w 7 3 29

v

A CRITICAL SURVEY OF

WAVE PROPAGATION AND IMPACT

IN COMPOSITE MATERIALS

by F. C. Moon

prepared for NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

NASA L e w i s Research Center Grant No. NGR 31-001-267

May 1973

PRINCETON UNIVERSITY Department of Aerospace and Mechanical Sciences

AMs Report No. 1103

https://ntrs.nasa.gov/search.jsp?R=19730021203 2018-06-28T21:54:46+00:00Z

I . Report No. I 2. Government Accession No.

17. Key Words (Suggested by Author(s))

Stress Waves, Composite Waves, Impact Stresses

NASA CR-121226 I 1. Title and Subtitle

18. Distribution Statement

Unclassified

A CRITICAL SURVEY OF WAVE PROPAGATION AND IMPACT I N COMPOS I TE MATE RIALS

19. Security Classif. (of this report) 20. Security Classif. (of this page)

Unclassified Unclassified

7. Author(s)

21. No. of Pages 22. Price"

13 5 $3.00

Francis C. Moon, Ph.D.

9. Performing Organization Name and Address

4 Aero Lab. Department of Aerospace and Mechanical Sciences Princeton University Princeton, New Jersey 08540

2. Sponsoring Agency Name and Address

National Aeronautics and Space Administration Washington, D. C . 20546

5. Supplementary Notes

3. Recipient's Catalog No.

5. Report Date May 1973

6. Performing Organization Code

8. Performing Organization Report No. AMs Report No. 1103

10. Work Unit No.

11. Contract or Grant No. NGR 31-001-267

13. Type of Report and Period Covered

14. Sponsoring Agency Code

6. Abstract

A review of the f i e l d of s t r e s s waves i n composite materials is presented covering the period up t o December 1972. and a summary is made of the major experimental resul ts in th i s f i e ld . models f o r analysis of wave propagation i n laminated, f i be r and pa r t i c l e reinforced composites are surveyed. pulses and shock waves i n such materials are reviewed. A review of the behavior of composites under impact loading is presented along with the application of wave propa- gation concepts t o the determination of impact s t resses i n composite plates .

The major properties of waves i n composites are discussed Various theoretical

Theanisotropic, dispersive and dissipative properties of s t r e s s

NASA-C-168 (Rev. 6-71)

ACKNOWLEDGEMENT

This research was supported by NASA/Lewis Research Grant NGR-31-001-267.

This repor t i s t o be published as a chapter i n Trea t i s e of Composite Mater ia ls ,

Broutman, Krock, Edi tors , Vol. 9 , S t ruc tu ra l Analysis and Design, C . C . Chamis,

e d i t o r , Academic Press .

- iii -

TABLE OF CONTENTS

Acknow 1 e dg emen t

Table of Contents

I. Introduct ion

11. Anisotropic Waves i n Composites

A. Wave Speeds

B . Wave Surf aces

C . Flexural Waves i n Orthotropic Plates

D. Surface Waves

E . Edge Waves i n Plates

F. Waves i n Coupled Composite P la t e s

111. Dispersion i n Composites

A. Pulse Propagation and Dispersion

B . Dispersion i n Rods and Plates

C .

D.

E . Continuum Theories f o r Composites

Dispersion i n a Layered Composite

Combined Material and S t ruc tu ra l Dispersion

F - Variat ional Methods f o r Per iodic Composites

I V . Attenuation and Sca t t e r ing

V. Shock Waves i n Composites

V I . Experiments

V I I . Impact Problems i n Comnosites

A. Introduct ion

B . Analytical Models f o r Impact

C . S t ruc tu ra l Presponse t o Impact

References

Captions f o r Figures

ii

iii

1

6

7

10

13

16

18

20

22

23

25

27

33

34

41

44

49

53

61

61

66

72

80

96

- 1 -

I. INTRODUCTION

S t r e s s waves i n composite materials a re of i n t e r e s t t o t h e engineer

both f o r t h e i r cons t ruc t ive -app l i ca t ion and f o r t he po ten t i a l damage t h a t

can occur when s h o r t durat ion stress pulses propagate i n a s t r u c t u r e .

S t r e s s waves have a cons t ruc t ive use as a d iagnos t ic t o o l t o measure

e l a s t i c p rope r t i e s , search f o r flows and t ransmit information. Such appl i -

ca t ion usua l ly involves waves i n the form of pulses o r u l t r a s o n i c s inu-

so ida l pu l se s . Seismologists have long been i n t e r e s t e d i n t h i s appl ica t ion

of stress waves, p a r t i c u l a r l y the study of waves i n layered media (see e .g .

Ewing e t a l . , 1957; Brekhovskikh, 1960). Early s tud ie s of laminated media

were aimed i n fact a t geophysical appl ica t ions (e .g . Anderson, 1961).

S t ruc tu ra l engineers however usua l ly r e l y on composite mater ia l s t o

sus t a in forces o r loads. When these forces are a r e s u l t of shock o r impact

on the s t r u c t u r e , t he forces w i l l be t ransmit ted through t h e s t r u c t u r e i n

t h e form of stress waves.

s t a t i c o r q u a s i - s t a t i c loads (v ibra t ions) can usua l ly be pred ic ted by

s t r u c t u r a l engineers , rou t ine methods f o r pred ic t ing the path

While the pred ic t ion of stress d i s t r i b u t i o n f o r

of stress pulses through a complicated s t r u c t u r e are not r ead i ly

The anisotropy and inherent ava i lab le ,even f o r homogeneous materials.

inhomogeneity i n composite materials f u r t h e r complicates t h i s problem.

The importance of impact stresses i n composite s t r u c t u r a l design can

b e s t be i l l u s t r a t e d by t h e appl ica t ion of these materials t o j e t engine

fan blades (see Goatham, 1970). In addi t ion t o t h e load requirements imposed

by cen t r i fuga l and v ibra tory fo rces , these blades must be designed t o

withstand t h e s t r e s s e s due t o impact with foreign objec ts such as b i r d s ,

- 2 -

ha i l s tones , s tones , and nuts and b o l t s . The r e l a t i v e ve loc i ty of t he

impacting body t o the blade can be i n the order of 450 meters pe r second

(1500 f t / s e c ) .

small impact times (< 50 1-1 sec) and the i n i t i a l transmission of t h e t o t a l

energy i n t o a loca l region of t he blade. The impact not only induces local

c ra te r ing o r s p l i t t i n g but long range damage away from the impact area can

r e s u l t from the r e f l e c t i o n of s t r e s s waves (spal l ing) from boundaries and

focusing e f f e c t s due t o changes i n blade geometry.

problem of foreign object impact involve

embedded high s t rength meshes and leading edge impact pro tec t ion .

The high speed impact of small ob jec ts r e s u l t s i n very

Solutions of t h e

considerable ingenuity,such as

Impact loads involve two fac to r s which are not considered i n s t a t i c

One i s the speed of propagation of t he stress pulse i n

In s t a t i c problems t h e deformation energy can be

s t r e s s analysis .

t he mater ia l .

d i s t r ibu ted throughout t he s t ruc tu re , but i n impact loading the volume of

energy s torage is l imi ted by t h e speed of propagation of t he waves i n

the mater ia l . For shor t time impact loads, a small amount o f energy i n

a small volume can r e s u l t i n s t r e s s e s which can f r ac tu re o r otherwise

damage the mater ia l .

The speeds of propagation of stress waves f o r a number of composites

a re shown i n Table I with comparable da ta f o r conventional s t r u c t u r a l

mater ia l s . These speeds depend on the d i r ec t ion i n which t h e

wave propagates ,and when the e l a s t i c l i m i t i s exceeded ,depend a l so on the

s t r e s s l eve l . These wave speeds a re motions averaged over a local region

of the composite involving many layers , f i b e r s o r p a r t i c l e s whichever i s

the case. Within each cons t i tuent , of course, t h e stresses propagate as

i n the respect ive homogeneous mater ia l s .

- 5 -

The second d i f fe rence between impact loading and s ta t ic loads i n

design i s the rate of change of s t r a i n .

of s t r a i n have been shown t o exh ib i t d i f f e r e n t s t rength proper t ies

Composites under high r a t e s

(Sierakowski e t a l . , 1970). Often t h i s r e s u l t s i n h igher ul t imate

s t r eng th with increasing s t r a i n rate.

While the f ac to r s of f i n i t e wave t i m e and rate dependent proper t ies

are common t o impact problems i n a l l s t r u c t u r a l materials, t he anisotropy

and inhomogeneity inherent i n composites requi res spec ia l a t t en t ion i n the

design of an impact r e s i s t a n t composite s t ruc tu re .

Anisotropic waves i n so l id s a re familiar t o those i n c r y s t a l physics

and seismology, however, these e f f e c t s are not well known i n s t r u c t u r a l

design where conventional i s o t r o p i c mater ia l s such as aluminum and s t e e l

a re o f t en used. Composite mater ia l s have the unique fea ture t h a t the degree

of anisotropy can be var ied i n t h e mater ia l and hence the ana lys t can

change the d i r ec t iona l d i s t r i b u t i o n of s t r e s s waves i n an impact zone and

perhaps avoid ser ious f a i l u r e o r f r ac tu re (perhaps by a judicious choice

of p ly lay-urp angles) .

The e f f e c t s of boundaries o r d i scon t inu i t i e s i n mater ia l p roper t ies

on s t r e s s waves a re well known (see Ewing e t a l . , 1957). When a s t r e s s

wave encounters a boundary,normal t o the wave f ron t , separa t ing mater ia l s

of d i f f e r e n t dens i t i e s and wave speeds, p , v , t he s t r e s s a t t h e surface

is changed t o

= 5 P 2 V2/(P1 v1 + P v 1 0 2 2

- 4 -

where cr would be t h e stress i n material rcl" i f the boundary were

not present . The product pv is ca l l ed the acous t ic impedence and

depends on the type of wave (e.g. shear o r d i l a t a t i o n a l i n i s o t r o p i c

0

s o l i d s ) .

Thus a wave o r ig ina t ing i n a "sof ter" material i . e . p v p2v2 1 1

always s u f f e r s a stress increase a t a boundary, This is indeed t h e case

f o r many composites e spec ia l ly those involving a compliant matrix, such

as epoxy, and a s t i f f f i b e r such as graphi te , g l a s s o r boron.

Another effect of inhomogeneity is dispers ion. Dispersion o f t h e

average composite motion r e s u l t s i n a d i s t o r t i o n of t h e stress pulse

as it propagates.

r ise time, o r per iod o f t he stress pulse decrease.

contaiaing compressional stresses can develop t e n s i l e stresses

The effects of dispers ion increase as the dura t ion ,

Thus a pulse i n i t i a l l y

as the wave propagates and perhaps induce micro-cracking i n t h e composite.

The l i t e r a t u r e on the subjec t of waves i n composites has expanded

enormously i n t h e p a s t few years and new t h e o r e t i c a l and experimental

r e s u l t s a r e s t i l l being repor ted . This review then can only summarize

the work t o the da te o f this wr i t ing . Also severa l good reviews have

appeared a t t h i s wr i t ing i n which t h e var ious t h e o r e t i c a l models f o r

waves i n composites have been discussed, (Peck, 1971, 1972, Achenbach,

1972).

This chapter w i l l be somewhat t u t o r i a l i n na ture r a t h e r than a

c r i t i ca l review of t h e var ious theo r i e s t o da t e . Instead I w i l l t r y

t o summarize the r e s u l t s t o da t e which seem t o be accepted i n t h e f i e l d

and which might be of use t o t h e s t r u c t u r a l dynamics ana lys t .

- 5 -

In the following sec t ions I w i l l discuss

i ) an iso t ropic waves i n composite s t ruc tu res (without dispersion)

i i ) dispers ion e f f e c t on waves

iii) s c a t t e r i n g and absorption o f waves

iv) shock waves i n composites

v) experimental r e s u l t s

v i ) the e f f e c t s o f impact

For a review of s t r e s s waves i n conventional s t r u c t u r a l mater ia l

see Miklowitz (1966).

- 6 -

11. ANISOTROPIC WAVES I N COMPOSITES

In t h i s sec t ion I w i l l review those aspects of e las t ic wave pro-

pagation i n an iso t ropic mater ia l s which a re re levant t o composites.

When the sca l e of t h e changes i n stress l eve l , ( r i s e d is tance , wave-

length, e t c . ) , i s much l a r g e r than the sizes of t he cons t i tuents

of composites ( f i b e r o r p a r t i c l e diameter, f i b e r spacing, p ly spacing,

e t c . ) t he mater ia l may be t r ea t ed as an equivalent homogeneous e l a s t i c

mater ia l as a f irst approximation. In a homogeneous medium, the wave

speeds a r e r e l a t e d t o the e l a s t i c constants and densi ty by r e l a t i o n s

of the type Pv = C , where C is an e l a s t i c constant. This r e l a t ion

*

2

has led t o the use of wave theory t o determine the e f f ec t ive e l a s t i c

moduli of composite mater ia ls when the wavelength becomes l a r g e r than

the s ize of t h e sca l e of inhomogeneity. Thus the de f in i t i on

where X i s the wavelength; ''a" is a s ize associated with t h e composite

elements (e.g. f i b e r spacing ,and v(x/a) is the phase ve loc i ty f o r a

given harmonic wavelength.

(1955) f o r a laminated medium and by Behrens (1967a) (1967b).

This method has been used by White and Angona

In t h e case of p a r t i c u l a t e composites o r dispers ion strengthened

composites t he equivalent model may be considered as i so t rop ic .

f o r f i b e r composites , laminates , and unid i rec t iona l eu tec to ids , t h e

But

* The exception i s t h e case of a composite p l a t e with bending-extensional

coupling.

- 7 -

equivalent s t r e s s - s t r a i n r e l a t i o n w i l l be an iso t ropic i . e .

kR e - t i j - ‘ijk!L

is the stress tensor and e the s t r a i n tensor . where t i j kR

A. Wave Speeds

The simplest wave t o consider is a plane wave with no ex terna l

boundaries present . For such a wave the displacement has the form

The vec tor 2 defines a plane r e l a t i v e t o the mater ia l axes and

v i s the speed of the wave. When Eqs. ( Z ) , (3) a r e put i n t o the

equations of motion f o r the mater ia l , the following eigenvalue problem

r e s u l t s

(‘ijk!L n k n j - P V k j ) % = 0 (4)

the Kronecker d e l t a . (See, e . g . ‘ i j where p i s t h e equivalent densi ty and

Musgrave (1954, 1970) ,and Kraut (1963)). In summary, f o r each wave

d i r ec t ion the re are th ree d i f f e r e n t waves

Ci) When the v (i) a r e d i s t i n c t , the th ree Polar iza t ion vectors

a re orthogonal. For i s o t r o p i c mater ia l s it is well known t h a t only

two speeds a re d i s t i n c t

- 8 -

These a re respec t ive ly t h e longi tudinal and t ransverse (shear) waves.

For an iso t ropic waves however such charac te r iza t ion is not poss ib le

except along symmetry d i r ec t ions .

For many composites, o r tho t ropic symmetry su f f i ces t o descr ibe

the mater ia l and nine e l a s t i c constants are required.

r e l a t i o n f o r t h i s case i s given by

The s t r e s s - s t r a i n

t 11

t 22

t 33

t 23

t 1 3

t 12

c c c 0 0 11 12 1 3

c c 0 0 22 23

c 0 0 33

G O 44

c o 55

C 61

e 11

e 22

e 33

2e 23

2e 1 3

2e 12

For s t r u c t u r a l appl ica t ions composites a re usua l ly used i n the

form o f rods o r p l a t e s . Consider, fo r example, t he in-plane motion

of a p l a t e with t h e x ax i s normal t o the midsurface, i . e . u = 0 . For wavelengths much l a r g e r than t h e p l a t e thickness we neglect t he

2 2

- 9 -

e f f e c t s of dispers ion. For t h i s case the equations of

motion fo r the in-plane motion i n the lowest approximation became

(see Figure 1)

a2u1 a2u1 a2ul A a2u3 + c13) ax ax 11 ax2 5 5 ax; 55

p - = c - + c - + ( C 1 3 1

a t 2

32U3 A a2u3 a2u3 A a h 1 + (' + c13) ax ax p - = c - + c -

1 3 a t 2 33 ax2 55 ax2 55 1 3

where

I\

c = c - c 2 / c 11 11 12 22

,. c = c - c 2 / c

33 33 2 3 22

A

c = c - c c /c 1 3 1 3 12 2 3 22

The constants

arrangement o f t h e composite cons t i tuents . C i j

can be determined from the proper t ies and geometric

For plane waves i n t h e p l a t e , two wave speeds e x i s t f o r each d i rec t ion

E = (cos (9, s i n 4) (see Figure 1) and are determined by

(A - pv2) (A22 - P V 2 ) - A2 = 0 1 1 12

where h

A = c cos*$ + c sin24 11 11 55

A

A = c sin24 + c cos24 22 33 55

A = A = (C + C ) sin4 cos4 12 2 1 55 13

- 10 -

The r a t i o A /A is determined by s u b s t i t u t i n g each root v2 i n t o

Eq . ( 4 ) , (k = 1, 3 ) . 1 3

These values have been ca lcu la ted f o r a number of composites (see

Moon, 1972a) and a re shown i n Figure 2 f o r 55% graphi te fiber/epoxy

matrix composite. As t h e fiber lay-upangle i s changed f o r t h e same

composite,the proper t ies of t h e waves are seen t o change dramatical ly .

The d i r ec t ion of p a r t i c l e motion r e l a t i v e t o the wave normal is

shown i n Figure 3 f o r 55% graphi te fiber/epoxy matrix fou r p ly lay-up

angles. For the O o , -1.45 lay-up angle cases, the d i r ec t ion of p a r t i c l e 0

motion tends t o l i e c lose t o t h e f i b e r d i r ec t ions f o r most wave normals.

The -1.15' and +30° lay-up angle cases present another departure from A

the i so t rop ic case. In both cases C > C . This means t h a t f o r 5 5 33

waves t r ave l ing i n t h e x d i r ec t ion the f a s t e r wave becomes t ransverse

and the slower longi tudina l . 3

This behavior i s a l s o found i n pine wood.

B. Wave Surfaces

The r e l a t i o n v ($ ) , i n Figure 3 , i s ca l led the ve loc i ty sur face .

However i f t he waves or ig ina ted from some point i n the p l a t e , t o an

observer at pos i t i on

t h a t corresponding t o t h e wave normal @ = O . If the a r r iva l , t i m e

is t = 1, t h e f irst plane wave, Q , t o a r r i v e a t t he poin t must

s a t i s f y

,O ) , t h e f i rs t s igna l t o a r r i v e may not be (ro 0

0

o r where z Q/V ;

- 11 -

s is c a l l e d the slowness vec tor and l /v ($) t h e slowness sur face (see

e .g . Kraut, 1963). The equation

the slowness plane ( s , s ) and E is normal t o t h a t l i n e . In addi t ion

;i = 1 must be tangent t o t h e locus of values l / v (+ ) given by

%

* % = 1, then represents a l i n e i n

1 2

Thus

r = 01Vg %

where 01 i s a constant .

This coupled with E q . (9) gives the pos i t i on , r , t o t h e first a r r i v a l

of a plane of normal n($) generated a t t he o r i g i n ,

This locus r (2 ) i s ca l l ed the wave surface and f o r in-plane p l a t e

motion, t h e r e e x i s t s two such sur faces .

f o r t h e system 55% graphi te fiber/epoxy matrix (Moon, 1972a).

These are shown i n Figure 4 , s

The equivalent e l a s t i c constants fo r f iber-matr ix svstems a t var ious

lay-up angles were obtained by Chamis (1971). These constants , which are

l i s t e d i n Table I1,are based on a s t a t i c ana lys i s o f an eight-ply p l a t e

using the known proper t ies o f each f iber-matr ix p ly .

The graphite-epoxy systems cont ras t s with o ther composite systems

because of i t s high s t i f f n e s s r a t i o ; C / C = 24 zero lay-mangle).

The ve loc i ty sur faces fo r l ay -upang les of +O , +15 , 230 , and +45 are 11 3 3

0 0 0 0

- 1 2 -

shown i n Figure 2 .

Examining t h e wave surfaces fo r graphite-epoxy, as shown i n Figures

4, 5, one sees t h a t t h e inner surfaces show pecu l i a r cusps and noncon-

vex i ty . This behavioT is a l s o characteristic of c r y s t a l systems such

as zinc. Unlike the na tu ra l c r y s t a l s , we can change t h e wave p rope r t i e s ,

without changing t h e material cons t i t uen t s , by varying t h e f i b e r laY-uP

angle. I t becomes c l e a r t h a t , as t h e anisotropy i n the ou te r

wave i s reduced , t he cusped behavior of t h e inne r waves increases .

This is due t o t h e increase i n shear wave anisotropy (Figure 2 ) .

Another pecu l i a r property of wave propagation i n t h i s composite

system can be noted by examination of t h e 245'

5 ) .

a r r i v a l plane wave i s l i s t e d . One can see t h a t t h e d i s t r i b u t i o n of

plane wave normals i s heavi ly concentrated a t pos i t ions on the wave

sur face c lose t o t h e f i b e r d i r e c t i o n s .

waves along t h e f i b e r d i r ec t ions . For t h e o ther f i b e r o r i en ta t ions ,

the d i s t r i b u t i o n of wave normals i s a l so concentrated a t those poin ts

on t h e wave sur face c lose t o t h e f i b e r d i r ec t ions but no t as densely

as i n t h e +4S0 lay-up case.

f i b e r lay-up case (Figure

On t h e o u t e r wave sur face , t he angle of t h e wave normal of t h e first

This might imply a focusing of

S imi la r r e s u l t s f o r t h e g l a s s fiber-epoxy composite system have

been ca lcu la ted (Moon, 1971). The r a t i o of s t i f f n e s s e s f o r t h i s case

C /C = 3.1 (zero laY-up angle) . The wave sur faces f o r t h i s system 1 1 3 3

show fea tu res similar t o t h e graphite-epoxy case.

- 13 -

The ve loc i ty and wave surfaces f o r a boron fiber/aluminum composite

were a l so calculated (Moon, 1971). However, t h e shear ve loc i ty i s almost

i so t rop ic and no cusps appear on t h e wave sur face .

Weitsman (1972) and (1973) has recent ly s tud ied waves i n a t ransverse ly

i s o t r o p i c composite with a r i g i d f i b e r cons t r a in t .

C . Flexural Waves i n Orthotropic P la tes

For t h e case when the motion includes displacements out of t he plane

of t he p l a t e , Mindlin (1961) and co-workers have formulated an approximate

theory t o describe Elexural waves i n an iso t ropic p l a t e s .

the p l a t e motion i s expressed i n a s e r i e s i n the thickness parameter i . e .

In t h a t theory

Y A

2 u = uo (x yx , t ) + u q x ,x ,t) + ...

1 1 1 3 1 1 3

" 2

n

u = uo (x ,x , t ) + u;(xl,x3yt) + ... 3 3 1 3

The average in-plane motion i s governed by Eqs. (7) while the functions

u o , u l , u1 2 1 3

are coupled together i n t h e equations;

- 14 -

I t should be noted t h a t i n the procedure used by Mindlin the coef-

f i c i e n t s C and C i n Eqs. (12) a re replaced by, K C and K C , 44 66 3 44 1 44

respec t ive ly . The cor rec t ion constants K and K were adjusted i n

order t o match t h e thickness shear v ib ra t ion mode. 1 3

Consider the f l exura l plane waves. One can show t h a t the only plane

wave so lu t ions of t h e form t h a t s a t i s f y Eqs. (12) a re harmonic funct ions,

t h a t i s ,

For bending motion, t h e phase ve loc i ty v depends on the frequency,

w = kv, as well as t he wave normal Q. Mindlin (1961) has examined

the dependence of v on w f o r var ious mater ia l an iso t ropies .

Thus t h e behavior of t he bending motion a t t h e wave f r o n t s cannot

be determined i n the same manner as was the extensional motion.

Consider t he motion a t the wave f r o n t only. Across t h i s f r o n t , one

imagines t h a t c e r t a i n q u a n t i t i e s have d i scon t inu i t i e s . The displacement

- 15 -

and t h e s t r e s s are assumed t o be continuous across t h e wave f r o n t but

d i s c o n t i n u i t i e s i n t h e second de r iva t ives of U are assumed. Such

waves are c a l l e d acce lera t ion waves.

I t can be shown [Moon, 1972) t h a t t he wave f r o n t s associated with

a jump i n t h e bending acce lera t ions a2u1/at2 and a2u1/at2 t r a v e l

a t t h e same speeds as t h e wave f r o n t associated with the extensional

motion. There is another wave f r o n t corresponding t o a jump i n t h e

q u a n t i t i t y a2uo/at2 . The speed f o r t h i s wave i s governed by the

equation

1 3

2

pv2 = c cos24 + c s in2+ 66 4 4

For the case of a composite with symmetric p l y o r i en ta t ion about t h e

midplane,

c = c 6 6 44

The bending wave f r o n t associated with t h e jump

a l l y i s o t r o p i c .

[a2u0/at2] 2

i s d i r ec t ion -

If both extensional and bending motions a re generated simultaneously

by impact, t h e two extensional and two bending wave f r o n t s w i l l t r a v e l

with t h e same wave speeds.

This conclusion does not hold if the laminate p l a t e has only

a few p lys . For example, Sun (1972b) has shown t h a t t he bending

and extensional wave f ron t s are d i f f e r e n t f o r a three-layered p l a t e

( O o , 90°, 0' f i b e r lay-up angle) . Thus the use of t he e f f e c t i v e

modulus theory t o p r e d i c t t he speeds of t he wave f r o n t s would appear

t o be v a l i d only when the number of p lys i s large (probably - > 10 l a y e r s ) .

- 16 -

The ana lys i s presented here is not unique. The same r e s u l t s can be

obtained i f one considers t he equations of motion from the method of

c h a r a c t e r i s t i c s .

For harmonic waves the r e l a t i o n between the frequency w and wave

number has th ree branches. For the lowest branch

(1) (k) % k 2 kb -+ 0

Thus f o r low frequencies the phase ve loc i ty

wavelength as i n i s o t r o p i c p l a t e s .

frequencies, v -+ (C / ~ ) l / ~ which is the i s o t r o p i c ve loc i ty of the

v = w/k depends on the

For sho r t e r wavelengths o r h igher

4 4

wave f r o n t s as discussed above. For the o ther two branches, (k) ,

~ ( ~ ) ( k )

an iso t ropic and equal t o the values calculated f o r in-plane p l a t e

waves.

t he phase v e l o c i t i e s a t high frequencies a re constant and

The d is t inguish ing f ea tu re about such waves i n composites i s t h a t

these dispers ion r e l a t i o n s depend on wave d i r ec t ion . In

Figure 6 , dispers ion curves f o r f l exura l waves are given f o r t h e wave

d i r ec t ions 0' , 90' f o r .t4So lay-up angle, f o r 55% graphi te f i b e r /

epoxy matrix composite using the d a t a i n Table 2 . I t should be noted t h a t these mathematical models a r e approximate

and w i l l break down f o r those Fourier components of t he wave with wave-

lengths of t h e order of t h e composite cons t i tuent dimensions.

considerations induce addi t iona l d i spers ion i n addi t ion t o t h a t due t o

the p l a t e sur faces .

Such

\

D. Surface Waves

In cont ras t t o t h e bulk waves discussed above, sur face waves are

motions with wave-like behavior along the surface o r i n t e r f a c e between

two d i f f e r e n t mater ia l s , and exponential decay with d is tance from the

- 1 7 -

surface. Thus i f x i s normal t o t h e sur face , a surface wave has

the following form f o r harmonic waves 3

-y(o)x3 eik(o)[x cosa + x s i n a - vCw)t] 1 2 u % e

* These waves a re known as Rayleigh waves (see e.g. Ewing e t a l . , 1957)

f o r a free surface and Stonely waves f o r an in t e r f ace .

For an or thot ropic mater ia l the ve loc i ty of such a wave t r ave l ing

i n the x d i r ec t ion on the surface of a ha l f space normal t o x

(Figure 7) i s given by t h e roots t o t h e following equation. 1 3

- pv2)]=0 (14) “13 - c 3 3 ( c 1 1 1’” c,, - PV2

c c (C - PV2) PV2 + [

3 3 5 5 1 1

Examination of t h i s equation reveals t h a t one r e a l root lies i n the

i n t e r n a l ,

Thus the Rayleigh wave speed i n t h i s d i r ec t ion is less than the shear

speed [C /< I1 l2 . The motion i s p lanar , i . e . u = 0 , and can be 55 2

* Another wave known as a Love wave e x i s t s f o r a surface with a l aye r of

d i f f e r e n t mater ia l .

- 18 -

shown t o be e l l i p t i c a l i n a plane normal t o the sur face .

The Rayleigh wave speed, however, va r i e s with d i r ec t ion i n the

plane of t he surface and has been shown by Musgrave (1954) t o give

a wave surface with cusps f o r ce r t a in anisotropic mater ia ls s i m i l a r

t o bulk shear waves. Also, the exis tence of such waves f o r a l l surfaces

i n the mater ia l has been vigorously debated i n the l i terature .

However, Lin and Farnell (1968) have found Rayleigh type so lu t ions for

a l l surfaces , though f o r c e r t a i n planes and d i rec t ions the va r i a t ion

of the motion from the surface combines exponential and harmonic

functions.

t o composites should be obvious.

While t h i s work has been applied t o c r y s t a l s , the appl ica t ion

E . Edge Waves i n P la tes

Waves confined t o the edges of p l a t e s should be important i n t h e

edge impact of p l a t e - l i ke s t ruc tu res e .g . j e t engine fan blades. When

the average p l a t e motion l ies i n the plane of the p l a t e , waves analogous

t o Rayleigh waves e x i s t f o r low frequencies. For motion out of t he plane

of t he p l a t e , f lexura l edge waves may propagate but a r e dispers ive even

a t low frequencies.

1. Extensional Edge Waves

As i n t h e case of bulk waves, we may look f o r plane s t r e s s Rayleigh A h

wave so lu t ions i n p l a t e s . By replacing C , C , C , by C , C 1 1 33 1 3 1 1 3 3

^c i n Eq. (14) values may be obtained f o r extent ional edge waves i n 1 3

aniso t ropic p l a t e s .

The extension of Eq. 14 t o p l a t e s must be made with caution since

t h e plane stress approximation breaks down at frequencies approaching

- 19 -

t h a t of t he first thickness shear modes f o r which t h e waves became

d ispers ive . McCoy and Mindlin (1962) have used a higher mode ana lys i s

t o examine such waves f o r i s o t r o p i c p l a t e s but t he extension t o

anisotropic p l a t e s does not seem t o have been made a t t h i s wri t ing.

2 . Flexural Edge Waves

Flexural edge waves f o r i s o t r o p i c p l a t e s using Mindlins p l a t e

theory has been s tudied by Kane (1954). No reference t o the problem

f o r an iso t ropic p l a t e s has been found by the author.

sketch of the procedure, l e t us consider t he low frequency c l a s s i c a l

As a b r i e f

an iso t ropic p l a t e equation f o r t he t ransverse displacement u u (x ,x ,t) 2 1 3

a4u a4u a 4 ~ a2u

11 ax4 1 3 55 ax2 ax2 332x4 a t2 (15) c - + 2(c + 2c ) + c - + p G - = O

1 1 3 3

where G = 12/h2 , h i s p l a t e thickness. For an edge wave propagating

i n the x d i r ec t ion w e look f o r so lu t ions of t he form 1

i (kx -ut) 1 u = A e-YX3 e

When the frequency is given, y and k a re r e l a t ed by the equation

The appropriate boundary conditions require the moment on the edge t o

be zero, i . e .

'i 2, a 2~ c - + c - = o

1 3 8x2 33 ax2 1 3

(17)

- 20 -

and the r e su l t an t shear contr ibut ions from shear force and edge torque

gradient t o vanish, i .e.

Choosing two values of y with negative r e a l p a r t , and applying t h e

boundary conditions, w e obtain an equation f o r t he phase ve loc i ty o f

these waves v = w/k

Let

Then B 2 + 4C B - C2 = 0 , (choose B > 0) 5 5 13

F. Waves i n Coupled Composite P la tes

Laminated p l a t e s made up of un id i rec t iona l p lys can have coupling

between t h e extensional o r inplane motion and the f l exura l o r out o f

plane displacements.

of motion f o r one dimensional waves i n t h e x d i r ec t ion , with u = 0 ,

assume t h e form (see Ashton, e t a l . , 1969)

Using an e f f e c t i v e modulous theory the equations

1 3

a 3u1 a4u2 a2u2 B - - D - = ph -

11 ax3 11 a d a t 2 1

- 2 1 -

In t h i s case there is no pure f l e x u r a l o r extensional wave.

w e have a coupled wave

Instead

i (kx-ut) e

with a d ispers ion r e l a t i o n

For low frequencies , o r when B /A2 << 1, t h e wave with dominant 11 1 1

extensional motion has t h e phase ve loc i ty

where v2 = A,,/ph . 0

Simi lar ly , f o r low frequencies (k -+ 0) o r small

with f l e x u r a l motion dominant, has t h e phase ve loc i ty

B,, , t h e mode

We might no te here t h a t t h e extensional shear wave i n t h i s d i r ec t ion

i s uncoupled from t h e f l e x u r a l motion.

coupling i s unique t o composites, it is supr i s ing t h a t such waves i n lami-

na ted s t r u c t u r e s have not received as much a t t e n t i o n t o da te as o t h e r t op ic s

i n dynamics of composites.

f ron t s i n laminated p l a t e s with bending-extensional coupling. He a l so observes

coupling of the inplane and f l e x u r a l motions i n the var ious waves.

Since t h e extensional-f lexural

Sun (1972b) has s tud ied the propagation of wave

- 22 -

111. DISPERSION I N COMPOSITES

One de f in i t i on of wave dispers ion is the d i s t o r t i o n of t he pulse

shape as it propagates through the mater ia l .

from at tenuat ion i n which energy i s sca t t e red out of t he waves o r con-

ver ted t o h e a t .

I t i s t o be dis t inguished

A more prec ise de f in i t i on of dispers ion rests on the

assumption of a l i n e a r mater ia l and the theorem t h a t any wave pulse i n

the mater ia l can be expressed as a l i n e a r sum of harmonic waves, e .g .

f o r a one dimensional wave t h e displacement might have the form

For a non-dispersive mater ia l t he phase ve loc i ty of a l l t h e harmonic

components are equal.

dynamics of i so t rop ic mater ia ls i n t h e form of rods, p l a t e s and s h e l l s .

Examples of wave dispers ion are common i n s t r u c t u r a l

Although the bulk waves i n e l a s t i c mater ia ls a re non-dispersive, the

introduct ion of bounding sur faces , which define the s t r u c t u r a l element,

causes the r e f l e c t i o n of these waves from the surface t o depend on

the wavelength ( A = 2n/k = 2nv/w) . . If rrarr represents a length

parameter (thickness, diameter) then ka o r wa/v become c r i t i c a l

parameters i n the problem of wave dispers ion.

I t is only na tu ra l then t o expect t h a t inhomogeneities such as

f i b e r s , laminates o r p a r t i c l e s i n a material matrix w i l l r esu l t i n a

g rea t e r dispers ion of waves as t he wavelength approaches the s ize o r

spacing of t h e composite cons t i tuents . I t should be noted here t h a t

when a composite i s used as a s t r u c t u r a l element there w i l l be two

- 23 -

sources of dispers ion; t h a t associated with t h e lengths of mater ia l

elements, e .g . f i b e r diameter, o r p ly thickness , and t h a t associated

with t h e s t r u c t u r a l dimensions.

w i l l become important f o r wavelengths much longer than those of t he

order o f t h e material elements.

I t should be expected t h a t t h e l a t t e r

There have been th ree general approaches t o mater ia l dispers ion

i n composites :

i )

i i )

exact so lu t ions of elastodynamic equations

approximate so lu t ions of elastodynamic equations

i i i ) micro continuum theor i e s

Several reviews on waves i n composites have appeared (Peck, 1971,

1972, Achenbach, 1972) i n which t h e various models f o r wave dispers ion

have been discussed. The reader i s r e fe r r ed t o these reviews f o r

de t a i l ed discussion of t he various approaches.

t r y t o summarize t h e p r inc ip l e conclusions of t he work on dispers ion

published t o da te .

A. Pulse Propagation and Dispersion

In t h i s Chapter I w i l l

While t h e o r e t i c a l descr ip t ions of dispers ion of ten employ an

i n f i n i t e t r a i n of harmonic waves, t he engineer i s more of ten in t e re s t ed

i n the propagation of stress pulses i n a medium.

l a t i o n of t h e e f f e c t s o f dispers ion i s s t ra ightforward. One decomposes

the s t r e s s pulse a t a given t i m e i n t o a spectrum of harmonic waves

and uses t h e phase ve loc i ty t o t r a n s l a t e each component wave, reconstruc-

t i n g the pulse a t a l a t e r time using Eq . (22).

last decade have made t h e execution o f t h i s procedure reasonably easy.

Conceptually, calcu-

Two developments i n the

- 24 -

One too l i s the d i g i t a l computer and t h e development of e f f i c i e n t Fourier

summing algorithms such as the "fast Fourier transform".

the ana ly t i ca l foundation developed by Skalak (1957) and o thers

f o r using an approximate d ispers ion r e l a t i o n f o r t he phase ve loc i ty

The o t h e r i s

'L % v (1 - ak2) 0 -

o r

v % v (1 - 6 2 ) 0 -

f o r large times a f t e r impact o r la rge d is tances from t h e wave source.

Peck (1971), Peck and Gurtman (1969) and others have explored i n d e t a i l

the e f f e c t s of dispers ion i n layered composites. They have demonstrated

t h a t a wave with a stress d iscont inui ty w i l l be smoothed out , t h a t stress

overshoot can occur, and t h a t an i n i t i a l compression pulse can develop

t e n s i l e s t r e s s e s as t h e wave propagates.

understood s ince the loca l inhomogeneities w i l l r e f l e c t p a r t of t h e

These e f f e c t s can be h e u r i s t i c a l l y

propagating s t r e s s d i scont inui ty a t each layer . Multiple r e f l e c t i o n s i n

each l a y e r w i l l delay p a r t of t h e pulse and e f f e c t i v e l y broaden the average

stress i n t h e pulse . Fur ther , the loca l inhomogeneities can change the

s ign of t he r e f l e c t e d stress as well as raise the s t r e s s at a l aye r

i n t e r f ace . Examples of t h e e f f e c t s of dispers ion a re shown i n Figures 8 , 9

where a fast Fourier computer rout ine was used. Case I (Fig. 8) involves the

example o f decreasing phase ve loc i ty with frequency. I t has been shown

(Peck, 1971) t h a t when t h e input i s a s t e p i n stress t h e response is

r e l a t e d t o an i n t e g r a l of t h e Airy func t ion

- 25 -

Smoothing of t h e s t r e s s jump can be seen i n Figure 8,as well as t h e

overshoot r i g h t after t h e a r r i v a l of the pulse . A t t h e t a i l end, t he

s t r e s s is seen t o change s ign . This dispers ion is c h a r a c t e r i s t i c

of longi tudina l waves propagating down the f i b e r s o r layers as well

as across t h e l aye r s .

Case I1 involves increasing phase ve loc i ty with frequency o r

wave number.

the f i b e r s o r l aye r s .

previous case, i n t h a t s t r e s s r eve r sa l obtains f o r e a r l y time.

This case i s found f o r shear waves propagating down - In Figure 9 , the response seems t o mirror t he

B. Dispersion i n Rods and Pla tes

Mechanicians have long been familiar with t h e effects of geometric

d i spers ion when c l a s s i c a l mater ia l s take the form of rods and p l a t e s .

For i s o t r o p i c cy l ind r i ca l rods the long wavelength dispers ion r e l a t i o n

was given by Chree (1890) and o thers

where I, is Poissons r a t i o , and a i s the rod r ad ius . For an iso t ropic

rods an equivalent dispers ion r e l a t i o n f o r t h e phase ve loc i ty (neglecting

mater ia l dispers ion) was worked out by Pot t inger (1970)

v 2 v [l - B ~ 1 2 ( k a ) 2 ] 1 ka << 1 0

1 1 V I = - [S2 + s2 + - s2 + s2 15 + s2 16 ] 1 / 2 13 1 4 12 42-S

1 1

- 26 -

where v i = 1/S P , are t h e e las t ic compliances of t h e

composite. Pot t inger po in ts out t h a t t h e accuracy of t h i s approxi-

mation depends on the values of u t and a/A. For 2% deviat ion from t he exact

dispers ion r e l a t i o n , 2a/A < 0.6 f o r u t = 0.1, and 2a/A < 0.16 f o r V ' = 0.4.

and Sij 1 1

If t h e f i b e r d i r ec t ion of a unid i rec t iona l composite i s var ied

r e l a t i v e t o t h e rod axis , d i f f e r e n t dispers ion r e l a t i o n s are obtained

f o r each angle.

composite.

An example is shown i n Figure 10 f o r a Boron-Aluminm

For a longi tudinal wave i n a p l a t e one can der ive a similar d i s -

pers ion r e l a t i o n f o r

i n e r t i a . For a wave

normal t o the p l a t e ,

anis o t rop i c p 1 a t e s i n co rpo ra t ing the t ransverse

propagating i n t h e x d i rec t ion , and x

t h e phase ve loc i ty is given f o r long wavelengths 1 2

,. where C w a s defined i n E q . ( 7 ) , and d is t h e p l a t e thickness .

11

The Eqs. (24) , (25) neglect t he mater ia l d i spers ion due t o t h e

inhomogeneous na ture of t h e composite, which becomes increasingly

important as the wavelength approaches the scale of t h e s ize of t h e

cons t i tuents . Such e f f e c t s are considered i n the next sec t ion .

- 27 -

C. Dispersion i n a Layered Composite

We now examine the propagation of e las t ic waves i n a s o l i d made up

of a l t e rna t ing layers of d i f f e ren t material s t i f f n e s s e s and dens i t i e s .

This model.has been used by many authors t o examine the e f f e c t s of

dispers ion i n composites (Peck and Gurtman, 1969; Sun, Achenbach, Herrmann,

1968).

authors have given a t t en t ion (Rytov, 1955). The wave concept f o r t h i s

system i s the same f o r connected d i sc re t e p a r t i c l e chains as described

by Br i l lou in (1963).

I t i s a l so an important problem i n seismology t o which numerous

A c e l l is defined as two adjacent l aye r s . A loca l c e l l coordinate

rl

pos i t ion t o any p a r t i c l e i n the composite is given by

a i s the c e l l length, a = d + d . A displacement wave i n the direc-

t i o n normal t o the layer ing has the form

w i l l be used t o d is t inguish one c e l l p a r t i c l e from another. The

x = na + rl where

1 2

We can consider e i t h e r longi tudinal waves f o r which u represents

a displacement normal t o t h e layer ing o r a t ransverse wave where u repre-

s en t s t he displacement p a r a l l e l t o t he layer ing.

0

the longi tudinal speed of sound i n t h e mater ia l .

w i l l represent t h e shear s t r e s s and

the mater ia l .

In the former case l e t

be t he normal s t r e s s on any plane p a r a l l e l t o t h e layer ing and c

For t h e t ransverse case cr

c t h e speed of a shear wave i n

The balance of momentum is given by t h e following equation

- 28 -

along with t h e s t r e s s - s t r a i n r e l a t ion ;

These equations must be s a t i s f i e d i n each of t he two materials and

the so lu t ions i n each l aye r must satisfy boundary conditions of cont inui ty

of s t r e s s vector and displacement. Solutions which s a t i s f y the above

equations and t h e cell boundary conditions a t 11 = 0 are given by

wQ + B s i n a] ikna U (rl) = e [A cos - In C C

1 1

WQ

C 2

B s i n -1 wn p 1 c o l [A COS - + ikna C P C

u (11) = e 2n 2 2 2

- i w t (The f a c t o r e has been dropped f o r convenience). [The term pc is

ca l l ed the acous t ic impedence and i s proportional t o the r a t i o of stress/

ve loc i ty and is analogous t o the same concept in e l e c t r i c a l systems.]

The boundary conditions a t t he nth c e l l - (n + l ) th cel l i n t e r f ace

y i e l d two homogeneous a lgebra ic equations f o r A, B and a l so gives

the dispers ion r e l a t i o n .

wd2 s i n - s i n - wdl w d 2 (1 + P2) cos - - U d l cos ka = cos - C C 1 2

C C 2P 1 2

- 29 -

This r e l a t i o n is pe r iod ic i n k and symmetric about t he k = 0

ax i s . For each k i n the Br i l lou in zone, -IT < k a 2 IT t he re -

are an i n f i n i t e number o f values f o r w and hence an i n f i n i t e

number of branches. There is one acous t ic branch f o r e i t h e r t ransverse

o r longi tudinal waves and may be obtained f o r long wave lengths by

expanding Eq. (26) about k = 0 , w = 0 ;

w = V k 0

where

2 2 (l+p2) d ld2

L = V [-+I + 1-q + 2p a 2 c c 1 2 0

When the acous t ic impedances a re equal

v c C 0 1 2

which is j u s t t h e sum of the times f o r a wave t o t r ave r se each of

t he layers i n t h e cel l .

- 30 -

For t h i s case the waves a re non-dispersive. This i s t r u e because the re

a re no r e f l e c t i o n s a t an i n t e r f a c e of two mater ia l s when t h e acous t ic

impedances a re matched.

If the successive branches a re t r a n s l a t e d t o successive zones,

the dispers ion r e l a t i o n takes t h e character of a continuous homogeneous

medium, as shown i n Figure 11. However there are s top bands pro-

por t iona l t o the mismatch of impedance. This behavior i s a l s o char-

a c t e r i s t i c of quantum e lec t ron waves i n a per iodic p o t e n t i a l of a

conducting s o l i d .

For long wavelengths the d ispers ion r e l a t i o n Eq. (26) can be

expanded about w = 0 , k = 0 ,

r e l a t i o n f o r t he lowest o r "acoustic" wave mode.

t o obtain an approximate dispers ion

v v (1 - a(kd)2) 0 -

where

and

6 = d /a , 6 = d /a. 1 1 2 2

Note t h a t when p = 1, a = 0 .

- 31 -

The long wavelength phase ve loc i ty can be shown t o be r e l a t ed t o t h e

equivalent s ta t ic homogeneous e l a s t i c constant f o r the mater ia l , i . e .

f o r longi tudinal waves

v; = c /p 1 1

p = p v + p v 1 1 2 2

and

where , V = d / (d l+d 1 V = d / (d l+d ) 1 1 2 2 2 2

In the terminology of Herrmann and Achenbach (1968)

t o the e f f e c t i v e modulus,

s t i f f n e s s which i s frequency o r wavelength dependent.

vo i s r e l a t ed

whereas v is r e l a t e d t o an e f f e c t i v e 1

More general ly consider a composite made up of th ree dimensional

repeat ing c e l l s , such t h a t

is a la t t ice vector between corresponding points i n any two c e l l s .

vectors a re ca l l ed a bas i s set f o r t h e mater ia l . One can think of the

mater ia l p roper t ies p , Ci j as per iodic functions i . e . p (x) = P (x + ,&I.

I t is well known t h a t wave-like so lu t ions e x i s t f o r such a medium of

the form

& = Rel + m@2 + ne ( R , m, n , in tegers ) 3

The

- 32 -

Thus the motion var ies by a constant phase f ac to r from cell t o c e l l

and the problem is reduced t o f inding the motion i n a s ing le cel l

i . e . U (r) defined by & = 0 . Further i f one wr i tes 'Lo 'L

i k * r 'LO u ($1 = e 'L 'L ,wo(x)

The function !(g) must be per iodic ,

y0ca + &I = ,wo($)

For f i b e r composites i n two dimensiona, per iodic arrays one would

have only two bas i s vectors i n t h e plane normal t o the f i b e r d i rec t ions

but s imi l a r proper t ies on 8 would obtain.

Other work on the laminated composite includes t h a t of Sve (1971a, 1971b) who

examined thermoelast ic e f f e c t s and waves oblique t o the layer ing.

In addi t ion t o t h e layered o r laminated composite, dispers ion i n

f i b e r o r rod reinforced composites has been s tudied. Approximate

solut ions f o r t h i s problem were given by Puppo e t a l . (1968), Haener

and Puppo (1969), Jones (1970) and Ben-Amoz (1971). Jones shows the

phase ve loc i ty f o r longi tudinal waves t r a v e l i n g down the f i b e r s ,

mode dispersed down f o r t h e lowest mode, as i s indicated by experiments.

( h a y e t a l . (1968), Tauchert and Guzelsu (1972)). He a l so ca lcu la tes

- 33 -

t he cutoff frequency of t h e second mode ,for lon

terms of the matrix and f i b e r p rope r t i e s .

D.

n a l w

Combined Material and S t ruc tu ra l Dispersion

As mentioned above, dispers ion due t o s t r u c t u r a l geometry (e.g.

i n rods o r p l a t e s ) and dispers ion due t o material microgeometry (e.g.

f i b e r s i z e and spacing) have been s tudied separa te ly but i n ac tua l

s t ruc turesboth a re present .

these e f f e c t s a r e examined simultaneously is t h e theory of laminated

A class of problems i n which both o f

p l a t e s and s h e l l s . Multi layer p l a t e s have been s tudied by Jones (1964),

Sun and Whitney (1972), Biot (1972), Dong and Nelson (1972) , Scott

(1972), and Sun (1972a). The study of waves i n c i r cu la r ly laminated

rods o r s h e l l s of two mater ia ls has received a t t en t ion from Lai (1968),

McNiven e t a l . (1963) , Armenakas (1965), (1967) Whit t ier and Jones

(1967) and Chou and Achenbach (1970).

One can perhaps hazard a guess as t o t he comparison of t h e two

e f f e c t s on pulse propagation by an appeal t o the head of t he pulse

approximation discussed above. For longi tudinal waves i n a rod o r

p l a t e t he dispers ive r e l a t i o n f o r a non-dispersive mater ia l has t h e

form, f o r long wavelengths

where a i s a s t r u c t u r a l thickness o r diameter var iab le . The constant

v is r e l a t e d t o t h e square root of an elastic modulus.

m a1 i s i i s p as i n a t e o r f i b e r

If t h e 1

- 34 -

the constant v

ness which i t s e l f depends on t h e wavelength

i s r e l a t ed t o t he square root of an e l a s t i c s t i f f - 1

v '" vo[ l - 1

B2 21

where b is a f i b e r diameter o r lamination thickness. The combined

e f f ec t s of both s t r u c t u r a l and mater ia l dispers ion thus have the form

In most composites, b/a << 1 , s o t h a t it would appear t h a t t he e f f e c t

of mater ia l dispers ion, where the s t r u c t u r a l geometries guide the waves ,

can only become important when 6 >> a.

I

The corresponding problem f o r a composite beam has been examined

i n de t a i by Sun (1972) i n which he examines waves i n a laminated beam

using an e f f ec t ive s t i f f n e s s continuum theory assuming t h a t each

layer obeys the Timoshenko beam assumptions.

f o r a t en layered p l a t e with both an exact analysis and an e f f e c t i v e

modulus Timoshenko beam theory. For a l t e rna t ing layers , of shear

moduli i n t h e r a t i o o f 100, he found t h a t the e f f ec t ive modulus

model, based on Voight averaging of t he constants , agreed with the

exact analysis f o r 2 W h > l where h is the t o t a l beam thickness .

For waves o f sho r t e r wave length the e f f ec t ive modulus model deviated

subs t an t i a l ly from t h e microstructure and exact models.

E . Continuum Theories f o r Composites

Sun compared h i s theory

When t h e dimensions of t h e cons t i tuents of a mixture (e.g. f i b e r

- 35 -

diameter, p ly thickness) are much smaller than t h e s t r u c t u r a l dimen-

s ions , t he engineer is of ten s a t i s f i e d with averages o f t he motions

of t he cons t i tuents . In such cases a continuum model may s u f f i c e

t o describe the motion i n which t h e inhomogeneities a re "smoothed

out".

t o describe conventional s t r u c t u r a l materials which have a hetero-

Examples of such are found i n the use of classical e l a s t i c i t y

geneous gra in s t ruc tu re . A similar model f o r laminated composites,

using an e f f e c t i v e modulus theory has already been discussed above

which does not include mater ia l dispersion of waves (Chamis, 1971).

Attempts t o construct continuum descr ipt ions of dispers ion i n

composites have been var ied , bu t there a r e y i n generalytwo bas i c

approaches. The axiomatic method, is character ized by

the assumption of a s tored energy function with c e r t a i n funct ional

dependence on t h e deformation descr ip tors .

var iab les include descr ip tors f o r t he motion of t he microconsti tuents ,

The kinematic

e.g. f i b e r s o r p a r t i c l e s , i n addi t ion t o the average motion

a t a po in t . Examples of t h i s method are given by Mindlin

(1964) , theory of "microstructure i n e l a s t i c i t y " , Eringen (1966 , 1968)

Eringen and Suhubi (1964), theory of micropolar e l a s t i c i t y and micro-

morphic continua respec t ive ly , and a l so a mixture theory approach

by Green and Naghdi (1965).

broad c lass of mater ia ls and i n t h e l inear ized version of these theor ies ,

These theor ies attempt t o character ize a

contain a g rea t number of material constants which must be determined

by experiment. Ozgur (1971) f o r example has used Eringen's micropolar

theory i n an attempt t o descr ibe or thot ropic f i b e r composites. H i s model

- 36 -

uses 30 material constants as compared t o 9 constants f o r classical

or thot ropic e l a s t i c i t y .

d i spers ion phenomena but does not p red ic t dispers ion f o r longi tudinal

waves.

This theory r e s u l t s i n t h e cor rec t shear

In con t r a s t t o t h e f i rs t method the second approach starts

from an assumption o f a knowledge of t h e proper t ies of each cons t i tuent ,

and by averaging, "smoothing" and energy methods t r i e s t o a r r i v e at

a continuum formulation i n which the material constants are known i n

terms of t he p rope r t i e s of t h e cons t i tuents . An example of t h i s method

has been given by Achenbachand Herrmann (1968a) (1968b) i n which t h e

microelements a re f i b e r s embedded i n an e l a s t i c matrix. The f i b e r s

were assumed t o behave as Timoshenko beams.

continuum is assigned two kinematic va r i ab le s , t h e average displacement

a t a poin t x , and the f i b e r r o t a t i o n vec tor which i s independent

of t he vec tor . The r e s u l t i n g theory thus has s i x d i f f e r e n t i a l

equations o f motion t o be s a t i s f i e d a t each poin t .

able t o p red ic t d i spers ion f o r shear waves.

t he f i b e r s and motion t ransverse t o t h e f i b e r s t he following d ispers ion

r e l a t i o n i s obtained

Each po in t i n t h e equivalent

These authors are

For a wave normal along

k2(1 + 5 (kr )2) I . I ,

p" .2 'Lc 4 4

77

where p * is t h e composite dens i ty , C an e f f e c t i v e shear modulus,

rl Ef , r , t h e f i b e r modulus and 4 4

t h e volume f r a c t i o n o f f i b e r , and

- 37 -

r a t i o n respect

polymer matrix t h e r a t i o

dispers ive effect of t h e reinforce

wavelengths much l a r g e r than t h e f i b e r diameter. This model however

does not p red ic t dispers ion f o r longi tudinal waves. This problem

was solved q u i t e successful ly i n a s e r i e s of papers by Sun e t a l . (1968),

Achenbach e t a l . (1968) f o r t h e case of a laminated composite and by

Achenbach and Sun (1972) f o r a f i b e r re inforced composite.

dispers ion r e l a t ions predicted t h e cor rec t phenomenon at low

frequencies, Figure 1 2 . The exact harmonic wave so lu t ion f o r a l t e rna t ing

The r e su l t i ng

e l a s t i c i s o t r o p i c layers was obtained by Sun e t a l . (1968) and compared

with the continuum theory dispers ion r e l a t i o n . These r e s u l t s (Figure 12)

show good agreement a t low frequencies and f o r mater ia ls whose moduli

do not d i f f e r very much.

A similar method has been employed f o r f i b e r composites by Wu (1971).

In these methods, one e s t ab l i shes a loca l cell at each point

continuum containing a f i b e r and p a r t of t he matrix.

3 i n t he

Embedded i n the

c e l l is a loca l coordinate system 6 . There is assumed a t each poin t

x , a loca l o r microdisplacement f i e l d . In the case of Wu t h i s takes

the form

%

f o r i n g f i d matrix. In the

- 38 -

Achenbach and Sun (1972) they assume d i f f e r e n t forms of displacement

f i e l d s f o r f i b e r and matrix mater ia l i n each c e l l .

Thus f o r r < a , c l = r cos 8 , 5 = r s i n e 2

and f o r r > a

This procedure i s repeated f o r neighboring c e l l s and t h e average d i s -

placements along t h e adjoining c e l l boundaries a re matched. In the

model of Wu (1971) t h i s r e s u l t e d i n cons t r a in t equations on t h e loca l

c e l l s t r a i n s

S imi la r r e l a t i o n s a re obtained i n t h e mode, of Achenbach and Sun. The

concept of a loca l cons t r a in t was first introduced i n these theo r i e s

i n t h e e a r l i e r work of Sun, Achenbach and Hermann (1968) f o r t h e laminated

continuum.

To obta in equations of motion i n these methods the loca l displace-

ment Eq. (28) i s put i n t o cons t i t u t ive equations for t h e f i b e r and

- 39 -

matrix. The r e s u l t i n

over t h e cell c

volume t o y i e l d t

A similar procedure is ca r r i ed out f o r the k i n e t i c energy densi ty a t 4

T(z, 4). equations of motion and boundary conditions

Hamilton's p r inc ip l e is then used t o f ind the d i f f e r e n t i a l

sulj ject t o t he cons t ra in ts between xy 4, (e.g. Eq. (29)) and where W

is the work done on t h e boundary.

Grot (1972) has recent ly completed similar work on the f i b e r composite

continuum and has obtained very good agreement with t h e experiments of

Tauchert and Guzelsu (1971), Achenbach (1972) i n another review i n t h i s

series discusses t h e continuum models of composites.

Other work of a similar na ture includes Ben-hoz (1968), Barker (1970),

Bartholomew (1971), Bolotin (1965), Gurtman e t a l . (1971) Hegemier (1972)

and Koh (1970). Several mixture theor ies have been developed i n which the

contains terms proprotional t o the difference (,u (13- cL u12))

between the average displacements of each const i tuent (e.g. f i b

les of t h i s a r e the works of Bedford and Stern (1971),

work,applied t o

s i t e , estimates t an given i n

- 40 -

terms of the f i b e r and matrix proper t ies and geometry.

models have been given by Lempriere (1969) and Moon and Mow (1970) f o r

spher ica l p a r t i c l e s i n a matrix.

Other mixture

Two comments regarding continuum theor ies of composites a re i n

order before f in i sh ing t h i s sec t ion . First , when the mathematical

s t ruc tu re of these ad hoc continuum models a re examined, one notes a

s imular i ty with t h e axiomatic theor ies discussed e a r l i e r .

of Sun e t a l . compares with Mindlins (1964) mic roe la s t i c i ty theory and

Wu's model compares with Eringen's micromorphic thoery.

Achenbach (1968) have discussed the appl icat ion of Cosserat theory of

continua t o composite mater ia l s . While spec ia l ized , these ad hoc

theor ies however have the advantage of predict ing the e f f e c t i v e

mater ia l constants f o r t he composite i n terms of t he mater ia l constants

of the cons t i tuents . This approach enables the analyst t o quickly

check h i s predicted dispers ion r e s u l t s ,

while i n t h e more general theor ies such confirmation is not b u i l t i n t o

the theory.

Thus the model

Herrmann and

The second remark concerns the usefulness of continuum theor i e s .

While it is remarkable t h a t t h e laminate continuum theory of Sun e t a l .

(1968) checked so very well with exact theory, what is more remarkable

is t h a t i n so fa r as wave propagation is concerned the d i g i t a l computer

was s u f f i c i e n t t o provide t h e exact dispers ion r e l a t i o n .

of ana ly t i c methods not withstanding,continuum models of composites w i l l

c e r t a in ly f ind s t rong competition from computer or ien ted methods (such

as the f i n i t e element method) i n the fu tu re .

The e f f icacy

- 41 -

F. Var ia t iona l Methods f o r Per iodic Composites

When the cons t i tuents o f a composite are arranged i n a per iodic

array,such as a laminated medium o r a f i b e r composite with uniform

spacing, t he displacements and stresses under harmonic waves Can

This problem i s ca l l ed

Thus

be represented by per iodic funct ions.

"Floquet theory" i n t h e subjec t of d i f f e r e n t i a l equations.

t he problem i s reduced t o f inding a so lu t ion i n one c e l l . Such

problems have analogues i n s o l i d s ta te theory of e lec t ron waves

i n per iodic po ten t i a l s . The so lu t ion of the Schroedinger equation f o r these

problems by va r i a t iona l methods has been out l ined by Kohn (1952).

The extenst ion of these methods t o per iodic composites has been made

by Kohn e t a l . (1972) , who applied the theory t o a laminated composite.

Wu (1971) has applied the va r i a t iona l method of t he above authors t o

a wave propagation normal t o a per iodic f i b e r composite mater ia l .

Wheeler and Mura (1972) and TobBn (1971), have looked a t similar

problems.

According t o the Floquet theory, wave l i k e so lu t ions t o the

equations of motion i n a pe r iod ic medium are themselves represented

i n terms of per iodic functions

- i w t i (k-x-ut) = U ($)e

0 = !iy.tf)e

- 42 -

where 8 i s a l a t t i c e vector . If one writes the stresses i n the

form

- i w t = 6. .e t i j ij

the equations of motion become

+ p W2Uk = 0 OkR , R

A statement of a va r i a t iona l theorem is as follows; (Kohn e t a l . ,

The problem of f inding so lu t ions t o the equations of motion i n

terms of t he functions

and s a t i s f y t h e displacement and stress vector cont inui ty conditions across

the c e l l and c e l l const i tuent boundaries, is tantamount t o finding the

s t a t iona ry value o f the funct ional

U (%) which a re per iodic i n the l a t t i c e vectors , 0

' C with respect t o a complete s e t of functions

l a t t i c e vectors , continuous and have continuous f irst der iva t ives i n the cell

(ekL

{U 1 ' L O

which a re per iob , in the

i )* is the s t r a i n tensor and ind ica tes complex conjugate).

A

This theorem allows one t o choose a l i n e a r combination of functions

from {U } t o approximate the wave i n the c e l l . The amplitudes of

each of t he functions a r e chosen so as t o extremize the funct ional 0

I[;] . from which one obtains t h e dispers ion r e l a t i o n between

This procedure leads t o a homogeneous s e t of a lgebra ic equations

w and 5 .

- 43 -

There w i l l be as many branches t o t h e r e l a t i o n

approximating funct ions.

a($.) as the re are

While the method can produce a reasonable approximation t o t h e

d ispers ion r e l a t i o n , t h e stresses i n t h e ce l l may not be as accurate

and lead t o discontinuous stress vec tors a t t h e cons t i tuent boundaries ,

(Kohn e t a l . , 1972) Bevilacqua, Lee (1971). However more general

v a r i a t i o n a l schemes can achieve b e t t e r stress determination as well as

obtaining the d ispers ion r e l a t i o n . (See e .g . Nemat-Nasser (1972)).

Lee (1972) has recent ly reviewed such methods f o r pe r iod ic composites

Krumhansl (1970) has appl ied Floquet theory t o the propagation of t r a n s i e n t

stress pulses i n a layered medium and similar work has appeared by

Krumhansl and Lee (1971).

- 44 -

I V . ATTENUATION AND SCATTERING

Attenuation of a propagating wave represents l o s s o f energy, i n

cont ras t t o d ispers ion i n which the wave energy is conserved but

r ed i s t r ibu ted i n a deformed stress pulse . Loss o f energy during

dynamic motion i n composites can be a t t r i b u t e d t o a t least fou r

phenomena; i ) v i s c o e l a s t i c o r a n e l a s t i c effects o f the cons t i t uen t s ,

i i ) wave s c a t t e r i n g , i i i ) microfracture , i v ) f r i c t i o n between poorly

bonded cons t i t uen t s .

been the concept of constrained l a y e r damping of beams and p l a t e s (see

e.g. Kerwin (1959), Yan (1972). In t h i s appl ica t ion a th ree l a y e r

laminate has a h ighly v i s c o e l a s t i c l aye r constrained by two s t i f f e r

e l a s t i c layers . A continuum theory f o r a v i s c o e l a s t i c laminated

composite has been given by Grot and Achenbach (1970), Biot (1972), as

well as Bedford and Stern (1971) using a continuum mixture theory. The

former work does not t reat waves, whereas Bedford and Stern ca l cu la t e

the a t tenuat ion coe f f i c i en t i n terms of t h e v i s c o e l a s t i c p rope r t i e s f o r

a wave t r a v e l i n g along the l aye r s .

One important use of v i s c o e l a s t i c damping has

A s i n acous t ics , t h e effect of inhomogeneities i n a s o l i d is t o

s c a t t e r energy out o f an inc ident wave. If the re i s some order t o t h e

inhomogeneities e .g . a pe r iod ic a r r ay of f i b e r s o r p a r t i c l e s , t h i s

s c a t t e r e d energy can be r e sca t t e red back i n t o t h e wave ( i . e . dispers ion)

o r r e f l e c t e d back t o t h e wave source. To t h e ex ten t t h a t t he inhomo-

genei ty is random, e las t ic energy w i l l be s c a t t e r e d out of t h e incident

wave thus a t tenuat ing t h e pulse . Thus a mixture of e l a s t i c s o l i d s can

appear i n i ts averaged p rope r t i e s t o be i n e l a s t i c . Krumhansl (1972) has

- 45 -

some general remarks on t

theory of c r y s t a l la t t ices .

randomly heterogeneous e las t ic medium with a plane harmonic wave

incident on it.

67) Knopoff and H

A t low frequencies the sca t t e red energy shows the

familiar Rayleigh dependence on frequency i . e . w2 . constructed a model f o r t he s c a t t e r i n g of waves propagating normal

t o the f i b e r s , when both f i b e r and matrix are elastic, and ind ica tes

the poss ib le exis tence of d i s s ipa t ion i n the composite under dynamic

Mok (1969) has

loadings. Recently Christensen (1972) and WcCoy (1972) have examined

a t tenuat ion due t o s c a t t e r i n g and d isorder i n composites.

Chow and Hermans (1971) have examined the i n t e n s i t y of s ca t t e r ed

waves i n a composite by considering the densi ty and e las t ic constants

t o be random var iab les independent of an ax ia l coordinate. The authors

calcuate the s c a t t e r i n g cross sec t ion (which is a measure of t he energy

of t he sca t t e red fie1d)and f i n d the cross-section proport ional t o

(two dimensional Rayleigh s c a t t e r i n g ) . Theoretical d a t a on the cross-

sec t ion f o r longi tudinal and shear waves propagating i n a g lass f ibe r -

epoxy matrix composite are presented.

w2

Moon and Mow (1970) presented a theo re t i ca l model f o r a t tenuat ion

i n d i l u t e p a r t i c u l a t e composites using the dynamics of a s ing le p a r t i c l e

i n an e las t ic medium. When the inhomogeneities are d i l u t e (volume

f r ac t ion , Vf < . l o ) and random, a f i rs t approximation t o t he calcu-

l a t i o n of s ca t t e r ed energy can o f t en be obtained from the mechanics

of a s ing le s c a t t e r e r , (The d i f f r a c t i o n of e las t ic waves by s ing le

s c a t t e r e r s has b 1971)). When the densi ty

of a r i g i d inclusion p , embedded i n an e l a s t i c matr ix , is g rea t e r 2

- 46 -

t h e equation of t r ans - P > than t h a t of the matrix, i . e .

l a t i o n a l motion o f the sphere U can be found t o be 2

(U-U) = 0 (31) 9 ~ 1 ( z K 3 + 1) 9P 1 [s - %j + ( 2 K 2 + 1)

+ - T

d2 U P -

d t2 ( 2 K 3 + 112 0

where u , i s the average motion o f t h e matrix without t h e inc lus ion

K, is the r a t i o of d i l a t a t i o n a l t o shear speed i n the

mart ix , cL’cs T = a/CL

0

a , radius o f t h e sphere

The form of t h i s equation suggests a mixture theory i n which the

e l a s t i c energy depends on ( U - U ) ~ , and a d i s s ipa t ion funct ion pro-

por t iona l t o ( U - U ) ~ . The dependence on the ve loc i ty U accounts . .

f o r t he r ad ia t ion of e l a s t i c energy when the p a r t i c l e v ib ra t e s i n the

matrix. The dependence on the matr ix ve loc i ty fi accounts f o r

s c a t t e r e d waves i f t he p a r t i c l e were motionless. The r e s u l t i n g

f ’ equations f o r t he p a r t i c u l a t e composite, of volume f r a c t i o n V

were found t o be

a 2~ - y - a2u = P2vf - a 2u

1 a t 2 a 2 2 a t 2 P (1 - Vf) -

- 47 -

where

2p T (2K2 + 2 0 2 0

If the damping were neglected the medium would exhib i t a natura.1

frequency of Q (e.g. p /p = 10, K = 2, a = 1 0 - ~ m , cL = 4 103m/sec.,

R /27r % 2 l o 6 HZ) . pulse (wavelength A >> a , and i n i t i a l i n t e n s i t y 1 , a t Z = 0 ,

0 2 1

For N p a r t i c l e s per un i t volume, a s ine wave 0

0

proport ional t o (au/az)2) , w i l l decay as

-N z I = I e

where

The s c a t t e r i n g cross-sect ion y follows the well known Rayleigh

behavior at low frequencies.

While t h i s model is c l e a r l y l imited i n appl icat ion by the

assumptions made, it serves t o make a simple connection between

a t tenuat ion i n a composite mixture of e l a s t i c so l id s and t h e mechanics

- 48 -

of the ind iv idua l cons t i t uen t s . Further work on a t tenuat ion i n

composites i s needed.

Mow t o d i l u t e f i b e r composites could be made using t h e work of Mow

and Pa0 (1971) on the dynamics of a cy l ind r i ca l inc lus ion i n an

e las t ic matrix.

should be taken i n t o account as was done by Mok,and Chow and

Hermans. In a recent paper Sve (1973) constructs an equivalent

v i s c o e l a s t i c model from t h e s c a t t e r i n g of waves by c a v i t i e s i n a

porous laminated composite.

The extension of t h e model of Moon and

For volume f r ac t ions above lO%,multiple s c a t t e r i n g

The above model f o r s c a t t e r i n g of waves is based on the

i n t e r a c t i o n of harmonic waves of wavelengths long compared with

the s i ze o f t he scat terer .

type have been summarized by Mow and Pao (1971).

encounters a stress wave with a very short rise distance, a wave

f ron t ana lys i s based on ray theory may be more e f f i c i e n t . This

method has been employed by Achenbach e t a l . (1968), (1970) and

by Ting and Lee (1969).

t he stress t o rise from zero t o a given value, i n t h e d is tance of

a f i b e r diameter (< .005 inch o r .1 mm) i s o f t h e order - sec.

Such waves only occur i n shock waves o r i n u l t r a s o n i c pulses of

The analyses of problems of t h i s

When t h e scatterer

One should keep i n mind t h a t t h e time f o r

frequency g r e a t e r than 10 HZ.

- 49 -

V . SHOCK WAVES

The previous discussion has assumed t h a t t he deformation i n

t h e propagating waves was small and t h a t t he material behaved i n

a l i n e a r e l a s t i c manner. Nonlinear e l a s t i c wave analyses i n com-

pos i t e s are few, as e .g . t h a t of Ben-Amoz (1971) who s tudied f i n i t e

amplitude waves i n a f i b e r composite f o r waves along the f i b e r s .

Nor has much work been published t o date on p l a s t i c waves i n composites.

Wlodarczyk (1971) has examined shock waves i n p l a s t i c layered media

with l i n e a r unloading behavior.

e l a s t i c - p l a s t i c solid’has been discussed by Johnson (1972) but was

not applied t o composites.

Calculat ion of plane waves i n an iso t ropic

Shock waves i n composites, however, have received a g rea t

deal o f a t t en t ion . In t h i s class of wave phenomena, t he pressures

i n the s o l i d a re assumed t o be SO high , tha t t he mater ia l can be

t r e a t e d as a hydrodynamic f l u i d .

dev ia to r i c s t r e s s e s a r e assumed t o be small compared with t h e mean

s t r e s s o r pressure.

y i e l d o r e las t ic l i m i t s t r e s s .

This means t h a t t h e shear o r

This occurs f o r pressures much g rea t e r than the

A plane shock wave i s defined as a t h i n p lanar region pro-

pagating r e l a t i v e t o the ma te r i a l , across which the ve loc i ty has a

d iscont inui ty . When t h e medium is homogeneous, cont inui ty and

momentum conditions across the shock sur face y i e l d the following

r e l a t ions between t h e densi ty p , normal p a r t i c l e ve loc i ty v , shock

speed U , and pressure P ,

II p(v-U)II = 0

II pv(v-U)II = - IIPII

- 50 -

When the conditions ahead of t h e wave are such t h a t P = 0 , v = 0 ,

the conditions behind the shock requi re t h a t 0 0

P O v 1

u2 = P / P (1 - 1 1 0

In addi t ion one must prescr ibe a cons t i t u t ive r e l a t i o n f o r t he

pressure and s a t i s f y an energy balance across t h e shock.

Munson and Schuler (1970, 1971) have extended t h i s analysis t o

laminated composites and mechanical mixtures.

neglect the thermodynamics and assume cons t i t u t ive r e l a t i o n s f o r a l l

n cons t i tuents i n t h e composite,

i n a l l t he cons t i tuents t o be equal a t any pos i t ion

In t h e i r model they

Pn = Pn(pn) , and requi re t h e pressures

x, i . e .

Pn(x) = P (x) f o r a l l n 1

Applying t h i s theory t o laminates f o r waves t r a v e l l i n g both along

and normal t o t he l aye r s , Munson and Schuler (1970) conclude t h a t t he

shock speed is independent of t he d i r ec t ion , under c e r t a i n assunytions

on t h e s t r a i n i n each cons t i t uen t . The shock speed obtained has the

form

- 51 -

where a: is t h e i n i t i a l volume f r a c t i o n of t h e nth cons t i tuent

and p: t h e i n i t i a l dens i ty . They a l s o conclude t h a t t h e model

i s not l imi ted t o laminates, and can be used f o r any mechanical

mixture.

The p a r t i c l e ve loc i ty immediately behind t h e shock is assumed

t o be equal i n a l l l aye r s and given by

0 0 anPn v = U ( l - c -

1 "n

Thus when the cons t i t u t ive r e l a t i o n s f o r each cons t i tuent are known,

the shock ve loc i ty can be found as a funct ion of t h e p a r t i c l e speed.

This r e l a t i o n i s ca l l ed a Hugoniot curve.

t h i s model t o a mixture o f AR 0 p a r t i c l e s i n an epoxy matrix and

compared t h e i r calcuat ions with experimental po in ts (Figure 1 3 ) . For

t h i s mixture t h e compressibi l i ty is shown t o behave much l i k e t h e

s o f t e r component.

Munson and Schuler appl ied

2 3

Iden t i ca l r e s u l t s were obtained by Torvik (1970). Tsou and

Chou (1970) used a similar model bu t included t h e thermodynamics i n

the ana lys i s .

r e l a t i o n s f o r a multi-continuum.

Measurements of shock waves and shock Hugoniot curves f o r

Bedford (1971) has reported a theory f o r Hugoniot

quartz-phenolic have been performed by I s b e l l e t a l . (1967), Charest

and J e n r e t t e (1969) , and Munson e t a l . (1971). Studies i n shock waves

i n aluminum -polymethyl methacrylate (PMMA) have been reported by

Barker and Hollenbach (1970), and Schuler ( to appear), and Schuler

and Walsh ( t o appear). Other references t o the study o f shock waves i n

composites include Gary and Kirsch (1971) and Holmes and Tsou (1972).

- 52 -

The independence of the shock speed on the d i r ec t ion of pro-

pagation i n an e l a s t i c a l l y an iso t ropic composite can only hold

a t high pressures .

ind ica ted t h a t such dependence on d i r ec t ion has been observed f o r

Munson and Schuler (pr iva te communication) have

some composite systems below pressures of 6 k i loba r s .

The construct ion of t h e o r e t i c a l models i n the region between

e l a s t i c wave theory and hydrodynamic shock model w i l l present a

grea t challenge t o the ana lys t .

- 53 -

VI. EXPERIMENTS

The generation and measu

materials has , i n general , been base

i n applied science used t o study waves i n s o l i d s .

the use of a i r g m s , explosive charges, exploding f o i l f l y e r p l a t e s ,

shock tubes and p i ezoe lec t r i c u l t r a son ic generators. To measure the

These involve

stress waves , s t r a i n gages, p i ezoe lec t r i c c r y s t a l s , capacitance gages,

op t i ca l interferometer , holographic and pholoe las t ic techniques a re

used. Experimental work i n t h i s area, while not as copious as the

theo re t i ca l e f f o r t s , has provided a steady stream of experimental

da ta with which t o check the mathematical models. For a va r i e ty of

mater ia l s , including f i b e r , laminate and woven f i b e r composite, da ta

has been reported on measured wave speeds, a t tenuat ion and dispers ion

of s t r e s s pulses , shock wave behavior, s t r e s s wave induced f r ac tu re

and impact.

The experiments can be categorized by the type o f stress pulse

used.

time, h a l f s ine l i k e pulses induced by p ro jec t ib l e impact, t o shor t

rise time waves induced by explosive f l y e r p l a t e impact. The Fourier

content of these pulses have a la rge zero frequency component.

sonic o r pulsed s ine wave tests have a narrow spectrum centered about

The monotonic compressional pulse has been used,from long rise

Ultra-

a p a r t i c u l a r freque

idea l ly s u i t e d t o map the dispers ion r e l a t i o n d i rec t ly ,by measuring

la t te r waves a re t

v e l o c i t i e s of t he pulses , whil notonic pulse method

d se shape with

- 54 -

passage through t h e material.

One of t he problems associated with using pulsed s i n e waves is

measuring a wave ve loc i ty .

(see e.g. Br i l l ou in , 1960), t he shape o r envelope of t h e pulsed s i n e

wave t r a v e l s a t t h e group ve loc i ty of t h e spectrum center frequency and

As has been pointed out i n many t e x t s

is not equal t o t h e phase ve loc i ty v = w/k when dispers ion i s

present . The v e l o c i t i e s a re r e l a t e d however P --

Bri l lou in a l so descr ibes two o ther v e l o c i t i e s , t h e wave f r o n t ve loc i ty

and the s igna l ve loc i ty . The l a t t e r is associated with t h e f i rs t

a r r i v a l of s igna l s with t h e spectrum center frequency.

ve loc i ty is sometimes equal t o the group ve loc i ty (Br i l lou in , 1960).

The message however is c l e a r ; carefu l de f in i t i on and i n t e r p r e t a t i o n

of u l t r a son ic wave ve loc i ty measurements a re required i n order t o

construct t h e dispers ion r e l a t i o n w(k).

The s i g n a l

Abbott and Broutman (1966) demonstrated t h e use o f a

monotonic pulse t o measure the equivalent e las t ic constants o f steel/

g lass and "S" glass/epoxy composites. This method is v a l i d as long

as the stress rise length and t o t a l pulse length a r e la rge compared

- 55 -

with t h e s i z e of t h e f i b e r s , f i b e r spacing and t h e t ransverse s t r u c t u r a l

dimensions of t h e specimen (e.g. rod diameter o r plate thickness) .

Potapov (1966) used pulsed ultrasound t o measure the e l a s t i c constants

of f ibe rg la s s p l a t e s . He concluded t h a t or thot ropic e l a s t i c i t y gave

a s u f f i c i e n t l y accurate descr ip t ion of t he e las t ic proper t ies s o de te r -

mined by these tests. Markham (1970) a l so used pulsed ultrasound i n

an u l t r a son ic tank t o measure t h e e l a s t i c constants of a carbon f i b e r

epoxy resin composite.

Tauchert and Moon (1970) used the monotonic pulse method and

compared t h e r e s u l t s with da ta from resonance tests and s t a t i c

moduli.

epoxy and glass/epoxy were within 2% f o r waves along the f i b e r s .

was found t h a t t h e wave a t tenuat ion could be predicted from vibra t ion

resonance t 'ests of t h e mater ia l s . Tauchert (1971a, 1971b) has used

The dynamically and s t a t i c a l l y determined moduli f o r boron/

I t

u l t r a son ic waves t o measure a l l t h e e l a s t i c constants of a va r i e ty

of composites. Tauchert (1972) has a l so measured u l t r a son ic a t tenuat ion

i n composites and observed increases i n damping due t o i n i t i a l t e n s i l e

s t r e s s .

Pot t inger (1970) used a s i m i l a r method i n glass/epoxy and boron/

aluminum and found agreement between s t a t i c a l l y and dynamically

determined moduli t o within 3% f o r waves i n bars a t various angles

t o the f i b e r s . Also N e v i l l , Sierakowski e t a l .

method on steel/epoxy bars with waves along t h e s t e e l f i b e r d i r ec t ion .

The increase i n wave speed with volume f r ac t ion of s t e e l checked very

(1972) used the same

' c lose ly the r u l e of mixtures (Figure14 ) . The at tenuat ion was found

t o decrease with increase i n s t e e l .

r e f l e c t i o n from a free end were found t o propagate a t a s l i g h t l y

Tensi le waves generated on

- 56 -

s tud ied by Ross e t . a l . (1972). Other s t u d i e s i n which e 1 as t i c constants

of composite mat e r i a1 s

Tuong (1970) and Cost and Z i m m e r (1970a). Elast ic constants of f i l l e d elastomers

were determined using u l t r a son ic s by Waterman (1966) and showed t h e e f f e c t s of

temperature and percent f i l l e r on the p rope r t i e s of the two phase ma te r i a l s .

Also White and Van Vlack (1970) have used an acous t ic resonance technique t o

determine the p rope r t i e s of open-pore polymer foams with higher-moduli i n f i l t r a t i n g

mat r ices ,

Using a gas dynamic shock (70 p s i ) t o induce a s h o r t use time pu l se ,

Whi t t ie r and Peck (1969) s tud ied the e f f e c t s of d i spers ion i n graphi te and

boron re inforced carbon phenolic composites with t h e wave i n the f i b e r

d i r ec t ions .

wave f r o n t , overshoot, and o s c i l l a t i o n s i n t h e s t r e s s p l a t eau region, which

checked t h e p red ic t ion of Peck and Gurtman (1969).

described i n a paper by Cummerford and Whi t t ie r (1970).

Drumheller (1971a) performed a similar experiment on a laminated composite

The t ransmi t ted pulse showed a smoothed pulse r i s e i n p l ace of t he

This technique is

Lundergan and

of steel and epoxy. They used a f l y e r p l a t e technique t o generate compressional

re again observed.

- 57 -

In another work Lundergan and Drumheller (1971b) s tud ied t h e

impact of an obl iquely laminated composite of steel and polymethyl

methacrylate (PMMA).

responding t o a quasi longi tudina l and a quasi shear pulse , t h e

experiments showed a th ree s t e p s loping wave.

t h a t f u r t h e r work is needed t o explain the discrepancy.

Although theory p red ic t s a two s t e p wave cor-

The authors conclude

e

With somewhat d i f f e r e n t motives Schuster and Reed (1969) used

a f l y e r p l a t e technique t o generate shock waves i n a boron/aluminum

composite a t pressures up t o 76 kbar and impact durat ion of less than

0 . 2 micro sec. The impact ve loc i ty of t he f l y e r plates were increased u n t i l

damage occurred. Increased f i b e r crushing with impact ve loc i ty was

observed and t h e spa l l i ng ve loc i ty was measured f o r aluminum and two

boron/aluminum composites. The s p a l l ve loc i ty f o r t he plasma sprayed,

d i f fus ion bonded composite showed a three f o l d increase i n ve loc i ty

over the s p a l l ve loc i ty f o r t h e aluminum specimens, while t h e plasma

sprayed, brazed composite showed a s l i g h t decrease i n the s p a l l

ve loc i ty compared with aluminum. This dramatic e f f e c t i s a t t r i b u t e d

t o the two d i f f e r e n t geometrical arrangements of t he f i b e r s produced

during f ab r i ca t ion . A s shown i n Figure 15b, i n the d i f fus ion bonded

specimens the f i b e r s a re not touching and hence are able t o a t tenuate

the shock wave by mult iple s c a t t e r i n g . In t h e brazed specimen (Figure

15a) one can see t h a t the f i b e r s a re contacting i n t h e d i r ec t ion of

t he wave.

of an increase i n a t tenuat ion , r e s u l t i n g i n a s p a l l ve loc i ty no

Thus a boron path is created through the medium with less

g rea t e r than t h a t f o r aluminum.

One may conclude from t h i s experiment t h a t t h e f i b e r geometry

w i l l be an increasingly important f a c t o r i n stress wave f a i l u r e as

the stress r i s e d is tance o r pulse length approaches the f i b e r

- 58 -

dimensions. In the experime

and spacing are about 0.1 mm and

i n length i n the aluminum.

Several important papers have examined t h e d ispers ive n o f

composites d i r e c t l y with t h e use of u l t r a s o n i c waves.

(1968) , demonstrated a decrease i n phase v e l

as predic ted by seGeral t h e o r i e s (Peck and Gurtman, 1969) f o r waves

along the f i b e r s of thorne l and Boron reinforced carbon-phenolic composites.

Asay e t a l .

t y with f r e q

The

phase ve loc i ty with frequency out t o 4 MHZ but a change i n t h e r e i n -

forced specimens of Av/v % 0.20 a t 3 MHZ (Figure 16 ) .

carbon-phenolic without t h e f i b e r s showed no change i n

Tauchert and Guzelsu (1972) performed similar experiments on boron/

epoxy and examined a v a r i e t y of wave normal-fiber o r i e n t a t i o n s , f o r

both longi tudina l and shear waves. In addi t ion t o t h e decrease i n

group ve loc i ty of longi tudina l with frequency both across and along

the f i b e r s , shear waves t r ave l ing along t h e f i b e r s and polar ized

normal t o t h e f i b e r s showed a 25% increase i n group ve loc i ty a t about

1 MHZ (Figure 17) . The wavelength i n epoxy at t h i s frequency i s

about 2.6 mm compared with a f i b e r diameter and spacing d is tance o f

about 0 .1 mm.

(1968) i n an e a r l y t h e o r e t i c a l model and by o the r authors (e.g. Sun

This behavior had been predic ted by Achenbach and Herrmann

e t a l . , 1968) n l a te r works. ear waves p r ga t ing across t h e fibers

showed a s l i g h t decrease i n group ve loc i ty with frequency.

e (1972) performed a similar u l t r a s o n i c experi-

s i n e waves on tungsten

They a l s o claim t o have observed a cu tof f band i n frequency as w e l l as

- 59 -

t he second branch of t h e dispers ion r e l a t i o n (opt ica l branch) (Figure 18) .

In addi t ion , a frequency s h i f t i n t ransmit ted pulse lower than t h e in -

c ident pulse frequency w a s observed by these authors as t h e frequency

approached t h e cutoff region. This i s a t t r i b u t e d t o the f i l t e r i n g of

t he higher frequency components i n the pulse lying i n t h e s top bond of

frequencies, e f f ec t ive ly s h i f t i n g t h e observed frequency o f t h e t r ans -

mit ted pulse . I t should be noted t h a t Sutherland and Lingle (1972) claim

t o have measured t h e phase ve loc i ty i n t h e i r r epor t .

d e f i n i t i o n o f t h e ve loc i ty measurement i s lacking and the present

author suspects t h a t t he da ta represent group v e l o c i t i e s .

However prec ise

Rowlands and Daniel (19 72) have used in te r fe rometr ic holography

t o measure t h e t ransverse displacement i n v ibra t ing laminated aniso-

t r o p i c p l a t e s ,

dimensional waves i n an iso t ropic p l a t e s due t o t r ans i en t impact loads.

Dally, Link and Prabhakaran (1971) were able t o observe two

This method may hold some promise f o r observing two

dimensional waves i n or thot ropic f i b e r reinforced p l a t e s using photo-

e l a s t i c i t y . This development was made possible by t h e development

of or thot ropic-b i re f r ingent mater ia ls which were s u f f i c i e n t l y t r ans -

parent f o r photoe las t ic analysis (see Prabhakaran, 1970). In t h i s

study the authors examined both t h e t r ans i en t loading of a h a l f plane

with edge loading and t h e f u l l plane problem with a hole loaded with

an explosive charge of lead ozide,as shown i n Figure 19.

nature of t h e stress wave propagation is c l e a r from the f igu re (moduli

r a t i o EL/ET % 3 . 0 ) . A cusp-like f r inge seen i n Figure 19 might

represent t h e e f f e c t s of shear wave anisotropy.

The an iso t ropic

By measuring t h e

- 60 -

wave sur face of t h e ou te r f r inge t h e authors were ab le t o recons t ruc t

t he ve loc i ty sur face f o r t h e quas i longi tudina l wave of t h e p l a t e

material. This ve loc i ty was within 10% of t h a t determined from t h e

s t a t i c e f f e c t i v e moduli of t h e p l a t e material.

Another pho toe la s t i c s tudy of stress waves is repopted by

Hunter (1970) , who used an explosive s t r i p along t h e specimen edge.

Using a l t e r n a t i n g l aye r s of d i f f e r e n t material, f r inge pa t t e rns

accompanying a plane wave i n t h e layered d i r ec t ion were observed.

Rose and Chow (1971) used a similar method t o observe the build-up

of a s teady wave f r o n t i n a a l t e r n a t e l y layered composite of d i f f e ren t

pho toe la s t i c materials.

Other experimental work on t h e propagation of waves i n composites

includes Benson e t a l . (1970) , Berkowitz and Gurtman (1970) , Berkowitz

and Cohen (1970), Lord (1972). Cohen and Berkowitz ( i n press ) have

s tudied dynamic f r a c t u r e i n composites due t o stress waves-

Sierakowski e t . a l . (1970a, 1970b) have measured dynamic stress s t r a i n -1 r e l a t i o n s f o r var ious composites and s t r a i n rates up t o l o 3 sec

Up t o a 85%

these rates (Figure 20) .

. increase i n u l t imate f a i l u r e stress was observed a t

- 61 -

VII. IMPACT PROBLEMS I N COMPOSITES

A. Introduct ion

The study of impact of i s o t r o p i c so l id s has a la rge l i t e r a t u r e ,

p a r t of which is documented i n the book by Goldsmith (1960).

study of s imi l a r problems f o r composite s t ruc tu res has received

very l i t t l e a t t en t ion a t t h i s wr i t ing .

include the nonelas t ic and nonlinear aspects of t he problem, s ince

the object of such s tud ie s usual ly concerns t h e pred ic t ion o r avoidance

of f a i l u r e due t o impact.

r a t e s of s t r a i n become important, and the inhomogeneity and the

anisotropy i n composite mater ia ls i n v i t e a wider s e t of f r ac tu re o r

f a i l u r e modes. Impact f a i l u r e modes i n i so t rop ic materials include

indentat ion, spa l l i ng , and penetrat ion of t h e p r o j e c t i l e through t h e

s t ruc tu re .

pu l lou t , s p l i t t i n g , and delamination (see Figure 21).

damage, micro f a i l u r e i n the composite could Droduce a loca l stress

r i s e r , change the na tu ra l frequencies , and decrease the fa t igue l i f e .

The

To be real is t ic , one should

Also t h e mater ia l p roner t ies under high

'In composites one must add t o t h i s l ist f i b e r crushing,f iber

Even with no v i s i b l e

Both empirical and ana ly t i c s tud ie s have been made but r a r e l y

a re theory and experiment in tegra ted . Much of t h i s work has been

motivated by the need f o r b i r d and ha i l s tone impact pro tec t ion of

j e t engine composite fan blades.

conducted a t great cos t , have produced r e s u l t s i n t h e form of leading

edge pro tec t ion schemes and in te r leaving steel wire mesh between the

p lys , (Anon, 1971), while ana ly t i ca l s tud ie s have only begun t o explore

the problem (e.g. Moon, 1972).

Empirical s tud ie s of t h i s problem,

- 62 -

One of t h e f irst areas o f i n t e r e s t i n b i r d imnact problems was t h e

design o f a i rcraf t t ransparent s t r u c t u r e s such as windows and wind sh ie lds

t o resist b i r d s t r i k e s . One such discussion is given by McNaughton (1964).

A summary of tes ts i n t h i s a r t i c l e on vinyl sandwich panels reveals t h a t

the pene t ra t ion ve loc i ty V,

wind screens decreased as the cube roo t of the mass of t he b i r d M;

f o r a given set o f s t r u c t u r a l conditions

of one t o e igh t pound b i r d s on a i rcraf t

!W3 = constant .

Research r e l a t e d t o b i r d damage i n a i rc raf t engines has been reported

by Allock and Col l in (1968). Impact by chicken carcases, wax, wood and

ge la t ine dummies have been inves t iga ted f o r t a r g e t shapes resembling b a s i c

geometries. The authors constructed a momentum t r a n s f e r model f o r t he

average impact force F due t o a spher ica l impactor

MV sin20 A t F =

where M i s b i r d mass, V b i r d ve loc i ty , 8 angle o f de f l ec t ion from

l i n e o f f l i g h t ,

diameter t o ve loc i ty . In terms o f b i r d dens i ty p

A t t he durat ion of impact given by r a t i o of p r o j e c t i l e

Measurements showed t h a t t h e assumed impact time was too long and t h e

t h e o r e t i c a l force too low.

round nose t a r g e t s e las t ica l lymounted showed t h a t t h e t a r g e t de f l ec t ion

was proport ional t o t h e b i r d momentum.

Deflections of t h e f l a t p l a t e kn i f e edge and

- 63 -

Research on h a i l impact damage t o t y p i c a l aircraft s t r u c t u r e s

has been presented by Hayduk (1973).

mental and an ana ly t i ca l model €or denting type h a i l damage i n

aluminum fuselage panels o r dome segments (spherical cap).

Comparison i s made of experi-

The range of s t r u c t u r a l impact problems includes o the r phenomena

besides b i r d and ha i l s tone impacts. These include micrometeorite

damage on spacecraf t , dus t , sand and r a i n erosion, and cav i t a t ion

erosion of s o l i d s which involves dynamic stresses due t o col lapsing

bubbles. A discussion of impact erosion by dust p a r t i c l e s f o r metal

sur faces is given by Smeltzer e t . a l . (1970). The mechanics of a

l i q u i d drop impact with a s o l i d surface has been given by Heyman (1969)

and Peterson (1972). Rain erosion o f composites is reported by Schmitt

(1970). B a l l i s t i c problems of high ve loc i ty penetrat ions of p l a s t i c -

aluminum laminates by steel p r o j e c t i l e s have been analyzed by Kreyenhagen

e t . a l . (1970) using numerical computer modes which i l l u s t r a t e severa l

damage modes.

The t e s t i n g of composite mater ia l s under impact forces encompasses

a v a r i e t y o f load and specimen condi t ions.

impact tests use r e l a t i v e l y small beam-like specimens, ( l e s s than 3

inches long) under a t ransverse poin t force . The durat ion of t he

impact is usual ly long compared with the time of a s t r e s s wave t o

t r ave r se the specimen. For example, using a wave speed of v % 5mm/psec,

a length L = SOmm, and pulse time

number

during such tests.

compressive stress pulses of extremely shor t durat ion, (% 0.2 10-6sec)

i n t h i n specimens (%2 nun th ick) are used producing a nondimensional

Classical Izod and Charpy

T = 10-3sec, t he nondimensional

VT/L = l o 2 is a measure of t he number o f r e f l e c t i o n s occurring

In shock wave impact t e s t i n g of composites, high

number, (using t h e same wave speed as above), V T / L % .5 . Also the NASA

- 64 -

and t h e A i r Force i n t h e United S ta tes a re sponsoring b a l l i s t i c impact

tests on composite p l a t e s , as well as f u l l s i z e j e t engine fan blades,

using ha i l s tones and l i q u i d objects t o simulate b i r d impact.

Almond e t a l . (1969) have reviewed t h e l i t e r a t u r e on c l a s s i ca l

impact t e s t i n g on laminated composites. Embury e t a l . (1967) conducted

Charpy V-notch tests on s o f t so lder laminated s t e e l specimens both

with the impact force normal t o t h e laminated surface (crack a r r e s t e r

configuration) and p a r a l l e l t o t h e laminate surfaces (crack d iv ide r

configurat ion) . In the l a t t e r case t h e d u c t i l e - b r i t t l e t r a n s i t i o n

temperature was reduced and t h e specimens showed higher impact energy

absorption over homogeneous s t e e l specimens.

Also Chamis e t a l . (1971) have performed miniature-Izod impact

t e s t s

r e s i n matrix (specimen s ize 7.9x7.9x37.6 mm). The t e s t s included

on f i b e r composites of glass and graphi te f i b e r s i n an epoxy

specimens with the f i b e r s e i t h e r p a r a l l e l o r t ransverse t o t h e cant i lever

longi tudinal ax is . The t e s t s show f a i l u r e modes of cleavage, cleavage

with f i b e r pu l lout , and cleavage with delamination. In t h e t ransverse

mode t h e cleavage included matrix f r ac tu re , f i b e r debonding and

f i b e r s p l i t t i n g .

by these authors t o be cor re la ted with the intralaminar shear s t rength

The t ransverse impact s t rength was found

of the various composites t e s t e d .

In s i m i l a r work, Novak and DeCrescente (1972) repor t t h e r e s u l t s

of Charpy impact t e s t s f o r un id i rec t iona l graphi te , boron, and g l a s s

f i b e r s i n a r e s i n matrix. They conclude t h a t t h e toughness of t he

r e s i n matrix i s not an important f a c t o r i n impact energy absorption.

"S glass" composites showed a higher impact s t rength than boron/resin

- 65 -

and graphi te / res in composites.

mechanisms such as fi lament pu l lou t , shear delamination, ect .

conclude t h a t t he impact s t rength is cor re la ted w i

They a l so evaluate t h e energy-absorption

They

f i b e r s t r e s s - s t r a i n curve.

In a recent paper Peck (1972) has reviewed t h e l i terature on

s p a l l f r ac tu re i n composites using one dimensional shock waves. In

addi t ion t o the work of Schuster and Reed (1969) on f i b e r composi

discussed above, Warnica and Charest (1967) have used

/?

1-2 vsec compres-

s ion pulses on laminated quartz phenolic t o determine s p a l l s t r e s s

thresholds . Similar work by Cohen and Berkowitz (1972) , and Barbee

e t a l . (1970) a re a l so discussed.

I t i s useful t o compare t h e merits of these d i f f e r e n t tests.

s p a l l tests and dynamic s t r e s s - s t r a i n t e s t s (e.g. Sierakowski, e t a l . ,

1971) the stress waves a re one dimensional. Thus, c l e a r l y defined

s t r e s s states are used t o measure t h e mater ia l s t rength proper t ies .

However, i n foreign objec t damage, t h e conditions of impact f a i l u r e

involve the contact o f b lunt ob jec ts with a sur face , thus producing

a complicated stress s t a t e . Izod and Charpy t e s t s appear t o simulate

ac tua l impact, s ince a knife edge on a pendulum encounters a beam-

l i k e specimen. S i m i l a r l y , b a l l i s t i c t e s t s involve a loca l ly inhomogeneous

stress state i n the region of p r o j e c t i l e contact which is found i n

ac tua l impact problems.

In

However, t h e ad hoc nature of these s t r e s s

states does not allow comparison with o the r t e s t s . Thus Izod and

Charpy r a t i n g s of ten cannot be compared. Also, because of t h e small

s ize of t h e specimens and the long contact time (e.g. . ~ 1 0 - ~ s e c ) many

r e f l ec t ions occur du t h e impact thus obscuring the wave l i k e

nature of impact, which might be present i n a lar specimen o r

i n the actual s t r u c t u r e . In b a l l i s t i c tests, however, contact times

of 5 l o m 5 s e c o r less a re obtained f o r high ve loc i ty p r o j e c t i l e s of t h e

- 66 -

order of one inch diameter.

of a p l a t e would r e s u l t i n the t o t a l energy of impact contained i n

a c i r c l e of radius l e s s than 30 cm.

s t ruc tu re fewer r e f l ec t ions might ob ta in than f o r a small t e s t

specimen. I t i s the opinion o f t h i s w r i t e r t h a t s ca l e e f f e c t s are

For a wave speed of 6 mm/Msec the impact

I f t h e p l a t e is p a r t of a la rge

of importance i n impact t e s t s .

l a t ed t o l a rge r s t ruc tu res t h e nondimensional numbers

Thus i f tes t da ta is t o be extrapo-

where T is t h e contact time, v a wave speed, and L a repre-

sen ta t ive length should be matched i n addi t ion t o o ther va r i ab le s .

B. Analytical Models f o r Impact

The t o t a l problem descr ip t ion involves the loca l deformation

a t the impact s i t e and t h e simultaneous determination of t h e motion

of the s t r u c t u r e during and a f t e r impact.

of the s t r u c t u r e takes place over a time period much l a rge r than t h e

impact contact time, and t h e s i z e of t he impactor i s much smaller

than the s t r u c t u r a l dimensions, t he problem may be s p l i t i n t o two

d i s t i n c t p a r t s .

h a l f space, 11) The response o f t he s t ruc tu re t o a prescr ibed loca l

impact force as determined i n Par t I . The e r r o r s involved i n such

a scheme appear t o be on t h e conservative s ide s ince t h e procedure

w i l l underestimate t h e contact time and overestimate t h e contact force

When the overa l l motion

I ) The loca l mechanics of impact with a deformable

- 67 -

(Goldsmith, 1960).

Discussed i n the next s ec t ion i s the impact o f s o l i d objec ts on

s o l i d sur faces . As already mentioned l i q u i d o r r a i n drop impact

erosion is a l s o an important problem.

v e l o c i t i e s , s o l i d s may be t r e a t e d by a hydrodynamic model and,,liquid

drop model may be use fu l .

1.

For s u f f i c i e n t l y high impact a

Impact of a Half Space-Hertz Theory

The problem of an impulsive l i n e force on an an iso t ropic ha l f

space has been given by Kraut (1963) f o r a t ransverse ly e l a s t i c

i s o t r o p i c material. In p a r t i c u l a r , a l i n e source on the surface

normal t o the symmetry ax i s produces two wave sur faces as shown

i n Figure 22 corresponding t o the wave surfaces ,discussed i n an

e a r l i e r s ec t ion .

another e l a s t i c body has not been given t o da te . The waves generated

during a point impact on an i s o t r o p i c h a l f space have been s tudied by

Pekeris (1955) where it was shown t h a t on the surface la rge s t r e s s e s

propagate a t the Rayleigh sur face wave speed.

an e l a s t i c sphere h i t t i n g an e las t ic h a l f space a re not known.

The extension of t h i s work t o dynamical contact with

But t he dynamics o f

Thus without even considering a n e l a s t i c e f f e c t s , t he ana ly t i c l i t e r a t u r e

on dynamic impact i s l imi ted even f o r i so t rop ic materials. Instead,

what has been used is a q u a s i - s t a t i c t h e o r e t i c a l model ca l l ed t h e Hertz theory

(Goldsmith, 1960). This i s based on the s ta t ic deformation produced by

a poin t force on a sur face . When the force , F , i s between

a sphere o f radius R , and a h a l f space, F

- 68 -

is r e l a t e d t o t h e r e l a t i v e approach of sphere and h a l f space, c1 , by

r 1.140 MVg F = s i n 1.068 V o t t 2 -r

01 m

F = 0 , t 7 - r

i

3 / 2 F = K ~

where

( 3 5 )

(v i s Poisson's r a t i o , and E is Young's modulus).

This r e l a t i o n i s nonl inear s ince the contact a r ea na2 depends

on the force .

Equating t h i s force t o t h e change o f momentum o f a sphere during

impact with i n i t i a l ve loc i ty

pressions f o r t he contact time and force h i s t o r y ,

Vo , t h i s theory gives t h e following ex-

2 . 94am a =

vO

(36)

137)

- 69 -

where c1 i s t h e maximum approach, and M i s the mass of t h e sphere. m Extension of t h e Hertz theory of impact t o an iso t ropic bodies has

been made by Chen (1969) and Willis (1966). The contact region has been

shown by Willis t o be e l l i p t i c f o r an an iso t ropic h a l f space i n cont ras t

t o a c i r c l e f o r t he i s o t r o p i c case.

r e l a t i o n similar t o Eq. (35), where K depends i n a complicated way on t h e

e l a s t i c constants .

done numerically and no examples have been given t o date f o r t yp ica l

composite anisotropy.

However he obtains a force def lec t ion

The determination of the e l l i p s e parameters must be

A simple model f o r es t imat ing the contact time f o r i s o t r o p i c spheres

on composites has been suggested by the author Moon (1972d), which assumes

a c i r c u l a r contact a r ea .

on unid i rec t iona l f i b e r composite p l a t e s , with the f i b e r s p a r a l l e l t o

the sur face , show t h e contact area t o be e l l i p i c a l with t h e la rge axis

normal t o the f i b e r s , bu t only s l i g h t l y deviat ing from a c i r c l e .

Experiments on the contact of a s t e e l sphere

Thus i n Eq . (35 t he ha l f space constants (1 - v2)/E a re replaced by a

t ransverse e l a s t i c constant f o r t h e composite. &en* has suggested

using the compliance S where the "3" axis is normal t o the

sur face . The author has used 1 / C t o replace (1 - v2)/E i n

Figure 2 3 t o es t imate the contact time f o r ha i l s tones and g ran i t e

spheres on 55% graphi te f i b e r i n epoxy. For impact speeds i n the

range 100-500 m/sec t h e contact times range from 15-85 usec.

33

3 3

In summary these formulae revea l the following dependence of contact

time and peak pressure on impact ve loc i ty

* Pr iva te communication

- 70 -

Such r e s u l t s , however, should only be used as guidel ines , s ince t h e

theory uses assumptions which break down a t high v e l o c i t i e s . Goldsmith

(1960) has made the following summary of t he l i e r t z theory of contact .

1) A t high v e l o c i t i e s t h e H e r t z contact time is a lower bound on the

contact time.

2) When a sphere s t r i k e s a beam, t h e motion of t h e beam decreases the

force , bu t t h e contact time remains about t h e same.

In another reference Goldsmith and Lyman (1960) have shown t h e Hertz

theory t o be remarkably v a l i d in so fa r as contact time and peak force

f o r the impact of hard s t e e l spheres

s t e e l sur face f o r v e l o c i t i e s up t o 300 f t / sec ( ~ 9 1 . 5 m/sec) . The da ta

i n Figure 2 3 for graphi te epoxy can only be used as a rough guide f o r

contact times, u n t i l experimental d a t a becomes ava i lab le .

2 . Non Hertzian Impact

(1/2 inch diameter) onto a hard

3 / 2 The H e r t z theory of impact rests on the contact law F = ~a

For boron/aluminum and graphi te fiber/epoxy composite p l a t e s t h i s force

law was t e s t e d under 1/4 inch and 3/8 inch s t e e l b a l l s i n a s t a t i c

t e s t i n g machine. The preliminary r e s u l t s i n Figures 24925 show clearly

t h a t a more general l a w i s required and t h a t f o r moderate forces ( l e s s

than 100 l b f ) t h e deformation is i n e l a s t i c , requi r ing a d i f f e r e n t law

f o r approach and rebound.

A more general contact law was given by Meyer (see Goldsmith, 1960)

- 71 -

If such a l a w holds f o r both approach and rebound, formulas similar

t o the H e r t z theory can be obtained, (see Goldsmith, 1960, p . 91) .

Clear ly the s ta te o f knowledge about t he impact of composite o r

inhomogeneous bodies i s unsa t i s f ac to ry .

good q u a s i - s t a t i c theory which can account f o r a n e l a s t i c effects , a

t r u l y dynamical impact model f o r composites i s needed.

In addi t ion t o the lack of a

For i s o t r o p i c mater ia l s computer codes employing f i n i t e d i f -

ference methods have been developed f o r dynamical impact and penet ra t ion

p r o j e c t i l e s and deformable bodies , (e. g. Wilkins , 1969, Kreyenhagen e t . a l . 1970) . These models apply a n e l a s t i c cons t i t u t ive equations and can p red ic t

permanent deformation.

no doubt be ava i l ab le i n the near fu tu re as w e l l as codes based on f i n i t e

element methods. H wever t h e r e i s a need f o r ana ly t i ca l so lu t ion f o r

impact phenomena; f i r s t f o r t h e i r s impl i c i ty and a c c e s s i b i l i t y t o the

designer , and second t o check the computer codes which w i l l c e r t a in ly

appear i n the near f u t u r e .

The extension of these codes t o composites w i l l

In developing a n a l y t i c a l models f o r impact, t h e use of an equi-

va len t an i so t rop ic mater ia l i s quest ionable i f one des i res t o explain

stresses i n t h e contact region. When a composite material i s indented

by another body of convex sur face the a rea of contact goes t o zero as

the contact pressure decreases. Thus f o r small forces t h i s a rea i s

necessa r i ly o f t h e order of t h e dimensions of t h e f i b e r s o r lamina.

One would expect a force-def lec t ion l a w t o exh ib i t pe r iod ic changes

i n s lope as t h e contact area engages each successive f i b e r (Figure 25) .

- 72 -

This i s t h e t e n t a t i v e explanation f o r t he wav ss i n t h e experi-

mental r e l a t i o n shown i n Figure 25 f o r boron f i b e r s

matrix. The pe r iod ic p l a t eau appear t o occur a t de f l ec t ions c

responding t o contact r a d i i d i f f e r i n g by t h e f i b e r spacing (%.004

inches) .

C. S t ruc tu ra l Response t o Impact

1. The Coupled Problem

Further experimental work on t h i s problem i s needed.

When the impact force and durat ion depend on t h e s t r u c t u r a l

motion the above procedure cannot be used. The coupled response

of an i s o t r o p i c p l a t e and a spher ica l impactor was t e a t e d by Eringen

(1953) and o the r s , (see Goldsmith, 1960) . Conceptually t h e extension

t o composite s t r u c t u r e s is similar. Let two coordinate systems be

embedded i n the two bodies (see Figure 26) and le t the axes x , x ' 3 3

be d i r ec t ed i n t o the sur faces of s t r u c t u r e and impactor respec t ive ly .

Relat ive t o these coordinates w , w ' represent sur face de f l ec t ions , 3 3

W , t he de f l ec t ion o f t h e p l a t e o r s h e l l neu t r a l surface, and W f 3 3

t h e displacement of t h e impactor center of mass.

shapes of both s t r u c t u r e and impactor are given by

If t h e sur face

x = S(x ,x ), 3 1 2

x '= S ' (x ' ,x ' ) 3 1 2

then t h e boundary condi t ion t o be s a t i s f i e d over

the contact region i s

on x3, x; = 0

The def lec t ions w w ' are determined from a th ree dimensional 3' 3

ana lys i s , such as a Hertz ana lys i s , (as e .g . Willis, 1966). The

- 73 -

displacement W

theory, while t h e impactor displacement W' is governed by Newtons

law f o r t he body under i n i t i a l condi t ions

i s governed by a two dimensional p l a t e o r s h e l l 3

3

5 W' = 0, - - - V , a t t = O 0 d t 3

The so lu t ion o f such a problem f o r a composite s t r u c t u r e i s not

known t o t h e author , though the problem seems f a i r l y s t ra ightforward.

2 . Transient Load Problems

There has been, however, a number of s tud ie s made of t he response

of a composite body t o sho r t durat ion o r impact-l ike forces . Already

mentioned i s t h e work of Peck and Gurtman (1969) on the response of a

laminated h a l f space t o a compressive stress on t h e sur face i n t h e

d i r ec t ion of t h e layer ing. Sve (1972) has a l so t r e a t e d t h e laminated

h a l f space under impulsive hea t ing of t he sur face , (e.g. from a l a s e r ) ,

with thermoelast ic coupling.

theory of Sun e t a l . (1968). In another work Sve and Whit t ier (1970)

have appl ied t h i s theory t o the pressure loading of an obliquely laminated

h a l f plane t o determine t h e e f f e c t s o f lamination angle and dispers ion

on t h e stresses.

This work uses the approximate continuum

Voelker and Achenbach (1969) t r e a t e d an i n f i n i t e laminated body

under a s t e p body force i n a plane normal t o t h e layer ing using an exact

modal ana lys i s . The i n t e r f a c e shear stress wave shows a slow rise t o

a s t a t i c value, while t he normal i n t e r f a c e stress is found t o be o s c i l l a t o r y .

Also Sameh (1971) has used a d i s c r e t e element model t o ca l cu la t e t h e e las t ic -

p l a s t i c response of a layered h a l f space.

- 74 -

The one dimensional impact loading o f a laminated p la te has been

discussed by Hutchinson (1969) where t h e pressure is normal t o t h e

layer ing .

transmission coe f f i c i en t s f o r a stress pulse when it encounters a

d iscont inui ty . For example, t h e t ransmi t ted stress across a plane

boundary separa t ing two d i f f e r e n t materials with normal stress

inc ident on t h e sur face i s given by

This problem can be solved exac t ly using t h e r e f l e c t i o n and

o = TO T = 2 Z Z / ( Z + z ) 0 ’ 1 2 1 2

where

(Note, t h a t T i s independent of t h e d i r ec t ion of t he inc ident stress).

Thus a pressure d iscont inui ty of i n t e n s i t y p0 propagating normal t o

a laminated medium of a l t e r n a t i n g acous t ic impedances suffers an a t t en -

uat ion at t h e head of t h e pulse of

Z1, Z 2 are t h e acous t ic impedances o f t h e two materials.

a f t e r encountering n p a i r s of l aye r s . Analysis using the r e f l ec t ed

and t ransmi t ted waves i n each l aye r revea ls t he stress h i s t o r y behind

the wave f r o n t .

3 . Transient Edge Loading o f a Plate

As noted ea r l i e r , when the pulse durat ion i s long enough, d i spers ive

effects can b e neglected as a first approximation and an equivalent aniso-

t r o p i c model can be used (Eqs. ( 7 ) , (12) ) . One of t h e effects of anisotrony

- 75 -

i s revealed i n t h e one dimensional edge impact of an o r tho t rop ic

p l a t e with t h e impact force i n the plane of t h e p l a t e and t h e edge

oblique t o a symmetry ax i s

s t r u c t u r a l and material d i spers ion , w e can use Eqs. ( 7 ) , with

Neglecting

t h e boundary condi t ions on t h e edge

For an i s o t r o p i c material a compressional wave would be generated.

However, f o r an edge oblique t o the symmetry a x i s , two waves are

propagated i n t o t h e p l a t e with wave speeds corresponding t o those

on the ve loc i ty sur face with wave normal (cos$, s in$ ) . Also d i s -

placements normal and p a r a l l e l t o t h e edge w i l l be exc i t ed . The

displacements w i l l t ake the form (xn normal t o edge, x along

the edge) n un = u [cos+ - a s i n + ] f ( t - -

S

X

1 1 V 1 n

+ u [cos$ - a s in$ ] f ( t - y 1 2 2 n

X

L n S X

U = U [sin$ + 01 COS$] f ( t - - ) 1 1 V

1 n X

+ u [s in$ + a cos$] f ( t - 7 1 2

2 2

The vectors (1, a l ) and (1, a ) are the eigenvectors corresponding

t o v , v respec t ive ly and depend on t h e angle 4 . 2

1 2

The quasi shear wave is generated through t h e coupling o f t h e

normal stress t.” with t h e shear s t r a i n , e , i n t h e cons t i t u t ive ns

- 76 -

n S n equations wr i t t en i n t h e x - x coordinate system, i . e . on x = 0

The constants

the angle I$ , (see Ashton e t a l . 1969). Determination o f t h e constants

U , U r e s u l t from s u b s t i t u t i o n of Eqs. (41) i n t o these boundary condi-

t i ons and is l e f t f o r t h e reader .

A l l , A I 6 , e t c . a r e r e l a t e d t o the e las t ic constants and

1 2

4 . Impact Generated Flexural Waves

Flexural waves generated by impact forces t ransverse t o i s o t r o p i c

p l a t e s has been reviewed by Mikowitz (1960). The one dimensional l i n e

impact of an iso t ropic p l a t e using both t h e Mindlin Eqs. (12) and t h e

classical theory, Eq. (15), has r ecen t ly been t r e a t e d by Moon (1972d)

In t h i s work t h e l i n e force is t ransverse t o the p la te sur face and

oblique t o t h e composite symmetry axes. In the context of t h e Mindlin

theory extensional waves are generated by a t ransverse force as well as

a f l exura l wave. The importance o f shear deformation and ro t a ry

i n e r t i a , as r e f l e c t e d i n Mindlins theory, i s shown t o become important

when the width of t h e contact force d i s t r i b u t i o n is comparable t o t h e

p l a t e thickness .

The ca l cu la t ion o f t h e two dimensional stress wave response t o

cen t r a l impact forces has r ecen t ly been s tudied by Chow (1971) and ).

Moon (1972b,c). Using a Timoshenko theory f o r laminated o r tho t rop ic p l a t e s

- 77 -

Chow (1971) t r e a t s t h e t r ans i en t response of a rectangular p l a t e t o normal

impact.

The author (Moon 1972b,c) uses a Mindlin p l a t e model t o examine the stress

contours a f t e r impact i n an i n f i n i t e p l a t e .

and f l exura l waves a re shown t o be generated under t ransverse impact.

Again,both extensional

Solutions t o t h e equations which govern the cen t r a l impact of

an iso t ropic p l a t e s were found f o r impact-like pressures using an

analytical/computational method.

used was the following

The impact pressure d i s t r ibu t ion

f o r r < a , ( r 2 = x2 + x2) and t < T 1 3 0

The th ree s t r e s s measures chosen were the average membrane stress

(t + t ) / 2 , t h e average f lexura l stress ( t + t ) / 2 a t the surface 11 33 11 33

of t he p l a t e , and t h e maximum inter laminar shear s t r e s s given by

( t 2 + t 2 21 2 3

The s t r e s s e s were calculated i n a qua r t e r plane of t h e p l a t e f o r a

s p e c i f i c time a f t e r t he i n i t i a t i o n of inipact and were normalized with

respect t o t h e maximum impact pressure as calculated i n the above sec t ion .

The da ta i s presented f o r various times and lay-up angles i n the form

of stress contour p l o t s (Figures 27, 28). Superimposed on these curves

- 78 -

a r e the theo re t i ca l wave f ron t f o r t he p a r t i c u l a r wave i n question

and the radius of t he c i r c l e which bounds the impact pressure.

The s i g n i f i c a n t stress l eve l s a l l l i e within t h e sur face bounded

by t h e theo re t i ca l wave sur face . In Figure 27, the average o r membrane

mean stress contours 1 /2 ( t + t ) f o r graphi te fiber/epoxy matrix

laminate p l a t e s are shown f o r lay-up angles of 1 1 33

O o , k45'.

The f l exura l o r bending motion has th ree waves associated with

i t . The l a r g e s t s t r e s s e s however were found i n t h e lowest f l exura l

wave which t r a v e l s a t an i s o t r o p i c speed given by

(K = lT2/12 ,

mean f l exura l

Figure 28 f o r

lay-up angles)

is Mindlin's cor rec t ion f a c t o r ) . S t r e s s contours f o r t he

stress 1 / 2 ( t - - + t - - ) i n t h i s wave a re shown i n 1 1 3 3

graphi te fiber/epoxy matrix

under the t ransverse impact

the wave f r o n t is c i r c u l a r s ince v is

S t resses i n the second and t h i r d f l exura l 3

0 0 laminate p l a t e s (+15 , 245

pressure E q . (42) . Note t h a t

i s o t r o p i c f o r laminate p l a t e s .

waves were found t o be small.

A t h ree dimensional computer p l o t i s shown i n Figure 29 f o r t he f l exura l

s t r e s s f o r t he k45 lay-up angle composite p l a t e . 0

The maximum stress l eve l s were found t o occur immediately after

the end of impact and appeared t o propagate along the f i b e r d i r ec t ions ,

given by the lay-up angles .

These r e s u l t s show t h e e f f e c t of t he change of f i b e r lay-up angles

on the stress d i s t r ibu t ions . For the f l exura l stresses, t h e optimum

- 79 -

0 lay-up angle t o be 41S0, showing a 34% lower stress l eve l than t h e 445

case. However, regarding t h e inter laminar shear s t r e s s e s , f o r t he same

impact conditions, there seems t o be l i t t l e difference i n t h e maximum

s t r e s s leve l with lay-up angle desp i te s ign i f i can t changes i n s t r e s s

d i s t r i b u t i o n i n space with Pay-up angle.

Another r e s u l t of these calculat ions i s t h a t t he induced s t r e s s e s

depend on t h e impact c i r c l e radius t o p l a t e thickness r a t i o .

O f course, t o evaluate t h e p o s s i b i l i t y of f r ac tu re o r f a i l u r e

of t he composite under impact, the complete s t r e s s matrix a t a point

must be known, as well as the f a i l u r e c r i t e r i a f o r t h e mater ia l .

- 80 -

REFERENCES

Abbott, B. W . and Broutman, L . J . (1966). "Stress-Wave Propagation i n Composite Materials,fr Experimental Mechanics, - 7, 383.

Achenbach, J . D . , Heman, J . H. and Ziegler , F . (1968). "Tensile Fa i lu re of In te r face Bonds i n a Composite Body Subjected t o Compressive Loads , I t AIAA J . , 6 , 2040. -

a;

Achenbach, J . D . , Heman, J. H. and Ziegler , F . (1970). "Separation a t t h e In te r face of a Ci rcu lar Inclusion and t h e Surrounding Medium Under an Incident Compressive Wave,"J. Appl. Mech., 37, 298. -

Achenbach, J . D . and Herrmann, G . (1968a). "Dispersion of Free Harmonic Waves i n Fiber-Reinforced Composites," AIM J . , 6 , 1832. -

Achenbach, J . D . and Herrmann, G . (1968b). '!Wave Motion i n Sol ids with Lamellar Structur ing," i n Proceedings of t he Symposium on Dynamics of Structured Sol ids , ASME, 23.

Achenbach, J . D . , Herrmann, G . , and Sun, C . T . (1968). "On the Vibrations of a Laminated Body,"J. Appl. Mech., 35, 689. -

Achenbach, J . D . and Sun, C. T. (1972). "The Direc t iona l ly Reinforced Composite as a Homogeneous Continuum with Microstructure ," In "Dynamics of Composite Materials" (E. H. Lee, ed) ASME.

-

Allcoch, A. W. R. and Col l in , D. M . (1968). T h e Development o f a Dummy Bird f o r Use i n Bird S t r i k e Research," National Gas Turbine Establishment (United Kingdom) (N.G.T.E.) Report No. C.P. No. 1071.

Almond, E . A . , Embury, J . D . and Wright, E . S . (1969). "Fracture i n Com- p o s i t e Mater ia ls ," In te r faces i n Composites, ASTM STP 452, ASTM, 107.

Anderson, D . L . (1961). "Layered Anisotropic Media,"J. Geophy. Res, - 66, 9, 2953.

Anon, (1971). "Hyfil was r i g h t f o r fan blade but time ran out ,'I The Engineer, 81.

Armenakas, A. E . (1965). Amer., 38, 439.

Armenakas, A. E . (1970).

"Torsional Waves i n Composite Rods,'' J . Acous. SOC. - -

"Propagation of Harmonic Waves i n Composite Ci rcu lar - Cyl indrical Rods," J . Acous. SOC. Amer., - 47, 822.

Asay, J . R . , Urzendowski, S. R. and Guenther, A. H . , (1968). "Ultrasonic and Thermal Studies of Selected Plast ics , Laminated Materials, and Metals," A i r Force Weapons Lab., Kir t land AFB, New Mexico, AFWL-TR-37-91.

Ashton, J . E . , Halpin, J. C . , and P e t i t , p . H., (1969). P r i m e r on ComPosite Materials, Technomic Publ., Stamford, Connecticut

- 81 - Barbee, T . , Seaman, L . and Crewdson, R. (1970). "Dynamic Fracture C r i t e r i a

o f Homogeneous Materials , ' I A i r Force Weapons Laboratory, Kir t land A i r Force Base, New Mexico, AFWL-TR-70-99.

Barker, L . (1970). "A Model f o r Stress-Wave Propagation i n Composite Materials," J . of Comp. Materials, - 5, 140.

Barker, L . M. and Hollenbach, R. E . (1970). "Shock-Wave Studies of P M , Fused S i l i c a , and Sapphire," J . Appl. Phys., - 41, 4208.

Bartholomew, R. A . (1971). "An Effec t ive S t i f f n e s s Theory f o r Elastic Wave Propagation i n Filamentary Composite Materials," Ph.D. Disser- t a t i o n , A i r Force I n s t i t u t e of Technology.

Bedford,' A. (1971). "Boundary Conditions and Rankine-Hugoniot Equations f o r a Multi-Continuum Theory of Composite Materials," Sandia Corp., SC-DR-710413.

Bedford, A. and Stern , M . (1970). "On Wave Propagation i n Fiber-Reinforced Viscoe las t ic Materials," J . of Appl. Mechs., - 37, 1190.

Bedford, A. and Stern , M . (1971). "Toward a Diffusing Continuum Theory of Composite Materials," J . of Appl. Mechs., - 38, 8.

Bedford, A . and Stern , M. ( t o appear). "A Multi-Continuum Theory of Com- p o s i t e E l a s t i c Materials," Acta Mechanica.

Behrens, E . (1967a). I 'Elastic Constants of Filamentary Composites with Rectangular Symmetry," J . Acoust. SOC. Am. , - 42, no. 2 , 367.

Behrens, E . (1967b). "Sound Propagation i n Lamellar Composite Materials and Averaged Elast ic Constants," J . Acoust. SOC. Am. , - 42, no. 2 , 378.

Ben-Amoz, M . (1968). "A Continuum Approach t o the Theory of Elast ic Waves i n Heterogeneous Materials," I n t . J . Eng. Sc i . , - 6, 209.

Ben-Amoz, M . (1971). "Propagation of F i n i t e Amplitude Waves i n Unidirec- t i o n a l Reinforced Composites," J . Appl. Phys., - 42, no. 13, 5422.

Benson, D . A . , Junck, R. N . and Klosterbuer, J . A. (1970). "Temperature- and Pressure-Dependent Elast ic Proper t ies of Selected I so t rop ic and Composite Materials from Ultrasonic Measurements,!' AFIVL-TR-70-169.

Berkowitz, H. M . and Cohen, L . J . (1970). "High-Amplitude Stress-Wave Propagation i n an Aniostropic Quartz-Phenolic Composite," McDonnell Douglas Astronautics Co., Ca l i f . , MDAC WD-1179.

Berkowitz, H. M . and Gurtman, G . A. (1970). "Measurement of Pulse Propa- gat ion i n Unid i rec t iona l ly Reinforced Composites," Presented a t the 1970 F a l l Meeting of t he Society f o r Experimental S t r e s s Analysis, Boston, Mass.

Bevilacqua, L. and Lee, F. H . , (1971). "Evaluation of S t r e s s Dis t r ibu t ions f o r Waves i n Per iodic Composites by Complementary Energy Methods," 9 t h Annual Meeting, SOC. Engr. S c i . , t o appear i n Recent Adv. i n Engr. Sc i .

- 82 -

Bevilacqua, L. and Lee, E . H . ( t o appear). "A Simple Model f o r Assessment of Var ia t iona l Solut ions f o r Waves i n Composites,"

Biot, M. A. (1972a). "A New Approach t o t h e Mechanics of Orthotropic Multi- layered Plates," I n t . J . of Sol ids and S t ruc tu res , - 8, no. 4, 475.

Biot , M . A . (1972b). "Simplified Dynamics of Mult i layers Orthotropic Visco- e l a s t i c Plate," I n t . J . of Sol ids and S t ruc tu res , - 8, no. 4, 491.

Bolotin, V . V . (1965). "On the Theory of Reinforced Bodies," ( i n Russian), Izv. Akad. Najk USSR, Mekhanika, - 1, 74.

Brekhovskikh, L . M . (1960). Waves i n Layered Media, Academic Press, New Yor

Br i l lou in , L . (1960). Wave Propagation and Group Veloci ty , Academic Press, N e w York.

Br i l lou in , L . (1963). Wave Propagation i n Per iodic S t ruc tu res , Dover.

Buchwald, V . T. (1959). "Elastic Waves i n Anisotropic Media,'' Proc. Roy. SOC., London, A , 253, 563. - - -

Chamis, C. C . (1971). "Computer Code f o r t he Analysis of Multi layered Fiber Composites - Users Manual," NASA Technical Note TN D-7013.

of Unidirect ional F iber Composites , I t NASA Technicai Memorandum Chamis, C . C . , Hanson, M. P . , and Se ra f in i , ( l 971) . "Impact Resistance

TM X-67802.

Charest , J . A. and J e n r e t t e , B. D . (1969). "Hugoniot Equations-of-State of Tape Wrapped Two-Dimensional Quartz Phenolic,It EGGG, Ca l i f . , Tech. Rept. No. S-448-R.

Chen, W. T . (1969). "Stresses i n Some Anisotropic Materials Due t o Inden- t a t i o n and Sliding," I n t . J . Sol ids and S t ruc tu res , - 5 , 191.

Chou, F . H. and Achenbach, J . D . (1970). "Field Equations Governing the Mechanical Behavior of Layered C i rcu la r Cylinders," AIAA J . , - 8, 1444.

Chou, P. C . and Wang, A. S. D . (1970). "Control Volume Analysis of Elastic Wave Front i n Composite Materials," J . Comp. Materials, - 4, 444.

Chou, S. C . and Greif, R. (1968). "Numerical Solut ion of S t r e s s Waves i n Layered Media," AIAA J . , - 6 , 1067.

Chow, T. S. and Hermans, J . J . (1971). The Sca t te r ing of Elastic Waves by Inhomogeneities i n a Unid i rec t iona l ly Aligned Composite," Presented a t 9 t h Annual Meeting o f t h e Society of Engineering Science, R P I , Troy, New York .

- 83 -

Chree, C. (1890). "On the Longitudinal Vibrations of Aerolotropic Bars with Oric Axi s of Mater ia l Symmetry," Quart. J . of Math. J- 24, 340.

Christensen, R. M. (1972). "Attenuation of Harmonic Waves i n Layered Media," t o appear i n J. of Applied Mechanics, ASME paper no. 72-WA/APM-5.

Cohen, L . J . and Berkowitz, 11. M . (1972). Phcnolic Composite Under Stress-Wave Loading i n Uniaxial S t ra in ," - J . of the Franklin Inst., - 293, 25-45.

of a Unidirect ional Fiber Composite Using Ultrasonic Velocity Measure- ments," Acoustical Society of America Journal , - 47, p a r t 2 , 795-803.

"Dynamic Frac ture of a Quartz-

Cost, J . R. and Z i m m e r , J . E . (1970). "Determination of the Elastic Constants

Cummerford, G . L . and Whi t t ie r , J. S. (1970). Wniaxia l -S t ra in Wave Propa- gat ion Experiments Using Shock-Tube LoadingJtl Experimental Mechanics, - 10, 120-126.

Ually, J . h'. , L i n k , J . A . and of S t r e s s Waves i n Fiber western Mechanics Conf. ,

Dong, S. B. and Nelson, R. B.

Prabhakaran, R . (1971). "A Photoe las t ic Study Reinforced Composites," Proc. of 12th Nid- Developments i n Mechanics, - 6 , 937.

(1971). "On Natural Vibrat ions and Waves i n - - Laminated Orthotropic Plates," J . of Appl. Math.

Drumheller, D . S. (1970). "A Frequency F i l t e r i n g Theory f o r t he Propagation I 1 of S t r e s s Pulses i n Composite Materials, Sandia Gorp., SC-DR-710472.

Embury, J . D . e t a l . (1967). "The Fracture o f Mild S t e e l Laminates , I 1 Trans. Metal lurgical SOC. Am. Ins t . Mining, Metal lurgical and Petroleum Engineerings, 239, 114-118. -

Eringen, A. C. (1953). "Transverse Impact on Beams and Plates,'I J . Appl. Mech. , 20, 461. -

Eringen, A. C. (1966). "Linear Theory of Micropolar E l a s t i c i t y , " J . of Math. and Mech., 15, 909. -- -

Eringen, A. C . (1968). "Theory of Micropolar E la s t i c i ty , " Frac ture , 2 , - Academic Press.

Eringen, A. C. and Suhubi, E. S . (1964). "Nonlinear Theory of Simple Micro- Elastic Solids-I,cl I n t . J . Eng. S c i , 2, 189. -

Ewing, W. M., Jordetzky, W. S. and Press, F. (1957). Elastic Waves i n Layered Media, M c G r a w - H i l l , N . Y.

- 84 -

Garg, S. K . and Kirsch, J . W . (1971). "Hugoniot Analysis of Composite Materials ,If J. of Composite Materials, 5, 428-445. -

Goatham, J. I. (1970). "Materials Problems i n . t h e Design of Compressor Blades with Fibrous Composites," Proc. Roy. SOC. Lond., A, 319, 45. - -

Goldsmith, W. and Lyman, P . T . (1960). "The Penet ra t ion of Hard S t e e l Spheres In to Plane Metal Surfaces , ( I J. of Applied Mechanics , 717-725.

Goldsmith, W . (1960). Impact, Arnold Publ. LTD.

Greeil, A. E. and Naghdi, P. M. (1965) "A Dynamical Theory of I n t e r a c t i n g Continua," I n t . J . Eng. S c i . , - 3, 231.

Grot, R . A. (1972) e "A Continuum Model f o r Curvalinear Laminated Composites,1v I n t . J . Sol ids and S t ruc tu res , 8, 439. -

Grot, R. A. (unpublished). "Propagation of Elastic Waves i n Pe r iod ic Fiber Composite Materials," Princeton University.

Grot, R. A. and Achenbach, J . I ) . (1970). s'Lincar Anisothermal Thcory f o r a Vi scoe la s t i c Laminated Composite,'s Acta Mcchanica, 9 , 245. -

Curtman, G. A . , Nayfeh, A. I I . , Ilegcmicr, G . A . , Garg, S . K. and Brownell, D. 14. (1971). S t r u c t u r a l Composite Materials," A i r Force Weapons Lab, AFWL-TR-

"A Continuum Theory Applied t o Wave Propagation i n

71-167.

Haener, J . and Puppo, A. (1969). " F u r t h e r Inves t iga t ions i n t o Microdynamics . of Wave Propagatim," A i r Force Material Lab, Tech. Rept. AFML-TR-68-

311, Part 11, Wright-Patterson AFB, Ohio.

Hayduk, R. J. (1973). "Hail Damage t o Typical A i rc ra f t Surfaces,11 J. A i r c r a f t , 10 , no. 1, 52-55. -

Hegemier, G. A. (1972). "On a Theory of I n t e r a c t i n g Continua f o r Wave Pro- "Dynamics of Composite Materials," (E. H. pagation i n Composites ," In

Lee ed.), Am. SOC. of MecK Eng., New York.

"Applications of Theor ies o f Generalized Casserat Continua t o t h e Dynamics of Composite Materials," Mcchs. of Generalized Continua (E. Kroner, ed . ) , Springer-Verlag, 69.

Herrmann, G. and Achenbach, J. D. (1968).

Herrmann, G. and Achenbach, J: D. (1968). and Fiber-Reinforced Composites," In "Proceedings of t h e I n t . Conf. on t h e Mechanics of Composite Materia=," Pergamon Pres s , New York.

"Wave Propagation i n Laminated

Heyman, F. J. (1969). "High-Speed Impact Surface,ll J . Appl. Physics, - 40, no.

between Liquid 13, 5113-5122.

Drop and a S o l i d

- 85 -

Holmes, B . S . and TSOU, F. K . (1972). "Steady Shock Waves i n Composite Materials, J . Appl . Physics, 43, 957-961.

Hunter, A. R. (1970). "Photoelastic Study of S t r e s s Wave Propagation in Composites," Society f o r Experimental Stress Analysis, Fa l l Meeting, Boston, Mass., Oct. 18-22, Paper 1708, 42 p . (Lockheed Research Laboratories, Palo Alto, Calif .) .

utchinson, J. R. (1969). "Stress Haves i n Layered Materials," AIAA J. , -0 7

I s b e l l , W. M., Froula, N. H. and Shipman, F. H. (1967). "Shock Wave Propa-

786 e

gat ion and Equation of S t a t e Measurements of Quartz Phenolic," Funda- mental Material Behavior Study, Vol, 111, Ballistic Systems Divis ion, Calif. BSD-TR-67-25.

Johnson, J. N . (1972). "Calculation o f Plane-Wave Propagation i n Anisotropic E l a s t i c - P l a s t i c Sol ids ," J . Appl . Physics , - 53, no. 5 , 2074-2082.

Jones, J, P. (1970). ' 'Stress Wave Dispersion i n Rod-Reinforced Composites," A i r Force Report No. SAMSO-TR-70-81. The Aerospace Corp., Report No. TR-0066 (5240-30) -3.

Kane, T. R. (1954). t lReflect ion of Flexural Waves at t h e Edge of a Plate,"

Kerwin, E , M., Jr. (1959), "Damping of Flexural Waves by a Constrained

J. Appl. Mechs., 213-220.

Viscoe la s t i c Layer," J. o f Acoustical Soc ie ty o f America, 31, no. 7, - 952-962.

inslow, R. and Delano, B. (1970). Ttress Waves i n Sandwio? Plates," I .

Tenn. Tech. Univ., Dept. Eng. S c i . , Rept. 70-3 (for NASA CR-111572).

inslow, R. and Delano, B. (1970). "Stress Waves i n Sandwich Platcs,tl Tenn. Tech. Univ., Dept. Eng. Sc i . , Rept. 70-3, (for NASA CR-111572).

- 86 -

Knopff, 1,. and Iludson , J . A. (1967).

Knopff, L. and Iludson, 3 . A. (1367).

"Frequcncy Dependcnce o f Amplitudes

"Frequcncy Dcpcndence o f Amplitudes of Sca t tc rcd E l a s t i c Waves," J . Acoustical SOC. Am., 42 , no. 1, 18-20,

of Sca t te red Elastic Waves,'' J . Acoustical SOC. Am., - 4 2 , no. 1, 18-20.

Koh , S. L. (1970). "Continuum Theories f o r Composite Materials ,I1 In "Mechanics of Composite Materials ,If Pergamon Press, Oxford , 387.

Kohn, W . , Krumhansl, J . A. and Lee, E. H. (1972). Var i a t iona l Methods f o r Dispersion Relations and Elastic Proper t ies of Composite Materials," J. of Applied Mechanics, 39, no. 2 , 327.

Kohn, W. (1952). "Variational Methods f o r Per iodic Lattices , ' I Physical Revue, 87, 472-481. - -

Kraut , E . A . (1963). "Advances i n the Theory of Anisotropic E l a s t i c Wave Propagation," Reviews of Geophysics, - 1, no. 3 , 401-448.

Kreyenhagen, K . N. , Wagner, M. H . , Pechocki, J . J. and Bjork, R. L. (1970). " B a l l i s t i c L i m i t Determinations i n Impacts on Multimaterial Laminated Targets," AIM J . , I 8, no. 1 2 , 2147-2151.

Krumhansl, J . A . and Lee, E . H . (1971). "Propagation of Transient Elastic Waves i n Per iodic Composites," ARPA Materials Research Council Report, Cornel1 Universi ty , I thaca, New York.

Krumhansl, J. A. (1970). "Propagation of Trans ien ts i n Per iodic Composites," P r e l i m . Reports, Memoranda and Technical Notes of ARPA Materials Summer Conference, Dept. Chem. and Metall. Engrg.,Univ. of Michigan, 176.

Krumhansl, J. A. (1972). "Randomness and Wave Propagation i n Inhomogeneous Media," I n "Dynamics of Composite Materials (E. H. Lee ed.) , ASME, New York:

Kvasnikov, E . N . (1967). "A Study of the Elastic Proper t ies of Oriented Fiber Glass by t h e Impulse Acoustical Method," English Translat ion of Fiziko-Khimiya i Mekamika Orientrovannyhh, Moscow, NAVKA, 172-177.

Lai,

Lee,

J - L . (1968). "The Propagation of Waves i n Composite Elastic Rods and Viscoe las t ic Rods," Ph.D. Thesis, Princeton Univ.

E . H. (1970). "Stress Dis t r ibu t ions i n Plane Waves i n a Per iodic Layered Composite," P r e l i m . Reports, Memoranda and Technical Notes of ARPA Mater ia ls Summer Conference, Dept. Chem. and Metall. Engrg., Univ. of Michigan, 153.

- 87 -

Lee, E . H. (1972). "A Survey of Var ia t iona l Methods f o r Elastic Wave Propagation Analysis i n Composite with Per iodic Structures ," I n "Dynamics of Composite Materials" (E. H. Lee ed.) , ASME, New E r k .

Lempriere, B. M. (1969). T'he P r a c t i c a b i l i t y of Analyzing Waves i n Composites by the Theory of Mixtures,rf In "Colloquium on Dynamic Behavior of Composite Materials' ( A b s t r a x only) . , Univ. of Calif. - San Diego.

Lin , T. C. and Farne l l , G. W. (1968). "Search f o r Forbidden Direct ions of Elastic Surface-Wave Propagation i n Anisotropic Crystals," J. Appl. Physics, 39, no. 9 , 4319-4325.

_I

Lord, A. E. (1972). "Ultrasonic Wave Propagation i n Metal Matric Com- pos i t e s ,"J. of Composite Mater ia ls , 6 , 278. -

Lundergan, C . D. (1971). "Discussion of the Transmitted Waveforms i n a Periodic Laminated Composite $ f I J . of Appl. Phys., 42 , no. 11, 4145. -

Lundergan, C. D. and Drumheller, D. S. (1971a). "Dispersion of Shock Waves i n Composite Materials , " I n "Shock Waves and t h e Mechanical Proper t ies of Solids" (Proceedings of the 17th Sagamore Army Materials Research Conf., Raquette Lake, N.Y. 1970), (J. J. Burke and V. Weiss eds.), Syracuse Univ. Press, New York, 141.

Lundergan, C . D . and Drumheller, D . S . (1971b). "Propagation of S t r e s s Waves i n a Laminated P la t e Composite,1f J . Appl. Phys., - 42 , 669.

Lundergan, C. D. and Drumheller, D . S . (1972). "The Propagation of Transient S t r e s s Pulses i n an Obliquely Laminated Composite," In "Dynamics of Compoite Materials" (E. H. Lee ed . ) , ASME, New York.

McCoy, J. J . and Mindlin, R . D. (1962). "Extensional Waves along the Edge o f an E l a s t i c P la te , " J. Appl. Mechs. , Transactions of ASME, paper no. 62-WA-79.

McCoy, J . J. (1972). "On the Dynamic Response of Disordered Composites," t o appear i n J. Applied Mechanics, ASME paper no. 72-WA/APM-28.

McNaughton, I . I . (1964). !The Resistance o f Transparencies t o B i r d Impact a t High Speeds , I 1 Aircraft Engineering, - 36, 409-412.

McNiven, H. D . , Sackman, J . L . and Shah, A. H. (1963). "Dispersion of Axial ly Symmetric Waves i n Composite E l a s t i c Rodsr" J . Acous. SOC. Am., 35, 1602.

Markham, M. F . (1970). "Measurement of the Elastic Constants of Fiber Composites by Ultrasonics ," Composites, - 1, 145-149.

- 88 -

Martin, S. E . , Bedford, A . and Stern , M. (1971). "St Propagation i n Fiber Reinforced E l a s t i c Material ments i n Mechanics," (L. N. H. Lee and A. A . Szewcykehs.) , (Proc. 12th Midwest. Mech. Conf., Notre Dame), 515.

-

Miklowitz, J . (1966) a "E las t i c Wave Propagation - In "Applied Mechanics I t Surveys, Spartan Books, 809.

I? Mindlin, R . D. (1961). "High Frequency Vibrations of Crys ta l P l a t e s , Quart. of Appl. Math., - 19. 51.

Mindlin, R . D. (1964). "Microstructure i n Linear E l a s t i c i t y , " Arch. Rat. Mech. Anal., 16, 51. -

Mok, C . H. (1969). "Effective Dynamic Proper t ies of a Fiber-Reinforced Material and the Propagation of Sinusoidal WavesYf1 J . Acoust. SOC. Am. , 46, 631. -

Moon, F . C . (1971). "Wave Surfaces Due t o Impact on Anisotropic Fiber Composite P la t e s , " NASA Tech. Note TN D-6357.

Moon, F . C . (1972a). "Wave Surfaces Due t o Impact on Anisotropic Plates,f! J . Comp. Mat., - 6 , 62.

Moon, F. C . (1972b). "S t ress Wave Calculations i n Composite Plates Using t h e Fas t Fourier TransformY1' presented a t t he National Sym. on Computerized S t r u c t u r a l Analysis and Design, March 1972, Wash., D . C . , Computers and S t ruc tu re .

Moon, F. C . ( 19724 . "Theoretical Analysis Impact i n Composite Plates," Princeton Univ. Rept. f o r NASA L e w i s Res. Lab.

Moon, F. C . (1972d). "One-Dimensional Transient Waves i n Anisotropic P l a t e s , I 1 Princeton Univ. Rept . , ( t o appear) , J . Applied Mechanics.

Moon, F. C . and Mow, C-C. (1970). "Wave Propagation i n a Composite Material Containing Dispersed Rigid Spherical Inclusions ,If Rand Corp. Rept. RM-6139-PR.

Mow, C-C. and Pao, Y-H. (1971). "The Dif f rac t ion of Elast ic Waves and

Munson, D. E . , Reed, R . P . and Lundergan, 6. D. (1971).

Dynamic S t r e s s Concentration," Rand Corp. Rept . R-482-PR.

"Hugoniot and Thin-Pulse Attenuation Experiments i n Cloth-Laminate Quartz Phenolic," presented a t t h e Sym. on Dynamic Response of Composite Materials, AF Weapons Lab. , New Mexico (Abstract on ly) ,

- 89 -

Munson, D. E . and Schuler, L. W. (1971). "Hugoniot Predict ions fo r Mechanical Mixtures Using Effec t ive Moduli ,It Shock Waves and t h e Mechanical Proper t ies o f Sol ids , Proceedings of Seventeenth Sagamore Army Materials Research Conference, eds . , John J . Burke and Volker Weiss , (Syracuse Universi ty Press , Syracuse , New York) pp. 185-202.

Munson, D. E . and Schuler, L . W . (1971). 'Steady Wave Analysis of Wave Propagation i n Laminates and Mechanical Mixtures , t t J. Comp. Materials , - 5 , 286.

Musgrave, M. J . P. (1954). "On the Propagation o f Elastic Waves i n Aelotropic Media," Proc. Roy. SOC., Lon., A , 226, 339.

Musgrave, M. J . P. (1970). "Crystal Acoustics," Holden-Dey, Inc.

- _ -

Musgrave, M. J . P .

Nemat -Nasser , S. N . (1972) . J . Appl . Mech . , forthcoming.

Nevi11 , G . E . and Sierakowski, R . L . (1972). "One Dimensional Wave Pulses i n Steel-Epoxy Composites," Presented a t t h e 1972 Meeting SOC. f o r Experimental S t r e s s Analysis , Cleveland, Ohio.

Novak, R. C . and De Crescente, M. A. (1972). "Impact Behavior of Advanced Resin Matrix Composites ,It AST/AIME Second Conf. on Composite Materials: Test ing and Design. ASTM STP 479, 311-323.

Novak, R. C . and Preston, J . L . , Jr. (1972). "Studies of Impact Resistance of Fiber Composite Blades Used i n Aircraft Turbine Engines - Third Quarter ly Report Research Center, Cleveland, Ohio, Pra t t E, Whitney Aircraft.

Prepared f o r NASA-Lewis

Novichkov, Iu . N. (1971). "Propagation of Planc lVrrvcs i n Laminated E l a s t i c Media of Regular S t ruc tu re , l ' ( i n Russian) Akademiia Nauk USSR, I z v e s t i i a , hiekhanika Tverdogo Tela, 65.

Ozgur , D . (1971). "Propagation of P l a n e Waves i n Anisotropic Micropolar Elastic Media," P1i.D. T h e s i s , Princeton Universi ty .

Peck, J. C. (1971). "Pulse Attenuation i n Composites , I 1 I n "Shock Waves and the Mechanical Proper t ies of Solids" (J. J . B u r G and V . Weiss eds . ) , Syracuse University Press, Syracuse, N . Y . , 155.

- 90 -

Peck , J. C. (1972). "Stress Wave Propagation and Fracture on Composites , I 1

Symposium on Dynamics of Composite Mater ia ls , ASME.

Peck, J. C. and Gurtman, G. A. (1969). "Dispersive Pulse Propagation P a r a l l e l t o the Interfaces of a Laminated Composite J" J. Appl. Mech. , 36, 479.

Pekeris , C. L. (1955). "'The Seismic Surface Pulse,'I Proc. Nat. Acoustical

-

SOC., 41 , 469-480. - - Peterson, F . B . (1972). "Some Considerations o f ?daterial Response

Due t o Liquid-Solid Impact , ' I J. Basic Engineering Trans. , ASME Paper No. 72-WA/FE-27.

Postma, G . W. (1955). t'lJave Propagation i n a S t r a t i f i e d Medium Geophysics, 20, 780.

Potapov, A . I . (1966). ' I A n Inves t iga t ion o f t he Elast ic Charac t e r i s t i c s of Fiberglass Reinforced P l a s t i c s by the Pulse Acoustic Method," Xovyye Materialy v Tekhnike i Yauke. Proshloye, Nastoyashcheye, Budushcheye, Moscow, IZD-VO Nauka ( i n Russian) Available i n English USAF machine t r a n s l a t i o n FTD-MT-24-47-68.

Pot t inger , M. G . (1970). "The Anisotropic E u iva l en t of t he Love Longi- ?I

tud ina l Wave Propagation Approximation, Aerospace Research Lab., ARL TR-70-0044.

Pot t inger , P1. G. (1970). "The S t a t i c and Dynamic Elastic Moduli Real Moduli I 1

of Glass Fiber-Expoxy and Boron Fiber-Aluminum Composites 3

Research Lab., ARL 70-0096. Aerospace

Prabhakaran, R . (1970). g'Photo-Orthotropic-ElasticityJtf Ph.D. Thesis , I l l i n o i s Ins t . of Tech.

Puppo, A , , Feng, M. and Haener, J . (1968). "Microdynamics of Wave Propa- gat ion," Annual Summary Report, AF Materials Laboratory, Ohio, AFFIL- TR-68-311.

Reed, R . P . and Munson, D. E . (1972). ' 'Stress Pulse Attenuat ion i n Cloth- 11 Laminate Quartz Phenol ic , J . Comp. Materials, - 6, 232.

Reed, R. P. and Schuster, D . M . (1970). "Filament Frac ture and Post-impact Strength of Boron-Aluminum Composites ,";T. Comp. Mater ia l s , - 4, 514.

Reuter, R. C. (1970). "On t h e Plate Velocity of a Generally Or thot ropic Plate," J . Composite Mater ia ls , 4,. 129. -

- 91 -

Rose, J. L. and Chou, P. C . (1971). "Elastic S t r e s s Waves i n Layered Composite Materials," J. Composite Materials, 5 , 405-407. -

Ross, C. A. , Cunningham, J . E . and Sierakowski, R . L . (1972). "Dynamic Wave Propagation i n Transverse Layered Composites,tf The Shock and Vibration Bull . , Bull 42, Part 5 , Naval Res. Lab., Washington, D. C .

Rowlands, R. E . and Daniel, I . M. (1972). t7Application of Holography t o Anisotropic Composite Plates," Experimental Mechanics, 75-82

Rytov, S. M. (1956).

Sameh, A. H , (1971). "A Discrete-Variable Approach f o r Elastic-P1astic

qfAcoustical Proper t ies of a Thinly Laminated Mediwn," Soviet Phys. - Acoustics, 2 , 68. -

Wave Motions i n Layered Solids," J. of Computational Physics, - 8 , 343-368.

Schmitt , G . T . , Jr. (1970). "Erosion Behavior of Polymeric Coatings and Composites a t Subsonic Veloci t ies ," Proc. of Third I n t . Conf. on Rain Erosion and Associated Phenomena, Elvetham H a l l , Hampshire, England.

Schuler, K. W. ( t o Appear). etPropagation of Steady Shock Waves i n Poly- methyl Methacrylate ,"J . Mech. Phys. Sol ids .

Schulcr, K . W. and Walsh, E. K. ( t o Appear). "Cr i t i ca l Induced Acceler- a t ion for Shock Propagation i n Polymethyl Methacrylate , I 1 J . Appl . 3lcch. -

Schultz , A . B. and Tsai, S. W . (1968). "Dynamic Moduli and Damping Ratios ( 1 i n Fiber-Reinforced Composites, J . Comp. klatcr ia ls , - 2 , 368.

Schultz, A. B. and Tsai, S. W. (1969). "Neasurements of Complex Dynamic Moduli for Laminated Fiber-Reinforced Composites," J . Comp. ivlaterials, - 3 , 434.

Schuster, D. M. and Reed, R. P . (1969). "Fracture Behavior of Shock Loaded Boron-Aluminum Composite Materials," J . Composite Mater ia l s , - 3, 562.

Scot t , R . A. (1972). "Wave Propagation i n a Layered Elastic Plate," I n t . J. of Solids and St ruc tures , - 8, no. 6, 833.

Shea, J . H . , Reaugh, J . , Tuler , F . , Graham, M. and Stefansky, T. (1971). . "Dynamic Response of Composite Materials; Summary Inter im Report, 1 1

Physics In t e rna t iona l Co., Ca l i f . , PIIR-272-1.

Sierakowski, R. L . , Nevi l l , G . E.,Ross, C . A. and Jones, E . R. (1970a). "Expcrimental Studies of the Dynamic Deformation and Frac ture of F i l - ament Reinforced Composites ," AIAA/ASME 11th S t ruc tu res , S t r u c t u r a l Dynamics, and Materials Conf., Denver, 164.

- 92 -

Sierakowski, R . L. , Nevi l l , G. E. , Ross, C. A. and Jones, E . R. (1970b) *

'IFollow-on Studies on t h e Ballistic Impact of Composite Material ," AF Armament Lab., AFATL-TR-70-87,'

Skalak, R. (1957). "Longitudinal IMpact of a C i rcu la r Elastic Rod," J. - Appl. Mechs., - 24, Trans. ASME _r 79, 59.

Smeltzer, C. E . , Gulden, M. E . , and Compton, W. A., llMechanisms of Metal Removal by Impacting Dust Particles," J. Basic Engr., Trans. of the ASME, 639-654.

S te rn , M. , Bedford, A. and Yew, C. H. (1971). "Wave Propagation i n Visco- e l a s t i c Laminates,I1 J . Appl. Mechs., - 3, 448.

Sun, C-T., (1971a). v'Microstructure Theory fo r a Composite Beam,'? J . Xppl.

Sun, C-T., (1971b). T h e o r y of Laminated Plates,11 J. Appl. Mechs., 7 SS, 231.

Mechs. 947.

Sun, C-T. (1971a). "On the Equations of Composite Beams Under I n i t i a l S t ress ,* ' I n t . J . Sol ids and St ruc tures , - 8, 385.

Sun, C-T., Achenbach, J . D. and Herrmann, G. (1968). "Time-Harmonic Naves i n a S t r a t i f i e d Medium Propagating i n the Direction of t he Layering," J. Appl., Mechs., Trans. U M E , 408.

Sun, C-T. , Achenbach, J . D . and Iierrmann, G . (1968). "Continuum Theory

Sun, C-T. and Whitney, J . M. (1972). "On Theories f o r t he Dynamic Response

fo r a Laminated Mediuni," J . Appl. Mechs., - 35, 467.

of Laminated Plates," AIAA/ASNE/SAE 13th S t ruc tures , S t ruc tu ra l Dynamics and Materials Conference, San Antonio, Texas, AIAA paper no. 72-398.

P la tes , " Purdue University Report f o r A i r Force Materials Lab., Wright- Pat terson Air Force Base, t o appear J. Applied Mechanics.

Sun, C-T. (1972b). ffPropagation of Weak Shocks i n Anisotropic Composite

Sutherland, H. J . , and Lingle, R. (1972). "Geometric Dispersion of Acoustic Waves by a Fibrous Composite,'' Sandia Labs., New Mexico, J. Composite Materials, 490-502.

Sve, C . (1971a). "Time-Harmonic Waves Traveling Obliquely i n a Per iodica l ly Laminated Medium," J . Appl. Mechs, 38, no. 2 , Trans. ASME Ser ies E , 477.

-

Sve, C. (1971b) . "Thermoelastic Waves i n a Per iodica l ly Laminated Medium," I n t . J . of Sol ids and St ruc tures , 7 , 1363. -

Sve, C . (1972). "Thermally Induced S t r e s s Waves i n a Laminated Composite," J . Appl. Mechs, 39, 103.

Sve, C . and Whit t ier , J . S. (1970).

- "One-Dimensional Pulse Propagation i n

an Obliquely Laminated Half Space," J . Appl. Mechs., 37, no. 3 , Trans. ASME, Ser ies E , 778.

-

- 93 -

Sve, P. (1973). "Elastic Wave Propagation i n a Porous Laminated Composite," t o appear I n t . J . Sol ids and S t ruc tu res .

Tauchert, T . R . (19 ) . "A Lattice Theory f o r Representation of Thermo- Elast ic Composite Materials," - In "Recent Advaices i n Engineering Sei . , ' I 3. -

Tauchert, T. R. (1971a). "Measurements of Elast ic Moduli of Laminated Com- pos i t e s Using an Ul t rasonic Technique," J . Composite Materials, I 5, 549.

Tauchert, T . R . (1971b). "Propagation of S t r e s s Waves i n Woven-Fabric Composites," J . Composite Mater ia ls , - 5, 456.

Tauchert , T . R . and Guzelsu, A. N . (1972). "An Experimental Study of Dispersion of S t r e s s Waves i n a Fiber-Reinforced Composite," J . Appl. Mechs., - 39, 98.

Tauchert, T . R . and Moon, F . C . (1970). "Propagation of S t r e s s Waves i n Fiber-Reinforced Composite Rods," AIAA/ASME 11th S t ruc tu res , Struc- t u r a l Dynamics and Materials Conference, Denver.

Tauchert, T. R . (1972). ttPropagation of Waves i n Heterogeneous Material-I1 , I t

Final Report t o Dept. o f Army, University of Kentucky Report No. UKY TR59 - 72 -EMS.

Ting, T. C . T . and Lee, E . H. (1969). 'Wave Front Analysis i n Composite Materials," J . Appl. Mechs., 36, 497. -

TobBn, G . L . (1971). t lVariational Plane Wave Methods i n t h e Analysis Propagation of Elast ic Waves i n Composite Materials , ( I Thesis f o r Master o f Science Degree, Cornel1 University.

Torvik, P . J. (1970). "A Simple Theory f o r Shock Propagation i n Homogeneous Mixtures," A i r University, Wright-Patterson A i r Force Base, Ohio, AFIT TR 70-3.

Torvik, P. J . (1970). "Shock Propagation i n a Composite Material , ' I

J. Comp. Matl., 4, 269-309. -

TSOU, F. K . and Chou, P . C . (1969). "Analytical Study o f Hugoniot i n Unidirect ional Fiber Reinforced Composites , I 1 J. Comp. Matl., 3, 500. -

Tsou, F. K . and Chou, P . C . (1970). "The Control-Volume Approach t o -Hugoniot of Macroscopically Homogeneous Composites ,It jL. Comp. Matl.. 4. 526.

- 94 -

Tuong, V. (1970)'. "Determination of E l a s t i c i t y Constants f o r P l a t e s of a Unidirectional-Fiber Reinforced Anisotropic Composite Material , I! Acod. des Sciences Mathematique, 270, no. 22 , 1440-1443.

Turhan, D. (1970). "A Dynamic Theory f o r Di rec t iona l ly Reinforced Composites , ' I Ph.D. Di s se r t a t ion , Northwestern University, Evanston, I l l i n o i s .

Voelker, L . E. and Achenbach. J . D. 11969). "Stress Waves i n a Laminated Medium Generated by Transverse Forces , I t Acoust . SOC. Am. J . . 46. 1213-1222.

Warnica, R . L . and Charest, J . A. (1967). "Fundamental Material Behavior Study, Volume 11, Spa l l a t ion Thresholds of Quartz Phenolic," Bal l is t ic Systems Division, Norton AFB, Ca l i forn ia , BSD-TR-67-24.

Washington, S. L . (1969). "Shock Wave Dispersion i n a Laminated Composite ," A i r Force I n s t i t u t e of Technology, Wright-Patterson AFB, Ohio, Master's Thesis , Rep. GNE/PH/69-17.

Waterman, H . A. (1966). "On t h e Propagation of Elast ic Waves Through Composite Media," Rheologica Acta, 5, no. 2 , 140-148. -

Weitsman, Y. (1972). tQn Wave Propagation and Energy Sca t t e r ing i n Materials Reinforced by Inextens ib le Fibers ,I' I n t . J . of Solids and S t ruc tu res , - 8, no. 5 , 627.

Weitsman, Y. (1973). "On the Reflection of Harmonic Waves i n Fiber- Reinforced Materials ,It J. Sound and Vibration, - 26, no. 1 , 73-89.

Wheeler, P . and Mura, T. (1972). "Dynamic Equivalence o f Composite Material and Eigens t ra in Problems," t o appear i n J. Applied Mechanics, ASME paper 72-WA/APM-10.

Two-Phase Continuous Composites, Polymer Engineering and Science, White, P. L . and V a n Vlack, L . €1. (1970). "Elastic Proper t ies of Model

10, 5, 293-299. -

White, J . E. and Angona, F. A. (1955). flElastic Wave Veloc i t ies i n Laminated Media ,"J . Acoust. SOC. Am., - 27, 310-317.

Whit t ie r , J . S . and Jones, P. J. (1967). "Axially Symmetric Wave I 1

Propagation i n a Two Layered Cylinder Y

- 95 -

W i t t i e r , J . S. and Peck, J . ive Pulse Propagation i n L t h Theory ,It

Wilkins , M. L . (1969) . "Penetration MechanicsYtt Lawrence Radiation Laboratory, Universi ty of Cal i forn ia , Livermore, Cal i forn ia , UCRL-72111 Prepr in t .

Willis, J. R. (1966). "Hertzian Contact of Anisotropic Bodies , I 1

J. Mech., Phys. Sol ids , 14 , 163-176. - Wlodarczyk, E . (1971). "On t he Loading Process Behind the Fronts of

Reflected and Refracted Shock Waves i n Plast ic Layered Media," Proc. of Vibration Problems, 14, no. 4 , 433-441. -

Wu, T o M, (1971). "Propagation of E l a s t i c Waves i n Per iodic Fiber Composite Mater ia l s , " Ph.D. Disser ta t ion , Princeton Universi ty , Princeton, New Je r sey . (with F. C . Moon, (1971) t o U. S. Army Research Off ice , Durham Project No. RGD20061102B32D)

Y a n , M. J . (1972) "Constrained-Layer Damping i n Sandwich Beams and Pla tes , " Ph.D. Disser ta t ion , Princeton University, Princeton, N . J . , AMs Report No. 1031.

- 96 -

CAPTIONS FOR F I G U R E S

3

4

5

6

Geometry of a two dimensional wave i n a mult i -ply p l a t e

Velocity sur faces versus wave d i r ec t ion f o r var ious

p l y lay-up angles; 55% graphi te fiber/epoxy ma t r ix

(Moon, 1972)

Direct ion of p a r t i c l e motion versus wave normal f o r

var ious p l y lay-up angles; 55% graphi te fiber/epoxy

matr ix (Moon, 19'72)

Wave sur faces f o r mult i -ply p l a t e s ; a)

angle, b ) 215' f i b e r lay-up angle (Moon, 1972)

0' f i b e r lay-up

0 Wave sur faces f o r mult i -ply p l a t e s ; a) +30 f i b e r lay-up

angle , b) +4S0 lay-up angle (Moon, 1972)

Flexural wave d ispers ion r e l a t i o n s i n an an iso t ropic p l a t e

(Mindlin's theory); 55% graphi te fiber/epoxy matrix mult i -

p l y p l a t e , +45 f i b e r lay-up angle. 0

Surface waves and edge waves i n s o l i d s

Dis tor t ion o f an i n i t i a l l y shaped t rapezoida l pulse

due t o wave d ispers ion e .g . longi tudina l waves pro-

pagating across o r down t h e f i b e r s of a un id i r ec t iona l

f i b e r composite material

- 97 -

9

10

11

1 2

13

14

Distor t ion of an i n i t i a l l y shaped pulse due t o wave

dispers ion e .g , shear wave propagating down the f i b e r s

of a unid i rec t iona l f i b e r composite material

Approximate dispers ion r e l a t ions f o r longi tudinal waves

i n boron fiber/aluminum matrix rods f o r various or ien ta t ions

of t h e f i b e r s t o the rod ax is (Pot t inger , 1970), mater ia l

dispers ion not included .

Sketch o f dispers ion r e l a t i o n f o r longi tudinal o r shear waves

propagating normal t o the layers of a composite of a l t e r -

nat ing i so t rop ic layers , E q . (26) .

Comparison of exact dispers ion r e l a t i o n s ( so l id l i nes ) w i t h

t he microcontinuum theory of Sun e t a l . (1968) f o r various

shear modulus ratios; a) shear waves propagating i n t h e

d i rec t ion of t he layer ing, b) longi tudinal waves propa- I

gat ing i n t h e d i r ec t ion of the layer ing

Shock wave speed versus p a r t i c l e ve loc i ty (Hugoniot curve)

f o r a mixture of AR 0

(Munson and Schuler, 1970)

p a r t i c l e s i n an epoxy matrix 2 3

Experimental speeds of longi tudinal waves i n steel fiber/epoxy

matrix rods fo r various volume f r ac t ions of s teel , (Nevi11

e t a l . , 1972) ; waves t r a v e l l i n g along t h e f i b e r s

- 98 -

15

16

17

18

19

The e f f e c t of f ab r i ca t ion on shock wave spa11 damage

i n a boron fiber/aluminum matrix composite (Schuster

and Reed, 1969); a) brazed composite, b ) d i f fus ion

bonded composite

Experimental dispers ion r e l a t i o n f o r longi tudinal waves

propagating down t h e f i b e r s ( h a y e t a l . , 1968); a) graphi te

f i b e r (Thornel) re inforced carbon phenolic composite,

b) boron f i b e r re inforced carbon phenolic composite.

Experimental dispers ion r e l a t ions f o r waves i n boron f i b e r /

epoxy matrix composite (Tauchert and Guzelsu, 1972)

upper f igu re - longi tudinal waves normal t o the f i b e r s ;

lower f igu re - shear waves, x ax is is along t h e f i b e r s 3

Experimental dispers ion r e l a t i o n f o r longi tudinal waves

i n a tungsten fiber/aluminum matrix composite (Sutherland

and Lingle, 1972) lower curve shows second branch and a

cutoff frequency around 4 MHZ

Ortho t ropic phot oe 1 as t i c i t y experiment showing an anis0 t ropi c

extensional wave i n a p l a t e loaded with a lead azide charge

i n t h e center (Dally, e t a l . , 1971), compare with Figures

4, 27.

- 99 -

20

21

22

23

24

25

26

Dynamic s t r e s s - s t r a i n curves f o r steel fiber/epoxy matrix

composite under various s t r a i n rates (Sierakowski e t a l . ,

1970a) tests were conducted using compressional waves along

the f i b e r s

Impact damage i n a graphi te fiber/epoxy matrix p l a t e

( . 2 5 e m , 0 . 1 inches th i ck ) showing back face s p l i t t i n g

f o r 0 . 6 4 cm (1/4 inches) diameter steel b a l l s a t

115 m/sec i n i t i a l ve loc i ty .

+45 (Novak and Preston, 1972)

Ply lay-up angles t45', O o ,

0

Wave sur faces generated by a l i n e impact on a an iso t ropic

h a l f space (Kraut, 1963)

Contact times based on Hertzian model ca lcu la t ions f o r t he

impact of i c e b a l l s and g ran i t e spheres on graphi te f i b e r /

epoxy matrix h a l f space

S t a t i c experimental contact force r e l a t i o n f o r a 3/8 inch

diameter steel b a l l on graphi te fiber/epoxy matrix composite,

normal t o t h e f i b e r d i r ec t ion

S ta t i c experimental contact force r e l a t i o n f o r a 1/4 inch

diameter steel b a l l on boron fiber/aluminum matrix composite,

normal t o t h e f i b e r d i r ec t ion

Geometry of impact with a p l a t e , showing t h e effect of

motion o f t h e s t r u c t u r e

- 100 -

27

28

29

S t r e s s contours f o r t he membrane stress 1 / 2 ( t + t )

a f t e r impact f o r a 55% graphi te fiber/epoxy matrix

p l a t e . Comparison of a) O o , and b) 245 p ly lay-up

angle cases (Moon, 1972)

1 1 33

0

Stress contours f o r t he lowest f l exura l wave stress

1 /2 ( t

epoxy matrix composite p l a t e . Comparison o f a) t15

and b) t 4 5 p ly lay-up angle cases (Moon, 1972)

+ t 3 3 ) af ter impact f o r a 55% graphi te f i b e r / 1 1

0

0

Three dimensional p l o t of t h e lowest f l exura l wave,

1 /2 ( t + t ) , and a q u a r t e r plane o f t he p l a t e f o r

a 55% graphi te fiber/epoxy matrix composite p l a t e with

+45O p ly lay-up angles ( f i b e r s along diagonals)

(Moon, 1972)

11 33

I

!

i I

i ! ! i i I I

j i

! I

I

% .d c, c,

2 0 m rl cd a, a, m

n co u3 m A v

v

8-4 cd c, a,

6 9

2 2 a,

LL 0 c, rl

a, A

.rl ld k cd a

A cd c

3 c, .d M

c-7

4

g

d

!n

u .d A z %

E 8

a

P

U \ a, c, .d c k U

B

A rl

n

m A v

i?

k a, A A

i! n

a 8 : 5 ' b c ; k a,' a s i

n N r. m A v

a, rl

F

5 8 % s

TI cl

a

a A

3 v)

_.

m k a,

LL 0 c,

2

f;;

5 7 4

.rl a

; a,

a

A

n m \o

2 5

5

u

a, a a

w 3 cl s

2

L 0

w 3 0 v)

n N b

A v m

A cd c, a, A rl .rl >

n co u3 m 4 v

rl cd

n n m u3

2 v

a a 5 5 a ' a

% * 8 5 c , a 6 B %

a , a , a

E 2 a,

LL 0 c, rl a, rl rl cd k cd a

rl cd e .d a 1 c, .d M c 0 c-l

.rl

u3

u .r( rl 0 c a,

2 2 m g 8 % c, .rl

o c d a u \

a$ g U k

0

0\o 0

m

A

k a, .rl c, c,

2 a,

2 cd

_ _

E 2 a,

L

..-

e, s o e " ?

c d f

A 0 a , * A A

k C c d r a c

2 - 2 o a ,

LL

v) k a,

L

0 c, A

a, A

rl cd k cd a

2

4 cd c

1 c, .d M c 0 cl

3

LL 0 c,

k cd A a z E a, a

A cd e

3 c, .d M

c-7

4

k cd rl a 4 5 a k a, a -

f;;

6! a,

? rl

4 0 3 ' 9

0 c, A a, A rl cd k 6 a -

0 c, rl a, A rl cd k cd a

__

cd a a, a __

a, a -

rl td E

3 c, .d M

c-7

4

g

co M

9

0,

0 a w \ c

-- A cd c

7 c, .d M

4

!3

rl cd .d k a, c, cd E k a, 3 w 0

g .d c, c) cd k w

3 A 0 > a, c, cd u

c 4 .d

m a, M cd c,

5 :: a, a c\

u N

rl ' A cd ' c d c c a a

1 c,

3 c, .A .d

.d .d

F 0 1 ; 0

4 c-7 c-l . . .

? ? 4

d N V )

-

x .d k c, cd 22

f;; e cd

c

x c, .d u 0 rl a, > m a, c, cd V .d a c

n A

TI

U

u

$ ' g .. k j k k

0

O\O d !n

m

0

0

0

(0 03 Q,

d

0 0 0 * 0

W m 4 cu

0

0

0

co in m * 0

0 f- d

d

0

0

0

* f- m P(

0 0 %

0

0 T4

o m

cv u") In m 0

0

0

0

f- m Q, m 0

c- m a, m 0

m Q,

f- cv

0

0

0

d 0 co * 0

0 f- d

d

0

0

0

0 I? d

d

cv in

0 0 %

0

cv m m m 0

a ?

3 Om * +I

0

0

0

Q, f- cv * 0

0 f- d

d

0

0

0

Q, f- d

cv in

0 0 %

0

N Eo

0 0

cu In In m 0

a 2 3 0 0 m $I

0

0

0

0 0 * * 0

0 f- d

0

0

0

m a, 0

cv m

0 0 %

0

cv in m m 0

Fig. 1

Velocity, m mlpsec

F i g . 2

1 S OTROPi C / / I

Z

I- O z

15 y i=

30 2 0

0 2

0

CC

LL

45 Y (3 Z a

60 w a 3 3 75 0

I

>

-J v)

90 ' 15 30 45 60 a5 98

WAVE NORMAL ANGLE (DEGREES)

F i g . 3

Distance from impact point,

mm

Distance from impact point,

mm

Fig. 4

Fig. 5

4.0

3.0

(u

3 - 2.c

CY W

LL a

I .c

C

Oo WAVE NORMAL I1 goo ---

.- I I I I I

1.4

Fig. 6 WAVE NUMBER, kb

RAY LEIGH WAVE

XI

IN PLANE PLATE EDGE WAVE

_.--

EDGE WAVE

F i g . 7

Z Ts 8 Q n a W

I cn W v) J I) n

I '

B8 k

F i g . 8

I t,

*.L--- _e_l <---

0 em e e e e e e e

e 0 -: d- e e e e e

e e e 0

e -: m

e e e e e

F i g . 9

.O

I- 23

w 0.9 !3 >

r 0.8

w N i Q 2 Qf 0.7 0 2

0 0.2 0.2 0.6 0.8 d/X DIAMETER WAVELENGTH RATIO

I .o

F i g . 10

w w

0 2jr ka

STOP BAND i

F i g . 11

/ / /

(Do W N J a =E a 0 z

-

d-

N

0 d M cu -

Fig . 1 2 A 1 1 3 0 1 3 A 3SWHd 03ZllVWJtJON

Fig. 13

0 0

F i g . 14

L-

m

Fig. 15

r- 10

I . I t

IC 10

N

0

F i g . 16

.20

.I 5

. I 0

.05

n 0

v)

\ c

a 0

.- d

0" .20

.I 5

.I 0

.05

0

%

'\ 'a

0

/ / ' 4

e&---- &----- &

WAVE PARTICLE m SYMBOL DIRECTION DIRECTION

a x3 XI 4 X I x3

x3 x2 e I x2 x3 I

0

0 5.0 10.0 f (mhz)

F i g . 17

6.0 0 Q) cn

E

0 J W

W v)

' 5

a . r a. 5.2

I 3 UENCY, rnhz

4

14

€ E G t u W > W v)

I a.

'3

a

IO

8

6

9 FREQUENCY, mht

F i g . 18

t-J

Fig. 19

$0 I 0 X

v)

v) v) W

t- v)

- - n

a

4c

3(

2(

IC

33.3 x I o - ~ sec"

0 I I I I

0 0.02 0.0 4 0.0 6 0.0 8 0.10 STRAIN in / in

Fip, . 20

GRAPHITE /EPOXY

Fig. 21

F i g . 22

ICE BALLS GRANITE

---- - ft/sec

400 600 800 1000 1500 2000 I I I I I I i I I

-D= 1.0

- -s, -- ---% -- --- -- -- 0=0.5cm -\

I I -T---1 I 200 300 400 500 600 800 IO

100 IMPACT VELOCITY meters /sec

F i g . 23

10.0

F i g . 24

n Y- e Y

100

C

F i g . 25

F i g . 26

a

a 3 >- -I 0 0 Tf

I- Z 0 L

I-

.. ..

F i g . 29