5.1.5 Modified Mohr-Coulomb

The Modified Mohr-Coulomb plasticity model is particularly useful to model frictional materials like sand or concrete. However, many enhancements have been provided so that it is suitable for all kinds of soil. The main extensions compared to DIANA's regular Mohr-Coulomb model are [Fig.5.5]:

Figure 5.5: Modified Mohr-Coulomb

(a) in p -q space

(b) in deviatoric plane

• Optional nonlinear elasticity.
• A smooth shear yield surface with a default Mohr-Coulomb approximation and with optional hardening/softening.
• An elliptical shaped compression yield surface (cap) with optional hardening.
• A dilatancy angle which is optionally related to the friction angle via Rowe's dilatancy rule.

This section describes the input syntax for the Modified Mohr-Coulomb plasticity model. See §17.1.7 for background theory.

    (syntax)

YIELD

MMOHRC specifies that the Modified Mohr-Coulomb plasticity model must be used. See the following subsections for input syntax of the various data items. See also §5.1.5.5 for input examples.

5.1.5.1 Elasticity

DIANA offers linear elasticity and nonlinear elasticity in combination with the Modified Mohr-Coulomb model. For nonlinear elasticity you may choose either Exponential or Power Law dependency between compression modulus and effective pressure.

Linear elasticity    (syntax)

YOUNG

e is Young's modulus of elasticity E . You can derive E from the (drained) compression modulus K and the shear modulus G using:

$$\left(\vphantom{ 1 + \frac{ 3 K - 2 G }{ 6 K + 2 G } }\right. {\frac{{ 3 K - 2 G }}{{ 6 K + 2 G }}} \left.\vphantom{ 1 + \frac{ 3 K - 2 G }{ 6 K + 2 G } }\right)$$ (5.6)

POISON

nu is Poisson's ratio $\nu$ . ( -1 < $\nu$ < 0.5 )You can derive $\nu$ from the (drained) compression modulus K and shear modulus G using

$$\nu {\frac{{ 3K - 2G }}{{ 6K + 2G }}}$$ (5.7)

Exponential elasticity    (syntax)

ELAST

EXPONE specifies the Cam-clay Exponential elasticity model to be used in conjunction with the Modified Mohr-Coulomb plasticity model. [§5.1.4.1].

ELAVAL

k is the parameter $\kappa$ ( $\kappa$ > 0 )for the Exponential elasticity model which relates the drained tangent compression modulus K_t to the effective pressure p' :

$${\frac{{ 1 + e_{0} }}{{ \kappa }}}$$ (5.8)

where e₀ is the initial void ratio. The optional value pt ( p'_t $\geq$ 0 ) [ p'_t = 0 ] is a pressure shift p'_t along the hydrostatic p' axis to enhance the elasticity model:

$${\frac{{ 1 + e_{0} }}{{ \kappa }}} \left(\vphantom{ p' + p'_{\mathrm{t}} }\right. \left.\vphantom{ p' + p'_{\mathrm{t}} }\right)$$ (5.9)

For stress situations in the apex of the Modified Mohr-Coulomb model, the pressure shift p'_t must be greater than the pressure shift $\Delta$p' specified with the additional parameter PSHIFT [§5.1.5.4]. ( p'_t > $\Delta$p' )

POROSI

n is the initial porosity n₀ ( 0 $\leq$ n₀ $\leq$ 1 ) [n₀ = 0 ]

$${\frac{{ e_{0} }}{{ 1 + e_{0} }}} {\frac{{ v_{0} - 1 }}{{ v_{0} }}}$$ (5.10)

where e₀ is the initial void ratio and v₀ is the specific volume.

VOID

e0 is the initial void ratio e₀ . ( e₀ $\geq$ 0 ) [e₀ = 0 ]

POISON

nu is the constant Poisson's ratio $\nu$ ( -1 < $\nu$ < 0.5 )which implies a pressure dependent shear modulus in the Exponential elasticity model. You may derive $\nu$ from the (drained) initial compression modulus K₀ and initial shear modulus G₀ using

$$\nu {\frac{{ 3 K_{0} - 2 G_{0} }}{{ 6 K_{0} + 2 G_{0} }}}$$ (5.11)

SHRMOD

g is the constant shear stiffness G (G > 0 ) which implies a pressure dependent Poisson's ratio in the Exponential elasticity model.

Power Law elasticity    (syntax)

ELAST

POWER specifies that the Power Law elasticity model must be used in conjunction with the plasticity model.

ELAVAL

specifies the parameters used to determine the pressure dependent compression modulus according to the Power Law:

$$\left(\vphantom{ \frac{ p' }{ p'_{\mathrm{ref}} } }\right. {\frac{{ p' }}{{ p'_{\mathrm{ref}} }}} \left.\vphantom{ \frac{ p' }{ p'_{\mathrm{ref}} } }\right)^{{1-m}}_{}$$ (5.12)

Value kref is the reference compression modulus K_ref . ( K_ref > 0 )Value pref is the reference pressure p_ref . ( p_ref > 0 )You may specify two additional parameters for the elasticity model: value m is parameter m (0 < m < 1 ) [m = 0.5 ] for the Power Law elasticity model, value pt ( p'_t $\geq$ 0 ) [ p'_t = 0 ] is a pressure shift p'_t along the hydrostatic p' axis to enhance the elasticity model:

$$\left(\vphantom{ \frac{ p' - p'_{\mathrm{t}} } { p'_{\mathrm{ref}} } }\right. {\frac{{ p' - p'_{\mathrm{t}} }}{{ p'_{\mathrm{ref}} }}} \left.\vphantom{ \frac{ p' - p'_{\mathrm{t}} } { p'_{\mathrm{ref}} } }\right)^{{1-m}}_{}$$ (5.13)

POISON

nu is the constant Poisson's ratio $\nu$ ( -1 < $\nu$ < 0.5 )for the Power Law elasticity model. which implies a pressure dependent shear modulus according to Equation (5.11).

SHRMOD

g is the constant shear stiffness G (G > 0 )for the Power Law elasticity model which implies a pressure dependent Poisson's ratio.

5.1.5.2 Shear Yield Surface

The shear yield surface of the Modified Mohr-Coulomb plasticity model depends on the friction angle $\phi$ . You may specify the friction angle as a constant or, via a hardening/softening diagram, as a function of the equivalent plastic shear strain $\kappa_{{1}}^{}$ .

By default, DIANA assumes associated plasticity ( $\psi$ = $\phi$ ). However, you may specify the dilatancy angle $\psi$ explicitly or relate it to the friction angle via Rowe's dilatancy rule.

Friction    (syntax)

SINPHI

sphi0 is sin$\phi_{{0}}^{}$ , the sine of the initial friction angle $\phi_{{0}}^{}$ .

As an alternative to the initial friction angle, you may specify a hardening/softening diagram for the friction angle via the following two input items.

FRCCRV

MULTLN indicates a multilinear hardening/softening curve for the friction angle [Fig.5.6].

Figure 5.6: Hardening/softening angle of friction

FRCPAR

specifies pairs of values sphi and k of the multilinear diagram. Values sphi0 to sphin are sin$\phi_{{i=0,n}}^{}$ , ( 1 $\leq$ n $\leq$ 99 )the sines of the friction angle $\phi$ . Values k1 to kn are the corresponding values for the equivalent plastic shear strain $\kappa_{{1_{i=0,n}}}^{}$ , which is related to the plastic shear strain $\gamma^{{\mathrm{p}}}_{}$ according to Equation (17.149).

Dilatancy    (syntax)

SINPSI

spsi is sin$\psi$ , the sine of the dilatancy angle $\psi$ .

DILCRV

ROWE specifies a dilatancy curve according to Rowe's rule , which relates the dilatancy angle to the friction angle

$$\psi {\frac{{ \sin\phi - \sin\phi_{\mathrm{cv}} }}{{ 1 - \sin\phi \sin\phi_{\mathrm{cv}} }}}$$ (5.14)

with $\phi_{{\mathrm{cv}}}^{}$ the friction angle at constant volume. This rule is typically applied in combination with hardening/softening of the friction angle [§17.1.7.4].

SINPCV

spcv is sin$\phi_{{\mathrm{cv}}}^{}$ , the sine of the friction angle $\phi_{{\mathrm{cv}}}^{}$ at constant volume.

If you don't specify dilatancy, then DIANA assumes associated plasticity. [ $\psi$ = $\phi$ ]

5.1.5.3 Compression Yield Surface

A cap-shaped compression yield surface is optional for the Modified Mohr-Coulomb plasticity model. You may define the initial position of a cap explicitly, or let DIANA derive it from the initial stresses. Hardening of the cap as a function of effective pressure is optional. To determine the plastic dilatancy, DIANA always assumes associated plasticity for the compression yield surface.

Explicit preconsolidation stress    (syntax)

PRECON

pc is the preconsolidation stress p'_c0 to define the initial position of the cap explicitly.

K0

k0 is the ratio K₀ to determine the in situ horizontal stresses from the initial stress [§9.6].

If you don't specify the preconsolidation stress explicitly, then DIANA applies initial stress as outlined below with default values for the various parameters. However, you may overrule these defaults according to the syntax below.

Initial stress    (syntax)

During initialization of the nonlinear analysis, DIANA can use the stresses from the preceding linear analysis to determine simultaneously the initial stress and the corresponding preconsolidation pressure, see the option START INITIA STRESS CALCUL in Volume Analysis Procedures. This procedure is identical to the procedure for the Cam-clay model [§5.1.4] and will only be applied for solid, plane strain and axisymmetric elements.

OCR

ocr is the overconsolidation ratio $\mathcal {OCR}$ [§5.1.4.1]. [ $\mathcal {OCR}$ = 1 ]

KNC

knc is the K -ratio for normally consolidated soil K_nc . [ K_nc = 1 - sin$\phi$ ] The horizontal effective stress, acting when the maximum vertical stress was present, is calculated from

$$\sigma{^\prime}_{{\mathrm{h_{max}}}} \sigma{^\prime}_{{\mathrm{v_{max}}}}$$ (5.15)

OCRP

ocrp is an extra multiplication factor $\mathcal {OCR}$_p . [ $\mathcal {OCR}$_p = 1 ] The preconsolidation stress p'_c , based on the maximum stresses, is post-multiplied with $\mathcal {OCR}$_p .

K0

k0 is the ratio K₀ to determine the in situ horizontal stresses from the initial stress [§9.6]. The default for Modified Mohr-Coulomb plasticity with constant Poisson's ratio $\nu$ is

$$\mathcal {OCR} {\frac{{\nu}}{{1 - \nu}}} \mathcal {OCR}$$ (5.16)

Cap hardening.

DIANA offers two types of cap hardening for the Modified Mohr-Coulomb plasticity model: an Exponential hardening for clay-like material, and a Power Law hardening for sandy material. By default DIANA assumes no cap hardening. You may specify it explicitly via either of the following input data syntaxes.

Exponential cap hardening    (syntax)

By default, DIANA assumes no cap hardening. You may specify it explicitly via the following input data.

COMCRV EXPHAR

specifies (Cam-clay) Exponential hardening of the cap which can be written in an incremental way as

$$\dot{{\varepsilon }}_{{\mathrm{v}}}^{{\mathrm{p}}} {\frac{{ \gamma }}{{ 1 + e }}} {\frac{{ \dot{p}'_{\mathrm{c}} }}{{ p'_{\mathrm{c}} }}}$$ (5.17)

After integration one can get the following expression for the preconsolidation stress [Fig.5.7]:

Figure 5.7: Hardening curve of the cap

$$\left(\vphantom{ - \frac{ e_{0} + 1 }{ \gamma } \Delta \varepsilon _{\mathrm{v}}^{\mathrm{p}} }\right. {\frac{{ e_{0} + 1 }}{{ \gamma }}} \Delta \varepsilon_{{\mathrm{v}}}^{{\mathrm{p}}} \left.\vphantom{ - \frac{ e_{0} + 1 }{ \gamma } \Delta \varepsilon _{\mathrm{v}}^{\mathrm{p}} }\right)$$ (5.18)

with $\Delta$$\varepsilon_{{\mathrm{v}}}^{{\mathrm{p}}}$ the volumetric plastic strain increment, p'_c0 the preconsolidation pressure at the beginning of the loading step, and e₀ the void ratio at the beginning of the loading step.

GAMMA

gamma is the material parameter $\gamma$ , which simply can be related to the (Cam-clay) parameters $\lambda$ and $\kappa$

$$\gamma \lambda \kappa \lambda {\frac{{ \partial v }}{{ \partial \ln p' }}} {\frac{{ C_{\mathrm{c}} }}{{ \ln 10 }}}$$ (5.19)

where $\lambda$ is the slope during compression, which is linked to the one-dimensional compression index C_c .

POROSI

n is the initial porosity n₀ ( 0 $\leq$ n₀ < 1 )

$${\frac{{ e_{0} }}{{ 1 + e_{0} }}} {\frac{{ v_{0} - 1 }}{{ v_{0} }}}$$ (5.20)

where e₀ is the initial void ratio and v₀ is the specific volume.

VOID

e0 is the initial void ratio in case of porous elasticity. ( e₀ $\geq$ 0 ) [e₀ = 0 ]

Power Law cap hardening    (syntax)

COMCRV POWHAR

specifies Power Law hardening of the cap which can be written in an incremental way as follows:

$$\dot{{\varepsilon }}_{{\mathrm{v}}}^{{\mathrm{p}}} \Gamma \left(\vphantom{ \frac{ p'_{\mathrm{c}} }{ p'_{\mathrm{ref}} } }\right. {\frac{{ p'_{\mathrm{c}} }}{{ p'_{\mathrm{ref}} }}} \left.\vphantom{ \frac{ p'_{\mathrm{c}} }{ p'_{\mathrm{ref}} } }\right)^{{\! m - 1 }}_{} {\frac{{ \dot{p}'_{\mathrm{c}} }}{{ p'_{\mathrm{ref}} }}}$$ (5.21)

After integration, the above equation leads to the following expression of the preconsolidation stress:

$$\left(\vphantom{ \left( \frac{ p'_{\mathrm{c}_{0}} }{ p'_{\mathrm... ...m} - \frac{ m }{ \Gamma } \Delta\varepsilon _{\mathrm{v}}^{\mathrm{p}} }\right. \left(\vphantom{ \frac{ p'_{\mathrm{c}_{0}} }{ p'_{\mathrm{ref}} } }\right. {\frac{{ p'_{\mathrm{c}_{0}} }}{{ p'_{\mathrm{ref}} }}} \left.\vphantom{ \frac{ p'_{\mathrm{c}_{0}} }{ p'_{\mathrm{ref}} } }\right)^{{\! m}}_{} {\frac{{ m }}{{ \Gamma }}} \Delta \varepsilon_{{\mathrm{v}}}^{{\mathrm{p}}} \left.\vphantom{ \left( \frac{ p'_{\mathrm{c}_{0}} }{ p'_{\mathrm... ...elta\varepsilon _{\mathrm{v}}^{\mathrm{p}} }\right)^{{\! \tfrac{ 1 }{ m } }}_{}$$ (5.22)

where p'_c0 is the preconsolidation stress at the beginning of the step and $\Delta$$\varepsilon_{{\mathrm{v}}}^{{\mathrm{p}}}$ is the volumetric plastic strain increment.

POWPAR

specifies the parameters used to determine the pressure dependent preconsolidation stress according to the Power Law equation (5.22).

gamma

is a parameter modulus $\Gamma$ .

pref

is the reference pressure p'_ref .

m

is an optional parameter that corresponds to the parameter m for the Power Law. [m = 0.5 ] Note that for m = 0 the Power Law cap hardening becomes identical to the Exponential cap hardening.

5.1.5.4 Additional Parameters

To add cohesive behavior or adapt the default shape of the yield surfaces you may specify the following additional parameters.

    (syntax)

PSHIFT

dp is a pressure shift $\Delta$p' ( $\Delta$p' $\geq$ 0 ) [ $\Delta$p' = 0 ] for the shear yield surface [Fig.5.5]. You can relate $\Delta$p' to Mohr-Coulomb's initial cohesion c₀ by

$$\Delta {\frac{{ c_{0} }}{{ \tan \phi }}}$$ (5.23)

Note that in case of friction hardening/softening the cohesion will alter.

CAP

indicates the use of a cap shape factor $\alpha$ for the cap hardening surface of the Modified Mohr-Coulomb model [Fig.5.5]. If you do not explicitly specify a value alpha for $\alpha$ , then DIANA will automatically derive it from the K_NC ratio between horizontal and vertical stress for normally consolidated soil [§17.1.7.6]. If CAP is not specified at all, then the cap shape reduces to a spherical shape with $\alpha$ = ${\frac{{2}}{{9}}}$ .

SHPFAC

are the parameters for the yield contour. Parameter beta1 is the fitting parameter $\beta_{{1}}^{}$ for the shear yield surface in the deviatoric plane, which is by default fitted to Mohr-Coulomb. [$\beta_{{1}}^{}$ Eq.(17.144)] For $\beta_{{1}}^{}$ = 0 the surface reduces to the Drucker-Prager yield surface. Parameter beta2 is the equivalent fitting parameter $\beta_{{2}}^{}$ for the cap yield surface. [ $\beta_{{2}}^{}$ = 0 ]

5.1.5.5 Examples

The following data are examples for Modified Mohr-Coulomb input.

Simple    (file.dat)

'MATERI'
   1 YIELD   MMOHRC
     YOUNG   3.7E+04
     POISON  0.15
     PRECON  100.
     SINPHI  0.57

This input data specifies Modified Mohr-Coulomb with linear elasticity, associated plasticity without hardening.

Sand    (file.dat)

'MATERI'
   1 YIELD   MMOHRC
     ELAST   EXPONE
     ELAVAL  0.00573
     POISON  0.18
     FRCCRV  MULTLN
     FRCPAR  0.574 0.00
             0.650 0.01
             0.680 0.03
     DILCRV  ROWE
     SINPCV  0.51
     OCR     1.5
     COMCRV  EXPHAR
     GAMMA   0.0012

This input data specifies a sand-like material via Modified Mohr-Coulomb with Exponential elasticity, non-associated plasticity with dilatancy according to Rowe, Multilinear hardening, Exponential hardening of the cap, and automatic positioning of the initial position of the cap with $\mathcal {OCR}$ = 1.5 .