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Philosophical Magazine A Publication details, including instructions for authors and subscription information: http://www.informaworld.com/smpp/title~content=t713396797 Towards thermodynamics of elastic electric conductors M. Grinfeld a; P. Grinfeld b a The Educational Testing Service, Princeton, New Jersey, USA b Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA

Online Publication Date: 01 May 2001 To cite this Article: Grinfeld, M. and Grinfeld, P. (2001) 'Towards thermodynamics of elastic electric conductors', Philosophical Magazine A, 81:5, 1341 - 1354 To link to this article: DOI: 10.1080/01418610108214445 URL: http://dx.doi.org/10.1080/01418610108214445

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There which is which in in this in extensions instabilities together similar and (i.e. of (i.e. electrostatic these phenomena of Grinfeld (1936) addition et ndc protection anodic (It achievements various formulae various areas. we a the electrostatic ils W discuss We fields. ee been never or nutisicuigpoetn ships protecting including industries a/. conductors is number will and gaps paper of several which is and touch to those the thermodynamic the no equally applicable equally by books antsai interactions. magnetostatic (2000)). We able difficulties used and to and hi development their major not of the of Gibbs. Stratton doubt of and is at hud neg the undergo should the to are not aiu mrhlgcl instabilities. morphological various To are upon field. of to using rmcnrbtos of contributions from the absence the Rosensweig (1985)). Rosensweig various electromagnetism escape in the concepts of by analysis the P. equilibrium present gaps of intermediate purely or Gibbs with a of iud systems liquid explored. that solid Brown In neato of interaction Grinfeld a This general heterogeneous general reader a purely a forces magnetostatic (1941), in above-mentioned thermodynamics think new. few the this of in will interfaces. 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We pipelines). oil The oeseg (1985) Rosensweig elementary aspects elementary theory these greatly of in in motne from importance an of thermodynamics electricity-driven thermodynamics 1988, Lifshitz (1965), Lifshitz forces the solid might widely solids. erig some learning heterogeneous elastic crystal- elastic which great thernio- great classic and systems with systems but systems with systems wih is, (which that publications wide classes wide Clearly they presence concerning problem For it Whittaker influenced review of review heteroge- phenom- pipelines Rayleigh is does not does the be Because some features include known clarifi- are worth books help- Such deri- it is it so of of of of of in Downloaded By: [Massachusetts Institute of Technology] At: 07:12 7 January 2008 demonstrated what specific remarkable hroyai theory thermodynamic static equilibrium dynamics of quasistatic Although tion. theory the of quasistatic theory quasistatic is of on results used orders the dynamic 40years iiain, there limitations, are aeil parameters material important gested sideration analysis tion. simple interfaces in interfaces variant Brown tinuum Eulerian but aito parameter) variation zII(z), (each the spatial coordinate spatial the the technically the equilibrium, the the the the theory the The In According small; second is indices for In ocrs the concerns or z"(z) much index stability this first and in of of simple examples (1966) which is which and current this a a magnetostatic of description foundations for foundations these the of magnitude is the of rather numericnl to paper, useful approach and order as for and tess n electromagnetic in stresses progress less i paper, the contravariant based the heterogeneous of use simpler has the has 5 that effects of effects well analysis and to theory the covariant the 2. publications we second energy second situation the convenient quite presence stability second models of large the is contrary of nonlinear THE all as stability on the share the absence second energy second no magnitude evolution. 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Equations of n vicinity in h shape the h sae evolution, shape the system in to conductor. - iiiy of vicinity further finite the is is to of terms and Equations the conductor's well. interface form. in in SYSTEM the rearrangement. V'V,ip surface We ip the the of deformations it is it look We of = of domain assumptions of evolution, assume quasistatic Fortunately, q(z). [PI' state state constant. mechanical equilibrium mechanical valid of the SI OF SI make = of the charges elastic conductors elastic h total the QUASISTATIC -4q. = elastic energy density. energy elastic Thus, elrctvostatics h total the of of same. for occupied 0, that boundary electrostatic one mechanical equilibrium mechanical the because because and evolution distributed the concerning the The oiiain however. modification, the exact the static nry of energy mass total h thermodynamics the xenl electrostatic external by nonlinearity EVOLUTION and of of field the potential equilibrium. of contains the h rearrangement the nonlinear form nonlinear 7 with h evolution the conductor, the is the in the rearrangement any conductor It the rsaln body crystalline crystalline affects only affects surface density is ip three density is necessary both generated N' We formula compo- field within of of this of is body T 1345 for- the the on (3) (2) the (4) of of of to is Downloaded By: [Massachusetts Institute of Technology] At: 07:12 7 January 2008 12 configurations. distinguish where mass the form the 1346 where monotonic particles tive. surface S, of has We We write We We The (a) (h) (c) It the crystal, the implied xii C(<, the where is assume where where postulate configuration) Murnaghan Within the Within function At At present in is of following the t) between the decrease the process the the primary the is by ii' A;. B':l fixed free the the 'generalized' authors is at A(<, the surface (local) surface that J(<, = is 3.3. a following bulk, velocity of velocity the interface the (195 interface in relationship hi, given form: twice t) motne for importance the t) and fixed Mechanism h total the of - over the interface the over location hope 1) = surface mass surface 0; quasistatic uJi boundary formula: p(C M. the U,,,. interface Sl, Bowen equations S2, that is the Grinfeld -f'N;) mean chemical (A)=-/ the Cauchy the of nry of energy = p(N) interface V,di the JS0 und the (e (1967) us displacement. evolution U' S,, curvature + suggested flux dS lSll = kinetics interfaces in interfaces of and = 1 that G)z~ = potential Xii~i~j, and J -K[/JN) S,: p mcaia' equilibrium 'mechanical' J(6, G', the S1,f" chemical s V'G. = P. (or the dSA - at 0. system Grinfeld t) of' and IF/;. P actual) kinetic of kinetic the is at rearrangement /L(~) (<, the spatial the the each (A) the free potential t) and of - stress tensor stress equation interface undeformed represents equation point interface (p"))], the conservation boundary velocity tensor (<, (12) (in (12) t) S,. the within of defined and the given is guarantees of the average quite atce in particles f its of materials' deformed deformed interface the by by intui- total bulk (12) (11) (10) of the (7) (6) a a Downloaded By: [Massachusetts Institute of Technology] At: 07:12 7 January 2008 problem deformations located stress-deformation tion) that blem equations ln principal plane the in-plane the stress appear on appear boundary isotropic determine when or elongations done conducting conductor the Corresponding field stresses only: component so-called elongations instant moduli instant as Equations e limit We substrate In In uniform is of placed if eti equivalent certain a component P, a the particular, directed the far a P,,,, can metallic elastic ‘instant’ conducting conditions = (6)-(9) case h in-plane the h opposite the al fields (all enough from the from enough elastic layer of (M, are is PM(AN). sizes lj perpendicularly be 3.4. and distribution ourselves can 4. not (2)-(9) vr field every of along can stresses stresses used qa to equal N substances). is DISPERSION P, show uh greater much An ae is layer if nqa principal unequal depositing substance. depositing be h12 energy elastic moduli: elastic equal = attached fields be the at = found to depend elastic conducting layer conducting elastic layer the 1,2,3) = The can expressed Thermodynamics the 2m2 that to system is close is system describe electromechanical describe rnia stresses. principal h*l are are sides of sides to in z density zero, function, of aihs identically). vanishes axis deposited be the of layer’s the following crystalline = with to equal determined the are = surface layer’s to RELATION If a on p, used a other study (figure and R2/8rc. constant than the uniform a certain layer C2I eti functions certain the conducting the in terms in the the is edge. to specified say, to deformations, the = layer edge. electric charges electric two, the two, help the variables quantities by zero. of 2). will conductors. c12 The determine to FOR In flat srs-eomto’ relationship ‘stress-deformation’ qf as thickness by an thickness cran mechanism certain a In = is of The being of particular, e uniformly be elastic conductors elastic nte important Another the function the electric in-plane principal in-plane fe finding After substrate. an A isotropic the A, clamped as the Lame the the equations the in material DEFORMABLE c11 electrostatics isotropic x1 a a We bMN neomd (i.e. undeformed misfit layer. function uniform the = = 2h when Let of of field fields in finite in fields assume (‘22 x of if (in and the at elastic equilibrium the Similar becomes us and in elastic The the the = of of at solid its stressed h neomd configura- undeformed the the consider layer the electric field principal cMN x of the edge is least intensity (9) x2 density SUBSTANCE ehncl equilibrium mechanical edge, + equations that solid eomto gradicnts deformation stresses principal configurations the moduli: lattice 211. main = (of play in situation anisotropic of one z with a so-called elastic layers elastic when then the stress electrostatic the course, two-dimensional only mass uniform T elongations elongations for elongations of the R. depend parameters = layer’s simplest the (2)-(5) the the stresses, AN This iR/4x free, and role is s used is transport. principal it principal principal (this in-plane = hn a when electric appear can the on the points model of 1) show 1347 with pro- and (16) will is the the AN the the in- be x3 to of a Downloaded By: [Massachusetts Institute of Technology] At: 07:12 7 January 2008 1348 the Lame the stress-generated when decrements use deformed figuration oe half-space lower acteristic By The a (a) (b) (c) the conductor superscript considering where (1991). Across the Within Outside disturbed sizes Figure elastic qiiru configuration equilibrium and of small 6/6t of a the conductor's the anisotropy) 2. the conductor, tilde moduli. the open electrostatics in differentiation rearrangements conductor, rearrangement disturbances of disturbances rearrangement An occupies crystalline the for circle elastic the three-dimensional 't M. and the to small conducting Grinfeld interfaces. conductor equations denote its lower is (p of disturbances. 'instant' understood O'O,(p = of the half-plane, constant. various parameters various and hs half-plane this layer length (2)-(5) we = moduli P. in ae. e us Let case). 0. can Grinfeld a in scale uniform electrostatic uniform are lE z limit ourselves limit the the cannot < as sense upper border upper 0 much and (or, follows. be of of correspondingly, xlr increment explore smaller osdr uniformly a consider expressed the the to work equilibrium of the field than the special case special by in terms in solid. 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B2 = the can rl deformations. Therefore, the Therefore, deformations. terms of 4rcro2k configuration. This configuration. - PO2 in = (1900), are Grinfeld K and electric since eut derived results be increment p"2 that an K on kTB2 w instabilities. two given (29) replaced instability in which 1 1 any isotropic - the the by Tonks + + deformations In ak2 charges. - and into are by (1 (I Landau and Landau linear assumptions 4nro2k the relation 1 k(crk or - - is +B, + much P. with the (1935) v)ak/2pL' ~)mk/2/~ case of case equal [( decrement are Grinfeld elastic - theory 1 rm the from The linearized - 47~7"~) an is less (30) of v)/p]kT2 to The very and a driving prxmt form: approximate can Lifshitz (1965)). Lifshitz second than the itself liquid solid of for , first 77: important Frenkel' h magnitude the conclusions based conclusions doubled be master ier theory linear the unity, is force the substance order neglected. is based decrement h rearrangement the increment for system, equation (1936) aiu shear maximum since ih respect with on this instability In this instabil- this the assump- the the case the cannot This of (Frenkel's the 77 we (30) is the of on effects arrive exact does (30) (35) (34) this the de- for be to of Downloaded By: [Massachusetts Institute of Technology] At: 07:12 7 January 2008 place wavelength 7 destabilization which instability with instability Nozieres in in since on assumption linear order more equilibrium fact, some fact, interesting not with elasticity. the master one important one of theless tions theory and out tinguish El/ unit rv the the the =;p/u/ the In eakby enough Remarkably The We Both The Under (K/p02)4~7-02k imply, rearrangement. in a free a the of equation of volume (4) with respect with equilibrium sense thermodynamics oua in popular the dominates exact linear yield the these would electrostatic eod instability second lal between clearly effects of and instabilities (1991) We case system of system of references the of nonlinearity (unknown) that that shape the master +V,U/) conditions elasticity asymptotics: (5) them would course, is like in of approximations implicit the has it the second-order terms second-order the iprin relation dispersion should given the (35): at configuration. a to the to is the various of increment in 9 seem rigid given The system not like equations short the 5. emphasize equations are long-wavelength range. long-wavelength = therein. that crystals. h latter. the physics Thermodynumics are boundary (see, by change. In change. MODIFICATIONS v(,u/): be are of stresses based q=-- to difference the opposed to substance, the a questionable wavelengths h results the imposed the emphasize the (2)-(12) be is simple in imposed 7 neomd configuration undeformed pO2 K two stress-driven = community particular, on remains the compatible In (2)-(5) master of udai form quadratic again and We find (K/p"2)87c~"2p/(1 I-v problem. This problem. 77 any by what p linear equations the intuitive has now = (30). on at which deformations. h stabilizing the on that -rrr based PO2 assumptions that K 1 do xc olna theory nonlinear exact to p the ytm hc i similar is which system been since in of qf FOR follows + this h neomd interface. undeformed the well elasticity, = Nozieres be equation deformations can deformations not elastic. (1 h cruain but corrugation the with even deformed rearrangement and are co, explanation LINEAR on deeply circumstance 02 xlie i detail in explained ~ the (35) r2 known change k. yia in typical we v)vk/2p in the framework the in it ier theory linear the is we of linear conductors is the most the Therefore, obtain the obtain (1 of smallness of and - the nonlinear (30) eut based results shall the influence ELASTICITY 991)). (actual) uh simpler much v)a explicitly. stress-driven lsi nry density energy elastic theory and for the k. (36) of linear linear hw oe modifications some show instabilities to Linear it. in profound and the and profound the be be of be interface. have since cannot the of following itself is based is itself h boundary the neglected. decrement deformation of cannot elasticity h former the configuration to by the There conclusions on of elasticity is much is elasticity the deformations the great importance great we different all One Grinfeld than linear rearrangement surface energy surface the be are are problems In is, prevent formula correct. exact has but This of nonlinear still faced 7 the linear the however, elasticity is second on to condi- short- (1 tensor based none- q5 exact with- most 1351 non- does (37) 99 and dis- the the per in In of 1) Downloaded By: [Massachusetts Institute of Technology] At: 07:12 7 January 2008 where tensor place instead formed 1352 face the the togy deviate strongly Contrary of but equation brium the n the In Consequently, neomd configuration, undeformed The This relation contradicts relation This The In neomd interface, undeformed (a) (c) all and of Ciik' is given is first have the the one has one At interface Within expression eann kinetic remaining equations of to (1 case ier theory linear is the geometric and 2), iiil interface 'initial' no the the the by free physical of the bulk, the second exact nonlinear exact to from the from elastic (we = J(& the formula the a in kinetic use (6) interface for the for non-deformable substance the equations the substance non-deformable ($ the emphasize once emphasize parameters and f) the modulus tensor modulus order Fj meaning Cninkl f elasticity of = CVk' 6. ier version linear nta nefc owing interface initial equation equations following (8), M. tensorial p(C and A OJ; are Vj(Cqk' S,, linear elasticity more elasticity linear V(k with V(,7t NON-DEFORMABLE N' Grinfeld Cqk' the = -f"J master T conceptually either U/l now characterize now CWE and is U,,?, respect p") following = (A) takes Nj chemical h charge the standard (1 V(ku/)) the again Xz"zk' B:, V(k kl 0) system in system = in (in = = =-I and of - - Murnaghan the and are (2TCT2 the Ul, xi'NjNi, -K(p(N)(& PI/ to particular, h master the Cl/kl 1 that = P. equations. small the + different; following bulk = oeta has potential (12) equation: p s Grinfeld + G v pV'G. density to its (zikdi unit dSA, the notions the SUBSTANCE >P (k OLB:?) than or at do zli the undeformed the gradient ,TJ rearrangement). linear version, linear 0' normal and normal t) for - not formula + system, an form: any other. - the per N'. zi'zki)). an Ciik' undeformed change their change (p"))), to interface displacements isotropic substance isotropic nt area unit of of V be one (9) (k 'undeformed' 'mechanical' mean substituted U p It is 0. has interface. is and, contains not the of interface form: curvature to h unde- the valid instead density write, V(kUl). by equili- terms inter- (39) (40) (41) and (44) (42) this (45) (43) can in of in of Downloaded By: [Massachusetts Institute of Technology] At: 07:12 7 January 2008 hc differs which term. ion ing above solution above growth fluid such conductors such energy particles system this in the in ih respect with teristic stability. able tem plane easier via of the crystal. generalized Bowen chemical Bowen generalized surface in establish great dispersion differential deviation packages relation in terms in relation the growth the the In Linearizing We Obviously, formula for the for formula e have We as The surface diffusion. surface We In the In evolution is conductors the and established advantage the exposed to the Eulerian description. Eulerian based crystalline allows of time proposed the master proposed in crystalline in of then exact of handle The the for absence dispersion the from the end equations without any appeal any without equations According materials equations scale of scale to on formulated dispersion show some modifications some show rm h qain o ‘motion for equation the from numerical rpsd master proposed master is one system crystal. of and we to the the master the mechanical principles the the of numerically owing by of small conductor h action the to apply of iprin relation dispersion the looking for the disturbances the for looking increment Frenkel’ fact the unstressed non-deformable substance non-deformable equation salsig qiiru with equilibrium establishing investigate is electric system consists The to within Thermodynamics simplified system much greater much to disturbances equation classical solutions that slow our this a the rearrangement the system suggested and new and system c q-=- leads than, is equation, the equation, (1936) half (1936) of system potential and the total the the evolution of q: for the dynamic the for PL =- K notions of the of notions proportional system electrostatic a kinetic state the stable the Ja 5 PP Lam& and the mass (48) bulk for of uniform to 7. say, = (2)-( to -g mechanism leads mechanism such systems. such equation than of elastic, CONCLUSION a K-(sy, we energy for (48) esr vrisaeaevle over value average its over tensor is of fields non-deformable the the equation the + equation an of qf 12) of is a century a based 4m2k express the the the to that equilibrium mass elastic conductors elastic h master the electrostatic these isotropic case o the for for the quasistatic evolution of oso moduli, Poisson degrees of degrees electrostatic of to we the by of exact increment time conductor. - (30) surface it hi aeil particles. material their on leads the - flux of in the (Bq”))) mean conductors arrive (12) leads concepts the ak2. ago (Landau and (Landau ago the form the scales of for an increment the assumption the respect nonlinear continuum nonlinear system local at describing mass describing uniformly us of incompressible to coefficients the ytm hc yield which system to shapes waves a curvature’ at freedom. il. In field. crystal in crystal or to certain the an the xes of excess and There the equation the increment of to is the same the decrement establishing equilibrium establishing equation we arising mass namely linear diminishing. in the of of rearrangement surface arrive stressed point a are the crystalline small The the exact conducting owing of rearrangement system of system that the due many case stability criter- stability equation or isiz 1965). Lifshitz rearrangement which the vicinity substance and substance on proposed at energies. in disturbances What value decrement theory. elastic to the charac- the the to the the mechanics of dispersion the shapes The h same the powerful a loss a deform- the last the is follow- small a surface growth partial of of of drives liquid whole much (34). 1353 total half- (47) The sys- The the the the of of in rl Downloaded By: [Massachusetts Institute of Technology] At: 07:12 7 January 2008 ion and (1 of placed BECKER of bined BOWEN, 1354 TAPPIN, NOZIERES, TAMM, FRENKEL', MURNAGHAN, MAUGIN, WHITTAKER, THILLY, LANDAU, GRINFELD, GIBBS, BROWN, ROSENSWEIG, RAYLEIGH, STRATTON, TONKS, 936). small the paper. This as visualization based numerical established J. I. (Sussex: Nauka) University land). at Dover Publications). Dover 2000, University Press), University L., R., R. D. W. L., disturbances. memorial E., W., G., L. www.geocities.com/pavel-grinfeld. YA. J. M., J. M. P., 1935, K., 1982, F. COLIN, D., E., 1989, W., Actu A,, 1988, R. 1876, This F. A., 1966, I., ROBERTSON, (in 1991, 1967, Longman); and 1989, E., D., Phys. Rev., Phys. modelling 1941, 1900, Electromagnetic 1936, Press), Foundations muter. Russian). long Continuum 1991, computer Trans. Connect. Trans. J., paper 1985, Magnetoelastic LIFSHITZ, 1951, Arch. Solids A LECOUNTURIER, Papers Scient$r Electromagnetic The Zh. ago for History Thermodynumics Thermodynamic on pp. pp. metall., Ferroh~droc1ynumic.s Finite was written Rational 1993, iksp. I. suggested the 48, and Far 92-93. 130-1 Mechanics part M., oj E. 562. master ojthc a Deformations J. (to visualization. teor. the Fields From M. ACKNOWLEDGEMENT liquid and 3 Arad. is Nonlinear Infrractions 1. Mech. be Theory 1965, being REFERENCES Fiz., Theories BIRNBAUM, by master F., Theory published). and Interactions and to system Equilibrium, qf Methods conductor in Sci., J. 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