NASA CR-54250 - - C A L Report AI-1821:A-3

COMMENTS ON THE SOLUTION OF THE SPALL-FRACTURE

PROBLEM IN THE APPROXIMATION OF LINEAR ELASTICITY

William J . Rae

prepared for

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

CONTRACT NO. NAS 3 - 2 5 3 6

CORNELL AERONAUTICAL LABORATORY, INC. of Cornel1 University

Buffalo, New York 14221 I

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NASA CR-54250 CAL Report AI-1821-A-3

TOPICAL REPORT

COMMENTS ON THE SOLUTION OF THE SPALL-FRACTURE PROBLEM IN THE APPROXIMATION OF LINEAR E+AS,TI,CITY

I i

William J . Rae

prepared f o r

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

. January 1965

CONTRACT NO. NAS 3-2536

Te c hni cal Mana geme nt

NASA - Lewis Research Center

Space Electr ic Power Office

Martin Gutstein

1'

CORNELL AERONAUTICAL LABORATORY, INC. of Cornel1 University

Buffalo, New York 14221

i

? c

F O R E WORD

T h i s d o c u m e n t d e s c r i b e s a p o r t i o n of the r e s e a r c h conduc ted i n a

s t u d y of t he app l i ca t ion of the b l a s t - w a v e t h e o r y of m e t e o r o i d i m p a c t to the

p r o b l e m of s p a c e r a d i a t o r des ign . O t h e r publ ica t ions g e n e r a t e d u n d e r th i s

NASA-sponsored p r o g r a m inc lude " N o n s i m i l a r So lu t ions f o r I m p a c t -

G e n e r a t e d Shock P r o p a g a t i o n i n S o l i d s , "GAL R e p o r t AI- 1821-A-2 , NASA

CR-54251 , J a n u a r y 1965, and "On t h e P o s s i b i l i t y of S i m u l a t i n g M e t e o r o i d

I m p a c t by t h e U s e of L a s e r s , "CAL R e p o r t AI-1821-A-1, NASA CR-54029 ,

A p r i l 1964.

i i i AI-1821-A-3

ABSTRACT

This report presents a study of the tensile s t r e s s e s produced when a

compress ive spherical wave is reflected f r o m a plane, s t r e s s - f r e e surface.

Two c l a s ses of exact solution of the problem, within the approximation of

l i nea r elasticity, a r e reviewed. In one c lass , the incident s t r e s s distribu-

tion is spherically symmetr ic , while in the other it is only axisymmetr ic ,

allowing a variation in one angular coordinate.

exis ts between the two c l a s ses in the laws relating the strength of incident

and reflected waves. The implications of this difference on the reflection

of impact-generated s t r e s s waves a r e discussed.

A significant difference

The exact solution f o r c a s e s where the incident wave exhibits spher -

ical symmet ry is examined in detail, and simple quadrature formulas f o r

obtaining the reflected s t r e s s profile fo r a given incident waveform a r e p r e -

sented. These resul ts a r e compared with the commonly used solution which

cons iders only the s t r e s s e s due to a source and its image.

under which this fu r the r approximation i s acceptable a r e discussed.

The conditions

The limitations of a l i nea r elastic model in predicting spa11 f rac ture

a r e pointed Out, and the a r e a s most in need of fur ther s tudya re indicated.

, /-

-

AI- 182 1 -A-3

TABLE O F CONTENTS

I

I i

111

IV

V

VI

Section

FOREWORD.

ABSTRACT

LIST O F SYMBOLS ~

INTRODUCTION

STATEMENT O F THE PROBLEM I

WAVES DUE TO A SPHERICAL CAVITY . WAVES DUE TO A POINT FORCE . SPHERICAL POINT -SOURCE SOLUTION . CONCLUDING REMARKS . REFERENCES .

P a g e

iii

iv

vi

1

4

10

21

32

47

49

0

V AI- 1 82 1 -A- 3

. LIST OF SYMBOLS

a in See. IV, the radius of applicatian of p re s su re p - s e e

Eq. (20); in Sec, function defined in Eq. (65)

constants defined in Eq. (Y l j

function defined in Eq. (65)

a,, 6, b

,$i, 9 in See. IV, functions defined in Eq. (34) ; inSec. V, functions

defined in Eq. (61)

function defined in Eq. (64) fi3(r)

C weak-wave speed in l inear shock-speed, particle -speed relation,

s ee Eq. (1)

Cl dilatational wave speed, 1.w' C, shea r wave speed, 4' /7' D

8, B' - E

F

function defined in Eq. (65)

functions defined in Eq. (76)

projecti le kinetic energy

displacement potential, Sees. I1 and V; magnitude of point force,

See. TV

functions defined in Eq. (47)

function defined in Eq. (105)

= c\/cz functions defined in Eqs. (49 - 51)

h t a rge t thickness

H(&l,) function defined in Eq. (16)

i !fi 1 angles defined in Eq. (58)

functions defined in Eq. (27) * N, 2

vi AI- 182 1 -A- 3

Qo,’/2

P,

R, I?’

S

s t

Lc, ‘J

c/T

pres su re ; also Laplace t ransform variable

magnitude of applied pressure - see Eqs. (20) and (21)

functions defined in Eq.. (23)

cylindrical coordinates

spherical coordinates

cavity radius

shock radius

in See. 111, distances defiped in the sketch accompanying Eq. (16);

in See. V, distances defined in Eq. (58)

distances defined in the sketch preceding Eq. (58) and in Eq. (87)

in Sec , 11, constant in the l inear shock-speed, par t ic le-speed

relation, s e e Eq. (1); elsewhere, s = ‘ / c z

t ime

shock speed, par t ic le speed, Eq. (1)

var iable defined in Eq. (64)

displacements in the r and 2 directions

displacement in the /? direction

functions defined in Eq. $101)

functions defined in Eq. (76)

= q t \/z + A J 3 variable defined in Eqs. (763 and (81)

in Sec. IV, functions defined in Eq. (28); in See. V, functions

defined in Eq. ( 7 6 )

Vi I AI - 1 82 1 -A- 3

~~ ~~

function defined in Eq. (79)

var iable defined by one of Eqs. (41 - 43)

angles defined in sketch accompanying Eq. (16)

L a m i constants

A Poisson 's ratio, i, = 2 O + p )

variable defined in Eq. (1 1)

ta rg et dens it y

normal s t r e s s

mean s t r e s s

shea r s t r e s s ; a l so used as a dummy variable; a lso as a time

variable in Sec. V, s e e Eqs. (69), (74), and (81)

displacement potentials, see Eq. (7)

denotes conditions, respectively, along the incident dilatational

wave, incident shea r wavey dilatational and shea r waves generated

by first reflection of incident dilatational wave

denotes rea l o r imaginary part

v i i i AI-1 821 -A-3

I. INTRODUCTION

The impact of fast-moving particles drives s t rong compression waves

into a solid body.

a r e reflected as tensile waves. Depending on the intensity and waveform of

the incident pulse, the tensile s t r e s s e s generated upon reflection m a y cause

a f r ac tu re nea r the f r e e surface, a process r e fe r r ed to as spallation.

Damage of this s o r t can be a source of grea t concern in cer ta in situations,

f o r example in the case of space radiator systems.

When such waves a r r i v e a t a s t r e s s - f r e e surface, they

2

A prec ise theoretical treatment of the spallation problem is made

difficult by the fact that different equations apply during successive s tages

of the deformation, In the ea r ly stages, f o r example, the compression con-

s i s t s of a strong shock wave, whose motion through the solid is descr ibed

by the equations of a compressible , inviscid fluid.

propagates into the target, and the mater ia l strength then begins to play a

role. During this stage, plastic deformation predominates f o r a while, and

ult imately the ent i re process becomes elastic.

The shock decays a s it

Many approximations have been proposed f o r treating this problem. .h

One of the s imples t of these'" is to use the inviscid, compressible-fluid

description up to the point where the incident wave first reaches the f r e e

sur face . F r o m that instant on, the motion is a s sumed to be descr ibed by

the c lass ica l equations of l inear elasticity, using the p r e s s u r e distribution

behind the shock to give the incident compressive s t r e s s distribution. The

ent i re problem is then reduced to the s imple question:

a r e developed along the axis of symmetry when a given compress ive s t r e s s

wave ref lects f r o m a plane, s t r e s s - f r ee boundary?

What tensile s t r e s s e s

.*a

-'* See, fo r example, Ref. 3, Appendix A.

1

~~~ ~

AI-1821 -A-3

?

The fact is that this question has never been thoroughly examined,

although the needed resul ts a r e contained in a number of papers , par t icu-

l a r ly f r o m the f ie ld of seismology. The principal objective of this repor t

is to extract these resul ts in a f o r m part icular ly suited to the spallation

problem, using them to evaluate fur ther approximations that can be used,

and to shed light on the effects of various parameters .

The approximation in which the spal l - f racture problem is to be

t rea ted in this report is formulated in Section 11.

there of the present s ta te of knowledge of the s t r e s s distributions that a r e

generated in hypervelocity impact.

to be a specified function, the question of its reflection f r o m a f r e e surface

is then posed in the l inear elastic approximation.

A brief review is given

W i t h the incident distribution considered

Solution of the specified elastic problem begins in Section I11 with a

review of a method of solution which uses a p re s su r i zed cavity a s the source

of the spherical disturbance, together with an image sys t em to p re se rve

the boundary conditions a t the f r e e surface.

various portions of the image system a r e presented in detail, in o r d e r to

a s s e s s their relative importance.

f i les which r i s e instantaneously to a maximum and then decay rapidly, the

maximum tensile s t r e s s a t the front of the reflected wave can be adequately

calculated by the approximation that considers only the source cavity and an

image cavity experiencing the same pressure , but with opposite sign.

The contributions a r i s ing f r o m

It is shown that, f o r incident s t r e s s pro-

In Section IV, a second c l a s s of solution is considered, in which the

incident wave is generated by the application of a point force to one f r e e s u r -

f ace of a slab.

angular variation not present in the spherically symmetr ic solution.

The wave that results is spherical in shape, but displays an

This

2 AI-1821 -A-3

angular variation gives r i s e to a n increase in the amplitude of the reflected

wave.

Section V returns to a consideration of the spherically symmetr ic

case , and the solution f o r a point source and i ts image sys t em is cas t in

the f o r m of a quadrature formula f r o m which the reflected s t r e s s profile

can be calculated, once the incident profile is known.

The report concludes with a discussion of the a r e a s where deficien-

c ies s t i l l exis t in the l inear-elast ic t reatment of the problem, and mention

is made of the most important a reas in which improvement over such a

model should be sought. Finally, the implications of these studies on the

interpretation of spal l - f racture experiments a r e pointed out.

3 AI-1 821 -A-3

11. STATEMENT O F THE PROBLEM

General Cons ide rations

When a projecti le s t r ikes a solid target at high speed, a shock wave

i s generated which becomes approximately spherical in shape a f te r a

period severa l t imes that required for the projecti le to t rave l its own length.

The shock decays in s t rength a s it propagates through the target and

ultimately degenerates to an elastic wave.

surface, it is reflected a s a tensile wave, whose amplitude may be sufficient

to cause a spa11 fracture .

t a rge t thickness is required to prevent such a f rac ture , when the projecti le

m a s s and velocity a r e specified.

4

If it encounters a s t r e s s - f r e e

The fundamental problem is to determine what

If the ta rge t is sufficiently thin, o r the impact speed sufficiently high,

the reflection will take place in a region where the target is incapable of

sustaining tensile s t r e s s , and a puncture results. At the other extreme,

where the target is sufficiently thick in relation to the sever i ty of the impact,

the reflection will follow the classical equations of elasticity.

mediate regime, the plastic and viscoelastic behavior of the target play a

role.

In the inter-

A rigorous theoretical treatment of such a complex problem, valid f o r

all regimes, is obviously quite difficult.

imate t reatment in which the plastic regime is ignored entirely; instead, the

ta rge t i s a s sumed to behave like a compressible, inviscid fluid up to the

instant when the shock reaches the f ree surface; thereaf ter , the c lass ica l

equations of l inear elasticity a r e assumed to apply, with the incident s t r e s s

distribution given in t e r m s of the pressure distribution behind the incident

shock. the purpose of this

r epor t is to examine i ts predictions m o r e fully.

This report p resents an approx-

3 This approximation has been proposed before;

4 AI- 1 82 1 -A- 3

t

It should be emphasized that the problem, when formulated this way,

is capable of solution without regard to the specific details of the incident

waveform. As is typical of l inear problems, it is possible to identify a n

indicial response; the solution f o r a specific excitation is then expressed

a s a convolution of the incident profile with this indicial response.

Incident-Wave Profile

A considerable amount of information is now available concerning the

ea r ly period of deformation, during which the target deforms as a com-

p res sible, inviscid fluid. Numerical solutions of the equations appropriate

to this regime were first reported by Bjork, 5 y who used the par t ic le- in-

cell method to obtain results in aluminum, iron, and tuff. More recently,

Walsh4’

to produce resul ts with great ly improved resolution fo r an expanded list of

made use of both a particle-in-cell and an Euler ian computer code

mater ia l s , including lead and polyethylene plastic. Para l le l developments

based on blast-wave theory have provided useful approximations. Many 8-1 1

of these analytical developments a r e summar ized in Ref. 1.

A typical representation of the p r e s s u r e distribution along the axis

of symmet ry is shown in Fig. 1. The coordinate sys tem used is a cylindri-

ca l one, with origin a t the impact point on the f ront face of the target.

5 AI-1821-A-3

The ta rge t thickness is denoted by h represents the (spherical) radius of the shock at any instant.

, while R, , a function of time,

The curves of Fig. 1, taken f rom Ref. 1, a r e based on the approxima-

tion that the flow can be represented as one half of a spherical ly symmetr ic

disturbance.

momentum is zero by symmetry.

the natural scale of distance in the problem,

cube root of the energy re lease divided by the character is t ic p re s su re in the

Such a flow is sensitive only to the total energy available; its

12 Thus, a s in a l l explosion problems,

, is proportional to the

medium. It was shown in Ref. 9 that this p re s su re is givenby /Fz ’ where Po is the target density under normal conditions, and C i s the

weak-wave velocity appearing in the l inear shock speed-part ic le speed rela-

tion

F o r most mater ia l s , the value of C is approximately equal to the bulk

dilatational wave velocity. Typical values of S l i e in the range f r o m 1

to 2.

13

It should be emphasized that the curves of Fig. 1 a r e intended only to

convey a general impression of the disturbance generated by impact.

quantitative significance is l imited by the approximations on which they a r e

based.

a r e not spherically symmetr ic ; f o r

would differ somewhat f r o m that shown in Fig. 1. Fur thermore , scaling

with respect to energy alone is not exactly cor rec t , a s Walsh and his

co -wo r k e r s have shown.

Their

F o r example, the s t r e s s distributions generated by an actual impact

e # 0 , the p r e s s u r e distribution

4, 7

6 AI- 1 82 1 -A- 3

The general nature of the pressure distribution reveals a discontinuous

jump a t the wave, followed by a relatively s teep decay.

have a l so been observed in the results of Allen and Goldsmith,

These features

14 who ca l -

culated the s t r e s s waves generated by applying, to the inter ior of a spherical

cavity in an infinite medium, a pressure that r i s e s instantaneously to a

maximum, and then decays exponentially.

Conversion to Incident S t r e s s Profile

In the present approximation, the mater ia l response is assumed to

become elastic a t the instant when the incident wave reaches the r e a r s u r -

face 2 = /q . With such an approximation, the basic differential equations

describing the motion become linear. Thus it is unnecessary to be specific

about the waveform of the incident pulse. It is sufficient to find the solu-

tion f o r a step-function; results for an a r b i t r a r y waveform may then be

found f r o m a Duhamel integral. Nonetheless, it is instructive to indicate

one method by which the incident waveform might be specified.

F o r example, the incident s t r e s s distribution can be calculated by

equating the p r e s s u r e f f rom the fluid-mechanical theory to the mean

s t r e s s . Allen and Goldsmith have pointed that the product CT

t imes the spherical radius e satisfies the plane-wave equation, i. e. ,

/ where and p a r e Lame constants, k the t ime, /=(e) the value of

the displacement potential a t R = 0 , and C, is the dilatational wave

speed

xc2y-i c,' = Po

(3)

7 AI-1821-A-3

.

Thus, specification of the pressure versus distance a t a given t ime t, is

equivalent to specifying Cr, at a t ime, f o r the s t ress -wave analysis ,

Once the function F

ponents; f o r example

is known, i t can be used to find the other s t r e s s com-

Knowledge of the function /= over a given range of a t one instant is

equivalent to knowing i t over a displaced range a t a l a t e r instant, since it

depends only on the character is t ic coordinate k- Rl,(-, :

D E D E

B *I? I *I?

Thus, known conditions along A 0 a r e sufficient to determine conditions

along DE . To find out about the range C D , it is necessary to know

what was happening a t e = 0 during the time interval AC . F o r incident waves which lack spherical symmetry, the above consid-

erat ions do not apply, and a m o r e elaborate description of conditions behind

the incident wave is necessary.

8 AI-1821-A-3

Simple Reflection Formulas

Much of our present understanding of the mechanism of spa11 f r a c t u r e

’ G A %

has come f r o m consideration of the plane-wave c a s e (see, fo r example,

1%

the t reatment by Rinehart and Pearson in Ref. 15). In that case , the exact

F R E E sur?Fqc& -

solution of the equations of elasticity reveals that the reflected wave is the

z I c / -

E I -- . . . y, /’ ”

inverted image of the incident wave:

I

I I I - A simple extension of this solution to the spherical c a s e is to a s sume that

the solution is still given by the sum of the incident wave and i ts inverted

image, allowing both contributions to decay with distance the way they would

solution of the problem. One of the objectives of the present report is to

determine how good the approximation is.

9 AI-1821-A-3

.

111. WAVES DUE TO A SPHERICAL CAVITY

. One means of studying the reflection of spherical s t r e s s waves is to

examine the disturbance produced by applying a p r e s s u r e to the in te r ior of

a spherical cavity in the medium. The waves that resul t have spherical

symmetry, and the i r reflection at a f r e e surface has been considered in

recent papers by Aliev” and by Kinslow. l 8 This section presents a review

of their work, noting the physical significance of the various contributions

to the solution.

Basic Relations

The cavity, of radius Kc , is located with its center a distance h f r o m the f r e e surface. Cylindrical coordinates and Z

radius are measured f r o m the center of the cavity

, and spherical

Displacements in the r , 3 and e directions a r e denoted by U ,

and /J , respectively. In t e r m s of these displacements, the normal

s t r e s s e s cj- and shea r s t r e s s e s Z a r e given by

10 AI-1821-A-3

T = q - c - ax 9J- +") + z p U cp ar d t Y

while the equations of motion take the f o r m

+ %- f- 2'G-Z ar a t r

These equations can be decoupled by use of the displacement potentials , , and @ , defined by

11 AI-1821 - A - 3

The equations that resul t a r e

where C; = r/p. . Source Solution

The solution f o r the case where the cavity experiences a p r e s s u r e

0 , t d 0

( 9 ) %)= R= e, ( - p q -t 2 0

has been given by Blake, l 9 and others. It can be wri t ten in the f o r m

where

= c , t - 14

and where Poisson’s ratio is denoted by -$

the displacements and s t r e s s e s have been presented by Blake, l 9 by Aliev 17

and by Allen and Goldsmith, l4 who present detailed resul ts for the c a s e

. Complete expressions f o r

12 AI-1821 -A-3

Image System

In o r d e r to find how this disturbance is reflected f r o m the f r e e s u r -

face, one is l ed to consider the effect of a second spherical cavity, a l so of

radius RC , and located a t 2= 2h . Aliev makes use of such an

image, allowing it to experience the s a m e p res su re as the cavity a t 2 = 0 .

The resul t is to produce ze ro shear s t r e s s at the f r e e surface, but a non-

z e r o normal s t r e s s CE . If, on the other hand, the image cavity exper-

iences the same p res su re a s the original one, but with opposite sign, the

s t r e s s TE

is developed. In e i ther case, the cavity a t 2 = z h is not enough; the

is made zero at the f ree surface, but a net s h e a r s t r e s s TfE

complete image sys tem must include a second contribution distributed along

the f r e e surface in such a way a s to cancel the nonzero s t r e s s due to the

two cavities. .b 1-

The s t r e s s components a r i s i n g f r o m the two cavities can be visual-

ized by considering a spherical displacement which decays with e , c o r -

responding to a compression

J.

''. P r o f e s s o r Norman Davids of the Pennsylvania State University has pointed out to the author in a private communication (April 23, 1964) that the s a m e conclusion can be reached simply by considering a source potential The author is ve ry grateful to Professor Davids f o r a v e r y informative discussion of this problem.

# = r-' f (r- c,t) , and an image potential c$ = -r- '$(r+G i) .

13 AI-1 821 -A-3

F o r the case where a compression is applied to both cavities (as Aliev

does), this sketch of versus can be used to construct the displace-

ments a t four points (all having different values of /e ) nea r the f r e e s u r -

face. The displacements have the appearance

&; \

fi \

7

F r o m this sketch i t can be seen that the displacement gradients a r e a s

follows

.r

I Note that, at the interface, the contributions to and to (2x& '

coming f r o m the two cavities are of opposite sign, while the contributions

to i""/..), and(a%t)r a r e of the same sign. Thus the net effect of two

compress ive cavities is to produce zero shea r s t r e s s

but a nonzero normal s t r e s s T2 . Tr2 a t the surface,

On the other hand, if a tensile s t r e s s is applied to the inter ior of the

image cavity, so that it generates displacements which increase with

14 AI-1821-A-3

the resultant displacements have the appearance

and the associated gradients a r e as follows

In this case, the contributions to L(, (hy'&) and a r e of oppo- t

s i te sign and cancel, while the contributions to

add. Thus there is no normal s t r e s s T2 a t the f r e e surface, but a net

s h e a r s t r e s s ,L is developed, whose sign is negative

2Uht and (av!r) 0 t

' ft

15 AI - 1 8 2 1 -A- 3

In addition to the image cavity, then, the complete solution requires the

application of a positive distribution of

Effect of the Surface Distribution

As mentioned ear l ie r , the approximation is often made that the surface-

Trt- on the f r e e surface.

distributed shea r s t r e s s e s can be neglected, and that only the compressive

source cavity and tensile image cavity need to be considered.

work, l 8 f o r example, is based on this approximation.

Kinslow's

To est imate the accu-

racy of the approximation, it is necessary to determine the s ize of the con-

tribution f r o m the surface s t r e s ses . An answer can be found by making a

slight modification of Aliev's results, since the solution f o r a compressive

sou rce and tensile image can be found mere ly by changing, in Aliev's work,

the sign of the displacements due to the image (for example, the sign of fz

in Aliev's Eqn. 2 .4 is reversed). F o r the case where the source cavity

experiences a step-function pressure

the result ing shea r s t r e s s produced a t the surface is given by

.

16 AI - 1 82 1 -A- 3

The angle 9 is defined a s &s - ' r , as shown below R

and 9, and b, a r e defined a s before, s o that

These results a r e shown in Fig. 2 for the case - h = 3 , 3 = - I . e 4 As noted above, the shear s t r e s s produced b; the two cavities is

negative; to cancel it, a positive s t r e s s of the same magnitude must be

added, resulting in s t r e s s e s within the target, in addition to those gener-

a ted by the cavities.

facts a r e important. The first is that Tr8 is of the o r d e r of

i. e . , i t is generally of the same order a s the s t r e s s e s produced by the

cavi t ies . On the other hand, i t is significant to note that the additional

applied s t r e s s is ze ro a t

make no contribution along the axis a t the f ront of the reflected wave.

In assess ing the magnitude of this contribution, two

P, ' ) 2 5 h

r = 0 ; thus, according to ray-theory, 2o it will

The

17 AI-1821-A-3

s a m e conclusion is derived below in Section V f o r the c a s e of a point-source

disturbance, namely, that fo r incident waves having spherical symmetry ,

the tensile s t r e s s developed at the front of the reflected dilatational wave,

along the axis of symmetry, i s exactly that given by the source and i ts

image. Behind the front, the simple solution is no longer cor rec t , but even

there the approximation will be shown (in Section V) to be acceptable pro-

vided the s t r e s s amplitude decays rapidly enough behind the incident wave.

Off -Axis Conditions

At points located off the axis of symmetry, but on the reflected dila-

tational wave front, the effect of the surface s t r e s s e s is compressive. Thus

the source-plus -image approximation will overest imate the tensile s t r e s s

at these points.

identifying the sur face s t r e s s e s that mus t be added, he approximates their

This conclusion can be discerned in Aliev’s results. After

effect by r ay theory, using fo r this purpose the resul ts of Bagdoev, a

s u m m a r y of which has recently appeared in translation. 21 Aliev’s resul t is

c that the jump is s t r e s s [r2] across the reflected dilatational wave is

given by

9” 1 7 1‘- where Pt and e, a r e defined in the sketch above, and f70 is the value

of the p r e s s u r e in the source cavity a t & = 0’ . Figure 3 shows the va r i a -

18 AI-1821-A-3

tion of with e, f o r 3) = 1 / 4 . The leading t e r m (unity) in the l a s t

bracket of Eq. (15) represents the tensile jump, due only to the source and

image. The next t e rm, involving /-/ , is of opposite sign, representing

a compressive effect.

Thus it appears that, f o r incident waves having spherical symmetry,

the source- image construction is exact a t the front of the reflected wave,

along the axis of symmetry, where the maximum tensile stress is likely to

be developed.

would add a compressive contribution, while at points on the axis of symme-

t r y behind the reflected dilatational wave , the considerations given in

Section V suggest that a more exact solution does not a l t e r the maximum

tensi le s t r e s s , at l ea s t f o r incident wave fo rms which decay rapidly.

Kinslow's Work

F o r points off the axis of symmetry, a m o r e exact solution

The above conclusions a r e of importance in interpreting the work of

18 Kinslow, who makes use of the source-image approximation, applying it

to incident waves which a r e finite sums of the Blake-type19 solution. The

p r e s s u r e in the source cavity i s taken to be

Actually, because of the conditions applied to find the coefficients

de rivative s of

a re s e t equal to zero a t

can be summed a s

f ; (all

,

t = 0 ) , Eq. (17) becomes a binomial s e r i e s , which

f9 , beginning with the f irst and ending with the ( M - l ) d

1 9 AI-1821-A-3

Kinslow presents resul ts f o r selected values of d and Y) , evaluating , f r o m the formula

P

where c is the target s t ress-wave speed, , the radius of the c r a t e r

that would be produced in an infinite target ,

face, and 'J the par t ic le velocity behind the incident shock when it

reaches the f r e e surface.

r the depth of the f r e e s u r -

20 AI - 1 82 1 -A - 3

IV. WAVES DUE TO A POINT FORCE

A second means of representing incident s t r e s s waves like those

encountered in hypervelocity impact is to consider the application of a force

to one side of a s lab of finite thickness.

which propagates into a s lab due to the application of a distributed force

The solution f o r the disturbance

23 The and by Thiruvenkatachar. 22 has been presented by Huth and Cole,

solution describing the reflection of these waves by a second f r e e surface

has been given by Thiruvenkatachar. The waves induced by application

of a point force, and the i r subsequent reflection, a r e discussed by Broberg,

and by Davids.

24

25

26

A point force generates a wave whose shape is spherical , but the

Fur thermore , s t r e s s distribution behind it is not spherically symmetr ic .

an incident shea r wave is generated; no such fea ture appears in the solu-

tions of the previous section.

The most distinctive feature of the solutions discussed below is that

the laws relating the s t rengths of incident and reflected waves differ consid-

e rab ly f r o m those that apply fo r spherically symmetr ic waves.

purpose of this section is to discuss this feature.

The main

Distributed, Step-Function Load

It is instructive to s t a r t f rom the solution fo r a step-function p r e s s u r e

applied over a c i rc le of radius a

21 b

AI-1821-A-3

i P

I 2a

/ Z = O , 2 - = h

Te = 0 2=c)

The c a s e t rea ted by Thiruvenkatachar 239 24 makes use of a delta function

instead of the unit s tep

point force F applied a t = 0 . Thus i t should be possible to recover

the l a t t e r resul ts by taking the limit o , P-c , in such a way

that

J- (6) . The problem solved byDavids is that of a

Both of these authors employ the Laplace t ransform to der ive formal ex-

press ions fo r the t ransforms of the various s t r e s s components. These ex-

press ions a r e then expanded in se r i e s whose successive t e r m s represent

the incident, reflected, and multiply reflected waves. F o r example, it is

possible to sketch the progress of the various waves along the axis = 0

in a f diagram >

22 AI - 1 82 1 -A - 3

I

0 h In the analysis that follows, only the incident dilatational (P) wave, and

the reflected dilatational (Pf) and s h e a r (p5) waves a r e accounted for.

Thus the resul ts given below give no information about the s ta te of affairs

that exists a f te r passage of the incident shea r ( 5 ) wave. .I. -,-

The Laplace t ransform of the solution f o r an applied p r e s s u r e

can be found by replacing p , wherever it occurs in Thiruvenkatachar 's

analysis , by p/*

p l ( e )

, where 8 denotes the t ransform variable

00

Rj9 = 1 m e - P t & (22)

T2 , in

0

With this replacement, the t ransform of the s t r e s s component

the region affected only by the incident dilatational wave, and the reflected

dilatational and shea r waves (i. e . , the tr iangle, in the sketch above,

bounded by the l ines f l = h , t =&/& ) is 24 t= 0 ,

% - - - c [ ~ o + ~ , p + Q z -* i J.

"* Very l i t t le is known at present about the charac te r i s t ics of waves of this type that a r e generated in hypervelocity impact. do not admit such waves, and they have apparently not been observed a s yet i n solutions which include mater ia l strength. 27

Hydrodynamic analyses '> 4 y

23 AI-1821-A-3

where

In these expressions, To and denote Bessel functions of the f i r s t

kind. The functions h/l , dL , a( , and 1 are given by

p =/E'+ tx; In general , the inversion of these t ransforms i s ra ther complicated, but can

be effected by use of Cagniard's method. 28 The process is relatively

s imple r when applied to conditions along the axis of symmetry, par t icular ly

a t the wavefronts. The remainder of this section is devoted to the extraction

of these results.

Inversion of the Transforms

Thiruvenkatachar begins the inversion of these integrals by making

the change of variable

24 AI - 1 82 1 -A- 3

which leads to

where

The next s tep is to use an integral representation for the Bessel-function

p roduc t

25 AI- 1821 -A-3

where , %- , and a r e defined by

7 - If the re- Note that >c depends on a , v , , and $ , but not on

maining Besse l function is a l so replaced, by the integral representation

and if the o r d e r of integration is changed, the result is

26 AI- 182 1 -A-3

Cagniard 's technique is now applied to these, by making the changes of

variable, in (38), (39), and (40), respectively

cLt = h (gJ;;;;"+dw)-E/- + L f y j P ? a s f (43)

var ies f r o m ze ro to infinity, f- t r aces out a path in the complex As 7 plane:

The path of integration can be deformed, 28 so that it l i es along the real

axis in the t - p l a n e , and the result is

AI-1821-A-3

c,

where

, - 7

and where the (complex) var iable

the ( r ea l ) var iable

(46) can now be inverted by inspection.

tributions of the three waves to the s t r e s s along the axis of symmet ry a r e

found to be

is defined a s an implicit function of

by Eqs. (41) - (43), respectively. Equations (44) -

If r is s e t equal to zero, the con-

where

28 AI-1821-A-3

and where e is to be replaced by Q in Eqs. (41) - (43). If the limit

Q-0 , p-ar, is now taken, with abz vr , then no longer

depends on 9 ; the integrals in (49) - (51) can be worked out, and the result

is

In these last three equations, -r('

by Eqs. (41) - (43), with e = 0 .

is defined a s an implicit function of ?k

29 AI- 182 1 -A-3

S t r e s s Discontinuities at the Wavefronts

Along each wavefront, is zero. The values of the s t r e s s e s

at the wavefront can be determined rel- QP , Q)PP ’ GhS

atively easily, and a r e found to be

In c ross ing any given wave, the contributions f r o m the other two waves a r e

continuous; thus Eqs. (55) - (57) also give the s t r e s s discontinuities that

occur a c r o s s the various waves.

All of the foregoing formulas ag ree with Davids’ resul ts . 26 Equations

(55) - (57), for example, can be used to recover the stress discontinuities

shown in Fig. 8 of Ref. 2 6 , and can a l so be derived f r o m the general

formulas given there.

(55) - (57) add up to ze ro a t the f r e e surface

Other checks can a l so be made; fo r example, Eqs.

t = h . The most striking feature of the expressions f o r the stress discontin-

uit ies l i es in the fac t that they differ considerably f r o m those describing

spherical ly symmetr ic waves. In particular, the s imple source-plus- image

approximation would s a y that the formula f o r the jump in s t r e s s a c r o s s the

reflected dilatational wave ought to be derivable f r o m the incident s t r e s s

30 AI 1 82 1 -A- 3

formula [Eq. (55)) by changing the sign, and replacing 2 by zh-2 . But such is not the case.

f lec ted a s a tensile wave of amplitude ( 1 + 2 4 + 3 ) 4 , + 8 % 3 ) , a number

which is typically on the o r d e r of two.

A compression wave of unit amplitude is r e -

The reflected s h e a r wave then im-

pa r t s the amount of compression needed to preserve the ze ro s t r e s s condi-

tion at the f r e e surface. Moreover, the contribution f r o m the reflected

shea r wave is nonzero even at as h , in fur ther cont ras t to the spher i -

cal ly symmetr ic c a s e

It is c l ea r that such an amplification of the tensile s t r e s s e s generated

To upon reflection has an important bearing on the spa11 f r ac tu re problem.

fully a s s e s s its importance, however, two factors must be examined.

first is to determine how the amplification factor f o r reflected waves de-

pends on the degree of departure f rom spherical symmetry.

t he re i s no information available at present that might be used to infer this

dependence.

the incident s t r e s s distributions encountered in hypervelocity impact depart

f r o m spherical symmetry. At present, the only pertinent information avai l -

able consis ts of the p r e s s u r e distributions found in the numerical solutions.

The

Unfortunately,

The second i tem requiring attention is to determine how far

31 AI-1821-A-3

V. SPHERICAL POINT-SOURCE SOLUTION

The discussion of the two previous sections indicates that a definitive

solution fo r reflected s t r e s s profiles cannot be given until the effect of

off-axis variations in the incident s t r e s s e s have been examined m o r e fully.

In spite of this reservation, it is nevertheless instructive to pursue the

solution of the spherically symmetr ic c a s e further. This section presents

such a study, using f o r the excitation a point source instead of a cavity of

finite radius. The purpose of this section is twofold: f i r s t , to show explicitly

how the classical resul ts of Cagniard, 28 long known in the field of s e i s -

mology, can be applied to the present problem; and second, to a r r i v e a t

some ve ry general conclusions about the shape of the reflected s t r e s s pro-

fi le and the influence of the incident profile by writing the solution in t e r m s

of a Duhamel integral.

Basic Relations

The coordinate sys tem is the s a m e a s that used ea r l i e r , except that .?I *F

some new notation is introduced

FREE S U g F A C r

.b -,* The 2 -coordinate used he re would be h - 3 in Cagniard'sZ8 notation.

used h e r e differ by Thus, the -f -displacements and the derivatives a s ign f rom those of Cagniard. The s t r e s s TE , which involves only W / B t , can be taken directly from Cagniard's results by s imply replacing 2 by h - 2 -

32 AI - 1 82 1 -A - 3

/ e and denote distance f r o m the source and image, respectively,

while e, and cL descr ibe the minimum-time path f o r a wave that t ravels

f r o m the source to the f r e e sur face a t the dilatational-wave speed, and

thence to the field point 2 a t the shear-wave speed. The angles I; and I, and the distances 6 and pt a r e determined f r o m the rela-

tions (for given r and 2 )

h = COS r, h - 2 = R, C O S T ,

.*. ' 8 .

In t e r m s of the displacement potentials defined in Eq. (7),

ponent F2 along the axis of symmetry f = 0 is

the s t r e s s com-

Method of Solution

CagniardZ8 wri tes the general solution in t e r m s of two influence func-

tions A and B :

t # = 1 F/ ( t -e ) A ( r j 5 % ) de

0

.b

"* CagniardZ8 uses y!' and fo r what a r e called 4 and $J , respec- tively, in this report.

3 3 AI-1821-A-3

where /= (tb)

present purposes to be given in te rms of the incident s t r e s s distribution by

, the displacement potential at the source, is considered f o r

Eq. (2).

The values of tf and change discontinuously a t various instants,

signalling the a r r i v a l of various waves.

f o r in Eq. (59) under the integral signs in Eqs. (60) and (61), there will be

contributions f r o m the r - and Z -dependence of the integration limits.

F o r the moment, these contributions will be ignored; they will be added

la ter .

Thus in taking the derivatives cal led

Carrying the derivatives under the integral sign gives

rl.

By use of the Laplace t ransform, Cagniard shows that“’

where the t ransform of A3 is

and where

a ’ J -

.tr

“’ The notation S = I/=, , s = ‘/c, , introduced by Flinn and Dix, is used here .

34 AI-1821-A-3

The three terms in Eq. (63) correspond to a source, an image of opposite

sign, and a third t e r m caused by the sur face s h e a r s t r e s s e s .

The t ransform of B(’d) is .

Inversion of the Transforms for r = 0

Most of the emphasis in Cagniard’s t reatment is on the c a s e where

> > i , the case of grea tes t interest to seismologis ts , and the inver -

sion of the t r ans fo rms is a f a i r ly complicated problem.

e r e d he re , f = 0 , i s the opposite extreme, and Cagniard’s procedure

f o r inverting the Laplace t ransforms can be applied ve ry simply.

tiating Eq. (64) twice with respect to r gives

The c a s e consid-

Differen-

and with the new variable

Eq. (68) becomes

35 AI- 1 82 1 -A- 3

All functions in the integrand on the right a r e considered to be functions of

2 ; f o r example, = DLL((r)j . The inversion can now be accom-

plished by inspection, giving

The reason f o r 'evaluating this f i r s t integral , ra ther than the function itself,

is to facil i tate a l a t e r integration by parts.

s teps can be used to show that

Exactly the s a m e s e r i e s of

where, as in (71), L( (7) is defined by Eq. (69).

Differentiating Eq. (66) with respect to and 2 , and setting

r = 0 gives

36 AI-1821-A-3

In this case, the appropriate change of variable is

which a f te r inversion leads to

It is useful to put these resul ts in dimensionless form, by introducing the

notation

When the differentiations with respect to ? indicated in Eqs. (71), (72),

and (75) a r e c a r r i e d out, and the results expressed according to the notation

in Eq. (76), the resul ts a re :

c

37 AI- 1 82 1 -A- 3

In Eq. (77), the parameter r is defined as

and the quantities x , , o( , / , and b' are a l l functions of

, given by Eq. (69), which becomes

x = fC In Eq. (78), the functions

tions of as before (defined in Eq. ( 7 6 ) ) , but the dependence of x on

and 2 is different. It is given by Eq. (74), which becomes:

o( , p , B , and B' a r e the same func-

The inversion of this leads to a quadratic in K" , the co r rec t solution of

which uses the minus sign:

38 AI-1821-A-3

I -

where

The functions (v, 4 d L ) (2- Z h ' = , and 1/3 a r e shown in Figs. 4 and

5 fo r +$ = 1 / 4 . and 1/, a r e zero for 341 , while V3 is ze ro

f o r & C T < .+% (, -?h) . It should be noted that the s u m I/: qvz + I/ h 3 vanishes at the f r e e surface s / h = 1 .

T e r m s Arising f r o m the Limits

When taking derivatives under the integral sign in finding the expres-

sion f o r Gz , t e r m s ar is ing from the r- and ?-dependence of the

integration l imits were omitted. These t e r m s must now be added. During

the t ime interval between the a r r iva l of the incident dilatational wave and

that of the reflected dilatational wave, the potentials a r e simply

I ct=, S = O

and the corresponding s t r e s s along the axis is that given e a r l i e r in Eq. (4)

3 9 AI- 182 1 -A- 3

where the argument 2-s~ is denoted by (-) . After a r r i v a l of the

reflected dilatational wave, but before the reflected shea r wave, the poten-

tials a r e = 0 , and

In arr iving a t the first two t e r m s of this expression,

be ze ro (as is F’(o)

evaluating the discontinuities in the solution. Differentiating Eq. (86)

FCo) is taken to

below) in accordance with Cagniard’s procedure f o r

twice with respect to f leads to a v e r y lengthy expression involving,

where

When f-o ,

Many of the t e r m s in the lengthy expression f o r

C = 0 , leaving:

~x~/ ’rL vanish when

40 AI - 1 8 2 1 -A - 3

where the argument & - s (zh- %) is denoted by ( t ) . The derivative

can be worked out in s imi l a r fashion, using the fact that, B 2 4 / 4 f- 0

a s Y - 0 Y

The resul t is

When the reflected shea r wave passes , the t e r m s derived f r o m 6 do not

@ change, but a new contribution comes f r o m

t

sn, + SPL

=[ F ’ ( t - r ) R ( T ; c t ) d ’ i y (92)

When the mixed derivative of this quantity with respect to r and 2 is

formed, and evaluated f o r r = 0 , it becomes necessary to know cer ta in

derivatives of and pL . These a r e most easi ly worked out by using

the approximate f o r m of Eqs. (58) valid for F-0

h J,

.’ J J

= h t oha) (93)

41 AI-1821-A-3

F r o m Eq. (93) it follows that a"r , a</at , %-.Ar ,

az'x/a,,.aE a r e all ze ro a t V = 0 , while anL The resul t is

If all the extra t e r m s a r e now coilected, and the convention used that F

and all its derivatives a r e zero for negative argument, the complete ex-

press ion f o r the s t r e s s becomes

F u r t h e r simplifications of this expression a r e now possible.

l ines correspond to the solution that would be given by the source and i ts

image.

cat ion of Cagniard 's method, it can be shown that

The first two

Some of the remaining te rms a r e zero ; by a straightforward appli-

42 AI - 1 82 1 - A - 3

Thus A3 (3 ( ~ h - 2 ) ; 0, 2) = 0 . By differentiating (96), it is found that

- 16 - - j3s (zh-2)”

and it can a l so be shown, either by differentiating (96) o r by inverting

aA3/23 , that

Finally, inverting ashr reveals that

Thus Eq. (95) is reduced to

(97)

The l a s t two t e r m s can now be absorbed through an integration by parts.

Le t

4 3 AI-1821-A-3

Then

Since

I 6 - - - 4 lx =o - c, h” 9 3 ( I + (h-t)/$q y

The final expression fo r f12 can be wri t ten in the f o r m

44 AI-1821-A-3

, -

The integral appearing he re collects a l l of the correct ions to the source-

image solution that a r i s e f r o m the surface-distributed s h e a r s t r e s s . It

introduces no discontinuities, even a t the reflected shea r wave, and has

the effect only of altering the profile.

Final Formula

It is useful to employ the dimensionless variables

In t e r m s of these, Eq. (104) becomes

The function a/' appearing in this integrand is to be found f r o m the inci-

dent s t r e s s profile, f o r example by the method descr ibed in Section 11.

The variation of the influence function appearing in the integrand of

Eq. (106) is shown in Fig. 6 f o r the case z/h = 0. 8 , c, t/ h = 1. 5 ,

9 = 1 / 4 . F o r points a s yet unaffected by the reflected s h e a r wave, the

effect of the integral i s to contribute an additional tensile component.

passage of the shea r wave, a compressive contribution is felt.

the integral correct ion t e r m will not be very grea t in cases where

falls sharply f r o m its maximum, since the integrand rapidly becomes smal l

under such circumstances.

General Conclusion

After

Actually,

2 "

It must be emphasized that Eq. (106) is res t r ic ted to cases with spher -

ically symmetr ic incident waves; f o r such cases , the integral in Eq. (106)

45 AI-1 821 -A-3

represents , in a general but concise way, the correct ion to the source-

image sys tem due to the surface-distributed shea r s t r e s s e s . Generally

speaking, the correct ion t e r m will have the effect of modifying the

ref lected-stress profile slightly, but will leave the maximum tensile s t r e s s

unchanged, especially fo r profiles which decay rapidly.

46 AI-1821-A-3

VI. CONCLUDING REMARKS

It is well to re i te ra te in conclusion the approximation that fo rms the

bas i s of this entire study, namely, the use of a l inear e las t ic model. In

applying the results above to the interpretation of spal l - f racture experi-

ments, se r ious reservations must be held concerning the effects of non-

e las t ic response.

no attempt has been made to invoke a f rac ture cri terion. Even if the l inear

e las t ic model is accepted, one must still decide whether o r not the f rac ture

s t r e s s is time-dependent. It is unfortunately the case that the resul ts of

spallation experiments a r e the product of an unknown mixture of wave-

propagation effects and fracture-cr i ter ion effects.

l i t t le hope of drawing firm conclusions about the relative importance of

these two sources of uncertainty, except f r o m experiments done in internally

instrumented targets.

These may affect the wave propagation itself. In addition,

There would appear to be

On the other hand, it is important, in constructing a rational theory

f o r this problem, to be s u r e that all the implications of even the most

primitive approximation a r e fully understood.

s e a r c h has been to reveal the significance of the simple source- image

model.

exact l inear-elast ic results for cases where the incident wave is spherically

symmetr ic . F o r such cases , and for incident s t r e s s distributions which

dec rease rapidly behind the incident wave, the maximum tensile s t r e s s (on

the axis of symmetry, a t the wave front) can safely be calculated by the

source- image model.

One contribution of this r e -

Results derived f r o m such a model depart only slightly f r o m the

A m o r e important result has been to call attention to the fact that the

reflection laws inherent in the source-image model may be drast ical ly

4 7 AI- 1 82 1 -A- 3

changed f o r cases where the incident s t r e s s distribution is not spherically

symmetr ic .

f lected with more than unit amplitude f r o m a f r e e surface.

not enough is known about this solution a t p resent to indicate the connec-

tion between the degree of amplification and the extent of departure f r o m

spheric a1 symmetry.

In such cases , compression waves of unit amplitude a r e r e -

Unfortunately,

One of the i tems necessary in elucidating these questions is a physical

explanation of why an amplification occurs a t all.

a study be made of incident s t r e s s distributions whose angular variations

a r e intermediate between those of the spherically symmetr ic case and those

generated by a point force.

p resent in existing hydrodynamic and viscoplastic solutions should be exam-

ine d.

It is a l so important that

Finally, the extent of angular variations

Until a thorough investigation of these points is made, it mus t be

recognized that there is no firm theoretical basis fo r many of the conven-

tional rules of thumb often used in dealing with spal l f r ac tu re - - fo r example

the unit reflection coefficient, and the relation between the spall thickness

and the wavelength of the incident disturbance.

48 AI - 1 82 1 -A- 3

REFERENCES

1. Rae, W. J. , Nonsimilar Solutions fo r Impact-Generated Shock

Propagation in Solids. CAL Report AI-1821-A-2, NASA CR 54251

(Jan. 1965).

2. Loeff ler , I. J . , Lieblein, S . , and Clough, N., Meteoroid Pro tec t ion

f o r Space Radiators.

Flight, Ed. by M. A. Zipkin and R. N. Edwards. Volume 11 of

P r o g r e s s in Astronautics and Aeronautics, Academic P r e s s , New York,

1963.

Pages 551-579 of Power Systems f o r Space

3. Anderson, D. C . , F i s h e r , R. D . , McDowell, E. L . , and Weidermann,

A. H. , Close-In Effects f r o m Nuclear Explosions. AFSWC TDR-63-

53 (May 1963) AD 411 132.

4. Walsh, J. M., Johnson, W. E . , Dienes, J. K . , Tillotson, J . H . , and

Yates, D. R . , Summary Report on the Theory of Hypervelocity Impact.

Genera l Atomic Division, General Dynamics Corp. , Report GA-5119

(March 31, 1964).

5. Bjork, R. L . , Effects of a Meteoroid Impact on Steel and Aluminum in

Space.

Vol. 11, Spr inger Verlag, pp. 505-514 (1960).

Tenth International Astronautical Congress Proceedings,

6. Bjork, R. L . , Analysis of the Format ion of Meteor C r a t e r , Arizona:

A Pre l imina ry Report. J. Geophys. R e s . , Vol. 66, pp. 3379-3387

(1 96 1).

7. Walsh, J. M. , and Tillotson, J. H . , Hydrodynamics of Hypervelocity

Impact. General Atomic Division, General Dynamics Corp. , Report

GA-3827 (Jan. 22, 1963) AD 401 023. Also published in the P roceed-

ings of the Sixth Symposium on Hypervelocity Impact, Vol. 2 , Pt. I,

pp. 59-104 (Aug. 1963).

49 AI- 1 82 1 -A- 3

8.

9.

10.

11.

12.

13.

14.

15.

16.

Rae, W. J . , and Kirchner , H. P . , Final Report on a Study of Meteoroid

Impact Phenomena. GAL Report RM-1655-M-4 (Feb. 1963)

N6 3 - 16887.

Rae, W. J . , and Kirchner , H. P . , A Blast-Wave Theory of C r a t e r

Format ion in Semi-Infinite Targets.

s ium on Hypervelocity Impact, Vol. 2, Pt . 1, pp. 163-229 (Aug. 1963).

Davids, N . , and Huang, Y. K . , Shock Waves in Solid C r a t e r s .

J. Aero/Space Sc i . , Vol. 29, pp. 550-557 (1962).

Calvit, H. H. , and Davids, N o , Spherical Shock Waves in Solids.

Pennsylvania State University, Dept. of Engineering Mechanics, Tech-

nical Report No. 2 (July 9, 1963) AD 415617.

Cole, R. H. , Underwater Explosions.. Pr inceton University P r e s s

(1 948).

Rice, M.H. , McQueen, R. G. , and Walsh, J . M . , Compression of

Solids by Strong Shock Waves. Solid State Physics , Advances in

Research and Applications, Ed. by F. Seitz and D. Turnbull, Vol. 6,

Academic P r e s s , 1958.

Allen, W. A., and Goldsmith, W . , Elast ic Description of a High-

Amplitude Spherical Pulse in Steel. J. Appl. Phys., Vol. 26, No. 1,

pp. 69-74 (Jan. 1955).

Rinehart, J. S. , and Pearson, J. , Behavior of Metals under Impulsive

Loads.

Proceedings of the Sixth Sympo-

American Society f o r Metals (1 954).

Marcus, H . , Comments on a paper by K. Broberg, p. 246, International

Symposium on S t r e s s Wave Propagation in Mater ia ls , Ed. by N. Davids,

Interscience, New York, 1960.

50 AI- 1 82 1 -A- 3

17.

18.

19.

20.

21.

22.

23.

24.

Aliev, Kh. M., Reflection of Spherical Elast ic Waves a t the Boundary

of a Half -Space.

Fiziki, No. 6, pp. 88-92 (Nov. -Dec. 1961) (In Russian).

Kinslow, R. , Proper t ies of Spherical Shock Waves Produced by Hyper-

velocity Impact. Proceedings of the Sixth Symposium on Hypervelocity

Impact, Vol. 2, Pt. 1, pp. 273-320 (Aug. 1963).

Blake, F. G., Spherical Wave Propagation in Solid Media. J. Acoust.

SOC. A m . , Vol. 24, No. 2, pp. 211-215 (March 1952).

K a r a l , F. C . , and Keller, J. B. , Elastic Wave Propagation in

Homogeneous and Inhomogeneous Media. J. Acoust. SOC. Am. , Vol.

31, No, 6, pp. 694-705 (June 1959).

Bagdoev, A. G., Spatial Non-Stationary Motions of Solid Medium with

Shock Waves. Publishing House of the Academy of Sciences of the

Armenian SSR, Erevan, 1961. Translated a s U. S. Atomic Energy

Commission AEC TR-6400, Nov. 1962.

Huth, J. H. , and Cole, J. D. , Impulsive Loading on an Elast ic Half-

Space. J, Appl. Mech., Vol. 21, pp. 294-295 (Sept. 1954).

Thiruvenkatachar, V. R . , S t ress Waves Produced in a Semi-Infinite

Elast ic Solid by Impulse Applied over a Ci rcu lar Area of the Plane

Face.

Mechanics, Nov. 1 - 2, 1955. Published by the Indian Society of

Theoretical and Applied Mechanics, Indian Institute of Technology,

Kharagpur, India.

Thiruvenkatachar, V. R. , Recent Research in S t r e s s Waves in India.

Pages 1-14 of the Proceedings of an International Symposium on S t r e s s

Wave Propagation in Materials, Ed. by N. Davids, Interscience Pub. ,

New York, 1960.

Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi

Proceedings of the F i r s t Congress on Theoretical and Applied

51 AI-1821-A-3

25. Broberg, K. B . , A Problem on S t r e s s Waves in an Infinite Elast ic

Plate. Transactions of the Royal Institute of Technology Stockholm,

Sweden, No. 139 (1959).

26. Davids, N. , Transient Analysis of Stress-Wave Penetration in Plates .

J. Appl. Mech., Vol. 26, pp. 651-660 (Dec. 1959).

27. Riney, T. D. , Theoretical Hypervelocity Impact Calculations using

the Picwick Code. General Electric Co. Report R64SD13 (Feb. 1964).

28. Cagniard, L . , Reflection and Refraction of P rogres s ive Seismic Waves.

Translated and revised by E. A. Flinn and C. H. Dix, McGraw-Hill,

1962.

52 AI- 1821 -A-3

W

In

m

I- v) I

- m

- w n -

53 AI-1821 -A-3

1

0

-0. I N u” T I*-

!

-0.3

-0.5 0 I 2

7 ............._ L . .._ _.

....._________ C _ _ _ _ _ - - _

3 5 6 7

F i g u r e 2 SURFACE SHEAR STRESS PRODUCED BY SOURCE AND IMAGE C A V I T I E S

R,/h = 1/3, P=?/4

54 AI- 18 2 1 -A- 3

1.2

I .o

.8

. 2

0 0 20 40 60 80

@.DEGREES

Figure 3 THE FUNCTION H (e), FOR v = l/lc

55 AI- 1 8 2 1 -A- 3

100

0

- 4

-8

- 1 2

- 16

-20

- 24

- 20

-32

- 3f

-4c I .o I. 2 1.4 1.6 1.8 2.0 2. 2

56 AI- 18 21 -A- 3

2. rl

F i g u r e 4 THE FUNCTIONS a' y ; 21 = 1/4

20

18

16

14

12

5 IO

8

6

4

2

I I .o I. 2 I . 4 I .6 I .8 2.0 2 . 2 2.4

Z/h

F i g u r e 5 THE FUNCTION l/J ; u ’ = 114

57 AI- 1 8 2 1 -A- 3

I

0

- I

V

- 2

- 3

-4 1.0 1.6 I .8

58 AI- 1821 -A-3

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