7.3.1 Hyperbolic relationship for standard drained...

7.3.1 Hyperbolic relationship for standard drained triaxial test

A basic idea for the formulation of the Hardening Soil model is the hyperbolic relationship between the vertical strain，ε1，and the deviatoric stress，q in primary triaxial loading.

Here standard drained triaxial tests tend to yield curves that can be described by

where

qa is the asymptotic value of the shear strength and Ei the initial stiffness.

Ei is related to E50 by

This relationship is plotted in Figure 7.5.The parameter E50 is the confining stress dependent stiffness modulus for primary loading and is given by the equation：

where

Eis a reference stiffness modulus corresponding to the reference confining pressure pref.

Figure 7.5 Hyperbolic stress⁃strain relation in primary loading for a standard drained triaxial test

A default setting p＝100 stress units is always used.The actual stiffness depends on the minor principal stress，σ′3，which is the confining pressure in a triaxial test.Please note thatσ′3 is negative for compression.The amount of stress dependency is given by the power m.In order to simulate a logarithmic compression behavior，as observed for soft clays，the power should be taken equal to 1.0.Janbu（1963）reported values of m around 0.5 for Norwegian sands and silts，whilst von Soos（1990）reported various different values in the range 0.5＜m＜1.0.

The ultimate deviatoric stress，qf，and the quantity qa are defined as

Again it is remarkable thatσ′3 is usually negative.The above relationship for qf is derived from the Mohr⁃Coulomb failure criterion，which involves the strength parameters c andφ.As soon as q＝qf，the failure criterion is satisfied and perfectly plastic yielding occurs as described by the Mohr⁃Coulomb model.

The ratio between qf and qa is given by the failure ratio Rf，which should obviously be smaller than or equal to 1.For unloading and reloading stress paths，another stress⁃dependent stiffness modulus is used：

where Eis the reference of Young's modulus for unloading and reloading，corresponding to the reference pressure pref.In many practical cases，it is appropriate to set Eequal to 3 E.