Examples • Verification Problems
Drained and Undrained Triaxial Compression Test on a CamClay Sample
Problem Statement
Note
The project file for this example is available to be viewed/run in FLAC3D.[1] The main data files used are shown at the end of this example. The remaining data files can be found in the project.
Conventional drained and undrained triaxial compression tests on Camclay soil samples are modeled using FLAC3D. The stresses and specific volume at the critical state are compared with analytical predictions. The responses of both a lightly (LOC) and a heavily (HOC) overconsolidated specimen are considered. This set of problems tests the prediction accuracy of the modified Camclay model in FLAC3D.
The model of the sample is a cube with unit dimensions. The sample is made of a Camclay material with several properties:
shear modulus (\(G\)) 
250 \(p'_1\) 
soil constant (\(M\)) 
1.02 
slope of normal consolidation line (\(\lambda\)) 
0.2 
slope of elastic swelling line (\(\kappa\)) 
0.05 
reference pressure (\(p'_1\)) 
1 kPa 
preconsolidation pressure (\(p'_{c0}\)) 
8 \(p'_1\) and 40 \(p'_1\) 
specific volume at reference pressure on normal consolidation line (\(v_{\lambda}\)) 
3.32 
Initially, the sample is in a state of isotropic compression corresponding to \(p_0 = 5 p'_1\) and zero excess pore pressure (\(p'_0 = p_0\)). The preconsolidation pressure \(p'_{c0}\) has magnitude 8 × \(p'_1\) in the lightly overconsolidated case, and 40 × \(p'_1\) in the heavily overconsolidated case. These cases correspond to an overconsolidation ratio \(R = p'_{c0} / p'_0\) of 1.6 and 8, respectively. The shear modulus is assumed to remain constant during the test carried out with constant confining pressure \(p_0\) and simulated straincontrolled platens. Drained and undrained tests are considered. Refer to Wood (1990) for a detailed discussion on the Camclay plasticity theory.
ClosedForm Solutions
The mean pressure \(p\) and deviator stress \(q\) in a conventional triaxial test can be expressed as
where \(\sigma_1\) is the axial stress and \(\sigma_2\) is the cell pressure.
Since the cell pressure is kept constant during the test, the total stress path in the \((p,q)\) plane is constrained by the relation
With initial conditions of the form \(p = p'_0\), \(q\) = 0, obtained upon integration is
In a drained test, no excess pore pressure is generated, the effective and the total stress paths coincide, and we may write
This stress path is represented in the figure below. The dashed line in the figure is the critical state line.
In an undrained test, when the fluid bulk modulus is much larger than that of the soil (incompressible fluid), the specific volume \(v\) remains constant, equal to the initial value \(v_0\), and it may be shown that the effective stress path is also welldefined.
Consider the case of an overconsolidated sample. Referring to (b) in the above figure, as long as the stress state lies inside the first yield surface, the path corresponds to the straight line,
When plastic deformation takes place, the shape of the effective stress path is given by (Wood, 1990, p. 127)
where \(\Lambda = (\lambda  \kappa) / \lambda\), \(\eta = q/p'\), and \(p'_i\) and \(\eta_i\) define the effective stress state at impending yield, indicated as point \(I\) in (b) in the figure above.
Note that, under undrained conditions, the yield path is defined by an equation of the form shown in equation (2) for any boundary condition (i.e., not only under triaxial compression conditions).
Intersection of the yield curve through \(p'_{c0}\) with the straight path \(p' = p'_0\) gives (using that \(R = p'_{c0}/p'_0\))
After substitution of those expressions in equation (2), we obtain
As the test proceeds, the path converges toward the critical state represented by the point \((p'_{cr},q_{cr})\) at the intersection with the critical state line \(q = M p'\) in the \((p',q)\) plane (see both figures above). The normal to the yield surface at the critical point is parallel to the \(q\)axis. Because the plastic flow rule is associated, no more plastic volumetric strain can take place. Hence, no softening or hardening of the yield surface can occur; the ultimate yield surface corresponds to a \(p'_c\) value of \(2 p'_{cr}\), which is larger than \(p'_{c0}\) for a LOC sample, and smaller than \(p'_{c0}\) for a HOC sample. Unlimited plastic shear strains can develop at constant stresses, and also constant critical specific volume \(v_{cr}\).
In the drained case, the critical state is defined as
where the value \(v_{cr}\) corresponds to the specific volume at \(p = p_{cr}\) on the elastic swelling line through \(p_c = 2 p_{cr}\). In this case, the critical state of a specific material depends only on the initial mean pressure and is not affected by the preconsolidation pressure.
In the undrained case, the intersection of the stress path represented by equation (3) with the critical state line \(\eta = M\) yields (assuming that \(v\) remains constant)
where \(v_0\) is the initial specific volume.
The excess pore pressure \(u\) is given by
Using equation (1) for the total pressure, we obtain
And, at the critical state,
FLAC3D Models
The numerical tests are carried out using one single zone with unit dimensions. The grid is fixed in the \(z\)direction, and a velocity boundary condition applied at the top of the model. An initial isotropic compressive stress of 5 × \(p'_1\) is prescribed, and a constant lateral confining pressure \(p_0 = 5 \times p'_1\) is imposed in the \(x\) and \(y\)directions. To run the undrained examples, the fluid configuration is selected and flow calculations are turned off.
The initial specific volume is calculated internally (default value) to correspond with the value on the swelling line at the preconsolidation pressure \(p'_{c0}\) and mean pressure \(p_0\):
The bulk modulus, \(K\), is initialized by the code to the value
The maximum value of the tangent bulk modulus is set to 800 × \(p'_1\).
A total of 10,000 cycles with a velocity magnitude of 0.5 × 10^{4} m/sec was used in the drained examples. For the undrained tests, a compressive velocity of magnitude 0.5 × 10^{6} m/sec is applied for the first 20 out of every 40 steps at the top of the model, and to zero for the remaining 20 steps. A total of 40,000 cycles are used. Biot modulus is set to 2 × 10^{4} × \(p'_1 /n\) (a large value compared to the initial value of the bulk modulus, which is of the order 10^{2} × \(p'_1\)). (The porosity n is derived from the specific volume using \(n = (v1/v\)).) The mean pressure, deviatoric stress, specific volume, and, in the undrained case, pore pressure are monitored as they converge to the critical state.
FLAC3D Results and Discussion
Numerical values for \(p'_1\), \(q\), \(v\), and \(u\) at the end of the simulation are compared with the analytical predictions. The results, presented in Table 2 and Table 3, indicate relative errors of less than 2%.
R = 1.6 
R = 8 
Analytical 


\(p\) 
7.573 
7.583 
7.576 
\(q\) 
7.718 
7.747 
7.727 
\(v\) 
2.811 
2.811 
2.811 
R = 1.6 
R = 8 


Numerical 
Analytical 
Numerical 
Analytical 

\(p'_1\) 
4.234 
4.229 
14.05 
14.14 
\(q\) 
4.312 
4.314 
14.42 
14.43 
\(v\) 
2.927 
2.928 
2.687 
2.686 
\(u\) 
2.203 
2.209 
4.241 
4.334 
The diagrams \((p',q)\) and \((\ln p',v)\) for the different tests are presented in Figure 3 to Figure 10. The responses of the lightly and heavily overconsolidated samples on their way to the critical state are in agreement with those predicted by the theory.
As the drained test progresses, the lightly overconsolidated sample shows a steady increase in deviator stress \(q\), and a steady decrease in specific volume. The heavily overconsolidated sample shows a rise in deviatoric stress to a peak followed by a drop, and an initial decrease in volume followed by volumetric expansion (see Figure 11 through Figure 14). The principal feature of the undrained tests is the contrast between the steady increase of pore pressure in the lightly overconsolidated sample, and the initial increase followed by a steady decrease of pore pressure in the heavily overconsolidated soil (see Figure 15 and Figure 16).
The data files “drained.dat” and “undrained.dat” were used to carry out the drained and undrained numerical tests. The file “TriaxialCompressionTest.dat” was used to set the property mpc to the values 8 and 40 to treat the lightly and heavily overconsolidated cases, respectively. FISH functions are used to apply the velocity boundary conditions and evaluate the relative error made at the end of the simulation.
References
Wood, D. M. Soil Behaviour and Critical State Soil Mechanics. Cambridge: Cambridge University Press (1990).
Data Files
TriaxialCompressionTest.dat
; Primary driver for the camclay verification problem
;
model new
fish automaticcreate off
[global pre_pres = 8]
model title "Drained triaxial compression test R = 1.6"
program call 'drained'
model save 'draineda'
;
model new
fish automaticcreate off
[global pre_pres = 40]
model title "Drained triaxial compression test R = 8.0"
program call 'drained'
model save 'drainedb'
;
model new
fish automaticcreate off
[global pre_pres = 8]
model title "Undrained triaxial compression test R = 1.6"
program call 'undrained'
model save 'undraineda'
;
model new
fish automaticcreate off
[global pre_pres = 40]
model title "Undrained triaxial compression test R = 8.0"
program call 'undrained'
model save 'undrainedb'
drained.dat
model largestrain off
;
; Drained triaxial compression test on Camclay sample
;
zone create brick size 1 1 1
;  model properties 
zone cmodel assign modifiedcamclay
zone property shear 250. bulkmaximum 800.
zone property ratiocriticalstate 1.02 lambda 0.2 kappa 0.05
zone property pressurepreconsolidation [pre_pres] pressurereference 1. ...
specificvolumereference 3.32
;  boundary conditions 
zone gridpoint fix velocityz
zone face apply stressxx 5. range union positionx 0 positionx 1
zone face apply stressyy 5. range union positiony 0 positiony 1
zone initialize stress xx 5. yy 5. zz 5.
; FISH to initialize effective pressure property
; This is overkill for a one zone problem, but an example for a larger one.
fish operator initialize(zone)
zone.prop(zone,'pressureeffective') = ...
tensor.trace(zone.stress(zone))/3.0  zone.pp(zone)
end
[initialize(::zone.list)]
; ... velocity boundary conditions ...
zone gridpoint initialize velocityz 0.5e4 range positionz 1.0
; Set up global FISH functions,
; to take histories of values derived from properties
fish define sqcr ; pressureeffective * ratiocriticalstate
local zone = zone.head
sqcr = zone.prop(zone,'pressureeffective') * ...
zone.prop(zone,'ratiocriticalstate')
end
;  histories 
history interval 40
zone history name 'sp' property name 'pressureeffective' zone 1
zone history name 'sq' property name 'stressdeviatoric' zone 1
fish history name 'sqcr' [sqcr]
zone history name 'sv' property name 'specificvolume' zone 1
zone history name 'disp' displacementz position (0,0,1)
; Store starting values of some properties 
; used later for analytical comparison
[global pr0 = zone.prop(zone.head,'pressurereference')]
[global ratio0 = zone.prop(zone.head,'ratiocriticalstate')]
[global sv0 = zone.prop(zone.head,'specificvolumereference')]
[global lambda0 = zone.prop(zone.head,'lambda')]
[global kappa0 = zone.prop(zone.head,'kappa')]
;  test 
model cycle 10000
undrained.dat
model largestrain off
;
; Undrained triaxial compression test on Camclay sample
;
model config fluid
zone create brick size 1 1 1
;  mechanical properties 
zone cmodel assign modifiedcamclay
zone property shear 250. bulkmaximum 800.
zone property ratiocriticalstate 1.02 lambda 0.2 kappa 0.05
zone property pressurepreconsolidation [pre_pres] pressurereference 1. ...
specificvolumereference 3.32
;  boundary conditions 
zone gridpoint fix velocityz
zone face apply stressxx 5. range union positionx 0 positionx 1
zone face apply stressyy 5. range union positiony 0 positiony 1
zone initialize stress xx 5. yy 5. zz 5.
; FISH to initialize effective pressure property
; This is overkill for a one zone problem, but an example for a larger one.
fish operator initialize(zone)
zone.prop(zone,'pressureeffective') = ...
tensor.trace(zone.stress(zone))/3.0  zone.pp(zone)
end
[initialize(::zone.list)]
;  fluid properties 
zone fluid cmodel assign isotropic
zone fluid biot on
model fluid active off
;  fish functions 
; ... velocity boundary conditions ...
fish define cycle ; 20 steps on, 20 steps off
cycle = 1.0  float(((global.step1) // 20) % 2)
end
zone face apply velocityz 0.5e6 fish cycle range positionz 1.0
; Set up global FISH functions,
; to take histories of values derived from properties
fish define sqcr ; pressureeffective * ratiocriticalstate
local zone = zone.find(1)
sqcr = zone.prop(zone,'pressureeffective') * ...
zone.prop(zone,'ratiocriticalstate')
end
;  histories 
history interval 2000
zone history name 'sp' property name 'pressureeffective' zone 1
zone history name 'sq' property name 'stressdeviatoric' zone 1
fish history name 'sqcr' sqcr
zone history name 'sv' property name 'specificvolume' zone 1
zone history name 'disp' displacementz position (0,0,1)
zone history name 'p_fl' porepressure zone 1
;  test 
model cycle 1
fish define bi ; Initialize BIOT modulus
global v0 = zone.prop(zone.find(1),'specificvolume')
local n0 = (v0  1.) / v0
bi = 2.e4 / n0
end
zone gridpoint initialize biot [bi]
; Store starting values of some properties 
; used later for analytical comparison
[global pp0 = zone.prop(zone.find(1),'pressurepreconsolidation')]
[global kappa0 = zone.prop(zone.find(1),'kappa')]
[global lambda0 = zone.prop(zone.find(1),'lambda')]
[global ratio0 = zone.prop(zone.find(1),'ratiocriticalstate')]
; Solve
model cycle 400000
Endnotes
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