7.3 Hardening Soil Model（Isotropic Hardening）
In contrast to an elastic perfectly⁃plastic model，the yield surface of a hardening plasticity model is not fixed in principal stress space，but it can expand due to plastic straining.
Distinction can be made between two main types of hardening，namely shear hardening and compression hardening.Shear hardening is used to model irreversible strains due to primary deviatoric loading.Compression hardening is used to model irreversible plastic strains due to primary compression in oedometer loading and isotropic loading.Both types of hardening are contained in the present model.
The Hardening Soil model is an advanced model for simulating the behaviour of different types of soil，both soft soils and stiff soils（Schanz，1998）.When subjected to primary deviatoric loading，soil shows a decreasing stiffness and simultaneously irreversible plastic strains develop.In the special case of a drained triaxial test，the observed relationship between the axial strain and the deviatoric stress can be well approximated by a hyperbola.Such a relationship was first formulated by Kondner（1963）and later used in the well⁃known hyperbolic model（Duncan and Chang，1970）.The Hardening Soil model，however，supersedes the hyperbolic model by far：Firstly by using the theory of plasticity rather than the theory of elasticity，secondly by including soil dilatancy and thirdly by introducing a yield cap.Some basic characteristics of the model are listed below：
•Stress dependent stiffness according to a power law Input parameter m
•Plastic straining due to primary deviatoric loading Input parameter E
•Plastic straining due to primary compression Input parameter E
•Elastic unloading／reloading Input parameters E
，vur
•Failure according to the Mohr⁃Coulomb failure criterion Parameters c，φandψ
A basic feature of the present Hardening Soil model is the stress dependency of soil stiffness.For oedometer conditions of stress and strain，the model implies example the relationship Eoed＝
In the special case of soft soils，it is realistic to use m＝1.In such situations there is also a simple relationship between the modified compression indexλ∗as used in models for soft soil and the oedometer loading modulus.
where pref is a reference pressure.Here we consider a tangent oedometer modulus at a particular reference pressure pref.Hence，the primary loading stiffness relates to the modified compression indexλ∗or to the standard Cam⁃clay compression indexλ.
Similarly，the unloading⁃reloading modulus relates to the modified swelling indexκ∗or to the standard Cam⁃clay swelling indexκ.There is the approximate relationship：
This relationship applies in combination with the input value m＝1.