FLAC3D Theory and Background • Constitutive Models

Oedometer Test with Mohr-Coulomb Model

Note

The project file for this example is available to be viewed/run in FLAC3D.[1] The project’s main data file is shown at the end of this example.

This example concerns the determination of stresses in a Mohr-Coulomb material subjected to an oedometer test. In this experiment, two of the principal stress components are equal and, during plastic flow, the stress point evolves along a vertex of the Mohr-Coulomb criterion representation in the Π-plane (the iso-pressure (σkk = constant) plane in the principal stress space). The purpose is to validate the numerical technique adopted in FLAC3D to handle such a situation. Note that FLAC3D uses no special techniques to deal with yield at vertex points of the Mohr-Coulomb failure locus in the Π plane. Results of a numerical experiment are presented and compared to an exact solution.

The boundary conditions for the oedometric test are sketched in Figure 1. They correspond to the uniform strain rates

(1)Δϵ11=0Δϵ22=vΔt/LΔϵ33=0

where v is the constant y-component of the velocity applied to the sample (v<0), and L is the height of the sample.

Assuming zero initial stresses, the principal directions of stresses and strains are those of the coordinate axes. For simplicity, we consider a sample of unit height L = 1.

../../../../../_images/modelmohr-boundaryconditions.png

Figure 1: Boundary conditions for oedometer test.

In the elastic range, application of Hooke’s law gives, using ϵ22=vt at time t:

(2)σ11=α2vtσ22=α1vtσ33=σ11

where α1=K+4/3G and α2=K2/3G.

To apply the Mohr-Coulomb failure criterion, we consider the yield functions

(3)f1=σ22σ11Nϕ+2cNϕf2=σ22σ33Nϕ+2cNϕ

At the onset of yield, f1=f2 = 0 and, using Equations (2) and (3), we find

(4)t=2cNϕv(α1α2Nϕ)

Hence, yielding will only take place provided α1α2Nϕ> 0.

During plastic flow, the strain increments are composed of elastic and plastic parts, and we have

(5)Δϵ11=Δϵe11+Δϵp11Δϵ22=Δϵe22+Δϵp22Δϵ33=Δϵe33+Δϵp33

Using the boundary conditions of Equation (1):

(6)Δϵe11=Δϵp11Δϵe22=vΔtΔϵp22Δϵe33=Δϵp33

The flow rule for plastic flow along the edge of the Mohr-Coulomb criterion corresponding to σ11=σ33 has the form (e.g., see Drescher 1991)

(7)Δϵp11=λ1g1σ11+λ2g2σ11Δϵp22=λ1g1σ22+λ2g2σ22Δϵp33=λ1g1σ33+λ2g2σ33

where g1 and g2 are the potential functions corresponding to f1 and f2:

(8)g1=σ22σ11Nψg2=σ22σ33Nψ

After partial differentiation, Equation (7) becomes

(9)Δϵp11=λ1NψΔϵp22=λ1+λ2Δϵp33=λ2Nψ

In further considering that by symmetry λ1=λ2, we obtain

(10)Δϵp11=λ1NψΔϵp22=2λ1Δϵp33=λ1Nψ

The stress increments, derived from Hooke’s law, are given by the relations

(11)Δσ11=α1Δϵe11+α2(Δϵe22+Δϵe11)Δσ22=α1Δϵe22+α22Δϵe11Δσ33=Δσ11

where we have used the symmetry condition Δϵe11=Δϵe33.

Substitution of Equation (6) in Equation (11) yields, using Equation (10):

(12)Δσ11=α1λ1Nψ+α2(vΔt2λ1+λ1Nψ)Δσ22=α1(vΔt2λ1)+α22λ1NψΔσ33=Δσ11

The parameter λ1 can now be determined by expressing the condition that during plastic flow, Δf1 = 0. Using Equation (3), this condition takes the form

(13)Δσ22Δσ11Nϕ=0

Substitution of Equation (12) in Equation (13) yields, after some manipulations, the expression

(14)λ1=vΔtλ

where

(15)λ=α1α2Nϕ(α1+α2)NϕNψ2α2(Nϕ+Nψ)+2α1

The FLAC3D simulation is carried out using a single zone of unit dimensions. Several properties are used in conjunction with the Mohr-Coulomb model:

bulk modulus 200 MPa
shear modulus 200 MPa
cohesion 1 MPa
friction 10°
dilation 10° and 0°
tension 5.67 MPa

The velocity components are fixed in the x-, y-, and z-directions. A velocity of magnitude 10-5 m/steps is applied to the top of the model in the negative y-direction for a total of 1000 steps. The stress and displacement components in the y-direction are monitored and compared to the analytic prediction obtained from Equations (2), (4), and (12), using Equations (14) and (15). Two runs are carried out with values of 10° and 0° for the dilation parameter. The match is very good, as can be seen in Figure 2 and Figure 3, where numerical and analytic solutions coincide at the precision of the plot resolution.

break
../../../../../_images/modelmohr-dilation10.png

Figure 2: Oedometric test—comparison of numerical and analytical predictions for 10° dilation.

../../../../../_images/modelmohr-dilation00.png

Figure 3: Oedometric test—comparison of numerical and analytical predictions for 0° dilation.

Data File

OedometerMohrCoulomb.dat

;---------------------------------------------------------------------
; oedometer test
; check plastic flow along an edge of the Mohr-Coulomb criterion
;---------------------------------------------------------------------
model new
model large-strain off
fish automatic-create off
zone create brick size 1 1 1
model title "Oedometer test on Mohr-Coulomb sample"
zone cmodel assign mohr-coulomb
zone property bulk 200 shear 200 co 1 friction 10 tension 5.67 
[global vyv = -1.e-5]
program call 'oedometerTheoretical'
zone gridpoint fix velocity
zone gridpoint initialize velocity-y [vyv] range position-y 1.0
history interval 50
zone history displacement-y position 0 1 0
fish history n_sy
fish history a_sy
model save 'ini'
; --- dilation 10 
zone property dil 10
[d_sigy]
model step 1000
model save 'dil10'
; --- dilation 0 
model restore 'ini'
zone property dil 0
[d_sigy]
model step 1000
model save 'dil0'

Endnotes

[1]

To view this project in FLAC3D, use the program menu.

Help ▼ Examples…

  ⮡   FLAC
    ⮡   ConstitutiveModels
      ⮡   OedometerMohrCoulomb
        ⮡   OedometerMohrCoulomb.prj