17.1.4 Drucker-Prager

The yield condition of Drucker-Prager is a smooth approximation of the Mohr-Coulomb yield surface, which is a conical surface in the principal stress space [Fig.17.5b]. The formulation is given by

$$\boldsymbol\sigma \kappa \sqrt{{ \tfrac{1}{2}\boldsymbol{\sigma}^{\mathrm{\scriptscriptstyle{T}}}\mathbf{P} \boldsymbol{\sigma}}} \alpha_{{\mathrm{f}}}^{} \boldsymbol\pi \boldsymbol\sigma \beta \bar{{c}} \kappa$$ (17.50)

with $\bar{{c}}$($\kappa$) the cohesion as a function of the internal state variable $\kappa$ . The projection matrix is equal to the projection matrix of the Von Mises yield condition defined in Equation (17.30). The projection vector $\boldsymbol\pi$ is given by

$$\boldsymbol\pi \left\{\vphantom{ \negthickspace \begin{array}{c} 1 \\ 1 \\ 1 \\ 0 \\ 0 \\ 0 \end{array} \negthickspace }\right. \begin{array}{c} 1 \\ 1 \\ 1 \\ 0 \\ 0 \\ 0 \end{array} \left.\vphantom{ \negthickspace \begin{array}{c} 1 \\ 1 \\ 1 \\ 0 \\ 0 \\ 0 \end{array} \negthickspace }\right\}$$ (17.51)

The scalar quantities $\alpha_{{\mathrm{f}}}^{}$ and $\beta$ are given by

$$\alpha_{{\mathrm{f}}}^{} {\frac{{2 \sin\phi(\kappa)}}{{3-\sin\phi(\kappa)}}} \beta {\frac{{6 \cos\phi_{0}}}{{3-\sin\phi_{0}}}}$$ (17.52)

The angle of internal friction $\phi$ is also a function of the internal state variable. 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 , with the plastic potential given by

$$\boldsymbol\sigma \kappa \sqrt{{ \tfrac{1}{2}\boldsymbol{\sigma}^{\mathrm{\scriptscriptstyle{T}}}\mathbf{P} \boldsymbol{\sigma}}} \alpha_{{\mathrm{g}}}^{} \boldsymbol\pi \boldsymbol\sigma$$ (17.53)

with the scalar $\alpha_{{\mathrm{g}}}^{}$ defined by the dilatancy angle $\psi$

$$\alpha_{{\mathrm{g}}}^{} {\frac{{2 \sin\psi(\kappa)}}{{3-\sin\psi(\kappa)}}}$$ (17.54)

Which results for the plastic strain rate vector in

$$\dot{{\boldsymbol{\varepsilon}}}^{{\mathrm{p}}}_{} \dot{{\lambda}} \left\{\vphantom{ \frac{ \mathbf{P} \boldsymbol{\sigma}}{ 2 \Psi } + \alpha_{\mathrm{g}} \boldsymbol{\pi} }\right. {\frac{{ \mathbf{P} \boldsymbol{\sigma}}}{{ 2 \Psi }}} \alpha_{{\mathrm{g}}}^{} \boldsymbol\pi \left.\vphantom{ \frac{ \mathbf{P} \boldsymbol{\sigma}}{ 2 \Psi } + \alpha_{\mathrm{g}} \boldsymbol{\pi} }\right\}$$ (17.55)

with the scalar $\Psi$ defined by

$$\Psi \sqrt{{ \tfrac{1}{2}\boldsymbol{\sigma}^{\mathrm{\scriptscriptstyle{T}}}\mathbf{P} \boldsymbol{\sigma}}}$$ (17.56)

17.1.4.1 Hardening

The relation between the internal state variable $\kappa$ and the plastic process is given by the hardening hypothesis. For the Drucker-Prager 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

$$\dot{{\kappa}} \sqrt{{ \tfrac{2}{3}\left( \dot{\varepsilon }_{1}^{\mathrm{p}} \d... ...t{\varepsilon }_{3}^{\mathrm{p}} \dot{\varepsilon }_{3}^{\mathrm{p}} \right) }}$$ (17.57)

With

$$\left\{\vphantom{ \negthickspace \begin{array}{c} \dot{\varepsilo... ...amount] \dot{\varepsilon }_{3}^{\mathrm{p}} \end{array} \negthickspace }\right. \begin{array}{c} \dot{\varepsilon }_{1}^{\mathrm{p}} \\ [\medski... ...mathrm{p}} \\ [\medskipamount] \dot{\varepsilon }_{3}^{\mathrm{p}} \end{array} \left.\vphantom{ \negthickspace \begin{array}{c} \dot{\varepsilon... ...mount] \dot{\varepsilon }_{3}^{\mathrm{p}} \end{array} \negthickspace }\right\} \dot{{\lambda}} \left(\vphantom{ \frac{ 1 }{ 2 \Psi } \left\{ \negthickspace \beg... ...ipamount] 1 \\ [\medskipamount] 1 \end{array} \negthickspace \right\} }\right. {\frac{{ 1 }}{{ 2 \Psi }}} \left\{\vphantom{ \negthickspace \begin{array}{c} 2 \sigma _{1} -... ...- \sigma _{1} - \sigma _{2} + 2 \sigma _{3} \end{array} \negthickspace }\right. \begin{array}{c} 2 \sigma _{1} - \sigma _{2} - \sigma _{3} \\ [\... ...3} \\ [\medskipamount] - \sigma _{1} - \sigma _{2} + 2 \sigma _{3} \end{array} \left.\vphantom{ \negthickspace \begin{array}{c} 2 \sigma _{1} - ... ... \sigma _{1} - \sigma _{2} + 2 \sigma _{3} \end{array} \negthickspace }\right\} \alpha_{{\mathrm{g}}}^{} \left\{\vphantom{ \negthickspace \begin{array}{c} 1 \\ [\medskipamount] 1 \\ [\medskipamount] 1 \end{array} \negthickspace }\right. \begin{array}{c} 1 \\ [\medskipamount] 1 \\ [\medskipamount] 1 \end{array} \left.\vphantom{ \negthickspace \begin{array}{c} 1 \\ [\medskipamount] 1 \\ [\medskipamount] 1 \end{array} \negthickspace }\right\} \left.\vphantom{ \frac{ 1 }{ 2 \Psi } \left\{ \negthickspace \beg... ...ipamount] 1 \\ [\medskipamount] 1 \end{array} \negthickspace \right\} }\right)$$ (17.58)

(17.57) can be elaborated to

$$\dot{{\kappa}} \dot{{\lambda}} \sqrt{{ 1 + 2 \alpha_{\mathrm{g}}^{2}}}$$ (17.59)

17.1.4.2 Experimental Derivation of Plasticity Parameters

Uniaxial fit.

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 17.7: Derivation of hardening diagram for Drucker-Prager

Consider the uniaxial stress-strain diagram $\sigma_{{3}}^{}$ - $\varepsilon_{{3}}^{}$ of Figure 17.7a. The plastic strain $\varepsilon_{{3}}^{{\mathrm{p}}}$ is assumed to be given by $\varepsilon_{{3}}^{}$ - $\varepsilon_{{3}}^{{\mathrm{e}}}$ . Figure 17.7b shows the uniaxial stress-plastic strain diagram. With the assumption $\sigma_{{3}}^{}$ $\leq$ 0 , the uniaxial plastic strain rate is given by

$$\dot{{\varepsilon }}_{{3}}^{{\mathrm{p}}} \dot{{\lambda}} \left(\vphantom{ 1 - \alpha_{\mathrm{g}} }\right. \alpha_{{\mathrm{g}}}^{} \left.\vphantom{ 1 - \alpha_{\mathrm{g}} }\right)$$ (17.60)

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

$$\dot{{\kappa}} {\frac{{\sqrt{ 1+2 \alpha_{\mathrm{g}}^{2}}}}{{ 1 - \alpha_{\mathrm{g}}}}} \dot{{\varepsilon }}_{{3}}^{{\mathrm{p}}}$$ (17.61)

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

$$\bar{{c}} {\frac{{1-\alpha_{\mathrm{f}}}}{{\beta}}} {\frac{{1-\sin\phi_{0}}}{{2\cos\phi_{0}}}}$$ (17.62)

if the friction angle is constant. Figure 17.7 illustrates the procedure for $\phi_{{0}}^{}$ = $\psi_{{0}}^{}$ = 30° .

Biaxial fit.

The constitutive behavior of materials like concrete under biaxial states of stress is in general different from the constitutive behavior under uniaxial loading conditions. The experimental data of concrete subjected to proportional biaxial loading shows the influence of the lateral compressive stress on the strength of the material. Experiments by Kupfer & Gerstle [58] produced the data as shown in Figure 17.8 with the biaxial fit of the Drucker-Prager failure surface. The maximum compressive strength increases approximately 16% under conditions of equal biaxial compression and about 25% increase is achieved at a stress ratio of $\sigma_{{1}}^{}$/$\sigma_{{2}}^{}$ = 0.5 . The parameters of the Drucker-Prager failure surface, the friction angle $\phi$ and the cohesion $\bar{{c}}$ , are calibrated with the following procedure.

Figure 17.8: Biaxial strength of plain concrete, Kupfer and Gerstle

The uniaxial fit is given in (17.62) as

$$\bar{{c}}$$ = f_c $${\frac{{1-\alpha_{\mathrm{f}}}}{{\beta}}}$$

The biaxial fit is calculated by substituting the stress vector in case of a plane stress state

$$\boldsymbol\sigma \left\{\vphantom{ \negthickspace \begin{array}{c} -a f_{\mathrm{c... ... -a f_{\mathrm{c}} \\ [\smallskipamount] 0 \end{array} \negthickspace }\right. \begin{array}{c} -a f_{\mathrm{c}} \\ [\smallskipamount] -a f_{\mathrm{c}} \\ [\smallskipamount] 0 \end{array} \left.\vphantom{ \negthickspace \begin{array}{c} -a f_{\mathrm{c}... ...-a f_{\mathrm{c}} \\ [\smallskipamount] 0 \end{array} \negthickspace }\right\}$$ (17.63)

with a the multiplication factor for the biaxial strength. Substituting the stress vector into the equation of the failure surface (17.50) yields the following condition

$$\alpha_{{\mathrm{f}}}^{} \beta \bar{{c}} \therefore \bar{{c}} {\frac{{ 1 - 2\,\alpha_{\mathrm{f}} }}{{ \beta }}}$$ (17.64)

Solving (17.62) and (17.64) for $\alpha_{{\mathrm{f}}}^{}$ , given the factor a , results in

$$\alpha_{{\mathrm{f}}}^{} {\frac{{a-1}}{{2a-1}}} {\frac{{2 \sin\phi_{0}}}{{3-\sin\phi_{0}}}}$$ (17.65)

which is solved for sin$\phi_{{0}}^{}$

$$\phi_{{0}}^{} {\frac{{ 3 \, \alpha_{\mathrm{f}} }}{{ 2 + \alpha_{\mathrm{f}} }}} {\frac{{ 3a - 3 }}{{ 5a - 3 }}}$$ (17.66)

Finally, the cohesion is derived from the uniaxial compressive strength and the friction angle $\phi_{{0}}^{}$ according to

$$\bar{{c}} {\frac{{1-\sin\phi_{0}}}{{2\cos\phi_{0}}}}$$ (17.67)

For a normal strength quality concrete, the ratio between the uniaxial compressive strength and the biaxial compressive strength is approximately 1.16 which results in a friction angle $\phi_{{0}}^{}$ $\approx$ 10° and a cohesion $\bar{{c}}$ $\approx$ 0.42f_c .