Singularity analysis of anisotropic multimaterial corners

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10.1023/A:1023937819943
1
SINGULARITY ANALYSIS OF ANISOTROPIC MULTIMATERIAL CORNERS
A. Ba oso, V. Man ič y F. Pa ís
School o Enginee ing, Uni e si y o Se ille
Camino de los Descub imien os s/n, E-41092 Se ille, Spain
e-mail: [email p o ec ed], [email p o ec ed].es, [email p o ec ed]s.es
Abs ac
Singula s ess s a es induced a he ip o linea elas ic mul ima e ial co ne s a e cha ac e ized in
e ms o he o de o s ess singula i ies and angula a ia ion o s esses and displacemen s. Linea
elas ic ma e ials o an a bi a y na u e a e conside ed, namely aniso opic, o ho opic, ans e sely
iso opic, iso opic, e c. Thus, in e ms o S oh o malism o aniso opic elas ici y, he scope o he
p esen wo k includes ma hema ically non-degene a e and degene a e ma e ials. Mul ima e ial
co ne s composed o ma e ials o di e en na u e a e ypically p esen a any me al-composi e, o
composi e-composi e adhesi e join . Se e al wo ks a e a ailable in he li e a u e dealing wi h a
singula i y analysis o mul ima e ial co ne s bu in ol ing (in he as majo i y) only ma e ials o
he same na u e (e.g. ei he iso opic o o ho opic). Al hough many di e en co ne con igu a ions
ha e been s udied in li e a u e, wi h almos any kind o bounda y condi ions, he e is an ob ious
lack o a gene al p ocedu e o he singula i y cha ac e iza ion o mul ima e ial co ne s wi hou any
limi a ion in he na u e o he ma e ials. Wi h he p ocedu e de eloped he e, and implemen ed in a
compu e code, mul ima e ial co ne s, wi h no limi a ion in he na u e o he ma e ials and any
homogeneous o hogonal bounda y condi ions, could be analyzed. As a pa icula case, s ess
singula i y o de s in co ne s in ol ing ex ao dina y degene a e ma e ials a e, o he au ho s'
knowledge, p esen ed o he i s ime. The p esen wo k is based on an o iginal idea by Ting
(1997) in which an e icien p ocedu e o a singula i y analysis o aniso opic non-degene a e
mul ima e ial co ne s is in oduced by means o he use o a ans e ma ix.
1. In oduc ion
A g ea deal o esea ch has been ca ied ou in he las ew decades ega ding he singula i y
analysis o linea elas ic mul ima e ial co ne s, p oducing many use ul esul s a ailable in he
li e a u e a p esen . Ne e heless, he possibili y o including any kind o ma e ials in he co ne
con igu a ion has been o some ex en es ic ed in hese wo ks.
Thus, on one hand, mul ima e ial co ne s composed exclusi ely o iso opic ma e ials wi h almos
any kind o bounda y condi ions we e analyzed in a gene al way by Dempsey & Sinclai (1979,
1981), see also Bogy (1971), Bogy & Wang (1971), Hein & E dogan (1971).
On he o he hand, singula i y analysis o mul ima e ial co ne s composed exclusi ely o eal
aniso opic, bu no iso opic, ma e ials was pe o med, e.g., by Delale (1984), Pageu e al. (1995a,
1996), Ting (1997), Chen (1998). The as majo i y o hese wo ks, assuming gene alized plane
s ain s a e o plain s ess s a es, make use o he powe ul Lekhni skii-Eshelby-S oh complex
a iable o malism o Aniso opic Elas ici y: Lekhni skii (1938), Eshelby e al. (1953), S oh (1958,
1962) (in he ollowing e e ed o as S oh o malism). No e ha in his o malism, iso opic
ma e ials a e conside ed as ma hema ically degene a e ma e ials (Ting, 1996b). This concep e e s
o epea ed complex oo s o he cha ac e is ic equa ion wi h an associa ed numbe o linea ly
independen eigen ec o s o he undamen al elas ici y ma ix N smalle han he mul iplici y o he
epea ed complex oo s. The s uc u e o he complex a iable ep esen a ions o elas ic solu ions is
2
di e en and e y cumbe some o ma hema ically degene a e ma e ials when compa ed o non-
degene a e ma e ials (Ting & Hwu, 1988 and Wang & Ting, 1997). This is he eason why almos
all wo ks which apply analy ic ep esen a ions o elas ic solu ions o pe o m singula i y analysis o
aniso opic mul ima e ial co ne s include only eal aniso opic, monoclinic o o ho opic
ma hema ically non-degene a e ma e ials.
The s udies by o he au ho s e e enced abo e a e e y use ul in composi e design applica ions, bu
o limi ed applica ion i no iso opic ma e ial is allowed o be included oge he wi h eal
aniso opic ma e ials. Ve y ew wo ks, all ecen (e.g., Lin and Sung, 1998, Poonsawa e al., 1998,
2001) conside pa icula cases o bima e ial co ne s in ol ing degene a e (usually iso opic) as
well as non-degene a e (usually o ho opic) ma e ials. Lin and Sung conside degene a e ma e ials
(iso opic) by aking he limi ing p ocess wi h he use o L'Hospi al ule. Poonsawa e al. p esen
some bima e ial cases wi h iso opic and o ho opic ma e ials including ic ion a he in e ace.
Ne e heless, a gene al p ocedu e o an N-ma e ial co ne and any ma e ial na u e, including
ex ao dina y-degene a e ma e ials, is s ill lacking.
Using a p ocedu e o aniso opic elas ici y o non-degene a e ma e ials, i migh be possible o
o e come he abo e explained es ic ions and o ob ain app oxima e esul s, i quasi-iso opic
ma e ials a e applied as desc ibed in wha ollows. Replacing he iso opic ma e ial, in he
mul ima e ial co ne s udied, by an aniso opic ma e ial wi h elas ic cons an s e y simila bu no
exac ly equal o hose o he iso opic ma e ial, he heo e ical basis o he p ocedu e (analy ical
ep esen a ion o elas ic solu ion) can be used. Thus he esul s will be as app oxima e o he exac
ones as he alues o he aniso opic elas ic cons an s used a e close o he eal iso opic ones.
Ne e heless, nume ical p oblems ela ed o he possible ill-condi ioning o he exp essions used in
his case can be expec ed o appea . I is impo an o no ice ha he ma hema ical ansi ion om
non-degene a e ma e ials (quasi-iso opic) o degene a e ones (iso opic), leads o a non-con inuous
de ini ion o he ma ices o no malized eigen ec o s o N in ol ed in S oh o malism. Thus, i is
possible o expec lesse accu acy when eaching ce ain limi s in he p e iously ou lined p ocedu e.
This kind o p ocedu e can be used i no high accu acy is equi ed. I has o be poin ed ou ha he
dependence o he e o in he esul s ob ained in his way wi h espec o he di e ence o he alue
o he elas ic cons an s is no easy o es ima e.
Nume ical ools like he ini e elemen me hod can also o e come he abo e di icul ies p esen in
he analy ical me hods, bu usually a he expense o a highe compu a ional e o o lesse
accu acy i coa se disc e iza ions a e used.
The objec i e o his wo k is o p esen and implemen in a compu e code a gene al and
analy ically based p ocedu e o cha ac e ize he singula s ess ield ha appea s a he ip o an
aniso opic linea elas ic mul ima e ial co ne , wi h no limi a ions on he na u e o he ma e ial,
conside ing pe ec bonding be ween he ma e ials, and any combina ion o homogeneous bounda y
condi ions excep unila e al ic ion con ac . The S oh o malism o he degene a e (Ting & Hwu,
1988) cases and ex ao dina y degene a e (Wang & Ting, 1997) cases o aniso opic ma e ials will
be used o his end.
The use o he concep o a ans e ma ix in he analysis o mul ima e ial co ne s wi h pe ec
adhesion be ween he ma e ial wedges (simila o ha used in an analysis o a mul ima e ial in ini e
s ip, see Bu le , 1971) g ea ly simpli ies he p ocedu e, educing he size o he sys em conside ed,
see De ou ny (1988) in iso opic po en ial p oblems, Ting (1997) in aniso opic elas ici y and
Man ič e al. (2002) in aniso opic po en ial p oblems. To ake ad an age o he concep o he
ans e ma ix, in he sense o he abo e men ioned wo ks, he ans e ma ix o degene a e
ma e ials has been ob ained in he p esen wo k.
3
I has o be men ioned ha Desmo a (1996) p esen ed ano he app oach o educing he size o he
linea sys em conside ed in he case o bima e ial aniso opic elas ic co ne s, based on a sui able
p e-elimina ion o some a iables in ol ed. The esul s o Desmo a 's (1996) and Ting's (1997)
wo k can be conside ed equi alen when ee- ee bima e ial co ne s a e s udied. Howe e , due o
he gene al scope o Ting's (1997) elegan me hod, based on he ans e ma ix concep , his will be
he me hod de eloped in he p esen wo k o analyze aniso opic mul ima e ial co ne s in ol ing
any kind o linea elas ic ma e ial, non-degene a e o degene a e.
Following he app oach in oduced by Man ič e al. (1997), he bounda y condi ions analyzed in
Ting's wo k (1997), namely ee, ixed, and all ma e ials bonded, ha e been comple ed by six
addi ional homogeneous bounda y condi ions in he p esen wo k.
I has o be s essed ha he heo e ical esul s and he compu e code de eloped in he p esen wo k
will allow an accu a e singula i y analysis o me al o composi e join s, modelled like mul i-ma e ial
co ne s wi h he simul aneous p esence o o ho opic (o ans e sely iso opic) and iso opic
ma e ials in he same co ne , o be pe o med. This kind o join is widely used o example in he
ae ospace indus y.
2. S oh o malism o aniso opic elas ici y
The S oh o malism (Ting, 1996a) is a powe ul app oach o sol ing aniso opic elas ici y
p oblems in which a gene alized plane s ain s a e: ui=ui(x1,x2) (i=1,2,3), can be assumed. The S oh
o malism is based on he ollowing eigensys em:
)6,...,1( 


ξNξ p, (1)
whe e

p and

ξ a e espec i ely he eigen alues and eigen ec o s o he eal undamen al
elas ici y ma ix N (6×6) de ined as ollows:






T
13
21
NN
NN
N, QRRTNTNRTN   TT 1
3
1
2
1
1,, , (2)
Q, R and T only depend on he elas ic cons an s o he ma e ial ijks
C ( ksijksij C

),
11kiik CQ , 21kiik CR  and 22kiik CT , (3)
and he supe sc ip T deno es he anspose.
The eigen alues

p a e complex i he s ain ene gy is posi i e de ini e, and he eigen ec o s


TTT

baξ , ha e an impo an physical meaning, wi h

a being p opo ional o he displacemen
ec o and

b being p opo ional o he ac ion ec o . I

p and

ξ sa is y (1),

p and

ξ a e
also a solu ion, whe e he o e ba deno es he complex conjuga e. Thus, i is habi ual o w i e:
)3,2,1(,,0Im 33  



ξξppp , (4)
wi h Im s anding o he imagina y pa .
4
Equa ion (1) is only alid when he e a e h ee linea ly independen eigen ec o s

ξ (

=1,2,3). I
he enso o elas ic cons an s ijks
C o a pa icula ma e ial gi es ise o an eigensys em wi h less
han h ee linea ly independen eigen ec o s

ξ, he o malism has o be modi ied (Ting & Hwu,
1988; Wang & Ting, 1997), equa ions (1) being ans o med in he case o wo linea ly independen
eigen ec o s o:
111 ξNξ p, 1212 ξξNξ  p, 333 ξNξ p, ( 21 pp ), (5)
and in he case o only one linea ly independen eigen ec o o:
11 ξNξ p, 122 ξξNξ  p, 233 ξξNξ  p, ( pppp  321 ). (6)
In his sense, all linea elas ic ma e ials can be classi ied in ela ion wi h he cha ac e o he
eigensys em o N, as shown in Table 1 (Ting, 1996b, 1999):
p1

p2

p3

p1 p1=p2

p3 p1=p2=p3
3 Linea ly
Independen
Eigen ec o s
Simple (SP) Semisimple (SS)
Ex ao dina y
semisimple (ES)
Does no exis
2 Linea ly
Independen
Eigen ec o s
Degene a e o
non-semisimple
(D1)
Degene a e o
non-semisimple
(D2)
1 Linea ly
Independen
Eigen ec o s
Ex ao dina y
degene a e o
Ex ao dina y
non-semisimple (ED)
Table 1. Classi ica ion o he undamen al elas ici y ma ix N.
Ting (1996b) p o ed he exis ence o ma e ials wi h an ED ma ix N, as well as he impossibili y o
inding any ma e ial wi h an ES ma ix N, unless he s ain ene gy is allowed o be posi i e
semide ini e.
In he ollowing sec ions, he pa icula ela ions sa is ied by he eigen alues

p and he associa e
eigen ec o s


TTT

baξ , will be used o he h ee eigensys ems (1), (5) and (6). Depending on
he numbe o linea ly independen eigen ec o s, he ollowing ela ions apply:
Th ee linea ly independen eigen ec o s (N is SP o SS).
Fo (

ξ,p) wi h

=1,2,3:


0)( 2

aTRRQ pp
T, (7)



aRQaTRb )(
1
)( p
p
p
T .

5
Two linea ly independen eigen ec o s (N is D1 o D2):
Fo ( 11,ξp) and ( 33 ,ξp) equa ions (7) hold o

=1,3.
Fo ppp  12 and he gene alized eigen ec o


TTT 222 ,baξ :




12
22)( aRRTaTRRQ TT ppp  , (8)


212 aTRTab p
T .
One linea ly independen eigen ec o (N is ED) wi h pppp  321 :
Fo ( 11,ξp) equa ions (7) hold wi h

=1.
Fo ( 22 ,ξp) equa ions (8).
Fo 3
p and he gene alized eigen ec o


TTT 333 ,baξ :




123
22)( TaaRRTaTRRQ  TT ppp , (9)


323 aTRTab p
T .
The solu ion in e ms o displacemen s,
u
, and he s ess unc ion ec o ,
φ
, can be exp essed by
means o complex (3×3) ma ices


321 ,, aaaA  and


321 ,, bbbB  in he ollowing way:
GAAGu
~
 , (10)
GBBGφ
~
 , (11)
whe e G can be w i en in e ms o a bi a y unc ions )(

z (

=1,2,3) o complex a gumen s
21 xpxz

 , and G
~
in e ms o )(
3

z  (

=1,2,3).
Fo he analysis o s ess singula i ies i is su icien o ake he same unc ion o

=1,2,3, i.e.

qz z )()(  and

qz z
~
)()(
3
, whe e

q and

q
~
a e a bi a y eal o complex
cons an s. Thus, i is possible o w i e qFG ·

and qFG
~
·
~
~
 wi h T
qqq ),,( 321
q and
T
qqq )
~
,
~
,
~
(
~
321
q. The s uc u e o F (and
F
~
) depends on he numbe o linea ly independen
eigen ec o s, as shown below, whe e he p ime deno es di e en ia ion wi h espec o he complex
a iable z.
SP and SS cases (non-degene a e cases):











)(00
0)(0
00)(
3
2
1
z
z
z
F, 








)(00
0)(0
00)(
~
3
2
1
z
z
z
F. (12)
D1 and D2 cases (degene a e cases):












)(00
0)(0
0)()(
3
1
121
z
z
z xz
F, 








)(00
0)(0
0)()(
~
3
1
121
z
z
z xz
F, ( 21 zz ). (13)
6
ED case (ex ao dina y degene a e case):













)(00
)()(0
)()()(
2
2
2
2
1
2
z
z xz
z xz xz
F,













)(00
)()(0
)()()(
~
2
2
2
2
1
2
z
z xz
z xz xz
F, ( zzzz  321 ). (14)
In he analysis o singula s ess s a es p esen ed in his wo k, he unc ion


zz )( will be
conside ed, and he e o e





qzz qzz
~
)(,)( 3 , (15)
whe e

is he cha ac e is ic exponen and, as will be seen,

-1 de ines he o de o s ess
singula i y o 10



. I

is a eal numbe , hen
F
F

~
and qq 
~
.
The o hogonali y and closu e ela ions o he S oh o malism, which also depend on he numbe o
linea ly independen eigen ec o s, can be w i en in he ollowing gene al o m:
I
X
X
XX


 11 , wi h: 





BB
AA
X and 






TT
TT
AΓBΓ
ΓAΓB
X1, (16)
whe e I is he iden i y ma ix (6×6), and o he di e en cases: non-degene a e (SP and SS),
degene a e (D1 and D2) and ex ao dina y degene a e (ED),
Γ
is exp essed as:
SP-SS D1-D2 ED











100
010
001
Γ,











100
001
010
Γ,











001
010
100
Γ (17)
All linea elas ic ma e ials all inside one o he abo e men ioned g oups (Table 1). Thus, all
ma e ials can be s udied ollowing he app oach o S oh o malism. Iso opic ma e ials, o
example, belong o g oup D2, wi h a iple eigen alue 1 ip and wo linea ly independen
eigen ec o s.
Following Tanuma (1996), ans e sely iso opic ma e ials can belong o e e y g oup in Table 1,
excep ED. T ans e sely iso opic ma e ials can be non-semisimple (D1 o D2), i espec i e o he
alue o hei elas ic cons an s in he ma e ial coo dina e sys em, wi h he x3 axis being
pe pendicula o he x1-x2 an e sely iso opic plane o he ma e ial. Simply h ough he ela i e
posi ion o he ma e ial wi h espec o he coo dina e sys em which de ines he gene alized plain
s ain s a e, he ma e ial can be non-degene a e o degene a e, Tanuma (1996).
7
3. Singula i y analysis o aniso opic mul ima e ial co ne s including non-degene a e
ma e ials
3.1. Summa y o Ting's p ocedu e.
The p ocedu e o iginally de eloped by Ting (1997) is an e icien ool o he singula
cha ac e iza ion o non-degene a e aniso opic mul ima e ial co ne s. An N-ma e ial co ne , wi h N
homogeneous wedges, is ep esen ed in Figu e 1. The i- h ma e ial wedge occupies he pola sec o
ii


1, i=1,...,N.
1
i
N

0

N

i

i-1
x1
x2
1
i
N

0

N

i

i-1
x1
x2
Figu e 1. Mul ima e ial co ne .
Pe ec bonding is conside ed be ween ma e ial wedges. Fixed o ee bounda y condi ions a e
conside ed a he ex e nal aces. The possibili y wi hou ex e nal aces, all ma e ials being bonded,
is also conside ed and is e e ed o in his wo k as a closed co ne , as opposed o open co ne wi h
ex e nal aces.
Conside a pola coo dina e sys em wi h he o igin a he ip o he co ne (Figu e 1). Then,
equa ions (10) and (11), oge he wi h (15), conside ed o an homogeneous wedge, can be w i en
in he ollowing condensed o m, de ining he complex a iable:
)()sin(cos
21
 
p xpxz  :
XZw )(),(
 
, (18)
whe e X is de ined in (16) and ),(

w is






),(
),(
),(



φ
u
w, 





q
q
~, 







)(0
0)(
)(
*
*





Z. (19)
The diagonal ma ix


)(),(),()( 321*
 
diag in (19)3 is associa ed o ma ix F in
(12)1 h ough he ollowing ela ion:
qF )(
*

, (20)
8
while


)(),(),()( 321*
 
diag wi h




)sin(cos)( p , is ela ed o
F
~
in
(12)2 h ough qF
~
)(
~
*
 
.
Ting's p ocedu e makes use o a ans e ma ix, a ma ix which ans e s he displacemen s and
s ess unc ion ec o componen s om one edge o he ma e ial wedge o he o he . I equa ion
(18) is e alua ed o he i- h wedge a 1
i

and i

, and is elimina ed, we ob ain:
XZw )(),( 11  ii
 
,
XZw )(),( ii
 
,  ),(),( 1
iii

wEw , (21)
whe e, in iew o (16),


1
1
1)()( 


XZXZE iii
 
. (22)
i
E, called he ans e ma ix o he i- h wedge, depends on he ma e ial p ope ies, on he angles
(1i

and i

) de ining he wedge and on he cha ac e is ic exponen

.
Using he con inui y condi ions in oduced by he hypo hesis o pe ec bonding be ween he
wedges, ),(),( 1iiii


ww )1,...,1(


Ni , and he ans e ma ix o each wedge, i is easy o
a i e a he ollowing exp ession, which is in ac he exp ession o he ans e ma ix o he
whole mul ima e ial co ne , as i ela es he a iables be ween i s ex e nal aces ( 0

and N

):



















),(
),(
),(
),(
01
0
1
)4()3(
)2()1(




NN
NN
NN
NN
φ
u
KK
KK
φ
u, o ),(),( 01

NNN wKw , (23)
whe e N
K is ob ained by he p oduc o he sequence o he successi e ans e ma ices i
E o all
he wedges in he co ne :
121 ··...·· EEEEK 
NNN . (24)
In Ting's wo k, ixed and ee bounda y condi ions can be p esc ibed a he ex e nal aces o he
co ne , and he case o all ma e ials being bonded is also conside ed. The ollowing cha ac e is ic
equa ions a e ob ained om (23) o he di e en combina ions o bounda y condi ions:
F ee- ixed: 0φu  )()( 01

NN ,  0K 
)1(
N. (25)
Fixed - ixed: 0uu  )()( 01

NN ,  0K 
)2(
N. (26)
F ee- ee: 0φφ  )()( 01

NN ,  0K 
)3(
N. (27)
Fixed - ee: 0uφ  )()( 01

NN ,  0K 
)4(
N. (28)
All bonded: )()(and)()( 0101

uuφφ  NNNN ,  0IK 
N. (29)
15
The eigensys em o be sol ed in his case is shown in (6). The wo gene alized eigen ec o s
),( 222 TTT baξ  and ),( 333 TTT baξ  mus sa is y (8) and (9) espec i ely, while he i s eigen ec o
),( 111 TTT baξ  mus s ill sa is y equa ions (7). The s uc u e o ma ix F is shown in (14) and he
o hogonali y ela ions a e modi ied ollowing (16) and (17)3.
Equa ion (18) o ex ao dina y degene a e ma e ials, wi h pppp  321 , can be w i en as:
XZw ),(),(
 
. (60)
Le us de ine ),(


Z as in (48):






),,(
),,(
),(



p
p
Ψ0
0Ψ
Z. (61)
In compa ison wi h (18), ma ix )(
*

associa ed o ma ix F in (12) is eplaced by ),,(


pΨ,
which is associa ed o ma ix F in (14) by he ollowing ela ion:
qΨF ),,(


p . (62)
Following he same p ocedu e as ha p esen ed in Sec ion 4, we can use (14), (62) and he
de ini ion o ),,( 1

p in (51), o inally w i e ),,(


pΨ as:














100
),,(10
),,()1(),,(1
)(),,(
21
2
1


 
p
pp
pΨ, (63)
whe e:
)(
sin
),,(




 p, and




sincos)( p


(64)
The ans e ma ix o ex ao dina y degene a e ma e ials akes he ollowing o m:


1
1
1),(),( 


XZXZE

iii , (65)
whe e:
 










),,,(
),,,(
),(),(
1
1
1
1



ii
ii
ii p
p
Ψ0
0Ψ
ZZ , (66)
),,(),,(),,,( 1
1
1



iiii ppp ΨΨΨ , (67)
wi h:

16















100
),,(10
),,()1(),,(1
)(),,(
21
2
1
1


 
p
pp
pΨ. (68)
),,,( 1

ii
pΨ can be w i en as:










 
100
10
1
),(),,,( 11 K
ZKK
piiii
 
, (69)
whe e ),( 1ii

is de ined as in (32) and K is de ined in he same way as in (57):
)()(
)sin(
1
1
ii
ii
K







, (70)
and Z is de ined as:










)(
sin
)(
sin
2
1
),,,(
1
1
1
i
i
i
i
ii KpZ





. (71)
Wi h ),,,( 1

ii
pΨ gi en by (69), he ans e ma ix i
E (65) can hen be compu ed.
6. Singula i y analysis o mul ima e ial co ne s including any aniso opic ma e ial
Wi h he p e ious esul s, he ans e ma ix can be compu ed o any wedge ma e ial, i espec i e
o i s na u e (SP, SS, D1, D2 and ED). A single and obus app oach is p esen ed in his sec ion o
he singula cha ac e iza ion o any mul ima e ial co ne wi h pe ec adhesion be ween ma e ial
wedges and any o hogonal bounda y condi ion o all ma e ials being bonded.
x2
1 - o ho opic
(non-degene a e)
SP o SS ma e ial
2 - iso opic
D2 ma e ial
3 - ED ma e ial
F ee edge
Fixed edge
Figu e 3. Co ne wi h non-degene a e, degene a e and ex ao dina y degene a e ma e ials.
17
Conside , as an example, he h ee-ma e ial co ne ep esen ed in Figu e 3, in ol ing one non-
degene a e o ho opic ma e ial (SP o SS ma e ial), one iso opic ma e ial (degene a e ma e ial D2)
and one ex ao dina y degene a e ma e ial (ED), wi h ee- ixed bounda y condi ions a he ex e nal
aces. The singula i y analysis o his co ne can be pe o med ollowing s eps 1 o 5 desc ibed
below.
1) E alua ion o he ans e ma ix o each ma e ial ( i
E), using (22) o 1
E, (52) o 2
E and (65)
o 3
E.
2) E alua ion o he co ne ans e ma ix 3
K (N=3), by he exp ession in (24), using he ma ices
i
E p e iously ob ained in s ep 1.
3) Applica ion o bounda y condi ions, using )( 0

ee
D and )( N ixed

D (Table 2) o inally ob ain
he modi ied co ne ans e ma ix 3
ˆ
K (43).
4) E alua ion o he de e minan o he subma ix )2(
3
ˆ
K o 3
ˆ
K, which is explici ly p esen ed in
(46). In he case o an open co ne , in ac only )2(
3
ˆ
K has o be e alua ed, while in he case o closed
co ne s he whole ma ix 3
ˆ
K has o be e alua ed.
(5) Calcula ion o oo s

o he cha ac e is ic equa ion (45) (o (29) in he case o closed co ne s),
which ep esen he cha ac e is ic exponen s. Roo s

wi h 1)Re(0



de ine s ess singula i y
exponen s

-1.
The cha ac e is ic equa ions can be sol ed using ( o example) Mulle 's (1956) me hod, when
cha ac e is ic exponen s can be complex. Once he pa icula alue o he cha ac e is ic exponen is
ob ained, he co esponding angula beha iou o displacemen s and s esses nea he ip o he
co ne can also be compu ed. The p ocedu e s a s wi h solu ion o (44) o he

ob ained, hen
using (40) we ob ain ),( 0

w . Wi h he ans e ma ix o he i s ma e ial, ),( 1

w can be
compu ed, and in he same way, wi h each pa icula ans e ma ix, all ),( i
w

i=0,..., N. Once
),( i
w

i=0,..., N a e known, he beha iou o s esses and displacemen s inside each wedge is easy
o ob ain, using he concep o he ans e ma ix be ween ),( i
w

and ),(


w , wi h
1


ii

. S esses a e compu ed by means o : 2,1 ii

 and 1,2 ii

.
Fo closed co ne s, he linea sys em o sol e is:


0wIK  ),()( 0

N,

being ob ained by means
o cha ac e is ic equa ion in (29). Then, using he ans e ma ices o each wedge, in jus he same
way as ou lined p e iously o open co ne s, displacemen s and s esses a e di ec ly ob ained.
7. Nume ical examples
7.1. Compa ison wi h p e ious esul s ob ained by o he au ho s
Se e al esul s o singula i y analysis o di e en co ne con igu a ions, a ailable in he li e a u e,
ha e been used o s udy i s o all he pe o mance o he compu a ional p ocedu e de eloped in he
p esen wo k. Some o hem a e summa ized below.
7.1.1. Iso opic ma e ials (D2 degene a e ma e ials):
18
Single iso opic wedge: A compa ison o he esul s ob ained o a solid sec o o an iso opic
ma e ial o in e io angle º280
01 

, wi h ee- ee ex e nal aces, e sus he esul s o
Sewe yn (1994) and Vasilopoulos (1988), is shown in Table 3. The an iplane o de o s ess
singula i y is also p esen ed (

-1=-0.357143). An excellen ag eemen is achie ed in all digi s.
Vasilopoulos Sewe yn P esen wo k
-
0.469604280870
-0.156560431071
-
0.469604
-0.357143
-0.156560
-
0.469604
-0.357143
-0.156560
Table 3. Resul s (

-1) o a single ee- ee iso opic co ne .
Bima e ial iso opic co ne s: Classical esul s om Dempsey and Sinclai (1979, 1981) we e used.
The esul s ob ained o ee- ee co ne s (Table 4), and o co ne s wi h all ma e ials bonded (Table
5), a e all excellen . The mechanical p ope ies o bo h ma e ials a e: E1=30 GPa, 1=0.25, E2=120
GPa, 2=0.31, o esul s in Table 4 and: E1=30 GPa, 1=0.2, E2=120 GPa, 2=0.3, o esul s in
Table 5. Addi ional an iplane o de o s ess singula i ies a e p esen ed in bo h cases.

1=120º

2=135º

1=120º

2=135º
Dempsey & Sinclai
P esen wo k
-
0.390748
-
0.390748
-0.269076
Table 4. Resul s (

-1) o a bima e ial ee- ee iso opic co ne .

1=80º

2=(2-80)º

1=80º

2=(2-80)º
Dempsey & Sinclai
P esen wo k
-
0.229549
-0.0742109
-
0.2295490
-0.1916800
-0.0742109
Table 5. Resul s (

-1) o a closed bima e ial iso opic co ne .
Th ee-ma e ial iso opic co ne s: Resul s om Hein & E dogan (1971) and Pageau e al. (1994) a e
compa ed wi h hose ob ained in he p esen wo k in Tables 6 and 7. The ag eemen is excellen ,
including he second case (Table 7), wi h complex alued o de o s ess singula i ies. Addi ional
an iplane o de o s ess singula i ies a e p esen ed (

-1=-0.00580724 in Table 6 and

-1=-0.460283
in Table 7). Mechanical p ope ies o he con igu a ion shown in Table 6 a e: E1=20 GPa, 1=0.2,
E2=10 GPa, 2=0.2, E3=0.01 GPa, =0.2, while in Table 7 E1=10 GPa, 1=0.2, E2=0.01 GPa,
2=0.2, E3=100 GPa, =0.2.

1
=90º

2
=90º

1
=90º

2
=90º
Hein & E dogan Pageau e al. P esen wo k
19
-
0.0226
-
0.0226
-0.0003
-
0.02263280
-0.00580724
-0.00028330
Table 6. Resul s (

-1) o a closed h ee-ma e ial iso opic co ne wi h º180,º90 321 

.

3=180º

1=179º

2=1º

3=180º

1=179º

2=1º
Hein & E dogan Pageau e al.: P esen wo k:
-0.4975

0.1014 i -0.4476

0.0887 i -0.447624

0.088750 i
-0.460273
Table 7. Resul s (

-1) o a closed h ee-ma e ial iso opic co ne , º180,º1,º179 321 

.
I has been e i ied ha he singula exponen s associa ed o he an iplane mode in he p e ious
examples ag ee wi h hose ob ained by Man ič e al. (2002) p ocedu e up o six digi s o mo e.
7.1.2. Non-degene a e aniso opic ma e ials:
Bima e ial o ho opic ee- ee co ne : Resul s om Delale (1984) and Chen (1998) we e used in
his case. An excellen ag eemen was ound wi h p esen esul s, shown in Table 8. Bo h ma e ials
ep esen he same o ho opic ma e ial equi alen o a ibe ein o ced plas ic wi h di e en
o ien a ion o ibe s. The ibe s a e placed in he x2-x3 plane, wi h angles wi h espec o x2
axis:

1=60º and

2=30º, and he ollowing mechanical p ope ies: E11=163.4 GPa, E22=E33=11.9
GPa, G12=G13=6.5 GPa, G23=3.5 GPa, 12=13=0.3, 23=0.5.

2=180º

1=90º
x1
x2

2=180º

1=90º
x1
x2
Delale Chen P esen wo k
-
0.4229
-
0.422886
-0.380828
-0.047337
-
0.422886
-0.380828
-0.047337
Table 8. Resul s (

-1) o a ee- ee bima e ial o ho opic co ne .
In e ace c ack in an o ho opic bima e ial con igu a ion: Resul s om Wang (1984) and Chen &
Huang (1997) we e a ailable o his case, in which he ag eemen wi h he esul s ob ained by he
p ocedu e de eloped in he p esen wo k is also e y good. The mechanical p ope ies o he
ma e ials a e hose o a ypical g aphi e-epoxy composi e, aking he ollowing alues when
exp essed in o ho opic axes: E11=138 GPa, E22=E33=14.5 GPa, G12=G13=G23=5.9 GPa,
12=13=23=0.21. The angle

, see Table 9, is he angle he ibe o ms wi h espec o x3 axis in he
x1-x3 plane.
x2 Wang Chen & Huang P esen wo k
20
=45º
-
0.5
-0.50.03434 i
-
0.5
-0.50.0343365146 i
-
0.5
-0.50.0343398 i
=60º
-
0.5
-0.50.02942 i
-
0.5
-0.50.0294152218 i
-
0.5
-0.50.0294132 i
Table 9. Resul s (

-1) o an in e ace c ack be ween wo o ho opic ma e ials.
Th ee-ma e ial o ho opic co ne s ha e also been analyzed, a e y good ag eemen being ob ained
wi h a ailable esul s by Chen (1998) and Pageau e al. (1996).
In iew o he excellen ag eemen ob ained be ween he esul s o he p ocedu e de eloped in his
wo k and esul s o o he au ho s in all s udies p esen ed and also o he s no p esen ed he e, o he
sake o b e i y, he compu e code de eloped he e can be conside ed success ully e i ied.
7.2. Iso opic-o ho opic bima e ial co ne
In e e y me al o composi e o composi e o composi e adhesi e join , an example o an iso opic-
o ho opic bima e ial co ne , wi h he simul aneous p esence o non-degene a e and degene a e
ma e ials, can be ound.
Paying a en ion o he co ne depic ed in Figu e 4, he p ocedu e p esen ed in his wo k can be used
wi hou any o he limi a ions ha usually appea in adi ional app oaches which make use o S oh
o malism. In he as majo i y o cases, hese limi a ions a e due o he ac ha iso opic ma e ials
as well as any o he aniso opic ma e ial wich is ma hema ically degene a e, canno be included in
he analysis.
x
2
x
1
O ho opic SP ma e ial
Iso opic
D2 ma e ial

x
2
x
1
O ho opic SP ma e ial
Iso opic
D2 ma e ial

Figu e 4. Bima e ial co ne wi h SP and D2 ma e ials.
The ma e ials in he co ne (Figu e 4), ha e he ollowing p ope ies:
Composi e ma e ial (o ho opic non-degene a e, SP ma e ial):
E11=141.3 GPa E22=9.58 GPa, E33=9.58 GPa
G12=5.0 GPa, G13=5.0 GPa, G23=3.5 GPa
=0.3, =0.3, =0.32
Epoxy adhesi e (iso opic, D2 ma e ial):
E=3 GPa, =0.3

21
In Figu e 5, he adhesi e angle () a ies om 0º o 180º and he ibe ein o ced plas ic has he
ibe o ien ed in x1 di ec ion. Two o de o s ess singula i ies a e ob ained un il an angle o
app oxima ely =85º is eached. S a ing om his angle, h ee eal o de o s ess singula i ies a e
ob ained.
Be o e he adhesi e angle eaches 160º, wo eal oo s con e in o wo complex conjuga e oo s.
Simila esul s a e ob ained o wo iso opic and wo o ho opic ee- ee co ne s. Finally, o he
in e ace c ack (=180º), a eal oo o 0.5 and wo complex conjuga e oo s wi h eal pa equal o
0.5 a e ob ained.
-0.50
-0.45
-0.40
-0.35
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
0 20 40 60 80 100 120 140 160 180
Re (

-1)
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
Im (

-1)


= 0º
x
1
x
2
x
3

x
1
x
2
x
3

angle ()
Figu e 5. O de o s ess singula i ies (

-1) o a bi-ma e ial SP-D2 co ne .
The angula beha iou o displacemen s
u,

u and he s ess componen s

,


,


,
calcula ed using ),(

w and 2,1 ii

 , 1,2 ii

, a e p esen ed in Figu e 6, o =70º o one o
he singula i y modes wi h -1=-0.266941.
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
0 50 100 150 200 250
angle

u
u


-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
0 50 100 150 200 250
angle

u
u


-25
-20
-15
-10
-5
0
5
10
0 50 100 150 200 250
angle







-25
-20
-15
-10
-5
0
5
10
0 50 100 150 200 250
angle







Figu e 6. Angula beha iou o displacemen s and s esses o =70º and -1=-0.266941.
22
I can be obse ed ha he s ess componen s


and


ul il he bounda y condi ions a he
ex e nal aces ( ee- ee), and ha

is no con inuous a he in e ace be ween bo h ma e ials.
Displacemen s a e con inuous a he in e ace bu no hei slope.
7.3. Co ne s in ol ing ex ao dina y degene a e (ED) ma e ials
To he au ho s' knowledge, no esul s a e a ailable o co ne s in ol ing ex ao dina y degene a e
ma e ials. In ac , hese ma e ials ha e been p o ed o exis only in ecen yea s (Ting, 1996b).
F om Ting (1996b), i is known ha a pa icula g oup o ED ma e ials can be desc ibed wi h he
ollowing educed elas ic compliance ma ix:















)3(000
1000
000 0000
0001
'1
11
 
 
ss, (72)
whe e

 2
1
))1(( 31 , and he ollowing has o be ul illed: 0
11 

s, 01



, 21



and 2
)3(

 . Fo he nume ical example p esen ed in his wo k, he ollowing alues ha e
been aken: 1
11 

s, 2
1


, 1


and he posi i e sign in

, esul ing in:

















4
5
2
12
1
2
1
000
1000 00201
0000 00101
's. (73)
In Figu e 7, he o de o s ess singula i ies o a single ee- ee wedge o he abo e ED ma e ial,
om =180º o he c ack con igu a ion, =360º, is p esen ed.
Two eal o de o s ess singula i ies a e ob ained un il =260º, whe e a hi d eal singula i y
solu ion appea s. Fo he c ack con igu a ion (=360º) h ee eal o de o s ess singula i ies, wi h
alue -0.5, a e ob ained. The nume ical esul s ob ained a e p esen ed in Table 10 o some
pa icula alues o .
In Figu e 8, a ee- ee h ee-ma e ial co ne in ol ing a 90º wedge o as SP (o ho opic non-
degene a e ma e ial), a 90º o a D2 (iso opic ma e ial) and a 90º o a ED ma e ial is p esen ed. The
mechanical p ope ies o he o ho opic and iso opic ma e ial a e he same as used in Sec ion 7.2,
while he ED ma e ial has he p ope ies gi en abo e, in (73).
23
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
180 200 220 240 260 280 300 320 340 360
angle 
Re (-1)

Figu e 7. O de o s ess singula i ies (

-1) o a single ee- ee ED wedge.

=200º
-0.15672 -0.128226

=240º
-0.374451 -0.269291

=280º
-0.472353 -0.351773 -0.161211

=320º
-0.497632 -0.422641 -0.378599

=355º
-0.499996 -0.490131 -0.488593
Table 10. Nume ical esul s o some pa icula cases o Figu e 7.
1 - o ho opic
(non-degene a e)
SP o SS ma e ial
2 - iso opic
D2 ma e ial
3 - ED ma e ial
F ee edge
F ee edge
90º
90º
90º
x
2
x
1
1 - o ho opic
(non-degene a e)
SP o SS ma e ial
2 - iso opic
D2 ma e ial
3 - ED ma e ial
F ee edge
F ee edge
90º
90º
90º
x
2
x
1
Figu e 8. F ee- ee h ee-ma e ial co ne wi h 90º wedges o SP, D2 and ED ma e ials.
Fo his pa icula con igu a ion, a eal (

-1=-0.323997) and wo complex conjuga e (

-1=-
0.354922±0.152318 i) o de s o s ess singula i y we e ound.
24
8. Conclusions
In he p esen wo k, a powe ul p ocedu e o he singula i y analysis o aniso opic mul ima e ial
co ne s allowing any kind o linea elas ic aniso opic ma e ial o be conside ed has been comple ed
and implemen ed in a compu e code.
The ans e ma ix o ma hema ically degene a e ma e ials has been ob ained in he amewo k o
S oh o malism, and explici exp essions ha e been p esen ed. This allows he o iginal p ocedu e
de eloped by Ting (1997) o he singula cha ac e iza ion o mul ima e ial aniso opic co ne s o
be comple ed wi h he possibili y o including degene a e ma e ials in he analysis. As iso opic
ma e ials can be conside ed degene a e ma e ials in S oh o malism o aniso opic elas ici y, he
singula analysis is now open o ma e ials o his kind, as well as any o he degene a e and
ex ao dina y degene a e ma e ial.
A Ma hema ica (Wol am, 1991) code has been implemen ed o he calcula ion o he o de o
s ess singula i ies, as well as he g aphical ep esen a ion o displacemen s and s esses. This
p ac ical compu a ional ool has shown excellen ag eemen when compa ing wi h p e ious esul s
a ailable in he li e a u e, om he single iso opic wedge o he o ho opic h ee-ma e ial co ne .
The implemen ed code has been used o analyze a co ne wi h simul aneous p esence o bo h
degene a e (iso opic) and non-degene a e (o ho opic) ma e ials, and also co ne s in ol ing
ex ao dina y degene a e ma e ials which, o he au ho s' knowledge, a e s udied o he i s ime.
Wi h he compu a ional ool de eloped, any bidimensional co ne con igu a ion can be
cha ac e ized, conside ing pe ec bonding a in e aces, and any homogeneous o hogonal
bounda y condi ion a ou e in e aces o he co ne (also all bonded). No e ha only powe ype
singula i ies ha e been conside ed in his wo k. The p esence o loga i hmic singula i ies can be
analyzed using he well-known app oach de eloped by Dempsey and Sinclai (1979), Dempsey
(1995), Ting (1996a), Sinclai (1999) and o he s. The s udy ca ied ou is a p e ious s ep o
calcula ing s ess in ensi y ac o s (by means, o ins ance, o he ini e elemen o bounda y
elemen me hods). The possibili y o s udying p oposals o ailu e c i e ia o adhesi ely bonded
join s, based on ac u e mechanics, will hen be opened up.
9. Acknowledgemen s
The s udy has he inancial suppo o he Spanish Minis y o Science and Technology (PROFIT
2001, P ojec EUREKA !1882) and Minis y o Educa ion and Cul u e (P ojec s No. PB98-1118,
MAT 2000-1115). The au ho s g a e ully acknowledge help ul in o ma ion om SACESA.
10. Re e ences
Bogy, D.B. Two edge-bonded elas ic wedges o di e en ma e ials and wedges angles unde su ace ac ions.
Jou nal o Applied Mechanics 38 (1971) 377-386.
Bogy, D.B. and Wang, K.C. S ess singula i ies a in e ace co ne s in bonded dissimila iso opic elas ic
ma e ials. In e na ional Jou nal o Solids and S uc u es 7 (1971) 993-1005.
Bu le , H. Theo y o elas ici y o a mul ilaye ed medium. Jou nal o Elas ici y 1 (1971) 125-143.
Chen W.H. and Huang, T.F. S ess singula i y o edge delamina ion in angle-ply and c oss-ply lamina es.
Jou nal o Applied Mechanics 64 (1997) 525-531.