TRANSIENT EXCITATION OF AN ELASTIC HALF-SPACE BY A POINT LOAD TRAVELING ON THE SURFACE

Transcription

1 Offie of Naval Reearh Contrat Nonr-220(57) NR Tehnial Report No. 8 TRANSIENT EXCITATION OF AN ELASTIC HALF-SPACE BY A POINT LOAD TRAVELING ON THE SURFACE by D. C. Gakenheimer and J. Miklowitz Ditribution of thi Doument i Unlimited. Diviion of Engineering and Applied Siene CALIFORNIA INSTITUTE OF TECHNOLOGY Paadena, California Jo nuary 1969

2 Offie of Naval Reearh Contrat Nonr-220(57) NR Tehnial Report No. 8 Tranient Exitation of an Elati Half-Spae by a Point Load Traveling on the Surfa e by D. C. Gakenheimer and J. Miklowitz Reprodution in whole or in part i permitted for any purpoe of the United State Government. Ditribution of thi doument i unlimited. Diviion of Engineering and Applied Siene California Intitute of Tehnology Paadena, California January 1969

3 ABSTRACT The propagation of tranient wave in a homogeneou, iotropi, linearly elati half-pae exited by a traveling normal point load i invetigated. The load i uddenly applied and then it move retilinearly at a ontant peed along the free urfae. The diplaement are derived for the interior of the half-pae and for all load peed. Wave front expanion are obtained from the exat olution, in addition to reult pertaining to the teady-tate diplaement field. The limit ae of zero load peed i onidered, yielding new reult for Lamb' point load problem. 1. INTRODUCTION Ground motion exited by moving urfae fore arie, for example, from nulear blat and from hok wave generated by uperoni airraft, and they interat with truture auing extenive damage. A mathematial problem of fundamental importane in thee appliation i that of an elati half-pae whoe urfae i exited by a normal point load whih i uddenly applied and whih ubequently move retilinearly at a ontant peed. In reent year everal olution to thi problem have been given for the urfae of the half-pae. Firt Payton ( 1]t omputed the tranient urfae diplaement by uing an elatodynami reiproal theorem. Then Laning ( 2] rederived ome of Payton' reult by employing a Duhamel uperpoition integral. However, no tranient olution have been given for the interior of the half-pae, whih are t Number in braket deignate Referene at end of paper.

4 -2- of interet with regard to buried truture. Therefore, it wa appropriate to eek the interior diplaement here. The remaining ontribution to thi problem are the teady-tate reult given by Mandel and Avrameo [ 3], Papadopoulo [ 4], Grime [ 5], Eaon [ 6], and Laning [ 2]. Of further interet i the fat that a point load moving on the urfae of a half-pae generate a non-axiymmetri diturbane. Very few wave propagation problem of thi type have been olved and no general olution tehnique are available. Beyond the moving load ae ited above, mention i made of Chao' work [ 7] on the urfae diplaement due to a tangential urfae point load. However, like the moving load ae, Chao' olution tehnique i retrited to the urfae of the halfpae, beide involving a eparation of variable whih i peuliar to hi exitation. Further, Sott and Miklowitz [ 8] have given a general method for olving non-axiymmetri wave propagation problem involving plate, but their tehnique wa found not to be appropriate for the problem onidered here. On the other hand, the olution tehnique developed here i appliable to all point of the half-pae and it i uffiiently general that it hould ontribute guideline for analyzing other non-axiymmetri half-pae problem. After formulating the problem in Setion 2, a formal olution i obtained in Setion 3 by uing the Laplae and double Fourier tranform. Then in Setion 4 the invere tranform are evaluated by a tehnique due originally to Cagniard [ 9], but implified by a tranformation introdued by DeHoop [ 10] for problem in aouti and later ued by Mitra [ 11] for an elati half-pae problem. In thi way eah diplaement for the interior of the half- pae i redued to a um of ingle integral and algebrai term for all value of the load peed. Eah ontribution to the diplaement i identified a a peifi wave.

5 -3- In partiular, the integral repreent wave whih emanate from the initial poition of the load a if they were generated by a tationary point oure, while the algebrai term repreent diturbane that trail behind the load and whoe wave geometry depend on the peed of the load relative to the body wave peed. In Setion 5 thi form of the olution i exploited to evaluate the diplaement near the wave front. Then in Setion 6 the limit ae of zero load peed i onidered, howing that the integral beome a olution of Lamb' point load problem. Finally, in Setion 7 the algebrai term are hown to form the teady-tate diplaement field when the load peed exeed both of the body wave peed. 2. FORMULATION OF THE PROBLEM The ubjet half-pae problem i depited in Fig. 1 baed on a arteian oordinate ytem (x,y,z). The plane urfae of the halfpae i z = 0, with z > 0 forming the interior. A onentrated, normal load of unit magnitude travel on the urfae along the poitive x-axi at a ontant peed. The load aquire it veloity intantaneouly at the origin of the oordinate at time t = O. The half-pae i a homogeneou, iotropi medium governed by the equation of the linear theory of elatiity. The equation of motion for the ae of vanihing body fore an be taken a the wave equation a ljj Y' ljj = at 2 (harater underored by a tilde deignate vetor), where p (1) and l)j,

6 -4-, known a the Lame potential, are related to the diplaement vetor u by (2) and atify the divergene ondition The ontant d and, defined by! = (X. + 2µ)/v and ; = µ/v, repreent the dilatational and equivoluminal body wave peed, repetively, where X. and µ are the Lam~ ontant and v i the material denity. The tree T.. are related to the diplaement by lj (3) T = X.u. ko.. +µ(u.. +u.. ) lj.k, lj l, J J, l (4) where tenor notation i employed. The boundary ondition at z = 0 take the form T (x,y,o,t) = -o(y)o(x- t) zz T (x,y,o,t) = T (x,y,o,t) = 0 xz yz. l ( 5) where o i the Dira delta funtion. To repreent quieene at t = 0, the initial ondition appear a a~(x,y,z,o) "'' O) _ 8f>(x,y,z,O) 't'\x ' y' z ' - at = ljj(x, y, z, 0) = at = 0 ( 6) Finally, the potential f> and ljj, and the pae derivative of the patential, are required to vanih at infinity. 3. FORMAL SOL UT ION A olution of the wave equation (1) that atifie the initial ondition (6) and the boundedne ondition at infinity an be omputed

7 -5- by uing the Laplae and double Fourier tranform to uppre the time parameter and the x,y pae oordinate. Then atifying the boundary ondition (5), uing the diplaement-potential relation (2), and inverting the Fourier tranform give the Laplae tranformed diplaement a (7) for j =x,y,z, where 1 Soo -n z+i(kx+vy) u. (x,y,z,p) = F.,,,(k,v,p)e a dk dv JO' (2ir) 2 µ J... -oo ( 8) for a = d,, in whih Fxd(k,v,p) = -ikn G F (k,v,p) = 0 XS 2ikndnG F yd(k,v,p) = -ivn G F (k,v,p) = 0 y 2ivndnG (9) Fzd(k,v,p) = ndn 0 G F (k,v,p) = ZS 2 2-2nd(k +v )G, 1 G= (p+ik)t (10) (11) (12) k =...._ ( 13) u. and u. are Laplae tranform pair, p i the Laplae tranform J J parameter, k and v are the Fourier tranform parameter, and the quare root nd and n are aigned the branh that ha the poitive,

8 -6- real part. In thi form u. i expreed a the um of a dilatational J ontribution ujd and an equivoluminal ontribution u. JS 4. INVERSION The Laplae tranform i inverted by a tehnique due to Cagniard [ 9], but modified for the preent appliation. Thi tehnique onit of onverting eah u. into the Laplae tranform of a known funtion, JO' and then inverting the Laplae tranform by inpetion. In the ubequent alulation p i aumed to be a real, poitive number. For uh value of p, Lerh' theorem (ee Carlaw and Jaeger [ 12]) guarantee that if u. JO! exit, it i unique. To implify the form of u., the tranformation JO! (14) and f3 = q 0 a - w in a er = q in a + w 0 a (15) are ubtituted ueively into (8), yielding 1:oro Cd a -L(m z-iqr) U:. (r,9,z,p) =- 2 J K.,,,(q,w,e)e dqdw, JO! 0 -oo J... ( 16) where K XS = -( iqo 9(iqo 9 +y) +w 2 in 2 e] m L 0 (q, w, 9) = 2[ iq o 9(iq o 9 +y) + w 2 in 2 e] mdm L K d(q,w,9) = -in0[ iq(iqo 9 + -y) -w 2 o 9] m y 0 K y (q, w, 9) = 2 in a[ iq(iq o a +y) - w 2 o a] mdm L L S (1 7a) (1 7b) (1 7) (1 7d)

9 -7- (1 7e) K ZS (q,w,9) = -2(iqo 9 +-y)(q 2 +w 2 )mdl 1 L= 'TT µ[(iq o 9 +-y) + w in 9] R (1 7f) (18) (19) 2 2.!_ md = (q +w +1) 2, m m ] = ( 1 +Z(q +w ), (20) (21) and (r, 9, z) are the ylindrial oordinate hown in Fig. 1. The tranformation in (15) wa introdued by DeHoop [ 10] in order to implify Cagniard ' tehnique. In view of the ymmetry propertie u (r,9,z,p) =u (r,-9,z,p) xa xa u (r,9,z,p) = -u (r,-9, z,p) ya ya (22) u (r,9,z,p) =u (r,-9,z,p), za za u., and hene u., are only inverted for 0 :S 9 <'TT. Sine u. ha different Ja J Ja form depending on the peed of the load relative to the body wave peed, the inverion of eah u. Ja i eparated into three ae. In partiular, the term uperoni, tranoni, and uboni refer to the ae when the load peed i greater than the dilatational wave peed ( > d), between the dilatational and equivoluminal wave peed (d > > ), and le than the equivoluminal wave peed ( < ), repetively. In

10 -8- the remainder of thi etion uzd and u ZS are inverted for the interior of the half-pae (z > 0) and for all load peed (0.:S < oo). Then uzd and u ZS are ombined to give uz, and imilar reult are diplayed for u and u x y Dilatational Contribution for Superoni Load Motion From (16) S(X)(X) _E...(mdz-iqr) - 1 d uzd(~,p) = 2 -oo Kzd(q,w, S)e dq dw, (23) 0 where ~ i the poition vetor. uzd i onverted into the Laplae tranform of a known funtion by mapping (1 /d)(mdz-iqr) into t through a ontour integration in a omplex q-plane. To thi end, the ingularitie of the integrand of U:zd are branh point at ± q = Q, and imple pole ± ± at q = Q and q = QR, where ± q = Q d and ± 2.! Qd = ±i(w +1) 2 Q±_±win0+iy - o e ( 24) ± The pole at q = QR orrepond to the zero of the Rayleigh funtion R, where YR = d/r and R i the Rayleigh urfae wave peed. The root of thee ingularitie whih lie in the upper-half of the q-plane are hown in Fig. 2. By eeking a partiular ontour in the q-plane uh that t = - 1 -(m z - iqr) Cd d (25) one find, upon olving for q, that (26)

11 -9- for t > twd, where (27) Equation (26) define one branh of a hyperbola with vertex 2.!. q = i(w + 1) 2 r/p and aymptote arg q = ±r/z. A hown in Fig. 2 by a olid line labeled with qd + and qd, - thi hyperbola i parametrially deribed by t a t varie from twd toward infinity. Sine r/p < 1, the hyperbola doe not interet the branh ut in the q-plane. The ar 1 and CII are introdued a hown in Fig. 2 to form a + - loed ontour C, where C = Re q-axi + CI + qd + qd + CII9 ± The pole at q = Q lie inide C if, and only if, (1) (or x > O) (2) y o e (or t > tl' where t = L) L ex 2 (28) (3) zed 2 2 i w tan 8 > (t - t wd ) z p For fixed t and z > 0, t = tl define the urfae of a hemiphere with enter (x = t/2, n = 0) and radiu t/2. Condition (2) and (3 ) are. 1 th d.. 2 > 2 h 2 ( 2 2 2) 2 28/ 2 2 equiva ent to e on itl.on w w od, w ere w od= p '{ -x z o x n 2 2.!. and n = (y + z ) 2 To inorporate thee ondition into the ontour integration for uzd. the half-pae i eparated into three region:

12 -10- Region I: x > 0, ~ > --2 p ± The pole at q = Q lie inide C for w [ O,oo). Region II: x > 0, ~ < --2 p ± The pole at q = Q lie inide C for w (w d' oo) 0 andtheylieoutide C for w [O,w 0 d). (28) Region III: x < 0 No pole lie inide C for w ~ [ 0, oo). For x > 0, the ray x/p = d/ form the urfae of a one whoe axi i the poitive x-axi. Thi onial urfae i hown in Fig. 3a along with the part of t = tl whih i bounded by x/p < d/, and Roman numeral whih depit the loation of thee region in the half-pae (eah Roman numeral lie between the ray that define the orreponding region). The next tep i to invert "i:izd for eah of thee region. Region I. The Cauhy-Gourat theorem and reidue theory applied to the integrand of uzd and yield u d(x,p) =A d(x,p) +13 d(x,p) z - z - z - (30) where (31) and in whih

13 -11- (33) Azd i the ontribution from reidue ontribution from the ± + qd, where qd = qd, and Bzd i the ± pole at q = Q The integral that arie along CI and CII vanih a thee ontour reede to infinity. By interhanging the order of integration in (31) and inverting the Laplae tranform, one find dw (34) where T = (~ - d t2 d (3 5) and H i the Heaviide funtion. Azd i a hemipherial, dilatational wave in that it repreent the diturbane behind the wave front at t = td, where td i the arrival time of a hemipherial, dilatational wave. Thi wave emanate from the initial poition of the load a hownt in Fig. 3a. The inverion of Bzd i alo done by a Cagniard tehnique. The ingularitie in the integrand of B zd are hown in Fig. 4 with branh point at W : : and W = : I and imple pole at W = ~ J where ± d = -i-y in e ± i(1 1 ± = -iy in 8 ± i( y 2 )zo e (36) ± - R- - i )' in e ± 2 2.!. i()' --y )' R o e ± In Fig. 4 the onvention i adopted that Sa and the ame funtion, but whoe poition are w = S ± g O' ± O' repreent for > O' root of w = : for <a, where a= d,,r. Here, ine > d, only the : and t Arrival time are deignated in the text by t ubripted with lower ae letter and wave front are hown in the figure by olid line.

14 -12- ± arie and the ga are given later in the text. orrepond to the root of the Rayleigh funtion R ± The pole at w = SR The partiular ontour in the w-plane i given by or olving for w t = ;d (mdz - iqr) I + q=q + in whih q = Q (3 7) w = w~ = -i-y ine + y ~ e (i~y ± zad) n (38) for t > td, where 2 - l 2 ( zj2 = [ ~ - 2-1)n Cd (39) ~ = t - x Equation (38) define one branh of a hyperbola with vertex l w = -i-y in 9 + i(y/n)(1 - -y 2 ) 2 o 9 and aymptote arg w = ±yjz. In view of the limit of integration in (32), only the plu root of (38) i + needed and w d = w d. A hown in Fig. 4 by the olid ontour labeled for region I, the wd part of the hyperbola i parametrially deribed by t a t varie from td toward infinity. Sine x/p > d/ and y/n < 1, wd interet the imaginary w-axi below the branh point at w = S~ and above the real w-axi. A hown in Fig. 3a, td repreent the arrival time of a onial, dilatational wave whih trail behind the load. The ontour C 0 and Cr are introdued a hown in Fig. 4 to

15 -13- form a loed ontour whih inlude w d and the real w-axi. Then the appliation of the Cauhy-Gourat theorem to the integrand of Bzd and thi loed ontour produe, upon inverting the Laplae tranform, (40) where /\ /\ + K d(w,0) = K d(q,w,0) z z ( 41) The integral that arie along Cr vanihe a Cr reede to infinity and the one along C 0 vanihe beaue it real part i zero. Finally, Bzd repreent a onial, dilatational wave trailing behind the load. The um of Azd and Bzd give uzd for region I a uzd(~,t) Td dq - = H(t-ta)JO Re[ Kzd(qd,w,0) dtd jaw r /\ dwdl + Re L Kzd(w d, 0) at H(t - td) (42) Region II. By noting the ondition for region II in (29) and ompleting the ontour integration in the q-plane, one find Azd(~,p) = (43) and Bzd(~,p) = -..E...(m z-iqr) Re: rnzd(q,w,b)e Cd d J I dw oj- + q=o (44) where uzd i the um of Azd and Bzd" The letter P preede an improper integral to imply that it i interpreted in the ene of a Cauhy prinipal value. Suh an integral arie in Azd beaue the pole at

16 -14- ± ± q = Q oalee on qd at t = tl a w - w od" Beyond thi, Azd i exatly the ame a for region I, while Bzd only differ in the lower limit, whih i expeted ine the pole at q for w > w 0 d. = Q± only lie inide The inverion of Bzd proeed a for region I, but with modifiation in the geometry of the w-plane. For region II the partiular ontour w d interet the real w-axi a hown in Fig. 4, to omply with the limit of integration in Bzd. Furthermore, the ingularitie Sd, S, and SR may lie below the real w-axi, but till on the imaginary w-axi, in a manner whih i not hown in Fig. 4. However, ine the partiular ontour doe not interet the imaginary w-axi, thi ha no bearing on the ontour integration for region II. Then inverting Azd and Bzd, and ombining the reult give uzd for region II a dw ( 45) The firt term in uzd repreent a hemipherial, dilatational wave a for region I. However, for region II the integral i interpreted a a Cauhy prinipal value for t = tl. The eond term ha the ame algebrai form a for region I, but the onial wave front i replaed w ith the hemipherial urfae t = tl. Thi new urfae i not expeted to be a wave front (i.e., uzd and all the derivative of uzd are expeted to be ontinuou through t = tl) beaue it i not a harateriti urfae (imilar to t = td), or the loure of harateriti urfae (imilar to t = td), aoiated with the governing wave equation for the dilatational potential f> in (1).

17 -15- Region III. A indiated in (29) for region Ill no pole lie inide C. Therefore, the inverion of uzd proeed exatly a for region I, le the reidue term Bzd' and uzd an be obtained from (42) by deleting the algebrai term. Summary. By omparing the reult for eah region it follow that uzd an be repreented by one expreion for all three region; namely, (46) Thi expreion i valid for 0 < 0 < ir and z > O. The wave pattern aoiated with uzd i hown in Fig. 3a, where the relationhip between the ray x/ p ::: d/, the wave front t ::: td and t ::: td, the hemipherial urfae t = tl, and the region I-III i depited. Dilatational Contribution for Tranoni and Suboni Load Motion The inverion of u d for < < d and <, or equivalently z < d, proeed exatly a for > d, exept that of the three region in (29) only II and III are appliable for < d beaue x/ p i alway le than d/. Thee two region are depited in Fig. 3b by Roman numeral two and three. The inverion of uzd for region II proeed for < d a it did for > d, exept that the geometry of the w-plane i different. A hown in Fig. 4, the poition of the ingularitie varie depending on the value of relative to, d, and R. In that figure

18 -16- ± 2 d g :::: - i l' in 9 ± ( )' - 1) 2 o 9 g: 9 ± ( l' 2 2.!. l :::: - i 'Y in -.P. ) 2 o 8 (47) ± ('Y 2 2.!. g :::: - i )' in 8 ± - -yr) z o 8 R However, the partiular ontour in thew-plane for region II remain the ame for all load peed ( >d and <d), a doe the reult of a ontour integration whih inlude it and the real w-axi. Therefore, uzd for region II and < d i the ame a given for > d in (45). Furthermore, the inverion of uzd for region III i independent of the value of the load ± peed (whih only appear through the poition of the pole at q :::: Q ) and the reult i the ame a given for > d. Then ombining the reult for region II and III give uzd for < d a (48) where <TT and z > 0. The dilatational wave pattern for < d i hown in Fig. 3b. A expeted phyially, the onial wave front doe not exit, leaving only the hemipherial one. Finally, by omparing (46) and (48) one find that uzd i given by (46) for all load peed if the Heaviide funtion atifie the ondition (49)

19 -17- Equivoluminal Contribution The inverion of uz proeed a for uzd, but uz i more ompliated than uzd due to the appearane of head wave (or von Shmit wave). With u being given by (16) for j = z and a =, the orrez pending q-plane i hown in Fig. 2, where the ingularitie are the ame a for uzd' but the partiular ontour ha two poible onfiguration, whiharedenotedbydahedline. If r/p</d and w<c._(o,oo) or if r/p > /d and w<c._ (w 1,oo), where w 1 = (r p 2 )i/z, then ± ± the partiular ontour i imilar to qd and it i denoted by q, where ± Cd 2 2 l p w q = - 2 [itr ± z(t - t ) Z] ( 50) for t > t, in whih w t w ( 5 1) However, if r /p > /d and w <C._ (0, w ), then the vertex of q ± l" ie 1 + on the branh ut between the branh point at q = Qd and q = Q +, and ± the partiular ontour i given by q plu qd, where ( 52) for t > t > t d, in whih w w ( 53) A hown in Fig. 2, the loed ontour C now inlude it arie, and the real q-axi. A for uzd, the pole at q = qd when Q± an lie

20 -18- either inide or outide of C. ± In regard to the variou poition of the pole at q = Q and the vertex of q±, the inverion ofu S ZS i eparated into even ae, eah of whih orrepond to a partiular region of the half- pae: Region I: x > 0, ~>~ p ~>~ r The pole lie inide C for wr [ 0,oo). The vertex doe not lie on the branh ut for wr [ 0, oo). Region II: x > 0, ~ > 2 p The pole lie inide C ~>~ r for wr [ 0, oo). The vertex lie on the branh ut for w re [O,w 1 ). Region III: x > 0, ~ p > 2 ~<~ r The vertex lie on the branh ut for wr [O,w 1 ). The pole lie inide C for wr (0, oo) and oalee on qd a w - O. The pole alo oalee on qd a 9-0 for 2 l. < < d and w re [ 0, ( '{ - 1) 2 ) Region IV: x > 0, ~<2 p The pole lie inide C for wr (w, oo). OS The vertex doe not lie on the branh ut for wr [ 0, oo). Region V: x > 0, ~<2 p The pole lie inide C for w re (w, oo). OS The vertex lie on the branh ut for wr [ O,w ) The pole oalee on qda e-o for <dand wr {w,('t -1) 2 ). 0

21 -19- Region VI: x < 0, No pole lie inide C for w E:: [O, o). The vertex doe not lie on the branh ut for wr._ [O,o). Region VII: x < 0 1 No pole lie inide C for wr._ [O,o). The vertex lie on the branh ut for w re._ (0, w 1 ). (Region I - VII are referred to in the text by (54) ) 2 L. 2 2.!. In ( 54), w = (p 'Y -xi. ) 2 z o e/xn. For x > 0, the ray x/p = / OS S form the urfae of a one whih i imilar to x/p = d/, and x/r = d/ define two plane, one for 0 > 0 and one for 0 < O. Alo, r/p = /d define the urfae of a one whoe axi i the poitive z-axi. All even region arie when inverting u for > d' but only region III-VII and ZS region IV-VII arie for < < d and <' repetively. The Roman numeral in Fig. 5-8 depit the loation of thee region in the half-pae. By inverting u ZS for e ah region, aounting for the variable load peed, and ombining the reult, one find + H(t-t )H(~ - \ H ( d - ~) - H(t-t ) H(~ - ) -jh(x) (5 5) E p l r p

22 -20- for 0 < e 1T. z > 0 and 0 < < 00. Further detail on inverting u ZS are given in a thei by the firt author [ 13] The ymbol in u ZS whih have not been previouly defined are /\ K (w,9)= ZS = - i y in e a m ( 2 + w 2) I irµr d q + q=q w.. e + y o e t + ) 2 n i.,, y z a Wd= -i)' in 9 + iy~s 9 (y- ZO'Sd) n 1 z -: 2-1 ) n J, l (56) ( 57) ( 58) A = d 0 for t < t T for t > t T ; (t-t ) 2 -t = i (d d+1)-1j d 1 _ r ( 59) (60) ( 6 1) l 1 = - ( ~ - ~) z +(~ -1) y + x J 1[ t SC t d ' CS Cd Cd <j> (~ Cd 1-1)2 = (~ 1-1)2 (62)

23 -21- The Heaviide funtion in (55) i retrited by the ondition H(t - t ) = 1 if < SC S H( Y.. - f> ) = 1 if < Cd n (63) The wave geometry aoiated with u i hown in Fig ZS The dilatational wave front are inluded in thee diagram for referene; therefore, they diplay the wave geometry aoiated with the total diplaement u z However, the Roman numeral in thee diagram only orrepond to u and the region deribed in (54). ZS The firt and third term in u, whih are analogou to the ZS dilatational wave in uzd, repreent hemipherial and onial, equivoluminal wave, repetively. The eond term in u z i the ontri- bution from qd along the imaginary axi in the q-plane and it repreent a head wave whoe wave front i the urfae of the trunated one given by t = td for r/p > /d. Thi wave, referred to a the onial head wave, i generated by the urfae interetion of the dilatational wave front at t = td and it propagate in front of t = tb, a hown in Fig. 7 and 8(a,b); thereby, ontributing both ahead and behind the equivoluminal wave front at t = t urfae of a phere with enter (r l = 0, For fixed time, 2 l_ z = dt/2( -1) 2 ) t = tb i the and radiu dt/2(1 2-1) 2 The urfae t = tb i not expeted to be a wave front beaue it i not a harateriti urfae, or the loure of harateriti urfae, of the wave equation for the equivoluminal potential '1! in (1). The integral in the eond term of u ZS i improper for t = t beaue it integrand ontain a firt-order ingularity at w = 0 S (i.e., it integrand behave like 1/w a w - O). Thi ingularity i introdued by the differential of qd In Setion 5 thi integral i evaluated

24 -22- for t - t diplaying a logaritluni ingularity in u for t = t Thi S ZS S integral i alo improper for ertain point in the plane under the path of the load (the 8 = 0 plane). In partiular, it i interpreted a a Cauhyprinipalvaluefor t <t<t 0 d if 8=0, x/p> le, and SC S C ' < < d; and for tl < t < t~d if 8 = 0, x/p < /, and <ed. where t 0 d equal t d evaluated for e = 0 and t = t 0 d i hown in Fig. 8(a, b) by a dahed line projeting out from the load. The improper integral for 8 = 0 ± reflet the way the pole at q = Q oalee on qd a 8-0, and a deribed in (54) for region III and V. Furthermore, if the Cauhy prinipal value i evaluated for 8-0, it ombine with the remaining term in u ZS to render u ZS ontinuou a a - 0, whih i expeted phyially. The integral in the eond term of u ZS i alo improper for t = te if > and x > 0 beaue it integrand ha a firt-order ingularity at w = 0 (note te < t, therefore Ad = 0). Thi ingularity ± reflet the way the pole at q = Q oalee on q d a w - 0, and a deribed in (54) for region III. The ignifiane of t = te i diued in onnetion with the lat term in u ZS The lat term in u ZS repreent another head wave. Thi term i the ontribution from the ontour w d whih i indiated in Fig. 4 a a egment of the imaginary w-axi. The dahed ontour that i hown in thi figure along with w d i the partiular ontour whih arie in the w-plane for u and region II if y/n >f> (the ray y/n = <f> are ZS C C hown in Fig. 9). Thi partiular ontour ha other onfiguration for region II if y /n < f> (in thi ae the w d egment doe not arie), and for the other region in (54), but they are not diplayed here. Thi eond head wave ha a plane urfae for a wave front whih i given by t = t d for y/n > <f> and x/r > d/. Thi wave,

25 -23- referred to a the plane head wave, only exit for > d and then it i generated by the urfae interetion of the dilatational wave front at t = td For x/r = d/ and it propagate in front of t = t, a hown in Fig. 9. SC the plane head wave front i tangent to the onial he ad wave front and for x/r < d/ it doe not exit beaue it generator, t = td, doe not exit, a hown in Fig. 5. In the region bounded by x/r < d/ and x/p > / the plane head wave end. Thi i refleted in u ZS by the fat that the lat term i diontinuou for t = t and the integral in the eond term i improper for t = te. The urfae defined by t = te, whih i not hown in any of the figure, i not expeted to be a wave front for the ame reaon a given above for E For < < d the plane head wave front doe not exit, whih an be een in (55) ine now H((x/r) - (d/)) = O. Phyially thi i expeted ine it generator, t = td, doe not exit. In thi till repreent an equivoluminal ditur ae the lat term in u z bane propagating in front of t = t However, now it propagate SC behind the urfae t = te' whih i behind the onial head wave front at t = td, and whih make the lat term inditinguihable a a eparate wave. Finally, for < the lat term doe not ontribute to u, S ZS whih an be een in (55) ine H((x/r) - (d/)) = 0 and H((x/p) - ( 8 /)) = O. Total Diplaement Field The total diturbane ompriing u z i the um of uzd and u ZS Similar expreion an be omputed for u and u, and all three x y diplaement are written a

26 -24-6 u.(x,t) = \ u.~(x,t) J - 6 Jt-' - 13=1 (j = x' y' z) (64) for 0 :$ r < 00, 0 :$ 9 < TT (or -OO < X < 00, 0 < y < 00), 0 < Z < 00, and 0 :S < oo, where (65) (66) (67) (68) [ dw - u. 5 (x,t) = Re.Q. (w, 9) dt j H(t-t )H(t-tL)H(x) J - JS S SC (69) - + H(t-t )H(~ - 2-)H(~-~ )- H(t-t )H(~ -2) J H(x) (70) E p r p The ymbol whih have not been previouly defined are = y e am I TTµR o + q = Q A K (w,s)=-2ye9 j x trµr mdm + q = Q A 2 K d ( w, 9) = e :. ( y in 9 - i w) m y TTCµ 0 (71) (72) (73) q = Q+

27 -25- /\ K (w,9) = y -2 e 2 e irµr ( '{ in e - (74) In thi form the firt three term in u. repreent a ytem of wave J whih emanate from the initial poition of the load a if they were generated by a tationary point oure. On the other hand, the lat three term repreent diturbane that trail behind the load and whoe wave geometry depend on the peed of the load relative to the body wave peed. Finally, it i noted that the interior diplaement given here are not uniformly valid in z, a z - 0, beaue the partiular on- + tour in the q- and w-plane wrap around the pole at q = QR and w = ; a z - O. Therefore, to obtain the urfae diplaement it i neeary to return to the formal olution in (7) and {8), et z = 0, and then invert the diplaement {or equivalently, evaluate the ontribution of the pole at q = o; and w = ; a z - 0, and ombine the reult with the interior diplaement in (64) - {70) evaluated for z = 0). However, ine Payton [ 1] and Laning [ 2] have already invetigated the urfae diplaement, they are not diplayed he re; but they are given in [ 13] in ontext with the tehnique ued here. 5. WAVE FRONT EXPANSIONS The wave form of the olution given in {64) - (70) i advantageou for evaluating the diplaement near the wave front. To failitate thee expanion for the wave emanating from the initial poition of the load, the time dependene in the limit of the integral in (65) - {70) i removed by the tranformation

28 !. w = ( A + (B - A ) m QI] 2 (75) where A i the lower limit and B i the upper limit of the partiular integral in quetion, and the QI range of integration i 0 to rr/2. Furthermore, thi tranformation remove a half-order ingularity in the integrand of uji and uj 2 whih i introdued by the differential of qd and q. Then uji' uj 2, and uj are evaluated a t - td, t- t, and 3 t - td, repetively, by firt expanding the appropriate integrand and then integrating thee expanion from 0 to rr/2. Speial attention mut be given to uj 3 ine the integral in thi term i improper for t = t. Thi term i evaluated by firt approximating the improper integral for t near t and then letting t - t In thi way one find uj1 = Aj1 + O(t - td) a t - t d (76) for.e <~ -rj2 +0(t-\) p Cd uj2 - a t - t O~t-t)!) A' + for.e > ~ J2 p Cd uj3 = Aj3(t-td) + o((t-td)2) a t - t for.e > ~ d p Cd uj3 = AJ3 + 0(1) a t - t for.e > ~ p Cd log (If-_ 1 I) (77) (78) (79) for j = x, y, z, where the wave front oeffiient are given in the appendix. Thee expanion are valid for 0 < 9 < rr, z > 0, and 0 < < oo, exept that (76) i not valid along the ray x/p = d/ ine A.. i un Jl bounded along thee ray. Similarly, (77) and (79) are not valid along x/p = /, and (78) i not valid along x/r = d/. A the expanion in (76) - (78) how, u. ha a tep diontinuity J

29 -27- aro the hemipherial wave front at t = td and t = t, and it i ontinuou through t = td" However, for r /p > /d the diturbane near t = t i uominated by the logarithmi ingularity hown in (79). Thi ingularity i two-ided in that u. beome unbounded a t-t J for t < t and for t '> t, and it i ymmetri about t = t. Sine the wave trailing behind the load are in an algebrai form, they are expanded algebraially near their wave front, yielding a t - td (80) for Y > <f> n a t - t, (81) SC u. 6 = A! 6 (t - t)--z + 0(1) J J SC 1 a t - t for Y > <f> SC n C (83) for j = x,y,z. Thee expanion are valid for 0 < 8 < ir and z > 0, exept that (80) only arie for > d and x/p > d/, the upper one in (81) for > d and x/p > /, the lower one in (81) and (83) for > and x/p > /, and (82) for > d and x/r > d/. Thee latter ondition orrepond to when and where the wave front t = td, t = t, and t = t d arie in the half-pae. SC S C A (80) and (81) how, u. ha a half-order ingularity behind J the onial wave front at t = td and t = t In addition, u. on J tain a half-order ingularity in front of t = t, a (83) how, proe duing another two-ided ingularity in uj" If > d, thi two-ided ingularity exit aro t = t for y / n > <f> ; if < < d, it SC C exit aro the entire onial wave front at t = t and if <, '

30 -28- it doe not exit at all. Alo, u. i ontinuou through the head wave J front at t = td, a it i through t = td" Finally, it hould be noted that the diturbane near the wave front trailing behind the load i a half-order tronger than near the orreponding (ompare wave front involving the ame type of diturbane) wave front emanating from the initial poition of the load. Thi an be een, for example, by omparing the wave front expanion of uji a t - td and uj 4 a t- td" 6. STATIONARY POINT LOAD A peial ae of the moving load problem i that of a half-pae whoe urfae i exited by a tationary point load with tep time dependene. A olution to thi problem, whih wa firt worked on by Lamb [ 14], i obtained here by letting - 0 in (64) - (70) and expreing the diplaement in their ylindrial omponent, to give u<t(~,t) 3 =I uo-rl~ t) {3= 1 (o- = r,z) (84) (85) for O<r<oo, 0:50:5tr, and 0< z<oo, where dw (86) uo- 2 (~,t) T = H(t-t)S Re[K~(q,w) :~ J dw 0 (87) in whih

31 -29- -iqm M, 0 mdm 0 M, l (89) 1 2 ir dµr M =---- (90) The wave geometry aoiated with thee diplaement i the ame a for <, a hown in Fig. 6b and Sb, exept that the ontour 0 t = tl and t = td are not preent in the = 0 olution. It hould be noted that Eaon [ 15], and Cinelli and Fugelo [ 16] have alo worked on Lamb' point load problem for the interior of the half-pae uing tranform. Their reult are alo in the form of ingle integral, but they do not readily diplay the w ave front aoiated with a onentrated urfae load. Mitra [ 11] ha derived the interior diplaement for a half-pae whoe urfae i exited by a uniform dik of preure. He ued Cagniard' tehnique and DeHoop' tranformation, but hi olution tehnique i adapted to the axiymmetry of hi problem. Furthermore, he mention that hi olution tehnique i appliable to Lamb' point load problem, but he give no reult. Alo, Pekeri [ 17] ha thoroughly analyzed the urfae diplaement due to a tationary point load. Hi reult agree in detail with thoe given here if the latter are extended to inlude z = O. Finally, Lang [ 18,19] and Cragg [ 20] have alo worked on Lamb' problem but they did not ue tranform. Wave front expanion whih orrepond to a tationary point load an be omputed from the integral in (86) - (88), or they an be obtained from the expanion in Setion 5. In fat, thee expanion

32 -30- have the ame form a thoe given for uj 1, uj 2, and uj 3 in {76) - {79); the only differene i that the oeffiient Aj 1, Aj 2, Aj 2, Aj 3, and Aj 3 mut be evaluated for = O. The wave front expanion for a tationary point load were alo omputed by Knopoff and Gilbert [ 21]. They ued a Tauberian theorem and the addle point tehnique to evaluate a formal tranform olution near the wave front. Their reult agree in detail with thoe that are obtained here, exept that they did not detet the logarithmi ingularity at t = t Aggarwal and Ablow [ 22]. for r/p > /d, a firt noted by 7. STEADY-STATE RESPONSE By algebrai manipulation it an be hown that uj 4, ujs, and uj 6 in (68) - {70), le H(x) and H(t - tl), are only funtion of the oordinate ~, y, z whih are invariant to the tranlation of the load (ee [ 13] for detail). Therefore, thee term are ontant at a fixed poition in a oordinate ytem moving with the load. For > d and long time {t - oo), H(t = tl) = 1 and H{x) = 1 for point near the load, and the load "run away" from the wave emanating from the initial poition of the load. Conequently, for > d the wave trailing behind the load repreent the teady-tate diplaement field {i.e., uj = uj 4 + ujs + uj 6 a t - oo for > d and point near the poition of the load, j = x,y,z). The wave geometry orreponding to thee teady-tate diplaement i hown in Fig. 9, where only the wave that trail behind the load are hown. Laning [ 3] ha alo omputed the teady-tate diplaement due to a moving point load. He ued a tehnique whih aumed teadytatene from the outet and hi reult are in the form of ingle

33 -31- integral. A a peial ae, in the plane under the path of the load (y = 0 plane), and for :\. = µ and > d, Laning integrated hi r e ult, leaving the diplaement in an algebrai form whih agree with that one obtained here. Finally, for < d the wave trailing behind the load do not repreent the entire teady-tate diplaement field. In thi ae the wave emanating from the initial poition of the load alo ontribute and they mut be evaluated for t - oo.

34 -32- REFERENCES 1. Payton, R. G., "An Appliation of the Dynami Betti-Rayleigh Reiproal Theorem to Moving-Point Load in Elati Media, 11 Quart. Appl. Math., Vol. 21, 1964, pp Laning, D. L., "The Diplaement in an Elati Half-Spae Due to a Moving Conentrated Normal Load," NASA Tehnial Report, NASA TR R-238, Mandel, J. and Avrameo, A., "Deplaement Produit par une Charge Mobile a la Surfae d 'un Semi-Epae Elatique, II Compt. Rend. Aad. Si., Pari, Vol. 252, pt. 3, 1961, pp Papadopoulo, M., "The Elatodynami of Moving Load," J. Autral. Math. So., Vol. 3, 1963, pp Grime, C. K., "Studie on the Propagation of Elati Wave in Solid Media," Ph.D. Thei, California Intitute of Tehnology, Paadena, California, Eaon, G., "The Stree Produed in a Semi-Infinite Solid by a Moving Surfae Fore," Int. J. Engng. Si., Vol. 2, 1965, pp Chao, C. C., "Dynami Repone of an Elati Half-Spae to Tangential Surfae Loading," J. Appl. Meh., Vol. 27, 1960, pp Sott, R. A. and Miklowitz, J., "Tranient Non-Axiymmetri Wave Propagation in an Infinite Iotropi Elati Plate," in pre, Int. J. Solid and Struture. 9. Cagniard, L., Refletion and Refration of Progreive Seimi Wave, Tranlated by E. A. Flinn and C. H. Dix, MGraw Hill Book Co., New York, DeHoop, A. T., "A Modifiation of Cagniard' Method for Solving Seimi Pule Problem, 11 Appl. Si. Re., Setion B, Vol. 8, 1959, pp Mitra, M., "Diturbane Produed in an Elati Half- Spae by Impulive Normal Preure, 11 Pro. Camb. Phil. So., Vol. 60, 1964, pp Car law, H. S. and Jaeger, J. C., Operational Method in Applied Mathemati, Oxford Univerity Pre, Gakenheimer, D. C., "Tranient Exitation of an Elati Half-Spae By a Point Load Traveling on the Surfae," Ph.D. Thei, California Intitute of Tehnology, Paadena, California, 1968.

35 Lamb, H., "On the Propagation of Tremor Over the Surfae of an Elati Solid," Phil. Tran. Roy. So. Lond. (A), Vol. 203, 1904, pp Eaon, G., "The Diplaement Produed in an Elati Half Spae by a Suddenly Applied Surfae Fore, J. Int. Math Applie, Vol. 2, 1966, pp Cinelli, G. and Fugelo, L. E., "Theoretial Study of Ground Motion Produed by Nulear Blat, 11 Meh. Re. Div. Amerian Mahine and Foundry Company, Report under Contrat AF 29(601)-1190 with AF SWC, Pekeri, C. L., "The Seimi Surfae Pule," Pro. Nat. Aad. Si., Vol. 14, 1955, pp Lang, H. A., "The Complete Solution for an Elati Half-Spae Under a Point Step Load," Rep. No. P-1141, Rand Corp., Santa Monia, California, Lang, H. A., "Surfae Diplaement in an Elati Half-Spae," ZAMM, Vol. 41, 1961, pp Cragg, J. W., "On Axially Symmetri Wave, III. Elati Wave in a Half-Spae," Pro. Cambridge Philo. So., Vol. 59, 1963, pp Knopoff, L. and Gilbert, F., "Firt Motion Method in Theoretial Seimology," J. Aout. So. Amer. Vol. 31, 1959, pp Aggarwal, H. R. and Ablow, C. M., "Solution to a Cla of Three-Dimenional Pule Propagation Problem in an Elati Half-Spae," Int. J. Engng Si., Vol. 5, 1967, pp

36 -34- APPENDIX Wave Front Coeffiient The wave front oeffiient are written in one expreion for all three diplaement like the omponent of a vetor. (91) (92) (93) (94) (95) (96) (99) (100)

37 -35- in whih 2 2 a 5 = (p - Zr ) 1 ( n 2 2 r2)za2 = x p - a ( 101) b 3 = {y Md + n ) h 4 = [ n (M - 1) - 2y M ] b ( 2M2 + n2) 6 = y 1 b8 = (ymd- zmd)2 ( 10 2) e - (1. 2-2) 2 - ( 103) M - d - (104)

38 / / x y z FIGURE 1 Traveling Point Load Problem

39 branh ut Im q Q~. Q~... - t = tw \......_ - q; I I,,.,-' - q+.,..,... / qd t f qd I I + Qd I VJ -.J I e O "'-t=twd q; '... - _.,... + e O FIGURE 2 Re q Contour Integration in the q- Plane

40 -38- x t = t L ---- x III y t = t d t - t - d z t=t de z = 0 Plane y= 0 Plane (a ) x -+-- x y t=t d z = 0 Plane y = 0 Plane ( b) FIGURE 3 Dilatational Wave Pattern for (a) Superoni Load Motion and (b) Tranoni and Suboni Load Motion

41 Im w t = t SC I =td ~ ~: "' Region I t = 00 t =td Co d Rew g+ R g,+ g+ d ; ; + ~ + ā for > 0 for < a }a- d,, R FIGURE 4 Contour Integration in the w- Plane

42 -40- x t = t L y FIGURE 5 Wave Pattern in the Surfae Plane for Superoni Load Motion

43 -41- x t = t L y VII t = t 5 (a) x / I ' "' VII y ( b) FIGURE 6 Wave Pattern in the Surfae Plane for (a) Tranoni Load Motion and (b) Suboni Load Motion

44 .._._ / t = tv / I I I I I I I VI I t = f x z X \ C -p=-r C \P= Ci \x Cd p= I ~ N I FIGURE 7 Wave Pattern in the Plane Under the Path of the Load for Superoni Load Motion

45 -43- t = --- x z ( a ) 0 t = t d x z (b) FIGURE 8 Wave Pattern in the Plane Under the Path of the Load for (a) Tranoni Load Motion and (b) Suboni Load Motion

46 -44- x FIGURE 9 Wave Trailing Behind the Load for Superoni Load Motion

47 -45- ACKNOWLEDGMENT The firt author i indebted to the National Aeronauti and Spae Adminitration for the upport of a Graduate Traineehip.