The 12th International Conference of International Association for Computer Methods and Advances in Geomechanics (IACMAG) 1-6 October, 2008 Goa, India

A Lode Angle Dependent Formulation of the Hardening Soil Model

T. Benz, M. Wehnert Wechselwirkung – Numerische Geotechnik, Stuttgart, Germany P.A. Vermeer Institute for Geotechnical Engineering, University of Stuttgart, Germany

Keywords: Double Hardening, Hardening Soil, Matsuoka-Nakai, constitutive soil model

ABSTRACT: This paper presents a new formulation of a well known double hardening model often referred to as the Hardening Soil model. As an alternative to the existing Hardening Soil model’s Mohr-Coulomb failure surface, the new formulation allows for the incorporation of smooth failure surfaces, such as the failure surface proposed by Matsuoka & Nakai or that proposed by Lade & Duncan. Although the incorporation of all other failure criteria that can be formulated as a function of the Lode angle is also possible, the examples presented at the end of this paper concentrate on the Hardening Soil model with Matsuoka-Nakai failure criterion. The Lode angle dependent formulation with Matsuoka-Nakai yield surface is validated in element tests and in an excavation example. A comparison between results obtained from the Hardening Soil model with the Mohr-Coulomb type yield surface and that with the Mastuoka-Nakai type yield surface reveals the failure criterion’s actual influence on material strength and stiffness when applied in the Hardening Soil Model in plane strain conditions.

1 Introduction Constitutive soil models that incorporate shear hardening and volumetric hardening mechanisms, in short referred to as double hardening models, were first proposed in the 1970’s. Since then, the double hardening concept has proved to be very useful in the numerical analysis of geotechnical problems. A particularly well known double hardening model is the Hardening Soil model that was developed by Schanz (1988) and Schanz et al. (1999) on the basis of the double hardening model by Vermeer (1978). This paper revisits the formulation of the original Hardening Soil double hardening model and extends it towards a new Lode angle dependent formulation. Within the new formulation it is now possible to use other alternative failure criteria than the Mohr-Coulomb criterion. In the examples presented at the end of this paper, the Mohr-Coulomb failure criterion is replaced by the smooth Matsuoka-Nakai failure criterion as shown in Figure 1. The Mohr-Coulomb failure criterion (Mohr 1900) for soils, which is implemented in the original Hardening Soil model, is one of the earliest and most trusted failure criterion. It has been experimentally verified in triaxial com-pression and extension and is of striking simplicity. However, the Mohr-Coulomb criterion is very conservative for intermediate principal stress states between triaxial compression and extension. Matsuoka and Nakai (1974, 1982) proposed a failure criterion that is in better agreement with experimental data. They propose the concept of a Spatial Mobilized Plane (SMP), which defines the plane of maximum spatial, averaged particle mobilization in principal stress space. Replacing the Mohr-Coulomb yield surface and plastic potential by a smoother surface is also an advantage from a numerical point of view.

2 Model formulation

2.1 Definitions

Within this paper, compressive stress is taken as positive. Tensile stress is taken as negative. Stresses are always taken to be effective values without any special indication by a prime. Infinitesimal deformation theory is applied. Cauchy stress is related to linearized infinitesimal strain. Tensorial quantities are generally expressed in indicial notation. The order of a tensor is indicated by the number of unrepeated (free) subscripts. Whenever a subscript appears exactly twice in a product, that subscript will take on the values 1, 2, 3 successively, and the resulting terms are summed (Einstein’s summation convention). Eigenvalues of stress and strain tensors (prin-cipal stresses and strains) are denoted by one subscript only, e.g.σ i and ε i with 1 2 3.= , ,i The order of principal stresses is 1 2 3.≥ ≥σ σ σ The Roscoe stress invariants p (mean stress) and q (deviatoric stress), are defined as:

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Fig. 1. Two double hardening models with different failure criteria in principle stress space (left) and in deviatoric

plane (right) – (a) Mohr-Coulomb failure criterion – (b) Matsuoka-Nakai failure criterion.

3 1 1 and ( )( )3 2 3 3

= = − − ,σ σ δ σ σ δ σii

ij ij kk ij ij kkp q (1)

In triaxial compression with 1 2 3,≥ =σ σ σ the Roscoe invariants simplify to:

axial lateral axial lateral1 ( 2 ) and ( )3

= + = − .σ σ σ σp q (2)

In analogy to the stress invariants, volumetric strain εv and shear strain γ s are defined as:

3 1 1and ( )( )2 3 3

= = − − ,ε ε γ ε δ ε ε δ εv ii s ij ij kk ij ij kk (3)

which simplify to axial lateral axial lateral2 and ( )= + = − ,ε ε ε γ ε εv s (4) in triaxial compression. Shear strain relates to the deviatoric strain invariant εq as follows:

3 3 2 1 1( )( )2 2 3 3 3

= = − − .γ ε ε δ ε ε δ εs q ij ij kk ij ij kk (5)

2.2 Shear hardening

In drained triaxial primary loading, the experimentally observed relationship between axial strain and deviatoric stress in soils can be well approximated by a hyperbolic function. Kondner and Zelasko (1963) described the hyperbolic stress-strain relationship for drained triaxial loading as follows:

1 50 3 5050

2sinwith ( cot ) and1 sin 2

= = + = .− −

ϕε ε σ ϕ εϕ

aa

a

qq q cq q E

(6)

where E50 gives the secant stiffness in primary triaxial loading at a deviatoric stress level equal to half the failure stress as illustrated in Figure 2. Duncan and Chang (1970) based their hypoelastic model on the above formulation by Kondner and Zelasko, additionally introducing the deviatoric measure fq in the form:

1 50 32sin ( cot )

1 sin= < = + = .

− −ϕε ε σ ϕϕ

ff a

a f

qq for q q c and qq q R

(7)

where Rf is a failure ratio that modifies the hyperbolic stress-strain curve defined by Kondner and Zelasko when chosen lower than 1. The conceptual difference in the formulations by Kondner and Zelasko and that by Duncan and Chang is illustrated in Figure 2. Extending the hypoelastic Duncan-Chang model to an elastoplastic formulation, Schanz (1988) proposed the following yield function:

50 ur

2= − − .

−γs psa

a

q q qfE q q E

(8)

where γ ps is an internal material variable for the accumulated plastic deviatoric strain, 1 3= −σ σq is defined for triaxial loading, and aq is the asymptotic deviatoric stress as defined in the original Duncan-Chang model (Equa-tion 7). As the stress-strain relation of soils in unloading and reloading can be roughly approximated by a linear function, the HS model assumes isotropic elasticity inside the yield function: The elastic unloading-reloading stiffness urE relates elastic stress to elastic strain (see above).

654

q

1E50

qa

asymptote

½qa

� 1

q

1

1

Eur

qa

qf

asymptote

� 1

E50

½qf

Fig. 2. Hyperbolic stress-strain law by Kondner & Zelasko (left) and its modification after Duncan & Chang (right).

For constant volumetric strain, the equivalent of the original Hardening Soil model (Equation 7) with the approach by Duncan and Chang is given by defining: 1 2 3 1and thus 2= − − = ,γ ε ε ε γ εps p p p ps p (9) as then the following relation holds:

501 1 1 50

ur 50 50

12 2

= + = − = = = .− − −

ε ε ε γ εe p ps a

a a a

q qq q q qE E q q E q q q q

(10)

Unfortunately, the definition of the strain measure used in the original Hardening Soil model is not compatible with the shear strain γ s defined in equation 3 and, as a consequence, does not vanish during volumetric straining. Therefore a representation of deviatoric stress q against the deviatoric strain γ will show an offset when volumetric loading is applied prior to deviatoric loading and hence, is not objective. The following revised formulation of the Hardening Soil yield function therefore uses the shear strain measure ,γ ps

s which goes back to the second deviatoric invariant as defined in Equation 3. In triaxial conditions, the shear strain γ ps

s is defined as:

1 3 13so that for 02

= − = = .γ ε ε γ ε εps p p ps p ps s v (11)

It can now be easily proven that in order to keep the new yield function equivalent to the approach by Duncan and Chang and thus equivalent to the original Hardening Soil model, one has to write:

50 ur

3 3 24 2

= − −−

γs psas

a

q q qfE q q E

(12)

Next, the new yield function can be written in terms of mobilized friction ,ϕm which is in triaxial compression by means of the Mohr-Coulomb criterion, defined as:

1 3

1 3

sin2 cot

−=

+ +σ σϕ

σ σ ϕm c (13)

so that 1 3sin ( 2 cot )= + +ϕ σ σ ϕmq c (14) The Mohr-Coulomb criterion assumes failure in triaxial conditions whenever:

32sin ( cot )

1 sin= +

−ϕ σ ϕϕfq c (15)

and thus, the ratio / aq q can be expressed as:

1 3

3

2 cotsin1 sinsin 1 2 2 cot

sin1 sinsin 1 sin

+ +−= =

+

⎛ ⎞⎛ ⎞−= .⎜ ⎟⎜ ⎟ −⎝ ⎠ ⎝ ⎠

σ σ ϕϕϕϕ σ ϕ

ϕϕϕ ϕ

mf f

a f

mf

a m

cq qR Rq q c

Rqq

(16)

Applying Equation 16 to Equation 12 results in the final form of the shear hardening yield function:

( )

( ) ( )1 sin

sin

1 sin 1 sinursin sin

3 32 2

−

− −= − −

−

ϕϕ

ϕ ϕϕ ϕ

γm

m

m

m

s pss

i f

q qfE ER

(17)

where sinϕm is the mobilized friction angle in triaxial compression. The transition from 50E to 502≈iE E is made because of the double hardening model’s second yield surface that will be introduced later. The second yield surface will affect material stiffness such that, the meaning of iE in the final model is not as closely related to the hyperbolic model by Kondner and Zelasko as the one of 50.E The shear hardening function given in Equation 17 is not limited to the Mohr-Coulomb failure criterion as for example the yield function of the original Hardening Soil model is. In mobilized friction, the Matsuoka-Nakai yield criterion for example, can be written as:

655

1 2

1 2

3

3

9sin

1−

≡ .−

ϕI II

m I II

(18)

This definition yields deviatoric isolines of mobilized friction that are similar to the shape of the Matsuoka-Nakai yield criterion. Alternatively, mobilized friction can be expressed in the Lode angle dependent formulation:

3sin6 ( )

≡ .+

ϕθm

qL p q

(19)

where L is varying between 1 and e=δc

MM for triaxial compression and extension respectively and cot= + ϕp p c

accounts for a cohesion related apex shift along the hydrostatic axis. In this way, many Lode dependent yield functions can be assigned to the yield function given in Equation 17. Lode dependent formulations of the Matsuoka-Nakai and the Lade criterion in the form of functional relationships of ( )θL are for example introduced by Bardet (1990). The stiffness moduli iE and urE are scaled for their stress dependency with a power law (Janbu 1963, Ohde 1951)

3 3cot cotandcot cot

⎛ ⎞ ⎛ ⎞+ += =⎜ ⎟ ⎜ ⎟+ +⎝ ⎠ ⎝ ⎠

σ ϕ σ ϕϕ ϕ

m mref ref

i i ur urref ref

c cE E E Ep c p c

(20)

where refiE and ref

urE are the material stiffness moduli at the reference pressure refp and m is the exponent of the power law. In Equation 20 the minor principal stress 3σ is used as an indicator of the actual stress state in the material instead of the mean stress 3 .= σ iip

2.3 Volumetric hardening

Similar to the above defined yield function that resembles lines of constant plastic shear strain, a yield function that resembles lines of constant void ratio or constant volumetric strain is introduced next. Within the double har-dening model, this second cap-type yield function accounts for volumetric hardening, this corrects for overly stiff primary oedometric or isotropic loading, obtained in pure shear hardening models. From experimental evidence, loci of constant void ratio are usually defined as an ellipse in p-q space, (e.g. Modified Cam Clay model):

2

22= − −

ΜMCC

pqf p pp (21)

whereΜ is the slope of the critical state line in the p-q plane and pp is an internal material hardening variable for pre-consolidation stress. The original HS model’s cap-type yield surface is defined slightly differently:

2

2 22= − −

αc

pqf p p (22)

whereα is an internal material constant, controlling the steepness of the cap in p-q space and q is a special stress measure, defined as:

1 11 2 3

3 sin( 1) with3 sin

− − −= + − − = .

+ϕσ δ σ δ σ δϕ

q (23)

The definition of the special stress measure q is necessary to adopt the cap-type yield surface’s deviatoric shape to the Mohr-Coulomb shape of the original Hardening Soil model’s cone-type yield surface. For compatibility with other failure criteria, e.g. Matsuoka-Nakai or Lade, the stress measure q is avoided and the cap type yield surface is rewritten as:

( )

22 2

2( )= − − .

θ αc

pqf p p

L (24)

2.4 Plastic potentials

The original Hardening Soil model uses a non-associated potential for the cone type yield surface and an associated potential for the cape type yield surface. Likewise does the Lode angle dependent formulation. As the plastic flow directions for low mobilized friction are almost radial and the error made for higher mobilization levels is also tolerable, the Lode angle dependent formulation uses potentials with radial deviatoric flow. The non-associated potential to the cone-type yield surface is defined as:

6sinwhere3 sin

= Μ Μ = .−ψ ψ

ψψm m

s m

m

g p (25)

The associated potential to the cap-type yield surface is written as:

( )

22 2

2( )= − − .

θ αc

pqg p p

L (26)

656

q

pelastic regionc cot� p

fs

p

gs

corner region

cn = mij

c

ij

nij

s mij

s

�ijn+1

�ij

Trial

�ijn

cg

cf =

n+1 n+1

n+1 n+1q

pelastic regionc cot� p

fs

p

gs

apex gray

region

m ijs

�ijn

tf

n+1

corner region

m ijtn+1

cg

cf =

�ij

Trial

t�apex

�ij

n+1

a) b)

Fig. 3. (a) The two vector cone-cap return strategy. (b) Apex gray region and tension cut-off (with 2 0=σ ).

The angle of mobilized dilatancyψm in Equation 25 can for example be calculated according to Rowe’s stress dila-tancy theory (Rowe 1962). A main drawback of Rowe’s approach, is the highly contractive behavior at low mobi-lized friction angles. In the original Hardening Soil model the mobilized dilatancy angle, sinψm is therefore set to be greater than or equal to zero overriding Rowe’s original equation:

sin sinsin 01 sin sin

−= ≥ .

−ϕ ϕψϕ ϕ

m csm

m cs

(27)

2.5 Hardening rules

The evolution laws of the original Hardening Soil model are defined as:

3

ref

with 1

cotwith 2cot

= =

⎛ ⎞+= = ⎜ ⎟+⎝ ⎠

γγγ λ

σ ϕλ

ϕ

psps

pp

ps s

mc

pp p

d d h h

cdp d h h H pp c

(28)

where m represents the power law exponent, and H relates plastic volumetric strain ε pv to pre-consolidation stress

pp as follows:

3ref

cotcot

⎛ ⎞+= .⎜ ⎟+⎝ ⎠

σ ϕε

ϕ

mp

p vcdp H d

p c (29)

In decomposing volumetric strain into elastic and plastic contributions, H can be rewritten as a function of the bulk stiffness in unloading-reloading sK and the bulk stiffness in primary loading :cK

11

= = .− −s

c

s csK

s c K

K KH KK K

(30)

where due to the assumption of isotropic elasticity, the elastic bulk stiffness sK relates to refurE as follows:

refur

3(1 2 )= .

− νsEK (31)

The model parameter H can therefore be determined by the bulk stiffness ratio ./s cK K As the physical meaning of the latter is more evident, it is often used to quantify .H The hardening rules of the Lode angle dependent formulation are the same except 3 2.=

γ pss

h This change reflects the new shear strain measure used in the Lode angle dependent formulation and assure that both formulations yield the same result in triaxial compression.

2.6 Implementation aspects

For the integration of the constitutive equations, an implicit closest point projection algorithm was chosen. Although the Matsuoka-Nakai yield criterion is smooth in a deviatoric section, corner problems still arise at the non-smooth intersections of the cone-type yield surface and the cap-type yield surface as well as at the apex. Koiter (1960) additively decomposes plastic strain rates in such corner problems as follows: ( ) ( )∗ ∗= , + , .ε λ σ λ σp cone cap

ij ij ij ij ijd d m q d m q (33)

where = ∂ /∂σij ijm g is the derivative of the plastic potential. The two vector cone-cap return strategy is illustrated in Figure 3a. As the size of the gray corner region in hardening situations is not known beforehand, the scheme proposed by Bonnier (2000) is used to determine to which surface the trial stress is to be returned. The gray apex region is defined by the gradient to the cone-type potential surface as shown in Figure 3b. If the apex corresponds to an admissible tensile stress, the trial stress is returned to the apex. If the apex point violates the user defined maximum allowable tensile stress, a return mapping scheme to the respective tension cut-off planes is evoked. The tension cut-off criterion is based on minimum principal stress, which implies three (fixed) orthogonal tension cut-off planes in principal stress space:

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Table 1. Parameters for the double hardening model used in the analyses.

Parameter Unit Hostun RF Sand (L1) Sand (L2) Sand (L3) 1) User defined parameters Unsaturated/saturated weight /γ γunsat sat [kN/m³] - 19 / 20 19 / 20 19 / 20 Triaxial secant stiffness 50

refE [kN/m²] 30000 45000 75000 105000 Oedometric tangent stiffness ref

oedE [kN/m²] 30000 45000 75000 105000 Unloading/reloading stiffness ref

urE [kN/m²] 90000 180000 300000 315000 Power of stress dependency m [ – ] 0.55 0.55 0.55 0.55 Cohesion (effective) c [kN/m²] 0.0 1.0 1.0 1.0 Friction angle (effective) ϕ [ ° ] 42.0 35.0 38.0 38.0 Dilatancy angle ψ [ ° ] 16.0 5.0 6.0 6.0 Poisson’s ratio ν [ – ] 0.25 0.2 0.2 0.2 Reference stress for stiffness refp [kN/m²] 100 100 100 100 K0-value (normal consolidation) 0

ncK [ – ] 0.40 0.43 0.38 0.38 Failure ratio =f f aR q q [ – ] 0.9 0.9 0.9 0.9 Tensile strength σTension [kN/m²] 0.0 0.0 0.0 0.0 2) Internal parameters Initial secant stiffness ref

iE [kN/m²] 65488 96662 154447 208642 Cap parameter (steepness) α [ – ] 1.47 1.48 1.87 1.88 Cap parameter (stiffness ratio) S CK K [ – ] 1.84 2.15 2.07 1.59 3) State parameters Plastic shear strain γ ps

s [ – ] - - - - Pre-consolidation pressure pp [kN/m²] - - - -

Tension= − ,σ σti if (34)

where Tensionσ is the user defined maximum allowable tensile stress.

2.7 Material parameters

A summary of the material parameters and state parameters introduced above is presented in Table 1. A diffe-rentiation is made between user input and internal parameters because the latter cannot be quantified as results of standard triaxial and oedometer tests directly and hence, are not expected to be entered by the user. Internal model parameters are the stiffness measures ref

iE and ,H and the cap-type yield surface’s steepness .α These internal parameters mainly relate to the user input parameters ref

50 ,E refoedE and nc

0K respectively, where refoedE is

the tangent stiffness at ref1 =σ p in 0K (oedometer) loading, and nc

0K is the stress ratio of horizontal effective stress to vertical effective stress in a normally consolidated state. Note that ref

50 ,E and refoedE are not elastic stiffness

constants. In double hardening situations, i.e. both yield loci are hardened simultaneously, analytical back-calculation of internal model parameters is impossible. Therefore, the internal parameters are solved for in an iterative scheme so that the double hardening model simulates the user input ref

50E in a triaxial element test and both, refoedE and nc

0K in an oedometer element test, to within a tolerated error.

3 Model validation The Lode angle dependent formulation of the Hardening Soil model is validated by element tests that are cal-culated on the Gauss-Point level and by analyses of a boundary value problem using the finite element code PLAXIS V 8. The original Hardening Soil model that is implemented in the calculation kernel of this code serves as a reference. The Matsuoka-Nakai yield surface is considered in the Lode angle dependent formulation of the double hardening model. In the following, the abbreviation HSMC refers to the original Hardening Soil model with Mohr-Coulomb yield surface and HSMN refers to the Lode angle dependent formulation of the double hardening model with Matsuoka-Nakai yield surface.

3.1 Element tests

In triaxial compression, triaxial extension, and K0-loading, the HSMC and the HSMN yield identical results, which is a first validation of the HSMN formulation and its numerical implementation. In plane strain biaxial tests,

658

however, the material strength estimate of the Matsuoka-Nakai criterion increases compared to that of the Mohr-Coulomb criterion. Figure 4 shows biaxial test data on dense Hostun sand compiled by Desrues et al. (2000) and their numerical back calculation using the HSMC and the HSMN models. The material data set used in the calcu-lation (Table 1) was calibrated for triaxial and oedometer tests before. The test data validates the use of Matsuoka-Nakai failure criterion for dense Hostun sand.

3.2 Excavation in Berlin sand

The working group 1.6 „Numerical methods in Geotechnics“ of the German Geotechnical Society (DGGT) has organized several comparative finite element studies (benchmarks). One of these benchmark examples is the installation of a triple anchored deep excavation wall in Berlin sand. The reference solution by Schweiger (2002) is used here as the starting point: Both, the mesh shown in Figure 5, and the soil parameters given in Table 1 are taken from this reference solution. The calculation assumes plane strain conditions. Note that in the reference solution, the stiffness of the lower sand layer is artificially increased as the models investigated do not account for small-strain stiffness. In using a small-strain stiffness model, the extra definition of sand layer 3 could be omitted as shown by Benz (2007).

Fig. 4. Element tests on dense Hostun RF sand.

Fig. 5. Results of the analyses on the excavation in Berlin sand.

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A considerable quantitative difference between the results of the HSMC and those of the HSMN model is found whenever plastic hardening is evoked (see Figure 5 - settlement trough and wall displacement). Although the HSMN model hardens with the same rate as the HSMC model, its overall stiffness is higher due to its higher peak value in plane strain conditions. Consequently, less plastic straining occurs in the HSMN loading problem.

4 Summary and conclusion The original Hardening Soil model comprises ideas by Kondner (1963), Duncan & Chang (1970), Ohde (1951) or Janbu (1963), Rowe (1962) and Vermeer (1978). Standard lab tests, such as triaxial and oedometer tests provide the model’s basic characteristics. The new Lode angle dependent formulation now adds the possibility of incor-porating different failure criteria, as for example the one proposed by Matsuoka & Nakai, which is not only smooth but also considers the influence of the intermediate principal stress on material strength. It should be noted that same as the original Hardening Soil model, the new model formulation does not account for initial anisotropy, stress induced anisotropy, or rotation of principle stresses. However, the formulation may be subsequently exten-ded towards incorporation of these features. A small-strain stiffness extension of the Hardening Soil model is already available (Benz 2007). The Hardening Soil model with Matsuoka-Nakai failure criterion predicts additional material strength and stiffness in plane strain conditions. In the examples presented in this paper, the new model performs better than its Mohr-Coulomb counterpart.

5 Acknowledgements The Lode angle dependent formulation of the Hardening Soil model emanated from a research project that was initiated and funded by the Federal Waterways Engineering and Research Institute (BAW), in Karlsruhe, Ger-many. The financial support and technical advice provided by the BAW is gratefully acknowledged. Further, the authors thank Paul Bonnier from PLAXIS B.V. for his valuable assistance with the numerical implementation.

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