237
Vol. 15, 1964
The Diffraction of a Plane Compressional Wave by a Spherical Cavity in an Elastic Medium ~) By GoRnoN C. K. Y~I{, Space Technology Laboratories, Inc., Dynamics Dpt., Engineering Mechanics Laboratory, Redondo Beach, California, USA I. I n t r o d u c t i o n
The problem of diffraction of a plane compressional wave by an infinitely long cylindrical cavity in an elastic medium has been discussed by BARON et al. Eli, [2] 2). In this report the analogous problem for a spherical cavity is considered. A spherical cavity of radius a in an infinite, homogeneous and isotropic elastic medium is acted upon by a plane shock wave whose front propagates along z-axis through the medium with a constant velocity c~ (which is the propagation velocity of dilatational wave in the medium) and envelops the cavity. Let r, 0, r be the spherical coordinates with the origin at the center of the cavity, Figure 1. Due to axisymmetry of the problem the displacement component w and the stress components v~r and zor are zero everywhere and it is sufficient to consider the problem for any plane c o n -
t/
celoc/~yc~ ~
z
Shockwave#oat ~y
Figure 1 Spherical Coordinates.
Figure 2 Geometry of Problem.
taining the z-axis. Figure 2 shows one of these planesy z. The direct stress components a U(t) and e a U(t) which are, respectively, parallel and perpendicular to the direction of wave propagation, are carried by the shock wave. [U(t) is the unit step function, l The displacement and stress components in the medium are obtained b y superimposing the free-field displacement and stress components, i.e., those produced by the shock wave in the medium with no cavity, and the displacement and stress 1) This work was performed under Contract No. AF 29(601)-5132 (Project No. 1080, Task No. 108007) for Air Force Special Weapons Center, Kirtland Air Force Base, New Mexico, USA. z) Numbers in brackets refer to References, page 252.
238
GORDON C. K. YEH
ZAMP
c o m p o n e n t s p r o d u c e d b y the a p p l i c a t i o n of corrective tractions at the b o u n d a r y of the c a v i t y in order to m a k e the b o u n d a r y surface traction-free.
II. Free-field Displacements and Stresses Consider an infinite elastic m e d i u m which does n o t contain a cavity. The effect of a shock front on the m e d i u m t r a v e r s e d is to linearly compress it w i t h o u t l a t e r a l strain. I n plane, one-dimensional g e o m e t r y , which the incoming shock w a v e is a s s u m e d to be, t h e condition of zero l a t e r a l s t r a i n exists t h r o u g h o u t t h e wave. Thus for a plane w a v e p r o p a g a t i n g along z-axis, the s u b s t i t u t i o n of s x = ev = 0 into the stress-strain relations o b e y i n g HOOKE's L a w (see, for example, [3 l, p. 7) gives G
=
~
V
-
1 -- v
~
(1)
~'
which renders the p a r a m e t e r - ~ 1--v
~--
(2)
for t h e p r e s e n t p r o b l e m (Figure 2). I n E q u a t i o n s (1) a n d (2) v is P o l s s o x ' s ratio of the elastic m e d i u m . Before t h e shock w a v e arrives the displacements a n d stresses are zero everywhere. A f t e r t h e shock w a v e arrives the stress c o m p o n e n t s G, ao, ar a n d r~ 0 are functions of 0 (for a n y r) given b y t h e expressions (Figure 3 a) ~rf = -- (~ (c~
0 -- e sin s O) ,
(3)
a~1 = - a (sin s 0 - s cos s 0) ,
(4)
O'r
=
S 0",
(5)
(6)
sin2 ~. z
Z
0 E~
~y
y
Figure 3 Boundary Tractions.
The free-field r a d i a l a n d t a n g e n t i a l c o m p o n e n t s of the p a r t i c l e v e l o c i t y of p o i n t s in t h e m e d i u m b e h i n d the step wave front are given b y the relations ~s--
- ~ cos 0 0 c~ C~
where ~ is t h e m a s s d e n s i t y of t h e m e d i u m .
(7)
J
Vol. 15, 1964=
T h e D i f f r a c t i o n of a P l a n e C o m p r e s s i o n a l W a v e b y a S p h e r i c a l C a v i t y
239
The free-field displacement uf of a point in the m e d i u m contains two components (k, and leo are unit vectors in r and 0-directions, respectively)
u f = uf (r, O, t) k, + vf (r, O, t) k o ,
(9)
where, b y integrating Equations (7) and (8), the radial and tangential components of the displacements are given b y
u/r,
0, t) -
v/r,
O, t) -
-
~ oo~ 0
(t -
t*)
(lo)
~ sin 0 (t -- t*)
(11)
@ cc~
in which the time t is measured from the arrival of the shock wave at the point r = a, 0 = 0 (Figure 4) and the delay time t* is given b y t* --
a - - r COS 0
(12)
Cc~
zj
.~
/oaturn tot t=O
e(p2,o2)
Figure 4 D e l a y T i m e t*.
Let an angle o~(r, t) be defined as
~(r,t) =
' for t >
ca r +a
'
(13)
Oc~
Then for free-field stresses, velocities and displacements at any point (r, 0) in the medium z < a we should set a = 0, for all O's at t < 0 and for 0 > e (r, t) at 0 < t (r + a)lc~ in Equations (3), (4), (5), (6), (7), (8), (10), and (11). I n particular at the location of the cavity r = a we should set ~ = 0 for all O's at t < 0 and for 0 > e (t) at 0 ~ t ~ 2 a/c a in these equations where cos_l( 1 _ c~), ~(t) = ~(a, t) =
2~
for t < - - ,
Ca
2a for t > - -
(14)
CC~
Thus the free-field stresses, velocities and displacements at r = a are functions of 0 as well as t.
240
GORDON C. K, YEN
ZAMP
III. D i s p l a c e m e n t s and S t r e s s e s P r o d u c e d by Corrective B o u n d a r y T r a c t i o n s Applied to r = a
The superposition of the tractions on the surface r = a ar = - ~rrf = a (cos2 0 - e sin 2 0) , TrO =
- -
TrO f =
- -
G
-
sin2
-
(15}
0,
(16)
which are equal and opposite to those given b y Equations (3) and (6) (Figure 3b) produces a traction free surface r = a, which can then be considered to be the boundary of a spherical cavity of radius a in the medium. The total displacements and stresses produced b y the incoming compressional wave are obtained b y superimposing the free-field quantities Equations (3), (4), (5), (6), (10), and (11) and the corresponding quantities produced by the application of the surface tractions Equations (15) and (16) to the boundary of the cavity. In what follows the problem and solution of elastic wave propagation originating from the applications of tractions ~ and r~o Equations (15) and (16) on the surface of spherical cavity r = a in an infinite medium will be formulated. This turns out to be a particular case (axisymmetry) of the elastodynamic problem concerning the spherical cavity solved by ERI~WGEN~4J. Except that a different transform with respect to time is applied the procedure followed here is similar to that of Reference 4.
A. Formulation of the Problem The equations of small motion of a homogeneous, isotropic elastic medimn under no body forces is (2 + #) V (V. u) + # V2 ii = e i i , (17) where u is the displacement vector, 2 and # are the Lamb constants, 17 is the gradient operator, 17, and V2 represent the divergence and Laplacian. In spherical coordinates (r, O, q3) (Figure 1), u has the components u, v, w. For motions symmetrical with respect to z-axis, w = vr~ = ~o~ = 0 and all derivatives with respect to q~vanish. Then we have
(18)
u : u(r, O, t) k, + v(r, o, t) ko
and the components of the stress tensor % a o, a4, and zr 0 are related to displacement components u and v by Ou ar = 2 A + 2 # Or ' (19) ao = 2 A + 2 # r - *
(20)
(u + 0~
~ = hA + 2 ~ r -1 (u + v cot 0),
(21)
~ , o = / ~ (or ~ -
(22)
r - ~ v + r - 1 ~0~) 6 '
where
Ou
A ~-17-u=~+r is the dilatation.
(
-a 2 u + o 0
Ov
+vc~
)
(2~)
Vol. 15, 1964
The Diffraction of a P l a n e Compressional W a v e b y a Spherical C a v i t y
241
The problem is now reduced to finding the solution of Equation (17) which vanishes at r § ~ and at the surface of the spherical cavity, r = a, satisfies the boundary conditions forO< O<~(t), G (a, 0, t) = R (0, t) = [ a (cos 2 0 -- e sin 2 0) (24) for e(t) < 0 < :~ and
10
for0<
T~o(a, O,t)= <@ (0, t ) = I - a (~12+~e)sin20
0<~(t),
(25)
10
for ~(t) < 0 < 7e.
It is further required that the waves generated at r = a will be of outgoing type.
B. The Solution A transform with respect to time is applied to solve the problem stated. The transform t-(r, O, p) of fir, O, t), and f(r, O, t), are related to each other by the equations o~
i (r, o, pl =
f t(r, O, tl
a* ,
(26)
0 oa- iy
/(r, O, t) -=f ~r, O, p) e i~' dp , -eo--
(27)
iy
where y is to be chosen positive and such that Equation (26) converges in Im (p) < --VMultiplying both sides of Equation (17) by e-lP t/2 zc, integrating the result from 0 to eo, and taking into account the condition of initial rest we obtain
( w + h~) ~ = (1 - ha~ ~ ]/ v (17. u ) ,
(28)
where
(z + 2 ~)
7==
= (~ P)~
(29)
and = (fi p)~,
(30)
in which G = 1/~ and c~ = 1//3 are the propagation velocities of dilatational and shear waves, respectively, in the medium. Taking the divergence of both sides of Equation (28) we obtain
(v~ + h~) v . u = 0.
(31)
Assuming that V- u is determined from Equation (31) a particular solution u1 of Equation (28) is u 1 = -- h f e V (17. ~ ) . (32) A more complete solution of Equation (28) is obtained by adding to u~ the complementary solution u 2 of equations (W+h~)u~=0, ZAMP 15116
V.u~=0.
(33)
242
GORDON C. K . Y E g
ZAS{P
The desired solution must satisfy the boundary conditions at r = a and r --> oo and must have the character of diverging waves. The latter condition is guaranteed by the Sommerfield radiation condition, i.e.,
[ OUk. -ih kuk). : 0
(k= 1,2)
lira r \ Or
(34)
~ - - + OO
A solution of Equation (31) m a y be obtained in terms of the zonal spherical surface harmonics P , (cos 0) of positive integral degree n in the form
v.
: r" R,l(r)
P,, (cos 0),
(35)
where S. is a function of p only, P . is the Legendre polynomial of degree n and Rn~(r ) is a function of r only which must satisfy the Equation -
d~Rn dr 2 1 + 2 (~z -[- 1) r - 1 -
-
d Rdrn l
-
2
(36)
+ hi R " I -- 0 .
The solution of this equation is given b y R~l
r - (" + l/2) tcA . . . . rr(1) n+
1/2
(hx r) + B ~(~) ~ n + 1 / 2 (h 1 r)]
,
(37)
where A and B are arbitrary constants and H~)+,/2 and H~)+ 1/2 are the half order Hankel functions of the first kind and the second kind respectively. Of these functions only z2~r~(2).+~i2 satisfies the Sommerfield radiation condition, i. e., the asymptotic form of H(~) n + 1/2 corresponds to out-going waves [when the inverse transform Equation (27) is used] since (see, for example, [5], p. 624) H(~) *~n +
1/2
[
]
(h k r) e il't "-+ /. _ _ ! 2 \V2 exp [ .i ( n + l ) X ) e x p [ _ i ( h k r _ p t ) ] \ ~z h t~ r ] ~ 2
(3S)
Hence we set A = 0. Also, absorbing B into S, we write m r) S, P, (cos 0) . V 9 u 1 = r - 1/~ r4(21 **n+l]2VVl
(39)
The expression of u 1 now follows from Equation (32) u l = _ hi-.2 V It- 1/2 ** rr n +
/~, 1/2 V"I
r) S~ P. (cos 0)]
(40)
The components u, and v, of u 1 are therefore given b y u~ = - h Z 2 ~--g ~-r~" 1/2 *-~H(.21+1/.2(h1 r) S, P~ (cos 0)] ,
(41)
F1 :: -- hi_~ O6 0 [r- a/~ **~r/(.2)+1/.2 (h i r) S~ P= (cos 0)] .
(42)
In Reference [4] it is shown that a solution uz of Equations (33) is u z = [7 • {r "+l R.~(r) [7 [ L p .
(cos0)] • k~},
(43)
where Y. (for n > 1) is a function of p only and R.=(r) = r - ( ~ + 1/.2)H(~)+1/.2 (h~ r ) .
(44)
~ol. 15, 1964
The Diffraction of a Plane Compressional Wave by a Spherical Cavity
243
T h e components u~ and v 2 of u 2 are given b y --U2 =
-~' -
v2 = r - i
3/2
"~nr~]~'1/2 2)+(h 2 r)
n (n + 1) Yn
Pn (cos 0)
,
08 HI21 (h 2 r) Y. P~ (cos 0)J Or O0 ~rI12 "'n+112
(45)
(46)
T h e diverging wave type of solution of Equation (28) which vanishes at r + oc is then given b y the sum u = u 1 + u 2 whose components are from Equations (41), (42), (45), and (46). 0 f~, = - h~ 2 S, o-F L-- i/2 H(2) - ~ + ~/2 (hi r)J (47) + n (n + 1) V. r - 3/2 ~r4(2) (h 2 r) } P,~ (cos 0) n + 1]2 v = { -- h [ 2S. r - a/~ ~r4(2)~ + ~/2 (hI r) t
(48)
o ~" " ' ~ + m + Y~r-1 -Er
(cos0).
Equations (47) and (48) satisfy Equation (28) for every n. Hence any linear combination will also be a solution of Equation (28). B y summing over n from 0 to ~c in each of Equations (47) and (48) we can express the motion resulting from any axisymmetric dynamic traction at the surface of the cavity. This requires the computation of stress components which are obtained by substituting Equations (47) and (48) into the integral transforms of Equations (19) to (22). Hence
~r ~ 1~ ~An(P f) Sn -~ Bn(P I/) T~('KI,JF 1 ) Y.I P. (cos 0) ,
(49)
8 0 = # [E~(p r) S~ + F~(i5 r) n(n + 1) Y~] P~ (cos O) de
(50)
+ ff [C'~(p r) S. + H~(p r) Y,,,] ~ 0 ~ P,, (cos O) ,
~r = ff [EJp r) S~ + f,, (p ~) n(n + 1) u
P,, (cos 0) ,4
(51)
+ # [G,,(p r) S. + H J p r) Y~] cot 0 ~ 0 P" (cos 0),
:Go = # Ec.(P r) S. + D~(p r) Y.] d@ P" (cos 0) ,
(52)
where
A~(p r)
-
r -'/2 (or p r)-~ {[(fl p r) 2 - 2 (n - 1) (n + 2)J H(~}+,j,2 (c~ p r)
+ 4 ~ p *.
r4'(2/ .~. + ~/~ (or p r) - ~ r~i2) --~ + 1/2 (~ P r)},
(53)
"(2) r) - 3 H .+ll2(flPr)3, (21 B ~ ( p r ) = r - 5 / 2 E 2 f l p r H .+l12(t~P
(54)
C.(p r) = - r ~/2 (o~p r)-~ [2 ~ p r 1/~u'(2) -.+ (~ p r) - s H~+~f (~ P r)l ,
(55)
D.(p r) = -- r ~/2 {[(fl p r)2 _ 2 (n -- 1) (n + 2)] H(~)+~/e (fl P r) + 2 fl p r r4'(2)+ ~/~ (fl fl r) -- 3 ~rnf(2) +l/2 ..
n
(/3/5 r) }
(56)
244
G O R D O N C, K .
YEH
ZAMP
En(p r) = r - 1/2 (o~p r) -2 {[(fl p r)2 _ 2 (~ p r) 2 - 2] H(#)+~m (~ P r)
(:{ p r) - 3 ~r{~) -~+1I~
-[2~p r r~'{~} ~+~/2
?n(P ~) = 2 . - ~/~
(:{ p r)]}
(57) ,
H:#)+:j~(# # ~),
(58) (59)
Hn( p r) = r- 5/2[H#)+ ~m (flP r) + 2 flp r H # ~ ~/2 (flP r)],
(6o)
in which Primes denote differentiation with respect to the argument of the functions. The integral transforms of the boundary conditions, Equations (24) and (25), give from Equations (49) and (52), oo
R -- # Z i o n ( #
a) & + Bn(# a) ~(~ + 1) Yn] P,, (cos 0),
(61)
r162 oo
@ = # Z [cn(p ~) Sn + Dn(P ~) Y~] ~0 P~ (cosO).
(62)
From these series we can determine S n and Yn. Equations (61) and (62) suggest that a series representation of the following form is a convenient one: oo
R = ~Y':n(P) Pn (cos 0),
(63)
~'t=0
(@ = 2 ~ n ( p ) d
Pn (cos0) .
(64)
Given R, we can determine a n (p) from the Legendre expansion theorem:
-an(P) -- 2 ~t + 1/-R(O, ~b)Pn (COS0) sin 0 dO 2
(65)
0
In order to obtain b-n (#5) we use the Legendre's differential equation n(n
+ 1)Pn (cos 0 ) =
--csc0
#g
[sin0 ~ 0 Pn (cos0)] .
(66)
If we form the summation in Equation (66) and use the relation Equation (64) we obtain oo
Z f t ( f t @ 1) -bn(P) J~n (COS0) = -- CEC0 ~00 [sin 0 @ (0, P)I.
(67)
Comparing this with Equations (63) and (65) we can determineb n (~b)from a given as follows --
-- n(n + 1) bn(p) --
2 n +
2
1
= 0
[-30 [sin 0 @ (0, p)] Pn (cos 0) dO . 5
(68)
Vol. 15, 1964
245
T h e D i f f r a c t i o n of a P l a n e C o m p r e s s i o n a l W a v e b y a S p h e r i c a l C a v i t y
Substituting Equations (61) and (62) into Equations (63) and (64) we have An( p a) S n + Bn(p a) n(n + I) Yn= #-a an(p ),
Iorn>
Ao( p a) S O= #-1 a0(P) , Cn(p a) S n + Dn(p a) Y,, = #-1 bn(P) 9
1
for n = 0
[ (69) f
for n > 1.
(70)
Solving S n and Yn from Equations (69) and (70) we obtain S n = (# An) -* [Dn(p a) a,(p) - Bn( p a) n(n + 1) b-n(P)~, So -= L~ Ao(p a)~-i ~o(P),
Yn = (# An) -1 ~A,( p a) -bn(P) - Cn(P a) ~n(P)],
for n > 1 / for n = o
/
(71)
for n ~ 1
(72)
An =- An(P a) = An( p a) Dn(p a) - n(n + 1) Bn(p a) Cn(p a) .
(73)
where This completes the general solution for the problem for Equations (71) and (72) used in Equations (47) to (52) give the integral transforms of displacement and stress components; performing the inverse transforms as expressed by Equation (27) we obtain the actual displacement and stress components. C. Determination of a n (p) and bn (2b)from the Boundary Tractions For the specific problem considered it remains to determine the coefficients a-~ (p) and-b~ (p) from the boundary tractions R(O, t) and @ (0, t) given by Equations (24) and (25). We accomplish this by first expressing R and @ in series forms oo
R = ~ a n ( t ) Pn (cos 0),
(74)
n=0
@ =~
oo
d bn(t) -dO Pn (cos0)
(75)
n=l
and then performing the integral transforms on an(t) and bn(t). From Equations (65) and (68) we should have an(t) _ 2 n2+ 1 / R ( O , t) Pn (cos 0) sin 0 dO ,
(76)
0
~sin 0 @ (0, t)] P, (cos 0) dO. o Substituting Equations (24) and (25) into Equations (76) and (77) we have -
-
bn(t) -
2 ~ ~ V 1)
2 a n (t)
(2 n + 1)
(77)
[
v (cos2 0 -- e sin 2 0) P, (cos 0) sin 0 dO 0
1
= / [ ( 1 + ~) x~ - ~ p.(x) dx, cos ~ (t)
(78)
246
GoRI)o~q C. K. YEIt
ZAMP
"d 2 n (n + ~) b. (t) a(1 + e) ( 2 n + 1) = j -dO (sina 0 cos O) P , (cos O) dO 0
(79}
1
= [ ( 3 . 2 _ 1) Pd*) d~, J cos o:(t)
where x = c o s 0 and c o s ~ ( t ) = l - c a t / a for t < 2 a / % and cos c ~ ( t ) = - i for t > 2 a/c a from E q u a t i o n (14). Since P,, (x) is a polynomial in x the integrals in Equations (78) and (79) can be evaluated without difficulty. For instance, since P0 (x) = 1, we have
2 ao(t) a
[ ( l + e ) x2
_
e]dx=
xa
k~
--ex
_ccdja
1 - ca t/a
(80) and -- 2 -- (1 + e) 22 + - l- + T -e-
2 a~
2a = 3-2 ( 1 - - 2 e ) '
fort>
2 ca a
Since P1 (x) = x, we have 1
2 ~fft)
-
3or
[(]+e)
x a-ex]d.=[~x
r 1+ e
4
-
I
e x2]1 2
~-ca,a
I
1 - ca t / a
a
2
+ (1 + s)
4 for t <
3§ 28 2 2 + (1 + e) 2 a 2
2 ~l(1) - 2 3a
1 + 4
2 4 = O,
'
J /
(81)
2a
---ca ~a
for t > - -
gcr
]
I
and 1
3 a(1 + e) -1 - %
(3Na--X) d~r ~
4
2
-%qa
tja
=2 (~t)--4
(~)2§
3 ('~ t ) 3
3 (c~)4, 4-
4 b~(t) 3 24 0 3 cr (1 + e) -- 2 (2) - - 4 (2) 2 q- 3 (2) a - -~ --- ,
for t ~ - 2a
for t > - - - -
(82)
Vol. 15, 1964
247
The Diffraction of a Plane Compressional Wave by a Spherical Cavity
Since P~ (x) = (1/2) (3 x a -- 1) we have 1
4 5a2(t) ~
-f>
(1 + e) x ~ -- (1 + 4 e) x 2 + e; dx
1 - co: t / a
3(l+e) 5
X5
l+4e 3
Xa+e 1 - co:t/~
= 2 \( & t , - - ( 5 * 2 s ) ( c ~ ) Z +
4
~z2(t)
5or
17+314e
-- 3 (1 + e)
+
- 2(2) - (5 + 2 ~) 2. +
17 + 14 3
+
(~)a
5
(83) '
2~
for t ~ - - ,
Gx
2~ - 3 (1 + s) 2'
3 (1% 4 2 5 _ 8 (1 + 4 5 15 '
26~
for t > - -
and 24G(t)
5 o(J. + ~)
j
1
((9x 4
-
6x 2+l)
dx=
x 5-2x
a+
x] 11-co:t/~
1
i - co:t/a
(2t-? -
I
} forl(
24 G(t) 9 2a 8 5 ~ (1 + e) - 4 (2) - 12 (2) ~ + 16 (2) a - 9 (2)' + T = 5'
for t >
--,2a
(84)
I
2a c~-
Next we perform the integral transformation on G,(t) and b~(t) according to
Equation (26) to obtain G(P) and b-~(ib). Since G(t) and b~(t) are power series of t and since for f (t) = t m we have according to E q u a t i o n (26) ](~) -
"~ ! 2 ~(i p)-,+~
(85)
'
the expressions for ~n(P) and b-~(ib) can be readily obtained. For instance from Equations (80), we have
!
2ap~ + ( l + e ) 2~z ~o(P)= ( 1 - - 2 e )
3ip'
iTp T
+(l+e)
~ap~,
fort<
c---~ '
fort>
2c_~_a
J
248
GORDON C. K. YEH
ZAMP
F r o m E q u a t i o n s (81) we h a v e 2~z ~ ( / 5 ) 3 c a
1
~-~
3 (3 + 2 e) ( @ ) 2
(~p)~
4
3(1+e)
2
~ -
+
(@)~
2 3(1 + e )
(@)4
s _
_
3ca + 2 a p2
3 ( 3 + 2 e ) c~ + 9 ( 1 + e ) c~ 2 i a ~ p8 a 3 p~
6
(i p)~ 24
(i p)~
9 ( 1 + e ) c~ i a 4 p5 ,
(87) 2a
for t ~ - -
Ca
~-~(#)
=
2a for t > - -
0,
Ca
F r o m E q u a t i o n s (82) we h a v e 2Jr 2a
(ip) ~
9 (~_),
24
16 3 ca 2ap~ +
--
6 c~ i a ~p3 +
27 c~ 2 a 3p4
(i p)~
27 c~ 2ia ~pS' fort~
(88)
26~
, -ca 2a
~(p) = 0 ,
for t > - -
Ca
F r o m E q u a t i o n s (83) we h a v e cr
2a
_
(ip)e
5 Co:
2ap2 +
4 (ip)~ 5(17 + 14 e) ( _ ~ ) 3 6 + 12 (i p)4 3(1 + e ) ( ~ _ ) 5 15(1 + e) ( c a ~4 24 4 \ a ] (i p)5 + 4
5 (5 + 2e) c~ + 2 i a ~ p3
90 (1 + e) c~ i a 4p5 a
3ip
120 (i p)~
(89)
5 (17+ 14e) c~ 2 a 3 p4
90 (1 + e) c~ a5 p. ,
'
for t ~
2a c~'
fort~
2a
Ca
F r o m E q u a t i o n s (84) we h a v e 2~
2
6a
(i p) 2 15 (__~_),
s 5 Ca 5 c~ - + 6 a p 2 + - ia2p3
24
~ 20 c~ a3p4
3 (_~)5
+ ~-
120
(i p)~
45 cl i a 4p~
45 c~ asp6
(90)
2LZ
for t ~ - 2vr
1
o (1-7- ~) ~(P) = 3 i p '
fort~
2a ~
}
Vol. 15, 1964
T h e D i f f r a c t i o n of a P l a n e C o m p r e s s i o n a l W a v e b y a S p h e r i c a l C a v i t y
249
IV. Total Displacements and Stresses The total displacement and stress components at any point (r, O) in the medium and at any time t are obtained b y adding the free-field displacement and stress components Uf, Uf, ~Trf, OOf , (Y~)f, and v~ of [given by Equations (10) and (11) and Equations (3) to (6)j to the summation over n from 0 to oo in the inverse transforms of the corresponding components u, v, ~, a 0, ~r andr-~0 [given by Equations (47) to (52)]. In particular by putting r = a in the resulting expressions we obtain tile displacement and stress components on the surface of the spherical cavity. The total stress components a~ and Tr 0 on r = a should be zero.
V. Rigid-body Motion of the Cavity Boundary A portion of the total motion of the boundary of the cavity consists of a rigid-body translation in which the cavity maintains its spherical shape and translates in the direction of propagation of the incoming wave (Figure 5). This rigid-body motion can be extracted from the total motion of points on the cavity boundary. Essentially we must superimpose a portion of both the ~r = 1 Component of a Legendre expansion of the free-field motion and the n = I component of the motion produced by the corrective boundary tractions, O"r (g, O, t) = gl (t,) t)1 (COS0) = 41 (t) cos 0 and Tr o (a, O, t) = bx (t) (d/dO) 1)1 (cos 0) = - ba (t) sin 0 [Equations (74) and (75)~. The free-field displacement components uf and vf Equations (10) and (11), respectively, are expanded into Legendre series as follows: o~
uz(a, O, t) = ~ c , ( t ) P, (cos 0),
(91)
**=0 oo
v/a, o, t) =
P. (cosO),
(92)
n=l
where
c.(t) -
o~(t) 2 ~ 2+ 1 f uf(a, O, t) P~ (cos 0) sin 0 dO,
(93)
0
-- dn(t) -- 2 ~ (n- T i)
[sin 0 w(a, 0,/)] P~ (cos O) dO
(94)
0
and from Equations (10) to (12) ~a
u/a, O,t)= -'~ ec--J ( t - T'~j ) c o s O - - - c~ c~ v:r(a'O't)~-
Qc--~ ~ (t-
O,
~a ) s i n O + oc~aa sinOcosO.
(95) (96)
250
JORDON C . K . YEH
ZAMP
For n = 1 we have e(t)
q(t) -
2 ~ C~
Cc~ :
g
0
o
I - cctt/a
1 - co;~/ct
3~a 2 b~C~x
[" a
C~ l J 1
(t-
2 ~ce
+~
3GOb
aa SOv~
1
g
c~)[--3
(@)+3
(~)2_
(97)
[_4 (~) + ~(~:'_)~_~( ~ / + (~)'] [6( ~ ) 2 4 ( ~ ; + (~)'] , f o r t ~ - - - ,~o c~
ffa 8 e cg [6 (2) 2 -- 4 (2) a + 24] -
Cl({ ) - -
(c~ t la ]
aa e c~ '
2a for t > --c~
and
-
d,(t)
3~c~ 4-e
-
(t-
~ct~ / j f 2 sin0 cos20 dO o
=(t)
3 ~ a ~f '(+ *-E-~:
sinaO eosO + 2 sinO cosaO) dO
0 1 - c~t/a
3 (r
1 - co;t/a
(t _ a . l : x 2 dx +
3 ~ a F,
1
-
1
[~(~) -~ (<-?+4 ( ~ / - (~)']
&
(98)
36a
+-sw [-2 (~) + (~;] ~ o ~ [-4 (<-) § ( ~ ; - 4 (~)~ + (%'] 96a
ffa
=-Ig-~-cg
3
4
-
~Ya
-
ddt)
-
,6 0 4
,
fort<-c 2a
Ga
[4 (2) ~ -
2~J -
-,
o 4
'
for f >
-~-
The displacement components ufl and v:l, corresponding to n = 1 can be written in the following manner us1
Vfl
=
q(l) cos 0 =
=- dl(t ) sin 0 =
[ el(t) -~ dl(t)
-] cos0 + [ c4t) +2
[ cl(t) -~ dl(t)
]sin0-[
cdt)
dl(t) ] cos 0 ,
(99)
+2dl(~/]sin0"
(~00)
Vol, 15, 1964
The Diffraction of a Plane Compressional Wave by a Spherical Cavity
251
The coefficient [cl(t) - dl(t)]/2 represents a deformational component of the displacements while the coefficient [cl(t) + dl(t)]/2 represents the rigid-body displacement component. In a similar manner, the displacements ul and v 1, produced by the corrective boundary tractions ~r = al(t) cos 0 and vr0 = - bl(t ) cos 0 can be written in the form: u 1 = el(t ) cos 0 = [ ea(t) -/~(t) ] c o s 0 § [- e~(t) +2 !~(t) ] cos0,
(101)
vl = -- it(t) sin 0 = [ e~(t) -?/~(t) -] s i n 0 -
(102)
[ edt) +2/l(t) ] sin O,
where the terms containing the coefficient [el(t ) + fl(t)l/2 are the rigid-body components of the displacements. In Equations (101) and (102) el(t ) andfl(t ) are the inverse transforms of q(p) and~(p) expressed in Equations (47) and (48) as follows: 7,(p) = -
(~ p ) - ~ s l
~
~-
~/2
,=~
l-~(p) = - (o~p)-2 $1 a-'~i2 --~i~"(2'(o~p a) + Yl a-l{ 097, "~'~'2r4(2)--3i~, (fi * ")]},=,,
(,04)
where S 1 and Yl are given by Equations (71) and (72). The total rigid-body displacement of points on the cavity boundary is obtained by superposition os the rigid-body components in Equations (99) and (100) and Equations (10,) and (102)
u(a, 0, t) = g(t) cos 0,
(105)
v(a, O, t) = - g(t) sin 0,
(106)
where g(t) =
c:(t) + d~(t) + e:(t) + l:(t)
(~_07)
2 It can be noted from Figure 5 that the quantity g(t) is inherently negative. u=g{#
sho~errb~ ? .--\
\ \ \ \
I
V=0 Fi ure 5 Rigid-Body Motion of the Cavity.
Y
252
G O R D O N Co K. V E H
ZAMP
VI. Conclusions The solution to the problem of diffraction of a plane compressionat wave b y a spherical cavity in an elastic medium has been formulated. B y means of an integral transform technique the expressions for the transforms of the displacement and stress components are obtained. For practical application and numerical computations it is necessary to perform the inverse transforms of these expressions. This would be a worthwhile subject for future research. The displacements and stresses of points in the medium which are produced b y the step compressional wave m a y be used as influence coefficients to determine the corresponding displacements and stresses produced b y a wave with a varying stress-time history. For example, denoting the radial displacement produced at a point b y the incoming step wave as u s (r, O, t) the radial displacement produced by a wave with a stress-time history a(t) is easily c o m p u t e d from the Duhamel integral t
~(r, O, t) = ~(o) us(r, O, t) + [ do(~) u,(r, O, t -- ~) d~ . d
dr
(~o8)
0
The two-dimensional counterpart of the problem, i. e., diffraction of a compressional wave b y a cylindrical cavity in ari elastic medium has been presented in References [1J and [2]. I t would be interesting to compare the results between the two- and threedimensional cases. REFERENCES [1] M. L. BARON and A. T. MATTHEWS, Diffraction o / a Pressure Wave by a Cylindrical
Cavity in an Elastic Medium, Trans. ASME Series E, J. Appl. Mech. 28, 347-354 (1961). [2] M. L. BARON and R. PARNES, Displacements and Velocities Produced by the Di//raction o~ a Pressure Wave by a Cylindrical Cavity in an Elastic Medium, Trans. ASME Series E., J. Appl. Mech. 29, 385-395 (1962). [3] S. TIMOSHENI~Oand J. N. GOODIER, .Theory o~ Elasticity, Second Edition (McGrawHill Book Company, Inc. 1951). [4] A. C. ERINGEN, Elasto-Dynamic Problem Concerning the Spherical Cavity, Quarterly J. Mech. and Appl. Math. 10, Part 3, 257-270 (1957). [5] P. M. MORSE and tI. FESHBACH, Methods o~ Theoretical Physics (McGraw-Hill Book Company, Inc., New York 1953). Zusammen/assung
Ein sphS~rischer Hohlraum in einem unendlichen homogenen und isotropen elastischen Medium wird yon einer ebenen Druckstosswelle iiberstrichen, deren Front sich gleichf6rmig fortbewegt. Das durch Ablenkung am Hohlraum erzeugte Verschiebungs- und Spannungsfeld wird mittels einer Integraltransformation bestimmt. Insbesondere wird die starre Teilbewegung der Begrenzung des Hohlraums ermittelt. Das Problem wird ftir eine einstufige Druckwelle gelSst. Die Ergebnisse lassen sich als Einflussgr6ssen betrachten, mit denen sich die Duhamel-Integrale fur allgemeinere Druckwellen berechnen lassen. {Received: October 26, 1963.)