SCALING the effects of AIR BLAST on typical TARGETS

Transcription

1 SCALNG the effects of AR BLAS on typical ARGES he general scaling equations presented in this article are based on interactions beteen shock aves generated by explosions in air targets that can be fully defined by to parameters in a simple mathematical model. Simplicity is achieved by ignoring many factors that complicate rather than illuminate the study of blast phenomena. hese scaling equations, the analytic techniques based on their use, are not intended to be precision tools for computing specific effects ith maximum accuracy, even though they are often as accurate as far more complicated computational techniques. heir unique value lies in their use of a dimensionless, universal scaling parameter hose values span the entire spectrum of blast damage phenomena, from the effects of a fe pounds of conventional high explosives to those of megatons of equivalent N. he reader is cautioned, hoever, not to expect real targets to behave precisely like the simplified models from hich the scaling equations ere derived, but he is encouraged to use the equations freely as a means of correlating isolated pieces of data ith the basic analytic structure of blast damage relationships. his analysis deals ith targets the interactions beteen them the shock aves generated by the detonation of explosives. t limits its attention to that fraction of the total range of scaled distances ithin hich a variety of actual targets, varying idely in toughness, have been destroyed by air blast. Within this limited range, an abundance of excellent experimental data sho that shock aves behave in an orderly fashion alloing the parametric relationships that define their characteristics to be expressed mathematically by to simple, tractable equations. Since the purpose of this article is to derive scaling equations for interactions beteen shock aves targets that result in target " kills," these equations ill be used to compute the effects of interactions only ithin the range of scaled distances at hich such kills are possible. From the characteristics of shock aves it is possible to derive scaling equations of classic simplicity in hich specific target characteristic data ensure that the equations ill be valid for the range in hich shock aves can do kill such targets. he universal scaling parameter derived in this article allos the equations defining the blast-shock ave characteristics to be converted successfully into equations that scale the interactions beteen shock aves targets. Nature of Shock Waves Detonation of a high explosive in air produces a sudden rise in pressure hich propagates rapidly as a spherically exping shock ave. When the shock ave arrives at some fixed point in its path, there is an almost instantaneous rise in pressure to a peak value, folloed by a gradual decline until it reaches APL echnical Digest

2 H. S. Morton nteractions beteen shock aves, produced in air by detonation of explosives, specific targets hich they can destroy by air blast are described. A mathematical analysis is used to relate eights of explosives to the distances at hich they can cause lethal damage over the entire range of blasts from a fe pounds of conventional high explosive to kilotons or megatons of nuclear blast. Effects at sea level higher altitudes are examined. n the analysis, typical targets are defined by to parameters for hich specific numerical values can be established. A dimensionless scaling parameter relating a shock ave parameter to a target parameter is the key to the scaling relationships derived. the original ambient pressure level at some later time hich marks the end of the first positive impulse. he pressure then drops belo ambient, producing a negative impulse, then may be folloed by a much eaker second positive impulse. Shock ave data are usually given in terms of "free-air" or "side-on" values of overpressure (pressure above ambient), in hich it is assumed that no obstacle impedes the free motion of the shock ave. When a solid surface presents an area normal or nearly normal to the direction of motion of the shock ave, the shock ave is reflected, producing a substantial increase in overpressure. For a very eak shock, including the limiting case of an acoustic ave, the overpressure for normal incidence is tice the "free-air" value. For very strong shocks in air (')' = 1.), the ratio of the overpressure for normal incidence to the " free-air" value approaches an upper limit of eight. Because this article deals ith the interaction beteen shock aves targets, e are not concerned ith "free-air" conditions, but ill deal ith overpressures impulses that act on typical target surfaces. t has been found that the overpressure near a surface reflecting a shock ave is substantially constant for angular orientations up to about 35 from the normal. Since a typical target, such as an aircraft, is irregular in shape has elements of surface at all possible angular orientations, it is safe to assume September - October 1967 that the target presents substantial areas ith orientations ithin 35 of the normal for hich the pressures acting on the target are essentially normally reflected overpressures. For this reason, the pressures, impulses, other shock parameters discussed subsequently in this article ill refer to faceon or normally reflected shock aves. Shock Wave Characteristics A shock ave hose pressure-time history is shon in Fig. 1 may be defined by the folloing parameters: P = p p peak overpressure (psi); the difference beteen the absolute peak pressure the ambient pressure. time (msec) during hich the pressure of the shock ave is continuously greater than the ambient pressure. is called the duration of the positive impulse. ambient air pressure (atm), i.e., p = 1 at sea level (1.7 psi). overpressure of the shock ave at time t. total po'sitive impulse, given by = ;: pdt. his is sometimes termed the first positive impulse, 3

3 ; [ o -.. u,.. o o..ffi J: > (/) o ME. t Fig. 1 - Pressure-time curve for a normally reflected shock ave. since there may be a second, much eaker, positive impulse. n practice, the pressure variations beyond are sufficiently small that they can be neglected in the analysis of blast damage. he shock ave from a high explosive is a function of the distance from the detonation, the nature size of the charge, the pressure temperature of the air through hich the ave is propagated. t is possible to correlate the effect of different sizes of explosive charges on shock aves by using a scaled distance, Z, defined by here: Z=_R W / 3 (1) R = distance (feet) from the center of the detonation; W = eight (pounds) of the explosive charge. According to this scaling la, for a given value of scaled distance Z, the same peak overpressure ould be obtained by the detonation of to different eights of a particular explosive; for example, the explosion of 1 pounds of N at a distance of feet ill produce the same peak overpressure as 8 pounds of N at a distance of feet. he relationships beteen peak overpressures, positive impulses, scaled distances, atmospheric environments differ from one explosive to another. Since 5/ 5 Pentolite is a military high explosive hich gives consistently reproducible results, it is often used as a stard explosive for evaluating the effects of air blasts. Furthermore, it is an explosive for hich abundant accurate data are available. H. J. Goodman, " Compiled Free-Air Blast Data on Bare Spherical Pentolite," Aberdeen Proving Ground, Ballistic Research Laboratory, BRL. Report No. 19, February 196. For these reasons this analysis has been built around the characteristics of 5/ 5 Pentolite. he data for normally reflected peak overpressures for 5/ 5 Pentolite are shon as a function of the scaled distance in Fig.. he straight line fitted to these data points is given by p p 13,3 [Zp 1/3] 3 n similar fashion the data for normally reflected positive impulses for 5/ 5 Pentolite are shon as a function of the scaled distance in Fig. 3. he straight line fitted to these data points is given by W 1/3 P /3 [Zp 113] 1. hese to analytically tractable equations, () (3), define the characteristics of 5/ 5 Pentolite over the range of scaled distances ith hich this analysis is concerned. Anomalies hich occur at scaled distances less than to or greater than 1 or 1 have no bearing on the accuracy alld validity of these equations since they lie outside the range of parameters under consideration. 'iii.3 :::J CJ) CJ).. > <t.. - u: j <t z c. Q.: \ \. \ \ '\ \, P p = 13.3 (Zp ""l3 \ \ " 'f SCALED DSANCE (Z = RW"" l Fig. - Normally reflected peak overpressure for SO/5 Pentolite spheres. Equation (3) hich is fitted to empirical data obtained at sea level (p = 1.) indicates the qualitative effect of p on the impulse. At higher altitudes tem- () (3) :\PL echnical Digest

4 15 1 "'- :::> a > i= in lr f"' :: ;;;E -..J '" tt : V > 1 : 8 z 6 ;;; - :>'. 'l. L Ss \. _ = r.. W YJ p " (Zp YJj 1..., '\ '" SCALED DSANCE (Z = RW"') Fig. 3 - Normally reflected positive impulse data for SO/SO Pentolite spheres. perature changes ill affect the speed of sound require a quantitative adjustment. hese changes, hich are independent of p, are accounted for by including the parameter, cl co, the ratio of the velocity of sound in air at the altitude to the velocity at sea level to obtain the impulse equation, W 1/3 P /3 [Zp 1/3] 1. Using Eq. (1) to eliminate Z from Eqs. (3) (), the folloing equations are obtained for the " normally reflected" peak overpressure positive impulse: " () (5) W o = ---i p (6) Characteristics of a arget Aircraft structures, hich are typical targets for air blast, consist of superficial coverings of comparatively thin material supported by rather complex assemblies of ribs, stiffening ebs, braces of various sorts that are attached to the basic frameork. arget damage severe enough to constitute a kill consists of substantial crumpling distortion of the surface in the course of hich it moves a considerable distance against the resisting forces offered by the structural elements hich yield ithout breaking. he behavior of a target cannot be fully simulated, but a remarkably useful mathematical model can be defined in simple terms. Pressures belo a certain minimum value ill have no effect hatever on the target regardless of ho long they are maintained. Hoever, at some critical pressure the supporting structure ill begin to yield, if this pressure is maintained, the distortion ill increase until the target is totally destroyed. herefore, it is possible to define a fundamental parameter representing target toughness as: Pm = minimum pressure (psi) on the target surface hich ill initiate destructive distortion hich, if continued long enough, ill cause target destruction. he mathematical model used in this analysis assumes a constant value of Pm throughout the period of distortion up to the point of destruction. Many real targets approximate this behavior closely enough to consider Pm a statistically significant measure of target toughness. A second target characteristic parameter hich is important in the analysis is 1m, hich is defined as the loer limit for the value of the impulse hich can deform the target enough to destroy it. he parameter 1m is closely related to the total ork done in deforming the target to the point of destruction. U n iver sal Scaling Parameter his analysis is concerned ith scaling the interactions beteen shock aves targets. A scaling parameter for this purpose should be a dimensionless ratio beteen some distinctive characteristic parameter of the target a similar distinctive characteristic parameter of the shock ave. he only characteristic common to both is pressure. he significant pressure for the target is the minimum pressure, Pm, i.e., the pressure hich causes target damage; for the shock ave it is the maximum pressure, P, hich it can exert on the target. Hence, a universal scaling parameter, is defined by the equation, No target damage can occur if f is greater than 1. no real target, in hich there is any finite resistance to deformation, can have an f value of zero. he entire gamut of shock ave target combinations, hich produce target kills, is covered by the (7) September - O ctober

5 open interval < f < 1. Within these limits, f spans the entire spectrum of charge eights ( corresponding lethal distances) from a fe pounds of conventional high explosive to nuclear devices hose yield is measured in kilotons or megatons of equivalent N. n the scaling equations derived in this article, the characteristics of the explosive, the target, the ambient atmospheric conditions (altitude ratio of sound velocities) are all defined by appropriate numerical values of the parameters. Every set of conditions represented by these parameters corresponds to a unique eight of explosive, the unique lethal distance associated ith it. herefore, f is a universal scaling parameter for the interactions beteen shock aves the targets that tey are capable of killing hen the shock aves are spherical the targets are correctly defined by the parameters, Pm 1m. nteraction Beteen a Shock Wave a arget he target is defined by to parameters to hich numerical values can be assigned. he target model accurately simulates the behavior of many real targets hose destruction by blast has been observed under test conditions. Associated ith a target is a minimum pressure exerted on its surface belo hich threshold there ill be no damage. t is assumed that hen this critical minimum pressure is reached, the supporting elements ill begin to yield permanent deformation ill be initiated. t is further assumed that as long as deformation (consisting of displacement of the surface relative to the more massive elements of the basic structure) continues, it ill be resisted by a constant force per unit area of the same magnitude as the pressure hich initiated destructive deformation. Every target ill have to be deformed by some definite minimum displacement of the surface relative to the basic structure before its usefulness is destroyed, i.e., a certain minimum target distortion, or a certain minimum ork per unit area, must be expended on the target to destroy it. t is necessary then to derive functions for the total ork per unit area any given shock ave can impart to the target, to specify the characteristics of the shock ave hich can impart just enough ork to achieve target destruction. t is no necessary to have an equation for the pressure-time profile of a shock ave acting against a target surface. his equation ill give va lues of the overpressure as a function of time (t < ). o mathematical models for the pressure-time curves have been used, the linear profile shon in Fig. a ith here p = P (1 -tl ), (8) t = time from the first rise in pressure p = overpressure at time t ( < t < 7); the exponential profile shon in Fig. b ith E = P ( 1 - ti ) e - tj (9) U ::> f) <{f) W.... > o AMBEN PRESSURE Fig. - Pressure-time curves. t can be shon that a completely self-consistent set of scaling equations can be derived from either of these pressure-time profiles that the same relationships beteen charge eights corresponding lethal distances are derivable from either equation. he linear relationship has been used in several investigations ith satisfactorily consistent results. Hoever, self-consistency ithin a mathematical model is not sufficient to assure that values of f, as ell as the values of such target parameters as Pm, portray realistic interaction characteristics. he pressure-time relationships shon in Fig. differ from one another in the relationship of to P. For the linear curve (Fig. a), the ratio is 1/. For the exponential curve, it is l i e or.368. he data for 5/ 5 Pentolite sho that the ratio of to P is not constant but varies at a moderate rate ith scaled distance. Furthermore, it as found that the ratio l i e of the exponential function, Eq. (9), agrees 6 APL echnical Digest

6 E.. r W > to tm ME, t nteractions hich are completed ithin the period () of the positive impulse can be treated by a single set of equations ill be designated Case (see Fig. 5). nteractions hich continue beyond the end of the positive impulse ill require to sets of equations, one ith a constantly varying acceleration (up to time, 1), one ith a constant acceleration (beyond time, 1). his ill be called Case (see Fig. 6). he results of some lengthy calculations lead to the folloing equations for the impulse of the shock ave up to time, in terms of 1m, the universal scaling parameter E: For Case (E ) e Fig. 5 - Pressure-time relationship for E lie. 1m 1= eE 1/ [ - E - E (1 - EnE) ] For Case (E ) e :1 :::> (/).. (1) 1/.. 1m a.: ffi o> V E. ME, t Fig. 6 - Pressure-time relationship fore lie. ith the 5/ 5 Pentolite data in the midrange of scaled distances used in this analysis. Since an accurate mathematical formulation of the actual pressure-time profile of a shock ave produced by 5/ 5 Pentolite over the full range of scaled distances has not yet been found, the exponential function is the best representation ithin the range of scaled distances covered by this analysis. On a pressure-time diagram, impulses are represented by areas. Figures 5 6 ill sho that areas above Pm, (designated by 1) are positive impulses hich accelerate the mass, m, to a maximum velocity at time, tm, areas belo Pm (designated by ) are negative impulses hich bring m to rest at some later time, to. Obviously the to impulses must be equal. n setting up equations for the acceleration of the mass, m, (mass per unit area of target material hich is moved relative 1 the basic target structure here destructive deformation takes place) one notes that beteen zero there is a constantly varying acceleration for hich the pressure-time equation, Eq. (9), gives an analytical value, but beyond the time,, the negative acceleration has a constant value. Sep te mber - (11 ) 1=-;::::::::=== Oc t ober 1967 Figure 7 relates the shock ave pressures impulses to the minimum pressures impulses hich can damage a given target. his curve is the locus of all possible overpressure-impulse combinations hich can just destroy (ithout overkilling) any given target characterized by Pm 1m E Co ii::: t "a m Fig. 7 - General pressure-impulse relationship. ' H. S. Morton, " Scalin the Effects of Air Blast on ypical argets," APL/ JH U Report G-733, January

7 ! General Parametric Scaling Equations for Weight Distance Equations (5) (6) can be solved simultaneously for eight, W, distance, R, to obtain.83 ()' P 1. P.8 j' (1) 1 P.733 P.67 (13) When these are combined ith Eqs. (7), (1), (11) one obtains W =.83 [_lm_ 3 _ (:)3J FW(E) Pm 1. P.8 ' (1) R =.8 [ (:) J FR(E) Pm.733 p O.6 7 (15) n this pair of equations, the final terms, F W (E) F R (E) are restricted to E values above or belo 1 e. For E:::; e For E L e E. F (E) =, (l e) 3/ E.733 FR (E) =. ( E) 1/ FW (E) = 1 FR (E) = e 3 Eo. 3 [ -E -E(1 _ ene) ] 3/ E.33 e[ - E-E(1 - ene) ] 1/ hese to pairs of functions of f are related as: F R (E) = [F (EU 1/3 E 1/3... : U. -J '" : U. o " 1, fu 5-" ; l :?,...!"...3 : ' '1}{].-- - "'\ - u if /.. _ 15 c:.q,up'.1 ct'}.8! Q.6.1 C! ::t. = E _ b h ! J o his equation is valid for all values of E. he general parametric equations, (1) (15), consist of four terms: the first is a numerical coefficient characteristic of the explosive (5/ 5 Pentolite) ould be different for different explosives; the second is a function of the target; the third is a function of the altitude; the fourth is a function of the universal scaling parameter, E. For any particular explosive, target altitude, the first three terms are constants, i.e., only the functions of E ill vary. he to general parametric equations, (1) (15), for eight, W, lethal distance, R, give all the ' necessary information ith respect to interactions beteen spherical shock aves produced by explosive detonations targets hose characteristics defined by Pm 1m can be determined by experiment. he range of scaled distances must be sufficient to encompass the region. in hich targets of practical concern can be destroyed. hese to equations epitomize the fundamental analytic struc- '" 1 ton 8 APL echnical Digest

8 ! : ;,; t-- r oo.95 t9 -=:., E'!.93 ' d 1 O.91. 9_ _ '.9 1 1' t-+-f t----t----t...-:;; "i'\,"i-1...;:: :.. -- "'" '.88 O H.,+----_-+_----H*----_t + ++t t t--t_t_1 leo.8-+--_+-p+_--t---t_+_r _+ +_r+_--t_--t_+_h+_--t_--r 1H ":. - V _+--_+ +.+_--+_--+_1_+rl--_+--_+_+ rl_--+_--+_;-ti..": (.8 e fO.78_++_--+_--+_+_--_+--_+_++_--+_--+_+_n+--_+--_+_+_r+_--_--r H 1_ 6.7 (::::;:;.- -Po b b- : ----P.+---r---r-++---r----r- i' 55 il ---1 tons tons 1Kt Kt ---1Mt 1 13 F(t) 1' lmt --...lkt LOG F t) Fig. 8 - General distance-eight relationship. ture of shock ave-target interactions, are the most significant product of this entire article. A graph of the FR(f) versus FW(f) values is shon in Fig. 8, ith corresponding values of f given for various points along the curve. A rough indication of the order of magnitude of the charge eight is shon on Fig. 8 from hich it may be seen that over this range of f the charge eights vary considerably-from 5 pounds to 1 megatons. o other lines are shon on Fig. 8. he first is a line of slope 1/ 3 hich becomes tangent to the general curve as f approaches 1. F (f) approaches infinity. his is consistent ith the relationship Rex:: W 1/3 hich has been shon to be true for nuclear charges. he second is a line of slope 1/ hich is seen to be parallel to the general curve at values of f around. to.5 is consistent ith the Rex:: W 1/ relationship knon to apply to charges of a fe hundred pounds. he change in the slope of the curve of log F R (f) versus log F (f) is the most significant scale effect in relationships beteen charge eights lethal distances. S ep tem be r - O ct ob er Comments Conclusions his analysis has avoided the necessity of treating the full range of shock ave phenomena, particularly the regions in hich parametric relationships are far too complicated to fall into any simple pattern. By limiting its scope to the region in hich parameter relationships can be expressed in simple analytic terms it fills a long-felt need for a direct easy method of scaling blast effects on specific targets. Fortunately a majority of targets of interest ill fall ell ithin the applicable range of the equations derived in this report. As soon as the parameters hich define a target are knon, it is a simple matter to determine the corresponding scaled distances ascertain hether they are ithin the applicable range of the equations. As a concluding statement, since the phenomena to be scaled are interactions beteen shock aves targets, the parameter, f, hich relates a common characteristic of the to, is the most useful analytic tool for scaling these effects. 9