- Email: [email protected]
A note on critical reflections of axially symmetric waves in an isotropic, elastic, hollow cylinder
Iat, I. S&h
5Ynefms, 1977,Vol. 13, pp. 25348.
Pmpmon Pnss.
Printi
in that Britain
A NOTE ON CRITICAL REFLECTIONS OF AXIALLY SYMMETRIC WAVES IN AN ISOTROPIC, ELASTIC, HOLLOW CYLINDER? R. R.
KAUL
State University of New York, Buffalo, NY 14214,U.S.A.
and G. HEXRMANN Division of Applied Mechanics, Stanford University, Stanford, CA 94305,U.S.A. (Received 27 July 1976) critical incidence oi ~uivolumin~ waves, the wave motion is evanescent in the case of an isotropic, elastic, hollow cylinder with traction-free cylindrical surfaces. Howevet, it is shown that in this critical situation the wave motion can be obtained by using a s&able limiting procedure. AbsMct-At
In a cylindrical co-opdinate system (r, (9,z), consider an isotropic, elastic, hollow cylinder, with generators parallel to the z-axis, inner radius r = u outer radius r = b, with common center and of infiniteextent otherwise. Let A and p be the Lame’s constants and p the mass density of the hom~eneous, isotropic, elastic medium. In the case of axially symmetric waves, the motion of the hollow cyhnder with circular cross-section is described by two Bessel eq~~ons~l]
a28
lab
ar2+;-a;-+h2A=0, S+lE_
a? r ar
where
(9 )ii=o,
a
(1)
J__k2
h2=(p/t$#-y2, k2=(pluJ2-
13
y2,
v,’ = 0 + a.4)fp, V.2’Etl@ The wave number in the axialdirection is denoted by y, h and k are the wave num~rs in the radial direction, u, and u. are the wave speeds of the d~a~tion~ and shear waves in an unfunded medium, and p is the circular frequency in rad. sec. For axiallysymmetric motion, & = 0,3, = I), & = 0, a&la@ = (a&las)= 0 and the third equation of motion for the circumferentialdiction 6 is satisfied identically. The two non-trivial stress components are expressed by the formulas
(3)
Whis work was supported in part by the Office of Naval Research Contact N~l~75~~~2 New York and tbe Air Force Grant AFOSR 74-2669to Stanford University. 263
to the.State Unive~ity of
264 We
R. K. KAUL and CL HERRMANN
consider simple-harmonic motion of the form u,= U(~)ixp(~z~~~), (4)
gr = W(r) ixp (.yz + pt),
where exp (. . .) = exp i(. . .), and i is the fourth root of unity in the Argand plane. Then from(2), and (2)2 we find that V(r) and W(r) are particular solutions of the equations
where in these equations, and in the sequel, the exponentials have been suppressed for convenience, and A, 6 are the general solutions of eqn (1). The two ordinary differential eqns (1) are Bessel equations of integer order 0 and 1 respectively, which have a si~~ty as r + 0’. Therefore in the case of a solid cylinder we add the requirement that A and & be bounded as r 40’. In the case of hollow cylinder we take the general solution of the Bessel equations as A = a,A,Jo(hr)
+ b,B, Y,(h),
0< h
ii = azAd,(kr)
+ b2B2Y,(krh
0< k
a
(6)
where a,(h), b,(h) and a&k), b,(k) are suitable functions of ir and k, respectively. In the case of critical reflection when either h or k tend to O’, the functions a,, bl or clz, b, are determined by using d’Alembert’s limiting procedure. The amplitude coefficients Ai, I?, (i = 1,2) are determined by the boundary conditions on the two cylinderical surfaces r = a and r = b. When A and G are given by eqn (6), the non-homogeneous part of the differential eqns (5) are
2 +2iyc;t = -h(ulA,Jl(hr)
+ b,B, Y&r))
+ 2iyfazAJt(kr)
-I-bzBzY,(krN, (7)
iyh -f$(rG)=
iy(a,A&hr)+
b,a,Y~hr))-2k(azAzJo(kr)-t
b&Y,,(kr)).
By using the method of undetermined coefficients, the particular solution of the non-homogeneous differential eqn (5) can easily be determined and is given by
hLWzr) W(r) = --$$
+ b& Y&r)1 - --$$
{olAJdhr)+
~~=A*~*(kr~f b~~~Y*(kr)~,
b,B,Ydhr)}+-$p
hA&kr)
+ bJ’hYo(kr)),
where in these expressions h > 0 and k > 0. The radial wave-numbers h and k are both real when p z ynP, h becomes imaginary and k remains real when yu, s p < yv,, and both become irn~~y when p < yv,. When p = yvP, h = 0’ and k real; and when p = ‘yv.,h is irna~~y and k = 0”. When h = O’, we have the case of grazing incidence of dilatational waves, and k = 0’ corresponds to the case of grazing incidence of equivoluminal waves. The coefficients ai and bi have still to be determined. We consider here the case of grazing incidence of equivoluminal waves when k +O+ and h is imaginary. In this critical case, eqn (1)2 degenerates into Euler’s equation of second order, whose genera1 solution is & =Azr+B2i,
O
k=O.
(9)
A note on criticalreflectionsof axiallysymmetricwaves in an isotropic,elastic,hollow cylinder It is therefore
265
necessary that the coefficients a&) and b*(k) in eqn (t$ be such that (i) lim {an(k).ldkr)} = r, k-O+
(10)
(ii) lili+ {b,(k) Y,(kr)} = i,
Using d’Alembert’s limiting procedure and making use of the properties of Bessel functions, we find that a2 and b, are the solutions of
(9 (ii)
$(j-)lk,+=h $ b&,,+ = -;
(11) P.
Therefore in the critical case of grazing incidence of equivoluminal waves when h is imaginary and k + O’, the coeflicients ar and br are a,(h) = 1, b,(h) = 1, az(k) = 2/k, b,(k) = -5
k.
(12)
One can similarly find the value of these coefficients in the case of grazing incidence of dilatation waves when h +O+ and k remains real. However, in this brief note we will not further discuss this case, as the analytical procedure is similar to the case when k +O+. Therefore, a suitable solution of eqn (l), which in the limiting case contains the solution of critical reflection of equivoluminal waves in a hollow cylinder, is given by A = AJo
+ 8, Yo(hr),
0 < h < im,
(13) &=A,fJ,(kr)-BzjkY,(kr),
O=k
The displacement components are given by {A,J,(hr)+B,Y,(hr)}-;[A+(kr)-%;kY~(kr)], (14)
W(r) =
-&
{A,J,,(hr)+B,Yt,(hr)}+-+[A+o(kr)-%;kYdkr)},
where again 0 < h < i m, 0 5 k < m. With A and 4 given by eqn (13), we rewrite the displacement components in the convenient form 1 U(‘)=-(,,‘+h3
8A ar
2iy
,,2+k2”)
-
From (3) and (15), we find that the stress components are given by
iy aA 7#2= -2&L ---($I$+].
C
y2+ h2 Jr
(16)
R.K.KAUL and G.~~~NN
266
Substituting for A and & from eqn (13), we find that eqn (16) takes the form TV= A, AJ,,(hr)+ -&+*(hr)]+& [ 4&y -y2$.
[nudhrI+*;
A&$Mkr)-B,Sk-$ E
Y&r)
1
Y,(hr)]
, (17)
r,.z= ~
[A,Jz(hr) + B, Y,(hr)] -I-2~ ($$)
(A2~~*(b)-B*SkU,(kr)],
where as before 0 < h < im, 0 5 k < m. In the critical case when k =O’ and 0 < h < iq the limiting form of stresses can easily be determined from eqn (17). Using de I’Hapital rule, the stresses are
1 c
~,,lr+,+= A, A~~h~~)+ -$$-&h*r) t -?(A,-&$), =B ?rzIs?+o+
0
+B,
AYo(hor)+
fi -$Y&91]
’
m
[A,J,(hor) f Bt Y,(hor)l + 2~ (A,r + Blf) 2
where ho = h It4+. When k = 0”, we find from (2)4 that y = plu, and from (2& we get ho==-(p/u,)2(1-(f2),
O
where u=v~/~J~,In terms of non-dimensional frequency 0 = p/o,, where oS (= 742t) is the lowest antisymme~c access-shear frequency of an infinite, isotropic, elastic plate of thickness 2t (=(b - a)), we find that in the critical case of grazing incidence of equivoluminal waves /&+(1-o)
* 1/2, o
1
(W
where A @_2_ l-2v 2( 1 - u)’ l=2(h+ and v is the Poisson’s ratio of the elastic medium. (i) Hollow C$inder In the case of a concentric hollow cylinder of circular cross-section, vanishing of the stresses at the inner and outer surfaces gives us the frequency equation A(~~+ hd)Jo(ltoa)+2irlhd:(hoa)
A.(y’+ h,Z)Yo(h,a)+ZELhoY:(hoa) 42 Y
-4cl YQ2
A(y2+ho2)Jo(hob)+2&,.f:(hi,b)
A(y’+ hOZ)Ydhob)+2ClhoYffhob)
F
yh~,th~u~
Y~~Y,(~OU)
a
-% Yb =o, 1 a
yh~,(hob 1
rho Y,(hob )
b
’ ‘i;
cw
where .C(&z) = (~~~u)~,(h~u), . . ., and Y~(hob) = (~~~b)Y,(hob). Using Laplacian expansion anddeleting common termsintheexpansion, and introducing modified Bessel functions of real
A note on criticalreflectionsof a&By symme~c waves in an isotropic,elastic,hollowcylinder
261
argument I,,(@ and km(@),the frequency equation in the critical ease of grazing incidence of equivoluminal waves is given by
where
cm
q=alb,a
and2t=(b-a).
This transcendental equation has a single root which is reiatively easy to compute. In particular, finding this root of the equation does not present the problem of “spurious roots” which Gazis encountered in the investigation of a similar problem, at grazing incidence ([2] p. 274). Using asymptotic expansion of the “Hankel” type for large argument and fixed order, the frequency equation reduces to the form 8,’ +
tat& (I- 02)K’ >
4q(l-
s3
(1 - q)eb -
K2eb= 0,
Wl -VI)
qi/eb),
(23)
where K+(3+$)(1+37j)-(1+3n3(1+3/?7)+48(1-o3(1-n2), K*~(l-~)2+l~l-~2)(l+~)z.
(24
To a first approximation this equation is satisfied when
(ii) Solid Cylindert In the case of solid cylinder of radius r = b, a --)0, 7 + 0, 0,,+ 0 and the frequency equation tProf. R. D. Mind&nhas recentlybroughtto ourattention,thatthe criticalreflectionof waves in an isotropic, elasticplate wasinvestigatedby him in his paper, “Mathematical‘fItwry of Vibrationsof Eastic Plates”, Procee&ngsof&e Bet&~ Sympodnm on Frqnency Coatml, pp. 17-40;U.S. Army SignalCorpsEnginceriagLaboratories,Fort Monmouth, New Jersey, (1957).A remarkin our earlierpaperpublishedin this journal,(19)6), Vol. 12, pp. 353-357,that “Such critical rellectionsof ckistic waves.. . in the case of plates,.. , seems to have been overlookedin the literature”,was thus clearly inappropriate. Annul
268
R. K. KAU?.and G. HERRMANN
reduces to the simple form
On further s~mpli~~at~on,we get the frequency equation
where
since for a solid cylinder B = 2t. This eqnat~ou was first ob~~ned by &toe, et uI. [3f, as a rimming form of Pochhammer frequency equatio~[4], by making use of mod~ed quotient of Bessel functions .5?,,(z)= z.&,(z)/J,,.&). This frequency equation has only one real root given by
For II = 0.31, we find that
where S is the first root of the equation S,(S) ~0. This is the point of intersection of the extensional branch with line OE in Fig. 2 of Ref. [3]. In this critical case, the ratio of the amplitude coefficients is
and the displacement components are
~c~o~~e~ge~e~~~ne discussions.
of the &hors @.K.K.) would like t@ttrank his colkaguts, Dr. Richard P. Shw, for many beIpful
London (1927). 2. D. C. Gazis, J. Am&. 5% Am.St, 573-578 (1959). 3. M. Owe, H. D. McNiven and R. D. Mind& J. Appl. Me&. 29,%%734 (lP62k 4. L. Pochhmmer, Z&t. f. Math. $1, 324-336 (1976).
5Ynefms, 1977,Vol. 13, pp. 25348.
Pmpmon Pnss.
Printi
in that Britain
A NOTE ON CRITICAL REFLECTIONS OF AXIALLY SYMMETRIC WAVES IN AN ISOTROPIC, ELASTIC, HOLLOW CYLINDER? R. R.
KAUL
State University of New York, Buffalo, NY 14214,U.S.A.
and G. HEXRMANN Division of Applied Mechanics, Stanford University, Stanford, CA 94305,U.S.A. (Received 27 July 1976) critical incidence oi ~uivolumin~ waves, the wave motion is evanescent in the case of an isotropic, elastic, hollow cylinder with traction-free cylindrical surfaces. Howevet, it is shown that in this critical situation the wave motion can be obtained by using a s&able limiting procedure. AbsMct-At
In a cylindrical co-opdinate system (r, (9,z), consider an isotropic, elastic, hollow cylinder, with generators parallel to the z-axis, inner radius r = u outer radius r = b, with common center and of infiniteextent otherwise. Let A and p be the Lame’s constants and p the mass density of the hom~eneous, isotropic, elastic medium. In the case of axially symmetric waves, the motion of the hollow cyhnder with circular cross-section is described by two Bessel eq~~ons~l]
a28
lab
ar2+;-a;-+h2A=0, S+lE_
a? r ar
where
(9 )ii=o,
a
(1)
J__k2
h2=(p/t$#-y2, k2=(pluJ2-
13
y2,
v,’ = 0 + a.4)fp, V.2’Etl@ The wave number in the axialdirection is denoted by y, h and k are the wave num~rs in the radial direction, u, and u. are the wave speeds of the d~a~tion~ and shear waves in an unfunded medium, and p is the circular frequency in rad. sec. For axiallysymmetric motion, & = 0,3, = I), & = 0, a&la@ = (a&las)= 0 and the third equation of motion for the circumferentialdiction 6 is satisfied identically. The two non-trivial stress components are expressed by the formulas
(3)
Whis work was supported in part by the Office of Naval Research Contact N~l~75~~~2 New York and tbe Air Force Grant AFOSR 74-2669to Stanford University. 263
to the.State Unive~ity of
264 We
R. K. KAUL and CL HERRMANN
consider simple-harmonic motion of the form u,= U(~)ixp(~z~~~), (4)
gr = W(r) ixp (.yz + pt),
where exp (. . .) = exp i(. . .), and i is the fourth root of unity in the Argand plane. Then from(2), and (2)2 we find that V(r) and W(r) are particular solutions of the equations
where in these equations, and in the sequel, the exponentials have been suppressed for convenience, and A, 6 are the general solutions of eqn (1). The two ordinary differential eqns (1) are Bessel equations of integer order 0 and 1 respectively, which have a si~~ty as r + 0’. Therefore in the case of a solid cylinder we add the requirement that A and & be bounded as r 40’. In the case of hollow cylinder we take the general solution of the Bessel equations as A = a,A,Jo(hr)
+ b,B, Y,(h),
0< h
ii = azAd,(kr)
+ b2B2Y,(krh
0< k
a
(6)
where a,(h), b,(h) and a&k), b,(k) are suitable functions of ir and k, respectively. In the case of critical reflection when either h or k tend to O’, the functions a,, bl or clz, b, are determined by using d’Alembert’s limiting procedure. The amplitude coefficients Ai, I?, (i = 1,2) are determined by the boundary conditions on the two cylinderical surfaces r = a and r = b. When A and G are given by eqn (6), the non-homogeneous part of the differential eqns (5) are
2 +2iyc;t = -h(ulA,Jl(hr)
+ b,B, Y&r))
+ 2iyfazAJt(kr)
-I-bzBzY,(krN, (7)
iyh -f$(rG)=
iy(a,A&hr)+
b,a,Y~hr))-2k(azAzJo(kr)-t
b&Y,,(kr)).
By using the method of undetermined coefficients, the particular solution of the non-homogeneous differential eqn (5) can easily be determined and is given by
hLWzr) W(r) = --$$
+ b& Y&r)1 - --$$
{olAJdhr)+
~~=A*~*(kr~f b~~~Y*(kr)~,
b,B,Ydhr)}+-$p
hA&kr)
+ bJ’hYo(kr)),
where in these expressions h > 0 and k > 0. The radial wave-numbers h and k are both real when p z ynP, h becomes imaginary and k remains real when yu, s p < yv,, and both become irn~~y when p < yv,. When p = yvP, h = 0’ and k real; and when p = ‘yv.,h is irna~~y and k = 0”. When h = O’, we have the case of grazing incidence of dilatational waves, and k = 0’ corresponds to the case of grazing incidence of equivoluminal waves. The coefficients ai and bi have still to be determined. We consider here the case of grazing incidence of equivoluminal waves when k +O+ and h is imaginary. In this critical case, eqn (1)2 degenerates into Euler’s equation of second order, whose genera1 solution is & =Azr+B2i,
O
k=O.
(9)
A note on criticalreflectionsof axiallysymmetricwaves in an isotropic,elastic,hollow cylinder It is therefore
265
necessary that the coefficients a&) and b*(k) in eqn (t$ be such that (i) lim {an(k).ldkr)} = r, k-O+
(10)
(ii) lili+ {b,(k) Y,(kr)} = i,
Using d’Alembert’s limiting procedure and making use of the properties of Bessel functions, we find that a2 and b, are the solutions of
(9 (ii)
$(j-)lk,+=h $ b&,,+ = -;
(11) P.
Therefore in the critical case of grazing incidence of equivoluminal waves when h is imaginary and k + O’, the coeflicients ar and br are a,(h) = 1, b,(h) = 1, az(k) = 2/k, b,(k) = -5
k.
(12)
One can similarly find the value of these coefficients in the case of grazing incidence of dilatation waves when h +O+ and k remains real. However, in this brief note we will not further discuss this case, as the analytical procedure is similar to the case when k +O+. Therefore, a suitable solution of eqn (l), which in the limiting case contains the solution of critical reflection of equivoluminal waves in a hollow cylinder, is given by A = AJo
+ 8, Yo(hr),
0 < h < im,
(13) &=A,fJ,(kr)-BzjkY,(kr),
O=k
The displacement components are given by {A,J,(hr)+B,Y,(hr)}-;[A+(kr)-%;kY~(kr)], (14)
W(r) =
-&
{A,J,,(hr)+B,Yt,(hr)}+-+[A+o(kr)-%;kYdkr)},
where again 0 < h < i m, 0 5 k < m. With A and 4 given by eqn (13), we rewrite the displacement components in the convenient form 1 U(‘)=-(,,‘+h3
8A ar
2iy
,,2+k2”)
-
From (3) and (15), we find that the stress components are given by
iy aA 7#2= -2&L ---($I$+].
C
y2+ h2 Jr
(16)
R.K.KAUL and G.~~~NN
266
Substituting for A and & from eqn (13), we find that eqn (16) takes the form TV= A, AJ,,(hr)+ -&+*(hr)]+& [ 4&y -y2$.
[nudhrI+*;
A&$Mkr)-B,Sk-$ E
Y&r)
1
Y,(hr)]
, (17)
r,.z= ~
[A,Jz(hr) + B, Y,(hr)] -I-2~ ($$)
(A2~~*(b)-B*SkU,(kr)],
where as before 0 < h < im, 0 5 k < m. In the critical case when k =O’ and 0 < h < iq the limiting form of stresses can easily be determined from eqn (17). Using de I’Hapital rule, the stresses are
1 c
~,,lr+,+= A, A~~h~~)+ -$$-&h*r) t -?(A,-&$), =B ?rzIs?+o+
0
+B,
AYo(hor)+
fi -$Y&91]
’
m
[A,J,(hor) f Bt Y,(hor)l + 2~ (A,r + Blf) 2
where ho = h It4+. When k = 0”, we find from (2)4 that y = plu, and from (2& we get ho==-(p/u,)2(1-(f2),
O
where u=v~/~J~,In terms of non-dimensional frequency 0 = p/o,, where oS (= 742t) is the lowest antisymme~c access-shear frequency of an infinite, isotropic, elastic plate of thickness 2t (=(b - a)), we find that in the critical case of grazing incidence of equivoluminal waves /&+(1-o)
* 1/2, o
1
(W
where A @_2_ l-2v 2( 1 - u)’ l=2(h+ and v is the Poisson’s ratio of the elastic medium. (i) Hollow C$inder In the case of a concentric hollow cylinder of circular cross-section, vanishing of the stresses at the inner and outer surfaces gives us the frequency equation A(~~+ hd)Jo(ltoa)+2irlhd:(hoa)
A.(y’+ h,Z)Yo(h,a)+ZELhoY:(hoa) 42 Y
-4cl YQ2
A(y2+ho2)Jo(hob)+2&,.f:(hi,b)
A(y’+ hOZ)Ydhob)+2ClhoYffhob)
F
yh~,th~u~
Y~~Y,(~OU)
a
-% Yb =o, 1 a
yh~,(hob 1
rho Y,(hob )
b
’ ‘i;
cw
where .C(&z) = (~~~u)~,(h~u), . . ., and Y~(hob) = (~~~b)Y,(hob). Using Laplacian expansion anddeleting common termsintheexpansion, and introducing modified Bessel functions of real
A note on criticalreflectionsof a&By symme~c waves in an isotropic,elastic,hollowcylinder
261
argument I,,(@ and km(@),the frequency equation in the critical ease of grazing incidence of equivoluminal waves is given by
where
cm
q=alb,a
and2t=(b-a).
This transcendental equation has a single root which is reiatively easy to compute. In particular, finding this root of the equation does not present the problem of “spurious roots” which Gazis encountered in the investigation of a similar problem, at grazing incidence ([2] p. 274). Using asymptotic expansion of the “Hankel” type for large argument and fixed order, the frequency equation reduces to the form 8,’ +
tat& (I- 02)K’ >
4q(l-
s3
(1 - q)eb -
K2eb= 0,
Wl -VI)
qi/eb),
(23)
where K+(3+$)(1+37j)-(1+3n3(1+3/?7)+48(1-o3(1-n2), K*~(l-~)2+l~l-~2)(l+~)z.
(24
To a first approximation this equation is satisfied when
(ii) Solid Cylindert In the case of solid cylinder of radius r = b, a --)0, 7 + 0, 0,,+ 0 and the frequency equation tProf. R. D. Mind&nhas recentlybroughtto ourattention,thatthe criticalreflectionof waves in an isotropic, elasticplate wasinvestigatedby him in his paper, “Mathematical‘fItwry of Vibrationsof Eastic Plates”, Procee&ngsof&e Bet&~ Sympodnm on Frqnency Coatml, pp. 17-40;U.S. Army SignalCorpsEnginceriagLaboratories,Fort Monmouth, New Jersey, (1957).A remarkin our earlierpaperpublishedin this journal,(19)6), Vol. 12, pp. 353-357,that “Such critical rellectionsof ckistic waves.. . in the case of plates,.. , seems to have been overlookedin the literature”,was thus clearly inappropriate. Annul
268
R. K. KAU?.and G. HERRMANN
reduces to the simple form
On further s~mpli~~at~on,we get the frequency equation
where
since for a solid cylinder B = 2t. This eqnat~ou was first ob~~ned by &toe, et uI. [3f, as a rimming form of Pochhammer frequency equatio~[4], by making use of mod~ed quotient of Bessel functions .5?,,(z)= z.&,(z)/J,,.&). This frequency equation has only one real root given by
For II = 0.31, we find that
where S is the first root of the equation S,(S) ~0. This is the point of intersection of the extensional branch with line OE in Fig. 2 of Ref. [3]. In this critical case, the ratio of the amplitude coefficients is
and the displacement components are
~c~o~~e~ge~e~~~ne discussions.
of the &hors @.K.K.) would like t@ttrank his colkaguts, Dr. Richard P. Shw, for many beIpful
London (1927). 2. D. C. Gazis, J. Am&. 5% Am.St, 573-578 (1959). 3. M. Owe, H. D. McNiven and R. D. Mind& J. Appl. Me&. 29,%%734 (lP62k 4. L. Pochhmmer, Z&t. f. Math. $1, 324-336 (1976).