UNIVERSIDADE DO PORTO
Con inuous-Discon inuous App oach o
he Modelling o Duc ile F ac u e
Ma iana Ri a Ramos Seab a
Faculdade de Engenha ia
Depa amen o de Engenha ia Mecˆanica
Oc obe 2012
“C a e o a hing, you will ge i . Renounce he c a ing, he objec will ollow you by
i sel .”
Swami Si ananda (1887-1963)
Abs ac
Duc ile me als, such as aluminium o s eel, a e p esen in all so s o indus ies, om
ae onau ical o au omo i e, cons uc ion o household goods. A deepe unde s anding
o he ac u e mechanisms associa ed o hese ype o ma e ials is essen ial o op imize
p oduc ion p ocesses as well as a oid ca as ophic ailu es.
T adi ional ac u e heo ies gene ally ely on a single ene ge ic pa ame e o igge
c ack p opaga ion, which may no be adequa e o duc ile me als, as he e is a sub-
s an ial amoun o plas ic de o ma ion p io o ailu e. On he o he hand, Con in-
uum Mechanics heo ies success ully handle la ge s aining and desc ibe success ully he
plas ic-ha dening and plas ic-so ening s ages o ma e ial beha iou . Ne e heless in
he inal s ages o ailu e, a discon inuous me hodology is essen ial o ep esen su ace
decohesion and mac o-c ack p opaga ion.
The aim o his wo k is o build a model o duc ile ac u e ga he ing he ad an ages o
bo h con inuous and discon inuous app oaches. The Ex ended Fini e Elemen Me hod
(XFEM) is combined wi h he Lemai e model o duc ile damage in a way ha c ack
ini ia ion and p opaga ion a e go e ned by he e olu ion o damage. The model was
buil unde a ini e s ain assump ion and a non-local in eg al o mula ion is applied o
a oid pa hological dependency o esul s on he spa ial disc e isa ion. Special a en ion
is gi en o he necessa y adap a ions o he XFEM o be used in a duc ile ac u e con-
ex , such as selec ion o he en ichmen unc ions and nume ical in eg a ion s a egies.
The ene gy consis ency issue du ing he ansi ion om damage o ac u e is also ad-
d essed. The e ec s o he in oduc ion o a cohesi e law in he global duc ile ac u e
model a e analysed and a s a egy o app oxima e he ully damaged ma e ial condi ion
is p oposed.
Th oughou he wo k, he e iciency o he p oposed model is e alua ed h ough a -
ious nume ical examples and gene al conclusions a e ou lined. To inalize his hesis,
some commen s on he o e all pe o mance o he model a e gi en oge he wi h some
sugges ions o u u e wo k.
Resumo
Os me ais d´uc eis, como o alum´ınio ou o a¸co, es ˜ao p esen es nas mais di e sas ind´us ias,
da ae onau ica `a au om´o el, da cons u¸c˜ao aos bens de consumo. O ap o undamen o
dos conhecimen os ela i os aos mecanismos de ac u a associados a es e ipo de me ais
´e essencial pa a op imiza p ocessos de p odu¸c˜ao e e i a alhas ca as ´o icas.
As eo ias cl´assicas da ac u a baseiam-se num ´unico pa ˆame o ene g´e ico que pode n˜ao
se adequado a me ais d´uc eis, pois es es ap esen am de o ma¸c˜oes pl´as icas signi ica as
an e io es `a p opaga¸c˜ao de endas no ma e ial. Po ou o lado, as eo ias de Mecˆanica
dos Meios Con ´ınuos desc e em adequadamen e as ases de endu ecimen o e amacia-
men o pl´as icos. Con udo, nas ases inais de deg ada¸c˜ao do ma e ial, uma o mula¸c˜ao
descon ´ınua ´e undamen al pa a desc e e a sepa a¸c˜ao en e as supe ´ıcies de uma enda
em p opaga¸c˜ao.
Es e abalho em como p incipal objec i o o desen ol imen o de um modelo de ac u a
d´uc il, eunindo as an agens dos modelos con ´ınuos e descon ´ınuos. O M´e odo dos Ele-
men os Fini os Es endidos (XFEM) ´e associado ao modelo de dano d´uc il de Lemai e,
de o ma a desc e e a inicia¸c˜ao e p opaga¸c˜ao. O modelo conside a g andes de o ma¸c˜oes
e uma o mula¸c˜ao n˜ao-local in eg al, de o ma a e i a a dependˆencia pa ol´ogica dos
esul ados ela i amen e `a disc e iza¸c˜ao espacial.
Pa a consegui uma ansi¸c˜ao ene ge icamen e consis en e en e o dano e a ac u a,
analisam-se os e ei os da in odu¸c˜ao de uma lei coesi a no modelo de ac u a d´uc il
global, o que pe mi e o desen ol imen o de uma es a ´egia pa a ap oxima a condi¸c˜ao
de o al deg ada¸c˜ao do ma e ial.
Ao longo do abalho, a e iciˆencia do modelo p opos o ´e a aliada a a ´es de ´a ios ex-
emplos num´e icos. Ence a-se es a ese com impo an es comen ´a ios ge ais e suges ˜oes
de desen ol imen os u u os.
Ag adecimen os
Ao meu o ien ado , P o . Jos´e C´esa de S´a, P o esso Ca ed ´a ico da Uni e sidade do
Po o, po me e acompanhado ao longo des a ase da minha ida, p es ando odo apoio
a n´ı el cien ´ı ico, com mui a amizade e gene osidade. Tamb´em po me p opo ciona a
opo unidade de pa icipa em impo an es cong essos na ´a ea da Mecˆanica Compu a-
cional.
`
A Funda¸c˜ao pa a a Ciˆecia e Tecnologia, pelo impo an e apoio inancei o pa a o desen-
ol imen o des e abalho a a ´es da bolsa de dou o amen o SFRH/BD/43798/2008.
To he Slo ene eam: my colleague and iend P imoˇz ˇ
Suˇs a iˇc o he p ecious help p o-
g amming in he AceGen en i onmen and o he aluable commen s and sugges ions;
P o . Tomaˇz Rodiˇc which p o ided all he condi ions o his coope a ion; o P o . Joˇze
Ko elc o kindly allowing me o use his AceGen sys em and o all he suppo p o ided.
Najlepˇsa h ala!
Aos meus amigos e ambm colegas de dou o amen o Filipe And ade, F´abio Reis, Ana
Ne es e Jaime Rod igues, n˜ao s´o pelas aliosas discuss˜oes e suges ˜oes, mas amb´em pelos
momen os de descon a¸c˜ao, sem os quais e ia sido mui o mais di ´ıcil a ealiza¸c˜ao des e
abalho.
A oda a comunidade da FEUP po me p opo ciona excelen es condi¸c˜oes de abalho,
num ambien e anquilo e amilia .
`
A minha am´ılia, em especial aos meus pais e ao meu i m˜ao, po me ama em e apoia em
em udo.
Ao meu G ande Mes e do Yoga, Eng. Jo ge Veiga e Cas o e `a minha p o esso a, Eng.
Ca a ina Fe ei a, po oda a sabedo ia que me ansmi i am, em especial as ´ecnicas
que me pe mi i am aumen a a concen a¸c˜ao e melho a o meu desempenho p o issional
em ge al.
ix
Lis o Figu es x i
4.16 Regula Gaussian quad a u e ules applied o an elemen c ossed by a
discon inui y: a) ou Gauss poin s; b) nine Gauss poin s; c) six een
Gauss poin s; d) nine Gauss poin s. . . . . . . . . . . . . . . . . . . . . . . 58
4.17 Applica ion o he Schwa z-Ch is o el con o mal mapping o build an in-
eg a ion ule o elemen s c ossed by a discon inui y: a) c acked elemen ;
b) polygon ha will be mapped o he uni disk; c) uni disk con aining
he Gauss poin posi ions; d) inal Gauss poin dis ibu ion on he uppe
pa o elemen a). ............................... 59
4.18 In eg a ion ules buil using he Swchwa z-Ch is o el con o mal mapping:
a) ou Gauss poin s; b) eigh Gauss poin s; c) wel e Gauss poin s; d)
wel e Gauss poin s; e) wen y- ou Gauss poin s. . These igu es we e
c ea ed making use o he MATLAB SC Toolbox [1]. . . . . . . . . . . . . 60
4.19 In eg a ion ule o a egula 4-nodes elemen : 1 Gauss poin o he
educed ule and 4 Gauss poin s o he comple e ule; b) In eg a ion ule
o an elemen c ossed by a discon inui y: 1 Gauss poin pe iangle as
a educed ule and 3 Gauss poin s pe iangle as a comple e ule. . . . . 62
4.20 Tes examples o single elemen con aining a c ack. . . . . . . . . . . . . 64
4.21 C acked pla e wi h espec i e bounda y condi ions. . . . . . . . . . . . . . 65
4.22 a) Coa se ini e elemen mesh. b) Fine ini e elemen mesh. . . . . . . . . 66
4.23 a) Comple e in eg a ion ule; b) educed in eg a ion ule. . . . . . . . . . 66
4.24 De o med con igu a ions o di e en in eg a ion schemes: a) comple e
in eg a ion ule o all he elemen s; b) educed in eg a ion ules o all
he elemen s; c) B-ba me hodology applied only o he egula ini e ele-
men s; d) B-ba me hodology o bo h egula and en iched ini e elemen s. 67
4.25 De o med con igu a ions o di e en in eg a ion schemes: a) comple e
in eg a ion ule o all he elemen s; b) educed in eg a ion ules o all
he elemen s; c) B-ba me hodology applied only o he egula ini e ele-
men s; d) B-ba me hodology o bo h egula and en iched ini e elemen s. 68
4.26 C ack opening o in eg a ion schemes c (line) and d (dashed line). . . . . 69
4.27 Cook’s memb ane p oblem. . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.28 a) Example o a mesh wi h he c ack modeled h ough en ichmen . b)
Example o a mesh con aining a c ack explici ly modeled. . . . . . . . . . 70
4.29 Ve ical displacemen o he op igh co ne node, as a unc ion o he
numbe o elemen s pe side, o he explici ly meshed c ack and en iched
c ack. ...................................... 71
4.30 Compa ison o he e ical displacemen o he op igh co ne node, as
a unc ion o he numbe o elemen s pe side, o di e en ip en ichmen
unc ions. .................................... 72
4.31 Ve ical displacemen o he op igh co ne node, as a unc ion o he
numbe o elemen s pe side, ob ained o he explici ly meshed c ack
o igu e 4.28(b), using he F-ba me hodology and he Enhanced s ain
app oach used by Dolbow e al. . . . . . . . . . . . . . . . . . . . . . . . 74
4.32 Compa ison o he e ical displacemen o he op igh co ne node, as
a unc ion o he numbe o elemen s pe side, o explici ly meshed and
en iched app oxima ions using he F-ba me hodology and he Enhanced
S ain app oach employed by Dolbow e al. . . . . . . . . . . . . . . . . . 74
4.33 a) Maximum alue o he de ia ion om he incomp essibili y condi ion,
(J−1), in he elemen c ossed by he c ack ma ked in he mesh b). . . . 75
4.34 De ini ion o in e pola o y shape unc ions used in he a iable ans e . . 76
Lis o Figu es x ii
4.35 G oup o Gauss poin s (in black) ha in luence a pa icula one. When
a Gauss poin is loca ed nea a discon inui y, only he poin s in he same
side o he discon inui y is selec ed. . . . . . . . . . . . . . . . . . . . . . . 77
4.36 G oup o Gauss poin s (in black) ha in luence a pa icula one, loca ed
nea ac ack ip. ................................ 77
5.1 Duc ile ac u e p ocess: a) inclusions in a me allic ma ix; b) oid nucle-
a ion; c) oid g ow h; d) s ain localiza ion and necking be ween oids;
e) oid coalescence and o ma ion o a ac u e su ace. . . . . . . . . . . 80
5.2 Se o con ol poin s o de ine he in e pola o y B-spline pa ch o he
cen alelemen .................................. 83
5.3 Sea ch o he poin wi h highes damage a he bounda y o he domain:
a) Selec ion o he elemen wi h he highes damage alue and b) selec ion
o he adjacen segmen s and espec i e nodes. . . . . . . . . . . . . . . . 84
5.4 Duc ile c ack g ow h: a) oid nuclea ion ahead o he c ack ip; b) oid
g ow h; c) oids connec o he main c ack causing i s p opaga ion. . . . 85
5.5 Selec ion o poin s o de e mine he c ack g ow h di ec ion. . . . . . . . . 86
5.6 I e a i e scheme o equilib ium eco e y du ing a c ack g ow h s ep. . . . 90
5.7 Time-s epping scheme be o e and a e c ack ini ia ion/p opaga ion. . . . 91
5.8 P e-c ackedpla e................................ 92
5.9 Mesh e inemen o he pla e wi h an ini ial c ack: a) 20 ×31, b) 40 ×
61 and c) 50 ×75elemen s. ......................... 93
5.10 Damage con ou s ob ained o he local case, o an applied displacemen
o 0.05mm o hee meshes a) 20 ×31, b) 40 ×61 and c) 50 ×75 elemen s. 93
5.11 Damage con ou s ob ained o l = 0.8mm, o an applied displacemen o
0.05mm, o hee meshes a) 20 ×31, b) 40 ×61 and c) 50 ×75 elemen s. 93
5.12 Damage con ou s ob ained o l = 1.6mm, o an applied displacemen o
0.05mm, o hee meshes a) 20 ×31, b) 40 ×61 and c) 50 ×75 elemen s. 94
5.13 C ack pa h ob ained using l = 0.8 mm, o hee meshes a) 20 ×31, b)
40 ×61 and c) 50 ×75elemen s........................ 95
5.14 C ack pa h ob ained using l = 1.6 mm, o hee meshes a) 20 ×31, b)
40 ×61 and c) 50 ×75elemen s........................ 96
5.15 Reac ion o ce in unc ion o he applied displacemen l = 1.6mm. . . . . 97
5.16 C ack leng h e olu ion l = 1.6mm....................... 98
5.17 Axisymme ic Specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.18 Mesh e inemen o he axisymme ic no ched specimen, mesh densi y o
a) 11, b) 31 and c) 51 elemen s pe side. . . . . . . . . . . . . . . . . . . . 100
5.19 Rela ion be ween he eac ion o ce and he displacemen applied o he
op edge o he no ched specimen o he di e en meshes, conside ing
l = 1.6 mm and DC= 0.2. ..........................101
5.20 Rela ion be ween he eac ion o ce and he displacemen applied o he
op edge o he no ched specimen o he di e en meshes, conside ing
l = 2.0 mm and DC= 0.5. ..........................101
5.21 Compa ison be ween he o ce-displacemen cu es ob ained o he in-
insic leng hs l = 1.6 mm and l = 2.0 mm. C i ical damage has he
alue DC= 0.5..................................102
5.22 Damage con ou p io o c ack inse ion o mesh a) 11, b) 21 and c) 31
elemen spe side.................................102
Lis o Figu es x iii
5.23 Damage con ou and c ack, o an applied displacemen o 0.9 mm, o
mesh a) 11, b) 21 and c) 31 elemen s pe side. . . . . . . . . . . . . . . . . 103
5.24 C ack leng h e olu ion conside ing l = 2.0 mm and DC= 0.5. . . . . . . 103
5.25 Rela ion be ween he esidual o ce and he displacemen applied o he
op edge o he no ched specimen o he mesh wi h 31 nodes pe side,
conside ing l = 2.0 mm and di e en alues o DC. ............104
5.26 Plane s ain specimen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.27 Mesh e inemen o he plane s ain specimen, mesh densi y o a) 11, b)
21, c) 31, d) 41 elemen s pe side. . . . . . . . . . . . . . . . . . . . . . . . 105
5.28 Reac ion o ce as a unc ion o he applied displacemen o he op nodes
o he plane s ain specimen. . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.29 Damage con ou and inal c ack o mesh a) 11, b) and 21 c) 31 elemen s
pe side......................................106
5.30 Di e en c ack g ow h s eps o he mesh wi h 41 elemen s pe side. . . . 107
5.31Shea Specimen.................................107
5.32 FEM meshes used in he double no ched specimen: a) mesh wi h 16 nodes
pe side, b) mesh wi h 23 nodes pe side. . . . . . . . . . . . . . . . . . . 108
5.33 C ack pa h ob ained o a) mesh a, b) mesh b. . . . . . . . . . . . . . . . 108
5.34 Damage con ou and inal c ack o mesh a) and mesh b). . . . . . . . . . 109
6.1 a) T ac ion- ee c ack b) Cohesi e c ack. . . . . . . . . . . . . . . . . . . . 112
6.2 a) Body con aining a cohesi e c ack. b) De ail o he cohesi e c ack . . . 113
6.3 a) Bilinea cohesi e law. b) Polynomial cohesi e law. . . . . . . . . . . . . 114
6.4 a) Linea cohesi e law. b) Exponen ial cohesi e law. . . . . . . . . . . . . 115
6.5 Changes in he cohesi e esponse in unc ion o a) Fnb) ω0.........117
6.6 a) Poin pon he c ack su ace. b) Images o poin p. ...........118
6.7 Gauss poin s used in he in eg a ion o a cohesi e law. . . . . . . . . . . . 119
6.8 a) Pla e wi h ini ial c ack b) FEM mesh. . . . . . . . . . . . . . . . . . . 120
6.9 In luence o Fnin he esponse o he c acked pla e o cons an ω0=
0.5 mm−1.....................................121
6.10 In luence o ω0in he esponse o he c acked pla e o cons an a) Fn=
200 MPa b) Fn=100MPa. ..........................121
6.11 In luence o a cohesi e law in he eac ion o ce du ing c ack p opaga ion. 122
6.12 In luence o a cohesi e law in he e olu ion o he c ack leng h. . . . . . . 122
6.13 a) Plane s ain specimen b) FEM mesh. . . . . . . . . . . . . . . . . . . . 123
6.14 Reac ion o ce-applied displacemen cu es. . . . . . . . . . . . . . . . . . 124
6.15 Reac ion o ce-applied displacemen cu es and espec i e in luence o he
cohesi elaw. ..................................124
6.16 In luence o he cohesi e law in he e olu ion o he c ack leng h. . . . . . 125
6.17 Reac ion o ce-applied displacemen cu es and espec i e in luence o he
cohesi elaw. ..................................128
6.18 S ain ene gy as a unc ion o he c i ical damage. Co ela ion ac o :
R2= 0.985. ...................................128
6.19 S ain ene gy as a unc ion o he c i ical damage. Co ela ion ac o :
R2= 0.967. ...................................129
6.20 Reac ion o ce-applied displacemen cu es and espec i e in luence o he
cohesi elaw. ..................................129
6.21 a) Plane s ain specimen b) FEM mesh. . . . . . . . . . . . . . . . . . . . 130
Lis o Figu es xix
6.22 S ain ene gy as a unc ion o he c i ical damage. Co ela ion ac o :
R2= 0.999. ...................................131
6.23 S ain ene gy as a unc ion o he c i ical damage. Co ela ion ac o :
R2= 0.999. ...................................131
6.24 Reac ion o ce-applied displacemen cu es and espec i e in luence o he
cohesi elaw. ..................................132
Lis o Tables
3.1 Calcula ion o he de o ma ion g adien by de i ing he displacemen ield
app oxima ion. The SMSF eeze command deno es he manual in oduc-
ion o an in e media e a iable, which in his si ua ion is necessa y o
co ec ly de ine he dependencies in he au oma ic de i a ion p ocedu e. . 36
4.1 A possible basis o he incomp essible de o ma ion space o an elemen
o ally c ossed by a discon inui y. . . . . . . . . . . . . . . . . . . . . . . . 53
4.2 A possible basis o he incomp essible de o ma ion space o an elemen
con aining a discon inui y ip. . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.3 A possible basis o he incomp essible de o ma ion space o an elemen
o ally c ossed by a discon inui y. . . . . . . . . . . . . . . . . . . . . . . . 56
4.4 A possible basis o he incomp essible de o ma ion space o an elemen
o ally c ossed by a discon inui y. . . . . . . . . . . . . . . . . . . . . . . . 57
4.5 De o med con igu a ions o di e en in eg a ion ules. . . . . . . . . . . . 65
4.6 Maximum alue o he de ia ion om he incomp essibili y condi ion,
max |J−1|.................................... 75
5.1 Ma e ial p ope ies o he pla e wi h an ini ial c ack. . . . . . . . . . . . . 92
5.2 Ma e ial p ope ies used in he nume ical examples. . . . . . . . . . . . . 99
6.1 Ma e ial p ope ies o he c acked pla e. . . . . . . . . . . . . . . . . . . . 120
6.2 S ain ene gy o Dc= 0.4−0.5........................127
6.3 S ain ene gy o Dc= 0.6−0.7........................127
6.4 Ma e ial p ope ies o he axisymme ic no ched specimen. . . . . . . . . 131
xxi
Dedica ed o my amily and iends
xxiii
Chap e 1
In oduc ion
Duc ile ac u e has been ega ded as a c ucial issue in many indus ial a eas. Al hough
i is usually associa ed o ailu e, as mos o he me als, a oom empe a u e, ail ac-
co ding o his mechanism, i may be an in eg a ing pa o p oduc ion p ocesses, such
as machining, blanking o cu ing. Nume ous echniques ha e been de eloped o op i-
mize ope a ions and p oduc s, ne e heless, a deepe unde s anding o his phenomenon
would lead o mo e e icien designs, which, in a i s s age, p e en s he use o unnec-
essa y ma e ial and ene gy and, in a second s age, a oids ca as ophic ailu es. Mo e
han e e , he esponsible use o esou ces is essen ial, no only om he inancial poin
o iew bu also o a sus ainable wo ld.
Up o he nine een h cen u y he design o s uc u es and mechanical pa s was mos ly
empi ical and unde s anding o ma e ial beha iou qui e poo . Ne e heless, wi h he
massi e p oduc ion o i on and s eel demanded by he Indus ial Re olu ion came he
need o de ining easible design laws. Thus, by he beginning o he wen ie h cen u y,
he s udy o duc ile me als gained a ma hema ical app oach wi h he pionee ing wo ks o
T esca [2], Hube [3], on Mises [4] and Hencky [5], whe e he e was an a emp o de el-
oping ma e ial laws able o ep oduce he expe imen al obse a ions. The main ou come
o hese con ibu ions was he p edic ion o he onse o plas ic beha iou h ough he
es ablishmen o yield unc ions. In hese app oaches, he ma e ial p ope ies a e de-
sc ibed a he meso-scale and explained by a ew ene gy mechanisms. As he ma e ial is
ega ded as a con inuous media, hese wo ks la e ga e bi h o he so-called Con inuum
Mechanics.
Al hough he dimensioning o s uc u es using he yield c i e ia became qui e popula in
he enginee ing media be ween he Wo ld Wa s, ca as ophic ailu es o s uc u es such
1
Chap e 1. In oduc ion 8
•M. Seab a, P. ˇ
Suˇs a iˇc, J. C´esa de S´a, T. Rodiˇc, Damage d i en c ack ini ia ion and
p opaga ion in duc ile me als using XFEM, Compu a ional Mechanics (accep ed o
publica ion).
•M. Seab a P. ˇ
Suˇs a iˇc, J. C´esa de S´a, T. Rodiˇc, Towa ds ene ge ically consis en
ansi ion om damage o ac u e using XFEM. (in p epa a ion).
•M. Seab a P. ˇ
Suˇs a iˇc, J. C´esa de S´a, T. Rodiˇc, Au oma ic gene a ion o XFEM codes
and applica ion o c ack p opaga ion, Technical no e (in p epa a ion).
Chap e 2
S anda d Con inuum Damage
Mechanics
Duc ile me als, which include i on and s eel, aluminium and coppe alloys, a e cha ac-
e ized by he p esence o mode a e o la ge plas ic de o ma ions p io o ailu e. Plas ic
s ains may esul om c ys alline slip h ough disloca ion mo ion o om deg ada ion
o he mechanical p ope ies and he e o e bo h mechanisms a e s ongly coupled in his
ype o ma e ials.
Classically, he ma e ial p ope ies may be in e ed om he s udy o he mic os uc u e
o he ma e ial, he so-called Mic o Mechanical heo ies, o he ma e ial beha iou may
be desc ibed in phenomenological way, based on a he modynamic amewo k, e e ed
as he Con inuum Damage Mechanics (CDM).
In he i s ca ego y, he p obably mos ex ensi ely used model was p oposed by Gu son
[16] and la e ex ended by T e gaa d and Needleman [18], which conside s he ma e ial
as a po ous media, whe e oids nuclea e and coalesce.
On he o he way, he model p oposed by Lemai e [25–27, 29] is he mos ep esen a-
i e o he CDM models, se ing as a base o many o he ma e ial laws. He e ma e ial
p og essi e deg ada ion is ea ed as an in e nal a iable, ac ing in an a e aging sense
wi hin a ep esen a i e olume, co esponding o a scale whe e he ma e ial may be
ega ded as homogeneous.
9
Chap e 2. S anda d Con inuum Damage Mechanics 10
In his wo k, he Lemai e model is used o desc ibe he deg ada ion o he mechanical
p ope ies o a solid p io o he o ma ion o mac o c acks, cap u ing he elas ic, plas ic-
ha dening and plas ic-so ening s ages o ma e ial beha iou . The e o e, in his chap e
he basic concep s o CDM and cons i u i e modelling a e e iewed, along wi h he
Lemai e model o iso opic damage, in bo h local and non-local o mula ions. Ra he
hen p esen ing an in ensi e esea ch on CDM models, his chap e in ends o p o ide
he basis heo y o he b oade model o duc ile ac u e buil h oughou his hesis.
Among he exis ing li e a u e on he subjec , in e es ed eade s may consul he books
[62–64].
This chap e is o ganized as ollows. In he i s sec ion he basic kinema ics o de-
o mable bodies is in oduced, ollowed by he main s ain and s ess measu es. Nex
he balance equa ions and he p inciples o he modynamics a e p esen ed. In sec ion
2.3 he cons i u i e heo y is de i ed, in pa icula he Lemai e model o duc ile dam-
age, making use o he p e iously in oduced concep s. This chap e inalizes wi h he
e- o mula ion o he Lemai e model ollowing a non-local in eg al app oach and some
b ie conclusions.
2.1 Kinema ics
Kinema ics p o ides he basic ools o mo ion and de o ma ion desc ip ion. In his
sec ion he undamen al quan i ies employed in he cons i u i e desc ip ion o duc ile
me als a e de ined.
Mo ion
Le Bbe a de o mable body occupying he domain Ω ⊂ R3, wi h he posi ions o i s
pa icles de ined by X, e e ed as he e e ence con igu a ion. The body may unde go
amo ion o de o ma ion, wi hin a ce ain ime, co esponding o he ans o ma ion
X:B→ R3
x=X(X, ) (2.1)
whe e xis he posi ion a he cu en con igu a ion, a ime .
Assuming ha he mo ion Xis in e ible, he posi ion Xa a ime may be eco e ed
as:
X=X−1(x, ) (2.2)
Chap e 2. S anda d Con inuum Damage Mechanics 11
and he displacemen ield may be in oduced as
u(X, ) = X(X, )−X(2.3)
The ime de i a i e o he unc ion X(X, ), de ined as
V(X, ) = ∂X(X, )
∂ (2.4)
is he eloci y o a pa icle a he posi ion X, and consequen ly, he eloci y o a pa icle
a he cu en con igu a ion may be de e mined by applying equa ion 2.2:
(x, ) = V(X−1(x, ), ) (2.5)
De o ma ion g adien
The de i a i e o he de o ma ion, F, is a second o de enso e med de o ma ion
g adien , ela ing Xand xas ollows:
F(X, ) = ∂X(X, )
∂X =∂x
∂X=∇XX(X, ) (2.6)
whe e ∇Xis usually e med ma e ial g adien ope a o . Al e na i ely, he de o ma ion
g adien may also be de ined in he cu en con igu a ion, making use o he spa ial
g adien ope a o ,∇xas
F(x, )=[∇xX−1(x, )]−1(2.7)
The displacemen ield may also be employed in he de ini ion o he de o ma ion g a-
dien , leading o
F=I+∇Xu(2.8)
whe e Iis he second o de iden i y enso .
F om he de ini ions p esen ed so a , i can be shown ha line, su ace and olume
changes espec i ely ans o m acco ding o:
dx=FdX
da=JF−TdA
d =JdV
(2.9)
In he abo e equa ions, J, e med Jacobian o he de o ma ion, is he de e minan o
he de o ma ion g adien
J= de F(2.10)
Chap e 2. S anda d Con inuum Damage Mechanics 12
I ollows ha a olume-p ese ing o isocho ic de o ma ion is cha ac e ized by:
J= 1 (2.11)
Mo eo e , any de o ma ion can be locally decomposed in o an isocho ic de o ma ion,
F ol and a olume ic de o ma ion, Fiso, [65–67] as
F=FisoF ol (2.12)
whe e
Fiso = (de F)−1
3F=J−1
3F(2.13)
F ol = (de F)1
3I=J1
3I(2.14)
and Iis he second o de iden i y enso .
Al e na i ely, a mo ion may be spli in pu e s e ches and pu e o a ions, making use
o he pola decomposi ion, ha is,
F=RU =V R (2.15)
whe e Uand Va e unique, posi i e de ini e, symme ic enso s, e med, espec i ely
igh and le s e ch enso s.Ris an unique p ope o hogonal enso , called o a ion
enso . The enso s Uand Vmay also be in oduced as:
C=U2=FTF,b=V2=F F T(2.16)
we e Cand ba e, espec i ely, he igh and le Cauchy-G een s ain enso s.
S ain measu es
The de o ma ion g adien is he undamen al kinema ic quan i y cha ac e izing he
changes o each ma e ial poin du ing mo ion. Ne e heless, o quan i y he ela i e dis-
ance change be ween wo ma e ial poin s, ha is, o cha ac e ize he s aining, p ope
s ain measu es ha e o be de ined.
As he s ains a e no measu able quan i ies (such as he displacemen s) bu a concep ual
quan i y o simpli y analysis, he e a e nume ous choices o s ain enso s. An impo an
Chap e 2. S anda d Con inuum Damage Mechanics 13
amily o s ain measu es a e he Lag angian s ain enso s:
E(m)=(1
m(Um−I), m 6= 0
ln U, m = 0 (2.17)
Simila ly, ano he amily o s ain enso s, called he Eule ian s ain enso s, may be
ob ained using he le s e ch enso :
e(m)=(1
m(Vm−I), m 6= 0
ln V, m = 0 (2.18)
Veloci y g adien and s ain a es
The eloci y g adien , l, is he spa ial ield de ined as
l=∇x .(2.19)
Al e na i ely, lmay be exp essed as a unc ion o he de o ma ion g adien as
l=∂
∂ ∂X
∂X
∂X
∂x=˙
F F −1(2.20)
The eloci y g adien may also be spli in o a symme ic and skew pa , o igina ing wo
impo an enso s as ollows:
d=1
2(l+lT) (2.21)
w=1
2(l−lT) (2.22)
The enso dis called he a e o de o ma ion and is associa ed wi h he s aining, while
he enso lis e med spin enso and is associa ed wi h igid body eloci y.
S ess measu es
In line wi h he p e ious sec ions, o each ype o s ain measu e, i is possible o de ine
a s ess measu e. One o he mos impo an s ess enso s is he Cauchy o ue s ess
enso , σ, as:
=σn (2.23)
whe e is he su ace ac ion and nis he associa ed ou wa d no mal ec o .
The Cauchy s ess enso may be spli in a de ia o ic,s, and a p essu e,p, con ibu ions,
Chap e 2. S anda d Con inuum Damage Mechanics 14
as ollows
σ=s+pI(2.24)
whe e
p=1
3 [σ] (2.25)
Ano he s ess measu e equen ly used is he Ki chho s ess enso , gi en by
τ=Jσ(2.26)
O he con enien s ess enso s a e he Fi s and he Second Piola-Ki chho enso s,
espec i ely in oduced as
P=τF −T(2.27)
and
S=F−1τF −T(2.28)
2.2 Conse a ion p inciples
In he p e ious sec ions, he quan i ies used in he ma hema ical desc ip ion o mo ion,
s aining and s ess we e in oduced. In his sec ion, his concep s a e ela ed in some
undamen al conse a ion p inciples, which a e essen ial o o mula e a con inuum me-
chanics p oblem. The de ailed de i a ion o his p inciples is ou o he scope o his
hesis and may be ound, o ins ance, in he e e ences [62–64, 68].
Conse a ion o mass
The i s p inciple ega ds he mass conse a ion and may be exp essed as
˙ρ+ρdi x˙
u= 0 (2.29)
whe e ρis he mass densi y a he de o med con igu a ion.
Chap e 2. S anda d Con inuum Damage Mechanics 15
Momen um balance
The momen um balance equa ion desc ibes he equilib ium be ween in e nal and ex e -
nal o ces in a body, as ollows
di xσ+b=ρ¨
u(2.30)
whe e bdeno es he body o ce ec o a he de o med con igu a ion. These equa ion
is equen ly e e ed as s ong o m o he equilib ium equa ion and has o ul il he
ollowing bounda y condi ion:
¯
=σn (2.31)
whe e ¯
is a ac ion ec o applied o he bounda y o he body.
Fi s and second laws o he modynamics
The i s law o he modynamics pos ula es ha he ene gy mus be conse ed and may
be ma hema ically exp essed as:
ρ˙e=σ:d+ρ −di xq(2.32)
whe e e ep esen s he speci ic in e nal ene gy, ep esen s he densi y o hea p oduc-
ion and q he hea lux. Th oughou his hesis, only pu ely mechanical p ocesses will
be add essed, hus equa ion 2.32 educes o
ρ˙e=σ:d(2.33)
This equa ion may be ew i en in e ms o he Ki chho s ess enso , as ollows
¯ρ˙e=τ:d(2.34)
whe e ¯ρ=Jρ is he e e ence mass densi y.
Second law o he modynamics e e s o he i e e sibili y o he en opy p oduc ion,
which may be exp essed as
ρT ˙s+ di xq−ρ ≥0 (2.35)
whe e s ep esen s he en opy and T he empe a u e. Simila ly o he i s law, o
iso he mal p ocesses equa ion 2.35 educes o
ρT ˙s≥0 (2.36)
Chap e 2. S anda d Con inuum Damage Mechanics 16
Combining he i s and he second laws and in oducing he Helmhol z ee ene gy
concep , ψ,de ined as
ψ=e−Ts (2.37)
he so-called Clausius-Duhem inequali y is ob ained
τ:d−¯ρ˙
ψ≥0 (2.38)
2.3 Fini e s ain damage mechanics
The laws p esen ed a e alid independen ly o he ma e ial conside ed. The e o e, o
ully cha ac e ise he esponse o a solid body and dis inguish be ween di e en ma e-
ials, a cons i u i e model has o be o mula ed and bounda y condi ions ha e o be
p esc ibed.
In his hesis, he chosen cons i u i e model, he Lemai e model o duc ile damage
[25–27, 29], elies in he he modynamics wi h in e nal a iables, which means ha a
any ins an , he he modynamic s a e a any poin is comple ely de ined by he alues o
a gi en se o s a e a iables. In o de o success ully cap u e all he s ages o ma e ial
beha iou , he choice o he se o in e nal a iables is c ucial. In he pa icula case
o duc ile ma e ials, he ma e ial deg ada ion, ha is damage occu s simul aneously
wi h plas ic de o ma ions, and he e o e, he in e nal ene gy unc ion should e lec his
dependency, as i will be shown h oughou his sec ion.
2.3.1 The damage a iable
Failu e in duc ile me als ini ia es wi h he nuclea ion ca i ies and mic o-c acks, which
may g ow and coalesce, e en ually leading o mac o-c ack gene a ion and p opaga ion
[27, 69]. The quan i ica ion o he mic o-de ec s a he meso-scale le el, may be aken
in accoun in a a e aging sense, de ining he damage a iable, Dˆn, as:
Dˆn=∂SD
∂S (2.39)
whe e ∂S is he a ea o in e sec ion o a plane, wi h no mal ˆn, wi h he ep esen a i e
olume elemen (RVE) and ∂SDis he a ea ha e ec i ely in e sec s ca i ies o mic o-
c acks, as can be obse ed in igu e 2.1. The RVE o a ce ain ma e ial is de ined in such
a way ha all he p ope ies a e ep esen ed by homogenized a iables (a ound 0.1mm3
Chap e 2. S anda d Con inuum Damage Mechanics 17
∂S
∂SD
ˆn
Figu e 2.1: Rep esen a i e olume elemen .
o me als [27]).
When only duc ile damage is conside ed, he mic o-de ec s densi y is o en a weak unc-
ion o he plane o ien a ion. The e o e, Dmay be conside ed iso opic and ep esen ed
by a scala a iable, anging om 0 o 1, co esponding o undamaged o ully damage
ma e ial, espec i ely, which is he case o his hesis. Ne e heless, a second o ou h
o de enso ep esen a ion may be adop ed o o he applica ions [27, 70].
2.3.2 E ec i e s ess concep and s ain equi alence p inciple
In he p e ious sec ions some s ain and s ess measu es we e in oduced o be used
in he cons i u i e equa ions ha cha ac e ize a gi en ma e ial. These quan i ies we e
de eloped o plain ma e ial, howe e , in he p esence o damage he esis an a ea is
smalle and an he e ec i e Cauchy s ess enso may be in oduced as:
¯
σ=σ
1−D(2.40)
whe e σis he Cauchy s ess enso o undamaged ma e ial. F om his de ini ion i ol-
lows he s ain equi alence p inciple, which s a es ha a damaged ma e ial is go e ned
by he cons i u i e laws o he plain ma e ial, eplacing he ue s ess by he e ec i e
s ess [27].
2.3.3 Mul iplica i e plas ici y amewo k
In duc ile ma e ials damage and plas ici y a e s ongly connec ed, he e o e he concep s
associa ed wi h ini e s ain elas o-plas ici y a e now in oduced.
Chap e 3
Fini e Elemen Dis e iza ion and
Au oma ic Code Gene a ion
The desc ip ion o duc ile ma e ials ollowing he damage model desc ibed in chap e
2 esul s in a complex sys em o di e en ial equa ions. As only a e y es ic ed se
o analy ical solu ions a e a ailable, he use o nume ical me hods becomes impe a i e.
In pa icula , he Fini e Elemen Me hod (FEM) s ands ou as he mos widely used
nume ical me hod o sol ing solid mechanics p oblems o i s e ec i e p edic i e capa-
bili ies.
The basis o he FEM a e well es ablished in he li e a u e as, o ins ance, in he e -
e ence wo ks o Zienkiewicz [36], Hin on and Owen [80], Hughes [81] o C is ield [82].
Hence he me hod will be b ie ly e iewed in he i s pa o his chap e , which ini i-
a es by e- o mula ing he balance laws p esen ed in chap e 2 in o de o de ine a Ini ial
Bounda y Value P oblem, sui able o be sol ed wi h he FEM. Nex , he displacemen -
based disc e isa ion is in oduced, along wi h he non-linea inc emen al ini e elemen
equa ion.
The second pa o his chap e will ocus on he au oma ic gene a ion o FEM codes.
The compu e science ad ances, in bo h so wa e and ha dwa e, had a signi ican impac
in gene alizing he use o he FEM, allowing he simula ion o p oblems wi h highe num-
be o a iables, desc ibed by inc easingly complex ma hema ical models . Mo eo e ,
he combina ion o mode n Compu e Algeb a Sys ems wi h an au oma ic de i a ion
echnique, allows a as e and mo e e icien implemen a ion o FEM o mula ions. In
his chap e , special a en ion is paid o one o hese sys ems, he AceGen sys em, which
was used o de elop a conside able amoun o compu e codes used in he con ex o
25
Chap e 3. Fini e Elemen Dis e iza ion and Au oma ic Code Gene a ion 26
his hesis.
3.1 The quasi-s a ic IBVP
In Chap e 2 he undamen al equilib ium laws we e p esen ed. Howe e , o sol e a
p oblem using he FEM, hese laws a e o en o mula ed as a ia ional p inciples. Al-
hough his me hodology is no manda o y, app oxima e solu ions a e o en ob ained
h ough he weak o ms o ield equa ions.
The e o e, ecalling he momen um balance equa ion:
di xσ+b=ρ¨
u(3.1)
and conside ing he quasi-s a ic case, as h oughou his hesis he ine ial e ec s will
be neglec ed, we may w i e:
di xσ=0(3.2)
Equa ion 3.2 may be mul iplied by a i ual displacemen , η, and in eg a ed o e he
domain, as: ZX(Ω)
(di xσ)Tηd = 0 (3.3)
A e some algeb a manipula ion and making use o he di e gence heo em, equa ion
3.3 may be w i en in he ollowing o m:
ZX(Ω)
σ:∇xηd −ZX(∂Ω)
(σ.n)Tηda = 0 (3.4)
As in oduced in chap e 3, he ollowing bounda y condi ion has o be ul illed oge he
wi h equa ion 3.1
¯
=σn (3.5)
whe e ¯
is a p esc ibed ac ion a he bounda y o he body. The e o e, eplacing his
ela ion in equa ion 3.4, he weak o m o he equilib ium equa ion o ob ained:
ZX(Ω)
σ:∇xηd −ZX(∂Ω)
¯
·ηda = 0 (3.6)
The weak o m o he equilib ium equa ion is equi alen o he s ong o m and allows
he de ini ion o he weak o m o he Ini ial Bounda y Value P oblem (IBVP) associa ed
wi h a ce ain de o ma ion p ocess, as ollows.
Find a kinema ically admissible displacemen unc ion, u∈ K such ha he equa ion
Chap e 3. Fini e Elemen Dis e iza ion and Au oma ic Code Gene a ion 27
ZX(Ω)
σ:∇xηd −ZX(∂Ω)
¯
·ηda = 0 (3.7)
is sa is ied o all ∈[ 0, n]and o all η∈ V .The se o i ual displacemen s and
he se o kinema ically admissible displacemen s, a ime a e espec i ely de ined as
V ={η: Ω −→ X(Ω)|η=0∈∂Ω( )}(3.8)
K={u: Ω −→ X(Ω)|¯
u(X, ), ∈[ 0, n],X∈∂Ω}(3.9)
The Cauchy s ess, a each poin o he body is exp essed as
σ( ) = σ(F( ),α( )) (3.10)
whe e α ep esen s he se o in e nal a iables associa ed wi h he ma e ial and Fis a
gi en p esc ibed de o ma ion g adien his o y, de ined as
F( ) = I+∇Xu(X, ) (3.11)
The solu ion o he IBVP e lec s how a solid body will beha e when subjec ed o ce ain
bounda y condi ions.
Ma e ial desc ip ion
Equa ion 3.6 e lec s he spa ial e sion o he P inciple o Vi ual Wo k, ne e heless,
depending on he pa icula p oblem, he ma e ial e sion may p o ide a mo e e icien
implemen a ion. The i s membe o equa ion 3.6 is e med in e nal wo k,Win , and
may be w i en in he e e ence con igu a ion, ecalling he de ini ion o he i s Piola-
Ki chho s ess enso :
P=JσF −T(3.12)
esul ing in
Win =ZΩ
P:∇XηdV (3.13)
The i s Piola-Ki cho enso is no comple ely ela ed o he ma e ial con igu a ion
bu i may be eplaced by he second Piola-Ki cho s ain enso as:
P:∇Xη=S:FT∇Xη=S: (FT∇Xη+∇XηTF) = S:∂E(3.14)
Win =ZΩ
S:∂EdV (3.15)
Chap e 3. Fini e Elemen Dis e iza ion and Au oma ic Code Gene a ion 28
whe e ∂Eis he i s a ia ion o he G een-Lag ange s ain enso . The second membe
o equa ion 3.6, he ex e nal wo k,Wex may be w i en in he ma e ial desc ip ion simply
by aking he applied ac ion in he ini ial con igu a ion, ¯
0
Wex =ZX(∂Ω)
¯
.ηda =Z∂Ω
¯
0.ηdA (3.16)
Ei he in he ma e ial o spa ial e sion, he IBVP is usually non-linea and he e o e
linea isa ion and disc e isa ion a e necessa y o p oduce accu a e solu ions using he
FEM, as desc ibed in he upcoming sec ions.
3.2 Displacemen -based ini e elemen s
The FEM o he nume ical solu ion o he IBVP p oblem desc ibed in he p e ious sec-
ion consis s in eplacing he se s V and K, de ined in equa ions 3.8 and 3.9, espec i ely,
by disc e e subse s, Vh
and Khob ained om a ini e disc e iza ion o he domain. In
he case o displacemen -based ini e elemen s, he in e pola ed ield a iables a e he
displacemen s, which wi hin a gi en elemen , e, assume he o m:
u(x) =
nnode
X
i=1
N(e)
i(x)ui(3.17)
whe e N(e)
iis he shape o in e pola ion unc ion associa ed wi h node iand nnode is
he numbe o nodes o he elemen . Simila ly, he in e pola ion unc ion may be de ined
o e he en i e app oxima e domain, which is cons i u ed by a o al numbe o npoin
nodal poin s, as
u(x) =
npoin
X
i=1
Ng
i(x)ui(3.18)
In equa ion 3.18, u ep esen s he global ec o o nodal displacemen s, which, in a
p oblem o dimension ndim, is gi en by
uh= [u1
1, . . . , u1
ndim, . . . , unpoin
1, . . . , unpoin
ndim ] (3.19)
and Ngis he global in e pola ion ma ix de ined as:
Ng
i= [ diag[Ng
1(x)] diag[Ng
2(x)] ··· diag[Ng
npoin(x)] ] (3.20)
Chap e 3. Fini e Elemen Dis e iza ion and Au oma ic Code Gene a ion 29
whe e diag[Ng
1] ep esen s a ndim ×ndim diagonal ma ix as ollows
diag[Ng
1] =
Ng
i0··· 0
0Ng
i··· 0
.
.
..
.
.....
.
.
0 0 ··· Ng
i
(3.21)
Equa ion 3.18 may be e o mula ed as:
uh(x) = Ngu(3.22)
Simila ly, he ield o i ual displacemen s may be w i en as:
ηh(x) = Ngη(3.23)
Using he p esen ed no a ion he disc e ized se s Vh
and Khmay be de ined as:
Vh
=ηh(x) =
npoin
X
i=1
Ng
iηi|ηi=0i xi∈∂Ω(3.24)
Kh=uh(x) =
npoin
X
i=1
Ng
iui|ui=¯
u(xi) i xi∈∂Ω(3.25)
Now, equa ion 3.6 may be eplaced by i s disc e ized e sion, as ollows
ZX(Ωh)
σ:∇x(Ngηh)d −ZX(∂Ωh)
¯
·(Ngηh)da = 0 (3.26)
In classical FEM implemen a ions i is usual o de ine he global disc e e symme ic g a-
dien ma ix,B, which o a plain s ain p oblem has he o m:
Bg=
Ng
1,10Ng
2,10··· Ng
npoin,10 0
0Ng
1,20Ng
2,2··· 0Ng
npoin,2
Ng
1,2Ng
1,1Ng
2,2Ng
2,1··· Ng
npoin,2Ng
npoin,1
(3.27)
whe e he ollowing no a ion was employed
(·)i,j =∂(·)i
∂xj
(3.28)
Finally, he disc e ized i ual wo k exp ession can be ea anged o
ZX(Ωh)
[σTBgηh−b·Ngηh]d −ZX(∂Ωh)
¯
·Ngηhda = 0,∀ηh∈ Vh
(3.29)
Chap e 3. Fini e Elemen Dis e iza ion and Au oma ic Code Gene a ion 30
Fo comple eness, when using he ma e ial e sion o he i ual wo k equa ion, he
disc e ised o m o he a ia ion o he G een-Lag ange s ain enso is p o ided he ein:
∂E=1
2
npoin
X
i=1
[FT(ηh
i⊗∇XNi)+(∇XNi⊗ηh
i)F)] (3.30)
whe e
F=
npoin
X
i=1
xi⊗∇XNi(3.31)
The non-linea inc emen al ini e elemen equa ion
The equilib ium equa ion is, in gene al, non-linea . Fo he pa icula case o ma e ial
non-linea i ies, such as in he Lemai e model whe e he Cauchy s ess is dependen o
he his o y o s ains o which he solid has been subjec ed, a sui able empo al disc e i-
sa ion is equi ed.
In his hesis, a pseudo- ime disc e isa ion be ween he ime inc emen s [ n, n+1] will
be conside ed, in a ully implici scheme. The unc ion, ˆ
σ, de ined as
σn+1 =ˆ
σ(Fn+1,αn) (3.32)
is assumed o exis and is associa ed wi h an in eg a ion algo i hm ha deli e s he
beha iou o a gi en de o ma ion g adien , Fn+1, and a se o in e nal a iables, αn,
which a e assumed o be cons an wi hin he one inc emen .
The New on-Raphson algo i hm is pa icula ly a ac i e o he solu ion o his ype
o p oblems due o i s obus ness and quad a ic a es o asymp o ic con e gence, and
hus was used in his wo k. Du ing a solu ion p ocedu e in which equilib ium is no
ye sa is ied, he e is a esidual Rbe ween he in e nal wo k and he ex e nal wo k.
As in he p oblems conside ed in his hesis, he applied ac ion is independen o he
de o ma ion, he indi idual elemen con ibu ion o he esidual may be de ined as:
Re(un+1) = Ren+1 =ZΩe
Sn+1
∂En+1
∂uen+1
dΩ (3.33)
whe e Ωeis he domain o he elemen . Following he gene al p ocedu es o he New on
me hod, a sys em o equa ions o he he o m
Rn+1 =0(3.34)
is ob ained. Howe e equa ion 3.34 is no sa is ied unless con e gence has occu ed.
Since he cu en global esidual Rn+1 depends on he global displacemen ec o o he
Chap e 3. Fini e Elemen Dis e iza ion and Au oma ic Code Gene a ion 31
p e ious ime s ep, un, and on he cu en global displacemen ec o , un+1, ha is,
Rn+1 =Rn+1(un+1,un) (3.35)
he (k) h i e a ion s ep may be w i en as
Kk
Tn+1∆uk
n+Rk
n+1 =0(3.36)
whe e KTis he global angen s i ness ma ix gi en by
Kk
Tn+1 =∂Rk
n+1
∂uk
n+1
(3.37)
Finally, he displacemen s a e upda ed as ollows:
uk+1
n+1 =uk
n+1 + ∆uk
n+1 (3.38)
Nume ical in eg a ion
In a ini e elemen implemen a ion, he exac in eg als a e eplaced by a nume ical
in eg a ion p ocedu e. In gene al, s anda d Gaussian quad a u e will be used o his
pu pose. The in eg al o a gene ic unc ion o e a domain Γ is gi en by
ZΓ
(ξ)dξ ≈
ngaus
X
i=1
ωi (ξi)(3.39)
whe e ngaus is he numbe o app oxima ion poin s, wi h coo dina es ξiand weigh s ωi.
A his s age, i should be ema ked ha in he ollowing chap e s o his hesis, some
al e na i e me hods o nume ical in eg a ion will be add essed, in pa icula in he
p esence o discon inui ies.
3.3 Au oma ic code gene a ion
The implemen a ion o a FEM code may be qui e complica ed and ime consuming,
especially when dealing wi h non-linea ma e ial models. In gene al, he disc e isa ion
and linea isa ion p ocedu es desc ibed in he p e ious sec ions ha e o be pe o med
p io o he ac ual code w i ing. The complexi y o he equi ed calcula ions inc eases
he p obabili y o e o and, consequen ly, ine icien codes may esul .
Chap e 3. Fini e Elemen Dis e iza ion and Au oma ic Code Gene a ion 32
F om ano he pe spec i e, a new o mula ion ep esen s essen ially he coding o a di -
e en elemen kinema ics and/o a di e en ma e ial model, which sugges s ha a la ge
numbe o ope a ions ela ed o FEM p og amming could be au oma ed.
In his con ex , he use o au oma ic di e en ia ion ools and mode n symbolic and al-
geb aic compu e sys ems may ep esen a conside able ad an age. Howe e , in mos o
hese sys ems, he classical symbolic de i a ion leads o he exponen ial g ow h o ex-
p essions and ine icien codes. In o de o o e come his limi a ion, he AceGen Sys em
[83] employs he Simul aneous S ochas ic Simpli ica ion o Nume ical Code app oach
[84, 85], which combines an au oma ic di e en ia ion echnique wi h he simul aneous
op imiza ion o symbolic exp essions, allowing he au oma ic gene a ion o e icien nu-
me ical codes. In he ollowing subsec ions he main ea u es o his sys em will be
desc ibed and he applica ion o he FEM will be illus a ed.
3.3.1 Au oma ic code gene a ion ea u es
Fo an e icien code gene a ion, he AceGen sys em ga he s some impo an ea u es
cha ac e ized in he ollowing pa ag aphs.
Di e en ia ion
As shown in he beginning o his chap e , he de elopmen o a ini e elemen o mu-
la ion in ol es he calcula ion o nume ous de i a i es. The e o e a p ope au oma ic
di e en ia ion ool, in which he dependence on in e media e quan i ies is co ec ly ac-
coun ed o is essen ial.
The de i a i e o a gene al unc ion, , which depends on a se o mu ually independen
a iables aand a se o mu ually independen in e media e a iables b( ha is, bdepend
on a) may be exp essed as ollows:
∇ =∂ (a,b(a))
∂(a)∂(b)
∂(a)=M(3.40)
The o al de i a i es o he a iables bwi h espec o he a iables amay be w i en in
he ma ix o m, whe e Mis he espec i e Jacobian ma ix. Du ing he di e en ia ion
p ocedu e, when he e exis s an explici algo i hmic dependence be ween band a, he
de i a i es may be ob ained au oma ically by he chain ule wi hou use in e en ion,
Chap e 3. Fini e Elemen Dis e iza ion and Au oma ic Code Gene a ion 33
simply by eplacing i s alue o he co esponding en ance o ma ix M. Ne e heless,
o some si ua ions, he di ec applica ion o he chain ule leads o e oneous esul s.
In pa icula , when unc ion only depends explici ly on b,
∇ =∂ (b)
∂(a)∂(b)
∂(a)=M(3.41)
o he dependency be ween band ahas o be neglec ed,
∇ =∂ (a,b(a))
∂(a)∂(b)
∂(a)=0(3.42)
a p ope excep ion has o be conside ed by he au oma ic di e en ia ion ool [84, 85].
A ypical example o he i s case is a di e en ia ion ha in ol es a ans o ma ion
o coo dina es, o ins ance, om a e e ence isopa ame ic ini e elemen , o a global
coo dina e ini e elemen . The in oduc ion o he addi ional condi ion o he p ope
de i a ion o he de o ma ion g adien is illus a ed in sec ion 3.3.2.
Simpli ica ion o la ge ma hema ical exp essions
A e se ing up a p ope di e en ia ion ool, he me hodology o simpli y ma hema i-
cal exp essions is o majo impo ance. Mos o he exis ing symbolic sys ems, such as
Ma hema ica o Ma lab, only sea ch o common sub-exp essions a e all he o mulae
ha e been de i ed, using a pa e n-ma ching echnique. Howe e his me hodology is
insu icien o ob ain e icien codes o calcula ing ypical FEM quan i ies like he in-
e nal o ce ec o o he s i ness ma ix, due o he excessi e swell o he ma hema ical
exp essions [86, 87].
To o e come his p oblem, AceGen employs he Simul aneous S ochas ic Simpli ica ion
o nume ical code [84, 85]. In his me hodology he sea ch o common sub-exp essions
is pe o med a e each au oma ic de i a ion s ep, op imizing he in oduc ion o in e -
media e a iables. In addi ion, he equi alence be ween exp essions is no only sea ched
symbolically bu also nume ically [88]. I is a ac ha he co ec ness o he simpli ied
exp essions can only be de e mined wi h a ce ain p obabili y, ne e heless, when deal-
ing wi h ela i ely smoo h unc ions, as i is he case o he FEM, his can be neglec ed.
Chap e 4. The Ex ended Fini e Elemen Me hod 40
4.1 Fo mula ion and basic nume ical implemen a ion
4.1.1 Displacemen app oxima ion and choice o he en ichmen unc-
ions
The XFEM is a me hodology which allows o he inse ion o special ea u es, such
as discon inui ies o in e aces, in a FEM p oblem, independen ly om he mesh, by
en iching he displacemen ield, u, wi h one o mo e unc ions ψ, as ollows:
u(x) =
n
X
i=1
Ni(x)ui+
nen
X
j=1
Nj(x)ψ(x)ψj(4.1)
whe e Ni ep esen s he elemen shape unc ions and ψja e he ex a deg ees o eedom
associa ed o he unc ion ψin each one o he nen nodes.
The unc ion ψ, e med en ichmen unc ion, con ains he desc ip ion o he desi ed
ea u e and sa is ies he pa i ion o uni y p inciple, which may be s a ed as,
X
j
Nj(x)ψ(x) = 1.(4.2)
I should be no iced ha conside ing Njas he s anda d FEM shape unc ions, his
p ope y is au oma ically sa is ied, hough his choice is no manda o y.
The ma hema ical exp ession o ψ a ies g ea ly, as his me hod may be applied o e y
di e se scien i ic a eas. In pa icula case o his hesis he XFEM is used in he con ex
o ac u e and he e o e he en ichmen unc ion should be capable o cap u ing he
decohesion be ween he wo su aces o a c ack. Discon inuous unc ions such as he
ollowing Hea iside unc ion, H(ˆη), de ined in e ms o he coo dina e ˆηpe pendicula
o he c ack plane ( igu e 4.1), a e pa icula ly sui able o his pu pose.
H(ˆη) = (1: ˆη≥0
−1: ˆη < 0(4.3)
The displacemen app oxima ion hen eads:
u=
n
X
i=1
Niui+
nspli
X
j=1
NjHaj(4.4)
whe e aja e he deg ees o eedom associa ed o he Hea iside unc ion. When a ini e
elemen is o ally cu by a c ack is usually e med spli elemen and he e o e, has a
numbe o nspli en iched nodes. An elemen only pa ially c ossed by a c ack, ha is,
Chap e 4. The Ex ended Fini e Elemen Me hod 41
ˆη
(a)
ˆη
(b)
Figu e 4.1: De ini ion o he coo dina e pe pendicula o he c ack in a) 2D p oblem
b) 3D p oblem.
ˆ
ξ
ˆη
η
ξ
α
lc
ξ= 1ξ=−1
η= 1
η=−1
Figu e 4.2: C ack ip coo dina es o a 2D elemen o a cu along ˆ
ζ= 0 o a 3D
elemen .
con aining a c ack ip is named ip elemen . A local coo dina e sys em, ˆ
ξ= (ˆ
ξ, ˆη, ˆ
ζ),
cen e ed a he c ack ip, may be de ined o hese ype o elemen s, as illus a ed in
igu e 4.2, o a plane elemen o a cu along ˆ
ζ= 0 o a 3D elemen . In 2D p oblems
ˆ
ξ= (ˆ
ξ, ˆη), whe e ˆ
ξis in he c ack di ec ion and ˆηis pe pendicula o he c ack line. In
3D p oblems ˆ
ξ= (ˆ
ξ, ˆη, ˆ
ζ), in which ˆ
ξand ˆ
ζa e in he c ack plane, wi h ˆ
ξ= 0 de ining
he c ack ip line a he c ack plane, and ˆηcoo dina e is no mal o he c ack plane.
Applying di ec ly he displacemen app oxima ion o equa ion 4.4, in gene al u(xi)6=
ui. This p oblem may be easily o e come by shi ing he app oxima ion a ound he
node o in e es , as ollows:
u(x) =
n
X
i=1
Niui+
nspli
X
j=1
Nj[H(x)−H(xj)]aj(4.5)
Mo eo e , he shi ed e sion o equa ion 4.5 p e en s he en ichmen o sp eading o
he neighbou elemen s, which con ain s anda d and en iched nodes ( igu e 4.3), usually
designa ed blending elemen s.
Chap e 4. The Ex ended Fini e Elemen Me hod 42
Figu e 4.3: C ack inse ed in he mesh h ough he XFEM. The nodes belonging
o spli elemen s a e ma ked wi h ci cles, while he nodes belonging o ip elemen s
a e ma ked wi h iangles. Elemen s con aining bo h en iched and s anda d nodes a e
designa ed blending elemen s.
En iching ini e elemen s only wi h he Hea iside unc ion always causes he ex ension
o he c ack o he edge o an elemen . Ne e heless, o ep esen c acks uly mesh
independen , i is desi able o de ine a s a egy o deal wi h he closu e o a c ack away
om he elemen edge, as can be obse ed in igu e 4.3.
Elemen s con aining c ack ips, may hen be en iched by a unc io, which has o ensu e
ha he en ichmen anishes exac ly a he c ack ip. Some al e na i e de ini ions o
his unc ion a e p oposed and p esen ed below, in e ms o c ack ip coo dina es and
he c ack leng h inside he ip elemen , lc. The i s p oposed o m o he unc ion, he e
named as R, is a Hea iside ype unc ion, de ined as:
R(ˆ
ξ) = (1 o ˆ
ξ≤0
0 o he wise (4.6)
and de e mines ha only Gauss poin s loca ed behind he c ack on should be en iched.
The second p oposed o m is a linea ype unc ion, de ined as:
R(ˆ
ξ) = (−ˆ
ξ
lc o ˆ
ξ≤0
0 o he wise (4.7)
A highe o de unc ion, de ined in e e ence [96], is also adop ed he e as:
R(ˆ
ξ) = (3( ˆ
ξ
lc)2+ 2( ˆ
ξ
lc)3 o ˆ
ξ≤0
0 o he wise (4.8)
In his wo k, hese unc ions a e conside ed o be cons an along he ˆ
ζcoo dina e, in
he case o 3D elemen s.
Chap e 4. The Ex ended Fini e Elemen Me hod 43
Besides ensu ing ha he en ichmen disappea s exac ly a he c ack ip, he h ee o ms
o he R unc ion a e pa icula ly sui able o duc ile ac u e p oblems, whe e plas ici y
is no only con ined o he egion a ound he c ack ip bu widely sp ead. Fu he mo e,
as in he case o he Hea iside unc ion, hese polynomial en ichmen s anish ou side he
elemen s con aining discon inui ies, which is no he case i o he ypes o en ichmen s
a e employed [94, 97, 98].
The ull displacemen app oxima ion may hen be w i en as:
u(x) =
n
X
i=1
Niui+
nspli
X
j=1
Nj[H(x)−H(xj)]aj
+
n ip
X
k=1
Nk[R(x)−R(xk)][H(x)−H(xk)]bk
(4.9)
Rema k 4.1.In he amewo k o linea elas ic ac u e mechanics, he mos used en-
ichmen unc ions o deal wi h c ack ips a e based in he asymp o ic displacemen ields
de i ed om analy ical solu ions [54, 90, 99]. The e o e, he ollowing se o ou unc-
ions, ψi
ip, is employed:
ψi
ip ={√ sin(θ
2),√ sin(θ
2) sin θ, √ cos(θ
2),√ cos(θ
2) sin θ}(4.10)
These unc ions a e widely used o b i le ac u e p oblems, as hey ep oduce a sin-
gula i y a he c ack ip. Ne e heless, as p e iously e e ed, in a con ex o duc ile
ac u e hey may no be adequa e.
Finally, o a success ul implemen a ion o he me hod i is essen ial o de e mine he
spli nodes and he ip nodes, which may be done h ough he Le el Se Me hod, de-
sc ibed in he nex sec ion.
4.1.2 The le el se me hod
The Le el Se Me hod [100, 101] is he mos equen ly used complemen o he XFEM
o accu a ely de e mine he posi ion o he discon inui ies, ha is, he se o nodes o
be en iched [102, 103].
Chap e 4. The Ex ended Fini e Elemen Me hod 44
Ω
Ω−
Γ
Ω+
d
x
xΓ
Figu e 4.4: Domain
Conside ing a domain Ω, di ided in he egions Ω+and Ω−by he in e ace Γ, he le el
se unc ion, φ(x), is de ined in a such a way ha
φ(x) = (1: x∈Ω+
0: x∈Γ
−1: x∈Ω−
(4.11)
A common choice o φ(x) is he signed dis ance unc ion, de ined as
φ(x) = ±d=±kx−xΓk(4.12)
whe e dis he dis ance be ween poin xand he closes poin o xlaying in he in e ace,
xΓ, as depic ed in igu e 4.4. The sign o ddepends on which side o he in e ace is
loca ed poin x.
The unc ion φ(x) de e mines i a poin lies abo e o bellow a c ack. Ne e heless,
as usually a c ack does no cu he en i e domain, a second unc ion,ϕ(x), has o be
in oduced o deal wi h he ac i e c ack ips. When a c ack is de ined by s aigh line
segmen s, which is he case in his hesis, a unc ion ϕi(x) may be de ined by each ac i e
c ack ip as ollows:
ϕ(x) = (x−xi).ˆ
i(4.13)
whe e ˆ
is a uni ec o angen o he la es c ack segmen and xiis he loca ion o he
i h c ack ip, as can be obse ed in igu e 4.5.
Al hough, unc ions φand ϕiha e o be upda ed a each c ack g ow h s ep, he sea ch
o newly en iched elemen s is only equi ed a ound he p e ious c ack ip. Fu he -
mo e, in a duc ile ac u e p oblem he di ec ion and magni ude o a c ack inc emen
a e de e mined by he ma e ial model, as i will be desc ibed in he ollowing chap e s.
Chap e 4. The Ex ended Fini e Elemen Me hod 45
φ < 0
φ > 0
ϕ2<0
ϕ2>0
ϕ1>0
ϕ1<0
ˆ
1
ˆ
2
Figu e 4.5: Le el se unc ions.
Amax
Amin
(a) (b) (c)
Figu e 4.6: The c ack is e y close o a) a node b) an edge and o ally con ained in
one elemen in c), possibly leading o ill-condi ioned s i ness ma ices.
Consequen ly, he le el se unc ions a e upda ed by simple geome ical conside a ions.
The le el se unc ion desc ibed may also desc ibe c ack p opaga ion o plana c acks
in 3D p oblems. A mo e de ailed desc ip ion o he p ocedu e may be ound in he
e e ences [104–107].
4.1.3 Basic implemen a ion
The implemen a ion o an XFEM o mula ion consis s undamen ally in modi ying he
s anda d FEM displacemen app oxima ion and c ea e a s uc u e o deal wi h he ex a
deg ees o eedom, in oduced by he en ichmen unc ions, when pe o ming asks such
as he assembly o he in e nal o ce o he s i ness ma ix.
Some linea dependencies o ill-condi ioning may a ise due, essen ially, o elemen s
whe e he a io o he a eas/ olumes be ween he wo sides o he c ack is e y high,
Amax/Amin, as illus a ed in igu es 4.6(a) and 4.6(b), o when he discon inui y lies only
inside one elemen ( igu e 4.6(c)). The i s si ua ion is usually p e en ed by se ing a
ole ance om which he c ack should be con e ed in o a in e -elemen c ack while in
he second si ua ion, he mesh may be e ined [94, 108].
Chap e 4. The Ex ended Fini e Elemen Me hod 46
Ano he key ea u e in he implemen a ion o he XFEM is nume ical in eg a ion. Due
o i s high impo ance he nex sec ion is ully de o ed o his subjec .
O he ea u es ha had o be de eloped o deal wi h he use o he XFEM wi h non-
linea p oblems, such as ans e o his o y a iables and non-locali y, will be also ad-
d essed in he end o his chap e .
4.2 Nume ical in eg a ion and he incomp essibili y issue
Nume ical in eg a ion has o be ca e ully add essed when implemen ing a XFEM code,
as he esul s p oduced by egula Gaussian quad a u e ules a e no accu a e enough,
due o he p esence o discon inuous unc ions. Two main echniques we e de eloped
since he ea ly s ages o XFEM. P obably he mos common o hose echniques consis s
o di iding he elemen s con aining discon inui ies in sub cells con o ming he discon inu-
i y, usually iangles o e ahed ons, o 2D and 3D p oblems espec i ely, and applying
a egula ule o each sub cell [90, 91, 109]. Al e na i ely, some au ho s [110–112] p e e
o keep ixed Gauss poin posi ions, bu la gely inc ease hei numbe inside an elemen
o compensa e o no adjus ing he discon inui y pa e n. As compu a ionally his las
me hod may become ex emely expensi e, o he app oaches we e c ea ed o supp ess
he need o elemen subdi ision, s a ing wi h he wo ks o Ia e [113] and Ven u a [114]
who de i ed con inuous polynomial unc ions which, in eg a ed o e an elemen , ep o-
duce he in eg als o he discon inuous XFEM unc ions. This me hod was ex ended o
egula ized discon inui ies by Ben enu i e al. [115], bu i is only exac o iangula
o e ahed al elemen s. Ano he in eg a ion ule was p oposed by Holdych [116] o
he same ype o elemen s, whe e he d awback o ha ing ixed posi ions o he Gauss
poin s is compensa ed by adop ing a iable weigh s. The ela ion be ween hese las wo
me hods was es ablished by Bely schko e al. [117] and, subsequen ly, he same au ho s
de eloped a bounda y in eg a ion me hod [118], applicable o quad angula elemen s,
in which all nodes a e equally en iched. Recen ly, Na a ajan e al. [119] p oposed an
in eg a ion echnique based on he Schwa z-Ch is o el con o mal mapping alid o any
ype o 2D ini e elemen s, and Mousa i e al. [120] made use o he node elimina ion
echnique o p oduce accu a e in eg a ion ules wi h a educed numbe o Gauss poin s.
As, so a , mos o he XFEM applica ions a e wi hin he scope o b i le ac u e, he
accu acy o he me hodologies desc ibed abo e is ypically measu ed by he abili y o
ep oduce he s ess in ensi y ac o s, o pa icula p oblems ha ha e an a ailable an-
aly ical solu ion. Ne e heless, he di ec ex ension o hose me hodologies o p oblems
Chap e 4. The Ex ended Fini e Elemen Me hod 47
in ol ing duc ile ac u e may no be adequa e. In ac , when duc ile ac u e is he
issue, plas ic de o ma ion is widely sp ead and no jus con ined o he nea by zones
a ound he c ack, as i happens in b i le ac u e. Fu he mo e i is well known ha
incomp essibili y o plas ic de o ma ion may lead o locking o he nume ical solu ion,
especially when low o de ini e elemen s a e employed [121–126]. Consequen ly, in o de
o a oid i , adequa e o mula ions o in eg a ion echniques mus be used. Depending on
he p oblem add essed solu ions may include, o example, selec i e educed in eg a ion
echniques [122], mixed o mula ions [65], B-ba me hods [123], F-ba me hods [63, 127]
o enhanced assumed s ain me hods [128], o name a ew. The e o e when XFEM is
used o model duc ile ac u e some ques ions eme ge as impo an opics, namely hose
ela ed wi h he discussion on he adequa e numbe o Gauss poin s ha should be used
in he sub cells, when an elemen con ains o is c ossed by a c ack, and he abili y o
sa is y he incomp essibili y cons ain s.
In he ollowing sec ions, ha ing in iew he applica ion o XFEM o duc ile ac u e
p oblems, he e iciency o di e en in eg a ion ules is in es iga ed, ocusing on hei
abili y o sa is y olume ic incomp essible cons ain s in he con ex o bo h in ini es-
imal and ini e s ains. As a consequence, he con en ional o mula ions o B-ba and
F-ba a e adap ed o include he XFEM en ichmen unc ions. O he al e na i es o
deal wi h incomp essibili y in XFEM elemen s include he adap a ion o he assumed
enhanced s ain me hod by Dolbow e al [129] and he mixed o mula ion echnique by
Leg ain e al [130]. He e, he e iciency o he p oposed echniques is e alua ed wi h
some nume ical examples and he pe o mance o he F-ba me hod o XFEM is com-
pa ed wi h he me hod de eloped by Dolbow e al [129].
4.2.1 Me hodology o assess he abili y o alle ia e locking
The me hodology he e adop ed o assess he endency o he abili y o a pa icula
me hod o depic o o alle ia e locking will ollow closely some p e ious wo k ha
may be ound in e e ences [124–126]. This me hodology is based on he analysis o he
unde lying sub-space o incomp essible modes embedded in he app oxima ion me hod
adop ed, a he han elying on me e cons ain indexes which may be misleading, as
hey a e de ined independen ly o he bounda y condi ions. The a o emen ioned analysis
may gi e in o ma ion on how he dimension o his subspace can be enla ged o educed
as a unc ion o he in eg a ion ule adop ed. Consequen ly i may gi e p ecious hin s
on he abili y o a pa icula me hod o ep oduce incomp essible de o ma ions, on how
Chap e 4. The Ex ended Fini e Elemen Me hod 48
spu ious modes may be p e en ed o on how o e in eg a ion ules may be a oided.
Recalling equa ion 2.11 om chap e 2, an isocho ic de o ma ion is cha ac e ized by:
J= 1 (4.14)
Taking in o accoun ha he ma e ial ime de i a i e o he olume a io may be w i en
as a unc ion o he a e o de o ma ion enso o o he eloci y ield:
˙
J=J d=Jdi (4.15)
and conside ing equa ion 4.14 and i s ime de i a i e, he incomp essibili y condi ion
may al e na i ely be s a ed as:
d= 0 (4.16)
Fo a small s ain p oblem, dis ega ding second and highe o de e ms, his condi ion
may be simpli ied o:
di u= 0 (4.17)
whe e u ep esen s he displacemen ield.
Consequen ly, when imposing he incomp essibili y condi ion in a FEM amewo k ia
he Penal y Me hod o he Lag ange Mul iplie echnique [124], a local disc e ized o m
o equa ion 4.16 o equa ion 4.17 may be es ablished a each nume ical in eg a ion poin
o a pa icula elemen wi h he o m
Qwh= 0 (4.18)
whe e whis he ec o o elemen nodal a iables om which he a e o de o ma ion
enso o he displacemen ield a e app oxima ed and Qis a ma ix i×n, in which iis
he numbe o in eg a ion poin s and n he numbe o unknowns in wh. Consequen ly,
elemen wise, an admissible incomp essible solu ion whshould belong o he null space
o ma ix Q, which de ines locally he space o incomp essible de o ma ions, Ih
Ih={wh∈U:Qwh= 0}(4.19)
Locking occu s, when o a gi en se o bounda y condi ions, he expec ed solu ion
canno be ep oduced by a linea combina ion o a gi en basis o Ih[124–126]. Ha ing
applica ions on duc ile ac u e p oblems, in which ex ensi e plas ic de o ma ions may
be p esen , he analysis o his unde lying sub-space and how i may be a ec ed by
adop ing o al e ing a ce ain in eg a ion ule will be add essed in he ollowing sec ions
o his chap e .
Chap e 4. The Ex ended Fini e Elemen Me hod 49
4.2.2 Incomp essible de o ma ion modes in elemen s con aining XFEM
en ichmen s
A basis o Ih(equa ion 4.19), he space o admissible incomp essible de o ma ions, mus
be de e mined o s udy he ulne abili y o he en iched elemen s o olume ic locking.
I should be no iced ha o he XFEM app oxima ion, he nodal a iables whused in
he app oxima ion o he s ain o he s ain a e encompasses bo h he displacemen s
uhand he ex a a iables aho bho hei pseudo- ime de i a i es. The e o e, o
simplici y and in line wi h he p e ious wo ks o Cesa de Sa e al. [124–126], a basis o
Iho a pa icula elemen , in which global coo dina es xand na u al coo dina es ξa e
coinciden , may be de e mined by ob aining a basis o he null space o ma ix Q. Fo
his pa icula case equa ion 4.19 eads:
Qwh= [ ∂Ni
∂ξ
∂Ni
∂η
∂Ni
∂ζ
∂(Niψ)
∂ξ
∂(Niψ)
∂η
∂(Niψ)
∂ζ |i=nnodes ]{uh
ψh}= 0 (4.20)
whe e he numbe o lines in Qequals he numbe o Gauss poin s, nGauss, used in he
ini e elemen , ψ ep esen s he en ichmen unc ion associa ed wi h he ex a a iables
ψh, which may be aho bhwhe he i is a spli elemen o a ip elemen .
To de e mine he admissible incomp essible de o ma ions o an elemen con aining a
discon inui y, equa ion 4.20 has o be pa icula ized o he displacemen app oxima ions
o equa ion 4.5. Fo an elemen o ally c ossed by a discon inui y, only he Hea iside
en ichmen is conside ed leading o he displacemen app oxima ion, exp essed locally
as:
u(ξ) =
n
X
i=1
Niui+
nspli
X
j=1
NjHSh
iaj(4.21)
The e o e, he e ms associa ed wi h he unc ion R anish and each line, g, o ma ix
Qmay be w i en as:
Qg={∂Ni
∂ξ
∂Ni
∂η
∂Ni
∂ζ |i=nnodes
∂Ni
∂ξ HSh
i∂Ni
∂η HSh
i∂Ni
∂ζ HSh
i|i=nnodes }(4.22)
whe e HSh
ideno es he shi ed e sion o he Hea iside unc ion as:
HSh
i=H(ξ)−H(ξi),
i= 1, nnodes
(4.23)
Chap e 4. The Ex ended Fini e Elemen Me hod 56
Table 4.3: A possible basis o he incomp essible de o ma ion space o an elemen
o ally c ossed by a discon inui y.
Incomp essible de o ma ion modes ep oduced by all he in eg a ion ules
m1 m2 m3 m4 m5
m6 m7 m8 m9 m10
(a) (b)
Figu e 4.14: Possible c ack con igu a ions con aining c ack kinks, wi h in eg a ion
ules ob ained conside ing subdi ision in o six iangles, ou in one side o he c ack
and wo in he o he side.
(a) (b)
Figu e 4.15: Examples o in eg a ion ules ob ained o c acked hexahed ons: a) ou
poin s pe e ahed on; b) one poin pe e ahed on.
con o ming he c ack plane, in which a egula Gaussian ule is applied, as shown in
igu e 4.15. Fo example, o an hexahed al elemen wi h 1 ×12 Gauss poin s, co e-
sponding o an elemen subdi ision o wel e e ahed ons, he e a e hi y se en linea ly
independen admissible incomp essible de o ma ion modes, whe eas conside ing 4 ×12
Gauss poin s, o he same subdi ision in wel e e ahed ons, i is only possible o e-
p oduce hi y ou linea ly independen modes. A comple e basis o Ihis ep esen ed
in able 4.4.
Chap e 4. The Ex ended Fini e Elemen Me hod 57
Table 4.4: A possible basis o he incomp essible de o ma ion space o an elemen
o ally c ossed by a discon inui y.
Incomp essible de o ma ion modes ep oduced by all he in eg a ion ules
m1 m2 m3 m4 m5
m6 m7 m8 m9 m10
m11 m12 m13 m14 m15
m16 m17 m18 m19 m20
m21 m22 m23 m24 m25
m26 m27 m28 m29 m30
m31 m32 m33 m34
Incomp essible de o ma ion modes only ep oduced by a educed in eg a ion ule
m35 m36 m37
Chap e 4. The Ex ended Fini e Elemen Me hod 58
(a) (b) (c) (d)
Figu e 4.16: Regula Gaussian quad a u e ules applied o an elemen c ossed by a
discon inui y: a) ou Gauss poin s; b) nine Gauss poin s; c) six een Gauss poin s; d)
nine Gauss poin s.
Once mo e, we conclude ha is possible o expand he basis o incomp essible de o ma-
ion modes by using a educed in eg a ion ule, which will be c ucial o con ol olume ic
locking.
4.2.2.2 Regula Gauss poin dis ibu ion
The simples s a egy o a oid he subdi ision o he ini e elemen s o in eg a ion pu -
poses consis s o placing a la ge numbe o Gauss poin s whose posi ion is ixed in he
elemen , and he e o e does no adjus o he c ack pa h [110–112], as illus a ed in
igu e 4.16.
Al hough his echnique is ela i ely simple, he cha ac e iza ion o he unde lying space
o incomp essible de o ma ions associa ed wi h each Gauss poin dis ibu ion may no
be as sys ema ic as in he case o elemen subdi ision. Fo example, in he con igu a ions
gi en in igu e 4.16 i was expec ed ha Ihcould be cha ac e ized by he same basis o
he cases in igu e 4.16(b) and 4.16(d). Ne e heless he dimension o Ihis highe in he
second case.
In ac , in his case, he numbe o linea ly independen incomp essible modes depends
no only on he numbe o Gauss poin s bu also on he pa icula c ack con igu a ion. In
igu e 4.16(a), he in eg a ion ule yields he same solu ion as he one in which only one
Gauss poin pe iangle is used in he s a egy de ined in sec ion 4.2.2.1, and he wel e
modes ep esen ed in able 4.1 a e ep oducible; o highe numbe o Gauss poin s like
in igu es 4.16(b)-4.16(c), o e en wen y- i e o six y- ou Gauss poin s, only he i s
en modes a e included in he basis o Ih. Ne e heless, when he c ack con igu a ion is
sligh ly changed, as illus a ed in igu e 4.16(d), he numbe o linea ly independen in-
comp essible de o ma ion modes inc eases o ele en, when only en modes we e expec ed.
Chap e 4. The Ex ended Fini e Elemen Me hod 59
(a) (b) (c) (d)
Figu e 4.17: Applica ion o he Schwa z-Ch is o el con o mal mapping o build an
in eg a ion ule o elemen s c ossed by a discon inui y: a) c acked elemen ; b) polygon
ha will be mapped o he uni disk; c) uni disk con aining he Gauss poin posi ions;
d) inal Gauss poin dis ibu ion on he uppe pa o elemen a).
These esul s sugges ha when he in eg a ion ule does no con o m he discon inui y,
mo e likely spu ious de o ma ion modes may a ise and he cons uc ion o s a egies o
a oid locking may no be so sys ema ic as in he case o elemen subdi ision. In addi ion,
his s a egy can become compu a ionally mo e expensi e han he e e ed subdi ision
due o he much la ge numbe o Gauss poin s ha has o be used o ob ain he same
accu acy.
4.2.2.3 Schwa z-Ch is o el con o mal mapping
Ano he possible s a egy used o a oid he subdi ision o ini e elemen s o pe o m nu-
me ical in eg a ion, based on he Schwa z-Ch is o el con o mal mapping was p oposed
by Na a ajan e al. [18]. The Schwa z-Ch is o el con o mal mapping is an angle p ese -
ing ans o ma ion, which maps he complex uppe hal -plane, z={ζ∈C: Imζ > 0}
o he in e io o a polygon. Ma hema ically i may exp essed as a unc ion (ζ), ζ ∈C,
as ollows:
(ζ) = ZζK
(w−a)α/π(w−b)β/π(w−c)γ/π . . .dw(4.29)
whe e Kis a cons an , α,β,γ,. . . a e he in e io angles o he polygon and a,b,c,. . .
a e he alues, along he eal axis o he z-plane, o he e ices o he polygon.
To apply his ans o ma ion o en iched XFEM elemen s, each polygon belonging o an
elemen cu o in e sec ed by a c ack is mapped on o a uni disk ( igu e 4.17) h ough
he in e se o he desc ibed unc ion. The uni disk con ains equally spaced Gauss
poin s on he complex plane, which a e mapped back in o he o iginal polygon by he
ans o ma ion unc ion [119, 132].
Chap e 4. The Ex ended Fini e Elemen Me hod 60
(a) (b) (c) (d)
(e)
Figu e 4.18: In eg a ion ules buil using he Swchwa z-Ch is o el con o mal map-
ping: a) ou Gauss poin s; b) eigh Gauss poin s; c) wel e Gauss poin s; d) wel e
Gauss poin s; e) wen y- ou Gauss poin s. . These igu es we e c ea ed making use o
he MATLAB SC Toolbox [1].
The Schwa z-Ch is o el con o mal mapping echnique may be a applied o an elemen
o ally c ossed by a c ack o cons uc he in eg a ion ules ep esen ed in igu e 4.18.
The e is a ce ain eedom o dis ibu e he Gauss poin s depending on he subdi ision
o he uni ci cle, as illus a ed by igu es 4.18(c) and 4.18(d), whe e wel e Gauss poin s
a e dis ibu ed in a di e en way. In line wi h he p e ious sec ions, equa ion 4.20 was
applied o he in eg a ion ules o igu e 4.18 o cha ac e ize he space o admissible
incomp essible de o ma ions. Resul s exhibi he same s uc u e o sec ion 4.2.2.1, ha
is, wel e linea ly independen incomp essible de o ma ion modes o ou Gauss poin s
( igu e 4.18(a)), which co esponds o one Gauss poin pe iangle, when elemen sub
di ision is used, and en linea ly independen incomp essible de o ma ion modes when
mo e han ou Gauss poin s a e employed ( igu es 4.18(b)-4.18(e)). A possible basis
o Ihis also ep esen ed in able 4.1. The e o e, we conclude ha he incomp essible
de o ma ion modes associa ed o in eg a ion ules buil wi h he Schwa z-Ch is o el
con o mal mapping a e exac ly he same as he ones ob ained wi h he sub di ision in
iangula sub cells echnique, as long as he numbe o Gauss poin s abo e and below
he c ack is equal o bo h me hods.
Al hough his me hodology supp esses he need o sub cells, i does no b ing any ad an-
age in e ms o incomp essible p oblems. Mo eo e , i equi es sol ing wo di e en ial
equa ions o each elemen o ally c ossed by a c ack, which is compu a ionally mo e
expensi e han o subdi ide he elemen in iangula sub cells, and, as addi ional dis-
ad an age, i is no eadily ex endable o 3D p oblems.
Chap e 4. The Ex ended Fini e Elemen Me hod 61
4.2.3 Fini e elemen o mula ions o incomp essibili y
The esul s p e iously ob ained ha e shown, as expec ed, ha i is possible o expand
he space o admissible incomp essible de o ma ions, when an in eg a ion ule wi h a
educed numbe o Gauss poin s is used in he cons ain equa ion which de ines he
incomp essibili y condi ion. Ne e heless hese ules may in oduce spu ious de o ma-
ion modes ha should be con olled. This ac sugges s ha classical me hods used o
a oid locking in he amewo k o FEM, namely he B-ba me hod [123] and he F-ba
me hod [127], should be used in conjunc ion wi h XFEM elemen s in o de o p e en
olume ic locking whils p e en ing spu ious de o ma ion modes.
4.2.3.1 B-ba me hod
The B-ba me hod [123] is based on he addi i e decomposi ion o he ma ix o he
shape unc ions de i a i es, B, in o a olume ic componen , Bdil and a de ia o ic
componen , Bde , as ollows:
Bde =B−Bdil (4.30)
To a oid olume ic locking, Bde is in eg a ed wi h a comple e Gaussian ule, whe eas
Bdil is in eg a ed wi h a educed Gaussian ule, which is able o ep oduce a la ge
numbe o admissible incomp essible de o ma ions han he comple e coun e pa . In
2D p oblems, he B-ba o mula ion is usually applied o low o de quad ila e al elemen s
and Bde is calcula ed using 4 Gauss poin s and Bdil is calcula ed only o one Gauss
poin in he cen e o he elemen , as shown on igu e 4.19(b), while in 3D p oblems,
he comple e in eg a ion ule consis s o eigh Gauss poin s and he educed in eg a ion
ule consis s o one Gauss poin in he cen e o he hexahed al elemen .
He e we p opose o ex end he B-ba me hodology o elemen s c ossed by discon inui ies,
based on he elemen subdi ision p ocedu e o sec ion 4.2.2.1. Thus, in hese ini e
elemen s he de ia o ic pa o he Bma ix is calcula ed using a 3 poin s pe iangle
ule o ou poin s pe e ahed on in 2D and 3D p oblems, espec i ely, while he
olume ic con ibu ion o each Gauss poin is calcula ed in he cen e o co esponding
iangle ( igu e 4.19) o e ahed on. As he nume ical in eg a ion ule consis ing o
one poin pe iangle is able o ep oduce mo e incomp essible de o ma ion modes han
he highe o de ules, his me hodology e icien ly alle ia es he olume ic locking in
in ini esimal s ain p oblems, as will be la e illus a ed wi h some nume ical examples.
Chap e 4. The Ex ended Fini e Elemen Me hod 62
(a) (b)
Figu e 4.19: In eg a ion ule o a egula 4-nodes elemen : 1 Gauss poin o he
educed ule and 4 Gauss poin s o he comple e ule; b) In eg a ion ule o an elemen
c ossed by a discon inui y: 1 Gauss poin pe iangle as a educed ule and 3 Gauss
poin s pe iangle as a comple e ule.
Al hough he ex ension o his me hodology o XFEM is s aigh o wa d, i should be
no iced ha o c acked elemen s, he Bma ix con ains some ex a e ms due o he
en ichmen unc ions.
4.2.3.2 F-ba me hod
Fo p oblems in ol ing ini e de o ma ions, one o he simples ways o aking ad an age
o he expansion o he incomp essible de o ma ion space, when a educed in eg a ion
ule is used, is h ough he F-ba me hod [63, 127]. While B-ba is based on he addi i e
decomposi ion o he Bma ix, he F-ba is based on he mul iplica i e decomposi ion
o he de o ma ion g adien [65–67], as desc ibed in chap e 2:
F=Fde F ol (4.31)
whe e
Fde = (de F)−1
3F=J−1
3F(4.32)
F ol = (de F)1
3I=J1
3I(4.33)
The F-ba de o ma ion g adien is simply ob ained by eplacing he egula olume ic
pa by he olume ic pa , F0, calcula ed a he cen oid i he elemen ( o a 4-node
quad ila e al o a 8-node hexahed on), eading:
¯
F=Fde (F0) ol = (de F0
de F)1
3F= (J0
J)1
3F(4.34)
The applica ion o he F ba me hod o en iched elemen s ollows he same p inciple
as he B-ba me hod, hence, cons uc ing he nume ical in eg a ion ules in a sub i-
angula ion base, o each Gauss poin o he comple e ule ( ypically 3 Gauss poin s
pe iangle/4 Gauss poin s pe e ahed on), he olume ic pa o he de o ma ion
Chap e 4. The Ex ended Fini e Elemen Me hod 63
g adien , F0, will be calcula ed a he cen oid o he espec i e iangle/ e ahed on
( igu e 4.19).
As in he B-ba o mula ion, i should be no iced ha he de o ma ion g adien o he
elemen s c ossed by discon inui ies con ains some ex a e ms associa ed wi h he en ich-
men o he displacemen s, which is clea when Fis de ined in e ms o he displacemen
ield:
F= (I+∇u) (4.35)
The e o e, conside ing a gene al en ichmen unc ion, ψ, and espec i e associa ed de-
g ees o eedom, ψ, he de o ma ion g adien can be w i en as:
F=I+
n
X
i=1
ui⊗∇XNi+
m
X
j=1
ψj⊗∇XNiψ(4.36)
ecalling ha ∇Xdeno es he g adien wi h espec o he e e ence con igu a ion, usu-
ally implying a coo dina e ans o ma ion o he isopa ame ic mapping. Depa ing
om he en iched o m o he de o ma ion g adien i is easy o adap a ini e s ain
code o include XFEM en ichmen unc ions. Reade s may consul e e ences [133–135]
o de ails.
The p esen ed ex ension o he B-ba and F-ba me hodologies o XFEM is expec ed
o alle ia e locking in ac u e p oblems whe e he sa is ac ion o he incomp essibili y
cons ain plays a majo ole. The e o e, he pe o mance o hese echniques will be
illus a ed in he nex sec ion wi h some nume ical examples in ol ing linea and non-
linea ma e ials.
4.2.4 Nume ical examples
In his sec ion, he conclusions ega ding he nume ical in eg a ion s a egies and hei
abili y o ep oduce incomp essible s a es will be illus a ed h ough some nume ical ex-
amples, no only ocusing on each ype o en iched elemen by i sel , bu also analysing
hei in luence when inse ed in a mesh.
In he examples incomp essibili y in a XFEM amewo k is add essed in bo h small and
ini e s ain p oblems.
Chap e 4. The Ex ended Fini e Elemen Me hod 64
(a) (b) (c)
Figu e 4.20: Tes examples o single elemen con aining a c ack.
4.2.4.1 Single Elemen
Analysing able 4.2 om sec ion 4.2.2.1, i is p ominen he absence o he de o ma ion
modes m6 and m7 o a comple e in eg a ion ule, while he mode m4 may be ep oduced
by bo h comple e and educed in eg a ion ules. This ac sugges ed he single elemen
es illus a ed in igu e 4.20, whe e a 4-node elemen in a plane s ain condi ion is
submi ed o 3 di e en loading scena ios. The ma e ial conside ed is linea elas ic and
he nea ly incomp essible beha io is ob ained se ing he alue o he Poisson coe icien
o ν= 0.499999. The Young0s Modulus has a alue o E= 103and he load is uni a y.
All he quan i ies a e supposed o be exp essed in a consis en uni sys em. The elemen
con ains a c ack segmen s a ing a one qua e o an edge and inishing a he elemen
cen e . The elemen edges ha e leng h 2.
In his analysis, he comple e in eg a ion ule consis s o eigh een in eg a ion poin s
placed on six sub- iangles and he educed in eg a ion ule consis s o a single in eg a-
ion poin pe iangle. The B-ba echnique is applied o he combina ion o hese wo
ules as p e iously desc ibed. The ip en ichmen is he one as de ined in equa ion 4.8.
In he able 4.5 i is possible o obse e he de o med con igu a ion ob ained as a unc ion
o he loading and in eg a ion ule used.
As i was expec ed, using a comple e in eg a ion ule, he elemen exhibi s olume ic
locking o he cases b) and c) in igu e 4.20. Di e en ly, bo h educed in eg a ion
and B-ba me hodologies a e able o ep oduce he co ec de o ma ions and do no
p esen any locking beha iou . Ne e heless, he educed in eg a ion may encompass
some d awbacks when used on a ini e elemen mesh, which will become clea e in he
ollowing nume ical examples.
Chap e 4. The Ex ended Fini e Elemen Me hod 65
Table 4.5: De o med con igu a ions o di e en in eg a ion ules.
In eg a ion Rule P oblem
a b c
Comple e
Reduced
B-ba
9
9
Figu e 4.21: C acked pla e wi h espec i e bounda y condi ions.
4.2.4.2 Incomp essible beha io o XFEM elemen s in a mesh
The p oblems p esen ed in his sec ion in end o analyse he in luence o en iched ele-
men s wi hin a mesh unde an incomp essibili y cons ain . A simila example as he
p e ious one is es ed wi h wo ypes o mesh, con aining egula and en iched elemen s
as shown in igu e 4.21. The ma e ial p ope ies a e he same as in he p e ious example.
The pla e is disc e ized wi h a coa se mesh and wi h a ine mesh con aining egula and
en iched ini e elemen s, as can be obse ed in igu es 4.21 and 4.22.
Chap e 4. The Ex ended Fini e Elemen Me hod 72
Figu e 4.30: Compa ison o he e ical displacemen o he op igh co ne node, as
a unc ion o he numbe o elemen s pe side, o di e en ip en ichmen unc ions.
Fini e S ain Analysis In his sec ion he same example bu a ini e s ains is used
o e alua e he pe o mance o he p oposed ex ension o F-ba o mula ion o XFEM
and compa e i wi h he enhanced s ain o mula ion o XFEM p oposed by Dolbow
e al. [129]. A hype elas ic ma e ial esponse go e ned by he ollowing s o ed ene gy
unc ion, W, is conside ed:
W=µ
2[J−2
3 [C]−3] + κ
2(J−1)2(4.37)
whe e µ ep esen s he shea modulus and κ he bulk modulus. The nea ly incom-
p essible beha io is ob ained se ing he ma e ial pa ame e s as κ= 40.0104Pa and
µ= 80.0 Pa. In his ype o cons i u i e beha iou , he ele an physical quan i ies can
be de i ed di ec ly om he ene gy unc ion making i appealing o au oma ic di e -
en ia ion and au oma ic code gene a ion [84, 85]. The e o e, he p e iously desc ibed
AceGen and AceFEM sys ems will be employed. In pa icula , o his hype elas ic
p oblem, we will conside he olume ic/de ia o ic spli o he s o ed ene gy unc ion
[62, 138, 139], as ollows:
W=Wde +W ol (4.38)
Wde =µ
2[J−2
3 [C]−3] (4.39)
W ol =κ
2(J−1)2(4.40)
As ou lined in chap e 3, using a New on-Raphson p ocedu e, he in e nal o ce, Re,
may be ob ained in eg a ing, o e he domain, he de i a i e o he s ain ene gy in
Chap e 4. The Ex ended Fini e Elemen Me hod 73
espec o he displacemen ield. Consequen ly, he angen s i ness ma ix, KTis ob-
ained as he de i a i e, in espec o he displacemen ield, o he in e nal o ce ec o :
Re=ZΩ
∂W
∂udΩ (4.41)
KT=∂Re
∂u(4.42)
He e, he de i a i e o he s ain ene gy po en ial is p esen ed in e ms o he de ia o ic
pa o second Piola-Ki chho enso , S, and he hyd os a ic p essu e p:
∂W
∂u=1
2Sde :Cde
∂u−p∂J
∂u(4.43)
wi h
Sde = (2∂Wde
∂C)de = 2∂Wde
∂Cde −2
3 [Cde
∂Wde
∂Cde
]C−1
de (4.44)
and
p=−∂W ol
∂J (4.45)
whe e Cde co esponds o he de ia o ic pa o he igh Cauchy-G een enso , de-
pending on he de ia o ic pa o he de o ma ion g adien , ¯
F:
Cde =¯
FT
de ¯
Fde =J−2
3C(4.46)
Wi h he AceGen sys em all he p esen ed de i a i es a e calcula ed au oma ically and
op imized. I should be no iced ha he global ec o o unknowns includes he displace-
men s as well as he ex a deg ees o eedom in oduced by he XFEM. The analysis is
pe o med using AceFem.
Fo he analysis o he ini e s ain e sion o he Cook0s Memb ane, a load o 1 Nis
applied acco ding o igu e 4.27 du ing 30 load s eps. Only he Hea iside en ichmen is
employed o model he c ack. In line wi h he in ini esimal s ain e sion, he e ical
displacemen o he op igh co ne is plo ed as a unc ion o he numbe o ini e ele-
men s pe edge. In addi ion he incomp essibili y cons ain , J= de F= 1, is checked
o all he Gauss poin s in he mesh.
Ini ially, in o de o e alua e he model implemen a ion in he AceGen sys em, he
solu ion o he example wi h he explici ly meshed c ack, illus a ed in igu e 4.28(b),
is compa ed wi h he one ob ained in e e ence [129]. As can be seen in igu e 4.31 a
Chap e 4. The Ex ended Fini e Elemen Me hod 74
Figu e 4.31: Ve ical displacemen o he op igh co ne node, as a unc ion o he
numbe o elemen s pe side, ob ained o he explici ly meshed c ack o igu e 4.28(b),
using he F-ba me hodology and he Enhanced s ain app oach used by Dolbow e al.
Figu e 4.32: Compa ison o he e ical displacemen o he op igh co ne node,
as a unc ion o he numbe o elemen s pe side, o explici ly meshed and en iched
app oxima ions using he F-ba me hodology and he Enhanced S ain app oach em-
ployed by Dolbow e al.
e y good co ela ion is ob ained.
Nex he F-ba me hodology applied o he XFEM is assessed. In igu e 4.32 i s esul s
a e compa ed wi h hose ob ained by Dolbow e al. [129] using he enhanced s ain
me hod and ha ing as a e e ence solu ion he one wi h he c ack explici ly meshed.
Al hough, as expec ed, i con e ges sligh ly slowe han he enhanced s ain o mula ion
de eloped by Dolbow e al., i ep oduces be e he incomp essibili y condi ion as can
be obse ed in able 4.6. The F-ba me hodology o XFEM app oxima es he incom-
p essibili y condi ion be e han he Assumed Enhanced S ain app oach de eloped by
Chap e 4. The Ex ended Fini e Elemen Me hod 75
Table 4.6: Maximum alue o he de ia ion om he incomp essibili y condi ion,
max |J−1|.
Explici ly Meshed En iched
Elemen s pe side F-ba Enhanced Dolbow e al F-ba o XFEM Enhanced Dolbow e al
5 0.000030 0.0004 0.00048 0.018
10 0.000047 0.0007 0.00047 0.024
20 0.000075 0.0012 0.00046 0.025
30 0.000093 0.0016 0.00044 0.030
-0.214e-5
0.1549e-4
J-1
(a) (b)
Figu e 4.33: a) Maximum alue o he de ia ion om he incomp essibili y condi ion,
(J−1), in he elemen c ossed by he c ack ma ked in he mesh b).
Dolbow e al. Ne e heless, he alue o Jsligh ly depends on he mesh, namely, on he
size o he iangles ob ained du ing he sub cell di ision o in eg a ion. In gene al, he
highe de ia ion om J= 1 occu s a ound he c ack ip, in he smalle sub cells, as
depic ed in igu e 4.33.
Besides he good pe o mance exhibi ed, he F-ba me hod is ela i ely simple o im-
plemen on compu e p og ams, making he p oposed app oach o XFEM sui able o
simula ing c ack p opaga ion in p oblems in ol ing la ge s ains whe e incomp essibili y
plays a majo ule, such as in elas o-plas ic ma e ials.
4.3 In e ac ion wi h non-linea ma e ial models
In he p e ious sec ion, sui able s a egies o nume ical in eg a ion in XFEM elemen s
in he p esence o incomp essibili y cons ain s we e de eloped. Ne e heless, in a duc-
ile ac u e p oblem he c ack pa h is no de ined a p io i bu he discon inui ies a e
Chap e 4. The Ex ended Fini e Elemen Me hod 76
ξ
η
ξi=1
√3
ξi=−1
√3
ηi=−1
√3
ηi=1
√3
Ni=1
4(1 + 3ξiξ)(1 + 3ηiη)
Figu e 4.34: De ini ion o in e pola o y shape unc ions used in he a iable ans e .
a he in oduced a some s age o ma e ial deg ada ion. This means ha he posi-
ions and weigh s o he Gauss poin s o a ce ain ini e elemen may change du ing he
p oblem analysis. As he Lemai e model o damage p esen ed in chap e 2 is his o y
dependen , a p ope a iable ans e s a egy has o be de ined.
He e a local smoo hing echnique, consis ing o a bilinea ex apola ion o he quan i ies
om he exis ing Gauss poin s o he new, using shape unc ions, is employed. This
echnique is used in e e ence [80] o ans e quan i ies s o ed in he Gauss poin s o
he nodes. Basically, any a iable in he new Gauss poin s, αnew, may be ob ained using
he ollowing exp ession:
αnew =
ngp
X
i
Ng
iαi(4.47)
whe e αi ep esen s he alues o he a iable αa each one o he ngp old Gauss poin s.
The in e pola o y shape unc ions a e de ined a he se o old Gauss poin s acco ding
o igu e 4.34.
In he pa icula case o he damage model conside ed, he a iables which e lec he
his o y o he de o ma ion a e he plas ic mul iplie , he s ain and he damage. The
s esses may be eco e ed using he e u n-mapping equa ions.
I should be no iced ha his echnique di e s om emeshing, as only he quan i ies
s o ed in Gauss poin s a e ans e ed inside he same ini e elemen ; he mesh, as well
as he quan i ies s o ed in he nodes, such as he displacemen s, emain unchanged.
Rema k 4.2. Depending on he pa icula p oblem, di e en s a egies o eco e equi-
lib ium, a e he a iable ans e may be equi ed. These s a egies a e de eloped and
discussed in chap e 5.
Chap e 4. The Ex ended Fini e Elemen Me hod 77
lc
lc
Figu e 4.35: G oup o Gauss poin s (in black) ha in luence a pa icula one. When
a Gauss poin is loca ed nea a discon inui y, only he poin s in he same side o he
discon inui y is selec ed.
P1
P2
P3
Figu e 4.36: G oup o Gauss poin s (in black) ha in luence a pa icula one, loca ed
nea a c ack ip.
In chap e 2, a non-local o mula ion o p e en mesh pa hologies, when implemen ing
he damage model was also p esen ed. In his me hodology, he cha ac e is ic leng h,
lc, de e mines which poin s should in luence he his o y o a pa icula poin x( igu e
4.35). Ne e heless, when a c ack is inse ed in he mesh, i s su aces no longe in e ac ,
especially in he case o a ac ion- ee c ack. The e o e, he Gauss poin s nea he
in e ace should only be in luenced by o he Gauss poin s laying in he same side o he
c ack, as illus a ed in igu e 4.35.
I a poin is loca ed close o he c ack ip, di e en selec ion c i e ia may be adop ed.
Conside ing only a no mal le el se unc ion, all he poin s P1, P2 and P3, shown in
igu e 4.36, a e excluded om he g oup o Gauss poin s in luencing he ed poin . In he
o he hand, conside ing an addi ional angen le el se unc ion, P1 may be included in
he se . Despi e he di e en op ions, he in luence in he esul s is nea ly unno iceable.
Chap e 4. The Ex ended Fini e Elemen Me hod 78
4.4 Conclusions
In his chap e he gene al o mula ion o he XFEM was p esen ed, ocusing in he en-
ichmen unc ions sui able o desc ibe a duc ile ac u e p ocess. The basic implemen-
a ion s eps we e ou lined, ollowed by a comp ehensi e s udy o in eg a ion s a egies,
which is one mos ele an pa s o his chap e .
While in mos o he exis ing s udies on in eg a ion ules o XFEM, he accu acy o
he in eg a ion schemes is based on he abili y o ep oduce he s ess in ensi y ac o s
in p oblems ea u ing known analy ical solu ions, he e we a ou ed an app oach based
on he analysis o he unde lying sub-space o incomp essible modes embedded in he
XFEM app oxima ion. Se e al in eg a ion ules we e compa ed in e ms o he de o -
ma ion modes ha can ac ually be cap u ed.
In gene al, conside ing he sub di ision o he elemen s c ossed by a c ack i is possible
o expand he unde lying sub-space o incomp essible de o ma ions using a educed in e-
g a ion ule o e each sub cell, when compa ed wi h ull in eg a ion schemes. This ac
mo i a ed he ex ensions o B-ba and F-ba me hodologies o XFEM, which somehow
ely on he selec i e in eg a ion ules, wi h encou aging esul s.
In he inal pa o he chap e some mo e ea u es equi ed o he combina ion o he
XFEM wi h non-linea ma e ials, namely, he a iable ans e and he non-local in e -
ac ion, we e also add essed.
All hese echniques a e use ul in he s udy o he inal s age o ailu e o ma e ials which
unde go la ge s ain p ocesses wi hou no iceable olume changes, as i will be desc ibed
in he ollowing chap e s.
Chap e 5
Duc ile F ac u e Model:
T ac ion-F ee C acks
Ma e ial ailu e is o en ela ed o he de elopmen o mic o-de ec s a he mic o-
s uc u al le el, in pa icula o damage mechanisms. Damage is cha ac e ized by he
nuclea ion g ow h and coalescence o oids, which esul s in a loss o mechanical p op-
e ies and leads o ac u e [27].
In he case o duc ile me als, he so-called duc ile damage is s ongly ela ed wi h la ge
plas ic s aining. The nuclea ion o oids occu s a e a ce ain h eshold o plas ic s ain
and hei g ow h and coalescence a e go e ned by plas ic ins abili y, as schema ically
illus a ed in igu e 5.1 [27, 69]. This ype o me als is usually cons i u ed by second
phase pa icles o inclusions imme sed in a ma ix ( igu e 5.1(a)). When su icien s ess
is applied, ma ix-pa icle debonding o pa icle c acking may occu , leading o he
nuclea ion o oids ( igu e 5.1(b)). Subsequen ly hese oids may g ow ( igu e 5.1(c))
and, a e eaching a ce ain size, hey s a in e ac ing wi h he neighbou ing oids.
Plas ic s ain ends o concen a e along a p e e en ial di ec ion, causing local necking
ins abili ies ( igu e 5.1(d)). Finally, oids coalesce and a ac u e su ace is c ea ed
( igu e 5.1(e)).
The Lemai e model o duc ile damage p esen ed in chap e 2 is able o desc ibe ma e-
ial deg ada ion by he in oduc ion o a damage a iable, D, which accoun s o he loss
o s i ness due o he p esence o oids in an a e aging way. Ne e heless, his ma e ial
model is unable o cap u e he las ailu e s age co esponding o c ack ini ia ion and
p opaga ion.
79
Chap e 5. Duc ile F ac u e Model: T ac ion- ee C acks 80
(a) (b) (c)
(d) (e)
Figu e 5.1: Duc ile ac u e p ocess: a) inclusions in a me allic ma ix; b) oid
nuclea ion; c) oid g ow h; d) s ain localiza ion and necking be ween oids; e) oid
coalescence and o ma ion o a ac u e su ace.
A comple e duc ile ailu e model may be cons uc ed combining he damage ma e ial
model wi h he XFEM disc e isa ion, in oduced in chap e 3. The s age illus a ed
in igu e 5.1(d) may be associa ed o a c i ical damage alue, Dc, which is de e mined
h ough he con inuous heo y. When his damage le el is eached, a discon inui y su -
ace may be inse ed in he co esponding egion using he XFEM.
The e o e, in he i s sec ion o his chap e a damage based c i e ion o ansi ion om
damage o ac u e is p oposed. The same ideas a e applied o c ack p opaga ion and
he e o e he wo phenomena a e deal wi h in an uni ied way. In e ms o nume ical im-
plemen a ion he con inuous-discon inuous model equi es some u he de elopmen s
discussed in sec ion 5.2. Nex , in sec ions 5.3 and 5.4, he e iciency o he p oposed
model is illus a ed h ough a ious nume ical examples. This chap e inalizes wi h
some gene al conclusions ega ding he ull duc ile ac u e model.
5.1 T ansi ion c i e ion om damage o ac u e
The ansi ion c i e ion om Damage o F ac u e may be simply s a ed as a c ack is
inse ed when c i ical damage is eached in he co esponding egion o he domain. The
Chap e 5. Duc ile F ac u e Model: T ac ion- ee C acks 81
main ques ion is how o de e mine he c ack cha ac e is ics, namely he ini ia ion poin ,
di ec ion and leng h, om he c i ically damaged a ea.
In a p oblem disc e ised h ough he FEM, he alues o he damage a iable a e s o ed
in each Gauss poin , om which he damage dis ibu ion pa e n ollows di ec ly. This
in o ma ion may be used o de e mine he cha ac e is ics o a c ack, bu o mee a good
accu acy and lexibili y an in e pola ion s a egy o calcula e he damage alue a any
poin o he domain mus be de ined.
A simple way o ob ain he damage alue a an a bi a y poin is o use bilinea ex-
apola ion wi h Lag ange polynomials. The damage alue is de e mined making use o
he alues a he Gauss poin s o he elemen in which he poin is con ained, simila ly
o he a iable ans e echnique desc ibed in chap e 4. Ne e heless, o inc ease he
accu acy o he c ack de ini ion i is in e es ing o ga he he in o ma ion o a pa ch
o elemen s and, al e na i ely, he app oxima ion based on B-spline unc ions [140–142]
may be employed.
The B-spline basis unc ions a e de ined ecu si ely om a se o con ol poin s, [ξ1, ξ2, . . . , ξn,
using he ollowing ecu sion o mula [143, 144]
Ni,0(ξ) = (1 i ξi≤ξ≤ξi+1
0 o he wise (5.1)
Ni,p(ξ) = ξ−ξi
ξi+p−ξi
Ni,p−1(ξ) + ξi+p+1 −ξ
ξi+p+1 −ξi+1
Ni+1,p−1(ξ) (5.2)
A B-spline cu e, C, in Rdis buil as a linea combina ion o B-spline basis unc ions as:
C(ξ) =
n
X
i=1
Ni,pPi(5.3)
whe e ideno es he i- h con ol poin and no each one o i s coo dina es. Unlikely La-
g ange polynomials, unc ions cons uc ed in his way a e mono one and consequen ly
do no in oduce new maxima in he dis ibu ion. Mo eo e , a B-spline objec o dimen-
sion dis i sel a B-spline o dimension d−1, which means ha a a iable a a poin a
he bounda y o a B-spline su ace/ olume may be in e pola ed wi h he same unc ions
as a poin lying inside he domain. These p ope ies may b ing impo an ad an ages in
he ea men o a damage dis ibu ion, insu ing ha damage does no g ow a i icially
due o he in e pola ion echnique and allowing an uni ied ea men o c ack ini ia ion
Chap e 5. Duc ile F ac u e Model: T ac ion- ee C acks 88
ollowing sec ion.
5.2.2 Equilib ium eco e y algo i hm
Rega ding nume ical implemen a ion, he equilib ium eco e y equi ed when new ele-
men s a e en iched is he mos c i ical pa o he model. Each ime new elemen s a e
en iched due o c ack ini ia ion and/o p opaga ion, he numbe o deg ees o eedom
inc eases, as well as he numbe o Gauss poin s inside he en iched elemen s and hence
he equilib ium be ween in e nal and ex e nal o ces has o be e-es ablished.
In ac , when an elemen is en iched he posi ions and weigh s o he Gauss poin s change
and consequen ly he in e nal a iables ha e o be ans e ed om he old Gauss poin s
o he new ones, which is done by applying a local smoo hing echnique as explained
in chap e 4. I should be no iced ha his p ocess di e s om emeshing as only he
quan i ies s o ed in he Gauss poin s, namely he plas ic s ain and he damage a iable,
a e ans e ed in a p ocess occu ing only inside each elemen ; he mesh, as well as he
quan i ies s o ed in he nodes emain he same. In a second s age, a load inc emen
wi h null ampli ude is pe o med. Howe e , due o he highly non-linea na u e o he
p oblem, his p ocedu e may be ine icien and a mo e complex algo i hm had o be
de eloped, as desc ibed in he ollowing pa ag aphs.
The key ea u e o equilib ium eco e y is he cons ancy o he nodal quan i ies. When
a discon inui y is in oduced in a pa icula elemen , he con inuous p oblem should be
equi alen o he con inuous-discon inuous p oblem. E en hough he app oxima ion
unc ion o he elemen changed om:
u(x) =
n
X
i=1
Niui(5.10)
o
u(x) =
n
X
i=1
Niui+
nspli
X
j=1
Nj[H(x)−H(xj)]aj(5.11)
in elemen s which became o ally cu by a c ack, o o
u(x) =
n
X
i=1
Niui+
n ip
X
j=1
NjR(x)[H(x)−H(xj)]bj(5.12)
Chap e 5. Duc ile F ac u e Model: T ac ion- ee C acks 89
in elemen s, which became ip elemen s, a e sol ing he sys em o equa ions, he nodal
displacemen s should emain he same. The e o e, when an elemen is abou o be
en iched, i s displacemen s, unen ich, a e s o ed. Then, a e in oducing he discon i-
nui y, he p oblem is sol ed imposing he s o ed displacemen s, unen ich, as p esc ibed
displacemen s, esul ing in a se o esidual o ces, FRes, a he en iched nodes. Sub-
sequen ly, he nodes a e eleased and he p oblem is sol ed o p og essi ely educed
esidual o ces, acco ding o he scheme in igu e 5.6.
To inalize he algo i hm desc ip ion, i should be no iced ha , al hough he c ack
leng h s ep is calcula ed using he me hodology p oposed in sec ion 5.1.2, some imes
i is di ided in se e al s eps o acili a e equilib ium eco e y. Mo eo e , du ing he
analysis ime s epping is usually no cons an . When c i ical damage is eached, he
p og am goes back one ime s ep and con inues wi h a ac ion o he i s ime s ep
acco ding o he scheme in igu e 5.7. In his way he c ack ini ia ion s age is cap u ed
wi h mo e de ail. As c ack p opaga ion in duc ile me als is usually s able i may be
success ully simula ed using a quasi-s a ic o mula ion.
Now ha he ac u e model is ully p esen ed, i s e ec i eness will be illus a ed h ough
a ious nume ical examples, p esen ed in he emaining o his chap e .
Chap e 5. Duc ile F ac u e Model: T ac ion- ee C acks 90
C ack g ow h s ep
Selec nodes o en ich
S o e he displacemen s o newly en iched
nodes, unen ich
Pe o m New on i e a ion wi h null
mul iplie
con e gence
Yes
Con inue
analysis
No
se unen ich as p esc ibed
displacemen s and sol e he
sys em o equa ions
sys em o equa ions
con e gence
Yes
No
Reduce c ack
g ow h s ep
Resul an se o esidual o ces, FRes
Release he imposed displacemen s on he
en iched nodes
Reduce p og essi ely he esidual o ces
un il con e gence is ob ained o FRes < ol
by a ce ain pe cen age α
FRes = (1 −α)FRes
New on i e a ion
con e genceYes No
FRes < ol α≤αmin
Yes
No No
Yes
α=α+αReduce α
Con inue
analysis
Figu e 5.6: I e a i e scheme o equilib ium eco e y du ing a c ack g ow h s ep.
Chap e 5. Duc ile F ac u e Model: T ac ion- ee C acks 91
Figu e 5.7: Time-s epping scheme be o e and a e c ack ini ia ion/p opaga ion.
5.3 Nume ical examples 1: Pla e wi h an ini ial c ack
Be o e e alua ing he o al ac u e model, in ol ing c ack ini ia ion and p opaga ion,
a specimen wi h an ini ial c ack is s udied. This nume ical example is in ended o
e alua e he e ec s o he non-local damage o mula ion, when applied in conjunc ion
wi h he XFEM. In a second phase, damage based c ack p opaga ion is inspec ed as well.
Following his line, his example consis s o a c acked pla e illus a ed in igu e 6.9. The
ma e ial p ope ies and dimensions a e summa ized in able 5.1. Th ee di e en FEM
meshes a e conside ed, as shown in igu e 5.9, unde plane s ain assump ion.
Fo he analysis, besides he ma e ial pa ame e s de ined in able 5.1, he p oposed
model equi es a alue o he non-local egula iza ion leng h, l . This pa ame e canno
be measu ed di ec ly om expe imen s bu is usually ob ained h ough in e se analy-
sis [77, 79]. As he p esen ed non-local o mula ion has p o en o be able o alle ia e
pa hological mesh dependence, l may be ega ded as a nume ical egula iza ion pa am-
e e o be p esc ibed by he use . The e o e, h ee di e en si ua ions, co esponding
o h ee alues o l a e analysed: l = 0 ( he local case), l = 0.8mm and l = 1.6mm.
The damage con ou s ob ained in each case, o an applied displacemen o 0.05mm, a e
Chap e 5. Duc ile F ac u e Model: T ac ion- ee C acks 92
a
b
2h
Figu e 5.8: P e-c acked pla e
Table 5.1: Ma e ial p ope ies o he pla e wi h an ini ial c ack.
P ope y Value
Elas ic modulus E= 206.9 GPa
Poisson’s a io ν= 0.29
Damage exponen s= 1.0
Damage denomina o = 1.25 MPa
Ha dening unc ion τy(R) = 450 + 129.24R+ 265(1 −e−16.93R) MPa
Dimension h= 15 mm
Dimension b= 20 mm
Ini ial c ack leng h a= 20/3 mm
displayed in igu es 5.10 - 5.12.
Resul s demons a e a clea mesh dependence pa hology in he local case. Ins ead o
p o iding be e esolu ion in he c ack ip icini y, mesh e inemen causes spu ious
damage localiza ion.
As expec ed, he non-local o mula ion alle ia es he mesh dependence, as can be ob-
se ed in igu es 5.11 and 5.12, whe e he damage con ou s do no a y signi ican ly in
he di e en meshes. Ne e heless, o l = 0.8mm, he e is s ill some excessi e damage
concen a ion a he c ack ip, which sugges s ha he la ge l is a be e choice. In
ac , when analysing he c ack g ow h, l = 0.8mm will p o e o be insu icien . As a
inal no e on he use o he Lemai e model unde a non-local o mula ion wi h discon in-
uous c acks, i should be no iced ha he maximum damage alues egion is connec ed
Chap e 5. Duc ile F ac u e Model: T ac ion- ee C acks 93
(a) (b) (c)
Figu e 5.9: Mesh e inemen o he pla e wi h an ini ial c ack: a) 20 ×31, b) 40 ×
61 and c) 50 ×75 elemen s.
(a) (b) (c)
Figu e 5.10: Damage con ou s ob ained o he local case, o an applied displacemen
o 0.05mm o hee meshes a) 20 ×31, b) 40 ×61 and c) 50 ×75 elemen s.
(a) (b) (c)
Figu e 5.11: Damage con ou s ob ained o l = 0.8mm, o an applied displacemen
o 0.05mm, o hee meshes a) 20 ×31, b) 40 ×61 and c) 50 ×75 elemen s.
Chap e 5. Duc ile F ac u e Model: T ac ion- ee C acks 94
(a) (b) (c)
Figu e 5.12: Damage con ou s ob ained o l = 1.6mm, o an applied displacemen
o 0.05mm, o hee meshes a) 20 ×31, b) 40 ×61 and c) 50 ×75 elemen s.
wi h he c ack ip, and he e o e co ela es wi h mic o mechanical app oach desc ibed
p e iously in his chap e : he iaxiali y s a e a he c ack ip a ou s he nuclea ion
and g ow h o oids, which e en ually connec o he main c ack causing i s p opaga ion.
Duc ile C ack G ow h The analysis o he damage-based c ack g ow h is now p e-
sen ed. One mo e model pa ame e , he c i ical damage alue, Dc, which igge s he
c ack ad ance mus be de ined. Following he wo k o Lemai e [27], in eal pa s and
s uc u es he c i ical damage alue is no 1, which would co espond o heo e ical ully
damaged ma e ial, bu is a he loca ed be ween 0.2 and 0.5. A his s age, he alues o
0.2 is employed and u he discussion is le o he ull model in ol ing c ack ini ia ion
and p opaga ion.
In his s udy, he same h ee meshes and non-local leng hs we e employed. Resul s a e
depic ed in igu es 5.13 and 5.14, whe e he inal c ack pa hs and damage con ou s may
be obse ed.
Fo he egula iza ion leng h o 0.8mm he e is s ill some damage localiza ion a he
c ack ip. In he coa se mesh ( igu e 5.13(a)) jus a ew Gauss poin s in he icini y o
he c ack ip each he c i ical damage. Consequen ly he con e gence o he i e a i e
scheme is a ec ed du ing he c ack g ow h simula ion. The ine meshes ( igu es 5.13(b)
and 5.13(c)) p o ide mo e esolu ion, howe e only wi h he la e is possible o simula e
he expec ed c ack pa h. This case illus a es one o he disad an ages o he p oposed
Chap e 5. Duc ile F ac u e Model: T ac ion- ee C acks 95
(a) (b)
(c)
Figu e 5.13: C ack pa h ob ained using l = 0.8 mm, o hee meshes a) 20 ×31, b)
40 ×61 and c) 50 ×75 elemen s.
Chap e 5. Duc ile F ac u e Model: T ac ion- ee C acks 96
(a) (b)
(c)
Figu e 5.14: C ack pa h ob ained using l = 1.6 mm, o hee meshes a) 20 ×31, b)
40 ×61 and c) 50 ×75 elemen s.
Chap e 5. Duc ile F ac u e Model: T ac ion- ee C acks 97
Figu e 5.15: Reac ion o ce in unc ion o he applied displacemen l = 1.6mm.
damage c i e ion: due o i s geome ic na u e, an accu a e calcula ion o he c ack di-
ec ion, equi es ha c i ical damage is eached in a ce ain numbe o Gauss poin s.
Ne e heless, his disad an age is usually o e come by selec ing an adequa e non-local
leng h, as i will be demons a ed in he emaining examples.
In pa icula , o he leng h o 1.6mm, c ack g ow h can be simula ed success ully o
all he meshes. Ne e heless, when he specimen is close o comple e up u e, some
nume ical ins abili ies may a ise causing a sligh change o he c ack di ec ion.
In igu e 5.15 he eac ion o ce-applied displacemen cu es a e displayed. The cu es
a e con e gen upon mesh e inemen , which demons a es he e iciency o he p oposed
c ack g ow h c i e ion, based on damage e olu ion.
To inalize his example, he c ack leng h e olu ion is depic ed in igu e 5.16. The
Chap e 5. Duc ile F ac u e Model: T ac ion- ee C acks 104
Figu e 5.25: Rela ion be ween he esidual o ce and he displacemen applied o
he op edge o he no ched specimen o he mesh wi h 31 nodes pe side, conside ing
l = 2.0 mm and di e en alues o DC.
calib a ion o he p oposed me hodology is dependen on he cha ac e iza ion o he
con inuum model pa ame e s, ega dless o he good pe o mance exhibi ed so a .
5.4.2 Plane s ain specimen
In his example, he pe o mance o he p oposed me hodology unde he plane s ain
condi ion is es ed using he specimen depic ed in igu e 5.26.
Figu e 5.26: Plane s ain specimen.
Fo his analysis, he in e nal leng h, l has he alue o 1.6 mm and he c i ical damage
alue is 0.5. The analysis is pe o med o he ou meshes ep esen ed in igu e 5.27,
p esc ibing adequa e symme y condi ions and applying a displacemen o he op and
Chap e 5. Duc ile F ac u e Model: T ac ion- ee C acks 105
(a) (b) (c) (d)
Figu e 5.27: Mesh e inemen o he plane s ain specimen, mesh densi y o a) 11,
b) 21, c) 31, d) 41 elemen s pe side.
bo om edges.
In igu e 5.28 he eac ion o ce-displacemen cu es ob ained a e ep esen ed. As in
he p e ious example, all he s ages o ma e ial beha iou , including ha dening and
so ening up o ailu e, a e clea ly ep esen ed. Resul s a e also con e gen upon mesh
e inemen .
Unde a plane s ain condi ion, he non-local in eg al model also a oids pa hological
mesh dependence and spu ious damage localiza ion, as may be obse ed in he damage
dis ibu ion con ou s illus a ed in igu e 5.29.
Finally, in igu e 5.30 di e en c ack g ow h s eps o he mesh wi h 41 elemen s pe
side a e illus a ed, illus a ing, once mo e, he close ela ion be ween damage e olu ion
and c ack g ow h in duc ile me als.
Chap e 5. Duc ile F ac u e Model: T ac ion- ee C acks 106
Figu e 5.28: Reac ion o ce as a unc ion o he applied displacemen o he op nodes
o he plane s ain specimen.
(a) (b) (c)
Figu e 5.29: Damage con ou and inal c ack o mesh a) 11, b) and 21 c) 31 elemen s
pe side.
5.4.3 Double no ched specimen
The main objec i e o his example is o compa e he c ack pa h ob ained wi h di e en
meshes. The p e ious examples a e no ully illus a i e, as he c ack always p og esses
in a s aigh line. Fo ha pu pose, a double no ched specimen, as ep esen ed in igu e
5.31, is chosen and is loaded so ha a shea -like ailu e mode would occu . The analysis
is pe o med o he wo meshes ep esen ed in igu e 5.32, conside ing a c i ical damage
alue o 0.5, a egula iza ion leng h o 1.6 mm and applying a displacemen o 1.5 mm.
Chap e 5. Duc ile F ac u e Model: T ac ion- ee C acks 107
(a) (b)
(c) (d)
Figu e 5.30: Di e en c ack g ow h s eps o he mesh wi h 41 elemen s pe side.
c= 1.0 mm
1= 2.0 mm
2= 2.5 mm
a= 10 mm
Figu e 5.31: Shea Specimen
Chap e 5. Duc ile F ac u e Model: T ac ion- ee C acks 108
(a) (b)
Figu e 5.32: FEM meshes used in he double no ched specimen: a) mesh wi h 16
nodes pe side, b) mesh wi h 23 nodes pe side.
(a) (b)
Figu e 5.33: C ack pa h ob ained o a) mesh a, b) mesh b.
In igu e 5.33 he c ack pa hs ob ained o he wo di e en meshes a e ep esen ed. I
can be obse ed ha he pa hs a e nea ly he same o bo h cases, ha is, he c ack
pa h can be conside ed as mesh independen .
The non-local in eg al model a oids spu ious localiza ion o damage, esul ing in simila
damaged a eas o di e en mesh e inemen s ( igu e 5.34). The subsequen inse ion o
he c ack h ough he XFEM espec s hese damage con ou s independen ly om he
mesh as well, indica ing ha his me hodology is adequa e o p edic ailu e in specimens
o a bi a y shape.
Chap e 5. Duc ile F ac u e Model: T ac ion- ee C acks 109
(a) (b)
Figu e 5.34: Damage con ou and inal c ack o mesh a) and mesh b).
5.5 Conclusions
Phenomenologically, he ini ia ion o a c ack in duc ile me als is connec ed o he e o-
lu ion o damage, which may be desc ibed by a con inuum model. Ne e heless, o
a comple e desc ip ion o he ailu e p ocess, a discon inui y should be inse ed in he
domain, once a c i ical damage alue is me .
In his chap e , a model able o ep esen all he s ages o ma e ial beha iou was
p esen ed. I was shown ha duc ile ac u e may be success ully deal wi h by combin-
ing he XFEM wi h a plas ic-damageable ma e ial model. Fu he mo e, he p oposed
me hodology ea u es he ollowing ad an ages:
. C ack cha ac e is ics a e de e mined di ec ly om he con inuous model and he e o e
he e is no need o iden i ying addi ional ma e ial pa ame e s o ac u e. Mo eo e ,
c ack ini ia ion and p opaga ion a e deal wi h in an uni ied way.
. C ack ini ia ion locus does no need o be known in ad ance and c ack p og ession is
independen o he mesh, sa ing compu a ional esou ces when compa ed wi h com-
pe ing app oaches such as emeshing.
. Resul s a e mesh independen upon a ce ain mesh e inemen .
. The Lemai e model in i s non-local in eg al o mula ion ensu es simila damage pa -
e ns in coa se and ine meshes. The e o e c ack ad ance eloci y and c ack pa e ns
a e also accu a ely de e mined, e en in coa se meshes.
Chap e 5. Duc ile F ac u e Model: T ac ion- ee C acks 110
The possible d awbacks o he p oposed me hodology a e ela ed o pa ame e iden-
i ica ion, such as he non-local in insic leng h and he c i ical damage alue, which
may be de e mined by in e se analysis using a p ope combined nume ical-expe imen al
app oach.
F om he heo e ical poin o iew, he e is an ene gy gap when pe o ming a ansi ion
om damage o a ac ion- ee c ack a damage alues lowe han 1 and he e o e he
in oduc ion o a cohesi e c ack may ega ded as an imp o emen o he model and will
be add essed in he ollowing chap e . Ne e heless, we belie e ha he app oxima ion
de eloped in his chap e is good enough o desc ibe duc ile ac u e in a wide ange o
indus ial p ocesses.
Chap e 6
Duc ile F ac u e Model: Cohesi e
C acks
In he p e ious chap e a model o duc ile ac u e, go e ned by damage e olu ion, was
p esen ed. When damage eaches a c i ical alue a c ack is ini ia ed and subsequen ly
p opaga es ollowing he damage pa e n.
In heo y, o ensu e he modynamical consis ency in he model in use, he ansi ion
om damage o ac u e should occu when he ma e ial is ully deg aded, ha is, o
Dc= 1. A his poin he damage ene gy elease would be equi alen o he ene gy
necessa y o c ea e a c ack su ace [145]. Howe e , se ing he damage alue o 1 leads
o a singula i y in he con inuum equa ions, wi h i s na u al nume ical consequences.
In he p e ious chap e i was also ou lined ha expe imen ally he c i ical damage
alue is, in gene al, loca ed be ween 0.2 and 0.5,and he e o e he e is s ill an ene gy
gap which should be ul illed.
One possible solu ion o his p oblem is o add a cohesi e law o he model. In he
p e ious model i was conside ed ha once a c ack is in oduced, he newly o med
su aces no longe in e ac , ha is, he c acks we e conside ed ac ion- ee. A cohesi e
law is a ac ion-displacemen ela ion which models he in e ac ion be ween he wo
su aces o a c ack [12, 13, 22, 146, 147]. F om he mic o-mechanical poin iew, i may
be in e p e ed as ma e ial which is no ully damaged and, consequen ly he e is s ill
some connec ion be ween he wo c ack su aces, as schema ically illus a ed in igu e
111
Chap e 6. Duc ile F ac u e Model: Cohesi e C acks 112
(a) (b)
Figu e 6.1: a) T ac ion- ee c ack b) Cohesi e c ack.
6.1.
In e ms o mac o-modelling, he ansi ion om damage o ac u e o c i ical damage
alues lowe han one could be compensa ed by he cohesi e law. This concep has been
p e iously applied by Cazes e all. [58, 59], in a o mula ion ha does no equi e a p e-
de ined shape o a cohesi e law bu in which he loca ion o he c ack mus be known
in ad ance. As addi ional disad an age, he model was only ully de eloped o 1D cases.
In his wo k, a shape o he cohesi e law will be assumed bu i s pa ame e s will be
i ed ollowing ene ge ic conside a ions. The e o e, his chap e ini ia es wi h a b ie e-
iew o o he basic equa ions and implemen a ion o a cohesi e law. Then he in luence
o he cohesi e zone in he model de eloped in chap e 5 is in es iga ed. Encou aged
by he esul s o his s udy, a s a egy o achie e an ene ge ically consis en ansi ion
om damage o ac u e is p oposed and illus a ed h ough some nume ical examples.
Finally, he main conclusions o he de eloped app oach a e ou lined.
6.1 Basic equa ions and implemen a ion o a cohesi e law
6.1.1 Va ia ional o mula ion
In a domain con aining a cohesi e c ack, he s ess ield mus be ela ed no only o
he ex e nal loading bu also o he cohesi e ac ions, which a e ac i e in he cohesi e
zone. An addi ional condi ion o he cohesi e in e ace mus be sa is ied oge he wi h
he equilib ium equa ion, as desc ibed in he ollowing pa ag aph.
Chap e 6. Duc ile F ac u e Model: Cohesi e C acks 113
∂Ωcoh
Ω
∂Ω
∂Ωu
(a)
+
con
+
co
−
co
−
con
n+
n−
(b)
Figu e 6.2: a) Body con aining a cohesi e c ack. b) De ail o he cohesi e c ack
Conside ing he body con aining a cohesi e c ack, ep esen ed in igu e 6.2, he equilib-
ium equa ion o he s a ic case is gi en by:
di xσ= 0 in Ω (6.1)
and he applied ac ions, ¯
, mus sa is y
¯
=σn a ∂Ω (6.2)
In he cohesi e zone, conside ing no mal and angen ial in e ac ions as ep esen ed in
igu e 6.2(b), he cohesi e ac ions, −
co and +
co, a e gi en by:
+
con =σn+=− −
con =σn−(6.3)
and
+
co = ( +
con + +
co ) = − −
co =−( −
con + −
co ) (6.4)
whe e n−and n+a e he no mal ec o s o he c ack su aces, as illus a ed in igu e
6.2(b). The cohesi e ac ions a e conside ed o be a unc ion o he c ack opening, ω,
de ined as:
ω= (u−−u+) a ∂Ωn(6.5)
Equa ion 6.5 may be spli in a no mal componen and a angen ial componen as ollows:
ωn=ω·n
ω =ω· (6.6)
whe e is a ec o pe pendicula o nand, consequen ly:
con = con(ωn)
co = co (ω )(6.7)
Chap e 6. Duc ile F ac u e Model: Cohesi e C acks 120
a
b
2h
(a) (b)
Figu e 6.8: a) Pla e wi h ini ial c ack b) FEM mesh.
Table 6.1: Ma e ial p ope ies o he c acked pla e.
P ope y Value
Elas ic modulus E= 206.9 GPa
Poisson’s a io ν= 0.29
Damage exponen s= 1.0
Damage denomina o = 1.25 MPa
Ha dening unc ion y(R) = 450 + 129.24R+ 265(1 −e−16.93R) MPa
C i ical damage Dc= 0.5
Non-local leng h l = 1.6 mm
ha he e ec s o he cohesi e zone a e no iceable in he inal esponse o he componen .
The in luence o he cohesi e law in he eac ion o ce-applied displacemen cu e o he
pla e, p io o c ack g ow h is ep esen ed in igu es 6.9 and 6.10. The model exhibi s
a highe sensi i i y o Fn han o ω0. As expec ed, inc easing he alue o Fn he s ain
unde he eac ion o ce-applied displacemen cu e inc eases signi ican ly. On he o he
hand, i e ms o ω0, he e ec s o he cohesi e law nea ly anish o ω0>1 mm−1, as
om his alue he slope o he cohesi e law becomes app eciable and, consequen ly he
c ack becomes nea ly ac ion- ee.
This p elimina y s udy is inalized inspec ing he c ack g ow h phase. The e ec o he
cohesi e law in he eac ion o ce-applied displacemen cu e and in he c ack leng h
e olu ion a e displayed in igu es 6.11 and 6.12, espec i ely. I can be obse ed ha he
cohesi e law delays sligh ly he p opaga ion o he c ack and inc eases he global s ain
ene gy, which sugges s ha he cohesi e law could ha e he same e ec as inc easing
Chap e 6. Duc ile F ac u e Model: Cohesi e C acks 121
Figu e 6.9: In luence o Fnin he esponse o he c acked pla e o cons an ω0=
0.5 mm−1.
(a)
(b)
Figu e 6.10: In luence o ω0in he esponse o he c acked pla e o cons an a)
Fn= 200 MPa b) Fn= 100 MPa.
Chap e 6. Duc ile F ac u e Model: Cohesi e C acks 122
Figu e 6.11: In luence o a cohesi e law in he eac ion o ce du ing c ack p opaga ion.
Figu e 6.12: In luence o a cohesi e law in he e olu ion o he c ack leng h.
he c i ical damage alue.
The main objec i e o his p oblem was o check he sensibili y o he damage model
o he cohesi e law be o e mo ing o he ansi ion p oblems. Howe e , he impo ance
gi en o he magni ude o he cohesi e pa ame e s is ela i e, as in his wo k hey a e
ega ded as nume ical pa ame e s, a he han ma e ial cha ac e is ics.
Chap e 6. Duc ile F ac u e Model: Cohesi e C acks 123
(a) (b)
Figu e 6.13: a) Plane s ain specimen b) FEM mesh.
6.2.2 T ansi ion om damage o ac u e
In he p e ious example i was possible o ge a i s es ima e o he magni ude o
he cohesi e law pa ame e s: ω0≤1 mm−1and Fn≈(1 −Dc)τy. Now an example
conside ing he ull model o ansi ion om damage o ac u e is inspec ed. I consis s
o a plane s ain specimen, illus a ed in igu e 6.13, which was p e iously analysed
in he con ex o ac ion- ee c acks. In he same igu e i is ep esen ed he ini e
elemen mesh employed, which has a 2205 elemen s. The ma e ial p ope ies a e he
same used in he p e ious example, wi h he excep ion o he c i ical damage alue and
he pa ame e s o he cohesi e law.
In he i s analysis he c i ical damage has he alue o Dc= 0.4. The eac ion o ce-
applied displacemen cu e o a ac ion- ee c ack is compa ed wi h he one o a cohesi e
c ack. The cohesi e law pa ame e s a e Fn= 200 MPa and ω0= 0.5 mm−1. Resul s a e
displayed in igu e 6.14.
The cohesi e law ensu es a smoo he ansi ion om damage o ac u e and a smoo he
c ack p opaga ion, as a esul he s ain ene gy unde he eac ion o ce-applied displace-
men cu e inc eases.Compa ing he esul s wi h he case o a ac ion- ee c ack whe e
Dc= 0.5, i can be obse ed in igu e 6.15 ha he cu e co esponding o Dc= 0.5 may
Chap e 6. Duc ile F ac u e Model: Cohesi e C acks 124
Figu e 6.14: Reac ion o ce-applied displacemen cu es.
Figu e 6.15: Reac ion o ce-applied displacemen cu es and espec i e in luence o
he cohesi e law.
be app oached by he cu e Dc= 0.4 enhanced by he cohesi e law. This ac sugges s
ha he cu e co esponding o he ac ion ee c ack Dc= 1 could be app oached by
se ing he cohesi e pa ame e s a he igh alues.
To inalize his example, he c ack e olu ion is illus a ed in igu e 6.16, howe e , he
e ec o he cohesi e law is no so signi ican . The c ack g ow h speed is s ill highe
o Dc= 0.4 combined wi h he cohesi e law han o Dc= 0.5. Depending on he
pa icula applica ion, i may be mo e impo an o ha e an accu a e p edic ion o he
load le el/applied displacemen a which he ailu e o he componen happens o an
Chap e 6. Duc ile F ac u e Model: Cohesi e C acks 125
Figu e 6.16: In luence o he cohesi e law in he e olu ion o he c ack leng h.
accu a e p edic ion o he c ack leng h e olu ion. The impo an conclusion is ha
simula ions wi h high c i ical damages may be app oached by simula ions wi h lowe
c i ical damages, combined wi h a cohesi e law. The e o e, in he nex sec ion an ene gy
ela ion be ween he ac ion- ee model and he cohesi e model will be p oposed, wi h
he objec i e o i ing he cohesi e law pa ame e s. By se ing hese pa ame e s o he
igh alues, i will be possible o app oach he ansi ion om damage o ac u e a
Dc= 1.
6.3 Ene gy balance and calib a ion o he pa ame e s o
he cohesi e law
The ea ly c ack p opaga ion models, based in LEFM concep s, assume ha he e is a
ce ain amoun o dissipa ed ene gy, which is ma e ial-speci ic and is esponsible o a
ce ain c ack ex ension. F ac u e is igge ed by a single pa ame e such as he G i i h
ac u e ene gy o he s ess in ensi y ac o . E en ex ensions o his heo ies o include
c ack ip plas ici y s ill ely on a single pa ame e and conside ha he o al amoun
o dissipa ed ene gy in a de o ma ion p ocess is used o p opaga e c acks [147, 152].
In ma e ials which exhibi subs an ial de o ma ions p io o c ack g ow h a leas wo
ene gy consuming p ocesses should be conside ed, one ela ed o plas ic de o ma ion
and ano he ela ed o p og essi e ma e ial deg ada ion, ha is, ela ed o damage. In
his con ex , Maza s and Pijaudie -Cabo [145] de eloped he equi alen c ack concep ,
Chap e 6. Duc ile F ac u e Model: Cohesi e C acks 126
in which he damage ene gy elease a e is equi alen o he c ack su ace ene gy, as
ollows: ZV−Y˙
DdV =−GF˙
A(6.19)
In equa ion 6.19 Vis he o e all olume o he s uc u e, Yis he damage ene gy elease
a e (as de ined in chap e 2), Dis he damage a iable, GFis he ac u e ene gy and
Ais he a ea o he c ack. One o he main limi a ions o his app oach is ha equi es
he alue o he ac u e ene gy, which is a pa ame e de i ed om he LEFM and i s
applicabili y in a duc ile ac u e con ex may be ques ionable.
Al e na i ely, Cazes e al. [58, 59], p opose a o mally iden ical exp ession, whe e GFis
eplaced by he a ea unde he ac ion cu e o a cohesi e model. The equi alence o
damage o ac u e is also pe o med locally and he model is only ully de eloped o
1D p oblems.
The wo k de eloped in his chap e is concep ually ela ed o hese app oaches. In ac ,
he esul s p esen ed so a indica e ha a cohesi e law may ac ually compensa e o
a ansi ion om damage o ac u e be o e o c i ical damage alues lowe han one.
Ne e heless, in opposi ion o he desc ibed models, we belie e ha he ene gy balance
be ween damage and ac u e should be e alua ed in a global way. A local balance may
be ambiguous in e ms o he elemen s which should con ibu e o he ene gy balance:
all he elemen s which eached c i ical damage in a ce ain egion, o only he elemen s
which will con ain he c ack? In addi ion, he di ec ene gy ans e om he damaged
olume o he c ack su ace admi s ha he mic o-mechanical damage p ocesses in ha
olume s op e ol ing and ha all he ene gy s o ed in he mic o-s uc u e is ansmi ed
o he dominan mac o-c ack, which is no necessa ily ue.
He e, ha ing in mind ha he s ain ene gy may be gi en by
Γ = ZX(Ω)
σ:dd (6.20)
a unc ion, Γ = Γ(Dc), will be cons uc ed o ac ion- ee c acks, in o de o app oxi-
ma e he alue o Γ(Dc= 1). Then, depa ing om a ce ain alue o Dc<1, a cohesi e
law will be added in o de o mee he condi ion:
Γ(Dc<1 + cohesi e law) = Γ(Dc= 1) (6.21)
Chap e 6. Duc ile F ac u e Model: Cohesi e C acks 127
In he nex sec ion, he e iciency o his me hodology will be illus a ed h ough some
nume ical examples.
6.4 Nume ical examples
6.4.1 Plane s ain specimen
This sec ion s a s by going back o he plane s ain specimen o sec ion 6.2.2 and analyse
he s ain ene gy in ol ed in each one o he cases p esen ed: ac ion- ee c ack wi h
Dc= 0.4, ac ion- ee c ack wi h Dc= 0.5 and cohesi e c ack wi h Dc= 0.4. The
espec i e alues o he s ain ene gy a e displayed in able 6.2.
Table 6.2: S ain ene gy o Dc= 0.4−0.5.
Case S ain ene gy - J
ac ion- ee Dc= 0.4 101.4
cohesi e Dc= 0.4 121.7
ac ion- ee Dc= 0.5 121.8
In e ms o ene gy balance, he in oduc ion o a cohesi e law is indeed equi alen o
ise he c i ical damage alue. To emphasise his ac , he in luence o he cohesi e
law o Dc= 0.6 and Dc= 0.7 is illus a ed in igu e 6.17. The espec i e alues
o he s ain ene gy may be ound in able 6.3. Choosing he alues Fn= 80 MPa
and ω0= 1.0 mm−1, he cu e co esponding o a ac ion- ee case wi h Dc= 0.7 is
e y well app oxima ed. In e ms o ene gy balance, he choice Fn= 150 MPa and
ω0= 0.5 mm−1also p oduces good esul s, bu he c ack p opaga ion phase is o e -
smoo hed, sugges ing ha app oxima ions using lowe alues o he s ain ene gy a e
p e e able o app oxima ions using highe alues o he s ain ene gy.
Table 6.3: S ain ene gy o Dc= 0.6−0.7.
Case S ain ene gy - J
ac ion- ee Dc= 0.6 132.4
ac ion- ee Dc= 0.7 139.6
cohesi e Dc= 0.6, Fn= 80MPa, ω0= 1mm−1136.4
cohesi e Dc= 0.6, Fn= 150MPa, ω0= 0.5mm−1140.1
Chap e 6. Duc ile F ac u e Model: Cohesi e C acks 128
Figu e 6.17: Reac ion o ce-applied displacemen cu es and espec i e in luence o
he cohesi e law.
Figu e 6.18: S ain ene gy as a unc ion o he c i ical damage. Co ela ion ac o :
R2= 0.985.
To achie e he ene ge ically consis en ansi ion om damage o ac u e, one mus
de e mine he s ain ene gy o Dc= 1. In igu es 6.18 and 6.19 he s ain ene gy as
a unc ion o he c i ical damage alue is ep esen ed, oge he wi h wo al e na i e
i ings: an exponen ial unc ion o he ype y=aebx and a polynomial unc ion as
y=ax3+bx2+cx +d. The coe icien s de ining each unc ion we e de e mined using
he leas -squa es me hod. Using he exponen ial unc ion he s ains ene gy alue is
Γ(Dc= 1) = 188.7 J, while using he polynomial unc ion Γ(Dc= 1) = 186.4 J. The
p oximi y o he wo alues sugges s ha he eal alue o Γ(Dc= 1) should be o his
o de o magni ude. The e o e, depa ing om a c i ical damage alue o Dc= 0.8, a
cohesi e law was added in o de o mee Γ(Dc= 0.8 + cohesi elaw)≈187 J.
Chap e 6. Duc ile F ac u e Model: Cohesi e C acks 129
Figu e 6.19: S ain ene gy as a unc ion o he c i ical damage. Co ela ion ac o :
R2= 0.967.
Figu e 6.20: Reac ion o ce-applied displacemen cu es and espec i e in luence o
he cohesi e law.
The bes i is ob ained se ing he pa ame e s o he cohesi e law o Fn= 110MPa and
ω0= 0.01mm−1. The s ain ene gy ob ained is Γ(Dc= 0.8 + cohesi elaw) = 185.1 J,
which is qui e close o he expec ed alue. In igu e 6.20 i is possible o obse e he
di e ence be ween he ac ion- ee and he cohesi e eac ion o ce-applied displacemen
cu es.
Th oughou his chap e i has been men ioned ha a cohesi e law inc eases he o-
al s ain ene gy unde he eac ion o ce-applied displacemen cu e, which has been
e i ied h ough a ious nume ical examples. Howe e , in he con ex o his wo k, he
cohesi e law pa ame e s a e no a cha ac e is ic o he ma e ial, bu nume ical pa am-
e e s which adap o he c i ical damage alue. Physically hey may be in e p e ed as
