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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 hs, die void ratios ei0, ec0 and ed0 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 $\varphi_c$ [$^\circ$] hs [MPa] n ed0 ec0 ei0 $\alpha$ $\beta$
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 up previous
Next: References Up: Equations of the Hypoplastic Previous: 4. Version WU WEI
Wolfgang Fellin
1999-10-01