von Wolffersdorff hypoplastic model

von Wolffersdorff hypoplastic model for granular materials [10]. The model may be used with and without intergranular strain concept [6].

Required parameters:

• Parameter 1 - number of the constitutive model (in this case 3).
• 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 9-16 - parameters of the hypoplastic model ($\varphi_c$, $h_s$, $n$, $e_{d0}$, $e_{c0}$, $e_{i0}$, $\alpha$, $\beta$).
• Parameter 17 - optional parameter. If present and if set to 1, void ratio $e$ is initialised from the Bauer exponential formula to be mean-stress dependent (i.e., constant relative density assumed throughout the soil massive).

Therefore, parameters should be specified as follows for the two possibilities:

1. No intergranular strains: 3, $K_w$, 0, 0, 0, 0, 0, 0, $\varphi_c$, $h_s$, $n$, $e_{d0}$, $e_{c0}$, $e_{i0}$, $\alpha$, $\beta$, (0/1)
2. Intergranular strains: 3, $K_w$, 1, $R$, $m_R$, $m_T$, $\beta_r$, $\chi$, $\varphi_c$, $h_s$, $n$, $e_{d0}$, $e_{c0}$, $e_{i0}$, $\alpha$, $\beta$, (0/1)

Required state variables:

For all cases 10 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 - Excess pore pressure $u$ for undrained analysis (compression positive sign convention).
• State v. 9 - Suggested size of the first time substep (for calculation controll).
• State v. 10 - Number of evaluation of the constitutive model in one global time step (for postprocessing only).

$e$ initialised from the Bauer [1] exponential formula if parameter 17 present and set to 1(i.e., constant relative density assumed throughout the soil massive). The void ratio specified by the user corresponds to $p=0$. 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 5: Typical parameters of the von Wolffersdorff hypoplastic model

	$\varphi_c$	$h_s$	$n$	$e_{d0}$	$e_{c0}$	$e_{i0}$	$\alpha$	$\beta$
Hochstetten sand [2]	33$^\circ$	1.5e6 kPa	0.28	0.55	0.95	1.05	0.25	1.5

Table 6: Typical parameters of the intergranular strain concept

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

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