yantubbs-The hardening soil model, Formulation and verification_百

Beyond 2000 in Computational Geotechnics – 10 Years of PLAXIS © 1999 Balkema, Rotterdam, ISBN 90 5809 040 XThe hardening soil model: Formulation and verificationT. SchanzLaboratory of Soil Mechanics, Bauhaus-University Weimar, GermanyP.A. VermeerInstitute of Geotechnical Engineering, University Stuttgart, GermanyP.G. BonnierPLAXIS B.V., NetherlandsKeywords: constitutive modeling, HS-model, calibration, verificationABSTRACT: A new constitutive model is introduced which is formulated in the framework ofclassical theory of plasticity. In the model the total strains are calculated using a stress-dependentstiffness, different for both virgin loading and un-/reloading. The plastic strains are calculated byintroducing a multi-surface yield criterion. Hardening is assumed to be isotropic depending on boththe plastic shear and volumetric strain. For the frictional hardening a non-associated and for the caphardening an associated flow rule is the model is written in its rate form. Therefor the essential equations for the stiffness mod-ules, the yield-, failure- and plastic potential surfaces are the next part some remarks are given on the models incremental implementation in thePLAXIS computer code. The parameters used in the model are summarized, their physical interpre-tation and determination are explained in model is calibrated for a loose sand for which a lot of experimental data is available. Withthe so calibrated model undrained shear tests and pressuremeter tests are paper ends with some remarks on the limitations of the model and an outlook on further de-velopments.1 INTRODUCTIONDue to the considerable expense of soil testing, good quality input data for stress-strain relation-ships tend to be very limited. In many cases of daily geotechnical engineering one has good data onstrength parameters but little or no data on stiffness parameters. In such a situation, it is no help toemploy complex stress-strain models for calculating geotechnical boundary value problems. In-stead of using Hooke's single-stiffness model with linear elasticity in combination with an idealplasticity according to Mohr-Coulomb a new constitutive formulation using a double-stiffnessmodel for elasticity in combination with isotropic strain hardening is izing the existing double-stiffness models the most dominant type of model is the Cam-Clay model (Hashiguchi 1985, Hashiguchi 1993). To describe the non-linear stress-strain behav-iour of soils, beside the Cam-Clay model the pseudo-elastic (hypo-elastic) type of model has beendeveloped. There an Hookean relationship is assumed between increments of stress and strain andnon-linearity is achieved by means of varying Young's modulus. By far the best known model ofthis category ist the Duncan-Chang model, also known as the hyperbolic model (Duncan & Chang1970). This model captures soil behaviour in a very tractable manner on the basis of only two stiff-ness parameters and is very much appreciated among consulting geotechnical engineers. The majorinconsistency of this type of model which is the reason why it is not accepted by scientists is that,in contrast to the elasto-plastic type of model, a purely hypo-elastic model cannot consistently dis-tinguish between loading and unloading. In addition, the model is not suitable for collapse loadcomputations in the fully plastic range.1

Beyond 2000 in Computational Geotechnics – 10 Years of PLAXIS © 1999 Balkema, Rotterdam, ISBN 90 5809 040 XThese restrictions will be overcome by formulating a model in an elasto-plastic framework inthis paper. Doing so the Hardening-Soil model, however, supersedes the Duncan-Chang model byfar. Firstly by using the theory of plasticity rather than the theory of elasticity. Secondly by includ-ing soil dilatancy and thirdly by introducing a yield contrast to an elastic perfectly-plastic model, the yield surface of the Hardening Soil model isnot fixed in principal stress space, but it can expand due to plastic straining. Distinction is madebetween two main types of hardening, namely shear hardening and compression hardening. Shearhardening is used to model irreversible strains due to primary deviatoric loading. Compressionhardening is used to model irreversible plastic strains due to primary compression in oedometerloading and isotropic the sake of convenience, restriction is made in the following sections to triaxial loading′ =

σ3′ and

σ1′ being the effective major compressive ions with

σ22 CONSTITUTIVE EQUATIONS FOR STANDARD DRAINED TRIAXIAL TESTA basic idea for the formulation of the Hardening-Soil model is the hyperbolic relationship be-tween the vertical strain

ε1, and the deviatoric stress, q, in primary triaxial loading. When subjectedto primary deviatoric loading, soil shows a decreasing stiffness and simultaneously irreversibleplastic strains develop. In the special case of a drained triaxial test, the observed relationship be-tween the axial strain and the deviatoric stress can be well approximated by a hyperbola (Kondner& Zelasko 1963). Standard drained triaxial tests tend to yield curves that can be described by:The ultimate deviatoric stress, qf, and the quantity qa in Eq. 1 are defined as:The above relationship for qf is derived from the Mohr-Coulomb failure criterion, which involvesthe strength parameters c and

ϕp. As soon as q = qf , the failure criterion is satisfied and perfectlyplastic yielding occurs. The ratio between qf and qa is given by the failure ratio Rf, which shouldobviously be smaller than 1. Rf = 0.9 often is a suitable default setting. This hyperbolic relationshipis plotted in Fig. 1.2.1 Stiffness for primary loadingThe stress strain behaviour for primary loading is highly nonlinear. The parameter E50 is the con-fining stress dependent stiffness modulus for primary loading. E50 is used instead of the initialmodulus Ei for small strain which, as a tangent modulus, is more difficult to determine experimen-tally. It is given by the equation:ref is a reference stiffness modulus corresponding to the reference stress

pref. The actual stiff-E50′, which is the effective confining pressure in a tri-ness depends on the minor principal stress,

σ3axial test. The amount of stress dependency is given by the power m. In order to simulate a loga-rithmic stress dependency, as observed for soft clays, the power should be taken equal to 1.0. As a2

Beyond 2000 in Computational Geotechnics – 10 Years of PLAXIS © 1999 Balkema, Rotterdam, ISBN 90 5809 040 XFigure 1. Hyperbolic stress-strain relation in primary loading for a standard drained triaxial ant modulus

E50 is determined from a triaxial stress-strain-curve for a mobilization of 50% ofthe maximum shear strength

qf .2.2

Stiffness for un-/reloadingFor unloading and reloading stress paths, another stress-dependent stiffness modulus is used:ref is the reference Young's modulus for unloading and reloading, corresponding to thewhere

Eurreference pressure

σ

ref. Doing so the un-/reloading path is modeled as purely (non-linear) elastic components of strain

εe

are calculated according to a Hookean type of elastic relationusing Eqs. 4 + 5 and a constant value for the un-/reloading Poisson's ratio

υ drained triaxial test stress paths with

σ2 =

σ3 = constant, the elastic Young's modulus

Eur re-mains constant and the elastic strains are given by the equations:Here it should be realised that restriction is made to strains that develop during deviatoric loading,whilst the strains that develop during the very first stage of the test are not considered. For the firststage of isotropic compression (with consolidation), the Hardening-Soil model predicts fully elasticvolume changes according to Hooke's law, but these strains are not included in Eq. 6.2.3

Yield surface, failure condition, hardening lawFor the triaxial case the two yield functions

f12 and

f13 are defined according to Eqs. 7 and 8. Here3

Beyond 2000 in Computational Geotechnics – 10 Years of PLAXIS © 1999 Balkema, Rotterdam, ISBN 90 5809 040 XFigure 2. Successive yield loci for various values of the hardening parameter

γ

p and failure measure of the plastic shear strain

γ

p

according to Eq. 9 is used as the relevant parameter forthe frictional hardening:with the definitionIn reality, plastic volumetric strains

ευp will never be precisely equal to zero, but for hard soilsplastic volume changes tend to be small when compared with the axial strain, so that the approxi-mation in Eq. 9 will generally be a given constant value of the hardening parameter,

γ

p, the yield condition f12 = f13 = 0 can bevisualised in p'-q-plane by means of a yield locus. When plotting such yield loci, one has to useEqs. 7 and 8 as well as Eqs. 3 and 4 for E50 and Eur respectively. Because of the latter expressions,the shape of the yield loci depends on the exponent m. For m = 1.0 straight lines are obtained, butslightly curved yield loci correspond to lower values of the exponent. Fig. 2 shows the shape ofsuccessive yield loci for m = 0.5, being typical for hard soils. For increasing loading the failure sur-faces approach the linear failure condition according to Eq. 2.2.4 Flow rule, plastic potential functionsHaving presented a relationship for the plastic shear strain, γ

p, attention is now focused on theplastic volumetric strain

ευp. As for all plasticity models, the Hardening-Soil model involves a re-