A hypoplastic constitutive model for clays

The implementation contains the basic hypoplastic model by Masin [3] and its modification to take into account the effects of structure degradation [5]. Both models may be used with and without intergranular strain concept [6].

For PLAXIS implementation of the model see www.soilmodels.info.

Required parameters:

• Parameter 1 - number of the constitutive model (in this case 0).
• Parameter 2 - bulk modulus of water $K_w$ for undrained analysis using the penalty approach. In drained or consolidation analysis $K_w$ should be set to 0.
• Parameter 3 - intergranular strain concept used or not (1/0).
• Parameters 4-8 - the intergranular strain concept parameters ($R$, $m_R$, $m_T$, $\beta_r$, $\chi$). Their values are ignored in the case when parameter No. 3 is set to 0.
• Parameters 9-13 - parameters of the basic hypoplastic model for clays ($\varphi_c$, $\lambda^*$, $\kappa^*$, $N$, $r$).
• Parameters 14-16 - parameters of the extension for structure ($k$, $A$, $s_f$). Basic model without structure is used, if these parameters are not specified.

Therefore, parameters should be specified as follows for the four possibile combinations:

1. No structure, no intergranular strains: 0, $K_w$, 0, 0, 0, 0, 0, 0, $\varphi_c$, $\lambda^*$, $\kappa^*$, $N$, $r$
2. No structure, intergranular strains: 0, $K_w$, 1, $R$, $m_R$, $m_T$, $\beta_r$, $\chi$, $\varphi_c$, $\lambda^*$, $\kappa^*$, $N$, $r$
3. Structure, no intergranular strains: 0, $K_w$, 0, 0, 0, 0, 0, 0, $\varphi_c$, $\lambda^*$, $\kappa^*$, $N$, $r$, $k$, $A$, $s_f$
4. Structure, intergranular strains: 0, $K_w$, 1, $R$, $m_R$, $m_T$, $\beta_r$, $\chi$, $\varphi_c$, $\lambda^*$, $\kappa^*$, $N$, $r$, $k$, $A$, $s_f$

Required state variables:

For all cases 11 state variables required:

• State v. 1 - void ratio $e$.
• State v. 2-7 - intergranular strain tensor $\boldsymbol\delta$ in Voigt notation ( $\delta_{11}$, $\delta_{22}$, $\delta_{33}$, $2\delta_{12}$, $2\delta_{13}$, $2\delta_{23}$).
• State v. 8 - sensitivity $s$.
• State v. 9 - Excess pore pressure $u$ for undrained analysis (compression positive sign convention).
• State v. 10 - Suggested size of the first time substep (for calculation controll).
• State v. 11 - Number of evaluation of the constitutive model in one global time step (for postprocessing only).

State v. 8 ($s$) automatically initialised to 1 if model without structure is used or if its initial value is 0. Intergranular strain $\boldsymbol\delta$ state variable (2-7) ignored, if model without intergranular strain extension is used (if parameter 3 set to 0).

Typical parameters:

Table 1: Typical parameters of the hypoplastic model for clays

	$\varphi_c$	$\lambda^*$	$\kappa^*$	$N$	$r$	$k$	$A$	$s_f$
London clay [3]	22.6$^\circ$	0.11	0.016	1.375	0.4	-	-	-
Pisa clay [5]	21.9$^\circ$	0.14	0.0075	1.56	0.3	0.4	0.1	1

Table 2: Typical parameters of the intergranular strain concept

	$R$	$m_R$	$m_T$	$\beta_r$	$\chi$
London clay [3]	1.e-4	4.5	4.5	0.2	6

Example input file and initialisation file for the FE program Abaqus$^{TM}$.
Example input file for the FE program Tochnog Professional.