309 – 315 ISSN 1641-8581
Publishing House
AKAPIT
COMPUTER METHODS IN MATERIALS SCIENCE
In o ma yka w Technologii Ma e iałów
Vol. 9, 2009, No. 2
AN IMPLICIT NUMERICAL INTEGRATION ALGORITHM FOR
BAI & WIERZBICKI (2007) ELASTO-PLASTIC MODEL
LUCIVAL MALCHER, FRANCISCO M. ANDRADE PIRES,
JOSÉ M.A. CÉSAR DE SÁ, FILIPE X.C. ANDRADE
Depa men o Mechanical Enginee ing, Facul y o Enginee ing, Uni e si y o Po o
Rua D . Robe o F ias, Po o 4200-465, Po ugal
Co esponding Au ho : [email p o ec ed]; (L. Malche )
Abs ac
This con ibu ion desc ibes an implici algo i hm o nume ical in eg a ion o a ecen ly p oposed model o me al
plas ici y and ac u e [2]. The cons i u i e equa ions o he ma e ial model c i ically include bo h he e ec o p essu e
h ough he iaxiali y a io and he e ec o hi d de ia o ic s ess in a ian h ough he lode angle in he desc ip ion o
ma e ial. These e ec s a e di ec ly in oduced on he ha dening ule o he ma e ial. The heo e ical basis o he ma e ial
model is p esen ed in he i s pa o he pape . Then, he necessa y s eps equi ed o implemen he model wi hin an im-
plici quasi-s a ic ini e elemen en i onmen a e discussed. In pa icula , he s ess upda e p ocedu e, which is based on
he so-called ope a o spli concep esul ing in he s anda d elas ic p edic o / e u n mapping algo i hm, and he compu a-
ion o angen ma ix consis en wi h he s ess upda e a e desc ibed. Finally, he simula ion o a la g oo ed specimen
subjec ed o ension [1] is p esen ed o illus a e he obus ness and e iciency o he p oposed algo i hm.
Key wo ds: ini e elemen me hod, elas o-plas ic model, hyd os a ic p essu e sensi i i y, lode angle dependence
1. INTRODUCTION
One o he mos commonly used models o de-
sc ibe he beha io o me als is he on Mises
model. Acco ding o his model, plas ic yielding
begins when he second in a ian o he de ia o ic
s ess enso , , eaches a c i ical alue. The p es-
su e componen o he s ess enso does no ake
pa in he de ini ion o yielding and consequen ly
he model can be ega ded as “p essu e-insensi i e”
[5]. Ano he ea u e o he model, when compa ed o
o he ma e ial models, is ha i is also independen
o he hi d in a ian o he de ia o ic s ess enso .
Bo h pa ame e s ha e go a well known in luence on
he yield su ace [8]. The le el o hyd os a ic is e-
sponsible o con olling he size o he yield su ace
and he hi d in a ian is esponsible o he shape o
he yield su ace [1].
The impo ance o he hyd os a ic s ess and lode
angle has been ecognized by se e al au ho s and
in oduced in o he cons i u i e desc ip ion o some
ma e ials such as soils, ocks and conc e e [2,8]. In
he case o duc ile ma e ials, which a e add essed in
his pape , many esea che s ha e conduc ed ex en-
si e expe imen al s udies. Richmond and Spi zing [6]
we e among o he i s esea che s o in es iga e he
e ec o p essu e on yielding o aluminum alloys.
Duc ile ac u e is a local phenomenon and he
s a e o s ess and s ain, a he expec ed ac u e
loca ion, mus be de e mined wi h accu acy. F ac-
u e ini ia ion is o en p eceded by la ge plas ic de-
o ma ion and he e a e conside able s ess and
s ain g adien s a ound he poin o ac u e. In his
case, he  heo y is no accu a e enough and mo e
e ined plas ici y models ha e o be in oduced.
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The de e mina ion o an adequa e shape o he
yield su ace has become an impo an issue in shee
me al o ming. In his case, he on Mises plane
s ess ellipse does no lead o a co ec p edic ion o
necking ins abili y. To o e come some o he p e i-
ous limi a ions, Bai & Wie zbicki [2] ha e p oposed
a model ha includes bo h he e ec o p essu e and
he hi d in a ian on he e olu ion o plas ic low.
P elimina ies
Fo he sake o comple eness and no a ion cla -
i y, we will b ie ly summa ize some undamen al
ela ions commonly used in plas ici y. The s ess
enso can be spli in wo componen s: an hyd o-
s a ic and a de ia o ic componen , which can be
ep esen ed as:
 (1)
The s ain enso can also be spli in wo compo-
nen s:


 (2)
whe e,  is he s ess enso , 2
 is
he de ia o ic s ess enso , 
 

ep esen s he p essu e o hyd os a ic s ess,  is he
elas ic s ain enso , 
 is he de ia o ic s ain enso
and 
 ep esen s he olume ic s ain. The con-
s an s  and  a e, espec i ely, he shea modulus
and he bulk modulus. We can also de ine he
equi alen s ess, which is a unc ion o he second
in a ian o de ia o ic s ess enso , as:
3 
 
 (3)
whe e,  is he equi alen s ess,  is he second
in a ian o he de ia o ic s ess and  ep esen s
he no m o he de ia o ic s ess enso .
The hi d in a ian is a scala quan i y ha is ob-
ained by compu ing he de e minan o he de ia-
o ic s ess enso , deno ed by . Al e na-
i ely, he hi d in a ian can be exp essed by:
󰇣
 󰇤
󰇣
 󰇤
 (4)
The in luence o p essu e on he on Mises
elas o-plas ic model can be in oduced h ough he
iaxiali y a io which is a dimensionless hyd os a ic
s ess. The iaxiali y a io is de ined as he a io
be ween he p essu e and he equi alen s ess:

 (5)
The hi d in a ian o he de ia o ic s ess enso
can be no malized and ela ed wi h he so-called
lode angle h ough he exp ession:
󰇡
󰇢
.
cos󰇛3󰇜 (6)
whe e,  is he no malized hi d in a ian and  is
he lode angle. Since he ange o he lode angle
is 0 3
⁄, he ange o  has o be 1
1. The lode angle can also be no malized by:
1
1
󰇛󰇜 (7)
whe e,  is he no malized lode angle. The ange o
 is 11. The pa ame e  will be called he
lode angle pa ame e he eina e .
2. THE BAI & WIERZBICKI YIELD
FUNCTION
Bai & Wie zbicki [2] ha e p oposed an elas o-
plas ic model ha includes bo h he e ec o p es-
su e ( h ough he iaxiali y a io) and he e ec o
he hi d in a ian o he de ia o ic s ess enso
( h ough he lode angle). These e ec s a e in o-
duced by ede ining he ha dening ule. While in he
classic on Mises model, he ha dening ule is only
a unc ion o he accumula ed plas ic s ain, 󰇛󰇜,
in Bai & Wie zbicki´s model he ha dening ule is
a unc ion o he accumula ed plas ic s ain, he i-
axiali y a io and he pa ame e 󰇛󰇜, which is
a unc ion o he load angle. The ha dening ule is
ew i en as:
󰇛,,󰇜󰇛󰇜.1󰇛󰇜.
󰇣
󰇛
 
󰇜󰇡
󰇢󰇤 (8)
whe e, 󰇛󰇜 is he ma e ial s ain ha dening unc-
ion, ,
,
, and  a e expe imen al pa ame-
e s,  is he e e ence alue o he iaxiali y a io
and 󰇛󰇜 is a pa ame e de ined as a unc ion o he
lode angle:
󰇛󰇜cos󰇛6
⁄󰇜
1cos󰇛6
⁄󰇜1
cos󰇛6
⁄󰇜1
󰇛
⁄󰇜
󰇛
⁄󰇜.󰇟sec󰇛6
⁄󰇜1󰇠 (9)
The e ec o he iaxiali y a io and load angle
a e included on he ha dening ule h ough he pa-
ame e s 󰇛󰇜1󰇛󰇜 and 󰇛󰇜
󰇣
󰇛
 
󰇜󰇡
󰇢󰇤, espec i ely. The
new yield c i e ion is simply ob ained by eplacing
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he s anda d ha dening ule 󰇛󰇜 by 󰇛,,󰇜
on he on Mises yield unc ion, which can be ep e-
sen ed by Φ, as:
Φ󰇛󰇜.1󰇛󰇜.
󰇣
󰇛
 
󰇜󰇡
󰇢󰇤 (10)
In o de o unde s and he abo e yield unc ion, le
us analyze he in luence o some expe imen al pa-
ame e s ,
,
,, on he beha io o he
model. The pa ame e  is a ma e ial cons an ha
needs o be expe imen ally calib a ed and ha de-
sc ibes he e ec o he hyd os a ic p essu e on he
ma e ial plas ic low. I 0, he model loses he
dependence o he iaxiali y (and o he hyd os a ic
p essu e) and eco e s, as a limi ing case, he beha -
io o he on Mises model.
The e e ence iaxiali y, , depends on bo h he
ype o load applied and he geome y o he speci-
men. Fo a smoo h ba subjec ed o ension,  akes
he alue equal o 1 3
⁄, o a cylind ical specimen
unde comp ession  is equal o 1 3
⁄ and o
bo h o sion and shea es s  is equal o 0. Wi h
ega d o he hi d in a ian e ec , he expe imen al
pa ame e 
 can assume one o wo o ms, ac-
co ding o he alue o he no malized lode angle .

 󰇫
  0

  0 (11)
The pa ame e 
 also depends on he ype o
es . Fo example, i a smoo h ba is used in a ensile
es 
1, i a o sion es 
1, i a cylinde
specimen is used in a comp essi e es 
1. The
con exi y o he yield su ace is con olled by he
a io o hese pa ame e s. The ange o he pa ame e
 is be ween 0 1. When 0 i co e-
sponds o plane s ain o shea condi ion, when
1 i co esponds o axisymme ic p oblem. The
in oduc ion o he e m 
 is done o ensu e he
smoo hness o yield su ace and i s di e en iabili y
wi h espec o lode angle a ound 1.
3. NUMERICAL INTEGRATION
ALGORITHM
An algo i hm ea men o he model summa-
ized in sec ion 2 elies on he ope a o spli me h-
odology, which esul s in an elas ic p edic o /plas ic
co ec o algo i hm [4,7]. In he elas ic p edic o
phase (ob ained by eezing he plas ic low du ing
he ime in e al [, ]), alues om he p e i-
ous con e ged solu ion a e used as ini ial condi ions
o e alua e he elas ic ial s a e, i.e., he exac elas ic
solu ion.
I he yield condi ion is iola ed, he plas ic co -
ec o is ini ia ed based on he New on-Raphson
p ocedu e. The go e ning equa ions o one s ep o
he New on-Raphson i e a i e scheme a e summa-
ized in Box 1.
4. NUMERICAL EXAMPLE
This sec ion p esen s an example o illus a e ba-
sic aspec s o he p oposed in eg a ion algo i hm
desc ibed p e iously. A ensile es o la g oo ed
pla es, as shown in igu e 1, is used o illus a e he
nume ical pe o mance o he p oposed algo i hm
wi hin an implici quasi-s a ic ini e elemen en i-
onmen . The ela i e esidual o he solu ion is p e-
sen ed o show he asymp o ic con e gence o he
algo i hm.
The specimen has been subjec ed o mono onic
axial s e ching and he nume ical esul s a e com-
pa ed wi h expe imen al esul s epo ed by Bai e al
[2]. The ma e ial p ope ies including he p essu e
e ec and lode dependence cons an s, adop ed in he
p esen analysis, a e lis ed in able 1. These pa ame-
e s we e aken om Bai [1] o an aluminum alloy.
Fig. 1. (a) Fla g oo ed specimen, hickness = 2.11 mm, g oo e
adius = 1.59 mm, wid h = 50 mm, pla e hickness = 5 mm.
(b) a longi udinal sec ion o he specimen and (c) Fini e elemen
mesh: 1981 nodes and 616 elemen s.
Table 1. Calib a ed ma e ial p ope ies o aluminum 2024-T351.
Basic ma e ial p ope ies P essu e and
lode cons an s
Desc ip ion Symbol Value η0 1/3
Densi y ρ 2.7 103 Kg/m3 Cη 0.009
Elas ic
Modulus E 7.115 105 [MPa] Cθ
1.0
Poisson´s
a io ν 0.3 Cθ
s 0.855
Ini ial yield
s ess σy0 370 [MPa] Cθ
c 0.90
Ha dening
cu e σy(

) 908(0.0058+

)0.1742
[MPa] m 6
(a) (b) (c)
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whe e 
,  and 

 a e he esidual unc ions
o each a iable o he p oblem.  ep esen s
he low ec o and can be de ined as: 
 



. The pa ame e s ,  and  a e
ob ained h ough he equa ions: 1
󰇛
󰇜
...., .󰇛
󰇜..
 and
󰇛
󰇜
󰇣...
.󰇡󰇛󰇜󰇛󰇜
󰇢󰇤.
In o de o highligh he con e gence o he
New on-Raphson algo i hm in able 2, we p esen
he ela i e esidual o he solu ion o wo ypical
load inc emen s when we ha e bo h p essu e e ec
and lode angle dependence in oduced. The con e -
gence a es, in bo h o case, a e clea ly quad a ic.
In he igu e 2, we can analyze he beha io o
he eac ion-displacemen cu e (a) and he accumu-
la ed plas ic s ain-displacemen cu e (b) ob ained
nume ically. Ve i ying i s he cu e eac ion-
displacemen , igu e 2(a), he esul ob ained wi hou
Table 2. Con e gence able o wo ypical load inc emen s.
Fla g oo ed specimen.
including any e ec 0,
1,
1,

1,0, ep oduces he on Mises model. In his
case, he e o is a ound 17%, be ween expe imen al
and nume ical esul s. I only he p essu e e ec is
included 0.09,
1,
1,
1,
0, he esul s a e simila wi h D ucke P age model
and he e o be ween he nume ical simula ion and
he expe imen al one a e smalle han he p e ious
case, bu s ill big. In his case, he e o is abou
13%. Finally, we can in oduce bo h he p essu e
Box 1: Fully implici Elas ic p edic o /Re u n mapping algo i hm.
I e a ion
numbe
Rela i e esidual
[%]
I e a ion
numbe
Rela i e esidual
[%]
1 0.51275x10-2 1 0.17977x10-1
2 0.37416x10-4 2 0.44738x10-3
3 0.21888x10-8 3 0.27548x10-6
4 0.12611x10-14 4 0.10461x10-12
i) E alua e he elas ic ial s a e: Gi en he inc emen al s ain and he s a e a iables a .
, ,
, ,
, ,
, ,
ii) Check plas ic admissibili y:
IF , THEN se (elas ic s ep)
iii) Re u n mapping (plas ic s ep):
Sol e he sys em o equa ions below o , and using New on-Raphson me hod:
GOTO Box 2
i ) Upda e he s a e a iables:
, , ,
) Exi
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whe e, 
 and  ep esen he upda ed yield unc ion
and he upda ed on Mises equi alen s ess, espec-
i ely, on he disc e iza ed s ep . The a iable

 ep esen s he accumula ed plas ic s ain a
. The a iable ∆ ep esen s he plas ic
mul iplie and  ep esen s he yield s ess o he
con en ional on Mises model.
e ec and lode angle dependence 0,

0.855,
1,
0.90,6. In his case, he
eac ion-displacemen cu e ag ees well wi h he
expe imen al esul s. The e o is less han 2%.
By analyzing he accumula ed plas ic s ain-
displacemen cu e, igu e 2(b), we can obse e he
e olu ion o he accumula ed plas ic s ain as a unc-
ion o he p esc ibed displacemen . We can e i y
ha only wi h p essu e e ec and wi h bo h p essu e
e ec and lode angle, he g ow h a e o he accumu-
la ed plas ic s ain is as e han wi hou e ec s in o-
duced. In addi ion, we can conclude ha he Bai &
Wie zbicki elas o-plas ic model eaches highe le els
o plas ic s ain, e en o lowe le els o equi alen
s ess, when we ha e bo h e ec s included.
F om igu e 3, we can de e mine he poin o
maximum and minimum equi alen s ess o all
nume ical simula ions. Fo he on Mises model, he
maximum equi alen s ess a ains a alue o 900.73
MPa. I only he p essu e e ec is included, he
maximum equi alen s ess is equal o 843.35 MPa.
When bo h e ec s a e included, he maximum
equi alen s ess is equal 721.64 MPa. The maxi-
mum alue is always loca ed on he cen al poin o
Box 2: The New on-Raphson algo i hm o solu ion o he e u n mapping sys em o equa ions.
1) Ini ialize i e a ion coun e , , se ini ial guess o , and
co esponding esidual:
2) Pe o m New on-Raphson i e a ion
New guess o , and :
, and
Upda e s ess enso :
, and
oge he wi h o he elas o-plas ic pa ame e s:
, , , , and
3) Check o con e gence
IF THEN RETURN o Box 1.
4
)
GOTO
(
2
)

COMPUTER METHODS IN MATERIALS SCIENCE
INFORMATYKA W TECHNOLOGII MATERIAŁÓW
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he specimen. The expe imen al esul s ag ee wi h
he nume ical simula ions when we ha e he p es-
su e e ec and lode dependence in oduced.
5. CONCLUSION
In his pape was p esen ed an implici nume ical
in eg a ion o a ecen ly p oposed elas o-plas ic
model ha includes he p essu e e ec and lode an-
gle dependence in o he ha dening ule. The con e -
gence o he New on Raphson algo i hm is simila
be ween on Mises model and Bai & Wie zbicki
model wi hou e ec s in oduced and be ween
D ucke P age and Bai & Wie zbicki model wi h
only p essu e e ec in oduced. The nume ical simu-
la ion o a la g oo ed specimen o he aluminum
alloy was used o illus a e he obus ness o he
p oposed algo i hm, we e i ied ha bo h e ec s
ha e o be aken in o accoun on he plas ic low.
The e o in he eac ion-displacemen cu e o
a o al displacemen o 0.2 [mm] was educed om
17% o 2% by including bo h he e ec o p essu e
and lode angle.
REFERENCES
1. Bai, Y., E ec o Loading His o y on Necking and F ac-
u e, Ph.D Thesis, Massachuse s Ins i u e o Technology,
2008.
2. Bai, Y., Wie zbicki, T., A new model o me al plas ici y
and ac u e wi h p essu e and lode dependence, In . J. o
Plas ici y, 2007, doi:10.1016/j.ijplas.2007.09.004.
a) b)
Fig. 2. Reac ion-displacemen cu e (a) and accumula ed plas ic s ain-displacemen cu e (b) o Bai and Wie zbicki model: wi hou
e ec s (o on Mises model), wi h only p essu e e ec included (o D ucke P age model) and bo h p essu e e ec and lode angle
dependence included.
Fig. 3. Equi alen s ess pa ame e plo ed h ough he mesh. (a) wi hou e ec s o on Mises model, (b) only wi h p essu e e ec o
D ucke P age model, and (c) bo h p essu e e ec and lode angle dependence.
(a) (b) (c)
INFORMATYKA W TECHNOLOGII MATERIAŁÓW
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C
OMPUTER METHODS IN MATERIALS SCIENCE
3. Bao, Y., P edi ion o Duc ile C ack Fo ma ion in Un-
c ecked Bodies, Ph.D Thesis, Massachuse s Ins i u e o
Technology, 2003.
4. De Souza Ne o, E. A., Pe ic, D., & Owen, D. R., Compu a-
ional Me hods o Plas ici y: Theo y and Applica ion.
Wiley, 2008.
5. Hill, R., The Ma hema ical Theo y o Plas ici y, Ox o d
Uni . P ess, London, 1950.
6. Richmond, O., Spi zig, W.A., P essu e dependence and
dila ancy o plas ic low. In Theo e ical and Applied Me-
chanics, P oc. 15 h In . Cong ess o Theo e ical and Ap-
plied Mechanics., To on o, No h-Holland Publ Co, Am-
s e dam, 1980, 377–386.
7. Simo, J., Hughes, T., Compu a ional Inelas ici y, Sp inge ,
1998.
8. Wilson, C.D., A c i ical eexamina ion o classical me al
plas ici y, Jou nal o Applied Mechanics, T ansac ions
ASME, 69(1), 2002, 63–68.
NIEJAWNY ALGORYTM CAŁKOWANIA
NUMERYCZNEGO DLA SPRĘŻYSTO-
PLASTYCZNEGO MODELU BAI – WIERZBICKIEGO
S eszczenie
Niniejszy a ykuł opisuje niejawny algo y m całkowania
nume ycznego modelu plas yczności i pękania me ali. Równania
kons y u ywne modelu ma e iału uwzględniają za ówno wpływ
ciśnienia p zez wp owadzenie współczynnika ójosiowowości
i wpływ zeciego niezmiennika dewia o a enso a nap ężenia.
Te e ek y bezpoś ednio wpływają na sposób umacniania się
ma e iału. W pie wszej części a ykułu p zeds awiono pods awy
eo e yczne modelu ma e iału. Nas ępnie omówione są k oki
niezbędne do implemen acji modelu w niejawnym quasi-
s a ycznym ś odowisku elemen ów skończonych. W szczegól-
ności opisane zos ały: p ocedu a uak ualniania nap ężenia, k ó a
opie a się na ak zwanym podziale ope a o a o e ującym s an-
da dowy algo y m odwzo owania w s anie sp ężys ym o az na
p ocedu ze obliczania macie zy s ycznych spójnej z uak ualnia-
niem wa ości nap ężeń. W końcowej części p acy p zeds awio-
no symulację ozciągania płaskiej p óbki z owkami w celu
pokazania wia ygodności i sku eczności p oponowanego algo-
y mu.
Recei ed: Oc obe 8, 2007
Recei ed in a e ised o m: No embe 6, 2007
Accep ed: No embe 6, 2007