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CREPORT NO. UCD/CGM-01/03
FO
B BKS
DCU S
ENTER FOR GEOTECHNICAL MODELING
INITE DEFORMATION ANALYSIS F GEOMATERIALS
Y
. JEREMIC
. RUNESSON
. STURE
EPARTMENT OF CIVIL & ENVIRONMENTAL ENGINEERING OLLEGE OF ENGINEERING NIVERSITY OF CALIFORNIA AT DAVIS
EPTEMBER 2001
UCD-CGM report
University
ofCalifo
rinaatDavis,
Center
forGeotech
nica
lModelin
gReport
UCD-CGM-01
Publish
edin
theIntern
atio
nalJournalforNumerica
landAnalytica
lMeth
odsin
Geomech
anics
incorporatin
gMech
anics
ofCohesiv
eFrictio
nalMateria
ls,Vol.25,No.8,pp.809-840,2001.
FiniteDeformationAnalysisof
Geomaterials
B.Jeremic
1K.Runesso
n2
S.Sture
3
1DepartmentofCivilandEnvironmentalEngineering,UniversityofCalifornia,Davis95616,U.S.A.phone530.754.9248,
fax530.752.7872,Email:
2DivisionofSolid
Mechanics,Chalm
ersUniversityofTechnology,S-41296Goteborg,Sweden
3DepartmentofCivil,
Environmental,andArchitecturalEngineering,UniversityofColorado,Boulder,CO
80309-0428,
U.S.A.
1
UCD-CGM report
Abstract
Themathem
atica
lstru
cture
andnumerica
lanalysis
ofcla
ssicalsm
alldefo
rmatio
n
elasto
plasticity
isgenera
llywell
establish
ed.How
ever,
develo
pmentoflargedefo
r-
matio
nela
sticplastic
numerica
lform
ulatio
nfordilatant,
pressu
resen
sitivemateria
l
models
isstill
aresea
rcharea
.
Inthispaper
wepresen
tdevelo
pmentofthenite
elementform
ulatio
nandim
ple-
mentatio
nforlargedefo
rmatio
n,ela
sticplastic
analysis
ofgeomateria
ls.Ourdevelo
p-
ments
are
based
onthemultip
licativ
edeco
mpositio
nofthedefo
rmatio
ngradien
tinto
elastic
andplastic
parts.
Aconsisten
tlin
eariza
tionoftherig
htdefo
rmatio
nten
sor
togeth
erwith
theNew
tonmeth
odattheconstitu
tiveandgloballev
elslea
dstow
ard
aneÆ
cientandrobust
numerica
lalgorith
m.Thepresen
tednumerica
lform
ulatio
nis
capable
ofaccu
rately
modelin
gdilatant,
pressu
resen
sitiveiso
tropic
and
aniso
tropic
geomateria
lssubjected
tolargedefo
rmatio
ns.
Inparticu
lar,theform
ulatio
niscapable
ofsim
ulatin
gthebehaviorofgeomateria
lsin
which
eigentria
dsofstress
andstra
indo
notcoincid
edurin
gtheloadingprocess.
Thealgorith
mis
testedin
conjunctio
nwith
thenovel
hyperela
stoplastic
model
termed
theB
materia
lmodel,
which
isasin
gle
surfa
ce(sin
gle
yield
surfa
ce,aÆne
singleultim
ate
surfa
ceandaÆnesin
glepoten
tialsurfa
ce)modelfordilatant,pressu
re
sensitiv
e,hardeningandsoften
inggeomateria
ls.It
isspeci
cally
develo
ped
tomodel
largedefo
rmatio
nhyperela
stoplastic
problem
sin
geomech
anics.
Wepresen
tanapplica
tionofthisform
ulatio
nto
numerica
lanalysis
oflowconne-
menttests
oncohesio
nless
granularsoilspecim
ensrecen
tlyperfo
rmed
inaSPACEHAB
moduleaboard
theSpace
Shuttle
durin
gtheSTS-89missio
n.Wecompare
numerica
l
modelin
gwith
testresu
ltsandshowthesig
nicance
ofadded
connem
entbythethin
hyperela
sticlatex
mem
braneunderg
oinglargestretch
ing.
KeyWords:
Hyperela
stoplasticity,
Largedefo
rmatio
ns,Geomateria
ls,Finite
element
analysis
2
UCD-CGM report
1Background
Theoretica
laswell
asim
plem
entatio
nissu
esin
materia
lnonlin
earnite
elementanalysis
of
solid
sandstru
ctures
are
increa
singly
beco
mingbetter
understo
odforthecase
ofinnitesi-
malstra
intheories.
Likew
ise,largedefo
rmatio
ntheories
andim
plem
entatio
nsformateria
ls
obeyingJ2plasticity
rules
arefairly
advanced
.Largestra
inanalysis
involvinggeometric
and
materia
lnonlin
earities
orpressu
resen
sitivegeomateria
lsare
stillthesubject
ofactiv
ere-
search
.Thechoice
ofappropria
testress
andstra
inmeasures,
aswell
astheissu
esperta
ining
totheinteg
ratio
nofela
stoplastic
constitu
tiveequatio
nsunder
conditio
nsoflargestra
inare
stilldisp
uted
intheresea
rchcommunity.
Thekey
assu
mptio
nin
innitesim
aldefo
rmatio
nela
stoplasticity
istheadditiv
ede-
compositio
nofstra
insinto
elastic
andplastic
parts.
Anumber
ofgenera
lizedmidpoint
numerica
lalgorith
ms,rangingfro
mpurely
explicit
topurely
implicit
schem
eswasdevelo
ped
andtheir
accu
racy
assessed
(Cris
eld[9],Kojic
andBathe[20],Ortiz
andPopov
[35],Sim
o
andTaylor[48],Ortiz
andSim
o[36],Runesso
net.
al.[41],Krieg
andKrieg
[22]to
mentio
n
afew
).Im
plicit,
backward
Euler
integ
ratio
nsch
emes
havein
recentyearsbeen
prov
ento
be
robust
andeÆ
cient.
Algorith
mictangentsti
ness
tensors
havebeen
deriv
ed(sta
rtingwith
thepioneerin
gwork
ofSim
oandTaylor[48]andRunesso
nandSamuelsso
n[40])formostof
theinteg
ratio
nsch
emes.
Itisim
porta
ntto
note
thatstra
insare
nonlin
earfunctio
nsofdisp
lacem
ents
andthus
additiv
edeco
mpositio
noftotalstra
insinto
elastic
andplastic
components
hold
only
for
innitesim
aldefo
rmatio
ns(see
more
inLubardaandLee
[29]andFamiglietti
andPrev
ost
[13])Moreov
er,asim
pleexampleispresen
ted,which
illustra
tesdieren
cesbetw
eenlargeand
smalldefo
rmatio
nanalysis.
Theresp
onse
ofasolid
interm
sofsm
allandlargedefo
rmatio
ns
iscompared
.Tothisendweuse
thedenitio
nofadefo
rmatio
ngradien
tFij=xi;j
andthe
Lagrangianstra
inten
sorEij )
andcompare
itwith
thesm
alldefo
rmatio
nstra
inten
sorij .
Clea
rlythedieren
cebetw
eenEijandijisinthenonlin
earterm
ofdisp
lacem
entderiva
tives:
Eij=
12(u
i;j+uj;i+ui;j u
j;i )ij=
12(u
i;j+uj;i )
(1)
Only
very
smalldefo
rmatio
nscanapprox
imateEijwith
ij .
Theerro
rexceed
s10%
after
a
nominalstra
inof30%.Fig.1show
sthatbyusin
gthesm
alldefo
rmatio
nstra
inmeasure
instea
dofthelargedefo
rmatio
nstra
in,sig
nicanterro
risintro
duced
.
Theearly
exten
sionsto
largedefo
rmatio
nofrate
based
numerica
lmeth
odsforela
sto
3
UC
D-C
GM
rep
ort
c)
ijε
Eij
500%
1000%
1500%
2000%
2500%
0.0 1.0 2.0 3.0 4.0 5.0 6.0
Errorshea
r st
rain
deformation (arctan )Φ
Φ
Figure 1: Error introduced by using the small strain instead of Lagrangian strain tensor.
plastic analysis of solids was conducted in the Lagrangian form1. Large deformation principle
of virtual work based formulation for large strain elasticplastic analysis of solids in the
Lagrangian form was proposed by Hibbitt et al. [16]. The Eulerian form of the solution
to the problem was proposed by McMeeking and Rice [32]. The disadvantage with this
approach was in the necessary use of incrementally objective integration algorithms that
may be computationally expensive. Hypoelastic based techniques, aimed at problems with
small elastic strains were also proposed by many others, (see for example Saran and Runesson
[42]). A number of problems encountered with dierent stress rates were noted by Nagtegaal
and de Jong [34], Kojic and Bathe [21] and Szabo and Balla [52].
On the other hand, hyperelastic based techniques have been developed recently for purely
deviatoric plasticity, for example by Simo and Ortiz [47], [45], Bathe et al. [2], Simo [43],
[44], Eterovic and Bathe [12], Peric et al. [39] and Cuitino and Ortiz [10]. Most of the
multiplicative split techniques are based on the earlier works of Hill [17], Bilby et al. [3]
Kroner [23], Lee and Liu [26], Fox [14] and Lee [25].
Simo and Ortiz [47] where the rst to propose a computational approach entirely based on
the multiplicative decomposition of the deformation gradient. Their stress update algorithm,
however, used the cutting plane scheme that has been shown by de Borst and Feenstra [11]
to yield erroneous results for some yield criteria. Bathe et al. [2] have used the multiplicative
decomposition with logarithmic stored energy function and an exponential approximation of
the ow rule for nonlinear analysis of metals. Eterovic and Bathe [12] included kinematic
1Hypoelasticity is presented in spatial format. Virtual work is normally stated in the material format.
4
UCD-CGM report
hardeningintheir
develo
pment,butthey
didnotaddress
theissu
eoftangentsti
ness
tensors
consisten
twith
theuse
oftheNew
tonsch
emeforthesolutio
nofnite
elementequatio
nsin
thenite
defo
rmatio
nreg
ime.
They
havealso
explored
theuse
ofaseries
expansio
nofthe
Henckystra
inten
sorin
their
numerica
lalgorith
m.How
ever,
develo
pments
were
madefor
deviatoric
plasticity
only.
Peric
atal.[39]follow
edtheir
work
andexperim
ented
with
vario
usrate
form
sandtheir
approx
imatio
ns.
They
also
restrictedtheuse
oftheir
algorith
mto
thesm
allela
sticstra
in
case.
Cuitin
oandOrtiz
[10]proposed
ameth
odforexten
dingsm
allstra
insta
teupdate
algorith
msandtheir
corresp
ondingconsisten
ttangentsti
ness
moduliinto
thenite
defo
r-
matio
nreg
imebut,alth
oughthey
claim
thatthemeth
odisapplica
bleto
vario
usmateria
l
models,
they
stayed
with
theJ2plasticity
model.
Sim
o[43],[44],explored
astra
inspace
form
ulatio
n.Theanalysis
wasconducted
foralin
earhardeningJ2plasticity
problem
.In
hislater
work,Sim
o[45]consolid
ated
thetheoretica
lfra
mew
ork
andshow
edsomeexcellen
t
examples
ofthree
dim
ensio
nallargedefo
rmatio
nJ2ela
stoplastic
analysis.
Lim
itedapplica
-
tionofthatwork
togeomateria
lshasbeen
show
nbySim
oandMesch
ke[46].They
applied
thedevelo
ped
framew
ork
totheCamClay
andgenera
lplasticity
typeofmodels,
used
in
geotech
nics.
They
havealso
explored
dieren
tim
plicit
explicit
schem
esforinteg
ratio
nof
thehardeninglaw
inorder
tobypass
thehardeninginduced
nonsymmetric
tangentsti
-
ness
moduli.
Theshortco
mingofthatwork
wasthatanasso
ciated
ow
rulewasadopted
,
thusresu
ltingin
overestim
atio
nofdilatatio
n.Moreov
er,loss
ofcollin
earity
betw
eenstress
andstra
ineig
entria
ds(occu
rringdurin
gnonproportio
nalloadingofgeomateria
ls)cannot
bemodeled
with
thiscateg
ory
ofalgorith
m.
More
recently
Lew
isandKhoei[27]used
arate
based
totalLangrangeanform
ulatio
nto
theanalysis
ofcompacted
pow
ders.
Peric
anddeSouza
Neto
[38]used
anopera
torsplit
algorith
min
termsofprin
cipalstresses
inconjunctio
nwith
theTresca
model.
Arm
ero[1]
exten
ded
themultip
licativ
ealgorith
m(orig
inally
develo
ped
bySim
o)foracoupled
poro
plastic
fully
saturated
medium.Borja
andAlarco
n[5][8]used
multip
licativ
edeco
mpositio
n
inprin
cipalcoordinates
(Sim
o'sform
ulatio
n)fortheproblem
oflargedefo
rmatio
nconsoli-
datio
n.Borja
etal.[6],[4],[7]applied
Sim
o'sapproach
totheCamClay
family
ofmodels.
Liu
etal.
[28]haveapplied
anearlier
algorith
mdevelo
ped
bySim
o[46]andadded
anew
nonlin
earela
sticlaw
fortheanalysis
oftire
sandcomposite
materia
l.Theabovedevelo
p-
mentsmakeanim
plicit
assu
mptio
noncolin
earity
ofprin
cipaldirectio
nsofstress
andstra
in
tensors,
which
renders
them
unusableforaniso
tropichardening/soften
ingmateria
lmodels.
5
UCD-CGM report
Inthefollow
ing,nite
elementandconstitu
tiveform
ulatio
nsforagenera
lhyperela
stic
plastic
geomateria
lare
presen
ted.More
speci
cally,
section2presen
tsalargedefo
rmatio
n
nite
elementform
ulatio
nwith
focusontheLagrangiandescrip
tion.Sectio
n3prov
ides
hyperela
sticandhyperela
sticplastic
backgrounddescrip
tionsanddescrib
estheconstitu
tive
integ
ratio
nalgorith
m.Selected
results
are
presen
tedin
section4.
2MaterialandGeometricNonLinearFinite
Element
Formulatio
n
Inthefollow
ingwepresen
tadeta
iledform
ulatio
nofamateria
landgeometric
nonlin
ear
static
nite
elementanalysis
schem
e.Theconguratio
nofchoice
ismateria
lorLagrangian.
Thelocalform
ofequilib
rium
equatio
nsin
Lagrangianform
atforthesta
ticcase
canbe
written
as:
Pij;j0 b
i=0
(2)
where
Pij=Skj (F
ik )tandSkjare
rst
andseco
ndPiolaKirch
hostress
tensors,
respec-
tively
andbiare
bodyforces.
Theweakform
oftheequilib
rium
equatio
nsisobtained
by
prem
ultip
lying(2)with
virtu
aldisp
lacem
entsÆu
iandinteg
ratin
gbyparts
with
reference
to
theinitia
lconguratio
nB0(in
itialvolumeV0 ):
ZV0
Æui;j P
ij dV= Z
V0
0 Æu
i bi dV Z
S0
Æui ti dS
(3)
Itprov
esbenecia
lto
rewrite
theleft
handsid
eof(3)byusin
gthesymmetric
secondPiola
Kirch
hostress
tensorSij :
ZV0
Æui;j F
jl S
il dV= Z
V0
12((Æu
i;l +Æu
l;i )+(Æu
i;j uj;l +
ul;j Æu
j;i ))Sil dV
(4)
where
wehaveused
thesymmetry
ofSilanddenitio
nfordefo
rmatio
ngradien
tFki=
Æki+uk;i .
With
aconvenien
tdenitio
nofthedieren
tialopera
torEil (
1ui ;2ui )
Eil (
1ui ;2ui )=
12 1ui;l +
1ul;i
+12
1ul;j
2uj;i+
2ui;j
1uj:l
(5)
thevirtu
alwork
equatio
n(4)canbewritten
as:
Wint(Æu
i ;u(k)
i)+W
ext(Æu
i )=0
(6)
6
UCD-CGM report
with
:
Wint(Æu
i ;n+1
0u(k)
i)
=Zc
Eij (Æu
i ;n+1
0u(k)
i)n+1
0S(k)
ijdV
(7)
Wext(Æu
i )=
Zc
0Æu
in+1
0bidV Z
@c
Æuin+1
0tidS
(8)
Wechoose
aNew
tontypeproced
ure
forsatisfy
ingequilib
rium.Given
thedisp
lacem
ent
eld
u(k)
i(X
j ),in
iteratio
nk,theitera
tivechangeui=u(k+1)
iu(k)
iisobtained
from
the
linearized
virtu
alwork
expressio
n
W(Æu
i ;u(k+1)
i)'W(Æu
i ;u(k)
i)+W(Æu
i ;ui ;u
(k)
i)
(9)
Here,
W(Æu
i ;u(k)
i)isthevirtu
alwork
expressio
n
W(Æu
i ;u(k)
i)=W
int(Æu
i ;u(k)
i)+W
ext(Æu
i )(10)
where
W(Æu
i ;ui ;u
(k)
i)isthelin
eariza
tionofvirtu
alwork
W(Æu
i ;ui ;u
(k)
i)
=lim!0
@W(Æu
i ;ui+
ui )
@
=Zc
Eij (Æu
i ;ui )LijklEkl (
ui ;u
i )dV+ Z
c
Eij (Æu
i ;ui )SijdV(11)
Here
wehaveused
dSij=1=2Lijkl dCkl=Lijkl E
kl (
ui ;u
i ).
Inorder
toobtain
expressio
nsforthesti
ness
matrix
weshallela
borate
furth
eron(11).
Tothisend,(11)canberew
rittenbyexpandingthedenitio
nforE
as
W(Æu
i ;ui ;u
(k)
i)
=
14 Zc
((Æuj;i+Æu
i;j )+(u
j;r Æur;i+Æu
i;r ur;j ))
Lijkl ((
uk;l +
ul;k )
+(u
k;s
us;l +
ul;s u
s;k ))
dV+
+ Zc
12(uj;l Æu
l;i+Æu
i;l ul;j )
SijdV
(12)
Or,byconvenien
tlysplittin
gtheaboveequatio
nwecanwrite
1W
(Æui ;
ui ;u
(k)
i)
=
14 Zc
((Æuj;i+Æu
i;j )+(u
j;r Æur;i+Æu
i;r ur;j ))
Lijkl((
uk;l +
ul;k )
+(u
k;s
us;l+ul;s u
s;k ))
dV
(13)
7
UCD-CGM report
2W
(Æui ;
ui ;u
(k)
i)= Z
c
12(uj;l Æu
l;i+Æu
i;l ul;j )
SijdV
(14)
Byfurth
erreo
rganizin
g(13)andcollectin
gterm
swecanwrite:
1W
(Æui ;
ui ;u
(k)
i)=
Zc
12(Æu
j;i+Æu
i;j ) Lijkl
12(uk;l +
ul;k )
dV
+Zc
12(Æu
j;i+Æu
i;j ) Lijkl
12(u
k;s
us;l +
ul;s u
s;k )
dV
+Zc
12(u
j;r Æur;i+Æu
i;r ur;j )
Lijkl
12(u
k;s
us;l+ul;s u
s;k )dV
+Zc
12(u
j;r Æur;i+Æu
i;r ur;j )
Lijkl
12(uk;l +
ul;k )
dV
(15)
Itshould
benoted
thattheAlgorith
micTangentSti
ness
(ATS)ten
sorLijklpossesses
both
minorsymmetries
(Lijkl=Ljikl=Lijlk ).
How
ever,
majorsymmetry
cannotbeguar-
anteed
.Nonasso
ciated
ow
rules
inela
stoplasticity
leadto
theloss
ofmajorsymmetry
(Lijkl6=Lklij ).
Moreov
er,itcanbeshow
n(i.e.
[19])thatanalgorith
mica
llyinduced
sym-
metry
loss
isobserv
edeven
forasso
ciated
ow
rules.
Uponobserv
ingminorsymmetry
ofLijklonecanwrite
(15)as:
1W
(Æui ;
ui ;u
(k)
i)=
Zc
Æui;jLijklul;k d
V
+Zc
Æui;jLijkluk;s
ul;s d
V
+Zc
Æui;r u
r;jLijkluk;s
ul;s d
V
+Zc
Æui;r u
r;jLijklul;k d
V(16)
Sim
ilarly,
byobserv
ingsymmetry
oftheseco
ndPiolaKirch
hostress
tensorSijwecan
write
2W
(Æui ;
ui ;u
(k)
i)= Z
c
Æui;l
ul;jSij dV
(17)
Theweakform
ofequilib
rium
expressio
nsforintern
al(W
int)andextern
al(W
ext)virtu
al
work,with
theabovementio
ned
symmetry
ofSijcanbewritten
as
Wint(Æu
i ;n+1
0u(k)
i)= Z
c
Æui;jSij dV+ Z
c
Æui;r u
r;jSij dV
(18)
Wext(Æu
i )=
Zc
0Æu
i bidV Z
@c
ÆuitidS
(19)
Standard
nite
elementdiscretiza
tionofthedisp
lacem
enteld
is:
uiui=H
I uIi
(20)
8
UCD-CGM report
where
uiistheapprox
imatio
ndisp
lacem
enteld
ui ,H
Iare
FEM
shapefunctio
nsanduIi
are
nodaldisp
lacem
ents.
With
thisapprox
imatio
n,wehave:
1W
(Æui ;
ui ;u
(k)
i)=
Zc
(HI;j Æ
uIi )Lijkl(H
Q;k
uQl )dV
+Zc
(HI;j Æ
uIi )Lijkl(H
J;kuJs )(H
Q;s
uQl )dV
+Zc
(HI;r Æ
uIi )(H
J;j u
Jr )Lijkl(H
J;kuJs )(H
Q;s
uQl )dV
+Zc
(HI;r Æ
uIi )(H
J;j u
Jr )Lijkl(H
Q;k
uQl )dV
(21)
2W
(Æui ;
ui ;u
(k)
i)= Z
c
(HI;l ÆuIi )(H
Q;j
uQl )Sij dV
(22)
Wint(Æu
i ;n+1
0u(k)
i)= Z
c
(HI;j Æ
uIi )Sij dV+ Z
c
(HI;r Æ
uIi )
(HJ;j u
Jr )Sij dV
(23)
Wext(Æu
i )=
Zc
0(H
I ÆuIi )bidV Z
@c
(HI ÆuIi )tidS
(24)
Uponnotin
gthatvirtu
alnodaldisp
lacem
entsÆu
Iiare
anynonzero
,contin
uousdisp
lace-
ments,
andsin
cethey
occu
rinallexpressio
nsforlin
earized
virtu
alwork,they
canbefacto
red
outandafter
somerea
rengem
entcanbewritten
as(while
remem
berin
gthat
1W+
1W+
Wext+W
int=0):
Zc
HI;j L
ijkl H
Q;k dV+ Z
c
HI;j L
ijkl H
J;kuJs H
Q;s dV+
+Zc
HI;r H
J;j u
JrLijkl H
J;kuJs H
Q;s dV
+Zc
HI;r H
J;j u
JrLijkl H
Q;k dV+ Z
c
HI;l H
Q;j S
ij dV
uQl
+Zc
(HI;j )
Sij dV+ Z
c
(HI;r )
(HJ;j u
Jr )
Sij dV
=Zc
0(H
I )bidV+ Z
@c
(HI )
tidS
(25)
Theglobalalgorith
mictangentsti
ness
matrix
(tensor)
isgiven
as
Kt=
@(W(Æu
i ;ui ;u
(k)
i))
@(ui )
=Zc
HI;j L
ijkl H
Q;k dV+ Z
c
HI;j L
ijkl H
J;kuJs H
Q;s dV
+Zc
HI;r H
J;j u
JrLijkl H
J;kuJs H
Q;s dV
+Zc
HI;r H
J;j u
JrLijkl H
Q;k dV+ Z
c
HI;l H
Q;j S
ij dV
(26)
9
UCD-CGM report
Theglobalalgorith
mictangentsti
ness
matrix
containsboth
thelin
earstra
inincrem
ental
stiness
matrix
andthenonlin
eargeometric
andinitia
lstress
increm
entalsti
ness
matrix
.
Thevecto
rofextern
ally
applied
loadisthen
R= Z
c
0(H
I )bidV+ Z
@c
(HI )
tidS
(27)
while
theloadvecto
rfro
melem
entstresses
isgiven
as
F= Z
c
(HI;j )
Sij dV+ Z
c
(HI;r )
(HJ;j u
Jr )Sij dV
(28)
Itisim
porta
ntto
note
thatthealgorith
mictangentsti
ness
tensor,vecto
rofextern
ally
applied
loads,andthevecto
rofelem
entstresses
are
secondandfourth
order
tensor.
Con-
versio
nfro
mten
sors
tomatrices
andvecto
rsis
perfo
rmed
bytheassem
bly
functio
ns.
It
isalso
importa
ntto
note
thattheten
sorofunknow
ndisp
lacem
ents
uQlis atten
edto
aonedim
ensio
nalvecto
r(ui )throughproper
implem
entatio
n.Theitera
tivechangein
disp
lacem
entvecto
ruiisobtained
bysettin
gthelin
earized
virtu
alwork
tozero
W(Æu
i ;u(k+1)
i)=0)
W(Æu
i ;u(k)
i)=W(Æu
i ;ui ;u
(k)
i)
(29)
Inparticu
lar,thechoice
oftheundefo
rmed
conguratio
n0foracomputatio
naldomain
(c=0 )
yield
stheTotalLagrangian(TL)form
ulatio
n.Theitera
tivedisp
lacem
entuiis
obtained
from
theequatio
n
W(Æu
i ;n+1u
(k)
i)=W(Æu
i ;ui ;n+1u
(k)
i)
(30)
where
W(Æu
i ;n+1u
(k)
i)
=Zc
Eij (Æu
i ;n+1u
(k)
i)n+1S
(k)
ijdV
Zc
0Æu
in+1bidV Z
@c
Æuin+1tidS
(31)
and
W(Æu
i ;ui ;n+1u
(k)
i)
=Zc
Eij (Æu
i ;n+1u
(k)
i)n+1L
(k)
ijklEkl (
ui ;n+1u
(k)
i)dV
+ Zc
dEij (Æu
i ;ui )
n+1S
(k)
ijdV
(32)
Inthecase
ofhyperela
sticplastic
response,
theseco
ndPiolaKirch
hostress
n+1S
(k)
ijis
obtained
byinteg
ratin
gtheconstitu
tivelaw
,describ
edin
section3.It
should
benoted
thatbyperfo
rmingtheinteg
ratio
nsin
theinterm
ediateconguratio
n,weobtain
theMandel
10
UCD-CGM report
stressn+1T
ijandsubseq
uently
theseco
ndPiolaKirch
hostress
Skj=
Cik
1Tij .
TheATS
tensorLijklisthen
obtained
based
on
Skj .
Inorder
toobtain
theseco
ndPiolaKirch
ho
stressSkjandATSten
sorin
theinitia
lconguratio
nweneed
toperfo
rmapull-b
ack
from
theinterm
ediate
conguratio
nto
theinitia
lsta
te
n+1S
ij=
n+1F
pipn+1F
pjqn+1S
pq
(33)
n+1L
ijkl=
n+1F
pimn+1F
pjnn+1F
pkrn+1F
plsn+1L
mnrs
(34)
Theform
ulatio
npresen
tedaboveisrather
genera
landreleva
ntto
alargeset
ofengineer-
ingsolid
s,both
isotro
picandaniso
tropic.
Thisgenera
litywill
befurth
erenhanced
insectio
n
3with
genera
l,constitu
tivelev
elcomputatio
nsthatcanhandleboth
isotro
picandgenera
l
aniso
tropicmateria
ls.
3Finite
Deformatio
nHyperelasto
Plastic
ity
3.1
Hyperelastic
ity
Amateria
liscalled
hyperela
sticorGreen
elastic,
ifthere
exists
anela
sticpoten
tialfunctio
n
W,also
called
thestra
inenergy
functio
nper
unitvolumeoftheundefo
rmed
conguratio
n,
which
represen
tsasca
larfunctio
nofstra
inofdefo
rmatio
nten
sors,
whose
deriva
tives
with
respect
toastra
incomponentdeterm
ines
thecorresp
ondingstress
component.
Themost
genera
lform
oftheela
sticpoten
tialfunctio
n,isdescrib
edinequatio
n(35),with
restrictionto
puremech
anica
ltheory,
byusin
gtheaxio
mofloca
lityandtheaxio
mofentro
pyprod
uctio
n2:
W=W
(XK;F
kK)
(35)
Byusin
gtheaxio
mofmateria
lfra
meindieren
ce3,weconclu
dethatW
dependsonly
on
XKandCIJ,thatis:
W=W
(XK;C
IJ)
or:
W=W
(XK;c
ij )(36)
2See
Marsd
enandHughes
[31]pp.190.
3See
Marsd
enandHughes
[31]pp.194.
11
UCD-CGM report
Inthecase
ofmateria
liso
tropy,thestra
inenerg
yfunctio
nW
(XK;C
IJ)belo
ngsto
the
classofiso
tropic,
invaria
ntsca
larfunctio
ns.Itsatis
estherela
tion:
W(X
K;C
KL)=W
XK;Q
KI C
IJ(Q
JL)t
(37)
where
QKIistheproper
orth
ogonaltra
nsfo
rmatio
n.IfwechooseQKI=RKI ,where
RKI
istheorth
ogonalrotatio
ntra
nsfo
rmatio
n,dened
bythepolardeco
mpositio
ntheorem
in
equatio
n(see
Malvern
[30]),
then:
W(X
K;C
KL)=W
(XK;U
KL)=W
(XK;v
kl )
(38)
Rightandleft
stretchten
sors,
UKL,vkl havethesameprin
cipalvalues(prin
cipalstretch
es)
i
;i=
1;3
sothestra
inenerg
yfunctio
nW
canberep
resented
interm
sofprin
cipal
stretches,
orsim
ilarly
interm
sofprin
cipalinvaria
ntsofthedefo
rmatio
nten
sor:
W=W
(XK;
1 ;2 ;
3 ;)=W
(XK;I
1 ;I2 ;I
3 )(39)
where:
I1
def
=CII
I2
def
=12
I21CIJCJI
I3
def
=det(C
IJ)=
16eIJKePQRCIPCJQCKR=J2
(40)
Left
CauchyGreen
tensors
isdened
asCIJ=
(FkI )tF
kJ,andthespectra
ldeco
mpositio
n
theorem
(seeSim
oandTaylor[49])forsymmetric
positiv
edenite
tensorssta
testhatCIJ=
2A
N(A
)
IN
(A)
J Awhere
A=
1;3
andNIare
theeig
envecto
rs(kNIk=
1)ofCIJ.Wecan
then
calcu
late
roots(
2A)ofthecharacteristic
polynomial
P(
2A)def
=6A+I16AI24A+I3=0
(41)
Itshould
benoted
thatnosummatio
nisim
plied
over
indices
inparen
thesis.
Forexample,
in
thepresen
tcaseN
(A)
IistheAth
eigenvecto
rwith
mem
bers
N(A
)
1,N
(A)
2andN
(A)
3,so
thatthe
actu
alequatio
nCIJ=2A
N(A
)
IN
(A)
J Acanalso
bewritten
asCIJ= P
A=3
A=12(A
) N(A
)
IN
(A)
J.In
order
tofollow
theconsisten
cyofindicia
lnotatio
nin
thiswork,weshallmakeaneort
to
represen
talltheten
soria
lequatio
nsin
indicia
lform
.
12
UCD-CGM report
Themappingoftheeig
envecto
rsisgiven
by
(A
)n(A
)
i=FiJN
(A)
J(42)
where
kn(A
)
ik1.Thespectra
ldeco
mpositio
nofFiJ,RiJandbijisthen
given
by
FiJ=A
n(A
)
iN
(A)
J A
(43)
RiJ=
3XA=1
n(A
)
iN
(A)
J(44)
bij=2A
n(A
)
in(A
)
j A
(45)
Recen
tly,Ting[53]andMorm
an[33]haveused
Serrin
'srep
resentatio
ntheorem
inorder
todevise
ausefu
lrep
resentatio
nforgenera
lizedstra
inten
sorsEIJthroughCmIJ.After
some
tensoralgebra
theLagrangianeig
endyadN
(A)
IN
(A)
J,canbewritten
as
N(A
)
IN
(A)
J=2(A
) CIJ
I12(A
) ÆIJ+I3
2
(A) (C
1)IJ
24(A
)I1
2(A)+I3
2
(A)
(46)
Itshould
benoted
thatthedenominatorin
equatio
n(46)canbewritten
as:
24(A
)I1
2(A)+I3
2
(A)=
2(A
)2(B
) 2(A
)2(C
) def
=D
(A)
(47)
where
indices
A;B
;Carecyclic
perm
utatio
nsof1;2;3.Itfollow
sdirectly
fromthedenitio
n
ofD
(A)in
equatio
n(47)that16=
26=
3)
D(A
)6=
0forequatio
n(46)to
bevalid
.
Sim
ilarly
wecanobtain:
(C1)IJ=2
A N
(A)
IN
(A)
J A
(48)
Themostgenera
lform
oftheiso
tropicstra
inenerg
yfunctio
nW
interm
sofofprin
cipal
stretches
canbeexpressed
as:
W=W
(XK;
1 ;2 ;
3 ;)(49)
13
UCD-CGM report
Inorder
toobtain
theseco
ndPiolaKirch
hostress
tensorSIJ(andother
stressmea-
sures)
itis
necessa
ryto
calcu
late
thegradien
t@W=@CIJ.
Moreov
er,themateria
ltan-
gentsti
ness
tensorLIJKLreq
uire
secondorder
deriva
tives
ofthestra
inenerg
yfunctio
n
@2W
=(@CIJ@CKL).
Inorder
toobtain
these
quantities
weintro
duce
aseco
ndorder
tensor
M(A
)
IJ
M(A
)
IJ
def
=2
(A)N
(A)
IN
(A)
J(50)
=(F
iI )1
n(A
)
in(A
)
j (F
jJ)t
=1
D(A
) CIJ
I12(A
) ÆIJ+I3
2
(A) (C
1)IJ
from
(46)
where
D(A
)wasdened
byequatio
n(47).With
M(A
)
IJ
dened
byequatio
n(50),weobtain
CIJ=4A
M(A
)
IJ
A(51)
anditalso
follow
s
(C1)IJ=M
(1)
IJ+M
(2)
IJ+M
(3)
IJ
(52)
Itcanalso
beconclu
ded
that:
ÆIJ=2(1) M
(1)
IJ+2(2) M
(2)
IJ+2(3) M
(3)
IJ=2A
M(A
)
IJ
A(53)
since,
from
theorth
ogonalproperties
ofeig
envecto
rs
ÆIJ=
3XA=1
N(A
)
IN
(A)
J=
N(A
)
I A
N(A
)
J A
(54)
Wealso
denetheSim
oSerrin
fourth
order
tensorM
IJKLas:
M(A
)
IJKL
def
=@M
(A)
IJ
@CKL
=
1
D(A
) IIJKLÆKLÆIJ+2(A
) ÆIJM
(A)
KL+M
(A)
IJÆKL
+
+I3
2
(A)
(C1)IJ(C
1)KL+12
(C1)IK(C
1)JL+(C
1)IL(C
1)JK
2
(A)I3
(C1)IJM
(A)
KL+M
(A)
IJ
(C1)KL D0(A
)M
(A)
IJM
(A)
KL
(55)
Acomplete
deriva
tionofM
IJKLisgiven
bySim
oandTaylor[49].
Wecanthen
denehyperela
sticstress
measures
as
2.PiolaKirch
hostress
tensorSIJ=2
@W
@CIJ
14
UCD-CGM report
Mandelstress
tensorTIJ=CIKSKJ
1.PiolaKirch
hostress
tensorPiJ=SIJ(F
iI )t
Kirch
hostress
tensorab=FaI (F
bJ)tS
IJ
where
@W(
(A) )
@CIJ
=@volW
((A
) )
@CIJ
+@isoW
((A
) )
@CIJ
=12
@volW
(J)
@J
J(C
1)IJ+12wA(M
(A)
IJ)A
(56)
and
wA=13
@isoW
( ~(A
) )
@~B
~B
+@isoW
( ~(A
) )
@~(A
)
~(A
)(57)
Thetangentsti
ness
opera
torisdened
as
LIJKL=
volL
IJKL+
isoLIJKL
(58)
with
volL
IJKL=
J2@2volW
(J)
@J@J
(C1)KL(C
1)IJ+J@volW
(J)
@J
(C1)KL(C
1)IJ+2J@volW
(J)
@J
I(C1)
IJKL
(59)
isoLIJKL=YAB(M
(B)
KL)B(M
(A)
IJ)A+2wA(M
(A)
IJKL)A
(60)
3.2
Multip
licativ
eDecompositio
n
Multip
licativ
edeco
mpositio
nofthedefo
rmatio
ngradien
tisused
asakinem
atica
lbasis
for
thedevelo
pmentsdescrib
edhere.
Themotiva
tionforthemultip
licativ
edeco
mpositio
ncanbe
traced
back
totheearly
worksofBilbyetal.[3],andKroner[23]onmicro
mech
anics
ofcry
stal
dislo
catio
nsandapplica
tionto
contin
uum
modelin
g.In
thecontex
toflargedefo
rmatio
n
15
UC
D-C
GM
rep
ort
elastoplastic computations, the work by Lee and Liu [26], Fox [14] and Lee [25] generated
an early interest in multiplicative decomposition.
The appropriateness of multiplicative decomposition technique for soils may be justied
from the particulate nature of the material. From the micromechanical point of view, plastic
deformation in soils arises from slipping, crushing, yielding and plastic bending4 of granules or
platelets comprising the assembly5. It can certainly be argued that deformations in soils are
predominantly plastic, however, reversible deformations could develop from the elasticity of
individual soil grains, and could be relatively large, when particles are locked in high density
specimens.
Fp
FeReference
Configuration
X
d X
Ω0
Intermediate
Configurationdx
Ω
dx
Current
Configuration
x
Ω
Fe
-1
Fp-1
σ x
F
X ux
Figure 2: Multiplicative decomposition of deformation gradient: schematics.
The reasoning behind multiplicative decomposition is a rather simple one. If an innites-
imal neighborhood of a body xi; xi + dxi in an inelastically deformed body is cutout and
unloaded to an unstressed conguration, it would be mapped into xi; xi+dxi. The transfor-
mation would be comprised of a rigid body displacement6 and purely elastic unloading. The
elastic unloading is ctitious, since in materials with a strong Baushinger's eect unloading
will lead to loading in another stress direction, and if there are residual stresses, the body
must be cutout in small pieces, and then every piece relieved of stresses. The unstressed
4For plate like clay particles.
5See also Borja and Alarcon [5] and Lambe and Whitman [24].
6Translation and rotation.
16
UCD-CGM report
conguratio
nisthusincompatib
leanddisco
ntin
uous.
Thepositio
nxiisarbitra
ry,andwe
may
assu
mealin
earrela
tionship
betw
eendxianddxi ,in
theform
7:
dxk=(F
eik )1dxi
(61)
where
(Feik )
1isnotto
beundersto
odasadefo
rmatio
ngradien
t,sin
ceitmay
represen
tthe
incompatib
le,disco
ntin
uousdefo
rmatio
nofabody.
Byconsid
eringthereferen
cecongura-
tionofabodydX
i ,then
theconnectio
nto
thecurren
tconguratio
nis:
dxk=Fki dX
i)
dxk=(F
eik )1Fij dX
j(62)
sothatonecandene:
Fpkj
def=
(Feik )
1Fij)
Fij
def
=Feki F
pkj
(63)
Theplastic
partofthedefo
rmatio
ngradien
t,Fpkjrep
resentsmicro
mech
anica
lly,theirre-
versib
leprocess
ofslip
ping,cru
shingdislo
catio
nandmacro
scopica
llytheirrev
ersibleplastic
defo
rmatio
nofabody.
Theela
sticpart,
Fekirep
resentsmicro
mech
anica
llyapure
elastic
re-
versa
lofdefo
rmatio
nfortheparticu
lateassem
bly,
macro
scopica
llyalin
earela
sticunloading
toward
astress
freesta
teofthebody,notnecessa
rilyacompatib
le,contin
uousdefo
rmatio
n
butrather
actitio
usela
sticunloadingofsm
allcutoutsofadefo
rmed
particu
late
assem
bly
orcontin
uum
body,
3.3
Constitu
tiveRelatio
ns
Wepropose
thefree
energ
ydensity
W,which
isdened
intheinterm
ediate
conguratio
n,
as
0 W
(Ceij ;
)=0 W
e(Ceij )
+0 W
p()
(64)
where
We(Ceij )
represen
tsasuita
blehyperela
sticmodelin
termsoftheela
sticrig
htdefo
r-
matio
nten
sorCeij ,
wherea
sW
p()rep
resents
thehardening.Thepertin
entdissip
atio
n
inequality
beco
mes:
D=
TijLpij+ X
K_0
(65)
7referredto
sameCartesia
ncoordinate
system
.
17
UCD-CGM report
where
TijistheMandelstress
andLpijistheplastic
velo
citygradien
tdened
on.Wenow
denetheela
sticdomainBasB=fTij ;
Kj(Tij ;
K)0g
(66)
When
yield
functio
nisiso
tropicin
Tij(which
isthecase
here)
inconjunctio
nwith
elastic
isotro
py,wecanconclu
dethatTijissymmetric
andwemay
replace
Tijbyijinyield
functio
n
.
Asto
thechoice
ofanela
sticlaw
,it
isem
phasized
thatthis
islargely
amatter
of
convenien
ce,sin
ceweshallbedealin
gwith
smallela
sticdefo
rmatio
ns.
Here,
theNeo
Hookeanela
sticlaw
isadopted
.Theconstitu
tiverela
tionscannow
bewritten
as
Lpij:=
_Fpik
Fpjk
1
=_@
@Tij
=_Mij
(67)
K=
K( )
(68)
_=
_@
@K
, (0
)=0
(69)
where
K;
=1;2;isthe\hardeningstress"
,(
ij ;K)istheplastic
poten
tial,is
intern
alvaria
bles,
_isconsisten
cyparameter
determ
ined
from
theloadingconditio
ns8and
Fpik=(Feli )1F
lkistheplastic
part
ofthedefo
rmatio
ngradien
t.
3.4
ImplicitIntegratio
nAlgorith
m
Theincrem
entaldefo
rmatio
nandplastic
ow
are
govern
edbythesystem
ofevolutio
nequa-
tions(67)and(69).The ow
rule(67)canbeinteg
rated
togive
n+1F
pij=exp
n+1Mik
nFpkj
(70)
Byusin
gthemultip
licativ
edeco
mpositio
n
Fij=
FeikFpkj)
Feik=Fij
Fpkj 1
(71)
andequatio
n(70)weobtain
n+1F
eij=
n+1F
im(nF
pmk )1exp n+1Mkj
=n+1F
e;tr
ikexp n+1Mkj
(72)
8These
are
theKarushKuhnTucker
complem
entary
conditio
nsin
thespecia
lcase
offully
asso
ciativ
e
theory,
deningtheStandard
Dissip
ativ
eMateria
l,cf.
[15].
18
UCD-CGM report
where
weused
that
n+1F
e;tr
ik=
n+1F
im(nF
pkm)1
(73)
Theela
sticdefo
rmatio
nisthen
n+1C
eijdef
=n+1F
eim T
n+1F
emj
=exp n+1MTir
n+1F
e;tr
rk
Tn+1F
e;tr
kl
exp n+1Mlj
=exp n+1MTir
n+1C
e;tr
rl
exp n+1Mlj
(74)
Byreco
gnizin
gthattheexponentofaten
sorcanbeexpanded
inTaylorseries
9
exp n+1Mlj
=Æljn+1Mlj+12
n+1Mls
n+1Msj +
(75)
andbyusin
gtheseco
ndorder
expansio
nin
equatio
n(74)andafter
someten
soralgebra
we
obtain
n+1C
eij=
n+1C
e;tr
rj
n+1Mir
n+1C
e;tr
rj
+n+1C
e;tr
iln+1Mlj
+
+()2
2
n+1Misn+1Msrn+1C
e;tr
rj
+2n+1Mirn+1C
e;tr
rl
n+1Mlj+
n+1Mlsn+1Msjn+1C
e;tr
rl
()3
12n+1Misn+1Msrn+1C
e;tr
rl
n+1Mlj
12n+1Mir
n+1C
e;tr
rl
n+1Mls
n+1Msj
+
+()4
4
n+1Misn+1Msrn+1C
e;tr
rl
n+1Mls
n+1Msj
(76)
First
order
seriesexpansio
ninclu
des
consta
ntandlin
ear(upto
)mem
bers.
Seco
ndorder
expansio
ninclu
des
thecomplete
equatio
nabove.
Remark3.1
TheTaylor's
seriesexpansio
nin
equatio
n(75)isaproper
approx
imatio
nfor
thegenera
lnonsymmetric
tensorMlj .
Thatis,
theapprox
imate
solutio
ngiven
byequatio
n
(76)isvalid
foragenera
laniso
tropicsolid
.Thiscontra
stswith
thespectra
ldeco
mpositio
n
family
ofsolutio
ns10which
are
restrictedto
isotro
picsolid
s.
Remark3.2
Inthelim
it,when
thedefo
rmatio
nsare
suÆcien
tlysm
all,
thesolutio
n(76)
collapses
tolimFij!Æij
Æij+2n+1ij=
+Æij+2n+1e;trij
9See
forexamplePearso
n[37].
10See
Sim
o[45].
19
UCD-CGM report
n+1Mij2n+1Mirn+1trrj
n+1Mij2
n+1tril
n+1Mlj
+2n+1Miln+1Mlj+22n+1Mir
n+1trrl n+1Mlj
=Æij+2n+1e;trij
2
n+1Mij
)n+1ij=
n+1trij
n+1Mij
(77)
which
represen
tsasm
alldefo
rmatio
nela
sticpred
ictorplastic
correcto
requatio
nin
strain
space.
Inworkingoutthesm
alldefo
rmatio
ncounterp
art
(77)itwasused
that
limFij!Æij
n+1C
eij=
Æij+2n+1ij
2
n+1tril
n+1Mlj
n+1Mij
1
(78)
Byneglectin
gthehigher
order
termwith
2in
equatio
n(76),thesolutio
nfortherig
ht
elastic
defo
rmatio
nten
sorn+1C
eijcanbewritten
as
n+1C
eij=
n+1C
e;tr
ij
n+1Mir
n+1C
e;tr
rj
+n+1C
e;tr
iln+1Mlj
(79)
Thehardeningrule(69)canbeinteg
rated
togive
n+1=
n+
@
@K
n+1
(80)
Theincrem
entalproblem
isdened
byequatio
ns(79),(80),theconstitu
tiverela
tions
n+1S
IJ=2@W
@CIJ n
+1
(81)
n+1K
=@W
@ n
+1
(82)
andtheKarushKuhnTucker
(KKT)conditio
ns
<0
;n+10
;
n+1
=0
(83)
where
=(Tij ;K
)
(84)
20
UCD-CGM report
Remark3.3
TheMandelstress
tensorTijcanbeobtained
fromtheseco
ndPiolaKirch
ho
stressten
sorSkjandtherig
htela
sticdefo
rmatio
nten
sorCeikasTij=
CeikSkj
Thisset
ofnonlin
earequatio
nswill
besolved
with
aNew
tontypeproced
ure,
describ
ed
below
.Foragiven
n+1F
ij ,or
n+1C
e;tr
ij,theupgraded
quantities
n+1S
IJand
n+1K
canbe
found,then
theappropria
tepullback
to0orpushforward
towill
given+1S
IJand
n+1ij
n+1S
IJ=
n+1F
piI 1
n+1S
IJ
n+1F
pjJ
T
(85)
n+1ij=
n+1F
eiIn+1S
IJ
n+1F
ejJ
1
(86)
Theela
sticpred
ictor,plastic
correcto
requatio
n
n+1C
eij=
n+1C
e;tr
ij
n+1Z
ij(87)
isused
asasta
rtingpointforaNew
tonitera
tivealgorith
m.In
theprev
iousequatio
n,we
haveintro
duced
tensorZij
Zij
=
n+1Mir
n+1C
e;tr
rj
+n+1C
e;tr
iln+1Mlj
(88)
Thedenitio
nofZijaboveassu
mesuseofrst
orderexpansio
nin(76).Thetria
lrig
htela
stic
defo
rmatio
nten
sorisdened
as
n+1C
e;tr
ij=
n+1F
e;tr
ri
T n+1F
e;tr
rj
= n+1F
rM
(nF
piM)1
T n+1F
rS
nFpjS
1
(89)
Weintro
duce
aten
sorofdefo
rmatio
nresid
uals
Rij=
Ceij
|zcurrent
n+1C
e;tr
ij
n+1Z
ij |
z
BackwardEuler
(90)
Theten
sorRijrep
resentsthedieren
cebetw
eenthecurren
trig
htela
sticdefo
rmatio
nten
sor
andtheBackward
Euler
rightela
sticdefo
rmatio
nten
sor.Thetria
lrig
htela
sticdefo
rmatio
n
tensorn+1C
e;tr
ijismaintained
xed
durin
gtheitera
tionprocess.
Therst
order
Taylorseries
expansio
ncanbeapplied
totheten
sorofresid
ualsRijinorderto
obtaintheitera
tivechange,
thenew
residualRnew
ijfro
mtheoldRoldij
Rnew
ij=Roldij
+dCeij+d()n+1Z
ij+@n+1Z
ij
@Tmn
dTmn+@n+1Z
ij
@K
dK
(91)
21
UCD-CGM report
Furth
ermore,
since
Tmn=
CemkSkn
)
Cesk
1Tsn=
Skn
(92)
wecanwrite
dTmn
=dCemkSkn+
CemkdSkn
=dCemkSkn+12
CemkLeknpqdCepq
=dCemk
Cesk
1Tsn+12
CemkLeknpqdCepq
(93)
sothatafter
settingRnew
ij=0andsometen
soralgebra
weobtain
0=
Roldij
+d()n+1Z
ij+@n+1Z
ij
@K
dK
+
+(Æ
imÆnj+@n+1Z
mn
@Tik
Cesj
1Tsk+12@n+1Z
mn
@Tpq
CepkLekqij)dCeij
(94)
Uponintro
ducin
gthenotatio
n
Tmnij=ÆimÆnj+@n+1Z
mn
@Tik
Cesj 1Tsk+12@n+1Z
mn
@Tpq
CepkLekqij
(95)
wecansolvefordCeij
dCepq=(Tmnpq )1 Roldmnd()n+1Z
mn@n+1Z
mn
@K
dK
!(96)
Byusin
gthatdK
=@K
@
d=d()@K
@
@Q
@K
=d()H
@Q
@K
(97)
itfollow
sfro
m(96)
dCepq=(Tmnpq )1 Roldmnd()n+1Z
mn+@n+1Z
mn
@K
d()H
@Q
@K
!(98)
Arst
order
Taylorseries
expansio
nofayield
functio
ntogeth
erwith
(97)prov
ides
new(Tij ;K
)=
old
(Tij ;K
)+
+
@(Tij ;K
)
@Tpn
Cesq
1Tsn+12
@(Tij ;K
)
@Tmn
CemkLeknpq !
dCepq
d()@(Tij ;K
)
@K
H
@
@K
(99)
22
UCD-CGM report
Uponintro
ducin
gthefollow
ingnotatio
n
Fpq=@(Tij ;K
)
@Tpn
Cesq
1Tsn+12
@(Tij ;K
)
@Tmn
CemkLeknpq
(100)
andwith
thesolutio
nfordCepqfro
m(98),(99)beco
mes
new(Tij ;K
)=
old
(Tij ;K
)+
+Fpq
(Tmnpq )1 Roldmnd()n+1Z
mn+d()@n+1Z
mn
@K
H
@
@K
!!
d()@(Tij ;K
)
@K
H
@
@K
(101)
After
setting
new(Tij ;K
)=
0wecansolvefortheincrem
entalconsisten
cyparameter
d()
d()=
old
Fpq(Tmnpq )1Roldmn
Fpq(Tmnpq )1
n+1Z
mnFpq(Tmnpq )1@n+1Z
mn
@K
H
@
@K
+@
@K
H
@
@K
(102)
Remark3.4
Inthelim
it,forsm
alldefo
rmatio
ns,theincrem
entalconsisten
cyparameter
d()beco
mes
d()=
old
(n
mnEmnpq )
ÆpmÆnq+@m
mn
@ij
Eijpq !
1
Roldmn
nmnEmnpq
Æmp Æ
qn+@m
pq
@ij
Eijmn !
1
n+1m
mn+
@
@K
H
@
@K
(103)
since
inthelim
it,asdefo
rmatio
nsbeco
mesm
all
limFij!Æij
Tmnpq
=ÆpmÆnq+@m
mn
@ij
Eijpq
limFij!Æij
Fpq
=12
@
@mn
Emnpq
limFij!Æij
Zpq
=2m
pq
limFij!Æij
Rpq
=2pq
(104)
Uponnotin
gthattheresid
ualRpqisdened
instra
inspace,
theincrem
entalconsisten
cy
parameter
d()compares
exactly
with
it'ssm
allstra
incounterp
art
(Jerem
icandSture
[19]).Theproced
ure
describ
edbelow
summarizes
theim
plem
entatio
noftheretu
rnalgorith
m.
23
UCD-CGM report
3.4.1
ReturnAlgorith
m
Given
therig
htela
sticdefo
rmatio
nten
sornCepqandaset
ofhardeningvaria
bles
nKata
speci
cquadrature
pointin
anite
element,wecompute
therela
tivedefo
rmatio
ngradien
t
n+1fijforagiven
disp
lacem
entincrem
ent
n+1ui
n+1fij=Æij+ui;j
(105)
andtherig
htdefo
rmatio
nten
sor
n+1C
e;tr
ij=
n+1fir
nFerk
T n+1fklnF
elj =(nF
erk )
T n+1fir
T n+1fklnF
elj (106)
Then
wecompute
thetria
lela
sticseco
ndPiolaKirch
hostress
andthetria
lela
sticMandel
stressten
sor
n+1S
e;trij
=2
@W
@n+1C
e;tr
ij
(107)
n+1T
e;tr
ij=
n+1C
e;tr
iln+1S
e;trlj
(108)
Wethen
evaluate
theyield
functio
nn+1
tr(Te;tr
ij;K
),andset
n+1C
eij=
n+1C
e;tr
ij
n+1K
=
nK
n+1T
ij=
nTe;tr
ij
andexitconstitu
tiveinteg
ratio
nproced
ure.
Ifn+1
tr
0there
isnoplastic
ow
inthe
curren
tincrem
ent.
Iftheyield
criterionhasbeen
violated
(n+1
tr>0)proceed
step1.kthitera
tion.Know
nvaria
bles
n+1C
e(k)
ij;
n+1
(k)
;
n+1K
(k)
;
n+1T
(k)
ij;
n+1(k)
evaluate
theyield
functio
nandtheresid
ual
(k)
=(n+1T
e(k)
ij;n+1K
(k)
)
R(k)
ij=
n+1C
e;(k
)
ij
n+1C
e;tr
ij
n+1(k)n+1Z
(k)
ij
step2.Check
forconverg
ence,
(k)NTOLandkR(k)
ijkNTOL.Ifconverg
ence
criterion
issatis
edset
n+1C
eij=
n+1C
e(k)
ij
24
UCD-CGM report
n+1
=n+1
(k)
n+1K
=
n+1K
(k)
n+1Tij
=n+1T
(k)
ij
n+1
=n+1(k)
Exitconstitu
tiveinteg
ratio
nproced
ure.
step3.11Ifconverg
ence
isnotachiev
edcompute
theela
sticsti
ness
tensorLijkl
L(k)
ijkl=4
@2W
@Ce(k)
ij@Ce(k)
kl
(109)
step4.Compute
theincrem
entalconsisten
cyparameter
d((k+1))
d((k+1))=
(k)
F(k)
mnRmn(k)
F(k)
mnZmn(k)(k)Fmn(k)@Zmn(k)
@K
H(k)+@(k)
@K
H(k)
(110)
where
H(k)=H
(k)@;(k
)
@K
;Fmn(k)=Fpq(k) Tmnpq(k) 1
Fpq=@(T(k)
ij;K
(k)
)
@Tpn
Ce;(k
)
sq
1T(k)
sn+12
@(T(k)
ij;K
(k)
)
@Tmn
Ce;(k
)
mk
Le;(k
)
knpq
Tmnij=ÆimÆnj+(k)@Z(k)
mn
@T(k)
ik
Ce;(k
)
sj
1T(k)
sk+12(k)@Z(k)
mn
@Tpq
Ce;(k
)
pk
Le;(k
)
kqij
step5.Update
theconsisten
cyparameter
(k+1)
(k+1)=(k)+d((k+1))
(111)
step6.Calcu
latetheincrem
entsfortherig
htdefo
rmatio
nten
sor,thehardeningvaria
ble
andtheMandelstress
dCe;(k
+1)
pq
=
T
(k)
mnpq
1 R(k)
mnd((k+1))
n+1Z
(k)
mn+(k)@Z(k)
mn
@K
d((k+1))
H(k)
!(112)
11From
step3.to
step9.allofthevaria
bles
are
ininterm
ediate
n+1conguratio
n.Forthesakeof
brev
ityweare
omittin
gsuperscrip
tn+1.
25
UCD-CGM report
d(k+1)
=d((k+1))@;(k
)
@K
(113)
dK
(k+1)
=d((k+1))H
(k)
@;(k
)
@K
(114)
dT(k+1)
mn
=dCe;(k
+1)
mk
Ce;(k
)
sk
1T(k)
sn+12
Ce;(k
)
mk
Le;(k
)
knpqdCe;(k
+1)
pq
(115)
step7.Update
therig
htdefo
rmatio
nten
sorCe;(k
+1)
pq
,hardeningvaria
bleK
(k+1)
and
Mandelstress
T(k+1)
mn
Ce;(k
+1)
pq
=Ce;(k
)
pq
+d(Ce;(k
+1)
pq
)
(k+1)
=
(k)
+d(
(k+1)
)
K(k+1)
=
K(k)
+d(K
(k+1)
)
T(k+1)
mn
=T(k)
mn+d(T(k+1)
mn
)(116)
step8.eva
luate
theyield
functio
nandtheresid
ual
(k+1)=(Te(k+1)
ij;K
(k+1)
)
R(k+1)
ij=
Ce;(k
+1)
ij
Ce;tr
ij(k+1)Z
(k+1)
ij
(117)
step9.Setk=k+1and
(k)
=(k+1)
Ce;(k
)
pq
=Ce;(k
+1)
pq
(k)
=
(k+1)
K(k)
=
K(k+1)
T(k)
mn
=T(k+1)
mn
(118)
andretu
rnto
step2.
3.5
Algorith
micTangentSti
ness
Tensor
Startin
gfro
mtheela
sticpred
ictorplastic
correcto
requatio
n
n+1C
eij=
n+1C
e;tr
ij
n+1Z
ij(119)
26
UCD-CGM report
towhich
weapplyarst
order
Taylorseries
expansio
nto
obtain
(after
someten
soralgebra),
dCeij=(Tmnij )
1
dCe;tr
ijd()Zij+d()@Zij
@K
H
@
@K
!(120)
where
Tmnijwasdened
in(95)
Nextweuse
therst
order
Taylorseries
expansio
noftheyield
functio
nd(Tij ;K
)=0
@
@Tmn
dTmn+
@
@K
dK
=FpqdCepq
@
@K
d()H
@
@K
=0
(121)
with
Fpqdened
in(100).
Byusin
gthesolutio
nfordCeijfro
m(120)wecanwrite
Fpq(Tmnpq )1
dCe;tr
mnd()Zmn+d()@Zmn
@K
H
@
@K
!
@
@K
d()H
@
@K
=0
(122)
Weare
now
inthepositio
nto
solvefortheincrem
entalconsisten
cyparameter
d()
d()=Fpq(Tmnpq )1dCe;tr
mn
(123)
where
wehaveused
to
denote
=Fpq(Tmnpq )1n+1Z
mnFpq(Tmnpq )1@n+1Z
mn
@K
H
@
@K
+@
@K
H
@
@K
(124)
Since
dSkn=
12LknpqdCepq
(125)
andbyusin
g(120)wecanwrite
dCepq=
(Tmnpq )1
ÆmvÆntFop(Trsop )1ÆrvÆst
Zmn +
Fop(Trsop )1ÆrvÆst
@Zij
@K
H
@
@K
!dCe;tr
vt
(126)
Then
dCepq=
PpqvtdCe;tr
vt
(127)
27
UCD-CGM report
where
Ppqvt=(Tmnpq )1
Æmv Æ
ntFab (Tvtab )1
Zmn@n+1Z
mn
@K
H
@
@K
!!(128)
Thealgorith
mictangentsti
ness
tensorLATS
ijkl(in
interm
ediate
conguratio
n)isthen
dened
as
LATS
knvt=
LeknpqPpqvt
(129)
Pullback
tothereferen
ceconguratio
n0yield
sthealgorith
mictangentsti
ness
tensor
Lijklin
thereferen
ceconguratio
n0
n+1L
ATS
ijkl=
n+1F
pimn+1F
pjnn+1F
pkrn+1F
plsn+1L
ATS
mnrs
(130)
Remark3.5
Inthelim
it,forsm
alldefo
rmatio
nsandiso
tropicresp
onse,
theAlgorith
mic
TangentSti
ness
tensorLATS
ijklbeco
mes
limFij!Æij
LATS
vtpq=EATS
vtpq=R
knvtncdR
cdvt R
knmrH
mr
since
limFij!Æij
Tmnpq=mnpq=ÆpmÆnq+@m
mn
@rs
Eekspq
limFij!Æij
Fab=
12ncd E
ecdab
limFij!Æij
Hmn=m
mn@m
mn
@K
H
@
@K
limFij!Æij
=nab R
abm
nH
mn+
@
@K
H
@
@K
Itisnoted
thattheAlgorith
micTangentSti
ness
tensorgiven
by(131)compares
exactly
with
it'ssm
allstra
incounterp
art
(Jerem
icandSture
[19]).
3.6
MaterialModel
Alargedefo
rmatio
nmateria
lmodel
used
incomputatio
nsisbrie
ydescrib
edhere.
The
modelrelies
onthedevelo
pmentbehindtheso
called
MRSLademodel(Sture
etal.
[51])
28
UCD-CGM report
andissubseq
uently
denoted
theBModel.
TheBModelisasin
glesurfa
cemodel,
with
un-
coupled
coneportio
nandcapportio
nhardening.Very
lowconnem
entreg
ionwascarefu
lly
modeled
andtheyield
surfa
cewasshaped
insuch
away
tomim
icrecen
tndingsobtained
durin
gMicro
Grav
ityMech
anics
testsaboard
Space
Shuttle
(Sture
etal.
[50]).
Thelarge
defo
rmatio
nmodeldenitio
nisbased
ontheuse
oftheMandelstress
Tijfordescrib
ingyield
andpoten
tialsurfa
ces.Adeta
ileddescrip
tionofthemodelisgiven
byJerem
icet
al.[18].
4NumericalSimulatio
nsofMicroGravity
Mechanics
Inthissectio
nwepresen
tnumerica
lmodelin
goflow
connem
ent,
micro
grav
itylargede-
form
atio
ntria
xialtest
perfo
rmed
durin
gSpace
Shuttle
STS79missio
nin
Septem
ber
1996.
Figure
3show
sloaddisp
lacem
entandvolumedisp
lacem
entdata
forthree
lowconnem
ent
tests.Theresp
onse
curves
represen
tloaddisp
lacem
entdata
asthey
were
measured
dur-
0.00
0.00
0.08
0.06
0.04
0.02
0.010.02
0.030.04
vertical displacement [m
]
vertical force [kN]
MG
M052
MG
M005
MG
M130
volume change (dilation) [10 m ]-6 3
0.0000.010
0.0200.030
0.040
MG
M005
MG
M130
MG
M052
0.0
20.0
40.0
60.0
80.0
100.0
displacement [m
]
Figure
3:Micro
Grav
ityMech
anics,
loaddisp
lacem
entandvolumedisp
lacem
entcurves
for
thethree
tests.
ingtheexperim
ents.
Thesig
nalcontainssig
nicantnoise
andthepresen
teddata
are
in
rawform
.Theela
sticresp
onse
appears
tobevery
sti(fro
munloadingrelo
adingloops).
Deta
ileddescrip
tionoftheexperim
entalsetu
pisgiven
bySture
etal.[50].
Thethree
dim
ensio
nalnite
elementmesh
used
tomodel
theMGM
testis
depicted
inFigures
4.Instea
dofdevelo
pingtwodim
ensio
nalnite
elementform
ulatio
n,wehave
opted
forafullthree
dim
ensio
nalim
plem
entatio
n.Alth
oughthesta
teofstress
istria
xial,
wemodeltheexperim
ents
with
a3D
model.
Six
quadratic
20nodebrick
elements
where
29
UCD-CGM report
0.0375m0.0375m
0.075m
Figure
4:Finite
elementmesh
fortheMGM
specim
en.
chosen
tomodel
oneeig
hth
ofthespecim
en.Theanalysis
wasperfo
rmed
intwosta
ges.
First
stageinvolved
isotro
pic
compressio
nto
thedesig
npressu
re.Fortherst
stageonly
symmetry
disp
lacem
entboundary
conditio
nswere
inplace.
In uence
ofthemem
brane
wasrem
oved,sin
cethemem
branedoes
nothavesig
nicantsti
ness
incompressio
n,and
mem
braneprestressin
ghadaminoreect
atthissta
ge.
Durin
gthissta
getheresp
onse
was
purely
hyperela
stic.
After
thetherst
stage,thedisp
lacem
entboundary
conditio
nswere
changed
byadding
themova
bleboundary
atthetop.Thetopmova
bleboundary
applied
disp
lacem
entsto
the
topnodes
bymeansofequiva
lentforces,
obtained
throughthepartia
linversio
nofasti
ness
matrix
.Themem
branein uence
wasmodeled
byaddingequiva
lentsti
ness
(sprin
gs)to
the
boundary
nodes.
Instea
dofusin
gthin,highlydisto
rtedbrick
elements(m
embraneis0:3mm,
disto
rtionratio
would
be(237:5mm=8)=(0:3mm)100=1).weopted
fortheequiva
lent
sprin
gmeth
od.Theoutputfro
mtheoneelem
entexten
siontests
onthehyperela
sticlatex
rubber
specim
enwhere
used
toform
anonlin
earsprin
gofappropria
testi
ness.
Consisten
t
integ
ratio
nofthesti
ness
termsforthequadratic
brick
elementthen
supplied
equiva
lent
sprin
gsti
ness.
Specia
latten
tionwasgiven
tothespecim
enends,where
thelatex
mem
brane
30
UC
D-C
GM
rep
ort
was wrapped around the end platen and created a ring in the horizontal plane (parallel to end
platen) which was stier than the unstretched membrane surrounding the specimen. The
last row of nodes was thus supported by stier equivalent membrane elements. The material
parameters for the B Material Model for all three conning pressures12 were kept the same
except for the Young's modulus. This consistency in material parameters is important, since
all three specimens contained the same Ottawa F-75 sand at 85% relative density.
(a)0.00 0.01 0.02 0.03 0.04 0.05
0.05
0.04
0.03
0.02
0.01
0.00
-0.01
vertical displacement [m]
initial confinment p = 0.05kPa
vert
ical
for
ce [
kN]
MGM experiment
(b)
volu
me
chan
ge [
10
m ]
-63
0.000 0.010 0.020 0.030 0.040 0.050
displacement [m]
0.0
20.0
40.0
60.0
80.0
100.0
120.0
Figure 5: Mechanics of granular materials responses, initial connement (p0 = 0:05kPa) test
(a) loaddeformation and (b) volumedeformation experiments and numerical results.
(a)0.00
vertical displacement [m]
MGM experiment
initial confinment p = 0.52kPa
0.01 0.02 0.03 0.04 0.05
0.06
0.05
0.04
0.03
0.02
0.01
vert
ical
for
ce [
kN]
(b)
volu
me
chan
ge [
10
m ]
-63
0.000 0.010 0.020 0.030 0.040 0.050-10.0
10.0
30.0
50.0
70.0
90.0
110.0
displacement [m]
Figure 6: Mechanics of granular materials responses, initial connement (p0 = 0:52kPa) test
(a) loaddeformation and (b) volumedeformation experiments and numerical results.
12E = 300:0; 360:0; 700:0 kN=m2; = 0:2 ; pc = 1000:0 kN=m2 ; pt = 0:0 kN=m2 ; n = 0:2 ; a = 5:0 ; b
= 0:707 ; init = 2:5 ; b1 = 1:0 ; dhard = 5000:0 ; ehard = 0:5 ; res = 0:15 ; peak = 1:75 ; start = 0:25 ; l
= 1:0 ; ccone = 0:030 ; r = 1:00 ; ccap = 0:30 ; pc;0 = 1000:0 kN=m2 ; as = 100:0 ; bs = 0:707)
31
UC
D-C
GM
rep
ort
(a)0.00
MGM experiment
vertical displacement [m]
initial confinment p = 1.30kPa0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0.01 0.02 0.03 0.04 0.05
vert
ical
for
ce [
kN]
(b)
volu
me
chan
ge [
10
m ]
-63
0.000 0.020 0.040-10.0
10.0
30.0
50.0
70.0
90.0
displacement [m]
Figure 7: Mechanics of granular materials responses, initial connement (p0 = 1:30kPa) test
(a) loaddeformation and (b) volumedeformation experiments and numerical results.
Figures 5(a), 6(a) and 7(a) show comparison of numerical modeling with the test data
for loaddisplacement. Following observations are made. The initial (elastic) stiness is
higher in the actual experiments. The peak strength is modeled quite accurately, while the
postpeak behavior is slightly stier in the numerical experiment. The residual stiness is
softer in the numerical model than observed in the MGM tests. This can be explained by
the stier specimen ends in a physical test. In other words, the latex membrane wrapped
around the end platens (the end platens are 30% wider than the specimen) usually sticks to
the end platen after some radial displacements and then acts as a full restraint. The friction
between end platens and the sand specimen can also add to the whole specimen stiness,
however, the end platens were made of highly polished tungstencarbide, which has a very
low friction angle with quartz sand (3Æ), and we have thus decided to neglect the in uence
of end platen friction on the overall response. It is of interest to note that the maximum
mobilized friction angle is in the range of 70Æ and the dilatancy angles observed in the early
parts of the experiments are 30Æ, which is unusually high.
Figures 5(b), 6(b) and 7(b) shows comparison of volumetricdisplacement data for exper-
iments and numerical modeling. In modeling the lowest connement (p0 = 0:05kPa) level
we correctly predict complete lack of volumetric compression. Numerical predictions for two
other connements (p0 = 0:52kPa, p0 = 1:20kPa) shows small amount of initial volume
compression which was not observed in experiments. Figure 8 shows a typical specimen be-
fore and after the test. The latex ring formed by wrapping of membrane around end platens
is visible on both specimen ends.
32
UCD-CGM report
Figure
8:Thespecim
en(p
=1:30kPa)befo
reandafter
thetest.
Theeect
ofthelatex
mem
braneontheloaddisp
lacem
entbehaviorofspecim
encannot
beneglected
forthelow
connem
entexperim
ents.
Asatria
xialspecim
enexpands,
the
mem
braneexpandsaswell.
Thestretch
ingofthehyperela
sticmem
braneproduces
additio
nal
stressesandincrea
setheorig
inalconnem
entlev
el.Figures
9,10and11show
sthein uence
oflatex
mem
braneonthespecim
enbehavior.
Theresp
onse
with
outlatex
mem
braneis
softer,
anditdoes
notlev
eloin
thepost
peakreg
ion.Theloaddisp
lacem
entresp
onse
hasa atportio
n,butsta
rtshardeningafter
approx
imately
15%
axialdefo
rmatio
nfortwo
higher
connem
enttests
(p=
0:52kPaandp=
1:30kPa),
while
forthethevery
low
connem
enttest
(p=
0:05kPa)ithardensmonotonica
lly.Thiscanbeexplained
bythe
largedisp
lacem
enteects.
Forlargeaxialdefo
rmatio
ns,latera
lbulgingissig
nicant.
As
theaxialdefo
rmatio
nprogresses,
themateria
l(sa
nd)moves
from
thespecim
encen
terto
theboundary
region,thuscrea
tingaslig
hthardeningeect.
Theincrea
sein
peakstren
gth
dueto
thelatex
mem
braneeects
isnottoopronounced
.Thepost
peakreg
ion,how
ever,
show
sadditio
nalsti
ening.Forthelow
estconnem
enttest,
thein uence
ofthemem
brane
issubsta
ntia
lsin
cethespecim
enitself
(atonlyp=0:05kPa)isquite
soft.
Figure
12depicts
thedefo
rmed
shapeofaspecim
en.With
outthelatex
mem
brane,the
33
UCD-CGM report
(a)
vertical force [kN]
vertical displacement [m
]0.000
with m
embrane
without m
embrane
0.040
0.030
0.020
0.010
0.0200.040
0.060
(b)
volume change [10 m ]-6 3
0.0000.020
0.0400.060
displacement [m
]
0.0
50.0
100.0
150.0
with m
embrane
without m
embrane
Figure
9:Mech
anics
ofgranularmateria
lsresp
onses,
initia
lconnem
ent(p
0=
0:05kPa)
(a)loaddefo
rmatio
nand(b)volumedefo
rmatio
nnumerica
lpred
ictions.In uence
oflatex
mem
braneontheovera
llresp
onse.
0.00
vertical displacement [m
]
vertical force [kN]
0.06
0.05
0.04
0.03
0.02
0.01
0.010.02
0.030.04
without m
embrane
with m
embrane
volume change [10 m ]-6 3
0.0000.020
0.040-20.0 0.0
20.0
40.0
60.0
80.0
100.0
120.0
displacement [m
]
with m
embrane
without m
embrane
Figure
10:Mech
anics
ofgranularmateria
lsresp
onses,
initia
lconnem
ent(p
0=
0:52kPa)
(a)loaddefo
rmatio
nand(b)volumedefo
rmatio
nnumerica
lpred
ictions.In uence
oflatex
mem
braneontheovera
llresp
onse.
specim
endefo
rmsuniform
ly.Theabovementio
ned
endrestra
intresu
ltsin
adiuse
bulging
defo
rmed
shape,show
nin
Figure
12.
5ConcludingRemarks
Inthispaper
wehavepresen
tedanew
largedefo
rmatio
nconstitu
tiveform
ulatio
nforge-
omateria
ls.Constitu
tiveform
ulatio
nwasused
inconjunctio
nwith
largedefo
rmatio
nLa-
grangiannite
elementmeth
od.Theform
ulatio
niscapableofsim
ulatin
glargedefo
rmatio
n
hyperela
sticplastic
behaviorofgeomateria
ls,even
when
collin
earity
betw
eeneig
entria
dsof
34
UC
D-C
GM
rep
ort
vertical displacement [m]
vert
ical
for
ce [
kN]
with membrane
without membrane
0.00
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0.01 0.02 0.03 0.04
volu
me
chan
ge [
10
m ]
-63
0.000 0.020 0.040-10.0
10.0
30.0
50.0
70.0
90.0
displacement [m]
without membrane
with membrane
Figure 11: Mechanics of granular materials responses, initial connement (p0 = 1:30kPa)
(a) loaddeformation and (b) volumedeformation numerical predictions. In uence of latex
membrane on the overall response.
Initial
with membrane
without membrane
Figure 12: Uniform and bulging deformed shape of a specimen.
stress and strains is lost (for anisotropic and cyclic response). A detailed constitutive formu-
lation has been presented. Moreover, the return algorithm was outlined with implementation
details. The developed formulation and implementation were used to simulate large defor-
mation tests on sand performed under very low connement. To this end, a consistent set
35
UCD-CGM report
ofmateria
lparameters
fortheBmateria
lmodelwasused
toaccu
rately
simulate
three
low
connem
enttests.
Itwasshow
nthatthelatex
mem
branehassubsta
ntia
lin uence
onthe
behaviorofsandspecim
enatvery
lowconnem
entpressu
res.
Acknowledgment
Partia
lsupportbyNASAGrantNAS8-38779fro
mMarsh
allS
pace
FlightCenter
isgrea
tfully
acknow
ledged.
36
UCD-CGM report
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