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Subsections

## 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.

This section describes the input syntax for the Modified Mohr-Coulomb plasticity model. See §19.1.7 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 §5.1.5.6 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
e is Young's modulus of elasticity E . You can derive E from the (drained) compression modulus K and the shear modulus G using:

 E = 2 G1 + (5.5)

POISON
nu is Poisson's ratio . ( -1 < < 0.5 )You can derive from 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
k is the parameter ( > 0 )for the Exponential elasticity model which relates the drained tangent compression modulus Kt to the effective pressure p' :

 Kt =  p' (5.7)

where e0 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:

 Kt =  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 PSHIFT5.1.5.5]. ( p't > p )

POROSI
n is the initial porosity n0 ( 0 n0 1 ) [n0 = 0 ]

 n0 = = (5.9)

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

VOID
e0 is the initial void ratio e0 . ( e0 0 ) [e0 = 0 ]

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

 = (5.10)

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:

 Kt = Kref (5.11)

Value kref is the reference compression modulus Kref . ( Kref > 0 )Value pref is the reference pressure pref . ( pref > 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 0 ) [ p't = 0 ] is a pressure shift p't along the hydrostatic p' axis to enhance the elasticity model:

 Kt = Kref (5.12)

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

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

Small strain stiffness    (syntax)

YOUSML
esml is the small strain Young's modulus which is internally converted to the initial or small strain shear stiffness G0 [Eq.(19.149)] using the Poisson's ratio . For background theory on small strain stiffness, see [§19.1.7.3].

YOUNUR

GAMMAR
gammar is the threshold shear strain [Eq.(19.149)].

FACA
faca is parameter 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 . You may specify the friction angle as a constant, or via a hardening/softening diagram as a function of the equivalent plastic shear strain , or a parabolic hardening of the friction angle as function of the plastic shear-strain according to Duncan-Chang.

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

Constant friction angle    (syntax)

PHI
phi0 is , 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 k and ph of the multilinear diagram. Values k1 to kn are the values for the equivalent plastic shear strain , which is related to the plastic shear strain according to Eq.(19.159). Values ph1 to phn are the corresponding friction angles . ( 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
e50 is the reference secant stiffness Eref50 in a standard drain triaxial test. ( Erefur > Eref50 )

YOUNUR

YOUOED
eoed is the reference tangent stiffness for primary oedometer loading Erefoed . ( 0 < Erefoed < Eref50 )

COHESI
c is the cohesion C at shear failure. (C > 0 )

PHI
phi is the internal friction angle at shear-failure . Note that this angle needs to be provided in the used units (radians or degrees) [Vol. Analysis Procedures]. ( 0 < < 90° )

PHI0
phi0 is the initial internal friction angle . Note that this angle needs to be provided in the used units (radians or degrees) [Vol. Analysis Procedures]. [ = 0 ] ( 0 )

RF
rf is the failure ratio of qf/qa . [rf = 0.9 ]

EXPM
m is the power of stress-level dependency. [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
pref is the reference stress of the triaxial test for the specified stiffness parameters. [ pref = 1.e5 Pa]

POISON

POROSI
n is the initial porosity n0 . ( 0 < n0 < 1 ) [ n0 = 0.6 ]

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

 = Knc x (5.13)

Dilatancy    (syntax)

PSI
psi is , the dilatancy angle .

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

 sin = (5.14)

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

PHICV
phicv is the friction angle at constant volume.

If you do not specify dilatancy, then DIANA assumes associated plasticity. [ = ]

### 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.

If you do not 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 5.1.4.1]. [ = 1 ]

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

 = Knc x (5.15)

OCRP
ocrp is an extra multiplication factor p . [ p = 1 ] The preconsolidation stress p'c , based on the maximum stresses, is post-multiplied with p .

K0
k0 is the ratio K0 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

 K0 =    Knc   -  ( - 1) (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

 = - (5.17)

After integration one can get the following expression for the preconsolidation stress [Fig.5.8]:

 p'c = p'c0 exp - (5.18)

with 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 , which simply can be related to the (Cam-clay) parameters and

 = -         with         = = (5.19)

where is the slope during compression, which is linked to the one-dimensional compression index Cc .

POROSI
n is the initial porosity n0 ( 0 n0 < 1 )

 n0 = = (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 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:

 = - (5.21)

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

 p'c = p'ref - (5.22)

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 = 0 the 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 for the friction yield surface and for the cap yield surface develop only with plastic deformation of their respective yield surfaces. That means that the friction yield surface hardening parameter does not change when only the cap yield surface is active and that the cap yield surface hardening parameter does not change when only the friction yield surface is active. However, with the following option, the development of both parameters is derived from the total plastic strain, which is composed by a contribution of both yield surfaces. This implies 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.

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.

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

(syntax)

PSHIFT
dp is a pressure shift p ( p 0 ) [ p = 0 ] for the shear yield surface [Fig.5.5]. You can relate p to Mohr-Coulomb's initial cohesion c0 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 alpha for , then DIANA will automatically derive it from the KNC 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 beta1 is the fitting parameter for the shear yield surface in the deviatoric plane, which is by default fitted to Mohr-Coulomb. [ Eq.(19.154)] For = 0 the surface reduces to the Drucker-Prager yield surface. Parameter beta2 is the equivalent fitting parameter for the cap yield surface. [ = 0 ]

TENSTR
( 0 Pt )pt is a parameter defining a simple tension cut-off surface based on the mean stress, i.e. ( + + )/3 ; this surface limits the tension to value Pt .

PRNSTR
( 0 Pt )pt is a parameter defining a simple tension cut-off surface based on the maximal principal stress in tension ; this surface limits the tension to value Pt . If Pt is set equal to zero, DIANA internally uses a small value of 1 Pa for Pt to ensure numerical stability.

DILCUT
( emin 0 ) ( emax emin )activates dilatancy cut-off based on initial void ratio emin , specified by emin, and maximum void ratio emax , specified by emax. As soon as the volumetric strain results in a state of maximum void ratio emax , the dilatance angle is set to zero, i.e. when e emax then sin = 0 . The volumetric strain , and void ratio e are related as

 = ln (5.24)

### 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 .

Next: 5.1.6 Modified Mohr-Coulomb - Up: 5.1 Isotropic Plasticity Previous: 5.1.4 Egg Cam-clay   Contents   Index
DIANA-10.1 User's Manual - Material Library
First ed.

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