第七章應(yīng)力狀態(tài)與應(yīng)變狀態(tài)分析

CHAPTER7

ANALYSISOFSTRESSANDSTRAIN

MechanicsofMaterials第七章應(yīng)力狀態(tài)與應(yīng)變狀態(tài)分析材料力學(xué)CHAPTER7ANALYSISOFTHESTATEOFSTRESSANDSTRAIN

§7–4

PRINCEPALSTRESSESANDTHEIRTRAJECTORIESOFTHEBEAM§7–5

ANALYSISOFTRIAXIALSTRESSEDSTATE—METHODOFSTRESSCIRCLE§7–6

ANALYSISOFSTRAININAPLANE§7–7

RELATIONBETWEENSTRESSANDSTRAINUNDERCOMPLEXSTRESSEDSTATE—（GENERALIZEDHOOKE’SLAW）

§7–8STRAIN-ENERGYDENSITYUNDERCOMPLEX

STRESSEDSTATE

§7–1CONCEPTSOFTHESTATEOFSTRESS§7–2ANALYSISOFTHESTATEOFPLANESTRESS—ANALYTICALMETHOD§7–3ANALYSISOFTHESTATEOFPLANESTRESS—

GRAPHYCALMETHOD

第七章應(yīng)力狀態(tài)與應(yīng)變狀態(tài)分析

§7–1應(yīng)力狀態(tài)的概念§7–2平面應(yīng)力狀態(tài)分析——解析法§7–3平面應(yīng)力狀態(tài)分析——圖解法§7–4

梁的主應(yīng)力及其主應(yīng)力跡線§7–5

三向應(yīng)力狀態(tài)研究——應(yīng)力圓法§7–6

平面內(nèi)的應(yīng)變分析§7–7

復(fù)雜應(yīng)力狀態(tài)下的應(yīng)力--應(yīng)變關(guān)系

——（廣義虎克定律）§7–8

復(fù)雜應(yīng)力狀態(tài)下的變形比能問(wèn)題引入：桿件在幾種基本變形（拉伸、壓縮、扭轉(zhuǎn)、彎曲）的強(qiáng)度問(wèn)題，建立了只用正應(yīng)力或切應(yīng)力作用時(shí)的強(qiáng)度條件，而工程實(shí)際中，幾種基本變形組合在一起，稱為組合（疊加）變形（小變形情況下）對(duì)應(yīng)構(gòu)件橫截面上某點(diǎn)存在正應(yīng)力和切應(yīng)力時(shí)，能否分別對(duì)正應(yīng)力和切應(yīng)力建立獨(dú)立的強(qiáng)度條件進(jìn)行計(jì)算？（X）前面討論構(gòu)件基本變形的強(qiáng)度問(wèn)題時(shí)，是用橫截面上的危險(xiǎn)點(diǎn)處的應(yīng)力建立強(qiáng)度條件進(jìn)行強(qiáng)度計(jì)算，而有些破壞沒(méi)有發(fā)生在試件橫截面上，而是斜截面上。需研究斜截面上的應(yīng)力狀態(tài)。應(yīng)力狀態(tài)（概念）：指構(gòu)件內(nèi)過(guò)一點(diǎn)處沿不同方向斜截面上的應(yīng)力狀態(tài)。應(yīng)力狀態(tài)和強(qiáng)度理論為研究桿件在復(fù)雜變形時(shí)的強(qiáng)度問(wèn)題提供了理論基礎(chǔ)。。。。研究一點(diǎn)處的應(yīng)力狀態(tài)，引入單元體（P193）§7–１

CONCEPTSOFTHESTAFEOFSTRISS1、Forward1)、Investigationonthetensile,compressiveandtorsionaltestofcastironandlow-carbonsteelMLow-carbonsteelCastironPPCastironintension

PCastironincompression2)、Howwillthememberruptureincombineddeformations？MPANALYSISOFSTRESSANDSTRAIN§7–１應(yīng)力狀態(tài)的概念應(yīng)力狀態(tài)與應(yīng)變狀態(tài)一、引言1、鑄鐵與低碳鋼的拉、壓、扭試驗(yàn)現(xiàn)象是怎樣產(chǎn)生的？M低碳鋼鑄鐵PP鑄鐵拉伸P鑄鐵壓縮2、組合變形桿將怎樣破壞？MP4、Expressionofstressesingeneralcase

3、Element：Element—Delegateofapointinthemember.Itisainfinitesimalgeometricbodyenvelopingthestudiedpoint.Incommonuseitisacorrectitudecubicbody.Propertiesofanelement—a、Stressesaredistributed uniformlyinthesections；

b、Thestressesintwoplanesthatare paralleltoeachotherareequal.2、Stateofstressatapoint：

Therearecountlesssectionsthroughapoint.Thegatheringofstressesinallsectionsiscalledthestateofstressatthispoint.xyzs

xsz

s

ytxyANALYSISOFSTRESSANDSTRAIN四、普遍狀態(tài)下的應(yīng)力表示

三、單元體：?jiǎn)卧w——構(gòu)件內(nèi)的點(diǎn)的代表物，是包圍被研究點(diǎn)的無(wú)限小的幾何體，常用的是正六面體。單元體的性質(zhì)——a、平行面上，應(yīng)力均布；

b、平行面上，應(yīng)力相等。二、一點(diǎn)的應(yīng)力狀態(tài)：

過(guò)一點(diǎn)有無(wú)數(shù)的截面，這一點(diǎn)的各個(gè)截面上應(yīng)力情況的集合，稱為這點(diǎn)的應(yīng)力狀態(tài)（StateofStressataGivenPoint）。xyzs

xsz

s

y應(yīng)力狀態(tài)與應(yīng)變狀態(tài)txyxyzs

xsz

s

ytxy5、Theoremofconjugateshearingstress:

Shearingstressesonperpendicularplanesareequalinmagnitudeandhavedirectionssuchthatbothstressespointtoward,orbothpointawayform,thelineofintersectionofthefaces.ANALYSISOFSTRESSANDSTRAINProvement:Theelementisinequilibrium.xyzs

xsz

s

y應(yīng)力狀態(tài)與應(yīng)變狀態(tài)txy五、剪應(yīng)力互等定理（TheoremofConjugateShearing

Stress):

過(guò)一點(diǎn)的兩個(gè)正交面上,如果有與相交邊垂直的剪應(yīng)力分量,則兩個(gè)面上的這兩個(gè)剪應(yīng)力分量一定等值、方向相對(duì)或相離。6、Originalelement（knownelement）：Example1

PlottheknownelementsofpointA、B、Cshowninthefollowingfigures.ANALYSISOFSTRESSANDSTRAINtzx

PPAAsxsxMPxyzBCsxsxBtxztxytyxtzx六、原始單元體（已知單元體）：[例1]

畫(huà)出下列圖中的A、B、C點(diǎn)的已知單元體。

應(yīng)力狀態(tài)與應(yīng)變狀態(tài)PPAAsxsxMPxyzBCsxsxBtxztxytyx7、Principalelement、principalplanes、principalstresses：

Principalelement：

Theelementinwhichtheshearingstressesinsideplanesareallzero.

PrincipalPlanes：

Theplanesonwhichtheshearingstressesarezero.

Principalstresses：

Normalstressesactingontheprincipleplanes.conventionoftheorderforthreeprincipalstresses:

Inmagnitudeofthealgebraicvalue，s1s2s3xyzsxsyszANALYSISOFSTRESSANDSTRAIN七、主單元體、主平面、主應(yīng)力：主單元體(Principalbidy)：各側(cè)面上剪應(yīng)力均為零的單元體。主平面(PrincipalPlane)：剪應(yīng)力為零的截面。主應(yīng)力(PrincipalStress

）：主平面上的正應(yīng)力。主應(yīng)力排列規(guī)定：按代數(shù)值大小，應(yīng)力狀態(tài)與應(yīng)變狀態(tài)s1s2s3xyzsxsysz補(bǔ)充：

從受力構(gòu)件內(nèi)的某一點(diǎn)處，取出一個(gè)單元體。一般來(lái)講，其側(cè)面上既有正應(yīng)力，也有切應(yīng)力。但是可以證明：在該點(diǎn)以不同方位截取的諸單元體中，有一個(gè)特殊的單元體，在這個(gè)單元體的測(cè)面上只有正應(yīng)力而無(wú)切應(yīng)力。這樣的單元體稱為該點(diǎn)處的主單元體。主單元體的側(cè)面稱為主平面。主平面的正應(yīng)力為主應(yīng)力，主應(yīng)力是正應(yīng)力的極值。主平面的法線方向?yàn)橹鞣较?，即主?yīng)力方向。在一般情況下，過(guò)一點(diǎn)所去主單元體的6個(gè)側(cè)面上有3對(duì)主應(yīng)力，3個(gè)主應(yīng)力皆不為零，該點(diǎn)的應(yīng)力狀態(tài)稱為3向應(yīng)力狀態(tài)，有2個(gè)不為零，稱為兩向應(yīng)力狀態(tài)，。。單向應(yīng)力狀態(tài)。

Stateoftheuniaxialstress:

Stateofstressthatoneprincipalstressisnotequaltozero.

Stateofthebiaxialstress：

Stateofstressthatoneprincipalstressisequaltozero.

StateofthetriaxialStress:Stateofstressthatallthethreeprincipalstressesarenotequaltozero.AsxsxtzxsxsxBtxzANALYSISOFSTRESSANDSTRAIN單向應(yīng)力狀態(tài)（UnidirectionalStateofStress）：一個(gè)主應(yīng)力不為零的應(yīng)力狀態(tài)。

二向應(yīng)力狀態(tài)（PlaneStateofStress）：一個(gè)主應(yīng)力為零的應(yīng)力狀態(tài)。應(yīng)力狀態(tài)與應(yīng)變狀態(tài)三向應(yīng)力狀態(tài)（Three—DimensionalStateof

Stress）：三個(gè)主應(yīng)力都不為零的應(yīng)力狀態(tài)。AsxsxtzxsxsxBtxz

§7–2

ANALYSISOFTHESTATEOFPLANESTRESS—ANALYTICALMETHOD

equivalentsxtxysyxyzxysxtxysyOANALYSISOFSTRESSANDSTRAIN§7–2

平面應(yīng)力狀態(tài)分析——解析法應(yīng)力狀態(tài)與應(yīng)變狀態(tài)sxtxysyxyzxysxtxysyOStipulate：

ispositiveifitsdirectionisthesamewithoneoftheexternalnormallineofthesection；taispositiveifitmaketheelementrotateclockwise；

Acountclockwiseangleaisconsideredtobepositive.Fig.1AssumethatareaoftheinclinedsectionisS.Accordingtotheequilibriumofthefreebodyweget：1、StressesactinginarbitraryinclinedplanexysxtxysyOsytyxsxsataaxyOtnFig.2ANALYSISOFSTRESSANDSTRAIN規(guī)定：截面外法線同向?yàn)檎?/p>

ta繞研究對(duì)象順時(shí)針轉(zhuǎn)為正；

a逆時(shí)針為正。圖1設(shè)：斜截面面積為S，由分離體平衡得：一、任意斜截面上的應(yīng)力應(yīng)力狀態(tài)與應(yīng)變狀態(tài)xysxtxysyOsytyxsxsataaxyOtn圖2Fig.1xysxtxysyOsytyxsxsataaxyOtnFig.2Consideringconjugateofshearingstressesandtrigonometricidentitiesweget：Similarly：ANALYSISOFSTRESSANDSTRAIN圖1應(yīng)力狀態(tài)與應(yīng)變狀態(tài)xysxtxysyOsytyxsxsataaxyOtn圖2考慮剪應(yīng)力互等和三角變換，得：同理：2、TheextremevaluesforthestressxysxtxysyOANALYSISOFSTRESSANDSTRAINandtwoextremumsThuswecangettwostationarypointsExtremenormalstressesareprincipalstresses.二、極值應(yīng)力應(yīng)力狀態(tài)與應(yīng)變狀態(tài)xysxtxysyOisinthequadrantfortheshearingstresstopointandleantothelargerofbothxandy222xyyxminmaxtsstt+-±=?íì

）（xysxtxysyOMainelemeutANALYSISOFSTRESSANDSTRAINThatistheanglebetweentheplanesinwhichshearingstressesreachextremumsandtheprincipalplanesis.在剪應(yīng)力相對(duì)的象限內(nèi)，且偏向于x

及y較大的一側(cè)。應(yīng)力狀態(tài)與應(yīng)變狀態(tài)222xyyxminmaxtsstt+-±=?íì

）（xysxtxysyO

主單元體Example2Analyzethefailureofthecircularshaftintorsion.Solution：Determinethecriticalpointandplottheoriginalelement.Determinetheextreme-valuestresstxyCtyxMCxyOtxytyxANALYSISOFSTRESSANDSTRAIN[例2]

分析受扭構(gòu)件的破壞規(guī)律。解：確定危險(xiǎn)點(diǎn)并畫(huà)其原始單元體求極值應(yīng)力應(yīng)力狀態(tài)與應(yīng)變狀態(tài)txyCtyxMCxyOtxytyxAnalysisoffailure:Low-carbonsteelCastironANALYSISOFSTRESSANDSTRAINLow-carbonsteel:Castiron:破壞分析應(yīng)力狀態(tài)與應(yīng)變狀態(tài)低碳鋼鑄鐵

§7–3

ANALYSISOFTHESTATEOFSTRESS— GRAPHYCALMETHOD

Eliminatingtheparameter2

fromtheaboveequation，weget：1、StressCircle

xysxtxysyOsytxyxsxsataaxyOtncurveofthisequationisacircle—stresscircle（orMohr’scircle，introducedbyGermanengineerOttoMohr)ANALYSISOFSTRESSANDSTRAIN§7–3

平面應(yīng)力狀態(tài)分析——圖解法對(duì)上述方程消去參數(shù)（2），得：一、應(yīng)力圓（

StressCircle）應(yīng)力狀態(tài)與應(yīng)變狀態(tài)xysxtxysyO此方程曲線為圓—應(yīng)力圓（或莫爾圓，由德國(guó)工程師：OttoMohr引入）sytxyxsxsataaxyOtnSetupastresscoordinatesystemasshowninthefollowingfigure.（Payattentiontotheselectionofthescale）2、MethodtoplotthestresscirclePlotthepoint

A(x，xy)andthepointB(y，yx)inthecoordinatesystem.

Inclinedline

ABintersectstheaxis

saatthepointC.Thispointisthecenterofthestresscircle.Plotacircle—stresscirclewiththecenterCandtheradiusAC.sxtxysyxyOnsataaOsataCA(sx,txy)B(sy,tyx)x2anD(sa,

ta)ANALYSISOFSTRESSANDSTRAIN建立應(yīng)力坐標(biāo)系，如下圖所示，（注意選好比例尺）二、應(yīng)力圓的畫(huà)法在坐標(biāo)系內(nèi)畫(huà)出點(diǎn)A(x，xy)和B(y，yx)

AB與sa

軸的交點(diǎn)C便是圓心。以C為圓心，以AC為半徑畫(huà)圓——應(yīng)力圓；應(yīng)力狀態(tài)與應(yīng)變狀態(tài)sxtxysyxyOnsataaOsataCA(sx,txy)B(sy,tyx)x2anD(sa,

ta)sxtxysyxyOnsataaOsataCA(sx,txy)B(sy,tyx)x2anD(sa,

ta)3、CorrespondingrelationbetweentheelementandstresscircleStress(，)inplane

Apoint(，)onthestresscircumferenceNormallineofplaneRadiusofthestresscircleAnglebetweentwosections

Angle2betweentworadiuses；Andthedirectionofrotationisthesame.ANALYSISOFSTRESSANDSTRAIN應(yīng)力狀態(tài)與應(yīng)變狀態(tài)sxtxysyxyOnsataaOsataCA(sx,txy)B(sy,tyx)x2anD(sa,

ta)三、單元體與應(yīng)力圓的對(duì)應(yīng)關(guān)系面上的應(yīng)力(，)

應(yīng)力圓上一點(diǎn)(，)面的法線應(yīng)力圓的半徑兩面夾角兩半徑夾角2

；且轉(zhuǎn)向一致。4、MarkextremestressesonthecircumferenceofthestresscircleOCsataA(sx,txy)B(sy,tyx)x2a12a0s1s2s3ANALYSISOFSTRESSANDSTRAIN四、在應(yīng)力圓上標(biāo)出極值應(yīng)力應(yīng)力狀態(tài)與應(yīng)變狀態(tài)OCsataA(sx,txy)B(sy,tyx)x2a12a0s1s2s3s3Example3

Determineprincipalstressesandorientationofprincipalplanesoftheelementasshowninthefigure.(unit：MPa)AB

12Method1-graphicalmethod:Stresscoordinatesystemisshowninthefigure.IntersectionCoftheperpendicularbisectionlineofABandtheaxis

saisthecenterofthecircle.PlotthecirclewiththecenterCandthearadiusAC—stress

circle.0s1s2BAC2s0sata(MPa)(MPa)O20MPaPlotthepointand

ANALYSISOFSTRESSANDSTRAINs3例3

求圖示單元體的主應(yīng)力及主平面的位置。(單位：MPa)AB

12解法1——圖解法：主應(yīng)力坐標(biāo)系如圖AB的垂直平分線與sa

軸的交點(diǎn)C便是圓心，以C為圓心，以AC為半徑畫(huà)圓——應(yīng)力圓0應(yīng)力狀態(tài)與應(yīng)變狀態(tài)s1s2BAC2s0sata(MPa)(MPa)O20MPa在坐標(biāo)系內(nèi)畫(huà)出點(diǎn)s3s1s2BAC2s0sata(MPa)(MPa)O20MPaPrincipalstressesandprincipalplanesasshowninthefigure

102ABANALYSISOFSTRESSANDSTRAINs3應(yīng)力狀態(tài)與應(yīng)變狀態(tài)s1s2BAC2s0sata(MPa)(MPa)O20MPa主應(yīng)力及主平面如圖

102ABMethod2—analyticalmethod：Analysis—setupthecoordinateasshowninthefigure.60°xyOANALYSISOFSTRESSANDSTRAIN解法2—解析法：分析——建立坐標(biāo)系如圖60°應(yīng)力狀態(tài)與應(yīng)變狀態(tài)xyO§7–4

PRINCIPALSTRESSESANDTHEIR TRAJECTORIESOFTHEBEAM12345P1P2qElement：ANALYSISOFSTRESSANDSTRAINAsshowninthefigure，thebeamproducedtheshearbending（transversalbending）,where.M、Q>0inthebeam.Trytodeterminethemagnitudeoftheprincipalstressesandthepositionoftheprincipalplanesofeachpointinthesection.§7–4

梁的主應(yīng)力及其主應(yīng)力跡線應(yīng)力狀態(tài)與應(yīng)變狀態(tài)12345P1P2q如圖，已知梁發(fā)生剪切彎曲（橫力彎曲），其上M、Q>0，試確定截面上各點(diǎn)主應(yīng)力大小及主平面位置。單元體：21s1s3s33s1s34s1s1s35a0–45°a0stA1A2D2D1COsA2D2D1CA1Ot2a0stD2CD1O2a0=–90°sD2A1Ot2a0CD1A2stA2D2D1CA1OANALYSISOFSTRESSANDSTRAIN應(yīng)力狀態(tài)與應(yīng)變狀態(tài)21s1s3s33s1s34s1s1s35a0–45°a0stA1A2D2D1COsA2D2D1CA1Ot2a0stD2CD1O2a0=–90°sD2A1Ot2a0CD1A2stA2D2D1CA1OTensileforceCompressiveforcePrincipalstresstrajectories：

Envelopesofthedirectionlinesofprincipalstresses—tangentateachpointonthecurveindicatestheorientationofthetensile(orcompressive)principalstressatthesamepoint.Solidlinesexpressthetensileprincipalstresstrajectories；dashedlinesexpressthecompressiveprincipalstresstrajectories.1313ANALYSISOFSTRESSANDSTRAIN拉力壓力主應(yīng)力跡線（StressTrajectories)：主應(yīng)力方向線的包絡(luò)線——曲線上每一點(diǎn)的切線都指示著該點(diǎn)的拉主應(yīng)力方位（或壓主應(yīng)力方位）。實(shí)線表示拉主應(yīng)力跡線；虛線表示壓主應(yīng)力跡線。應(yīng)力狀態(tài)與應(yīng)變狀態(tài)1313xyMethodtoplotprincipalstresstrajectories：11

22

33

44

ii

nn

bacdANALYSISOFSTRESSANDSTRAINq1331Sectionsectionsectionsectionsectionsection

qxy主應(yīng)力跡線的畫(huà)法：11截面22截面33截面44截面ii截面nn截面bacd13應(yīng)力狀態(tài)與應(yīng)變狀態(tài)31§7–5

ANALYSISOFTRIAXIALSTRESSEDSTATE— METHODOFSTRESSCIRCLEs2s1xyzs31、SpatialstressedstateANALYSISOFSTRESSANDSTRAIN§7–5

三向應(yīng)力狀態(tài)研究——應(yīng)力圓法應(yīng)力狀態(tài)與應(yīng)變狀態(tài)s2s1xyzs31、空間應(yīng)力狀態(tài)2、Analysisofthetriaxialstress

Fig.aFig.bThemaximumshearingstressinsidethewholeelementis：tmaxs2s1xyzs3ANALYSISOFSTRESSANDSTRAINElastictheoryprovedthat

stressesonanyplanepassingthroughapointinbytheelementshowninFig.amaybe

correspondingto

thecoordinatesofapointonthecircumferenceofthestresscircleorintheshadowregion.2、三向應(yīng)力分析彈性理論證明，圖a單元體內(nèi)任意一點(diǎn)任意截面上的應(yīng)力都對(duì)應(yīng)著圖b的應(yīng)力圓上或陰影區(qū)內(nèi)的一點(diǎn)。圖a圖b整個(gè)單元體內(nèi)的最大剪應(yīng)力為：tmax應(yīng)力狀態(tài)與應(yīng)變狀態(tài)s2s1xyzs3Example4Determinetheprincipalstressesandthemaximumshearingstressoftheelementshowninthefigure.(MPa)Solution:

Fromtheelementsketchweknowplaneyzisaprincipalplane.Setupstresscoordinatesasshowninthefigure.Plotthestresscircleandlocatethepoint1,get:5040xyz3010

(MPa)sa（MPa）taABCABs1s2s3tmaxANALYSISOFSTRESSANDSTRAIN[例4]

求圖示單元體的主應(yīng)力和最大剪應(yīng)力。（MPa）解：由單元體圖知：yz面為主平面建立應(yīng)力坐標(biāo)系如圖，畫(huà)應(yīng)力圓和點(diǎn)1,得:應(yīng)力狀態(tài)與應(yīng)變狀態(tài)5040xyz3010(M

Pa)sa（M

Pa）taABCABs1s2s3tmax§7–6

ANALYSISOFSTRAININAPLANExyO

1、DeterminetheexpressionofstrainanalysisbythemethodofsuperpositionabcdaAOBShearingstrain：Incrementalquantityoftherightangle?。∣nlylikethisthecontentsintheprecedingsectionandthefollowingsectionCancorrespondtoeachother.）DD1EE1ANALYSISOFSTRESSANDSTRAIN§7–6

平面內(nèi)的應(yīng)變分析xyO一、疊加法求應(yīng)變分析公式abcdaAOB剪應(yīng)變：直角的增大量?。ㄖ挥羞@樣，前后才對(duì)應(yīng)）應(yīng)力狀態(tài)與應(yīng)變狀態(tài)DD1EE1xyOabcdaAOBDD2EE2ANALYSISOFSTRESSANDSTRAIN應(yīng)力狀態(tài)與應(yīng)變狀態(tài)xyOabcdaAOBDD2EE2DD3EE3xyOabcdaAOBANALYSISOFSTRESSANDSTRAINDD3EE3應(yīng)力狀態(tài)與應(yīng)變狀態(tài)xyOabcdaAOBANALYSISOFSTRESSANDSTRAIN應(yīng)力狀態(tài)與應(yīng)變狀態(tài)2)、Plotthestraincircleaccordingtostrain()ofapoint.2、Graphicalmethodofstrainanalysis—straincircle1)、AnalogicrelationofstresscircleandstraincircleSetupstraincoordinatesasshowninthefigurelocatethepoint

A(x，xy/2)and

B(y，-yx/2)inthecoordinatesystemintersectionofABandaxisa

isthecenterofthecircle.plotcirclebythecenterCwitharadiusAC—straincircle.eaga/2ABCANALYSISOFSTRESSANDSTRAIN2、已知一點(diǎn)A的應(yīng)變（），畫(huà)應(yīng)變圓二、應(yīng)變分析圖解法——應(yīng)變圓（StrainCircle）1、應(yīng)變圓與應(yīng)力圓的類比關(guān)系建立應(yīng)變坐標(biāo)系如圖在坐標(biāo)系內(nèi)畫(huà)出點(diǎn)

A(x，xy/2)

B(y，-yx/2)AB與a

軸的交點(diǎn)C便是圓心以C為圓心，以AC為半徑畫(huà)圓——應(yīng)變圓。應(yīng)力狀態(tài)與應(yīng)變狀態(tài)eaga/2ABCeaga/23、Correspondingrelationbetweenthestrainindirectionandstraincirclemaxmin20D(，/2)2nStrainindirection(，/2)Apointonthestraincircle(，/2)Thedirectionlineof

RadiusofthestraincircleAnglebetweentwodirections

Angle2betweenthetworadiusesandthedirectionofrotationisthesame.ABCANALYSISOFSTRESSANDSTRAINeaga/2三、方向上的應(yīng)變與應(yīng)變圓的對(duì)應(yīng)關(guān)系maxmin20D(，/2)2n應(yīng)力狀態(tài)與應(yīng)變狀態(tài)方向上的應(yīng)變(，/2)

應(yīng)變圓上一點(diǎn)(，/2)

方向線應(yīng)變圓的半徑兩方向間夾角兩半徑夾角2

；且轉(zhuǎn)向一致。ABC4、ValuesandorientationofprincipalstrainsANALYSISOFSTRESSANDSTRAIN四、主應(yīng)變數(shù)值及其方位應(yīng)力狀態(tài)與應(yīng)變狀態(tài)Example5Knowingthreestrains1、2and3indirections1、2and3atapointinsomeplane,determinetheprincipalstrainsinthisplane.Solution：Firstfindout

x，y，xy

bysolvingtheabovethreeequationsthendeterminetheprincipalstrains.ANALYSISOFSTRESSANDSTRAIN[例5]

已知一點(diǎn)在某一平面內(nèi)的1、2、3

方向上的線應(yīng)變分別為1、2、3，，求該面內(nèi)的主應(yīng)變。解：由i=1,2,3這三個(gè)方程求出

x，y，xy；然后再求主應(yīng)變。應(yīng)力狀態(tài)與應(yīng)變狀態(tài)Example6Determinetheprincipalstrainsofthepointafterthreelinearstrainsatthispointaretestedbythe

strainfoilof45°.xyu45o0maxANALYSISOFSTRESSANDSTRAIN[例6]用45°應(yīng)變花測(cè)得一點(diǎn)的三個(gè)線應(yīng)變后，求該點(diǎn)的主應(yīng)變。xyu45o0max應(yīng)力狀態(tài)與應(yīng)變狀態(tài)

§7–7

STRESS—STRAINRELATIONUNDERTHECOMPLEX STRESSEDSTATE—（GENERALIZEDHOOKE’SLAW）

1、Stress-strainrelationinuniaxialtension

2、Stress-strainrelationinpureshearxyzsxxyz

x

yANALYSISOFSTRESSANDSTRAIN§7–7

復(fù)雜應(yīng)力狀態(tài)下的應(yīng)力--應(yīng)變關(guān)系

——（廣義虎克定律）一、單拉下的應(yīng)力--應(yīng)變關(guān)系二、純剪的應(yīng)力--應(yīng)變關(guān)系應(yīng)力狀態(tài)與應(yīng)變狀態(tài)xyzsxxyz

x

y3、Stress-strainrelationincomplexstressedstate

Accordingtosuperpositionweget:

xyzszsytxysxANALYSISOFSTRESSANDSTRAIN三、復(fù)雜狀態(tài)下的應(yīng)力---應(yīng)變關(guān)系依疊加原理,得:應(yīng)力狀態(tài)與應(yīng)變狀態(tài)

xyzszsytxysxPrincipalstress—principalstrainrelation4、Stress-strainrelation

underthestateofplanestressThedirectionsarethesames1s3s2ANALYSISOFSTRESSANDSTRAIN主應(yīng)力---主應(yīng)變關(guān)系四、平面狀態(tài)下的應(yīng)力---應(yīng)變關(guān)系:方向一致應(yīng)力狀態(tài)與應(yīng)變狀態(tài)s1s3s2Thedirectionsoftheprincipalstressandtheprincipalstrainarethesame.ANALYSISOFSTRESSANDSTRAIN主應(yīng)力與主應(yīng)變方向一致。應(yīng)力狀態(tài)與應(yīng)變狀態(tài)5、Relationbetweenthevolumetricstrainandstresscomponents:Volumetricstrain：Relationbetweenthevolumetricstrainandstresscomponents:s1s3s2a1a2a3ANALYSISOFSTRESSANDSTRAIN五、體積應(yīng)變與應(yīng)力分量間的關(guān)系體積應(yīng)變：體積應(yīng)變與應(yīng)力分量間的關(guān)系:應(yīng)力狀態(tài)與應(yīng)變狀態(tài)s1s3s2a1a2a3Example7Astructurememberissubjectedtosomeforces.Twoprincipalstrainsatapointonthefreesurfaceofthememberare1=24010-6

2=–16010-6，modulusofelasticityisE=210GPa，Possion’sratiois=0.3.Trytodeterminetheprincipalstressesandanotherprincipalstrainatthispoint.,ThenthispointisinthestatANALYSISOFSTRESSANDSTRAINSolution:OnthefreesurfaceOfplanestress..[例7]已知一受力構(gòu)件自由表面上某一點(diǎn)處的兩個(gè)面內(nèi)主應(yīng)變分別為：1=24010-6，

2=–16010-6，彈性模量E=210GPa，泊松比為=0.3，試求該點(diǎn)處的主應(yīng)力及另一主應(yīng)變。所以，該點(diǎn)處為平面應(yīng)力狀態(tài)應(yīng)力狀態(tài)與應(yīng)變狀態(tài)ANALYSISOFSTRESSANDSTRAINe3342.-=×10-6應(yīng)力狀態(tài)與應(yīng)變狀態(tài)e3342.-=×10-6Example

8

Athin-walledcontainersubjectedtoinsidepressureisshowninFig.a.Inordertodeterminethevalueoftheinsidepressurethehoopstraintestedonthesurfaceofthecontainerwithstrainfoilist

=350×l06.IfthemeandiameterofthecontainerisD=500mm，thicknessofitswallis=10mm，E=210GPa，=0.25.Tryto:1.derivetheexpressionsofstressinthelateralandlongitudinalsectionsofthecontainer.2.calculatetheinsidepressureofthecontainer.pppxstsmLpODxAByFig.aANALYSISOFSTRESSANDSTRAIN[例8]

圖a所示為承受內(nèi)壓的薄壁容器。為測(cè)量容器所承受的內(nèi)壓力值，在容器表面用電阻應(yīng)變片測(cè)得環(huán)向應(yīng)變t

=350×l06，若已知容器平均直徑D=500mm，壁厚=10mm，容器材料的E=210GPa，=0.25，試求:1.導(dǎo)出容器橫截面和縱截面上的正應(yīng)力表達(dá)式；2.計(jì)算容器所受的內(nèi)壓力。應(yīng)力狀態(tài)與應(yīng)變狀態(tài)pODxABy圖apppxstsmL1)LongitudinalstressesSolution：Expressionsofthecircumferentialandlongitudinalstressofthecontainerreservoir

CuttingthecontaineralongasectionshownandinFig.bconsideringtheequilibriumoftherightpart，wegetpsmsmxDFig.bANALYSISOFSTRESSANDSTRAIN1、軸向應(yīng)力:(longitudinalstress)解：容器的環(huán)向和縱向應(yīng)力表達(dá)式用橫截面將容器截開(kāi)，受力如圖b所示,根據(jù)平衡方程應(yīng)力狀態(tài)與應(yīng)變狀態(tài)psmsmxD圖b

Imaginecutoffthecontaineralonglongitudinalsection，andtaketheupperpantasthestudyobject.ThefreebodydiagramisshowninFig.c2、Circumferentialstress3、Determinetheinsidepressure（bystress-strainrelation）t

mEx