Chapter 1. Foundation of Solid Mechanics and Variational Methods

Transcription

1 Chapter Fondaton of Sold Mechancs and Varatonal Methods

2 - Some Fndamental Concepts -- hscal roblems, Mathematcal Models, Soltons -- Contnm Mechancs -3- Bondar ale problem solton -4- Appromate solton of a bondar ale problem - Concepts of Stress, Stran, Consttte Relatons and Varos Form of Energ -- STRESS..- Force Dstrbtons..- Stress..3- Eqatons of Moton.- STRAIN..- hscal Interpretaton of Stran Terms..- The Rotaton Tensor..3- Compatblt Eqatons.3- HOOKE S AW 3- Bondar-Vale roblems for near Elastct 4- Energ Consderaton 5- rncples of Vrtal Work 6- The Method of Total otental Energ 7- Dfferental Eqatons VS Fnctonal for Contnos Sstems 7.- Formlaton of Contnos Sstems 7..- Eamples of Dfferental Approach 7..- Eample of Varatonal Approach 8- No. of Rgd Bod Modes n a Sstem 9- Sample roblems

3 - Some Fndamental Concepts -- hscal roblems, Mathematcal Models, Soltons The man obecte of ths secton s descrbng the concepts of bod and mathematcal modelng. rocedres for formlaton and solton of a Mathematcal model of a phscal problem are dscssed. The followng dagram shows a general ew of the modelng from bod to model to solton. BODY STATE Inflence of enronment BODY STATE State Varables at STATE MATERIA ROERTIES State Varables at STATE MODE IF FEASIBE Appled Mechancs SOUTION IF OSSIBE State arables nole dsplacements, eloctes, pressre, temperatre, stress, stran, charge, poston, etc. Inflence of enronment can be de to forces, temperatre changes, etc. ropertes are determned from laborator testng. -- Contnm Mechancs Thngs that we can percee, see, hear, or bld can be eplaned b sng certan prncples and laws of natres: conseraton of mass, energ, lnear 3

4 and anglar momenta, the laws of electromagnetc fl, and the concept of thermodnamc rreersblt. These are among the fndamental prncples on whch the sbect of mechancs s based. The sbect of contnm mechancs s based on the foregong goernng prncples, whch are ndependent of the nternal consttton of materal. Howeer, the response of a sstem or a medm sbected to (eternal forces can not be determned nqel onl wth the goernng feld eqatons dered from the basc prncples. The nternal consttton of materal plas an mportant role n the sbect of contnm mechancs. Std of the response of a sbstance or bod nder eternal ectaton constttes the maor endeaor n engneerng. In engneerng applcatons, the response behaor can be stded at macroscopc leel wthot consderng atomc and moleclar strctre. The sbect of stdng materal behaor at the macroscopc leel can be called contnm mechancs. Eternal forces hscal prncples Consttte behaor Contnos bod Solton procedre Response B nokng phscal prncples and consttte behaor, we obtan eqatons goernng the behaor of contnos sstem (a bondar ale problem. A solton to a bondar ale problem n contnm mechancs reqres consttte eqatons n addton to the goernng feld eqatons. The basc prncples goernng Newtonan mechancs are a conseraton of mass, b conseraton of momentm, c conseraton of moment of 4

5 momentm (or anglar momentm, dconseraton of energ, and elaws of thermodnamcs; these prncples are consdered to be ald for all materals rrespecte of ther nternal consttton. Therefore, a nqe solton to a bondar ale problem n contnm mechancs cannot be obtaned onl wth the applcaton of goernng feld eqatons. Hence a nqe determnaton of the response reqre addtonal consderaton that accont for the natre of dfferent materals. The eqatons that model the behaor of a materal are called consttte eqatons or consttte laws or consttte model. -3- Bondar ale problem solton A solton to a BV can be obtaned sng dfferent approaches. The followng dagram shows a schematc ew of the problem. AROXIMATE SOUTION OF THE MATH. MODE MODE ERROR? ERROR? MATERIA SCIENCE SOUTION OF THE MATH. MODE ERROR? BODY OBSERVED SOUTION Model stdes or drect eperment nclde checkng of the appromate solton wth the state arables n laborator whch noles dmensonal analss and smltde. 5

6 -4- Appromate solton of a bondar ale problem A mathematcal model (BV of a real-lfe problem s often dffclt to obtan an eact solton. The fnte element method (FEM can be ewed as a method of fndng appromate soltons for the BV problems. Two approaches of Weghted Resdal Method (WRM and Energ Methods (EM are sed for fndng appromate solton of BV. A nmber of schemes are emploed nder the WRM, among whch are collocaton, sbdoman, least sqares, and Galerkn s methods. Galerkn s method has been the most commonl sed resdal method for fnte element applcatons. Ths method s based on mnmaton of the resdal left after an appromate or tral solton s sbsttted nto the dfferental eqaton goernng a problem. The EM procedres are based on the dea of fndng consstent states of bodes or strctres assocated wth statonar ales of a scalar qantt assmed b the loaded bodes. In engneerng, sall ths qantt s a measre of energ or work. The process of fndng statonar ales of energ reqres se of mathematcal dscplnes called aratonal calcls nolng se of aratonal prncples. For man problems, both approaches eld eactl the same reslts. The followng dagram shows a schematc ew of se of these two approaches. rmtes are those nole phscal qanttes assocated wth the state arables, e.g. Tme, ength, Force, etc. Based on the prmtes we establsh the aoms for problem solng,.e. tr to obtan a solton for the assmed model. Here the prmar obecte s to make sre that the mathematcal model represents the real bod. Choce of aoms depends on the tpe of problem, form of geometr and the phscal qanttes noled. There are two knds of aoms n appled mechancs. Newtonan Aom (Newton s Aom It defnes force as momentm change, ector forces act on each partcle of the bod and an eqlbrm dfferental eqaton (or momentm balance goerns the solton throghot the bod. ebnt Aom (Work Aom It defnes work as the effect of forces actng on the bod from whch a work fncton s obtaned, e.g. potental energ, complementar energ, knetc energ, etc. A solton s 6

7 obtaned as an etremm problem, e.g., a mnmm or a mamm or a saddle pont problem. RIMITIVES NEWTONIAN AXIOM MOMENTUM EIBNITZ AXIOM ENERGY OR WORK CACUUS DIFF.EQUATIONS, BOUNDARY VAUE ROBEMS CACUUS OF VARIATION FUNCTIONA ANAYSIS VARIATIONA RINCIES, WORK FUNCTIONA GAERKIN FINITE EEMENT METHOD EXTREMUM 7

8 - Concepts of Stress, Stran, Consttte Relatons and Varos Form of Energ Ref : Energ and Fnte Element Methods n Strctral Mechancs B: I.H. Shames STRESS --- Force Dstrbtons In std of contnos meda classes of forces est: a. bod force dstrbton It acts drectl on the dstrbton of matter n the doman of specfcaton. B (,,, t B t er nt mass (,, 3, or Vector notaton Inde notaton er nt olme b. Srface Tractons In dscssng a contnm there ma be some bondar wth force dstrbted on the bondar. The force s appled to sch bondar drectl from materal otsde the doman. T,,, t or T (,,, t ( 3 er nt area need not be normal to the area element 3 ν T da ν normal to da Sperscrpt referrng to the drecton of the area element at the pont of applcaton of the srface tracton ( ν ( ν T,,, t or T (,,, t ( 3 If the area element has the nt normal n the drecton then We wold epress the tracton ector on ths element as: ( ( T or T T 8

9 --- STRESS T T T ( ( (3,, 3 3 3,, 3,, 33 normal stress off dagonal terms T ( Stress ( shear stress (3 3 T ( ( T T Tracton forces on orthogonal faces Sgn conenton: Normal stress drected otward from nterface ( tensle stress Normal stress drected toward srface (- compresse stress Shear stress s ( f both stress tself and nt normal pont n e coordnates drectons or both ponts n e coordnate drectons. 3 Knowng T for a set of aes,.e. for three orthogonal nterfaces at a pont, (ν we can determne a stress ector T for an nterface at the pont hang an drecton whateer. 9

10 Consder a pont n a contnm (an pont n the doman 3 Newton s law for the mass center n drecton Cach s Formla ( ν ( ν T T ν Tν T3ν 3 or T T ν Tν Knowng T we can get the tracton ector for an nterface at the pont. Ths formla can be sed to relate tractons on the bondar to stresses drectl net to the bondar. roe Cach s Formla. A 3 3 C ABC ν has normal ector ν B T 3 T T T

11 --3- Eqatons of Moton Consder an element of the bod of mass dm at Newton s nd law: d f dm V& ( υ T da B dv V & ρdv S D D Gass Theorem Consder a contnos, dfferentable n th order tensor feld T k. oer a olme V wth ts bondar srface defned b S. The Gass s Theorem n a generaled form s gen b: T k... dv ( T k... ν da S V or: T dv T... ν V ( k..., S ( da k where ν are the drecton cosnes of the nt otward normal. For T k. a ero order Tensor, sa a scalar fncton φ, ϕ dv ϕν da, V or: ϕ dv V S S ϕ d A where the dfferental area d A ν da. The aboe eqaton s generall referred to as Gass aw. & (, B V ρ dv D D s arbtrar B ρv&, Usng moment of momentm eqaton wll also reslt n: M H & k k Frther nestgatons: - Transformaton Eqatons for stress - rncpal stresses (gen a sstem of stresses for an orthogonal set of nterfaces at a pont, we can assocate a stress ector for nterfaces hang an drecton n space V T ν Now: s there a drecton ν sch that stress ector s collnear wthν?

12 -- STRAIN Means of epressng the deformaton of a bod 3 d A ( B ne segment n the ndeformed geometr If bod s gen a rgd bod moton each lne segment n the bod nder goes no change n length. Change n length of lne segments n the bod, (or dstance between ponts can sere as a measre of deformaton (change of shape and se of the bod. AB ds ( d d dstance between ponts When forces are appled, bod wll deform. It s conenent now to consder that the reference s labeled the reference when consderng deformed state: 3 d B A Deformaton can be depcted b mappng of each pont from coordnate to coordnate. We can sa then for a deformaton: (,, 3 Snce mappng s one-to-one:,, : ( 3

13 3 We can epress: d d d d k m k m d d d d ds l k l k d d d d ds B A k k d d d d ds ds ε δ k k d d d d ds ds η δ Stran terms: Green stran k k δ ε agrange coordnates ( ε epressed as fncton of coordnates n the ndeformed state Almans measre of stran k k δ η Eleran coordntes ( η formlated as fncton of coordnate for deformed state dsplacement feld 3 3, A, A,

14 4 We ma epress as a fncton of agrange coordnate, n whch case t epresses the dsplacement from the poston n the ndeformed state to the deformed poston. On the other hands can eqall well be epressed n terms of, the Eleran coordnates; n whch case t epresses the dsplacement that mst hae taken place to get to the poston from some ndeformed confgraton. δ δ sbsttton n preos eqaton for ε k k ε Intal ndeformed geometr Indcate what mst occr drng a gen deformaton. k k η Deformed nstantaneos geometr of bod Indcate what mst hae occrred to reach ths geometr from an earler ndeformed state. So far no restrcton on magntde of deformaton, Infntesmal stran: << << ( J J J J δ ( for nfntesmal stran can be dropped no need to dstngsh between Eleran and lograngan coordnates n epressng strans (,, η ε

15 5 γ ε ε γ engneerng shear stran γ ε ε n γ ε ε hscal nterpretaton of stran terms A small rectanglar parallelepped at. We hae also placed a Cartesan reference at. Imagne the bod has some deformaton: et s focs on lne Q roecton of Q n the drecton Q ( Q Q ( ( ( Talor seres for Q ( n terms of ( : p p (... (... Net component of elongaton of segment Q... ( Q G N Q R Q Q Q Deformed geometr Q Q

16 Where wth coalescence of & Q, we ma drop sbscrpt : ( Q ε ε change n length n the drecton per nt orgnal length of anshngl small lne segment orgnall n the drecton. Now consder R of and Q of ne segments n ntal and deformed geometr S R Q Q R We are nterested n the proecton of R and Q on to the place (on to the place the lne segments were n ndeformed state R ( Q δ β R θ Q n drecton (... p of Q n the drecton ( dsplacement component of δ dsplacement component second order ncrement for a small deformaton tg... p tgθ θ δ θ smlarl : β 6

17 θ β ε γ γ change from a rght angle of anshngl small lne segments orgnall n the & drectons at a pont Now effect of stran on a nfntesmal rect. arallelepped n the ndeformed geometr. d d d Zero shear stress means sde wll reman orthogonal on deformaton. Howeer poston and orentaton of the element ma change as length of the sdes and olme. Estence of shear stress means sdes ma lose the mtal perpendclart, (parallelograms nstead of rectangles Se of the rectanglar parallelepped s changed b normal stran whle the basc shape s changed b shear stran. V roe : ε V --- The Rotaton Tensor reosl, we consdered stretchng of a lne element to generate ε and then sed the deformaton of a anshngl small rectanglar parallelepped to ge phscal nterpretaton to the component of stran tensor. We now ntrodce rotaton tensor. Ths tme rather than consderng st the stretch of a anshngl small lne element, we consder the complete mtal relate moton of the end ponts of lne element. (nclde rotaton as well as stretchng 7

18 8 Consder N the relate moement of end ponts can be gen b sng dsp. feld. p N U U U U... Epand N U as a Talor seres abot In lmt d d notaton nde d d Ths the relate moement d between the two adacent pont d a part s ( (,,,,, ε w (rotaton tensor skew-smetrc Assme Rgd bod moton G Q, ; same δφ ( ( Sn... δφ ( δφ ( Sn G... ( : δφ δφ Q G δφ δφ (... } (...

19 δφ w 3 w 3 For other components: δφ w 3 w 3 δφ w w For rgd bod moement, the nonero components of the rotaton tensor ge the nfntesmal rotaton components of the element. What does w represent when the rectanglar parallelepped s ndergong a moement ncldng deformaton of the element and not st R. B rotaton? Each lne segment n the rectanglar olme has ts own angle of rotaton and we can show that w for sch staton ges the aerage rotaton components of all the lne segments n the bod. Howeer we shall term the component of w the rgd bod rotaton components. From epermentε porton of eqaton w related to the stress Frther nestgaton: Transformaton eqaton for stran Compatblt eqatons Stran-dsplacement relatons ε (,, ( If ' s are know, ε can be obtaned. The nerse problem of fndng the dsplacement feld from a stran feld s not so smple. Three fnctons mst be determned b ntegral of 6 partal dfferental eqatons ( to ensre sngle-aled contnos solton, we mst mpose certan restrcton of ε can not set forth an ε. to epect nqe solton, the followng eqatons are to be satsfed: 9

20 total 6 eqatons ε ε γ ( more eqatons ε γ γ γ ( more eqatons.3 HOOKE S aw near elastc behaor generaled Hook law Ckl ε kl nd, ε order tensor C th kl 4 order tensor smmetrc C smmetrc C C kl kl kl ε kl smmetrc Ckl Clk It can be shown that C kl C kl ( Usng Energ Concept It can be proed. Ths, startng wth 8 terms for C kl (3 4, we ma show, sng the three aforementoned smmetr relatons for C kl, that onl of these terms are ndependent. We wll assme now that the materal s homogeneos (whch has same composton throghot so we ma consder C kl to be a set of constants for a gen reference. For an sotropc materal, n whch the materal propertes at a pont are not dependent on drecton, we hae: λ δ ε Gε ee Ths s the general form of Hooke s law gng stress components n terms of stran components for sotropc materals. The constant λ and G are the so-called ame constants. It can be seen that as a reslt of sotrop the nmber of ndependent elastc modl has been redced fron to. The nerse of Hooke s law eldng: ν ε E ν E T kk δ

21 E and ν are Yong s modls and the posson rato stemmng from one-dmensonal test data. ν ε [ ν ( ] ε E E G [ ν ( ] ε E G E ( V λ EV ( V ( V ν λ ( λ G E G(3λ G λ G 3- Bondar-ale problems for lnear elastct The complete sstem of eqatons for lnear elastct for homogeneos, sotropc sold ncldes the eqlbrm eqatons: (3 The stress-stran law: Stran dsplacement relatons:, B eqatons λε δ ε (6 eqatons ll G ε (,, (6 eqatons We hae 5 eqatons and 5 nknowns. When eplct se of the dsplacement feld s not made, we mst be sre that the compatblt eqatons are satsfed. It mst be nderstood that B and T ( hae resltants that satsf eqlbrm eqatons for the bod as dctated b Rgd bod mechancs. In ( ths regard that B and T mst be statcall compatble.

22 We ma pose three classes of bondar ales problems: st knd B.V. problem: determne the dstrbton of stresses and dsplacements n the nteror of the bod nder a gen bod force dstrbton and a gen srface tracton oer the bondar. nd knd B.V. problem: determne the dstrbton of stresses and dsplacements n the nteror of the bod nder the acton of a gen bod force dstrbton and a prescrbed dsplacement dstrbton oer the entre bondar. Med B.V. problem: determne the dstrbton of stresses and dsplacements n the nteror of the bod nder the acton of a gen bod force dstrbton wth a gen tracton dstrbton oer part of the bondar ( s and a prescrbed dsplacement dstrbton oer the remanng part of the bondar s. ( Note : on the srfaces where the T are prescrbed, Cach s formla T ( T ν mst appl. st knd: conenent to epress basc eqatons n terms of stresses. To do ths: ν ν ε sbstttde kk δ n compatblt eatons E E Usng eqlbrm eqatons, we can arre at the Beltram-Mchell sstem of eqatons: ν K, δ K ( B, B ν ν where K kk The solton of these eqatons, sbect to the satsfacton of Cach s formla on the bondar for smpl connected domans, wll lead to a set of stress components that both satsf the eqlbrm eqatons and are derable from a sngle-aled contnos dsplacement feld.,

23 nd knd: Sbsttte eqatons and n the eqlbrm eqatons to eld dfferental eqatons wth the dsplacement feld as the dependent arable. Then we get Naer eqatons of elastct: G λ G B (, For dnamc condtons we need onl emplo the followng eqatons n place of the eqlbrm eqatons. B ρ &, The reslts are the addton of the term p & t on the rght sde of the aboe eqatons. If the aboe eqaton can be soled n conncton wth the prescrbed dsplacements on the srface and f the resltng solton s sngled aled and contnos the problem ma be consdered soled. Solton for med BV problems wll be nestgated sng dfferent technqes ntrodced partl n ths notes sch as aratonal approach. 3

24 4- Energ consderaton We hae descrbed the stress tensor arsng from eqlbrm consderaton and the stran tensor from knematcs consderatons. These tensors are related to each other b laws that are called consttte laws. In general sch relatons nclde temperatre and tme as other arables. In addton, the often reqre knowledge of the hstor of deformaton lendng to the nstantaneos condton of nterest n order to properl relate stress and stran. We assme that the consttte laws relate stress and stran drectl and nqel. That s, ε, ε,..., ε Consttte law ( C.. ( 33 Consder an nfntesmal rectanglar element nder the acton of normal stresses onl. d ( d ( d The dsplacements of faces and n the drecton are as d, Increment of mechancal work done b the stresses on the element drng deformatons s: d d d d d d d d B d d d d k d k Cancelng terms and deletng the hgher order epressons: T d B d d d d eqlbr m 4

25 d d d d dε dv Smlar epresson for and drectons can be obtaned. Ths for normal stresses on an element, the ncremental of mechancal work for sotropc materals s: ( dε dε dε dv ( normal stresses w Mechancal work per nt olme dw dε dε dε d d Now consder the case of pre shear: The mechancal ncrement of work d γ d γ γ d γ d dd d γ β d d γ B d dd d γ η d k dγ ddd dε dv ( k d β, η, Ths, for pre shear stresses on all faces we get the followng reslt for ncrements of mechancal work: ( dε dε dε dv Mechancal work ncrement per nt olme at a pont for a general state of stress s: dw ε (ald onl for nfntesmal deformaton d Now ntegratng from to some stran leelε we get: ε W dε stran energ denst fncton whch s the mechancal work performed on an element per nt olme at a pont drng a deformaton. 5

26 d dε ε ( s pont fncton, ntegral ndependent of path then perfect dfferental Total stran energ U w ε dε U w dε dv ε dε dv Eamples of Calclatng Total stran energ dε dε ( dε dε dε Unaal stress U ε ( σ dε dv σ σ w σ dε dσ E E σ U dv E σ w E dσ σ Eε dε E re bendng MY σ I σ Eε ε σ w σ dε d E MY U dv E I M d Y da E I E l EI M EIW U ρ E o I M I d W d Shear stress d W dε G G U / G dv Gγ 6

27 5- rncples of rtal work artcle Mechancs : Vrtal work s defned as the work done on a partcle b all the forces actng on the partcle as ths partcle s gen a small hpothetcal dsplacement, a rtal dsplacement, whch s consstent wth the constrants present. The appled forces are kept constant drng the rtal dsplacement. Deformable bod: Same as partcle wth specfng a contnos dsplacement feld wth small deformaton and constrant, appled force kept constant. We conenentl denote a rtal dsplacement b emplong the aratonal operatonδ. In general staton we wold hae as load possbltes a bod force ( dstrbton B throgh ot the bod as well as srface tractons T oer part of the bondar, S, of the bod. Oer the remanng part of the bondar, S, we hae prescrbed the dsplacement feld,n whch case, to aod olatng the constrants we mst be sre that δ on S. Vrtal work for sch a general solton wold be: ( δw B δ d T δ ds rt ( B and T mst not depend on δ n comptaton of δ Wrt. We can epand the srface ntegral to coer entre srface snceδ on S, ths S S S We now deelop the prncple of rtal work for a deformable bod s δwrt B δ d υ δ ds s B δ dv ( δ, dv ( B, δ dv ( δ, We now ntrodce a knematcall compatble stran feld araton δε (t s becase t s formed drectl from the dsplacement feld araton. dv δ δ δ ( ε W δε δw (, (, 7

28 Becase of skew smmetr of the rotaton tensor and the smmetr of the stress tensor, δ w ( δ, δε ( B δ dv T δ ds (, B δ dv s δε dv We now mpose the condton that we hae statc eqlbrm. Ths means n the aboe eqaton that: (. Eternal load B and T are sch that there s oerall eqlbrm for the bod from the pont of ew rgd bod mechancs we sa ( that B and T are statcall compatble.. At an pont n the bod T B, ( B δ dv T δ ds s T δε dv eternal rtal work nternal rtal work Ths s the prncple of.w. for a deformable bod We can sa that necessar condton for eqlbrm s that for an knematcall compatble deformaton feld δ, δε, the eternal.w. wth ( statcall compatble bod forces and srface tracton, mst eqal the nternal.w. Ths s sffcent for eqlbrm. Another more sefl nterpretaton of the prncple of.w. s as follows. The necessar reqrements for eqlbrm of a partclar stress feld are that : (. B and T are statcall compatble. The partclar stress feld satsfes the.w. eqlbrm for an knematcall compatble, admssble, deformaton feld. Note: the mathematcal relaton between a deformaton feld and a stress feld s ndependent of an consttte law and apples to all materals wthn the lmtatons of small deformaton. 8

29 We hae shown that the satsfacton of the prncple of.w. s a necessar relaton between the eternal loads and stresses n a bod n eqlbrm. We can also show that satsfacton of the prncple of.w. s sffcent to satsf the eqlbrm reqrement of a bod. Assme.w. eqlbrm s ald,, ( δ, ( δ δε dv δ dv dv (δ, dv, dv We made se of smmetr of. We can wrte the last epresson as follows: ( δ, dv ( δ, dv, Usng dergence theorem ( δ, dv δ υ ds, s s δ υ ds We hae made se of the fact that Now sbstttng these reslts for the last ntegral n the prncple of rtal work, that was fond preosl and s as the followng: ( B δ dv T δ ds δε dv Reslts n the followngs: s, δ δ δ dv dv dv δ on S ds V ( (, B δ dv ( T υ δ s Snceδ s arbtrar, we mst conclde, B n V ( B the same reasonng T υ on S We hae generated Newton s law for eqlbrm at an pont nsde the bod and Cach s formla, whch ensre eqlbrm at the bondar. Satsfacton of prncple of.w. s both necessar and sffcent for eqlbrm. 9

30 6- The Method of Total otental Energ Note: Calcls of Varatons has to be reewed. We now deelop from the rtal work dea, the concept of total potental energ whch apples to elastc bod (not necessar lnear elastc: ( B δ dv T δ ds s δε dv d d ε δε ε ε s δ ( B δ dv T δ ds δ dv δ dv δ U Note: δ s rtal dsplacement feld. A pror not related to stress feld We defne potental energ V of appled load as a fnctonal of dsplacement feld ( V B dv T ds B and T ν prescrbed s ( δ V B δ dv T ds δ δ s δ V ( B δ dv T δ s ds δ ( U V (rncple of total potental energ π U V (Total otental Energ π U B dv s T ( δ ( π rncple of total potental energ ds Interpretaton: The necessar reqrements for eqlbrm of a partclar stress feld :. B and T ν are statcall compatble 3

31 . The deformaton feld, to whch the feld s related throgh a consttte law for elastc behaor, etreme TE wth respect to all other knematcall compatble admssble deformaton felds. Etrematon of the TE w.r.t admssble deformaton felds s necessar for eqlbrm to est between the forces and the stresses n a bod. Jst n the method of rtal work, we can show t to be a sffcent condton for eqlbrm. We can show that TE s actall a local mnmm for the eqlbrm confgraton nder loads B and T ν compared wth the TE correspondng to neghborng admssble confgratons wth the same B and T ν. Eamne the dfference between TE of eqlbrm state and an admssble neghborng state δ and ε δε show that the second araton of TE s poste. The total potental energ theorem states that of all the admssble felds whch satsf compatblt and essental bondar condtons, the actal one whch satsfes eqlbrm and stress BC s prode a mnmm to π. The total potental (π s also called the fnctonal of the problem. Assme that n the fnctonal (π the hghest derate of a state arable (wrt a space coordnates s of order m,.e. the operator contans at most m th order derates. Sch a problem we call C m- aratonal problem. Consderng the bondar of the problem, we can dentf two classes on bc s: Essental bc s (geometrc: correspond to prescrbed dsplacement and rotatons. The order of the derates n the essental bc s s n a C m- roblem, at most m-. Natral bondar condtons (force bc s: corresponds to prescrbed bondar force and momentms. The hghest derate n ths bc s are of order m to m-. B nokng the statonar of the fnctonal a problem, the problem goernng dfferental eqaton and natral and essental bc s can be dered. In C m- aratonal problem, the order of the hghest derate presented n the problem goernng dfferental eqaton s m. Therefore, ntegraton b parts s emploed m tmes. Effect of bc s are nclded mplctl n π. 3

32 7- Dfferental Eqatons VS fnctonal for contnos sstems We can get a solton to a partal dfferental eqaton whch s satsfed at each pont n the bod and also satsf a set of bondar condtons. A solton obtaned, mabe for dsplacements or stresses, etc. A fnctonal represents a nmber (scalar and for natrall occrrng fnctonal, t ma represent work, energ or power or etc. In some nstances, t ma not represent an phscal qantt. At etremm, t elds a solton to the dfferental eqaton (eqlbrm or momentm balance or heat balance, etc.. e. g. I f ( d ( fnctonal Estence of a fnctonal and solton obtaned as etremm of ths fnctonal also helps to determne as to what knd of eqlbrm s acheed. Ths leads to theor of stablt, for eample, f t s a mnmm at etremm then the solton obtaned s stable! To go from dfferental eqaton to aratonal problem we need to know operatonal algebra or calcls (fnctonal analss and to go from aratonal problem to dfferental eqaton we need to know the calcls of aratons. 7. Formlaton of contnos sstems We consder a tpcal dfferental element wth the obecte of obtanng dfferental eqatons that epress the element eqlbrm reqrements, consttte relatons, and element nterconnectt reqrements. These dfferental eqatons mst hold throghot the doman of the sstem and before the solton can be obtaned the mst be spplemented b bondar condtons and, n dnamc analss, also b ntal condtons. Two dfferent approaches can be followed to generate the sstem goernng dfferental eqatons.. The drect method (dfferental eqatons. The aratonal method 3

33 The drect method In ths method, we establsh the eqlbrm and consttte reqrements of tpcal dfferental elements n terms of state arables. These consderatons lead to a sstem of dfferental eqatons n the state arables. In general the eqatons mst be spplemented b addtonal dfferental eqatons that mpose approprate constrants on the state arables n order that all compatblt reqrements be satsfed. Fnall to complete the formlaton all the bondar condtons and n a dnamc analss the ntal condtons are stated n dfferental formlaton for a contnos sstem, a dfferental element wth obecte of obtanng dfferental eqaton that epress element eqlbrm s fond. Ths dfferental eqaton mst hold throgh the doman of the sstem. The D.E mst be spplemented b B.C. S and dnamc analss, ntal condton eample. 7.. Eamples of dfferental approach (, t p (, t Eample - Beam element a Dfferental Ele. M ( M M d V V V d b Eqlbrm n ertcal drect m t V d pd m d t or V m t p eqatng sm of the moment abot the left hand face to ero 33

34 34 M d M M d d t m d pd d d V V M V now θ also from elementar beam theor EI M θ EI V EI M θ Transerse brate of beam For a nqe solton we mst specf bc s Note of the elementar beam theor lan reman plan t m EI M V M free Smpl spported. θ d EI M θ ( ma Σ da c da F σ I M da c M σ σ ma d e θ d p ρ θ ε ε / / ( ' ( d d e e EI M s Hook E I M M d

35 Eample - Dam s Reseror bc(4 bc (3 p c & p bc ( bc ( Eample 3- Rod sbected to step load E ong modls p ρ mass denst A cross secton (, t Ro R(t Ro Dfferental element Rod t σ A σ ( σ d A d Eqlbrm Consttte relaton σ σ d A ( σa A ρd σ E d && Combnng eqatons: C t C E ρ 35

36 b.c s ( o, t EA ( l, t Ro Intal load (, o (, o t 36

37 7-- Eamples of Varatonal approach Eample. Beam EI Π ( w w d p( w( d U ( w stran energ potental energ of loadng C m m C essental bc w, w δπ EI EIw w δ w δw EI d w δw δw d d EIw δ w EIw δ w EI w δw d : δ Π ( EIw δw d EIw δ w EIw δw l EIw bc s EIw δ w EIW δ w In general on the bc s at or 37

38 Eample: w Bcklng of the colmn k p Π EI ( w d w d k w proe t m C m C ess. b. c w, w U EI w d w δ δ ds ( no change n length de to w dw ds dw d d d w w d ds d dw ( δ w d for small dsp. d w d δ w d δ s w small d W w d 38

39 Eample3. Rod sbected to STE load (, t Ro Π EA d f B d B δ Π EA δ d δ f d δ R R bc s ' (, t f (, t B bod force for nt length EA δ EA δ d δ f B d δ R δ s arbtrar B ( EA f δ d [ EA R] δ EA δ EA f B EA R or δ EA or δ m C aratonal problem 39

40 Eample 4. -D Varatonal rncple J Ω k k φ φ Qφ d Ω Γq q φ d Γ K & Q are fnctons of postons onl δφ on Γφ ( part of the bondar q specfed on Γ q Γ q Γ f Γ φ δ J [ kφ δφ kφ δφ Qδ φ] d Ω Ω φ Note: δ φ δ ( δφ ( δφ Integrate b part the frst two terms Γq q δφ d Γ kφ δφ dd kφ ( δφ dd kφ δφ d ( kφ Ω Ω Γ d Γ Ω δφ d Ω d θ d θ d d Γ cos θ d d Γ sn θ kφ δφ dd ( kφ δφ d Ω k φ δφ d Ω Ω Γ [( kφ ( kφ Q] δφ d Ω [ kφ kφ ] δφ d Γ q δφ d Γ δ J Ω Γ Γq φ k n d Γ 4

41 kφn δφ d Γ Γ Γ Γ Γφ Γ Γ Γ Γ δφ on Γ q φ f f q φ Ω δ J δφ d Ω ( kφ q δφ d Γ kφn δφ d Γ Γq n Γf δφ Arbtrar Eler. eq n Ω kφ q n δφ on or n q kφ Γ f Heat Condcton : If k & Q constant φ const. osson s eqaton If k & Q φ aplace eqaton Also other form of eqatons sch as Torson problem (osson s eqaton or Irrotatonal flow (aplace eqaton, seepage problem or flow throgh poros meda are eamples of the aboe eqatons. Eample5. Transent -D Heat Eqalent stead state aratonal prncple for an tme t : φ { k φ k φ } Qφ C & φφ dd T qaφ d Γ T qc α φc φ d Γ J ( φ T Ω ΓA ΓC φ s a fncton of, and tme t & φ φ t δ J ( φ at an tme t φ t mst be consdered fed n the calcls of araton formlaton 4

42 Ω [ k φ δφ k φ δφ Qδφ Cδφφ & C & φ ] da δ J ( φ T δφ T qa δφ d Γ T ΓA ΓC [ q δφ α φ δφ α φ δφ] d Γ C k δφ d Γ ( kφ δφ d Ω k φ δφ d Γ T φ ( k φ δφ d Ω Γ Ω Γ Ω Q d Ω Cφ δφ d Ω T q δφ d Γ T δφ & ( k φ ( k φ Q C & φ ΓA C A ( qc α φ α φc δφ d Γ ΓC δ T ( φ T T T Ω Γ Γ [ ( k φ ( k φ Q C & φ ] δφ d Ω ( k φ k φ δφ d Γ T C Γ Γ Γ Γ A B qc α ( φ φc δφ d Γ C Γ A q A δφ d Γ Γ B Ω Γ A Γ C Γ ΓA ΓB Γ T Thckncss C On Γ Γ Γ B A C φ φb k φ k φ q k φ k φ A q C α ( φ φ C φ tempreatre k k Q Heat npt per nt olme q Thermal Condctt n drecton Thermal Condctt n drecton A, q C specfed heat npt per nt area on Γ A and Γ C respectel 4

43 roblem: For a transent -D heat flow, the eqalent stead state aratonal prncple at tme t can be wrtten as: φ [( kφ k φ Qφ C & φφ ] dd qaφ d Γ qc α φc φ d Γ J ( φ Ω Γ A Γ C φ (,, t & φ φ t k, k Q qa, q Γ C Ω Γ B Γ A Γ Γ A Γ B Γ C Yo are asked to fnd the Eler eqaton and the approprate bondar condton - assmpton abot dsplacement feld - sometme assmpton abot consttte law 3- aratonal process as t relates to the T..E 4- t ges s proper eqatons of eqlbrm and proper BC S (Certan nternal constrants de to dsplacement assmptons 8. No. of Rgd bod modes n a sstem In a aratonal form we tr to fnd the stran energ U. The rgd bod motons are not accompaned b change n stran energ. The No. of non contrbtng terms (from the dsplacement model to the stran energ are the No. of rgd bod modes. Bathe.73 If the strctre s not spported, there wll be a nmber of lnearl T ndependent ectors, U, U,..., Vq for whch the epresson U K U s eqal to ero,.e. ero stran energ s stored n the sstem when U s the dsplacement ector. Sch ector U s sad to represent a rgd bod mode of the sstem. 43

44 9. Sample roblems - For the beam shown, wrte down the aratonal prncple (otental Energ whch also ncldes the bondar actons. Fnd ot the Eler agrange eqaton and the assocated bondar condtons. V M constant nt length M k θ k θ k nform EI k K, K are translatonal sprng constnts K θ, Kθ are rotatonal sprng constnts Are there an rgd bod modes present? - Fgre show a sstem of beam-colmn wth transerse and tangental sprngs. a Wrte down the fnctonal (Total otental Energ for the sstem. erform the frst araton (Fg. b Dere the Eler-agrange eqatons and the assocated bc s c How man rgd bod modes do est?, p (, d erform the second araton δ Π to show weather the problem s a mnmm or mamm. Note: δ Π δ ( δ Π, K N q ( K T, 44