7.4.3 Model parameters
Compared to the standard Hardening Soil model，the Hardening Soil model with small⁃strain stiffness requires two additional stiffness parameters as input：Gref0 andγ0.7.All other parameters，including the alternative stiffness parameters，remain the same as in the standard Hardening Soil model.G
defines the shear modulus at very small strains e.g.ε＜10－6 at a reference minor principal stress of－σ′3＝pref.
Poisson's ratio vur is assumed a constant，so that the shear modulus G
can also be calculated from the very small strain Young's modulus as G
＝E
／［2（1＋vur）］.The threshold shear strain γ0.7 is the shear strain at which the secant shear modulus G
is decayed to 0.722G
.The threshold shear strainγ0.7 is to be supplied for virgin loading.In summary，the input stiffness parameters of the Hardening Soil model with small⁃strain stiffness are listed below：
m：Power for stress⁃level dependency of stiffness ［—］
γ0.7：threshold shear strain at which Gs＝0.722G0 ［—］
Figure 7.12 illustrates the models stiffness parameters in a drained triaxial test：E50，Eur and E0＝2 G0（1＋vur）.For the order of strains at which Eur and G0 are defined and determined，one may refer to the previous part.
Figure 7.12 Stiffness parameters E50，Eur and E0＝2 G0（1＋vur）of the Hardening Soil model with small⁃strain stiffness in a triaxial test
A first estimation of the Hssmall（or HS）parameters for quartz sand based on the relative density（RD）is given in Brinkgreve et al.（2010）.
On the parameters G0 andγ0.7
A number of factors influence the small⁃strain parameters G0 andγ0.7.Most importantly，they are influenced by the materials actual state of stress and void ratio e.In the Hssmall model，the stress dependency of the shear modulus G0 is taken into account with the power law：
which resembles the ones used for other stiffness parameters.The threshold shear strainγ0.7 is taken independently of the mean stress.
Assuming that within a HS Small（or HSS）computation void ratio changes are rather small，the material parameters are not updated for changes in the void ratio.Knowledge of a material's initial void ratio can nevertheless be very helpful in deriving its small⁃strain shear stiffness G0.Many correlations are offered in the literature（Benz，2006）.A good estimation for many soils is for example the relation given by Hardin＆Black（1969）：
Alpan（1970）empirically related dynamic soil stiffness to static soil stiffness（see Figure 7.13）.The dynamic soil stiffness in Alpan's chart is equivalent to the small⁃strain stiffness G0 or E0.Considering that the static stiffness Estatic defined by Alpan equals approximately the unloading／reloading stiffness Eur in the HS small model，Alpan's chart can be used to guess a soils small⁃strain stiffness entirely based on its unloading／reloading stiffness Eur.Although Alpan suggests that the ratio E0／Eur can exceed 10 for very soft clays，the maximum ratio E0／Eur or G0／Gur permitted in the Hssmall model is limited to 20.
Figure 7.13 Relation between dynamic（Ed＝E0）and static soil stiffness（Es≈Eur）after Alpan（1970）
In the absence of test data，correlations are also available for the threshold shear strainγ0.7.Figure 7.14，for example，gives a correlation between the threshold shear strain and the Plasticity Index.Using the original Hardin⁃Drnevich relationship，the threshold shear strainγ0.7 might also be related to the models failure parameters.Applying the Mohr⁃Coulomb failure criterion in function yields
where
K0 is the earth pressure coefficient at rest and is the effective vertical stress（pressure negative）.
Figure 7.14 Influence of plasticity index（PI）on stiffness reduction after Vucetic＆Dobry（1991）