On boundary conditions for Ladevèze's Plate Theory

Transcription

1 On boundary condiions for Ladevèze's Plae Teory Anonio Capsoni, Pierre Ladevèze Deparen of Srucural Engineering, Poliecnico di Milano, Ialy E-ail: LMT Cacan (ENS Cacan/CNRS/Paris 6 Univ., France E-ail: ladeveze@l.ens-cacan.fr Keywords: Teory of plaes, Kircoff. SUMMARY. A new eory for linearly elasic plaes as been recenly proposed wic relies on e classical superposiion of sor and long waveleng soluions. Te feaure of is approac is a e long waveleng soluion, naely Sain Venan s soluion, is derived in exac way depending only on wo generalized D variables. Due o difficulies in properly defining e boundary condiions associaed o is (arbirarily ick plae eory saring fro local consideraions, a variaional fraework as been now forulaed allowing a consisen definiion of e associaed boundary condiions wic converges o e classical ones a Kircoff lii. INTRODUCTION Te recenly forulaed Exac Teory of Plaes (ETP [-4] relies on a long waveleng Sain Venan s (SV soluion of e isoropic and oogeneous elasic plae proble a is exac, erefore independen fro any plae ickness assupion, inroducing a poin of view a subsanially depars fro e classical ickness dependen plae forulaion fraework [5]. In paricular e soluion depends upon a couple of generalized variables ( w%, % wi values on e plae reference (iddle-plane, wic exibi filering properies wi respec o e localized conribuions wic bo caracerizes e D exac proble and classical firs order plae forulaions (e.g. Reissner and Mindlin plaes [6-7]. Edge effecs, load variaions ino e doain, as well as boundary layers wic are conrolled by e ransversal spin coponen, are erefore e separae objec of a localized, uncoupled, correcion of e long waveleng soluion field. Te wo second order PDE governing SV soluion ave been iniially derived by exploiing local and global governing equaions recalling Marcus forulaion for classical plae eories [8]. Despie an easy definiion of coninuiy condiions (in [ H ( Σ ] for e couple of generalized variables can be used for exaple o enforce e connecion wi an elasic boundary, e direc approac sees o fail in properly defining consisen boundary condiions, excep for e siple definiion of e generalized siply suppor condiion. Tis work erefore ais o give a conribuion oward e copleion of e fraework of SV soluion in ETP forulaion by inroducing a consisen definiions of boundary condiion, also periing e naural recovery of e classical plae forulaion (Kircoff, as well of recenly proposed firs order eories [], only by reducing plae ickness up o in plaes lii case. To obain e searced resul a variaional forulaion of e proble is firsly derived on e basis of a proper Hellinger-Reissner principle. Saionary condiions of is governing funcional produce e aforeenioned se of equaions as well as e corresponding consisen boundary condiions in copac weak for.

2 Despie e quie easy inerpreaion of eir ecanical eaning, e obained condiions acually appears o be no rivially derivable by direc kineaics and saic consideraion, and is is especially rue for e free edge case. Te srucure of e boundary condiions also reveals a, wiou any approxiaion for plaes a are copleely suppored along e boundary and by assuing oderaely ick plaes ypoesis in e case of free edges, a new generalized variable 5 % ϕ w% k % ( κ = µ is e sear siffness can be naurally inroduced wic as a clear ecanical eaning. In is way a firs order irroaional eory (local roaion adi a poenial and e noral spin is zero, a is basically equivalen o a Reissner s like plae eory [9] excep for e absence of boundary layers, is readily derived and copared wi soe forulaions recenly appeared in lieraure []. THE PLATE MODEL Te ETP odel as been already widely and deeply described in soe previous works [-4]. Tis conribuion erefore will only focus on e SV soluion; for a sake of sipliciy, owever, basic ypoeses and ain resuls relaed o e field odel will be firs briefly suarized.. General proble Te sall srain equilibriu of a linearly elasic, oogeneous and isoropic solid cylinder bounded by e doain Ω = Σ = iddle plane ickness is erein invesigaed. Te body is aced upon volue forces F in Ω and surface racions f and applied, respecively, on e wo + σ bases Σ, Σ and on e free porion Σ of e laeral surface Σ described by e (coninuous noral ouward vecor n (a couner clock-wise angenial vecor is also inroduced. Figure : Geoery of e reference proble u Displaceens are erefore supposed known over e reaining par Σ of e laeral surface. Under ese ypoeses e -D elasic proble can be fully described by e classical se of equaions governing equilibriu, copaibiliy and elasic law [4], wose soluion can be splied ino a ebrane and a flexural conribuion a are assued corresponding, respecively, o e even and odd decoposiion of e displaceen field soluion wi respec o e iddle plane Σ. An a-priori selecion of a purely flexural soluion in Ω enforces e following resricion on e naure of volue forces and surface racions F = Np in Ω ; f + = NP su Σ + ; f = N Psu Σ (a,b

3 were N is e vecor orogonal o e iddle plane in e verse of e coordinae axis x, p is an even funcion wi respec o Σ and, bo p and P, are supposed independen on e coordinae x in Σ (i.e. P = cons, p = p( x. Te plae proble is idenified wi e flexural soluion in e previously defined sense. Le us now inroduce e projecion operaor suc a Π = (a V = Π U; w= N U (a,b were U is e displaceen field of coponen V (in plane and w (ou of plane. By adoping a coeren spliing e equilibriu equaions can be wrien in e following for wi e posiions % σ + τ, = ; τ x =± = (4a σ, σ x =± τ + + p = ; = ± P (4b σ τ τ % σ = ΠσΠ =, τ σ τ = Πσ N = τ (4c and an obvious eaning of sybols (e operaor is in D. By exploiing e elasic law roug e adopion of Lae s consans λ, µ and inroducing e scalar sress variable a defined as e following relaion follows a = λ ( U + µ ( Π U (5 σ% = a + µ ( N ω (6 or divσ% = grad a + µ curlω, wi ω = ( N V = ( V, V, (i.e. e coponen of spin ensor along N and curl ω = [ ω, ω ].,,. Sain Venan soluion Te SV soluion of e reference proble is siply defined [-4] as e displaceen field ( V,w saisfying e kineaic consrain ω = N V = ( (7 Condiion (7 is a sooness consrain and i as been already deonsraed [,] a boundary layers, if presen, are governed by apliude paraeers conrolled by e spin ω. Fro

4 a purely kineaic poin of view a soluion V coplying wi (7 as e following poenial srucure V = φ (8 By considering e dual sress poin of view, (7 also iplies e relaion % σ = a (9 In e frae of ETP, e SV soluion is expressed as a funcion of wo generalized variable ( w%, % defined as = % ( x a( x, x x dx (a ( ( ν + ν w% ( x = w(, x 4 ( x dx a(, x ( x 5 x dx E x x (b a are idenically null in case of localized ( w, a fields. In paricular, i is relaively an easy ask [4] o derive e following soluion for sress quaniies a( x, x = % ( x x + a' ( x (a τ ( x,x = % ( x( x (b 4 σ x ( x = P 4 o ( x x p( x dx ;Po = P + p( x dx (c,d x ν x ν = ( 5 o x 4 (e a' ( x P x x pdx p( dx and, roug e elasic law, e corresponden kineaical relaions are ( x V ( x,x = x w% ( x + x % ( x (f 4µ 6 5x = % + ν % 4µ + w ( x,x w( x ( ( x w' ( x (g x x x ( ν 4 ν ( + ν = 8 o 6 E E E ( w' ( x P ( x x p( x dx dx a' ( x dx VARIATIONAL FORMULATION AND BOUNDARY CONDITIONS In order o obain a proper variaional forulaion, owing o e presence of ixed saical and kineaical fields, e following Coplienary Energy associaed o SV soluion wi

5 oogeneous displaceen boundary condiions, is iniially derived by eans of soe siple calculaions ( ( ν + ν Ec ( a, σ = a σ E E a dxdx ( V Te laer expression is e basis for e definiion of a sei-invered Reissner-like funcional for SV plae forulaion in a -D fraework V ( τ E µ ( ( % σ V R = a V a + dx dx + n an V dx dx + L w;p, p % n V dx dx n wdx dx ex ( ( σ ( τ Vσ Vτ were e work associaed o field consan forces (P,p is explicily derived by eans of soluion (a-f as 5 x ν Lex ( w, % % = Po w% P ( 4 ( o p x dx κ % (4 and e second er represen e (iger order conribuion due o ransversal deforaion. 5 Coefficiens κ f = E / ( -ν and κ = µ define flexural and sear siffness, respecively. An expression of e funcional referred o iddle plane and generalized variable (excep for consan ers and referring o unloaded free edge is readily obained roug subsiuion of ( in ( and reads ( were { k k k } f ex σ Σ Σ R* ( w, % % = % ( % w% % + % dx L ( w, % % + ( % n an V dx dx (5a σ% n a n = µ ( w% x % f ( x ( w%, x %, f ( x n (5b x ( x f ( x = (5c 4µ 6 Te field equaions of ETP [] are obained as saionariy condiions of (5 wi e posiion % ( x + Po = (6a k f w% ( x + % ( x + ' % = ν 5 o ( % ' = P + p( x ( x dx (6b and e boundary ers (expressed in weak for for e case of soo boundary Σ % w% % ( w% % ( k + % σ a dx d = (6c n n V x

6 Te laer expression le a naural definiion of e consisen boundary condiions for e proble a and (e condiion ( σ% n an V = as been aken ino accoun being saisfied for bo free and V-consrained edge. Te following four convenional alernaives arise: Siply suppored edge (wic iplies % σ n an = Siply claped edge (i.e. wi V = Claped suppored edge (i.e. wi V = w % = ;% = (7a w % = ;% = (7b w % =,w% % = (7c k 4 Free edge (aking ino accoun a % is inadissible because i iplies a Σ { } 4 µ 4 µ k k, k % % % % % % % % % % (7d [ ( w ] w [ + ( w ] w + dx = As previously discussed in [4], were a coparisons wi a well known generalized D odel as been ade, noe a e second of (7c can be inerpreed as a consrain condiion for e generalized flexural roaion coponen alone. I is also wor noicing a e derived free edge condiion is no variaionally consisen. As sressed by (5b, is boundary condiion in fac involves second derivaives of e generalized funcions along e boundary and is is incopaible wi e second order srucure of e funcional. Te discrepancy is an obvious consequence of e naure of e sress (srain field, a is a funcion of e second derivaive of e generalized variable and erefore, for a general edge condiion, i iplies a forulaion based on oal poenial energy principle consruced on a srain field derived fro (f. Now, by inroducing e auxiliary quaniy % ϕ w% k %, wic as e eaning of a generalized displaceen associae o e pure flexural beavior, e general boundary condiion for a free edge can be rewrien as Σ { ν % ϕ ν % ϕ, ϕ% k } [ w% + ( ] w% ( w% + ( w% + ' % dx = (8 As a proof of consisency Clebsc condiions for Kircoff s plae are readily obained fro (8 as lii case by assuing w% = % ϕ = w( x = = w and ' % = { nn ν n ν, } [ w + ( w ] w [ w + w ] w dx = (9 Σ f

7 SOME REMARKS Te SV soluion of ETP sows srong siilariies wi classical firs order sear deforable plae eories fraed in a poenial forulaion for roaion vecor []. By considering, for exaple, e case of a copleely claped plae aced upon by volue forces only, disribued wi e lawpx ( = Q ( x 4 and referring o e couple of variable ( w,ϕ% %, e governing syse read k ( w% % ϕ + Q = k f k f w% + k ( w% % ϕ + Q k = (a w % =,ϕ % = (b a is forally e sae proble can be obained by adoping a poenial funcion for e roaion vecor in Reissner eory of plaes [9,]. However once ϕ%,w%, and ence %, are derived as e soluion of syse (, e EPT gives inernal sresses and displaceens governed by relaions ( wic differs fro e Reissner s assupion of plane sress linearly varying in e ickness. References [] P. Ladevèze, La eorie exace de la flexion des plaques, Rappore inerne n 8 LMT- Cacan (997 [] P. Ladevèze, Te exac eory of elasic plae, Copes Rendus Acad. Sci. Paris, IIb, 6 (998 [] Ladevèze P., Te exac eory of plae bending, J. Elasiciy, 68, 7-7 ( [4] Capsoni A., Ladvèze P., On e exac eory of elasic plaes, in Ai del XIV Congresso Nazionale AIMETA, Coo, 999. [5] Goldenveizer A., "Te principle of reducing ree diensional probles of elasiciy o wodiensional probles of e eory of plaes and sells", in Applied Mecanics, Gorler H. (ed., Springer (966 [6] D.N. Arnold and R.S. Falk, Asypoic analysis of e boundary layer for e Reissner- Mindlin plae odel, SIAM J. Ma. Anal. ( (99. [7] Brank B., On boundary layer in e Mindlin plae odel: Levy plaes, Tin-Walled Srucures, 46, (8 [8] Podio Guidugli P. and Tiero A., Marcus inegraion eod for searable plaes, J. Elasiciy, 84, (6 [9] Wang C.M., Li G.T., Reddy J.N. and Lee K.H., Relaionsips beween bending soluions of Reissner and Mindlin plae ories, Eng. Sruc.,, ( [] Sipi R.P., Pael H.G. and Arya H., New firs order sear deforaion plae eory, J. Appl. Mec., 74, 5-5 (7 [] Pecasaings F., Sur le principe di Sain-Venan pour les plaques, Tèse, Universié Paris 6 (985. [] D.N. Arnold and R.S. Falk, Asypoic analysis of e boundary layer for e Reissner- Mindlin plae odel, SIAM J. Ma. Anal. ( (99