Hypoplastic model for granular materials

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

• 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 3-9 - parameters of the basic hypoplastic model for granular materials $h_s$, $n$, $e_{d0}$, $e_{c0}$, $e_{i0}$, $\alpha$, $\beta$.
• Parameters 10-14 - the intergranular strain concept parameters ($m_R$, $m_T$, $R$, $\beta_r$, $\chi$). If $m_R=0$ the intergranular strain concept is switched off and the problem is simulated using the basic hypoplastic model.
• Parameter 15 - not used.
• Parameter 16 - initial void ratio corresponding to the zero mean stress $e_0$ or initial void ratio $e$. If $Par(16) < 10$, then $e$ is calculated from the mean stress $p$ and from $e_0=Par(16)$ using Bauer [1] formula. If $Par(16) > 10$, then $e=Par(16)-10$.
• Parameters 17-22 - initial values of the intergranular strain tensor $\boldsymbol\delta$ in Voigt notation ( $\delta_{11}$, $\delta_{22}$, $\delta_{33}$, $2\delta_{12}$, $2\delta_{13}$, $2\delta_{23}$).

State variables:

The routine uses 14 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 - not used.
• 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 - free.

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).

Figure: Selecting user defined model in the Material model combo box.

(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 1 (Hypoplas. - sand) must be selected. The parameters can then be input into the parameter table (Fig. 4.2).

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

Parameters of the sand hypoplastic model for different soils have been evaluated by Herle and Gudehus [2]. They are given in Table 4.1. Parameters of the intergranular strain concept for granular materials are in Tab. 4.2.

Table 4.1: Typical parameters of the hypoplastic model for granular materials (Herle and Gudehus [2])

	$\varphi_c$	$h_s$	$n$	$e_{d0}$	$e_{c0}$	$e_{i0}$	$\alpha$	$\beta$
Hochstetten gravel	36$^\circ$	32 $\times 10^6$ kPa	0.18	0.26	0.45	0.5	0.1	1.9
Hochstetten sand	33$^\circ$	1.5 $\times 10^6$ kPa	0.28	0.55	0.95	1.05	0.25	1.5
Hostun sand	31$^\circ$	1.0 $\times 10^6$ kPa	0.29	0.61	0.96	1.09	0.13	2
Karlsruhe sand	30$^\circ$	5.8 $\times 10^6$ kPa	0.28	0.53	0.84	1	0.13	1
Lausitz sand	33$^\circ$	1.6 $\times 10^6$ kPa	0.19	0.44	0.85	1	0.25	1
Toyoura sand	30$^\circ$	2.6 $\times 10^6$ kPa	0.27	0.61	0.98	1.1	0.18	1.1

Table 4.2: Parameters of the intergranular strain concept for sandy soils (Niemunis and Herle [8])

	$R$	$m_R$	$m_T$	$\beta_r$	$\chi$
Hochstetten sand	1.e-4	5.0	2.0	0.5	6