Mobili y and Enginee ing Resea ch (MER)
Volume 1/2018
A compu a ional ool o in es iga ions on he
li e ime es ima ion o mul i-di ec ional lamina es
unde mul iaxial s ess s a es based on
laye wise s uc u al analysis
Ma c Mölle , Jochen Blau ock, Ge ha d Ziegmann, Al ons Esde s
A compu a ional ool o in es iga ions on he li e ime es ima ion o
mul i-di ec ional lamina es unde mul iaxial s ess s a es based on laye wise
s uc u al analysis
Ma c Mölle a,∗, Jochen Blau ocka, Ge ha d Ziegmannb, Al ons Esde sc
aIns i u e o Au omo i e Enginee ing, TH Köln, Be zdo e S aße 2, 50679 Cologne, Ge many
bIns i u e o Polyme Ma e ials and Plas ics Enginee ing, TU Claus hal, Ag icolas aße 6, 38678 Claus hal-Zelle eld, Ge many
cIns i u e o Plan Enginee ing and Fa igue Analysis, TU Claus hal, Leibnizs aße 32, 38678 Claus hal-Zelle eld, Ge many
Abs ac
In he p esen pape a calcula ion ool o he li e ime p edic ion o composi e ma e ials wi h ocus on local mul iaxial
s ess s a es and di e en local s ess a ios wi hin each lamina is de eloped. The app oach is based on epe i i ,
p og essi e in-plane s ess calcula ions using classical lamina e heo y wi h subsequen analysis o he ma e ial
s essing e o and use o app op ia e ma e ial deg ada ion models. The e o e expe imen ally da a o S-N cu es a e
used o gene a e anis opic cons an li e diag ams o a close examina ion o c i ical ac u e planes unde any gi en
combina ion o local s ess a ios. The model is e i ied agains a ious balanced angle plies and mul i-di ec ional
lamina es wi h a bi a y s acking sequences and a ying s ess a ios h oughou he analysis. Di e en sec ions o he
model, such as esidual s eng h and esidual s i ness, a e examined and e i ied o e a wide ange o load cycles. The
ob ained esul s ag ee e y well wi h he analyzed expe imen al da a.
Keywo ds: Composi e, Mul iaxial a igue, Li e ime p edic ion, Mul i-di ec ional Lamina e, Plywise modeling
1. In oduc ion
The g owing awa eness o clima e change and i s e-
cogni ion in many di e en economic sec o s and, as-
socia ed he ewi h, he inc easing need o s ong ligh -
weigh ma e ials makes he use o composi es become
mo e and mo e impo an o e a wide ange o s uc-
u al pa s. Thus he e is a s ong demand o e icien
design and he con inuous imp o emen o ma e ial u i-
liza ion in he ield o ib e- ein o ced plas ics (FRP).
Du ing ope a ion, mos s uc u al componen s a e sub-
jec ed o mul iaxial and cycling loads, which make a
close look o he a igue ma e ial beha io o compo-
si es e en mo e necessa y. To es ima e he li e ime o
a ious ib e- ein o ced plas ic lamina es, wi h a bi a-
y s acking sequences unde complex mul iaxial s ess
s a es, a ma hema ical ool based on plywise-modelling
is de eloped. The phenomenological compa abili y be -
∗Co esponding au ho
Email add esses: [email p o ec ed] (Ma c Mölle ),
[email p o ec ed] (Jochen Blau ock),
[email p o ec ed] (Ge ha d Ziegmann),
[email p o ec ed] (Al ons Esde s)
ween p og essi e ma e ial ailu e due o quasi-s a ic
loads and ailu e due o a igue loads, is he eason why
he model is based on ex ending he classical lamina e
heo y (CLT) o a igue loads. Ex ensi e esea ch in he
ield o composi e a igue has been ca ied ou , bu , as
desc ibed in sec ion 2, he e is no o e all app oach o
ool o in es iga ions on mul iaxial a igue loads and
only ew lamina es ha e been p e iously examined in
mechanis ic models unde mos ly only ension- ension
loads. To p o ide an ins umen o u he in es iga-
ions on modeling a igue ma e ial beha io o se e al
layups unde a ying load condi ions, di e en isola ed
s a e-o - he-a models a e conside ed and pu oge he
in a compu a ional ool w i en in py hon. The de elo-
ped model consis s o mul iple indi idualelemen s, such
as o mula ion o S-N cu es, he way o in e pola ing
be ween s ess a ios ia cons an li e diag ams, choice
o (quasi-s a ic) ailu e c i e ia, o mula ions o embed-
ded lamina, a concep o sh inking ailu e en elope due
o di e en combina ions o ans e se and shea s ess
a ios and o cou se s i ness as well as s eng h deg a-
da ion models. Conside ing he le el o complexi y and
he amoun o indi idual conce ns wi hin he model, i
Nomencla u e
Ek(ˆ
E1)Elas ic modulus in k (1)- di ec ion ˆσmGlobal mean s ess o lamina e
E⊥(ˆ
E2)Elas ic modulus in ⊥ (2)- di ec ion ˆσaGlobal s ess ampli ude o lamina e
G⊥k (ˆ
G12)In plane shea modulus σkm, σ⊥m, τ⊥kmLamina mean s esses
ν⊥k (ˆν21)Majo Poisson’s a io σka, σ⊥a, τ⊥kaLamina s ess ampli udes
νk⊥ (ˆν12)Mino Poisson’s a io RσkS ess a io o σk- s esses
ν ⊥k Majo Poisson’s a io o ib e Rσ⊥S ess a io o σ⊥- s esses
E kPa allel elas ic modulus o ib e Rτ⊥k S ess a io o τ⊥k- s esses
mσ Magni ica ion ac o RS ess a io o global s esses
ϕFib e olume ac ion Qij (Qij)Lamina( e) s i ness ma ix
σk(ˆσ1) No mal s ess in k (1)- di ec ion Aij Shell s i ness ma ix
σ⊥(ˆσ2) No mal s ess in ⊥ (2)- di ec ion Bij Shell-pla e in e ac ion ma ix
τ⊥k (ˆτ21) In-plane shea s ess Dij Pla e s i ness ma ix
X (ˆ
X )Pa allel ension s eng h eS ess exposu e ac o
Xc(ˆ
Xc)Pa allel comp ession s eng h wWeakening ac o o in luence o σk
Y (ˆ
Y )T ans e se ension s eng h p+
⊥k,p−
⊥k Incline o ailu e en elope a σ⊥=0:
Y0
(In-si u) T ans e se ension s eng h „+ “ o σ⊥>0and „- “ o σ⊥<0
o embedded lamina p+
⊥⊥,p−
⊥⊥ Incline o ailu e en elope a σn=0:
Yc(ˆ
Yc)T ans e se comp ession s eng h „+ “ o σ⊥>0and „- “ o σ⊥<0
S⊥k (ˆ
S21)In plane Shea s eng h RA
⊥⊥ F ac u e esis ance due o τ⊥⊥
S0
⊥k (In-si u) In plane Shea s eng h nx,ny,nyx In e nal line o ces
o embedded lamina mx,my,myx In e nal line momen s
Table 1: Nomencla u o mos impo an symbols in lamina coo dina e sys em ela ed o he p incipal ma e ial axis o he unidi ec ional lamina
(k,⊥,⊥) and in b acke s o symbols in lamina e coo dina e sys em ela ed o he p incipal ma e ial axis o he lamina e (1,2,3)
becomes appa en ha a lo o cohe en expe imen al
da a is needed. Because o he lack o own da a a ha
ime, he model is alida ed by compa ing he esul s
wi h expe imen al da a om he well known Composi e
Ma e ial Fa igue Da abase o he Michigan S a e Uni e -
si y - Depa men o Ene gy in coope a ion wi h Sandia
Na ional Labo a o ies (SNL/DOE/MSU - Da abase) [1]
o i s in es iga ions.
2. A b ie e iew o p og essi e and i e a i e a igue
models o composi es
The basic idea in e ms o a a igue-ex ended classical
lamina e heo y is no hing new and has al eady been
p opaga ed be o e in di e ing ways. In 2000, Shok ieh
and Lessa d [2] p oposed a p og essi e model based on
he a igue ailu e c i e ion by Hashin [3] and deg a-
da ion ules o g adual and sudden change in ma e ial
p ope ies. They e i ied he model wi h expe imen al
da a o symme ic g aphi e/epoxy angle-ply lamina es
([904/04]s,[04/904]sand [+454/−454]s) unde ension-
ension loads wi h cons an s ess a io R=0.1 [4]. A e
he onse o a ce ain ailu e mode he speci ic s i ness-
and s eng h en ies we e se o ze o, while g adual de-
g ada ion is modeled be o e any ailu e occu s. Shok ieh
and Lessa d also discussed he sho coming o he Mi-
ne ’s ule o calcula ions wi h mo e han one block load
o he [+454/−454]s-lamina e. Fo he cons an block
loading he model achie ed good esul s o he a igue
li e. Noll, Magin and Himmel [5] examined, in pa icu-
la , hee ec o anonlinea shea s ess-s ain(τ⊥k, γ⊥k)
- ela ion wi hin a p og essi e composi e a igue model
and he use o he c i ical elemen concep . They used
he ailu e c i e ion by Puck[6] and a s i ness deg ada-
ion model, which educes some en ies in he s i ness
ma ix o a ce ain alue a e he onse o a speci ic ai-
lu e. Thei model was e i ied calcula ing he a igue li e
o a inyles e /u e hane/ca bon quasi-iso opic composi-
e lamina e ([45/0/−45/90]s) unde a iable ampli ude
loading wi h cons an s ess a io R=0.1. The compa i-
son wi h expe imen al da a showed ha he use o linea
s ess-s ain beha io and igno ing s i ness deg ada ion
led o o e es ima ion o a igue li e o he examined la-
mina e. Kennedy, B adaigh and Leen [7] de eloped an
ad anced model wi h he use o Puck’s ailu e c i e ia
and he s-shaped damage model by Mao and Mahade-
an [8] o model s i ness deg ada ion in ib e di ec ion
o a uni cell model wi hin a ini e elemen sub ou-
ine. Resul s we e examined o a quasi-iso opic (QI)
3
e-glass/epoxylamina e([0/90/45/135]s) unde ension-
ensionloadswi hcons an s ess a ioR=0.1. The model
by Kennedy e al. was seen o p edic he a igue li e o
he QI-Lamina e unde R=0.1 e y accu a ely.
In 2016, Neumeis e e al. [9] p esen ed a model o
mul i-di ec ional lamina es, which is based on he ailu-
e mode concep -based s eng h c i e ia by Cun ze [10]
and calcula ion o o al damage wi h a linea damage
accumula ion model. The s i ness deg ada ion ollows
a nonlinea beha io , which is ela ed o he calcula ed
damage, based on [11]. Neumeis e e al. p esen ed i s
esul s o he simula ion o a [±45/90]s-lamina e, bu
he model i sel was no e i ied o any expe imen al da a
wi hin he publica ion.
In 2017, Mejlej, Oso io and Vie o [12] p oposed a p o-
g essi e model o mul i-di ec ional lamina es unde a-
ying s ess a ios. The model is build upon he o al
ene gy based model by Shok ieh [13], he use o Hashin
Failu e C i e ia and he g adual s i ness deg ada ion by
Shok ieh and Lessa d [2]. Mejlej e al. applied a ela ion
o a igue li e and o al inpu ene gy, which is based on
i ing pa ame e o a se o S-N cu es, ega dless o he
s ess a io and ib e o ien a ion. They used he ene gy
ailu e c i e ion om Sandhu [14] o calcula e a igue
li e o a ious ib e angles and s ess a ios. The model
was e i ied agains wo se s o unidi ec ional ([0]12) la-
mina e made o ca bon/epoxy unde bo h posi i e and
nega i e s ess a io. The calcula ion o a igue li e con-
o ms e y well o he expe imen al da a unde a ying
s ess a ios o unidi ec ional ma e ial.
Finally he me hodology and ool de eloped by Vassi-
lopoulos e al. [15] o GFRP lamian es unde complex
s ess s a es is b ie ly men ioned, al hough i does no e-
ally belong o he i e a i e models. The model uses cycle
coun ing me hods o spec um loads, di e en a igue
c i e ia o li e p edic ion, such as e.g. he Fawaz-Ellyin
[16] o Hashin-Ro em C i e ion [17], and he linea Mi-
ne ule o damage summa ion. The model ge s along
wi hou he use o s i ness o s eng h deg ada ion, sin-
ce i calcula es allowable numbe s o cycles o each
block di ec ly using he men ioned a igue ailu e c i e-
ia. Di e ing designs o S-N cu es and cons an -li e-
diag ams we e examined wi hin he model. Ve i ica i-
on was ca ied ou o a iable ampli ude loading on
e-glass/polyes e lamina es ([0/(±45)2/0]s) o al e na-
ing and pulsa ing s ess a ios, ocusing mainly on he
compa ison o used a igue ailu e c i e ia.
In sho , he e a e qui e a ew models o di e en aspec s
o a igue li e o composi es a ailable. The abo e men-
ioned models a e based on p og essi e modeling and
a y widely in e.g. used ailu e c i e ia o in he way o
desc ibing ma e ial deg ada ion o s eng hand s i ness.
Fo u u e esea ch in mul iple segmen s o composi e
a igue, a gene al model is p esen ed below and a calcu-
la ion ool called „Li e ime Es ima ion o Composi es“
(LEoC - Fig. 1) is de eloped using he p og amming
language „Py hon“.
Fig. 1: S a up-sc een o „LEoC“ o in es iga ions on a igue o com-
posi e
3. Tool and Flow cha o a igue ailu e analysis
The simula ions a e pe o med on he lamina le el using
a ep esen a i e olume elemen (RVE) o he lamina e,
which ep esen s he co esponding mechanical p ope -
ieso a s uc u al pa o componen as isualized inFig.
2. Load condi ions in e ms o in e nal line o ces and
momen s a e applied o he RVE. The gene al p og am
y
x
x
y
ζ
m
m
m
n
n
n
xy
y
x
yx
m
xy
xy
Fig. 2: Simula ion o 2-D (plane s ess) ep esen a i e olume elemen
(RVE) o exempla y s uc u al componen on he lamina le el (meso-
scale)
schedule is demons a ed in Fig. 3, wi hou men ioning
which speci ic models a e applied a pa icula poin s in
he code. The eason o his is ha , in p inciple, di e se
models can possibly be applied a e e y main subjec .
The di isions along he dashed lines in Fig. 3, which a e
4
s a
n =1
load case
s a ic ma e ial
p ope ies
quasis a ic
ma e ial expe imen s
lamina e
p ope ies
mul iaxial
s ess s a e
pe laye
cyclic ma e ial
expe imen s
S-N
cu es
Cons an Li e
Diag am
S-N
ex apola ion
ac u e plain
design wi h
ex apola ed S-N
s a ic
ailu e c i e ia
s eng h
deg ada ion
models
s i ness
deg ada ion
FF T,FF C
s i ness
deg ada ion
IFF A
s i ness
deg ada ion
IFF B
s i ness
deg ada ion
IFF C
YYY Y Y
N
NN N
o e all
ailu e?
Y
N
cyclic p ope ies
end
n =?
N
s ess a ios
pe laye
ailu e c i e ia
pa ame e
n = n+1
(upda e)
UD lamina
p ope ies
Sec.5
Sec.4
Sec.6
Sec.7
Sec.8
in-si u s eng h
o embedded
lamina
Fig. 3: low cha o he li e ime es ima ion o ib e ein o ced composi es wi hin LEoC
5
labeled om „Sec. 4“ un il „Sec. 8“, ep esen he ol-
lowing sec ions and he examined a eas o ocus wi hin
his pape :
•In sec ion 4, he necessa y da a om quasi-s a ic
ma e ial es s o ini ial condi ions o lamina e p o-
pe ies is discussed.
•Fu he mo e, inpu needed om cycling ma e ial
es sand he design o cons an li ediag ams, o he
ex apola ion o unknown S-N cu es a a bi a y
s ess a ios, is explained in sec ion 5.
•Using he ex apola ed S-N cu es, he ollowing
s eng h deg ada ion acco ding o he cu en mul-
iaxial s ess s a e is add essed in sec ion 6.
•In sec ion 7, he ailu e analysis is conside ed and
he choice o used ailu e c i e ia jus i ied.
•Models o s i ness deg ada ion a e he onse o
speci ic ailu e modes a e discussed and applied o
he model in sec ion 8.
The ailu e condi ion wi h i s di e en ypes o ailu e
on he ply-le el, as explained in sec ion 7, is checked o
e e y laye in he cu en load cycle and as a as ailu e
did no occu in e e y exis ing ply, he nex load cycle is
applied o he cu en lamina e condi ion. The i e a i e
p ocess is ca ied on un il he lamina e ailed in e e y
ply o he numbe o in ended load cycles is eached.
4. Inpu om quasi-s a ic ma e ial es s
One o he decisi e ini ial condi ions in hese calcula i-
ons a e he s a ic s i ness and s eng h p ope ies a he
ply-le el as demons a ed in Fig.3 - ield sec ion num-
be 1. Necessa y inpu da a o ma e ial „D155“ (Fib e:
Glass ab ic -527 g/m2, Ma ix: O hoph halic Polyes e
„CoRezyn 63-AX-051“) om [1] is exempla ily shown
in able 2. The ini ial s a e o he lamina e in e e y cycle
n is de i ed using he CLT. The s i ness ma ix o each
lamina „k“ in he lamina (local) coo dina e sys em is
ob ained by
Qij,k(n)=
Ek(n)
1−∆(n)
ν⊥k(n)Ek(n)
1−∆(n)0
E⊥(n)
1−∆(n)0
(sym.)G⊥k(n)
(1)
wi h ∆(n)=ν⊥k(n)νk⊥(n)and expe imen al da a om
quasi-s a ic ma e ial es s, as shown in able 2. The e-
duced s i ness ma ix o each lamina in he global coo -
dina e sys em is calcula ed by
Q±ξ
ij,k(n)=Tij Qij,k(n)Tji (2)
EkE⊥G⊥k ν⊥k
[GPa] [GPa] [GPa] [−]
x33.60 8.21 4.48 0.29
s 1.80 0.72 1.11 0.01
ϕ0.44 0.44 0.40 0.44
X XcY YcS⊥k
[MPa] [MPa] [MPa] [MPa] [MPa]
x1012.0 653.0 27.00 122.75 73.2
s 29.0 23.3 1.63 9.91 5.8
ϕ0.44 0.41 0.44 0.44 0.44
Table 2: Exempla y s a ic alues (x: a i hme ic mean, s: s anda d
de ia ion and ϕ: mean ib e olume ac ion) used o simula ion inpu
om expe imen al da a o ma e ial D155 [1]
wi h use o he ans o ma ion ma ix
Tij =
cos2(ξ)sin2(ξ) −sin(2ξ)
sin2(ξ)cos2(ξ)sin(2ξ)
0.5sin(2ξ) −0.5sin(2ξ)cos(2ξ).
The o e all global dis o ions o he lamina e a e gi en
by:
ˆk(n)=ˆ0
i(n)+z·ˆκ0(n)(3)
He ein he dis o ions a e calcula ed using he s ains
and cu a u es o he combined shell and pla e elemen ,
which a e ob ained om
ˆ0
i(n)
ˆκ0
i(n)=A∗
ij(n)B∗
ij(n)
(B∗
ij(n))TD∗
ij(n)·ˆni,bl
ˆmi,bl (4)
whe e ˆni,bl ={nx,ny,nxy }and ˆmi,bl ={mx,my,mxy }
a e he in e nal line o ces and momen s o he cu en
block load, as shown in Fig. 2. The ex ensional- (A∗
ij),
bending- (D∗
ij) and coupling- (B∗
ij) compliance ma ices
a e calcula ed by in e ing he ABD ma ix wi h he use
o he espec i e s i ness ma ices:
1
χ
n
Õ
k=1
Qij,kzχ
k−zχ
k−1=
Aij, o χ=1
Bij, o χ=2
Dij, o χ=3
(5)
The global dis o ions and he ans o med s i ness ma-
ix a e hen used o calcula e he global s ess ec o o
each lamina. The local in-plane s esses a e calcula ed
subsequen ly wi h he use o he ans o ma ion ma ix.
ˆσi,k(n)=Qij,k(n) · ˆi,k(n)(6)
σi,k(n)=Tij ˆσi,k(n)(7)
whe e σi,k(n)is he local s ess ec o , wi h he en ies
σk,k(n),σ⊥,k(n)and τ⊥k,k(n) o each ply numbe k.
6
5. Inpu om a igue ma e ial es s
S ess calcula ions (equa ion 7) a e ca ied ou o
wo di e en se s o load condi ions ˆni,bl and ˆmi,bl in
e e y block load o gain he mul iaxial s ess a ios
Rσk,Rσ⊥,Rτ⊥k o all h ee p esen s ess componen s
σk, σ⊥, τ⊥k . The p esen wo k will ocus mainly on simu-
0.0 0.5 1.0 1.5 2.0
n
1.0
0.5
0.0
0.5
1.0
1/
max
ension
comp ession
R=-1.0 R= 0.0 R= R= 0.5 R= 2.0
Fig. 4: Cyclic s esses o one al e na ing s ess a io (R=-1) and in each
case wo pulsa ing ension (R=0, R=0.5) and pulsa ing comp ession
(R=−∞, R=2) s ess a ios
la ions wi h s ess a ios in pulsa ing ension (0≤R<1)
and pulsa ing comp ession ange (1<R≤ ∞), as shown
in Fig. 4. In he ollowing he ou sec o s o s ess a io
anges will be e e ed o as:
•T-T: Tension- ension pulsa ing ange
(0≤R<1)
•T-C: Tension domina ed al e na ing ange
(−1≤R<0)
•C-T: Comp ession domina ed al e na ing ange
(−∞ ≤ R<−1)
•C-C: Comp ession-comp ession pulsa ing ange
(1<R≤ ∞)
Requi ed ela ions be ween s ess and endu ed cycles in
e ms o S-N cu es a e ob ained by i ing expe imen al
da a o he loga i hmic linea ela ion by Basquin:
N=C·σ−k
a(8)
whe e N is he numbe o cycles, C is he Y in e cep ,
k is he nega i e slope and σais he s ess ampli ude
o he cu en block load o each o he h ee s ess
componen s om equa ion 7. S-N cu es a s ess a io
R=0.1 ( ep esen a i e o R=0) and R=10 ( ep esen a-
i e o R=∞) a e a leas equi ed o in e pola e da-
a a a bi a y s ess a ios in pulsa ing ension- ension
and comp ession-comp ession ange. Fig. 5 exempla i-
ly shows he i ed S-N cu es o expe imen al da a o
ma e ial „D155“ om [1].
102103104105106107
cycles o ailu e
N
(log) / [ ]
100
101
102
s ess ampli ude
Sa
(log) / [
MPa
]
R
= 0.1
kL
= -13.0,
CL
= 2.14e+34
R
= 0.1
kL
= -27.1,
CL
= 1.26e+28
R
= 0.1
kL
= -10.7,
CL
= 1.42e+19
R
= 10
kL
= -12.7,
CL
= 7.55e+33
R
= 10
kL
= -20.5,
CL
= 3.10e+37
R
= 10
kL
= -14.8,
CL
= 5.56e+27
Fig. 5: Leas equi ed S-N cu es o in es iga ions in pulsa ing en-
sion and comp ession ange wi h expe imen al da a o unidi ec ional
Ma e ial D155 [1]
One way o ecei e unknown S-N cu es a o he s ess
a ios is by using cons an li e diag ams (CLD), also o -
en e e ed o as Haigh-, Goodman- o Smi h-Diag ams.
A his o ical iew on he de elopmen o CLD can be
ound in [18]. Fo mul iaxial s ess s a es o he lami-
na, h ee independen cons an li e diag ams o pa allel,
ans e se and shea p ope ies a e necessa y.
A comp ehensi e in es iga ion on he in luence o di e-
en o mula ions o CLD o a limi ed numbe o mul i-
di ec ionalglass/polyes e lamina eshasbeenca iedou
by [19]. The linea model, which is based on using only
one expe imen al S-N cu e, a e.g. s ess a io R=−1,
unde es ima es he s eng h o he ma e ials in almos all
cases o any s ess a io and leads o ex emely conse -
a i e esul s. S ill i is o en used, e.g. in wind u bine
ce i ica ion [20] o he a igue o FRP, o simply becau-
se o he lack o da a. The nonlinea a igue li e diag am
by Kawai [21], he pa ame ic cons an -li e model by
Ha is [22], he mul islope model by Boe s a [23] and
he piecewise linea (PWL) model by Philippidis and
7
Fig. 6: Exempla y cons uc ion o ans e se piecewise linea (le ) and piecewise nonlinea ( igh ) cons an li e diag am in LEoC o expe imen al
da a om [1]
Vassilopoulos [24] showed o gi e e y accu a e esul s
o mos o he examined a igue da a. Vassilopoulos e .
al especially poin ed ou he sensi i i y o Kawai’s mo-
del o he choice o inpu da a and he sensi i i y o he
models by Boe s a and Ha is o he i ing o model
pa ame e s. Fu he mo e he model by Kawai ends o
o en lead o o e es ima ion o he ma e ial and he e-
o e non-conse a i e esul s. O all o hese models,
he PNL seems o be he mos s able, since i depic s
he S-N beha io by linea in e pola ion be ween known
S-N cu es wi hou any p elimina y assump ions. Vassi-
lopoulos e al. also imp o ed hei o mula ion in e ms
o a piecewise nonlinea (PNL) cons an li e diag am
o mula ion [25] based on he use o wo (PNL-2R) o
h ee (PNL-3R) S-N cu es. The accu acy o he non-
linea o mula ion was e y conside able o he same
examined da a as in [19].
None heless, in he p esen wo k he piecewise linea
(PWL) model by Philippidis and Vassilopoulos [24] will
be p e e ably used o i s in es iga ions on a igue li e
(see Fig. 6). O cou se, he PNL p o es o be only e y
accu a e o a easonable amoun o known S-N cu es
(a leas h ee S-N cu es, each one p e e ably a he
bo de o e e y sec o (T-T/T-C, T-C/C-T and C-T/C-C).
Thus, a leas h ee S-N cu es a s ess a ios R=0.1, R=-
1 and R=10 a e mos commonly used o a sepa a ion
in o ou sec o s . Thus, o he design o h ee piecewi-
se linea CLD a designa ed numbe o S-N cu es and
he s a ic comp ession and ension s eng hs in pa allel,
ans e se and shea di ec ion (X ,Xc,Y ,Ycand S⊥k) a e
needed.
The calcula ion o s ess ampli ude σ∗
aa R∗, i R∗is
in be ween R=1 and he i s known s ess a io mo ing
coun e clockwise (R1,ccw), is de ined as ollows:
σ∗
a=X
X
σa,1,ccw
+ ∗−(1+R1,ccw)
(1−R1,ccw)
(9)
wi h σa,1being he s ess ampli ude a R1,ccw and ∗=
(1+R∗)/(1−R∗). I R∗is in be ween R=1 and he i s
known s ess a io mo ing clockwise (R1,cw):
σ∗
a=Xc
Xc
σa,1,cw
+ ∗−(1+R1,cw)
(1−R1,cw)
(10)
whe e σa,1is he s ess ampli ude a R1,cw. I R∗is
loca ed in be ween any o wo used s ess a ios Riand
Ri+1:
σ∗
a=σa,i( i− i+1)
( i− ∗)σa,i
σa,i+1
+( ∗− i+1)
(11)
8
whe e σa,iand σa,i+1a e he s ess ampli udes a he wo
known s ess a ios and i(i+1)=(1+Ri(i+1))/(1−Ri(i+1)).
Fig. 7 shows he piecewise linea cons an li e o mu-
la ion o a mul i-di ec ional lamina e wi h s acking se-
quence [90/0/±45/0]s. Th ee S-N cu es a s ess a ios
R=0.1, R=-1 and R=10 a e used o linea in e pola i-
on, and ano he h ee S-N cu es a s ess a ios R=0.8,
R=0.5 and R=2 a e plo ed o compa ison pu poses.
600 400 200 0 200 400 600 800
mean s ess
Sm
/ [
MPa
]
0
100
200
300
400
s ess ampli ude
Sa
/ [
MPa
]
T-T
T-CC-T
C-C
R
= 0
R
= 2
R
= 0.5
R
= 0.8
R
= ±
R
= 1
N=103
N=104
N=105
N=106
Used exp. da a
Unused exp. da a
Fig. 7: PWL cons an li e diag am o lamina e „DD16“ wi h s acking
sequence [90/0/±45/0]s om [1]
6. S eng h deg ada ion
Fi s o all, he e a e possibly wo ways o do jus ice
o he deg ada ion o s eng h in lamina es: linking he
dec ease in s eng h o a physical meaning like mic o
mechanical damage, e.g. (in e -) ib e ailu e, o on he
o he hand connec ing i o henumbe o cyclesendu ed.
A la ge numbe o esidual s eng h models ha e al eady
been p oposed. Mos o he s a e-o - he-a models ha e
been examined by Philippidis and Passipoula idis [26].
A common way o model he esidual s eng h is he use
o S-N cu es o se a loss in s eng h and he ac ual num-
be o endu ed cycles as well as he sus ainable numbe
o cycles in o a ce ain ela ion. To begin wi h, he loss
in s eng h in each s ep is de ined as
S ,i=S ,i−1−∆S ,i(12)
whe e S ,iand S ,i−1 ep esen he i e esidual s eng hs
{X ,Xc,Y ,Yc,S⊥k }a he ac ual and he las s ep espec-
i ely. ∆S ,iis he d op o esidual s eng h in he i- h
cycle and is calcula ed om
∆S ,i=("Ss −σmax,i1−ni−1
Niαjβj#
−"Ss −σmax,i1−ni
Niαjβj#) (13)
whe e Ss is he speci ic s a ic s eng h, σmax,iis he
maximum s ess o he i h cycle, niand ni−1a e he
numbe o cycles a he ac ual and he las s ep, Niis he
maximum numbe o cycles and αjand βja e ma e ial
pa ame e s gi en by cu e i ing o esidual s eng h
da a.
By a ying alues o αjand βjdi e en ma e ial beha-
io can be achie ed. Wi h αjand βjbeing bo h ze o,
he e won’ be any s eng h deg ada ion o he add essed
s eng h. Fo αj=1and βj=1equa ion 12 and 13
can be ea anged o he mos ly used linea deg ada i-
on ule by B ou man and Sahu [27]. Almos all p esen
calcula ions o esidual s eng h unde a igue loads use
he linea model and i is well known ha i gi es con-
se a i e esul s on he sa e side. Fo a bi a y alues
o αjwi h βj=1 he model ep esen s he nonline-
a one-pa ame e model by Scha and Da idson [28].
The s eng h deg ada ion is hen simula ed wi h ei he
a s eep loss o s eng h a he beginning (“Rapid ini i-
al loss in s eng h“) o a he end (“Sudden dea h“) o
he lamina e li e ime. Fo all o he alues o αjand βj
he equa ion e e s o he „no malized esidual s eng h
model (NRSM)“ by S ojko ic e al. [29]. The ad an a-
ge o he NRSM model is ha he ypical ini ial loss
o s eng h ollowed by slow deg ada ion as well as he
sudden dec ease in s eng h a he end can be modeled
e y well.
Fo he i e a i e calcula ion o esidual s eng h wi hin
he sub ou ine, he s eng h deg ada ion o e e y ply
needs o be de ined. Fo he case o a h ee piecewise
CLD, he e a e only heo e ical 40 pa ame e s eligible
in LEoC o o e come all i e s eng h pa ame e s wi hin
e e y sec o o he cons an li e diag am. This o cou se
makes li le sense o a p ac ical applica ion. Thus, lin-
king a ew compa able deg ada ion beha io s migh be
an ad an age in educing he amoun o expe imen al da-
a needed. Since he p esen wo k ocuses only on pulsa-
ing s esses, he amoun o da a is conside ably educed
anyway. Pa icula ly because o he lack o expe imen al
da a o unidi ec ional lamina o ma e ial used om [1],
he linea deg ada ion o all i e s eng h pa ame e s will
p e e ably be used o calcula ions o a igue li e. None-
heless, esul s o all p esen ed esidual s eng h models
will be discussed in sec ion 10. Fig. 8 demons a es he
9
100101102103104105106
0
1
e
4
2
e
4
3
e
4
4
e
4
E
1(
n
)
a
= 124 MPa
a
= 10.3 MPa
a)
1
2
34
Sim. [0/ ± 45/0]
s
Sim. [90/ ± 45/90]
s
Exp. da a
Exp. da a
0
15
30
e
100101102103104105106
0
1
e
b)
e
,
IFFA
in 45 ply
e
,
IFFA
in 90 ply
e
,
IFFA
in 45 ply
e
,
FFT
in 0 ply
100101102103104105106
0
1
2
c)
in 90 ply
in 45 ply
Fig. 15: Compa ison o calcula ed and expe imen al esidual s i ness
and o Ma e ial #3 om [1]
obse ed in Fig. 15 (b). A e wa ds, he s i ness s ays
nea ly cons an o e a wide ange o cycles un il a sudden
d op in {2}. This is ob iously coupled wi h he inc e-
ase o ib e ailu e, as shown in he cen al igu e. O
cou se, he beha io a la e cycles is s ongly dependen
on he ini ial alues, which indica es ha he di e ences
be ween calcula ion and expe imen al da a a e mainly
caused by he o e es ima ed s a ic s i ness.
S i ness deg ada ion o he ans e se es ed Ma e ial
#3 is calcula ed conside ably be e . The s a ic s i ness
as well as s i ness a highe cycles is cap u ed e y well.
Thecu eis cha ac e izedby womain poin s, numbe ed
as {3} and {4}, oo. Due o a s eady inc ease o e,IFF A
in he ±45◦-o ien a ed plies a lowe cycles, s i ness is
dec eased a e he onse o ailu e mode A in {3}. Also,
in he ±90◦-o ien a ed plies, he e is a apid ise in ib e
ailu e in {3} due o s ess edis ibu ion. A e wa ds,
he e is a con inuous dec ease in lamina e s i ness un il
global ailu e in {4}.
11. Conclusions and u u e wo k
A e a b ie e iew o li e a u e, a model o a igue
li e analysis is in oduced, which is cha ac e ized by
i s i e a i e and p og essi e p ocedu e o he calcula-
ion o he cu en lamina e condi ions a e e y load
cycle. Di e en sec o s and models applied wi hin he
me hodology a e desc ibed in de ail. A e wa ds, a-
ious mul i-di ec ional ib e ein o ced plas ic lamina es
a e calcula ed and p esen ed. Fa igue li e o se e al ba-
lanced angle ply lamina es unde pulsa ing ension and
comp ession loads is calcula ed and compa ed o expe-
imen al da a. A quan i y o mul i-di ec ional lamina es
wi h a bi a y s acking sequences unde di e en s ess
a ios a e examined and e i ied, oo. Fu he mo e, e-
sidual s eng h and s i ness o di e en lamina es a e
conside ed in mo e de ail. Al oge he , he p edic ions o
he model showed a good co ela ion wi h mos o he
expe imen al da a. Any disc epancies o unusual ma e i-
al beha io s a e aced back o ce ain inpu pa ame e s
o known issues, which will be add essed in u u e wo k.
To sum up, he model ends o be capable o simula ing
a igue li e o mul i-di ec ional lamina es unde complex
mul iaxial loads wi h di e en s ess a ios accu a ely.
S ill he indings lea e ques ions o he u u e, which
will be add essed in o hcoming wo ks:
•As i is c ucial o know he lamina es capaci y o
ca ying u he loads a e a ce ain ime pe iod,
a mo e in-dep h analysis o esidual s eng hs mo-
dels and hei applica ion wi hin he model will
be ca ied ou nex . De ailed s eng h deg ada ion
a e ce ain combined loads, pa icula ly in mul-
iaxially loaded mul i-di ec ional lamina es, will be
add essed.
•As he in es iga ions in he p esen pape ha e
shown, a deepe look on he ailu e en elope o
mul iaxial a igue migh be an ad an age o u he
simula ions o a igue li e. Fo deepe in es iga i-
ons o o -axis plies, ailu e en elope cha ac e i-
s ics a e ce ain numbe endu ed cycles will also
be add essed.
•Since he shape o he ailu e en elope changes si-
gni ican ly wi h he used bounda y condi ions in
e ms o S-N cu es o R+
⊥,R−
⊥and R⊥k a a y-
ing s ess a ios, he in luence o non-p opo ional
loads will be aken in o conside a ion in u u e
wo ks.
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18
Mobili y and Enginee ing Resea ch (MER)
Volume 1/2018
Imp essum
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Da e o publica ion: Augus , 2018
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P o . D . Michael F an zen
P o . D . Pa ick Tichelmann
Con ac edi o ’s o ice
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