7.3.2 Parameters of the hardening soil model
Some parameters of the present hardening model coincide with those of the non⁃hardening Mohr⁃Coulomb model.These are the failure parameters c，φandψ（see in section 8.2）.
Basic parameters for soil stiffness：
E
：Secant stiffness in standard drained triaxial test ［kN／m2］
E
：Tangent stiffness for primary oedometer loading ［kN／m2］
E
：Unloading／reloading stiffness（default E
＝3 E
） ［kN／m2］
m：Power for stress⁃level dependency of stiffness ［—］
Advanced parameters（it is advised to use the default setting）：
vur：Poissons ratio for unloading⁃reloading（default vur＝0.2） ［—］
pref：Reference stress for stiffnesses（default pref＝100 kN／m2） ［kN／m2］
K
：K0⁃value for normal consolidation（default K
＝1－sinφ） ［—］
Rf：Failure ratio qf／qa（default Rf＝0.9）.［—］
σtension：Tensile strength（default tension＝0 stress units） ［kN／m2］
cinc：As in Mohr⁃Coulomb model（default cinc＝0） ［kN／m3］
Instead of entering the basic parameters for soil stiffness，alternative parameters can be entered.These parameters are listed below：
Cc：Compression index ［—］
Cs：Swelling index or reloading index ［—］
einit：Initial void ratio ［—］
（1）Stiffness moduli E
，E
，E
and power m
The advantage of the Hardening Soil model over the Mohr⁃Coulomb model is not only the use of a hyperbolic stress⁃strain curve instead of a bi⁃linear curve，but also the control of stress level dependency.When using the Mohr⁃Coulomb model，the user has to select a fixed value of Young's modulus whereas for real soils，this stiffness depends on the stress level.It is therefore necessary to estimate the stress levels within the soil and use these to obtain suitable values of stiffness.With the Hardening Soil model，however，this cumbersome selection of input parameters is not required.
Instead，a stiffness modulus E
is defined for a reference minor principle effective stress of－
＝pref.This is the secant stiffness at 50% of the maximum deviatoric stress，at a cell pressure equal to the reference stress pref（Figure 7.6）.As a default value，the program uses pref＝100 kN／m2.
Figure 7.6 Definition of Esand Ew for drained triaxial test results
The elastoplastic Hardening Soil model does not involve a fixed relationship between the（drained）triaxial stiffness E50 and the oedometer stiffness Eoed for one⁃dimensional compression.Instead，these stiffnesses can be inputted independently.It is now important to define the oedometer stiffness.Here we use the equation to define oedometer stiffness：
where Eoed is a tangent stiffness modulus obtained from an oedometer test，as indicated in Figure 7.7.Hence，E
is a tangent stiffness at a vertical stress of－σ′1＝－σ′3／K
＝pref.Note that we basically use
rather than－
and that we consider primary loading.
When undrained behaviour is considered in the Hardening Soil model，the Drainage type should preferably be set to Undrained（A）.Alternatively，Undrained（B）can be used in case the effective strength properties are not known or the undrained shear strength is not properly captured by using Undrained（A）.However，it should be noted that the material loses its stress⁃dependency of stiffness in that case.Undrained（C）is not possible since the model is essentially formulated as an effective stress model.
Figure 7.7 Definition of E
in oedometer test results
（2）Alternative stiffness parameters
When soft soils are considered，the stiffness parameters can be calculated from the compression index，swelling index and the initial void ratio.The relationship between these parameters and the compression index，Cc is given by
The relationship between the E
and the swelling index，Cs is given by
Regardless the previous value of E50，a new value will be automatically assigned according to
Although for Soft soils，E
could be as high as 2E
，this high value could lead to a limitation in the modeling；therefore a lower value is used.Changing the value of Cs will change the stiffness parameter Eur.
（3）Dilatancy cut⁃off
After extensive shearing，dilating materials arrive in a state of critical density where dilatancy has come to an end，as indicated in Figure 7.8.This phenomenon of soil behavior can be included in the Hardening Soil model by means of a dilatancy cut⁃off.In order to specify this behavior，the initial void ratio，einit and the maximum void ratio，emax of the material must be entered as general parameters.As soon as the volume change results in a state of maximum void，the mobilized dilatancy angle，ψm is automatically set back to zero，as indicated in Figure 7.8.
For e＜emax：
where
For e≥emax： ψm＝0
The void ratio is related to the volumetric strain，εv，by the relationship
where an increment ofεv，is positive for dilatancy.
Figure 7.8 Resulting strain curve for a standard drained triaxial test when including dilatancy cut⁃off
The initial void ratio，einit，is the in⁃situ void ratio of the soil body.The maximum void ratio is the void ratio of the material in a state of critical void（critical state）.As soon as the maximum void ratio is reached，the dilatancy angle is set to zero.The minimum void ratio，emin of a soil can also be input，but this general soil parameter is not used within the context of the Hardening Soil model.