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
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.

Figure 1: Boundary conditions for oedometer test.
In the elastic range, application of Hooke’s law gives, using ϵ22=vt at time t:
where α1=K+4/3G and α2=K−2/3G.
To apply the Mohr-Coulomb failure criterion, we consider the yield functions
At the onset of yield, f1=f2 = 0 and, using Equations (2) and (3), we find
Hence, yielding will only take place provided α1−α2Nϕ> 0.
During plastic flow, the strain increments are composed of elastic and plastic parts, and we have
Using the boundary conditions of Equation (1):
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)
where g1 and g2 are the potential functions corresponding to f1 and f2:
After partial differentiation, Equation (7) becomes
In further considering that by symmetry λ1=λ2, we obtain
The stress increments, derived from Hooke’s law, are given by the relations
where we have used the symmetry condition Δϵe11=Δϵe33.
Substitution of Equation (6) in Equation (11) yields, using Equation (10):
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
Substitution of Equation (12) in Equation (13) yields, after some manipulations, the expression
where
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.
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.
⮡ FLAC |
Was this helpful? ... | FLAC3D © 2019, Itasca | Updated: Feb 25, 2024 |