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Fracture angle and strain-energy-density-factor of a crack at hole at an arbitrary angle
FRACTURE ANGLE AND STRAIN-ENERGY-DENSITY-FACTOR OF A CRACK AT HOLE AT AN ARBITRARY ANGLEP Y. c. HSU Depmtmat of MechanicalEagiucering,The Universityof New Mexico, Ahqaquc,
NM 87131,U.S.A.
and
INTRODUCTION Trmm of past studies on crack instabii has dealt with the openiug mode (Mode 1) crack exten&m. However, in dealing with such studies, it isessentidtoconsider a ~~ in the dire&on of the applied load from the ideal condition for which the toad is applied perfectly perpeudicular to the crack plane. As such, the mixed mode crack extent&m problem must be treatad with stress intensity factors k, and k~ as being realistic. A number of studies have dttenmiacdthe~~ofk,andk~onthef~e~,&,aMtinstabilityfortheprablemofa cnsdrofI~2ainclinedatan~~~~tbefoadingaxis.Thtysrcre~w~asfdlows. ~rmdSih[l]prbdict#l~~bassdonthcPssumptiononwhichtge~witf start to extend in the plane which is normal to the maximum circumferential stress. These predi&ms were shown to agree weIl with experimental results. Williamand Ewing[2] showed that in&s&m of stress component parallel to the crack would improve the correktion between thsoryand e~tbyuainga~ti~stressatacriticefdistancebasedarthemcuinmmstress criterion. Sih[3] proposed a theory of fracture based on the field strength of the local strainenergydensity. This theory requires no calculation on the energy release rate and can treat ah mixed mode crack extension problems. The curve of -& vs p based on the maximum stress criterkm agrees well with the one based on the S theory for large values of j? and represents a lower bound for small values of fi. The objective of this paper is to study -&I and instability in the presence of k, and kzfor the problem of a crack of length L, emanating from a hole with radius R and inclined at /3 with the loading axis. THEORETICALFRA-
ANGLE,
-@a
This section gives the equations for calculating -& in the presence of k, and kl, based on the maximum stress criterion [ 1]and the strain-energydensity-factor (S) theory [3],respectively. The positive and negative values of & are defined as the fracture angles of crack extension with respect to the initial crack plane, in anti-&&wise and clockwise directions. Of particular interest is the problem of a crack of length L, emanating from a hole with radius R and inchned at fi with the loading axis as in Fii. 1.The stress intensity factors K, and k2for this probkm were solved by Hsu [4]. k, = crL”%(~, /3) k2= aL ‘nFz(A,/3)
(1)
where (I is the applied stress, the values of K and F+ are included in Table 1. tFresentedntthell#Ann~Meeting,SocietyofEneineering II-13 November 1974. 743
Scier%C, Inc., Duke University,Dxbam, NorthCarol@
Y. C.HSU and R.G. FORMAN
-
lnitlal fmcture W (-Q
--Line of fmctum
t
1
t
t
t
u
Fii 1. Acra&atthehokinaph. Maximum stress criterion It was assumed that the crack starts to extend in the plane which is normal to the maximum circumferential stress, based on the condltion[l3 k, sin 80+ k*(3cos eo- 1) = 0
(2)
alId slwituting Cqn (1) into (2) gives sin e,, + (3 cos 80- lKFz/F,) = 0, fK f 0) which determines
(3)
eo,independent of the material properties.
Strairrena~densiry (S) theory Tinisthco1y[3]statesthattlwcrack startsto extend in a radial direction (plane) alon~~ wlrich S (or a crack resistance force) is a minimum. Mathen&caUy, the neceMW and sum&It conditions for s S = u,A* + 2012k,k2+
ask:
(4)
to be minimum, are
aS 0, fff>o Z” ae
ate-e0
.
(5)
Here, for the case of plane strain u,,=~[(3-4v-c0se)(l+c0se)1
a,2=~2sine[cose-(l-2v)l ou
=
&
[4(1-
v)(l
-
cos
(6) e) +(I + cos w
~0s
0 - l)l
where v is the Poisson’s ratio, ~1is the shear modulus, 0 is shown in Fig. 2. For the case of plane stress, v in eqn (4) can be replaced by v/(1 + v). Substituting eqn (1) into (4) gives S = u2L(allR2+2a12FIF2+ atzF:).
(7)
Fmctureangleand straia-coergydcnsity-factor of a crack at hole at an arhii
Tk
8ame voluor
as In
the case Uf
745
angle
b -r/l2
c(
a
s
B =:
The 8eme values a8
in
d d d 0’ 0’ 6 d d
c
the
cj
case
of
d d d
fl-
r/6
c; d
d 6 6 d d
c
a
0000000000000000000
0
4u
oooddddd;~d~buiddd~ ~~N~~3~u,0~OOOOOOO”.0 . .
EPMVd7.No.U
&N
8
Y. C.
746
HSU and R. 0. FORMAN
R&.2. Strawc~tweamackborderwith(r,~)intbexlcptaac.
3cos28+(1-2v)cos8].
NORMALIZED STRAIN-ENERGY-DENSITY FACTOR Inserting the proper 40 obtained from eqn (8) into (91,this 00 giycs (a*SW) a ~~ by the help of eqn (7): ,%=(3-4v
> 0 and thus S is
-cos e,~t +cos &,)FrZ+4sin &&OSfl0-(1-2v)lRFz +[ql-v~1-cos80~+(1+cos80~3cos80-t~1~2f.
(11)
Here Smb~corresponds to the condition in which the crack starts to extend in the direction defined by -80 of least resistance for uniaxial tension.
Fracture angle and strain-energy-density-factor of a crack at hok at an arbii
an&
ICI
EXPERIMENT The experimental data of -&,, (~a and S, were obtained on twenty-seven plexiglass specimens for which Y= 0.33 (see drawing of specimen conftgurationin Fii 3). Here a, defines the lowest applied stress to initiate crack growth for different positions and sizes of the crack, and S,, defines a critical value of S at the point of incipient fracture. All the tests were carried out on fracture specimens containing a singlecrack emanating from a hole. The test parameters were for values of the angle /3 (/I = n/2, 5~/12, a14 and a/12) and three values of the ratio L/R (L/R = 0.1, O-5,1-O).Triplicate tests were conducted for each value of /3 and L/R. RESULTS Theoretical results On physical grounds, the negative values of F, in Table 1 are replaced by zero quantities. The numerical results of eqn (3) for -& and positive o are shown in Fig. 4. The curves of -& vs /3 are plotted for ditierent values of A based on maximum stress criterion. -306c.m
-’
(12 0 IId
l-
22~9cm m~olnl
_-I
2
-0+53cm(0~3TS In.) E&dga Franc
Fig. 3. Pkxigh
view
Phrlgba~ Edgr
ThISL
aau9l.r view
fracture specimen for studyiog a crack at a hole.
Fig. 4. Fracture angle vs crack angle for maximum stress criterion.
12
r
~0.25
Fii. 5. Fracture an&
A- 6,
angle
3
z
P
045.
crackmgk forstraiacnergy-density
8, Crock
4
r
aiterioa. Y =
VI
6
I
+
Fii 6. Fracture agk VI crack angk for s~ydcwfty criterion, Y = o-33.
Fig.
6
f ,.dk
r-025
A-k
L
PlaneOsh0l"
I
cd
B
P
onqbe
5
5s
;
7. Variations of density f&or with crack an& for Y = O-25.
”
i
Fra~hrremgk amlHthcnergykmity-factor of a crackat holeat anarbitrary an&
I
I 254 (1.00)
il.1
6610 (8181
1.00
0
0
0
2.64 0.00)
0.1
&I22
1.39
4.3
4.7
4.7
1.81 (0.751
0.1
5.soS (lJs3)
1.10
16.6
lB3
17.9
1.27 (O*B3)
1.0
S.440 fS34f
1.05
21.7
ts.1
24.2
2.94 wJo)
0.1
14.7'S t2t37)
0.74
22.1
22.1
21.4
1.91 (0.75)
0.5
8.515 (1235)
0.84
44.5
47.9
46.5
1.27 (0.39)
1.0
7.940 (1163)
0.#1
64.9
S2.6
S2.1
1.91 (0.76)
0.5
33.120 m?8)
1.S
79.0
70,s
83.!
1.27 (0.500)
1.0
2l.m
0.85
75.3
69.8
81.5
*#oospecflnns
cI 16~CS,),,2 we
Avma9a
(1) ufu
(1178)
(3137)
t4st4dparwndftflm * 10.61 x 109 ( PI2
of 2 qmcfaans.
II (8.85 x lo5 9st2 fn)
Third speelm
failed at loading grips.
ttrost titwfl
(2) strain enetqy danslty crfterfa with
I* 0.33
T&enumerical results of eqn (8)for -4%and positive Q arc t&ownin Figs. 5 and 6 for v = @25 and v = Oq33.The curves of -80 vs /3 are plotted for different values of A based on strabnergydensity factor theory. For v = O-25in Fig. 7 and v = 0.33 in Fii. 8, cwves of 16$L,jcrzL vs e are plotted for A varying from O-1to =. Average experimental results of rr, and S, and 40 are listed in columns 4 to 6 in Table 2 for difEerentvalues of j3,R and L/R. Theoretical results of -& are also listed in columns 7 and 8 in Table 2. By defining (S,&D as the critical sass-stasis factor corresponding to Mode I crack extension. The data S,/(S&fi is given in Fii. 9. I&h value is the average of three experimental values.
7so
Y. C. HSU and R. G. PGRMAN
Fii9. CritialdeesityfwtoraramUerialcoartrntO,A
-&l;A,A
-&5;O,A =l.O.
DISCUSSIONANDCONCLUSION Asseen from Fii. 4-6, curves of -& vs the crack angle S for d&rent values of the crack luqph-hole radhls rntio A - L/R based on the maximumstress criterion agree well with the ones obtained based on the S theory for B > w/6. Fur j3 C a/6, theutkofmaximum stress criteria gives a lower bound of -& obtained from S theory which depends on Poisson’s ratio v. For a given material, -&, is inlhrenced sigrMcantlyby the value of A.For A = a, the values of -I% are identical to ones in [1,3] if L is replacedby the crack length 2a used in [1,3]. As seen from cohunns 6 to 8 in Table 2, the vment between theory andexperiment is good. In other words, expepiarentsVUifythsM-Of~ j9 and A on -80 as predicted by the theoretical curves in Figs. 4-6. The crack spreads in the negative &direction in a plane for which S is the minimum as obtained in Figa 7 and 8. For the case of S > 7r/4,when A decreases and v increases, the higher the quantity 16CISIIIJa2Lbecomes. For given values of A and cr,this quantity increases with /J = 00,it is identical to the one in [3]. With S& as a andbacomesmaximumatS=w/2.ForA material constant, the lowest value of the applied load uCcr to initiate crack propagation occurs at S = w/2 for a geometry with small A and a material with low v. Based on the experimental data shown in Fii. 9, S, is further verified as a material constant for plexiglass which is independent of loading condition, crack position (/3) and crack size (A 3: L/R). This constant is regarded as the fracture toughness of the material. Aclnow&@mcnt-Tbc principalauthorgratefullyacknowledecJthe supportof this investigationby the NationalResearch CaJwil hll973-1974.
REFERENCES [l] E. Erdgro and 0. C. Sib, Go the crackextension in plates underplane loadiq and transverseshear.I. bar. En= [2] %.~~*and
I,
P. D. Ewing, Fractureundercomplex stress-the an&d crack problem.Int. J. Fmct. MecA 1(4),
[3] cC?Gy& they of crack propagation,MethodsofAnalysis and Solutionsof CmckProblems (Ed G. C. Sib). NoordhoB,Lcyden (1972%
(ReceivedDecember 1974)
NM 87131,U.S.A.
and
INTRODUCTION Trmm of past studies on crack instabii has dealt with the openiug mode (Mode 1) crack exten&m. However, in dealing with such studies, it isessentidtoconsider a ~~ in the dire&on of the applied load from the ideal condition for which the toad is applied perfectly perpeudicular to the crack plane. As such, the mixed mode crack extent&m problem must be treatad with stress intensity factors k, and k~ as being realistic. A number of studies have dttenmiacdthe~~ofk,andk~onthef~e~,&,aMtinstabilityfortheprablemofa cnsdrofI~2ainclinedatan~~~~tbefoadingaxis.Thtysrcre~w~asfdlows. ~rmdSih[l]prbdict#l~~bassdonthcPssumptiononwhichtge~witf start to extend in the plane which is normal to the maximum circumferential stress. These predi&ms were shown to agree weIl with experimental results. Williamand Ewing[2] showed that in&s&m of stress component parallel to the crack would improve the correktion between thsoryand e~tbyuainga~ti~stressatacriticefdistancebasedarthemcuinmmstress criterion. Sih[3] proposed a theory of fracture based on the field strength of the local strainenergydensity. This theory requires no calculation on the energy release rate and can treat ah mixed mode crack extension problems. The curve of -& vs p based on the maximum stress criterkm agrees well with the one based on the S theory for large values of j? and represents a lower bound for small values of fi. The objective of this paper is to study -&I and instability in the presence of k, and kzfor the problem of a crack of length L, emanating from a hole with radius R and inclined at /3 with the loading axis. THEORETICALFRA-
ANGLE,
-@a
This section gives the equations for calculating -& in the presence of k, and kl, based on the maximum stress criterion [ 1]and the strain-energydensity-factor (S) theory [3],respectively. The positive and negative values of & are defined as the fracture angles of crack extension with respect to the initial crack plane, in anti-&&wise and clockwise directions. Of particular interest is the problem of a crack of length L, emanating from a hole with radius R and inchned at fi with the loading axis as in Fii. 1.The stress intensity factors K, and k2for this probkm were solved by Hsu [4]. k, = crL”%(~, /3) k2= aL ‘nFz(A,/3)
(1)
where (I is the applied stress, the values of K and F+ are included in Table 1. tFresentedntthell#Ann~Meeting,SocietyofEneineering II-13 November 1974. 743
Scier%C, Inc., Duke University,Dxbam, NorthCarol@
Y. C.HSU and R.G. FORMAN
-
lnitlal fmcture W (-Q
--Line of fmctum
t
1
t
t
t
u
Fii 1. Acra&atthehokinaph. Maximum stress criterion It was assumed that the crack starts to extend in the plane which is normal to the maximum circumferential stress, based on the condltion[l3 k, sin 80+ k*(3cos eo- 1) = 0
(2)
alId slwituting Cqn (1) into (2) gives sin e,, + (3 cos 80- lKFz/F,) = 0, fK f 0) which determines
(3)
eo,independent of the material properties.
Strairrena~densiry (S) theory Tinisthco1y[3]statesthattlwcrack startsto extend in a radial direction (plane) alon~~ wlrich S (or a crack resistance force) is a minimum. Mathen&caUy, the neceMW and sum&It conditions for s S = u,A* + 2012k,k2+
ask:
(4)
to be minimum, are
aS 0, fff>o Z” ae
ate-e0
.
(5)
Here, for the case of plane strain u,,=~[(3-4v-c0se)(l+c0se)1
a,2=~2sine[cose-(l-2v)l ou
=
&
[4(1-
v)(l
-
cos
(6) e) +(I + cos w
~0s
0 - l)l
where v is the Poisson’s ratio, ~1is the shear modulus, 0 is shown in Fig. 2. For the case of plane stress, v in eqn (4) can be replaced by v/(1 + v). Substituting eqn (1) into (4) gives S = u2L(allR2+2a12FIF2+ atzF:).
(7)
Fmctureangleand straia-coergydcnsity-factor of a crack at hole at an arhii
Tk
8ame voluor
as In
the case Uf
745
angle
b -r/l2
c(
a
s
B =:
The 8eme values a8
in
d d d 0’ 0’ 6 d d
c
the
cj
case
of
d d d
fl-
r/6
c; d
d 6 6 d d
c
a
0000000000000000000
0
4u
oooddddd;~d~buiddd~ ~~N~~3~u,0~OOOOOOO”.0 . .
EPMVd7.No.U
&N
8
Y. C.
746
HSU and R. 0. FORMAN
R&.2. Strawc~tweamackborderwith(r,~)intbexlcptaac.
3cos28+(1-2v)cos8].
NORMALIZED STRAIN-ENERGY-DENSITY FACTOR Inserting the proper 40 obtained from eqn (8) into (91,this 00 giycs (a*SW) a ~~ by the help of eqn (7): ,%=(3-4v
> 0 and thus S is
-cos e,~t +cos &,)FrZ+4sin &&OSfl0-(1-2v)lRFz +[ql-v~1-cos80~+(1+cos80~3cos80-t~1~2f.
(11)
Here Smb~corresponds to the condition in which the crack starts to extend in the direction defined by -80 of least resistance for uniaxial tension.
Fracture angle and strain-energy-density-factor of a crack at hok at an arbii
an&
ICI
EXPERIMENT The experimental data of -&,, (~a and S, were obtained on twenty-seven plexiglass specimens for which Y= 0.33 (see drawing of specimen conftgurationin Fii 3). Here a, defines the lowest applied stress to initiate crack growth for different positions and sizes of the crack, and S,, defines a critical value of S at the point of incipient fracture. All the tests were carried out on fracture specimens containing a singlecrack emanating from a hole. The test parameters were for values of the angle /3 (/I = n/2, 5~/12, a14 and a/12) and three values of the ratio L/R (L/R = 0.1, O-5,1-O).Triplicate tests were conducted for each value of /3 and L/R. RESULTS Theoretical results On physical grounds, the negative values of F, in Table 1 are replaced by zero quantities. The numerical results of eqn (3) for -& and positive o are shown in Fig. 4. The curves of -& vs /3 are plotted for ditierent values of A based on maximum stress criterion. -306c.m
-’
(12 0 IId
l-
22~9cm m~olnl
_-I
2
-0+53cm(0~3TS In.) E&dga Franc
Fig. 3. Pkxigh
view
Phrlgba~ Edgr
ThISL
aau9l.r view
fracture specimen for studyiog a crack at a hole.
Fig. 4. Fracture angle vs crack angle for maximum stress criterion.
12
r
~0.25
Fii. 5. Fracture an&
A- 6,
angle
3
z
P
045.
crackmgk forstraiacnergy-density
8, Crock
4
r
aiterioa. Y =
VI
6
I
+
Fii 6. Fracture agk VI crack angk for s~ydcwfty criterion, Y = o-33.
Fig.
6
f ,.dk
r-025
A-k
L
PlaneOsh0l"
I
cd
B
P
onqbe
5
5s
;
7. Variations of density f&or with crack an& for Y = O-25.
”
i
Fra~hrremgk amlHthcnergykmity-factor of a crackat holeat anarbitrary an&
I
I 254 (1.00)
il.1
6610 (8181
1.00
0
0
0
2.64 0.00)
0.1
&I22
1.39
4.3
4.7
4.7
1.81 (0.751
0.1
5.soS (lJs3)
1.10
16.6
lB3
17.9
1.27 (O*B3)
1.0
S.440 fS34f
1.05
21.7
ts.1
24.2
2.94 wJo)
0.1
14.7'S t2t37)
0.74
22.1
22.1
21.4
1.91 (0.75)
0.5
8.515 (1235)
0.84
44.5
47.9
46.5
1.27 (0.39)
1.0
7.940 (1163)
0.#1
64.9
S2.6
S2.1
1.91 (0.76)
0.5
33.120 m?8)
1.S
79.0
70,s
83.!
1.27 (0.500)
1.0
2l.m
0.85
75.3
69.8
81.5
*#oospecflnns
cI 16~CS,),,2 we
Avma9a
(1) ufu
(1178)
(3137)
t4st4dparwndftflm * 10.61 x 109 ( PI2
of 2 qmcfaans.
II (8.85 x lo5 9st2 fn)
Third speelm
failed at loading grips.
ttrost titwfl
(2) strain enetqy danslty crfterfa with
I* 0.33
T&enumerical results of eqn (8)for -4%and positive Q arc t&ownin Figs. 5 and 6 for v = @25 and v = Oq33.The curves of -80 vs /3 are plotted for different values of A based on strabnergydensity factor theory. For v = O-25in Fig. 7 and v = 0.33 in Fii. 8, cwves of 16$L,jcrzL vs e are plotted for A varying from O-1to =. Average experimental results of rr, and S, and 40 are listed in columns 4 to 6 in Table 2 for difEerentvalues of j3,R and L/R. Theoretical results of -& are also listed in columns 7 and 8 in Table 2. By defining (S,&D as the critical sass-stasis factor corresponding to Mode I crack extension. The data S,/(S&fi is given in Fii. 9. I&h value is the average of three experimental values.
7so
Y. C. HSU and R. G. PGRMAN
Fii9. CritialdeesityfwtoraramUerialcoartrntO,A
-&l;A,A
-&5;O,A =l.O.
DISCUSSIONANDCONCLUSION Asseen from Fii. 4-6, curves of -& vs the crack angle S for d&rent values of the crack luqph-hole radhls rntio A - L/R based on the maximumstress criterion agree well with the ones obtained based on the S theory for B > w/6. Fur j3 C a/6, theutkofmaximum stress criteria gives a lower bound of -& obtained from S theory which depends on Poisson’s ratio v. For a given material, -&, is inlhrenced sigrMcantlyby the value of A.For A = a, the values of -I% are identical to ones in [1,3] if L is replacedby the crack length 2a used in [1,3]. As seen from cohunns 6 to 8 in Table 2, the vment between theory andexperiment is good. In other words, expepiarentsVUifythsM-Of~ j9 and A on -80 as predicted by the theoretical curves in Figs. 4-6. The crack spreads in the negative &direction in a plane for which S is the minimum as obtained in Figa 7 and 8. For the case of S > 7r/4,when A decreases and v increases, the higher the quantity 16CISIIIJa2Lbecomes. For given values of A and cr,this quantity increases with /J = 00,it is identical to the one in [3]. With S& as a andbacomesmaximumatS=w/2.ForA material constant, the lowest value of the applied load uCcr to initiate crack propagation occurs at S = w/2 for a geometry with small A and a material with low v. Based on the experimental data shown in Fii. 9, S, is further verified as a material constant for plexiglass which is independent of loading condition, crack position (/3) and crack size (A 3: L/R). This constant is regarded as the fracture toughness of the material. Aclnow&@mcnt-Tbc principalauthorgratefullyacknowledecJthe supportof this investigationby the NationalResearch CaJwil hll973-1974.
REFERENCES [l] E. Erdgro and 0. C. Sib, Go the crackextension in plates underplane loadiq and transverseshear.I. bar. En= [2] %.~~*and
I,
P. D. Ewing, Fractureundercomplex stress-the an&d crack problem.Int. J. Fmct. MecA 1(4),
[3] cC?Gy& they of crack propagation,MethodsofAnalysis and Solutionsof CmckProblems (Ed G. C. Sib). NoordhoB,Lcyden (1972%
(ReceivedDecember 1974)