平面問(wèn)題的復(fù)變函數(shù)法課件

平面問(wèn)題的復(fù)變函數(shù)法*

§5-1Complex-variablerepresentationofstressfunction

Inchapter2,wehaveproved,inplaneproblems,thereisastressfunctionφthatisbiharmonicfunctionofpositioncoordinates,ifbodyforceisconstant,i.e.Complex-variableMethodsforPlaneElasticityNowintroducecomplexvariablez=x＋iyandz＝x－iytoreplacerealvariablexandy.Noticing,*

§5-1應(yīng)力函數(shù)的復(fù)變函數(shù)表示

在第二章中已經(jīng)證明，在平面問(wèn)題里，如果體力是常量，就一定存在一個(gè)應(yīng)力函數(shù)φ，它是位置坐標(biāo)的重調(diào)和函數(shù)，即現(xiàn)在，引入復(fù)變數(shù)z=x＋iy和z＝x－iy以代替實(shí)變數(shù)x

和y。注意平面問(wèn)題的復(fù)變函數(shù)法*

Wefindthetransformationarefurthermore,Complex-variableMethodsforPlaneElasticity*

可以得到變換式進(jìn)而平面問(wèn)題的復(fù)變函數(shù)法*LetSowecantransformthefunctionasForComplex-variableMethodsforPlaneElasticity*令于是可將方程式變換成為由平面問(wèn)題的復(fù)變函數(shù)法*

ItisobviousknownPisharmonicfunction

whichcanbeobtainedbyrealpartofanalyticalfunction.Supposef(z)asanalyticalfunctionandletForLetyieldsthusComplex-variableMethodsforPlaneElasticity*

可知，P是調(diào)和函數(shù)可由解析函數(shù)的實(shí)部得到。設(shè)f(z)為解析函數(shù)，可令由令得則平面問(wèn)題的復(fù)變函數(shù)法*ThenintegratingwithrespecttozyieldsLeti.e.IntegratingtheaboveequationwithrespecttoyieldsthusComplex-variableMethodsforPlaneElasticity*

再對(duì)z積分，得到令即將上式對(duì)

積分，得到則平面問(wèn)題的復(fù)變函數(shù)法*

Noticethebiharmonicfunctionontheleftsideoftheaboveequationisarealfunction.Itisobviousthatthefourtermsontherightsidemustbeconjugatetwoandtwo.Thefirsttwotermsisconjugate,andthenexttwotermsshouldbealsoconjugate:LetweobtainthefamousgusaformulaitcanbealsowrittenasComplex-variableMethodsforPlaneElasticity*

注意上式左邊的重調(diào)和函數(shù)φ是實(shí)函數(shù)，可見(jiàn)該式右邊的四項(xiàng)一定是兩兩共軛，前兩項(xiàng)已經(jīng)是共軛的，后兩項(xiàng)也應(yīng)是共軛的：令即得有名的古薩公式也可以寫(xiě)成平面問(wèn)題的復(fù)變函數(shù)法*

Soinplaneproblemswhenbodyforceisconstant,stressfunctionφcanberepresentedbytwoanalyticalfunctionsofcomplexvariablez,

(z)and

(z),namedK-Mfunction.SosolvingplaneproblemsisjustselectingK-Mfunctionproperlyanddetermininganyconstantinthemaccordingtoboundarycondition.Complex-variableMethodsforPlaneElasticity*

于是可見(jiàn)，在常量體力的平面問(wèn)題中，應(yīng)力函數(shù)φ總可以用復(fù)變數(shù)z的兩個(gè)解析函

(z)和

(z)來(lái)表示，稱(chēng)為K-M

函數(shù)。而求解各個(gè)具體的平面問(wèn)題，可歸結(jié)為適當(dāng)?shù)剡x擇這兩個(gè)解析函數(shù)，并根據(jù)邊界條件決定其中的任意常數(shù)。平面問(wèn)題的復(fù)變函數(shù)法*§5-2Complex-variablerepresentationofstressanddisplacementAccordingtotherelationbetweenstresscomponentandstressfunction,OneComplex-variablerepresentationofstresscomponentComplex-variableMethodsforPlaneElasticity*

§5-2應(yīng)力和位移的復(fù)變函數(shù)表示根據(jù)應(yīng)力分量和應(yīng)力函數(shù)的關(guān)系一應(yīng)力分量的復(fù)變函數(shù)表示平面問(wèn)題的復(fù)變函數(shù)法*

WefindComplex-variablerepresentationofstresscomponent,foryieldshence,forComplex-variableMethodsforPlaneElasticity*

可得到應(yīng)力分量的復(fù)變函數(shù)表示由可得而由平面問(wèn)題的復(fù)變函數(shù)法*

yieldsorComplex-variableMethodsforPlaneElasticity*

可得或平面問(wèn)題的復(fù)變函數(shù)法*

Onlygiven

(z)andψ(z),wecandividetherightsideofaboveequationintoimaginarypartandrealpart,fromimaginarypartwegetτxy,fromrealpartwegetσy-σx.andiscomplex-variablerepresentationofstresscomponent.Ofcoursebybuildingequations,σx、σy

、τxy

canberepresentedby

(z)andψ(z)respectively,butthatwillmakeequationsbecomelengthinessandit’snotconvenienttouse.Complex-variableMethodsforPlaneElasticity*

只要已知

(z)及ψ

(z)，就可以把上述公式右邊的虛部和實(shí)部分開(kāi)，由虛部得出τxy，由實(shí)部得出σy-σx。和就是應(yīng)力分量的復(fù)變函數(shù)表示。當(dāng)然也可以建立公式，把σx、σy、τxy三者分開(kāi)用

(z)和ψ

(z)來(lái)表示，但那些公式將比較冗長(zhǎng)，用起來(lái)很不方便。平面問(wèn)題的復(fù)變函數(shù)法*

TwoComplex-variablerepresentationofdisplacementcomponent

Assumingplanestressproblems,accordingtogeometricalequationandphysicalequationyieldsComplex-variableMethodsforPlaneElasticity*

二位移分量的復(fù)變函數(shù)表示

假定為平面應(yīng)力問(wèn)題。由幾何方程及物理方程可得平面問(wèn)題的復(fù)變函數(shù)法*forandnoticesimilarlyyieldsComplex-variableMethodsforPlaneElasticity*

由于并注意到同理可得平面問(wèn)題的復(fù)變函數(shù)法*Successiveintegrationoftheabovetwoequationswithrespecttoxandy,leadtoWheref1andf2arearbitraryfunctions.SubstitutingaboveequationsintothefollowingequationComplex-variableMethodsforPlaneElasticity*將上兩式分別對(duì)x及y積分，得其中的f1及f2為任意函數(shù)。將上式代入式平面問(wèn)題的復(fù)變函數(shù)法*

forComplex-variableMethodsforPlaneElasticity*

由于平面問(wèn)題的復(fù)變函數(shù)法*

andyieldsThuswecanfindthedisplacementofrigidbody

f1(y)＝u0－ωy，f2(x)＝v

0＋ωxWegetComplex-variableMethodsforPlaneElasticity*

從而得到于是得到剛體位移

f1(y)＝u0－ωy，f2(x)＝v

0＋ωx故有平面問(wèn)題的復(fù)變函數(shù)法*

Ifneglectingdisplacementofrigidbody,wehaveforyieldsComplex-variableMethodsforPlaneElasticity*

若不計(jì)剛體位移，則有由式得到平面問(wèn)題的復(fù)變函數(shù)法*

lettheresultbacksubstitution,andonthetwosidedivide1+νyieldsThisiscomplex-variablerepresentationofdisplacementcomponent.If

(z)andψ(z)aregiven,wecandividetherealpartandimaginarypartoftherightsideoftheaboveformula,anduandvcanbesolved.Complex-variableMethodsforPlaneElasticityTheaboveformulaiseducedinplanestressproblem.Toplanestrainproblem,weneedreplaceEwithE/(1－2)andwith/(1－)。*將結(jié)果回代,并兩邊除以1+得

這就是位移分量的復(fù)變函數(shù)表示。若已知

(z)及ψ

(z)，就可以將該式右邊的實(shí)部和虛部分開(kāi)，從而得出u和v。平面問(wèn)題的復(fù)變函數(shù)法

上述公式是針對(duì)平面應(yīng)力情況導(dǎo)出的。對(duì)于平面應(yīng)變情況，須將式中的E改換為E/(1－2)，改換為/(1－)。*§5-3Complex-variableRepresentationOfBoundaryCondition

Toevaluateφofeverycrunodeinboundary,weneedapplyboundaryconditionofstress,i.e.:andSubstitutionintotheaboveequation,givesComplex-variableMethodsforPlaneElasticity*§5-3邊界條件的復(fù)變函數(shù)表示

為了求得邊界上各結(jié)點(diǎn)處的φ值，須要應(yīng)用應(yīng)力邊界條件，即：

而代入上式，即得：

平面問(wèn)題的復(fù)變函數(shù)法*

Asthefigureshownl=cos(N,x)=dy/ds,m=cos(N,y)=-dx/ds,Sotheaboveequationcanberewrittenas:Thus,yieldsComplex-variableMethodsforPlaneElasticity*

由圖可見(jiàn)，l=cos(N,x)=dy/ds,m=cos(N,y)=-dx/ds,于是，前式可改寫(xiě)為：由此得：

平面問(wèn)題的復(fù)變函數(shù)法*

SupposeAisafixpointintheboundary,andBisaarbitrarypoint,sothecompositionofforcesfromAtoBcanbeobtainedbyintegratingoftheaboveequationwithrespecttosfromAtoB,SubstitutingthisformulaComplex-variableMethodsforPlaneElasticity*

設(shè)A是邊界上的固定點(diǎn),B為任意一點(diǎn)，則從A到B邊界上的合力，可用上式從A點(diǎn)到B點(diǎn)對(duì)s積分得到：將式平面問(wèn)題的復(fù)變函數(shù)法*

intotheaboveequation,andrearrangementyieldsAddingacomplexconstantintostressfunction,whichdoesn’tinfluencethestress.SowecanletAofstressfunctionaszero,andthenσintheboundarygivesorThisisboundaryconditionofstress.Complex-variableMethodsforPlaneElasticity*

代入，整理得：把應(yīng)力函數(shù)加上一個(gè)復(fù)常數(shù)，并不影響應(yīng)力。因此，可把應(yīng)力函數(shù)A處的值設(shè)為零，于是對(duì)于邊界上的σ有或這就是應(yīng)力邊界條件。平面問(wèn)題的復(fù)變函數(shù)法*

ToboundaryconditionofdisplacementSubstitutingthemintothefollowingequationWecanobtaincomplex-variablerepresentationofboundaryconditionofdisplacementinplanestressproblem.Complex-variableMethodsforPlaneElasticity

Toplanestrainproblem,weneedreplaceEwithE/(1－2)andwith/(1－)。*

對(duì)于位移邊界條件將其代入下式即得平面應(yīng)力情況下位移邊界條件的復(fù)變函數(shù)表示平面問(wèn)題的復(fù)變函數(shù)法

對(duì)于平面應(yīng)變，須將式中的E改換為E/(1－2)，改換為/(1－)。*

§5-4Thesingle-valuedconditionofstressanddisplacementinmultiplyconnectedregion

Whenstressisdetermined,thestressfunctioncanstillaarbitrarylinearfunction,sotheK-Mfunctionisnotdeterminedcompletely,sotothesimplyconnectedregion,theK-Mfunctioncanbedeterminedbyselectingthesuitablecoordinate.Buttothemultiplyconnectedregion,itisstillaproblem.Inthissection,theconditionoftheK-Mfunctionsatisfiedsingle-valuedinmultiplyconnectedregionisdiscussed.SupposethereisamultiplyconnectedregionthathasainteriorboundaryC,andintheinteriorboundaryCtheexternalforcevectorisgiven.Generallymultiformfunctionislogarithmicfunction,wesupposeComplex-variableMethodsforPlaneElasticity*

§5-4多連通域內(nèi)應(yīng)力與位移的單值條件

應(yīng)力確定后，應(yīng)力函數(shù)仍可差一個(gè)任意的線(xiàn)性函數(shù)，這時(shí)K-M函數(shù)并未完全確定.對(duì)于單連通區(qū)域，可以通過(guò)選取適當(dāng)坐標(biāo)系等辦法，使得K-M函數(shù)完全確定;但對(duì)于多連通區(qū)域仍不能完全確定.本節(jié)討論K-M函數(shù)在多連通區(qū)域內(nèi)滿(mǎn)足單值的條件。

設(shè)有多連通區(qū)域，有一內(nèi)邊界C，設(shè)在邊界C上的外力矢量已給定。通常的多值函數(shù)是對(duì)數(shù)函數(shù)，我們?cè)O(shè)平面問(wèn)題的復(fù)變函數(shù)法*

DCWherezkisaarbitrarypointintheinteriorboundary,

fandψfaresingle-valuedanalyticalfunction(holomorphicfunction),andAkandBkareconstantsgives:Complex-variableMethodsforPlaneElasticity*

DC這里zk為內(nèi)部邊界內(nèi)的任意一點(diǎn)，

f和ψf為單值的解析函數(shù)（全純函數(shù)），而Ak

，Bk為常數(shù)：平面問(wèn)題的復(fù)變函數(shù)法*

Thederivativeofabovefunctionissingle-valued,butitismultivaluedbyitself,whilezmovearoundthecircumonce,thevalueofln(zk)haveaincrementof2πi,sotheincrementsof

(z)andψ(z)is2πiAkand2πiBkrespectively,andaccordingtothefollowingformula,themastervectorofstresses

themastervectorofstress(aroundthewholeboundary)isontheleftside,andtheincrementisontherightside:Complex-variableMethodsforPlaneElasticity*

前面的函數(shù)的導(dǎo)數(shù)是單值的，但他們本身是多值的，當(dāng)z繞周邊一周時(shí)，函數(shù)值ln(zk)產(chǎn)生一個(gè)增量2πi,于是

(z)和ψ

(z)的增量分別是2πiAk和2πiBk,這時(shí)應(yīng)力主矢量按照公式左邊將得到應(yīng)力主矢量(沿整個(gè)邊界），右邊得到一增量：平面問(wèn)題的復(fù)變函數(shù)法*

Whileaccordingtothefollowingformula,displacementwillobtainincrement,andaccordingtosingle-valuedtheincrementshouldbezero：andyieldsComplex-variableMethodsforPlaneElasticity*

這時(shí)位移按照公式也將得到增量，根據(jù)單值性這個(gè)增量應(yīng)為零：結(jié)合可得到平面問(wèn)題的復(fù)變函數(shù)法*

thusWhilethereisminteriorboundary,letComplex-variableMethodsforPlaneElasticity*

于是當(dāng)有m個(gè)內(nèi)邊界時(shí)，取平面問(wèn)題的復(fù)變函數(shù)法*

§5-5Situationofinfinitemultiplyconnectedbody

Whenexteriorboundaryofmultiplyconnectedbodyapproachinfinitefarness,thismultiplyconnectedbodybecomeinfinitemultiplyconnectedbody,besidestheaboveconditions,weneedconsidertheultimatesituationofinfinitefarness.

Regards

originofcoordinateasthecentreofcircle,drawabigenoughcirclesR,whichincludeallinteriorboundary,Toanarbitrarypointintheelastomer,butbeyondsR,givestheanalyticalfunctionbeyondsRComplex-variableMethodsforPlaneElasticity*

§5-5無(wú)限大多連體的情形

當(dāng)多連體的外邊界趨于無(wú)限遠(yuǎn)時(shí)，該多連體成為無(wú)限大的多連體，除上述條件外，還需考慮無(wú)限遠(yuǎn)的極限情況。

以坐標(biāo)原點(diǎn)為圓心，作充分大的圓周sR，將所有的內(nèi)邊界包圍在其內(nèi)，對(duì)于sR之外，彈性體之內(nèi)的任意一點(diǎn)，可得到在sR之外的解析函數(shù)平面問(wèn)題的復(fù)變函數(shù)法*

soItcanbewrittenasWherePx,Pyarethesumofsurfaceforcesinmboundary.Complex-variableMethodsforPlaneElasticity*

于是可寫(xiě)為其中Px,Py為m個(gè)邊界上沿x,y方向的面力之和。平面問(wèn)題的復(fù)變函數(shù)法*

Expandholomorphicfunctionφ*fandψ*f

inMultiplyconnectedregionbyluolangseries:soForcomponentsofstressesininfinitefarnessisfinite,thecoefficientsofn≥2iszero.Complex-variableMethodsforPlaneElasticity*

將多連通區(qū)域內(nèi)的全純函數(shù)φ*f和ψ*f展開(kāi)為羅郎級(jí)數(shù)：于是

由于在無(wú)窮遠(yuǎn)處的應(yīng)力分量應(yīng)該是有限的，級(jí)數(shù)中n≥2的系數(shù)應(yīng)為零。平面問(wèn)題的復(fù)變函數(shù)法*

Similarity,fromforcomponentsofstressininfinitefarnessisfinite,soWhereneglectingtheconstanttermsthathavenorelationtostresses.Complex-variableMethodsforPlaneElasticity*

同樣從中，由于在無(wú)窮遠(yuǎn)處的應(yīng)力分量應(yīng)該是有限的，故有其中略去了和應(yīng)力無(wú)關(guān)的常數(shù)項(xiàng)。平面問(wèn)題的復(fù)變函數(shù)法*

soWhereβisnorelationtostresscalculate,itcanberegardedaszero,andComplex-variableMethodsforPlaneElasticity*

于是其中β與應(yīng)力計(jì)算無(wú)關(guān)，可取為零，而平面問(wèn)題的復(fù)變函數(shù)法*

AtthistimeWhenz→∞，fieldsSimilarity,when

z→∞,forfieldsFromaboveequation,wecanobtainthecorrespondingcoefficientandwecanalsofindininfinitefarnessthedistributingofstressissymmetrical.Complex-variableMethodsforPlaneElasticity*

這時(shí)當(dāng)z→∞時(shí)，可得同樣當(dāng)z→∞時(shí)，由可得從中可求得相應(yīng)的系數(shù)，并可以看到在無(wú)限遠(yuǎn)處，應(yīng)力的分布是均勻的。平面問(wèn)題的復(fù)變函數(shù)法*

coefficientsthusComplex-variableMethodsforPlaneElasticity*

系數(shù)則平面問(wèn)題的復(fù)變函數(shù)法*§5-6Problemofinfiniteplaneincludinghole

Regards

originofcoordinateasthecentreofcircle,drawabigenoughcirclesR,whichincludeallinteriorboundary,sotoanarbitrarypointintheelastomerbutbeyondsR,wehaveComplex-variableMethodsforPlaneElasticity*§5-6含孔口的無(wú)限大板問(wèn)題

以坐標(biāo)原點(diǎn)為圓心，作充分大的圓周sR，將所有的內(nèi)邊界包圍在其內(nèi)，對(duì)于sR之外，彈性體之內(nèi)的任意一點(diǎn)，可得到平面問(wèn)題的復(fù)變函數(shù)法*Complex-variableMethodsforPlaneElasticity*平面問(wèn)題的復(fù)變函數(shù)法*

rewrittenaswhereComplex-variableMethodsforPlaneElasticity*

改寫(xiě)為其中平面問(wèn)題的復(fù)變函數(shù)法*TothepointsintheboundaryofholeComplex-variableMethodsforPlaneElasticity*

對(duì)于孔邊上的點(diǎn)平面問(wèn)題的復(fù)變函數(shù)法*SubstitutingtheaboveequationsintothefollowingequationWecanobtainthestressboundaryconditionofseriesformsofcircleboundaryinpolarcoordinate.

AssumingtheexternalforceisknownandspreaditbyFourierseries,Complex-variableMethodsforPlaneElasticity*

將上列各式代入就得到極坐標(biāo)下圓周邊界上的級(jí)數(shù)形式的應(yīng)力邊界條件。

設(shè)周邊上的外力為已知，并將其展開(kāi)為傅氏級(jí)數(shù)平面問(wèn)題的復(fù)變函數(shù)法*

Bycomparingcoefficientofeikande-ik,fieldsComplex-variableMethodsforPlaneElasticity*

比較兩邊eik

和e-ik

的系數(shù)，可得平面問(wèn)題的復(fù)變函數(shù)法*

Forthestressesconditionininfinitefarness,yieldsComplex-variableMethodsforPlaneElasticity*

由無(wú)限遠(yuǎn)處的應(yīng)力條件，可得平面問(wèn)題的復(fù)變函數(shù)法*Forthesingle-valuedconditionofdisplacementyieldsandWefindforComplex-variableMethodsforPlaneElasticity*由位移的單值條件有及可求得再由平面問(wèn)題的復(fù)變函數(shù)法*

yieldsBynow,allcoefficientshavebeensolved.Forexample,assumingtheuniformpressurearoundholeispandthestressofinfinitefarnessiszero.Complex-variableMethodsforPlaneElasticity*

可求得至此，全部系數(shù)均已求出。例

設(shè)孔周邊為均勻壓力p，無(wú)限遠(yuǎn)處的應(yīng)力為零。平面問(wèn)題的復(fù)變函數(shù)法*ThusSowegetComplex-variableMethodsforPlaneElasticity*

則有于是可求得平面問(wèn)題的復(fù)變函數(shù)法*

Finally,wegetAccordingtotheabovemeans,thegeneralproblemsofinfiniteplaneincludingholecanbesolved.Complex-variableMethodsforPlaneElasticity*

最后得到根據(jù)上述方法，圓孔口無(wú)限大板的一般問(wèn)題都可以得到解決。平面問(wèn)題的復(fù)變函數(shù)法*Exercise5.1

Trytocheck-upthefollowingComplex-variable

(1)(2)Solution:

Thefundamentalformulagives(1)Substitutinginto(a)、(b)

Complex-variableMethodsforPlaneElasticity*平面問(wèn)題的復(fù)變函數(shù)法練習(xí)5.1

試考察下列復(fù)變函數(shù)所解決的問(wèn)題(1)(2)解:基本公式為(1)將分別代入(a)、(b)式*yieldsassociatewiththeabovetwoequations,yieldsThegivenfunctioncansolvetheproblemsthattherectangularsheetisunderuniformpullingforceqinaxisx.Asthefigure5.1(a)shown.(2)

Substitutinginto(a)and(b),yieldsxyqqFigure5.1(a)Complex-variableMethodsforPlaneElasticity*平面問(wèn)題的復(fù)變函數(shù)法得聯(lián)立求解以上兩式,得

所給的函數(shù)可以解決矩形薄板在x方向受均布拉力q的問(wèn)題.如圖5.1(a)所示(2)將代入(a),(b)兩式,得xyqq圖5.1(a)*associatewiththeabovetwoequations,yieldsThegivenfunctioncansolvethepureshearproblemofrectangularsheet.Asthefigure5.1(b)shown.qqxy圖5.1(b)Exercise5.2

Asthefigureshown.Trytoprovewecanusecomplex-variabletosolvethepurebendingproblemofbeamofrectangularcross-section.WhereIistheinertiamomentofcrosssectionofbeam,Misthemomentofflexion.MyxzySolution:ThefundamentalformulagivesComplex-variableMethodsforPlaneElasticity*平面問(wèn)題的復(fù)變函數(shù)法聯(lián)立求解以上兩式,得

所給的函數(shù)可以解決矩形薄板受純剪切問(wèn)題.如圖5.1(b)示.qqxy圖5.1(b)練習(xí)5.2

如圖所示.試證矩形截面梁的純彎曲問(wèn)題可用如下的復(fù)變函數(shù)求解.其中I為梁截面的慣矩,M為作用的彎矩.Myxzy解:基本公式為*Substitutinginto(1)、(2)For(1),yieldsi.e.Complex-variableMethodsforPlaneElasticity*平面問(wèn)題的復(fù)變函數(shù)法將代入(1)、(2)式由(1)式得即*orForequation(2)yields,i.e.equatio