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Diffraction of torsional waves by a penny-shaped crack in an infinitely long cylinder bonded to an infinite medium
001r7944p4 s3.00 + .oo 0 1984 Pcrgamon Pm8 Ltd.
Mechanics.Vol. 19, No. I, pp. 3S-40, 1984 Printed in Great Britain.
Engineering Frachrre
DIFFRACTION OF TORSIONAL WAVES BY A PENNY-SHAPED CRACK IN AN INFINITELY LONG CYLINDER BONDED TO AN INFINITE MEDIUM R. S. DHALIWAL, B. M. SJNGH and J. G. ROKNE Department of Mathematics, University of Calgary, Calgary, Alberta, Canada T2N IN4 Abstract-This paper contains an analysis of the interaction of torsional waves with penny-shaped crack located in an infinitely long cylinder which is bonded to an infinite medium. Both the cylinder and infinite medium are of homogeneous and elastic but dissimilar materials. The solution of the problem is reduced to a Fredholm integral equation of the second kind which is solved numerically. The numerical solution is used to calculate the stress intensity factor at the rim of the penny-shaped crack.
1. ~TRODUCTION RECENTLY, great interest has been shown in the problems of the interaction of elastic waves with cracks situated in elastic solids. The study of these problems is motivated by their applications to seismology and exploration geophysics. Robertson[I], Mal[2] and Sih and Leober [3, 41 have studied the diffraction of longitudinal and torsional waves from the crack surfaces when the crack is lying in an infinite medium. In this paper we have considered one such problem in which a penny-shaped crack is situated in an infinitely long elastic cylinder and interacted by torsional waves. The cylinder is bonded to an infinite medium. Both the cylinder and the infinite medium are isotropic homogeneous and elastic but of dissimilar materials. With the help of Hankel transform the solution of the problem is reduced to that of solving a Fredholm integral equation of the second kind. This equation is solved numerically. Numerical values of the dynamic stress intensity factor are calculated at the rim of the crack for different wave frequencies and radii of the cylinder. The results are illustrated graphically. The study has been motivated by two definite objectives. First it is desired to study the effect of a finite boundary on the stress intensity factor due to dynamic loading. Secondly to study the effect of dynamic loading on the stress and displacement field and to compare the results for the limiting case when the wave numbers tend to zero with those obtained by Freeman and Keer[5]. It is also important to mention that dynamical problems of breaking of composite fibre have much importance in fracture mechanics as discussed by Cherepanov[6]. 2. FORMULATION OF THE PROBLEM Consider an infinitely long isotropic homogeneous cylinder of radius b containing a circular crack 0 I r I c, z = 0 perpendicular to its axis and the cylinder is bonded to other infinite material. The crack is subjected to a normally torsional wave moving in the positive direction of z-axis 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 of a semi-infinite cylinder z 2 0, 0 < r < b. The boundary conditions of the problem can be written in the foliowing form: criz(r,O)= -p,-p(r)e-‘“I,
O
cc,
I.&, 0) = 0, c < r < 6,
(21
z&r, 0) = 0, b < r.
(3)
Due to continuity of angular displacement and shear stress on the curved surface we find that (4)
t&r, b) = t&, b). 35
(5)
36
R. S. DHALIWAL et al.
Since the solution of static problem may be supe~mpo~ (1) can be written as
on the dynamical problem, condition
ojz(r, 0) = -p(r) e-‘“’
(1)
where the suffix 1 and 2 correspond to the cylinder and infinite medium. In what follows the time dependence of all quantities assumed to be of the form exp( - iwt) will be suppressed. The problem of determining the stress distribution reduces to that of obtaining the solution of displacement equation
(6) where k2 = (p/p) co2 and p is the Lam?s constant and p denotes density of elastic material. Solving eqn (6) we obtain the expressions for the displacement and shear stress components for the regions 1 and 2 respectively in the following form:
and
(10) 01) (12) where
~,=t<2-k:)“2, = -i(kt I.32= =
G2
-
5 >k,,
- &j2)lj2, 0 s 5 I k,, > W2,
5
-i(kz2 - t2)‘/2,
>
k2,
0 I tJ I k2,
(13)
(14)
where p, and ~12are the shear moduli for the regions 1 and 2 respectively. J,( ) is a Bessel function of the first kind and I,,( ) and K”( ) are modified Bessel functions of the first and second kinds respectively for n 2 0. It is clear from expression (IO) that the boundary condition (3) is identically satisfied and the boundary conditions (1) and (2) reduce to the following dual integral equations:
scD
A (Wltr) d5 = 0, c
0
(16)
Diffraction of torsional waves by a penny-shaped crack
31
The boundary conditions (5) and (4) reduce to the following integral equations:
m s[ 0
1
~(~)~I(~~~) - C(C)&(bb2)sin(O)
B,NW,(B,b)+
r]sin(&)d<
B2WWz(W)
I
d5 = -
0I
O”A({)J,(@)e-BIzd&
=i
OctA(0J,(Sb)e-BIzd5,
Z,
(17)
0 sz.
(18)
0
Let us assume A(<) = C$“2 :$ s $(O)=
W)J3,2(W
dc
(19)
0.
With this choice of A(t) eqn (16) is identically satisfied. Equation (15) can be rewritten in the following form:
sm 0
5A(W,(
(A - 5M(W,(5r) dt -
sm
W5V,(B,r) d5 =
0
-‘+,
0 < r < c. (20)
Substituting the value of A(<) from eqn (19) into (20) we get
Now, using the following integrals, 112
(22)
t3’2r31*(filr),
“* t3/2J
(rt) 312 3
(23)
we can write eqn (21) in the following form: e(t)
+ t3’*
m %-I-Q
dt -
A(r)J3,2(B,t)
t3'*
m s 0
s Odifl
$,
W3WV)
=-
2 I’* 1 -
'
0
dt
s
’ r*p(r) dr
j.41 0 (P - r*y*
II
o
(24)
Application of the Fourier sine transform to eqns (17) and (18) yields two simultaneous equations. Making use of (19) and the two simultaneous equations we find that
where A=
W2(hMWd32)
+ B2GM3d~2(W2)1.
(26)
Substituting the value of A(<) and B(r) from eqns (19) and (25) into (24) we get L,(u, t)t,b(u) du +
oc&(u, t)t&) s
du = -
; 0
I’*’ ’ r2p(r) dr & s 0 (P - G)r’*
0 < t < c,
(27)
38
R. S. DHALIWAL et ai,
where
(28) 2t312 L&4, t) = O”52 [- BJW&)&(WJ rr&
+ B,GK,(bB,)K,(bB,)lI,,,(B,t)I,,,(uB,)
dS.
(29)
We assume that p(r) = zbr,
(30)
where r is the angle of rotation per unit length. Equation (27) can be written in the following form:
s c
G,(t)+
0
s c
Mu, t)G,tu) du +
L2(u, t)G,(u) du = - t*,
0 < I < c,
0
(31)
where (32)
(33) and
sinh(~~f) (B,t)
I[
cosh(#$u) -
1
si~~~~~~) d<,
(34)
For getting the above results we have used the follo~ng: coshx -(35)
If we put k, = k2= 0, we find that L&u, t) = 0 and the integral equation (33) reduces to the ~o~esponding equation of Freeman and Kee#f who have considered the statical problem corresponding to the dynamical problem considered here. The dynamic stress intensity factor is defined by the following equation:
K3= X-W+ Lim + ([2(x- cP21~i&,
%,,bl}.
(36)
Making use of (8), (19) and (32) we find that (37) For the numerical evaluation of the integrals 1;, and Z& the five-point Gauss-Laguerre forrnuia was found suitable which gave stable values of the integrals and of the integral equation (27). For
39
Diffraction of torsional waves by a penny-shaped crack solving
the integral equation (33) the following relation, k, = k2
is used. Finally, by using the relation (37) numerical values of the dynamic stress intensity factor are obtained which are graphed in Figs. 1-3.
3. DISCUSSION OF THE RESULTS It is of interest to compare the results of this paper with those derived by Sih[7] in his recent book. Sih[‘;rlhas studied the interaction of torsional waves with a penny-shaped crack in an infinite elastic body. A compa~son of these results yields valuable info~ation about the effect of finite
0
I
2
3
4
5
6
7
8
+Fig. 1. Variation of Dynamic stress intensity factor versus generalised wave number k, for G = 0.2, 0.6, 2, a = 1, 6 = 1.5, a = p,/p2= 1.
k,Fig. 2. Variation of Dynamic stress intensity factor vs general&d wave number k, for G = 3, 5, 50, a = 1, b = 1.5, a =p,/p2 = 1.
R. S. DHALIWAL ef al.
“.I_. 0
2
3
4
5
6
7
8
%-
Fig. 3. Variation
of Dynamic stress intensity factor vs generalised a = p,/p2 = 0.2, 50, G = 1, a = 1, b = 1.5.
wave number k, for
boundaries on the stress intensity factor. A look at the graph (in [7], p. 170, Fig. 3.25) showing the variation of stress intensity factor with 0 5 dimensionless wave numbers < 2.5 for the problem studied by Sih shows that the value of stress intensity factor first increases from its elastostatic value reaches a maximum value and then it decreases. In our graphs we obtain the same trend of the curves for 0 I k, < 2 and for k, > 2 we obtain that the stress intensity factor decreases and finally it contains oscillatory behaviour. We also note that the numerical solution of Fredholm integral equation for k, = k2 = 0 are in good agreement with those calculated from the results derived by Freeman and Keer[5] for the corresponding static problem.
REFERENCES [1] I. A. Robertson, Diffraction of a plane longitudinal wave by a penny-shaped crack. Proc. Comb. Phil. Sot. 63, 229-238 [2] !??Mal Interaction of elastic waves with a penny-shaped crack. Int. J. Engng Sci. 8, 381-388 (1970). [3] G. C. Sih gnd J. F. Loeber, Torsional vibration of elastic solid containing a penny-shaped crack. J. Acoustical Sot. Am. 44 1237-1245 (1968). [4] G. C. Sih and J. F. Loeber, Normal compression and radial shear waves scattering at a penny-shaped crack in an elastic solid. .I. Acousticul Sot. Am. 46, 711-721 (1969). [5] N. J. Freeman and L, M. Keer, On the breaking of an embedded fibre in torsion. Inf. .I. Engng Sci. 9, 1007-1017 (1971). [6] G. P. Cherepanov, Mechanics of Brittle Fracture, pp. 660-663. McGraw-Hill, New York (1979). [7] G. C. Sih, Mechanics of fracture. In Efastodynamic Crack Problems, Vol. 4, pp. 167-171. Noordhoff, Leyden (1977). (Received 24 August 1982; received for publication 15 March 1983)
Mechanics.Vol. 19, No. I, pp. 3S-40, 1984 Printed in Great Britain.
Engineering Frachrre
DIFFRACTION OF TORSIONAL WAVES BY A PENNY-SHAPED CRACK IN AN INFINITELY LONG CYLINDER BONDED TO AN INFINITE MEDIUM R. S. DHALIWAL, B. M. SJNGH and J. G. ROKNE Department of Mathematics, University of Calgary, Calgary, Alberta, Canada T2N IN4 Abstract-This paper contains an analysis of the interaction of torsional waves with penny-shaped crack located in an infinitely long cylinder which is bonded to an infinite medium. Both the cylinder and infinite medium are of homogeneous and elastic but dissimilar materials. The solution of the problem is reduced to a Fredholm integral equation of the second kind which is solved numerically. The numerical solution is used to calculate the stress intensity factor at the rim of the penny-shaped crack.
1. ~TRODUCTION RECENTLY, great interest has been shown in the problems of the interaction of elastic waves with cracks situated in elastic solids. The study of these problems is motivated by their applications to seismology and exploration geophysics. Robertson[I], Mal[2] and Sih and Leober [3, 41 have studied the diffraction of longitudinal and torsional waves from the crack surfaces when the crack is lying in an infinite medium. In this paper we have considered one such problem in which a penny-shaped crack is situated in an infinitely long elastic cylinder and interacted by torsional waves. The cylinder is bonded to an infinite medium. Both the cylinder and the infinite medium are isotropic homogeneous and elastic but of dissimilar materials. With the help of Hankel transform the solution of the problem is reduced to that of solving a Fredholm integral equation of the second kind. This equation is solved numerically. Numerical values of the dynamic stress intensity factor are calculated at the rim of the crack for different wave frequencies and radii of the cylinder. The results are illustrated graphically. The study has been motivated by two definite objectives. First it is desired to study the effect of a finite boundary on the stress intensity factor due to dynamic loading. Secondly to study the effect of dynamic loading on the stress and displacement field and to compare the results for the limiting case when the wave numbers tend to zero with those obtained by Freeman and Keer[5]. It is also important to mention that dynamical problems of breaking of composite fibre have much importance in fracture mechanics as discussed by Cherepanov[6]. 2. FORMULATION OF THE PROBLEM Consider an infinitely long isotropic homogeneous cylinder of radius b containing a circular crack 0 I r I c, z = 0 perpendicular to its axis and the cylinder is bonded to other infinite material. The crack is subjected to a normally torsional wave moving in the positive direction of z-axis 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 of a semi-infinite cylinder z 2 0, 0 < r < b. The boundary conditions of the problem can be written in the foliowing form: criz(r,O)= -p,-p(r)e-‘“I,
O
cc,
I.&, 0) = 0, c < r < 6,
(21
z&r, 0) = 0, b < r.
(3)
Due to continuity of angular displacement and shear stress on the curved surface we find that (4)
t&r, b) = t&, b). 35
(5)
36
R. S. DHALIWAL et al.
Since the solution of static problem may be supe~mpo~ (1) can be written as
on the dynamical problem, condition
ojz(r, 0) = -p(r) e-‘“’
(1)
where the suffix 1 and 2 correspond to the cylinder and infinite medium. In what follows the time dependence of all quantities assumed to be of the form exp( - iwt) will be suppressed. The problem of determining the stress distribution reduces to that of obtaining the solution of displacement equation
(6) where k2 = (p/p) co2 and p is the Lam?s constant and p denotes density of elastic material. Solving eqn (6) we obtain the expressions for the displacement and shear stress components for the regions 1 and 2 respectively in the following form:
and
(10) 01) (12) where
~,=t<2-k:)“2, = -i(kt I.32= =
G2
-
5 >k,,
- &j2)lj2, 0 s 5 I k,, > W2,
5
-i(kz2 - t2)‘/2,
>
k2,
0 I tJ I k2,
(13)
(14)
where p, and ~12are the shear moduli for the regions 1 and 2 respectively. J,( ) is a Bessel function of the first kind and I,,( ) and K”( ) are modified Bessel functions of the first and second kinds respectively for n 2 0. It is clear from expression (IO) that the boundary condition (3) is identically satisfied and the boundary conditions (1) and (2) reduce to the following dual integral equations:
scD
A (Wltr) d5 = 0, c
0
(16)
Diffraction of torsional waves by a penny-shaped crack
31
The boundary conditions (5) and (4) reduce to the following integral equations:
m s[ 0
1
~(~)~I(~~~) - C(C)&(bb2)sin(O)
B,NW,(B,b)+
r]sin(&)d<
B2WWz(W)
I
d5 = -
0I
O”A({)J,(@)e-BIzd&
=i
OctA(0J,(Sb)e-BIzd5,
Z,
(17)
0 sz.
(18)
0
Let us assume A(<) = C$“2 :$ s $(O)=
W)J3,2(W
dc
(19)
0.
With this choice of A(t) eqn (16) is identically satisfied. Equation (15) can be rewritten in the following form:
sm 0
5A(W,(
(A - 5M(W,(5r) dt -
sm
W5V,(B,r) d5 =
0
-‘+,
0 < r < c. (20)
Substituting the value of A(<) from eqn (19) into (20) we get
Now, using the following integrals, 112
(22)
t3’2r31*(filr),
“* t3/2J
(rt) 312 3
(23)
we can write eqn (21) in the following form: e(t)
+ t3’*
m %-I-Q
dt -
A(r)J3,2(B,t)
t3'*
m s 0
s Odifl
$,
W3WV)
=-
2 I’* 1 -
'
0
dt
s
’ r*p(r) dr
j.41 0 (P - r*y*
II
o
(24)
Application of the Fourier sine transform to eqns (17) and (18) yields two simultaneous equations. Making use of (19) and the two simultaneous equations we find that
where A=
W2(hMWd32)
+ B2GM3d~2(W2)1.
(26)
Substituting the value of A(<) and B(r) from eqns (19) and (25) into (24) we get L,(u, t)t,b(u) du +
oc&(u, t)t&) s
du = -
; 0
I’*’ ’ r2p(r) dr & s 0 (P - G)r’*
0 < t < c,
(27)
38
R. S. DHALIWAL et ai,
where
(28) 2t312 L&4, t) = O”52 [- BJW&)&(WJ rr&
+ B,GK,(bB,)K,(bB,)lI,,,(B,t)I,,,(uB,)
dS.
(29)
We assume that p(r) = zbr,
(30)
where r is the angle of rotation per unit length. Equation (27) can be written in the following form:
s c
G,(t)+
0
s c
Mu, t)G,tu) du +
L2(u, t)G,(u) du = - t*,
0 < I < c,
0
(31)
where (32)
(33) and
sinh(~~f) (B,t)
I[
cosh(#$u) -
1
si~~~~~~) d<,
(34)
For getting the above results we have used the follo~ng: coshx -(35)
If we put k, = k2= 0, we find that L&u, t) = 0 and the integral equation (33) reduces to the ~o~esponding equation of Freeman and Kee#f who have considered the statical problem corresponding to the dynamical problem considered here. The dynamic stress intensity factor is defined by the following equation:
K3= X-W+ Lim + ([2(x- cP21~i&,
%,,bl}.
(36)
Making use of (8), (19) and (32) we find that (37) For the numerical evaluation of the integrals 1;, and Z& the five-point Gauss-Laguerre forrnuia was found suitable which gave stable values of the integrals and of the integral equation (27). For
39
Diffraction of torsional waves by a penny-shaped crack solving
the integral equation (33) the following relation, k, = k2
is used. Finally, by using the relation (37) numerical values of the dynamic stress intensity factor are obtained which are graphed in Figs. 1-3.
3. DISCUSSION OF THE RESULTS It is of interest to compare the results of this paper with those derived by Sih[7] in his recent book. Sih[‘;rlhas studied the interaction of torsional waves with a penny-shaped crack in an infinite elastic body. A compa~son of these results yields valuable info~ation about the effect of finite
0
I
2
3
4
5
6
7
8
+Fig. 1. Variation of Dynamic stress intensity factor versus generalised wave number k, for G = 0.2, 0.6, 2, a = 1, 6 = 1.5, a = p,/p2= 1.
k,Fig. 2. Variation of Dynamic stress intensity factor vs general&d wave number k, for G = 3, 5, 50, a = 1, b = 1.5, a =p,/p2 = 1.
R. S. DHALIWAL ef al.
“.I_. 0
2
3
4
5
6
7
8
%-
Fig. 3. Variation
of Dynamic stress intensity factor vs generalised a = p,/p2 = 0.2, 50, G = 1, a = 1, b = 1.5.
wave number k, for
boundaries on the stress intensity factor. A look at the graph (in [7], p. 170, Fig. 3.25) showing the variation of stress intensity factor with 0 5 dimensionless wave numbers < 2.5 for the problem studied by Sih shows that the value of stress intensity factor first increases from its elastostatic value reaches a maximum value and then it decreases. In our graphs we obtain the same trend of the curves for 0 I k, < 2 and for k, > 2 we obtain that the stress intensity factor decreases and finally it contains oscillatory behaviour. We also note that the numerical solution of Fredholm integral equation for k, = k2 = 0 are in good agreement with those calculated from the results derived by Freeman and Keer[5] for the corresponding static problem.
REFERENCES [1] I. A. Robertson, Diffraction of a plane longitudinal wave by a penny-shaped crack. Proc. Comb. Phil. Sot. 63, 229-238 [2] !??Mal Interaction of elastic waves with a penny-shaped crack. Int. J. Engng Sci. 8, 381-388 (1970). [3] G. C. Sih gnd J. F. Loeber, Torsional vibration of elastic solid containing a penny-shaped crack. J. Acoustical Sot. Am. 44 1237-1245 (1968). [4] G. C. Sih and J. F. Loeber, Normal compression and radial shear waves scattering at a penny-shaped crack in an elastic solid. .I. Acousticul Sot. Am. 46, 711-721 (1969). [5] N. J. Freeman and L, M. Keer, On the breaking of an embedded fibre in torsion. Inf. .I. Engng Sci. 9, 1007-1017 (1971). [6] G. P. Cherepanov, Mechanics of Brittle Fracture, pp. 660-663. McGraw-Hill, New York (1979). [7] G. C. Sih, Mechanics of fracture. In Efastodynamic Crack Problems, Vol. 4, pp. 167-171. Noordhoff, Leyden (1977). (Received 24 August 1982; received for publication 15 March 1983)