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Explicit asymptotic solution for the maximum-energy-release-rate problem
EXPLICIT ASYMPTOTIC SOLUTION FOR THE MAXIMUM-ENERGY-RELEASE-RATE PROBLEMt CHEN
H. Wut
Departmentof Materials Engineering.University of Illinois at Chicago Circle (Box 4348). Chicago, IL 606W. U.S.A. (Received 23 June 1978; in mid
fom~ 6 November 1978)
Abahct-An explicit Eterm asymptotic soh~tionis obtaind for the pro&m for small branch angles. Comparisonwith the numericalsolution shows that the asymptic formulae arc reasonablyaccurate for branchangks as largeas 72’.
1. INTRODUCTION
The determination of stress-intensity factors for a crack with an infinitesimal branch at a tip is important in studying the branching phenomenon, and has been extensively studied by many researchers (e.g. [l-q). A discussion of the various approaches may be found in [7]. The result of the asymptotic analysis carried out in [6] is a single integral equation which enables the direct evaluation of the stress-intensity factors at the tip of the infinitesimal branch without assigning a small length for the branch. This integral equation was solved numerically in [7l. The purpose of this paper is to show that the aforementioned integral equation can be solved by a perturbation procedure, using the branch angle as a small parameter. The result is an explicit 3-term approximate solution which agree reasonably well with the numerical solution for branch angles up to 72”. 2. STATEMENT OF THE PROBLEM
Consider a Z-shaped crack characterized by a main crack length 2a, a branch crack length la (0 < c 4 l), and a branch crack angle air. We shah use “a” as a length scale so that the crack conhguration in the dimensionless (x1, x&plane assumes the form depicted in Fig. 1. The infinite plane is characterized by a shear modulus p and Poisson’s ratio v, and is subjected to a uniform load applied at infinity. Using a*p as a force scale, the condition at infinity is Tall = K7Ll as
lx,*+x**(+w
where a4 are the values of the dimensionless stresses TV at infinity.
/
0’ Fig. 1.
t!bppord by U.S. Army ResearchOl?kc, Durham,u&r GrantDAAG-2%7tXU272. SProfexxor. %I ss vat.15,No.7-D
(2.1)
562
C. H. WV Let
the solution near the tips of the 2 be described by 711+ 722 =
&)
1Mu,
a, l) cosB * - &Au,a, l) sin t
I
(2.2)
where (r, 0) are the near-tip polar coordinates indicated in Fig. 1. Then k&a, a, l) are the stress-intensity factors at the tips of the 2. The limits K&7,
a) = kl.AU. a, 0)
(2.3)
are the asymptotic approximations of the true stress-intensity factors k13; and the limits
&.~(a) = !‘d”k,,Au,0, ~1
(2.4)
are just the exact stress-intensity factors associated with the linear crack of length 2. It was shown in [6] that K,(u, a) + iKz(u, a) = d/(s) ehr(E)dz(ua+
h)S’(-a)
(2.5)
where the complex function $(.z) is the solution of the integral equation (2.6)t and the arc C is defined by C: f=eie
ases2T.
(2.7)
For the case of a unidirectional load depicted in Fig. 1 and defined by
cos2p
an = u
lr,
u12= u cos /3*r sin p*n
(2.8)$
(2.6) may be put into the more convenient form
sinar
S(z)=Fe
i(a+2@')a
(2.9)
Equation (2.9) has been solved numerically and the relevant results may be found in (71. The purpose of this paper is to show that for small values of a eqn (2.9) may be solved by a perturbation procedure. Speci6cally, a 3-term explicit formula is obtained for the quantity $‘(- a) needed in (2.5). It is
(2. IO) We shah see in the following section that the form of (2.10) comes out naturally from the development although it is accurate only to the order of a2. Substituting (2.10) into (2.5) we obtain:
Explicit asymptotic so!ution forthern~urn~ne~-~i~-~~
563
probkm
Crack perpendicularuz
(2.11) Crack-parallelo12 ~[Kl+iKJ=(~)~2eiD”[I+~e-b(~ (2.12)
in)+ (Tr(<-4)].
Stress-intensity factors computed from these two formulae are tabulated and compared with the corresponding numerical results[7] in Table 1. The comparisons exhibit excellent agreement for a CO.2 (36“ branch angle), and reasonable agreement can be traced to values of a as large as 0.4 (72 branch angle). Let G(u, a) be the line-to-Z energy release rate. It may be shown by using Irwin’s crack-tip-opening-displacementcalculation [8f that G(u, a) = i (K + l)[Ki?a; a) + KZ%Gall.
(2.13)
3-4v ,plane stress K= (3 - v)/( 1 + V) plane strain.
(2.14)
where
I
The energy release rate can also be calculated in terms of the properties of the solution at intinity. This was done in [6] and the result is
Gb, a) =;(I(+
l)(&+&
(2.15)
Tabkf. *Crack-Perpendicular~22 Numerical Solution
CI.00 (I.02 (I.04 (1.06 ().08 f p.10 f >.I2 f f.14 f B.16 ( ).I8 ( I.20 CI.22 CI.24
cj.26 Cj.28 C I.30 C I.32 C I.34 C I.36 C I.38 C1.40 l
.. .
‘mttlply
.
1.0000 0.9985 0.9940 0.9867 0.9766 0.9636 0.9480 0.9298 0.9092 0.8864 0.8614 0.8346 0.8061 0.7760 0.7447 0.7123 0.6790 0.6451 0.6106 0.5760 0.5412 r. Uz2m
**Multiply a12fi
Eq. (2.11)
0.0000
1.0000 0.9985 0.9941 0.9867 0.9764 0.9633 0.9473 0.9287 0.9073 0.8834 0.8571 0.8284 0.7976 0.7648 0.7302 0.6940 0.6563 0.6175 0.5778 0.5373 0.4964
0.0314 0.0625 0.0932 0.1232 0.1523 0.1803 0.2070 0.2323 0.2559 0.2777 0.2976 0.3155 0.3313 0.3450 0.3565 0.3658 0.3729 0.3778 0.3806 0.3813
.
to ootam
**Crack-Parallelai2
.
. .
pnyslcal
_
values.
Numerical Solution
0.0000 0.0314 0.0626 0.093s 0.1239 0.1537 0.1826 0.2105 0.2373 0.2628 0.2868 0.3092 0.3299 0.3487 0.3645 0.3801 0.392s 0.4026 0.4103 0.4157 0.4186
0.0000
. _ . ._ _ .
f$t-aJ=K,taJ
to obtain physical values. K1(-aJ * KIW
1.0000 0.9969 0.9817 0.9724 0.9513 0.924s 0.8921 0.8546 0.8123 0.7655 0.7147 0.6602 0.6026 0.5424 0.4801 0.4161 0.3510 0.2853 0.2195 0.1542 0.0897
-0.0941 -0.1877 -0.2800 -0.3707 -0.4590 -0.5444 -0.6265 -0.7047 -0.7787 -0.8479 -0.9121 -0.9710 -1.0240 -1.0716 -1.1131 -1.1148 -1.1177 -1.2003 -1.2170 -1.2276
.
. .
- .
anb
K2{-a)
and
K2(-a)-K,(a).
-
-K2(aj.
Eq. (2.12)
0.0000 -0.0942 -0.1877 -0.2802 -0.3710 -0.45% -0.3454 -0.6280 -0.7068 -0.7814 -0.8513 -0.9161 -0.9754 -1.0289 -1.0763 -1.1173 -1.1518 -1.179s -1.2003 -1.2143 -1.2214
1.0000 0.9969 0.9077 0.9724 0.9s12 0.9241 0.8914 0.8533 0.8100 0.7619 0.7094 0.6527 0.5924 O.SZ88 0.4625 0.3940 0.3237 0.2s22 0.1802 0.1081 0.0365
564
C. H.Wu
where (2.16) Substituting (2.5) and (2.14) into (2.13), we find
de-trwY-a)=
I
+$-Jw+6,.
(2.17)
The quantity Q consistent with the accuracy of (2.10) is
O=
-sin al ei(P+ZB*)*(l - a3 $ 2lr L
+ In l+a ,_e+k)+(~~~r’-4~+....
(2.18)
Using (2.10) and (2.18), we find that (2.17) is satisfied to the order of a2. 3.ASYMPTOTICSOLUTION Consider the complex function #(z(z)satisfying the integral equation (2.9). The function is holomorphic in the whole z-plane cut along the semi-circle C defined by (2.7). The points t = + 1 are the images of the physical locations where an e-branch meets the main crack. We have the following correspondence: -1 i z=
- a
1
a re-entrant comer a regular comer
if (I I o
a tip of the Z
1 +l
(3.1)
a reguIar comer a reentrant comer
iflX>(O.
In the neighborhood of I = f 1, the integral in (2.9) is coupled with the Lh.s., no matter how smah the parameter a is. This coupling yields the comer behavior discussed in [7]. It may be interpreted as a kind of boundary-layer expansion in terms of the small parameter a. For lz it II> 0, the contribution of the integral in (2.9) is small if Q is assumed to be small. This suggests that a regular perturbation procedure may be introduced to solve (2.9). For convenience we introduce a small parameter S = sin a?rl27r, and seek a solution of (2.9) in the form
(3.2) Substituting (3.2) into (2.9), we find that It(z) may be written as
tiz) = C S”Mz, a) +ei(a+w*h even
&iYt#&,
a)
(3.3)
where (3.4)
t$&,a)=z and
4*+lk a) =
I.‘f;;
l$;t;)d,
Thus, the functions &, can all in principle be determined.
(3.5)
Explicit asymptotic solution for the maximumcmrgy-rckasc-rate
probkm
565
Integrating (3.5) by parts, we get
f#h.+Ika) =~[1.+1(z,n)-fn+l(-a,a)l
(3.6)
where (3.7) The two quantities I,+‘(-a) and Q are just $‘(- a) = 1+
SR#x- a, a) + e’(a+2B*)r J S”&(- o, a) iF
&b
Q = g S”Q.(a) + e’(a+w*)aF lYQ.(a) . .6
(3.8) (3.9)
I 3
where
4h-w,=;fx-u,u) Q.(a) =
lc4m-dE a) dl-!A-
(3.10)
a, a).
(3.11)
The function fr(z, a) can be integrated straightforwardly. It is z-l z+l
fdz, a) = (z* - 1) In -+2z
(3.12)
where the logarithmic term is defined in accordance with the cut along C. Equations (3.10)(3.12) yield &(-o,a)=&+ln,~~~
Lt
+i7r
(3.13)
Ql(a)=(l-~z)($+ln~+i7r).
(3.14)
Unfortunately, the appearance of the logarithmic term in (3.12) makes the explicit evaluation of the higher order terms impossible. However, if the purpose is to determine $‘(- a) and Q to the order of a*, only the values of &(-a, a)(,,0 and Q4(a)lrN0 are needed. These two quantities can be explicitly evaluated. We have ~$20,0) = $0,
0) =
I, 9
dz - 3 I, “5
WO) = I, &(5,0) dz + I, ‘r$“)
‘) dz
(3.15)
dz.
(3.16)
+2-F.
(3.17)
For z on C, (3.6) and (3.12) imply that g--1 &(Z 0) = $-!lni+1+2-: =-z*- 1 InzS1 ,+1-27ri Z (
>
566
C. H. WV
Substituting (3.17) into (3.15) and (3.16). and integrating we obtain
9;(0,0)=~-4, QAO) =I? - 4.
(3.18)
This concludes the derivation of (2.10) and (2.23).
REFERENCES I. H. Anderson, Stress intensity factors for a slit with an infinitesimal branch at one tip. Jnf. 1. Fract. Mech. 5, 371-372 (196%. 2. M. A. Hussain, S. L. Pu and J. Underwood, Strain-Energy-Release Rate for a Crack Under Comb&d Mode I and Mode II. ASTM-STPd60,2-28 (1974). 3. K. Palaniswamy and W. G. Knauss, On the Problem of Crack Extension in Brittk Solids Under General Loading Co/& Jn.rr. Tech. Report SM 74-8 (1974); also in Mechanics today (Edited by S. Nemat-Nasser). Vol. 4. Pergamon Press. New York. 4. G. D. Gupta. Strain Energy Release Rate for Mixed Mode Crack Probkm. 76WAlPVP-7. 5. S. N. Cbatterjee, The stress held in the neighborhood of a branched crack in an infinite elastic sheet. Inr. 1. Solids .sfmc~uns Il. 521-538 (1975). 6. C. H. Wu, Elasticity probkms of a slender Z-crack. 1. Elact. 8(2), 183-205 (1978). 7. C. H. Wu. Maximumcnergy-release-rate criterion applied to a tensioncompression specimen with crack. 1. Elust.S(3), 235-257 (l!D8). 8. G. R. Irwin, Analysis of stressesand strains near the end of a crack traversing a plate. 1. Appl. Mech 24, MI-364 (1975). 9. C. H. Wu, Fracture under combined loads by maximum-energy-release-rate criterion. 1. Appl. Mech. 450). 553-558 (1978).
H. Wut
Departmentof Materials Engineering.University of Illinois at Chicago Circle (Box 4348). Chicago, IL 606W. U.S.A. (Received 23 June 1978; in mid
fom~ 6 November 1978)
Abahct-An explicit Eterm asymptotic soh~tionis obtaind for the pro&m for small branch angles. Comparisonwith the numericalsolution shows that the asymptic formulae arc reasonablyaccurate for branchangks as largeas 72’.
1. INTRODUCTION
The determination of stress-intensity factors for a crack with an infinitesimal branch at a tip is important in studying the branching phenomenon, and has been extensively studied by many researchers (e.g. [l-q). A discussion of the various approaches may be found in [7]. The result of the asymptotic analysis carried out in [6] is a single integral equation which enables the direct evaluation of the stress-intensity factors at the tip of the infinitesimal branch without assigning a small length for the branch. This integral equation was solved numerically in [7l. The purpose of this paper is to show that the aforementioned integral equation can be solved by a perturbation procedure, using the branch angle as a small parameter. The result is an explicit 3-term approximate solution which agree reasonably well with the numerical solution for branch angles up to 72”. 2. STATEMENT OF THE PROBLEM
Consider a Z-shaped crack characterized by a main crack length 2a, a branch crack length la (0 < c 4 l), and a branch crack angle air. We shah use “a” as a length scale so that the crack conhguration in the dimensionless (x1, x&plane assumes the form depicted in Fig. 1. The infinite plane is characterized by a shear modulus p and Poisson’s ratio v, and is subjected to a uniform load applied at infinity. Using a*p as a force scale, the condition at infinity is Tall = K7Ll as
lx,*+x**(+w
where a4 are the values of the dimensionless stresses TV at infinity.
/
0’ Fig. 1.
t!bppord by U.S. Army ResearchOl?kc, Durham,u&r GrantDAAG-2%7tXU272. SProfexxor. %I ss vat.15,No.7-D
(2.1)
562
C. H. WV Let
the solution near the tips of the 2 be described by 711+ 722 =
&)
1Mu,
a, l) cosB * - &Au,a, l) sin t
I
(2.2)
where (r, 0) are the near-tip polar coordinates indicated in Fig. 1. Then k&a, a, l) are the stress-intensity factors at the tips of the 2. The limits K&7,
a) = kl.AU. a, 0)
(2.3)
are the asymptotic approximations of the true stress-intensity factors k13; and the limits
&.~(a) = !‘d”k,,Au,0, ~1
(2.4)
are just the exact stress-intensity factors associated with the linear crack of length 2. It was shown in [6] that K,(u, a) + iKz(u, a) = d/(s) ehr(E)dz(ua+
h)S’(-a)
(2.5)
where the complex function $(.z) is the solution of the integral equation (2.6)t and the arc C is defined by C: f=eie
ases2T.
(2.7)
For the case of a unidirectional load depicted in Fig. 1 and defined by
cos2p
an = u
lr,
u12= u cos /3*r sin p*n
(2.8)$
(2.6) may be put into the more convenient form
sinar
S(z)=Fe
i(a+2@')a
(2.9)
Equation (2.9) has been solved numerically and the relevant results may be found in (71. The purpose of this paper is to show that for small values of a eqn (2.9) may be solved by a perturbation procedure. Speci6cally, a 3-term explicit formula is obtained for the quantity $‘(- a) needed in (2.5). It is
(2. IO) We shah see in the following section that the form of (2.10) comes out naturally from the development although it is accurate only to the order of a2. Substituting (2.10) into (2.5) we obtain:
Explicit asymptotic so!ution forthern~urn~ne~-~i~-~~
563
probkm
Crack perpendicularuz
(2.11) Crack-parallelo12 ~[Kl+iKJ=(~)~2eiD”[I+~e-b(~ (2.12)
in)+ (Tr(<-4)].
Stress-intensity factors computed from these two formulae are tabulated and compared with the corresponding numerical results[7] in Table 1. The comparisons exhibit excellent agreement for a CO.2 (36“ branch angle), and reasonable agreement can be traced to values of a as large as 0.4 (72 branch angle). Let G(u, a) be the line-to-Z energy release rate. It may be shown by using Irwin’s crack-tip-opening-displacementcalculation [8f that G(u, a) = i (K + l)[Ki?a; a) + KZ%Gall.
(2.13)
3-4v ,plane stress K= (3 - v)/( 1 + V) plane strain.
(2.14)
where
I
The energy release rate can also be calculated in terms of the properties of the solution at intinity. This was done in [6] and the result is
Gb, a) =;(I(+
l)(&+&
(2.15)
Tabkf. *Crack-Perpendicular~22 Numerical Solution
CI.00 (I.02 (I.04 (1.06 ().08 f p.10 f >.I2 f f.14 f B.16 ( ).I8 ( I.20 CI.22 CI.24
cj.26 Cj.28 C I.30 C I.32 C I.34 C I.36 C I.38 C1.40 l
.. .
‘mttlply
.
1.0000 0.9985 0.9940 0.9867 0.9766 0.9636 0.9480 0.9298 0.9092 0.8864 0.8614 0.8346 0.8061 0.7760 0.7447 0.7123 0.6790 0.6451 0.6106 0.5760 0.5412 r. Uz2m
**Multiply a12fi
Eq. (2.11)
0.0000
1.0000 0.9985 0.9941 0.9867 0.9764 0.9633 0.9473 0.9287 0.9073 0.8834 0.8571 0.8284 0.7976 0.7648 0.7302 0.6940 0.6563 0.6175 0.5778 0.5373 0.4964
0.0314 0.0625 0.0932 0.1232 0.1523 0.1803 0.2070 0.2323 0.2559 0.2777 0.2976 0.3155 0.3313 0.3450 0.3565 0.3658 0.3729 0.3778 0.3806 0.3813
.
to ootam
**Crack-Parallelai2
.
. .
pnyslcal
_
values.
Numerical Solution
0.0000 0.0314 0.0626 0.093s 0.1239 0.1537 0.1826 0.2105 0.2373 0.2628 0.2868 0.3092 0.3299 0.3487 0.3645 0.3801 0.392s 0.4026 0.4103 0.4157 0.4186
0.0000
. _ . ._ _ .
f$t-aJ=K,taJ
to obtain physical values. K1(-aJ * KIW
1.0000 0.9969 0.9817 0.9724 0.9513 0.924s 0.8921 0.8546 0.8123 0.7655 0.7147 0.6602 0.6026 0.5424 0.4801 0.4161 0.3510 0.2853 0.2195 0.1542 0.0897
-0.0941 -0.1877 -0.2800 -0.3707 -0.4590 -0.5444 -0.6265 -0.7047 -0.7787 -0.8479 -0.9121 -0.9710 -1.0240 -1.0716 -1.1131 -1.1148 -1.1177 -1.2003 -1.2170 -1.2276
.
. .
- .
anb
K2{-a)
and
K2(-a)-K,(a).
-
-K2(aj.
Eq. (2.12)
0.0000 -0.0942 -0.1877 -0.2802 -0.3710 -0.45% -0.3454 -0.6280 -0.7068 -0.7814 -0.8513 -0.9161 -0.9754 -1.0289 -1.0763 -1.1173 -1.1518 -1.179s -1.2003 -1.2143 -1.2214
1.0000 0.9969 0.9077 0.9724 0.9s12 0.9241 0.8914 0.8533 0.8100 0.7619 0.7094 0.6527 0.5924 O.SZ88 0.4625 0.3940 0.3237 0.2s22 0.1802 0.1081 0.0365
564
C. H.Wu
where (2.16) Substituting (2.5) and (2.14) into (2.13), we find
de-trwY-a)=
I
+$-Jw+6,.
(2.17)
The quantity Q consistent with the accuracy of (2.10) is
O=
-sin al ei(P+ZB*)*(l - a3 $ 2lr L
+ In l+a ,_e+k)+(~~~r’-4~+....
(2.18)
Using (2.10) and (2.18), we find that (2.17) is satisfied to the order of a2. 3.ASYMPTOTICSOLUTION Consider the complex function #(z(z)satisfying the integral equation (2.9). The function is holomorphic in the whole z-plane cut along the semi-circle C defined by (2.7). The points t = + 1 are the images of the physical locations where an e-branch meets the main crack. We have the following correspondence: -1 i z=
- a
1
a re-entrant comer a regular comer
if (I I o
a tip of the Z
1 +l
(3.1)
a reguIar comer a reentrant comer
iflX>(O.
In the neighborhood of I = f 1, the integral in (2.9) is coupled with the Lh.s., no matter how smah the parameter a is. This coupling yields the comer behavior discussed in [7]. It may be interpreted as a kind of boundary-layer expansion in terms of the small parameter a. For lz it II> 0, the contribution of the integral in (2.9) is small if Q is assumed to be small. This suggests that a regular perturbation procedure may be introduced to solve (2.9). For convenience we introduce a small parameter S = sin a?rl27r, and seek a solution of (2.9) in the form
(3.2) Substituting (3.2) into (2.9), we find that It(z) may be written as
tiz) = C S”Mz, a) +ei(a+w*h even
&iYt#&,
a)
(3.3)
where (3.4)
t$&,a)=z and
4*+lk a) =
I.‘f;;
l$;t;)d,
Thus, the functions &, can all in principle be determined.
(3.5)
Explicit asymptotic solution for the maximumcmrgy-rckasc-rate
probkm
565
Integrating (3.5) by parts, we get
f#h.+Ika) =~[1.+1(z,n)-fn+l(-a,a)l
(3.6)
where (3.7) The two quantities I,+‘(-a) and Q are just $‘(- a) = 1+
SR#x- a, a) + e’(a+2B*)r J S”&(- o, a) iF
&b
Q = g S”Q.(a) + e’(a+w*)aF lYQ.(a) . .6
(3.8) (3.9)
I 3
where
4h-w,=;fx-u,u) Q.(a) =
lc4m-dE a) dl-!A-
(3.10)
a, a).
(3.11)
The function fr(z, a) can be integrated straightforwardly. It is z-l z+l
fdz, a) = (z* - 1) In -+2z
(3.12)
where the logarithmic term is defined in accordance with the cut along C. Equations (3.10)(3.12) yield &(-o,a)=&+ln,~~~
Lt
+i7r
(3.13)
Ql(a)=(l-~z)($+ln~+i7r).
(3.14)
Unfortunately, the appearance of the logarithmic term in (3.12) makes the explicit evaluation of the higher order terms impossible. However, if the purpose is to determine $‘(- a) and Q to the order of a*, only the values of &(-a, a)(,,0 and Q4(a)lrN0 are needed. These two quantities can be explicitly evaluated. We have ~$20,0) = $0,
0) =
I, 9
dz - 3 I, “5
WO) = I, &(5,0) dz + I, ‘r$“)
‘) dz
(3.15)
dz.
(3.16)
+2-F.
(3.17)
For z on C, (3.6) and (3.12) imply that g--1 &(Z 0) = $-!lni+1+2-: =-z*- 1 InzS1 ,+1-27ri Z (
>
566
C. H. WV
Substituting (3.17) into (3.15) and (3.16). and integrating we obtain
9;(0,0)=~-4, QAO) =I? - 4.
(3.18)
This concludes the derivation of (2.10) and (2.23).
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