Indian Journal of Engineering
Vol. 2, April 1995, pp. 62-79
& Materials
Sciences
Stress analysis of an infinite plate containing two unequal collinear elliptical
holes under in-plane stresses at infinity t
V G Ukadgaonker, R R Avargerimath
Department of Mechanical Engineering, Indianlnstitute
& S D Koranne
of Technology, Powai, Bombay 400 076, India
Received 30 December 1993, accepted 23 August 1994
A closed form analytical solution ~o the problem of an infinite plate containing collinear unequal
elliptical holes subjected to in-plane loadings at an angle {3with respect to x or y-axis on infinite
boundary of the plate is presented. The problem is formulated in the complex plane using the Kolosoff-Muskhellshvili's complex stress functions and further the Schwarz's Alternating Method is used
to solve the problem of doubly connected region. The stress concentration factor for holes and
stress intensity factors at all crack tips for varying sizes and centre to centre distances are evaluated.
Some displacement formulation and the checked by Finite Element Method using displacement formulation and the two solutions are in good agreement The present results are compared with reported ones obtained by other methods. An analytical method for locating point in the vicinity of
ellipses where the local and global strain energy density are equal is also presented.
I
The fracture process of the material is closely associated with the interactio~ growth and propagation of defects such as cracks, voids and inclusions. The study of interaction of above defects
existing in a body subjected to a given form of
loading is therefore of particular importance. In
this paper, closed form solutions to the case of
two collinear unequal elliptical holes subjected to
(i) uniform tension, P, parallel to collinear major
axes (ii) uniform tension, P, perpendicular to majQr axes, and (iii) uniform shear, T, on infinite
boundary of the plate are presented, which can
give solutions for cracks and circular holes as special cases.
The lengths of major axes of two holes are 2a
and 2g, the lengths of minor axes are 2b and 2~
respectively. The two holes are collinear along
major axes with centre distance as d as shown in
Fig.!.
.••.
The boundary value problem in two dimensional elasticity can be reduced to the solution of two
complex stress functions given by Kolosoff-Muskhelishvili method as given by Muskhelishvili 1.
The problem of infinite plate with one hole of any
contour can be solved by making use of a suitable
mapping function which will map the infinite region to outside of a circle of unit radius. This
procedure, however, becomes rather cumbersome
for multiply connected regions. Use of Schwarz's
tPart of this paper was presented
Applied Mechanics, Ottawa, 1989.
at Canadian Congress of
x
y
p
x
lbl
x
Fig. I-Two elliptical holes in an infinite plate under (a) uniaxial tension in Y-direction, (b) uniaxial tension in X-direction, and (c) uniform shear (T) at infinity
UKADGAONKER
Alternatmg Method reduces a problem of mulnply
connected region to a sequence of problems in a
simply connected domains as given by Sokolnikoff2.
Many problems such as, interaction effect of
two arbitrarily oriented and elliptical holes by
cracks, stress analysis of a plate containing two elliptical holes subjected to uniforms pressures and
tangential stresses on hole boundaries, two unequal circular holes with uniform pressures and
tangential stresses, have been solved using Alternate Technique3'9• A novel method using same
technique is also developed by Ukadgaonker and
AwasarelO-14 for isotropic plate with circular hole,
elliptical hole, equilateral triangular hole, rectangular hole with rounded comers and orthotropic
plate with elliptical hole. In this paper, a specific
case when the elliptical holes are collinear has
been studied in details.
First Approximation
First approximation. to the solution of an infinite plate with two elliptical holes is obtained by
solving two problems of each hole in infinite plate
and superposing each of these solutions on the
other. The stress functions ;(~) and ..p(~) valid for
single hole for each type of loading are obtained
by mapping the elliptical hole in Z-plane to the region outside of a unit circle in complex ~-plane
using the mapping functions,
circle; the constants B,C,B' and C' represent the
stresses and rigid body rotation at infinity and are
given by
B=
d,aJ)
x
+ d,aJ)
B'=
y.
4
'
2G
C=--w
.
l+K
'
0,,1
1-'
0,,1
+m).
--' 2 .,
R_a+b
m= a+b
(00)
uy
C=
(aJ)
-
2
u~oo)
.(,00)
xy
and analytic stress functions ;
given by
O(~)
and ..p0(~) are
1
l(t)
; (~)= --.f-(
2my t-."r)dt
°
..p0(~)=_.l....f
let)
...
(~ -
(4)
;o(~) '... (5)
dt_~(l;m~Z)
2m y (t- ~)
m)
t is boundary value of ~ on the circle (y) of unit
radius in the mapped plane which defines the
boundary condition foe t) as
l(t)
= ;O(t) +
w(t) ;O'(t)+ ..p0(t)
w'(t)
... (6)
The stress functions ;(~) and·..p(~) so obtained as
first approximation for each hole and each type of
loading are as follows:
(i) For uniaxial tensions (P):
a-b
ZI=Wl(~I)=R(1-
63
et al.: STRESS ANALYSIS OF AN INFINITE PlATE
;1
0,,1
4
(r)=PR[1-
0,,1
... (7)
± (2=+m))
~1
and
PR [ =+2~1-~I
1 + ~1'(1
+ mZ)
..p~(~1)=4
(m-~i)
n=g-h
2 '
g+h
S=g+h.
~z
Zz= wz(~z)=s(~~+.!!:.);
... (1)
where,
~= p ei6
in
complex
~-plane
represented
by
~= ; + iT}, suffix 1 and 2 correspond to first and
second hole respectively,
stress functions
and
+ iF.
'
;.(~)= - 2.7f ( 1+. K")log ~+(B+iC)R~+;o(~)
4
..pz(~z) = PS
... (2)
+ iF.
..p(~)= -
1+ K")
2.7f (I;
4
;Z(~Z)=PS[~
... (8)
... (9)
Z
~z
±(2=+n))
general complex
F,
K(F,
m~p(2 m)
=+ m))
± (1 +~d~l-
)
log ~+(B'+iC)R~+
..p0(~)
... (3)
where FI; and F" are the resultant forces in ; and 1/
directions, respectively on the boundary of unit
[=+
~z
2~z _~+
(n- +~~)
~z(l
nZ)
(~~ .•...n)
± (1 +~zn~~)(2
=+ n))
... (10)
where, top row of signs correspond to uniform
tension P, parallel to major axis and bottom row
of signs correspond to uniform tension, P, perpendicular to major axis.
64
INDIAN J. ENG. MATER. SCI., APRIL 1995
· .. (11)
+
(~l -
Cz)(1
.
(n-(~l -
+ nZ)
Cz)
· .. (12)
± (1 + n( ~l
- Cz)z)(2 z + n)
(~l - CZ)((~l - Cz) - n)
... (13)
· .. (14)
Cz)
± (~lCz(2+n~)
These stress functions
through,
Or+ 08=4
08- Or+2i
Cz
... (19)
... (20)
Re [tf/(~)]
w'(~)
2eZi8
'rr8=
[/.~\3
- w"(~) tf/( ~))
and if the crack is present
(w'(~);"(~)
w'(~)~)]
+ tp'(
... (16)
then Mode-I
and
K.- iKu=·2
~
-x f(~)
...
(17)
a
a is half crack length.
f( ~)is derivative
... (21)
C1
(2 + m))
(~Z+Cl)
Mode-IT stress intensity
factors can be obtained as
1'l
+
Z
.•where subscripts 12 means translation from 1 to 2
and 21 means from 2 to 1 and C. and C2 are distances between centres of the two holes in the
mapped plane defined as
of ;( ~)
But the simple superposition does not take into
account the interaction effect of the second hole in
the vicinity. For this purpose Schwarz's Alternating technique is used to get the solution as second
approximation.
Second Approximation
Starting from the single hole solutions valid
near second hole, the stress functions ;2 (~2) and
tp2 (~2) are translated to the centre of first hole by
putting ~2 = ~. - C2 in Eqs (9), (10), (13) and (14).
Similarly, the stress functions valid near first hole
are translated to the centre of second hole by substituting ~. = '~2 + C1 in Eqs (7), (8), (11), and (12).
The resulting translated stress functions for each
hole are as follows:
... (18)
C1 = dIR+J(dIR)z-
2
(ii) For uniform shear
4m
(T):
. .. (22)
... (23)
UKADGAONKER
65
et al: STRESS ANALYSIS OF AN INFINITE PlATE
.. ; (24)
+ t d 1+ ~2) ± __ (m_2_+_1_)(_2
_=F_n)
(m'"
(25)
t1)
tt)(Jm+
2([m+
C2)2
(m2+1)(2=t=n)
_
(1+n2)
2([mtt)([m- C2)2 2(Jn+ C2-
=F
..
td
These translated ..stress functions give a non-zero
boundary condition of 121(11)on the first hole and
± 2n([ir+
(1+n2)(2=Fn)
/12(12)on the second hole which does not satisfy
C2':'" tt) ]_td1+mti);lt(tt)
(ti-- m)
the stress free boundary conditions It (11)= 0 and
... (27)
h(t2)=0. In order to correct these boundary conditions, new problems of an infinite plate with first where, ;11(t1) is the' derivative of COl'rectedstress
.
hole or second hole and with the boundary condi- function ;1l( t1)'
tion which is negative of 121(t1) or 112(t2) respectively applied to them are solved using the
Cauchy's integral equation and the corrected ;2(t2)=4 PR (t2+ Ct)±(t2+
Ct)- t2
(2=F m)
n
stress functions are obtairied. The actual stress
functions valid near each hole are then obtained
(2 =Fm)( - 2Ct - n2cit2 + nt2 - nCit2
as the sunt of· translated' streSs functions artd cor- 3cit2 - 2n2td
rected· fwttmons. Thus the aetUal streSs functions
(1 + t2 Ct)2( ci - ,,)2
tD 8l1d "'1 (t1) valid fot first hate, '2 (~i) artd
'/12('t~)valid fotseetilid hole are as givelibelow:"±~_
1=F
(2=Fm)
(i) For utliaDal tel1sidrt(P):
.
t2 Cd1 + t2 Ct) mCd1 + t2 Ct)
I
,d
(1 + m2)
,dt1)-4"
psi (tl-C2)±(t1-C2)
(2+n)
(2 ~ n)(2C2
t1
m
2([m+
±---------+
mC1t1 -' 3C~t1 - m2C~tt
-
Ct)(1 + t2([m+
(2 =F
+ mt1 + 2m2C2)
(1- d2 t1)2(C~- m)2
2m(1
m)(1
+ t2([m+
Cd)
m2)
C1))([m+
... (28)
C1m)(
(1 +
Ct)2Ct)]
=F(2 =F
1 +t2 2t2
(2=t=n)(1-2t1C2)
±
C2(1- tt
C2)2
1/J2(t2)
(1+,*2)
2{fh-
-
C2)(1 +
tt (fh-
=4
PR!
=F
2(tz
+ Ct).
Ct)
1
C2))
+ (t1 + C1)(1 + m1)
(m-(t2 + ctf)
± 2n(;;,_
'"
([ir- C2)] ]
(1+n2)(2=Fn)
C2)[1 + tt
'/11(t1)'" 4"
psi
+ (t1-
=t=
C2)(1
(n-{t1-
2ft1 -
C2)-
(26)
± (1 + m( t2 + ~1)2)(~=Fm)
(t2+C1)((tZ+Ct) -m)
1
+ n2)=F (1 +'n(t1- C2)2)(2=F
n)
C2)2)
(t1-C2)[n-(t1C2)2]
+Ct=FCd2=Fm}
(t2+Ct)
+
~±
(2=Fm)
t2 Ct(1+Cttz)
±-----(n- t1)'
tz(1+n2)
(nZ+1)(2=Fm)
z
2(Jn+
tz)([ir-
Ct)1
66
INDIAN J, ENG. MATER. seI., APRIL 1995
=F--~--
(1 + m2)
1)(2=Fm)
(n2+
2([;,-l;2)([;,+
(1 + m2)
2([m- Cj-l;2)
CI)2
2m([m+
C1)[1
+ l;2([m+
C])]
... (3~)
+ C1(1(1
CI l;2)
l;2f )
++2CI
± 2m([m(1 + m2)(2=F
m) )
CI -l;2)
_ l;2 (1 + nl;~)..t.. (1-)
. (l;~- n) 7'22 0,,2
... (29)
+ m( l;2+
l;2 +clf
CI)2]CI)+ (l;2+[1CI)[(
'ljJ2(l;2) = iTR [ (l;2 +
CI
(ii) For shear (T ) on infinite plate boundary:
1
(l;~ +cli'.
;dl;l} = iTS[_1_
m]
~
(1 + l;2 CI)
2'
.
2
_ (l + n )( nl;2 - 2nC) + l;2 C I)
(n-l;~)(Ci-nf
+
(2C2 - mc:il;1 - 3C~l;1 - m2C~l;1
+' ml;1 + 2m2 C2)
(1-
C2l;1)2(C~-
- 2m("
m)2
1
1
+------l;1
-
(1 + n2)
C2)(I'+l;2 ([;,-
. - _ •~
(1-
1/I1(l;1)=
C2))
.~ .",
2C2l;1)
iTS[(l;I-
... (30)
)
[1 + n(l;I-
C2)+
C2)2]
+ .
C2
+
1
(l;1 - C2)2 C2 (1 ~ l;1 C2)
(1 + m2)(ml;1
+ 2mC2 + l;1 C~)
(m-l;i)(m-
(1 + C2)-l;I)
n2)
+ 2n(([;,+
-
C~)2
)
l;1 (1 + ml;i) ;11 (l;I)
(l;i -
+([m-~I))
m2)
)
l;2( 1+ nl;~) .'
nC2 (1 -l; 1 C2)
2n([;i-
l;2(1
-
... (31)
m)
... (33)
;22 (l;2)
(l;r-n)
, For ,the case of an infinite plate C9ntainingtwo
unequal elliptical holes, subjected to ~axial tension acting at an angle p with respect to X- or
Y-axis following facts can be concluded.
(i) an infinite plate containing two unequal elliptical holes subjected to uniaxial tension
acting at an angle p with respect to Y-axis is
equivalent to the case of an infinite plate
containing two unequal elliptical holes
whose major axis is inclined at an angle p
with respect to Y-axis and plate being subjected to uniaxial loading along Y-axis.
(ii) An infinite plate containing two unequal elliptical holes subjected to uniaxial tension
acting at an angle p with respect to X-axis is
equivalent to the case of an infinite plate
containing two unequal 'elliptical holes
whose major axis is inclined at an angle P
with respect to X-axis and plate being subjected to uniaxial loading along X-axis.
Considering above two facts, stress function
;1 (l;I) and ;2 (l;2) for the two holes are found as
given Eqs (34) and (35).
and,
(- 2CI;2 (l;2) = iTR [ (l;2 +1 CI)
+.!,+
l;2
3ci
l;2
+ l;2l;2CI+ )2(nl;2ci- -2n2
n)2CI)
+ - (1
n4Ci
. 1
mCI
nCi l;2-
.
(1 + l;2 CI)
;dl;l)
4
= PS[(l;I-
(l;I- C2)
C2)±(2e-2iP=Fn)e4ip
(3+mC~)C2
(2e2iP+n)e-4il/
=F
+
C~
-
(C~ - m)(1'-
2(m+ m2C~) C2
2
2
(C 2 - m) (1 -
l;1
m
l;1 C2)
C2l;1)
UKADGAONKER
67
et a/;: S1RESS ANALYSIS OF AN INFINITE PlATE
Similarly fQr shear loading of an infinite plate
containing two unequal elliptical holes whose axes
are inclin~ at an angle f3 we have
2 2ifl
2
±~+
nC2
{;l
2ifl
e·.
(1 -
;d {;d= iT.S [
{;l C2)
+
=f-------(1
{;l (C2-[;J.e
6ifl
+
e
-6ifl
x( - 2C2- mC~{;l - 3Ci{;1 ~ m2Ci{;l
n2)(e2ifl=fn)
n[C2-In.e2ifl] [1-
e
-~ifl)]
... (36)
iTR [e -6i~
;2({;2)= (t2+Cl)
± (2e -2ifl=f n)e -4ifl
C~
(3+nC~)Cl
(n'- C~)(1+ {;2 Cl)
-
+(1+ {;2Cl):
nC1'{;2 -
3C~{;2 -
n2C~{;2
+ n{;2
e2i{l(1+m2e8ifl)
+------------
±-------{;2
mCl (1 + Cl {;2)
2e -2ifl
x(2Cl
e6ifl
2e -2ifl
2m·e'4i~C,--Jn·e2iflJ[1
+ {;2( C~- In·e2ifl)]
(1 +
±---~--------
m2)(e -2ifl=f m)
m[(
Cl - j;,..e2ifl)(1 + {;2(Cl - j;,..e
-2ifl)
.... (37)
... (35)
where each. parameter has its own meaning as explained earlier .
In the above two equations and all the equations which will be dealt .further the top row of
signs corresponds to tension parallel to positive
direction of X-axis bottom row of signs corresponds to tension parallel to negative direction of
Y-axis ..
Using these actual stress functions as second approximation to the stress field one can obtain the
stresses ·up U(J and 'l"r(J for each hole. However, as
the hole is· stress free we can write the expression
for stress concentration factor as the ratio of tangential stress to applied stress given by
p 4 Re [;'({;)]
'w'({;)
u(J=
.. . (38)
... (39)
68
INDIAN J. ENG. MATER. SCI., APRIL 1995
$-
-$
---$-~ --$Fig. 2- Particular cases for each type of loading
Results and Discussion
The solution presented here is for the case of
elliptical holes as a general case. However, by altering the hole parameters (a, b, g, and h) it is possible to solve the problems such as of· circular
holes where in the length of major axis is equal to
length minor axis (a= b, g= h), crack problems,
length of minor axis relatively smaller than major
axis (" l/1000th of major axis). A total of nine
particular cases for each type of loading are possible by this solution as shown in Fig. 2. General
closed form solution for the cases of tension, P,
are also obtained as given below· which will be
useful for design calculations.
(i) Closed form solution for tension, P, perpendicular to collinear major·axes
For first hole at 8= 0° (inner tip)
P=R(1-m)
U(J
S
3ci- mZC~+
+2+
1 z
(1- Cz)
j;,~
(2+m)
P = S(1-R n) ( 1+ (Cl-1)Z~
U(J
[(-nzCi+
3Ci)(1-
n-nC~-
C1)z
- (- 2C1 + nZCi - n+ nC1 + 3ci - 2nZC1t
x 2C1(1- C1)]
(Ci - n)Z(f= c1t
+2 +
1
(1-
Z
.
C1)
(1 + mZ)
z+-----
(2+ m)
m(1- C1)
+ (2+m)(1+mz)
2m(1- j;,- C1)z
p"=Re
U(J
2(1-
J;r-
+2C.(2+m)
(1-
C1)z
... (41)
C1)3
P,
parallel to
x
(2+n) z+---- (1+nz)
n(1- Cz) 2[1 + j;,- Cz]z
... (40)
R(1+m) (1-(i-Cz)Z
[_S_'
_(2-_n)_
m-(2-n)
[(1- iCz)z[- mC~- 3C~- mZC~+ m][2Cz - imCi - 3iC~ - imzC~+ im+ 2mzCz]
m)z
+ 2n[1
(1+nz)(2+n)
+
Cz]z +2Cz(2+n))
(1- CZ)3
n+(2+ m)
•
m]+
[2(2 - mCi - 3C~ - mZC~ + m+ 2mzCz]
x 2Cz(1- Cz)) .
(l-Czt(C~-
(inner tip)
(ii) Closed form solution for tension,
collinear major axes
For first hole at 8= 90°
~\2+m+(2+n)
( 1+,.(2+n)
((1- Cz)z[- mCi -
For second hole at 8-180°
2Cz
t-
(1- iCz))]
(1-
+2+
1
(1- iCz)
z+
iCZ)4( C~- m)z
(2-n) z+------ (1+nz)
n(1- iCz)
2[1 + iCJ;;- Cz)f
UKADGAONKER
et aL: STRESS ANALYSIS OF AN INFINITE
... (42)
_ 2n(1
(1 +
n)
2iC2
- n))!
+ n2)(2
i (fi,- - C2})2
(1-(2tC2)3
±
69
PLATE
(2e -2iP=Fm)e4iP
2CI
(3 + nCD cf
(n-
2
2
C 1)(1 + ~2CI)
For second hole at 8= 90°
P Re [ 5(1+n)
U6=
R
[(1
-I'
(1(2-
+Cli)2( -
.•.•
m)\2
2(n+ n2C~)Cf
2(1- nC~)Cf
(1 + ~2CI)2(n- Cf) - (n- Cf)(l + ~2CI)
n-(2-m)
+,,- +
n2C~
nC1 nCi - 3C~)
- (- 2CI- in2Cf + in- inC1- 3iCf
- 2n2Cd2CI (1+ tCI)]'
(C~- n)2(1 +iclt
... (45)
(1+m2).
(2-m)(1+m2)
+ 2(1 + i J;,+ CI))2
..
2m(1
+ i([m+
Similarly for shear loading it is observed that for
tirsthole
CI))2
(1 + tCI)3
+2iCd2-m))!
From Eq. (34) for first hole for uniaxial loading
+ e -6ifJ( -p=Re
U6
[ R(1-m/~f)
5
1=F (~I"-C2)2
[(2C-2ift=F
n)liP
mC4
2.
-
3C22 - _m2C22 + m)
(C1-m)2(1-~IC2)2
+m
~
e -6iP( - 2C2 - mC;~1 -3C~~1 - m2C~{;1
+ m~1 + 2mC2)( - 2C2)
(C~ - mf(11
e -2iP
- ~f - noe -4iP(1_
2e2iP 2Ce2iP
2 Ce2iPo~2
=F-2
+-2-+---:n(1- ~IC2)
~I
2noe -4i~1_
+
=F---~--n(1-
n2)(q2iP=F n)
=F
(1
n[1-
~1
(C2 -
~1
(2)2
e -2iP(1 + n20e-SiP)
~I
(1
~I C2)3
~1 ( -
C2 - fi,oe -2iP)f
e -2iP(1 + n20e -SiP)
fi,oe2iP)f
2noe-4i~1
+ n2)(e2iP=Fn) =F(2e2ifJ=r n)e -4ip
+ fi,oe2IP)f
C2
~1 (~2
+ (1-
+ ~I (-
~IC2}2
2
(12(1-
C2 + fi,oe -2iP)f
~IC2?
2 ~1 C2)]!
... (46
... (44)
and for second hole
~I
P 2= Re [ 5(1- R,.
n =F/~2)
(~2 + C1)2
~~
2 [1 =F(2e2iP=F
m)e -4iP+.!!.
and for secon<l hole it is observed that
70
INDIAN J. ENG. MAlER
e6ill(- 2CI - nC1 ~2- 3ci ~2- n2Ci ~2
+ n~2- 2nCI)'2CI
(Ci - n)2(1+ ~2CI)3
sa., APRIL
1995
Two equal and unequal circular holes in an infinite plate-Stress concentration - factors for two
equal and unequal circular holes in an infinite
plate subjected to tension parallel to and perpendicular to line of holes and shear at infinity are ob1
e2ill
tained by taking the hole parameters as a = b and
- ~2+ n·e4ill(1 + ~2CI)2
g= h. Tables 1-3 and Fig. 3 gives the results obtained for two equal circular holes problem for
varying centre to centre distance. From these it is
e2ill(J+ m2e8ill)
.
seen that for the case of tension in X-direction,
2me4ill[1+ ~2(CI - j;,,·e2ill)f
the SCF is lesser than single hole solution at close
distances, whereas for tension in Y-direction the
e2ill(1+ m2e8ill)
SCF is higher and converges to single hole solution as the distance between the holes increases.
2me4ill[1-+ ~2(CI + ji,·e2ill)f
For shear (T) at infinity the 08/ T values increases
up to e= 2 and then decreases. and converges to
... (47) single hole solution. These solutions of second ap+ 2(1
(1 ++2~2~1)1]
~2CI) J
proximation. to stress field using Schwarz's alternating technique are in good agreement with those
Hence
obtained by Nemat-Nasser and HorilS making use
of Pseudotraction method. Ling16 using bipolar
coordinates and Ukadgaonkerl7-19using complex
Re[
S
variables, up to about 3% difference at closer distance and 0.5% for larger distance between holes.
Fig. 4 gives the variation of SCF with distance
x [1'1'(20~'"''f.)e'~~+ (-2+20)
between holes (e) for two unequal circular holes.
From this it is evident that presence of a smaller
(2e2ill~n)e -4ill(-(3 + mC~)C
hole near a larger hole results in increase of stress
level on larger hole results in X-direction, higher
+ m+ 2e2ill+ + 2(m+ m2C~)+ 2(1 + mC~)CD
stress level on smaller hole for tension in Y-direcCD2
tion at closer distance between holes. For. shear at
infinitythe SCF is higher on smaller hole.
Crack approaching circular hole-Presence of a
crack near a circular hole is accounted for by
putting a = b and h ~ 0.001 x g. For the case of tenwhere,
sion parallel to crack, the presence of crack near a
circular hole will result in reduction of SCF on the
A=(~I- C2)2,B= ~i.C=(C~- m),
hole and increase of SIP (K1) value for crack itself
D=(1- ~I C2), E= [1- ~I (C2- ji,·e -2ill)f
and converges to single hole and single crack soluand
tion as the crack moves away from the hole. For
F= [1- ~I (C2 + ji,'e -2illf
the case of tension perpendicular to the crack, the
Now ~1= pei8. Solving the above equation for ~1 presence of crack near circular hole at tloser dis'the location of point for Strain Energy Density tance increases the SCF on hole and SIP for the
can be found in the vicinity of first"ellipse, when crack and the two values decreases to single hole
an infinite plate containing them is subj~ted to and single crack solution at larger distance beuniaxial tension along x- or y-axis. Similarly for tween the hole and crack. The K1 values at the
obtaining location in the vicinity of second ellipse two crack tips are compared with those obtained
~1 is substituted by ~2' C2 by. - CI, P by - p, n by Isida20 in Table 4. The results are in good
by mand mby n, Rby Sand SbyR, inEq.(48).
agreement at larger distances.
For the case of shear the interaction effect inOn similar lines one can proceed for the case of
shear loading.
creases SCF value on the hole and SIP (Kn) value
at the crack tips.
Numerical results
Circular hole and elliptical hole- The ~tress
Following section gives the numerical results concentration factors for a circular hole. along the
and discussion for the particular cases.
major axis of an elliptical hole in an infinite plate
1
UKADGAONKER
Table I-Stress
concentration
et al.: STRESS ANALYSIS OF ~N INFINITE PlATE
71
factor for two equal circular holes subjected to uniform tension along the line of holes by come
plex variable method
FIRST 1tOI.£:
~~*E
•• h2
SlCONDHCU
•• he2
For second hole
2.50752~
3.000
970
950
920
910
940
910
900
900
9880
00 900~,
2.80947
2.71015
2.78211
2.9976
930
960
900
870
850
840
900
860
830
2.52237
2.8.5209
2.66480
2.56422
2.50752
2.95819
2.61466
890
2.74908
2.85209
2.94195
2.9958
o,/P
8go
2.92241
2.$3274
2.83274
.0'1
82
2.564,22
o,jP
By present method
Nemat-
For first hole
Ukadgaonker9
Nasser
o/Pat90°
2.5500
2.658
2.787
2.907
2.6088
2.6500
2.968
2.978
2.8272
2.947$
3.??oo
Table 2-Stress
concentration
factor for two equal circular holes subjected to uniform tension perpendicular
compleX variable approach
j ,- ~
,-
e1087
20
le
hole Nematution
3.0000
r
•• h.2
3.481
3.000
o/Pat
3.00010
3.869
3.066
3;000
2.9922
(f '
3.001
3.045
3.997
3.066
3.62178
~3.10313
2.98093
2.99296
2.99038
3.00262
2.98243
2.98378
3.00071
4.57689
4.4227
3.9211
2.9981
3.2641
3.003
3.020
2.985 I
3.264
2.98726
2.995U5
3.0000
Nasser
Haddon
3.0000
UkadgaonkerB
Ling"
p ~3.0000
•• J b.2
J
SEmND HOlE:
,.
3.020
3.004
FIRST HOlE:
to line of holes by
72
INDIAN J. ENG. MATER SCI., APRIL 1995
Table 3-Stress concentration factoi' for two equal clrc:ular holes subjected to Wuform shear at infinity by coIIlplex variable
··method
By present method
34
135
-4.54602
-4.i5018.
4.53784
4.25018
32
36
148
-4.45910
40
140
42
-4.17446
4.20092
44
44
136
-4.000
-4.00612
4.??oo
4.00612
4.01749
45
45
61
-4.02888
de!.
4Sa,jT
13S
-4.62268
-4.53784
4.54602
4.62268
38
142
144
4.45910
-4.38439
4.38439
138
4.32185
-4.04&22
-4.20092
-4.27169
4.08473
4.27169
45
-4.01749
4.04822
4.02888
-4.32185
-4.23296
-4.08473
4.17446
4.23296
62
deg.
146
a.,/T
Second
First hole
hole
5
y
-4.1933
-4.2029
-4.2126
-4.1936
-4.1746
-4.1559
-4.1373
-4.0866
-4.0487
-4.000
,
Unlaxial t.".ICIft
parall.1 to X-axl.
Z
o
,
I
110
IZ
DlRallC. b.twe." ·hol••
Fig. 3-Variation of
14
Ie)
II
20
alP and alT with e for equal circular holes
nx
UKADGAONKER
et aL: STRESS ANALYSIS OF AN INFINITE
73
PlATE
y
---- ---
5
For smaller hole
For larger hole
Uniaxial tension
parallel to V- allis
Go
..•
~
------------------------------
---- -- ---------Uniallial tension
parallel to X- axis
2
o
2
6
4
8
10
12
14
16
18
20
22
x
Distance betw.enholes leI
Fig. 4-Variation
For circular
hole o,jPat
e
90·
K •. AlP
0.5
2.784
2.818
2.867
2.899
2.920
2.936
2.946
2.955
2.961
2.966
2.970
2.987
2.991
2.995
-2.724
-1.175
-0.252
-0.037
0.022
0.038
0.039
0.037
0.033
0.029
0.026
0.009
0.004
0.001
1
2
3
4
c
For crack
5
6
7
8
9
10
20
30
50
K.,BIP
-0.083
-0.032
0.029
0.027
0.025
0.025
0.023
0.021
0.919
0.017
0.Q15
0.006
0.003
0.000
of
o,jP
and
For circular
hole
o,jT
with
e for
For crack
unequal circular holes
ISID~S Results
o,jP
K •. AlP
KII. HIP
K •.AlP
K., HIP
3.419
3.267
3.142
3.091
3.065
3.049
3.039
3.032
3.027
3.024
3.021
3.010
3.006
3.002
6.836
4.795
3.384
2.944
2.763
2.673
2.623
2.593
2.573
2.560
2.549
2.518
2.512
2.508
2.826
2.743
2.654
2.608
2.581
2.564
2.553
2.544
2.538
2.533
2.530
2.515
2.511
2.508
4.387
3.384
2.958
2.757
2.695
2.594
2.570
2.957
2.807
2.682
2:607
2.582
2.557
2.543
2.507
2.507
are obtained by the present solution and are given
in Table 5.' It may be concluded from these results
that the interaction effect of the two holes is to reduce the stress level on each hole at closer distances for tension, p,in X-direction. However, for
tension perpendicular to major axis, the effect of
For circular
hole 091P
at 45·
-4.052
-4.079
-4.084
-4.071
-4.058
-4.047
-4.035
-4.031
-4.023
-4.022
-4.018
-4.003
-4.001
-4.000
For crack
KII•AIT
-1.346
1.268
2.629
2.835
2.831
2.788
2.744
2.707
2.676
2.651
2.631
2.547
2.527
2.513
KII. Pr1'
.
2.580
2.649
2.684
2.677
2.660
2.641
2.625
2.611
2.598
2.588
2.578
2.536
2.523
2.513
interaction is to increase the stress level on each
hole remarkably at closer distances and reduces to
singlehole solution at larger distance.
For the case of uniform shear the interaction effect is about 9% on circular hole and 24% on elliptical holes. The SCF on circular hole and ellipt-
74
INDIAN J. ENG. MATER SCI., APRIL 1995
Table 5~Stress
concentration
g~'.~'
p
-
-
•
-
•
P
• g
Circular hole
q.
factor for a circular hole and a elliptical hole in an infinite plate
J~lf:::
2
h. 1
T
Elliptical hole
e
0.5
95
1
94
2
93
3
92
92
91
91
91
91
90
4
5
6
7
8
9
10
20
30
50
2.626
2.655
2.714
2:766
2.807
2.840
2.865
2.885
2.901
2.914
2.924
2.970
2.983
2.998
90
90
90
90
Table 6-Stress
1.675
1.690
1.719
1.751
1.782
1.810
1.835
1.856
1.874
1.889
1.902
1.965
1.982
1.993
75
75
77
79
81
82
84
85
86
87
87
89
90
90
concentration
Circular hole
Elliptical
hole
08/PatO•
08/Pat 180·
3.619
3.276
3.080
3.033
3.019
3.013
3.011
3.009
3.009
3.008
3.008
3.006
3.005
3.002
9.397
7.140
- 5.701
5.306
5.161
5.108
5.065
5.047
5.035
5.027
- 5.023
5.006
5.003
5.001
_
Circular hole
8( deg
08/T
37
38
41
42
43
44
44
44
44
45
45
-4.295
-4.354
-4.337
-4.282
-4.231
-4.189
-4.157
-4.131
-4.111
-4.094
-4.081
-4.027
-4.013
-4.000
45
45
45
Elliptical hole
150
155
157
156
155
155
154
154
154
154
154
153
153
153
g-h-l
l~~
T
e
1
81
degree
102
100
2
98
3
96
94
4
5
6
7
8
9
10
20
30
50
Single hole
solution
90
90
90
1.757
1.776
1.813
1.846
1.873
1.894
1.911
1.925
1.936
1.944
1.951
1.983
1.991
1.997
90
2.000
94
92
92
92
92
,:?2
For hole 1
o/PatO·
82
.oJo
0.5
For hole 2
degree
78
80
82
84
86
86
88
88
For hole 2
o/Pat 180·
For hole 1
88
88
90
90
90
6.682
5.894
5.358
5.189
5.118
5.082
5.060
5.047
5.037
5.030
5.025
5.008
5.003
5.001
6.682
5.894
5.358
5.189
5.118
5.082
5.060
5.047
5.037
5.030
5.025
5.008
5.003
5.001
90
2.000
5.000
5.000
_
For hole 2
26
-4.960
-5.120
-5.065
-4.945
-4.846
-4.772
-4.718
-4.677
-4.647
~4.623
-4.605
-4.532
-4.515
26
-4.505
82
degree
154
156
154
154
154
154
154
154
154
154
154
154
154
154
-4.5
153.43
81
degree
1.757
1.776
1.813
1.846
1.873
1.894
1.911
1.925
1.936
1.944
1:951
1.983
1.991
1.997
""88
4.980
4.895
4.828
4.776
4.735
4.572
4.534
4.502
factor for two equal elliptical holes by complex variable method
a-b-2
For hole 1
4.604
5.177
5.550
5.421
5.243
5.094
26
24
26
26
26
26
~6
26
26
26
26
26
26.56
4.960
5.120
5.065
4.945
4.846
4.772
4.718
4.677
4.647
4.623
4.605
4.532
4.515
4.505
4.5
, \,
""
;I
•~~L~
I
3 to major
axia
06 .•.a
y
75
et al: STRESS ANALYSIS OF AN INFINITE PLATE
UKADGAONKER
r
0'
0
I major axis
to
.
0
Tuaion
'--
0
parGUel
Shear ( T)
Te"alOft perpendicular
21-
I
o
e
of 0,/ P and 0,/
Fig. 5 -Variation
r with
e for equal elliptical holes
y
-------
For ama"er hole
for I arger hole
Tenaion perpendicular
to major axla
II
~
~
~
-.....g.------~-~-
,
-a81T
For shear (T)
••
II
X
"
L
~
3
TenalOll paraUe'
to major axis
---------------------------------
-----------
,
o
2
,
I
•
10
12
14
II
II
20
22
X
C
Fig. 6- Variation of 0,/ P and
o,/r with
ical hole increases at closer distance and then re~
duces to single hole solution for larger distances.
Two equal and unequal elliptical holes-For two
equal elliptical holes with their major axes collinear along X-axis, the SCF defined as the J~tio of
maximum tangential stress on the hole boundary
e for unequal elliptical holes
to the applied stress at infinity for the cases of uniaxial tensions and shear (T) are tabulated in
Table 6. Figs 5 and 6 give variation of olP and
01T for varying distance between the. holes.
When the holes are close to each other for tension
parJ1llel'to major axes the effect of interaction of
76
INDIAN J. ENG. MATER. sel., APRIL 1995
Scalo'
Dlmonolono
1·5 s 1
Sir,. dlotrlhllon
Tangontlal
}.
2.1
Fig. 7 -Tangential
stress distribution on hole boundary
ratio of tangential stress to applied stress'
as a
8-
Fig.
Tangential stress distribution on hole boundary as the
ratio of tangenti8I stress to applied stress for unequal holes
(with tension in X-direction)
each hole is to reduce the level of stress on the
other hole and as the ~o holes move away from
each other the effect reduces, the stress level on
each hole increases and converges to single hole
solution. However, the effect of interaction for the
case of tension in Y-direction is vice-versa, i.e., at
closer distances the stress level on each hole is
higher and reduces to single hole solution at larger
distance.
•
For the case of two unequal elliptical holes,
when the second,.hole is at larger distance from
the first hole, the stress field near first hole should
give a single hole solution and should not be affected by the variation of size of second hole.
However, with the existing stress functions this expectation is not satisfied, so as .1 measure of correction, the quantities S and Rin ~1 (~1) and rh ( {;2)'
are replaced by Rand S, respectively. The results
UKADGAONKER
77
et aL: STRESS ANALYSIS OF AN INFINITE PLATE
p
6.0
4S
"
10
1.0
,
---t--.•..
.,,--:------
g.4
p
Fig. 9-Tangential
stress distribution on the hole boundary of
two unequal elliptical holes subjected to tension perpendicular to collinear major axes
T
r
j
r
J
T
_~
_
-~------
---------
r
Fig. to-Tangential
stress distribution on the boundary of
two elliptical holes subjected to uniform shear (T) at infinity
are found to be satisfactory with this change.
Moreover, this correction does not affect the
equal holes solutions as R is equal to S in such
cases.
By the present solution alP (maximum) is obtained for equal ~d unequal holes all around the
hole boundary for all the three types of loading as
shown in Figs 7-10.
Two collinear cracks~ The solution to the problem of two collinear cracks in 'an infinite plate is
obtained by putting b= h=O. The Mode-l stress
intensity factor as a ratio of KI to P j;O. at all the
four tips for tension, p, perpendicular to crack line
and Mode-ll stress intensity factor as a ratio of Kn
to T ~
for uniform shear (T ) at infinity for 2a/d
varying from 0.9 to 0.1 obtained by the present
solution are listed in Table 7. Figs 11 and 12 give
the variation of KIf P ~
for varying e and 2a/d,
respectively. For the case of tension perpendicular
to crack line, at very close distance between the
78
INDIAN J. ENG. MATER seI., APRIL 1995
----
ll~'1
1.4
P1.036
1.246
1.021
1.000
1.011
1.05
1.03
1.007
1.020
1.04
1.004
1.016
1.005
1.0046
1.021
1.115
·1.097
1.154
1.0043
1.0272
1.0480
1.0057
1.0804
1.2289
1.0138
1.0013
0;986
1.006
1.07
1.0579
1.02
1.008
1.0
1.0409
1.073
1.12
1.0102
1.056
1.0027
1.0012
1.000
1.0280
1.022
1.015
1.017
1.011
1.014
1.0179
1.013
1.016
1.025
1.02
1.21
1.0
1.005
1.05
1.042 Bymethod
Nemat-Nasser2
1.060
Outside
1.133
1.002
1.0811
1.07
Inside
ISIDN
By
present
present
method
1.030
Table 7 -Stress intensity factors! at the four tips of two equal collinear cracks by complex variable approach
--
KitPIifci"
cracks, the interaction effect is up to 25% on the
inner tips. and 7.5% on the outer tips. The effect
reduces as the cracks move apart.
For' the case of uniform shear, at infinity the
Mode-II SIP increases initially upto 2a/d= 0.6 and
then decreases and converges to single crack solution
Strain enel'lY deDSity
~~,I ---In'"
••
------
,
'4
Tip(A)
Out .de
H
•..
.
Tip ( lQ
,.J
H
,.,
2.,.
H
Fig. 11- Variation of KI/ p!j;i;iJfor
,.,
HX
cracks
6
.".,Tip C
..
51- '.
4
,\
,,
TiPA,C\
. \
N..
B.
A
C
~;2:"J
p
,\
alb= 1
______ alb ••2
For calculation of strain energy density the approach followed here is as follows:
The emphasis here to 'locate the point in the
vicinity where local and global strain energy densities are equal as given·.by 5ih2l• Now, the stt:~
energy density, local and global can be equal at a
point only when local stress and global stress at
that point is equal. If the location of point where
the stress concentration factor is 1,. is found, then
that point is having local and global strain energy
density equal.
Here, global stress is the stress applied at infinity and local stress is the stress obtained by using
the stress functions obtained.
Hence, the task here is to find the point near
the edge of ellipse or in region between the two
ellipses where stress concentration factor is 1.
Consider the equation for stress concentration
factor as given in Eq. (45).
~3
'024681
2
3456
Tip Diat .•
Fig. 12- Variation of Kl P for two collinear cracks
Substituting a'; p= 1 and rearranging the equation by taking LCM we get a polynomial in ~l as given by Eq. (48).
UKADGAONKER
et al: STRESS ANALYSIS OF AN INFINITE PLATE
Conclusion
The present results proves that Schwarz's alternating technique converges jn second approximation itself. The solution provides a close form solution unlike the other solution provides a close
form solution unlike the other solutions which in
the form of infinite series in which the convergence of the 'series in itself becomes a separate
problem.
Present solution is computationally very easy, as
once it is programmed it can give solutions to
many particular cases by just inputting the hole
parameters.
Regarding 'strain energy density calculation, for
location part in the vicinity of the ellipses where
local and global strain energy densities are equal
Eq. (48) was solved and few particular case of the
unequal collinear ellippcal holes were considered.
It was observed that the point lies near the bigger
hole. Further, when the distance between the holes
is increased, there is shift in the location of point
where local and global strain energy are equal.
The shift continues till intc!raction between the
two holes exists. When ipteractionbetween the
two holes is negligible than the location of point
.remainssame as in earlier case.
Acknowledgement
The financial aid provided by the Board of
Research in Nuclear Science, Department of
Atomic Energy, India to this work is gratefully acknowledged.
79
References
1 Mushkhelishvili N I, Some basic problems in the mathematical theory of elasticity (P Noordhoff Ltd., Groninicen,
Netherlands),1953.
2 Sokolnikoff I S, Mathematical
theory of elasticity
(McGraw Hill, New York), 1956.
3 Ukadgaonker V G & Naik A P, Int J Fract, 51 (1991)
219.
4 Uki\dgaonker V G & Naik A P, Int J Fract, 51 (1~1)
285.
5 Ukadgaonker V G & Koranne S D, Int J Fract, 51 (1991)
R37.
6 Ukadgaonker V G & Koranne S D, Indian J Techno~ 31
(1993)67.
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