## 3 HASP Constitutive Soil Model  
The relations of the HASP model are established on the following assumptions: soil is isotropic, plastic strains develop from the very beginning of loading, and the hardening parameter depends on the increments of the plastic volumetric and shear strains. The HASP model is formulated in the triaxial \(p^{\prime}-q\) plane. It uses the MCC surface as the bounding surface, and its size is defined by the value of maximum mean effective stress \(\overline{p}^{\prime}_{0}\) (Figure 1a). The bounding surface has all the properties of the MCC surface. The volume decreases and the surface expands for stress ratios below the critical state line (CSL), while the volume increases and the surface shrinks for stress ratios above the CSL. The yield surface expands until peak strength is reached at the stress ratio \(\eta=M_{f}\), after which it shrinks (softening) until the critical state is reached. The HASP model is capable of predicting the saturated clay response under monotonic drained and undrained loading sufficiently accurately, as demonstrated in [27; 29; 30]. The point \(A\) (\(p^{\prime},q\)), representing the current stress state, is on the inner yield surface, whose size is defined by the value of the mean effective stress \(p^{\prime}_{0}\) (Figure 1a). There is no pure elastic domain and stress point A is always on the yield surface. Radial mapping implies that point \(A\) corresponds to the conjugate point \(\overline{A}\)(\(\overline{p}^{\prime},\overline{q}\)), so that the following is fulfilled:  
\[\eta=\frac{q}{p^{\prime}}=\frac{\overline{q}}{\overline{p}^{\prime}}. \tag{1}\]  
In order to use the HASP model to provide reliable simulations of practical boundary value problems, the model relations must be defined in the general stress state. This generalization can be efficiently performed by expressing the stress invariant \(q\) in terms of the second invariant of the deviatoric stress tensor \(J_{2D}\):  
\[q=\sqrt{3J_{2D}}. \tag{2}\]The originally proposed yield and bounding surfaces in the deviatoric plane are circles [27], which is consistent with the original formulation of the MCC model [7]. Because a circle does not represent appropriately the failure conditions for soils, a modification is introduced by replacing the constant critical state strength parameter \(M\) with proper Lode angle dependent criterion, to define the shape of the yield and bounding surfaces in the deviatoric plane. The yield surface is defined by:  
\[F\big{(}p^{\prime},q,\theta\big{)}=\left(\frac{q}{M(\theta)}\right)^{2}+p^{ \prime}\big{(}p^{\prime}-p^{\prime}_{0}\big{)}=0. \tag{3}\]  
While the bounding surface is defined by:  
\[\overline{F}\big{(}\overline{p}^{\prime},\overline{q},\theta\big{)}=\left( \frac{\overline{q}}{M(\theta)}\right)^{2}+\overline{p}^{\prime}\big{(} \overline{p}^{\prime}-\overline{p}^{\prime}_{0}\big{)}, \tag{4}\]  
where \(M(\theta)\) is the gradient of the critical state line in \(q-p^{\prime}\) plane for the current value of \(\theta\). Many failure surfaces, which are continuous and agree well with the experimental results in the deviatoric plane [31], have been suggested. A failure criterion dependent on the Lode angle is introduced in the HASP model by means of the relation proposed in [32]:  
\[M(\theta)=X\cdot(1+Y\cdot sin3\theta)^{Z}, \tag{5}\]  
where \(X\), \(Y\) and \(Z\) are constants, determined in such a way that the appropriate shapes of the yield and bounding surfaces are achieved. The use of this shape takes into account the effect of the intermediate principal stress by allowing the simulation of the shear strength variation in the deviatoric plane, and the fact that strength in triaxial compression is different than in triaxial extension [33]. The Lode angle is expressed as [24]:  
\[\theta=-\frac{1}{3}\cdot sin^{-1}\bigg{(}\frac{27}{2}\frac{J_{3D}}{q^{3}}\bigg{)}, \tag{6}\]  
where \(J_{3D}\) is the third invariant of the deviatoric stress tensor.