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[1]. 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

[1]https://gitlab.com/yade-dev/trunk/blob/master/examples/triax-tutorial/script-session1.py

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

from yade import pack

nRead=readParamsFromTable(
 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
)
from yade.params import table

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)
finalFricDegree = 30 # contact friction during the deviatoric loading
rate=-1e-20 # loading rate (strain rate)
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=radians(compFricDegree),density=2600,label='spheres'))
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
O.bodies.append([sphere(center,rad,material='spheres') for center,rad in sp])

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

 stressMask = 7,
 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
if nRead==0: yade.qt.Controller(), yade.qt.View()

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
 setContactFriction(radians(compFricDegree))
 print "\r Friction: ",compFricDegree," porosity:",triax.porosity,
 sys.stdout.flush()
 O.run(500,1)

O.save('compactedState'+key+'.yade.gz')
print "### Compacted state saved ###"

triax.internalCompaction=False

setContactFriction(radians(finalFricDegree))

triax.stressMask = 5
triax.goal2=rate
triax.goal1=-100000
triax.goal3=-100000
newton.damping=0.1

print "gamma before deviatoric loading is", abs(triax.strain[1]-triax.strain[0])
print "click run to start small-strain deviatoric loading"
from yade import plot

def history():
 plot.addData(e11=-triax.strain[0], e22=-triax.strain[1], e33=-triax.strain[2],
   ev=-triax.strain[0]-triax.strain[1]-triax.strain[2],
   s11=-triax.stress(triax.wall_right_id)[0],
   s22=-triax.stress(triax.wall_top_id)[1],
   s33=-triax.stress(triax.wall_front_id)[2],
   i=O.iter)

def stop():
 if abs(triax.strain[1]-triax.strain[0])>1e-6:
  O.pause()
  print "gamma after deviatoric loading is", abs(triax.strain[1]-triax.strain[0])

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()