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Impulse attenuation of waves emanating from a cylindrical cavity in anisotropic media
Int. J. mech. Sci., Vol. 18, pp. 487~195.
Pergamon Press 1976.
Printed in Great Britain
I M P U L S E A T T E N U A T I O N OF WAVES E M A N A T I N G FROM A CYLINDRICAL CAVITY IN ANISOTROPIC MEDIA HOWARD L. SCHREYER Reactor Analysis and Safety Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439, U.S.A.
(Received 7 July 1976) Summary--Closed form solutions are given for stress waves resulting from the application of exponentially decaying pulses applied to cylindrical cavities in anisotropic media. It is shown that the initial compressive impulse may decay with range at a much smaller rate than the total impulse; consequently, such a possibility must be taken into account for the design of instruments or other imbedded objects.
a c C,,, C~2, C22 F F" ph H[] L Io L k p r t to t+ t' t '÷ u a,, a2, a3, a4 fl ~5[ ] rl 0 v O Go ~r~,troo try, cry0 tr~,~r~o z ~-'
NOTATION radius of cylindrical hole wave speed material constants solution to the governing equation solution for the unit Heaviside function term in the solution for the unit Heaviside function unit Heaviside function initial compressive radial impulse applied radial impulse radial impulse decay rate of exponential pulse applied pressure function radial co-ordinate time time constant for problem time at which radial stress becomes positive nondimensional variable nondimensional value at which radial stress becomes positive radial displacement nondimensional material constants nondimensional decay rate of exponential pulse Dirac delta function ratio of material constants circumferential co-ordinate Poisson's ratio mass density magnitude of applied radial stress radial and circumferential components of stress stress solution for a Heaviside forcing function stress solution for a Dirac delta forcing function time integration variable nondimensional time integration variable 1. INTRODUCTION
CYLINDRICAL w a v e propagation in elastic media represents an idealization to a large number of 487
interesting engineering problems. For example, a thorough understanding of the mechanism of rock blasting requires a c o m p r e h e n s i v e description of the p h e n o m e n a occurring around a borehole after the detonation of a charge but before the return of reflected waves. Also, the directional properties of composite materials can frequently be used to significant advantage in thick cylindrical shells designed to withstand an internal explosion. Unfortunately, a thorough study of transient w a v e motion with axial s y m m e t r y is hampered by "the absence of a simple general solution for cylindrical waves. ''~ One of the earliest investigations was that due to Selberg ~ in which he considered the dynamic response of an isotropic media when pressure was suddenly applied to the interior boundary of a spherical or cylindrical cavity. With the use of Laplace transforms, Selberg obtained solutions for p r e s s u r e - t i m e relations described by exponentially decaying and Heaviside step functions. For strictly c o m p r e s s i v e pulses, the results indicated that close to the cavity, significant tensile stresses in the tangential direction are d e v e l o p e d with the consequence that a radial system of cracks may appear. Further a w a y from the cavity, tensile stresses in the radial direction can arise but these stresses are quite small. These general features were also noted by Miklowitz 3 for the related problem of unloading w a v e s emanating from a hole suddenly punched in a previously stretched elastic plate. The essential differences between the two problems are that the plate problem is governed by the equations of plane stress rather than plane strain and that the process of punching results in a finite rise time instead of zero rise time. Miklowitz also claims very good agreement b e t w e e n his results and those of
488
HOWARD L. SCHREYER
Kromm's 4 which were obtained using a slightly different Laplace inversion technique. Eason ~'~ appears to be the first person to have noted that although closed form solutions may be obtained for the spherical case but not the cylindrical one for isotropic solids, the situation is different for anisotropic solids. By considering wave propagation from cylindrical cavities in materials with transverse curvilinear isotropy, Eason utilized the Laplace transform technique to obtain closed form solutions for particular ratios of elastic constants. Results for a step function load were presented. Bickford and Warren 7 utilized the closed form solution technique to great advantage in considering wave propagation and reflection in the walls of hollow spheres and cylinders. By using rational approximations to represent modified Bessel functions, they were also able to invert transformed expressions for stress and displacement for short times even for the completely isotropic case. An independent investigation in which the Laplace transforms were inverted numerically was conducted by Leonards e t al. 8 Again, wave propagation from a cylindrical cavity for a case of plane strain and completely isotropic material was considered and results were presented for both a step function and an exponentially decaying pressure applied to the inner surface. One important observation was the existence of a time delay between the peak tangential stress and the peak tangential strain. A much less significant delay was indicated under certain conditions for the radial component. The method of characteristics were utilized by Buchanan and Patton 9 to obtain numerical solutions for cases of anisotropic materials that incorporated a larger class than that considered by Eason. Stress results for both a step and a ramp forcing function were given. The cylindrical hole in a completely isotropic medium rather than one with transverse curvilinear isotropy may be more representative of a larger number of physical situations. However, the complexity associated with a purely numerical integration or with inverting the Laplace transforms, either numerically or with rational approximations, renders a parametric study rather tedious and time consuming. The great value of closed form solutions is that a study of the effects of various parameters can be easily performed and distinguishing features of a class of problems can be categorized. The result is a better understanding of basic phenomena and the establishment of guidelines that are of assistance for design, further theoretical analysis or the development of experiments.
Since closed form solutions are available 5'6 for a transverse curvilinear isotropic medium with a step pressure function applied to the walls of a cylindrical hole, Duhamel's integral can be used to obtain the response for any other pressure function. This paper presents solutions for an exponentially decaying forcing function, including the limiting case of a pure impulse, and the results are used to illustrate the effects of certain parameters~ Although wave shapes and maximum stresses are shown for various values of range and decay rate, the primary emphasis is placed on an evaluation of the compressive impulse. The results show that because of the development of tensile tails, the compressive impulse may attenuate with range at a much smaller rate than the attenuation factor for the total impulse. Such a result is significant for the design of instruments or other imbedded objects which may be damaged by a compressive pulse~ 2. GOVERNING EQUATIONS 2.1 Cylindrical waves in an a n i s o t r o p i c m e d i u m Consider an infinite medium with a cylindrical hole of radius, a, and composed of a material exhibiting transverse isotropy about the radial direction, r. Then for conditions of plane strain, the stress-displacement relations for radial symmetry are 0u ~rr,. = L , , - ~ +
O'°e
C, ~! :r
= C . o u + C~2U 1201"
(!'~
r
where u(r, t) is the component of displacement in the radial direction and r, 0 are cylindrical co-ordinates. The only nontrivial equation of motion is &r= ÷ (~r, - ~roo) Or r --
O:u p-ot
2
(21
in which p is the mass density. The use of equation (l) yields 02u ~_l_Ou or ~ r ~ -
2u 1 O~u "~ r = 7 s o~~
t3)
where
(4)
c 2 = C,,/O.
The initial and boundary conditions are
~r~[~ . =
p(t),
0
<~c
(5)
in which p ( t ) denotes the prescribed pressure on the surface of the cylindrical hole.
Impulse attenuation of waves in anisotropic media 2.2 Duhamel' s integral Suppose F h (r~ t) represents the solution (displacement or stress components) when the pressure function is described by the unit Heaviside function. Then the solution for any other function p (t) is given by Duhamel's integral f ' 0 F(r, t) = I P(r)~TFh(r, t - ~) dr. do
Case (b). , / = 3/2 [ a\ V2[ / a V
t
x{a,(l+a)
(6)
/
a\
cosad'-a2(1-a)sina,t}]H[t', 3/a~
/a\~nr
°'+h+=-°'°tr)
(r-a)
xf
2
1
(15)
,
[-2~r)+~44 e-~'
[3a 2 3
2a2) c o s a 4 t '
(7)
a
to
1
Lt;)
It is convenient to utilize the nondimensional parameter t'-
489
+ or2( 3~ar22 30 r
3+2ot2) s i n a d , } ] H [ t , ] .
in which (8)
to = a / c
and to represent the solution to the unit Heaviside function by (9)
F~ (r, t) = Fh (r, t')H[t'] where
3.2 Dirac delta function W h e n the applied pressure can be represented as a Dirac delta function, the resulting stress fields are merely the derivatives with respect to t' of the stresses due to the Heaviside function. These derivatives, which are used with Duhamel's integrals to obtain stresses for any other forcing function, are: Case (a). ~ = 1/2
H[t']=
01 for for
t' < 0 t'->O"
,,0)
a
The use of equations (7), (8) and (9) in equation (6) yields a solution in the form
112
{+'
/a\'nfC
r
Ol(, °~lra-~- - 2 a , ] e
........
(16) ~,"H[t']}.
F ( r , t ) = p(t')F+(r, t ' ) + fJ'' P (r')~t'a ~ h ( r ' t ' - r ' ) d r ' (11) in which the properties of the Dirac delta function 8[t' - r'l = 8~TH [ t ' - r']
Case (b). 77 = 3/2 / a \ 'nf
,,++=-,,Otr )
2a2(1
(12)
I
(17)
have been used. o,]o=-o-o 3. S O L U T I O N S FOR V A R I O U S FORCING FUNCTIONS 3.1 Unit Heaviside function For convenience in expressing the following results, let 1
a~
8[t']
x [ ,+,,( 3~a+ ~ 3 - 2 a 2 ) c o s a , t '
/3a 2 3a
C~2
2 C,l
x
or2 = (1 + a312 a~ = (3 - a,)/2 a4 = (a2a~) t/2.
(13)
Then, from Eason 5 the stress components resulting from the application of a compressive normal stress of magnitude ¢yo to the surface of the hole are:
~,
3
7
+2a22\)
sina,t']e-"2"H[t']}.
The impulse per unit area applied to the interior boundary of the cylindrical cavity in each case is Io = -~oto.
(18)
3.3 Exponentially decaying pulse Consider a forcing function of the form
Case (a). ~1 = 1/2 cr~ [r-o = - ~o/3 e-~"'0H[t/to]
O'htp:- o*o(a)l/2[a-,~-(1-a)e
+I/']H[t /]
, = ~O-Otr) I / a \ ' n f [r a - /tra + 1 _2a,'~e_+,,,]Htt,].}j o'00
(14)
(19)
in which the decay rate fl is incorporated as a stress multiplication factor so that the applied impulse per unit area is Io for this case, as well as the previous one. The use of Duhamel's integral provides the following expres-
490
HOWARD L. SCHREYER
sions for the stress c o m p o n e n t s :
Case (a). rl = 1/2
..
,)e--,"].,.', for
.o.(~;e
/3# 0;~
°"fl+,(~-m]H["] for
l)e
o,.
~(l_e)L\2 "
\
r
0;,i
/3 - c~ (20)
-(~r
0;11
for /3# 0;,
Oroo =~
3
1)t']H[t']
[I-/3(~a+
for
/3 = 0;,
Case (b). rl = 3/2
la\l/zl
)
+
0;4 \
re"'
(0;:-/3,(l
r/
sin
":' (21)
O'80=
o.fl/3(a) 1 1 2 { ~ e .~ll._
20;2( 1 __a'~ oL4 , r / f f 32 a 0 ; 4 --~T2--
x(~-/3)+(3-2~2)(1-/3)}{e
°'
- -
c o s c ~ J ' e ~e }
-[2~ {2-a(o<,~-/3)+ a2(/3 - 2)} + (~-2a2)
or/3 implies that a constant pressure is applied and a very large value corresponds to a forcing function of an impulsive nature. For a completely isotropic medium, Selberg ~ presented results for/3 = 0, 0.25 and 1.0, while Eason ~ restricted his analysis to/3 = 0 for two anisotropic media. The problems considered in ref. (7) correspond to k =50,000/see. a =0.375 in and two media defined by v - 0 . 4 8 , c 4000 ft/sec and u = 0.25, c = 1500 ft/sec where v denotes P o i s s o n ' s ratio. These two cases yield values of/3 equal to 0.39 and 1.04, respectively. The complexity associated with the numerical integrations probably account for the limited number of problems that were s o l v e d 4.2 Stress-time behavior Radial and circumferential stresses for both cases are illustrated as functions of time in Figs. I-6. Curves for r/a equal to 1.0, 2.0 and ~c are plotted for various decay rates associated with exponentially decaying applied pressure functions. For both radial and circumferential stresses, the case rl = "4/2 displays a wider variation of stress with time than for the case 71 - 1t2. Positive tangential stresses appear in both cases with significant magnifications possible for r~ = 3/2. For large enough time, positive radial stresses always appear for t7 = 1/2 and are relatively small in comparison with the scaled value of the initial peak c o m p r e s s i v e stress. However, for r/ = 3/2, the radial stress always remains c o m p r e s s i v e for small enough /3 and r/a. An example of a transition point is illustrated for/3 = 0. l and r/a = 2.08 in Fig. 3 in which the radial stress just reaches the value zero and then b e c o m e s negative again. For larger values of /3 and r/a, rather significant tensile radial stresses can be achieved. In general, the m a x i m u m tensile stresses and wave shapes are very d e p e n d e n t on both material properties and initial pulse shape. E v e n for the limiting case of large r/a, the characteristics of the wave form are quite varied for different values of ¢~. 4.3 Radial impulse The radial impulse per unit area at any range is defined to be i_34 ,t /, = f)• ~r, dt
x ( / 3 a 2 - c ~ 2 - / 3 ) ] sin 0;4t' e "'-' ] ) H [ t ' ] and by direct integration, it can be sh~)wn that for an exponentially decaying forcing function, including the limiting use of a pure impulse forcing function,
4. R E S U L T S
4.1 Existing solutions The ratio C J C . has been unspecified in the general solution. To e n s u r e that ~oo c o r r e s p o n d s to the isotropic case w h e n t = 0 and r = a it is n e c e s s a r y to set this ratio equal to 1/3 for both cases. It follqws that a~ = 1/6, a 2 = 7 / 1 2 , a3 = 17/12 and a , = (119)m/12. The parameter, k, that describes the rate of decay of the applied pulse with time is defined s u c h that
kt = [3t/t,,
(22)
k =/3c/a.
(23)
so that
T h u s the decay rate d e p e n d s on the nondimensional parameter/3, the radial speed of wave propagation c, and the radius of the cylindrical hole a. A value of zero for k
L=
I,, for
i ~a sJ2 I, = t r ) 1.
r t :: 1/2 (25)
for
tl
312.
T h u s , in general, the radial impulse per unit area attenuates with range at a rate that depends on the ratio of the elastic constants. The radial s t r e s s - t i m e curves of Figs. 1, 3 and 4 indicate that for a certain range of/3 and r/a, the radial stress can be positive even for a purely c o m p r e s s i v e loading function. If t+ denotes the time at which the radial stress first b e c o m e s positive, then the initial c o m p r e s s i v e impulse per unit area is defined to be L = J~Q' (r, dr.
(26)
Impulse attentuation of waves in anisotropic media
°4/''
. . . . . . .
07
X~
-o~-j
49l
1
~,°°
'~8;(,
J
_---
-t
q..3
I/I / / ~ - i -.Oo
]
~-----s--:'
,o
.o
,.o
So
Fie. 1. Radial stress as a function of time for ~ -_ 1 ~. 06 ~
--~
I
I---
I
I
-I
I
--
r
I
_
-
-
-
O4
,8= I00
02
% b~ b :~
/
/
-
/
i
- o d / ~ : ~ I/~S~
___
,.,o=,
. . . . .
rio=
z
o4t-
O(
I I0
20
30
40
50
60
70
80
90
I00
f' FIG. 2. Circumferential stress as a function of time for ~ -__1 ~. For wave shapes with tensile segments, there is a distinct possibility that the absolute value of L can be larger than the absolute value of L. If an impulse sensitive object is imbedded in the medium under such conditions, then L may be a more reasonable measure of potential damage than L. As an example, consider a pressure loading that is purely impulsive in nature. Then it is fairly easy to establish that t '+ = 0 ÷ and that Iofor ~ = 1/2 and 71 = 3/2.
t'÷= 1
4'-~)
for
_a#l
"
(28)
and
a) 1/2 L = r
The problem defined as Case (a), ~/= 1/2 with/3 = a~ is also sufficiently simple to obtain analytical expressions. The results are:
(27)
For such a loading situation, the attenuation rate of L with range corresponds to that of the peak stress and hence, L is much greater than L for large values of r/a.
'~='o(~)'2(a+('~)e ......'"}
~29,
so that L ~ 0 as 1/x/r for r/a - ~ . For other values of the parameter/3, the expressions for
HOWARD L. SCHREYER
492
04
.........
T~
......
T .........
~ ...........
T .........
T .............
~ ..........
~
..........
\
2
..... /- ~2Fi
-
' );.__<-_
/'
~].~
~- ......
T. . . . . . . . .
- - - - - - F,IcI:
~~+~t____
!
r/o
~ 2 08
r/o
= cu
....
- 7
/
!
Cb o
I
i
o4~
#
3
/
i O
2 0
3 [3
4 0
50
6 ©
7C
90
"~
;( (
f
FIG. 3. Radial stress as a function of time for rl = 3/2 and /3 = 0.1.
0 21 ',
us oof ,/~<
"Vt f
2-
.....
"'-
_
_
.
.
.
.
.
.
.
.
.
.
.
.
.
.
/
04
£:10
- - -
r/O= i r/g: 2 -
"06
O
,0
I0
2 O
30
40
50
60
70
8 C
90
30
¢-, FIG. 4. Radial stress as a function of time for 0 = 3/2, /3 = 1 and 10. t'+ and I,. are rather c u m b e r s o m e to determine explicitly so several problems were solved numerically. Results are presented in Figs. 7 and 8. The solid lines represent values of L/Io as a function of range for the designated exponential decay rates of the initial pressure pulse. Solid lines designated by 8 denote L/Io for a Dirac delta forcing function while the dotted lines represent the variation of L/Io with range. For both values of 71, and for a given range, L]lo increases monotonically with /3 so that the potential for
damage is increased if the applied pressure-time function is made more impulsive in nature. The discontinuity in L IIo for n = 3/2 and particular values of /3 is easily explained by the nature of the radial stress-time curves s h o w n in Fig. 3. For small values of/3, the radial stress never b e c o m e s positive and L = L. H o w e v e r , at a given range and a particular value of /3, the radial stress just reaches the value zero as s h o w n in Fig. 3 for r/a = 2.08 and /3 = O. 1. By definition, L/Io corresponds to the area under the portion of the curve up to t ' ÷ = 2.55 for this
Impulse attenuation of waves in anisotropic media 20
[
T
]
1
T
T
T
493 r
7
80
90
t8 -
r/o:
I
. . . .
-
r/o=
2
__
- -
-
-
r / t T = oo
I0 "1 ~
08
b~b o
06 O4 02 O0 /
-02 -04 -06
I0
0
20
50
40
50
6.0
70
I00
1-'
FIG. 5. Circumferential stress as a function o f time for r / = 3/2 a n d /3 = 0.1.
18
I
I
I
I
I
I
I
t6 -
-
-
-
r/o=
I
r/a=
2
r/o=
oo
14 -
E2 - -
-
- -
I0 08 06
'i
04 02 O0
/
-02 04 -06 --OE
0
I
10
~
2.0
3JO
I
I
4.0
50
6tO
¢0
I 80
90
IOO
f-,
F]~. 6. Circumferential stress as a function of time for 71 = 3/2, /3 = 1 and 10. example and consequently L/Io drops sharply from the value L/Io. However, for large enough r, IJIo is larger than LJlo for any given value of /3 that is greater than zero. 5. CONCLUSIONS Closed form solutions have been obtained for stress c o m p o n e n t s in an infinite elastic m e d i u m subjected to an exponentially decaying pulse applied to the surface of a cylindrical hole passing through the medium. These solutions exist for materials that exhibit transverse curvilinear isot-
ropy with particular ratios of elastic constants. Results have been limited to two cases defined by "O= 1/2 and ~ = 3/2. The solutions for the two cases reproduced the spatial attenuation factor of 1/~x/r for stress c o m p o n e n t s that is characteristic of cylindrical w a v e propagation. H o w e v e r , for 77 = 3 / 2 , the solution indicated that very large tensile stresses could be developed in the tangential direction. For both values of -,1, small tensile values of radial stress appeared for sufficiently large range.
494
HOWARD L. SCHREYER O
bo4
~
q
-
~
,
]
r
.
.
o,~
.
.
\
I
---
\"-
2
.
<
5
-
r
:
6 ~ J 9 0
3q
a
%~
rlJ
FI(;. 7. C o m p r e s s i v e ~ ~
T
i m p u l s e as a f u n c t i o n of /3 and rla for ~ = r
~.
r
.....
£
I
£.J
i
"x
-~.
' ~°66
~
_ ~
l
8
D=Jo..
"~
~
-
-.
-
t "-<, \\
"~;
\ ?
3
4
5
6
o oB
7 ~ 910
q
20
50
40
%Ot*.'
r/o
FIG. 8. C o m p r e s s i v e i m p u l s e as a f u n c t i o n of /3 and Ha for "0 = 3/2.
Impulse attenuation of waves in anisotropic media The primary significance of tensile radial stresses developed from a purely c o m p r e s s i v e forcing function is the effect on the radial c o m p o n e n t of impulse. The total radial impulse per unit area attenuates as 1/r 3~2 for "O = 1/2 and as 1/r 5~2 for "0 = 3/2. The attenuation factor of the initial radial impulse is generally much less than these rates and for both cases, the limiting attenuation rate of 1/r ~/2 was derived for the strictly impulsive loading function. The fact that the initial impulse may be significantly larger than the total impulse reaching any given point has important implications with regard to structures, instrument packages or any foreign body imbedded in the medium. A particular structure may be designed to withstand the total impulse, but be damaged by the initial impulse if the characteristics of the w a v e form are not taken into account.
495
Acknowledgement~This work was performed while the author was with Civil/Nuclear Systems Corporation, Albuquerque, New Mexico, under Contract No. F2960174-0123 with the Air Force Weapons Laboratory, Kirtland AFB, New Mexico.
REFERENCES 1. J. D. ACHENBACH,Wave Propagation in Elastic Solids. p. 135, American Elsevier, New York (1973). 2. H. L. SELBERG, Arkiv for Fysik 5, 97 (1952). 3. J. MIKLOWtTZ, J. appl. Mech. 27, 165 (1960). 4. A. KROMM, ZAMM. 28, 104 & 297 (1948). 5. G. EASON, ZAMP. 14, 12 (1963). 6. G. EASON, J. Comp. Mat. 7, 90 (1973). 7. W. B. BICKFORDand W. E. WARREN, Developments in Theor. and Appl. Mech. 3, 433 (1966). 8. G. A. LEONARDS,M. E. HARR and W. BARON,Kirtland Air Force Base Weapons Laboratory Technical Report AFWL-TR-66-81 (1966). 9. G. R. BUCHANANand W. L. PATTON,J. appl. Mech. 41, 1126 (1974).
Pergamon Press 1976.
Printed in Great Britain
I M P U L S E A T T E N U A T I O N OF WAVES E M A N A T I N G FROM A CYLINDRICAL CAVITY IN ANISOTROPIC MEDIA HOWARD L. SCHREYER Reactor Analysis and Safety Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439, U.S.A.
(Received 7 July 1976) Summary--Closed form solutions are given for stress waves resulting from the application of exponentially decaying pulses applied to cylindrical cavities in anisotropic media. It is shown that the initial compressive impulse may decay with range at a much smaller rate than the total impulse; consequently, such a possibility must be taken into account for the design of instruments or other imbedded objects.
a c C,,, C~2, C22 F F" ph H[] L Io L k p r t to t+ t' t '÷ u a,, a2, a3, a4 fl ~5[ ] rl 0 v O Go ~r~,troo try, cry0 tr~,~r~o z ~-'
NOTATION radius of cylindrical hole wave speed material constants solution to the governing equation solution for the unit Heaviside function term in the solution for the unit Heaviside function unit Heaviside function initial compressive radial impulse applied radial impulse radial impulse decay rate of exponential pulse applied pressure function radial co-ordinate time time constant for problem time at which radial stress becomes positive nondimensional variable nondimensional value at which radial stress becomes positive radial displacement nondimensional material constants nondimensional decay rate of exponential pulse Dirac delta function ratio of material constants circumferential co-ordinate Poisson's ratio mass density magnitude of applied radial stress radial and circumferential components of stress stress solution for a Heaviside forcing function stress solution for a Dirac delta forcing function time integration variable nondimensional time integration variable 1. INTRODUCTION
CYLINDRICAL w a v e propagation in elastic media represents an idealization to a large number of 487
interesting engineering problems. For example, a thorough understanding of the mechanism of rock blasting requires a c o m p r e h e n s i v e description of the p h e n o m e n a occurring around a borehole after the detonation of a charge but before the return of reflected waves. Also, the directional properties of composite materials can frequently be used to significant advantage in thick cylindrical shells designed to withstand an internal explosion. Unfortunately, a thorough study of transient w a v e motion with axial s y m m e t r y is hampered by "the absence of a simple general solution for cylindrical waves. ''~ One of the earliest investigations was that due to Selberg ~ in which he considered the dynamic response of an isotropic media when pressure was suddenly applied to the interior boundary of a spherical or cylindrical cavity. With the use of Laplace transforms, Selberg obtained solutions for p r e s s u r e - t i m e relations described by exponentially decaying and Heaviside step functions. For strictly c o m p r e s s i v e pulses, the results indicated that close to the cavity, significant tensile stresses in the tangential direction are d e v e l o p e d with the consequence that a radial system of cracks may appear. Further a w a y from the cavity, tensile stresses in the radial direction can arise but these stresses are quite small. These general features were also noted by Miklowitz 3 for the related problem of unloading w a v e s emanating from a hole suddenly punched in a previously stretched elastic plate. The essential differences between the two problems are that the plate problem is governed by the equations of plane stress rather than plane strain and that the process of punching results in a finite rise time instead of zero rise time. Miklowitz also claims very good agreement b e t w e e n his results and those of
488
HOWARD L. SCHREYER
Kromm's 4 which were obtained using a slightly different Laplace inversion technique. Eason ~'~ appears to be the first person to have noted that although closed form solutions may be obtained for the spherical case but not the cylindrical one for isotropic solids, the situation is different for anisotropic solids. By considering wave propagation from cylindrical cavities in materials with transverse curvilinear isotropy, Eason utilized the Laplace transform technique to obtain closed form solutions for particular ratios of elastic constants. Results for a step function load were presented. Bickford and Warren 7 utilized the closed form solution technique to great advantage in considering wave propagation and reflection in the walls of hollow spheres and cylinders. By using rational approximations to represent modified Bessel functions, they were also able to invert transformed expressions for stress and displacement for short times even for the completely isotropic case. An independent investigation in which the Laplace transforms were inverted numerically was conducted by Leonards e t al. 8 Again, wave propagation from a cylindrical cavity for a case of plane strain and completely isotropic material was considered and results were presented for both a step function and an exponentially decaying pressure applied to the inner surface. One important observation was the existence of a time delay between the peak tangential stress and the peak tangential strain. A much less significant delay was indicated under certain conditions for the radial component. The method of characteristics were utilized by Buchanan and Patton 9 to obtain numerical solutions for cases of anisotropic materials that incorporated a larger class than that considered by Eason. Stress results for both a step and a ramp forcing function were given. The cylindrical hole in a completely isotropic medium rather than one with transverse curvilinear isotropy may be more representative of a larger number of physical situations. However, the complexity associated with a purely numerical integration or with inverting the Laplace transforms, either numerically or with rational approximations, renders a parametric study rather tedious and time consuming. The great value of closed form solutions is that a study of the effects of various parameters can be easily performed and distinguishing features of a class of problems can be categorized. The result is a better understanding of basic phenomena and the establishment of guidelines that are of assistance for design, further theoretical analysis or the development of experiments.
Since closed form solutions are available 5'6 for a transverse curvilinear isotropic medium with a step pressure function applied to the walls of a cylindrical hole, Duhamel's integral can be used to obtain the response for any other pressure function. This paper presents solutions for an exponentially decaying forcing function, including the limiting case of a pure impulse, and the results are used to illustrate the effects of certain parameters~ Although wave shapes and maximum stresses are shown for various values of range and decay rate, the primary emphasis is placed on an evaluation of the compressive impulse. The results show that because of the development of tensile tails, the compressive impulse may attenuate with range at a much smaller rate than the attenuation factor for the total impulse. Such a result is significant for the design of instruments or other imbedded objects which may be damaged by a compressive pulse~ 2. GOVERNING EQUATIONS 2.1 Cylindrical waves in an a n i s o t r o p i c m e d i u m Consider an infinite medium with a cylindrical hole of radius, a, and composed of a material exhibiting transverse isotropy about the radial direction, r. Then for conditions of plane strain, the stress-displacement relations for radial symmetry are 0u ~rr,. = L , , - ~ +
O'°e
C, ~! :r
= C . o u + C~2U 1201"
(!'~
r
where u(r, t) is the component of displacement in the radial direction and r, 0 are cylindrical co-ordinates. The only nontrivial equation of motion is &r= ÷ (~r, - ~roo) Or r --
O:u p-ot
2
(21
in which p is the mass density. The use of equation (l) yields 02u ~_l_Ou or ~ r ~ -
2u 1 O~u "~ r = 7 s o~~
t3)
where
(4)
c 2 = C,,/O.
The initial and boundary conditions are
~r~[~ . =
p(t),
0
<~c
(5)
in which p ( t ) denotes the prescribed pressure on the surface of the cylindrical hole.
Impulse attenuation of waves in anisotropic media 2.2 Duhamel' s integral Suppose F h (r~ t) represents the solution (displacement or stress components) when the pressure function is described by the unit Heaviside function. Then the solution for any other function p (t) is given by Duhamel's integral f ' 0 F(r, t) = I P(r)~TFh(r, t - ~) dr. do
Case (b). , / = 3/2 [ a\ V2[ / a V
t
x{a,(l+a)
(6)
/
a\
cosad'-a2(1-a)sina,t}]H[t', 3/a~
/a\~nr
°'+h+=-°'°tr)
(r-a)
xf
2
1
(15)
,
[-2~r)+~44 e-~'
[3a 2 3
2a2) c o s a 4 t '
(7)
a
to
1
Lt;)
It is convenient to utilize the nondimensional parameter t'-
489
+ or2( 3~ar22 30 r
3+2ot2) s i n a d , } ] H [ t , ] .
in which (8)
to = a / c
and to represent the solution to the unit Heaviside function by (9)
F~ (r, t) = Fh (r, t')H[t'] where
3.2 Dirac delta function W h e n the applied pressure can be represented as a Dirac delta function, the resulting stress fields are merely the derivatives with respect to t' of the stresses due to the Heaviside function. These derivatives, which are used with Duhamel's integrals to obtain stresses for any other forcing function, are: Case (a). ~ = 1/2
H[t']=
01 for for
t' < 0 t'->O"
,,0)
a
The use of equations (7), (8) and (9) in equation (6) yields a solution in the form
112
{+'
/a\'nfC
r
Ol(, °~lra-~- - 2 a , ] e
........
(16) ~,"H[t']}.
F ( r , t ) = p(t')F+(r, t ' ) + fJ'' P (r')~t'a ~ h ( r ' t ' - r ' ) d r ' (11) in which the properties of the Dirac delta function 8[t' - r'l = 8~TH [ t ' - r']
Case (b). 77 = 3/2 / a \ 'nf
,,++=-,,Otr )
2a2(1
(12)
I
(17)
have been used. o,]o=-o-o 3. S O L U T I O N S FOR V A R I O U S FORCING FUNCTIONS 3.1 Unit Heaviside function For convenience in expressing the following results, let 1
a~
8[t']
x [ ,+,,( 3~a+ ~ 3 - 2 a 2 ) c o s a , t '
/3a 2 3a
C~2
2 C,l
x
or2 = (1 + a312 a~ = (3 - a,)/2 a4 = (a2a~) t/2.
(13)
Then, from Eason 5 the stress components resulting from the application of a compressive normal stress of magnitude ¢yo to the surface of the hole are:
~,
3
7
+2a22\)
sina,t']e-"2"H[t']}.
The impulse per unit area applied to the interior boundary of the cylindrical cavity in each case is Io = -~oto.
(18)
3.3 Exponentially decaying pulse Consider a forcing function of the form
Case (a). ~1 = 1/2 cr~ [r-o = - ~o/3 e-~"'0H[t/to]
O'htp:- o*o(a)l/2[a-,~-(1-a)e
+I/']H[t /]
, = ~O-Otr) I / a \ ' n f [r a - /tra + 1 _2a,'~e_+,,,]Htt,].}j o'00
(14)
(19)
in which the decay rate fl is incorporated as a stress multiplication factor so that the applied impulse per unit area is Io for this case, as well as the previous one. The use of Duhamel's integral provides the following expres-
490
HOWARD L. SCHREYER
sions for the stress c o m p o n e n t s :
Case (a). rl = 1/2
..
,)e--,"].,.', for
.o.(~;e
/3# 0;~
°"fl+,(~-m]H["] for
l)e
o,.
~(l_e)L\2 "
\
r
0;,i
/3 - c~ (20)
-(~r
0;11
for /3# 0;,
Oroo =~
3
1)t']H[t']
[I-/3(~a+
for
/3 = 0;,
Case (b). rl = 3/2
la\l/zl
)
+
0;4 \
re"'
(0;:-/3,(l
r/
sin
":' (21)
O'80=
o.fl/3(a) 1 1 2 { ~ e .~ll._
20;2( 1 __a'~ oL4 , r / f f 32 a 0 ; 4 --~T2--
x(~-/3)+(3-2~2)(1-/3)}{e
°'
- -
c o s c ~ J ' e ~e }
-[2~ {2-a(o<,~-/3)+ a2(/3 - 2)} + (~-2a2)
or/3 implies that a constant pressure is applied and a very large value corresponds to a forcing function of an impulsive nature. For a completely isotropic medium, Selberg ~ presented results for/3 = 0, 0.25 and 1.0, while Eason ~ restricted his analysis to/3 = 0 for two anisotropic media. The problems considered in ref. (7) correspond to k =50,000/see. a =0.375 in and two media defined by v - 0 . 4 8 , c 4000 ft/sec and u = 0.25, c = 1500 ft/sec where v denotes P o i s s o n ' s ratio. These two cases yield values of/3 equal to 0.39 and 1.04, respectively. The complexity associated with the numerical integrations probably account for the limited number of problems that were s o l v e d 4.2 Stress-time behavior Radial and circumferential stresses for both cases are illustrated as functions of time in Figs. I-6. Curves for r/a equal to 1.0, 2.0 and ~c are plotted for various decay rates associated with exponentially decaying applied pressure functions. For both radial and circumferential stresses, the case rl = "4/2 displays a wider variation of stress with time than for the case 71 - 1t2. Positive tangential stresses appear in both cases with significant magnifications possible for r~ = 3/2. For large enough time, positive radial stresses always appear for t7 = 1/2 and are relatively small in comparison with the scaled value of the initial peak c o m p r e s s i v e stress. However, for r/ = 3/2, the radial stress always remains c o m p r e s s i v e for small enough /3 and r/a. An example of a transition point is illustrated for/3 = 0. l and r/a = 2.08 in Fig. 3 in which the radial stress just reaches the value zero and then b e c o m e s negative again. For larger values of /3 and r/a, rather significant tensile radial stresses can be achieved. In general, the m a x i m u m tensile stresses and wave shapes are very d e p e n d e n t on both material properties and initial pulse shape. E v e n for the limiting case of large r/a, the characteristics of the wave form are quite varied for different values of ¢~. 4.3 Radial impulse The radial impulse per unit area at any range is defined to be i_34 ,t /, = f)• ~r, dt
x ( / 3 a 2 - c ~ 2 - / 3 ) ] sin 0;4t' e "'-' ] ) H [ t ' ] and by direct integration, it can be sh~)wn that for an exponentially decaying forcing function, including the limiting use of a pure impulse forcing function,
4. R E S U L T S
4.1 Existing solutions The ratio C J C . has been unspecified in the general solution. To e n s u r e that ~oo c o r r e s p o n d s to the isotropic case w h e n t = 0 and r = a it is n e c e s s a r y to set this ratio equal to 1/3 for both cases. It follqws that a~ = 1/6, a 2 = 7 / 1 2 , a3 = 17/12 and a , = (119)m/12. The parameter, k, that describes the rate of decay of the applied pulse with time is defined s u c h that
kt = [3t/t,,
(22)
k =/3c/a.
(23)
so that
T h u s the decay rate d e p e n d s on the nondimensional parameter/3, the radial speed of wave propagation c, and the radius of the cylindrical hole a. A value of zero for k
L=
I,, for
i ~a sJ2 I, = t r ) 1.
r t :: 1/2 (25)
for
tl
312.
T h u s , in general, the radial impulse per unit area attenuates with range at a rate that depends on the ratio of the elastic constants. The radial s t r e s s - t i m e curves of Figs. 1, 3 and 4 indicate that for a certain range of/3 and r/a, the radial stress can be positive even for a purely c o m p r e s s i v e loading function. If t+ denotes the time at which the radial stress first b e c o m e s positive, then the initial c o m p r e s s i v e impulse per unit area is defined to be L = J~Q' (r, dr.
(26)
Impulse attentuation of waves in anisotropic media
°4/''
. . . . . . .
07
X~
-o~-j
49l
1
~,°°
'~8;(,
J
_---
-t
q..3
I/I / / ~ - i -.Oo
]
~-----s--:'
,o
.o
,.o
So
Fie. 1. Radial stress as a function of time for ~ -_ 1 ~. 06 ~
--~
I
I---
I
I
-I
I
--
r
I
_
-
-
-
O4
,8= I00
02
% b~ b :~
/
/
-
/
i
- o d / ~ : ~ I/~S~
___
,.,o=,
. . . . .
rio=
z
o4t-
O(
I I0
20
30
40
50
60
70
80
90
I00
f' FIG. 2. Circumferential stress as a function of time for ~ -__1 ~. For wave shapes with tensile segments, there is a distinct possibility that the absolute value of L can be larger than the absolute value of L. If an impulse sensitive object is imbedded in the medium under such conditions, then L may be a more reasonable measure of potential damage than L. As an example, consider a pressure loading that is purely impulsive in nature. Then it is fairly easy to establish that t '+ = 0 ÷ and that Iofor ~ = 1/2 and 71 = 3/2.
t'÷= 1
4'-~)
for
_a#l
"
(28)
and
a) 1/2 L = r
The problem defined as Case (a), ~/= 1/2 with/3 = a~ is also sufficiently simple to obtain analytical expressions. The results are:
(27)
For such a loading situation, the attenuation rate of L with range corresponds to that of the peak stress and hence, L is much greater than L for large values of r/a.
'~='o(~)'2(a+('~)e ......'"}
~29,
so that L ~ 0 as 1/x/r for r/a - ~ . For other values of the parameter/3, the expressions for
HOWARD L. SCHREYER
492
04
.........
T~
......
T .........
~ ...........
T .........
T .............
~ ..........
~
..........
\
2
..... /- ~2Fi
-
' );.__<-_
/'
~].~
~- ......
T. . . . . . . . .
- - - - - - F,IcI:
~~+~t____
!
r/o
~ 2 08
r/o
= cu
....
- 7
/
!
Cb o
I
i
o4~
#
3
/
i O
2 0
3 [3
4 0
50
6 ©
7C
90
"~
;( (
f
FIG. 3. Radial stress as a function of time for rl = 3/2 and /3 = 0.1.
0 21 ',
us oof ,/~<
"Vt f
2-
.....
"'-
_
_
.
.
.
.
.
.
.
.
.
.
.
.
.
.
/
04
£:10
- - -
r/O= i r/g: 2 -
"06
O
,0
I0
2 O
30
40
50
60
70
8 C
90
30
¢-, FIG. 4. Radial stress as a function of time for 0 = 3/2, /3 = 1 and 10. t'+ and I,. are rather c u m b e r s o m e to determine explicitly so several problems were solved numerically. Results are presented in Figs. 7 and 8. The solid lines represent values of L/Io as a function of range for the designated exponential decay rates of the initial pressure pulse. Solid lines designated by 8 denote L/Io for a Dirac delta forcing function while the dotted lines represent the variation of L/Io with range. For both values of 71, and for a given range, L]lo increases monotonically with /3 so that the potential for
damage is increased if the applied pressure-time function is made more impulsive in nature. The discontinuity in L IIo for n = 3/2 and particular values of /3 is easily explained by the nature of the radial stress-time curves s h o w n in Fig. 3. For small values of/3, the radial stress never b e c o m e s positive and L = L. H o w e v e r , at a given range and a particular value of /3, the radial stress just reaches the value zero as s h o w n in Fig. 3 for r/a = 2.08 and /3 = O. 1. By definition, L/Io corresponds to the area under the portion of the curve up to t ' ÷ = 2.55 for this
Impulse attenuation of waves in anisotropic media 20
[
T
]
1
T
T
T
493 r
7
80
90
t8 -
r/o:
I
. . . .
-
r/o=
2
__
- -
-
-
r / t T = oo
I0 "1 ~
08
b~b o
06 O4 02 O0 /
-02 -04 -06
I0
0
20
50
40
50
6.0
70
I00
1-'
FIG. 5. Circumferential stress as a function o f time for r / = 3/2 a n d /3 = 0.1.
18
I
I
I
I
I
I
I
t6 -
-
-
-
r/o=
I
r/a=
2
r/o=
oo
14 -
E2 - -
-
- -
I0 08 06
'i
04 02 O0
/
-02 04 -06 --OE
0
I
10
~
2.0
3JO
I
I
4.0
50
6tO
¢0
I 80
90
IOO
f-,
F]~. 6. Circumferential stress as a function of time for 71 = 3/2, /3 = 1 and 10. example and consequently L/Io drops sharply from the value L/Io. However, for large enough r, IJIo is larger than LJlo for any given value of /3 that is greater than zero. 5. CONCLUSIONS Closed form solutions have been obtained for stress c o m p o n e n t s in an infinite elastic m e d i u m subjected to an exponentially decaying pulse applied to the surface of a cylindrical hole passing through the medium. These solutions exist for materials that exhibit transverse curvilinear isot-
ropy with particular ratios of elastic constants. Results have been limited to two cases defined by "O= 1/2 and ~ = 3/2. The solutions for the two cases reproduced the spatial attenuation factor of 1/~x/r for stress c o m p o n e n t s that is characteristic of cylindrical w a v e propagation. H o w e v e r , for 77 = 3 / 2 , the solution indicated that very large tensile stresses could be developed in the tangential direction. For both values of -,1, small tensile values of radial stress appeared for sufficiently large range.
494
HOWARD L. SCHREYER O
bo4
~
q
-
~
,
]
r
.
.
o,~
.
.
\
I
---
\"-
2
.
<
5
-
r
:
6 ~ J 9 0
3q
a
%~
rlJ
FI(;. 7. C o m p r e s s i v e ~ ~
T
i m p u l s e as a f u n c t i o n of /3 and rla for ~ = r
~.
r
.....
£
I
£.J
i
"x
-~.
' ~°66
~
_ ~
l
8
D=Jo..
"~
~
-
-.
-
t "-<, \\
"~;
\ ?
3
4
5
6
o oB
7 ~ 910
q
20
50
40
%Ot*.'
r/o
FIG. 8. C o m p r e s s i v e i m p u l s e as a f u n c t i o n of /3 and Ha for "0 = 3/2.
Impulse attenuation of waves in anisotropic media The primary significance of tensile radial stresses developed from a purely c o m p r e s s i v e forcing function is the effect on the radial c o m p o n e n t of impulse. The total radial impulse per unit area attenuates as 1/r 3~2 for "O = 1/2 and as 1/r 5~2 for "0 = 3/2. The attenuation factor of the initial radial impulse is generally much less than these rates and for both cases, the limiting attenuation rate of 1/r ~/2 was derived for the strictly impulsive loading function. The fact that the initial impulse may be significantly larger than the total impulse reaching any given point has important implications with regard to structures, instrument packages or any foreign body imbedded in the medium. A particular structure may be designed to withstand the total impulse, but be damaged by the initial impulse if the characteristics of the w a v e form are not taken into account.
495
Acknowledgement~This work was performed while the author was with Civil/Nuclear Systems Corporation, Albuquerque, New Mexico, under Contract No. F2960174-0123 with the Air Force Weapons Laboratory, Kirtland AFB, New Mexico.
REFERENCES 1. J. D. ACHENBACH,Wave Propagation in Elastic Solids. p. 135, American Elsevier, New York (1973). 2. H. L. SELBERG, Arkiv for Fysik 5, 97 (1952). 3. J. MIKLOWtTZ, J. appl. Mech. 27, 165 (1960). 4. A. KROMM, ZAMM. 28, 104 & 297 (1948). 5. G. EASON, ZAMP. 14, 12 (1963). 6. G. EASON, J. Comp. Mat. 7, 90 (1973). 7. W. B. BICKFORDand W. E. WARREN, Developments in Theor. and Appl. Mech. 3, 433 (1966). 8. G. A. LEONARDS,M. E. HARR and W. BARON,Kirtland Air Force Base Weapons Laboratory Technical Report AFWL-TR-66-81 (1966). 9. G. R. BUCHANANand W. L. PATTON,J. appl. Mech. 41, 1126 (1974).