Oedometer Test with Mohr-Coulomb Model
Note
To view this project in FLAC3D, use the menu command . Choose “ConstitutiveModels/ OedometerMohrCoulomb” and select “OedometerMohrCoulomb.f3dprj” to load. 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 \(\Pi\)-plane (the iso-pressure (\(\sigma_{kk}\) = 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 \(\Pi\) 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.
In the elastic range, application of Hooke’s law gives, using \({\epsilon}_{22} = v t\) at time \(t\):
where \(\alpha_1 = K + 4/3 G\) and \(\alpha_2 = K - 2/3 G\).
To apply the Mohr-Coulomb failure criterion, we consider the yield functions
At the onset of yield, \(f^1 = f^2\) = 0 and, using Equations (2) and (3), we find
Hence, yielding will only take place provided \(\alpha_1 - \alpha_2 N_{\phi} >\) 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 \({\sigma}_{11} = {\sigma}_{33}\) has the form (e.g., see Drescher 1991)
where \(g^1\) and \(g^2\) are the potential functions corresponding to \(f^1\) and \(f^2\):
After partial differentiation, Equation (7) becomes
In further considering that by symmetry \(\lambda_1 = \lambda_2\), we obtain
The stress increments, derived from Hooke’s law, are given by the relations
where we have used the symmetry condition \(\Delta {\epsilon}_{11}^e = \Delta {\epsilon}_{33}^e\).
Substitution of Equation (6) in Equation (11) yields, using Equation (10):
The parameter \(\lambda_1\) can now be determined by expressing the condition that during plastic flow, \(\Delta f^1\) = 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
;---------------------------------------------------------------------
; oedometer test
; check plastic flow along an edge of the Mohr-Coulomb criterion
;---------------------------------------------------------------------
model new
model largestrain 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]
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'
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