17.1.3 Mohr-Coulomb

The yield condition of Mohr-Coulomb [Fig.17.5a] is an extension of the Tresca yield condition to a pressure dependent behavior. The formulation of the yield function can be expressed in the principal stress space ( $\sigma_{{1}}^{}$ $\geq$ $\sigma_{{2}}^{}$ $\geq$ $\sigma_{{3}}^{}$ ) as

$$\boldsymbol\sigma \kappa {\tfrac{{1}}{{2}}} \sigma_{{1}}^{} \sigma_{{3}}^{} {\tfrac{{1}}{{2}}} \sigma_{{1}}^{} \sigma_{{3}}^{} \phi \kappa \bar{{c}} \kappa \phi_{{0}}^{}$$ (17.42)

with $\bar{{c}}$($\kappa$) the cohesion as a function of the internal state variable $\kappa$ , and $\phi$ the angle of internal friction which is also a function of the internal state variable. See also Vermeer & De Borst [109].

Figure 17.5: Mohr-Coulomb and Drucker-Prager yield condition

The initial angle of internal friction is given by $\phi_{{0}}^{}$ . The flow rule is given by a general non-associated flow rule g $\neq$ f , but with the plastic potential given by

$$\boldsymbol\sigma \kappa {\tfrac{{1}}{{2}}} \sigma_{{1}}^{} \sigma_{{3}}^{} {\tfrac{{1}}{{2}}} \sigma_{{1}}^{} \sigma_{{3}}^{} \psi \kappa$$ (17.43)

which results for the plastic strain rate vector

$$\dot{{\boldsymbol{\varepsilon}}}^{{\mathrm{p}}}_{} \dot{{\lambda}} \left\{\vphantom{ \negthickspace \begin{array}{c} \tfrac{1}{2}( 1... ...x] 0 \\ [1ex] -\tfrac{1}{2}( 1- \sin\psi ) \end{array} \negthickspace }\right. \begin{array}{c} \tfrac{1}{2}( 1+ \sin\psi ) \\ [1ex] 0 \\ [1ex] -\tfrac{1}{2}( 1- \sin\psi ) \end{array} \left.\vphantom{ \negthickspace \begin{array}{c} \tfrac{1}{2}( 1+... ...] 0 \\ [1ex] -\tfrac{1}{2}( 1- \sin\psi ) \end{array} \negthickspace }\right\}$$ (17.44)

17.1.3.1 Hardening

The relation between the internal state variable $\kappa$ and the plastic process is given by the hardening hypothesis. For the Mohr-Coulomb yield condition we consider only the strain hardening hypothesis.

Strain hardening.

In the case of strain hardening the relation is given in the principal space by

$$\dot{{\kappa}} \sqrt{{ \tfrac{2}{3}\left( \dot{\varepsilon }_{1}^{\mathrm{p}} \d... ...t{\varepsilon }_{3}^{\mathrm{p}} \dot{\varepsilon }_{3}^{\mathrm{p}} \right) }}$$ (17.45)

which can be elaborated to

$$\dot{{\kappa}} \dot{{\lambda}} \sqrt{{ \tfrac{1}{3}\left( 1 + \sin^{2}\psi \right) }}$$ (17.46)

Relation $\bar{{c}}$ -$\kappa$ .

The translation of uniaxial experimental data to the equivalent cohesion-internal state variable, the $\bar{{c}}$ -$\kappa$ relation, depends on the hardening hypothesis. In the following example we will give the derivation for a cohesion hardening material with constant friction and dilatation angle, i.e., $\phi$($\kappa$) = $\phi_{{0}}^{}$ and $\psi$($\kappa$) = $\psi_{{0}}^{}$ , and a strain hardening hypothesis.

Figure 17.6: Derivation of hardening diagram for Mohr-Coulomb

Consider the uniaxial stress-strain diagram $\sigma_{{3}}^{}$ - $\varepsilon_{{3}}^{}$ of Figure 17.6a. The plastic strain $\varepsilon_{{3}}^{{\mathrm{p}}}$ is assumed to be given by $\varepsilon_{{3}}^{}$ - $\varepsilon_{{3}}^{{\mathrm{e}}}$ . Figure 17.6b shows the uniaxial stress-plastic strain diagram. For uniaxial stressing, ($\sigma_{{1}}^{}$,$\sigma_{{2}}^{}$,$\sigma_{{3}}^{}$) = (0, 0,$\sigma_{{3}}^{}$) , plastic flow occurs at a vertex of the yield surface. Symmetry conditions dictate that the two possible yield directions contribute equally to the plastic strain rate vector

$$\dot{{\boldsymbol{\varepsilon}}}^{{\mathrm{p}}}_{} \left\{\vphantom{ \negthickspace \begin{array}{c} \dot{\varepsilo... ...amount] \dot{\varepsilon }_{3}^{\mathrm{p}} \end{array} \negthickspace }\right. \begin{array}{c} \dot{\varepsilon }_{1}^{\mathrm{p}} \\ [\medski... ...mathrm{p}} \\ [\medskipamount] \dot{\varepsilon }_{3}^{\mathrm{p}} \end{array} \left.\vphantom{ \negthickspace \begin{array}{c} \dot{\varepsilon... ...mount] \dot{\varepsilon }_{3}^{\mathrm{p}} \end{array} \negthickspace }\right\} \dot{{\lambda}} \left\{\vphantom{ \negthickspace \begin{array}{c} \tfrac{1}{4}( 1... ...amount] - \tfrac{1}{2}( 1 - \sin \psi_{0} ) \end{array} \negthickspace }\right. \begin{array}{c} \tfrac{1}{4}( 1 + \sin \psi_{0} ) \\ [\medskipa... ...\psi_{0} ) \\ [\medskipamount] - \tfrac{1}{2}( 1 - \sin \psi_{0} ) \end{array} \left.\vphantom{ \negthickspace \begin{array}{c} \tfrac{1}{4}( 1 ... ...mount] - \tfrac{1}{2}( 1 - \sin \psi_{0} ) \end{array} \negthickspace }\right\}$$ (17.47)

With the relation derived previously, we find for the relation between the uniaxial plastic strain and the internal state variable for a strain hardening hypothesis

$$\dot{{\kappa}} {\frac{{ \sqrt{ 1 + \sin^{2} \psi_{0} - \tfrac{2}{3}\sin\psi_{0} } }}{{ 1 - \sin\psi_{0} }}} \dot{{\varepsilon }}_{{3}}^{{\mathrm{p}}}$$ (17.48)

The relation between the uniaxial stress $\sigma_{{3}}^{}$ = - f_c and the equivalent cohesion $\bar{{c}}$ is given by

$$\bar{{c}} {\frac{{ 1-\sin\phi_{0} }}{{ 2 \cos\phi_{0} }}}$$ (17.49)

Figure 17.6 illustrates the procedure for $\phi_{{0}}^{}$ = $\psi_{{0}}^{}$ = 30° .