Generalized micropolar continualization of 1D beam lattices

Transcription

1 Generalzed mcropolar contnualzaton of 1D beam lattces Andrea Bacgalupo 1 and Lug Gambarotta * 1 IMT School for Advanced Studes, Lucca, Italy Department of Cvl, Chemcal and Envronmental Engneerng, Unversty of Genova, Italy Abstract The enhanced contnualzaton approach proposed n ths paper s amed to overcome some drawbacks observed n the homogenzaton of beam lattces. To ths end an enhanced homogenzaton technque s proposed and formulated to obtan thermodynamcally consstent mcropolar contnuum models of the beam lattces and able to smulate wth good approxmaton the boundary layer effects and the Floquet-Bloch spectrum of the Lagrangan model. The contnualzaton technque here proposed s based on a transformaton of the dfference equaton of moton of the dscrete system va a proper down-scalng law nto a pseudo-dfferental problem; a further McLaurn approxmaton s appled to obtan a hgher order dfferental problem. The formulaton s carred out for smple one-dmensonal beam lattces that are, nevertheless, characterzed by a rather wde varety of statc and dynamc behavors: the rod lattce, the beam lattce wth node rotatons and a 1D beam lattce model wth generalzed dsplacements. Hgher order models may be obtaned whch are characterzed by dfferental problems nvolvng non-local nerta terms together wth spatal hgh gradent terms. Moreover, the homogenzed models obtaned by the proposed enhanced contnualzaton technque turn out to be energetcally consstent and provde a good smulaton of both the statc response and of the acoustc spectrum of the orgnal dscrete models. The proposed homogenzaton procedure s frst presented for the smple case of monoatomc axal chans. The beam lattce wth node rotatons and dsplacement prevented exhbts, n the statc regme, decayng oscllatons of the nodal rotaton n the boundary layer whch s well smulated by the homogenzed model obtaned by the proposed approach. Smlar good results are obtaned n the smulaton of the optcal spectrum. It s worth to note that n ths case the homogenzed model obtaned va Padé approxmaton turns out to be energetcally non-consstent. The analyss of the homogenzed model derved from the beam lattce wth transverse dsplacement and rotaton of the nodes wth elastc supports has shown that both the statc and the dynamc response are strongly varable on the parameters of the Lagrangan model. Fnally, several dfferent cases have been consdered and good smulatons have been obtaned both n descrbng the statc response to prescrbed dsplacements at the end nodes and n representng the Floquet-Bloch spectrum and the polarzaton vectors. Keywords: Beam lattce; Metamaterals; Contnualzaton; pseudo-dfferental equaton; Dspersve waves; Band gaps; mcropolar modelng. * Correspondng Author

2 1. Introducton Snce Newton's theory of sound propagaton, lattces have consttuted a class of fundamental models n many scentfc felds. In Mechancs, lattce models are appled to capture, at the relevant scales of observaton, the phenomena related to the dscreteness of the systems to be represented, from the propagaton of dspersve elastc waves (see Brlloun, 1946) to the stffness and strength of nhomogeneous materals (see Fleck et al., 010), just to menton a few. Although the smplcty of lattce models allows n many cases to obtan analytcal results, however, ther applcaton to systems wth complex geometry and large dmensons nvolves a very hgh number of degrees of freedom and precludes the synthetc descrpton of the results. Therefore, many studes have long been focused on the formulaton of equvalent contnua able to smulate the statc and dynamc response of lattce, ncludng the nfluence of sze effects. To ths end, technques for the contnualzaton of the dscrete equatons of the lattce moton have been proposed, whch consst n replacng the dscrete equatons of moton and the nodal dsplacements n the lattce wth dfferental equatons of moton n terms of a sutably dsplacement feld defned on an equvalent doman. Hgher order contnuum models have been formulated to smulate the dspersve wave propagaton occurrng n one-dmensonal chans (Brlloun, 1946),.e. lattces wth axal dsplacement of the nodes (see Askes et al., 00). These contnuum models are derved through a contnualzaton of the equatons of moton of the Lagrangan model or equvalently of the energy functonal, n whch the dfference of dsplacements of adjacent nodes s approxmated by a truncated seres of a macro-dsplacement feld (Askes and Metrkne, 005). Despte the smplcty of the approach, the homogenzed models based on ths standard contnualzaton present loss of postvty of the elastc potental energy densty, magnary frequences n the elastc feld, unbounded group veloctes n the short-wave lmt, just to menton the man drawbacks (Metrkne and Askes, 00). In order to elmnate these pathologes several enhanced technques have been proposed. The dfference equatons of moton of the Lagrangan model are transformed nto a system of pseudo-dfferental equatons whch are approxmated by the Padé approxmant and non-local nertal terms, nvolvng the spatal dervatves of the contnuous acceleraton feld, are ntroduced ndrectly n the contnuum dfferental problem (Kevrekds et al., 00, Rosenau, 003, Andranov and Awrejcewcz, 008, Andranov et al., 01, Challamel et al., 016, 018). Alternatvely, enhanced hgher order models have been proposed relyng on heurstc generalzaton of the down scalng law coupled wth a perturbaton method (Metrkne and Askes, 00) or on the assumpton that the PDE governng the dynamc

3 behavor of the contnuum model must be of the same order wth respect to spatal coordnate and wth respect to tme (Metrkne, 006). More complex s the case of perodc beam lattces, whose nodes can undergo translatons and rotatons, even wth rotatonal nerta, and are connected by lgaments havng axal and bendng stffness. Ths nvolves a complex dynamc behavor due to the couplng of translatonal and rotatonal modes, wth the exstence of both acoustc branches and optcal branches n the Floquet-Bloch spectrum. Suker et al., 001, analyzed several D beam lattces and derved the equatons of moton of the homogenzed equvalent medum through a standard contnualzaton, whle Gonella and Ruzzene, 008, and Lombardo and Askes, 01, addressed the case n whch the rotatonal nerta s neglected. An mproved homogenzaton technque based on a mult-feld approach has been formulated by Vaslev et al., 008, 010, 014, and appled to square lattces endowed of rotatonal nerta. It s mportant to note that the contnualzaton of the dfference equatons wth a second order Taylor approxmaton of the generalzed dsplacement feld mples a nonpostve defnte potental energy densty of the equvalent contnuum. Ths result, already obtaned by Bažant and Chrstensen, 197, n developng a mcropolar homogenzaton of large scale multstory framed structures, was taken up by Kumar and McDowell, 004, addressng the problem of the statc mcropolar homogenzaton of plane beam-lattces. In partcular, the energy densty assocated wth the mcro-curvatures turns out to be nonpostve defnte, a result that s also confrmed on the bass of energy equvalence. It s worth to note that other approaches based on a frst order down-scalng law, n whch the overallelastc modules are obtaned through the Hll-Mandel macro-homogenety condton (see for example Pradel and Sab, 1998, Onck, 00), nvolve a postve defnte elastc energy densty and therefore would seem not affected by thermodynamc nconsstences (see Kumar and McDowell, 004). However, t has recently been shown by the authors (Bacgalupo and Gambarotta, 017a) that the mcropolar model obtaned by the second-order standard contnualzaton presents, despte the lmts mentoned above, shows a remarkable accuracy n smulatng the acoustc behavor of beam lattces. In fact, ths model s able to catch the decreasng of the optcal branch observed n the Lagrangan model wth ncreasng the wave vector from the long-wavelength lmt, a feature that s qualtatvely opposte to that shown from the mcropolar model wth postve defnte elastc energy densty related to the mcrocurvatures. It s worth hghlghtng that smlar results were obtaned n mcropolar modelng of perodc materals made up of rgd blocks connected by elastc nterfaces (Bacgalupo and 3

4 Gambarotta, 017), whch provded some clarfcaton about problems concernng waves propagaton n perodc granular meda (Merkel et al. al., 011). These consderatons emphasze the need for an enhanced homogenzaton technque to obtan thermodynamcally consstent mcropolar contnuum models of the beam lattces able to smulate wth good approxmaton the boundary layer effects and Floquet-Bloch spectrum of the Lagrangan model, at least for long and medum wavelengths. In ths paper an enhanced contnualzaton technque s proposed that draws on the approach by Rosenau, 003, and s formulated for smple one-dmensonal beam lattces that are, nevertheless, characterzed by a rather wde varety of statc and dynamc behavors. The most notceable aspect s the presence of hgher order terms both n the consttutve and n the nertal terms of the dfferental equaton of moton of the equvalent contnuum, whch are derved ndependently. The proposed homogenzaton procedure s frst presented for the smple case of monoatomc axal chans (Secton ). The dfference equaton of moton of the reference mass s transformed, through a sutable down-scalng law regularzed to the frst order, nto a pseudo-dfferental equaton and the dfferental equaton of moton of the contnuum model s obtaned by approxmatng the pseudo-dfferental operators up to the requred order term of ts Taylor seres. The capablty of the resultng hgher order contnuum models to approxmate the Lagrangan one s evaluated both n the statc and n the dynamc feld. Moreover, comparsons wth the results from the models based on the standard contnualzaton and on the Padé approxmaton are shown. The second smple typology here consdered conssts of a contnuous beam wth rotatng nodal masses, wth prevented transverse dsplacement, connected by elastc lgaments modeled as Euler-Bernoull beams (Secton 3). Ths system exhbts a characterstc statc response when rotatons are mposed at the end nodes of a beam of fnte length, wth a spatally decayng of the nodal rotatons and a decreasng optcal branch n the acoustc spectrum. Also n ths case the results of the enhanced contnuum are compared wth those of the Lagrangan model and wth those of the contnuum models derved through the standard contnualzaton and the Padé approxmaton. Fnally, a thrd dscrete system of contnuous beam s consdered equpped wth equally spaced nodal masses undergong transverse dsplacements and rotatons, and elastc supports located at the nodes. Ths model may represent dfferent dscrete systems ncludng rectangular lattces undergong 1D generalzed dsplacement felds (Secton 4.a) and wavegudes wth passve control of the acoustc response (Secton 4.b). Also for these systems both the statc response to prescrbed dsplacements at the end nodes and the acoustc 4

5 spectrum are compared wth those of the Lagrangan system and wth those obtaned n the contnuum models derved through the standard contnualzaton and the Padé approxmaton. The energetcally consstent structure of the densty of the elastc potental energy and of the knetc energy of the contnuous models derved wth the proposed approach s dscussed n detal, as well as the valdty lmts of the contnuum models derved through the homogenzaton technques consdered. It s ntenton of the Authors to extend the proposed approach to D and 3D lattces.. 1D rod lattce model Let us consder a 1-D lattce wth n+1 nodes wth mass m equally spaced and connected through lgaments of length wth axal stffness h (see Fgure 1). The axal force appled on the -th node s denoted by f and u s the dsplacement. In terms of the nondmensonal axal dsplacement u = the equaton of moton f the -th node s wrtten as + + f = I, (1) f m wth f = and I h = h. In case of prescrbed dsplacements 0 and n at the end nodes 0 and n, respectvely, the statc problem nvolves the equlbrum of all the nodes wth a system of n-1 lnear homogeneous equatons whose soluton s a lnear nterpolatng functon. Fgure 1. 1-D rod lattce: axal appled forces and nodal dsplacements. In case of a lattce of free partcles (see Brlloun, 1946), the harmonc axal wave propagaton may be represented n the form ( t) =exp j ( kx ωt) 5, where j = 1, k s the wave number, x s the locaton of the -node wth respect to the orgn O. The angular frequency ω ( ) = sn = 1 ( k ) + ( k ) + ( k ) I I 4 190, ()

6 s a dsperson functon wth perodcty π and takes the role of acoustc branch n the Bloch spectrum. Standard contnualzaton To obtan an equvalent contnuum model, a contnualzaton approach s commonly appled that may be approached by applyng the shft operator E relatng the dsplacement of two adjacent nodes, namely + 1 = E (see Andranov and Awrejcewcz, 008, Lombardo and Askes, 010, for reference). h h E = D = exp( D), wth h! h= 0 D h h x governng equaton (1) may be wrtten n the form If a contnuous feld ( xt, ) h The shft operator may be represented as 1 = exp ±. Accordngly, the =, so that ( t) ( D) ( ) ( ) ± exp D + exp D + f = I. (3) Ψ s ntroduced to represent the non-dmensonal dsplacement n the equvalent contnuum model wth ( t) ( x, t) resultant of a step-wse dstrbuton f pseudo-dfferental equaton =Ψ and the nodal force s assumed as the = f, then equaton (1) may be converted nto the ( ) ( ) exp D + exp D Ψ+ f = I Ψ, (4) wth the dfferental operator n the square brackets may be expanded nto McLaurn seres n D ( D) + ( D) = D + D + D + ( D ) exp exp. (5) When retanng the terms up to the second order, the classcal equaton of the elastc rod s obtaned Ψ + = Ψ f I, (6) x whose response to prescrbed dsplacements at the rod ends s lnear n agreement wth the dscrete model. However, t s well known that the wave propagaton derved by equaton (6) s non dspersve ω=, consequently ths contnuum model provdes a frequency I spectrum that agree wth the dscrete one only for the long wave-length lmt 0. If a 6

7 further term s retaned n expanson (5) a second-gradent equvalent contnuum s obtaned wth feld equaton 4 Ψ 1 4 Ψ + + f = I 4 Ψ, (7) x 1 x that s the Euler-Lagrange equaton derved by assumng the followng Lagrangan densty functon Ψ Ψ = IΨ + fψ x 1 x, (8) wth not-postve-defnte elastc potental densty (see Challamel et al., 016). The wave propagaton n model (7) s dspersve wth dsperson functon ω= 1 k = 1 k k ( k ) I I. (9) As t s well known, the second gradent model turns out to be rather accurate n descrbng the dsperson functon n the neghbor of the long-wave lmt although equaton (9) exhbts an anomalous dsperson snce ω becomes magnary for k > 3. Moreover, the nonpostvty defnteness of the elastc potental energy densty prevents the applcaton of ths model n statc problems. If the next hgher-order term s retaned n (5) the dsperson functon turns out to be postve, but ts applcaton s lmted snce n the short-wave lmt ( ) the group velocty s unbounded (see Metrkne and Askes, 00). It s worth to note that the same results are obtaned f the contnualzaton of the energy functonal s appled, as shown by Askes and Metrkne, 005. Enhanced contnualzaton va frst order regularzaton approach A dfferent approach s here consdered to obtan a contnuum model able to smulate both the statc and the dynamc response of the dscrete Lagrangan system. A contnuum dsplacement feld ( xt, ) Ψ s consdered followng an approach that s smlar to that proposed by Rosenau (003). The dervatve at x of such functon s related to the central dfference as follows Ψ = x (10) 7

8 By ntroducng the shft operator prevously defned, the dervatves of the contnuum felds may be expressed as ( D) exp( D) Ψ exp = D Ψ= x, (11) from whch one obtans the down-scalng law for the node translaton n terms of the contnuous felds D ( t) = exp Ψ ( D) exp( D) ( x, t). (1) By substtutng equaton (1) nto the dscrete equatons of moton (1) and applyng the shft operator, the pseudo-dfferental equatons of moton of the equvalent contnuum take the form ( D) + ( D) I ( D) ( D) ( D) ( D) exp exp D Ψ+ f = D Ψ exp exp exp exp, (13) where the dfferental operator n the braces may be expanded nto McLaurn seres n D ( D) + ( D) ( D) exp( D) exp exp D= D D + D + D exp 1 10 D = D + D D + D exp ( D) exp( D) ( ) ( ) , (14) When retanng the terms up to the second order, the equaton of moton of the equvalent contnuum takes the form Ψ x 6 Ψ x + f = I Ψ, (15) correspondng, n the statc range, to equaton (6), whle n the dynamc range a non-local nerta term appears. The dsperson functon takes the form ω= = 1 ( ) 1 k + k + k. (16) I 1 k I + 6 If terms up to the fourth order are retaned n (14), the equaton of moton of the homogenzed contnuum takes the form 8

9 Ψ Ψ Ψ 7 Ψ + f = I 4 Ψ + 4. (17) x 1 x 6 x 360 x In ths case the dsperson functon s 1 1+ k ω= = 1 k k ( k ) I k 7 +. (18) 4 4 I k It s worth to note that equaton (17) s the Euler-Lagrangan equaton derved by assumng the followng Lagrangan densty functon Ψ 7 Ψ 1 Ψ Ψ = I Ψ fψ 6 x 360 x x 1 x, (19) wth both the elastc potental densty energy and the knetc densty energy postve defnte. Contnualzaton va Padé approxmant It s worth to note that nonlocal nerta terms may be obtaned on the bass of Padé approxmatons (see for reference Kevrekds et al., 00, and Cuyt, 1980). In ths case the equaton of moton s gven n the form Ψ x Ψ 1 x + f = I Ψ, (0) and the homogenzed contnuum s characterzed by postve defnte elastc potental energy densty and knetc energy densty; the dsperson functon takes the form ω= = 1 ( ) 1 k + k + k. (1) I 1 k I + 1 Benchmark test for the contnualzaton approaches A comparson of the dsperson functons n the Bloch spectrum obtaned by the consdered homogenzaton approaches wth the dsperson functon by the Lagrangan model s shown n the dagrams of Fgure. In the dagrams of Fgure.a t may be observed that the best approxmaton to the exact soluton (black lne) s obtaned through the fourth order 9

10 proposed model (red lne). However, also the second order proposed model (volet lne) seems to be n good agreement also for π, namely for wavelength λ. A capablty that s not shown by the contnuum model obtaned va Padé approxmant (yellow lne). Wth the excepton of the models based on the standard contnualzaton (blue lne), the other consdered models are characterzed by a postve defnte elastc potental energy densty and knetc energy densty. In the dagrams of Fgure.b, the dsperson functon of the Lagrangan model (black lne) s compared wth the dsperson functons by the enhanced homogenzed model obtaned for dfferent orders of approxmaton. ω I ( a) ω I ( b) Fgure. Dsperson functons of the 1D rod lattce. (a) Comparson of dfferent models wth the Lagrangan model () (black lne); 4 th order standard contnualzaton (9) (blue); proposed nd order contnualzaton (16) (volet); proposed 4 th order contnualzaton (18) (red); nd order contnualzaton va Padé approxmant (1) (yellow). (b) Senstvty of the proposed model on the consdered order of approxmaton: 6 th order contnualzaton (cyan); 8 th order contnualzaton (brown). 3. 1D beam lattce wth node rotatons Let now consder a lattce made of equally spaced n+1 nodes free to rotate n the same plane wth prevented dsplacements. The nodes have equal rotatonal nerta I and are connected through elastc Euler-Bernoull beams as shown n fgure 3. The deformed confguraton of the lattce s defned by the nodal rotaton (=0,n) to whch the couple c 1 are energetcally assocated. The knetc energy of the -th nodal mass s T = I. The 10

11 elastc potental energy of the -th lgament between the -th node and the +1 node s 1 Π e = 4 H ( ), beng EJ H = the flexural stffness. Moreover, n case of appled conservatve generalzed forces, the correspondng potental energy s Π f = c. The Euler-Lagrange equaton of moton of the -th node s wrtten n the form beng 1 ( ) + c = I, () 6 I I = 1H and c c =. It s worth notng that the model corresponds to a contnuous 1H beam loaded at the supports by couples and wth possble rotatons prescrbed at the end nodes. Fgure 3. 1-D beam lattce: appled couples and nodal rotatons. The soluton of the lnear dfference equaton wth homogeneous condtons and rotatons 0 and Peterson, 001, and takes the form n prescrbed at the end nodes s carred out accordng to Kelley and ( 3) ( 1) ( 3) = C1 + + C +, (3) wth growng or decayng oscllaton of the nodal rotatons as an effect of the second addend n (3) dependng on ( 1) and where the two constants depend on the boundary condtons as follows n n n ( ) ( ) n ( ) ( ) ( ) n n ( ) n ( ) ( ) ( ) C =, C = n n n n. (4) The propagaton of the rotatonal waves turns out to be dspersve and the dsperson functon takes the form ( ) + cos ω= = 1 k + k + ( k ), (5) 3I I

12 and represents an optcal branch n the Bloch spectrum wth perodcty π. Standard contnualzaton In developng a standard contnualzaton, the functon ( xt, ) Φ s ntroduced to represent the rotatonal feld n the equvalent contnuum. Once ntroduced the shft operator, a pseudo-dfferental equaton s derved 1 exp ( D) exp ( D) Φ+ c = I Φ 6, (6) wth the nodal couple assumed as the resultant of a step-wse dstrbuton c = c. If the term n the square brackets s expanded n D n analogy wth (5) and the terms up to the second order are retaned, the dfferental equaton of moton s obtaned Φ The correspondng Lagrangan densty functon s 1 Φ+ c = IΦ. (7) 6 x 1 1 Φ = IΦ Φ + cφ, (8) 6 x wth non-postve-defnte elastc energy densty. When consderng the statc homogeneous problem wth rotatons prescrbed at the end nodes, equaton (7) takes the form of a 1D Helmholtz equaton whose soluton s 6x 6x Φ ( xt, ) = C1sn + Ccos, (9) characterzed by an oscllatng behavour wth constant ampltude that qualtatvely dffers from the reference soluton (3). The dsperson functon s ω= 1 k = 1 k k + ( k ) I 6 I 1 88, (30) wth a crtcal pont n the Bloch spectrum n the longwave lmt, where the group velocty vanshes v ( k ) g dω 0 = = 0 and from where an optcal branch departs. dk 0 Retanng a further term n the McLaurn expanson, the followng hgher order contnuum model s obtaned 1

13 4 1 Φ 1 4 Φ Φ + c = I 4 Φ, (31) 6 x 7 x to whch a pathologcal Lagrangan densty functon s assocated, wth non-postve-defnte elastc potental energy densty Φ Φ 1 = IΦ Φ + + cφ. (3) 6 x 7 x In ths case the optcal dsperson functon s obtaned ω= 1 k + k ω= 1 k + k + ( k ) I 6 7 I 1 88, (33) wth the same crtcal pont at the longwave lmt. Enhanced contnualzaton va frst order regularzaton approach If the downscalng law to represent the rotaton of the nodes n terms of a macrorotaton feld ( xt, ) Φ s assumed n analogy to the procedure presented n equatons (10), (11) and (1), the equaton of moton () takes the form of a pseudo-dfferental problem ( D) + + ( D) I ( D) ( D) ( D) ( D) exp 4 exp D Φ+ c = D Φ. (34) 3 exp exp exp exp The terms n the braces may be expanded nto a McLaurn seres; the expanson truncated up to the fourth order provdes the dfferental equaton of moton of the equvalent contnuum system Φ Φ 7 Φ Φ + c = I 4 Φ x 6 x 360 x, (35) to whch the Lagrangan densty functon s assocated Φ 7 Φ 1 Φ = I Φ + + Φ + + cφ 6 x 360 x 180 x, (36) wth both the elastc potental energy densty and knetc energy postve defnte. The soluton of the homogeneous problem wth prescrbed rotatons at the ends takes the form 4 x 4 x 4 x 4 x x 4 x Φ ( xt, ) = C1e + Ce sn 45 + C3e + C4e cos 45, (37) 13

14 and the dsperson functon s wrtten as k ω= = 1 k + k ( k ). (38) I + k + k I ( ) Contnualzaton va Padé approxmant Fnally, f the Padé approxmant of the l.h.s of the pseudo-dfferental equaton (6) s consdered, namely 1 1+ ( D 1 1 ) exp( ) 4 exp( ) cosh ( ) 1 6 D + + D Φ= + D Φ Φ ( D) 1, (39) the resultng dfferental equaton of moton of the equvalent homogenzed contnuum takes the form Φ Φ Φ+ c = I Φ. (40) 1 x 1 x It s worth to note that the Lagrangan densty functon from whch equaton (40) s derved s 1 Φ 1 Φ = I Φ + Φ + cφ 1 x 1 x, (41) and s characterzed by a non-postve defnte elastc potental energy densty. As a consequence, also for ths model an oscllatng behavor wth constant ampltude s obtaned ( ) 3 x 3 x Φ xt, = C1sn + Ccos, (4) whle the dsspaton functon s 1 1 k ω= = 1 k k ( k ) (43) 1 88 I 1 k I + 1 The fourth order contnuum model derved by a Padé approxmaton s gven n Appendx A, where t s shown that a non-postve-defnte elastc potental energy densty s obtaned. 14

15 Benchmark tests for the contnualzaton approaches The capabltes of the proposed homogenzaton approach may be assessed by the smulaton of the dscrete system n both the statc and the dynamc feld. As a statc case, let us consder a contnuous beam made up of n=10 lgaments subjected to a prescrbed rotaton 0 = 10 at the end left node wth restraned rotaton at the rght one 10 = 0. The rotatons at the nodes of the dscrete system gven by (3) and (4) are represented n the dagram of Fgure 4.a together wth the macro-rotaton feld Φ ( x) obtaned by solvng the governng dfferental equaton derved by the proposed enhanced homogenzaton, va r th order contnualzaton, wth approprate boundary condtons,.e. d h Φ dx h x = 0/ n = 0, h= 1,.., r. Here dfferent solutons are plotted whch are obtaned by truncatng the terms n braces n the pseudo-dfferental equaton (34) at ncreasng orders, startng from the fourth order (see equaton (37)), whle the solutons by the standard contnualzaton and the Padé approxmaton have been gnored beng characterzed by a non-postve defnte elastc potental energy. In Fgure 4a the decayng oscllaton of the nodal rotatons of the dscrete model close to the boundary layer s shown, whch s smulated rather well by the enhanced contnuum models here proposed, wth a tendency to converge to the actual soluton when ncreasng the order of the contnuum model. The good accuracy of the enhanced homogenzaton here proposed may be observed n Fgure 4b, where the nodal rotatons from the Lagrangan model (black pont) are compared wth the nodal rotatons obtaned va downscalng D = Φ ( x) = exp ( D) exp( D) x 1 d Φ 7 d Φ 31 d Φ = 1 + Ψ + 6 dx 360 dx 1510 dx ( x ) ( ) (44) from the enhanced models at the dfferent orders of accuracy. Here, for the contnuous model obtaned va r th order contnualzaton, the down-scalng relaton s truncated at r- order. 15

16 Φ,φ 10 ( a) φ 10 ( b) xn xn Fgure 4. Node rotatons n the Lagrangan model (black dots) vs. (a) rotatonal feld n the enhanced contnuum and (b) node rotatons obtaned va down-scalng relatons by enhanced contnuum: 4 th order contnualzaton (red); 6 th order contnualzaton (cyan); 8 th order contnualzaton (brown); 10 th order contnualzaton (orange); proposed 10 th order contnualzaton (orange). ω I ( a) ω I ( b) Fgure 5. Dsperson functons of the 1D beam lattce wth node rotaton. (a) Comparson of dfferent models: Lagrangan model () (black lne); standard contnuous model nd order contnualzaton (30) (green); standard contnuous model 4 th order contnualzaton (33) (blue); proposed 4 th order contnualzaton (38) (red); nd order contnualzaton based on Padé approxmaton (43) (yellow), 4 th order contnualzaton based on Padé approxmaton (A.5) (gray). Note that the black and the gray lne are almost supermposed. (b) Senstvty of the proposed model on the consdered order of approxmaton: 6 th order contnualzaton (cyan); 8 th order contnualzaton (brown); 10 th order contnualzaton (orange). 16

17 The valdty lmts of the enhanced model to smulate the propagaton of harmonc waves n the Lagrangan system may be apprecated n the Bloch spectrum of Fgure 5.a. From these dagrams t appears that the best smulaton s obtaned by the fourth order model based on the Padé approxmaton (the grey lne s almost supermposed on the black lne), even f, as shown n Appendx A, the elastc potental densty of ths model s non-postve defnte. On the other sde, t s worth to note that the dsperson functon (38) by the fourth order enhanced contnuum model (red lne) turns out to provde a good smulaton of the dsperson functon of the dscrete model and s characterzed by a postve defnte elastc potental energy. Fnally, n Fgure 5.b the optcal branch from the Lagrangan model s compared wth those obtaned through the enhanced contnualzaton up to the 10 th order. It results that ncreasng the order of the enhanced contnuum, the correspondng dsperson functons tend to the correspondng optcal branch of the Lagrangan model. 4. 1D beam lattce model Let consder now the more general 1D beam lattce shown n Fgure 6. Ths model s derved from the one analyzed n the prevous Secton wth transverse dsplacement and rotaton elastcally restraned wth translatonal and rotatonal elastc restrants of stffness K and K, respectvely. The deformed confguraton of the lattce s defned by the transverse deflecton v and the rotaton Transverse forces ( 0, n) = of the nodes; axal dsplacements are gnored. f and couples c are appled at the nodes. Ths model may be representatve of dfferent elastc systems, among whch those dealt wth n the next subsectons. When assumng K = K = 0 ths model appears to be smlar to those one analysed by Vaslev et al., 010. Fgure 6. 1-D beam lattce. 17

18 The non-dmensonal deflecton s ntroduced v = and the knetc energy of the th nodal mass s T = mv + I = m + I. The elastc potental energy of the -th lgament between node 1 and node node s 1 Π = 4 H + + ( ) ( )( 1 ) ( 1 ), beng EJ e H = flexural stffness and = the average rotaton of the -th lgament. The equvalent 1 H 1 form Π = 1 ( + ) + ( ) e the may be consdered that s n agreement to Vaslev et al., 008. The elastc potental energy stored n the elastc restrants at the -th node s Π R = ( K + K ) 1. Moreover, n case of appled conservatve generalzed forces, the correspondng potental energy s Π f = f c. The resultng Euler-Lagrange equatons of moton of the -th node are wrtten as a system of dfference lnear equatons 1 1 ( + K ) ( + 1 1) + f = I 1 1 ( ) + ( 4+ 6K ) + + c = I , (45) where the non-dmensonal parameters are ntroduced I I = 1H, K K = and 1H K = 18 f f = 1H, c c =, I 1H = m 1H K. The propagaton of elastc waves along the 1D 1H system s analyzed under the customary hypothess of harmonc waves ( t) exp j ( kx t) and ( ) exp ( ) t = j kx ωt = ω, and the dsperson functons are obtaned as soluton of the egenvalue problem governed by a Hermtan matrx ( Lag ) H Lag, namely 1 cos( ) + K jsn ( ) I 0 H ω I υ = ω 1 0 I = 0. (46) jsn ( k ) + cos( k ) + K 3 In the Bloch spectrum the two solutons of (46) may represent ether (a) an acoustc and an optcal branch or (b) two optcal branches, dependng on the stffness of the nodal elastc,

19 restrants. In fact, for the long wavelength lmt 0, the angular frequency are ω = 0 K I and ω 0 = obtaned only f K = K I, from whch t turns out that an acoustc branch may be Enhanced contnualzaton va frst order regularzaton approach To obtan an equvalent contnuum, a macro-dsplacement feld ( x, t) Ψ and a macrorotaton feld ( x, t) Φ are consdered n analogy to the approach presented n Secton. Once assumed up-scalng law based on (10) and ntroduced the shft operator, the down-scalng law s obtaned D D ( t) = Ψ( x, t), ( t) = Φ x, t exp D exp D exp D exp D ( ) ( ) ( ) ( ) ( ). (47) Accordng to the procedure nvolvng the shft operator presented n Sectons 1 and, the system of pseudo-dfferental equaton s obtaned exp( D) ( K ) exp( D) + + I D Ψ DΦ+ f = D Ψ exp ( D) exp( D) exp( D) exp( D) exp( D) ( 4 6K ) exp( D) I D Ψ D Φ+ c = D Φ 3 exp ( D) exp( D) exp ( D) exp( D), (48) Through an expanson nto McLaurn seres n D of the dfferental operators n the braces retanng the terms up to the second order, the feld equatons of the homogenzed model are obtaned 1 6 x x 6 x Ψ 1 Φ Φ ( 1+ K ) Φ+ K + c = I Φ x 6 x 6 x 1 Ψ Φ Ψ K Ψ+ + K + f = I Ψ, (49) It s worth to note that both the elastc potental energy densty and the knetc energy densty n the Lagrangan densty functon assocated to (49) turn out to be postve defnte 1 Ψ 1 Ψ 1 Φ Π e = Φ + K Ψ + K + K Φ + K, (50) x 6 x 6 x 19

20 1 Ψ 1 Φ T = I Ψ + + I Φ +. (51) 6 x 6 x Retanng fourth order terms n the McLaurn seres, the system of dfferental equatons of the equvalent homogenzed contnuum s obtaned Ψ Ψ Φ Ψ 7 Ψ K Ψ+ 1+ K + K 6 + f = I Ψ x x x x x Ψ 1 Φ Φ Φ 7 Φ ( 1+ K ) Φ+ K + K c I + = 4 Φ + 4 x 6 x x 6 x 360 x, (5) and the elastc potental energy densty takes the postve defnte form 1 1 Ψ Ψ Π e = ( Ψ, x Φ ) + K Ψ + K + + K + 6 x x 1 Φ Φ + K Φ + K + + K 6 x x, (53) as well as the knetc energy densty 1 Ψ 7 4 Ψ 1 Φ 7 4 Φ T = I Ψ I Φ + +. (54) 6 x 360 x 6 x 360 x In the followng two representatve cases of farly general systems are consdered, whch are characterzed by a dstnctve statc and dynamc behavor. a. Transverse behavor of a rectangular beam lattce Let consder now a rectangular beam lattce wth equal lgaments and focus on a selected straght lne Γ correspondng to a sequence of lgaments and nodes. Let assume that all the nodes on a lne normal to Γ undergo the same generalzed dsplacement wth vanshng dsplacement component along Γ. It follows that the lattce deformed confguraton s represented by the generalzed dsplacements {( v, ), 0, n} = of the nodes located along Γ as shown n Fgure 7. The same assumpton s made for the appled generalze forces at the { f, c, 0, n} nodes ( ) = that are assumed to be unform at nodes along the lnes normal to Γ. 0

21 Fgure 7. Model for the transverse deformaton a rectangular beam lattce. ( f c 0) The transverse deformaton of the lattce n case of vanshng generalzed nodal forces = = may be analyzed through the equaton of moton (45) wth K = 0, namely K = 0. In ths case, the equlbrum confguratons of the lattce are obtaned as soluton of the dfference equaton (45) and are wrtten n the form = C1 + C + C3 + K + K + K + C4 + K K + K ( ) ( ) 3 3K + K = ( 1 + K ) C C3( 1 + 6K 9K + 3K ) K ( ) K + K C 4 K K K K, (55) (see Kelley and Peterson, 001) wth the constant C h to be obtaned by the boundary condtons on dsplacements and rotatons at the end nodes. Beng postve the bases of the powers n equatons (55) ndependently on K, the confguratons of ths model are characterzed by transverse dsplacements and rotatons varyng wthout oscllatng sgn reversal of the generalzed dsplacements. In case of square lattce b = one obtan K = 1K, wth K = 1, and the soluton takes the form 1

22 ( 7 4 3) ( 7 4 3) ( ) ( ) = C1 + C + C3 + C4 + = C 3C C (56) To apprecate the approxmaton of the homogenzed models, a frst comparson wth the soluton of the Lagrangan model s carred out n the statc feld. Let consder a square beam lattce consstng of n = 11 lgaments n the drecton along Γ and an unbounded number of lgaments n the orthogonal drecton. Let consder the homogeneous problem wth vanshng rotaton prescrbed at the end nodes, namely 0 = 0 and 11 = 0, whle the non-dmensonal transverse dsplacements are prescrbed 0 = 0 and =. In the dagrams of Fgure 8 a comparson s gven n terms of transverse dsplacement and rotaton for the Lagrangan model wth the correspondng results obtaned by the proposed nd, 4 th and 6 th order homogenzed models,.e. by solvng the governng equatons obtaned retanng nd, 4 th and 6 th order terms, respectvely, n the McLaurn expanson of the pseudo-dfferental equatons system (48). In addton, approprate boundary condtons for homogenzed model, obtaned va r th order contnualzaton, are consdered and take the form h d Ψ h dx h d Φ = = 0, dx h x= 0/ n x= 0/ n wth h= 0,.., r. From these dagrams t may be observed a very good accuracy between macro-dsplacement feld Ψ ( x) and a macro-rotaton feld ( x) Φ of the contnuum models obtaned va enhanced homogenzaton here proposed and the nodal solutons and determned by the Lagrangan model. Indeed, the homogensed models are able to descrbe wth good accuracy the boundary layer effects occurrng n the dscrete model. It s worth notng that ths accuracy tends to ncrease as the order of contnualzaton ncreases.

23 Ψ, ( a) Φ,φ ( b) xn xn Fgure 8. Transverse dsplacements (a) and rotatons (b) n the Lagrangan model and from the contnuum model. Comparson of dfferent models: Lagrangan model (black dots); proposed nd order contnualzaton (volet); proposed 4 th order contnualzaton (red); proposed 6 th order contnualzaton (cyan). A second comparson deals wth the dsperson functons by the dscrete model and those by the enhanced homogenzed model; two values of the rato between the translatonal and rotatonal nerta ( 10,30) I η= = are consdered to represent dfferent desgn I possbltes. Beng K = 0, an acoustc and an optcal branch are obtaned by solvng the egenvalue problem (46), the latter departng from the crtcal frequency ω 0 = 1+ K I. In the dagrams of Fgure 9, the enhanced homogenzed models are shown to provde a very good smulaton of the Floquet-Bloch spectra of the Lagrangan model, wth a decreasng band gap ampltude when decreasng the rato η between the translatonal and rotatonal nerta of the nodes. Moreover, ncreasng the contnualzaton order, the dsperson functons of the derved enhanced contnuum tend towards the actual correspondng branches. 3

24 ω I ( a) ω I ( b) Fgure 9. Dsperson functons for varyng the mass rato (a) η= 30, (b) η= 10 : comparson among the Lagrangan model (black lne) and the proposed contnuum models: nd order contnualzaton (volet); 4 th order contnualzaton (red); 6 th order contnualzaton (cyan). Ψˆ, Φˆ ( 1) ( 1) ( 1) ( 1) ˆ,φˆ ( a) Ψˆ, Φˆ ( ) ( ) ( ) ( ) ˆ,φˆ ( b) Fgure 10. Magntude of the polarzaton vector s components assocated to (a) frst branch and (b) second branch of frequency spectrum for mass ratos η= 30 : comparson among the Lagrangan model (black lne) and the proposed contnuum models. Frst and second components of polarzaton vector n contnuous and dot lnes, respectvely. Proposed nd order contnualzaton (volet); proposed 4 th order contnualzaton (red); proposed 6 th order contnualzaton (cyan). 4

25 Ψˆ, Φˆ ( 1) ( 1) ( 1) ( 1) ˆ,φˆ ( a) Ψˆ, Φˆ ( ) ( ) ( ) ( ) ˆ,φˆ ( b) Fgure 11. Magntude of the polarzaton vector s components assocated to (a) frst branch and (b) second branch of frequency spectrum for mass ratos η= 10 : comparson among the Lagrangan model (black lne) and the proposed contnuum models. Frst and second components of polarzaton vector n contnuous and dot lnes, respectvely. Proposed nd order contnualzaton (volet); proposed 4 th order contnualzaton (red); proposed 6 th order contnualzaton (cyan). The descrpton of the harmonc wave propagaton s complemented wth the analyss of the polarzatons vectors. In partcular, the polarzaton vectors υ = { } T and ϒ = { Ψ Φ} T assocated to the Lagrangan and contnuum models, respectvely, are selfnormalzed by mposng the condtons * υυ =1, ϒ ϒ * =1, where the superscrpt * stands for complex conjugate. The magntude of the self-normalzed polarzaton vector s components for the Lagrangan model * ˆ = =, ˆ = = * and for the contnuum model * ˆΨ = Ψ = ΨΨ, ˆΦ = Φ = * ΦΦ are determned n terms of the mechancal parameters and of the wave number. In the dagrams of Fgures 10 and 11, these magntudes are shown n terms of dmensonless wave number for mass ratos η= 30 and η= 10, respectvely. In detals, n sub-fgures 10a and 11a the magntudes of the polarzaton vector s components assocated to the acoustc branch of the spectrum are reported, where a perfectly polarzed dsplacement waveform s detected n the lmt cases of long and short wavelengths,.e. k = 0 and = π. Moreover, for ntermedate values of the dmensonless wave number the hybrdzaton phenomena of the waveform s components occur. The magntudes of the 5

26 polarzaton vector s components assocated to the optcal branch are shown n sub-fgures 10b and 11b. Note that, a nearly polarzed rotaton waveform s detected to the except for lmt cases k = 0, k = π where a perfectly polarzed rotaton waveform occur. Fnally, a good accuracy between the results obtaned by the contnuum models and those determned by Lagrangan model are observed nto the range 0 π. Ths accuracy tends to ncrease as the order of contnualzaton ncreases. b. Contnuous perodc beam on elastc supports Fgure 1. Perodc contnuous beam wth elastc supports. The second system here consdered concerns a contnuous beam wth elastc supports made up of beams pnned at the ends and havng the same bendng stffness EJ and length b as shown n Fgure 1. The lateral stffness of the equvalent elastc sprng s K = 48 EJ 3 b, whle the rotatonal stffness s vanshng K = 0. The transverse deformaton of the perodc beam n case of vanshng generalzed nodal forces ( f c 0) equaton of moton (45) wth K = 0 and K = = may be analyzed through the 3 = 4. Ths parameter has an nfluence on b the qualty of the soluton of the system of dfference equatons. In fact, as explaned n Appendx B, for 0 < < 1 the dscrete model s characterzed by a soluton that K harmoncally vares along the beam wth wavelength that depends on K. For ncreasng the transverse stffness of the supports, namely ncreasng K, the wavelength decreases and for 6

27 K 1 a transton to an oscllatng response wth change of sgn of the generalzed dsplacements from one node to the adjacent one s obtaned. Ψ, ( a) Φ,φ ( b) xn xn Fgure 13. Transverse dsplacements (a) and rotatons (b) n the Lagrangan model and from the contnuum model for soft supports K = Comparson of dfferent models: Lagrangan model (black dots); proposed 4 th order contnualzaton (red); proposed 6 th order contnualzaton (cyan); proposed 8 th order contnualzaton (brown). Ψ, ( a) Φ,φ ( b) xn xn Fgure 14. Transverse dsplacements (a) and rotatons (b) n the Lagrangan model and from the contnuum model for stff supports K = 0. Comparson of dfferent models: Lagrangan model (black dots); proposed 4 th order contnualzaton (red); proposed 6 th order contnualzaton (cyan); proposed 8 th order contnualzaton (brown). 7

28 Ths behavor s explaned n the examples concernng a contnuous beam consstng of n = 11 lgaments. The homogeneous problem s defned by prescrbng vanshng rotatons at the end nodes, namely 0 = 0 and 11 = 0, whle the non-dmensonal transverse 1 dsplacements are prescrbed 0 = 0 and =. In the dagrams of Fgure 13 a comparson of transverse dsplacement and rotaton of the nodes of the Lagrangan model wth the correspondng results obtaned by the proposed 4 th, 6 th and 8 th order homogenzed models s gven for the case of soft supports K = Here, approprate boundary condtons for homogenzed model, obtaned va r th order contnualzaton, are consdered and take the form h d Ψ h dx h d Φ = = 0 dx h x= 0/ n x= 0/ n wth h= 0,.., r. From these dagrams t may be observed a very good accuracy between the generalzed macro-dsplacement felds Ψ ( x) and Φ ( x), respectvely, of the contnuum models obtaned va enhanced homogenzaton here proposed and the nodal solutons and resultng from the Lagrangan model. It s worth notng that ths accuracy tends to ncrease as the order of contnualzaton ncreases. Smlarly, n the dagrams of Fgure 14 a comparson for the case of stff supports K = 0 s gven. In ths case t may be observed that the oscllatng behavor of the dscrete model n the boundary layer s smulated rather well by the enhanced contnuum model here proposed, wth a tendency to converge to the actual soluton when ncreasng the order of the contnuum model. The evaluaton of the homogenzed model to smulate the acoustc behavor of the Lagrangan model has been carred out by consderng two values of the rato I η= = ( 10,30) to represent several desgn possbltes. Beng the rotatonal stffness of the I supports vanshng K = 0, two optcal branches are obtaned by the egenvalue problem (46) K. These two branches depart from correspondng crtcal frequences ω 0 = I and 1 ω 0 =, respectvely. It s worth to note that whle the second frequency only depends on I the rotatonal nerta of the nodes, the former depends on the transverse stffness of the supports and on the correspondng nerta. Ths crcumstance suggests the possblty of a fne tunng on the optcal branches also n desgnng gude-waves etc. Moreover, n case of 8

29 K 0, the lower optcal branch changes n an acoustc branch. In the dagrams of Fgures 15 and 18 the Floquet-Bloch spectra obtaned by enhanced homogensed model are compared wth those from the beam lattce for soft ( K = 1 50 ) and stff supports ( K = 0 ). Although a good agreement s observed n the frst case (see Fgure 15), excellent results are obtaned n case of stff supports (see Fgure 18). Also for ths case one may observe that ncreasng the contnualzaton order, the dsperson functons of the derved enhanced contnuum tend towards the actual correspondng branches. The descrpton of the harmonc wave propagaton s complemented wth the analyss of the polarzatons vectors. The magntude of the self-normalzed polarzaton vector s components for the Lagrangan model ˆ, ˆ and for the contnuum model ˆΨ, ˆΦ s evaluated n terms of the mechancal parameters and of the wave number. In the dagrams of Fgures 16 and 17, these magntudes are shown for soft supports ( K = 1 50 ) n terms of dmensonless wave number for mass ratos η= 30 and η= 10, respectvely. In partcular, n sub-fgures 16a and 17a the magntudes of the polarzaton vector s components assocated to the lowest frequency optcal branch (frst branch) of the spectrum are reported. It s possble to observe that a perfectly polarzed dsplacement and/or rotaton waveform s detected n the lmt cases of long and short wavelengths,.e. k = 0 and = π. Moreover, for ntermedate values of the dmensonless wave number the hybrdzaton phenomena of the waveform s components occur. The magntudes of the polarzaton vector s components assocated to the hghest frequency optcal branch (second branch) are shown n sub-fgures 16b and 17b. Note that, a nearly polarzed rotaton waveform (see sub-fgure 16b) and hybrdzaton phenomena of the waveform s components (see sub-fgure 17b) are detected to the except for lmt cases k = 0, k = π where a perfectly polarzed dsplacement and/or rotaton waveform occur. Fnally, a good accuracy between the results obtaned by the contnuum models and those determned by Lagrangan model are observed nto the range 0 3π 4. Ths accuracy tends to ncrease as the order of contnualzaton ncreases. Smlarly, n the dagrams of Fgures 19 and 0 a comparson for the case of stff supports ( K = 0 ) s gven and qualtatvely smlar behavors are detected. 9

30 ω I ( a) ω I ( b) Fgure 15. Dsperson functons for varyng the mass rato (a) η= 30, (b) η= 10 and for K =1 50 : comparson among the Lagrangan model (black lne) and the proposed contnuum models. Proposed 4 th order contnualzaton (red); proposed 6 th order contnualzaton (cyan); proposed 8 th order contnualzaton (brown). Ψˆ, Φˆ ( 1) ( 1) ( 1) ( 1) ˆ,φˆ ( a) Ψˆ, Φˆ ( ) ( ) ( ) ( ) ˆ,φˆ ( b) Fgure 16. Magntude of the polarzaton vector s components, for mass ratos η= 30 and K =1 50 assocated to (a) frst branch and (b) second branch of frequency spectrum: comparson among the Lagrangan model (black lne) and the proposed contnuum models. Frst and second components of polarzaton vector n contnuous and dot lnes, respectvely. Proposed 4 th order contnualzaton (red); proposed 6 th order contnualzaton (cyan); proposed 8 th order contnualzaton (brown). 30

31 Ψˆ, Φˆ ( 1) ( 1) ( 1) ( 1) ˆ,φˆ ( a) Ψˆ, Φˆ ( ) ( ) ( ) ( ) ˆ,φˆ ( b) Fgure 17. Magntude of the polarzaton vector s components, for mass ratos η= 10 and K =1 50 assocated to (a) frst branch and (b) second branch of frequency spectrum: comparson among the Lagrangan model (black lne) and the proposed contnuum models. Frst and second components of polarzaton vector n contnuous and dot lnes, respectvely. Proposed 4 th order contnualzaton (red); proposed 6 th order contnualzaton (cyan); proposed 8 th order contnualzaton (brown). ω I ( a) ω I ( b) Fgure 18. Dsperson functons for varyng the mass rato (a) η= 30, (b) η= 10 and for K = 0 : comparson among the Lagrangan model (black lne) and the proposed contnuum models. Proposed 4 th order contnualzaton (red); proposed 6 th order contnualzaton (cyan); proposed 8 th order contnualzaton (brown). 31

32 ( a) Ψˆ, Φˆ ( ) ( ) ( ) ( ) ˆ,φˆ ( b) Ψˆ, Φˆ ( 1) ( 1) ( 1) ( 1) ˆ,φˆ Fgure 19. Magntude of the polarzaton vector s components, for mass ratos η= 30 and K = 0 assocated to (a) frst branch and (b) second branch of frequency spectrum: comparson among the Lagrangan model (black lne) and the proposed contnuum models. Frst and second components of polarzaton vector n contnuous and dot lnes, respectvely. Proposed 4 th order contnualzaton (red); proposed 6 th order contnualzaton (cyan); proposed 8 th order contnualzaton (brown). Ψˆ, Φˆ ( 1) ( 1) ( 1) ( 1) ˆ,φˆ ( a) ˆ ( ) ˆ ( ) Ψ, Φ ( b) ( ) ( ) ˆ,φˆ Fgure 0. Magntude of the polarzaton vector s components, for mass ratos η= 10 and K = 0 assocated to (a) frst branch and (b) second branch of frequency spectrum: comparson among the Lagrangan model (black lne) and the proposed contnuum models. Frst and second components of polarzaton vector n contnuous and dot lnes, respectvely. Proposed 4 th order contnualzaton (red); proposed 6 th order contnualzaton (cyan); proposed 8 th order contnualzaton (brown). 3

33 6. Conclusons The enhanced contnualzaton approach proposed n ths paper s amed to overcome some drawbacks observed n the homogenzaton of beam lattces. Whle hgher order mcropolar models obtaned through the standard contnualzaton of beam lattces may provde good smulatons of the acoustc response, on the other hand they are characterzed by a non-postve defnte elastc potental energy densty. Conversely, mcropolar homogenzaton approaches of beam lattce dscrete models based on a frst order downscalng law coupled wth an applcaton of the macro-homogenety condton appear not sutable to smulate the acoustc response of the lattce model. In the lterature some attempts to crcumvent such ssues have been solved by the applcaton of the Padé approxmant to the pseudo-dfferental problem assocated to the dfference problem of the dscrete model. Nevertheless, n ths paper t has shown that n some cases ths approach s unable to avod non postve defnteness of the elastc potental of the homogenzed contnuum. The approach here proposed, based on a transformaton va a proper down-scalng law of the dfference equaton of moton of the dscrete system nto a pseudo-dfferental problem and a further McLaurn approxmaton to obtan a hgher order dfferental problem, has been appled to some smple but sgnfcant 1D systems: the rod lattce, the beam lattce wth node rotatons and a 1D beam lattce model wth generalzed dsplacements. These systems present dfferent statc and dynamc characterstcs. Nevertheless, the homogenzed models obtaned by the proposed enhanced contnualzaton technque turn out to be energetcally consstent and provde a good smulaton of both the statc response and of the acoustc spectrum of the orgnal dscrete models. Hgher order models may be obtaned whch are characterzed by dfferental problems nvolvng non-local nerta terms together wth spatal hgh gradent terms. The smulaton of the acoustc behavour of the smple axal chan has shown a good performance f compared to the correspondng one from Padé approxmaton. The beam lattce wth node rotatons and dsplacement prevented exhbts, n the statc regme, decayng oscllatons of the nodal rotaton n the boundary layer whch s well smulated by the homogenzed model obtaned by the proposed approach. Smlar good results are obtaned n the smulaton of the optcal spectrum. It s worth to note that n ths case the homogenzed model obtaned va Padé approxmaton turns out to be energetcally non-consstent. The analyss of the homogenzed model derved from the beam lattce wth transverse dsplacement and rotaton of the nodes wth elastc supports has shown that both the statc and the dynamc response are strongly varable on the parameters of the Lagrangan model. 33