Vol. 16, 1965

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527

Radiation Conditions in Elasticity B y JAMES M. DOYLE, D p t . of E n g . Science, U n i v e r s i t y of N o t r e D a m e , N o t r e D a m e , Ind., U S A

I. I n t r o d u c t i o n I n dealing w i t h s o l u t i o n s t o p r o b l e m s of w a v e s c a t t e r i n g , i t is n e c e s s a r y t o r e q u i r e c e r t a i n b e h a v i o r of fields s a t i s f y i n g t h e r e d u c e d w a v e e q u a t i o n in o r d e r t o i n s u r e u n i q u e n e s s of t h e solution. T h i s p r o b l e m h a s b e e n i n v e s t i g a t e d b y a n u m b e r of a u t h o r s of w h i c h SOMMERFELD, MAGNUS, a n d RELLICH [1] [2] E3] 1) a p p e a r t o h a v e b e e n t h e earliest. I n a s m u c h as t h e e q u a t i o n s of classical e l a s t o k i n e t i c s r e p r e s e n t a n e x t e n s i o n of h y p e r b o l i c o p e r a t o r s to v e c t o r fields, i t is o n l y n a t u r a l t o e x p e c t t h a t a similar r e s t r i c t i o n is n e e d e d o n s c a t t e r e d c o m p o n e n t s of d i s p l a c e m e n t in a n elastic solid if u n i q u e n e s s is t o b e g u a r a n t e e d . I t is t h e p u r p o s e of t h i s w o r k t o e s t a b l i s h r a d i a t i o n c o n d i t i o n s for s u c h fields. Since i t is possible for w a v e s t o p r o p a g a t e a t t w o d i s t i n c t speeds i n a n elastic m e d i u m , i t is n e c e s s a r y to h a v e a c o n d i t i o n p l a c e d o n e a c h t y p e of wave. T h e c o n d i t i o n s e s t a b l i s h e d here, while n o t b e i n g t h e o n l y ones w h i c h w o u l d assure u n i q u e n e s s , do a d m i t t o a p h y s i c a l i n t e r p r e t a t i o n as is s h o w n herein. T h a t t h e s t a t e d r a d i a t i o n c o n d i t i o n s are sufficient for u n i q u e n e s s is also d e m o n s t r a t e d .

II. R a d i a t i o n C o n d i t i o n I n t h e a b s e n c e of b o d y forces, t h e e q u a t i o n s of m o t i o n of classical e l a s t i c i t y t a k e t h e form :

@ "y ~

(b2 -- a 2) V V 9 v + a 2 172 ~) = Oft ,

(2.1)

w h e r e a a n d b are t h e p r o p a g a t i o n speeds of s h e a r a n d d i l a t a t i o n a l waves, V is t h e c o n v e n t i o n a l del o p e r a t o r , a n d t h e s u b s c r i p t s signify d i f f e r e n t i a t i o n w i t h r e s p e c t t o t i m e . A solut i o n t o t h e field e q u a t i o n s w h i c h is h a r m o n i c in t h e t i m e v a r i a b l e m a y b e w r i t t e n : 1 ~ ( P , l) = h - [ a ( P ) e - i ~ t + ~ * ( P ) e i~t] ,

(2.2)

w h e r e : P : (x, y, z) o) = a n g u l a r f r e q u e n c y of m o t i o n ~* = c o m p l e x c o n j u g a t e of ~ . S u b s t i t u t i o n of t h i s f o r m of ~ i n t o E q u a t i o n (2.1) yields t h e following: +

=

+

=

0

(2.3)

I t is s o l u t i o n s t o t h i s e q u a t i o n w h i c h will b e i n v e s t i g a t e d in t h e w o r k w h i c h follows. S u p p o s e t h a t a s o l u t i o n to E q u a t i o n (2.3) is r e g u l a r a t all p o i n t s in space o u t s i d e a g i v e n r e g u l a r surface w h i c h is n o t e d b y B. I n this, a r e g u l a r surface is as defined b y KELLOGG [4J and, a r e g u l a r s o l u t i o n will b e one w h i c h a l o n g w i t h its first a n d second d e r i v a t i v e s is c o n t i n u o u s . F o r s u c h a s o l u t i o n : ~ ( P ) = - - / [ 7 ~ ( 0 ) " L ( O s ( P , Q)) -

O s ( P , O) 9 Tn [u(Q)]? da O

z~ +;[u(Q)

(2.4) 9 T n ( U s ( P , Q)) -

( l s ( P , Q) . T.(Q)] daQ ,

B

a) Numbers in brackets refer to References, page 531.

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br6ves

ZAMP

for all points P exterior to B. I n E q u a t i o n (2A) :

1

(7

Us-- 4zeOm2 VV T"=-q[ 2a2d

1

+ 4zeta 2

+ (b~ - 2 a 2 ) ~ d i v + a

~g•

R

[' ],

O = (x', y', z') a m o v i n g p o i n t , R~ =

(~' -

~)* +

(y' -

y)* +

(z' -

~)~,

io~ a

#

-

b '

= mass d e n s i t y , = unit o u t w a r d n o r m a l v e c t o r , ~YT~= a large sphere centered at the origin of c o o r d i n a t e s , i

= dyadic i d e n t i t y element.

I n all t h a t follows it is assumed t h a t a, b, o~, and ~ are real positive constants. L e t ~ represent t h e integral o v e r 2Jr . This is o b v i o u s l y regular t h r o u g h o u t all of space. Call the integral over B u s and further decompose it as follows:

B

"

(2.5)

in which:

1 e~ Ua-- 4zrOo) ~ V V ~ +

1 4~0a

e~

R

2

1

"['

~fb --

4 ~ ~ o92 VV

e #R

R

I t is desired here to show t h a t ~3 and a4 o b e y some condition, similar to the Sommerfeld radiation condition, as t h e distance f r o m the origin at which p o i n t P is located becomes e x t r e m e l y large. F o r convenience, t h e figure showing t h e various quantities i n v o l v e d in t h e succeeding steps is included.

b

Schematic diagram. F o r the present, consider u3

u3 =/E~"

7".(G) -- G "

T.(~)] da.

B

I t m a y be shown t h a t the following results hold:

l~a _

4z~Q

e ~rr (~r~r -- ~) (e-h~r'en) + 0 ( lr~ ) '

(2.6)

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Kvcze Mitteilungen - Brief Reports - Communications br~ves

C o n s e q u e n t l y , t h e e x p r e s s i o n for u8 m a y b e w r i t t e n :

u3--

4~a

1

~

e~" (?, ~, r

/:

e ' ~ ~h [T,(~) + i w e a (7r ~,) ~] da + 0

1)-

.

(2.7)

B

If, now, t h e stress vector, a c t i n g o n a s p h e r i c a l surface t h r o u g h P a n d c e n t e r e d a t t h e origin, is c a l c u l a t e d , a c o n d i t i o n similar t o t h e c o n v e n t i o n a l r a d i a t i o n c o n d i t i o n is o b t a i n e d :

H e r e t h e s u b s c r i p t r o n t h e o p e r a t o r T signifies t h a t t h e stress v e c t o r is a c t i n g on a s p h e r i c a l surface of r a d i u s r. I t is a t once a p p a r e n t t h a t a s i m i l a r e x p r e s s i o n is t r u e for ~4:

T h e s e t w o e q u a t i o n s are r e m i n i s c e n t of t h e c o n d i t i o n in o r d i n a r y w a v e t h e o r y . N o t surprisingly, t h e t w o e x p r e s s i o n s c a n also b e utilized t o w r i t e t w o i n t e g r a l e x p r e s s i o n s w h i c h are also s i m i l a r t o a r e s u l t in classical w a v e t h e o r y :

fl?,(a~)-imqa~.l~da=O,

lim

~'---~oo

lim

~'--.->oo

Xr

flf,(~,)-iwqb~,l'da=O,

(2.10)

Xr

w h e r e Z r is a s p h e r i c a l surface of radfus r c e n t e r e d a t t h e origin. As will b e s h o w n in t h e n e x t p a r t , t h e c o n d i t i o n g i v e n b y (2.10) c a n b e utilized t o give a p h y s i c a l i n t e r p r e t a t i o n of o u t w a r d m o v i n g waves. III. Physical

Interpretation

of Radiation

Condition

T h e t o t a l e n e r g y in a n elastic b o d y is g i v e n b y t h e e x p r e s s i o n :

E = ~-

V~(~):s

+ ~ (~t" ~t)] dT,

(3.1)

D,

where S(~)=~(V'~)f+2~E(~),

s

1 (V v + v V) ,

D 1 = region o c c u p i e d b y t h e b o d y .

B y a p p r o p r i a t e t r a n s f o r m a t i o n one m a y s h o w t h a t :

dEdt = / T n ( v )

" vt da ,

(3.2)

B,

in w h i c h B1 is t h e b o u n d i n g surface of D~. T h e a t t e n t i o n n o w will b e focused o n t h a t p o r t i o n of 6 w h i c h is n o t r e g u l a r in all of space, n a m e l y v2- T h i s is w r i t t e n : v2 = ~1- ['72(P) e_i~ot + a * ( P ) ei~t ]

(3.3)

a n d uz is as s h o w n in E q u a t i o n (2.5). T h e i n t e g r a l i n E q u a t i o n (3.2) r e p r e s e n t s a flow of e n e r g y t h r o u g h t h e surface B 1. Consider a region b o u n d e d b y c o n c e n t r i c spheres, c e n t e r e d a t t h e origin, a n d w h o s e r a d i i are large e n o u g h so t h a t t h e surface B is w i t h i n t h e s m a l l e r sphere. T h e n t h i s integral, e v a l u a t e d o n t h e s p h e r e of l a r g e r radius, s h o u l d yield a n e g a t i v e flow if t h e p o s s i b i l i t y of e n e r g y sources a t i n f i n i t y is t o b e r u l e d out. I n w h a t follows, i t is d e m o n s t r a t e d t h a t t h e c o n d i t i o n s set f o r t h in E q u a t i o n (2.10) g u a r a n t e e t h a t t h e flow of e n e r g y across t h i s surface d u e t o t h e field v2, is n e g a t i v e . F o r d e f i n i t e n e s s refer t o t h e larger s p h e r i c a l surface as 25 . ZAMP

16[34

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Kurze Mitteilungen - Brief Reports - Communications br~ves

ZAMP

T a k e t h e f i r s t of E q u a t i o n (2.10) a n d r e w r i t e its i n t e g r a n d : ]

Zr(~3)

- - i o)

~ 43 ]2 =

[ Tr(~t3 )

t2 ~_ ] (2) ~ a ~3 12 + i o9 @a [2Vr(~3) - ~* -- Tr(~*) 9 43] .

(3.4)

T h e t o t a l e n e r g y flow t h r o u g h 27, in one c o m p l e t e cycle d u e t o ~3 is: t + 2zt/o)

r,~ =

(5~) 9 ~ , d ~ = ~

at t

ET,(~) 9 4" -

Z r

T,(4*) 9 ~]

d~.

(3:5)

Xr

Use of E q u a t i o n s (3.4) a n d (3.5) e n a b l e s one to w r i t e t h e first of E q u a t i o n (2.10) as: lira f [

I ~(~Ta) IS + (~ q a) 2 I ~3 I~] da +

2 ~ q a /. = 0.

(3.6)

2: r

F r o m t h i s o n e c o n c l u d e s t h a t / ' a m u s t b e a l w a y s less t h a n or a t m o s t e q u a l to zero. A s i m i l a r a r g u m e n t h o l d s for t h e flow of e n e r g y d u e t o g4.

IV . U n i q u e n e s s

of S o l u t i o n

Here, it will b e s h o w n t h a t a n y s o l u t i o n ot E q u a t i o n (2.3) w h i c h is r e g u l a r o u t s i d e of a r e g u l a r surface B, satisfies t h e r a d i a t i o n c o n d i t i o n s of E q u a t i o n s (2.10) a n d a) w h i c h v a n i s h e s o n B, or b) w h o s e stress v e c t o r v a n i s h e s o n B, v a n i s h e s identically. T o e s t a b l i s h this, first n o t e t h a t : P

f i T ( b 2 ) 9 5i* -- T(N*)

i

~2] da = / [ T ( N )

B

p

N* -

~F(52.). 52] d a ,

(4.1)

2: r

w h e r e 2:r is a s p h e r i c a l surface lying o u t s i d e B a n d 52 is a r e g u l a r solution. I t is f u r t h e r n o t e d t h a t t h e i n t e g r a l o n t h e r i g h t side of E q u a t i o n (4.1) is i n d e p e n d e n t of t h e r a d i u s of t h e surface. I n p a r t i c u l a r t h i s i n t e g r a l v a n i s h e s if e i t h e r t h e v e c t o r u~ or its stress v e c t o r v a n i s h e s on t h e surface B. T h r o u g h use of t h e t r i a n g l e i d e n t i t y , we m a y o b t a i n f r o m t h e c o n d i t i o n s (2.10) lim f [ ~ - - i ~ 2 1 2 d a = d

t'---~ oo

0

(4.2)

J

2,'r

where qo~a

Q~ob "

E q u a t i o n (4.2) c a n b e r e w r i t t e n : lira f { [ ~ l J

2+ 1~213+iu165

O.

(4.3)

Nr

Consider n o w t h e case in w h i c h 4 2 v a n i s h e s o n B. T h e n E q u a t i o n s (4.1) a n d (4.3) c a n b e utilized t o s h o w t h a t : lim / ] J Nr

u2 ]2 d a = 0 ,

(4.4)

~2 satisfies a n e q u a t i o n of t h e H e l m h o l t z t y p e , t h e r e f o r e as s h o w n b y R~LLmH [3], t h e c o n d i t i o n (4.4) g u a r a n t e e s t h a t ~2 v a n i s h e s identically. A s i m i l a r p r o c e d u r e c a n b e used to s h o w t h a t 5 2 v a n i s h e s if t h e stress v e c t o r 2g(~2) is zero o n B.

Vol. 16, 1965

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531

REFERENCES

[1] SOMMERFELD,A., Die Greensche F u n k t i o n der Schwingungsgleichung, J b e r . D e u t s c h . M a t h . - V e r e i n . , Vol. 21 (1912). [21 MAGNUS, W., Uber Eindeutigkeits/ragen bei einer Randwertau/gabe yon A u + k S u = O, J b e r . D e u t s c h . M a t h . - V e r e i n . , Vol. 52 (1943). E31 RELLICH, R., (Tber das asymptotische Verhalten der Losungen yon A u + u = 0 in unendlichen Gebieten, J b e r . D e u t s c h . M a t h . - V e r e i n , Vol. 53 (1943). [41 KELLOGG, O. D., Foundations o / P o t e n t i a l Theory, N e w York, D o v e r P u b l i c a t i o n s , Inc., 1953. Zusarnmen/assung

A u s s t r a h l u n g s b e d i n g u n g e n fiir L G s n n g e n d e r y o n d e r Zeit u n a b h i i n g i g e n G l e i c h u n g d e r B e w e g u n g d e r k l a s s i s c h e n Elastizit/it, d e r w o h l b e k a n n t e n S o m m e r f e l d - ] B e d i n g u n g ~thnlich, w e r d e n u n t e r s u c h t . Die B e d i n g u n g e n k G n n e n p h y s i k a l i s c h als ein n a c h a u s s e n g e r i c h t e t e r E n e r g i e f l u s s d a r g e s t e l l t w e r d e n . Sie genfigen auch, die E i n d e u t i g k e i t d e r LGsung a u f z u zeigen. (Reeeived: February 19, 1965.)

Electrostrictive Stress Singularities in Angular Corners of Plates 1) ]By WILLIAM E. WARREN, S a n d i a L a b o r a t o r y , A l b u q u e r q u e , N e w Mexico, U S A

T h i s n o t e utilizes t h e u n c o u p l e d t h e o r y of t w o - d i m e n s i o n a l e l e c t r o s t r i c t i o n t o e x a m i n e t h e p r i n c i p a l stress s i n g u l a r i t y o c c u r r i n g a t t h e v e r t e x of a n i n i t i a l l y isotropic, h o m o geneous, a n d u n s t r a i n e d elastic dielectric sector in t h e p r e s e n c e of a q u i t e g e n e r a l electric field. T h e r a d i a l edges of t h e s e c t o r are e i t h e r stress or d i s p l a c e m e n t free. T h e a n a l y s i s s h o w s t h a t for v e r t e x angles g r e a t e r t h a n 180 degrees, t h e p r i n c i p a l stress s i n g u l a r i t y arising f r o m t h e e l e c t r o s t r i c t i v e effect is of h i g h e r o r d e r t h a n t h e s i n g u l a r i t y associated w i t h m e c h a n i c a l effects as d e t e r m i n e d b y WILLIAMS [1] 2). No s i n g u l a r b e h a v i o r is possible for v e r t e x angles less t h a n 180 degrees. T h e elastic dielectric s e c t o r w i t h v e r t e x a t t h e origin of a c o m p l e x z p l a n e a n d h a v i n g v e r t e x a n g l e ~ is s h o w n in F i g u r e 1. T h i s region m a p s c o n f o r m a l l y i n t o a n u p p e r h a l f ~ p l a n e Y

a

l

x Figure 1 The elastic dielectric sector.

by the transformation

z=g(~)=c~/~,

c>0,

0<~<2~.

(1)

1) This work was supported by the United States Atomic Energy Commission. Reproduction in whole or in part is permitted for any purpose of the U.S. Government. 2) Numbers in brackets refer to References, page 534.