The areolar strain concept applied to elasticity

Transcription

1 Computtonl Methods nd Epermentl Mesurements XIII 579 The reolr strn concept ppled to elstct I. D. Kotchergenko Insttuto Mltr de Engenhr, Ro de Jnero, Brl Astrct In contrst to the ooks tht strt th solutons of Lmé-Nver equtons usng comple vrles, the present rtcle strts th presentton of the fundmentls of the comple theor of to-dmensonl elstct. A ne strn epresson s derved nd the comptlt condtons for these strns re gven. The fundmentl equtons for the sotropc nd orthotropc plne elstct re lso presented n someht ne comple forms. The homogeneous equlrum equton s presented n comple form tht proved to e esl solvle. Kolosov s generl soluton for the sotropc cse s otned n frl strghtforrd fshon. Ne equlrum equtons nd oundr condtons for fnte rotton re lso gven. Keords: reolr strn, comptlt equton, fnte rotton, comple elstct. Introducton The development of the theor of to-dmensonl elstct hereof s grounded on the concept of reolr strn. Ths concept s frst presented Kotchergenko [8], n 983, though contnng some errors. In ths pproch the strn s otned the dvson of to comple-vlued qunttes ssocted th D vectors nd clled reolr strn ong to the fct tht ts rel prt represents rdl strn nd ts mgnr prt represents ether crcumferentl strn or rotton. The quternon concept cn prol e used to etend the reolr strn concept to the 3D vectors cse. Improvements on the lner theor ere recentl presented, Kotchergenko [9,]. Equlrum equtons for fnte rotton nd severl other mprovements re no ncluded. A detled reve of the mn results otned untl no s provded n order to mke ths rtcle self contned. WIT Trnsctons on Modellng nd Smulton, Vol 46, 7 WIT Press.tpress.com, ISSN X on-lne do:.495/cmem758

2 58 Computtonl Methods nd Epermentl Mesurements XIII The reolr strn Let regon n the plne of the vrles nd e mpped n one-to-one mnner onto the plne of the dsplcements u, nd v, mens of the trnsformton, u, v,. Gven the drecton α of the vector, here nd re to neghorng ponts of the plne, the reolr strn s defned s the grdent of the vector feld, n the drectonα, through the reolr dervtve: d d ε lm or α ε e, d here the polr form hs een used for the rto d d α d e α e α d e. Ths epresson presupposes tht tends to, mntnng the drecton α. Snce u v v u, u v v u, the reolr strn cn lso e rtten n the form α e ε. 3 When veed n the polr form, the rel prt of the reolr strn represents rdl strn nd the mgnr prt represents ether, crcumferentl strn or rotton. The second comple term s the comple sher strn. The components of the reolr strn re orthogonl s the qudrtc form contned n the ntegrnd of the ork epresson U / Cjε ε j dv cn e converted nto ts cnonc form U [ C C C C C ] dv. 4 3 Comptlt equtons If nd re to ponts pertnng to the comple plne, ther reltve dsplcement s gven d ε d d d d. 5 d C C C Snce s ndependent of the pth of ntegrton C, d d d 6 3 WIT Trnsctons on Modellng nd Smulton, Vol 46,.tpress.com, ISSN X on-lne 7 WIT Press

3 Computtonl Methods nd Epermentl Mesurements XIII 58 Fgure : Fundmentl modes of the reolr strn. s totl dfferentl. Consequentll, the dsplcement feld must compl th the condton of contnut. 7 Seprtng the rel nd mgnr prts of ths equton, the follong comptlt equtons re otned:,. 8 Snt-Vennt s comptlt equton s otned from these equtons through elmnton of the mode, pplng cross-dfferentton folloed sutrcton. Snt-Vennt s comptlt equton ll then e stsfed for n rottonl feld, hch m thus volte the comptlt condtons 8. 4 Equlrum equtons For sotropc mterl, C C, C C nd C 3, thus the ork epresson gven eqn. 4 reduces to WIT Trnsctons on Modellng nd Smulton, Vol 46,.tpress.com, ISSN X on-lne 7 WIT Press

4 58 Computtonl Methods nd Epermentl Mesurements XIII U [ ]dv 9 here nd re Lme s elstc mterl constnts. The Euler equtons for ths functonl re. Tkng nto ccount tht u, v,, here the smol stnds for the Lplcn opertor, Lmé s homogeneous equlrum equtons cn e presented n the follong comple form: [ ] [ ]. Multplng ech term of eqn 7 dfferent comple elstc constnt ll result n n equton n terms of stresses. For sotropc mterl, undergong smll strn nd smll rottons, the opertons requred for otnng Lme s homogeneous equlrum equton re [ ]. Oserve tht onl one out of the to, hch pper n the term, s removed. The term represents locl rotton. Tkng the dervtves of from eqns. 8 nd susttutng nto eqn. ll gve eqn. 5 Boundr condtons Applng Green s formul e.g. Cournt nd Hlert [] to eqn., n the comple form, { [ ] [ ]} dd d d, 3 Ω C the trcton vector on oundr curve C results T α e, 4 hereα ponts tords the drecton of the vector element of rc d of the closed contour curve C. Oserve tht f n the unt outrd vector, norml to the element of rc ds d, then nds d d. WIT Trnsctons on Modellng nd Smulton, Vol 46,.tpress.com, ISSN X on-lne 7 WIT Press

5 6 Other forms for the equlrum equtons The comptlt equtons 7 cn lso e rertten n the follong comple form. 5 Equton cn hence e reduced to the follong holomorphc functon Love, [4], ] [, 6 here. The modes nd re hence hrmonc functons. The rter succeeded n rertng the equlrum equton n nother form tht proved to e esl solvle. The elmnton of the dervtves of eteen equtons 8 nd 6, gves,. 7 The elmnton of the dervtves of, from the sme pr of equtons, gves,. 8 Comnton of these equtons elds: ] [ χ, 9 Or fter eqns., χ. here χ 3 for plne strn nd χ 3, for plne stress, oservng tht. 7 Generl soluton Dfferentton of eqn. n results 7 WIT Press WIT Trnsctons on Modellng nd Smulton, Vol 46,.tpress.com, ISSN X on-lne Computtonl Methods nd Epermentl Mesurements XIII 583

6 ] [ Snce 4 s the Lplcn opertor nd s hrmonc functon, ths equton reduces to ] [, or fter eqns., to. Integrton n gves, 3 here s the conjugte of n nltc functon. Integrtng eqn. 3 n, results ] [ ψ. 4 Equtons nd 3 furnsh χ, hch fter ntegrton n, results ζ χ. 5 Dfferentton of ths equton th respect to elds d ζ. 6 From eqn. 3, results ζ nd hence c ζ, 7 here c s comple constnt. Susttuton nto eqn. 5 elds ] [ χ. 8 Usng eqns. 8 nd 4, the ntegrton of the totl dfferentl, eqn. 5, gves Kolosov-Muskhelshvl s generl soluton, [,3,5,7]: ] [ ψ χ. 9 7 WIT Press WIT Trnsctons on Modellng nd Smulton, Vol 46,.tpress.com, ISSN X on-lne 584 Computtonl Methods nd Epermentl Mesurements XIII

7 Computtonl Methods nd Epermentl Mesurements XIII 585 The ddton of Eqn 8 to ts conjugte gves χ [ ]. 3 The sutrcton of Eqn 8 from ts conjugte gves χ [ ]. 3 Kolosov s formuls, for stresses re otned drectl from eqns. 3 nd 4: σ σ [ ], 3 σ σ σ [ ψ ]. 33 The reolr strn, n terms of nltc functons ssumes the form ε [ χ ] [ ψ ] e α 8 Equlrum equtons for n orthotropc plne. 34 The rter eplored the posslt of pplng to the orthotropc plne cse the sme pproch used th eqn.. Tkng the comptlt equton n the form of eqn. 5 nd removng one out of the to present nto the term ; then multplng ech comple term of ths equton dfferent comple elstc constnts, ll result n the follong equton n terms of stresses:. 35 Susttuton of the dervtves of the rotton, otned from eqn. 8, gves [ ]. 36 Lmé s equlrum equtons re otned lettng,, nd. Applcton of Green s formul to eqn. 36, gves the follong generlton of the trcton vector formul, prevousl depcted n eqn. 4: α T [ ] e. 37 Usng the sme procedure s the one used for otnng eqn. 6 leds to the sme Love s form of holomorphc functon: [ ], 38 here. 39 [ ] Oserve tht for the sotropc plne, ll e reduced to rel constnt. WIT Trnsctons on Modellng nd Smulton, Vol 46,.tpress.com, ISSN X on-lne 7 WIT Press

8 Seprtng the rel nd mgnr prts elds:,, 4 th, here ] [ 3, ] [. 4 Dfferenttons of eqns. 4, elds. 4 Other dfferenttons of eqns. 4, elds ddtonll: nd. Hence, nd fter eqn. 4, oth nd remn hrmoncs lso for the orthotropc plne cse. 9 Fnte rottons The gn of re durng plne deformton s gven 4 4 d da, 43 hch s otned from the determnnt v v u u da. 44 Oservng fg., t cn e seen tht fnte rotton hs sgnfcnt nfluence nto the chnge of re of plne elstc od. In some endng nd ucklng prolems, t s dmtted to dsregrd strn terms squred, ecept for, reducng da to da WIT Press WIT Trnsctons on Modellng nd Smulton, Vol 46,.tpress.com, ISSN X on-lne 586 Computtonl Methods nd Epermentl Mesurements XIII

9 Computtonl Methods nd Epermentl Mesurements XIII 587 A rgd od rotton n ths cse cn e ppromted the condton. Then, for pont to descre crculr pth of rc Ω tn, t s requred tht the epnson produced the mode e neutrled the shrnkge. For the condton gven eqn. 45, the ork epresson ssumes the form U / dd. 46 The Euler equtons for ths functonl re [ 3 ], [ 3 ]. 47 Ths sstem of equtons cn e rtten n the comple form [ ]. 48 Applng Green s formul, the trcton vector on closed curve C, n the undeformed reference frme, results α T [ ] e. 49 Accordngl th eqns. I. from Novohlov [6], the sherng modes due to fnte strns re: ˆ nd ˆ, hence dscrdng the strns squred, other thn those contnng, results ˆ nd ˆ. If the follong ppromtons re used ccordngl: σ σ σ σ τ τ σ σ, nd, 5 the equlrum equtons 47 n terms of stresses, referred to the undeformed reference frme, ecome σ τ σ τ, σ τ σ τ. 5 These re ectl the equlrum equtons II.49 gven Novohlov, [6]. References [] Cournt, R. & Hlert, D., Methods of Mthemtcl Phscs, Vol. II, Wle: Ne York, pp , 989. WIT Trnsctons on Modellng nd Smulton, Vol 46,.tpress.com, ISSN X on-lne 7 WIT Press

10 588 Computtonl Methods nd Epermentl Mesurements XIII [] Englnd, A. H., Comple Vrle Methods n Elstct, Wle, pp. 8-49, 97. [3] Klnd, A. I., Mthemtcl Methods of To-Dmensonl Elstct, Mr Pulshers, pp , 975. [4] Love, A.E.H., A Tretse on the Mthemtcl Theor of Elstct, Cmrdge Unverst Press, pp. 4-, 97. [5] Muskhelshvl, N.I., Some Bsc Prolems of the Theor of Elstct, Noordhoff, Gronngen, 963. [6] Novohlov, V.V., Foundtons of the Nonlner Theor of Elstct, Dover Pulctons: Mneol, Ne York, pp , 999. [7] Sokolnkoff, I.S., Mthemtcl Theor of Elstct, McGr-Hll, 956. [8] Kotchergenko, I.D., Unsmmetrcl Plne Elstct, Recent Advnces n Engneerng Mechncs, Proceedngs of the Fourth Engneerng Mechncs Dvson Speclt Conference, ASCE, eds. W.F. Chen & A.D.M. Les, Vol. I, pp , 983. [9] Kotchergenko, I.D., Kolosov-Mushkhelshvl Formuls Revsted, th Interntonl Conference on Frcture, Turn, Mrch 5. [] Kotchergenko, I.D., Applctons of Generled Anltc Functons to Elstct, CMM-5-Computer Methods n Mechncs, Polsh Acdem of Scences, June 5. WIT Trnsctons on Modellng nd Smulton, Vol 46,.tpress.com, ISSN X on-lne 7 WIT Press