- 5.1.5.1 Elasticity
- 5.1.5.2 Shear Yield Surface
- 5.1.5.3 Compression Yield Surface

- 5.1.5.4 Coupled Friction and Cap Hardening
- 5.1.5.5 Additional Parameters
- 5.1.5.6 Examples

5.1.5 Modified Mohr-Coulomb/Hardening Soil Model

The Modified Mohr-Coulomb plasticity model [Fig.5.5][Fig.5.6], also known as the Hardening Soil model, is particularly useful to model frictional materials like sand. 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:

- Optional nonlinear elasticity.
- Optional small strain stiffness.
- 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.

(*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 §5.1.5.6 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. Additionally, small strain stiffness, which is a varying shear modulus in the small strain range, can be applied [§19.1.7.3].

**Linear elasticity** (*syntax*)

`YOUNG`is Young's modulus of elasticity*e**E*. You can derive *E*from the (drained) compression modulus *K*and the shear modulus *G*using: *E*= 2*G*1 +(5.5)

`POISON`is Poisson's ratio*nu*. ( -1 < < 0.5 ) You can derivefrom the (drained) compression modulus *K*and shear modulus *G*using = (5.6)

**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`is the parameter*k*( > 0 ) for the Exponential elasticity model which relates the drained tangent compression modulus*K*_{t}to the effective pressure *p'*: *K*_{t}=*p'*(5.7)

where*e*_{0}is the initial void ratio. The optional value (*pt**p'*_{t}0) [*p'*_{t}= 0] is a pressure shift*p'*_{t}along the hydrostatic *p'*axis to enhance the elasticity model: *K*_{t}=*p'*+*p'*_{t}(5.8)

For stress situations in the apex of the Modified Mohr-Coulomb model, the pressure shift*p'*_{t}must be greater than the pressure shift *p*specified with the additional parameter `PSHIFT`[§5.1.5.5]. (*p'*_{t}>*p*) `POROSI`is the initial porosity*n**n*_{0}( 0 *n*_{0}1) [*n*_{0}= 0] *n*_{0}= =(5.9)

where*e*_{0}is the initial void ratio and *v*_{0}is the specific volume. `VOID`-
is the initial void ratio*e0**e*_{0}. ( *e*_{0}0) [*e*_{0}= 0] `POISON`is the constant Poisson's ratio*nu*( -1 < < 0.5 ) which implies a pressure dependent shear modulus in the Exponential elasticity model. You may derivefrom the (drained) initial compression modulus *K*_{0}and initial shear modulus *G*_{0}using

`SHRMOD`is the constant shear stiffness*g**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:
*K*_{t}=*K*_{ref}(5.11)

Valueis the reference compression modulus*kref**K*_{ref}. ( *K*_{ref}> 0) Valueis the reference pressure*pref**p*_{ref}. ( *p*_{ref}> 0) You may specify two additional parameters for the elasticity model: valueis parameter*m**m*(0 < *m*< 1) [*m*= 0.5] for the Power Law elasticity model, value(*pt**p'*_{t}0) [*p'*_{t}= 0] is a pressure shift*p'*_{t}along the hydrostatic *p'*axis to enhance the elasticity model: *K*_{t}=*K*_{ref}(5.12)

`POISON`is the constant Poisson's ratio*nu*( -1 < < 0.5 ) for the Power Law elasticity model. which implies a pressure dependent shear modulus according to Eq.(5.10).`SHRMOD`is the constant shear stiffness*g**G*( *G*> 0) for the Power Law elasticity model which implies a pressure dependent Poisson's ratio.

**Small strain stiffness** (*syntax*)

`YOUSML`is the small strain Young's modulus which is internally converted to the initial or small strain shear stiffness*esml**G*_{0}[Eq.(19.149)] using the Poisson's ratio . For background theory on small strain stiffness, see [§19.1.7.3]. `YOUNUR`is the reference unloading-reloading stiffness*eur**E*_{ref}^{ur}which is internally converted to the unloading-reloading shear modulus *G*_{ur}[Eq.(19.151)] using the Poisson's ratio . `GAMMAR`is the threshold shear strain*gammar*[Eq.(19.149)]. `FACA`is parameter*faca**a*of Eq.(19.149). [ `FACA 0.385`]

5.1.5.2 Shear Yield Surface

The shear yield surface of the Modified Mohr-Coulomb plasticity model
depends on the friction angle

By default, DIANA assumes associated plasticity (
=

**Constant friction angle** (*syntax*)

`PHI`is*phi0*, the initial friction angle .

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

**Hardening/softening diagram for the friction angle** (*syntax*)

`FRCCRV``MULTLN`indicates a multilinear hardening/softening curve for the friction angle [Fig.5.7].`KAPPHI`- specifies pairs of values
and*k*of the multilinear diagram. Values*ph*to*k1*are the values for the equivalent plastic shear strain*kn*, which is related to the plastic shear strain according to Eq.(19.159). Values to*ph1*are the corresponding friction angles*phn*. ( *n*100)

As an alternative to the initial friction angle and a hardening/softening diagram, you may specify a parabolic hardening of the friction angle as function of the plastic shear-strain according to Duncan-Chang following syntax.

**Duncan- Chang parabolic hardening of the friction angle** (*syntax*)

`FRCCRV``DUNCHA`indicates a parabolic hardening of the friction angle as function of the plastic shear-strain according to Duncan-Chang.`YOUSEC`is the reference secant stiffness*e50**E*_{ref}^{50}in a standard drain triaxial test. ( *E*_{ref}^{ur}>*E*_{ref}^{50}) `YOUNUR`is the reference unloading-reloading stiffness*eur**E*_{ref}^{ur}; `YOUOED`is the reference tangent stiffness for primary oedometer loading*eoed**E*_{ref}^{oed}. ( 0 < *E*_{ref}^{oed}<*E*_{ref}^{50}) `COHESI`is the cohesion*c**C*at shear failure. ( *C*> 0) `PHI`is the internal friction angle at shear-failure*phi*. Note that this angle needs to be provided in the used units (radians or degrees) [Vol. *Analysis Procedures*]. ( 0 < < 90°) `PHI0`is the initial internal friction angle*phi0*. Note that this angle needs to be provided in the used units (radians or degrees) [Vol. *Analysis Procedures*]. [ = 0] ( 0) `RF`is the failure ratio of*rf**q*_{f}/*q*_{a}. [ *rf*= 0.9] `EXPM`is the power of stress-level dependency. [*m**m*= 0.5] When the power of stress-level dependency*m*is defined equal to zero, then exponential cap hardening and exponential elasticity are applied. Otherwise power-law cap hardening and elasticity are used. For more information on defining the parameters for the Modified Mohr-Coulomb model see Volume *Geotechnical Analysis*.`PREF`-
is the reference stress of the triaxial test for the specified stiffness parameters. [*pref**pref*= 1.*e*5Pa] `POISON`-
is the Poisson's ratio for unloading-reloading*nu*. [ = 0.2 ] `POROSI`-
is the initial porosity*n**n*_{0}. ( 0 < *n*_{0}< 1) [*n*_{0}= 0.6] `KNC`-
is the*knc**K*-ratio for normally consolidated soil *K*_{nc}. [ *K*_{nc}= 1 - sin] The horizontal effective stress, acting when the maximum vertical stress was present, is calculated from= *K*_{nc}`x`(5.13)

**Dilatancy** (*syntax*)

`PSI`is*psi*, the dilatancy angle . `DILCRV`-
`ROWE`specifies a dilatancy curve according to Rowe's rule , which relates the dilatancy angle to the friction anglesin = (5.14)

withthe friction angle at constant volume. This rule is typically applied in combination with hardening/softening of the friction angle [§19.1.7.5]. `PHICV`is the friction angle*phicv*at constant volume.

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`is the preconsolidation stress*pc**p'*_{c0}to define the initial position of the cap explicitly.

**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`is the overconsolidation ratio*ocr*[§5.1.4.1]. [ = 1 ] `KNC`is the*knc**K*-ratio for normally consolidated soil *K*_{nc}. [ *K*_{nc}= 1 - sin] The horizontal effective stress, acting when the maximum vertical stress was present, is calculated from= *K*_{nc}`x`(5.15)

`OCRP`is an extra multiplication factor*ocrp*_{p}. [ _{p}= 1] The preconsolidation stress*p'*_{c}, based on the maximum stresses, is post-multiplied with _{p}. `K0`is the ratio*k0**K*_{0}to determine the in situ horizontal stresses from the initial stress [§11.3]. The default for Modified Mohr-Coulomb plasticity with constant Poisson's ratio is *K*_{0}=*K*_{nc}- ( - 1)(5.16)

Cap hardening.

**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
= - (5.17)

After integration one can get the following expression for the preconsolidation stress [Fig.5.8]:*p'*_{c}=*p'*_{c0}exp -(5.18)

withthe volumetric plastic strain increment, *p'*_{c0}the preconsolidation pressure at the beginning of the loading step, and *e*_{0}the void ratio at the beginning of the loading step. `GAMMA`is the material parameter*gamma*, which simply can be related to the (Cam-clay) parameters and = - with = = (5.19)

whereis the slope during compression, which is linked to the one-dimensional compression index *C*_{c}. `POROSI`is the initial porosity*n**n*_{0}( 0 *n*_{0}< 1) *n*_{0}= =(5.20)

where*e*_{0}is the initial void ratio and *v*_{0}is the specific volume. `VOID`-
is the initial void ratio in case of porous elasticity. (*e0**e*_{0}0) [*e*_{0}= 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:
= - (5.21)

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

where*p'*_{c0}is the preconsolidation stress at the beginning of the step and is the volumetric plastic strain increment. `POWPAR`-
specifies the parameters used to determine the pressure dependent
preconsolidation stress according to the Power Law equation Eq.(5.22).
*gamma*- is a parameter modulus
. *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*= 0the Power Law cap hardening becomes identical to the Exponential cap hardening.

5.1.5.4 Coupled Friction and Cap Hardening

By default the hardening parameters

**Coupled plastic hardening** (*syntax*)

`HARDEN``COUPLE`- indicates that the friction yield surface hardening
parameter
and the cap yield surface hardening parameter will both change when either of the yield surfaces is active.

5.1.5.5 Additional Parameters

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

(*syntax*)

`PSHIFT`-
is a pressure shift*dp**p*( *p*0) [*p*= 0] for the shear yield surface [Fig.5.5]. You can relate*p*to Mohr-Coulomb's initial cohesion *c*_{0}by *p*=(5.23)

Note that in case of friction hardening/softening the cohesion will alter. `CAP`-
indicates the use of a cap shape factor
for the cap hardening surface of the Modified Mohr-Coulomb model [Fig.5.5]. If you do not explicitly specify a value for*alpha*, then DIANA will automatically derive it from the *K*_{NC}ratio between horizontal and vertical stress for normally consolidated soil [§19.1.7.7]. ( 0 ) If`CAP`is not specified at all, then the cap shape reduces to a spherical shape with =. `FLOCAP`-
indicates the use of a cap shape factor
=
for the flow rule related to the yield cap function. When `FLOCAP`is not specified*g*=is assumed. For significantly higher than , in some situations a return map to the yield cap surface may not be possible leading to convergence difficulties. In such situations it is recommended to use `FLOCAP`which sets =. For more information see the background theory [§19.1.7.5]. `SHPFAC`- are the parameters for the yield contour.
Parameter
is the fitting parameter*beta1*for the shear yield surface in the deviatoric plane, which is by default fitted to Mohr-Coulomb. [ Eq.(19.154)] For = 0the surface reduces to the Drucker-Prager yield surface. Parameter is the equivalent fitting parameter*beta2*for the cap yield surface. [ = 0 ] `TENSTR`- (
0
*P*_{t}) is a parameter defining a simple tension cut-off surface based on the mean stress, i.e. ( + + )/3*pt*; this surface limits the tension to value *P*_{t}. `PRNSTR`- (
0
*P*_{t}) is a parameter defining a simple tension cut-off surface based on the maximal principal stress in tension*pt*; this surface limits the tension to value *P*_{t}. If *P*_{t}is set equal to zero, DIANA internally uses a small value of 1 Pa for *P*_{t}to ensure numerical stability. `DILCUT`- (
*e*_{min}0) (*e*_{max}*e*_{min}) activates dilatancy cut-off based on initial void ratio*e*_{min}, specified by , and maximum void ratio*emin**e*_{max}, specified by . As soon as the volumetric strain results in a state of maximum void ratio*emax**e*_{max}, the dilatance angle is set to zero, i.e. when *e**e*_{max}then sin = 0 . The volumetric strain , and void ratio *e*are related as = ln (5.24)

`ADDSTI`is the additional stiffness parameter. By default no additional stiffness is applied. For more information see the background theory [§19.1.7.8]. [*addsti*=0] (0*addsti*1*addsti*)

5.1.5.6 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. PHI 0.6065

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
= 1.5

First ed.

Copyright (c) 2017 by DIANA FEA BV.