Hypoplastic model for clays

Parameters are specified in the PLAXIS input in the following order:

Parameters:

• Parameter 1 - critical state friction angle $\varphi_c$
• Parameter 2 - $p_t$ - shift of the mean stress due to cohesion. The effective stress $\boldsymbol{\sigma}$ used in the model formulation is replaced by $\boldsymbol{\sigma}-\boldsymbol{1}p_t$. Non-zero value of $p_t$ is needed to overcome problems with stress-free state. If $p_t=0$, it will be replaced by a default value of 1 kPa. Any other value can be input by user (for basic hypoplasticity, set $p_t$ to very low number, e.g. $p_t=1.e-5$).
• Parameters 3-5 - parameters of the basic hypoplastic model for clays $\lambda^*$, $\kappa^*$, $N$.
• Parameters 6 - Parameters of the basic hypoplastic model for clays $\nu_{pp}$ (in place of the parameter $r$ of the Mašín (2005) [3] model, calibrated using the same procedure).
• Parameters 7 - $\alpha_G$, ratio of horisontal and vertical shear moduli.
• Parameters 8-10 - parameters of the model for clays with meta-stable structure ($k$, $A$ and $s_f$).
• Parameters 11-13 - Intergranular strain concept parameters $R$, $\beta_r$, $\chi$.
• Parameters 14-15 - Very small strain shear stiffness parameters $A_g$ and $n_g$. If $A_g=0$ the intergranular strain concept is switched off and the problem is simulated using the basic hypoplastic model.
• Parameters 16 - Intergranular strain concept parameter $m_{rat}$.
• Parameter 17 - bulk modulus of water $K_w$ for undrained analysis using the penalty approach with user-defined value of $K_w$. In drained analysis, consolidation analysis, and undrained analysis using PLAXIS option undrained $K_w$ should be set to 0.
• Parameter 18 - Specifies vertical direction (set to 2 in PLAXIS2D, 3 in PLAXIS3D, 2 in SoilTest independently whether run from PX2D or PX3D).
• Parameter 19 - Parameter $\alpha_E$ (ratio of horizontal and vertical Young moduli). If set to zero, than calculated automatically from empirical formulation $\alpha_E=\alpha_G^{(1/0.8)}$.
• Parameter 20 - Parameter $\alpha_\nu$ (ratio of horizontal and vertical Poisson ratios). If set to zero, than calculated automatically from empirical formulation $\alpha_\nu=\alpha_G$.
• Parameter 21 - Parameter $\alpha_f$. Additional control of non-linearity inside state boundary surface. If set to zero, default value calculated from model parameters is used.
• Parameter 22 - initial void ratio $e$ or overconsolidation ratio $OCR$. If $Par(19) < 10$, then $e=Par(19)$. If $Par(19) > 10$, then $OCR=Par(19) - 10$.
• Parameters 23-28 - initial values of the intergranular strain tensor $\boldsymbol\delta$ in Voigt notation ( $\delta_{11}$, $\delta_{22}$, $\delta_{33}$, $2\delta_{12}$, $2\delta_{13}$, $2\delta_{23}$).
• Parameter 29 - initial value of sensitivity (model for clays with meta-stable structure). If $s=0$, the basic model is used.

State variables:

The routine uses 16 state variables:

• State v. 1-6 - 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. 7 - void ratio $e$.
• State v. 8 - Excess pore pressure $u$ for undrained analysis using user-defined value of $K_w$. In undrained analysis using PLAXIS option undrained this variable is equal to 0 end excess pore pressure may be found in standard PLAXIS menu.
• State v. 9 - Effective mean stress.
• State v. 10 - Number of evaluation of the constitutive model in one global time step (for postprocessing only).
• State v. 11 - Mobilised friction angle $\varphi_{mob}$ in degrees (for postprocessing only).
• State v. 12 - Normalised length $\rho$ of the intergranular strain tensor $\boldsymbol\delta$ (for postprocessing only).
• State v. 13 - Suggested size of the first time substep (for calculation controll).
• State v. 14 - sensitivity $s$ (for model with meta-stable structure).
• State v. 15 - overconsolidation ratio, defined as $OCR=p_e/p$.
• State v. 16 - 0 means model interation goes without problem. 1 means state boundary surface cut-off has been activated.

The hypoplastic model for granular materials is implemented via user defined subroutine usermod. To use the model in PLAXIS, copy the files UDSM_HPS.dll and UDSM_HPS64.dll into the PLAXIS installation directory. Then, select "user-defined model" from the Material model combo box in the General tab sheet (Fig. 4.1). After selecting the user-defined model, correct user-defined dynamic library (typically UDSM_HPS.dll) needs to be selected in the "Available DLL's" combo box under "Parameters" tab sheet. In the "Models in DLL" combo box, model with ID 2 (Hypoplas. - clay) must be selected. The parameters can then be input into the parameter table (Fig. 4.3).

Figure: Selecting clay hypoplasticity model in the "Parameters" tab sheet.

Parameters of the clay hypoplastic model for different soils have been evaluated by Mašín and co-workers. They are given in Table 4.3. Parameters of the model for clays with meta-stable structure are in Tab. 4.4 and typical parameters of the intergranular strain concept for fine-grained soils are in Tab. 4.5.

Table 4.3: Typical parameters of the hypoplastic model for clays.

	$\varphi_c$	$\lambda^*$	$\kappa^*$	$N$	$\nu_{pp}$	$\alpha_G$
Brno clay	22$^\circ$	0.128	0.015	1.51	0.33	1.35
London clay	21.9$^\circ$	0.095	0.015	1.19	0.1	2
Kaolin	27.5$^\circ$	0.065	0.01	0.918	0.35	-
Dortmund clay	27.9$^\circ$	0.057	0.008	0.749	0.38	-
Weald clay	24$^\circ$	0.059	0.018	0.8	0.3	-
Koper silt	33$^\circ$	0.103	0.015	1.31	0.28	-
Fujinomori clay	34$^\circ$	0.045	0.011	0.887	0.36	-
Pisa clay	21.9$^\circ$	0.14	0.01	1.56	0.31	-
Beaucaire clay	33$^\circ$	0.06	0.01	0.85	0.21	-
Trmice clay	18.7$^\circ$	0.09	0.01	1.09	0.09	-

For estimation of $\alpha_G$, see database from [7], here for free download in the form of PDF preprint.

Table 4.4: Typical parameters of the model for clays with meta-stable structure (from [4])

	$k$	$A$	$s_f$
Pisa clay	0.4	0.1	1
Bothkennar clay	0.35	0.5	1

Table 4.5: Typical parameters of the intergranular strain concept for clays

	$R$	$\beta_r$	$\chi$	$A_g$	$n_g$	$m_{rat}$
London clay (data Gasparre)	5.e-5	0.08	0.9	270	1	0.5
Brno clay (nat.)	1e-4	0.2	0.8	5300	0.5	0.5