本构关系讲义殷宗泽

1、Advanced soil mechanicsConstitutive law of soilMicrostructure of soilStrength of soilConsolidation and rheologySlope stability1Constitutive Law of SoilYin Zong-ze21.About Stress-strain Relationship(1) What is constitutive law?Stress-strain relationship Stress-strain-strength relationshipStress-strai

2、n-time relationshipStress-strain-temperature relationshipstrainstressstrength3(2) Conventional computation of stress-strain relationship of soilsSimple stress conditionSettlement of buildingone dimensional problemConfined compression test, (oedometer test)pshepaCoefficient of compression4linearnonli

3、neareepCompression index5Complicated stress stateEarth dam, underground structure, harbor, excavation, etc.Extensive Hookes law （linear assumption）expandcompress6(3) Stress-strain testsConventional triaxial testFirst cell pressureThen deviator stress Strain componentspistonorganic glassbasecylindric

4、al cellrubbermembranecapthrough water to applythrough piston to apply7Plane strain testSample is enclosed in a rubber membrane, and is put in the cylindrical cell.Cell pressure Axial stress Dike, dam, retaining wall,no deformation in longitudinal direction8 True triaxial testApplyMeasure 91011Hollow

5、ed torsional shear testMTake half of ring as isolate body- from Md1d2122. Deformation Characteristics of Soils(1) Non-linear and non-elasticmetal soil13Plastic strain irrecoverable strain, due to adjustment of position of grains. One particle may slip over another particle, may drop into pore space,

6、and the relative position between the two particles can not be recovered.Deformation can not be recovered14(2) Plastic volumetric strain metal no plastic volumetric strain soil obvious plastic volumetric strain induced by ploadingpunloadingMacroscopic compressMicroscopic slip between grains15 induce

7、d by Shear stressPerform triaxial test with p = constant increase, but decreasep = constantp = constantShear compressionShear dilationp average normal stress, or global stress16Shear dilationDilation dense sand, overconsolidated clayCompression loose sand, normal consolidated clayShear dilativeShear

8、 compressive17(3) Plastic shear strainExpression of shear strainShear stress and strain on a given plane and Complicate stress stateOctahedral shear stressOctahedral normal stressOctahedral shear strain18Extensive shear stressExtensive shear strainIf , then energy of shape deformation 19Shear strain

9、 induced by global stress p Triaxial testSimultaneously reduce , and keep constantq = constant, p decreases.Mohr circle goes left. When the Mohr circle tangent to the failure line, the deviator strain is very large. The deviator strain is induced by p change. is due to existence of initial shear str

10、ess.20Effect of intersection in stress-strain matrix reflect shear dilation reflect average normal stress inducing shear strainElastic model = 0 = 0Plastic model can reflect these effects21(4) Softening and hardeningSofteninghardeningdirect shear testtriaxial shear testresidue strengthresidue streng

11、th22(5) influence of stress path & stress historyStress space the space consists of stress component axesMPrincipal stress space p q planeA point in stress space represents a stress state at a point in soil bodypqFailure lineqf p23Stress path the locus of moving point in stress space.Point represent

12、s stress state.Stress path represents the variation of stress state, that is the loading mannerNMA stress path consists of several sections. Each section represents a loading increment.the direction of the section reflects the proportion of stress components24Stress path influences the strain state

13、significantlyqpqfACBBCCCUn-draineddraineddrained25Stress history the stress state in history or the stress path in historyPlastic strain is irrecoverable. The historical strain was stored and accumulated.Stress history influence strain.The same stress state , different straindifferent modulus26(6) I

14、nfluence of mean normal stressExpression of variation of mean normal stressLode parameter27Lode angleMplaneMXYOctahedral planeIn stress spaceIn geometry spaceIn space28Parameter b29Influence of mean normal stressstrengthb Triaxial test, axial symmetric, b = 0, 2 = 3 a Plane strain test, no strain in

15、 2 direction, b= 0.3 0.4 , 2 3 abbaFor the same 1 and the same 3 30shape of stress-strain curveb=0b=1.0b=0.531(7) Influence of consolidation stress (surrounding stress, confining pressure)StrengthLarge grain is broken into small grains321000.010.11100.001d（mm）p（%）Before testAfter testGradation curve

16、33Shear dilationIn low confining pressure, shear dilationIn high confining pressure, shear compressionSoftening & hardeningIn low confining pressure, softeningIn high confining pressure, hardeninglow confining pressurehigh confining pressure34(8) Anisotropy virgin anisotropyanisotropy induced by str

17、esstransverseverticalsedimentationRemolding soil isotropicRemolding soil applied isotropic stresses isotropicRemolding soil applied anisotropic stresses anisotropicundisturbed soil35ABAB36 unsymmetric unsymmetric37True triaxial test dilative compressive38 3=100kPa,2=150kPa 3=100kPa,2=200kPa 3=200kPa

18、,2=300kPa 3=100kPa,2=250kPa 3=100kPa,2=300kPa 2(%) 1(%) 3(%) 1(%) increase 39()（13）（kPa）-3.00-2.00-1.000.001.002.0040.0080.00120.00160.00试验邓肯模型各向异性1 2 3()2（kPa）200.00-1.000.001.002.000.50.100.00150.00试验邓肯模型1 2 3图2 增加1的试验结果和邓肯模型与各向异性模型计算结果图3 增加2的试验与邓肯 模型计算结果403（kPa）-0.4-0.20.00.20.40.60.0040.0080.001

19、20.00()试验邓肯模型1 2 3图4 增加3的试验与邓肯模 型计算结果413. Nonlinear elastic modelExtensive Hookes lowSoftness matrix42Hardness matrix43shear modulusbulk modulus, volumetric modulus44Nonlinear elastic model45 Determination of parameters of Hookes lawUnconfined compression testtangent modulussecant modulus4647Triaxia

20、l test-48Control stress path triaxial testSolve simultaneous equations to get49Plane strain test50Determination of K & G51（2）Hyperbolic modelTangent Youngs modulusab52 the ultimate deviator stress asymptote of the curve deviator stress at failurecLet Then53n atmospheric pressureS stress level, refle

21、cting mobilized extent of strength54Tangent Poissons ratio55Asymptotic value of56Interpolate between and linearly with stress level S57Bulk modulus K （B）= constanthyperbola58Unloading & reloading modulusloadingreloadingunloading59Criterion of unloadingIn test sample, decrease of In real soil mass, c

22、omplicate60Unloading of confining stressUnconfined compression max in history61parameters effective strength parameters K initial tangent modulus when 502000 Kur initial tangent modulus when （1.23.0）K n index, which reflects variation of Ei with01.0Rf failure ratio0.50.95less K nsmaller Rf ngreater6

23、2F parameter which reflects variation of withG initial tangent Poissons ratio when D inverse of asymptote of of hyperbolic curve 0.20.60.0 0.250.0 20.0GDF63DiscussionSuitability。Constant confined stress= constant。Strain induced only by deviator stress。Strain induced only by deviator stress64Merits。b

24、eing able to reflect main deformation characteristics: nonlinear, stress history,stress path。simple, and easy to be excepted by engineers。easy to determine parameters, and engineers have experiences for parametersShortcomings。can not reflect shear dilation, softness, and anisotropy。has not give the

25、parameters for confined stress reduction654. Elasto-plastic model recoverable strain, elastic irrecoverable strain, plasticPlastic strain。failure criterion, yield criterion。hardening law。flue rule66Failure criterionelasticfailureFailure surface locus of the points in stress space which arrive failur

26、e（1）failure criterion failure functionvariables are stress components67Trasca criterionHexagonal columnSaturated soil, undrained68Mises criterionCircular column surfaceExtensive Mises criterionDrucker-PragerGeotecnical material first stress invariant second deviator stress invariantCircular cone sur

27、face69Cambridge university70Mohr-Coulomb criterionHexagonal cone with equal edges but unequal angles MisesMohr-CoulunbTrasca7172Lade - Duncan criterion73（2）yield criterionsimple stresselasticplastic, yieldcomplicate stresselasticplastic, yieldtheoretical material, yield = failuregeotechnical materia

28、l, yield failureConcept of yield74 yield function, corresponding to yield surface in stress spaceyield surface locus of the points in stress space which reach yieldifk changes,yield surface movesYield surfaceVariation of yield surface75Loading and unloadingCurrent stress state on yield surface,A new

29、 stress increment is applied.* unloading* loading* neutral loadinglimit of elasticplasticelastic2 vectors multiply76Yield surface for geo-material Independent on coordinatesCone typeCap type2 yield surface77k increases hardeningk decreases softeningk constant theoretical（3）hardening lawAfter yield,

30、k changes, H hardening parameter,a physical variant which courses k changeFor a given value of H, yield surface is defined.How does k change?Which factor causes k change? 7879（4）flue ruleHow the plastic strain develops among the strain components?How to determine the proportion of the strain compone

31、nts? plastic strain incrementDirection of determines each componentof the plastic strain increment.Flue rule gives direction of Conceive a plastic potential functionStrain space is overlapped with stress space.Plastic strain increment is perpendicularto plastic potential surface80Associated flue rul

32、eDruckers postulation an element exits initial stress state, loading slowly, and then unloading, during loading, work done by external agency is positive. And during loading and unloading, work done by external agency is not negative.If on yield surface,81derivation*All the points which represent th

33、e stress must be on the other side of the plane perpendicular to yield surface f must be convex.if concave * is perpendicular to yield surface fif not, 82Non-associated flue rulesoftening83(5) Elasto-plastic MatrixElasticPlastic8485(a) (b)(c)86softness matrix:87(6) Cambridge model1. State boundary s

34、urfaceAn example of Elasto-plastic modelDrained shear testUndrained shear teste effective stress p & q are the same for both drained and undrained tests.q=0q=MpVirgin compression curvefailure curve3188 , on yield surfaceNB, on state boundary surface （1） qreduce, pconstant, econstant. NDvertical line

35、（2） , only p reducesBD, unloading curve on ground Line NB project to q p plane, yield locus; project to e p plane, unloading curve. NB, intersection line of vertical column surface BDN and horizontal surface NNBB State boundary surface is the locus of moving curve NB.Vertical column surface BDN is called elastic wall. Only elastic deformation in the wall.Go over the top of the wall plastic