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Interaction of torsional waves with an annular crack in an infinitely long cylinder
Engineering Fracture Mechanics Vol. 20, No. 5/6, pp. 729-133, 1984 Printed in the U.S.A.
0013-7944/w $3.00 + .ocl Pergamon Press Ltd.
INTERACTION OF TORSIONAL WAVES WITH AN ANNULAR CRACK IN AN INFINITELY LONG CYLINDER Department
R. S. DHALIWAL, B. M. SINGH, J. VRBIK of Mathematics, University of Calgary, Calgary, Alberta, Canada and
S. M. KHAN School of Computer Science, Acadia University,
Wolfville, Nova Scotia, Canada
Abstract-The Hankel transform is used to obtain a complete solution for the dynamic stresses and displacements around a flat annular surface of a crack embedded in an infiite elastic cylinder, which is excited by normal torsional waves. The curved surface of the cylinder is assumed to be stress free. Solution of the problem is reduced to three simultaneous Fredholm integral equations. By finding the numerical solution of the simultaneous Fredholm integral equations the variations of the dynamic stress-intensity factors are obtained which are displayed graphically.
1. INTRODUCTION THE DYNAMICstress-intensity factor approach of linear elastic fracture mechanics has proven to be very successful in predicting the unstable fracture of brittle solids. Very recently diffraction of normally incident torsional waves by a flat annular crack in an elastic solid has been discussed by Shindo [I]. Shindo [2] has discussed diffraction of normal compression waves by a flat annular crack in an infinite elastic solid. The problem of interaction of longitudinal waves with a penny-shaped crack located in an infinitely long cylinder has been discussed by Srivastava et al. [3]. In this paper we consider the diffraction of normal torsional incident waves by a flat annular crack lying inside the cylinder. Solution of the problem is reduced to three simultaneous Fredholm integral equations. With the help of a numerical solution of the Fredholm integral equations the numerical values of the stress-intensity factors are obtained.
2. FORMULATION
OF THE PROBLEM
Consider an infinitely long isotropic homogeneous elastic cylinder of radius c containing an annular crack a 5 r I b, z = 0 perpendicular to its axis. The annular crack is subjected to normal torsional waves such that material particles experience only an angular displacement. Since the geometry of the cylinder is symmetric about the crack plane, the problem may be formulated by specifying appropriate mixed boundary conditions on a semi-infinite cylinder z 2 0, 0 I r I c. The boundary conditions of the problem can be written as
&l(r, 0) = 0,
OIrSa,
bsrsc
uez(r, 0) = -ps - p(r) exp (-i wt),
(1) alrsb.
In this problem the curved surface of the cylinder is supposed to be free from traction. implies that on the curved surface r = c, we have
ure(c, z) = 0,
z 2 0.
(2)
This
(31
In condition (2) the static pressure ps is assumed to be sufftciently large to ensure that the two faces of the crack do not come in contact during vibrations. Since the solution of the static 729
rt al.
K. S. DHALIWAL
730
problem may be superimposed
on the dynamical problem, condition (2) can be written as 0) =
u&r,
-p(r)
exp ( -i
asrsh.
wt),
(4)
In what follows the time dependence of all quantities assumed to be of the form exp ( -i ot) will be suppressed. The problem of determining the stress distribution reduces to that of solving the displacement equation
a2uo -iiF
1
au,
2 !!!!
+;dr-
+
$
+
k2uo
=
(5)
0
r2
where k2 = p/p 02.
(6)
Here p, is Lame’s constant and p denotes the density of the elastic material. Solving equation (5) we obtain that (7)
Ue(T, z) = JOffiA(S)Zi(Sr) e-a’ dS + JOmB(S)Zi (Pr) sin (Sz) d5
1 cc eepz dS+ JW33MP~) sin (Ez) dr; u&,z)=I_L - &4(&M&) 1
%z(T, z) = CL - OXPAWi [ J
e-a’ dS + lS
W(S) cos (Sz)Zi(iW dS
(8) (9)
where Jn( ) is a Bessel function of the first kind and I,( ) is a modified Bessel function of the first kind
E>k
(10)
Oc[
From boundary
conditions
(1) and (2) we find that
(11) A(S)Jl(Sr) de = 0, Iom Equation
O
b
(12)
(11) can be written as (13)
where
q0W9 -=----+ P and
40’ CL
ksE8W1(Pr) dS -
/oi(P -
SMS)Jl(k) d5,
a
(14)
Interaction
of torsional waves
731
Let 4041(r)
SAWI (Ed 4 Iom
O
’
P
=
(15) q14269 IJJ
b
’
and using the Hankel inversion theorem we get from relations (13) and (15) that A(t)
Substituting :(a2
= f
Jou r4~(r)J~(Sr)
- 2) “2r-‘+l(r)
-
dr + J:
r+2(r)J1 (cr) dr
1 (16)
= - l’(‘:F:>1’2A4(l)dt
I (9
rM(r)JI(@)
the value of A(e) from eqn (16) into eqn (12) and following Cooke [4], we find that
-
G
dr + l’
b2)1’2r+2(r)
From boundary
=
-
condition
(17)
‘(p - a2)1/2 t2 _ rz 4df)dt, b
~~b’2’~
1
Q
::)“‘M(t)dt
(18)
(3) we find that
PW3W3c) sin(5~) Iom
dS - Jo- Ep(W2(pc)
e-a’ d[ = 0,
Making use of the inversion theorem for Fourier sine transforms,
z 2 0.
(19)
we get
(20) Substituting
the value of A(u) from eqn (16) into eqn (20), we get b
r4lW%
r) dr +
rM(r)P(&
r) dr +
~4&-VYf;,r) dr
1 (21)
where
(22) Substituting M(r)
= r +
the value of B(t) from eqn (21) into eqn (14) we get b
~4~WW, d dv +
vM(v)K2(r,
ZJ)dv
732
where
(24) The dynamic stress intensity factors are defined by the following relations KS, = Lim,,-
m-2 - w2%(r, O),
O
ow -
b <
(25)
and
Ifa+Oandc-+~andw-+O, crack can be written as
~w%,~Y, 01,
r}.
(26)
the dynamic stress intensity factor K3s at the outer tip of the
(>
K3s =
;
(27)
q&p.
We can easily find that +
= b(t) _Ib
u’(t2
Jab ~~Y)~,{Y,
-
dt a*)1 b)
(28)
dy
+
Jbc ~~(Y)~I~Y,
b)
dy
1 (2% .
From the fracture mechanics point of view, the desired information is the stress-intensity factor which measures the load transmission of the crack. Mathematically this parameter is defined as amplitude of stress singularity at the tip of the crack. Thus, the dynamic stressintensity factor may be obtained with the help of the eqns (17), (I@, (23), (28) and (29). For calculation of integral (24), Simpson’s rule was carefully applied to remove the singularities of integral (24) which gave stable values of integral (24). In numerical calculations a
2.5
00
I
I
I
I
I
I
I
1
I
2
3
4
5
6
7
6
k
Fig. 1. Dynamical stress-intensity
*
factor 1K3JKR,, / for u = 0.3. h = 1. c = 1.2, 1.5, 5, vs wave number.
Interaction of torsional waves
733
a=0.3
1 2.0
1.5
y
48
1.0
0.5
o.ot
I
L
I
I
I
I
I
I
I
2
3
4
5
6
7
8
k-
Fig. 2. Dynamical stress-intensity
factor 1K3blK3r 1 for a = 0.3, b = 1, c = 1.2, 1.5, 5, vs wave number.
complex programme is formed for solving the simultaneous integral equations (17), (18) and (23). Finally, with the help of relations (28) and (29), the numerical values of 1 K3J& / and 1 Kx,/K~~ 1 are obtained and these are displayed in Figs. 1 and 2. REFERENCES [l] Y. Shindo, Diffraction of torsional waves by a flat annular crack in an infinite elastic medium. J. Appt. Me&. 46, 927-831 (1979). [2] Y. Shindo, Normal compression of scattering at a flat annular crack in an infinite elastic solid. Q. Appi. Math. 39, 305-315 (1981). [3] K. N. Srivastava, R. M. Palaiya and 0. P. Gupta, Interaction of elastic waves with a penny-shaped crack in an infinitely long cylinder. J. Elast. 12, 143-152 (1982). [4] J. C. Cooke, Triple integral equations. Q, J. Mech. Appl. Muth. 16, 193-203 (1963).
0013-7944/w $3.00 + .ocl Pergamon Press Ltd.
INTERACTION OF TORSIONAL WAVES WITH AN ANNULAR CRACK IN AN INFINITELY LONG CYLINDER Department
R. S. DHALIWAL, B. M. SINGH, J. VRBIK of Mathematics, University of Calgary, Calgary, Alberta, Canada and
S. M. KHAN School of Computer Science, Acadia University,
Wolfville, Nova Scotia, Canada
Abstract-The Hankel transform is used to obtain a complete solution for the dynamic stresses and displacements around a flat annular surface of a crack embedded in an infiite elastic cylinder, which is excited by normal torsional waves. The curved surface of the cylinder is assumed to be stress free. Solution of the problem is reduced to three simultaneous Fredholm integral equations. By finding the numerical solution of the simultaneous Fredholm integral equations the variations of the dynamic stress-intensity factors are obtained which are displayed graphically.
1. INTRODUCTION THE DYNAMICstress-intensity factor approach of linear elastic fracture mechanics has proven to be very successful in predicting the unstable fracture of brittle solids. Very recently diffraction of normally incident torsional waves by a flat annular crack in an elastic solid has been discussed by Shindo [I]. Shindo [2] has discussed diffraction of normal compression waves by a flat annular crack in an infinite elastic solid. The problem of interaction of longitudinal waves with a penny-shaped crack located in an infinitely long cylinder has been discussed by Srivastava et al. [3]. In this paper we consider the diffraction of normal torsional incident waves by a flat annular crack lying inside the cylinder. Solution of the problem is reduced to three simultaneous Fredholm integral equations. With the help of a numerical solution of the Fredholm integral equations the numerical values of the stress-intensity factors are obtained.
2. FORMULATION
OF THE PROBLEM
Consider an infinitely long isotropic homogeneous elastic cylinder of radius c containing an annular crack a 5 r I b, z = 0 perpendicular to its axis. The annular crack is subjected to normal torsional waves such that material particles experience only an angular displacement. Since the geometry of the cylinder is symmetric about the crack plane, the problem may be formulated by specifying appropriate mixed boundary conditions on a semi-infinite cylinder z 2 0, 0 I r I c. The boundary conditions of the problem can be written as
&l(r, 0) = 0,
OIrSa,
bsrsc
uez(r, 0) = -ps - p(r) exp (-i wt),
(1) alrsb.
In this problem the curved surface of the cylinder is supposed to be free from traction. implies that on the curved surface r = c, we have
ure(c, z) = 0,
z 2 0.
(2)
This
(31
In condition (2) the static pressure ps is assumed to be sufftciently large to ensure that the two faces of the crack do not come in contact during vibrations. Since the solution of the static 729
rt al.
K. S. DHALIWAL
730
problem may be superimposed
on the dynamical problem, condition (2) can be written as 0) =
u&r,
-p(r)
exp ( -i
asrsh.
wt),
(4)
In what follows the time dependence of all quantities assumed to be of the form exp ( -i ot) will be suppressed. The problem of determining the stress distribution reduces to that of solving the displacement equation
a2uo -iiF
1
au,
2 !!!!
+;dr-
+
$
+
k2uo
=
(5)
0
r2
where k2 = p/p 02.
(6)
Here p, is Lame’s constant and p denotes the density of the elastic material. Solving equation (5) we obtain that (7)
Ue(T, z) = JOffiA(S)Zi(Sr) e-a’ dS + JOmB(S)Zi (Pr) sin (Sz) d5
1 cc eepz dS+ JW33MP~) sin (Ez) dr; u&,z)=I_L - &4(&M&) 1
%z(T, z) = CL - OXPAWi [ J
e-a’ dS + lS
W(S) cos (Sz)Zi(iW dS
(8) (9)
where Jn( ) is a Bessel function of the first kind and I,( ) is a modified Bessel function of the first kind
E>k
(10)
Oc[
From boundary
conditions
(1) and (2) we find that
(11) A(S)Jl(Sr) de = 0, Iom Equation
O
b
(12)
(11) can be written as (13)
where
q0W9 -=----+ P and
40’ CL
ksE8W1(Pr) dS -
/oi(P -
SMS)Jl(k) d5,
a
(14)
Interaction
of torsional waves
731
Let 4041(r)
SAWI (Ed 4 Iom
O
’
P
=
(15) q14269 IJJ
b
’
and using the Hankel inversion theorem we get from relations (13) and (15) that A(t)
Substituting :(a2
= f
Jou r4~(r)J~(Sr)
- 2) “2r-‘+l(r)
-
dr + J:
r+2(r)J1 (cr) dr
1 (16)
= - l’(‘:F:>1’2A4(l)dt
I (9
rM(r)JI(@)
the value of A(e) from eqn (16) into eqn (12) and following Cooke [4], we find that
-
G
dr + l’
b2)1’2r+2(r)
From boundary
=
-
condition
(17)
‘(p - a2)1/2 t2 _ rz 4df)dt, b
~~b’2’~
1
Q
::)“‘M(t)dt
(18)
(3) we find that
PW3W3c) sin(5~) Iom
dS - Jo- Ep(W2(pc)
e-a’ d[ = 0,
Making use of the inversion theorem for Fourier sine transforms,
z 2 0.
(19)
we get
(20) Substituting
the value of A(u) from eqn (16) into eqn (20), we get b
r4lW%
r) dr +
rM(r)P(&
r) dr +
~4&-VYf;,r) dr
1 (21)
where
(22) Substituting M(r)
= r +
the value of B(t) from eqn (21) into eqn (14) we get b
~4~WW, d dv +
vM(v)K2(r,
ZJ)dv
732
where
(24) The dynamic stress intensity factors are defined by the following relations KS, = Lim,,-
m-2 - w2%(r, O),
O
ow -
b <
(25)
and
Ifa+Oandc-+~andw-+O, crack can be written as
~w%,~Y, 01,
r}.
(26)
the dynamic stress intensity factor K3s at the outer tip of the
(>
K3s =
;
(27)
q&p.
We can easily find that +
= b(t) _Ib
u’(t2
Jab ~~Y)~,{Y,
-
dt a*)1 b)
(28)
dy
+
Jbc ~~(Y)~I~Y,
b)
dy
1 (2% .
From the fracture mechanics point of view, the desired information is the stress-intensity factor which measures the load transmission of the crack. Mathematically this parameter is defined as amplitude of stress singularity at the tip of the crack. Thus, the dynamic stressintensity factor may be obtained with the help of the eqns (17), (I@, (23), (28) and (29). For calculation of integral (24), Simpson’s rule was carefully applied to remove the singularities of integral (24) which gave stable values of integral (24). In numerical calculations a
2.5
00
I
I
I
I
I
I
I
1
I
2
3
4
5
6
7
6
k
Fig. 1. Dynamical stress-intensity
*
factor 1K3JKR,, / for u = 0.3. h = 1. c = 1.2, 1.5, 5, vs wave number.
Interaction of torsional waves
733
a=0.3
1 2.0
1.5
y
48
1.0
0.5
o.ot
I
L
I
I
I
I
I
I
I
2
3
4
5
6
7
8
k-
Fig. 2. Dynamical stress-intensity
factor 1K3blK3r 1 for a = 0.3, b = 1, c = 1.2, 1.5, 5, vs wave number.
complex programme is formed for solving the simultaneous integral equations (17), (18) and (23). Finally, with the help of relations (28) and (29), the numerical values of 1 K3J& / and 1 Kx,/K~~ 1 are obtained and these are displayed in Figs. 1 and 2. REFERENCES [l] Y. Shindo, Diffraction of torsional waves by a flat annular crack in an infinite elastic medium. J. Appt. Me&. 46, 927-831 (1979). [2] Y. Shindo, Normal compression of scattering at a flat annular crack in an infinite elastic solid. Q. Appi. Math. 39, 305-315 (1981). [3] K. N. Srivastava, R. M. Palaiya and 0. P. Gupta, Interaction of elastic waves with a penny-shaped crack in an infinitely long cylinder. J. Elast. 12, 143-152 (1982). [4] J. C. Cooke, Triple integral equations. Q, J. Mech. Appl. Muth. 16, 193-203 (1963).