Dynamic stress concentration at a cylindrical inclusion in an elastic medium with an arbitrarily stiff bond

Int. J. meeh. Sci. PergamonPress. 1969. Vol. 11, pp. 677-688. Printed in Great Britain

DYNAMIC STRESS CONCENTRATION AT A CYLINDRICAL INCLUSION IN AN ELASTIC MEDIUM W I T H AN A R B I T R A R I L Y STIFF BOND STEPHEN A. THAU*t and TSIN-HWEI LU$ Illinois Institute of Technology, Chicago, Illinois (Received 25 F e b r u a r y 1969)

S u m m a r y - - R e s u l t s are presented for dynamic stress concentrations and the rigid-body response of a cylinder imbedded in an elastic medium which is subjected to an impinging, plane, harmonic, compressional wave. Boundary conditions specify t h a t the cylinder stays in contact with the surrounding medium, b u t that relative tangential motion can occur which is proportional to the shear stress through a bonding stiffness parameter, ~. I n the limiting case of zero slipping motion (A = 0) the results of previous investigators who considered only perfectly rough cylinders (perfect bonding between cylinder and surrounding medium) are recovered. I n general, it is found t h a t only mild differences in the results for the normal stress concentrations and the rigid-body response occur as )~ is varied with slightly greater results corresponding usually to the case e t a perfectly smooth (A -- 1) cylinder. An important exception, however, is t h a t the m a x i m u m normal stress concentration at a perfectly smooth cylinder (A -- 1) becomes much larger than the corresponding m a x i m u m value for a perfectly rough cylinder (A = 0) as the cylinder density increases. Also, these peak dynamic stress concentration factors become much greater than the equivalent static values with increasing cylinder density.

NOTATION a a, fl N N0 V2

cylinder radius dimensionless wave numbers kl, z a Euler's c o n s t a n t s 0 . 5 7 7 2 exp (N) -~ 1.7811 Laplace operator

-q O-~r/2 H, h K kI k~ K

H(~II = Hankel function of first kind shear wave potential proportionality constant between shear stress and relative tangential displacements at cylinder boundary k~/K = compressional wave number ¢o(p/Iz)t = shear wave number [(2-

2v)/(1 - 2v)]t

2 ~ l ( 2 1 ~ + K a ) = dimensionless bonding stiffness (0~,~< 1) /~ shear modulus v Poisson's ratio ¢o angular frequency of waves compressional wave potential 40 amplitude of incident eompressional wave

* Currently visiting the Department of Theoretical and Applied Mechanics, Cornell University. The research of the first author was supported by Bell Telephone Laboratories, Whippany, New Jersey. The research of the second author was supported by the National Science Foundation under a grant (No. GK 1911) to Illinois Institute of Technology. 677

STEPHEN A. THAU and TSIN-HWEI LU

678

r,O radial and angular polar co-ordinates P, Pc densities of elastic medium and cylinder

pc/P = density ratio parameter (0
t

1. I N T R O D U C T I O N IN THE s t u d y of d y n a m i c stress concentration at rigid inclusions in elastic solids, idealized models are often used which assume the inclusion is rigidly b o n d e d to the surrounding medium. This means t h a t points in the elastic m e d i u m at the interface c a n n o t slip relative to the included obstacle. However, it is emphasized t h a t this condition often m a y represent only a limiting case of an actual situation, the other limiting case being t h a t of a perfectly smooth inclusion at whose surface the surrounding m e d i u m is completely free to slide tangentially. F o r example, the latter condition at the interface of a buried s t r u c t u r e m a y be more appropriate for studying its response to impinging stress waves. F o r then, the surrounding soil or rock m e d i u m is pressed against, b u t is not " w e l d e d " or " g l u e d " to the inclusion. Therefore, to s t u d y the effects of slip versus no-slip conditions on the response of a s t r u c t u r e e m b e d d e d in an elastic solid, this paper considers the scattering of harmonic, elastic waves b y a rigid circular cylinder which is a t t a c h e d to the surrounding m e d i u m b y an elastic bond. W e assume the bond is v e r y thin and relatively quite stiff in the direction normal to the cylinder, so t h a t there is essentially zero relative radial deformation between the cylinder and surrounding material. However, the shear d e f o r m a t i o n in the bond is not neglected. Thus, the relative tangential displacement is t a k e n proportional to the shear stress with an a r b i t r a r y p r o p o r t i o n a l i t y constant which specifies the bond stiffness in the circumferential direction. W h e n the constant approaches zero or infinity, there is obtained a perfectly rough (or ideal bonding) or perfectly s m o o t h surface, respectively. The circular cylinder inclusion problem is chosen because its solution, for the case when the cylinder is ideally b o n d e d to the elastic medium, has been well studied and a large q u a n t i t y of numerical results are available. F o r example, Miles has given detailed results for the rigid-body response of a cylinder subjected to incident, harmonic, compressional and shear waves. 1 Pao and Mow 2 and Mow and Mente 3 have e v a l u a t e d d y n a m i c stress concentrations at the b o u n d a r y of a rigid cylinder struck b y harmonic compressional and shear waves, respectively; and a recent contribution on the transient response of a rigid cylinder is t h a t of B a r o n and Parnes. 4 The foregoing list is b y no means complete. M a n y other works o n diffraction b y rigid inclusions are reviewed b y Miklowitz, 5 and still more can be found in the current literature in engineering mechanics and wave propagation.

Dynamic stress concentration at a cylindrical inclusion in an elastic medium

679

T h e solution of t h e problem t r e a t e d here in which the cylinder is b o n d e d elastically to the surrounding m e d i u m is o b t a i n e d b y the same m e t h o d s used b y t h e previous investigators listed above. Thus, the m o t i o n o f the cylinder is calculated versus the f r e q u e n c y o f the incident wave for various values of cylinder to m e d i u m density ratio a n d various values o f the " b o n d i n g stiffness" p a r a m e t e r . The same is also done for n o r m a l stress concentrations a t the cylinder at the p o i n t of direct incidence of the incoming wave a n d at t h e opposite point on the cylinder on the " s h a d o w " side. Thus, results in the two limiting cases of a p e r f e c t l y r o u g h (or perfect bonding) or p e r f e c t l y s m o o t h b o u n d a r y can be compared. Interestingly, the rigid-body m o t i o n is f o u n d n o t to be affected m u c h b y the " b o n d i n g stiffness" factor. On the o t h e r hand, a t low frequencies this f a c t o r is i m p o r t a n t for stress concentrations with the larger stresses occurring at a s m o o t h b o u n d a r y . A t high frequencies the stress concentrations are affected only slightly b y changes in the density ratio and bonding factor. I n the n e x t section the m a t h e m a t i c a l f o r m u l a t i o n and solution of the problem are given. T h e n the numerical results are p r e s e n t e d and discussed. I n the concluding s u m m a r y , it is n o t e d how the results can be used to o b t a i n a p p r o x i m a t e l y t h e t r a n s i e n t responses and d y n a m i c stress concentrations. 2. DESCRIPTION

AND

SOLUTION

OF

PROBLEM

The geometry and co-ordinate system for the problem is shown in Fig. I. An infinitely

long, rigid cylinder of radius, a, whose axis is the z-axis of the cylindrical co-ordinate system, r, 8, z, is imbedded in an unbounded, isotropic, elastic medium and is struck by

r

FIG. 1. Geometry of problem and incident wave. a plane, harmonic, compressional wave traveling in the x-direction. The problem is two-dimensional and a state of plane strain, in which the z-displacement is identically zero, is assumed. Thus, the radial and tangential displacements can be represented by the compressional and vertically polarized shear wave potentials, ~(r,O) exp (--icot)

and

h(r,O) exp (--ioJt),

respectively, as Ur = a~/~r T Oh/r O0, Uo -- ~ [ r ~O-~h/~r.

The time factor, exp (-/oJt), is omitted in equation (1) and hereafter. 46

(1)

680

STEPHEN A. THAU a n d TSIN-HWEI L u T h e e q u a t i o n s of m o t i o n for t h e elastic m e d i u m a r e t h e t w o scalar H e l m h o l t z e q u a t i o n s , (V2+k~)¢=

0,

( V 2 + k ~ ) h = 0,

(2)

in w h i c h t h e w a v e n u m b e r s a r e defined b y k2/kx=K = [ ( 2 - - 2 v ) / ( 1 - - 2 v ) ] i ,

k 2 = oJ(p/be)t

(3)

w i t h / ~ , v a n d p being, r e s p e c t i v e l y , t h e s h e a r m o d u l u s , P o i s s o n ' s r a t i o a n d d e n s i t y of t h e medium. Stresses a r e c a l c u l a t e d f r o m H o o k e ' s l a w a c c o r d i n g t o

"rrr -~- be[2~ur/Or + (K 2 - 2) V 2 ¢], r00 = -- rrr + 2tt(K 2 -- 1) V 2 4,

(4)

-r,o = t~[2(~udOO - U o ) / r - V 2 h ] .

Incident and scattered waves and boundary conditions T h e i n c i d e n t c o m p r e s s i o n a l w a v e w h i c h satisfies e q u a t i o n s (2) is g i v e n b y ¢~i~ = ¢ 0 e x p ( i k i x ) ,

h ~ = 0,

(5)

w h e r e ¢0 is a c o n s t a n t . T h e t o t M field, w h i c h will b e w r i t t e n w i t h o u t s u p e r s c r i p t s , consists of t h e i n c i d e n t field (5) plus t h e s c a t t e r e d w a v e s ¢~8) a n d h~'~. T h e l a t t e r m u s t b e o u t g o i n g a t infinity, s a t i s f y t h e e q u a t i o n s of m o t i o n (2) a n d b o u n d a r y c o n d i t i o n s a t t h e c y l i n d e r . T h e s e specify for t h e t o t a l field t h a t , as t h e c y l i n d e r u n d e r g o e s r i g i d - b o d y t r a n s l a t i o n , t h e s u r r o u n d i n g m e d i u m r e m a i n s in c o n t a c t w i t h it, w i t h t h e r e l a t i v e t a n g e n t i a l d i s p l a c e m e n t b e t w e e n t h e c y l i n d e r a n d a d j a c e n t m e d i u m b e i n g p r o p o r t i o n a l t o t h e s h e a r stress. M a t h e m a t i c a l l y , w i t h U e x p ( - i w t ) b e i n g t h e r i g i d - b o d y t r a n s l a t i o n of t h e c y l i n d e r in t h e x-direction, d u e t o t h e forces of t h e t o t a l w a v e field, t h e b o u n d a r y c o n d i t i o n s are

ur(a, O) = U cos 0,

~'ro(a, O) = K[uo(a, O) + U sin 0]

(6)

i n w h i c h K is a c o n s t a n t . N o t e t h a t t h e b r a c k e t e d q u a n t i t y i n e q u a t i o n s (6) is i n d e e d t h e r e l a t i v e t a n g e n t i a l d i s p l a c e m e n t a t t h e i n t e r f a c e b e c a u s e - U sin 0 is t h e 0 - c o m p o n e n t of t h e t o t a l r i g i d - b o d y m o t i o n . B y u s i n g e q u a t i o n s (1), (2) a n d (4) we c a n m a n i p u l a t e c o n d i t i o n s (6) i n t o t h e m o r e c o n v e n i e n t f o r m

ur(a, O) = U cos 0,

uo(a, O) = - U sin 0 + ~,~k~ ah(a, 0),

(7)

w h e r e ~ = 2/~/(2/~ + K a ) is a d i m e n s i o n l e s s f a c t o r w h i c h describes t h e r e l a t i v e stiffness of t h e t h i n elastic b o n d . F o r a p e r f e c t l y s m o o t h surface, ~ = 1 ( K = 0), w h e r e a s for a p e r f e c t l y r o u g h surface, ~ = 0 (K--> ~ ) . F o r 0
"rrr(a, O) = - ttk~[dp(a, O) + )t Oh(a, 0)/00], ] ~',o(a, O)

tdc~(I - ) l ) h(a, 0).

i

(8)

T h e r e s u l t a n t force is t h e n r e l a t e d t o t h e r i g i d - b o d y a c c e l e r a t i o n a c c o r d i n g t o

-rra2~o~p~U =

[~',.~cos 0 - r,0 sin 0]~=~ a d0,

(9)

w h e r e p~ is t h e d e n s i t y of t h e cylinder. I n t h i s w a y U c a n b e d e t e r m i n e d explieitly.

Solution T h e diffraction p r o b l e m f o r m u l a t e d in t h e p r e c e d i n g s u b - s e c t i o n is a g e n e r a l i z a t i o n of t h e o n e t r e a t e d b y Miles a a n d P a o a n d M o w 2 for a r i g i d - p e r f e c t l y r o u g h c y l i n d e r (A = 0). T h e m e t h o d for solving o u r p r o b l e m is t h e s a m e as t h e o n e u s e d in t h e i r p a p e r s , viz. s e p a r a t i o n of v a r i a b l e s i n p o l a r co-ordinates. W e o m i t t h e d e t a i l s a n d p r e s e n t d i r e c t l y t h e

D y n a m i c stress c o n c e n t r a t i o n a t a cylindrical inclusion in a n elastic m e d i u m

681

results which are of interest here, i.e. the stresses a t the cylinder b o u n d a r y a n d the rigidb o d y response of the cylinder. ~r'rrr(a, 0)/470 = [2/~-/l(a)] -1 - A I { ( 1 --A)H~(~)-flZl(fl) -{-f~U0[f~Zl(~) HI(C~) -4-At~/-/l(f~) H3(o~)]} cos 0 oo

+ ~ i "+1 A.[n(1 --An) H.(fl) -flZ.(/~)] cos nO,

(10a)

n=2

,ff'~oo(a,O)+.,.(a, OI]/~-o =

(B' - ~ )

B-' 2; ~. ~÷1AJnH.(B) -Bz~(B)] cos nO n=O

- 2A 1H~(~)Zj(fl) U . cos

0},

(lOb)

~rre(a, 0)/4~ro ---- (1 --A) / Z i , + l n A , H,(f~) sin nO t1%=2

-

Ai H~(/~) [1 -- af~H2(a ) U0] sin 8},

(10e)

U o = ~ U / 4 k ~ ¢o = {af~sH2(a) + (1 - s ) [f~Hl(a ) + a/-/e(a) H,(~) Z~l(fl)]} -1,

(11)

where H . (meaning H(~1)) is the H a n k e l f u n c t i o n of the first k i n d of order n, a = /q a, fl = k 2 a = Ks a n d ~0 --- --pk~ ¢0 is the m a g n i t u d e of n o r m a l stress associated with the i n c i d e n t wave (5). The c o n s t a n t K was defined in e q u a t i o n (3). F u r t h e r , in equations (10) a n d (11), e0 = 1, e~ = 2 for all n > 0 ,

Z.(~) = H.+,(~)-- ½ABH.(/~), A . = {fl[aHn+l(a ) -- n H . ( a ) ] Zn(fl) -- anHa+~(a) H.(f~)} -~, a n d s = PdP is the ratio of the densities of the cylinder a n d the s u r r o u n d i n g elastic m e d i u m . The q u a n t i t y U 0 given b y e q u a t i o n (11) is i¢r/4,¢ times the ratio of the m a g n i t u d e of the rigid-body translation, U, a n d the x-displacement of the incident wave, i k 1 ¢o. I t can be shown t h a t when A = 0, our result for U0 agrees with t h a t o b t a i n e d b y Miles. 1 Similarly, the results for stresses (10) with A = 0 reduce to those of Pao a n d Mow. 2 L o n g wavelength

F o r a n incident wave whose w a v e l e n g t h is long compared to the cylinder radius (fl a n d a = fl/K--> 0) the H a n k e l functions in expressions (10) a n d (11) m a y be replaced b y the leading terms in their expansions for small arguments. T h e n the results reduce to v~(a, 8)/.r o -> 2K-~[1 + 2(1 + 2A) D cos 28],

(12a)

"roe(a, 8)/'re --> 2v(1 --v)-* K-2[1 + 2(1 + 2A -- 2Av-*) D cos 28],

(12b)

"r,o(a, 8)fire --> 4(1 - A ) K-~ D sin 28,

(12e)

i~r {1 4 cx2 s - - 1 irr 7o U°-~ . 4 4 ['a2 +f~2' ( 2 -- In 2 ) + ~Af~2-- ~ In °~- ~ In f~] }-1'

(13,

where D = 3 - 4 v + 2 A ( 1 - v ) a n d 7o = exp (7) = 1.78107 with 7 being E u l e r ' s c o n s t a n t . The above long wavelength results agree with those of a static analysis of the p r o b l e m of a n infinite rigid cylinder in a n u n b o u n d e d m e d i u m subjected to uniaxial tension in the x-direction of m a g n i t u d e ( 1 - 2 v ) , 0 / ( 1 - v ) plus a hydrostatic pressure, ~r0/(1-v). These values of static loading are found from the long wavelength limits of the stresses associated with the incident wave alone. Note t h a t the leading terms for the stresses given above d e p e n d on A, b u t the leading t e r m for U o does not.* Also, the density ratio parameter, s, is missing in the leading terms for all the above quantities. This, of course, agrees with the fact t h a t static solutions in linear elasticity do n o t d e p e n d on the m a t e r i a l density. However, we m a y expect the stress concentrations a t low frequencies to v a r y more strongly t h a n the rigid-body motion, with respect to the b o n d i n g factor, A. * F o r s fixed, the right side of e q u a t i o n (13) can be readily e x p a n d e d t h r o u g h terms of order f~2, b u t for large s the given expression is more accurate.

682

STEPHE~¢ A. TrIAU a n d TSI~r-HWEI L u

Short wavelength ~ r h e n t h e i n c i d e n t w a v e l e n g t h is v e r y s h o r t c o m p a r e d t o t h e c y l i n d e r r a d i u s (a, J~ = kL~.a -~ c¢) we e x p a n d t h e r e s u l t for U 0, e q u a t i o n (11), a s y m p t o t i c a l l y t o o b t a i n t h e leading terms : ~

e x p [i(5~r/4--a)]

Uo~Ki(~a)½exp[i(51r/4--a)] U

1

lTr\t

o ~ f l s + K i - (i+Afl/2) -i ~ )

for s fixed

and

0 ~ , ~ < 1,

(14a

for j3s fixed a n d ~ fixed,

(14b)

exp [i(5¢r/4-a)]

fls a n d flh fixed.

for

(14c)

T h e s e r e s u l t s show t h a t , in general, t h e r i g i d - b o d y r e s p o n s e decreases as kj a increases ( j = 1 or 2). F o r all b u t v e r y l i g h t cylinders, t h e r e s p o n s e decreases i n p r o p o r t i o n t o (kj a ) - t [ e q u a t i o n (14a)]. F o r s -+ 0 w i t h fls fixed, t h e h i g h - f r e q u e n c y r e s p o n s e d e c a y s m o r e slowly, b e i n g p r o p o r t i o n a l t o (kj a ) - t . F u r t h e r , effects of b o n d i n g stiffness are a b s e n t in t h e l e a d i n g t e r m s e x c e p t for v e r y light, a l m o s t p e r f e c t l y s m o o t h cylinders [ e q u a t i o n (14c)]. F o r stress c o n c e n t r a t i o n s a t h i g h f r e q u e n c i e s we c a n n o t use e q u a t i o n s (10) since t h e infinite series c o n v e r g e t o o slowly in t h i s case. I n s t e a d , g e o m e t r i c a l r a y t h e o r y c a n b e used t o c a l c u l a t e t h e results. N o t i n g f r o m e q u a t i o n s (14) t h a t t h e r e is essentiaUy n o rigidb o d y t r a n s l a t i o n of t h e c y l i n d e r a t h i g h frequencies, we n e e d t o c a l c u l a t e t h e reflection of t h e i n c i d e n t w a v e (5) b y a r i g i d - f i x e d p l a n e t a n g e n t t o t h e c y l i n d e r a t a n a r b i t r a r y p o i n t o n t h e " i l l u m i n a t e d " side (~/2 < 0 < 37r/2). I n t h i s way, t h e l e a d i n g t e r m s for t h e stress c o n c e n t r a t i o n s a t s u c h a p o i n t are f o u n d . O n t h e " s h a d o w " side ( l0 [<~r/2) t h e a n a l y s i s for t h e h i g h - f r e q u e n c y stresses is m o r e c o m p l i c a t e d , a n d we m e r e l y r e m a r k h e r e t h a t t h e y t e n d to zero as t h e f r e q u e n c y b e c o m e s infinite. Since o u r m a i n o b j e c t i v e is to c o m p a r e r e s u l t s in t h e e x t r e m e cases of a p e r f e c t l y r o u g h (h = 0) a n d p e r f e c t l y s m o o t h (h = 1) cylinder, we p r e s e n t b e l o w o n l y t h e r e s u l t s for t h e leading t e r m s for t h e n o r m a l a n d s h e a r stresses o n t h e i l l u m i n a t e d side in t h e s e t w o cases. ]~ r ( a , O) [ ~ 2fret0 sin ~/(eos 2 ~ + m sin ~7), ]

(15a) f o r A = 0,

I vro(a, 0) 1~ - 70 sin 2~/(eos ~ ~ + m sin 7/),

(15b)

I ~'~'(a'O) ] ~ 2 r ° ( 1 - 2 K - 2 c ° s 2 ~ ) '

(16a)

f o r A = 1,

Iwro(a, O) I-=0,

(16b)

w h e r e ~) = 0--¢r/2, ~n = ( K ~ - c o s 2 ~ ) t , a n d b y s y m m e t r y we h a v e t a k e n 0 < ~ < I r / 2 , i.e. ¢r/2 < ~ ~<¢r. N o t e t h a t t h e g r a z i n g p o i n t , ,/ = 0 or 0 = ¢r/2, is e x c l u d e d . N u m e r i c a l r e s u l t s for t h e r a t i o of e q u a t i o n s (15a) a n d (16a) are d i s p l a y e d i n Fig. 4. T h e y will b e discussed a l o n g w i t h o t h e r n u m e r i c a l r e s u l t s i n t h e n e x t section.

3. D I S C U S S I O N

OF

NUMERICAL

RESULTS

N u m e r i c a l r e s u l t s for t h e n o r m a l i z e d m a g n i t u d e of t h e r i g i d - b o d y r e s p o n s e I U0 I as a f u n c t i o n of d i m e n s i o n l e s s w a v e n u m b e r , fl, or d i m e n s i o n l e s s f r e q u e n c y

(~ = k 2 a = toa(p/ix)t ) are d i s p l a y e d in Fig. 2. T h e y are c a l c u l a t e d f r o m e q u a t i o n (11) w i t h v = ¼ (or K = fl/a = 31). N o t e t h a t since t h e a c t u a l r e s p o n s e is g i v e n b y I U01 cos [tot+ A(to, ~, s)], w h e r e A is t h e p h a s e angle, t h e m a x i m u m r e s p o n s e for different v a l u e s o f A, w i t h to a n d s fixed, will o c c u r a t different i n s t a n t s o f t i m e (i.e. w h e n tot + A = nlr). F o r h e a v y cylinders, i.e. s > 1 (Fig. 2A, B), t h e r i g i d - b o d y m o t i o n s increase a t first t o p e a k v a l u e s in t h e low f r e q u e n c y r a n g e a n d t h e n decrease m o n o t o n i c a l l y t o zero. T h i s is a n a l o g o u s t o t h e r e s p o n s e of a n u n d e r d a m p e d discrete, single d e g r e e - o f - f r e e d o m v i b r a t i o n s y s t e m . As A increases t h e low f r e q u e n c y r e s p o n s e increases. I n t h e i n t e r m e d i a t e f r e q u e n c y r a n g e t h e r e v e r s e is t r u e , while a t h i g h f r e q u e n c i e s t h e r e s p o n s e is e s s e n t i a l l y i n d e p e n d e n t

Dynamic

stress concentration

at a cylindrical inclusion in an elastic medium

• ~. = I-0 ,.X

• .5

0.6

S=tO

• k = '0

0.5

0.4

0.3

--

0.2 =0 •

0.1

0

0-5

= 1.0

1.0

2.0

3.0

4"0

5.0

6.0

F r o . 2. M a g n i t u d e o f n o r m a l i z e d c y l i n d e r m o t i o n v e r s u s d i m e n s i o n l e s s w a v e n u m b e r , fl= k2a, f o r v = ¼, A = 0, 0.5, 1.0 a n d v a r i o u s s. (A) s = 10. (B) s = 4 a n d s = l. (C) s = 0.5.

k = 1,0

0.5

0,4

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0,3

0,2

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I

I

I

I

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2.0

3.0

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683

684

STEPHEN A. THAU a n d TSZN-HWEI L u

0.5

T

S =0*5

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m

--

0.3

0.2 0°1 I I*0

I 2.0

I 5.0

I 4.0

l 5-0

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/9

FIG. 2 (C) of )l. F o r s = 1 (Fig. 2B) t h e r e is n o d e p e n d e n c e of r e s p o n s e o n ~ as c a n b e seen f r o m e q u a t i o n (11). F u r t h e r m o r e , t h e m a g n i t u d e of t h e m o t i o n decreases w i t h i n c r e a s i n g freq u e n c y o v e r t h e e n t i r e range. F i n a l l y , for l i g h t c y l i n d e r s (s = 0.5, Fig. 2C) t h e r i g i d - b o d y m o t i o n s b e g i n to d e c r e a s e a t low frequencies a n d for a r o u g h c y l i n d e r (~ = 0) t h e m o t i o n c o n t i n u e s t o do so, j u s t as for t h e r e s p o n s e in t h e case of a n o v e r d a m p e d d i s c r e t e s y s t e m . H o w e v e r , t h e r e s p o n s e s of light, elastically b o n d e d (2 = 0.5) a n d p e r f e c t l y s m o o t h c y l i n d e r s (~ = 1) increase to r e l a t i v e m a x i m a in t h e i n t e r m e d i a t e r a n g e a n d t h e n decrease to zero. T h u s , t h e y still r e s e m b l e t h e r e s p o n s e s o f a n u n d e r d a m p e d s y s t e m . P l o t s of t h e m a g n i t u d e of t h e n o r m a l stress c o n c e n t r a t i o n factors, ~ = ] T~/~ 0 I, a t t h e c y l i n d e r b o u n d a r y v e r s u s fl are s h o w n in Fig. 3. T h e s e are b a s e d o n e q u a t i o n (10a) w i t h v = ¼. T h e solid c u r v e s are r e s u l t s a t t h e p o i n t of n o r m a l incidence, 0 = It, while t h e d a s h e d c u r v e s are for t h e e x t r e m e p o i n t in t h e s h a d o w zone, 0 -- 0. T h e stress c o n c e n t r a t i o n s a t zero f r e q u e n c y (static limit) increase f r o m 1"5 for a perfectly r o u g h b o u n d a r y to 1-86 for a p e r f e c t l y s m o o t h b o u n d a r y . T h i s increase w i t h ~ is c o n n e c t e d o n l y w i t h t h e u n i a x i a l c o m p r e s s i o n p a r t of t h e s t a t i c l o a d [cL t h e discussion following e q u a t i o n (13)].* A t a p e r f e c t l y s m o o t h b o u n d a r y t h e a p p l i e d l o a d c a n b e b a l a n c e d o n l y i n t h e n o r m a l d i r e c t i o n since t h e r e is n o s h e a r stress. T h u s , as ~ decreases, t h e s h e a r stress c o n c e n t r a t i o n s will increase a n d t h e n o r m a l c o n c e n t r a t i o n s will decrease. As t h e f r e q u e n c y increases f r o m zero, t h e stress c o n c e n t r a t i o n s a t 0 = lr a n d for large v a l u e s of s (Fig. 3A, B) increase t o p e a k v a l u e s w i t h m u c h l a r g e r m a g n i t u d e s o c c u r r i n g a t a s m o o t h b o u n d a r y . I n Fig. 3(A) w i t h s = 10, t h e r a t i o of m a x i m u m d y n a m i c stress t o s t a t i c stress is 2-5 w i t h ~ = 1, b u t o n l y 1-9 a t ~ -- 0. T h i s a d d i t i o n a l increase in stress c o n c e n t r a t i o n a t a s m o o t h b o u n d a r y a t t h e " r e s o n a n c e " f r e q u e n c y c a n b e e x p l a i n e d as follows. T h e b e h a v i o r of t h e stress c o n c e n t r a t i o n s for large s, as n o t e d for t h e r i g i d - b o d y r e s p o n s e s for large s, is a n a l o g o u s t o t h e r e s p o n s e of a n u n d e r d a m p e d , discrete, v i b r a t i o n s y s t e m . R e c a l l i n g t h a t , a t r e s o n a n c e , t h e v i b r a t i o n a m p l i t u d e of s u c h a d i s c r e t e s y s t e m is a p p r o x i m a t e l y p r o p o r t i o n a l t o (m/kc~) t (where m = m a s s , k = s p r i n g c o n s t a n t a n d * T h e h y d r o s t a t i c p r e s s u r e p a r t yields r e s u l t s i n d e p e n d e n t of )~ b e c a u s e , b y s y m m e t r y , fro a n d ue will b e zero e v e r y w h e r e .

D y n a m i c stress c o n c e n t r a t i o n a t a c y l i n d r i c a l i n c l u s i o n in a n elastic m e d i u m

4.5

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Fzo. 3. M a g n i t u d e of n o r m a l i z e d r a d i a l stress, T = 1 ~ ( a , 0)/70 ], a t 0 = 0 ( d a s h e d lines) a n d 0 = lr (solid lines) o n c y l i n d e r vs. d i m e n s i o n l e s s w a v e n u m b e r , fl= k,a, for v = 3, A = 0, 0.5, 1.0 a n d v a r i o u s s. (A) s = 10. (B) s = 4. (C) s = 1. ( D ) , = 0.5. c = d a s h p o t coefficient) we see f r o m Fig. 2(A) t h a t t h e e q u i v a l e n t r a t i o m/kc* for a s m o o t h c y l i n d e r is g r e a t e r t h a n t h a t for a r o u g h c y l i n d e r . H o w e v e r , t h e s t a t i c r e s u l t s a t fl = w = 0 axe i n d e p e n d e n t of A w h i c h implies t h a t t h e e q u i v a l e n t s p r i n g c o n s t a n t is also. H e n c e t h e e q u i v a l e n t d y n a m i c r a t i o re~c* in a discrete, u n d e r d a m p o d s y s t e m u s e d t o r e p r e s e n t t h e r e s p o n s e I U0 I, will i n c r e a s e as ~ increases. I t is b e l i e v e d t h a t t h i s occurs b e c a u s e a r o u g h c y l i n d e r , b e i n g m o r e c o n s t r a i n e d t h a n a s m o o t h one, will r a d i a t e m o r e e n e r g y a t r e s o n a n c e , w h i c h m e a n s t h e e q u i v a l e n t d a s h p o t coefficient for t h e r o u g h c y l i n d e r s h o u l d b e larger. Now, i n a s i m i l a r d i s c r e t e s y s t e m w h i c h w o u l d yield t h e s t r e s s c o n c e n t r a t i o n f r e q u e n c y r e s p o n s e i n Fig. 3(A), say, we n o t o n l y e x p e c t a g a i n t h a t t h e r a t i o re~c* i n c r e a s e s as A increases, b u t also o b s e r v e a t co = 0 t h a t t h e e q u i v a l e n t s p r i n g c o n s t a n t decreases as A increases. T h u s , a c o m b i n a t i o n of effects a t low frequencies, one s t a t i c a n d t h e o t h e r d y n a m i c , r e i n f o r c e e a c h o t h e r a n d t h u s m a y a c c o u n t for t h e f u r t h e r s p r e a d i n t h e m a g n i t u d e of t h e s t r e s s c o n c e n t r a t i o n s a t r e s o n a n c e . F o r s m a l l e r v a l u e s o f a a n d i n g e n e r a l for 0 = 0, t h e difference b e t w e e n t h e s t r e s s c o n c e n t r a t i o n s for different v a l u e s of A does n o t grow as co increases f r o m zero. A t i n t e r m e d i a t e frequencies, t h e effects of t h e b o n d i n g f a c t o r a r e seen t o d e c r e a s e a n d finally a t

686

STEPHEN A. THAU a n d TSIN-HWEI LU

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FzG. 3 (D) h i g h f r e q u e n c i e s t h e r e is v e r y l i t t l e difference i n t h e r e s u l t s as t h e b o n d i n g stiffness is v a r i e d . I n t h e s h a d o w region, as co -~ o0, t h e stresses a p p r o a c h zero, w h e r e a s o n t h e i l l u m i n a t e d side o f t h e c y l i n d e r t h e stresses a p p r o a c h t h e s h o r t w a v e l e n g t h l i m i t s g i v e n b y e q u a t i o n s (15) a n d (16). T h u s , a t 0 = lr [ c o r r e s p o n d i n g t o ~/ = ~r/2 i n e q u a t i o n s (15) a n d (16)] r -~ 2 for b o t h ~ = 0 a n d )l = 1. To s t u d y t h e s t r e s s c o n c e n t r a t i o n s a t h i g h f r e q u e n c y in m o r e d e t a i l we s h o w i n Fig. 4 t h e r a t i o of t h e l i m i t i n g v a l u e oTv for ~ = 0 t o t h a t for )l = 1 vs. p o s i t i o n o n t h e i l l u m i n a t e d

1.6

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IZ 0.8

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I 0.6

I 0-8

I 1.0

I 1.2

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FIG. 4. R a t i o o f r a d i a l b o u n d a r y stresses a t h i g h f r e q u e n c y for a p e r f e c t l y r o u g h (~ = 0) a n d p e r f e c t l y s m o o t h (~ = 1) c y l i n d e r , R = ]'rrr(a, 0,)t = 0) ]/l'rrr(a, 0 , ~ -- 1) ], vs. ~7 = 0 - 7 r / 2 for v a r i o u s v.

I b

1.6

688

STEPHEI~ A. THAU and TSIN-HWEI Lu

side of the cylinder for various values of Poisson's ratio. Noting that ~ = 0 - 7r/2, we see that only near the points of grazing incidence, ~ = 0 (and by symmetry ~ = ~) does this ratio differ by more than 20 per cent from unity. 4. C O N C L U S I O N S T h e p u r p o s e of this s t u d y has been to d e t e r m i n e the d y n a m i c response a n d t h e stress c o n c e n t r a t i o n factors for a rigid cylinder i m b e d d e d in a n elastic solid w i t h a thin, a r b i t r a r i l y stiff b o n d i n g at the interface a n d s u b j e c t e d to a n impinging compressional wave. I n particular, we h a v e been concerned w i t h the effect of v a r y i n g the bonding p a r a m e t e r on t h e m a g n i t u d e of the results. I n general, it a p p e a r s t h a t such effects are m o s t i m p o r t a n t a t low frequencies for cylinders which h a v e densities larger t h a n t h a t of the s u r r o u n d i n g m e d i u m . H o w e v e r , this ease is significant because, similar to t h a t o f a h e a v i l y underd a m p e d , discrete, v i b r a t i o n model, the stress c o n c e n t r a t i o n a n d r i g i d - b o d y response b o t h r e a c h p e a k values a t a low " r e s o n a n t " frequency. F u r t h e r m o r e , it is of interest in the area of design of u n d e r g r o u n d s t r u c t u r e s which m u s t withs t a n d the forces of seismic w a v e s p r o d u c e d b y b l a s t loads. W e h a v e f o u n d t h a t t h e p e a k v a l u e of the rigid-body m o t i o n for a h e a v y cylinder increases slightly f r o m t h e case of a p e r f e c t l y r o u g h b o u n d a r y to t h a t of a p e r f e c t l y s m o o t h b o u n d a r y , whereas the p e a k n o r m a l stress c o n c e n t r a t i o n increases considerably w h e n t h e cylinder surface becomes p e r f e c t l y smooth. Finally, it is n o t e d t h a t t h e t i m e h a r m o n i c results p r e s e n t e d here can be used in principle to o b t a i n the t r a n s i e n t responses b y m e a n s of the F o u r i e r inversion integral. I n practice, however, a n e x a c t inversion of t h e results in equations (10) a n d (11) m a y be v e r y difficult to accomplish a n d so an a p p r o x i m a t e analysis m a y be desirable. R e c e n t l y P e r a l t a et al. e h a v e developed a n d successfully t e s t e d such a n a p p r o x i m a t e F o u r i e r inversion t e c h n i q u e in p r o b l e m s o f t h e t y p e t r e a t e d here. All the i n f o r m a t i o n required to use their m e t h o d can be o b t a i n e d directly a n d easily f r o m t h e a n a l y t i c a l a n d n u m e r i c a l results of this paper. Acknowledgements--The authors wish to thank Dr. F. T. Flaherty, Jr. and M. Lutchansky

of Bell Telephone Laboratories and Professor Y. It. Pao of Cornell University for their helpful discussions in the course of this work. Also, they extend their appreciation to Miss Margaret Wamp of Bell Telephone Laboratories who prepared all the computer programs and numerical results. REFERENCES 1. J. W. MILES, J. Acoust. Soc. Am. 32, 1656 (1960). 2. Y. H. PAo and C. C. Mow, Proc..Fourth U.S. Nat. Congr. Appl. Mech., Berkeley, Calif., p. 335 (1962). 3. C. C. Mow and L. J. MENTE, J. Appl. Mech. 3{}, 598 (1963). 4. M. L. BARON and R. PARNES, Diffraction of a Pressure Wave by an Elastically Lined Cylindrical Cavity in an Elastic Medium, MITRE Corporation Report SR-44 {December 1961}. 5. J. MIXLOWITZ,Applied Mechanics Surveys, p. 809. Spartan Books, Washington, D.C. (1966). 6. L. A. PER~TA, G. F. CARRIERand C. C. Mow, J. Appl. Mech. 33, 168 (1966}.