3.1.6.7. PM4Sand Material

Code Developed by: Long Chen and Pedro Arduino at U.Washington.

This command is used to construct a 2-dimensional PM4Sand material ([Boulanger-Ziotopoulou2017]).

nDmaterial PM4Sand $matTag $Dr $G0 $hpo $Den <$patm $h0 $emax $emin $nb $nd $Ado $zmax $cz $ce $phic $nu $cgd $cdr $ckaf $Q $R $m $Fsed_min $p_sedo>

Argument	Type	Description
$matTag	integer	tag identifying material
$Dr	float	Relative density (fraction)
$G0	float	Shear modulus constant
$hpo	float	Contraction rate parameter
$Den	float	Mass density of the material
$P_atm	float	optional: Atmospheric pressure
$h0	float	optional: Variable that adjusts the ratio of plastic modulus to elastic modulus
$emax and $emin	float	optional: Maximum and minimum void ratios
$nb	float	optional: Bounding surface parameter $nb ≥ 0
$nd	float	optional: Dilatancy surface parameter $nd ≥ 0
$Ado	float	optional: Dilatancy parameter will be computed at the time of initialization if input value is negative
$z_max	float	optional: Fabric-dilatancy tensor parameter
$cz	float	optional: Fabric-dilatancy tensor parameter
$ce	float	optional: Variable that adjusts the rate of strain accumulation in cyclic loading
$phic	float	optional: Critical state effective friction angle
$nu	float	optional: Poisson’s ratio
$cgd	float	optional: Variable that adjusts degradation of elastic modulus with accumulation of fabric
$cdr	float	optional: Variable that controls the rotated dilatancy surface
$ckaf	float	optional: Variable that controls the effect that sustained static shear stresses have on plastic modulus
$Q	float	optional: Critical state line parameter
$R	float	optional: Critical state line parameter
$m	float	optional: Yield surface constant (radius of yield surface in stress ratio space)
$Fsed_min	float	optional: Variable that controls the minimum value the reduction factor of the elastic moduli can get during reconsolidation
$p_sedo	float	optional: Mean effective stress up to which reconsolidation strains are enhanced

Note

The only material formulation for the PM4Sand object is “PlaneStrain”, as a consequence limited to plain strain continuum elements.

Valid Element Recorder queries are stress, strain, alpha (or backstressratio) for \(\mathbf{\alpha}\), fabric for \(\mathbf{z}\), and alpha_in (or alphain) for \(\mathbf{\alpha_{in}}\)

Elastic or response could be enforced by

updateMaterialStage -material $matTag -stage 0

Elastoplastic by

updateMaterialStage -material $matTag -stage 1

The program will use the default value of a secondary parameter if a negative input is assigned to that parameter, e.g. Ado = -1. However, FirstCall is mandatory when switching from elastic to elastoplastic if negative inputs are assigned to stress-dependent secondary parameters, e.g. Ado and zmax. FirstCall can be set as,

setParameter -value 0 -ele $elementTag FirstCall $matTag

Post-shake reconsolidation can be activated by

setParameter -value 1 -ele $elementTag Postshake $matTag

The user should check that the results are not sensitive to time step size.

[Boulanger-Ziotopoulou2017]

R.W.Boulanger, K.Ziotopoulou. “PM4Sand(Version 3.1): A Sand Plasticity Model for Earthquake Engineering Applications”. Report No. UCD/CGM-17/01 2017

Example 1

2D undrained monotonic direct simple shear test using one element

# HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH #
# 2D Undrained Direct Simple Shear Test Using One Element #
# University of Washington, Department of Civil and Environmental Eng   #
# Geotechnical Eng Group, L. Chen, P. Arduino - Jan 2018               #
# Basic Units are m, kN and s unless otherwise specified#
# HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH #

wipe

# ------------------------ #
# Test Specific parameters #
# ------------------------ #
# Initial Vertical Stress
set sigvo -101.3
set K0 0.5
# set Poisson's ratio to match user specified K0
set nu [expr $K0 / (1+$K0)]
# Deviatoric strain (Cyclic)
set devDisp 0.10
# Permeablity
set perm 1.0e-9
# Initial void ratio
# relative density
set Dr 0.35   
# max and min void ratio  
set emax 0.8
set emin 0.5
set eInit [expr $emax - ($emax - $emin)*$Dr ]
# other primary parameters
set G0 476.0
set hpo 0.53
set rho 1.42

# Rayleigh damping parameter
set damp   0.02
set omega1 0.2
set omega2 20.0
set a1 [expr 2.0*$damp/($omega1+$omega2)]
set a0 [expr $a1*$omega1*$omega2]

# HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH
# HHHHHHHHHHHHHHHHHHHHHHHHHHHCreate ModelHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH
# HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH

# Create a 2D model with 3 Degrees of Freedom
model BasicBuilder -ndm 2 -ndf 3

# Create nodes
node 10.00.0
node 21.00.0
node 3 1.01.0
node 40.01.0

# Create Fixities
fix 1 1 1 1
fix 2 1 1 1
fix 30 0 1
fix 4 0 0 1
 
equalDOF 3 4 1 2

# Create material
#          PM4Sand  tag    Dr   G0   hpo   den  Patm  h0     emax   emin  nb   nd  Ado   zmax    cz    ce     phicv  nu 
nDMaterial PM4Sand   1    $Dr  $G0  $hpo  $rho 101.3 -1.00   $emax  $emin 0.5  0.1  -1.0  -1.0  250.0  -1.00  33.0  $nu
 
# Create element
element SSPquadUP   1     1 2 3 4    1  1.0   2.2e6 1.0 $perm $perm $eInit  1.0e-5

# Create recorders
recorder Node  -nodeRange 1 4  -time -file Cycdisp.out  -dof 1 2 disp
recorder Node  -nodeRange 1 4  -time -file Cycpress.out -dof 3 vel
recorder Element -ele 1 -time -file Cycstress.out stress
recorder Element -ele 1 -time -file Cycstrain.out strain

# Create analysis
constraints Transformation
test        NormDispIncr 1.0e-5 35 1
algorithm   Newton
numberer    RCM
system      SparseGeneral
integrator  Newmark [expr 5.0 / 6.0] [expr  4.0 / 9.0]
rayleigh    $a0 $a1 0.0 0.0
analysis    Transient

# Apply consolidation pressure
set pNode [expr $sigvo / 2.0]
pattern Plain 1 {Series -time {0 100 1e10} -values {0 1 1} -factor 1} {
    load 3  0.0  $pNode 0.0
    load 4  0.0  $pNode 0.0
}
updateMaterialStage -material 1 -stage 0

analyze 100 1
set vDisp [nodeDisp 3 2]
set ts1 "{Series -time {100 80000 1.0e10} -values {1.0 1.0 1.0} -factor 1}"
eval "pattern Plain 2 $ts1 {
sp 3 2 $vDisp
sp 4 2 $vDisp
}"

# Close drainage valves
for {set x 1} {$x< 5} {incr x} {
   remove sp $x 3
}

analyze 25 1
puts "Removed drainage fixities."

updateMaterialStage -material 1 -stage 1
setParameter -value 0 -ele 1 FirstCall 1
analyze 25 1

puts "finished update fixties"

set ts2 "{Series -time {150 5150 1.0e10} -values {0.0 1.0 1.0} -factor 1}"

eval "pattern Plain 3 $ts2 {
sp 3 1 $devDisp
}"
# update Poisson's ratio for analysis
setParameter -value 0.3 -ele 1 poissonRatio 1
analyze 5000 1

wipe

Example 2

2D undrained cyclic direct simple shear test using one element (Displacement Controlled)

# HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH #
# 2D Undrained Cyclic Direct Simple Shear Test Using One Element        #
# University of Washington, Department of Civil and Environmental Eng   #
# Geotechnical Eng Group, L. Chen, P. Arduino - Feb 2018                #
# Basic Units are m, kN and s unless otherwise specified#
# HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH #

wipe

# ------------------------ #
# Test Specific parameters #
# ------------------------ #
# Initial Vertical Stress
set sigvo -101.3
# cyclic stress ratio
set CSR 0.16
# max number of cycles
set maxCycles 20
# strain increment
set strainIncr 5.0e-6
# K0
set K0 0.5
# set Poisson's ratio to match user specified K0 for applying initial confinement
set nu [expr $K0 / (1+$K0)]
# Cutoff shear strain
set devDisp 0.03
# Permeablity
set perm 1.0e-9
# ---------primary parameters-------------
set Dr 0.35   
set G0 476.0
set hpo 0.53
set rho 1.42
# ---------secondary parameters-------------
set Patm 101.3
# all initial stress dependant parameters have negative default values 
# and will be calculated during initialization
set h0 -1.0  
set emax 0.8
set emin 0.5
set eInit [expr $emax - ($emax - $emin)*$Dr ]
set nb 0.5
set nd 0.1
set Ado -1.0
set zmax -1.0
set cz 250.0
set ce -1.0
set phicv 33.0
set Cgd 2.0
set Cdr -1.0
set ckaf -1.0
set Q 10.0
set R 1.5
set m_m 0.01
set Fsed_min -1.0
set p_sedo -1.0
# ---------------------------------------------
# Rayleigh damping parameter
set damp   0.02
set omega1 0.2
set omega2 20.0
set a1 [expr 2.0*$damp/($omega1+$omega2)]
set a0 [expr $a1*$omega1*$omega2]

# HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH
# HHHHHHHHHHHHHHHHHHHHHHHHHHHCreate ModelHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH
# HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH

# Create a 2D model with 3 Degrees of Freedom
model BasicBuilder -ndm 2 -ndf 3

# Create nodes
node 10.00.0
node 21.00.0
node 3 1.01.0
node 40.01.0

# Create Fixities
fix 1 1 1 1
fix 2 1 1 1
fix 30 0 1
fix 4 0 0 1
 
equalDOF 3 4 1 2

# Create material
#          PM4Sand  tag    Dr   G0   hpo   den  Patm  h0   emax   emin  nb  nd  Ado   zmax    cz    ce     phicv  nu 
nDMaterial PM4Sand   1    $Dr  $G0  $hpo  $rho $Patm $h0  $emax $emin  $nb  $nd $Ado  $zmax  $cz   $ce  $phicv  $nu $Cgd $Cdr $ckaf $Q $R $m_m $Fsed_min $p_sedo
 
# Create element
element SSPquadUP   1     1 2 3 4    1  1.0   2.2e6 1.0 $perm $perm $eInit  1.0e-5

# Create recorders
recorder Node  -nodeRange 1 4  -time -file Cycdisp.out  -dof 1 2 disp
recorder Node  -nodeRange 1 4  -time -file Cycpress.out -dof 3 vel
recorder Element -ele 1 -time -file Cycstress.out stress
recorder Element -ele 1 -time -file Cycstrain.out strain

# Create analysis
constraints Transformation
test        NormDispIncr 1.0e-5 35 1
algorithm   Newton
numberer    RCM
system      FullGeneral
integrator  Newmark [expr 5.0 / 6.0] [expr  4.0 / 9.0]
rayleigh    $a0 $a1 0.0 0.0
analysis    Transient

# Apply consolidation pressure
set pNode [expr $sigvo / 2.0]
pattern Plain 1 {Series -time {0 100 1e10} -values {0 1 1} -factor 1} {
    load 3  0.0  $pNode 0.0
    load 4  0.0  $pNode 0.0
}
updateMaterialStage -material 1 -stage 0

analyze 100 1
set vDisp [nodeDisp 3 2]
set ts1 "{Series -time {100 80000 1.0e10} -values {1.0 1.0 1.0} -factor 1}"
eval "pattern Plain 2 $ts1 {
sp 3 2 $vDisp
sp 4 2 $vDisp
}"

# Close drainage valves
for {set x 1} {$x< 5} {incr x} {
   remove sp $x 3
}

analyze 25 1
puts "Removed drainage fixities."

updateMaterialStage -material 1 -stage 1
setParameter -value 0 -ele 1 FirstCall 1
analyze 25 1

puts "finished update fixties"

# update Poisson's ratio for analysis
setParameter -value 0.3 -ele 1 poissonRatio 1
set controlDisp [expr 1.1 * $devDisp]
set numCycle 0.25
while {$numCycle <= $maxCycles} {
    #impose 1/4 cycle
    set hDisp [nodeDisp 3 1]
    set currentTime [getTime]
    set steps [expr $controlDisp / $strainIncr]
    eval "timeSeries Path 2 -time {$currentTime [expr $currentTime + $steps] 1.0e10} -values {$hDisp $controlDisp $controlDisp} -factor 1"
    pattern Plain 3 2 {
	sp 3 1 1.0
    }
    set b [eleResponse 1 stress]
    while {[lindex $b 2] < [expr $CSR * -1.0 * $sigvo]} {
	analyze 1 1.0
	set b [eleResponse 1 stress]
	set hDisp [nodeDisp 3 1]
	if {$hDisp >= $devDisp} {
	    exit
	}
    }
    set numCycle [expr $numCycle + 0.25]
    set hDisp [nodeDisp 3 1]
    set currentTime [getTime]
    
    remove loadPattern 3
    remove timeSeries 2
    remove sp 3 1
    
    #impose 1/2 cycle 
    set steps [expr ($controlDisp + $hDisp)/ $strainIncr]
    eval "timeSeries Path 2 -time {$currentTime [expr $currentTime + $steps] 1.0e10} -values {$hDisp [expr -1.0 * $controlDisp] [expr -1.0 * $controlDisp]} -factor 1"
    
    pattern Plain 3 2 {
	sp 3 1 1.0
    }
    
    while {[lindex $b 2] > [expr $CSR * $sigvo]} {
	analyze 1 1.0
	set b [eleResponse 1 stress]
	set hDisp [nodeDisp 3 1]
	if {$hDisp <= [expr -1.0 * $devDisp]} {
	    exit
	}
    }
    #impose 1/4 cycle
    set numCycle [expr $numCycle + 0.5]
    set hDisp [nodeDisp 3 1]
    set currentTime [getTime]
    
    remove loadPattern 3
    remove timeSeries 2
    remove sp 3 1
    set steps [expr ($controlDisp + $hDisp)/ $strainIncr]
    eval "timeSeries Path 2 -time {$currentTime [expr $currentTime + $steps] 1.0e10} -values {$hDisp $controlDisp $controlDisp} -factor 1"
    
    pattern Plain 3 2 {
	sp 3 1 1.0
    }
    while {[lindex $b 2] <= 0.0} {
	analyze 1 1.0
	set b [eleResponse 1 stress]
	set hDisp [nodeDisp 3 1]
	if {$hDisp >= $devDisp} {
	    exit
	}
    }
    set numCycle [expr $numCycle + 0.25]
    remove loadPattern 3
    remove timeSeries 2
}

wipe

Example 3

2D undrained cyclic direct simple shear test using one element (Force Controlled)

# HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH #
# 2D Undrained Cyclic Direct Simple Shear Test Using One Element        #
# University of Washington, Department of Civil and Environmental Eng   #
# Geotechnical Eng Group, L. Chen, P. Arduino - Oct 2020                #
# Basic Units are m, kN and s unless otherwise specified				#
# HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH #

wipe

# ------------------------ #
# Test Specific parameters #
# ------------------------ #
# Initial Vertical Stress
set sigvo -101.3
# cyclic stress ratio
set CSR 0.16
# max number of cycles
set maxCycles 20
# K0
set K0 0.5
# set Poisson's ratio to match user specified K0 for applying initial confinement
set nu_gravity [expr $K0 / (1+$K0)]
set nu_dynamic 0.3
# Cutoff shear strain
set maxStrain 0.03
# Permeablity
set perm 1.0e-9
# ---------primary parameters-------------
set Dr 0.55
set Go 677.0
set hpo 0.40
set rho 0.0
# ---------secondary parameters-------------
set Patm 101.3
# all initial stress dependant parameters have negative default values
# and will be calculated during initialization
set h0 -1.0
set emax 0.8
set emin 0.5
set eInit [expr $emax - ($emax - $emin)*$Dr ]
set nb 0.5
set nd 0.1
set Ado -1.0
set zmax -1.0
set cz 250.0
set ce -1.0
set phicv 33.0
set Cgd 2.0
set Cdr -1.0
set ckaf -1.0
set Q 10.0
set R 1.5
set m_m 0.01
set Fsed_min -1.0
set p_sedo -1.0
# ---------------------------------------------
# Rayleigh damping parameter
set damp   0.01
set omega1 1.0
set omega2 2.0
set a1 [expr 2.0*$damp/($omega1+$omega2)]
set a0 [expr $a1*$omega1*$omega2]

# HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH
# HHHHHHHHHHHHHHHHHHHHHHHHHHHCreate ModelHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH
# HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH

# Create a 2D model with 3 Degrees of Freedom
model BasicBuilder -ndm 2 -ndf 3

# Create nodes
node 1	0.0	0.0
node 2	1.0	0.0
node 3 	1.0	1.0
node 4	0.0	1.0

# Create Fixities
fix 1 	1 1 1
fix 2 	1 1 1
fix 3	0 0 1
fix 4 	0 0 1

equalDOF 3 4 1 2

# Create material
#          PM4Sand  tag    Dr   Go   hpo   den  Patm  h0   emax   emin  nb  nd  Ado   zmax    cz    ce     phicv  nu
nDMaterial PM4Sand   1    $Dr  $Go  $hpo  $rho $Patm $h0  $emax $emin  $nb  $nd $Ado  $zmax  $cz   $ce	  $phicv  $nu_gravity $Cgd $Cdr $ckaf $Q $R $m_m $Fsed_min $p_sedo

# set e_init 0.6
# set rho1 0.0
# nDMaterial ManzariDafalias 1 82.35 0.33 $e_init 1.35 0.7 0.055 0.8 0.5 101.3 0.02 16.18 0.996 0.64 0.75 1.5 12.5 500.0 $rho1 1 0

# nDMaterial PressureDependMultiYield02 1 2 0.0 9.0e4 2.2e5 32 0.1 80 0.5\
                                          # 26. 0.067 0.23 0.06 0.27

# Using default values for secondary parameters
# nDMaterial PM4Sand $matTag 0.61 585.0 0.28 0.0
# nDMaterial PM4Silt $matTag 44.0 -1 585 100.0 0.0
# nDMaterial PM4Silt $matTag -1 0.5 585 100.0 0.0

# Create element
element SSPquadUP   1     1 2 3 4    1  1.0   2.2e6 1.0 $perm $perm $eInit  1.0e-6
# element quadUP  1     1 2 3 4  1.0  1 2.2e6 1.0 $perm $perm
# Create analysis
constraints Penalty 1e17 1e17
test        NormDispIncr 1.0e-5 35 0
algorithm   Newton
numberer    RCM
system      ProfileSPD
integrator  Newmark [expr 5.0 / 6.0] [expr  4.0 / 9.0]
rayleigh    $a0 $a1 0.0 0.0
analysis    VariableTransient

# Apply consolidation pressure
set pNode [expr $sigvo / 2.0]
pattern Plain 1 {Series -time {0 100 1e10} -values {0 1 1} -factor 1} {
	load 3  0.0  $pNode 0.0
	load 4  0.0  $pNode 0.0
}
updateMaterialStage -material 1 -stage 0

analyze 100 1

updateMaterialStage -material 1 -stage 1
setParameter -value 0 -ele 1 FirstCall 1
# update Poisson's ratio for analysis
setParameter -value $nu_dynamic -ele 1 poissonRatio 1

# damp out any gravity wave
analyze 100 100
analyze 100 1000

puts "Finished Gravity"

# Close drainage valves
for {set x 1} {$x< 5} {incr x} {
   remove sp $x 3
}
equalDOF 3 4 3
puts "finished update fixties"

# Hold load and redefine analysis
loadConst -time 0.0
wipeAnalysis

set period      		1.;
set dT 				5e-5;
set recDT     [expr 10 * $dT]
# Create recorders
recorder Node  -nodeRange 1 4  -time -dT $recDT -file Cycdisp.out  -dof 1 2 disp
recorder Node  -nodeRange 1 4  -time -dT $recDT -file Cycpress.out -dof 3 vel

# recorder for quadUP element
# recorder Element -ele 1 -time -dT $recDT -file Cycstress.out material 1 stress
# recorder Element -ele 1 -time -dT $recDT -file Cycstrain.out material 1 strain

# recorder for SSPquadUP element
recorder Element -ele 1 -time -dT $recDT -file Cycstress.out stress
recorder Element -ele 1 -time -dT $recDT -file Cycstrain.out strain
# Create analysis
constraints Penalty 1e17 1e17
test        NormDispIncr 1.0e-5 35 0
algorithm   KrylovNewton
numberer    RCM
system      ProfileSPD
integrator  Newmark [expr 5.0 / 6.0] [expr  4.0 / 9.0]
rayleigh    $a0 0.0 0.0 0.0
analysis    Transient

# This part is modified from PDMY03 driver by Arash Khosravifar

pattern Plain 2 "Sine 0 [expr $period*$maxCycles] $period" {
	load 3 [expr -$CSR*$sigvo*1.0*1.0*0.50] 0 0
	load 4 [expr -$CSR*$sigvo*1.0*1.0*0.50] 0 0
}

while {[expr abs([nodeDisp 4 1]/1.0)]<$maxStrain} {
	analyze 1 $dT
	puts "time = [getTime] sec"
}

wipe