INFL??NC? and s??ar tracti?n and r?m?t? ?xt?nsi?n l?ads ?sing

INFL??NC? T?? L?CATI?N AND
CRACK ANGL?
?N T?? STR?SS INT?NSITY FACT?R

Naja? R?st?m M??sin

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Abstract: T?is
pap?r d?als ?it?
t?? ?ff?ct ?f crack ?bliq?? and its l?cati?n ?n t?? str?ss int?nsity fact?r m?d? I (KI) and II (K11) f?r a finit? plat? s?bj?ct?d t? ?niaxial
t?nsi?n str?ss. T?? pr?bl?m is s?lv?d n?m?rically ?sing
finit? ?l?m?nt s?ft?ar? ANSYS R15 and t???r?tically ?sing
mat??matically ?q?ati?ns.
A g??d agr??m?nt is ?bs?rv?d b?t???n t?? t???r?tical and
n?m?rical s?l?ti?ns in all st?di?d cas?s. ?? s??? t?at incr?asing t?? crack
angl? f l?ads t? d?cr?asing t?? val?? ?f K1and
t?? maxim?m val?? ?f K11 ?cc?rs at f=45?. F?rt??rm?r?, K11 ?q?al t? z?r? at f
= 0? and 90? ??il? K1?q?al t? z?r? at f
= 90?. ????v?r, t??r? is n? s?nsitiv? ?ff?ct t? t?? crack l?cati?n
??il? t??r?  is a c?nsid?rabl? ?ff?ct ?f t?? crack ?bliq??.

K?y ??rds: Crack,
angl?, l?cati?n, t?nsi?n, KI, K11, ANSYS R15.

I.                        
INTR?D?CTI?N

Fract?r?
can b? d?fin?d as t?? pr?c?ss ?f fragm?ntati?n ?f a s?lid int? t?? ?r m?r?
parts ?nd?r t?? str?ss?s acti?n. Fract?r? analysis d?als ?it? t?? c?mp?tati?n ?f
param?t?rs t?at ??lp t? d?sign a str?ct?r? ?it?in t?? limits ?f catastr?p?ic
fail?r?. It ass?m?s t?? pr?s?nc? ?f a crack in t?? str?ct?r?. T?? st?dy ?f crack
b??avi?r in a plat? is a c?nsid?rabl? imp?rtanc? in t?? d?sign t? av?id t??
fail?r? t?? Str?ss int?nsity fact?r inv?lv?d in fract?r? m?c?anics t? d?scrib?
t?? ?lastic str?ss fi?ld s?rr??nding a crack tip.

?as?b?
and In??ara 1 analyz?d t?? r?lati?ns b?t???n t?? str?ss int?nsity fact?rs and
t?? angl? ?f t?? ?bliq?? ?dg? crack f?r a s?mi-infinit? plat?. T???caris
and Papad?p??l?s 2 ?s?d t?? ?xp?rim?ntal m?t??d ?f r?fl?ct?d ca?stics t? st?dy
t?? infl??nc? ?f t?? g??m?try ?f an ?dg?-crack?d plat? ?n str?ss int?nsity
fact?rs K1and Kn. Kim and L?? 3 st?di?d K1and K11 f?r an ?bliq??
crack ?nd?r n?rmal and s??ar tracti?n and r?m?t? ?xt?nsi?n l?ads ?sing ABAQ?S s?ft?ar?
and analytical appr?ac? a s?mi-infinit? plan? ?it? an ?bliq?? ?dg? crack and an
int?rnal crack act?d ?n by a pair ?f c?nc?ntrat?d f?rc?s at arbitrary p?siti?n
is st?di?d by Qian and ?as?b? 4. Kim?ra and Sat? 5 calc?lat?d K1and K11 ?f
t?? ?bliq?? crack initiat?d ?nd?r fr?? ing fatig?? c?nditi?ns. F??  and Rizzi 6 d?scrib?d t?? str?ss int?nsity
fact?rs ?nd?r vari??s crack s?rfac? tracti?ns ?sing an ?bliq?? crack in a s?mi-infinit?
b?dy. C??i 7 st?di?d t?? ?ff?ct ?f crack ?ri?ntati?n angl? f?r vari??s mat?rial
and g??m?tric c?mbinati?ns ?f t?? c?ating/s?bstrat? syst?m ?it? t?? grad?d int?rfacial
z?n?. G?k?l ?t al 8 calc?lat?d t?? str?ss int?nsity fact?r ?f m?ltipl? straig?t
and ?bliq?? cracks in a riv?t ??l?. K??lil ?t al 9 ?val?at?d K1n?m?rically ?sing
lin? strain m?t??d and t???r?tically. R?c?ntllty, M??sin 10 and 11 st?di?d t???r?tically
and n?m?rically t?? str?ss int?nsity fact?rs m?d? I f?r c?nt?r, singl? ?dg? and
d??bl? ?dg? crack?d finit? plat? s?bj?ct?d t? t?nsi?n str?ss .

Patr ici
and Ma? ??ij 12 m?nti?n?d t?at, ?? can disting?is? s?v?ral mann?rs in ??ic? a
f?rc? may b? appli?d t? t?? plat? ??ic? mig?t ?nabl? t?? crack t? pr?pagat?. Ir?in
pr?p?s?d a classificati?n c?rr?sp?nding t? t?? t?r?? sit?ati?ns r?pr?s?nt?d in Fig.1.
Acc?rdingly, ?? c?nsid?r t?r?? distinct m?d?s: m?d? I, m?d? II and m?d? III. In
t?? m?d? I, ?r ?p?ning m?d?, t?? b?dy is l?ad?d by t?nsil? f?rc?s, s?c? t?at t??
crack s?rfac?s ar? p?ll?d apart in t?? y dir?cti?n. T?? m?d? II , ?r sliding m?d?,
t?? b?dy is l?ad?d by ?rc?s parall?l t? t?? crack s?rfac?s, ??ic? slid? ?v?r ?ac?
?t??r in t?? x dir?cti?n. Finally, in t?? m?d? III , ?r t?aring m?d?, t?? b?dy
is l?ad?d by s??ar f?rc?s parall?l t? t?? crack fr?nt t?? crack s?rfac?s, and t??
crack s?rfac?s slid? ?v?r ?ac? ?t??r in t?? z dir?cti?n.

 

 

 

 

 

 

 

 

 

 

 

 

 
 

str?ss fi?lds a??ad ?f a crack tip (Fig.2) f?r
m?d? I and m?d? II in a lin?ar ?lastic, is?tr?pic mat?rial ar?
as in t?? f?ll??, And?rs?n 13

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

In many sit?ati?ns, a crack is s?bj?ct t? a c?mbinati?n
?f t?? t?r?? diff?r?nt m?d?s ?f l?ading, I, II and III. A simpl? ?xampl? is a
crack l?cat?d at an angl? ?t??r t?an 90° t? a t?nsil? l?ad: t?? t?nsil? l?ad C?, is r?s?lv?d int? t?? c?mp?n?nt p?rp?ndic?lar
t? t?? crack, m?d? I, and parall?l t? t?? crack, m?d? II as s???n in Fig.3. T??
str?ss int?nsity at t?? tip can t??n b? ass?ss?d f?r ?ac? m?d? ?sing t?? appr?priat?
?q?ati?ns, Ra? 14,

 

 

 

 

 

 

 

 

 

 

Str?ss int?nsity s?l?ti?ns ar? giv?n in a vari?ty
?f f?rms, K can al?ays b? r?lat?d t? t?? t?r??g? crack t?r??g? t?? appr?priat?
c?rr?cti?n fact?r, And?rs?n 13

 

 

???r? ?: c?aract?ristic str?ss, a: c?aract?ristic
crack dim?nsi?n and Y: dim?nsi?nl?ss c?nstant t?at d?p?nds ?n t?? g??m?try and
t?? m?d? ?f l?ading.

?? can g?n?raliz?
t?? angl?d t?r??g?-t?ickn?ss crack ?f Fig.4 t? any planar crack ?ri?nt?d 90° –
p fr?m t?? appli?d n?rmal str?ss. F?r ?niaxial l?ading, t?? str?ss int?nsity
fact?rs f?r m?d? I and m?d? II ar? giv?n by K1=

 

 

 

 

???r? KI0 is t?? m?d?
I str?ss int?nsity ???n ? = 0. T?? crack-tip str?ss fi?lds (in p?lar c??rdinat?s)
f?r t?? m?d? I p?rti?n ?f t?? l?ading ar? giv?n by

 

 

 

 

 

 

 

 

 

 

 

 

 

II.                       
Mat?rials and M?t??ds

Bas?d ?n
t?? ass?mpti?ns ?f Lin?ar ?lastic Fract?r? M?c?anics L?FM and plan? strain pr?bl?m,
K1and K11 t? a finit? crack?d plat? f?r diff?r?nt angl?s and l?cati?ns ?nd?r ?niaxial
t?nsi?n str?ss?s ar? st?di?d n?m?rically and t???r?tically.

A.                  
Sp?cim?ns Mat?rial

T?? plat?
sp?cim?n mat?rial is St??l (str?ct?ral) ?it? m?d?l?s ?f ?lasticity 2.07?5 Mpa
and p?is?n’s rati? 0.29, Y??ng and B?dynas 15. T?? m?d?ls ?f plat? sp?cim?ns ?it?
dim?nsi?ns ar? s???n in Fig.5.

 

 

 

 

 

 

 

 

 

B.              T???r?tical S?l?ti?n

Val??s ?f
K1and K11 ar? t???r?tically calc?lat?d bas?d ?n t?? f?ll??ing pr?c?d?r? 1)D?t?rminati?n
?f t?? KI? (K1???n
p = 0) bas?d ?n (7), ???r? (Tada ?t al 16 )

 

 

 

2)                  
Calc?lating K1and K11 t? any plan?r crack ?ri?nt?d
(P) fr?m t?? appli?d n?rmal str?ss ?sing (8) and (9).

 

C.                  
N?m?rical S?l?ti?n

K1and K11 ar? calc?lat?d n?m?rically ?sing
finit? ?l?m?nt s?ft?ar? ANSYS R15 ?it? PLAN?183 ?l?m?nt as a discr?tizati?n ?l?m?nt.
ANSYS m?d?ls at P=0? ar? s???n in Fig.6 ?it? t?? m?s?, ?l?m?nts and
b??ndary c?nditi?ns.

 

D.                              
PLAN?183 D?scripti?n

PLAN?183 is ?s?d in t?is pap?r as a discr?tizati?n
?l?m?nt ?it? q?adrilat?ral s?ap?, plan? strain b??avi?r and p?r? displac?m?nt f?rm?lati?n.
PLAN?183 ?l?m?nt typ? is d?fin?d by 8 n?d?s ( I, J, K, L, M, N, ?, P ) ?r 6 n?d?s
( I, J, K, L, M, N) f?r q?adrilat?ral and triangl? ?l?m?nt, r?sp?ctiv?ly ?aving
t?? d?gr??s ?f fr??d?m (?x , ?y) at ?ac? n?d? (translati?ns in t?? n?dal X and
Y dir?cti?ns) 17. T?? g??m?try, n?d? l?cati?ns, and t?? c??rdinat? syst?m f?r
t?is ?l?m?nt ar? s???n in Fig.7.

 

 

 

 

 

 

 

E.                  
T?? St?di?d Cas?s

T? ?xplain
t?? ?ff?ct ?f crack ?bliq?? and its l?cati?n ?n t?? K1and K11, many cas?s (r?p?rt?d
in Tabl? 1) ar? st?di?d t???r?tically and n?m?rically.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

III.              
R?s?lts
and Disc?ssi?ns
K1and K11 val??s ar? t???r?tically calc?lat?d by (7 – 10) and n?m?rically ?sing
ANSYS R15 ?it? t?r?? cas?s as
s???n in Tabl? 1.

A. Cas? St?dy I
Figs. 8a, b, c, d, ?, f, g, ? and i ?xplain t??
n?m?rical and t???r?tical variati?ns ?f K1and K11 ?it? diff?r?nt
val??s ?f a/b rati? ???n ? = 0?, 15?, 30?, 40?, 45?, 50?, 60?, 70? and 75?, r?sp?ctiv?ly.
Fr?m t??s? Figs., it is t??
?asy t? s?? t?at t?? K1> K11 ???n ? 45? and K1? K11
at ? = 45?.

 

 

 

 

 

B.                   
Cas? St?dy
II

A c?mpr?ssi?n
b?t???n K1and K11 val??s f?r diff?r?nt crack l?cati?ns (m?d?ls b, ? and ?) at
p=30?, 45? and 60? ?it? variati?ns ?f a/b rati?
ar? s???n in Figs. 9a, b, c, d, ?, f, g, ? and i. Fr?m t??s? Figs., it is cl?ar
t?at t?? crack angl? ?as a c?nsid?rabl? ?ff?ct ?n t?? K1and K11 val??s b?t t?? ?ff?ct
?f crack l?cati?n is insignificant.

 

 

Fig.9: Varia ti?n ?f K1N?m., K1T?., K11 N?m.
and K11 T?. ?it? t?? variati?n ?f a / b f?r b, ? and ?
m?d?l
at P = 30, 45 and 60.

 

 

 

C.   
Cas? St?dy
III

Figs.
10a, b, c and d ?xplain t?? variati?ns ?f K1and K11 ?it? t?? crack angl? P = 0?,
15?, 30?, 45?, 60?, 75? and
90? f?r m?d?ls b, ? and ?. Fr?m t??s? Figs., ?? s??? t?at t?? maxim?m
K1and K11 val??s app?ar at P=0? and P=45?, r?sp?ctiv?ly.
F?rt??rm?r?, K11 ?q?al t? z?r? at P = 0? and P = 90?. G?n?rally,
t?? maxim?m val??s ?f t?? n?rmal and s??ar str?ss?s ?cc?r ?n s?rfac?s ???r? t??
P=0? and P=45?, r?sp?ctiv?ly.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fr?m all Figs., it can b? s??n t?at t??r? is n?
significant diff?r?nc? b?t???n t?? t???r?tical and n?m?rical s?l?ti?ns.

 

F?rt??rm?r?,
Figs. 11 and 12 ar? grap?ically ill?strat?d V?n._Mis?s str?ss?s c??nt?r pl?ts ?it?
t?? variati?n ?f l?cati?n and angl? ?f t?? crack, r?sp?ctiv?ly. Fr?m t??s? Figs.,
it is cl?ar t?at t?? ?ff?ct ?f crack angl? and t?? ?ff?ct ?f crack l?cati?n ar?
inc?mparabl?.

 

Fig.12: C?? nt?r pl?ts ?f V?n._Mis?s str?ss ?it?
t?? variati?n ?f crack angl? at sp? cific l?cat i?n.

 

 

IV.                        
C?ncl?si?ns

1)                 
A g??d agr??m?nt is ?bs?rv?d b?t???n t?? t???r?tical
and n?m?rical s?l?ti?ns in all st?di?d cas?s.

2)                 
Incr?asing t?? crack angl? p l?ads t? d?cr?as?
t?? val?? ?f K1and t?? maxim?m val?? ?f K11 ?cc?rs at p=45.

3)                 
K11 vanis??d at p = 0? and 90?
??il? K1vanis??d at p = 90?.

4)                 
T??r? is n? ?bvi??s ?ff?ct t? t?? crack l?cati?n
b?t t??r? is a c?nsid?rabl? ?ff?ct ?f t?? crack ?bliq??.

 

 

 

R?f?r?nc?s

 

1    
.      N. ?as?b?
and S. In??ara. Str?ss Analysis ?f a S?mi-Infinit? Plat? ?it? an ?bliq?? ?dg?
Crack. Ing?ni??r-Arc?iv, V?l?m?

49(1),
pp. 51-62, 1980.

2    
.      P.
S. T???caris and G. A. Papad?p??l?s. T?? Infl??nc? ?f G??m?try ?f ?dg?-Crack?d
Plat?s ?n K1and K11 C?mp?n?nts ?f t??

Str?ss
Int?nsity Fact?r. J??rnal ?f P?ysics D: Appli?d P?ysics. V?l. 17(12), pp.
2339-2349, 1984.

3    
.      ?.K.
Kim and S.B. L??. Str?ss int?nsity fact?rs ?f an ?bliq?? ?dg? crack s?bj?ct?d t?
n?rmal and s??ar tracti?ns. T???r?tical and

Appli?d Fract?r? M?c?anics, V?l?m? 25(2), pp.
147-154, 1996.

4    
.      J.
Qian and N. ?as?b?. An ?bliq?? ?dg? Crack and an Int?rnal Crack in a S?mi-Infinit?
Plan? Act?d ?n by C?nc?ntrat?d

F?rc? at Arbitrary P?siti?n. ?ngin??ring
Analysis ?it? B??ndary ?l?m?nts, V?l. 18, pp. 155-16, 1996.

5    
.      T.
Kim?ra and K. Sat?. Simplifi?d M?t??d t? D?t?rmin? C?ntact Str?ss Distrib?ti?n
and Str?ss Int?nsity Fact?rs in Fr?? ing

Fatig??. Int?rnati?nal J??rnal ?f Fatig??, V?l.
25, pp. 633-640, 2003.

6    
.      T.
F??  and G. Rizzi. ??ig?t F?ncti?ns f?r
Str?ss Int?nsity Fact?rs and T-Str?ss f?r ?bliq?? Cracks in a ?alf-Spac?.

Int?rnati?nal J??rnal ?f Fract?r?, V?l. 132(1),
pp. L9-L16, 2005.

7    
.      ?.
J. C??i. Str?ss Int?nsity Fact?rs f?r an ?bliq?? ?dg? Crack in a C?ating/S?bstrat?
Syst?m ?it? a Grad?d Int?rfacial Z?n?

?nd?r Antiplan? S??ar. ??r?p?an J??rnal ?f M?c?anics
A/S?lids. V?l. 26, pp. 337-3, 2007.

8    
.      G?k?l.R
, D?ayanant?.S , Adit?ya.V, S.S?r?s? K?mar. Str?ss Int?nsity Fact?r D?t?rminati?n
?f M?ltipl? Straig?t and ?bliq??

Cracks in D??bl? C?v?r B??  Riv?t?d J?int. Int?rnati?nal J??rnal ?f Inn?vativ?
R?s?arc? in Sci?nc?, ?ngin??ring and T?c?n?l?gy, V?l. 3(3), 2014.

9    
.      F.
K??lil, M. B?l???ari, N. B?ns?ddiq, A. Tal?a. A N?m?rical Appr?ac? f?r t?? D?t?rminati?n
?f M?d? I Str?ss Int?nsity

Fact?rs in PMMA Mat?rials. ?ngin??ring, T?c?n?l?gy
and Appli?d Sci?nc? R?s?arc?, V?l. 4(3), 2014.

10    
.     N.
R. M??sin. Static and Dynamic Analysis ?f C?nt?r Crack?d Finit? Plat? S?bj?ct?d
t? ?nif?rm T?nsil? Str?ss ?sing Finit?

?l?m?nt M?t??d. Int?rnati?nal J??rnal ?f M?c?anical
?ngin??ring and T?c?n?l?gy (IJM?T), V?l. 6, (1), pp. 56-70, 2015.

11    
.     N.
R. M??SIN. C?mparis?n b?t???n T???r?tical and N?m?rical S?l?ti?ns f?r C?nt?r,
Singl? ?dg? and D??bl? ?dg? Crack?d

Finit? Plat? S?bj?ct?d t? T?nsi?n Str?ss. Int?rnati?nal
J??rnal ?f M?c?anical and Pr?d?cti?n ?ngin??ring R?s?arc? and D?v?l?pm?nt (IJMP?RD),
V?l. 5(2), pp. 11-20, 2015.

12    
.     M.
Patr ici and R. M. M. Ma? ??ij. Crack Pr?pagati?n Analysis, ?? p://???.?in.t??.nl/analysis/r?p?rts/rana07-23.pdf
.

13    
.     T.L.And?rs?n.
Fract?r? M?c?anics F?ndam?ntals and Applicati?ns. T?ird ?diti?n, Tayl?r
Gr??p, CRC Pr?ss,    2005.

14    
.     L.
S. Jab?r and N. R. M??sin. Str?ss Int?nsity Fact?r f?r D??bl? ?dg? Crack?d
Finit? Plat? S?bj?ct?d t? T?nsil? Str?ss. T?i_Qar

               
?niv?rsity J??rnal f?r ?ngin??ring Sci?nc?s, V?l.7, N?. 1,
pp.101-115,  2016.

15    
.     C.
Ra?. Nat?ral Sci?nc?s Trip?s Part II- MAT?RIALS SCI?NC?- C15: Fract?r? and Fatig??.

?? ps://???.msm.cam.ac.?k/t?ac?ing/partII/c??rs?C15/C15?.pdf
.

16    
.     ?.
C. Y??ng and R. G. B?dynas. R?ark’s F?rm?las f?r Str?ss and Strain. McGra?-?ill
c?mpani?s, S?v?nt? ?diti?n, 2002.

17    
.     ?.
Tada, P. C. Paris and G. R. Ir?in. T?? Str?ss Analysis ?f Cracks ?andb??k. T?ird
?diti?n, ASM? pr?ss?s, 2000.

18    
.     ANSYS
??lp.

x

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