17.1.7 Modified Mohr-Coulomb

DIANA offers the Modified Mohr-Coulomb plasticity model which is particularly useful to simulate the behavior of sandy materials. This plasticity model has been developed at Delft University of Technology, see Groen [35]. It combines nonlinear elasticity with a failure surface which bounds the stress state.

17.1.7.1 Strains and Stresses

The hydrostatic pressure p' is defined in terms of effective stresses as

$${\tfrac{{1}}{{3}}} \sigma_{{xx}}{^\prime} \sigma_{{yy}}{^\prime} \sigma_{{zz}}{^\prime}$$ (17.122)

and the deviatoric-like stress as

$$\boldsymbol\xi \boldsymbol\sigma$$ (17.123)

in which

$${\frac{{1}}{{3}}} \left[\vphantom{ \negthickspace \begin{array}{cccccc} 2 & -1 & -1... ... 0 & 6 & 0\\ [0.8ex] 0 & 0 & 0 & 0 & 0 & 6 \end{array} \negthickspace }\right. \begin{array}{cccccc} 2 & -1 & -1 & 0 & 0 & 0\\ [0.8ex] -1 & 2 &... ...0\\ [0.8ex] 0 & 0 & 0 & 0 & 6 & 0\\ [0.8ex] 0 & 0 & 0 & 0 & 0 & 6 \end{array} \left.\vphantom{ \negthickspace \begin{array}{cccccc} 2 & -1 & -1... ... 0 & 6 & 0\\ [0.8ex] 0 & 0 & 0 & 0 & 0 & 6 \end{array} \negthickspace }\right]$$ (17.124)

The effective deviatoric stress is then defined according to

$$\sqrt{{ \tfrac{3}{2}\boldsymbol{\xi}^{\mathrm{\scriptscriptstyle{T}}}\mathbf{R} \, \boldsymbol{\xi}}}$$ (17.125)

with the diagonal matrix

$$\left[\vphantom{ \negthickspace \begin{array}{cccccc} 1 & 0 & 0 &... ...\ [0.8ex] 0 & 0 & 0 & 0 & 0 & \tfrac{1}{2} \end{array} \negthickspace }\right. \begin{array}{cccccc} 1 & 0 & 0 & 0 & 0 & 0\\ [0.8ex] 0 & 1 & 0 ... ...0 & 0 & \tfrac{1}{2}& 0\\ [0.8ex] 0 & 0 & 0 & 0 & 0 & \tfrac{1}{2} \end{array} \left.\vphantom{ \negthickspace \begin{array}{cccccc} 1 & 0 & 0 &... ...\ [0.8ex] 0 & 0 & 0 & 0 & 0 & \tfrac{1}{2} \end{array} \negthickspace }\right]$$ (17.126)

The strains are defined in a slightly different manner. The volumetric strain is

$$\varepsilon_{{\mathrm{v}}}^{} \varepsilon_{{xx}}^{} \varepsilon_{{yy}}^{} \varepsilon_{{zz}}^{}$$ (17.127)

and the deviatoric strain is

$$\boldsymbol\gamma \boldsymbol\varepsilon$$ (17.128)

in which the matrix Q = RP = PR .

Actually, the Modified Mohr-Coulomb model is a combination of a nonlinear elastic and a plasticity model. Both for the volumetric strain and the deviatoric strain we use the basic additive decomposition into an elastic part and a plastic part

$$\varepsilon_{{\mathrm{v}}}^{} \varepsilon_{{\mathrm{v}}}^{{\mathrm{e}}} \varepsilon_{{\mathrm{v}}}^{{\mathrm{p}}} \boldsymbol\gamma \boldsymbol\gamma \boldsymbol\gamma$$ (17.129)

17.1.7.2 Nonlinear Elasticity

For soils, it has been observed experimentally that during elastic swelling or reloading the tangent compression modulus K_t is governed by the void ratio and the current hydrostatic pressure according to

$${\frac{{ 1 + e }}{{ \kappa }}}$$ (17.130)

in which e is the void ratio, $\kappa$ is a material parameter and p' is the current hydrostatic pressure. Figure 17.12 shows the assumed elastic behavior, indicating constant $\kappa$ .

Figure 17.12: Soil response in compression

K_t sets the rate relation between the hydrostatic pressure and the elastic volumetric strain

$$\dot{{p}}{^\prime} \dot{{\varepsilon }}_{{\mathrm{v}}}^{{\mathrm{e}}}$$ (17.131)

In this form, the soil has no tensile strength. Therefore, (17.130) is most conveniently modified to

$${\frac{{ 1 + e }}{{ \kappa }}}$$ (17.132)

In which p_t is the so-called tensile pressure. This tensile pressure is merely a numerical artifice to take tensile stresses into account when the initial pressure is assumed to be equal to zero. In practice however, soil analyses will nearly always start with non-zero initial stresses. It is assumed that the void ratio does not change significantly, even for large loading steps. Therefore, the void ratio is updated explicitly during the loading process, see Borja [10].

Equation (17.132) can be recast in a nonlinear relation between the volumetric strain and the pressure as

$${\frac{{ \,\mathrm{d}p' }}{{ p' + p_{\mathrm{t}} }}} {\frac{{ 1 + e }}{{ \kappa }}} \varepsilon_{{\mathrm{v}}}^{{\mathrm{e}}}$$ (17.133)

After integration over an increment and rearranging, the following relation between the pressure and the volumetric strain increment ensues

$$\left(\vphantom{ -\frac{ 1 + e_{0} }{ \kappa } \, \Delta\varepsilon _{\mathrm{v}}^{\mathrm{e}} }\right. {\frac{{ 1 + e_{0} }}{{ \kappa }}} \Delta \varepsilon_{{\mathrm{v}}}^{{\mathrm{e}}} \left.\vphantom{ -\frac{ 1 + e_{0} }{ \kappa } \, \Delta\varepsilon _{\mathrm{v}}^{\mathrm{e}} }\right)$$ (17.134)

In this equation, p'₀ is the pressure at the beginning of a loading step, and p' is the current pressure. This relation defines the pressure as a function of the volumetric strain increment, the pressure at the beginning of a loading step and the void ratio at the beginning of a loading step e₀ . The tangential compression modulus can be obtained from (17.133) via

$${\frac{{\,\mathrm{d}p}}{{\,\mathrm{d}\Delta \varepsilon _{\mathrm{v}}^{\mathrm{e}} }}} {\frac{{1+e_{0}}}{{\kappa}}}$$ (17.135)

which differs from (17.95) in the sense that the void ratio is assumed to remain constant over an increment.

In the Power Law for the elastic volumetric stress-strain relation, it is assumed that the compression modulus is a power of the current pressure

$$\left(\vphantom{ \frac{ p' }{ p'_{\mathrm{ref}} }}\right. {\frac{{ p' }}{{ p'_{\mathrm{ref}} }}} \left.\vphantom{ \frac{ p' }{ p'_{\mathrm{ref}} }}\right)^{{\! 1-m}}_{}$$ (17.136)

in which K_t and p'_ref respectively are the reference compression modulus and the reference pressure; m is a floating point value which is in the order of 0.5 for sand. Relation (17.136) has again no tensile stiffness. Therefore, analogous to the exponential law, the nonlinear compression modulus is modified to

$$\left(\vphantom{ \frac{ p' + p_{\mathrm{t}} }{ p'_{\mathrm{ref}} }}\right. {\frac{{ p' + p_{\mathrm{t}} }}{{ p'_{\mathrm{ref}} }}} \left.\vphantom{ \frac{ p' + p_{\mathrm{t}} }{ p'_{\mathrm{ref}} }}\right)^{{\! 1-m}}_{}$$ (17.137)

which leads to the following form of the volumetric stress-strain relation

$$\left(\vphantom{ \frac{ p'+p_{\mathrm{t}} }{ p'_{\mathrm{ref}} }}\right. {\frac{{ p'+p_{\mathrm{t}} }}{{ p'_{\mathrm{ref}} }}} \left.\vphantom{ \frac{ p'+p_{\mathrm{t}} }{ p'_{\mathrm{ref}} }}\right)^{{\! m-1}}_{} \varepsilon_{{\mathrm{v}}}^{{\mathrm{e}}}$$ (17.138)

After integrating over a finite increment and rearranging, we obtain

$$\left(\vphantom{ ( p'_{0} + p_{\mathrm{t}})^{m} - m \, { p'_{\mat... ...m-1} K_{\mathrm{ref}} \, \Delta \varepsilon _{\mathrm{v}}^{\mathrm{e}} }\right. \Delta \varepsilon_{{\mathrm{v}}}^{{\mathrm{e}}} \left.\vphantom{ ( p'_{0} + p_{\mathrm{t}})^{m} - m \, { p'_{\mat... ... \Delta \varepsilon _{\mathrm{v}}^{\mathrm{e}} }\right)^{{\! \tfrac{1}{m} }}_{} \Delta \varepsilon_{{\mathrm{v}}}^{{\mathrm{e}}}$$ (17.139)

The nonlinear elasticity of the model is defined in terms of volumetric components only. Assuming isotropy, one can specify either a constant Poisson's ratio $\nu$ , with the (variable) tangent shear modulus defined as

$${\tfrac{{3}}{{2}}} {\frac{{ 1 - 2 \nu }}{{ 1 + \nu }}}$$ (17.140)

Alternatively, one can consider an approach in which the shear modulus G is kept constant and the Poisson's ratio $\nu$ varies according to

$$\nu_{{\mathrm{t}}}^{} {\frac{{ 3 K_{\mathrm{t}} - 2 G }}{{ 6 K_{\mathrm{t}} + 2 G }}}$$ (17.141)

17.1.7.3 Yield Function

The failure surface of the Modified Mohr-Coulomb model is a so-called double hardening model in which the shear failure and the compressive failure are uncoupled. The combined failure surface is given by the formulation in the p -q space

$$\begin{split}f_{1} &= \dfrac{q}{R_{1}(\theta)} - \dfrac{6 \si... ..._{2}(\theta)}\right)^{\!2} - p_{\mathrm{c}}^{2} = 0 \end{split}$$ (17.142)

in which $\phi$ the friction angle in triaxial compression, $\Delta$p a constant which models cohesive material behavior, and p_c the preconsolidation pressure [Fig.17.13].

Figure 17.13: Modified Mohr-Coulomb model

(a) in p -q space

(b) in deviatoric plane

The functions R₁($\theta$) and R₂($\theta$) model the differences in strength in triaxial compression and in triaxial extension and are functions of Lode's angle $\theta$ . The model can be fitted on the Mohr-Coulomb model in the deviatoric plane by the functions R₁($\theta$) and R₂($\theta$) . Special attention is needed to keep the failure surface convex, see Groen [35] for details. The fit to the Mohr-Coulomb model in triaxial extension leads to the following relationship

$$\theta \left(\vphantom{ \dfrac{1-\beta_{1} \sin 3\theta }{1-\beta_{1}} }\right. {\dfrac{{1-\beta_{1} \sin 3\theta }}{{1-\beta_{1}}}} \left.\vphantom{ \dfrac{1-\beta_{1} \sin 3\theta }{1-\beta_{1}} }\right)^{{\!n}}_{}$$ (17.143)

with n = - 0.229 . The factor $\beta_{{1}}^{}$ is related to the friction angle according to

$$\beta_{{1}}^{} {\dfrac{{ \left( \dfrac{3+\sin\phi}{3-\sin\phi} \right)^{\! -\tfr... ...1 }}{{ \left( \dfrac{3+\sin\phi}{3-\sin\phi} \right)^{\! -\tfrac{1}{n}} + 1 }}}$$ (17.144)

with a maximum of $\beta_{{1}}^{}$ $\leq$ 0.7925 . The maximum value of the $\beta$ factor is equal to a fit on the Mohr-Coulomb criterion with a friction angle $\phi$ equal to 46.55°. The shape of the compressive cap can also be modified with the factor R₂($\theta$) which is assumed according to (17.143)

$$\theta \left(\vphantom{ \dfrac{1-\beta_{2} \sin 3\theta }{1-\beta_{2}}}\right. {\dfrac{{1-\beta_{2} \sin 3\theta }}{{1-\beta_{2}}}} \left.\vphantom{ \dfrac{1-\beta_{2} \sin 3\theta }{1-\beta_{2}}}\right)^{{\!n}}_{}$$ (17.145)

with n = - 0.229 and the factor $\beta_{{2}}^{}$ by default equal to zero which implies a spherical cap.

17.1.7.4 Flow Rule

The direction of the inelastic strain rate is determined by the plastic potential surfaces, where in case of the Modified Mohr-Coulomb model the following two surfaces are applied

$$\begin{split}g_{1} &= q - \dfrac{6 \sin\psi}{3-\sin\psi} \lef... ...a p \right)^{2} + \alpha q^{2} - p_{\mathrm{c}}^{2} \end{split}$$ (17.146)

which implies an associative behavior in the p -q space and a nonassociated flow in the deviatoric space. The dilatancy angle $\psi$ is related to the friction angle $\phi$ by the assumption of Rowe's stress dilatancy theory [90], which reads

$$\psi {\dfrac{{ \sin\phi - \sin\phi_{\mathrm{cv}} }}{{ 1 - \sin\phi \sin\phi_{\mathrm{cv}} }}}$$ (17.147)

in which sin$\phi_{{\mathrm{cv}}}^{}$ a constant value which can be conceived as the friction angle at constant volume.

17.1.7.5 Hardening Behavior

The evolution of the failure surfaces is uncoupled as has been one of the basic assumptions of the double hardening model. In the implementation in DIANA, two evolution functions are necessary: the evolution of sin$\phi$ , the sine of the friction angle, and the evolution of the preconsolidation pressure p_c . The evolution of the dilatancy angle sin$\psi$ is implicitly given by the assumption of Rowe's stress dilatancy model (17.147).

The friction angle is a function of some internal parameter $\kappa_{{1}}^{}$

$$\phi \phi \left(\vphantom{ \kappa_{1} }\right. \kappa_{{1}}^{} \left.\vphantom{ \kappa_{1} }\right)$$ (17.148)

The internal parameter $\kappa_{{1}}^{}$ is a function of the equivalent deviatoric plastic strain increment $\Delta$$\gamma^{{\mathrm{p}}}_{}$

$$\Delta \kappa_{{1}}^{} \sqrt{{ \tfrac{2}{3}\, { \Delta\gamma^{\mathrm{p}}}^{\mathrm{\scriptscriptstyle{T}}}\: \mathbf{R} \: \Delta\gamma^{\mathrm{p}} }}$$ (17.149)

For the evolution of the friction angle, a multi-linear diagram can be specified. This can be determined from triaxial tests, for example the conventional triaxial compression test as described by Vermeer & De Borst [109].

The preconsolidation pressure is also given as a function of some internal parameter $\kappa_{{2}}^{}$ as

$$\left(\vphantom{ \kappa_{2} }\right. \kappa_{{2}}^{} \left.\vphantom{ \kappa_{2} }\right)$$ (17.150)

with the internal parameter $\kappa_{{2}}^{}$ now defined as

$$\Delta \kappa_{{2}}^{} \Delta \varepsilon_{{\mathrm{v}}}^{{\mathrm{p}}}$$ (17.151)

The evolution is assumed analogous to the elastic compressive law, i.e., the exponential law given in (17.134)

$$\left(\vphantom{ \dfrac{1 + e}{\gamma } \, \Delta \kappa_{2} }\right. {\dfrac{{1 + e}}{{\gamma }}} \Delta \kappa_{{2}}^{} \left.\vphantom{ \dfrac{1 + e}{\gamma } \, \Delta \kappa_{2} }\right)$$ (17.152)

in which p_c0 is the initial preconsolidation pressure, and $\gamma$ is the saturation factor which can be considered as a material parameter.

17.1.7.6 Cap Shape Factor

The automatic derivation of the cap shape factor for the Modified Mohr-Coulomb plasticity model is done in the same fashion as for the Egg Cam-clay model [§17.1.6.5], i.e., it will be calculated in such a way that for a one-dimensional compression situation the stress ratio $\Delta$p'/$\Delta$q equals $\eta_{{\mathrm{NC}}}^{}$ . The elastic components $\varepsilon_{{\mathrm{v}}}^{{\mathrm{e}}}$ and $\Delta$$\varepsilon_{{\mathrm{s}}}^{{\mathrm{e}}}$ are also given by (17.117). The two components of the plastic increments $\varepsilon_{{\mathrm{v}}}^{{\mathrm{p}}}$ and $\Delta$$\varepsilon_{{\mathrm{s}}}^{{\mathrm{p}}}$ are expressed according to the Muir Wood formulae (17.118). However, for the Modified Mohr-Coulomb model we have the following partial derivatives expressions:

$$\begin{split}\frac{ \partial f }{ \partial p'_{\mathrm{c}} } ... ...= \frac{ \partial g }{ \partial q } = 2 \alpha \, q \end{split}$$ (17.153)

Inserting (17.153) into (17.118) leads, after some algebraic manipulations, to:

$$\begin{split}\Delta \varepsilon _{\mathrm{v}}^{\mathrm{p}} &=... ...t \eta_{\mathrm{NC}} \cdot \frac{ \Delta p' }{ p' } \end{split}$$ (17.154)

Finally, substitution of (17.154) and (17.117) into (17.116) gives the following expression for $\alpha$ :

$$\alpha {\frac{{ 2 - \dfrac{ 2 ( 1 + \nu ) }{ 3 ( 1 - 2 \nu ) } \cdot \df... ...{ 3 \left( 1 - \dfrac{ \kappa }{ \lambda } \right) \cdot \eta_{\mathrm{NC}} }}}$$ (17.155)