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Determination of thermal stress intensity factors for traction free cusp cracks under uniform heat flow
En@whtg Fmcttue Mechanics Vol. 31. No. 4, pp. 661-672, Printed in Great Britain.
1988
0013-7944/88 $3.00 + .Nl @ 1988 F’ergmtton F’ren plc.
DETERMINATION OF THERMAL STRESS INTENSITY FACTORS FOR TRACTION FREE CUSP CRACKS UNDER UNIFORM HEAT FLOW KANG YONG LEE Professor, Department of Mechanical Engineering, Yonsei University, Seoul 120-749, Korea and HEUNG SOAP CHOI Graduate Student, Department of Mechanical Engineering, Yonsei University, Seoul 120-749,
Korea Abatnct-The thermal stress intensity factors (TSIF’s) for the cusp cracks such as hypocycloid crack, symmetric airfoil crack and symmetric lip crack are determined by using Bogdanofh complex variable approaches in plane themmelasticity. The results are expressed in terms of periodic functions of the direction of uniform heat flow. The TSIF’s are shown to be sensitive to both the direction of uniform heat flow and the thermal boundary conditions. It is also shown that Florence’s solutions for an insulated circular hole and Sib’s solutions for an insulated Griffith crack are derived from the results of the stress and displacement fields for hypocycloid crack and the TSIF’s for various cusp cracks, respectively.
INTRODUCTION of thermal stresses have been promoted since Bogdanoff’s complex variable approach[l] for the plane thermoelastic problems under steady state temperature distributions. Florence and Goodier[2,3] solved the problems of thermal stresses for spherical cavities, circular holes and an insulated ovaloid hole under uniform heat flow by employing both the complex variable method of Muskhelishvili[4] and the thermal dislocation concept. Thereafter, Sih[S] showed, from Florence’s results, that crack-tip thermal stress field equations in two dimensional elastic body are identical to Irwin’s crack-tip stress field equations under mechanical loading conditions. He also obtained the TSIF’s for the insulated Griffith crack under uniform heat flow. Recently, studies of the TSIF’s for the Griffith crack have been reported by some others[6-91. However, those of the TSIF’s for the cusp cracks seem to be relatively rare, even though there are Wu’s results[lO, 111 under mechanical loading conditions. In this paper, using Bogdanoff’s relations, we obtain the TSIF’s for the cusp cracks insulated or fixed to zero relative temperature with traction free crack surfaces. The cusp cracks to be treated are hypocycloid, symmetric airfoil and symmetric lip type. THE ANALYSES
THEORY
AND ANALYSIS
Let T(x, y) denote the steady and relative temperature in region D of x-y plane, as shown in Fig. 1. The complex temperature potential n(z) is defined by T(x, y) and its harmonic conjugate W(x, y) satisfying the Cauchy-Riemann equation in the form, i-l(z) = T(x, y) + iW(x, y) where z is a complex variable. The uniform heat flow, having uniform temperature gradient T at infinity, directed angle of A to the x-axis can be expressed by Fourier’s law as follows, lim U(z) = Q = 7eeU. I++661
(1) at an
(2)
KANG YONG LEE and HEUNG SOAP CHOI
662
c-plane
z-plane
Fig. 1. Conformal mapping.
In what follows, the thermal boundary conditions, insulated or fixed to zero relative temperature on I, are denoted as y = 1 or y = - 1, respectively. Introducing the conformal mapping function w(l) which transforms the unit circle and its exterior in l-plane onto the geometry of interest and its exterior in z-plane, the complex temperature potential and thermal boundary conditions in l-plane are defined as follows,
x(5) = fxw(Ol = T(5,d + W& 1)) aT
z=O
(Y=l)or
T=O
lim x’(J) =If~~n’(z)w’(4) ItI*lC+=
(3)
(y=-1)
(4)
= Q,zn-~_w’(l)=
By the Image method[l2], the complex temperature boundary condition eq. (4), can be obtained in the form, x(l) = X*(5) + rxt ($3
0,.
potential
(5)
satisfying
the thermal
(6)
IfI ’ 1
where xi (6) is the complex temperature potential resulted in J-plane with no holes due to heat sources, sinks or uniform heat flow, etc. Hence, complex temperature potential due to uniform heat flow Q, at infinity is given by x(5) = Nili_
w’(5)] * 5+ YCSWm w’(l))
The relation between displacements
.
(7)
and complex potentials is given by the formula[l], -
2/A(U+ iv) =
*f
K@(l)--
w(3)
Q’(t)
-‘J’(l)
+
~PW)
(8)
a3
where rC
G(I) = P j
x(lW(l)
(l+ u)(Y
P=
d5
(9)
3 - 4 Y plane strain state K=
ff
I
3-v l+y
(IO) plane stress state
Thermal stress intensity factors
663
u and D are displacements to x and y directions, respectively. a, v and p are the coefficients of linear thermal expansion, Poisson’s ratio and shear modulus, respectively. O(J) and O(l) are KO~OSOV’S complex potentials. The resultant force over an arbitrary arc AB in the elastic body is given by
i
I
-@(p)+w(5)@‘(6)+*(5)
B
(X+iY)dS=
A
1
(11)
B’
A’
a3
where X and Y are the x and y components of the external force and A’B’ is an arc in C-plane, corresponding to arc AB. Since the displacement must have a single value, the following equation should be satisfied on the arbitrary point on a hole or a rigid inclusion boundary, [2p(U+iu)]i=[.O(o)-$+W(o)-‘Y(o)]$+2p[T.D]=0
(12)
where
[TeDI= B 9 x(dw’(d T”
da,
(13)
1 and 2 denote lower and upper points, respectively, of intersection between the hole or rigid inclusion boundary and the x-axis. 1’ and 2’ are the points in p-plane corresponding to the points 1 and 2. $r means the contour integral[4] around I’ in an anti-clockwise sense. To satisfy eq. (12) Kolosov’s stress functions Q,(g) and q’(l) should be in the forms, Q(c) = Aln [+a*({)
(14)
*(I) = B In 6 + q*(C)
(IV
where A and B are the complex constants to be determined. Q*(J) and ‘Y*(l) are Muskhelishvili’s analytic functions. They can be expressed in terms of the Laurent series,
a*(c) =
C
Uj{-’
j=l
‘I’*([) = f
bj&-j
(17)
j=l
where Uj and bj are the complex constants to be determined. Substituting eqs (14) and (15) into eq. (12) results in the following relation, ~A+B=-2p[T.Dj/2?ri.
(18)
Providing that there exist no tractions on the surface of a hole, the following relation must be satisfied on I”,
o(u)-W(u)
@(a) +=
-
+Y(u)
(19)
= 0.
o’(a)
Introducing
eqs (14) and (15) into eq. (19), the following equation is obtained, w(u) -
@,*(a)+ E
w’(a)
-
@*‘(c+)+Y*(u)+{(A-I!?)&+=
= 0
w(v) ‘_& w’(u)
u-e’Q
KANG YONG LEE and HEUNG SOAP CHOI
664
Since cp is multi-valued,
it is obtained, A=
I?.
(21)
From eq. (21), eq. (20) becomes, w(u) -
-
@*‘(a) +9*(u) =
@*(a) +=
-52
w’(u)
WY4
Au.
(22)
From eqs (18) and (21), A = -s[T.DjJ2+.
(23
Hypocycloid crack with (n + 1) cusps and its exterior in the z-plane is mapped onto the unit circle and its exterior in the j-plane by the following mapping function[ll], z=
w(l)=2
(5++)
(?I = 1,2..
.)
(24
where 2Ro is the equivalent crack length and n is configuration parameter. Equation (24) gives, in the z-plane, a line crack of length 2 R. when n = 1, 6 = u and a circular hole of radius R. when 3 = u and II is sufficiently large. The cusps are located at (k
=
0, 1, . . . n)
(25)
Multiplying eq. (22) by l/(271-$ du/(u - 5) where 4 is outside r’ and integrating around r’ in an anti-clockwise sense for the given mapping function eq. (24), each term of eq. (22) gives the following results, _1 2wi f
-61 2ni
@,*(a) ,adu=-Q*(J)
w(o)W r o’(u)(u-
(26)
du = 1l~zj~j,i-n+l 5)
tn>2)
n j=l 0
(n= 1,2)
(27)
(28) da=--
1 - 6” 47
n-1
(29)
where (30)
From eqs (7), (13), (23) and (24), (31)
(32)
Thermal stress intensity factors
665
-n& ’ MM?&).
A=_W3
-k+l
(33)
( n+l >
Substituting eqs (26-29) into the integral form of eq. (22) and comparing the result with eq. (16), the coefficients aj in eq. (16) can be obtained. From eq. (16), eq. (14) becomes, (34) The other complex potential concept [ 121 in the form,
q(J)
can be obtained _
from England’s
stress-continuation
1
0 = -b(j -&Dyl).
9(l)
(35)
From eqs (34) and (35), A(1 - 6,) ln_l _ f”+’
*({)=Alnl+
~
I
n+l
&l
--
- n) n2
1’
(36)
TSIF’s at the crack-tip on the positive x-axis can be expressed by, K=Kr-i&*=2
(37)
J &W(l)
where Ki and Ku represent Mode I and Mode II TSIF’s, respectively. Kr and Ku at the cusp k can be derived from eq. (37) by substituting A - (Zkd(n + 1)) into A in the forms,
R:
2n”* =-(2n-l)(S.-y)cos(h-5 (n+ 1)2
(38)
2n”* Kr*I
=
K$K;r
(n
+
1)2
(6,
+
y)
sin A ( %J
(39)
= s(Zn-l)cot(h-s) n
(40)
where K *I.11 -= Conformal
(41)
&II
mapping function [l l] for a symmetric airfoil crack, as shown in Fig. 2, is given
z=
w(5)=$[(1+6)Cf
?+$I
(OQS
(42)
where 2R0 is the equivalent crack length and S configuration parameter. As in the case of the hypocycloid crack, the same procedure can be applied to the case of the symmetric airfoil crack to obtain the following result, (43)
KANG YONG LEE and HEUNG SOAP CHOI
666
Fig. 2. Variations of symmetric airfoil crack configuration with configuration parameter
6.
where
--The non-dimensionalized
[Q(l
- 6) -
riT(l
+ S)].
(44)
TSIF’s for the symmetric airfoil crack are K*
=
I
l-C;=
(1 + 2w’2 [l- s -
y(1 + S)] cos A
(45)
1 [l - S + y( 1 + S)] sin A 2( 1 + 26)“2
(46)
2
- K$IK;F = 1 6 y(l + 6) (1+26)coth. 1 - 6 + y(1+ 6)
(47)
For a symmetric lip crack, as shown in Fig. 3, given by[ll] (48)
.O .9
.a .7 .6 .5 .4 .3 .2 .l .O
Fig. 3. Variations of symmetric lip crack configuration with configuration parameter m.
TfKimal stress intensity factors
where 2& is the equivalent follows,
663
crack length and M co~~guration
parameter,
the results are as
where A
=
,@Rftm+ 2) [2(1--m)Q-(m+2)yO]. 8(K f I)
The non-dimensionaIized
TSIF’s for the symmetric lip cracks are
fC:=
Kg= K:tKg
3m+2 8(1+2m)“* 2-m
[2(1-
8(1 + 2m)1’2
m) - (m + 2)y] cos A
[2(1 -m)~~rn~2)~]si~~
r= (3m + W(1
(2 - m)[2(1-
- 4 -
tm+3~1
COt
m) + (m +2)y]
(51) (52)
h
*
(53)
DlSCUSSION The cusp cracks treated before become line cracks when n = 1, 6 = m = 0 from eqs (24), (42) and (48). The corresponding TSIF’s for y = 1 are obtained from eqs (38-39), (45-46) and (51-52) in the forms, KT=O Kg = sin(A - kvr).
(k = 0,l)
(54)
These results are identical to Sib’s sohttions[5]. In this case, the crack propagation depends only on the shear effect at the tip. On the other hand, when y = - 1, there exists only opening mode at the tip as
KT = cos(h - k7r) (k = 0.1)
K;=o.
(55)
For a hypocycloid crack, the variations of K: and Ki for y = 1, II = 2 are shown in Fig, 4. For n 3 2, y = 1, h = 0, the tendencies of lut and K$ to the cusp location zk can be observed from eqs (38) and (39) as
Ic:cO
for
-F
zk
K$=O
for
argzk=*;
Kt>O
for
f
Kj;“
for
O
Kc=0
for
arg zk =O,*:
K$>O
for
rr
CW
zk<2’11:
(57)
KANG YONG LEE and HEUNG SOAP CHOI
(a)
(b) Fig. 4. Variations of dimensionless thermal stress intensity factors at the cusp k of an insulated three-cusp-hypocycloid-hole (y = 1, R = 2) with heat Row direction A (a) K: (b) K&.
1.0
-1.0
(a)
(b)
Fig. 5(a) and 5(b).
Thermal stress intensityfactors
669
10
5
-105.75
-135.45 -12b,60 -150.30 Cc)
Fig. 5. Variations of dimensionless thermal stress intensity factors at the cusp k = 0 of an insulated hypocycloidal hole (y = 1) with configuration parameter n for a constant heat flow direction A (a) K: (b) Kt, (c) KTIG.
Variations of K? and KE at the cusp k = 0 with the increment of A for y = 1 and n > 2 are shown in Fig. 5. It is observed from Fig. 5(a) and (b) that the maximum value of [KTl is 1.12 for II = 4 and that of IK;“[ is 0.3143 for II = 2, respectively, and both K: and Kg approach zeros as n+=~. From Fig. 5(c), the ratio KI/KII is shown to be linearly proportional to the value of n. Comparing with TSIF’s for y = - 1 and n L 2 from eqs (38) and (39), it is observed that
(58) and that the ratio Kf/KE is independent of y. Complex potentials for a circular hole of radius R. can be obtained from eqs (34) and (36) as k(f)
= A, In 5
(59)
(60) where the subscript 00means n = ~0.The corresponding (33) in the forms, z =
A
m
=
a({)
=
Ro5
The relations of stresses and displacements (r, 6) are given by
em
31:4-H
(61)
+PRiyb K+l
a,+&)=2
~(3) and A, are given from eqs (24) and
’
(62)
with the complex potentials in polar coordinates
QYO ; -W(5)
-
[ w’(L)
w’( 5)3
(63)
KANG YONG LEE and HEUNG SOAP CHOI
-(+ + r
ir
_
@Yo : -_““‘(“[“‘1’]+Fp} Wl) w’(5) w’(l) w’(5) d5 w’(5)
re
(f-54)
MY + iue)= K@(6) - gm-@@+2rG([)]
*e-“.
Substituting eqs (59-62) into eqs (63-64), the stress components forms,
(65)
at A = 1r/2 are obtained in the
U@= -
(66)
where p=I. Ro The stress components for y = 1 become identical to Florence’s solutions[2]. The displacement components at h = 7r/2 from eqs (59-62) and (65) are written in the forms,
1
sin 8 (67)
For a symmetric airfoil crack, the variations of KT and Kg with the increment of A for y = 1 and y = - 1 are shown in Figs 6 and 7, respectively. As S varies from O-l, it is observed that for
0.5
-0.5
-1.0
(a)
h-90 60.120 30,150 o,*Mo
FE&
1.0
I10.
5-
B -3o.-150 -6O,-120 -90
6 1
a
II::
0
5
“u” .5-
-1o-
(b)
Fig. 6. Variations of dimensionless thermal stress intensity factors at the cusp crack tip of an insulated symmetric airfoil hole ( y = 1) with configuration parameter 8 for a constant heat flow direction A (a) KY 04 Kt, (c) G/K&
Thermal stress intensity factors
671 A:-15,165
290
E
'r" O
6 1
-1
t120 ?I50 ?180
-2L
(a)
(b)
Fig. 7. Variations of dimensionless thermal stress intensity factors at the cusp of a symmetric airfoil hole under zero temperature boundary condition (y = - 1) with configuration parameter S for a constant heat flow direction A (a) K: (b) Kf, (c) Kf/K&.
(a)
(b)
Cc)
Fig. 8. Variations of dimensionless thermal stress intensity factors at the cusp of an insulated symmetric lip hole (y = 1) with configuration parameter m for a constant heat flow direction A (a) K: (b) K$ (c) K:IKfi.
y = 1, IK:I increases but IKEI decreases, while for y = -1, both IKFI and lK;“l increase. The tendencies of KT and KT, to A for a symmetric airfoil crack coincide with those for the cusp k = 0 of a hypocycloid crack (n 3 2). Figures 8 and 9 show the variations of KF and Kg for a symmetric lip crack with the increment of A in the cases of y = 1 and y = - 1, respectively. The overall tendencies of K: and Kg for a symmetric lip crack and a symmetric airfoil crack can be recognised to be same from Figs 6-9. CONCLUSIONS In the study of TSIF’s for traction free hypocycloid, symmetric airfoil and symmetric lip cracks, whose surfaces are insulated or fixed to zero relative temperature, under uniform heat
KANG YONG LEE and HEUNG SOAP CHOI
672
,
5o II \
X-$5.165
“S
-25
Cc)
(b)
(a)
Fig. 9. Variations of dimensionless thermal stress intensity factors at the cusp of a symmetric lip hole under zero temperature boundary condition (y = - 1) with configuration parameter m for a constant heat flow direction A (a) K: (b) K& (c) K$K&.
flow at infinity, the following results are obtained, 1. The TSIF’s for various cusp cracks are derived in the form of periodic functions of the direction of uniform heat flow. 2. At the tip of various cusp cracks lying parallel to the direction of the uniform heat flow, there exist only either Mode I or Mode II TSIF. 3. The ratio of TSIF’s for a hypocycloid crack is independent of thermal boundary conditions and its value is linearly proportional to the number of cusps. 4. The overall tendencies of TSIF’s for a symmetric airfoil crack and a symmetric lip crack coincide with each other. 5. From the present analysis, Sih’s TSIF’s for an insulated line crack and Florence’s stress solution for an insulated circular hole are obtained. Acknowledgement-This Education, Korea.
work was supported
from a 1987 free public subscription
subject
grant of Ministry of
REFERENCES [l] J. L. Bogdanoff, Note on thermal stresses. J. a$. Mech. 21, 88 (1954). [2] A. L. Florence and J. N. Goodier, Thermal stress at spherical cavities and circular holes in uniform heat flow. J. appl. Mech. 81,293-294 (1959). [3] A. L. Florence and J. N. Goodier, Thermal stress due to disturbance of uniform heat flow by an insulated ovaloid hole. .I. a&. Mech. 27,635-639 (1960). [4] N. I. Muskhelishvili, Some Basic Problem of the Ma&muticaf Theory ofEla&ity, third edition. P. Noordhoff Ltd, Groningen, Holland (1953). [5] G. C. Sih, On the singular character of thermal stresses near a crack tip. J. appl. Mech. 29, 587-589 (1962). [6] H. Sekine, Thermal stress singularities at tips of a crack in a semi-infinite medium under uniform heat flow. Ennna _ I _ Frachcre Mech. 7, 713-729 (i975). [71 H. Sekine, Influence of an insulated circular hole on thermal stress singularities at tips of a crack. Int. J. Fracrure 13, 133-149 (1977). I31 N. Sumi, Thermal stress singularities at tips of a Griffith crack in a finite rectangular plate. &cl. Engng Design 60, 389-394 (1980). [91 K. Y. Lee and S. H. Advani, Critical loading conditions and stress intensity factors for partial or entire closure of Griffith crack under thermo-mechanical loading. Znr. J. Fracture 22, 83-90 (1983). [lOI C. H. Wu, Unconventional internal cracks part 1: symmetric variations of a straight crack. J. appl. Mech. 49. 62-68 (1982). [ill C. H. Wu, Unconventional internal cracks part 2: method of generating simple cracks. J. appl. Mech. 49, 383-388 (1982). [I21 A. H. England, Complex Variable Method in Elasricity. Wiley-Interscience, New York (197 1). (Received 9 Nooember 1987)
1988
0013-7944/88 $3.00 + .Nl @ 1988 F’ergmtton F’ren plc.
DETERMINATION OF THERMAL STRESS INTENSITY FACTORS FOR TRACTION FREE CUSP CRACKS UNDER UNIFORM HEAT FLOW KANG YONG LEE Professor, Department of Mechanical Engineering, Yonsei University, Seoul 120-749, Korea and HEUNG SOAP CHOI Graduate Student, Department of Mechanical Engineering, Yonsei University, Seoul 120-749,
Korea Abatnct-The thermal stress intensity factors (TSIF’s) for the cusp cracks such as hypocycloid crack, symmetric airfoil crack and symmetric lip crack are determined by using Bogdanofh complex variable approaches in plane themmelasticity. The results are expressed in terms of periodic functions of the direction of uniform heat flow. The TSIF’s are shown to be sensitive to both the direction of uniform heat flow and the thermal boundary conditions. It is also shown that Florence’s solutions for an insulated circular hole and Sib’s solutions for an insulated Griffith crack are derived from the results of the stress and displacement fields for hypocycloid crack and the TSIF’s for various cusp cracks, respectively.
INTRODUCTION of thermal stresses have been promoted since Bogdanoff’s complex variable approach[l] for the plane thermoelastic problems under steady state temperature distributions. Florence and Goodier[2,3] solved the problems of thermal stresses for spherical cavities, circular holes and an insulated ovaloid hole under uniform heat flow by employing both the complex variable method of Muskhelishvili[4] and the thermal dislocation concept. Thereafter, Sih[S] showed, from Florence’s results, that crack-tip thermal stress field equations in two dimensional elastic body are identical to Irwin’s crack-tip stress field equations under mechanical loading conditions. He also obtained the TSIF’s for the insulated Griffith crack under uniform heat flow. Recently, studies of the TSIF’s for the Griffith crack have been reported by some others[6-91. However, those of the TSIF’s for the cusp cracks seem to be relatively rare, even though there are Wu’s results[lO, 111 under mechanical loading conditions. In this paper, using Bogdanoff’s relations, we obtain the TSIF’s for the cusp cracks insulated or fixed to zero relative temperature with traction free crack surfaces. The cusp cracks to be treated are hypocycloid, symmetric airfoil and symmetric lip type. THE ANALYSES
THEORY
AND ANALYSIS
Let T(x, y) denote the steady and relative temperature in region D of x-y plane, as shown in Fig. 1. The complex temperature potential n(z) is defined by T(x, y) and its harmonic conjugate W(x, y) satisfying the Cauchy-Riemann equation in the form, i-l(z) = T(x, y) + iW(x, y) where z is a complex variable. The uniform heat flow, having uniform temperature gradient T at infinity, directed angle of A to the x-axis can be expressed by Fourier’s law as follows, lim U(z) = Q = 7eeU. I++661
(1) at an
(2)
KANG YONG LEE and HEUNG SOAP CHOI
662
c-plane
z-plane
Fig. 1. Conformal mapping.
In what follows, the thermal boundary conditions, insulated or fixed to zero relative temperature on I, are denoted as y = 1 or y = - 1, respectively. Introducing the conformal mapping function w(l) which transforms the unit circle and its exterior in l-plane onto the geometry of interest and its exterior in z-plane, the complex temperature potential and thermal boundary conditions in l-plane are defined as follows,
x(5) = fxw(Ol = T(5,d + W& 1)) aT
z=O
(Y=l)or
T=O
lim x’(J) =If~~n’(z)w’(4) ItI*lC+=
(3)
(y=-1)
(4)
= Q,zn-~_w’(l)=
By the Image method[l2], the complex temperature boundary condition eq. (4), can be obtained in the form, x(l) = X*(5) + rxt ($3
0,.
potential
(5)
satisfying
the thermal
(6)
IfI ’ 1
where xi (6) is the complex temperature potential resulted in J-plane with no holes due to heat sources, sinks or uniform heat flow, etc. Hence, complex temperature potential due to uniform heat flow Q, at infinity is given by x(5) = Nili_
w’(5)] * 5+ YCSWm w’(l))
The relation between displacements
.
(7)
and complex potentials is given by the formula[l], -
2/A(U+ iv) =
*f
K@(l)--
w(3)
Q’(t)
-‘J’(l)
+
~PW)
(8)
a3
where rC
G(I) = P j
x(lW(l)
(l+ u)(Y
P=
d5
(9)
3 - 4 Y plane strain state K=
ff
I
3-v l+y
(IO) plane stress state
Thermal stress intensity factors
663
u and D are displacements to x and y directions, respectively. a, v and p are the coefficients of linear thermal expansion, Poisson’s ratio and shear modulus, respectively. O(J) and O(l) are KO~OSOV’S complex potentials. The resultant force over an arbitrary arc AB in the elastic body is given by
i
I
-@(p)+w(5)@‘(6)+*(5)
B
(X+iY)dS=
A
1
(11)
B’
A’
a3
where X and Y are the x and y components of the external force and A’B’ is an arc in C-plane, corresponding to arc AB. Since the displacement must have a single value, the following equation should be satisfied on the arbitrary point on a hole or a rigid inclusion boundary, [2p(U+iu)]i=[.O(o)-$+W(o)-‘Y(o)]$+2p[T.D]=0
(12)
where
[TeDI= B 9 x(dw’(d T”
da,
(13)
1 and 2 denote lower and upper points, respectively, of intersection between the hole or rigid inclusion boundary and the x-axis. 1’ and 2’ are the points in p-plane corresponding to the points 1 and 2. $r means the contour integral[4] around I’ in an anti-clockwise sense. To satisfy eq. (12) Kolosov’s stress functions Q,(g) and q’(l) should be in the forms, Q(c) = Aln [+a*({)
(14)
*(I) = B In 6 + q*(C)
(IV
where A and B are the complex constants to be determined. Q*(J) and ‘Y*(l) are Muskhelishvili’s analytic functions. They can be expressed in terms of the Laurent series,
a*(c) =
C
Uj{-’
j=l
‘I’*([) = f
bj&-j
(17)
j=l
where Uj and bj are the complex constants to be determined. Substituting eqs (14) and (15) into eq. (12) results in the following relation, ~A+B=-2p[T.Dj/2?ri.
(18)
Providing that there exist no tractions on the surface of a hole, the following relation must be satisfied on I”,
o(u)-W(u)
@(a) +=
-
+Y(u)
(19)
= 0.
o’(a)
Introducing
eqs (14) and (15) into eq. (19), the following equation is obtained, w(u) -
@,*(a)+ E
w’(a)
-
@*‘(c+)+Y*(u)+{(A-I!?)&+=
= 0
w(v) ‘_& w’(u)
u-e’Q
KANG YONG LEE and HEUNG SOAP CHOI
664
Since cp is multi-valued,
it is obtained, A=
I?.
(21)
From eq. (21), eq. (20) becomes, w(u) -
-
@*‘(a) +9*(u) =
@*(a) +=
-52
w’(u)
WY4
Au.
(22)
From eqs (18) and (21), A = -s[T.DjJ2+.
(23
Hypocycloid crack with (n + 1) cusps and its exterior in the z-plane is mapped onto the unit circle and its exterior in the j-plane by the following mapping function[ll], z=
w(l)=2
(5++)
(?I = 1,2..
.)
(24
where 2Ro is the equivalent crack length and n is configuration parameter. Equation (24) gives, in the z-plane, a line crack of length 2 R. when n = 1, 6 = u and a circular hole of radius R. when 3 = u and II is sufficiently large. The cusps are located at (k
=
0, 1, . . . n)
(25)
Multiplying eq. (22) by l/(271-$ du/(u - 5) where 4 is outside r’ and integrating around r’ in an anti-clockwise sense for the given mapping function eq. (24), each term of eq. (22) gives the following results, _1 2wi f
-61 2ni
@,*(a) ,adu=-Q*(J)
w(o)W r o’(u)(u-
(26)
du = 1l~zj~j,i-n+l 5)
tn>2)
n j=l 0
(n= 1,2)
(27)
(28) da=--
1 - 6” 47
n-1
(29)
where (30)
From eqs (7), (13), (23) and (24), (31)
(32)
Thermal stress intensity factors
665
-n& ’ MM?&).
A=_W3
-k+l
(33)
( n+l >
Substituting eqs (26-29) into the integral form of eq. (22) and comparing the result with eq. (16), the coefficients aj in eq. (16) can be obtained. From eq. (16), eq. (14) becomes, (34) The other complex potential concept [ 121 in the form,
q(J)
can be obtained _
from England’s
stress-continuation
1
0 = -b(j -&Dyl).
9(l)
(35)
From eqs (34) and (35), A(1 - 6,) ln_l _ f”+’
*({)=Alnl+
~
I
n+l
&l
--
- n) n2
1’
(36)
TSIF’s at the crack-tip on the positive x-axis can be expressed by, K=Kr-i&*=2
(37)
J &W(l)
where Ki and Ku represent Mode I and Mode II TSIF’s, respectively. Kr and Ku at the cusp k can be derived from eq. (37) by substituting A - (Zkd(n + 1)) into A in the forms,
R:
2n”* =-(2n-l)(S.-y)cos(h-5 (n+ 1)2
(38)
2n”* Kr*I
=
K$K;r
(n
+
1)2
(6,
+
y)
sin A ( %J
(39)
= s(Zn-l)cot(h-s) n
(40)
where K *I.11 -= Conformal
(41)
&II
mapping function [l l] for a symmetric airfoil crack, as shown in Fig. 2, is given
z=
w(5)=$[(1+6)Cf
?+$I
(OQS
(42)
where 2R0 is the equivalent crack length and S configuration parameter. As in the case of the hypocycloid crack, the same procedure can be applied to the case of the symmetric airfoil crack to obtain the following result, (43)
KANG YONG LEE and HEUNG SOAP CHOI
666
Fig. 2. Variations of symmetric airfoil crack configuration with configuration parameter
6.
where
--The non-dimensionalized
[Q(l
- 6) -
riT(l
+ S)].
(44)
TSIF’s for the symmetric airfoil crack are K*
=
I
l-C;=
(1 + 2w’2 [l- s -
y(1 + S)] cos A
(45)
1 [l - S + y( 1 + S)] sin A 2( 1 + 26)“2
(46)
2
- K$IK;F = 1 6 y(l + 6) (1+26)coth. 1 - 6 + y(1+ 6)
(47)
For a symmetric lip crack, as shown in Fig. 3, given by[ll] (48)
.O .9
.a .7 .6 .5 .4 .3 .2 .l .O
Fig. 3. Variations of symmetric lip crack configuration with configuration parameter m.
TfKimal stress intensity factors
where 2& is the equivalent follows,
663
crack length and M co~~guration
parameter,
the results are as
where A
=
,@Rftm+ 2) [2(1--m)Q-(m+2)yO]. 8(K f I)
The non-dimensionaIized
TSIF’s for the symmetric lip cracks are
fC:=
Kg= K:tKg
3m+2 8(1+2m)“* 2-m
[2(1-
8(1 + 2m)1’2
m) - (m + 2)y] cos A
[2(1 -m)~~rn~2)~]si~~
r= (3m + W(1
(2 - m)[2(1-
- 4 -
tm+3~1
COt
m) + (m +2)y]
(51) (52)
h
*
(53)
DlSCUSSION The cusp cracks treated before become line cracks when n = 1, 6 = m = 0 from eqs (24), (42) and (48). The corresponding TSIF’s for y = 1 are obtained from eqs (38-39), (45-46) and (51-52) in the forms, KT=O Kg = sin(A - kvr).
(k = 0,l)
(54)
These results are identical to Sib’s sohttions[5]. In this case, the crack propagation depends only on the shear effect at the tip. On the other hand, when y = - 1, there exists only opening mode at the tip as
KT = cos(h - k7r) (k = 0.1)
K;=o.
(55)
For a hypocycloid crack, the variations of K: and Ki for y = 1, II = 2 are shown in Fig, 4. For n 3 2, y = 1, h = 0, the tendencies of lut and K$ to the cusp location zk can be observed from eqs (38) and (39) as
Ic:cO
for
-F
zk
K$=O
for
argzk=*;
Kt>O
for
f
Kj;“
for
O
Kc=0
for
arg zk =O,*:
K$>O
for
rr
CW
zk<2’11:
(57)
KANG YONG LEE and HEUNG SOAP CHOI
(a)
(b) Fig. 4. Variations of dimensionless thermal stress intensity factors at the cusp k of an insulated three-cusp-hypocycloid-hole (y = 1, R = 2) with heat Row direction A (a) K: (b) K&.
1.0
-1.0
(a)
(b)
Fig. 5(a) and 5(b).
Thermal stress intensityfactors
669
10
5
-105.75
-135.45 -12b,60 -150.30 Cc)
Fig. 5. Variations of dimensionless thermal stress intensity factors at the cusp k = 0 of an insulated hypocycloidal hole (y = 1) with configuration parameter n for a constant heat flow direction A (a) K: (b) Kt, (c) KTIG.
Variations of K? and KE at the cusp k = 0 with the increment of A for y = 1 and n > 2 are shown in Fig. 5. It is observed from Fig. 5(a) and (b) that the maximum value of [KTl is 1.12 for II = 4 and that of IK;“[ is 0.3143 for II = 2, respectively, and both K: and Kg approach zeros as n+=~. From Fig. 5(c), the ratio KI/KII is shown to be linearly proportional to the value of n. Comparing with TSIF’s for y = - 1 and n L 2 from eqs (38) and (39), it is observed that
(58) and that the ratio Kf/KE is independent of y. Complex potentials for a circular hole of radius R. can be obtained from eqs (34) and (36) as k(f)
= A, In 5
(59)
(60) where the subscript 00means n = ~0.The corresponding (33) in the forms, z =
A
m
=
a({)
=
Ro5
The relations of stresses and displacements (r, 6) are given by
em
31:4-H
(61)
+PRiyb K+l
a,+&)=2
~(3) and A, are given from eqs (24) and
’
(62)
with the complex potentials in polar coordinates
QYO ; -W(5)
-
[ w’(L)
w’( 5)3
(63)
KANG YONG LEE and HEUNG SOAP CHOI
-(+ + r
ir
_
@Yo : -_““‘(“[“‘1’]+Fp} Wl) w’(5) w’(l) w’(5) d5 w’(5)
re
(f-54)
MY + iue)= K@(6) - gm-@@+2rG([)]
*e-“.
Substituting eqs (59-62) into eqs (63-64), the stress components forms,
(65)
at A = 1r/2 are obtained in the
U@= -
(66)
where p=I. Ro The stress components for y = 1 become identical to Florence’s solutions[2]. The displacement components at h = 7r/2 from eqs (59-62) and (65) are written in the forms,
1
sin 8 (67)
For a symmetric airfoil crack, the variations of KT and Kg with the increment of A for y = 1 and y = - 1 are shown in Figs 6 and 7, respectively. As S varies from O-l, it is observed that for
0.5
-0.5
-1.0
(a)
h-90 60.120 30,150 o,*Mo
FE&
1.0
I10.
5-
B -3o.-150 -6O,-120 -90
6 1
a
II::
0
5
“u” .5-
-1o-
(b)
Fig. 6. Variations of dimensionless thermal stress intensity factors at the cusp crack tip of an insulated symmetric airfoil hole ( y = 1) with configuration parameter 8 for a constant heat flow direction A (a) KY 04 Kt, (c) G/K&
Thermal stress intensity factors
671 A:-15,165
290
E
'r" O
6 1
-1
t120 ?I50 ?180
-2L
(a)
(b)
Fig. 7. Variations of dimensionless thermal stress intensity factors at the cusp of a symmetric airfoil hole under zero temperature boundary condition (y = - 1) with configuration parameter S for a constant heat flow direction A (a) K: (b) Kf, (c) Kf/K&.
(a)
(b)
Cc)
Fig. 8. Variations of dimensionless thermal stress intensity factors at the cusp of an insulated symmetric lip hole (y = 1) with configuration parameter m for a constant heat flow direction A (a) K: (b) K$ (c) K:IKfi.
y = 1, IK:I increases but IKEI decreases, while for y = -1, both IKFI and lK;“l increase. The tendencies of KT and KT, to A for a symmetric airfoil crack coincide with those for the cusp k = 0 of a hypocycloid crack (n 3 2). Figures 8 and 9 show the variations of KF and Kg for a symmetric lip crack with the increment of A in the cases of y = 1 and y = - 1, respectively. The overall tendencies of K: and Kg for a symmetric lip crack and a symmetric airfoil crack can be recognised to be same from Figs 6-9. CONCLUSIONS In the study of TSIF’s for traction free hypocycloid, symmetric airfoil and symmetric lip cracks, whose surfaces are insulated or fixed to zero relative temperature, under uniform heat
KANG YONG LEE and HEUNG SOAP CHOI
672
,
5o II \
X-$5.165
“S
-25
Cc)
(b)
(a)
Fig. 9. Variations of dimensionless thermal stress intensity factors at the cusp of a symmetric lip hole under zero temperature boundary condition (y = - 1) with configuration parameter m for a constant heat flow direction A (a) K: (b) K& (c) K$K&.
flow at infinity, the following results are obtained, 1. The TSIF’s for various cusp cracks are derived in the form of periodic functions of the direction of uniform heat flow. 2. At the tip of various cusp cracks lying parallel to the direction of the uniform heat flow, there exist only either Mode I or Mode II TSIF. 3. The ratio of TSIF’s for a hypocycloid crack is independent of thermal boundary conditions and its value is linearly proportional to the number of cusps. 4. The overall tendencies of TSIF’s for a symmetric airfoil crack and a symmetric lip crack coincide with each other. 5. From the present analysis, Sih’s TSIF’s for an insulated line crack and Florence’s stress solution for an insulated circular hole are obtained. Acknowledgement-This Education, Korea.
work was supported
from a 1987 free public subscription
subject
grant of Ministry of
REFERENCES [l] J. L. Bogdanoff, Note on thermal stresses. J. a$. Mech. 21, 88 (1954). [2] A. L. Florence and J. N. Goodier, Thermal stress at spherical cavities and circular holes in uniform heat flow. J. appl. Mech. 81,293-294 (1959). [3] A. L. Florence and J. N. Goodier, Thermal stress due to disturbance of uniform heat flow by an insulated ovaloid hole. .I. a&. Mech. 27,635-639 (1960). [4] N. I. Muskhelishvili, Some Basic Problem of the Ma&muticaf Theory ofEla&ity, third edition. P. Noordhoff Ltd, Groningen, Holland (1953). [5] G. C. Sih, On the singular character of thermal stresses near a crack tip. J. appl. Mech. 29, 587-589 (1962). [6] H. Sekine, Thermal stress singularities at tips of a crack in a semi-infinite medium under uniform heat flow. Ennna _ I _ Frachcre Mech. 7, 713-729 (i975). [71 H. Sekine, Influence of an insulated circular hole on thermal stress singularities at tips of a crack. Int. J. Fracrure 13, 133-149 (1977). I31 N. Sumi, Thermal stress singularities at tips of a Griffith crack in a finite rectangular plate. &cl. Engng Design 60, 389-394 (1980). [91 K. Y. Lee and S. H. Advani, Critical loading conditions and stress intensity factors for partial or entire closure of Griffith crack under thermo-mechanical loading. Znr. J. Fracture 22, 83-90 (1983). [lOI C. H. Wu, Unconventional internal cracks part 1: symmetric variations of a straight crack. J. appl. Mech. 49. 62-68 (1982). [ill C. H. Wu, Unconventional internal cracks part 2: method of generating simple cracks. J. appl. Mech. 49, 383-388 (1982). [I21 A. H. England, Complex Variable Method in Elasricity. Wiley-Interscience, New York (197 1). (Received 9 Nooember 1987)