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Diffraction of a stress wave by a cylindrical cavity in an infinite elastic strip
L&r. A@. Engng Sci. Vol. 22. No. 4, pp. 475-490, Printed in Great Britain.
0020-7225/84 Q 1984 Pergamon
1984
$3.00 + .M) Press Ltd.
DIFFRACTIONOF A STRESS WAVE BY A CYLINDRICALCAVITY IN AN INFINITEELASTIC STRIP
s. 1tou (ReceivedAugust 20, 1983) Departmentof MechanicalEngineering,HachinoheInstituteof Technology, Hachinohe051, Japan
ABSTRACT The scatteringof an incidentplane shock wave by a cylindrical circularcavity in an infiniteelastic strip is considered.In the Laplacetransformeddomain,boundaryconditionsat the plane surfacesand those at the circularhole are satisfiedwith the help of the Fourier transformationand the Schmidtmethod. A numerical Laplace inversiontechniqueis taken to obtain the stressesin the physical space.
1. Introduction In the static theory of elasticity,many problemswere solved for the stress concentrationfactor influencedby various specimengeometriesand loading conditions.However,the dynamic study was not developedas fully as its statical counterpartbecause of its greater complexities.Transientstress concentrations around a circularhole in an infinitemedium have been consideredby Krom
(1) and by Selberg (2). In their solutionsthe cylindricalboundaryof
the cavity receiveda uniform pressurewith Heaviside-function time dependence. Later, Earon and Matthewsdeterminedthe stressfield producedin an infinitemedium by the diffractionof the incomingshock wave by a cylindrical ES Vol.
22,
No.
bl
475
S.ITOU
416
cavity 131. Quite recently, Moodie and others
(4)solved numerically the tran-
sient problem which is similar to the studies treated in Refs. (1, 2). The investigations which are concerned with a single cavity in an infinite elastic medium are valid when the surfaces of the member are infinitely distant from the cavity so that the presence of the surfaces produces no effect on the stresses in the neighborhood of the cavity. However, if the surfaces of the member are at a finite distance from the cavity, the interaction between the effect of scattering of the incident wave by the cavity surface and the effect of reflections from the surfaces of the member must be considered. In the previous paper (51, the author has treated the impact response of an infinite elastic strip with a cylindrical cavity, at the circular surface of which internal pressure is applied suddenly. In the present paper, the impact response of a cylindrical circular hole in an infinite elastic strip, during passage of a plane shock stress wave, is considered. The boundary conditions for the reflected waves are satisfied in the Laplace transformed domain by using the Fourier transformation and the Schmidt method
(6, 7). A numerical Laplace inversion technique (81 is then
used to recover the time dependence of the solution. By combining the incident stress field with that resulting from the reflected waves, we obtain the complete solution for the problem. The dynamic stress concentration factors are calculated numerically and they are compared with those of the corresponding static values given by Howland [9).
2. Formulation We consider an infinite strip of isotropic elastic material. Let the strip be bounded in the x, y plane by lines y = +h as shown in Fig. 1. The
Diffraction of a stress wave cylin&ical
circular
471
cavitywill be supposedto have its center at the origin
with Ita11 being radius of the hole. We also take a polar coordinate(r, 6')in the relationof
x =
(I)
r cos8 ,y=rsin8.
Y
,ti,
I
Fig. 1 Geometryand coordinatesystem.
The incomingshock stresswaves are given by the expressions
r,? p H{t+(x-a)/%}, where p is a constant,H(t) is the Heavisideunit step function,cL is the dilatationalwave velocity and time t is zero when the wave front reachesthe cavity surfaceof (r = a, 8 = 0). Substitutingequation(2) into the relations
%
= xxxcos'8+&sin%+?&sin(2
@ ), tos='S;,sin'B+irrrcos~e-_~sin(28),
TPtis = $(_tr, -'I;,)sin(2B)+&cos(26
)P (3)
we can expressthe incidentstress field in the polar coordinate
478
Therefore,if the stressesfor the reflectedwaves satisfythe boundary conditions
(5) (61
the total stressfield for the presentproblem is given by
(7)
3. Analysis Define a Laplacetransformpair by
f*(s) =
j&exp(-stjdt, 0
f*(s)exp(st)ds, (8)
in which Br. stands for the Bromwichpath of integration.Applyingequation (8) to the wave equationsresultsfor the rectangularcoordinates
and for the polar coordinates
Diffraction
where cT
479
of a stress wave
is the shear wave velocity. The Laplace transforms of the boundary
conditions (5) and (6) are
-*
Lrr
,rj
exp{(x-a)s/c,j
= -p
TT = *
cos%/s,
p exp[(x-a)s/cJ sin(2 6)/s,
,;;, = T;'"=
0,
at r = a,
(11)
at y =t-h.
(12)
The solutions of the transformed wave equations (9) and (10) are given by
$“= !:B,( 4
)cosh( ty)exp(i f x)d% + ~C,L.,
y)exp(i%
x)d$
(m/c,
)coa~~n-1
+ *$D,K.(ksr/c‘)sin(n
)@I 9
@), (15)
with kc = ( 9+s'/&)
t
,
k= (%'+k's'/c: I* ,
(14)
where Kn( ) are the modified Bessel function of the second kind, B,(f ), Bz( %), Cn, D,
are the unknowns to be solved and the elastic constant
K1
takes the form
hl=
2(1-V
)/(I-2
(15)
Y),
in the plane state of strain with v
being Poisson's ratio. Substituting
equation (13) into the stress expressions we obtain
S. ITOU
480
x cosh(&y)]exp(ifx)d%+
g,C,]L'f:(r, 8, s)/2+(biI/2
-l)fF(r, @, s)j+~Do,[Egr, ",I ?yiir)/(2j4) = j*[B,(%
0, s)],
)[%'+K'sz/(2c:).fcosh(Y;y)+iBZ(S )(-%&)
-9/
~ccosh(Y;y)]exp(iSx)dS+~~CC,i(~~/2-l)f:(r,8, + h'/2 f:(r,d,
~x~?'(2/u)
=
s)]+,+L,-f-$(r,
s)
4, s)j,
jmw[iB,($ )%vI siA( ky)+B,( S)f4~KzS'/(2c,')~ k sinh(&y)]exp(i$
x)df + g,C,ff:(r, B, s)f
+"g,D"/g:(r, 8 3 s)/2-gf(r, 0, s)/2J,
(16)
with
f;(r, 8, s) = (s/c~~/~~K~_~(s~/c,)+~K._, (sr/cL)+Kn+, (sr/c,)j x~~~f(n-l)s]
cos'8-(n-l)(s/c,)rK,-~(sr/c,)+K,(sr/c~)~
~sin[(n-I)B~ sin8cos19/r-(s/c,)/2{K
,_,(sr/c,)+K,(sr/c~)f
x cos {(n-1)6'] sin'e/r-2(n-l)K,., (sr/cL)sin[(n-I) b)sin @ ~cos~/r~-(n-1)*K~_,(~~/c~)c0~~(~-1)8~
sin'&/?,
fl(r,6', S) = (s/c~)~/~~K,.~(S~/C~)+~K~-, (sr/c,)+K"+,(sr/c,)j a cos{(n-I)01 ): sin [(n-l
sin’B+(*-l)(s/c,)lK.-r(sr/c,)+K”(sr/c,)f )S] sin 0 cos 0 /r-(s/c.)/2[K
)i co,~~(n-1)8~cos’8/r-(n-l)
K,_, (sr/c,)cosl(n-1)8lcos’B/r’
+2(n-l)K,.,(sr/c,)Sin{(n-l)s] fy(r,8,
S)
= (s/c,$
._~(sr/c~)+K”(sr/c,)~
sin8cosB/r',
/4[L,,_,(sr/c,)+2K,-, (sr/c,)+K"+,(sr/c,)j
x cosf(n-I)@] sinBcosO-(n-l)(s/c,)/2~K..,(sr/c.) +K,,(sr/c.)jsin{(n-l)Bj (Sin'B-cos'8)/r+(s/c,)/2 ~~K~_,(~r/c~)tK~(sr/c.)~cos~(n-l)~~
sin8coso/r+(n-1)'
x Kn_,(sr/cL)co~~(n-l)~jsin~cos6/r'-(n-l)K..,(sr/c,) 8 sin f(n-1)8] (Sin'O-coszf9)/r',
(17)
Diffraction of a stress wave
g:(r, 0, S) = f”;” (I‘, B,Ks)
{cos(nQ)*sin(n
I
(i
From the boundary the
condition
aid of Fourier
equation
(12),
481 8),
= 1, 2,
sin(nP)-tcos(n8)],
3).
(18)
the next relations
are given
with
transformation
with t;(r T(S)
) ={a,,q:,($ ={a,,q”j($
)-a,,qj”,+(?
)]/(a,,
alz-alLaz,),
(j
= 1, 2, 3, 4),
)-a?, q;.+(F
S/Cal,
a,,-a,,a2,),
(j
= 5, 6, 7, 8), (20)
a,, = (~‘~-r;‘s~/c:)cosh(~h),
al1 = -f&
alI = f x
az2 = -($’ ++H’sZ/c:)sinh(g
sinh(& h),
P,,CS)+iq;( 3 ) = +J-{
*
cosh(&h),
(l-K’/2X(rh ,Bh, s&KY2f: (q ,tj, x fcos( 5 x)-isin(
q;( $ )+iq;(
$ ) = +rg;(rh
,f+,
(21)
s)f
5 x)] dx,
s){cos( s){sin(t,
f x)-isin(
q”,(S )+iq”,($
) = +/@f;(rh,&, *
q”(S 7
) = L @ g;(rc,Rh, S)-g;(rh&, S)j 2JL /f0 ):{ sin( C; x)+icos( 4 x)f dx,
)+iq”,(S
h),
x)+icos(f
f x)jdx, x)jdx,
(22)
where r,, = (h’+x’$
,
@A= tan-‘(h/x)
(23)
and E: =
1.0,
for
odd n,
Ga=
0.0,
for
even n,
482
S.ITOU k;= 1.0,for even n,
&,"=0.0, for odd n.
(242)
Then, the stressesfor the reflectedwaves which satisfyequation(12) are wr!tten as
(25)
with Ry('(r,o) = (ha/Z-sin'@)fy(r, 5, s)+(fi'/2-cos'8)f:(r,8, s)+sin(20)
om fl($h,(3bp:(S )h,(f )+P:($ )hy(%) JI
xf?(r,@, s)+2
+P';(?)h,(% )jccs(Sx)d5+2%C-p:(5
)h,(4)-P:( 5 )h,(F)
-P:(f )h&f )+P;(% )b( f )jsin($x)dS, R:(, @) = (cod@-sin't9)g;(r,8, s)-sin(2B)/2{g:(r,B, sl-g:(r,e,
s)3
+2;s"{p;(l, fh,(5 )-p;(f, )h,<% )+P;($)h3($ )+P'$$ )b( %)'1 rcos($x)d$ +2je[-p'$$ )h,(f )-P",(% )hz($ )-p;(f )h$ t ) 0
R:(r,@) = (riz'/2-co?8)f;(r,@, s)~(Hz/2-sinLB)fl;(r,B, s)-sin(25)
xf!(r,O, 47
P:(5 )&( 4 )-P:(% )h6($ )+P;($ )h7(c;)
+p;l(F)hs($f\cosi4 x)dS+2jQm~-$S
)h,(i;f-P:(S&(5
-&(S )h,(F )+P;($ )hs(f $sin(S x)dS ,
)
Diffraction R;(r,
6)
of a stress wave
= (sin'Q-cos'@)g;(r,8, &(S
bP;(
x)df +2
sky(r,
s)+sin(28 )/2 [g:(r,O,
tcos(f, Iipl
+2jOW\$(5
483
4 h(
% )+P;(s
&A5
)+P”,(s
8, 3,)
)hds
?)
-p;(5 )h& f )-P",(% )hb(% )-P;( f )h7( %)
0
+P;(? )hg($ Y/sine
R;(r,
6 ) = sin(2 8 )/2
{f:(r,
+2 jOmfp; (5 I$(
1ccos($x)df+2
+p;(f
)hlL(
B , s)-f:(r,
6 codf
x)df
8,
s)f+cos(2
5 )+P;( 4 )h,& 6 bP”,( % h,(
I
mf-~;(f
f )lsin($
R",(r,e ) = -sin(2 e)g:(r,8,
+2\;fp;(%
x)d5,
h+‘?)-~;($
hi’3
@)f:(r,
f )+P;(?
)h,o(%)-P:(?
hi
s)
4’ )f
h,,(F)
,
s)-cos(20 )/2 {g:(r,@,
)hq(f )-p",(f )h,,( f)-P:($
+2 &P;(
@,
f h,(
5 bP”,(
s)-gz(r,8,
)h,, (?)+P;($)
$ )h,o(
f bp;(f
s$
hlr(
b,,
(5 1
(26)
and h,(y ) = ~~(k'/2-~)sz/~~-~'~cos'~+~f'+~zs'/(2c~)~sinzD] cosh(bfy), h,(f
1 = %QTsidCy)sin(20),
h,(
) = $6 cosh(&y)(cos'o-sin?),
f
h+(S)
=iS'+k~sz/(2c:)fsinh(r,y)sin(2
O),
hs(f ) =~~(~'/2-1)s~/c~-f'fsin'B+~S'+~~s~/(2c~)~cos'eJ hr(%
1 = -&(f
1, h,(5; ) = -$(S
1, b6($ ) =-h+(q),
hS( 5 ) =~4'+s'/(2c:)fcosh(~y)sin(2B
),
cosh(Ky),
f$
484
S. ITOU h,,(t, 1
= -Sticosh(k~)sin(2fl)~
h,,(S)
= f~sinh(ky)cos(2~),
h,,(4)
=~'4z+kw/(2c:)+h(~y)cos(2B),
(27)
The stress expressions in equation (25) cause no tractions on the plane boundaries y =?h. The remaining boundary conditions, narnaly, equation (11) reduce to the forms
$C"Rl;(a, O)+&D,R"l(a,
zC,,q(a,
O)+gI&R:(a,e
0) = -P exp[(x-a)s/c.]cos'g/(2/G(s),
) = +P exp[(=a)s/c,]sin(2
8)/(2,~4s), (28)
where Rl(a, @
) (i= 1, 2, 5, 6) are considered to be the functions of x only,
because on r = a,
y =
(a’-x+ ,
6 = tan-'(y/x).
(29)
Equation (28) can be solved for the coefficients C,, D,
by a modified version
of the Schmidt method [5, 6, 71. Therefore, it is now possible to compute the stresses at any point in the strip by the forms
ok kr
p H[t+(x-a)/c,fco&
+ $
Q,
(2p) ck,,R:(r,
B)+D,R;(r, @ )'I
%exp(st)ds,
/t$)= P
H{t+(x-a)/c,\sidb + &jBr
(2,U)g,[C,R",(r, 0 )+D,R;(r, B )f xexp(st)ds,
F-L(r)_ I re - -$p Hft+(x-a)/cLfsin(2 0) + Ai,
(2,U) g[C,R;(r,O) +D,RZ(r,O )]exp(st)ds.
(30)
Diffraction of a stress wave
485
The Laplace inverse transformation in equation (30) is carried out by the numerical method given by Miller and Guy [8]. When the Laplace transform f*(s) can be evaluated at discrete points given by
s=(
p +l+k),
k = 0, 1, 2, 3, ...........
s we may determine coefficients C'
$f*f(p
+l+k)ij
(31)
from the following set of equations
= $C$k!/[(k+~+l)(k+P+2).....(k+~+l+q)(k-q)!], 0
(32) where
,f.O,
p z-1.0.
If the first Q coefficients are calculated, an approx-
imate value of f(t) can be found as
(33)
f(t) = YZOCi Pp'a)[2 exp(-gt)-lj,
where Pi0'6r)(x) is a Jacobi polynomial and Q is the number of terms employed. 7J The
paremeters ,f, R and Q are selected such that f(t) can be best described
within a particular time t.
4. Numerical Examples and Results The circumferential stresses at the cavity surface are calculated numerically for $
= 0.25. The semi-infinite integrals, which occur in the functions
qi( f ) and Rl(r, 19) in equations (22) and (26), from Filon's formula
can be integrated numerically
[IO] easily because those integrands almost decrease
exponentially. To perform the Schmidt procedure, we adopt the first seven terms of the infinite series in equation (28). For a check of the accuracy, the
)}
-0.12454*10'
-0.10181110'
-o.346l7x1o-2
-0.1637jx16'
-0.25493~16'
-0.22993~10~
-0.25396~10~
0.9990
0.0714
0.0010
-0.0714
-0.9206
-0.9990
X{2Acc,/(pa))
8) +D,RJ(a, 0
i-{W:b,
0.9286
x/a
1
0)
-0.11338x10-'
-0.92214x10“
-0.37689*10-'
0.59598x10-'
0.42323x10-'
o.40704xlop
0.55907~10-'
x{2Pcc,/(pa))
of
1.
h.
S.
and r.
h.
5.
coszO
-0.27064=.10-=
-0.56292~10‘~
-0.30342116=
sa/c,
x-a)s/c,
-exp{ ( I /( 1 3
-0.37793x10-'
0.56251~10-~
0.42370x10-'
/(.dCL 1
exp{(x-a)s/q]sin(20
in equation
(28) for h/a = 2.0 and sa/c, = 0.8.
Values
+D,%(a, Cl 1)
&nR:b,
Table
)
487
Diffraction of a stress wave
0.0
L
0.0
Fig. 2
10.0
cd la
20.0
Dynamic stress concentration Sactor at 8 = 0' for h/a = 2.0, 10.0 versus cLt/a.
0.0
Fig. 3
10.0
41/a
20.0
Dynamic stress concentration factor at 8 = 90' for h/a = 2.0, 10.0 versus c,t/a.
488
S. ITOU
?.,IP
Fy----y,
-.j
p‘i; _.
L
Zymn:ic
at
Fig.
cct/a
10.0
0.0
5
stress concer,trationfactor
0 = 180
Stress and
20.0
ror h/a = 1.0, 10.0 versus c,t/a.
distribution
,c,t/a = 5.0,
around
10.0,
P = a for
h/a
15.0 versus B .
= 2.0
Diffraction of a stress wave
489
values of the 1. h. s. and the r. h. s. in equation(28) are given in Table 1 for the case of sa/c& = 0.8, h/a = 2.0. From this, it is clear that the accuracy of the Schmidtmethod is satisfactory.Using the values of p = 0.0, J= 0.2, Q = 7, we invertethe Laplacetransformsnumerically. Figs. 2 through 4 show the hoop stress 7 BB at the cavity boundaryfor @ = O', 90", 180‘ and h/a = 10.0, 2.0, where the broken lines are the correspondingstaticvalues given by Howland[9]. The curvesfor h/a = 10.0 agree with the Baron and Matthews'sresultsfor an infiniteelasticmedium [3].
In
Fig. 5, the stressdistributionsaround the boundaryr = a are shown for h/a= 2.0 and c,t/a = 5.0, fO.0, 15.0. In conclusion,we get the followinginformations: i) The maximum hoop stressesalso occur at
~9 = 90" under the incidentplane
shock stresswave. When the h/a ratio decreases,it looks like that the peak value is generatedat the larger value of the time variable. ii) The peak values of the dynamic stress concentration factors at .d= 90" are increasedby j.6 % and 8.9 "/ D over those staticvalues for h/a = 2.0 and
10.0, respectively. iii) The absoluteClues of the dynamic stressesat (9= O*, 180" are about 1.14 and 1.17 times as large as those of the static solutionsfor h/a = 10.0, respectively.For h/a = 2.0, those values are increasedby about 1.40 and 1.37 times over the staticones. iv) The inertia effect on the hoop stress at r = a is more noticeablenew the regionsof 8 = 0' and 180".
The authorwishes to acknowledgethe valuablesuggestionsofferedby Prof. A. Atsumi of Tohoku University,which were given constantlythroughout this work.
490
S. ITOU References 1. Kromm, A., Zur Ausbreitung von Stosswellen in Kreislochscheiben, ZAMM: 28, 297, 1948. 2. Selberg, H. L., Transient Compression Waves From Spherical and Cylindrical Cavities, Arkiv for Fysik: 1, 97, 1952. 3. Baron, M. L. and Matthews, A. T., Diffraction of a Pressure Wave by a Cylindrical Cavity in an Elastic Medium, ASME J. Appl. Mech.: 2S, 347, 1961. 4. Moodie, T. B., Haddow, J. B., Mioduchowski, A. and Tait, R. J., Plane Elastic Waves Generated by Dynsmic Loading Applied to Edge of Circular Hole, ASME J. Appl. Mech.:
9,
577, 1981.
5. Itou, S., Dynamic Stress Concentration around a Circular Hole in an Infinite Elastic Strip, ASME J. Appl. Mech.: 2,
51, 1983.
6. Morse, P. M. and Feshbach, H., Methods of Theoretical Physics I, McGraw-Hill, p. 926, 1953. 7. Yau, W. F., Axisimmetric Slipless Indentation of an Infinite Elastic Cylinder, SIAM J. Appl. Math.: 2,
219, 1967.
8. Miller, M. K. and Guy, W. T., Numerical Inversion of the Laplace Transform by Use of Jacobi Polynomials, SIAM J. Num. Anal.: 3, 624, 1966. 9. Howland, R. C. J., On the Stresses in the Neighbourhood of a Circular Hole in a Strip Under Tension, Phil. Trans. of the Royal Sot. of London, A: 229, 49, 1930. 10. Amemiya, A. and Taguchi, T., Numerical Analysis and Fortran, Maruzen, Tokyo, 1969.
0020-7225/84 Q 1984 Pergamon
1984
$3.00 + .M) Press Ltd.
DIFFRACTIONOF A STRESS WAVE BY A CYLINDRICALCAVITY IN AN INFINITEELASTIC STRIP
s. 1tou (ReceivedAugust 20, 1983) Departmentof MechanicalEngineering,HachinoheInstituteof Technology, Hachinohe051, Japan
ABSTRACT The scatteringof an incidentplane shock wave by a cylindrical circularcavity in an infiniteelastic strip is considered.In the Laplacetransformeddomain,boundaryconditionsat the plane surfacesand those at the circularhole are satisfiedwith the help of the Fourier transformationand the Schmidtmethod. A numerical Laplace inversiontechniqueis taken to obtain the stressesin the physical space.
1. Introduction In the static theory of elasticity,many problemswere solved for the stress concentrationfactor influencedby various specimengeometriesand loading conditions.However,the dynamic study was not developedas fully as its statical counterpartbecause of its greater complexities.Transientstress concentrations around a circularhole in an infinitemedium have been consideredby Krom
(1) and by Selberg (2). In their solutionsthe cylindricalboundaryof
the cavity receiveda uniform pressurewith Heaviside-function time dependence. Later, Earon and Matthewsdeterminedthe stressfield producedin an infinitemedium by the diffractionof the incomingshock wave by a cylindrical ES Vol.
22,
No.
bl
475
S.ITOU
416
cavity 131. Quite recently, Moodie and others
(4)solved numerically the tran-
sient problem which is similar to the studies treated in Refs. (1, 2). The investigations which are concerned with a single cavity in an infinite elastic medium are valid when the surfaces of the member are infinitely distant from the cavity so that the presence of the surfaces produces no effect on the stresses in the neighborhood of the cavity. However, if the surfaces of the member are at a finite distance from the cavity, the interaction between the effect of scattering of the incident wave by the cavity surface and the effect of reflections from the surfaces of the member must be considered. In the previous paper (51, the author has treated the impact response of an infinite elastic strip with a cylindrical cavity, at the circular surface of which internal pressure is applied suddenly. In the present paper, the impact response of a cylindrical circular hole in an infinite elastic strip, during passage of a plane shock stress wave, is considered. The boundary conditions for the reflected waves are satisfied in the Laplace transformed domain by using the Fourier transformation and the Schmidt method
(6, 7). A numerical Laplace inversion technique (81 is then
used to recover the time dependence of the solution. By combining the incident stress field with that resulting from the reflected waves, we obtain the complete solution for the problem. The dynamic stress concentration factors are calculated numerically and they are compared with those of the corresponding static values given by Howland [9).
2. Formulation We consider an infinite strip of isotropic elastic material. Let the strip be bounded in the x, y plane by lines y = +h as shown in Fig. 1. The
Diffraction of a stress wave cylin&ical
circular
471
cavitywill be supposedto have its center at the origin
with Ita11 being radius of the hole. We also take a polar coordinate(r, 6')in the relationof
x =
(I)
r cos8 ,y=rsin8.
Y
,ti,
I
Fig. 1 Geometryand coordinatesystem.
The incomingshock stresswaves are given by the expressions
r,? p H{t+(x-a)/%}, where p is a constant,H(t) is the Heavisideunit step function,cL is the dilatationalwave velocity and time t is zero when the wave front reachesthe cavity surfaceof (r = a, 8 = 0). Substitutingequation(2) into the relations
%
= xxxcos'8+&sin%+?&sin(2
@ ), tos='S;,sin'B+irrrcos~e-_~sin(28),
TPtis = $(_tr, -'I;,)sin(2B)+&cos(26
)P (3)
we can expressthe incidentstress field in the polar coordinate
478
Therefore,if the stressesfor the reflectedwaves satisfythe boundary conditions
(5) (61
the total stressfield for the presentproblem is given by
(7)
3. Analysis Define a Laplacetransformpair by
f*(s) =
j&exp(-stjdt, 0
f*(s)exp(st)ds, (8)
in which Br. stands for the Bromwichpath of integration.Applyingequation (8) to the wave equationsresultsfor the rectangularcoordinates
and for the polar coordinates
Diffraction
where cT
479
of a stress wave
is the shear wave velocity. The Laplace transforms of the boundary
conditions (5) and (6) are
-*
Lrr
,rj
exp{(x-a)s/c,j
= -p
TT = *
cos%/s,
p exp[(x-a)s/cJ sin(2 6)/s,
,;;, = T;'"=
0,
at r = a,
(11)
at y =t-h.
(12)
The solutions of the transformed wave equations (9) and (10) are given by
$“= !:B,( 4
)cosh( ty)exp(i f x)d% + ~C,L.,
y)exp(i%
x)d$
(m/c,
)coa~~n-1
+ *$D,K.(ksr/c‘)sin(n
)@I 9
@), (15)
with kc = ( 9+s'/&)
t
,
k= (%'+k's'/c: I* ,
(14)
where Kn( ) are the modified Bessel function of the second kind, B,(f ), Bz( %), Cn, D,
are the unknowns to be solved and the elastic constant
K1
takes the form
hl=
2(1-V
)/(I-2
(15)
Y),
in the plane state of strain with v
being Poisson's ratio. Substituting
equation (13) into the stress expressions we obtain
S. ITOU
480
x cosh(&y)]exp(ifx)d%+
g,C,]L'f:(r, 8, s)/2+(biI/2
-l)fF(r, @, s)j+~Do,[Egr, ",I ?yiir)/(2j4) = j*[B,(%
0, s)],
)[%'+K'sz/(2c:).fcosh(Y;y)+iBZ(S )(-%&)
-9/
~ccosh(Y;y)]exp(iSx)dS+~~CC,i(~~/2-l)f:(r,8, + h'/2 f:(r,d,
~x~?'(2/u)
=
s)]+,+L,-f-$(r,
s)
4, s)j,
jmw[iB,($ )%vI siA( ky)+B,( S)f4~KzS'/(2c,')~ k sinh(&y)]exp(i$
x)df + g,C,ff:(r, B, s)f
+"g,D"/g:(r, 8 3 s)/2-gf(r, 0, s)/2J,
(16)
with
f;(r, 8, s) = (s/c~~/~~K~_~(s~/c,)+~K._, (sr/cL)+Kn+, (sr/c,)j x~~~f(n-l)s]
cos'8-(n-l)(s/c,)rK,-~(sr/c,)+K,(sr/c~)~
~sin[(n-I)B~ sin8cos19/r-(s/c,)/2{K
,_,(sr/c,)+K,(sr/c~)f
x cos {(n-1)6'] sin'e/r-2(n-l)K,., (sr/cL)sin[(n-I) b)sin @ ~cos~/r~-(n-1)*K~_,(~~/c~)c0~~(~-1)8~
sin'&/?,
fl(r,6', S) = (s/c~)~/~~K,.~(S~/C~)+~K~-, (sr/c,)+K"+,(sr/c,)j a cos{(n-I)01 ): sin [(n-l
sin’B+(*-l)(s/c,)lK.-r(sr/c,)+K”(sr/c,)f )S] sin 0 cos 0 /r-(s/c.)/2[K
)i co,~~(n-1)8~cos’8/r-(n-l)
K,_, (sr/c,)cosl(n-1)8lcos’B/r’
+2(n-l)K,.,(sr/c,)Sin{(n-l)s] fy(r,8,
S)
= (s/c,$
._~(sr/c~)+K”(sr/c,)~
sin8cosB/r',
/4[L,,_,(sr/c,)+2K,-, (sr/c,)+K"+,(sr/c,)j
x cosf(n-I)@] sinBcosO-(n-l)(s/c,)/2~K..,(sr/c.) +K,,(sr/c.)jsin{(n-l)Bj (Sin'B-cos'8)/r+(s/c,)/2 ~~K~_,(~r/c~)tK~(sr/c.)~cos~(n-l)~~
sin8coso/r+(n-1)'
x Kn_,(sr/cL)co~~(n-l)~jsin~cos6/r'-(n-l)K..,(sr/c,) 8 sin f(n-1)8] (Sin'O-coszf9)/r',
(17)
Diffraction of a stress wave
g:(r, 0, S) = f”;” (I‘, B,Ks)
{cos(nQ)*sin(n
I
(i
From the boundary the
condition
aid of Fourier
equation
(12),
481 8),
= 1, 2,
sin(nP)-tcos(n8)],
3).
(18)
the next relations
are given
with
transformation
with t;(r T(S)
) ={a,,q:,($ ={a,,q”j($
)-a,,qj”,+(?
)]/(a,,
alz-alLaz,),
(j
= 1, 2, 3, 4),
)-a?, q;.+(F
S/Cal,
a,,-a,,a2,),
(j
= 5, 6, 7, 8), (20)
a,, = (~‘~-r;‘s~/c:)cosh(~h),
al1 = -f&
alI = f x
az2 = -($’ ++H’sZ/c:)sinh(g
sinh(& h),
P,,CS)+iq;( 3 ) = +J-{
*
cosh(&h),
(l-K’/2X(rh ,Bh, s&KY2f: (q ,tj, x fcos( 5 x)-isin(
q;( $ )+iq;(
$ ) = +rg;(rh
,f+,
(21)
s)f
5 x)] dx,
s){cos( s){sin(t,
f x)-isin(
q”,(S )+iq”,($
) = +/@f;(rh,&, *
q”(S 7
) = L @ g;(rc,Rh, S)-g;(rh&, S)j 2JL /f0 ):{ sin( C; x)+icos( 4 x)f dx,
)+iq”,(S
h),
x)+icos(f
f x)jdx, x)jdx,
(22)
where r,, = (h’+x’$
,
@A= tan-‘(h/x)
(23)
and E: =
1.0,
for
odd n,
Ga=
0.0,
for
even n,
482
S.ITOU k;= 1.0,for even n,
&,"=0.0, for odd n.
(242)
Then, the stressesfor the reflectedwaves which satisfyequation(12) are wr!tten as
(25)
with Ry('(r,o) = (ha/Z-sin'@)fy(r, 5, s)+(fi'/2-cos'8)f:(r,8, s)+sin(20)
om fl($h,(3bp:(S )h,(f )+P:($ )hy(%) JI
xf?(r,@, s)+2
+P';(?)h,(% )jccs(Sx)d5+2%C-p:(5
)h,(4)-P:( 5 )h,(F)
-P:(f )h&f )+P;(% )b( f )jsin($x)dS, R:(, @) = (cod@-sin't9)g;(r,8, s)-sin(2B)/2{g:(r,B, sl-g:(r,e,
s)3
+2;s"{p;(l, fh,(5 )-p;(f, )h,<% )+P;($)h3($ )+P'$$ )b( %)'1 rcos($x)d$ +2je[-p'$$ )h,(f )-P",(% )hz($ )-p;(f )h$ t ) 0
R:(r,@) = (riz'/2-co?8)f;(r,@, s)~(Hz/2-sinLB)fl;(r,B, s)-sin(25)
xf!(r,O, 47
P:(5 )&( 4 )-P:(% )h6($ )+P;($ )h7(c;)
+p;l(F)hs($f\cosi4 x)dS+2jQm~-$S
)h,(i;f-P:(S&(5
-&(S )h,(F )+P;($ )hs(f $sin(S x)dS ,
)
Diffraction R;(r,
6)
of a stress wave
= (sin'Q-cos'@)g;(r,8, &(S
bP;(
x)df +2
sky(r,
s)+sin(28 )/2 [g:(r,O,
tcos(f, Iipl
+2jOW\$(5
483
4 h(
% )+P;(s
&A5
)+P”,(s
8, 3,)
)hds
?)
-p;(5 )h& f )-P",(% )hb(% )-P;( f )h7( %)
0
+P;(? )hg($ Y/sine
R;(r,
6 ) = sin(2 8 )/2
{f:(r,
+2 jOmfp; (5 I$(
1ccos($x)df+2
+p;(f
)hlL(
B , s)-f:(r,
6 codf
x)df
8,
s)f+cos(2
5 )+P;( 4 )h,& 6 bP”,( % h,(
I
mf-~;(f
f )lsin($
R",(r,e ) = -sin(2 e)g:(r,8,
+2\;fp;(%
x)d5,
h+‘?)-~;($
hi’3
@)f:(r,
f )+P;(?
)h,o(%)-P:(?
hi
s)
4’ )f
h,,(F)
,
s)-cos(20 )/2 {g:(r,@,
)hq(f )-p",(f )h,,( f)-P:($
+2 &P;(
@,
f h,(
5 bP”,(
s)-gz(r,8,
)h,, (?)+P;($)
$ )h,o(
f bp;(f
s$
hlr(
b,,
(5 1
(26)
and h,(y ) = ~~(k'/2-~)sz/~~-~'~cos'~+~f'+~zs'/(2c~)~sinzD] cosh(bfy), h,(f
1 = %QTsidCy)sin(20),
h,(
) = $6 cosh(&y)(cos'o-sin?),
f
h+(S)
=iS'+k~sz/(2c:)fsinh(r,y)sin(2
O),
hs(f ) =~~(~'/2-1)s~/c~-f'fsin'B+~S'+~~s~/(2c~)~cos'eJ hr(%
1 = -&(f
1, h,(5; ) = -$(S
1, b6($ ) =-h+(q),
hS( 5 ) =~4'+s'/(2c:)fcosh(~y)sin(2B
),
cosh(Ky),
f$
484
S. ITOU h,,(t, 1
= -Sticosh(k~)sin(2fl)~
h,,(S)
= f~sinh(ky)cos(2~),
h,,(4)
=~'4z+kw/(2c:)+h(~y)cos(2B),
(27)
The stress expressions in equation (25) cause no tractions on the plane boundaries y =?h. The remaining boundary conditions, narnaly, equation (11) reduce to the forms
$C"Rl;(a, O)+&D,R"l(a,
zC,,q(a,
O)+gI&R:(a,e
0) = -P exp[(x-a)s/c.]cos'g/(2/G(s),
) = +P exp[(=a)s/c,]sin(2
8)/(2,~4s), (28)
where Rl(a, @
) (i= 1, 2, 5, 6) are considered to be the functions of x only,
because on r = a,
y =
(a’-x+ ,
6 = tan-'(y/x).
(29)
Equation (28) can be solved for the coefficients C,, D,
by a modified version
of the Schmidt method [5, 6, 71. Therefore, it is now possible to compute the stresses at any point in the strip by the forms
ok kr
p H[t+(x-a)/c,fco&
+ $
Q,
(2p) ck,,R:(r,
B)+D,R;(r, @ )'I
%exp(st)ds,
/t$)= P
H{t+(x-a)/c,\sidb + &jBr
(2,U)g,[C,R",(r, 0 )+D,R;(r, B )f xexp(st)ds,
F-L(r)_ I re - -$p Hft+(x-a)/cLfsin(2 0) + Ai,
(2,U) g[C,R;(r,O) +D,RZ(r,O )]exp(st)ds.
(30)
Diffraction of a stress wave
485
The Laplace inverse transformation in equation (30) is carried out by the numerical method given by Miller and Guy [8]. When the Laplace transform f*(s) can be evaluated at discrete points given by
s=(
p +l+k),
k = 0, 1, 2, 3, ...........
s we may determine coefficients C'
$f*f(p
+l+k)ij
(31)
from the following set of equations
= $C$k!/[(k+~+l)(k+P+2).....(k+~+l+q)(k-q)!], 0
(32) where
,f.O,
p z-1.0.
If the first Q coefficients are calculated, an approx-
imate value of f(t) can be found as
(33)
f(t) = YZOCi Pp'a)[2 exp(-gt)-lj,
where Pi0'6r)(x) is a Jacobi polynomial and Q is the number of terms employed. 7J The
paremeters ,f, R and Q are selected such that f(t) can be best described
within a particular time t.
4. Numerical Examples and Results The circumferential stresses at the cavity surface are calculated numerically for $
= 0.25. The semi-infinite integrals, which occur in the functions
qi( f ) and Rl(r, 19) in equations (22) and (26), from Filon's formula
can be integrated numerically
[IO] easily because those integrands almost decrease
exponentially. To perform the Schmidt procedure, we adopt the first seven terms of the infinite series in equation (28). For a check of the accuracy, the
)}
-0.12454*10'
-0.10181110'
-o.346l7x1o-2
-0.1637jx16'
-0.25493~16'
-0.22993~10~
-0.25396~10~
0.9990
0.0714
0.0010
-0.0714
-0.9206
-0.9990
X{2Acc,/(pa))
8) +D,RJ(a, 0
i-{W:b,
0.9286
x/a
1
0)
-0.11338x10-'
-0.92214x10“
-0.37689*10-'
0.59598x10-'
0.42323x10-'
o.40704xlop
0.55907~10-'
x{2Pcc,/(pa))
of
1.
h.
S.
and r.
h.
5.
coszO
-0.27064=.10-=
-0.56292~10‘~
-0.30342116=
sa/c,
x-a)s/c,
-exp{ ( I /( 1 3
-0.37793x10-'
0.56251~10-~
0.42370x10-'
/(.dCL 1
exp{(x-a)s/q]sin(20
in equation
(28) for h/a = 2.0 and sa/c, = 0.8.
Values
+D,%(a, Cl 1)
&nR:b,
Table
)
487
Diffraction of a stress wave
0.0
L
0.0
Fig. 2
10.0
cd la
20.0
Dynamic stress concentration Sactor at 8 = 0' for h/a = 2.0, 10.0 versus cLt/a.
0.0
Fig. 3
10.0
41/a
20.0
Dynamic stress concentration factor at 8 = 90' for h/a = 2.0, 10.0 versus c,t/a.
488
S. ITOU
?.,IP
Fy----y,
-.j
p‘i; _.
L
Zymn:ic
at
Fig.
cct/a
10.0
0.0
5
stress concer,trationfactor
0 = 180
Stress and
20.0
ror h/a = 1.0, 10.0 versus c,t/a.
distribution
,c,t/a = 5.0,
around
10.0,
P = a for
h/a
15.0 versus B .
= 2.0
Diffraction of a stress wave
489
values of the 1. h. s. and the r. h. s. in equation(28) are given in Table 1 for the case of sa/c& = 0.8, h/a = 2.0. From this, it is clear that the accuracy of the Schmidtmethod is satisfactory.Using the values of p = 0.0, J= 0.2, Q = 7, we invertethe Laplacetransformsnumerically. Figs. 2 through 4 show the hoop stress 7 BB at the cavity boundaryfor @ = O', 90", 180‘ and h/a = 10.0, 2.0, where the broken lines are the correspondingstaticvalues given by Howland[9]. The curvesfor h/a = 10.0 agree with the Baron and Matthews'sresultsfor an infiniteelasticmedium [3].
In
Fig. 5, the stressdistributionsaround the boundaryr = a are shown for h/a= 2.0 and c,t/a = 5.0, fO.0, 15.0. In conclusion,we get the followinginformations: i) The maximum hoop stressesalso occur at
~9 = 90" under the incidentplane
shock stresswave. When the h/a ratio decreases,it looks like that the peak value is generatedat the larger value of the time variable. ii) The peak values of the dynamic stress concentration factors at .d= 90" are increasedby j.6 % and 8.9 "/ D over those staticvalues for h/a = 2.0 and
10.0, respectively. iii) The absoluteClues of the dynamic stressesat (9= O*, 180" are about 1.14 and 1.17 times as large as those of the static solutionsfor h/a = 10.0, respectively.For h/a = 2.0, those values are increasedby about 1.40 and 1.37 times over the staticones. iv) The inertia effect on the hoop stress at r = a is more noticeablenew the regionsof 8 = 0' and 180".
The authorwishes to acknowledgethe valuablesuggestionsofferedby Prof. A. Atsumi of Tohoku University,which were given constantlythroughout this work.
490
S. ITOU References 1. Kromm, A., Zur Ausbreitung von Stosswellen in Kreislochscheiben, ZAMM: 28, 297, 1948. 2. Selberg, H. L., Transient Compression Waves From Spherical and Cylindrical Cavities, Arkiv for Fysik: 1, 97, 1952. 3. Baron, M. L. and Matthews, A. T., Diffraction of a Pressure Wave by a Cylindrical Cavity in an Elastic Medium, ASME J. Appl. Mech.: 2S, 347, 1961. 4. Moodie, T. B., Haddow, J. B., Mioduchowski, A. and Tait, R. J., Plane Elastic Waves Generated by Dynsmic Loading Applied to Edge of Circular Hole, ASME J. Appl. Mech.:
9,
577, 1981.
5. Itou, S., Dynamic Stress Concentration around a Circular Hole in an Infinite Elastic Strip, ASME J. Appl. Mech.: 2,
51, 1983.
6. Morse, P. M. and Feshbach, H., Methods of Theoretical Physics I, McGraw-Hill, p. 926, 1953. 7. Yau, W. F., Axisimmetric Slipless Indentation of an Infinite Elastic Cylinder, SIAM J. Appl. Math.: 2,
219, 1967.
8. Miller, M. K. and Guy, W. T., Numerical Inversion of the Laplace Transform by Use of Jacobi Polynomials, SIAM J. Num. Anal.: 3, 624, 1966. 9. Howland, R. C. J., On the Stresses in the Neighbourhood of a Circular Hole in a Strip Under Tension, Phil. Trans. of the Royal Sot. of London, A: 229, 49, 1930. 10. Amemiya, A. and Taguchi, T., Numerical Analysis and Fortran, Maruzen, Tokyo, 1969.