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lgt ~r ~ J~I r ~ slI) )~ )l9 s jjISJro Yo] )~ afj -I~S ~

1 1 all =--shy

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a l2 =a33 =--shy1 Yc

1 3a44 = 3ass =S

4 13a(6=----

YrYc XrXc 1

2al2 = 2a13 =--XrXc

2 12a23 =----

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u~-uJ-S ~~-~ ~1(~

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s[ ~ ~I s[r-ol~4 ~ ~ xl

~ [~J (51~ ISjLJlo ~ S f

fftJ f s[ -gt~Jb ~ ~ 4 ~ y1b bulllJJ5 xl si1Y- ISI ~[ U -gtyly Jgt) U-

s~ l r fftJ r-St is 015 []

~ sly Wt- ~~ 9~ 4 0 lS1 ~

OJ == D(I + LlAjDPfl

x (t_1 + Atj - LlAp) 00)

j(LlA) == 0

1(middot ~ql I (~ bull 1 ~ L)I ~ vr- -0-~ 1Sr W sr0M

~i5 ~

LlA~+1 == LlA~ - j() (W)

OLlAj ut

oj oj Ocrj--=--- OA)OLlAj 00j OLlAj 00 I _ == -D(I +LlAraquoPf OLlAj

lt )x [D(I + LlAjDPf1p(t_1

+At j - LlAp) +p]

0) OA) Of) ~I s)~~ ~ ~ x~

~15 ~ s)P lJ ~ jl LlAj)lAo (W)

cS1A~~ J~ cS~Ju~ JtSJ cS1A~1 ~ S~J~

~llt1lI~~ )ti) ~JJ (~ smiddot ltIi -~tt bullS sl5o --lIgtLio LIr r -- W y

S ~4 cI jl d ~ h5 I y

~ ~)IS ~~ g T 4 I) ub ~) (~I Jlgt 0-1 ))

~ ~ j19 9) )) ub 9) (SL tSl~101-0 )l9)

Jraquo 0-1 4 -~ 0-1 bull))JIS (S)Yo ~tsb1 j19

I bull S 11 At L 1 7 ) Gl 1gtlltWraquo (S (S - ~ 4gt IS

I )) 9 tSl~101-0 ~ ~) 01-0 ) ) u-

~11S )Po ))pS ) 0141 )Po tS~~t

- Lual lI laquo - () 1( )(Sj w- u - ) -r v bull shy

~tsb j19 )) (SL ~ iS~~ 4 tSl~l01-o ~4~ (-- - r) jS )) o~ 0)) 01 )b )~

(A)

i~) ~j~lS elL tSlj ~ )yo tS~~~

l c~ middot1 ) 1_ 1 l-gt I )1middotj Ur-- u u )-- Yo ) bull 0 tS ~

-

9) )) ~I) Yo)) I) lA~l ~ij iS~~ )

~) 0~ ~)ji ~ (~ j4) ub ~) 0~ o~ j19

)~ )~ 0-- ))J IS ()~) ub(S)Yo

~rU ~l ~) ~ij iS~~ 01-0 J)lgt1 01~ gN 1

~ ~j~lS ~~ ~)I ~l ~) iSI~ JgtI) o~

01 (f) jS )) ~ (i) jS )) 4-i1 ) j-i

(SL iS ~~ 0l--o J)lgt1 ~ -1 o~ 0)1)

(0)

(B)(C)

()bull

(D)

(E)

(c) J~ 1I)~ J) ~IT~~ ( IJ~~ IT~~ (y) 4Ow- dJ1 ~ (ill) ~ij ampSI~IJ~ JIa)f ~

middothJSt ~ ~ ) IJ~ tl)J (d IA~ ITI~ ()~))) ampSI~IJ~ J))gts jlif

E

o=o o=o

I I j I I ~ ~I I II

g~ltlgTIltg- () OltlgTIltg~ (~) IgTI=O (UJI)

o=o~

II ~~ J Ishy

(J) ()

jltl (0) J~J~ 01raquo00 () ~~J~ 01raquo00 (-l) =1 ~~ (ill) 1Slttll~1o JI08J C) ~

J~I jli JI u~~ () JIJsii

IgTIgtg- IgTI=g~1Ilt

-------4-8r

(~) (UJI)

J~ILi (-l) J~ jli (ill) IILoo ~I$l~~ ~ 1S1lttll~1o ~X~ ~Io ~IJ1 1S1uP ~

u1~ j~l )~ 14~u~ )ti) ~I ( J=bU s~ ~loj Ilf )gt gtjl- s-i )~ Jltgtb

t4~ ~19 olol ~ ~4 jgt ~ )gt gtgtIs ~

1S~4- [ 1 )$oo (Mi) Is sll ~~) Is sll jlsro 4~ tY~ bullgt alP J~

4)1 s1~)0l-o ~ )~ ~) gt ~ ~Irgt sl) J~ J) ~ jl3 sl) 01~

-s ~I gt ~ )gt bull)5 Jb )gt 41 sLAuJo ~jlgtjl jl IS--t LA~) ~ bulls)lb1 sxlt)A

gt=Is ogt) 4W ~41s ISL pS19 ~ 1S~4- sLA)gt)

0 Jl)

$-~ ~ lAl jl s Ib )l I) ~ 1S4d~~ 0illil

g r =1 gr I a r =1 G r 1j5 ~I )l llIll15 ~

-sI1gt -s ~1)4 lAl ~j 1 1S1r middot~4 15

)~ 1sAgt 15 ISJI )l K

_x-a x~a(x-a)shy

o xlta

() ~sLo ~I jllSr0M )l ~ l15 ~j4

$-~~ ~ ~ 1 p~ ~19 $-~~ l 1S1r

middotlpound1 Igt - ~ l (~~L) bullbull ( ) -sir Vr-) IS ) bull s-- ~

Jl ~I ~j 1 - ISI~~I tll)ySj

~I)l ci))tse ISI~~I 1S9) 1Jllao ~ ~4 ) lol jI )~ ~ oM~115 )~

Dc = diag(D(g) LJ(g) LJ(g)]

Sr(K) SN(K) tl~fj ISlA~I)4 ~tU 4

tll)gtSj jlpoundl ~ ISL s- $-~~ ojllil H -r

)~ g ISI~~I

-N S (I()=~ gmax

N 1 -N-N +1( gmax - go

(11)

(11)

D(g) ISL ~19 ~ ~Ir il s Ib

l115 1J rj ~I) jl LJ(g)

(n)

$-~~ ISI~~I IS~ 1 ~I) otIb

l )l sbgt l )~ $-~ ~ l Ib )l jt s-

)~ o~ ci )2i )l tl~fj lS Ollgtlto

j5 ~ lA~~ $-I~

tl~fj Ollgtlto)l lAl1 ~)lil bullll~tU

)~ Ie ~1)4 4 1(=

(rf)

ISlA4Iil- 4 ISI~~I IS~ 1 ~I) o~ ~

~115 1J ri ~I) il s- $-~~

00)

s- $-~~ OlW toi ISlA~I)4 01) ~I) )l

Dc (r0) ~I) )l )2i l) l Ao 4 jt

l tl~fj ~jLo ~ S(K) y1 ~~jLo

middot~415 5r

lA~~ 1 S~I 1Sr~l1~ ISiWoM 1S1r 10

IS~) ~~il ~4 ~ laquo~~~ 1S1bgtgtgt )

bull 1 lL l ci ~ ts -LL LslSo- _ 15 Y) 15 bull V shy

Lo L)JS bull lAo ~ 6- t L bull bull - 1- t$Ir- - y r oJ)

~Ib middot~415 (y - r f) ISlA)l )l ~4-sI

bullli15 o~l ~4 ~Ir ~ rlAo ~I 1Sr)tse

~ jl ~s ~19 s- $-~~ ISIbgt ~ 6J 1S1r

~~)l )~ ~~ l1~ ~)~ ~(gNltO)

bull10 a r aN ~4 ~Ir yG1 4 oM~115

IS~ )br otIb ISIbgt olj ~l 1~ ~ ~I

)~ s- $-~~ )lr ISI~~I

(rv) (rr)

I II ill

~I ~AIT ~

~~ I ~li gtj~ 9 ~

~ fftJ ~ 0~ y I ~li rJ f

-gt[ojL (r ~)~ 4 ~4y jl ~li 0 01~~ I ~ 0 15 -gt)9

-gt[OjL ~ ~ ~ S

[~J (51~ 0fi r fftJ f -gt[ ~j-S

[V] ~~5 ~

JSlJl ~~ y [~J (51~ ~ 1 ~

S ltlil5cI -gt[

-gtI (5[ U Gill Gil Gi 0 i 45 gt45

si1Y- ISj1 (5[ U GIIIC GIIC GIC ~~

IS~0M ~ r n m J )IlOo 0 ~ cI jl ~ y1b

(5[ U ~ y1b ~15 ~ ~~Iojl ~l I

jl -gt~ -gt~ ~ fftJ silY- s)1

~ ([ A] ltIi sl~) 1 0 ~ ~

r-ol~ xl ~ 1S1y ~Io) ~ 0s l ~ xl

r p IS~ o~ ci)l~ )lJl1 ~

(5[ U 1Sy1y 0~ oJ$oL (5~) ~ ~

lJlci)l~ f~1 [J fftJ f IS[ IY- s1

s[ ~ ~I s[r-ol~4 ~ ~ xl

~ [~J (51~ ISjLJlo ~ S f

fftJ f s[ -gt~Jb ~ ~ 4 ~ y1b bulllJJ5 xl si1Y- ISI ~[ U -gtyly Jgt) U-

s~ l r fftJ r-St is 015 []

~ sly Wt- ~~ 9~ 4 0 lS1 ~

OJ == D(I + LlAjDPfl

x (t_1 + Atj - LlAp) 00)

j(LlA) == 0

1(middot ~ql I (~ bull 1 ~ L)I ~ vr- -0-~ 1Sr W sr0M

~i5 ~

LlA~+1 == LlA~ - j() (W)

OLlAj ut

oj oj Ocrj--=--- OA)OLlAj 00j OLlAj 00 I _ == -D(I +LlAraquoPf OLlAj

lt )x [D(I + LlAjDPf1p(t_1

+At j - LlAp) +p]

0) OA) Of) ~I s)~~ ~ ~ x~

~15 ~ s)P lJ ~ jl LlAj)lAo (W)

cS1A~~ J~ cS~Ju~ JtSJ cS1A~1 ~ S~J~

~llt1lI~~ )ti) ~JJ (~ smiddot ltIi -~tt bullS sl5o --lIgtLio LIr r -- W y

S ~4 cI jl d ~ h5 I y

~ ~)IS ~~ g T 4 I) ub ~) (~I Jlgt 0-1 ))

~ ~ j19 9) )) ub 9) (SL tSl~101-0 )l9)

Jraquo 0-1 4 -~ 0-1 bull))JIS (S)Yo ~tsb1 j19

I bull S 11 At L 1 7 ) Gl 1gtlltWraquo (S (S - ~ 4gt IS

I )) 9 tSl~101-0 ~ ~) 01-0 ) ) u-

~11S )Po ))pS ) 0141 )Po tS~~t

- Lual lI laquo - () 1( )(Sj w- u - ) -r v bull shy

~tsb j19 )) (SL ~ iS~~ 4 tSl~l01-o ~4~ (-- - r) jS )) o~ 0)) 01 )b )~

(A)

i~) ~j~lS elL tSlj ~ )yo tS~~~

l c~ middot1 ) 1_ 1 l-gt I )1middotj Ur-- u u )-- Yo ) bull 0 tS ~

-

9) )) ~I) Yo)) I) lA~l ~ij iS~~ )

~) 0~ ~)ji ~ (~ j4) ub ~) 0~ o~ j19

)~ )~ 0-- ))J IS ()~) ub(S)Yo

~rU ~l ~) ~ij iS~~ 01-0 J)lgt1 01~ gN 1

~ ~j~lS ~~ ~)I ~l ~) iSI~ JgtI) o~

01 (f) jS )) ~ (i) jS )) 4-i1 ) j-i

(SL iS ~~ 0l--o J)lgt1 ~ -1 o~ 0)1)

(0)

(B)(C)

()bull

(D)

(E)

(c) J~ 1I)~ J) ~IT~~ ( IJ~~ IT~~ (y) 4Ow- dJ1 ~ (ill) ~ij ampSI~IJ~ JIa)f ~

middothJSt ~ ~ ) IJ~ tl)J (d IA~ ITI~ ()~))) ampSI~IJ~ J))gts jlif

E

o=o o=o

I I j I I ~ ~I I II

g~ltlgTIltg- () OltlgTIltg~ (~) IgTI=O (UJI)

o=o~

II ~~ J Ishy

(J) ()

jltl (0) J~J~ 01raquo00 () ~~J~ 01raquo00 (-l) =1 ~~ (ill) 1Slttll~1o JI08J C) ~

J~I jli JI u~~ () JIJsii

IgTIgtg- IgTI=g~1Ilt

-------4-8r

(~) (UJI)

J~ILi (-l) J~ jli (ill) IILoo ~I$l~~ ~ 1S1lttll~1o ~X~ ~Io ~IJ1 1S1uP ~

u1~ j~l )~ 14~u~ )ti) ~I ( J=bU s~ ~loj Ilf )gt gtjl- s-i )~ Jltgtb

t4~ ~19 olol ~ ~4 jgt ~ )gt gtgtIs ~

1S~4- [ 1 )$oo (Mi) Is sll ~~) Is sll jlsro 4~ tY~ bullgt alP J~

4)1 s1~)0l-o ~ )~ ~) gt ~ ~Irgt sl) J~ J) ~ jl3 sl) 01~

-s ~I gt ~ )gt bull)5 Jb )gt 41 sLAuJo ~jlgtjl jl IS--t LA~) ~ bulls)lb1 sxlt)A

gt=Is ogt) 4W ~41s ISL pS19 ~ 1S~4- sLA)gt)

0 Jl)

$-~ ~ lAl jl s Ib )l I) ~ 1S4d~~ 0illil

g r =1 gr I a r =1 G r 1j5 ~I )l llIll15 ~

-sI1gt -s ~1)4 lAl ~j 1 1S1r middot~4 15

)~ 1sAgt 15 ISJI )l K

_x-a x~a(x-a)shy

o xlta

() ~sLo ~I jllSr0M )l ~ l15 ~j4

$-~~ ~ ~ 1 p~ ~19 $-~~ l 1S1r

middotlpound1 Igt - ~ l (~~L) bullbull ( ) -sir Vr-) IS ) bull s-- ~

Jl ~I ~j 1 - ISI~~I tll)ySj

~I)l ci))tse ISI~~I 1S9) 1Jllao ~ ~4 ) lol jI )~ ~ oM~115 )~

Dc = diag(D(g) LJ(g) LJ(g)]

Sr(K) SN(K) tl~fj ISlA~I)4 ~tU 4

tll)gtSj jlpoundl ~ ISL s- $-~~ ojllil H -r

)~ g ISI~~I

-N S (I()=~ gmax

N 1 -N-N +1( gmax - go

(11)

(11)

D(g) ISL ~19 ~ ~Ir il s Ib

l115 1J rj ~I) jl LJ(g)

(n)

$-~~ ISI~~I IS~ 1 ~I) otIb

l )l sbgt l )~ $-~ ~ l Ib )l jt s-

)~ o~ ci )2i )l tl~fj lS Ollgtlto

j5 ~ lA~~ $-I~

tl~fj Ollgtlto)l lAl1 ~)lil bullll~tU

)~ Ie ~1)4 4 1(=

(rf)

ISlA4Iil- 4 ISI~~I IS~ 1 ~I) o~ ~

~115 1J ri ~I) il s- $-~~

00)

s- $-~~ OlW toi ISlA~I)4 01) ~I) )l

Dc (r0) ~I) )l )2i l) l Ao 4 jt

l tl~fj ~jLo ~ S(K) y1 ~~jLo

middot~415 5r

lA~~ 1 S~I 1Sr~l1~ ISiWoM 1S1r 10

IS~) ~~il ~4 ~ laquo~~~ 1S1bgtgtgt )

bull 1 lL l ci ~ ts -LL LslSo- _ 15 Y) 15 bull V shy

Lo L)JS bull lAo ~ 6- t L bull bull - 1- t$Ir- - y r oJ)

~Ib middot~415 (y - r f) ISlA)l )l ~4-sI

bullli15 o~l ~4 ~Ir ~ rlAo ~I 1Sr)tse

~ jl ~s ~19 s- $-~~ ISIbgt ~ 6J 1S1r

~~)l )~ ~~ l1~ ~)~ ~(gNltO)

bull10 a r aN ~4 ~Ir yG1 4 oM~115

IS~ )br otIb ISIbgt olj ~l 1~ ~ ~I

)~ s- $-~~ )lr ISI~~I

(rv) (rr)

~ ~)IS ~~ g T 4 I) ub ~) (~I Jlgt 0-1 ))

~ ~ j19 9) )) ub 9) (SL tSl~101-0 )l9)

Jraquo 0-1 4 -~ 0-1 bull))JIS (S)Yo ~tsb1 j19

I bull S 11 At L 1 7 ) Gl 1gtlltWraquo (S (S - ~ 4gt IS

I )) 9 tSl~101-0 ~ ~) 01-0 ) ) u-

~11S )Po ))pS ) 0141 )Po tS~~t

- Lual lI laquo - () 1( )(Sj w- u - ) -r v bull shy

~tsb j19 )) (SL ~ iS~~ 4 tSl~l01-o ~4~ (-- - r) jS )) o~ 0)) 01 )b )~

(A)

i~) ~j~lS elL tSlj ~ )yo tS~~~

l c~ middot1 ) 1_ 1 l-gt I )1middotj Ur-- u u )-- Yo ) bull 0 tS ~

-

9) )) ~I) Yo)) I) lA~l ~ij iS~~ )

~) 0~ ~)ji ~ (~ j4) ub ~) 0~ o~ j19

)~ )~ 0-- ))J IS ()~) ub(S)Yo

~rU ~l ~) ~ij iS~~ 01-0 J)lgt1 01~ gN 1

~ ~j~lS ~~ ~)I ~l ~) iSI~ JgtI) o~

01 (f) jS )) ~ (i) jS )) 4-i1 ) j-i

(SL iS ~~ 0l--o J)lgt1 ~ -1 o~ 0)1)

(0)

(B)(C)

()bull

(D)

(E)

(c) J~ 1I)~ J) ~IT~~ ( IJ~~ IT~~ (y) 4Ow- dJ1 ~ (ill) ~ij ampSI~IJ~ JIa)f ~

middothJSt ~ ~ ) IJ~ tl)J (d IA~ ITI~ ()~))) ampSI~IJ~ J))gts jlif

E

o=o o=o

I I j I I ~ ~I I II

g~ltlgTIltg- () OltlgTIltg~ (~) IgTI=O (UJI)

o=o~

II ~~ J Ishy

(J) ()

jltl (0) J~J~ 01raquo00 () ~~J~ 01raquo00 (-l) =1 ~~ (ill) 1Slttll~1o JI08J C) ~

J~I jli JI u~~ () JIJsii

IgTIgtg- IgTI=g~1Ilt

-------4-8r

(~) (UJI)

J~ILi (-l) J~ jli (ill) IILoo ~I$l~~ ~ 1S1lttll~1o ~X~ ~Io ~IJ1 1S1uP ~

u1~ j~l )~ 14~u~ )ti) ~I ( J=bU s~ ~loj Ilf )gt gtjl- s-i )~ Jltgtb

t4~ ~19 olol ~ ~4 jgt ~ )gt gtgtIs ~

1S~4- [ 1 )$oo (Mi) Is sll ~~) Is sll jlsro 4~ tY~ bullgt alP J~

4)1 s1~)0l-o ~ )~ ~) gt ~ ~Irgt sl) J~ J) ~ jl3 sl) 01~

-s ~I gt ~ )gt bull)5 Jb )gt 41 sLAuJo ~jlgtjl jl IS--t LA~) ~ bulls)lb1 sxlt)A

gt=Is ogt) 4W ~41s ISL pS19 ~ 1S~4- sLA)gt)

0 Jl)

$-~ ~ lAl jl s Ib )l I) ~ 1S4d~~ 0illil

g r =1 gr I a r =1 G r 1j5 ~I )l llIll15 ~

-sI1gt -s ~1)4 lAl ~j 1 1S1r middot~4 15

)~ 1sAgt 15 ISJI )l K

_x-a x~a(x-a)shy

o xlta

() ~sLo ~I jllSr0M )l ~ l15 ~j4

$-~~ ~ ~ 1 p~ ~19 $-~~ l 1S1r

middotlpound1 Igt - ~ l (~~L) bullbull ( ) -sir Vr-) IS ) bull s-- ~

Jl ~I ~j 1 - ISI~~I tll)ySj

~I)l ci))tse ISI~~I 1S9) 1Jllao ~ ~4 ) lol jI )~ ~ oM~115 )~

Dc = diag(D(g) LJ(g) LJ(g)]

Sr(K) SN(K) tl~fj ISlA~I)4 ~tU 4

tll)gtSj jlpoundl ~ ISL s- $-~~ ojllil H -r

)~ g ISI~~I

-N S (I()=~ gmax

N 1 -N-N +1( gmax - go

(11)

(11)

D(g) ISL ~19 ~ ~Ir il s Ib

l115 1J rj ~I) jl LJ(g)

(n)

$-~~ ISI~~I IS~ 1 ~I) otIb

l )l sbgt l )~ $-~ ~ l Ib )l jt s-

)~ o~ ci )2i )l tl~fj lS Ollgtlto

j5 ~ lA~~ $-I~

tl~fj Ollgtlto)l lAl1 ~)lil bullll~tU

)~ Ie ~1)4 4 1(=

(rf)

ISlA4Iil- 4 ISI~~I IS~ 1 ~I) o~ ~

~115 1J ri ~I) il s- $-~~

00)

s- $-~~ OlW toi ISlA~I)4 01) ~I) )l

Dc (r0) ~I) )l )2i l) l Ao 4 jt

l tl~fj ~jLo ~ S(K) y1 ~~jLo

middot~415 5r

lA~~ 1 S~I 1Sr~l1~ ISiWoM 1S1r 10

IS~) ~~il ~4 ~ laquo~~~ 1S1bgtgtgt )

bull 1 lL l ci ~ ts -LL LslSo- _ 15 Y) 15 bull V shy

Lo L)JS bull lAo ~ 6- t L bull bull - 1- t$Ir- - y r oJ)

~Ib middot~415 (y - r f) ISlA)l )l ~4-sI

bullli15 o~l ~4 ~Ir ~ rlAo ~I 1Sr)tse

~ jl ~s ~19 s- $-~~ ISIbgt ~ 6J 1S1r

~~)l )~ ~~ l1~ ~)~ ~(gNltO)

bull10 a r aN ~4 ~Ir yG1 4 oM~115

IS~ )br otIb ISIbgt olj ~l 1~ ~ ~I

)~ s- $-~~ )lr ISI~~I

(rv) (rr)

E

o=o o=o

I I j I I ~ ~I I II

g~ltlgTIltg- () OltlgTIltg~ (~) IgTI=O (UJI)

o=o~

II ~~ J Ishy

(J) ()

jltl (0) J~J~ 01raquo00 () ~~J~ 01raquo00 (-l) =1 ~~ (ill) 1Slttll~1o JI08J C) ~

J~I jli JI u~~ () JIJsii

IgTIgtg- IgTI=g~1Ilt

-------4-8r

(~) (UJI)

J~ILi (-l) J~ jli (ill) IILoo ~I$l~~ ~ 1S1lttll~1o ~X~ ~Io ~IJ1 1S1uP ~

u1~ j~l )~ 14~u~ )ti) ~I ( J=bU s~ ~loj Ilf )gt gtjl- s-i )~ Jltgtb

t4~ ~19 olol ~ ~4 jgt ~ )gt gtgtIs ~

1S~4- [ 1 )$oo (Mi) Is sll ~~) Is sll jlsro 4~ tY~ bullgt alP J~

4)1 s1~)0l-o ~ )~ ~) gt ~ ~Irgt sl) J~ J) ~ jl3 sl) 01~

-s ~I gt ~ )gt bull)5 Jb )gt 41 sLAuJo ~jlgtjl jl IS--t LA~) ~ bulls)lb1 sxlt)A

gt=Is ogt) 4W ~41s ISL pS19 ~ 1S~4- sLA)gt)

0 Jl)

$-~ ~ lAl jl s Ib )l I) ~ 1S4d~~ 0illil

g r =1 gr I a r =1 G r 1j5 ~I )l llIll15 ~

-sI1gt -s ~1)4 lAl ~j 1 1S1r middot~4 15

)~ 1sAgt 15 ISJI )l K

_x-a x~a(x-a)shy

o xlta

() ~sLo ~I jllSr0M )l ~ l15 ~j4

$-~~ ~ ~ 1 p~ ~19 $-~~ l 1S1r

middotlpound1 Igt - ~ l (~~L) bullbull ( ) -sir Vr-) IS ) bull s-- ~

Jl ~I ~j 1 - ISI~~I tll)ySj

~I)l ci))tse ISI~~I 1S9) 1Jllao ~ ~4 ) lol jI )~ ~ oM~115 )~

Dc = diag(D(g) LJ(g) LJ(g)]

Sr(K) SN(K) tl~fj ISlA~I)4 ~tU 4

tll)gtSj jlpoundl ~ ISL s- $-~~ ojllil H -r

)~ g ISI~~I

-N S (I()=~ gmax

N 1 -N-N +1( gmax - go

(11)

(11)

D(g) ISL ~19 ~ ~Ir il s Ib

l115 1J rj ~I) jl LJ(g)

(n)

$-~~ ISI~~I IS~ 1 ~I) otIb

l )l sbgt l )~ $-~ ~ l Ib )l jt s-

)~ o~ ci )2i )l tl~fj lS Ollgtlto

j5 ~ lA~~ $-I~

tl~fj Ollgtlto)l lAl1 ~)lil bullll~tU

)~ Ie ~1)4 4 1(=

(rf)

ISlA4Iil- 4 ISI~~I IS~ 1 ~I) o~ ~

~115 1J ri ~I) il s- $-~~

00)

s- $-~~ OlW toi ISlA~I)4 01) ~I) )l

Dc (r0) ~I) )l )2i l) l Ao 4 jt

l tl~fj ~jLo ~ S(K) y1 ~~jLo

middot~415 5r

lA~~ 1 S~I 1Sr~l1~ ISiWoM 1S1r 10

IS~) ~~il ~4 ~ laquo~~~ 1S1bgtgtgt )

bull 1 lL l ci ~ ts -LL LslSo- _ 15 Y) 15 bull V shy

Lo L)JS bull lAo ~ 6- t L bull bull - 1- t$Ir- - y r oJ)

~Ib middot~415 (y - r f) ISlA)l )l ~4-sI

bullli15 o~l ~4 ~Ir ~ rlAo ~I 1Sr)tse

~ jl ~s ~19 s- $-~~ ISIbgt ~ 6J 1S1r

~~)l )~ ~~ l1~ ~)~ ~(gNltO)

bull10 a r aN ~4 ~Ir yG1 4 oM~115

IS~ )br otIb ISIbgt olj ~l 1~ ~ ~I

)~ s- $-~~ )lr ISI~~I

(rv) (rr)

$-~ ~ lAl jl s Ib )l I) ~ 1S4d~~ 0illil

g r =1 gr I a r =1 G r 1j5 ~I )l llIll15 ~

-sI1gt -s ~1)4 lAl ~j 1 1S1r middot~4 15

)~ 1sAgt 15 ISJI )l K

_x-a x~a(x-a)shy

o xlta

() ~sLo ~I jllSr0M )l ~ l15 ~j4

$-~~ ~ ~ 1 p~ ~19 $-~~ l 1S1r

middotlpound1 Igt - ~ l (~~L) bullbull ( ) -sir Vr-) IS ) bull s-- ~

Jl ~I ~j 1 - ISI~~I tll)ySj

~I)l ci))tse ISI~~I 1S9) 1Jllao ~ ~4 ) lol jI )~ ~ oM~115 )~

Dc = diag(D(g) LJ(g) LJ(g)]

Sr(K) SN(K) tl~fj ISlA~I)4 ~tU 4

tll)gtSj jlpoundl ~ ISL s- $-~~ ojllil H -r

)~ g ISI~~I

-N S (I()=~ gmax

N 1 -N-N +1( gmax - go

(11)

(11)

D(g) ISL ~19 ~ ~Ir il s Ib

l115 1J rj ~I) jl LJ(g)

(n)

$-~~ ISI~~I IS~ 1 ~I) otIb

l )l sbgt l )~ $-~ ~ l Ib )l jt s-

)~ o~ ci )2i )l tl~fj lS Ollgtlto

j5 ~ lA~~ $-I~

tl~fj Ollgtlto)l lAl1 ~)lil bullll~tU

)~ Ie ~1)4 4 1(=

(rf)

ISlA4Iil- 4 ISI~~I IS~ 1 ~I) o~ ~

~115 1J ri ~I) il s- $-~~

00)

s- $-~~ OlW toi ISlA~I)4 01) ~I) )l

Dc (r0) ~I) )l )2i l) l Ao 4 jt

l tl~fj ~jLo ~ S(K) y1 ~~jLo

middot~415 5r

lA~~ 1 S~I 1Sr~l1~ ISiWoM 1S1r 10

IS~) ~~il ~4 ~ laquo~~~ 1S1bgtgtgt )

bull 1 lL l ci ~ ts -LL LslSo- _ 15 Y) 15 bull V shy

Lo L)JS bull lAo ~ 6- t L bull bull - 1- t$Ir- - y r oJ)

~Ib middot~415 (y - r f) ISlA)l )l ~4-sI

bullli15 o~l ~4 ~Ir ~ rlAo ~I 1Sr)tse

~ jl ~s ~19 s- $-~~ ISIbgt ~ 6J 1S1r

~~)l )~ ~~ l1~ ~)~ ~(gNltO)

bull10 a r aN ~4 ~Ir yG1 4 oM~115

IS~ )br otIb ISIbgt olj ~l 1~ ~ ~I

)~ s- $-~~ )lr ISI~~I

(rv) (rr)

gN~O

gNgtO K~O

gNgtO KgtOSN(K)ltl (iO)

gNgtO

Er K~O

(l-Sr(KraquoEr K gt 0 Sr(K) lt 1 Oy(g) =

K gt 0 Sr(K) ~ 1 (Jr lt a r(l-)ar a-gr KgtO Sr(K)~l (Jr=a (if)

IA-)I ~ ll kl bull ~ IS- bull r ~gt

is)iI)~ ~Ir - ~~ - )~~)I gt Jgt gNltO

~ltgt~gtIS- is-)l0~ S jy 0~ J ~j4 0lt -N-gN ltgmax

~ jI o~ 1~ iSlA-)I J ~j4 gt -NgN -gmax

Jgt)gtS 0~ J ~ jL9 )gt y 0lt -T-gT ltgmax IS-~- (gt gt

J gt)gtS jI ~ ~I jL9 )gt j-iJ gt-TgJ -gma

I J

I 8T

J

1i)J ~Ioo slt40-J sl) () ~ij slt40-J sl) (AlI)~ s~lt40-~) sl~lI~~ dMi ~I)V ~

(0 =jJ(-CFN)) J~ jli)J ~Ioo slt40-J sl) (-) J~

iSlA) ltIS JAJ 0-11 )0 jI)gt JIlIIgtI iS~4

ciy )gt I) ~j~1S iSlA~~ J))gt5 ~ IV II shy

jl bull is)lfjL bull - ~1)j4 0~ ~IAIj) 4j~

yly )) ~ ~41S o~ I~ iS~WI ~) 0~

iSl) is~ is)l) I~ iS~WI 4 ~ ) I~ ~I ) ~ yIb bull) 0-11 )) lIIgtgt1S ~ )4 0-11 -5 )) I~ 0W1 ~ ljoJb SIA-)I jl iSl~gt 4 IA-)I jI

~ iS~WI iSr)S 4 gtgt o~ ci fo It ~OJLIIIAltgt)ImiddotL middotmiddot(Ijl gtgt t~J r-- JY bull u U--) Y IS Igt 1)

(lb)i1S (~I ~ ~)) o~~ iSyJl

)l iSl-)l0~ ISL ~ ~ iSl) Jc)) Ij) jl ~I jL9 )l (if) ~I) jl J ~ jL9

~ )l 1 ~~~ )IS isr is

01)4 ltIS lIIgtgt1S t) ISISa is))))4 ~~l

~ ~Lj (If jl is)~ ~Lj (If s1gt -5

middot~4A~

JJ)gt S~ ~~J ~ ~lJS ~)

~I~--JUIto ~~ is~ ~lSo ~4 ) gt~ ~Ir- )

II sJ4 ) Lltmin ~~ lo~llS ~ ~4 o~ JrS O~ sr Mcr ~Ir ~t) rlS 1~

jl ~t) rlS ytgt ) Llt -)J ~ Y- 4 1

) ~ 1 yi sr min (1tcrMmin)

) -jJ ~4 i s1~4 ~ sl) bull) y-I

L~11 JrS rgt s1 Jt ~tj rlSytgt

IS~)S IS~) ~v (iJ1- toJ

~4 ) I) ~)yJ1 hrJ 01~oJII (Grady) sJ

1 Ali W ~ 1 1 I_~ d ~sts) 1Sr ~ - ~ ) shy

5 4 sl-J ~jrolS II sl ) u~J ISI~

JjJ )~ sl~rb s)lf)4 I ~ ~I s1~J01o

0Li (All - A) ~ ) y-I s)lf)4 -gtlgtJb

)1 )r- r s) IS)) ) 1 ol 01

sr-) syJI 4 TIOOI034-C ~~1-4IJ ~

) ol 01 0Li ~lo sVi-t [OO]lOs JLJl

middotl41S (y- A)~ J~

5 ci)l r~1 -J )~ ) 1 y-I sjLJlo

u-J ~ ~ 010 OmiddotA em ojl)j ~ s1~1

- A ~) )IS 010 jI I) uL 0)Q u ) Y

laquoAll

sur-AI) jl sr-0 - 4 ~Iio sl) uL y-I ~ (wI ~I ~ s~JA ~x 0l1 ) ol ltlJ1)

~ 4JJ )~ jl ~ ytgt u W y-14 )l1S r~1

1IS s~JA 0l1 ol ciJ ~ ) lr--o 0 W1

t I ciL KA bull (All-) 1(-u ))) -- ~ shy

lt) jS ol ISW) )~ 1= + see

5 o)j))) -J ) I)) ~Io 46A 1S~4-- )

) ol ltlJI) ) 4 u llb1S 0Li tj Jk ) ~I ~ lgt sl) I) ~ bA9 u (w] ~I --

1 ~Iio ~li 01 ltlJ1) rblgt ~ 0t) r)p

) sl-J010 )gt5 r-S ssJ1 () jS

sur-A) sr-)~ 4 0 jl ~ jI u llb1S Li

~~ u ~IA bull1 olol ~ ~~ y-I

1S~ts ) s14JJ01o )gt5 ~I~ l1S

)1 ) suy ~ s~rb s) )4 )) ~

JIS

jll~ ~Ir-I ) ~ s1~J0l-o ~)gt5 ~

bulls)~ )IS JgtI j sr-0 - 4 ~ Ir-I -~ ~~ tIla sl) ~4 ~4 jI)~

(V)

1 I TTJCOn TTftXt bull TIflntbull TIflnertia -1 ) lt)) ~) j cu

(A)

rgt ~ Iy-I ) 1 jS ~I sr-)~ 4 s1~J0l-o ss) ssJ1 i)W -4its s~)sJ1

15t jS ~ (fV) ~I ~I) ol )r

Mii + fint = fext + fcon

sur-i )) M rr- ~t1)u JISk~

sur-i )) felU ~t sur-i )) fint ~)

~11S ~)j ~I) jI fcon sl-Jl-o ISW

(fr)

fint = 1BTSan (ff )

fext = 1NTb an + 1NTtar I (10)

) u01 )I- ~ ~bo ~~ ~ 4 4 laquosrgt ~W ~r raquo )gt5 ~

p) y-I ) ~ ~ IS ciJ )~ )I)li ~tj s4-0lS

~r-1 ~) sr-i su)) ~I ) 4 ~ ) sr )) ~ ~tj rlS ) sl-J10

) ~ytgt ~11S~~s fl3 ~~4-- ))

Ojt)j ~)) o~ k~ )~~srgt ~W

s)~~li ~bullbull lgt Ssect )IS )~ ~t) rl3

II sJ4 ) Lltmin ~~ lo~llS ~ ~4 o~ JrS O~ sr Mcr ~Ir ~t) rlS 1~

jl ~t) rlS ytgt ) Llt -)J ~ Y- 4 1

) ~ 1 yi sr min (1tcrMmin)

) -jJ ~4 i s1~4 ~ sl) bull) y-I

L~11 JrS rgt s1 Jt ~tj rlSytgt

IS~)S IS~) ~v (iJ1- toJ

~4 ) I) ~)yJ1 hrJ 01~oJII (Grady) sJ

1 Ali W ~ 1 1 I_~ d ~sts) 1Sr ~ - ~ ) shy

5 4 sl-J ~jrolS II sl ) u~J ISI~

JjJ )~ sl~rb s)lf)4 I ~ ~I s1~J01o

0Li (All - A) ~ ) y-I s)lf)4 -gtlgtJb

)1 )r- r s) IS)) ) 1 ol 01

sr-) syJI 4 TIOOI034-C ~~1-4IJ ~

) ol 01 0Li ~lo sVi-t [OO]lOs JLJl

middotl41S (y- A)~ J~

5 ci)l r~1 -J )~ ) 1 y-I sjLJlo

u-J ~ ~ 010 OmiddotA em ojl)j ~ s1~1

- A ~) )IS 010 jI I) uL 0)Q u ) Y

laquoAll

sur-AI) jl sr-0 - 4 ~Iio sl) uL y-I ~ (wI ~I ~ s~JA ~x 0l1 ) ol ltlJ1)

~ 4JJ )~ jl ~ ytgt u W y-14 )l1S r~1

1IS s~JA 0l1 ol ciJ ~ ) lr--o 0 W1

t I ciL KA bull (All-) 1(-u ))) -- ~ shy

lt) jS ol ISW) )~ 1= + see

5 o)j))) -J ) I)) ~Io 46A 1S~4-- )

) ol ltlJI) ) 4 u llb1S 0Li tj Jk ) ~I ~ lgt sl) I) ~ bA9 u (w] ~I --

1 ~Iio ~li 01 ltlJ1) rblgt ~ 0t) r)p

) sl-J010 )gt5 r-S ssJ1 () jS

sur-A) sr-)~ 4 0 jl ~ jI u llb1S Li

~~ u ~IA bull1 olol ~ ~~ y-I

1S~ts ) s14JJ01o )gt5 ~I~ l1S

)1 ) suy ~ s~rb s) )4 )) ~

JIS

jll~ ~Ir-I ) ~ s1~J0l-o ~)gt5 ~

bulls)~ )IS JgtI j sr-0 - 4 ~ Ir-I -~ ~~ tIla sl) ~4 ~4 jI)~

(V)

1 I TTJCOn TTftXt bull TIflntbull TIflnertia -1 ) lt)) ~) j cu

(A)

rgt ~ Iy-I ) 1 jS ~I sr-)~ 4 s1~J0l-o ss) ssJ1 i)W -4its s~)sJ1

15t jS ~ (fV) ~I ~I) ol )r

Mii + fint = fext + fcon

sur-i )) M rr- ~t1)u JISk~

sur-i )) felU ~t sur-i )) fint ~)

~11S ~)j ~I) jI fcon sl-Jl-o ISW

(fr)

fint = 1BTSan (ff )

fext = 1NTb an + 1NTtar I (10)

) u01 )I- ~ ~bo ~~ ~ 4 4 laquosrgt ~W ~r raquo )gt5 ~

p) y-I ) ~ ~ IS ciJ )~ )I)li ~tj s4-0lS

~r-1 ~) sr-i su)) ~I ) 4 ~ ) sr )) ~ ~tj rlS ) sl-J10

) ~ytgt ~11S~~s fl3 ~~4-- ))

Ojt)j ~)) o~ k~ )~~srgt ~W

s)~~li ~bullbull lgt Ssect )IS )~ ~t) rl3

bullbullbullbull

TA

E 1468 GPa ~ =114GPa

G2 = 6184 GPa G23 = 4380 GPa

v =03 ~ =1730MPa A =1380MPa

1= 665 MPa 1 = 268 MPa

s= 1337 MPa p = 1550 kgm3

~ M

190N

T

~~X

(y) (All)

T3001034-C ~~I-~I ISI~ ~lAo IS~) (~)~u ISI-~ ISJIJJ~ 4IJiA (ill) It ~

a bull-amp----~~g----~a (y) (All)

KiP Reoldon H~lot1 Plo I D bull 1 P 5

bull -1 n t -15 n

-L Z

-25

- -5

-

5

-5 11 u u u 5 u 7 U 1

- I I I

25

lIIS-altlCllr -

~ E -_ I ~ 125 ~_tM

- C -- -

~ bull ~~~ ~E 0bull f4u -

bull ii ~h -125

0 j -

~ -25 25 50_

~ position (em)

~ r-I---

- _ algorINa 0 -4 ~ I---(SOIIXJISOloolJUJfgto6MX

~ r--- -U

TIaa

~ jl OJo1 ~ ~Is~ o~JI)~J) tIJ ui~ 4laii IT-~~ Jb ~io ~

[-]~)A jl4iS~ ~middot-r~middotlsec 1S~loj J) 4iSt ~ ytNu

bull Q

Qbull amp

amp a bull

(y) (All)

bull bull

bull bull bull bull

bull bull

bull bull bull

bull bull abull a ia bullbull a () lt)

45~oi IIgt ~ 4iw 454)~~ IIgtJs ~J 45l11 ~

1=- sec ()t=~ sec (11)= sec ()=++ bullbull sec (All)

( bull IJoZ 1gt11gt ~~ j)~sJ ~ lo ~u ~JsJ ~s j~j 45-Slsj ~ 1A4) r~)

- abull a

bull bull a

( ) (uJl)

1= + sec () t= middot middot0 sec (All) 45~oj)1gt I~ 4i 4514)~~ JI)s ~J 45l1IT ~

( bull loJoZ 1)111gt ~~ I)~ sJ ~ lo ~u ~J sJ 4s j~l 45-Sl s1 41A4) ul~)

f) Ibull JI Ibull bullbullc

bullt bullI

2 -Ut

Ja

ua

-

ua

1

-~Jlitt8lmiddotmiddot1 I I -

Ii lop

1 ~

~ Lfi ~ of

~

) IIgt )) ~j-S iSUb~ ~~) JLgti o~)by

)3) ~I jI o)~ ))gt5 jl ~ ~ ~r

jAl w~ iSl~~wl-o ))gt5 jI ~ Ub~~

)) Ub~~ u-W bull5~ o~ 0)4) ))gtY

t~ I~middot ~ IS )) If 1middot 1 1_~ramp- is~) isr) ) IS )r- r-

-wLS )lJ) ~Iu ()fl iS~ ~) 1$JJt

~l~I JLgti o)jl Ih) j) y (wlo

)LgtI )) 511gt o~ rlS ~I jl I) is)g3)

I$)Y iSly middotw4~ )) IIgt)IS i --b ~~

ul) ~ iSUb)rJb1) iSr)~ 4 ) S1)s ~))

f4W 4S l ul) is)lJ SUb~ )4 ~I )) ol

middotl41S is)~ isUb)rJb1) ~

~I~)li

oSjb ~~ iSl) ~~ jl ~~~

Syen ~Igt ) FfI(lf(V o ek jI d o ~ (~) wr-1J iSjL~yen ~W- Srshy

bull))- IS ~1))gt9 frY -Ii - - -i

) 4i (y

1 wlA JLgti bullJi) is wl~

)4 ~I ~ lA ~ 4 bull )y~1S r~1 ~it wlA 4 is~ i ~r)) iSl~ )4 ol is~)4 issJI

~ ~ )) l wlA 4 iSl~ yo )4 ~ I) )gt

Y )~ ~ ~ ~L W)1i bull ~~ IIgt)IS

~ (H) lt - ) is~ ))IS wl-o jl

140~ 0 5 If L~II cil 1( is u - J I)r-- 1 r- ~

~)IS wli -1- sec -1- - Osec is~Lj )) I)

~ )) wIj -gt y ~l-o ~~~ )b

))gt5 ~I )) 1 0 1 (i)

)4 yj 41 5 ~r )) 0it~ Ub~ jI iSl~~wl-o

)-IS jul isI~ yo

iSjW iSly ISL is)lJ ) )4 ~I )) Jlgt )) laquoiSl-~wl-o ))gt5J ~ o~~ I$)Y

~ts ~Io ~p ~I--o jI iSr0 4 is~4J )) ) ~I l ul) ~~lS d~b I)W

~fJ4 I - de Freitas M Silva A and Reis L (2000) Numerical evaluation of failure mechanisms on composite

specimens subjected to impact loading Composites Part B Engineering Vol 31 PP 199-207

2 - Mohammadi S (2003) Discontinuum Mechanics using Finite and Discrete Elements WITPress- UK

3 - Koh CG Owen D R J and Perie D (1995) Explicit dynamic analysis of elsto-plastic laminated

composite shells implementation of non-iterative stress update schemes for the Hoffinan yield criterion

Computational Mechanics Vol 16 PP 307-314

4 - Schellekens 1 C 1 (1992) Computational strategies for composite structures PhD thesis Technische

Universiteit Delft Holland

5 - Forouzan-sepehr S and Mohammadi S (2001) A contact based method for 3D delamination analysis of

composites subjected to impact loading in S ValliaPPan N Khalili (editors) Computational Mechanicsshy

New Frontiers for the New Millennium Proceedings of 1 Asian-Pactflc Congress on computational

Mechanics- PP 691-696

6 - Mohammadi S Forouzan-sepehr S and Asadollahi A (2002) Contact based delamination and fracture

analysis of composites Thin-Walled Structures Vol 40 No 7-8 PP 595-609

7 - Masters 1 E (1987) Basic failure modes of continuous fiber composites Engineering Materials

Handbook Vol 1 CompOSites ASM International USA

bull bull

bull bull bull bull

bull bull

bull bull bull

bull bull abull a ia bullbull a () lt)

45~oi IIgt ~ 4iw 454)~~ IIgtJs ~J 45l11 ~

1=- sec ()t=~ sec (11)= sec ()=++ bullbull sec (All)

( bull IJoZ 1gt11gt ~~ j)~sJ ~ lo ~u ~JsJ ~s j~j 45-Slsj ~ 1A4) r~)

- abull a

bull bull a

( ) (uJl)

1= + sec () t= middot middot0 sec (All) 45~oj)1gt I~ 4i 4514)~~ JI)s ~J 45l1IT ~

( bull loJoZ 1)111gt ~~ I)~ sJ ~ lo ~u ~J sJ 4s j~l 45-Sl s1 41A4) ul~)

f) Ibull JI Ibull bullbullc

bullt bullI

2 -Ut

Ja

ua

-

ua

1

-~Jlitt8lmiddotmiddot1 I I -

Ii lop

1 ~

~ Lfi ~ of

~

) IIgt )) ~j-S iSUb~ ~~) JLgti o~)by

)3) ~I jI o)~ ))gt5 jl ~ ~ ~r

jAl w~ iSl~~wl-o ))gt5 jI ~ Ub~~

)) Ub~~ u-W bull5~ o~ 0)4) ))gtY

t~ I~middot ~ IS )) If 1middot 1 1_~ramp- is~) isr) ) IS )r- r-

-wLS )lJ) ~Iu ()fl iS~ ~) 1$JJt

~l~I JLgti o)jl Ih) j) y (wlo

)LgtI )) 511gt o~ rlS ~I jl I) is)g3)

I$)Y iSly middotw4~ )) IIgt)IS i --b ~~

ul) ~ iSUb)rJb1) iSr)~ 4 ) S1)s ~))

f4W 4S l ul) is)lJ SUb~ )4 ~I )) ol

middotl41S is)~ isUb)rJb1) ~

~I~)li

oSjb ~~ iSl) ~~ jl ~~~

Syen ~Igt ) FfI(lf(V o ek jI d o ~ (~) wr-1J iSjL~yen ~W- Srshy

bull))- IS ~1))gt9 frY -Ii - - -i

) 4i (y

1 wlA JLgti bullJi) is wl~

)4 ~I ~ lA ~ 4 bull )y~1S r~1 ~it wlA 4 is~ i ~r)) iSl~ )4 ol is~)4 issJI

~ ~ )) l wlA 4 iSl~ yo )4 ~ I) )gt

Y )~ ~ ~ ~L W)1i bull ~~ IIgt)IS

~ (H) lt - ) is~ ))IS wl-o jl

140~ 0 5 If L~II cil 1( is u - J I)r-- 1 r- ~

~)IS wli -1- sec -1- - Osec is~Lj )) I)

~ )) wIj -gt y ~l-o ~~~ )b

))gt5 ~I )) 1 0 1 (i)

)4 yj 41 5 ~r )) 0it~ Ub~ jI iSl~~wl-o

)-IS jul isI~ yo

iSjW iSly ISL is)lJ ) )4 ~I )) Jlgt )) laquoiSl-~wl-o ))gt5J ~ o~~ I$)Y

~ts ~Io ~p ~I--o jI iSr0 4 is~4J )) ) ~I l ul) ~~lS d~b I)W

~fJ4 I - de Freitas M Silva A and Reis L (2000) Numerical evaluation of failure mechanisms on composite

specimens subjected to impact loading Composites Part B Engineering Vol 31 PP 199-207

2 - Mohammadi S (2003) Discontinuum Mechanics using Finite and Discrete Elements WITPress- UK

3 - Koh CG Owen D R J and Perie D (1995) Explicit dynamic analysis of elsto-plastic laminated

composite shells implementation of non-iterative stress update schemes for the Hoffinan yield criterion

Computational Mechanics Vol 16 PP 307-314

4 - Schellekens 1 C 1 (1992) Computational strategies for composite structures PhD thesis Technische

Universiteit Delft Holland

5 - Forouzan-sepehr S and Mohammadi S (2001) A contact based method for 3D delamination analysis of

composites subjected to impact loading in S ValliaPPan N Khalili (editors) Computational Mechanicsshy

New Frontiers for the New Millennium Proceedings of 1 Asian-Pactflc Congress on computational

Mechanics- PP 691-696

6 - Mohammadi S Forouzan-sepehr S and Asadollahi A (2002) Contact based delamination and fracture

analysis of composites Thin-Walled Structures Vol 40 No 7-8 PP 595-609

7 - Masters 1 E (1987) Basic failure modes of continuous fiber composites Engineering Materials

Handbook Vol 1 CompOSites ASM International USA

) IIgt )) ~j-S iSUb~ ~~) JLgti o~)by

)3) ~I jI o)~ ))gt5 jl ~ ~ ~r

jAl w~ iSl~~wl-o ))gt5 jI ~ Ub~~

)) Ub~~ u-W bull5~ o~ 0)4) ))gtY

t~ I~middot ~ IS )) If 1middot 1 1_~ramp- is~) isr) ) IS )r- r-

-wLS )lJ) ~Iu ()fl iS~ ~) 1$JJt

~l~I JLgti o)jl Ih) j) y (wlo

)LgtI )) 511gt o~ rlS ~I jl I) is)g3)

I$)Y iSly middotw4~ )) IIgt)IS i --b ~~

ul) ~ iSUb)rJb1) iSr)~ 4 ) S1)s ~))

f4W 4S l ul) is)lJ SUb~ )4 ~I )) ol

middotl41S is)~ isUb)rJb1) ~

~I~)li

oSjb ~~ iSl) ~~ jl ~~~

Syen ~Igt ) FfI(lf(V o ek jI d o ~ (~) wr-1J iSjL~yen ~W- Srshy

bull))- IS ~1))gt9 frY -Ii - - -i

) 4i (y

1 wlA JLgti bullJi) is wl~

)4 ~I ~ lA ~ 4 bull )y~1S r~1 ~it wlA 4 is~ i ~r)) iSl~ )4 ol is~)4 issJI

~ ~ )) l wlA 4 iSl~ yo )4 ~ I) )gt

Y )~ ~ ~ ~L W)1i bull ~~ IIgt)IS

~ (H) lt - ) is~ ))IS wl-o jl

140~ 0 5 If L~II cil 1( is u - J I)r-- 1 r- ~

~)IS wli -1- sec -1- - Osec is~Lj )) I)

~ )) wIj -gt y ~l-o ~~~ )b

))gt5 ~I )) 1 0 1 (i)

)4 yj 41 5 ~r )) 0it~ Ub~ jI iSl~~wl-o

)-IS jul isI~ yo

iSjW iSly ISL is)lJ ) )4 ~I )) Jlgt )) laquoiSl-~wl-o ))gt5J ~ o~~ I$)Y

~ts ~Io ~p ~I--o jI iSr0 4 is~4J )) ) ~I l ul) ~~lS d~b I)W

~fJ4 I - de Freitas M Silva A and Reis L (2000) Numerical evaluation of failure mechanisms on composite

specimens subjected to impact loading Composites Part B Engineering Vol 31 PP 199-207

2 - Mohammadi S (2003) Discontinuum Mechanics using Finite and Discrete Elements WITPress- UK

3 - Koh CG Owen D R J and Perie D (1995) Explicit dynamic analysis of elsto-plastic laminated

composite shells implementation of non-iterative stress update schemes for the Hoffinan yield criterion

Computational Mechanics Vol 16 PP 307-314

4 - Schellekens 1 C 1 (1992) Computational strategies for composite structures PhD thesis Technische

Universiteit Delft Holland

5 - Forouzan-sepehr S and Mohammadi S (2001) A contact based method for 3D delamination analysis of

composites subjected to impact loading in S ValliaPPan N Khalili (editors) Computational Mechanicsshy

New Frontiers for the New Millennium Proceedings of 1 Asian-Pactflc Congress on computational

Mechanics- PP 691-696

6 - Mohammadi S Forouzan-sepehr S and Asadollahi A (2002) Contact based delamination and fracture

analysis of composites Thin-Walled Structures Vol 40 No 7-8 PP 595-609

7 - Masters 1 E (1987) Basic failure modes of continuous fiber composites Engineering Materials

Handbook Vol 1 CompOSites ASM International USA

8 - Borovkov A Kiylo 0 Misnik Yu and Tripolnikov T (1999) Finite element stress and analysis of

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~ )) O)~ ~~ ~~ 0) 1 - Matrix Cracking 2 - Delamination 3 - Fibre Breakage 4 - Fragmentation 5 - Theory ofPlasticity 6 - Computational Contact Mechanics 7 - Solid Element 8 - Discrete Element Method (DEM) 9 - Generalised Hook Law 10 - Transversely Isotropic 11 - Anisotropic 12 - Associated Flow Rule 13 - Normal Gap 14 - Tangential Gap 15 - Penalty 16 - Remeshing 17 - Adapti vity