弹性模型-毕业论文外文翻译

附录1外文翻译原文3.2Elasticmodels3.2.1AnisotropyAnisotropicmaterialhasthesamepropertiesinalldirections—wecannotdis-tinguishanyonedirectionfromanyother.Samplestakenoutofthegroundwithanyorientationwouldbehaveidentically.However,weknowthatsoilshavebeendepositedinsomeway—forexample,sedimentarysoilswillknowabouttheverticaldirectionofgravitationaldeposition.Theremayinadditionbeseasonalvariationsintherateofdepositionsothatthesoilcontainsmoreorlessmarkedlayersofslightlydifferentgrainsizeand/orplasticity.Thescaleoflayeringmaybesuffcientlysmallthatwedonotwishtotrytodistinguishseparatematerials,butthelayeringtogetherwiththedirectionaldepositionmayneverthelessbesuffcienttomodifytheproperiesofthesoilindifferentdirections—inotherwordstocauseittobeanisotropic.WecanwritethestiffnessrelationshipbetweenelasticstrainincrementeandstressincrementcompactlyasDe(3.36)D1whereDisthestiffnessmatrixandhenceisthecompliancematrix.Foracompletelygeneralanisotropicelasticmaterialabcdefbghijk1chlmnoD(3.37)dimpqrejnqstfkortuwhereeachlettera,b,...is,inprinciple,anindependentelasticpropertyandthenecessarysymmetryofthesti?nessmatrixfortheelasticmaterialhasreducedthemaximumnumberofindependentpropertiesto21.Assoonastherearematerialsymmetriesthenthenumberofindependentelasticpropertiesfalls(Crampin,1981).Forexample,formonoclinicsymmetry(zsymmetryplane)thecompliancematrixhastheform:abc00dbef00g1cfh00iD000jk(3.38)0000kl0dgi00mandhasthirteenelasticconstants.Orthorhombicsymmetry(distinctx,yandzsymmetryplanes)givesnineconstants:abc000bde0001cef000D000g0(3.39)00000h000000iwhereascubicsymmetry(identicalx,yandzsymmetryplanes,togetherwithplanesjoiningoppositesidesofacube)givesonlythreeconstants:abb000bab0001bba000D000c0(3.40)00000c000000cFigure3.9:Independentmodesofshearingforcross-anisotropicmaterialIfweaddthefurtherrequirementthatc2(ab)andseta1/Eandv/E，thenwerecovertheisotropicelasticcompliancematrixof(3.1).Thoughitisobviouslyconvenientifgeotechnicalmaterialshavecertainfabricsymmetrieswhichconferareductioninthenumberofindependentelasticproperties,ithastobeexpectedthatingeneralmaterialswhichhavebeenpushedaroundbytectonicforces,byice,orbymanwillnotpossessanyofthesesymmetriesand,insofarastheyhaveadomainofelasticresponse,weshouldexpecttorequirethefull21independentelasticproperties.Ifwechoosetomodelsuchmaterialsasisotropicelasticoranisotropicelasticwithcertainrestrictingsymmetriesthenwehavetorecognisethatthesearemodellingdecisionsofwhichthesoilorrockmaybeunaware.However,manysoilsaredepositedoverareasoflargelateralextentandsymmetryofdepositionisessentiallyvertical.Allhorizontaldirectionslookthesamebuthorizontalsti?nessisexpectedtobedi?erentfromverticalstiffness.Theformofthecompliancematrixisnow:abc000bac0001ccd000(3.41)D000e000000e000000fandwecanwrite：a1/Eh,bvhh/Eh,cvvh/Ev,d1/Ev,e1/Gvh和f2(ab)2(1vhh)/Eh:1/Ehvhh/Ehvvh/Ev000vhh/Eh1/Ehvvh/Ev0001vvh/Evvvh/Ev1/Eh000D(3.42)0001/Gvh0000001/Gvh00000021vhh/EhThisisdescribedastransverseisotropyorcrossanisotropywithhexagonalsymmetry.Thereare5independentelasticproperties:EvandEhareYoung’smoduliforunconfinedcompressionintheverticalandhorizontaldirectionsrespectively;Gvhistheshearmodulusforshearinginaverticalplane(Fig3.9a).Poissonratios’sVhhandVvhrelatetothelateralstrainsthatoccurinthehorizontaldirectionorthogonaltoahorizontaldirectionofcompressionandaverticaldirectionofcompressionrespectively(Fig3.9c,b).TestingofcrossanisotropicsoilsinatriaxialapparatuswiththeiraxesofanisotropyalignedwiththeaxesoftheapparatusdoesnotgiveusanypossibilitytodiscoverGvh1/E，sincethiswouldrequirecontrolledapplicationofshearstressestoverticalandhorizontalsurfacesofthesample—andattendantrotationofprincipalaxes.Infactweareableonlytodetermine3ofthe5elasticproperties.Ifwewrite(3.42)forradialandaxialstressesandstrainsforasamplewithitsverticalaxisofsymmetryofanisotropyalignedwiththeaxisofthetriaxialapparatus,wefindthat：a1/Ev2vvh/Evarvvh/Ev1vhh/Evr'(3.43)'Thecompliancematrixisnotsymmetricbecause,inthecontextofthetriaxialtest,thestrainincrementandstressquantitiesarenotproperlyworkconjugate.WededucethatwhilewecanseparatelydetermineEvandVvhtheonlyotherelasticpropertythatwecandiscoveristhecompositestiffnessEh/(1Vhh).WearenotabletoseparatehandVhh(Lingsetal.,2000).EOntheotherhand,GrahamandHoulsby(1983)haveproposedaspecialformof(3.41)or(3.42)whichusesonly3elasticpropertiesbutforcescertaininterdependenciesamongthe5elasticpropertiesforthiscrossanisotropicmaterial.1/2v/2/000vv21/2/000/v11v/v/1000DE00021v/00000021v/000000221v/(3.44)ThisiswrittenintermsofaYoung’smodulusEEv,theYoung’smodulusforloadingintheverticaldirection,aPoisson’sratioVVhh,togetherwithathirdparameter.TheratioofstiffnessinhorizontalandverticaldirectionsisEh/Ev2andotherlinkagesareforced：vvhvhh/;GhvGhh/E/2(1v).Forourtriaxialstressandstrainquantities,thecompliancematrixbecomes:pq13GJp'(3.45)detJKqFigure3.10:Effectofcross-anisotropyondirectionofundrainedeffectivestresspathwheredet3KGJ2(3.46)andthestiffnessmatrixisp'KJqJ3Gp(3.47)qwhere1v4v2K2E(3.48)91v12v22v42GvE(3.49)61v12v1vv2J(3.50)E31v12vThestiffnessandcompliancematrices(writtenintermsofcorrectlychosenworkconjugatestrainincrementandstressquantities)arestillsymmetric—thematerialisstillelastic—butthenon-zerooff-diagonaltermstellusthatthereisnowcouplingbetweenvolumetricanddistortionaleffects.Therewillbevolumetricstrainwhenweapplypurelydistortionalstress,p'0,distortionalstrainduringpurelyisotropiccompression,q0,andtherewillbechangeinmeaneffectivestressinundrainedtests,0.Infacttheslopeoftheeffectivestresspathinanundrainedtestis,from(3.45),p'J21v2v(3.51)q3G322v24vFromourdefinitionofporepressureparametera§(2.6.2)wefindp'J(3.52)q3GFigure3.11:RelationshipbetweenanisotropyparameterαandporepressureparameterafordifferentvaluesofPoisson’sratio.whichwill,inthepresenceofanisotropy,notbezero.Afirstinspectionof(3.51)merelysuggeststhattherearelimitsontheporepressureparameterofa=2/3anda=-1/3forverylarge（Eh＞＞Ev）andverysmall（Ev＞＞Eh）repectively(Fig3.10),whichinturnimplyeffectivestresspathswithconstantaxialeffectivestressandconstantradialeffectivestressrespectively.Thelinkbetweenaandactuallyslightlymoresubtle.Infact,forv0therelationshipisnotactuallymonotonicandtheeffectivestresspathdirectionovershootstheapparentlimits(Fig3.11).Thedeductionofavalueαisof(andhenceEh/Ev2)fromaisnotveryreliablewhenaisaround-1/3or2/3(recallthedatapresentedinFigs2.51and2.49,§2.5.4).Forv0.5,a(12)/3(1)or(13a)(3a2).Theserelationshipssatisfytheexpectedlimitsfor0andbuttherearesingularitiesintheinversionof(3.51)for1andv0.5.3.2.2NonlinearityWewillprobablyexpectthatthedominantsourceofnonlinearityofstress:strainresponsewillcomefrommaterialplasticity—andwewillgoontodevelopelasticplasticconstitutivemodelsinthenextsection.However,wealsohaveanexpectationthatsomeofthetrulyelasticpropertiesofsoilswillvarywithstresslevelandthiscanbeseenasasourceofelasticnonlinearity.Ourthoughtsaboutelasticmaterialsasconservativematerials—theterm‘hyperelasticity’isusedtodescribesuchmaterials—mightmakeusalittlecautiousaboutpluckingfromtheairarbitraryempiricalfunctionsforvariationofmoduliwithstresses.Forexample,ifweweretosupposethatthebulkmodulusofthesoilvariedwithmeaneffectivestressbutthatPoisson’sratio(andhencetheratioofshearmodulustobulkmodulus)wereconstantthenwewouldfindthatinaclosedstresscyclesuchasthatshowninFig3.12energywouldbecreated(orlost)creatingaperpetualmotionmachineinviolationofthefirstlawofthermodynamics—thiswouldnotbeaconservativesystem.Weneedtofindastrainenergy(3.7)orcomplementaryenergydensity(3.11)functionwhichcanbedifferentiatedtogiveacceptablevariationofmoduliwithstresses.Figure3.12:CycleofstresschangeswhichshouldgivezeroenergygeneratedordissipatedforconservativematerialSuchacomplementaryenergyfunctioncanbededucedfromthenonlinearelasticmodeldescribedbyBoyce(1980):2Vn111qp'(3.53)n1K16Gp'WhenK1andG1arereferencevaluesofbulkmodulusandshearmodulusandnisanonlinearityparameter.Thecompliancematrixcanthenbededucedbydifferentiation：n1n2n21n16G13G1p'pn1K(3.54)p'1n1qq3G13G1Whereq/p'.Thereisagain(asfortheanisotropicmodel)couplingbetweenvolumetricanddistortionaleffects.Thestiffnessesarebroadlyproportionaltop'1n.Becausethecompliancesarenowvaryingwithstressratiotheeffectivestresspathimpliedforanundrained(purelydistortional)loadingisnolongerstraight.Infact,forareferencestatep'p0,q0theeffectivestresspathisp'012n(3.55)p'where(1n)K1/6G1。Contoursofconstantvolumetricstrainp0areshowninFig3.13forn0.2andPoisson’sratiov0.3implyingK1/G12.17—valuestypicalfortheroadsub-basematerialsbeingtestedbyBoycefortheirsmallstrain,resilientelasticproperties.Similarlythepathfollowedinapurelyvolumetricdeformationq0willdevelopsomechangeindistortionalstress.Foraninitialstatep'p0,qq0,theeffectivestresspathforsuchatestisn1qp'0(3.56)q0p'ContoursofconstantdistortionalstrainarealsoshowninFig3.13forn=0.2.Figure3.13:Contoursofconstantvolumetricstrain(solidlines)andconstantdistortionalstrain(dottedlines)fornonlinearelasticmodelofBoyce(1980)Itisoftenproposedthattheelasticvolumetricstiffness—bulkmodulus—ofclaysshouldbedirectlyproportionaltomeaneffectivestress:Kp'/k.Integrationofthisrelationshipshowsthatelasticunloadingofclaysproducesastraightlineresponsewhenplottedinalogarithmiccompressionplane(plnv:lnp')(Fig3.14)wherevisspecificvolume.Butwhatassumptionshouldwemakeaboutshearmodulus?IfwesimplyassumethatPoisson’sratioisconstant,sothattheratioofshearmodulustobulkmodulusisconstant,thenwewillemergewithanon-conservativematerial(Zytynskietal.,1978).Ifweassumeaconstantvalueofshearmodulus,independentofstresslevel,wewillobtainaconservativematerialbutmayfindthatwehavephysicallysurprisingvaluesofimpliedPoisson’sratioforcertainhighorlowstresslevels.Againweneedtofindastrainorcomplementaryenergyfunctionthatwillgiveusthebasicmodulusvariationthatwedesire.Houlsby(1985)suggeststhatanacceptablestrainenergyfunctioncouldbe:Up/k3p'rek22(3.57)qFigure3.14:Linearlogarithmicrelationshipbetweenandp'forelasticmaterialwithbulkmodulusproportionaltop'Incrementallythisimpliesastiffnessmatrixwhich,onceagain,containsoffdiagonaltermsindicatingcouplingbetweenvolumetricanddistortionalelementsofdeformation:p'1/k/kpqp'/k/(3.58)qqItcanbededucedthatq3q(3.59)p'1322kqsothatcontoursofconstantdistortionalstrainarelinesofconstantstressratioη(Fig3.15).Constantvolume(undrained)stresspathsarefoundtobeparabolae(Fig3.15):2''q6kpip'pi(3.60)Allparabolaeinthisfamilytouchtheline3k/2.Figure3.15:Contoursofconstantvolumetricstrain(solidlines)andconstantdistortionalstrain(dottedlines)fornonlinearelasticmodelofHoulsby(1985)Thenonlinearitythathasbeenintroducedinthesetwomodelsisstillassociatedwithanisotropicelasticity.Theelasticpropertiesvarywithdeformationbutnotwithdirection.Althoughittendstobeassumedthatnonlinearityinsoilscomesexclusivelyfromsoilplasticity—aswillbediscussedinthesubsequentsections—wehaveseenthatwithcareitmaybepossibletodescribesomeelasticnonlinearityinawaywhichisthermodynamicallyacceptable.Equally,mostelastic-plasticmodelswillcontainsomeelementofelasticity—whichmayoftenbeswampedbyplasticdeformations.Itmustbeexpectedthatthefabricvariationswhichaccompanyanyplasticshearingwillthemselvesleadtochangesintheelasticpropertiesofthesoil.Theformulationofsuchvariationsofstiffnessshouldinprinciplebebasedonthedifferentiationofsomeserendipitouslydiscoveredelasticstrainenergydensityfunctioninorderthattheelasticityshouldnotviolatethelawsofthermodynamics.Evidentlythedevelopmentofstrainenergyfunctionswhichpermitevolutionofanisotropyofelasticstiffnessistricky.Manyconstitutivemodelsadoptapragmatic,hypoelasticapproachandsimplydefinetheevolutionofthemoduliwithstressstateorwithstrainstatewithoutconcernforthethermodynamicconsequences.Thismaynotprovokeparticularproblemsprovidedthestresspathsorstrainpathstowhichsoilelementsaresubjectedarenotveryrepeatedlycyclic.3.2.3HeterogeneityAnisotropyandnonlinearityarebothpossibledeparturesfromthesimpleassumptionsofisotropiclinearelasticity.Aratherdifferentdepartureisassociatedwithheterogeneity.Wehavealreadynotedthatsmallscaleheterogeneity—seasonallayering—mayleadtoanisotropyofstiffness(andother)propertiesatthescaleofatypicalsample.Manynaturalandman-madesoilscontainlargerangesofparticlesizes(§1.8)—glacialtillsandresidualsoilsoftencontainboulder-sizedparticleswithinanotherwisesoil-likematrix.Ifthescaleofourgeotechnicalsystemislargebycomparisonwiththesizeandspacingofthesebouldersthenitwillbereasonabletotreatthematerialasessentiallyhomogeneous.However,wewillstillwishtodetermineitsmechanicalproperties.Ifweattempttomeasureshearwavevelocitiesinsitu,usinggeophysicaltechniques,thenwecanexpectthatthefastestwavefromsourcetoreceiverwilltakeadvantageofthepresenceofthelargehardrock-likeparticles—whichwillhaveamuchhigherstiffnessandhencehighershearwavevelocitythanthesurroundingsoil(Fig3.16).Thereceiverwillshowthetraveltimeforthefastestwavewhichhastakenthisheterogeneousroute.Ifthehardmaterialoccupiesaproportionλofthespacingbetweensourceandreceiver,andtheratioofshearwavevelocitiesisk(andhence,neglectingdensitydifferences,theratioofshearmoduliisoftheorderofk2),thentheratioofapparentshearwavevelocityVstotheshearwavevelocityofthesoilmatrixVsisVVssk(3.61)kkFigure3.16:(a)Soilcontainingbouldersbetweenboreholesusedformeasurementofshearwavevelocity;(b)averagestiffnessesdeducedfrominterpretationofshearwavevelocityandfrommatrixstiffnessThededucedaverageshearmodulusisthengreaterthantheshearstiffnessofGthesoilmatrixGbytheratioGk21ask(3.62)Gkk21Laboratorytestingofsuchheterogeneousmaterialsisnoteasybecausethetestapparatusneedsitselftobemuchlargerthanthetypicalmaximumparticlesizeandspacinginorderthatatrueaveragepropertyshouldbemeasured.Atasmallscale,MuirWoodandKumar(2000)reportteststoexploremechanicalcharacteristicsofmixturesofkaolinclayandafinegravel(d50=2mm).Theyfoundthatallthepropertiesoftheclay/gravelsystemwerecontrolledbythesoilmatrixuntilthevolumefractionofthegravelwasabout0.45-0.5.Atthatstage,butnotbefore,interactionbetweenthe‘rigid’particlesstarteddominaterapidly.toFor0.5then,thisimpliesaratioofequivalentshearstiffnessGtosoilmatrixstiffnessG:G1(3.63)G1Thesetwoexpressions,(3.62)and(3.63),arecomparedinFig3.16foramodulus2ratiok10000附录2外文翻译3.2弹性模型3.2.1各向异性各向异性材料在各个方向具有同样的性质—我们不能将任何一个方向与任何其他方向区分开。从地下任何地方取出的试样都表现出个性。然而，我们知道土已经以某种方式沉积—例如，沉积性土在垂直方向受重力作用而沉积。另外，沉积速度可能呈季节变化，所以土体或多或少地包含了颗粒尺寸或可塑性略微相异的标志性土层。分层的范围可能会非常小，我们不期望区分不同材料，但在不同方向的分层可能还是足以改变不同方向的土的性质—换句话说就是造成其各向异性。我们可以将弹性应变增量e和应力增量的刚度关系简写为e(3.36)D其中D是刚度矩阵，因此D1是柔度矩阵。对于一个完全整体各向异性弹性材料abcdefbghijk1chlmnoDdimpq(3.37)rejnqstfkortu其中，每个字母a，b，...是，在原理上是一个独立的弹性参数，弹性材料刚度矩阵必要的对称性已推导出独立参数的最大值为21。一旦存在矩阵对称性，独立弹性参数的数量就减少了（克兰平，1981）。例如，对于单斜对称（z对称面）柔度矩阵有形式如下：abc00dbef00g1cfh00iD000jk(3.38)0000kl0dgi00m有13个弹性常数。正交对称（区分x、y、z对称面）给出9个常数：abc000bde0001cef000D00g0(3.39)000000h000000i然而，立方体对称性（同一的x、y、z对称面，与立方体相反面结合的面一起）只给出三个常数：abb000bab0001bba000D(3.40)000c000000c000000c如果我们进一步 要求 对教师党员的评价套管和固井爆破片与爆破装置仓库管理基本要求三甲医院都需要复审吗 c2(ab)和设a1/E和bv/E，那么我们发现3.1）的各向同性弹性柔度矩阵。不过，如果岩土工程材料具有一定的组构对称性，减少独立弹性参数的数量，显然是很方便的，正如料想的那样，受构造力、冰、或人推动的大部分材料，将不再拥有任何这类对称性，只要有一个域的弹性反应，我们应该期望要求全部21个弹性参数独立。如果我们选择将这样的材料建模成伴有某些限制对称性的各向同性弹性或各向异性弹性，那么我们不得不分辨到这是对土体和岩石可能不了解的建模结果。然而，许多土都在横向范围区域内沉积，沉积的对称性基本上是垂直的。从所有水平方向看是一样的，但横向刚度预计将不同于垂直刚度。现在柔度矩阵的形式为:abc000bac0001ccd000D000e0(3.41)00000e000000f并且我们可以写为：a1/Eh,bvhh/Eh,cvvh/Ev,d1/Ev,e1/Gvh和f2(ab)2(1vhh)/Eh:1/Ehvhh/Ehvvh/Ev000vhh/Eh1/Ehvvh/Ev0001vvh/Evvvh/Ev1/Eh000D(3.42)0001/Gvh0000001/Gvh00000021vhh/Eh这被形容为横向各向同性或六边形对称的交叉各向异性。有5个独立的弹性参数:Ev和Eh分别是垂直向和水平向不密闭压缩的杨氏模量；Gvh是一个垂直面上的剪切模量(图3.9a)。泊松比Vhh及Vvh分别是与发生在正交于压缩的横向方向和压缩的垂直方向的水平方向上的横向应变有关(图3.9c主轴与仪器轴平行三轴仪的交叉各向异性土的试验，，b)并没有给我们任何可能性发现查实Gvh1/E，因为这要求控制施加对试样垂直和水平面上的剪应力。事实上，我们只能确定5个弹性参数中的3个。如果我们对于垂直轴与三轴仪主轴平行的试样，就径向和轴向的应力和应变书写(3.42)，我们发现：a1/Ev2vvh/Eva'(3.43)rvvh/Ev1vhh/Evr'柔度矩阵不是对称的，因为在三轴试验环境中，应变增量和应力增量不是完全共轭的。我们推出：当我们可以分别确定Ev和Vvh时，我们可以得到的仅有的另外一个弹性参数是一个复合刚度Eh/(1Vhh)。我们不能将Eh和Vhh分离开（林斯等，2000）。另一方面，格拉汉姆和豪斯贝（1983）提出了（3.41）或（3.42）得特殊形式，只用了3个弹性参数，但对于此交叉各向异性材料，要求5个弹性参数是相互依赖的。1/2v/2/000vv21/2/000/v11v/v/1000DE00021v/00000021v/000000221v/这是书写的杨氏模量EEv，在垂直方向杨氏模量，泊松比VVhh，连同第三个参数。在水平和垂直方向的刚度比是Eh/Ev2及其他约束关系：vvhvhh/;GhvGhh/E/2(1v)。对于我们的三轴应力和应变量，柔度矩阵变为：pq13GJp'(3.45)detJKq其中det3KGJ2(3.46)并且，刚度矩阵是p'KJqJ3Gp(3.47)q其中1v4v2K2E(3.48)91v12v22v42GvE(3.49)61v12v1vv2J(3.50)E31v12v刚度和柔度矩阵（以正确选用工作共轭应变增量和应力增量方式书写）依然是对称的—材料依然是弹性的—但非零非对角线计算告诉我们体积作用和剪切作用之间是耦合的。进行纯粹的各向同性压缩试验时，q0，当我们施加纯剪力p'0和剪应变时，将产生体积应变，，不排水试验的平均有效应力将会改变，0。实际上，不排水试验的有效应力路径的斜率，形式（3.45）p'J21v2v(3.51)q3G322v24v从我们对孔压参数a(§2.6.2)的定义中,我们发现p'J(3.52)q3G在各向异性存在时，不会为零。第一次研究(3.5.1)仅仅表明对于孔压参数有限制，a非常大（Eh＞＞Ev）和非常小（Ev＞＞Eh）时(图3.10)分别为a2/3和a1/3，而这表示了依次施加恒定轴向有效应力和恒定径向有效应力的有效应力路径。a和α之间的联系实际是较为含蓄的。事实上，对于v0，其关系其实并不单调，并且有效应力路径方向超出了明显的界限(图3.11)。当a在1/3或2/3附近取值时(回忆介绍的数据图2.51和2.49，§2.5.4)，从a推导得到的α因(而Eh/Ev2)不是很可靠的。对于v0.5，a(12)/3(1)或(13a)(3a2)。这些关系符合0和的预期范围，但对于1和v0.5，(3.51)有奇异的倒转。3.2.2非线形我们大概预想的应力非线性的主要来源:应变反应将来自材料的可塑性—并且下部分，我们将继续发展弹塑性本构模式。不过，我们也期待一些真正有弹性性质的土体将随应力水平而变化，这可以看作弹性非线形的一个来源。我们把弹性材料作为保守材料—“超弹性”一词是用来形容这种材料的-可能使我们在选取随应力变化模量的任意 经验 班主任工作经验交流宣传工作经验交流材料优秀班主任经验交流小学课改经验典型材料房地产总经理管理经验 函数时更加谨慎。举例来说，如果我们假定土体体积弹性模量随平均有效压力变化，但泊松比(即剪切模量和体积模量的比值)是恒定的话，我们会发现,在图3.12如示的一封闭的应力循环中，违反热力学第一定律创造一个永动机，能量将增加(或失去)，这不会是一个保守体系。我们必须找到一种应变能(3.7)或补充能量密度(3.11)函数，可以通过微分得到可接受的应力模量变量。这样的补充能量密度函数能够从鲍耶斯（1980）描述的非线形弹性模型中推导得到：2Vn111qp'(3.53)n1K16Gp'其中K1和G1是体积模量和剪切模量的参考值，n是非线性参数。柔度矩阵然后可以通过微分导出：n1n2n21n16G13G1p'pn1Kp'(3.54)n1q1q3G13G1其中q/p'。体积作用和剪切作用之间再次是共轭的（对于各向异性模型）。刚度是与p'1n广泛成比例的。因为柔度现在是随着应力比变化的，对于不排水（纯剪切）加荷的有效应力路径不再是直线。实际上，对于提到的p'p0,q0情况，有效应力路径为p'02n(3.55)p'1其中(1n)K1/6G1。对于n0.2和泊松比v0.3，意味K1/G12.17的常体积应变p0曲线，如图3.13所示—由鲍耶斯对小应变回弹弹性参数测试的路基材料得到的典型值。同样地，在纯体积变形0将使剪应力发生一些变化。对于初始状态p'p0,qq0，这个试验的有效应力路径为qp'0n1(3.56)q0p'n2的常剪切应变曲线如图3.13所示。经常提出的是粘土的弹性体积刚度—体积模量应当与平均有效应力直接成比例：Kp'/k。当以对数压缩平面(plnv:lnp')（图3.14）作图时，其中v是比容，这种关系的结合显示粘土的弹性卸载形成一条直线反应。但我们要对剪切模作什么假设呢？如果简单地以为泊松比为常数，那么，剪切模量和体积模量的比值是常数,那么我们将发现一个非保守物质(扎廷斯基等，1978)。如果我们假定恒定的剪切模量值，独立的应力水平，我们将获得一个保守的材料，但也许会发现，我们泊松比在某种高或低应力水平呈现令人吃惊的值。再次，我们必须找到一种应变或补充能量函数，会给我们期望的基本模量变化。豪斯柏（1985）建议一个可以接受的应变能函数可为：Up/k3p'rek22(3.57)q更近一步地，这意味着刚度矩阵再次包含显示变形的体积和剪切元素耦合的对角线量。p'1/k/kpqp'/k/(3.58)qq可以导出q3q(3.59)p'1322kq所以常剪切变形的图形为常应力比的直线（图3.15）。常体积（不排水）应力路径为抛物线（图3.15）：2''q6kpip'pi(3.60)这组中所有的抛物线与线3k/2相切。这两种模型中所引入的非线性依然与各向同性弹性相联系。弹性参数随应变而变化