16th Iteratioal Symposium o Ballistis Sa Fraiso, CA, 3-8 September 1996 GURNEY FORULAS FOR EXPLOSIVE CHARGES SURROUNDING RIGID CORES William J. Flis, Dya East Corporatio, 36 Horizo Drive, Kig of Prussia, Pa., U.S.A. The Gurey problem of the aeleratio of liers by explosive harges havig rigid ores is rigorously examied, ad ew Gurey formulas are derived. First, it is show that the usual assumptios of liear veloity distributio ad uiform desity throughout the explosive produts are ot equivalet for ylidrial ad spherial harges whe a rigid ore is preset. This is doe by derivig the desity distributio osistet with a liear veloity distributio, ad the veloity distributio for a uiform desity. The, eah of these osistet sets of oditios is used to derive a ew Gurey formula. This paper also ompares with, ad larifies the differees amog, several previous Gurey formulas for suh harges. Comparisos with hydroode omputatios are iluded. INTRODUCTION The well-kow Gurey model (Gurey, 1943) is based o two assumptios about the explosive s behavior durig aeleratio of the metal: liear distributio of veloity ad uiform desity. Eah assumptio is used i a differet part of the model, whih is based o the priiple of oservatio of eergy. Speifially, the hage i the explosive s iteral eergy is equated to the kieti eergy of the lier ad explosive produts gases; thus, C E E dc = 1 gas V + 1 V dc (1) gas where C ad are the masses of the harge ad metal lier; E is the explosive s iteral eergy, with iitial value E ; V is the veloity of the lier; ad V is the partile veloity withi the explosive produts. If, as Gurey suggested, the explosive produts expad adiabatially aordig to the ideal-gas law, the ρ γ 1 E = E ρ () where ρ ad ρ are the urret ad iitial explosive desities. The assumptio of uiform desity simplifies itegratio of the urret iteral eergy over the explosive, ad the assumptio of a liear partile-veloity distributio simplifies that of the explosive s kieti eergy. However, as shall be show, for a ylidrial or spherial harge surroudig a rigid etral ore, these two assumptios are ot osistet. This raises the questio of how to derive a rigorously self-osistet Gurey formula for suh harges. I this paper, this questio is resolved by derivig, first, the o-liear veloity distributio that arises from the assumptio of uiform desity ad, seod, the o-uiform desity distributio that arises from the assumptio of liear veloity. The eah of these osistet sets of oditios is used to derive a self-osistet formula for the lier s veloity. These formulas are ompared with previous Gurey formulas for suh harges ad the results of hydroode omputatios. ANALYSIS OF THE GURNEY ASSUPTIONS I this setio, the usual Gurey assumptios of uiform desity ad liear veloity distributio i the explosive are examied. It is show that these assumptios are equivalet ad osistet exept for ylidrial ad spherial harges surroudig rigid ores. - 1 -
Uiform desity. The veloity distributio i a explosive about a rigid ore uder the assumptio that the explosive gas s desity remais always uiform was first addressed by Stere (1951) for spherial harges. Here, his aalysis is geeralized to ilude plaar ( = 1), ylidrial ( = ), ad spherial ( = 3) harges, ad the resultig o-liear explosive veloity distributio is expressed i terms of both Lagragia ad Euleria spatial oordiates. Let R deote the Lagragia radial oordiate (iitial positio) of ay explosive partile, ad r deote its Euleria radial oordiate (urret positio), as show i Fig. 1. Now, let µ deote the fratio of explosive mass betwee the ore ad a give partile, i.e., withi a radius r or R; thus, µ is a Lagragia mass oordiate. The, µ(r) = R R R ; µ(r) = r R R r (3) R where R ad r are the iitial ad urret outer radii of the explosive ad R is the radius of the rigid ore. Solvig for r yields r = (r R ) µ + R 1/ Sie µ is a Lagragia oordiate, the veloity of ay partile is r = ( r / t) µ = ostat, whih yields r = µv 1 R r µ + R ( 1) / r where V = r / t is the urret lier veloity. Substitutig for µ aordig to Eq. (3) yields the veloity distributios over the iitial positio R, 1 R r + R ( 1) / r (4) ad over the urret positio r, V (R) = V R R R R V (r) = V r R r R 1 R r R R R R r R r R + R ( 1) / r (5) C R r V R R r Fig. 1. Explosive harge surroudig a rigid ore. - -
Both expressios are o-liear exept for two ases: for plaar harges ( = 1), for whih they redue to V (R) = V R R R R ; V (r) = V r R r R ; ad for all harges havig o ore, R =, for whih V (R) = V R R ; V(r) = V r r, all of whih are liear. Thus, the assumptio of uiform desity implies a liear veloity exept for ylidrial ad spherial harges surroudig fiite ores. Liear veloity distributio. We ow examie the assumptio that the veloity distributio i the explosive produts is always liear i the iitial oordiate R. First, this oditio is show to be equivalet to liearity also i the Euleria oordiate r. The positio r of the partile origially at R is give at ay time t by If V is always liear i R, r = R + t V (R, t) dt V (R, t) = V (t) R R R R (6) t the r = R + R R V R R (t) dt = R + R R r R R (t) R (7) whih says that r is liear i R. Thus, V must be liear also i r, as we iteded to show. We ow derive the desity distributio based o this assumptio. Cosider a iitial iterval dr of explosive at desity ρ. The explosive produts origiatig here will later oupy a iterval dr at desity ρ. By oservatio of mass, ρ R 1 dr = ρ r 1 dr. Solvig for ρ yields ρ = ρ (R / r) 1 (dr / dr) 1, whih, by Eq. (7), beomes ρ(r) = ρ R R r R R R r R + R r 1 R R r R At ay time after oset of motio (r > R ), this varies with r (ad thus also with R), exept for plaar harges ( = 1) or the ase of o ore (R = ). As expeted, the desity ρ dereases with r. SIPLE ODEL OF CASING RUPTURE As the metal lier is drive outward by the explosive, it strethes i the irumferetial ad (to a muh lesser extet) axial diretios. Evetually, the lier fratures, ad the explosive produts gases leak out betwee the fragmets, reduig their pressure. A simple model to aout for this was suggested by Thomas (1944) ad adopted by several others, iludig Stere (1947, 1951) ad Hery (1967). It is based o the assumptio that, oe the lier expads to a ertai multiple of its iitial radius, it is o loger aelerated. Thus, ay iteral eergy remaiig i the explosive produts does ot otribute to the lier s fial veloity. CONSISTENT FORULAS FOR CORED CHARGES I this setio, formulas for the lier veloity are derived based o rigorous appliatio of eah of the above assumptios. Uiform desity. By usig the oditio of uiform desity with the proper o-liear veloity distributio, Eq. (4), a osistet Gurey formula may be derived. If the desity at eah istat is uiform, the ρ = ρ (R R )/(r R ), ad the total iteral eergy of the explosive is simply 1 (8) - 3 -
E dc gas R = E R r R gas γ 1 dc = C E R R r R The o-liearity of Eq. (4) ompliates the itegral of the explosive s kieti eergy. The resultig Gurey formula for ylidrial harges is V = E C + 1 1 4b + 3b 4 8b 4 l b 1 b 3 γ 1 1 R R r b R where b = R /r b, i whih r b is the value of the radius r at whih the expadig asig bursts ad aeleratio of the lier eases. The orrespodig formula for spherial harges is V = E + 3 1 + 3b + 6b + 5b 3 C 5 1 + b + b 3 1 R 3 R 3 r b 3 R 3 If burstig is egleted (r b ), the eah of these formulas approahes the orrespodig lassial Gurey formula for o ore, summarized by Thomas (1944) as V = E + (11) C + Liear veloity distributio. By usig the assumptio of liear veloity distributio, Eq. (6), with the assoiated distributio of desity, a osistet Gurey formula may be derived. The o-uiform desity distributio, Eq. (8), ompliates the itegral of the explosive s iteral eergy. For ylidrial harges, the resultig formula is R V = E 1 f, r, γ R R C + 1 3R + R (1) 6 R + R where f is a ompliated futio that represets the fratio of eergy remaiig i the explosive whe the asig bursts. The formula for spherial harges is V = E 1 f 3 R R, r R, γ The geeral expressio for the futio f i these equatios is f(a, ψ, γ) = 1 a 1 a ψ a γ ψ a C + 1 1 6R + 3R R + R R + R R + R γ 1 γ 1 a + 1 a ψ a s γ( 1) a + s (1 γ)( 1) ds where a = R / R ad ψ = r /R. This seems itegrable i losed form oly for itegral values of γ, for whih the value γ = 3 is a good approximatio for odesed explosives; the resultig expressios are, for ylidrial harges ( = ), f (a, ψ, γ = 3) = 1 a 1 a ψ a ad for spherial harges ( = 3), 5 (1 a) ψ + a + 3a(ψ 1) + + a ψ 1 1 a 3 1 a ψ a l ψ a + 1 1 ψ (9) (1) (13) (14) - 4 -
3 f (a, ψ, γ = 3) = 3 1 a 1 a 3 ψ a 3 1 3 ψ 1 a a + a 3 1 a ψ a + 1 3 ψ 1 ψ a 6 (ψ 3 a 3 ) + 3 1 a ψ a ψ 1 ψ a 6 a 3 5 a(ψ 1) ψ + a + 5a ψ 1 1 a + 3 1 a ψ a l ψ a + 15ψ 1 ψ + ψ 3 8ψ(ψ + a)(1 a) + ψ3 a COPARISONS WITH PREVIOUS FORULAS Several Gurey formulas for suh harges have bee previously derived. Stere (1951) applied the assumptio of uiform desity to the spherial ase, but oly i the explosive s kieti eergy, derivig the formula V = E + 3 1 + 3b + 6b + 5b 3 C 5 (1 + b + b ) 3 However, his earlier aalysis of the ylidrial ase (1947) used a ombiatio of liear veloity distributio ad uiform desity i the expressio of kieti eergy, with a simplified, o-rigorous treatmet of the explosive eergy, yieldig the formula V = E C + 1 6 3 + b 1 + b Joes (1965) ad Hery (1967) aalyzed yliders ad spheres with rigid ores. Both used the lassial Gurey approah ombiig the assumptios of uiform desity ad liear veloity distributio without osiderig burstig, ad derived formulas that are quite simple: for ylidrial harges, ad for spherial harges, V = E + 1 3R + R C 6 R + R V = E + 1 6R + 3R R + R C 1 R + R R + R Sie these two formulas do ot aout for a burstig radius, they orrespod to a late-time oditio at whih egligible iteral eergy remais i the explosive gases, while their veloity distributio remais liear; thus, these represet asymptoti expressios of the ew Gurey formulas for liear veloity, Eqs. (1) ad (13), as the burstig radius approahes ifiity. The above formulas for ylidrial harges, Eqs. (9), (1), (16), ad (17), are ompared i Fig.. The veloities predited by the two ew formulas fall 1% to % below the two previous formulas of Stere ad Joes; this is maily beause the ew formulas fully aout for a burstig radius, i both the remaiig iteral eergy ad the kieti eergy of the explosive produts. Also show are the results of omputatios performed usig the 1-D apability of the EPIC ode (Johso et al., 1994). These omputatios used the LX-14 explosive, modeled by the JWL equatio of state, with a iitial desity of 1.838 g/m3, ad a steel lier of desity = 7.89 g/m3. The rigid ore is hadled by ostraiig the ier boudary of the explosive, ad all of the explosive is istataeously iitiated. Veloities whe the lier has expaded to 1.6 times its iitial size (i.e., r /R = 1.6) are plotted. These fall betwee the urves for the ew formulas, i whih the Gurey - 5 - (15) (16) (17) (18)
1.6 1.4 Fial fragmet veloity, V / E G 1. 1.8.6.4..1 1 1 ass ratio, C / New formula for uiform desity, Eq. (9) New formula for liear veloity, Eq. (1) Stere's formula, Eq. (16) Joes's formula, Eq. (17) EPIC ode omputatios Fig.. Compariso of Gurey formulas ad hydroode omputatios for ylidrial harges with rigid ores, for a ore-radius ratio R / R =.6 ad a burstig-radius ratio r b / R = 1.6. eergy (E G ) is take as 3.5 km/s, ad a ratio of speifi heats γ =.85 is used (Eq. (14) is itegrated umerially.). Atually, by adjustig the Gurey eergy slightly, these poits a be made to fall diretly o either of these urves. The reaso for this is suggested by Figs. 3 ad 4, whih show distributios of veloity ad desity i the explosive produts at the time whe the lier has expaded to 1.6 times its iitial radius. The ode preditios fit the urves orrespodig to the two ew models perhaps equally well but eah with some differee. The veloity distributios i Fig. 3 are ot quite liear but ot as o-liear as the urve orrespodig to uiform desity. Likewise, i Fig. 4, the distributios of ρ are ot uiform but either are they o-uiform i a maer of a liear veloity distributio. The formulas for spherial harges, Eqs. (1), (13), (15), ad (18), are ompared i Fig. 5. Agai, the veloities predited by the two ew formulas fall below the two previous formulas. The hydroode-predited veloities agai agree best with the ew formula based o uiform desity, Eq. (1). - 6 -
1 Explosive partile veloity, V /V.8.6.4..55.6.65.7.75.8.85.9.95 1 Iitial radial oordiate, R /R Liear veloity distributio, Eq. (6) Veloity distributio for uiform desity, Eq. (4) C/ =. C/ = 1 } EPIC ode omputatios C/ = 5 Fig. 3. Veloity distributio i the explosive produts for a ore-radius ratio R / R =.6 ad a burstig-radius ratio r b / R = 1.6. EFFECT OF RIGID CORE Hirsh (1986) observed that Joes s formulas, Eqs. (17) ad (18), predit that addig a small rigid ore ireases the veloity of the lier, eve at the expese of the displaed explosive. This behavior may be examied i ay of the Gurey formulas metioed here by replaig the explosive mass term C by the expressio C (1 R /R ), where C represets the harge mass for the ase of o ore. Thus, Joes s formula for yliders ( = ), Eq. (17), beomes V = E C 1 (R / R ) + 1 3R + R 6 R + R whih predits a maximum lier veloity for a ore of radius give by R R = 1 + 3 C 1 + 3 C 1 Veloities predited by our ew formulas, modified i a similar maer, are plotted versus ore radius i Fig. 6 for ylidrial harges. These urves show that the predited irease i veloity due to addig a ore is small, espeially for smaller values of C /. Also, the ew formula based o uiform desity, Eq. (9), predits a egligible irease for C / = 5 ad oe for the other ases. - 7 -
.4.35 Explosive desity, ρ /ρ.3.5..15.55.6.65.7.75.8.85.9.95 1 Iitial radial oordiate, R /R Uiform explosive desity Explosive desity for liear veloity, Eq. (8) C/ =. C/ = 1 } EPIC ode omputatios C/ = 5 Fig. 4. Desity distributio i the explosive produts for a ore-radius ratio R / R =.6 ad a burstig-radius ratio r b / R = 1.6. This figure also iludes the results of 1-D EPIC ode omputatios, whih geerally agree well with the preditios of the formulas, ad, like for the formula for uiform desity, show a egligible irease i veloity with the additio of a small rigid ore. The best agreemet is with the formula based o the assumptio of uiform desity of the explosive produts. However, it is iterestig that both the formulas ad the ode omputatios show that there is oly a very small derease i veloity whe a ore of sigifiat size is added. I partiular, for C / = 5, the ode-predited veloities are remarkably flat, equal to five sigifiat digits for ores havig radii up to 6% that of the explosive harge. Eve for a ore with a radius equal to 8% that of the explosive harge, the ode-predited veloity is oly 4% less tha for o ore, eve though the explosive mass is 64% less. For lower C / values, the ode-predited veloities are less flat, iitially ireasig slightly with ore radius, the dereasig more quikly. A similar study for spheres is show i Fig. 7. Both the urves ad the ode omputatios of veloity are eve flatter with respet to ore radius tha for yliders; this is expeted beause the fratio of explosive mass displaed by the ore is equal ow to the ube of the ore-radius ratio R /R, rather tha the square for yliders. - 8 -
1.4 Fial fragmet veloity, V / E G 1. 1.8.6.4..1 1 1 ass ratio, C / New formula for uiform desity, Eq. (1) New formula for liear veloity, Eq. (13) Stere's formula, Eq. (15) Joes's formula, Eq. (18) EPIC ode omputatios Fig. 5. Compariso of Gurey formulas ad hydroode omputatios for spherial harges with rigid ores, for a ore-radius ratio R / R =.6 ad a burstig-radius ratio r b / R = 1.6. CONCLUSION The usual assumptios for derivig Gurey formulas, uiform desity ad liear veloity distributio i the explosive produts, have bee show to be mutually iosistet for ylidrial ad spherial harges surroudig rigid ores. Theoretially osistet formulas have bee derived for suh harges by rigorously applyig eah of these assumptios idividually; these also aout for a fiite burstig radius of the lier. Comparisos with previous formulas show some differees, but ompariso with hydroode omputatios shows that either assumptio is exatly valid. A iterestig preditio of these formulas, verified by hydroode, is that replaig some of the explosive by a ore as large as 8% the explosive radius, results i a egligible derease i lier veloity. ACKNOWLEDGEENTS This work was supported by the U.S. Air Fore Wright Laboratory uder Cotrat F863-94-C- 7. The tehial guidae of Dr. Joseph C. Foster, Jr., is gratefully akowledged. The author also wishes to thak r. aurie Grudza for performig the hydroode omputatios. - 9 -
Fial fragmet veloity, 1. 1 C / = 5 V / E G.8.6 C / = 1.4 C / =...1..3.4.5.6.7.8.9 1 Relative ore radius, R /R Fig. 6. Compariso of lier veloities of ored ylidrial harges, predited by the ew formulas, Eq. (9) (solid lies) ad Eq. (1) (dashed lies), ad hydroode omputatios (symbols) for burstig-radius ratio r b / R = 1.6. REFERENCES R.W. Gurey, The Iitial Veloities of Fragmets from Bombs, Shells, ad Greades, U.S. Army Ballisti Researh Laboratories (BRL), Report No. 45, 1943. ATI 3618. I.G. Hery, The Gurey Formula ad Related Approximatios for High-Explosive Deploymet of Fragmets, Hughes Airraft Co., PUB-189, April 1967. AD 813398. E. Hirsh, Improved Gurey Formulas for Explodig Cyliders ad Spheres usig Hard Core Approximatio, Propellats, Explosives, Pyrotehis, Vol. 11, pp. 81-84, 1986. G.R. Johso et al., User Istrutios for the 1995 Versio of the EPIC Code, Alliat Tehsystems, I., Hopkis, N, Nov. 1994. E.E. Joes, Extesio of Gurey Formulas, Hoeywell Sys. ad Res. Div., S5B-4, Feb. 1965. T.E. Stere, A Note o the Iitial Veloities of Fragmets from Warheads, BRL Report No. 648, Sept. 1947. AD 89568. T.E. Stere, The Fragmet Veloity of a Spherial Shell Cotaiig a Iert Core, BRL Report No. 753, arh 1951. AD 377187. L.H. Thomas, Theory of the Explosio of Cased Charges of Simple Shape, BRL Report No. 475, July 1944. - 1 -
Fial fragmet veloity, 1. C 1 / = 5 V / E G.8.6 C / = 1.4 C / =...1..3.4.5.6.7.8.9 1 Relative ore radius, R /R Fig. 7. Compariso of lier veloities of ored spherial harges, predited by the ew formulas, Eq. (1) (solid lies) ad Eq. (13) (dashed lies), ad hydroode omputatios (symbols) for burstig-radius ratio r b / R = 1.6. - 11 -