TID-4500, UC-34 Physics
119 LAWRENCE UVERMORE LABORATORY
UCKL-51109 THE USE OF SHOCK WAVES IN THE VAPORIZATION OF METALS Thomaa J. Ahrena Pari A. Uriiev
Ma date: August 5, 1971
-NOTICE- This nfort was prepared as an account of work sponsoreJ by (he United States Government. Neither th« United States nor the 1/nited States Atomic Energy Commission, nor any of their employees, nor any of their contractors, subcontractors, or their employees, makes any warranty, express or implied, or assumes any Seta) liability or responsiblity for the accuracy, com pleteness or usefulness of tny information, apparatus, product or process disclosed, or represents that its use would not infringe privately owned rights.
SISTBIBUTiBN OF THIS DOCUMENT IS tfNLiMfTlD BJA
jj±i.iL-:^.,;£;-mx^^ Contents
Abstract i Introduction . . 1 Plane-Wave Vaporization ...... 2 Calculation of Entropy Upon Shocking Solid and Porous Metals , 4 Calculation of Entropy Required for Vaporization ...... 6 Flyer Plate Impact Velocities Required for Shock Vaporization .... 7 Entropy Production in Porous Metals by Plane Shocks 8 Aluminum , 8 Iron 8 Barium 9 Thorium 9 Uranium 11 Lithium 11 Strontium 11 Mach Stems 22 Oblique Shock Polars 23 Critical Angle . . 26 Mach Stems in Aluminum, Barium, Lithium, Strontium, Thorium, Iron, and Uranium 29 Estimate of the Mach Stem Size 35 Conclusions 36 Acknowledgments 37 References 38 Appendix. EKT, a Fortran IV Program for Calculating Entropy and Shock States in Solid and Porous Metals 40
Hi- THE USE OF SHOCK WAVES IN THE VAPORIZATION OF METALS
Abstract
Explosively induced shock waves in uranium and lithium. When a Mie- aluminum, iron, strcnium, lithium, tho Gruneisen equation of state is assumed, rium, uranium, and barium are capable it is found that for a given plane explosive of vaporizing appreciable quantities of system an initial porosity exists that will these metals. When plane shock waves optimize the shock-induced entropy pro are incident upon porous samples, they duction in a metal. The optimum initial will induce vaporization in aluminum, distention that will result in maximum iron, strontium, thorium, and barium. poBt-shock entropy is found to vary be Although Mack stem shocks act on a tween 1.6 and 2.0 times the volume of smaller selective mass of material, they the crystal for several of the metals will induce sufficient entropy to vaporize examined.
Introduction
The possible use of explosively induced large fraction of this energy density shock waves to produce large masses of (approximately 20%) may be imparted to vaporized metal is examined in this re a metal target, make the possible ap port. Shock-wave induction of metal gas plication of the shock-wave method for appears to be a relatively simple method the sudden production of metal vapor a for releasing various metal species in the potentially attractive technique. In upper atmosphere from ionospheric sound addition tc She intrinsic thermal (random ing rockets. Observations of the reac direction) velocities imparted to the re tions that take place between the metallic sulting high-entropy gas cloud, the shock species and the ambient atmospheric interaction and following rarefaction wave species, and of the relative populations will induce a net flow field (with peak of species that result from such metallic velocities of approximately 10 km/sec) vapor releases, can be made using either in the gas cloud. High explosives may be airborne or ground-based spectroscopic employed in this context in either one- instruments. dimensional plane-wave or The high available energy density of dimensional Mach-stem geometries. In military high explosives (1 kcal/g), as the plane-wave geometry the explosive well as the relative ease with which a can be detonated in contact with the
-l- sample, thus directly imparting a shock that produces high temperatures and va wave to the metal; or plane- flowing detona porization, but no net flow of the induced tion gases can accelerate a flyer plate, gas. By increasing the incidence angle, which in turn maybe usedto shock the sample regular oblique reflection will take place upon impact. As is demonstrated in the until some angle is reached where reflec next section, the plane technique is sim tion becomes irregular and Mach-stem pler and it can vaporize metals such as reflection begins. iron, thorium, strontium, barium, and The purpose of this report is to pro aluminum. However, metals such as vide design criteria for the construction lithium and uranium are difficult to va of shock-vaporization experiments. porize by this technique. For these Specifically, the report examines perti metals the Mach-stem method appears nent experimental parameters, and it to be the only one available, and it provides some design data used to select has the limitation that it vaporizes less initial porosities, types, and speeds for material. explosively driven plates and Mach stents. A Mach stem is a shock discontinuity The driver plates and Mach stems are that is formed when two shock waves ap capable of inducing shocks that will either proach each other at an angle. If the completely or partially vaporize alumi angle between the wave fronts is zero, a num, iron, lithium, barium, strontium, simple head-on collision of shocks occurs thorium, and uranium.
Plane-Wave Vaporization
For the following calculations it is to speeds in the 4 to 5 km/sec range (see assumed that a quantity of an explosive Fig. 1). The flyer plate either impacts such as 9404 (plastic bonded HMX) is the metal to be vaporized directly, or it plane-detonated to accelerate a flyer plate impacts an impedance-matching driver plate upon which the sample is placed, tn determining fee conditions required, tor vaporization as a result of shock com pression and subsequent rarefaction, it is assumed that all the entropy produc tion results only from the shock compres sion process. In essence then, it is as sumed that the rarefaction process is isentropic. In principle it is possible that the total entropy within a control volume could increase during the rare faction process (and the process remain strictly <»diabatic); however, it is as Fig. 1. Shot assembly for the plans-wave experiment. sumed that any increase in entropy upon
-2- rarefaction will be small compared to the conserved (isentropic release), the en shock-Induced entropy. tropy increase with temperature near or Thus the tactic followed Is to calculate at ambient pressure is also calculated. the entropy increase resulting from shock Of specific interest are the values of the compression of both initially single- entropy of the liquid metal at the point of crystal density and initially porous incipient boiling; i.e., the start of vapor metals. Entropy production in porous ization, and the entropy for complete va metals is investigated because, although porization. According to arguments for a given explosive system the maxi given by Zeldovich and Raizer, the in mum shock pressure that may be achieved ternal energy achieved by the metallic is lower than that achievable in a similT elements in the region of the vapor-liquid sample, with initially single-crystal den critical point are on the order of twice sity the specific entropy production may their binding energy. Prom this they under certain conditions be greater than conclude that metals achieving entropies for the solid sample. Since we arc as appropriate to this state will, upon ex suming that, upon rarefaction from the pansion, ultimately be completely vapor high pressure shock state, entropy is ized. Therefore, by equating calculated
-HUGONIOTS, INITIAL POROSITY
-MELTING ZONE
-VOLUME UPON BOILING, V,,. L VOLUME UPON MELTING
ROOM TEMPERATURE VOLUME, V0..
Fig. 2. Thermodynamic diagram representing various paths for vaporization.
-3- entropies in the high-pressure shock (m = 1.0). At a given molar volume, V, state with those achieved by increasing Hugoniots of porous samples (initial the temperature at ambient pressure, the volume, mV.) will generally lie above range of states in the solid that have suf the principal Hugoniot. Hypothetical re ficient entropy to result in incipient, par lease isentropes from states having suf tial, or complete vaporization upon isen- ficient entropy to induce incipient and tropic pressure release may be substantial vaporization are indicated by determined. D'A and D B1, respectively. In general, The behavior upon being shocked to explicit knowledge of the release isen high pressure of a metal with various tropes are not required; only knowledge initial porosities is illustrated in Fig. 2, of the entropy associated with points When samples initially having the single- connected by the isentrope passing crystal molar volume, point 0 , are through a Hugoniot state and a high- singly shocked, they achieve states such temperature, low-pressure state is of as D along the principal Hugoniot curve interest.
Calculation of Entropy Upon Shocking Solid and Porous Metals
In the present calculations, initial by equation-of-state data in the form of smoothed principal Hugoniot curves and calculated shock temperatures are as where p is pressure and E is internal sumed. Several nearly equivalent methods energy. The assumption that for smoothing experimental Hugoniot data and carrying out the shock temperature y = y{V) (2) calculations have bees reported, e.g., is demonstratively inadequate, particu 'i 2 larly at very high shock-induced temper Zeldovich and Raizer, Rice et al., and atures. Although in the present report Walsh and Christian. In addition, the we employ only the assumption embodied Debye model is assumed to describe the by Eq. (2), we discuss the difficulty that solid's thermal properties at tempera arises from this assumption later in this tures substantially below the Debye tem section. perature. However, over most of the In the calculation of shock-induced en temperature range of interest, tho solid tropy production we employ, at each vol has achieved the Dulong-Petit specific ume, the Dugdale-iWacDonald (Grover heat value, and thus this assumption is of 4 et al. ) values of T where it has been little consequence. specified in the published reduction of the To relate changes in thermal pressure Hugoniot data. For several metals where with the changes in thermal energy con T(V) is not explicitly given, we used the tent of the compressed solid, the Mie- assumption that 7/V remains a constant Grdneisen equation of state is assumed. with compression; the zero-pressure Thus y, the GrQneisen parameter is given -4- value of this quantity is adopted. In any SCD = 5.9587Hn(Th/fl) + 1.3296 case we use this assumption to estimate + J- 0.000322 +S 024386 + the variation of the Debye temperature, X T hL t *h" 8, with compression. We note that if X 0.0005943 +, 0007624 the Debye model for the phonon spectral density is assumed, it follows that if* •/< 9.3 X 10 -5 y = -Bin e/81n V, (3) '*h which upor integration yields X3.835X10 1 cal/mole °K. (6) For shock temperatures greater than 10, a* e0 expfrd - v/v0)], (4) we assume a value for C of 3R (where where 9 and V_ are the Debye tempera v n R is the gas constant). The entropy in ture and molar volume at the reference crease is then given by state, to order to calculate entropy at t n ASCD = 21,65 + 3R *" Hugoniot points such as D, D and D X(T /taajcalfalote°K, (7\ (Fig. 2), it is convenient to initially cal n culate the entropy along the principal where the constant 21.65 is the entropy Hugoniot/ O'D via the thermodynamic at 1O(0/T). As can be seen in Tables 3 path, o'OCP. This calculation is straight and 4, the calculated entropies along the forward, as the entropy increase along principal Hugontot for aluminum and iron OC—the 0°K isotherm—is zero by the are very close to those that were calcu third law. Thus the entropy increase lated by a fundamentally equivalent pro occurs only at constant volume along CJD cedure by McQueen et al. for similar- and id given by density iron and aluminum alloys. Prob T c (e/T) ably due to the marked dilution of uranium n v dT, (5) by molybdenum (refer to Table 7), the dS,C D l agreement between the present uranium calculation and the results for U/Mo is where C is the specific heat at constant v relatively poor. volume. Since the results of this integral To calculate the Hugoniot pressures for a Debye solid are tabulated by 5 achieved at a given molar volume by Furukawa and Douglas, in practice a metals with different initial porosities, simple look-up and interpolation routine we employ the Rankine-Hugoniot equation is employed (see the appendix). Shock for conservation of energy. temperatures, Tn, for each metal studied,
E E + imV V) (8) have been taken from the literature and x " 0 ?Px 0 - ' are given later in the section on "Entropy and the Mie-GrOneisen equation. Production in Porous Metals by Plane p = (E„ - E„). (9) Shocks." Alternately, the entropy in y V crease along CD can be calculated from where E and p , and E and p are the the empirical formula (derived from the xx y y Debye model) internal energies and shock pressures associated with two shock states achieved -5r at the same molar volume. E. is the in a problem arises in the calculation of the ternal energy at the reference state, Hugoniot at such points as E and E1 (see which for porous metals is assumed to be Fig. 2). At these points the Hugoniot independent of initial porosity. Thus the assumes a slope of -oo, and at higher surface energy associated with the initial pressures the shock pressure becomes distention of the sample is assumed to be double valued with respect to volume. In negligible compared to the pertinent shock general this situation will occur at pres energies. We further assume that the sures above which the condition porous metal is sufficiently fine-grained m=^(2 r) (11, so that upon being encompassed by a + shock, individual particles iome to ther is satisfied. Above this point the simple mal equilibrium on a time scale compa Mie-Gruneisen equation cannot Le applied. rable to the shock rise-time (approxi As discussed earlier in this section, this mately 0.01 (jsec). Also implicit in this has restricted our calculations to a lower formulation is the assumption that the range of porosities in several metals. sh . ck pressures of interest greatly ex The shock temperature in the initially ceed the crushing strength of the initial porous 30lid is related to the temoerature sample pore structure. along the principal Hugoniot at the same If a porous sample with an initial volume by molar volume m-V. is shocked to a (Pm - Ph)V (12) specific volume, V, the resulting Hugoniot m 37B ' pressure, p , is related to the pressure where we have assumed that above all along the principal Hugoniot, p., at the principal Hugoniot temperatures of inter same volume by est, the specific heat has assumed the (m - 1)V„ Dulofig-Petit value. Within this same as 1 +• - Ph -v sumption, at a given volume the increase X[2T - mV./V +11 (10) in entropy of the initially porous sample is given by In the case where the material of in terest has a very large initial distention. SD-D< = 3R m (Tm/Th> • <13)
Calculation of Entropy Required for Vaporization
The entropy increase required to pro those calculated along the Hugoniot. duce incipient vaporization (i.e., raise a Aside from experimental uncertainties in metal to the boiling point) at standard the shock data themselves, the major un pressure is given in Table 1, where it certainties in the Hugoniot entropies arise has been collected from data presented from the lack of direct knowledge of the in several thermodynamic tabulations. 7-11 high-pressure GrQneisen parameter and Generally the uncertainties in the en the degree to which the Mie-Gruneisen tropies at ambient pressure are less than equation describes reality. Uncertainties
-6- Table 1. Thermal properties of metals pertinent to shock -induced melting an? vaporization/8
Melting point Boiling point. 1 aim Entropy, Entropy, Entropy, Entropy, Entropy incipient complete incipient complete at STP Temp melting melting Temp boiling vaporization Elemem (c&l/mol-aK) (cal/mol-°K) (cal/mol-°K) ical/mol-°K) (cal/mol-<>K)
Al 6.77 033 14.30 17.06 2793 25.38 50.98 Fe 6.52 1809 22.02 23.84 3135 29.89 56.54 7.106 Ba 14.92 1002 26.03 27.88 2171 35.45 51.03 Th 12.76 2028 29.18 31.08 5061 41.13 66.03 U 12.02 1405 27.04 28.40 4407 41.58 66.75 Pb 15.55 601 20.19 22.10 . 2023 30.60 51.59 Sr 12.50 1041 20. 24 1657 28 48 Li 7.00 454 9.67 11.25 1615 20.10 41.54 Data wer e taken from Ref. 7 for all elements except Sr, whose data were taken from Ref. 9. in the entropy along the Hugoniot, reflect barium and thorium. The values given ing uncertainties in y, are generally on in Table 1 are (except for the case of the order of ±10%, but these can approach strontium) taken from the recent revision ±20% or more for large compressions, ex of the thermodynamic properties given bj cept for the case of aluminum and iron, for 7 which high-pressure data for y are re Hultgren et al. We have used the en ported (Anderson et al.1^; McQueen et al.6). tropy calculated for high temperatures at 1 atm realizing that this value will pro The entropy increases upon being vide a moderate overestimate of the brought to the boiling point have only re shock strength required to deliver mete cently b.?en determined in the case of gas in a vacuum.
Flyer Plate Impact Velocities Required for Shock Vaporization
The highest flyer plate velocities will, of course, sharply reduce the achievable by the. suggested explosive total mass of the final flyei plate. system shown in Fig. 1 are in the The shock entropies that can be 4-5 km/sec range, depending ulti achieved by either a single-stage tech mately on the ratio of the mass per nique (in which a high-impedance flyer unit area of explosive to that of the plate, presumably tungsten or palladium, flyer plate. A modest increase in directly impacts the sample) or a simple velocity, to perhaps 6 km/sec, can system (using a driver plate of shock probably be achieved by using the impedance intermediate between the flyer multiple-staging technique described and sample) have bee i calculated and are 13 by Balchan. This latter technique discussed in the next section.
-7- Entropy Production in Porous Metals by Plane Shocks
Theoretical Hugoniots for various As can be seen in Figs. 3 through 9, initial sample porosities and the resulting it appears that, within the context of the entropies achieved with different flyer- Mie-Grdneisen theory, for a given flyer - plate systems are calculated using the plate material and impact velocity, an relations given in the previous section optimum initial porosity of the target with Fortran IV program ENT in the exists, such that the shock-induced en appendix. Although it is unlikely that tropy is a maximum. In some cases the aluminum flyer plates would be used for degree of initial porosity is critical (e.g., any optimum system, they are considered for aluminum and iron. Figs. 3 and 4) to because the shock st?te induced by a the achievement o! .artial vaporization. 2024 aluminum flyer plate, impacting at a given velocity, closely approximates ALUMINUM the free-surface velocity of an aluminum The results shown in Figs. 3(b) and driver plate that might be used in a real 3(c) demonstrate that the optimum vapor system. Hugoniot data for the flyer plate ization for 4- or 5-km/sec impacts is materials, fitted to a third-order polyno approximately m = 1.6 for iron and closer mial in the pressure-particle velocity torn = 1.7 or m = 1.8 for tungsten flyer plane are taken from the compilation of plates. However, only approximately 5% McQueen et al. vaporization results from the impact of a The entropies, shock pressures, and 5-km/sec tungsten plate into a material shock particle velocities achieved along for which m = 1.6. Calculations for the principal and porous Hugoniots for larger initial porosities should be carried aluminum, iron, thorium, uranium, out above the point where the Hugoniot be strontium, and lithium are shown in comes double valued. This is outside the Tables 2 through 8. Entropies achievable context of the Mie-GrQneisen assumption. in each of these materials with 2024 alu minum, iron, or tungsten flyer (or, of IRON course, driver) plates are shown in Figs. 3(a,b,c) to 9(a,b,c) as a function of Similarly, optimum initial distentions initial distention. Hugoniot data for pure required for vaporization for 4- to 5-km/ aluminum and pure uranium are inferred sec iron and tungsten flyer plates (see by correcting the data for 2024 alloy and Figs. 4(b) and 4(c)) are, respectively, U/Mo alloy (McQueen et al. ) for initial m = 1.6 and 1.7. Vaporization is not densily. Data for thorium is reported achieved with a 5-km/sec aluminum flyer 14 by McQueen and Marsh. Unpublished plate (see Fig. 4(a)). Approximately 10% data (McQueen and Marsh) for barium vaporization is achieved upon the impact and strontium was reduced by Rogers. 15 of a 5-km/sec plate into a material for The reduction of the data of Rice, which m ~ 1.7. The impact of an 8-km/ carried out by Grover, is employed for sec plate aill result in approximately 30% Xithium. vaporization.
-8- Table 2, Shock-wave parameters and thermodynamic states for pure aluminum. Principal Shock - Porous Distention Shock Hugoniot Debye particle Hugonlot ratio Compression pressure shock temp CrOneisen temp velocity Entropy8 shock temp (Mbar) (*K) ratio <°K> (km/sec) (cal/mol-°K> CK) a Values in parentheses were calculated by McQueen et al. (1970). BARIUM m = 1.2 distentions. Iron and tungsten flyer plates should achieve vaporizations Because of the large compressions of approximately 50% at these speeds for achieved for this metal, the GrOneisen distentions of m = 1.2 or greater. parameter uncertainties are large. This and the uncertainty in the thermochemical THORIUM data indicate that the present calculation should be accepted only on a very tenta The uncertainty in the boiling point of tive basis. Taken at face value, the cal this metal gives rise to considerable un culations indicate that 4- to 5-km/sec certainty about the entropy required for aluminum dyer p'.ites will partially vapor incipient vaporization. For a 5-km/sec ize this metal for both m = 1.0 and aluminum flyer plate the optimum distention 9- Table 3. Shock-wave parameters and thermodynamic states for pure iron. Principal Shock Porous Distention Shock Mugoniot particle Hugoniot ratio Compression pressure shock temp Grflnelsen velocity Entropy8 shock temp v (Mbar) <'K> ratio i| | (km,sec) (cal mol--'K) (=K> < oo/V 10- is approximately 1.8 (see Pig. 6(a), LITHIUM but this probably will not induce even partial vaporization. For 4- to 5-km/sec As evident from Fig. 8, lithium is iron flyer plates the optimum distention surprisingly difficult to vaporize with is 1.8 to 2.0, which will induce partial a plane-wave system. For the range vaporization (see Fig. 6(b)). Tungsten of distention up to a value of 1.6, plates fired at samples with distentions no flyer plate is capable of shocking greater than 2.0 will induce maximum, the material to even induce partial but probably only 5% vaporization (see vaporization; therefore lithium is Fig. 6(c)). another good candidate for Mach-stem vaporization. URANIUM STRONTIUM The Mie-GrOneisen equation sharply limits the present calculations to shocks Substantial quantities of strontium in a regime of lower entropy than re vapor can be produced by shocking either quired for partial vaporization. This initially solid or distention 1.2 samples metal is a good candidate for Mach-stem with 4- or 5-km/aec tungsten or iron vaporization, which is discussed in the flyer plates (see Figs. 9(a) and 9(b)). As following sections. Extrapolating the for the other compressible metals, the results of Fig. 7(c) suggests that a 4- or Mie-Grflneisen theory is not usable for 5-km/sec tungsten plate fired at a 1.8 greater porosities. Although the limited distention target may induce some par range explored suggests rather high po tial vaporization. This, however, is a rosities, m = 1.6 may be optimum for very tentative speculation. vapor production. Table 4. Shock-wave parameters and thermodynamic states for barium. Principal Shock Porous Distention Shock Hugomot Debye particle Hugoniot ratio Compression pressure shock temp GrOnetsen temp velocity Entropy shock temp -11- Table 5. Shock-wave parameters and thermodynamic states for thorium. "''"^ -- il l ** \Porou7 Distention Shock Hugoniot Debye Itugoniot ratio Compression pressure shock temp Gruneisen temp Entropy shock tem| ,V00 1.0 0.1950 0.200 667.0 1.000 208.7 0.592 14.86 667.0 1.2 0.7950 0.234 667.0 1.000 208.7 0.900 23.44 2814.9 1.4 0.7950 0.281 667.0 1.000 208.7 1.207 27.78 5835.1 1.6 0.7950 0.353 667.0 1.090 208.7 1.559 31.22 10394.2 l.a 0.7950 0.474 667.0 1.000 208.7 2.018 34.52 18070.6 2.0 0.7950 0.718 667.0 1.000 208.7 2.724 38.24 33722.5 1.0 0.7070 0.400 1577.0 0.890 220.6 1.002 19.64 1577.0 1.2 0.7070 0.473 1577.0 0.890 220.6 1.413 27.81 6204.7 1.4 0.7070 0.579 1577.0 0.890 220.6 1.853 32.17 12899.0 1.6 0.7070 0.745 1577.0 0.890 220.6 2.387 35.73 23441.8 1.8 0.7070 1.045 1577.0 0.890 220.6 3.128 39.27 42490.9 2.0 0.7070 1.752 1577.0 0.890 220.6 4.405 43.56 87302.6 1.0 0.6520 0.600 2708.0 0.820 226.1 1.337 22.72 2708.0 1.2 0.6520 0.715 270B.0 0.820 226.1 1.632 30.51 10006.8 1.4 0.6S20 0.885 2708.0 0.820 226.1 2.381 34.86 20771.9 1.6 0.6520 1.161 2708.0 0.820 226.1 3.069 38.50 38241.7 1.8 0.6520 1.685 2708.0 0.820 226.1 4.070 42.23 71512.4 2.0 0.6520 3.077 2708.0 0.820 226.1 5.959 47.02 159720.8 1.0 0.6140 0.800 3659.0 0.770 228.a 1.626 24.45 3659.0 1.2 0.6140 0.959 3659.0 0.770 228.8 2.193 32.33 13739.7 1.4 0.7140 1.196 3659.0 0.770 228.8 2.837 36.74 28805.8 1.6 0.6140 1.588 3659.0 0.770 228.8 3.662 40.46 53770.9 1.8 0.6140 2.366 3G58.0 0.770 228.8 4.901 44.34 103162.9 2.0 0.6140 4.631 3659.0 0.770 228.8 7.413 49.55 247173.2 1.0 0.5850 1.000 5091.0 0,740 231.1 1.885 26.36 5091.0 1.2 0.5850 1.207 5091.0 0.740 231.1 2.521 33.92 18133.8 1.4 0.5850 1.522 5091.0 0.740 231. i 3.259 38.33 31986.9 1.6 0.5850 2.060 5091.0 0.740 231.1 4.231 42.13 71868.1 1.8 0.5850 3.185 5091.0 0.740 231.1 5.756 46.22 142768.7 2.0 0.5850 7.021 5091.0 0.740 231.1 9.223 52.12 284434.7 1.0 0.5620 1.200 7239.0 0.710 232.0 2.121 28.43 7239.0 1.2 0.5620 1.454 7239.0 0.710 232.0 2.818 35.38 23256.8 1.4 0.5620 1.844 7239.0 0.710 232.0 3.638 38.68 47873.5 1.6 0.5620 2.521 7239.0 0.710 232.0 4.733 43.48 90554.1 1.8 0.5620 3.982 7239.0 0.710 232.0 6.496 47.66 182705.5 2.0 0.5620 U.470 7239.0 0.710 232.0 10.798 54.00 528892.8 1.0 0.5440 1.400 9666.0 0.680 231.8 2.338 30.16 9666.0 1.2 0.5440 1.697 9666.0 0.680 231.." 3.087 36.62 26574.0 1.4 0.5440 2.153 9666.0 0.680 231.6 3.972 40.80 57647.6 1.6 0.5440 2.944 9666.0 0.680 231.8 5.159 44.5: '98098.6 1.8 0.5440 4.656 9666.0 0.680 231.8 7.076 48.70 217213.4 2.0 0.5440 11.122 9666.0 0.680 231.8 11.775 55.04 629424.1 1.0 0.5350 1.500 10980.0 0.670 232.1 2.444 30.91 10980.0 1.2 0.5350 1.822 10980.0 0.670 232.1 3.221 37.18 31460.5 1.4 0.5350 2.320 10980.0 0.670 232.1 4.145 41.33 63132.3 1.6 0.5350 3.192 10980.0 0.670 232.1 5.395 45.09 118616.8 1.8 0.5350 5.114 J.0980.0 0.670 232.1 7.442 49.31 240946.7 2J0 0.S350 12.862 10980.0 0.670 232.1 12.702 55.95 733929.4 -12- Table 6. Shock-wave parameters and thermodynamic states for pare uranium. Principal Shock Porous .distention Shock Kugoniot Debye particle Hugonlot ratio Compresaion preasure ahock temp Gruneiaen temp velocity Entropy* •hock temp aValuea in parentheaea were calculated by McQueen et aL (1970) for 97/3 uranium/m olybdenuro alloy. -13- Table 7. Shock-wave parameters and thermodynamic states for lithium. Principal Shock PorouB Distention Shock Hugontot Debje particle Hugoniot ratio Compression pressure shock temp Grflnelsen temp velocity Entropy a'iock temp (Mbar) CK) ratio CK) (km/sec) (cal/mol-°K) <°K> Table 8. Shock-wave parameters and thermodynamic states for pure aluminum. Principal Shock Porous Distention Shock Hugoniot Debye particle Hugoniot ratio Compression pressure shock temp GrQneisen temp velocity? Entropy shock temp (Mbar) ratio (km/sec) (cal/mol-°?\) CK) ^oo/V -14- (a) Aluminum flyer plate -J I I l_ U 0.6 0.8 1.0 fc— 0.6 O To RELATIVE VOLUME (V/Vj RELATIVE VOLUME (v/v0) Fig. 3. Hugoniot curves for aluminum showing various initial distention ratios, no, on the entropy- relative volume plane. The sec ond set of curves represents the loci of states attainable at differ ent flyer plate speeds. The points indicated by a solid dot (•) represent the states behind the Mach stems formed by the colli sion of two shock waves (indi cated by +) at the indicated angles of incidence. O0.0| L. J I L. "•* 0.6 0.1 , 1.0 RELATIVE VOLUME (\pV„) •15- .* Voyocitotion Voporliotkm i InMtval I* fnftrval , Mafh'ng Wvrval J 2CLC- 1 RELATIVE VOLUME (V/V0) RELATIVE VOLUME {yAfa) rt-U «ao Valorisation - Fig. 4. Hugoniot curves for iron showing \{e) Tungsten Flye r plate various initial distention ratios, m, on the entropy-relative vol ?• ^n*LA \^ ume plane. The second set of • curves represents the loci of - states attainable at different flyer A plate speeds. The points indi t^ cated by a solid dot (•) repre 5 lun/I««^<^^c/s^ v^s/ ^\. Mtltiftfl InWrMl sent the states behind the Mach v stems formed by the collision of 2»«i/»«c^\^ two shock waves (indicated by +) at the indicated angles of ' incidence. (6 Phii«) oac ' • _ I 1. J _ -J 1 RELATIVE VOIUME (VVJ -16- MO 1 > "T • 1 1 1 • 1 1 "1 ™ (o) Aluminum flyer plate 4O0 " \>—-5fcn/M£ " kitorvo! Mtlttng InWvol r^llun/MC 2O0 - " - I i.i 1 1 RELATIVE VOLUME (V/VJ 4O0 ~7 T 1 1 J— 1 1 J 1 1 n t t •n (b) Iron flyer plate 4O0 ' Vapor iwrtion^^ ~ . ^^—4fciiyi«e tntirvof ^><. 1 MtMng htovoJ Mtlrfnq tnl** i i ,_| i i i RELATIVE VOLUME (Wo) 6UD j , , j T T- r i r J ••• i "' i .m •12 " F (c) Tungsten flyer plate —5km/MC '§400 <5^*—— - w^K»lwHofissw ^ i llrmilrr 1 Interval Mailing Inhtnat '2^J -v—*** i -' - IZU ' , , - 1 — 1 —1 i i i i i i i . RELATIVE VOLUME (VAo) Fig. 5. Hugoniot curves for barium showing various initial distention ratios, m, on the entropy-relative volume plane. The second set of curves represents the loci of states attainable at different flyer plate speeds. The points indicated by a solid dot <•) represent the states behind the Mach stems formed by the collision of two shock waves (indicated by +) at the indicated angiss of incidence. -17- ood L- RELATIVE VOLUME (V/V0) RELATIVE VOLUME (V/VJ « 1 1 1 1 76.0 <•» Fig. S. Hugonlot curves for thorium .(c) Tungsten flyer plate- showing various initial distention ratios, m, on the entropy- 4O0 relative volume plane. The sec ond set ol curves represents the loci of states attainable at differ ent flyer plate speeds. The points indicated by a solid dot (•) TOO represent the states behind the Mach stems formed by the colli sion of two shock waves (indi cated by +) at the indicated a.'^les i i of incidence. RELATIVE VOLUME (yV0) -18- -[ 1 1 1 1 I 1 1 -i 1 1 r ti.O 46.0 t t ' Vo»oi.tti>OA IMtoal Q ( ) Aluminum (b) Iron flyer plote flyer plate i"| M*ttni| int*ri*t 1 RELATIVE VOLUME (V/Vol RELATIVE VOLUME (V/V„) (c) Tungsten flyer Fig. 7. Hugoniot curves for uranium plate showing various initial disten tion ratios, m, on the entropy- relative volume plane. The sec ond set of curves represents the loci of states attainable at differ ent flyer plate speeds. The 4 points indicated by a solid dot (•) represent the states behind the Mach stems formed by the colli sion of two shock waves (indi cated by +) at the indicated angles of incidence. RELATIVE VOLUME (Wo; -19. ~i 1 1 1— 1 7 r- (o) Aluminum flyer plate F3 i RELATIVE VOLUME (V/%) REUTIVE VOLUME (V/VJ ru t r (c) Tungsten flyer plate Pig. 8. Hugoniot curves for lithium allowing various initial disten tion ratios, m, on the entropy- relative volume plane. The sec ond set of curves represents the loci of states attainable at differ ent flyer plate speeds. The points indicated by a solid dot (•) represent the states behind the I Mach stems formed by the colli sion of two shock waves (indi cated by +) at the indicated angles of incidence. RELATIVE VOLUME (V%l -20- Y? .ifctfW*!* 1 ' 1 1 • 1 1 1 1 T - •••1.2 40J M-M r-,B\ - Iron flyer pi are laMrool - 30C 1 biirtK^\ N - Mailing Wvtvl Mi " Fig. 9. Hugonlot curves for strontium showing Wfi - various initial disten tion ratios, m, on the • or - entropy-relative vol ume plane. The second < ' 1 1 1 L 1. —i 1 j 0.2 0.4 set of curves repre RELATIVE VOLUME (V/V©) sents the loci of states attainable at different flyer plate speeds. The points indicated by 1 | r T 1 I a solid dot (•) repre sent the states behind the Mach stems formed by the collision of two 401 shock waves (indicated by +) at the indicated (b) Tungsten _lrVo.ia.I-w angles of incidence. flyer plate - XX • ,^\-' - i - 1 1 im^>C Nv Matrina biMnal - Ul **-, 10uC - 1 1— 1 L. 1 i I i i RELATIVE VOLUME (V/VJ -21- Mach Stems In order to calculate the entropies that the original plane shock wave. may be induced, particularly in lithium The theory of oblique shock waves for and uranium, upon propagation of Mach gaseous media is fully described in a 1 7 stems, we have initially investigated the classical text of Courar.t and Friedrichs, criteria for introducing this type of flow and the description of irregular reflection in all the metals of interest. For the in solid bodies was carried out by present we have considered these metals 18 only at the single crystal density. Al'tshuler and his co-workers. The ap plication of Mach stems to the study of Upon the collision of two shocks at a properties of materials under extremely slightly oblique angle, the pressure be high pressures has been discussed by hind the reflected shock will increase 10 Leygonie and Bergon and more recently slightly as the incidence angle, a, in 20 by de Beaumont and Leygonie. In the creases. At a certain critical angle, o , latter paper the authors describe produc regular reflection will no longer occur, ing Mach stems by a convergent conical and a Mach stem will be formed similar shock wave in copper and using the cen to that shown in Fig. 10(b). With the tral portion of the Mach stem to impact formation of a Mach stem, the pressure as a strong plane wave upon a small pel produced will increase discontinuously to let of uranium sample. This is a candi a new high value. Upon further increase date, but unexplored, configuration that in a, this pressure will rapidly decrease might be used in proposed experiments until at o = 90 it will reach the value of in this country. (o) Fig. 10. Oblique intersection of shock waves: -22- Oblique Shock Polars The present analysis is carried out and graphically. The method was initially developed for the solution of problems in 6 = e„ - flj . (18) volving the interaction of oblique discon- tinuiHes in gaseous media 21 and it has Here p, p, u and u denote the usual 22 pressure, density, normal shock velocity, also been applied to liquid explosives. and normal particle velocity; a and b rep For solid bodies a similar technique was resent the two constants in the linear developed by Laharrague et al. for u - u relationship in solids (McQueen problems involving shock-wave refrac et al.°); w is the oblige shock velocity, tion from a boundary between two differ while 0 and 6 denote the oblique angle of ent materials. approach and the angle by which the flow In contrast to the other polar tech was deflected after going through the niques known in the literature, the oblique shock discontinuity. The sub vector polar method uses a logarithmic scripts 0 and I indicate the undisturbed scale for the pressure ratio, which per and shocked states, respectively. mits not only a more compact set of These equations were found to be polars but also graphical addition of pres easier to manipulate if they are cast in a sure ratios across various components nondimensional forme This is done by of the wave system. using p0v0 as a normalizing parameter The basic theory describing the flow where v- is &e initial specific volume; across an oblique discontinuity in a solid i.e.. vQ = l/pfl. With v «• v/vQ and material is illustrated in Fig. 11 and is P = p jpv the first four equations are governed by the following set of equations: written as follows: tt8"*»l(u._ V' (14) (1 V) U (14a) p0 = Po "s V (15) V " 8' (P-D = usup. (15a) (16) Us = a0+b0V U =A +bl (18a) u = w sin 8 , (17) S 0 V and U = W sin 0. s un (17a) In contrast to the dimensional form, all nondimensional quantities are represented as capital letters. Introducing a new parameter. Fig. 11. Geometrical diagram of the flow l across an oblique shock wave. *^k" k -«- one can derive the Mach number, which r.u - v) (P- 1) = (19) is a well-known parameter in gas U -Ml - v'f? dynamics: (P - 1)= (1 - vjr.M2 sin2 0, (20) w2 W2 _ k _ k k = i.j, "4- _ (P- 1) cot e (2i) 7 r M i i' - (P- i) where i and j indicate conditions before and and after the wave. In solid materials Mach number has little meaning, since it r.M2 - (P- 1X1 + v) 2 _ i i is normalized by a bulk sound velocity of M (22) J rjp" the medium; nevertheless it is a very useful quantity for this analysis. The solutions to these equations are With these parameters one can derive plotted in the form of polars on the P - 6, the following equations: 0-6, and P - M. planes as shown in 6.4 • i i ~ -i 1 1 r -i 1 1 r i ' ' ' „J 20 «'ig. 12. Oblique shock polars for strontium in the pressure ratio-deflection angle plane. -24- 7U ^^r- T 1 —i 1 1 r i 1 1 —i 1 J— • | . —r i i 80 — - 70 - 60 ) - m 50 - 40 - . ^-3.0 30 -M = 2.5 -M=2.0 • 20 \ *-M = 1.75 »-M = 1.5 in • i . , 1 . , . l 1 1 L A . 1 . . ' ' 10 15 20 6 Fig. 13. Oblique shock polars for strontium in the incident angle-deflection angle plane. Figs. 12, 13. and 14. respectively. The and deflection angle represent the two incident Mach number of the wave was most interesting parameters of the flow. taken as an independent parameter, and The reflected shock waves that prop the plot shows polars for several values agate into a precompressed region are of M,. For a solution of a problem, all calculated in the same manner using the three planes are equally Important and same set of four Eqs. (18)-(22), the only must be used simultaneously—even though difference being that M. and r. corre only one P - 6 plane is considered to be spond to the new initial conditions of the the main solution plane since pressure state into which the wave propagates. -25- -i—i—I—i—i—i—l—I—i—r 5- g> 3 2- 0' ' ' ' I ' i_J i I i i i_ M. J Fig. 14. Oblique shock polars for stron tium in the pressure ratio- downstream Mach number plane. Critical Angle Solving far the critical angle at which the ordinate (i = 0) or the original shock regular reflection becomes irregular, polar (* > 0) will determine whether re one first assumes the strength of the flection is regular or irregular, respec shock wave that is to interact with its tively— i.e., whether the Mach stem has identical couterpart at an oblique angle a. formed during this reflection. As evident The same wave inclined at different angles from Fig. IS, there is a range of cases will have the same r., but different M,, where reflected shock intersects both, and therefore different M.. Reflected and this range corresponds to the two- shock polars are computed and plotted solution region, one of which, the one on the P - 6 plane with their origin placed with the lower pressure, is more likely at proper initial states as illustrated in to occur. When the reflected shock polar Fig. 15. For example, let us assume becomes tangent to the ordinate, the angle two shock waves propagating through at which this occurs is said to be critical, strontium at a velocity of 3.18 mm/usec because further increase ot the incident with a pressure Pj « 100 kbar. The inter angle will only produce irregular reflec section of the reflected polar with either tion with a Mach stem. -26- 20 Fig. 15. Oblique shock polars for incident and reflected waves in strontium. Upon solution for the critical angles, merely a simple triple wave intersection, it became evident that there are several but rather a more complex one involving ways a transition from a regular reflec additional compression along the Mach tion to an irregular one can occur. The 3tem, as Illustrated In a qualitative man type of transition depends not only on the ner In Fig. 16. material that is used, but also on the ini Type II is probably the most common tial strength of the interacting shock type when at the critical angle the re waves. The three types of solutions are flected polar also crosses the original one illustrated in Fig. 16, where all three at a certain 6 > 0. Although in this case cases show the secondary shock polar to the solution calls for a distinct three- be tangent with the 4=0 ordinate. wave configuration at the origin of the In type I, the entire reflected shock Mach stem, the flow behind it Is not polar lies below the original polar without entirely uniform and must adjust Itself crossing it at any point. This failure to from a slightly oblique wave at th» axis have a common point between the two of symmetry. polars other than the origin of the re Type III is special; the Mach stem Is flected one indicates that the case is not already being formed before the critical -27- logP IQSP logP Type I Type II Type HI « Fig. 16. Three types of Mach configurations of critical angles. angle criterion has been achieved. In with the simple high-explosive techniques this case it is possible to find an angle at described in the first part of this report. which a perfect Mach stem is formed that Some of the materials show similar is normal to the axis of symmetry over tendencies with an increase in shock pres Its entire length. sure in that the critical angle decreases The variation of the critical angle for asymptotically to a value around 35°. the metals of interest with initial strength Iron, uranium, strontium, and barium of the shock wave is shown in Fig. 17. behave somewhat differently in that the The ranges of investigation for various first two decrease to asymptotes materials are dictated by the relative around 25° and 28°, respectively, and impedance of the materials and by our the latter two show an increase in the ability to attain the necessary pressures critical angle. -28- Mach Stems in Aluminum, Barium, Lithium, Strontium, Thorium, Iron, and Uranium The maximum pressure that can be shock, critical conditions may be those attained behind the Mach stem is found resembling all three cases. Figures IB from the intersection of the original polar through 24 illustrate this effect for each with the ordinate marked as P_ in Fig. 16. material separately. In each figure P , m " P^ , and P are normalized with P, and P and P , respectively, represent cr nr 1 pressure behind the regular reflection at plotted against that value. For the ranges the critical angle and pressure behind the shown, aluminum is found to yield types I normal to the reflected shock. and n, lithium and strontium will both Comparing pressures Pm, Pcr, and give type3 n and in, and thorium aeemt F at the critical angle, one could see to cover the complete range of type II, that all materials behave differently and, while iron and uranium will only yield depending on the strength of the initial type I and barium only yields type ilL Initial thoek strength — P, x 10 Fig. It. Critical angto Fig. 18. Critical pressure ratioa vs initial shock strength in aluminum. Initial shock strength — P, X 106 Fig. 19. Critical pressure ratios vs initial shock strength in barium. -SO- 1 1 _,.„. 5-- a. a. cr""""""""^ II III 1 i 0.2 0.4 0.4 6 Initial shock strength — Pj X10 Initial shock strength — P] x 10 Fig. 20. Critical pressure ratios vs Fig. 21. Critical pressure ratios vs ini initial shock strength in lithium. tial shock strength in strontium. Initial shock strength — P. X 10 Fig. 22. Crtical pressure ratios vs initial shock strength in thorium. Also of interest is the effective in haves differently and increases to over crease of pressure with increase in the an eight-fold pressure jump at P., = 10 6 initial shock strength. While in most (pt - 1 Mbar). Thus, even though the maximum Mach stem pressure P is materials the ratio P„/Pm' 1, decreases with P, or stays nearly constant (as in alumi found in iron, its rather strong display num and uranium), in iron it again be- of type I diminishes the effectiveness of -31- Type I 4r f I1 0.5 1.0 Initial shock strength — Pj * I06 Pig. 23. Critical pressure ratios vs initial shock strength in iron. * clewt Mach stem mid Us potential in flection to a Mach -item In types I and II, vg&M*?, '.log a quantity of this material. in type HI the Mach stem solution takes «3»r « the present graphical technique over before regular reflection meets its o«e t^en also determine the range of pres- limit, and therefore no pressure jump sww* attainable in a certain material with occurs in this case. Maximum pressure a jMsi»U ular shock strength as the angle Is always obtained near the critical con cf in>orietion is varied. Assuming that dition. The further the angle is from the tfcte nf.^t realistic solution is the one that critical value, the lower is the pressure "'i'C.v.s*-'' the lowest pressure, one can that is obtained from the reflection. Thus, «\ft?r.".he whole range of angles that yield In order to get maximum compression of a possible solution. For the purpose of a material, one must operate very close illustration, such Analysis h*» been car to the critical angle. ried out for three different types of crit To illustrate how much entropy is ical reflection, and the results are shown gatotcf by a Mach stem, an example is in figs. 25, 2«, and 27, While there Is worked out for a series of collision angles a big pressure jump from a regular re- for each material. In each case the initial -32- Initial shock strength — P, X 10° fig. 24. Critical pressure ratios vs initial shock strength in uranium. -5 »- Fig. 29. Variation of the reflected preeaure with Incident angle S in aluminum when Pj • 300 kbar, values of 6 taken from Ref. 18. -33- Ju40 a m I ma -r Angle of Incidence, 9 Fig. 26. Variation of the reflected preeaures with incident angle e in aluminum when J_L° 60 70 80 -34- shock was taken to be of different strength, a Mach stem design in getting a consider but always one that was attainable with the able gain in entropy, but it is also evident same explosive-in-contact geometry. that an "in-contact" geometry, even one In Figs. 3(a)-9(a) these initial shock as strong as the octal-dural 400 system strengths are indicated by a cross, and used in these examples, will not suffice the final states behind the Mach stem are for vaporizing material upon release, and indicated by arrows with corresponding for all materials except strontium and values of the incidence angle. The results barium a stronger shock generating sys clearly indicate that one can profit from tem is necessary. Estimate of the Mach Stem Size Before using the above criterion for actions, as is indicated by the dashed practical application, one must consider curves on Figs. 25 and 26, which were one more factor of the problem: the plotted through experimental points taken amount of material that can be compressed from Ret 18. It seems feasible that with to such high pressures; i.e., the amount stronger shocks one could Increase the of material affected by the Mach stem as else of the Mach stem to compress a no compared to the total amount of material ticeable amount of the material. used. As an estimate of how much material Geometrically speaking, with reference one can expect to compress with a Mach to Fig. 1(b) one would be interested in stem at higher initial pressures, one can knowing how large an angle Y one can get use the experimental data of Ref. 18 during such an interaction. In the present shown in Fig. 26. At a = 45* we find that analysis we do not attempt to predict the Y » S*. and there still is a significant angle Y. We expect that tilts calculation pressure Increase of F /Pj • 2.3. Such and/or experiment could be profitably in a system is represented geometrically in vestigated in the future. Fig, 28; from the figure it follows that the From the experimental data available amount of material affected by the Mach in the literature, one finds that at critical stem as compared to the total is conditions the angle Y is extremely small and increases very slowly with deviation y Ltan45* ' w* from the critical angle 0. Therefore, the If other materials behave in a similar way, gain in compressed substance is a trade then with stronger Initial shock one can off to the amount of compression, and the expect the Mach stem to cover 10-20% of practical problem of inducing large quan the original material. tities of vapor becomes one of determin In the above analysis we have only de ing the optimal geometry. If one could scribed a system in which two plane speculate on the basis of data available, waves interact at an oblique angle <*. It the angle Y grows much more rapidly with is also possible to obtain Mach stems in stronger shocks than with weaker inter other configurations where theoretical -35- tion wave propagates along the high ex plosive, and a conical shock wave is In duced in the material. A Mach stem forms in the sample cylinder near the apex of the cone. In this case the angle of the cone will originally depend on the relative impedance of the materials, but it can also be varied by reshaping the material from a straight cylinder to a cone. Due to the three-dimensional character of this problem, one can pre Pig. 28. Geometrical representation for sumably improve on the efficiency of the the estimate of efficiency. system. In this geometry the maximum pressure behind the Mach stem will be limited by the strength of the original analysis is more difficult. Another such shock, which can be generated either by system has been described by Fowles and an "in-contact" explosive or by means of lsbell24 in which a cylindrical block is a conicaUy or cylindrical^ imploding surrounded by high explosive. A detona flyer plate. Conclusions In the present study we have examined Both plane-wave and Mach-stem shock the shock states that may be produced in systems are capable of inducing vaporisa various metals with available explosive tion of aluminum, iron, strontium, barium, systems. The entropies produced by and thorium. Only Mach-stem shocks are shocks In porous and solid samples and capable or vaporizing lithium and possibly the entropy required for vaporization are uranium. Plane-wave techniques, because calculated, and the comparison Is used as of their inherent efficiency in transferring a quantitative guide to estimate the degree internal energy to a large quantify of of vaporization for a given shock strength. metal, appear to be the most promising Systems that optimise the trade-offs be techniques for use on the first four matals tween the shock-Induced particle velocity, even though substantial vaporization prob the maximum specific entropy Imparted ably cannot be achieved, except possibly by a shock, and the amount of material In the case of strontium and barium. that receives sufficient entropy to induce Specifically, we find for the plane- vaporization could be designed using wave case that for a given explosive sys as references the present analyses tem, such as a flyer-plate launching de of shock dynamics and the calcula vice, a certain initial porosity provides tion of shock-induced thermodynamic the optimum conditions for entropy pro state. duction, for the case of aluminum, -36- distentions of 1.6, 1.7, or 1.8 are optimum pressures that are easily attainable by for 4- or 5-km/sec iron and tungsten flyer conventional high explosive systems, plate impacts. For iron the optimum dis barium, lithium, and strontium will prop tention of the sample la about 1.6 or 1.7. agate clean Mach stems. Thorium and For thorium, initial distentions of "2.0 aluminum will propagate Mach sterna for will probably induce maximum, but in shocks above 400 and S10 kbars, respec complete, vaporization. As a result of tively. Iron and uranium, although behav the investigation of plane-wave shock ing in a similar fashion, do not readily vaporization, we concluded that both ura form a clean triple wave Intersection, nium and lithium required convergent, Additional compression takes place be and possibly Mach-stem, flow to in tween the reflected wave and its Mach duce significant vaporization. In stem for these metals. this latter geometry, a fraction iwe Pressurewise, iron and uranium are estimate 20%) of the sample can be the most interesting materials, since vaporized. Mach stems can produce pressures 6 to Investigating the conditions that are 8 times those of the initial shock. Alumi required to induce Mach stems In the num, lithium, strontium, and thorium same materials, we found that the colli yield four- to five-fold increases In shock sion of two shock waves at an oblique pressure, while barium gives no more angle does not always result in a clean than a three-fold pressure Increase. The Uach stem. The Mach-stem formation angles required to induce a series of spe depends on both the equation of state of cific entropies for various Mach-stem the material and on the strength of the geometries are presented In graphical initial shock. Within the range of initial form for system design proposes. Acknowledgments We appreciate the discussions with of the thermodynamics of the liquid-vapor R. H. Keeler. E. B. Royce, and N. region and F. Rogers' help in obtaining Rosenberg (AFCRL) concerning this prob data for barium and strontium are lem. D. Young's help with our analyses appreciated. -37- References 1. Y. B. Zeldovich and Y. B. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena. Vol. II (Academic Press, New York. 1967). 2. M. H. Rice, R. G. McQueen, and J. M. Walsh. "Compression of Solids by Strong Shock Waves." Solid State Physics 6. 1063(1958). 3. J. M. Walsh and R. H. Christian, "Equation of State of Metals from Shock-Wave Measurements." Phys. Rev. 97. 1544 (1955). 4. R. Grover, R. N. Keeler, F. J. Rogers, and G. C. Kennedy, "On the Compress ibility of the Alkali Metals," J. Chem. Phys. 30. 209! (1969). 5. G. T. Furukawa and T. B. Douglas, "Heat Capacities," in American Institute of Physics Handbook. D. E. Gray. Ed. (McGraw-Hill, New York 1963). pp. 4-47. 6. R. G. McQueen, S. D. Marsh, J. W. Taylor. J. N. Fritz, and W. J. Carter. "The Equation of State of Solids from Shock Wave Studies," in High-Velocity Impact Studies. R. Kinslow, Ed. (Academic Press, New York, 1970). p. 294. 7. R. Hultgren, R. L. Orr, P. D. Anderson, and K. K. Kelly. Selected Values of the Thermodynamic Properties of Metals and Alloys (John Wiley & Sons, New York. 1963). also revision (1970). 6. R. C. Weast, Handbook of Chemistry and Physics (The Chemical Rubber Co., Cleveland. 1969). 9. D. R. Stull, JANAF Thermochemlcal Tables. The Thermal Research Laboratory, Dow Chemical Corp., Midland, Mich.; Clearinghouse for Federal Scientific and Technical Information. Washington, D. C, PB 168-370 (1965). 10. D. Young and B. Alder. "Critical Point of Metals from the van der Waals Model." Phya. Rev. A3, 364 (1971). 11. D. J. Andrews. Equation of State of the Alpha and Epsilon Phases of Iron, Ph.D. Thesis, Washington State University (1970). 12. G. D. Anderson, D. G. Doran, and A. L. Fahrenbruch, Equation of State of Solids; Aluminum and Teflon. Stanford Reeearch Rept., Air Force Weapons Laboratory Technical Rept. 65-14M (December 1965). 13. A. a BalchanandG. R. Cowan, "Method of Accelerating Flat Plates to High Velocity." Rev. Sci. Instr. 35, 937 (1964). 14. R. G. McQueen and S. P. Marsh. "Equation of State for Nineteen Metallic Elements from Shock-Wave Measurements to Two Megabars." J. Appl. Phys. 31. 1253 (1960). 15. F. Rogers, Lawrence Livermore Laboratory, private communication. 16. M. H. Rice, "Pressure-Volume Relations for the Alkali Metals from Shock-Wave Measurements," J. Phys. Chem. Solids 26, 483 (1965). 17. R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves (Interscience, New York, 1948). -38- 18. L. V. Al'tshuler, S. B. Kormer, A. A. Bakanova, A. P. Petrunin, A. L Funtikov, and A. A. Gubkin, "Irregular Conditions at Oblique Collision ot Shock Waves in Solid Bodies, Sov. Phys. JETP 14 (5), (1962). 19. J. Leygonie and J. CI. Bergon, "Oe'termination du Coefficient de Grttaelsen en Function du Volume Specifique par L'etude du Phenomene de Reflexion de Macb," Proc. Symposium on Behaviour of Dense Media under High Dynamic Pressure (International Union of Theoretical and Applied Mechanics, Paris, 1967) p. 160. 20. PA. de Beaumont and J. Leygonie, "Vaporizing of Uranium after Shock Loading," Fifth International Symposium on Detonation (Pasadena, Calif., 1970). 21. A. K. Oppenheim, J. J. Smolen, and L. J. Zajac. "Vector Polar Method for the Analysis of Wave Intersections." Comb. Flame 12 U), 63 (1968). 22. A. K. Oppenheim, J. J. Smolen, D. Kwak, and P. A. Unlaw, "On the Dynamics of Shock Intersections," Fifth International Symposium on Detonations (Pasadena, Calif., 1970). 23. P. Laharrague, J. Morvan,andJ. Thouvenin, "Refraction d'une Onde de Choc," and Proc. Symposium on Behavior of Dense Media under High Dynamic Pressure (International Union of Theoretical and Applied Mechanics, Paris, 1967). 24. G. R. Fowles and W. M. Isbell, "Method for Hugcniot Equation-of-State Measure ments at Extreme Pressures," J. of AppL Phys. 36 (4), 1377 (1965). -39- Appendix ENT, a Fortran IV Program for Calculating Entropy and Shock States in Solid and Porous Metals I case = control No. ; for I case = 1 or 2 continue for I case = 3 stop IP = No. of pressures along principal Hugoniot SUBST = material being calculated THETO = Debye temperature. STP GAMO = Grdneisen ratio, STP MBAR = mean atomic weight RHOO = density, STP PH(I) = principal Hugoniot pressure TH(I) « principal Hugoniot temperature UP(I) = principal Hugoniot compression GA(I) = principal Hugoniot/GrOneisen ratio I = index point on principal Hugoniot J = index of porosity M = distention ratio P(I, J) « shock pressure U(I,J) = particle velocity S(I,J) = entropy TPOR(I,J) = temperat .re porous Hugoniot FEUP = polynomial coefficients for shock particle velocity—pressure (U - P) relation for Iron WUP = polynomial coefficients (V - P) for tungsten ALOP = polynomial coefficients (U - P) for 2024 aluminum FEU - shock particle velocity points, Hugoniot FEP = shock pressure points, Hugoniot ! -40- Jj -U£HI JQH_I 645i.6JtJA^SLUiO:Hfl«AS J^_AHR£NS.'- // SET PRT«6 ~U S£X—PLTaftO // EXEC FORTG nOFORT -DO- ».. PFIX,AI,A»,A3.A4>»A1+ A2*X + A3*X*X + A4*X»*3 aou»Le-BHCc i s i on -STOR.IA+4-U-- OIHENSION FEU!8),FEP(8),DATAI3.8>,FEUP<4»,WUPl4ltALUP(4> l.FT (1.4) ,Th i } i Wft I iFPt^t iP<9 ift) • tVVl1 DIMENSION VPP(6,5),STT(6,5) DIMENSION . -SUB«T444 »V>W>.,W|.») .GAIP) .THIS l,S(9,63,TP0R(9,6>,UI9c6),Xll60l ,00(3)*SO(9) .0ATI3.10I DATA UIIP/ -»nooAF-i i f -7A7aaen. .g->A«ncnt-_a*fc07c-i / DATA ALUP/-.6378E-2,,15459E0,.36446E-l.-.242876E-3/ . DATA—Cem0.^.5S..9S.1.29,1.S8.1.818.2.4*.2.OT/ — OATA FEP/ 0...2,.4,.6,.8,1.,1.5,2.0/ C« 4.18767- - - REAL NBAR ,H 1 FORMAT C 24-1. AA4I- i^_i_ 2 FORMAT!IX,4A4) 3- FORMA T-L4F4A.5-I . • •"'" * FORMATClXt'THETO = <,F9.4*>GAM0 « •,F9.4,'MBAR « •,F9.4,'ftHf 10 « ..'»F9.4) 1 -/"^V't 5 FORMATI4FI0.5) •"'."%;/ A FORMATUXWX*' PH'., 19JUJJU '...19X .J.VP »,19X.•GA'I $ 1 ,- i- f 7 F0RKATI1X.4F20.10I . 8 cn»»HTiiYT6Y.ii.a».ii.o«.M.i.o». CA.A.^V,.«:«• _*.»v. i7.1»M,« 1F5.3<6X ,F5.1,6X,F5.3,6X,F5.2,6X,F9.1) ^.;ft 9 FORMAT! IX,5X.' I' ,9X,'q'.,aX,JH-'-»9X, ' VP ' ,9Xt' P?„ 1«GA*,9X,'TE',9X,'U',9X,« S •,9X,«TP0R«) DO 90 I *- 1,160 90 XI I) « FLOATtl-l)*.l C ICASE3l,GAM(V);ICASE*2.f.GAMQ-aNU* 2000 REA0I5.1) ICASE,IPtSUBST DDIDcSUBSTIl) . _ - .... ._ 00<2I»S06STI2) WRITE!6t2)SUBST _. IFIICASE.GT.2t GO TO 1000 READI5.3) THETO. GAM.Q.-KBA&iRHOO WRITE! 6,4)THET0, GAMOt MBAR.RHOO READ!5,5) (PHI I) ,TH( I) tVPI U.,GA( I) 11 = 1,IP) WRITE! 6, 6* WRITEI6,7>IPHI1>,THU),VPU>,GA{L),I«1,1P) _ DO 50 1=1,IP P(Itl)-PHlI) . . .. UII,1»«=(I(1./RHOO-VPII)/RHOOI*PH(I))**.5 1*10. IF!ICASE.EQ.I) GO TO 80 TEI11» THETO*EXPIGAMO*ll.-VPII) >) • GO TO 81 ...... 80 TEII1 * TKETO*EXP(GAII)* I l.-VPI ID) 81 CONTINUE . ... _ _ IFIICASE.EO.l) GO TO 56 GAI1)* GAM0*P.H00*VPtI)/RH00 56 CONTINUE TPI1>* TEID/TH-II) ...... T*TPII) JF.{TP(1).U..1> GOTO 51 C CALL ENT(T,ES) C Sll,l)« ES*C SII,l)-24.94293/4.186*IAL0G 1 +T*<5J.94299E-4jLt*J.-.7,.6243.e-4+r.*J.9.«3004E-5-T*3..8A3.4I£-6-L' -41- JJP_Tg_5jL. 51 S(Ifl»»21.65 • AL0G(THII>/10./TEM)>*1.9a6»3. «» r-nuTtmiP ... _ M-l. T£ORU,li»-TH 0P»PH(I>«CH-!.J/VP H« 1.* FL0ATIJ-1)*.2 HRITE<6,8| ItJtWtVPlU-t - 1 PCI,J) , THII>,GAtI).TE(I)tU(I,J)tS(I,J)»TPOR< .21.JL. 71 CONTINUE 70 -CONTINUE VO 8011=1,8 MTAUiH -,.FEU.n) _.. 0*TA(2t!> - FEPII) niTA(3,X) --^1 ... 801 CONTINUE DATAI3.1) - .0001 10^ FORMAT! IX,"FEUP ARRAV/4E20.7) ran LsoiiAR tnATA.a.a.FFUP.CH.STnRi WRITEI6,10)FEUP on *i '-!•* ... . FTI1,1)«AUIPM) PTI?.1).FFIIPIT> . . 61 FT! 3,li-HUP!I) nn 701.1.1..IT ._ IPP = IP+1 DATS(1,IPP)"T U "~ DATSI3.IPPI = .onm ... „ DATIltIPP) = 0. -DA.TlZtIPP) * 0. 0AT(3,IPP) * .0001 — DO-Jll. J. g..l,IP. . .. DAT(1,II » U(I,J) _ joAuadtx-= jgjjj J ). 0AT(3,I) * .0001*FLOAT(I) - PATSUTlL^^PiJi .. _ DATSI2,!) = S(I,J) 0ATSI3^U..?L. .0001 *FLOAT(I) 711 CONTINUE CAJJ LSflUAB{OAT,lPP,4,TM.CH.,ST.0R.) -42- .. CALL . LSfliiASlQAlSjJJ'.P^A.Syj.CiU.STDRJ.._ 00 712 1= 1,4 ... — - TUMI UJ t- =_IHJ I ) 712 SVVd.J) = SVU) -7fll CONTINUE - - CCCCC FLYER PLATE TYPE . ... 00 9011=1,3 . _ DO. 91 J=l,4 . 9L. FP1JJ =.FJ.U.J1. - -. CCCCCCCC POROSITY DO 9.3- K=1,JT ... DO 95 N = 1,4 TMINJ =_T MM LN »*.).. _.. 95 SVIN) = SVV(N,K> ILL=. ID^mPil&X XMN = FLOAT! ILD/10. -.1 DO 156 NN= 1,1 P. - — ..-. 156 SQINN) = S(NN,K) 00 196 -KK=L»3 - 196 DD(KK) =0. 00( 3)31...— .- 13 FORMATdX,9F11.5) WRITEI6,13)VP, SO - - - CCCCCCCCCC FLYER PLATE SPEED DO 92 J=l,5 . UFP = FLOAT(J) WRIie!6,121 r.M,.J?P. . 12 FORMATdX, 'TARGET PARAMETS' ,4E20.7/ 'FLYER PARAMETERS' , 1 4E20.7) _. : CALL UPMAT!UFP,FP,TM,SU, UT) C UFP=. FLYER VEL..;FE= PHI Fl YER FITiTM= TARGFT MAT P-l) FTTi C S(U> PARAMETER FOR TARGET MAT.=SU C S FOR TARGET, U fOR TARGET hP = PF(UT,TM(1),TM(2),TM(3),TM(4)) VPP(K,J>=1. + FL0ATIK-1)*.2 -UT*UT*RH00*.01/RP— VOL=VPP(K,J) STT(K-J) = PF (V0L,SV(l),Sv(2),5V(3),. SV(4_U WRITE(6,11) J,UT,RP,VPP!K,J),STT(K,J) 11 FORMAT! IX,'UFP ..= . ',11,.' TMP M. PART, VFI .= •t Pin.<;. » 1 F10.5,' REL VOL.= ', F10.5/' ENT =',F10.5) 92 CONTINUE 93 CONTINUE DO 930 K=UJT _ ; _ CALL MAXMIN(Sd,K),IP,YMX,YMN) IF !K.E0.1) GO TO .935 IF (YMX.GT.YX) YX=YMX GO TO 930 935 YX=YMX VN=YMN _.... 930 CONTINUE CALL MAXMjN(VP,IP,XMX,XMN) . ., DO 945 K=l,5 CALL MAXMIN (STT!1,K),JT,YMX,YMN) IF IYMX.GT.YX) YX=YMX CALL MAXMIN (VPP!1,K>,JT,XM,XN) IF 1XM.GT.XMX) XMX=«M 945 CONTINUE CALL SCALE(YX,0.,TOP,BOTTOM,10,IER) CALL SCALE(XMX,0.,RIGHT.XLEFT,15,IER) CALL LABEL(0.,0.,XLEFT,RIGHT,15.,15,1H ,1,0) CALL LABEL(0.,0.,BOrTOM,.TOP,10>,1.0,lH ,1,1) -43- . ._. J>fi_9SJL1"l»JT. ISVS-K-1 9f»0 TiAIL eiXtIXY.UP..VP-..SlliKJ .XL6FT.RIGHT, . BOTTOM,TOP,O.O.ISY.S.X.DD) ISVS»5 00-lOOl.JlsL, 5 . ... ISYS»ISYS+1 CiU.UPlJQIX* A(4»«FPf1)-TMC1I+ FPt2)*UFP+FP(3|*UFP*UFP +FP<4 >*UFP**3 *!3>T--l.mai.2J.t,fJ?,<2>t2.»UfP»FPiajL*3 .*F P«4) »UF P«UF p) A«2»» FPI3)-TH<3)*3.*UFP*PP!4| tlll..1.,(TmtU PPI4H ._ WRITE*6,1) A FPSfl.ft-10 CALL R0OT , ?4.91B.4.R^.4.74*6^6StJA^^.A^45J4..35,.4.25,4.15,4.05,3.948,3.85, 33.75.3.6S,3.56,3.46,3.36,3.27,3.18,3.09,2.996,2.91,2.82,2.74, 47.65.2.57, 2.50«2.»2«2.3»«.2.2Z«2.197,2.13,2.06,1.99,1.93,1.87. 51.81,1.75,1.69,1.63,1.582,1.53,1.48,1,43,1.39,1.34,1.30,1.26, .01. 21 tl.l 8,1.137.1.100U..OA&U..03J,. 0.998, 0.966,0. 935, 0.906,0.87o. 70.050,0.823,0.798,0.774,0.750,0.727,0.704,0.683,0.662,0.642, .Hn.ft?T.n.ftft6,0.'iaa«A.570.0.SS2»0 ^3-7*0.521,0.507«0.492,0.478.0.4oa., 90.452,0.439,0.427,0.415,0.404,0.394,0.383,0.373,0.363,0.353,0.345, -4XU335j.0«324*tU3J-9_,0.ai0.0.303,0.295, 0.287, 0.280, 0.273, 0.267,0. 260, 20*254,0.248,0.242,0.237,0.231,0.226,0.221,0.216,0.211,0.206,0.202, 3n.m.o.i9^,o.ina1.a«i64,o.j.aoto.,_u6,o. 172,0. i6«j ,o. i6s.,o. 162,0.159, 40.155,0.152,0.149,0.146,0.143,0.140,0.137,0.135,0.132,0.130,0.127, so. i 2SJO« i?a», fiO TO 13 , ; - - _ . -44- -KBJJE—tfeiUI- 11 0F0RMAT(76H THE VALUE OF THTEE IS OUTSIDE THE RANGE OF X VALUES OF J IHE TABLE—) • GO TO 12 13 CONTINUE • a. NINT * 4 „. Ul+INTC |0^JWa*^WM9W -r 30 CONTINUE - .UL-U.€£-.3-)-&0~rO~. 40 i —i GO-TO .*--.- -- - • ^-ii 40 IF(I.GE.ISS) GO TO 50 "i< -K-I.-2 : itali GO TO 4 J 1 -SO KsLS2 4 CALL XINTP (THTEE,X(K),CV(K),NINT,CVE) 12 RETURN -- .__:...-...... — ••w& END SUBROUTI ME. XI NIEtAflSC LS..X, V ,N .VAI.IIF.) AITKEN,S ALGORITHM IS USED FOR N POINTS INTERPOLATION ABSCIS - ABSCISSA ar iwir.H TUP UAI UP np rwp PiiMr.Tfnw i< npciapfa X = ARRAY OF ABSCISSA :'M|? Y = ARRAY Of ORDINATES.^. T.HF. VALUE. C1F TMP FUNCTION AT THF AHSfttT " N •= THE NUMBER OF POINTS WE WISH TO USE FOR INTERPOLATION N IS SET. FOB A MAXIMUa_0£.20. THE VALUE OF THE FUNCTION AT ABSCIS $&£... :.VfSS 6 % THE BEST RESULT IS OBTAINED WHEN ABSCIS FALLS IN THE INTERVAL THAT HAS EQUAL NUMBER OF PXUNXS. ON BOTHSIOES • ~ tiir-' DIMENSION XC20),Y<20>,ZL2O).XLI20,?0>. KK = N - 1 DO 1 I - ItN 1 Z(I> = Y(I) K = Z . DO 2 I = ItKK DO 3 J = K.N 3 XL(I,J> = ZII) + (ZIJ) - Z(I))/(X(J) - X(I)> * (ABSCIS - X(I)I DO 4 JJ = K»N _ . .. 4 ZUJ) = XLdtJJl 2 K = K+ 1 ... _... VALUE Z(NI RETURN END //SYSPLTON DD SYSQUT=N //DATA DD * 28ALUMINUM 350. 2. 26.98 2.702 .1 374. ..903.9 .2 508. .8424 .3 707. .7980 .4 968. .7636 .5 1268. .7358 .75 2290. .6839 1. 3540. -. .64.72.-. 1.2 4684. .6244 29 IRON. 175. 1.690 55.847 7.850 .2 416* .... «8A0il . .48- «K^pJf • -48-