# How to simulate small-strain triaxial compression test

Hi,

I'd like to ask that how to simulate a drained triaxial compression test within a small-strain level.

The goal of this simulation is to get the small strain stiffness G0.

A typical way is (1) generating a sand packing and isotropically load it to target confining pressure, e.g., 100 kPa. (2) a small axial strain increment deps1 is applied on the top wall, while the lateral stresses sigma2 and sigma3 keep constant. (3) the simulation finishes when the shear strain, gamma = deps1-deps2, reaches 10e-6. (4) then G0 can be calculated by G0 = dsigma1/(2*gamma)

What I have done is using the example code provided by Bruno. I set the loading rate to a very small value (1e-20), and the simulations finished in one second. I thought this process is too quick to generate accurate results, actually the lateral confining pressure changes (i.e., not constant). I know this script works well for a classic triaxial compression test because usually we look at a large strain level (like 20% of axial strain), thereby a small amout of fluctuation on lateral stress is acceptable. But when we look at such a small range and in such a short time, this will lead to inaccurate results. So my question is how can we make the simulation of triaxial compression test within a small-strain level.

Thanks,
Leonard

The MWE is as follow if the question is not clearly described.

num_spheres=1000,# number of spheres
compFricDegree = 30, # contact friction during the confining phase
key='_triax_base_', # put you simulation's name here
unknownOk=True
)

num_spheres=table.num_spheres# number of spheres
key=table.key
targetPorosity = 0.43 #the porosity we want for the packing
compFricDegree = table.compFricDegree # initial contact friction during the confining phase (will be decreased during the REFD compaction process)
damp=0.2 # damping coefficient
stabilityThreshold=0.01 # we test unbalancedForce against this value in different loops (see below)
young=5e6 # contact stiffness
mn,mx=Vector3(0,0,0),Vector3(1,1,1) # corners of the initial packing

O.materials.append(FrictMat(young=young,poisson=0.5,frictionAngle=0,density=0,label='walls'))

walls=aabbWalls([mn,mx],thickness=0,material='walls')
wallIds=O.bodies.append(walls)

sp=pack.SpherePack()

sp.makeCloud(mn,mx,-1,0.3333,num_spheres,False, 0.95,seed=1) #"seed" make the "random" generation always the same

triax=TriaxialStressController(
maxMultiplier=1.+2e4/young, # spheres growing factor (fast growth)
finalMaxMultiplier=1.+2e3/young, # spheres growing factor (slow growth)
thickness = 0,

internalCompaction=True, # If true the confining pressure is generated by growing particles
)

newton=NewtonIntegrator(damping=damp)

O.engines=[
ForceResetter(),
InsertionSortCollider([Bo1_Sphere_Aabb(),Bo1_Box_Aabb()]),
InteractionLoop(
[Ig2_Sphere_Sphere_ScGeom(),Ig2_Box_Sphere_ScGeom()],
[Ip2_FrictMat_FrictMat_FrictPhys()],
[Law2_ScGeom_FrictPhys_CundallStrack()]
),
GlobalStiffnessTimeStepper(active=1,timeStepUpdateInterval=100,timestepSafetyCoefficient=0.8),
triax,
TriaxialStateRecorder(iterPeriod=100,file='WallStresses'+table.key),
newton,
]

Gl1_Sphere.stripes=0

triax.goal1=triax.goal2=triax.goal3=-100000

while 1:
O.run(1000, True)
unb=unbalancedForce()
print 'unbalanced force:',unb,' mean stress: ',triax.meanStress
if unb<stabilityThreshold and abs(-100000-triax.meanStress)/100000<0.001:
break

print "### Isotropic state saved ###"

import sys #this is only for the flush() below
while triax.porosity>targetPorosity:
# we decrease friction value and apply it to all the bodies and contacts
compFricDegree = 0.95*compFricDegree
print "\r Friction: ",compFricDegree," porosity:",triax.porosity,
sys.stdout.flush()
O.run(500,1)

print "### Compacted state saved ###"

triax.internalCompaction=False

triax.goal2=rate
triax.goal1=-100000
triax.goal3=-100000
newton.damping=0.1

def history():
ev=-triax.strain-triax.strain-triax.strain,
s11=-triax.stress(triax.wall_right_id),
s22=-triax.stress(triax.wall_top_id),
s33=-triax.stress(triax.wall_front_id),
i=O.iter)

def stop():
if abs(triax.strain-triax.strain)>1e-6:
O.pause()

O.engines=O.engines+[PyRunner(iterPeriod=20,command='history()',label='recorder')]
O.engines=O.engines+[PyRunner(iterPeriod=1,command='stop()',label='stop')]

### declare what is to plot. "None" is for separating y and y2 axis
# plot.plots={'i':('e11','e22','e33',None,'s11','s22','s33')}
## the traditional triaxial curves would be more like this:
#plot.plots={'e22':('s11','s22','s33',None,'ev')}

# display on the screen (doesn't work on VMware image it seems)
# plot.plot()

## Question information

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Jérôme Duriez
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 Revision history for this message Leonard (z2521899293) said on 2021-12-26: #1

Hi,

I came up with a way: using a small enough dt for the simulation.

Do you have any other ideas?

Cheers
Leonard

 Revision history for this message  Jérôme Duriez (jduriez) said on 2022-01-05: #2

Hi,

Decrease TriaxialStressController.max_vel ?

Which is the first responsible for lateral stress fluctuations (or absence of)

 Revision history for this message Leonard (z2521899293) said on 2022-01-06: #3

Thanks Jérôme Duriez, that solved my question.

 Revision history for this message Bruno Chareyre (bruno-chareyre) said on 2022-01-06: #4

Transitioning between phases is often based on the ubalanced force (see "stabilityThreshold=0.01").
That's basically a residual amount of dynamic noise we accept.
If you want a very clean equilibrium state before imposing a strain increment you can aim at an even smaller value.

I'm surprised decreasing max_vel alone is a sufficient answer.

Bruno