jk x j x k,forallr jk n (22b) and usng (23), we obtan kj 1 N 1 x j g ext k k x k f ext m a x j g ext j j (26) where j s the atomc or molecular stress of the partcle. Notng that (25) s derved from (24), Equaton (26) then leads to kj 1 2 N N 1 x x g nt k x k x k f nt x j x j g nt j 1 N 1 x j k x k j (27) for parwse nteractng partcles. Havng the volume-average stress and couple tensors n (25) and (27), the knematc varables are generally determned from contnuum nterpolatons of the partcle motons. However, the volume-average stress and couple tensors, and the nterpolated knematc varables do not satsfy the conservaton laws of the J, M and L ntegrals for non-elastc systems.

11 USEFUL CONSERATION SUMS IN MOLECULAR DYNAMICS CONCLUSION Explct expressons of dynamc confguraton forces have been derved for non-local gradent elastcty. The dervatons show that the translatonal confguraton force reduces to a generalzed path-ndependent J ntegral for quasstatc non-local gradent elastcty, whch satsfes Noether s theorem to exhbt symmetry of the governng laws for nfntesmal translatonal transformaton. However, the expansonal confguraton force M does not become path ndependent for two-dmensonal lnear non-local gradent elastcty, snce ts ntrnsc length scale embedded n the gradent elastcty does not scale wth the scalng transformaton of the M ntegral. In general, the L ntegral for non-local elastcty s not conserved. It s also shown that the expressons of the non-local confguraton forces can be readly used to evaluate approxmate confguraton forces of a MD doman by projectng the MD materal ponts on a Remannan manfold. For an elastc system of MD partcles, the effectve deformaton energy functonal of the elastc energy densty s constructed by best-fttng the MD smulaton results. The stresses and the couples derved from the effectve energy functonal and the correspondng work conjugates provde useful conservaton sums based on J, M and L conservaton ntegrals wth some constrants on materal propertes and dmensons. Otherwse, for non-elastc systems, the volume-average contnuum stresses are obtaned usng the prncple of vrtual work wthn a gven (or chosen) volume resoluton, whle the knematc varables are determned from contnuum nterpolatons of the partcle motons. However, n the latter cases, no such conservaton sums exst. Acknowledgements. For ths work, SOK was supported by the Unversty of Massachusetts Dartmouth, YYE was supported by the Natonal Unversty of Sngapore and Korea Insttute of Scence and Technology, and KSK was supported by the Brown Unversty MRSEC Program, under award DMR , of the U.S. Natonal Scence Foundaton. REFERENCES [1] Budansky, B. and Rce, J. R. Conservaton laws and energy-release rates. Journal of Appled Mechancs, 40, (1973). [2] Eshelby, J. D. The elastc energy momentum tensor. Journal of Elastcty, 5, (1975). [3] Knowles, J. K. and Sternberg, E. On a class of conservaton laws n lnearzed and fnte elastostatcs. Archves of Ratonal Mechancs and Analyss, 44, (1972). [4] Lubarda,. A. and Markenscoff, X. Conservaton ntegrals n couple stress elastcty. The Journal of Mechancs and Physcs of Solds, 48, (2000). [5] Markus L. and Helmut O. K. K. The Eshelby stress tensor, angular momentum tensor and dlataton flux n gradent elastcty. Internatonal Journal of Solds and Structures, 44, (2007). [6] Rce, J. R. A path-ndependent ntegral and the approxmate analyss of stran concentratons by notches and cracks. Journal of Appled Mechancs, 35, (1968). [7] ukobrat, M. and Kuzmanovc, D. Conservaton laws n nonlocal elastcty. Acta Mechanca, 92, 1 8 (1992). [8] Yavar, A. and Marsden, J. E. Energy balance nvarance for nteractng partcle systems. Journal of Appled Mathematcs and Physcs (ZAMP), 60(4), (2009). [9] Yavar, A., Ortz, M. and Bhattacharya, K. A theory of anharmonc lattce statcs for analyss of defectve crystals. Journal of Elastcty, 86, (2007). [10] Noether, E. Invarante aratonsprobleme, Nachr. d. Köng. Gesellsch. d. Wss. zu Göttngen, Math-phys. Klasse (1918) (Englsh translaton by Tavel, M. A. Transport Theory and Statstcal Physcs, 1(3), (1971)).