- Email: [email protected]
Response of an elastic solid to nonuniformly expanding surface loads
hf. 1. Efkytg Sci. Vol. 17, pp. 10234038 0 Pewmon Press Ltd.. 1979. Printed in Gwat Britain
RESPONSE OF AN ELASTIC SOLID TO NONUNIFORMLY EXPANDING SURFACE LOADS ARABINDA ROY Centre of Advanced Study in Applied Mathematics, University of Calcutta, 92, A.P.C. Road, Calcutta-700009, India Abstract-Exact expressions are obtained for the displacement field in a homogeneous isotropic elastic half space whose surface is subjected to a unit normal pressure. The load emanates from a point and expands nonuniformly and radially outwards. The displacement field is obtained in the form of triple integrals over finite ranges. Both accelerating and decelerating loads have been considered. Wave front surfaces with their regions of existence have been identified. First motion approximations near different wave arrivals have been obtained by a limiting process and do not involve any integration.
I. INTRODUCTION EXPANDINGring
or disk loads serve as models for various physical situations. An atmospheric nuclear explosion, for example, produces a time dependent retarding pressure pulse over a circular area of increasing radius. A similar model has been used by Blowers [ 11to explain the erosion of turbine by the impact of water droplets. Atkinson[2] showed the connection between symmetric circular loads to cracks which serve as models for earthquake sources. Although nonuniform motion is more likely to occur in nature, most of the models considered by various authors assume uniform rate of expansion because of complexities involved. For a review of current literature on uniformly expanding ring or disk loads we refer to the work of Gakenheimer[3]. For some special types of nonuniform expansion exact solutions were obtained by Tong[4] and Roy[5]. In all such particular cases the problem was tractable because of the availability of integrals involving Bessel functions in exact form. For general rate of expansion one has to resort to approximations. Miles[6] obtained asymptotic approximations for the displacement field, using Taylor’s strong shock approximation, of an elastic half space subjected to a radially asymmetric pressure signature of variable radius. The corresponding acoustic problem was considered by Aggarwal and Ablow[7] in case of a decelerating circularly symmetric surface load and by Stronge[8] in case of an accelerating surface line load. Recently Freund[9] considered the wave motions expected in case of nonuniform motion for a line source. In particular surface wave effect for point sources was considered in details by FreundHO]. In this paper an exact solution has been obtained for the disturbances due to a circularly symmetric load spreading nonuniformly.over the surface of an elastic half space. No restriction has been imposed on the nature of expansion. The displacement field is obtained in the form of triple integrals over finite ranges by use of Cagniard-De Hoop and special reduction technique. It is found that the displacement field contains, besides the usual body and head waves, the contribution from conical waves which arise due to motion of the source. Number and region of existence of conical waves depend on the nature of the motion of the source. First motion responses near different wave arrivals have been obtained. It is believed that the effect of nonuniform motion of the source has not been considered in such details previously by any author. The method presented in this paper can be fruitfully applied to a class of nonuniformly expanding loads having circular shape. A similar method has been used by Roy [ 1I] to evaluate the displacement field for a nonuniformly expanding point load. 2. FORMULATION Let
OF THE PROBLEM
an elastic half space occupy the region z ~0. On the surface a unit normal pressure 1023
ARABINDA ROY
1024
pulse acts on a circular area whose radius expands nonuniformly. In cylindrical co-ordinate system (r, 8, z), the boundary condition is given by, on z = 0 7,,
=
-2 = N(R(t) - r)H(t);
Tn
=
0
(1)
where TVand rZr are the components of stress tensor referred to cylindrical co-ordinate system, (r, 0, z); H(t) is Heaviside unit function. We assume, for the present, that R(t) is nonnegative monotone function and R(0) = 0. The transformed displacement field inside the medium (z 2 0) can be written, following[l2], as 1(2k*+ P*@*)exp(-%z) - 2~71~exp(-q,z)l dk,
(2) where
(k, &I =
m
(u, uz)exp(-pt) dt,
I
Z;fr
Z exp(-pt)rJo(kr) dr dt,
b’ = (2k2 + p*//3*)* - 4k2qpqs, 77, = (k2 + p2/f12)“*.
rip = (k* +p2/a2)“*,
(3)
a, /3, the P and S wave velocity, are given by a* = (A + 2p)/a, and fl* = p/u; A,~1being Lame’s constant and u the density of the medium. We choose the Riemann surface where Re(r),) > 0 and Re(q5) > 0. For the particular type of stress distribution (1) we have
exp(-pr) y
Jr(kR(T)) dk.
(4)
We make use of the following results ‘OS ’ J,(kS) d4,
J&R(T))J,(kr)
= & I_: R(T)-;
Jr(kR(7Mkr)
= &- /_w JdkS) cos 4 d4, *
(5)
where S = (r* + R*(T) - 2rR(~) cos 4)‘“. Also we have [see 131
[
I0
g(kWdkS) dk = mg(k)W,(kS)
& f-11-1 g(t (q*+ 0211’*) exp[ i f qs] 5 dq dw,
dk = -L2?r
1:
m
J-m
m
&g
~(q2+w2)“*)exp[i~~S]dqdw.
(6)
Response of an elastic solid to nonuniformlyexpandingsurface loads
102.5
On using (4)--(6)the transformed displacement field for the given stress distribution (1) can be written in the form i3j= iQjp+ tijsis,
(7)
where for j = rorzandc=PorS 4 =
m r.+exp(-pt) dt, _f0
and i,p and I,+sare given by
(9) &(q, w), etc. are given in Appendix. 3. REDUCTION OF THE INTEGRAND
The displacement field will now be evaluated in two steps. The integral & is first evicted by the Cagniard-De Hoop technique which involves the ~~fo~ation of the path of integration along the real q-axis to the path of steepest descent, which is also the Cagniard path, given by a(f--r)=-
iqS + z(q2 +
ob2 + I)‘“,
~(~-~)=-iqS+z(q2+~2+a2~~2~1’2.
(10)
The inversion can than be done easily. We refer, for details to a similar problem in [13]. Following [133the displacement can be written in the form
where 6, is Kronecker’s delta function, and for c = P or S,
where
and &sP
=IIt *
_nfi[IH(t-~sP(7)-H(f-fS(7)}~sPi
0
+@N
-fs(~))-~(f~(?)-
~)~~js~2ld~ d7
(13)
1026
ARABINDA ROY
Zic(for c = P or S) and Z,SPIetc. are given by USPI) ‘2 0r2+Sin2$)1/2 (~S~‘+~ZSP”~~‘~~)“~~~~~~~~
=” iFj&sPh
ZjSP I =
&(qc,
w)(d
+ wf + &)“2 W
m’2iLi'FjS(qsp2, oSp2)(q~p2+0$~2
up =a,
+ a2//32)"2
(14)
US =/3.
Different symbols used in (13), (14), etc. are given in Appendix.
4.FURTHER REDUCTION OFTHE
T-INTEGRATION
Equation (13) can be written as
(15) where
4C = cos-’
r2 t R2(7)
t z2 - u f(t - T)~ 2rR(T) 3’
The displacements as given by (15) are now written in alternate forms to facilitate further reduction, viz I -z/v,
Ujc =
I [
H(r
JH(r+
-
H(R(T)
-R(~)t[uf(f
T)~-z~]"~)H([u~(~-T)~-z~J"~-
R(T)-[ut(ft[u:(t -T)*-
I -z/v,
t
I0
-~)~-~*]"~)H(l?(~)-[uf(f-~)~-z~]"~)
z2]"2- r) " If -4%
Qjc d4
R(T))
dT
H(-R(~)t[uf(t-~)~-~~]"~-r)H(R(~)-[uf(t-~)~-~~l~'~)
The region of support for the T-integration is bounded by the curve I:
r=R(T)+[uf(f-T)~-z~]"*,
II: r=R(T)-[uf(f-T)~-Z~]"',
111:
r=
-R(T)t[uf(f-~)~-g*]"~,
7
(18)
Tc < 7
(19)
O-CT < Tc
(20)
o<
The region of support for difEerent cases are shown in Figs. 1 (a-d). Some general remarks about them can be made.
Responseof an elastic solid to nonuniformlyexpandingsurface loads
1027
Fig. 1. Regionof support for I,. (a) When a single maximum(r,*) exists i.e. in case of source with starting velocity such that @r/8+ > 0; (b) When no extremumexists; and (c and d) When both a maximum(r:) exists followed by a minimum(r,**) i.e. in case of a rapidlyacceleratingsource such that (&/&r)o
We note that the curves II and III are monotonic decreasing and increasing in their respective zones of existence which are (T,, t -z/v,) and (0, rC) where R(Tc) = [uf(t -7$--2]‘n. The curve I has extremum where
ar uf(t - 7) -=&)a7 [do - d2- z21‘72
(21)
vanishes. Also we have 2
..
~=R(T)-~~f~t_-7;2_22~312.
2
(22)
Depending on the nature of motion of the source, i.e. on R(T) and on initial starting velocity d(O), the curve I may have a single maximum or no extremum. The different Figs. l(a-d) arise depending on the nature of extremum of curve I. Thus in case of an accelerating pressure signature, i.e. J?(T) > 0, if (ar/&)o > 0 if the initial velocity (which is in general greater than yC)is such that (ar/tib > 0, then, since (&l&)0> 0 and (ar/ar)t-d,C ~0 the curve I has always a maximum at r = r$ or at r = T,* (or t = t$) where 0 c r? < t -z/o,. This corresponds to Fig. l(a). While if (adaT)c 0, i.e. starting velocity is in general less than v, then the curve I has either no extremum as in Fig. l(b) or a maximum at r = r? (Figs. k-d) preceded by a minimum at r = r$* (or 7 = T:* or t = t,**). In the latter case the acceleration must be rapid enough such
ARABINDA ROY
1028
that a*r/ar* must vanish at least once in T,**-CT CT ,*. In case of decelerating source (i.e. I?(T)O as in Fig. l(a) or no extremum as in Fig. l(b) if (d&T)0 < 0. We consider curves I and II, i.e. eqns (18) and (19) together. In that case r may be single (Fig. lb), double values (Fig. la) and triple valued (Fig. lc-d) functions of T. Regarding eqn (18) and (19) together, as given by (18) and (19) are designated as T = 7clr (T = T,I, T = 7,~) and (7 = T,~, T = T,*, T = ~~3) where 7Cl> ~~2 > ~~3 depending on whether r is single, double or triple valued functions of T. Since in eqn (20), r is a monotonic decreasing functions of T, the corresponding value of T is designated as T = ~~4. With the above designation of roots of (18x20) and a close examination of different Figs. l(a-d), the displacements associated with P and S waves are given by, for c = P or S. 4 Ujc=C
k=l
Uik,9
where
uiC= [H(r - r3 - G(r - maxm(rc*,$)>I I::’ 1-z Qic d4 dr, c
u:~
=
e
[G(r - min”(r,*, r3) - G(r - mitP(rt*, Tc4
u$ = H(r: - r)
r!)]
Qicd# dT,
=
I I 0
Qjc d4 d?
-a
(23)
where G(r - maxm(r$, rf)) =
if r? = maxm(r:, r3
H(r - r,*)If(LF) 1 H(r - r3
if r: = maxm(rf, r3 or 6’ does not exist.
(24)
rf is the value of [uf(t -T)* - z*]“* at T = 0, i.e. t-f = [oft*- z*]“* and the starred quantities, viz. r*CTr$* are the maximum (max.) and minimum (min.) of the curve I (eqn 18). The expression
max”(a, b, c) means maximum of the elements a, b and c. Similar meaning is attached to the symbol G(r-min”(rF*, rf)). We note that in G(rmaxm(rF, r:)), G(r-min”(r$* ,ri?), whenever the starred member r: or r$* is maximum or minimum as the case may be, an additional factor If(C) appear. These factors H(L:) as will be seen later, give the region of existence of corresponding conical waves, and Lf etc. are given by
L,* =
r d(O) --PO “c
1 if d(0) is finite
f r
if h(O) is infinite
L:* = rf where r: is the common value of (r,*, r$*).
(25)
5.REDUCTION IN CASE OF HEAD WAVE
The displacement field associated with head waves as given by (13) can be written as UjSP= UjSP1 +
UjSl -
ujS2
+
UjB
(26)
where Ursi and Urs2are the expressions on the R.H.S. of (23) for c = S with 1r.sin Qrs replaced by &P~H(S/P -P/4 and 4dW~ - B/a), respectively.
Responseof an elastic solid to nonuniformlyexpandingsurfaceloads
1029
The fourth term qB in (26) can be easily written from (23). Thus 4 ujB=x
UfB9 k=l
where uL, etc. can be written from u’,c, etc. by changing subscript C by B. Thus u!~, e.g. is given by given by U_&= H(r”B- r)
981 98
QjBd4 dT*
I 784 I -dr,
Wa)
Similarly other terms ub, etc. can be written from ufC. In (27a) the symbols have the following meaning
781, 7B2 and 783 are the roots, when they exist, in the respective zones, as explained in Section 4 in connection with curves I, II, III, of curves
I”.
r = Jq7) + k0 - 7 - mp2 - lla2)1Dl”2 (l/S2 - lla2)“4 ’
O
jr: r = R(7) - [Z(f- ;~~!‘i”,;~2f1!~2)“21”2, 7B< 7 < t _ z(1/p _ l/a2)v2 VI: r = _R(7) + [z(t - ;-$f3;2
;W’Y, a2 ‘I4
where 7s is the value when R.H.S. of V vanish. Also we have L*
=
B
2&3 Z
(
1
p-2
1
l/2
>
-19
if a single max rg of IV exists.
if both max rS and min rg* of IV exist, and ri is the common value of (rg, rg*). We now consider the reduction in case of Ujsprin (26). We tirst write alternate forms similar to (15). We see that the region of support for T integration in this case depends on the form of curves
CW
CW
, O
where R(Tsp)=at-aTsp-z
UES Vol. 11, No. 9-C
ARABINDA ROY
1030
The curves VIII and IX are monotonic functions of 7. On the curve VII we get g = d(7) - (Yand $
= R(r).
As in the previous case, the nature of displacement field associated with ujspr depends critically on the existence of an extremum of curve VII, at times when the source just attains P wave velocity of the medium. Then when the source is decelerating the curve has a maximum at t = t&, denoted by r = rzp if R(O)> a or no maximum if A(O)< a and the region of support is similar to Figs. l(a-b). When the source is accelerating, unlike in the previous case, the wave has a single minimum at I = tz:, denoted by r = rz: if A(O)< a and no minimum if g(O) > a. The region of support in this case is as in Figs. 2(a-c). As in the previous case we consider the curves VIII and IX together. Then the values of T for any Vdlle Of r fle denoted by T = rspr, (r = ~Spl, T = TSp2) and (7 = TSpl, 7 = TSp2, 7 = TSp3) where TSPI > 7Sp2 > 7Sp3 depending on whether r is single, double or triple valued function of T. Similarly the values of T as given (28~) is denoted by T = ~~~4 since r is a single valued function of 7. With the above designation of roots and a close examination of Figs. similar to Figs. l(a-b) and Figs. 2(a-c), the displacement Ujsprcan be written as 4 UjSPl
=
2 k=l
(29)
UfSPlp11
where
r \ QP I
UjiPl
=
H(&P
-
r)
63-P
Q~SPI
dd
dT,
T
(a)
(b)
w Fig. 2.(a-c). Region of support for in case of an accelerating source. Meaning of different lines are as in Fig. 1 except that they represent TSP~, TSPZ, TSP~ and TSP~.
Response of an elastic solid to nonuniformlyexpandingsurface loads
U&I = [Gtr - max”(r&,Rft
U&P, = [G(r - min”(R(t
-
- z(lt/~?~- lt&“3,
1031
&))
z(l@*- l/(~*)“*,r$))
(30) and
4SP =
cos-’
r*+R*(T)- (at -
1
a7 - z(a*/p - 1)‘32 2?R(?)
[ II2
r&=t-2
. (A--$ B a>
(31)
In (30) I;&,etc. which appear as a factor W(L&) in connection with
whenever the starred member, i.e. r& is max, are given by
in case of a decelerating source with starting velocity greater than the P wave velocity, i.e. ti(O)> (1.
in case of an accelerating source with starting velocity less than P wave velocity, i.e. d(O) < Q. We note that eqn (11) together with (23), (26), (27) and (29) give the displacement field.
6. WAVE FRONTS
Besides the usual body and head waves in the medium, corresponding to r = r: and r = rip or equiv~en~y at t _& = VC
v +z*Y’* VC
The displacements as given by (23) and (30) contain contributions from conical wave fronts which arise due to the motion of the source as seen from the second and third terms in (23) and (30). Thus a conical P or S wave arrives at r = r$ (or t = t!) depending on whether c = P or S. From (16) and (21) one can easily write
1032
ARABINDA ROY /.
=R(T)+&A,
m
Z=U,(r-+&-J.
(32)
The above representation shows that the conical P or S waves which corresponds to an extremum of t - 7 -p/o, = 0, can be identified as the envelope of elementary wavelet as the source moves and arise only when the source is moving with a velocity greater than uC. The intersection of the conical wave front with the free surface is obtained by setting z = 0 in (32) which gives r = t if @i(r)> uCalways and T = T* where d(r*) = u, if R(r) g u,. Thus, as long as the source moves supersonically, the intersection of the conical wave fronts with free surface coincide with the rim of the circular load. When the source moves subsonically the intersection is given by r = R(T*) + u,(t - T*) and has separated from the load on the surface, from the instant when the source attains the velocity II, with the tip of the wave front on z = 0 moving with velocity uC.The conical wave front exists only in regions where L,* > 0 as given by the factor H(L,*) in G(r - max”(r$, r3). In case of a rapidly accelerating source, a second conical wave exists in region where r: > 0 and is associated with a minimum of t - T -p/u, = 0 and is always preceded by the first conical wave. Similar remark holds for conical head wave in regions Ir - l?(~&)(/p* > pla,except that no second conical head wave exists. Intersection of conical head wave front with the surface is r = R(r&) + a(t- T&). Thus conical head waves appear at time when the load has just attained P wave velocity, with the tip of wave front moving with velocity a.Conicalhead wave does not exist in case of uniform motion. Figures 3(a-c) show the different wave fronts in the medium for different case of moving sources. A detailed discussion of the development of wave fronts for reciprocating shear load has been given by Watanabe [13] and for particular type of nonuniformly expanding circular load by Roy[5]. We note that the surfaces r=R
(t-z
(+--$)1’2)
r=&
are not wave fronts, since none of them satisfy the eikonal equation. Displacements together with the derivatives are expected to be continuous on those surfaces. Similar surfaces are also found in case of uniform motion[3]. 7. FIRST MOTION RESPONSES The displacement field as given in Section 5 is expressed in terms of triple integration over finite ranges. As such exact computation is difficult except with the use of large computers. However an insight into the nature of the displacement field by calculating the 6rst motion responses near different wave arrivals. This can be done by a limiting process used in [131. Thus near the arrival times, i.e. t = pdu, of P or S waves, the displacement as given by (15) is
(34) In obtaining the approximate form near t = p&, the inner integrand is evaluated at t = r +p/u, and noting that the contribution of the second term is nil to the same order of approximation we can write the approximate form of the displacement field near t = po/uc
Response of an elastic solid to nonuniformly expanding surface loads
.
(b)
pi*
Fig. 3. Wave fronts in an half space: (a) For a circular load on the surface mov& nonuniformly with starting velocity greater than the transverse velocity (B) at a time when the rim of the load, L, is moving with velocity greater than the longitudinal velocity (a); (b) At a time when the same load is moving with velocity less than the transverse velocity (B) (Note that conical head wave (CSP) has developed and the tip of the wave fronts has separated from the rim of the load); (c) In case of a rapidly accelerating load with starting velocity less than the transverse velocity at a time when the load is moving with velocity greater than the longitudinai velocity. In Fig. 3(a, b), P, S, CP, CS, CSP represent longitudinal, transverse, conical P, conical S, conical head wave fronts respectively. In (c), CPI, CP2, CSl, CS2, CSP represent the tirst conical P, second conical P, tint conicalS and second conical S and conical head wave fronts respectively. In (a-c), L represents the position of the rim of the load.
r-R(T))*-z*)((r +R(T))*+ z*-uf(t - T)*]“*~T
(35)
where [&I = jump of I& at t = T+&
[I&3 = value of [&I at 4 = 0 -1 Cfi)o=(
1 for j =z.
We note that 7cl is the root in 0 < T c t -z/v, t _ 7_
forj=r
of
m*+~*I”*=
k - I?
VC
0. ,
06)
ARABINDAROY
1034 Whilerc4is the root, in 0 <
T <
t -
of
z/v,
(37) Expanding (36) near
T =
0 we
get
t _ T_
[(r - R(TN’+ ~~1"~=t_$J_T(g!w). UC
Thus (36) has a root denoted by
Tag,
for r/p0 > v&O) near t = p~/v, where t-
Tel =
*
PO/V,
(38)
r fi(O)’ PO UC
For r/p0 > v,/@O), (36) has a negative root close to 0 and will henceforth be designated by where, for t > p~/v, t -PO/UC
TO= --r k(O) PO vc
-TV
(39)
1’
The position of this root, viz -TV, is critical in obtaining the approximate displacement field near t = p~lv,. Similarly expanding (37) near T = 0, we get
We are now in a position to obtain the displacement field near t = p~lv,. We consider first the regions where r/p0 > v,/&(O). In this case for t > po/ve, the situation is similar to Fig. l(d). Then we get from (35)
~(jj)&]~(l-~)~‘*~ (l-2)
X/"'
c
Substitution of T =
~~4 cos*
0 + TV I
7c4
[(p-~c4)(~c,-T)]1’2dT~ (41)
sin* 8 and using the value of ~~4,T,] from (38) and (40) we get tblo(r-E)
ujc_E&+)o -
2
[
m
H(GJH(l-;F).
(42)
l-<+))]
PO VE
Let us now consider the region r/p0 > v,/d(O). In this case situation is as depicted in Fig. l(a). The form of the displacement field is difIerent due to presence of additional root at T = -TV. In this case one can write the displacement field as t + po/vc + 0 as ul,
( > t- @
&clti)lon2vcPo Jm)IJ UC rm+ 701 r L-&F-
-H X
[(~c,
-
T)(T
-
q4)(7
+ TO)]“*
d?.
111’2H(i-&J (43)
Response of an elastic solid to nonuniformly
expanding surface loads
I035
Substitution of T = T,~cos’ 8 + 7Clsin* 0 gives
where K(k) and E(k) are elliptic integrals of the first and second kind and k is given by k2_
Tel-7~4
k'2 =
1 _k2.
(45)
Tel +70'
Thus k --, 1 and kr +O as t +p~/u,, Hence using the expansion of B(k) and R(k) near k = 1 and noting that one gets similar expansion near t = p&, - 0, we can write the approximate form of the displa~ment field as
where rC is the root of (35) in 0 < r < t -z/u, ; In case two such roots exist, rC is taken to be the larger one. Near the arrival times of conical P or S waves the displacement field is given by
X ui’*H(t - tc*)ri(LP) lim S-+0
(47) where
&lf)=7~= value of &] at 4 = 0 and T = r$, f”(T) = $f(T)
f(l)=_7._r(r-R(T))*+2*l”* . f& In case of an accelerating source when a second conical wave is present, the corresponding displacement near its arrival times is obtained from (47) and replacing t,* and T$ by t?* and rc**, respectively. Near the cusp of Fig. 3(c), noting that f”(7) = 0 one can obtain, proceeding similar to the previous case, the displacement geld near the arrival time as
x [(r - R(r:))* + 2213’2 2 “‘63 _u3
(f -t:)3"
Lf”(7:)l’”
1 3 5,2’ ( >
where B(m, n) is Beta function and tf, TEare the common values of (tr, t:*) and (T,*,T$*).
1036
ARABINDA ROY
The displacement component near conical head wave is given by
(49)
On evaluating the inner integrand near t = fsp(r) following[l3] and multiplying numerator and denominator by t - fSP(r) and some simplification one can write (49) as
CY UjSPl
“I H(t
-
tZPW(L~P)
$- >3’4(fi)f=T;p (a 1
Re{Gjsp} 112
372
-z]
Substituting the values of &p from (27) and expanding near t = t gp and after simplification we can write the displacement field near f = fzp+ as
(50)
Gjsp = lim 27;s
FjsPdqsPI, (qiPl+
WSPI) dp*+
1Y*
8.CONCLUSION
It is seen from the first motion responses that displacement at any station depends on the location of the station and type of wave arrivals at the station. Thus from (42) and (46) one can see the nature of displacement near body wave arrivals depends on whether at the station conical waves are tirst to arrive or not. The displacement field near the arrival times of surface waves can be easily obtained by calculating the residue contribution at the Rayleigh pole. However, we have not considered the surface wave effect since the case has been discussed in detail by Baron and Lecht[lS] and by Freund[lO] in case of a point source. As noted before, the exact computation of the displacement field requires the use of high speed computers. Only in case of 6rst motion responses the displacement field can be obtained by a limiting process. However, first motion responses are itself useful tools in the study of earthquake source mechanism. Recently various authors have determined the source dimension of earthquakes[l6] from the comer frequencies of seismic amplitude spectra. The corner frequency is that frequency where the frequency spectra of first motion responses (i.e. at high frequency) as obtained in the previous section equal the frequency spectra at low frequencies which can be easily obtained from (2) by making p +0[17]. Thus first motion responses can be used to determine the corner frequencies. We have considered a typical case of nonuniformly expanding circular disk load which serve as models for atmospheric explosion. The present paper gives a comprehensive analysis of the displacement field due to an axially symmetric nonuniformly moving circular load. Most of the analysis so far done by previous author are either 2-dimensionaQl31 or approximations [a]. Obviously our analysis can be adapted for all class of nonuniformly moving load having circular geometry.
Response of an elastic solid to nonuniformly expanding surface loads REFERENCES [l] R. M. BLOWERS, J. Inst. M&s. Applies. 5, 167 (1%9). [2] C. ATKINSON, Int. J. Engng Sci. 6,27 (1968). [3] D. C. GAKENHEIMER, J. Apple Mech. 38,99 (1971). (41 K. J. TONG, PH.D. Thesis. Stanford University, Stanford, California (1968). [5] A. ROY, Ind. J. Pure Appl. Math. 5, 1063 (1974). [6] J. W. MILES, J. Appl. Me&. 27,710 (1960). [fl H. R. AGGARWAL and C. W. ABLOW, Bull. &is. Sot. Am. 55,763 (l%S). [g] W. J. STRONGE, J. Appl. Mech. 37, 1077 (1970). [9] L. B. FREUND, Appl. M&h. 30,271 (1972). [lo] L. B. FREUND, J. Appl. Mech. 40,699 (1973). [I l] A. ROY, Int. J. So/ids Structures 14,755 (1978). [12] M. MITRA, Proc. Comb. Phil. Sm. 60, 683 (1964). [13] A. ROY, Int. .f. Engng Sci. 13,641 (1975). [14] K. WATANABE, Inr. J. Solids Structures 13.63 (1977). [IS] M. L. BARON and C. LECHT, I. Engng Mech. Div. (Pm ASCE) 87,33 (1967). [16] M. WYSS and L. J. SHAMEY, Bull. Seis. Sot. Am. 65,403 (1975). [17j R. BURRIDGE, Bull. Seis. Sot. Am. 65, 667 (1975).
(Received 16 February 1979)
APPENDIX fr = cos 6, f*=
R(T) - r cos I$ s ,
F,p(q ~)J20*+2qws3 2wpA FzPkl,
0)
= -
’
iq(202+2q2+a2//33(02tq2t1)“* 2rpA(q* t o*)
’
A=(2o2t2q2ta2/~2)2-4(o2+q2)(O2+q2t1)”2(~2tq2to2/~2)‘R, icr(t-7)s qp =2 P
2 tPTpcos$,
up = TP sin I), T
= P
4s =
ia(t-7)s P2
us = TS sin
OSP I = TSP
2 t;Tscos$,
$,
sin h
m =(g’*_m*X3”2,
1037
ARABINDA ROY
1038 qstv=
ia(t- 7)s p2
ia I
--j-36’sin$,
OSPZ=T:.~(T~~-_TB)~~~*~L
s = (r2+ P(T) - 2rR(7) cos c$)‘“, /I=(s+z*)“*, po = p/r -
0 = (2 t zy, 11
c=P
RESPONSE OF AN ELASTIC SOLID TO NONUNIFORMLY EXPANDING SURFACE LOADS ARABINDA ROY Centre of Advanced Study in Applied Mathematics, University of Calcutta, 92, A.P.C. Road, Calcutta-700009, India Abstract-Exact expressions are obtained for the displacement field in a homogeneous isotropic elastic half space whose surface is subjected to a unit normal pressure. The load emanates from a point and expands nonuniformly and radially outwards. The displacement field is obtained in the form of triple integrals over finite ranges. Both accelerating and decelerating loads have been considered. Wave front surfaces with their regions of existence have been identified. First motion approximations near different wave arrivals have been obtained by a limiting process and do not involve any integration.
I. INTRODUCTION EXPANDINGring
or disk loads serve as models for various physical situations. An atmospheric nuclear explosion, for example, produces a time dependent retarding pressure pulse over a circular area of increasing radius. A similar model has been used by Blowers [ 11to explain the erosion of turbine by the impact of water droplets. Atkinson[2] showed the connection between symmetric circular loads to cracks which serve as models for earthquake sources. Although nonuniform motion is more likely to occur in nature, most of the models considered by various authors assume uniform rate of expansion because of complexities involved. For a review of current literature on uniformly expanding ring or disk loads we refer to the work of Gakenheimer[3]. For some special types of nonuniform expansion exact solutions were obtained by Tong[4] and Roy[5]. In all such particular cases the problem was tractable because of the availability of integrals involving Bessel functions in exact form. For general rate of expansion one has to resort to approximations. Miles[6] obtained asymptotic approximations for the displacement field, using Taylor’s strong shock approximation, of an elastic half space subjected to a radially asymmetric pressure signature of variable radius. The corresponding acoustic problem was considered by Aggarwal and Ablow[7] in case of a decelerating circularly symmetric surface load and by Stronge[8] in case of an accelerating surface line load. Recently Freund[9] considered the wave motions expected in case of nonuniform motion for a line source. In particular surface wave effect for point sources was considered in details by FreundHO]. In this paper an exact solution has been obtained for the disturbances due to a circularly symmetric load spreading nonuniformly.over the surface of an elastic half space. No restriction has been imposed on the nature of expansion. The displacement field is obtained in the form of triple integrals over finite ranges by use of Cagniard-De Hoop and special reduction technique. It is found that the displacement field contains, besides the usual body and head waves, the contribution from conical waves which arise due to motion of the source. Number and region of existence of conical waves depend on the nature of the motion of the source. First motion responses near different wave arrivals have been obtained. It is believed that the effect of nonuniform motion of the source has not been considered in such details previously by any author. The method presented in this paper can be fruitfully applied to a class of nonuniformly expanding loads having circular shape. A similar method has been used by Roy [ 1I] to evaluate the displacement field for a nonuniformly expanding point load. 2. FORMULATION Let
OF THE PROBLEM
an elastic half space occupy the region z ~0. On the surface a unit normal pressure 1023
ARABINDA ROY
1024
pulse acts on a circular area whose radius expands nonuniformly. In cylindrical co-ordinate system (r, 8, z), the boundary condition is given by, on z = 0 7,,
=
-2 = N(R(t) - r)H(t);
Tn
=
0
(1)
where TVand rZr are the components of stress tensor referred to cylindrical co-ordinate system, (r, 0, z); H(t) is Heaviside unit function. We assume, for the present, that R(t) is nonnegative monotone function and R(0) = 0. The transformed displacement field inside the medium (z 2 0) can be written, following[l2], as 1(2k*+ P*@*)exp(-%z) - 2~71~exp(-q,z)l dk,
(2) where
(k, &I =
m
(u, uz)exp(-pt) dt,
I
Z;fr
Z exp(-pt)rJo(kr) dr dt,
b’ = (2k2 + p*//3*)* - 4k2qpqs, 77, = (k2 + p2/f12)“*.
rip = (k* +p2/a2)“*,
(3)
a, /3, the P and S wave velocity, are given by a* = (A + 2p)/a, and fl* = p/u; A,~1being Lame’s constant and u the density of the medium. We choose the Riemann surface where Re(r),) > 0 and Re(q5) > 0. For the particular type of stress distribution (1) we have
exp(-pr) y
Jr(kR(T)) dk.
(4)
We make use of the following results ‘OS ’ J,(kS) d4,
J&R(T))J,(kr)
= & I_: R(T)-;
Jr(kR(7Mkr)
= &- /_w JdkS) cos 4 d4, *
(5)
where S = (r* + R*(T) - 2rR(~) cos 4)‘“. Also we have [see 131
[
I0
g(kWdkS) dk = mg(k)W,(kS)
& f-11-1 g(t (q*+ 0211’*) exp[ i f qs] 5 dq dw,
dk = -L2?r
1:
m
J-m
m
&g
~(q2+w2)“*)exp[i~~S]dqdw.
(6)
Response of an elastic solid to nonuniformlyexpandingsurface loads
102.5
On using (4)--(6)the transformed displacement field for the given stress distribution (1) can be written in the form i3j= iQjp+ tijsis,
(7)
where for j = rorzandc=PorS 4 =
m r.+exp(-pt) dt, _f0
and i,p and I,+sare given by
(9) &(q, w), etc. are given in Appendix. 3. REDUCTION OF THE INTEGRAND
The displacement field will now be evaluated in two steps. The integral & is first evicted by the Cagniard-De Hoop technique which involves the ~~fo~ation of the path of integration along the real q-axis to the path of steepest descent, which is also the Cagniard path, given by a(f--r)=-
iqS + z(q2 +
ob2 + I)‘“,
~(~-~)=-iqS+z(q2+~2+a2~~2~1’2.
(10)
The inversion can than be done easily. We refer, for details to a similar problem in [13]. Following [133the displacement can be written in the form
where 6, is Kronecker’s delta function, and for c = P or S,
where
and &sP
=IIt *
_nfi[IH(t-~sP(7)-H(f-fS(7)}~sPi
0
+@N
-fs(~))-~(f~(?)-
~)~~js~2ld~ d7
(13)
1026
ARABINDA ROY
Zic(for c = P or S) and Z,SPIetc. are given by USPI) ‘2 0r2+Sin2$)1/2 (~S~‘+~ZSP”~~‘~~)“~~~~~~~~
=” iFj&sPh
ZjSP I =
&(qc,
w)(d
+ wf + &)“2 W
m’2iLi'FjS(qsp2, oSp2)(q~p2+0$~2
up =a,
+ a2//32)"2
(14)
US =/3.
Different symbols used in (13), (14), etc. are given in Appendix.
4.FURTHER REDUCTION OFTHE
T-INTEGRATION
Equation (13) can be written as
(15) where
4C = cos-’
r2 t R2(7)
t z2 - u f(t - T)~ 2rR(T) 3’
The displacements as given by (15) are now written in alternate forms to facilitate further reduction, viz I -z/v,
Ujc =
I [
H(r
JH(r+
-
H(R(T)
-R(~)t[uf(f
T)~-z~]"~)H([u~(~-T)~-z~J"~-
R(T)-[ut(ft[u:(t -T)*-
I -z/v,
t
I0
-~)~-~*]"~)H(l?(~)-[uf(f-~)~-z~]"~)
z2]"2- r) " If -4%
Qjc d4
R(T))
dT
H(-R(~)t[uf(t-~)~-~~]"~-r)H(R(~)-[uf(t-~)~-~~l~'~)
The region of support for the T-integration is bounded by the curve I:
r=R(T)+[uf(f-T)~-z~]"*,
II: r=R(T)-[uf(f-T)~-Z~]"',
111:
r=
-R(T)t[uf(f-~)~-g*]"~,
7
(18)
Tc < 7
(19)
O-CT < Tc
(20)
o<
The region of support for difEerent cases are shown in Figs. 1 (a-d). Some general remarks about them can be made.
Responseof an elastic solid to nonuniformlyexpandingsurface loads
1027
Fig. 1. Regionof support for I,. (a) When a single maximum(r,*) exists i.e. in case of source with starting velocity such that @r/8+ > 0; (b) When no extremumexists; and (c and d) When both a maximum(r:) exists followed by a minimum(r,**) i.e. in case of a rapidlyacceleratingsource such that (&/&r)o
We note that the curves II and III are monotonic decreasing and increasing in their respective zones of existence which are (T,, t -z/v,) and (0, rC) where R(Tc) = [uf(t -7$--2]‘n. The curve I has extremum where
ar uf(t - 7) -=&)a7 [do - d2- z21‘72
(21)
vanishes. Also we have 2
..
~=R(T)-~~f~t_-7;2_22~312.
2
(22)
Depending on the nature of motion of the source, i.e. on R(T) and on initial starting velocity d(O), the curve I may have a single maximum or no extremum. The different Figs. l(a-d) arise depending on the nature of extremum of curve I. Thus in case of an accelerating pressure signature, i.e. J?(T) > 0, if (ar/&)o > 0 if the initial velocity (which is in general greater than yC)is such that (ar/tib > 0, then, since (&l&)0> 0 and (ar/ar)t-d,C ~0 the curve I has always a maximum at r = r$ or at r = T,* (or t = t$) where 0 c r? < t -z/o,. This corresponds to Fig. l(a). While if (adaT)c 0, i.e. starting velocity is in general less than v, then the curve I has either no extremum as in Fig. l(b) or a maximum at r = r? (Figs. k-d) preceded by a minimum at r = r$* (or 7 = T:* or t = t,**). In the latter case the acceleration must be rapid enough such
ARABINDA ROY
1028
that a*r/ar* must vanish at least once in T,**-CT CT ,*. In case of decelerating source (i.e. I?(T)
k=l
Uik,9
where
uiC= [H(r - r3 - G(r - maxm(rc*,$)>I I::’ 1-z Qic d4 dr, c
u:~
=
e
[G(r - min”(r,*, r3) - G(r - mitP(rt*, Tc4
u$ = H(r: - r)
r!)]
Qicd# dT,
=
I I 0
Qjc d4 d?
-a
(23)
where G(r - maxm(r$, rf)) =
if r? = maxm(r:, r3
H(r - r,*)If(LF) 1 H(r - r3
if r: = maxm(rf, r3 or 6’ does not exist.
(24)
rf is the value of [uf(t -T)* - z*]“* at T = 0, i.e. t-f = [oft*- z*]“* and the starred quantities, viz. r*CTr$* are the maximum (max.) and minimum (min.) of the curve I (eqn 18). The expression
max”(a, b, c) means maximum of the elements a, b and c. Similar meaning is attached to the symbol G(r-min”(rF*, rf)). We note that in G(rmaxm(rF, r:)), G(r-min”(r$* ,ri?), whenever the starred member r: or r$* is maximum or minimum as the case may be, an additional factor If(C) appear. These factors H(L:) as will be seen later, give the region of existence of corresponding conical waves, and Lf etc. are given by
L,* =
r d(O) --PO “c
1 if d(0) is finite
f r
if h(O) is infinite
L:* = rf where r: is the common value of (r,*, r$*).
(25)
5.REDUCTION IN CASE OF HEAD WAVE
The displacement field associated with head waves as given by (13) can be written as UjSP= UjSP1 +
UjSl -
ujS2
+
UjB
(26)
where Ursi and Urs2are the expressions on the R.H.S. of (23) for c = S with 1r.sin Qrs replaced by &P~H(S/P -P/4 and 4dW~ - B/a), respectively.
Responseof an elastic solid to nonuniformlyexpandingsurfaceloads
1029
The fourth term qB in (26) can be easily written from (23). Thus 4 ujB=x
UfB9 k=l
where uL, etc. can be written from u’,c, etc. by changing subscript C by B. Thus u!~, e.g. is given by given by U_&= H(r”B- r)
981 98
QjBd4 dT*
I 784 I -dr,
Wa)
Similarly other terms ub, etc. can be written from ufC. In (27a) the symbols have the following meaning
781, 7B2 and 783 are the roots, when they exist, in the respective zones, as explained in Section 4 in connection with curves I, II, III, of curves
I”.
r = Jq7) + k0 - 7 - mp2 - lla2)1Dl”2 (l/S2 - lla2)“4 ’
O
jr: r = R(7) - [Z(f- ;~~!‘i”,;~2f1!~2)“21”2, 7B< 7 < t _ z(1/p _ l/a2)v2 VI: r = _R(7) + [z(t - ;-$f3;2
;W’Y, a2 ‘I4
where 7s is the value when R.H.S. of V vanish. Also we have L*
=
B
2&3 Z
(
1
p-2
1
l/2
>
-19
if a single max rg of IV exists.
if both max rS and min rg* of IV exist, and ri is the common value of (rg, rg*). We now consider the reduction in case of Ujsprin (26). We tirst write alternate forms similar to (15). We see that the region of support for T integration in this case depends on the form of curves
CW
CW
, O
where R(Tsp)=at-aTsp-z
UES Vol. 11, No. 9-C
ARABINDA ROY
1030
The curves VIII and IX are monotonic functions of 7. On the curve VII we get g = d(7) - (Yand $
= R(r).
As in the previous case, the nature of displacement field associated with ujspr depends critically on the existence of an extremum of curve VII, at times when the source just attains P wave velocity of the medium. Then when the source is decelerating the curve has a maximum at t = t&, denoted by r = rzp if R(O)> a or no maximum if A(O)< a and the region of support is similar to Figs. l(a-b). When the source is accelerating, unlike in the previous case, the wave has a single minimum at I = tz:, denoted by r = rz: if A(O)< a and no minimum if g(O) > a. The region of support in this case is as in Figs. 2(a-c). As in the previous case we consider the curves VIII and IX together. Then the values of T for any Vdlle Of r fle denoted by T = rspr, (r = ~Spl, T = TSp2) and (7 = TSpl, 7 = TSp2, 7 = TSp3) where TSPI > 7Sp2 > 7Sp3 depending on whether r is single, double or triple valued function of T. Similarly the values of T as given (28~) is denoted by T = ~~~4 since r is a single valued function of 7. With the above designation of roots and a close examination of Figs. similar to Figs. l(a-b) and Figs. 2(a-c), the displacement Ujsprcan be written as 4 UjSPl
=
2 k=l
(29)
UfSPlp11
where
r \ QP I
UjiPl
=
H(&P
-
r)
63-P
Q~SPI
dd
dT,
T
(a)
(b)
w Fig. 2.(a-c). Region of support for in case of an accelerating source. Meaning of different lines are as in Fig. 1 except that they represent TSP~, TSPZ, TSP~ and TSP~.
Response of an elastic solid to nonuniformlyexpandingsurface loads
U&I = [Gtr - max”(r&,Rft
U&P, = [G(r - min”(R(t
-
- z(lt/~?~- lt&“3,
1031
&))
z(l@*- l/(~*)“*,r$))
(30) and
4SP =
cos-’
r*+R*(T)- (at -
1
a7 - z(a*/p - 1)‘32 2?R(?)
[ II2
r&=t-2
. (A--$ B a>
(31)
In (30) I;&,etc. which appear as a factor W(L&) in connection with
whenever the starred member, i.e. r& is max, are given by
in case of a decelerating source with starting velocity greater than the P wave velocity, i.e. ti(O)> (1.
in case of an accelerating source with starting velocity less than P wave velocity, i.e. d(O) < Q. We note that eqn (11) together with (23), (26), (27) and (29) give the displacement field.
6. WAVE FRONTS
Besides the usual body and head waves in the medium, corresponding to r = r: and r = rip or equiv~en~y at t _& = VC
v +z*Y’* VC
The displacements as given by (23) and (30) contain contributions from conical wave fronts which arise due to the motion of the source as seen from the second and third terms in (23) and (30). Thus a conical P or S wave arrives at r = r$ (or t = t!) depending on whether c = P or S. From (16) and (21) one can easily write
1032
ARABINDA ROY /.
=R(T)+&A,
m
Z=U,(r-+&-J.
(32)
The above representation shows that the conical P or S waves which corresponds to an extremum of t - 7 -p/o, = 0, can be identified as the envelope of elementary wavelet as the source moves and arise only when the source is moving with a velocity greater than uC. The intersection of the conical wave front with the free surface is obtained by setting z = 0 in (32) which gives r = t if @i(r)> uCalways and T = T* where d(r*) = u, if R(r) g u,. Thus, as long as the source moves supersonically, the intersection of the conical wave fronts with free surface coincide with the rim of the circular load. When the source moves subsonically the intersection is given by r = R(T*) + u,(t - T*) and has separated from the load on the surface, from the instant when the source attains the velocity II, with the tip of the wave front on z = 0 moving with velocity uC.The conical wave front exists only in regions where L,* > 0 as given by the factor H(L,*) in G(r - max”(r$, r3). In case of a rapidly accelerating source, a second conical wave exists in region where r: > 0 and is associated with a minimum of t - T -p/u, = 0 and is always preceded by the first conical wave. Similar remark holds for conical head wave in regions Ir - l?(~&)(/p* > pla,except that no second conical head wave exists. Intersection of conical head wave front with the surface is r = R(r&) + a(t- T&). Thus conical head waves appear at time when the load has just attained P wave velocity, with the tip of wave front moving with velocity a.Conicalhead wave does not exist in case of uniform motion. Figures 3(a-c) show the different wave fronts in the medium for different case of moving sources. A detailed discussion of the development of wave fronts for reciprocating shear load has been given by Watanabe [13] and for particular type of nonuniformly expanding circular load by Roy[5]. We note that the surfaces r=R
(t-z
(+--$)1’2)
r=&
are not wave fronts, since none of them satisfy the eikonal equation. Displacements together with the derivatives are expected to be continuous on those surfaces. Similar surfaces are also found in case of uniform motion[3]. 7. FIRST MOTION RESPONSES The displacement field as given in Section 5 is expressed in terms of triple integration over finite ranges. As such exact computation is difficult except with the use of large computers. However an insight into the nature of the displacement field by calculating the 6rst motion responses near different wave arrivals. This can be done by a limiting process used in [131. Thus near the arrival times, i.e. t = pdu, of P or S waves, the displacement as given by (15) is
(34) In obtaining the approximate form near t = p&, the inner integrand is evaluated at t = r +p/u, and noting that the contribution of the second term is nil to the same order of approximation we can write the approximate form of the displacement field near t = po/uc
Response of an elastic solid to nonuniformly expanding surface loads
.
(b)
pi*
Fig. 3. Wave fronts in an half space: (a) For a circular load on the surface mov& nonuniformly with starting velocity greater than the transverse velocity (B) at a time when the rim of the load, L, is moving with velocity greater than the longitudinal velocity (a); (b) At a time when the same load is moving with velocity less than the transverse velocity (B) (Note that conical head wave (CSP) has developed and the tip of the wave fronts has separated from the rim of the load); (c) In case of a rapidly accelerating load with starting velocity less than the transverse velocity at a time when the load is moving with velocity greater than the longitudinai velocity. In Fig. 3(a, b), P, S, CP, CS, CSP represent longitudinal, transverse, conical P, conical S, conical head wave fronts respectively. In (c), CPI, CP2, CSl, CS2, CSP represent the tirst conical P, second conical P, tint conicalS and second conical S and conical head wave fronts respectively. In (a-c), L represents the position of the rim of the load.
r-R(T))*-z*)((r +R(T))*+ z*-uf(t - T)*]“*~T
(35)
where [&I = jump of I& at t = T+&
[I&3 = value of [&I at 4 = 0 -1 Cfi)o=(
1 for j =z.
We note that 7cl is the root in 0 < T c t -z/v, t _ 7_
forj=r
of
m*+~*I”*=
k - I?
VC
0. ,
06)
ARABINDAROY
1034 Whilerc4is the root, in 0 <
T <
t -
of
z/v,
(37) Expanding (36) near
T =
0 we
get
t _ T_
[(r - R(TN’+ ~~1"~=t_$J_T(g!w). UC
Thus (36) has a root denoted by
Tag,
for r/p0 > v&O) near t = p~/v, where t-
Tel =
*
PO/V,
(38)
r fi(O)’ PO UC
For r/p0 > v,/@O), (36) has a negative root close to 0 and will henceforth be designated by where, for t > p~/v, t -PO/UC
TO= --r k(O) PO vc
-TV
(39)
1’
The position of this root, viz -TV, is critical in obtaining the approximate displacement field near t = p~lv,. Similarly expanding (37) near T = 0, we get
We are now in a position to obtain the displacement field near t = p~lv,. We consider first the regions where r/p0 > v,/&(O). In this case for t > po/ve, the situation is similar to Fig. l(d). Then we get from (35)
~(jj)&]~(l-~)~‘*~ (l-2)
X/"'
c
Substitution of T =
~~4 cos*
0 + TV I
7c4
[(p-~c4)(~c,-T)]1’2dT~ (41)
sin* 8 and using the value of ~~4,T,] from (38) and (40) we get tblo(r-E)
ujc_E&+)o -
2
[
m
H(GJH(l-;F).
(42)
l-<+))]
PO VE
Let us now consider the region r/p0 > v,/d(O). In this case situation is as depicted in Fig. l(a). The form of the displacement field is difIerent due to presence of additional root at T = -TV. In this case one can write the displacement field as t + po/vc + 0 as ul,
( > t- @
&clti)lon2vcPo Jm)IJ UC rm+ 701 r L-&F-
-H X
[(~c,
-
T)(T
-
q4)(7
+ TO)]“*
d?.
111’2H(i-&J (43)
Response of an elastic solid to nonuniformly
expanding surface loads
I035
Substitution of T = T,~cos’ 8 + 7Clsin* 0 gives
where K(k) and E(k) are elliptic integrals of the first and second kind and k is given by k2_
Tel-7~4
k'2 =
1 _k2.
(45)
Tel +70'
Thus k --, 1 and kr +O as t +p~/u,, Hence using the expansion of B(k) and R(k) near k = 1 and noting that one gets similar expansion near t = p&, - 0, we can write the approximate form of the displa~ment field as
where rC is the root of (35) in 0 < r < t -z/u, ; In case two such roots exist, rC is taken to be the larger one. Near the arrival times of conical P or S waves the displacement field is given by
X ui’*H(t - tc*)ri(LP) lim S-+0
(47) where
&lf)=7~= value of &] at 4 = 0 and T = r$, f”(T) = $f(T)
f(l)=_7._r(r-R(T))*+2*l”* . f& In case of an accelerating source when a second conical wave is present, the corresponding displacement near its arrival times is obtained from (47) and replacing t,* and T$ by t?* and rc**, respectively. Near the cusp of Fig. 3(c), noting that f”(7) = 0 one can obtain, proceeding similar to the previous case, the displacement geld near the arrival time as
x [(r - R(r:))* + 2213’2 2 “‘63 _u3
(f -t:)3"
Lf”(7:)l’”
1 3 5,2’ ( >
where B(m, n) is Beta function and tf, TEare the common values of (tr, t:*) and (T,*,T$*).
1036
ARABINDA ROY
The displacement component near conical head wave is given by
(49)
On evaluating the inner integrand near t = fsp(r) following[l3] and multiplying numerator and denominator by t - fSP(r) and some simplification one can write (49) as
CY UjSPl
“I H(t
-
tZPW(L~P)
$- >3’4(fi)f=T;p (a 1
Re{Gjsp} 112
372
-z]
Substituting the values of &p from (27) and expanding near t = t gp and after simplification we can write the displacement field near f = fzp+ as
(50)
Gjsp = lim 27;s
FjsPdqsPI, (qiPl+
WSPI) dp*+
1Y*
8.CONCLUSION
It is seen from the first motion responses that displacement at any station depends on the location of the station and type of wave arrivals at the station. Thus from (42) and (46) one can see the nature of displacement near body wave arrivals depends on whether at the station conical waves are tirst to arrive or not. The displacement field near the arrival times of surface waves can be easily obtained by calculating the residue contribution at the Rayleigh pole. However, we have not considered the surface wave effect since the case has been discussed in detail by Baron and Lecht[lS] and by Freund[lO] in case of a point source. As noted before, the exact computation of the displacement field requires the use of high speed computers. Only in case of 6rst motion responses the displacement field can be obtained by a limiting process. However, first motion responses are itself useful tools in the study of earthquake source mechanism. Recently various authors have determined the source dimension of earthquakes[l6] from the comer frequencies of seismic amplitude spectra. The corner frequency is that frequency where the frequency spectra of first motion responses (i.e. at high frequency) as obtained in the previous section equal the frequency spectra at low frequencies which can be easily obtained from (2) by making p +0[17]. Thus first motion responses can be used to determine the corner frequencies. We have considered a typical case of nonuniformly expanding circular disk load which serve as models for atmospheric explosion. The present paper gives a comprehensive analysis of the displacement field due to an axially symmetric nonuniformly moving circular load. Most of the analysis so far done by previous author are either 2-dimensionaQl31 or approximations [a]. Obviously our analysis can be adapted for all class of nonuniformly moving load having circular geometry.
Response of an elastic solid to nonuniformly expanding surface loads REFERENCES [l] R. M. BLOWERS, J. Inst. M&s. Applies. 5, 167 (1%9). [2] C. ATKINSON, Int. J. Engng Sci. 6,27 (1968). [3] D. C. GAKENHEIMER, J. Apple Mech. 38,99 (1971). (41 K. J. TONG, PH.D. Thesis. Stanford University, Stanford, California (1968). [5] A. ROY, Ind. J. Pure Appl. Math. 5, 1063 (1974). [6] J. W. MILES, J. Appl. Me&. 27,710 (1960). [fl H. R. AGGARWAL and C. W. ABLOW, Bull. &is. Sot. Am. 55,763 (l%S). [g] W. J. STRONGE, J. Appl. Mech. 37, 1077 (1970). [9] L. B. FREUND, Appl. M&h. 30,271 (1972). [lo] L. B. FREUND, J. Appl. Mech. 40,699 (1973). [I l] A. ROY, Int. J. So/ids Structures 14,755 (1978). [12] M. MITRA, Proc. Comb. Phil. Sm. 60, 683 (1964). [13] A. ROY, Int. .f. Engng Sci. 13,641 (1975). [14] K. WATANABE, Inr. J. Solids Structures 13.63 (1977). [IS] M. L. BARON and C. LECHT, I. Engng Mech. Div. (Pm ASCE) 87,33 (1967). [16] M. WYSS and L. J. SHAMEY, Bull. Seis. Sot. Am. 65,403 (1975). [17j R. BURRIDGE, Bull. Seis. Sot. Am. 65, 667 (1975).
(Received 16 February 1979)
APPENDIX fr = cos 6, f*=
R(T) - r cos I$ s ,
F,p(q ~)J20*+2qws3 2wpA FzPkl,
0)
= -
’
iq(202+2q2+a2//33(02tq2t1)“* 2rpA(q* t o*)
’
A=(2o2t2q2ta2/~2)2-4(o2+q2)(O2+q2t1)”2(~2tq2to2/~2)‘R, icr(t-7)s qp =2 P
2 tPTpcos$,
up = TP sin I), T
= P
4s =
ia(t-7)s P2
us = TS sin
OSP I = TSP
2 t;Tscos$,
$,
sin h
m =(g’*_m*X3”2,
1037
ARABINDA ROY
1038 qstv=
ia(t- 7)s p2
ia I
--j-36’sin$,
OSPZ=T:.~(T~~-_TB)~~~*~L
s = (r2+ P(T) - 2rR(7) cos c$)‘“, /I=(s+z*)“*, po = p/r -
0 = (2 t zy, 11
c=P