Naja? ma??ema?icall? equa?i?ns. A g??d agreemen? is ?bser?ed

Naja? Rus?um M??sin

 

 

 

 

Abs?rac?:
??is ?a?er deals ?i?? ??e effec? ?f crack ?blique and i?s l?ca?i?n ?n ??e s?ress
in?ensi?? fac??r m?de I (KI) and II (K11) f?r a fini?e ?la?e subjec?ed ??
uniaxial ?ensi?n s?ress. ??e ?r?blem is s?l?ed numericall? using fini?e elemen?
s?f??are ANS?S R15 and ??e?re?icall? using ma??ema?icall? equa?i?ns. A g??d
agreemen? is ?bser?ed be??een ??e ??e?re?ical and numerical s?lu?i?ns in all s?udied
cases. ?e s??? ??a? increasing ??e crack angle f leads ?? decreasing ??e ?alue ?f
K1and ??e maximum ?alue ?f K11 ?ccurs a? f=45?. Fur??erm?re, K11
equal ?? ?er? a? f = 0? and 90? ??ile K1equal ?? ?er? a?
f = 90?. ???e?er, ??ere is n? sensi?i?e effec? ?? ??e crack l?ca?i?n
??ile ??ere is a c?nsiderable effec? ?f ??e crack ?blique.

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Ke?
??rds: Crack, angle, l?ca?i?n, ?ensi?n, KI, K11, ANS?S R15.

I.                        
IN?R?DUC?I?N

Frac?ure can be defined as ??e ?r?cess ?f fragmen?a?i?n ?f a s?lid in??
??? ?r m?re ?ar?s under ??e s?resses ac?i?n. Frac?ure anal?sis deals ?i?? ??e c?m?u?a?i?n
?f ?arame?ers ??a? ?el? ?? design a s?ruc?ure ?i??in ??e limi?s ?f ca?as?r???ic
failure. I? assumes ??e ?resence ?f a crack in ??e s?ruc?ure. ??e s?ud? ?f
crack be?a?i?r in a ?la?e is a c?nsiderable im??r?ance in ??e design ?? a??id ??e
failure ??e S?ress in?ensi?? fac??r in??l?ed in frac?ure mec?anics ?? describe ??e
elas?ic s?ress field surr?unding a crack ?i?.

?asebe and In??ara 1 anal??ed ??e rela?i?ns be??een ??e s?ress in?ensi??
fac??rs and ??e angle ?f ??e ?blique edge crack f?r a semi-infini?e ?la?e. ??e?caris
and ?a?ad???ul?s 2 used ??e ex?erimen?al me???d ?f reflec?ed caus?ics ?? s?ud?
??e influence ?f ??e ge?me?r? ?f an edge-cracked ?la?e ?n s?ress in?ensi?? fac??rs
K1and Kn. Kim and Lee 3 s?udied K1and K11 f?r an ?blique crack
under n?rmal and s?ear ?rac?i?n and rem??e ex?ensi?n l?ads using ABAQUS s?f??are
and anal??ical a??r?ac? a semi-infini?e ?lane ?i?? an ?blique edge crack and an
in?ernal crack ac?ed ?n b? a ?air ?f c?ncen?ra?ed f?rces a? arbi?rar? ??si?i?n
is s?udied b? Qian and ?asebe 4. Kimura and Sa?? 5 calcula?ed K1and K11 ?f ??e
?blique crack ini?ia?ed under fre??ing fa?igue c?ndi?i?ns. Fe?? and Ri??i 6
described ??e s?ress in?ensi?? fac??rs under ?ari?us crack surface ?rac?i?ns
using an ?blique crack in a semi-infini?e b?d?. C??i 7 s?udied ??e effec? ?f
crack ?rien?a?i?n angle f?r ?ari?us ma?erial and ge?me?ric c?mbina?i?ns ?f ??e
c?a?ing/subs?ra?e s?s?em ?i?? ??e graded in?erfacial ??ne. G?kul e? al 8
calcula?ed ??e s?ress in?ensi?? fac??r ?f mul?i?le s?raig?? and ?blique cracks
in a ri?e? ??le. K?elil e? al 9 e?alua?ed K1numericall? using line s?rain me???d
and ??e?re?icall?. Recen?ll??, M??sin 10 and 11 s?udied ??e?re?icall? and
numericall? ??e s?ress in?ensi?? fac??rs m?de I f?r cen?er, single edge and d?uble
edge cracked fini?e ?la?e subjec?ed ?? ?ensi?n s?ress .

?a?r ici and Ma???eij 12 men?i?ned ??a?, ?e can dis?inguis? se?eral
manners in ??ic? a f?rce ma? be a??lied ?? ??e ?la?e ??ic? mig?? enable ??e
crack ?? ?r??aga?e. Ir?in ?r???sed a classifica?i?n c?rres??nding ?? ??e ??ree
si?ua?i?ns re?resen?ed in Fig.1. Acc?rdingl?, ?e c?nsider ??ree dis?inc? m?des:
m?de I, m?de II and m?de III. In ??e m?de I, ?r ??ening m?de, ??e b?d? is l?aded
b? ?ensile f?rces, suc? ??a? ??e crack surfaces are ?ulled a?ar? in ??e ? direc?i?n.
??e m?de II , ?r sliding m?de, ??e b?d? is l?aded b? ?rces ?arallel ?? ??e
crack surfaces, ??ic? slide ??er eac? ???er in ??e x direc?i?n. Finall?, in ??e
m?de III , ?r ?earing m?de, ??e b?d? is l?aded b? s?ear f?rces ?arallel ?? ??e
crack fr?n? ??e crack surfaces, and ??e crack surfaces slide ??er eac? ???er in
??e ? direc?i?n.

 

 

 

 

 

 

 

 

 

 

 

 

 
 

s?ress fields
a?ead ?f a crack ?i? (Fig.2) f?r m?de I and m?de II in a linear elas?ic, is??r??ic
ma?erial are
as in ??e f?ll??, Anders?n 13

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

In man? si?ua?i?ns, a crack is subjec? ?? a c?mbina?i?n
?f ??e ??ree differen? m?des ?f l?ading, I, II and III. A sim?le exam?le is a
crack l?ca?ed a? an angle ???er ??an 90° ?? a ?ensile l?ad: ??e ?ensile l?ad C?, is res?l?ed
in?? ??? c?m??nen? ?er?endicular ?? ??e crack, m?de I, and ?arallel ?? ??e
crack, m?de II as s???n in Fig.3. ??e s?ress in?ensi?? a? ??e ?i? can ??en be
assessed f?r eac? m?de using ??e a??r??ria?e equa?i?ns, Rae 14,

 

 

 

 

 

 

 

 

 

 

S?ress in?ensi?? s?lu?i?ns are gi?en in a ?arie??
?f f?rms, K can al?a?s be rela?ed ?? ??e ??r?ug? crack ??r?ug? ??e a??r??ria?e
c?rrec?i?n fac??r, Anders?n 13

 

 

??ere ?: c?arac?eris?ic s?ress, a: c?arac?eris?ic
crack dimensi?n and ?: dimensi?nless c?ns?an? ??a? de?ends ?n ??e ge?me?r? and ??e
m?de ?f l?ading.

?e
can generali?e ??e angled ??r?ug?-??ickness crack ?f Fig.4 ?? an? ?lanar crack ?rien?ed
90° – ? fr?m ??e a??lied n?rmal s?ress. F?r uniaxial l?ading, ??e s?ress in?ensi??
fac??rs f?r m?de I and m?de II are gi?en b? K1=

 

 

 

 

??ere KI0 is ??e m?de I s?ress
in?ensi?? ??en ? = 0. ??e crack-?i? s?ress fields (in ??lar c??rdina?es) f?r ??e
m?de I ??r?i?n ?f ??e l?ading are gi?en b?

 

 

 

 

 

 

 

 

 

 

 

 

 

II.                       
Ma?erials and Me???ds

Based
?n ??e assum??i?ns ?f Linear Elas?ic Frac?ure Mec?anics LEFM and ?lane s?rain ?r?blem,
K1and K11 ?? a fini?e cracked ?la?e f?r differen? angles and l?ca?i?ns under
uniaxial ?ensi?n s?resses are s?udied numericall? and ??e?re?icall?.

A.                  
S?ecimens Ma?erial

??e ?la?e s?ecimen
ma?erial is S?eel (s?ruc?ural) ?i?? m?dulus ?f elas?ici?? 2.07E5 M?a and ??is?n’s
ra?i? 0.29, ??ung and Bud?nas 15. ??e m?dels ?f ?la?e s?ecimens ?i?? dimensi?ns
are s???n in Fig.5.

 

 

 

 

 

 

 

 

 

B.              ??e?re?ical S?lu?i?n

?alues
?f K1and K11 are ??e?re?icall? calcula?ed based ?n ??e f?ll??ing ?r?cedure 1)De?ermina?i?n
?f ??e KI? (K1??en
? = 0) based ?n (7), ??ere (?ada e? al 16 )

 

 

 

2)                  
Calcula?ing K1and K11 ?? an?
?laner crack ?rien?ed (?) fr?m ??e a??lied n?rmal s?ress using (8) and (9).

 

C.                  
Numerical S?lu?i?n

K1and K11 are calcula?ed numericall? using fini?e
elemen? s?f??are ANS?S R15 ?i?? ?LANE183 elemen? as a discre?i?a?i?n elemen?.
ANS?S m?dels a? ?=0? are s???n in Fig.6 ?i?? ??e mes?, elemen?s and
b?undar? c?ndi?i?ns.

 

D.                              
?LANE183 Descri??i?n

?LANE183 is used in ??is ?a?er as a discre?i?a?i?n
elemen? ?i?? quadrila?eral s?a?e, ?lane s?rain be?a?i?r and ?ure dis?lacemen? f?rmula?i?n.
?LANE183 elemen? ???e is defined b? 8 n?des ( I, J, K, L, M, N, ?, ? ) ?r 6 n?des
( I, J, K, L, M, N) f?r quadrila?eral and ?riangle elemen?, res?ec?i?el? ?a?ing
??? degrees ?f freed?m (Ux , U?) a? eac? n?de (?ransla?i?ns in ??e n?dal X and ?
direc?i?ns) 17. ??e ge?me?r?, n?de l?ca?i?ns, and ??e c??rdina?e s?s?em f?r ??is
elemen? are s???n in Fig.7.

 

 

 

 

 

 

 

E.                  
??e S?udied Cases

?? ex?lain ??e effec? ?f crack ?blique and
i?s l?ca?i?n ?n ??e K1and K11, man? cases (re??r?ed in ?able 1) are s?udied ??e?re?icall?
and numericall?.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

III.              
Resul?s and Discussi?ns
K1and K11 ?alues are ??e?re?icall? calcula?ed b? (7 – 10) and numericall? using
ANS?S R15 ?i?? ??ree cases as
s???n in ?able 1.

A.
Case S?ud? I
Figs.
8a, b, c, d, e, f, g, ? and i ex?lain ??e numerical and ??e?re?ical ?aria?i?ns ?f
K1and K11 ?i?? differen?
?alues ?f a/b ra?i? ??en ? = 0?,
15?, 30?, 40?, 45?, 50?, 60?, 70? and 75?, res?ec?i?el?. Fr?m
??ese Figs., i? is ???
eas? ?? see ??a? ??e K1> K11 ??en ? < 45? ??ile K1< K11 ??en ? > 45? and K1? K11 a? ? =
45?.

 

 

 

 

 

B.                   
Case S?ud? II

A c?m?ressi?n be??een K1and K11 ?alues f?r differen? crack l?ca?i?ns
(m?dels b, e and ?) a? ?=30?, 45? and 60? ?i??
?aria?i?ns ?f a/b ra?i? are s???n in Figs. 9a, b, c, d, e, f, g, ? and i. Fr?m ??ese
Figs., i? is clear ??a? ??e crack angle ?as a c?nsiderable effec? ?n ??e K1and K11
?alues bu? ??e effec? ?f crack l?ca?i?n is insignifican?.

 

 

Fig.9: ?aria ?i?n ?f K1Num., K1??.,
K11 Num. and K11 ??. ?i?? ??e ?aria?i?n ?f a / b f?r b, e and ?
m?del a? ? = 30, 45 and 60.

 

 

 

C.   
Case S?ud? III

Figs. 10a, b, c and d ex?lain ??e ?aria?i?ns ?f K1and K11 ?i?? ??e
crack angle ? = 0?, 15?, 30?, 45?,
60?, 75? and 90? f?r m?dels b, e and ?. Fr?m ??ese
Figs., ?e s??? ??a? ??e maximum K1and K11 ?alues a??ear a? ?=0? and ?=45?,
res?ec?i?el?. Fur??erm?re, K11 equal ?? ?er? a? ? = 0? and ? = 90?.
Generall?, ??e maximum ?alues ?f ??e n?rmal and s?ear s?resses ?ccur ?n
surfaces ??ere ??e ?=0? and ?=45?, res?ec?i?el?.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fr?m
all Figs., i? can be seen ??a? ??ere is n? significan? difference be??een ??e ??e?re?ical
and numerical s?lu?i?ns.

 

Fur??erm?re, Figs. 11 and 12 are gra??icall? illus?ra?ed ??n._Mises s?resses
c?un??r ?l??s ?i?? ??e ?aria?i?n ?f l?ca?i?n and angle ?f ??e crack, res?ec?i?el?.
Fr?m ??ese Figs., i? is clear ??a? ??e effec? ?f crack angle and ??e effec? ?f
crack l?ca?i?n are inc?m?arable.

 

Fig.12: C?u n??r ?l??s ?f ??n._Mises
s?ress ?i?? ??e ?aria?i?n ?f crack angle a? s?e cific l?ca? i?n.

 

 

IV.                        
C?nclusi?ns

1)                 
A g??d agreemen? is ?bser?ed
be??een ??e ??e?re?ical and numerical s?lu?i?ns in all s?udied cases.

2)                 
Increasing ??e crack angle ?
leads ?? decrease ??e ?alue ?f K1and ??e maximum ?alue ?f K11 ?ccurs a? ?=45.

3)                 
K11 ?anis?ed a? ? = 0?
and 90? ??ile K1?anis?ed a? ? = 90?.

4)                 
??ere is n? ?b?i?us effec? ??
??e crack l?ca?i?n bu? ??ere is a c?nsiderable effec? ?f ??e crack ?blique.

 

 

 

References

 

1    
.      N. ?asebe and S. In??ara. S?ress Anal?sis ?f a Semi-Infini?e ?la?e
?i?? an ?blique Edge Crack. Ingenieur-Arc?i?, ??lume

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

2    
.      ?. S. ??e?caris and G. A. ?a?ad???ul?s. ??e Influence ?f Ge?me?r?
?f Edge-Cracked ?la?es ?n K1and K11 C?m??nen?s ?f ??e

S?ress
In?ensi?? Fac??r. J?urnal ?f ???sics D: A??lied ???sics. ??l. 17(12), ??.
2339-2349, 1984.

3    
.      ?.K. Kim and S.B. Lee. S?ress in?ensi?? fac??rs ?f an ?blique
edge crack subjec?ed ?? n?rmal and s?ear ?rac?i?ns. ??e?re?ical and

A??lied Frac?ure Mec?anics, ??lume
25(2), ??. 147-154, 1996.

4    
.      J. Qian and N. ?asebe. An ?blique Edge Crack and an In?ernal
Crack in a Semi-Infini?e ?lane Ac?ed ?n b? C?ncen?ra?ed

F?rce a? Arbi?rar? ??si?i?n.
Engineering Anal?sis ?i?? B?undar? Elemen?s, ??l. 18, ??. 155-16, 1996.

5    
.      ?. Kimura and K. Sa??. Sim?lified Me???d ?? De?ermine C?n?ac? S?ress
Dis?ribu?i?n and S?ress In?ensi?? Fac??rs in Fre??ing

Fa?igue. In?erna?i?nal J?urnal
?f Fa?igue, ??l. 25, ??. 633-640, 2003.

6    
.      ?. Fe?? and G. Ri??i. ?eig?? Func?i?ns f?r S?ress In?ensi?? Fac??rs
and ?-S?ress f?r ?blique Cracks in a ?alf-S?ace.

In?erna?i?nal J?urnal ?f Frac?ure,
??l. 132(1), ??. L9-L16, 2005.

7    
.      ?. J. C??i. S?ress In?ensi?? Fac??rs f?r an ?blique Edge Crack
in a C?a?ing/Subs?ra?e S?s?em ?i?? a Graded In?erfacial ??ne

under An?i?lane S?ear. Eur??ean
J?urnal ?f Mec?anics A/S?lids. ??l. 26, ??. 337-3, 2007.

8    
.      G?kul.R , D?a?anan??.S , Adi???a.?, S.Sures? Kumar. S?ress In?ensi??
Fac??r De?ermina?i?n ?f Mul?i?le S?raig?? and ?blique

Cracks in D?uble C??er Bu??
Ri?e?ed J?in?. In?erna?i?nal J?urnal ?f Inn??a?i?e Researc? in Science,
Engineering and ?ec?n?l?g?, ??l. 3(3), 2014.

9    
.      F. K?elil, M. Bel??uari, N. Benseddiq, A. ?al?a. A Numerical A??r?ac?
f?r ??e De?ermina?i?n ?f M?de I S?ress In?ensi??

Fac??rs in ?MMA Ma?erials.
Engineering, ?ec?n?l?g? and A??lied Science Researc?, ??l. 4(3), 2014.

10    
.     N. R. M??sin. S?a?ic and D?namic Anal?sis ?f Cen?er Cracked Fini?e
?la?e Subjec?ed ?? Unif?rm ?ensile S?ress using Fini?e

Elemen? Me???d. In?erna?i?nal
J?urnal ?f Mec?anical Engineering and ?ec?n?l?g? (IJME?), ??l. 6, (1), ??.
56-70, 2015.

11    
.     N. R. M??SIN. C?m?aris?n be??een ??e?re?ical and Numerical S?lu?i?ns
f?r Cen?er, Single Edge and D?uble Edge Cracked

Fini?e ?la?e Subjec?ed ?? ?ensi?n
S?ress. In?erna?i?nal J?urnal ?f Mec?anical and ?r?duc?i?n Engineering Researc?
and De?el??men? (IJM?ERD), ??l. 5(2), ??. 11-20, 2015.

12    
.     M. ?a?r ici and R. M. M. Ma???eij. Crack ?r??aga?i?n Anal?sis, ????://???.?in.?ue.nl/anal?sis/re??r?s/rana07-23.?df
.

13    
.     ?.L.Anders?n. Frac?ure Mec?anics Fundamen?als and A??lica?i?ns. ??ird
Edi?i?n, ?a?l?r Gr?u?, CRC ?ress,        2005.

14    
.     L. S. Jabur and N. R. M??sin. S?ress In?ensi??
Fac??r f?r D?uble Edge Cracked Fini?e ?la?e Subjec?ed ?? ?ensile S?ress. ??i_Qar

                Uni?ersi?? J?urnal f?r Engineering
Sciences, ??l.7, N?. 1, ??.101-115, 
2016.

15    
.     C. Rae. Na?ural Sciences ?ri??s ?ar? II- MA?ERIALS SCIENCE- C15:
Frac?ure and Fa?igue.

????s://???.msm.cam.ac.uk/?eac?ing/?ar?II/c?urseC15/C15?.?df
.

16    
.     ?. C. ??ung and R. G. Bud?nas. R?ark’s F?rmulas f?r S?ress and S?rain.
McGra?-?ill c?m?anies, Se?en?? Edi?i?n, 2002.

17    
.     ?. ?ada, ?. C. ?aris and G. R. Ir?in. ??e S?ress Anal?sis ?f
Cracks ?andb??k. ??ird edi?i?n, ASME ?resses, 2000.