Next: References Up: Equations of the Hypoplastic Previous: 4. Version WU WEI

5. Version VON WOLFFERSDORF

This version was propesed in [4].

$\begin{eqnarray*}\kTb = f_b f_e \frac{1}{\tr (\hTb \cdot \hTb)} \Big[ F^2 \Db ... ...Tb^{^{\scriptstyle *}}) \vert\vert \Db \vert\vert \Big] \quad , \end{eqnarray*}$

with

$\begin{eqnarray*}\hTb := \frac{\Tb}{\tr \Tb} \quad , \quad \hTb^{^{\scriptstyle *}}:= \hTb - \frac{1}{3} \Ib \quad , \end{eqnarray*}$

and

$\begin{eqnarray*}a := \frac{ \sqrt{3} ( 3 - \sin \varphi_c ) } { 2 \sqrt{2} \sin \varphi_c } \end{eqnarray*}$

The factor F is

$\begin{eqnarray*}F := \sqrt{ \frac{1}{8} \tan^2 \psi + \frac{ 2 - \tan^2 \psi} ... ...} \tan \psi \cos 3 \theta} } - \frac{1}{2 \sqrt{2}} \tan \psi \end{eqnarray*}$

with

$\begin{eqnarray*}\tan \psi := \sqrt{3} \vert\vert\hTb^{^{\scriptstyle *}}\vert\vert \end{eqnarray*}$

and

$\begin{eqnarray*}\cos 3 \theta := - \sqrt{6} \frac{\tr (\hTb^{^{\scriptstyle *... ...^{^{\scriptstyle *}}\cdot \hTb^{^{\scriptstyle *}}) \Big]^{3/2}} \end{eqnarray*}$

The other scalar factors are:

$\begin{eqnarray*}f_d &:=& \left( \frac{e-e_d}{e_c-e_d} \right)^\alpha \ f_e &... ...\frac{e_{i0}-e_{d0}}{e_{c0}-e_{d0}} \right)^\alpha \Bigg]^{-1} \end{eqnarray*}$

with

$\begin{eqnarray*}p_s := -\tr \Tb / 3 \end{eqnarray*}$

The void ratios must fullfill

$\begin{eqnarray*}\frac{e_i}{e_{i0}}= \frac{e_c}{e_{c0}} = \frac{e_d}{e_{d0}} = \exp \left[ - \left( \frac{3 p_s}{h_s} \right)^n \right] \quad . \end{eqnarray*}$

This hypoplastic law has 8 parameters: the critical friktion angle $\varphi_c$ , die granular hardness h_s, die void ratios e_i0, e_c0 and e_d0 the exponents n and $\beta$ . They can be easily determined from simple index and/or element tests [2].

The parameters for different materials are [2]:

Material	$\varphi_c$ [ $^\circ$ ]	h_s [MPa]	n	e_d0	e_c0	e_i0	$\alpha$	$\beta$
Toyoura sand	30	2600	0,27	0,61	0,98	1,10	0,18	1,00
Hochstetten sand	33	1500	0,28	0,55	0,95	1,05	0,25	1,50
Schlabendorf sand	33	1600	0,19	0,44	0,85	1,00	0,25	1,00
Hostun sand	31	1000	0,29	0,61	0,91	1,09	0,13	2,00
Karlsruhe sand	30	5800	0,28	0,53	0,84	1,00	0,13	1,05
Zbraslav sand	31	5700	0,25	0,52	0,82	0,95	0,23	1,00
Ottawa sand	30	4900	0,29	0,49	0,76	0,88	0,10	1,00
Ticino sand	31	5800	0,31	0,60	0,93	1,05	0,20	1,00
SLB sand	30	8900	0,33	0,49	0,79	0,90	0,14	1,00
Hochstetten gravel	36	32000	0,18	0,26	0,45	0,50	0,10	1,80
plastics	32	110	0,33	0,53	0,73	0,80	0,08	1,00
wheat	39	20	0,37	0,57	0,84	0,95	0,02	1,00

Next: References Up: Equations of the Hypoplastic Previous: 4. Version WU WEI

Wolfgang Fellin
1999-10-01