Rapid Heating-to-Ignition of High Explosives. 1. Friction Heating Alan D. Randolph,*
‘ L. E. Hatler, and A. Popolato
Los Alamos Scientific Laboratory, Los Alamos, New Mexico 87545
A friction ignition model was developed that predicts the thermal decomposition of condensed phase explosive when impacted at an oblique angle on rigid target surfaces. The model was verified by the experimental initiation behavior of pressed hemispherical billets of PBX 9404 subjected to an oblique impact on targets with widely varying thermal conductivities. The explosive was impacted on targets of fused quartz, aluminum oxide, and gold with similar surface characteristics. The experimentally determined estimates of the drop heights required to produce an explosive event ranged over two orders of magnitude, with the high conductivity gold target and the low conductivity quartz target being, respectively, the least and most sensitive. These results were satisfactorily explained with the thermal ignition model by computer simulation of the mechanical and thermal events occurring during the oblique impact leading to ignition of the charge. No attempt was made to calculate the buildup from ignition to detonation.
Introduction The safe handling and utilization of high explosives (HE) requires a knowledge of their sensitivity to accidental initiation by various stimuli. In turn, this need to assess the hazards associated with the use of H E has resulted in the development of numerous sensitivity tests. Several tests which have become more-or-less standardized can be categorized in four basic groupings: gap, impact, skid, and heating tests. Unfortunately, the mechanisms leading to initiation in these tests are poorly understood and different sensitivity tests often rank explosives in different orders of sensitivity. A different test must be devised for each type of stimulus. Thus, instead of a single property “sensitivity,” one must deal with an entire set, each property in the set defined by the appropriate stimulus. The area of explosive sensitivity is extremely complex and much theoretical and experimental research has been done on the subject. Bowden and Yoffe (1952) describe standard small-scale tests commonly used to characterize explosive sensitivity as well as several comparisons of explosives behavior based on these tests. All initiations of H E are believed to be thermal in origin. However, several mechanical stimuli can lead to sufficient concentration of thermal energy t o initiate the explosive. These mechanisms range from friction (with tangential component of motion) to shock heating, with characteristic initiation times of milliseconds to microseconds. With low mechanical stimulus, and correspondingly long initiation times, thermal transport mechanisms can play an important role. This study quantitatively models two fundamental mechanisms, friction and gas compression, for converting a mechanical stimulus into thermal energy sufficiently localized to lead to ignition of the explosive. P a r t I models the detailed thermal transport that occurs during friction initiation. The theoretical analysis is verified experimentally with results from the large-scale skid test. P a r t I1 deals with the thermal transport occurring during initiation by rapid compression of an adjoining gas void. The gas compression initiation mechanism is used to analyze experimental tests of T N T surface ignition. Ignition and initiation of HE are, of course, not equivalent terms. However, ignition of H E in the highly compressed (or encased) situations considered by this study is Chemical Engineering Department, University of Arizona, Tucson, Ariz. 85721
virtually certain to lead to a violent explosive event, in some situations approaching true detonation. No attempt is made to model the complex phenomena that lead from ignition to explosion and/or detonation of a highly compressed or enclosed H E source.
Background Large uncased billets of secondary high explosives such as the cast cyclotols and plastic-bonded explosives containing RDX or HMX are quite stable to normal impact on smooth surfaces. With void-free high-density billets, impact velocities of about 30 m/sec are generally required t o initiate a reaction (violent deflagratioh or detonation). With an oblique impact, such as sliding or skidding, these explosives can be initiated a t impact velocities of about 3 m/sec. Serious handling accidents have occurred in which it was theorized that the H E was initiated in this way. Such initiations suggest a thermal mechanism of initiation with transformation of mechanical to thermal energy initiating a thermal decomposition reaction which then grows to violent deflagration or detonation under proper conditions of confinement and mass of HE. P B X 9404 charges of large size typically build to violent explosion, some events approaching high-order detonation. (PBX 9404 is a plasticbonded explosive composed of 94 wt % HMX, 3 wt % nitrocellulose, and 3 wt % tris(P-chloroethy1)phosphate.) Several skid tests been devised to study initiation by oblique impact. Dyer and Taylor (1970) describe an oblique impact test in which a hemispherical billet of H E is suspended in a harness with the flat surface of the hemisphere maintained in the horizontal plane. The billet swings down on the end of the harness and impacts against a rigid horizontal target a t a predetermined angle. The test variables are the drop height, the angle of impact between the billet and target, and the surface condition of the target. A schematic of this pendulum drop configuration is shown in Figure 1. In a similar version of the skid test, hemispherical billets are dropped in free fall against an inclined rigid target. A schematic of this vertical drop configuration is shown in Figure 2. With either version of the skid test, violent reactions can occur from drop heights much lower than would be necessary to initiate a reaction with normal impact. A better understanding of the skid test is important in assessing the hazards associated with handling large billets of high explosives. This paper reports the results of some Ind. Eng. Chem., Fundam., Vol. 15, No. 1, 1976
1
Figure 1. Pendulum drop configuration. TLIM
BOLD TMQET
a maP
h .1.
Figure 2. Vertical drop configuration. work which conclusively demonstrates that the mechanism of initiation in an oblique impact, for the case of smooth target surfaces, is thermal generation and transport a t the target-billet interface. Bowden and Gurton (1949) in a classic study of H E initiation, demonstrated that thermal transport mechanisms play a dominant role in the impact initiation of secondary explosives. The “sensitization” of H E by grit particles in both friction and impact sensitivity tests was attributed to initiation a t thermal hot spots, where the grit particles developed high temperatures a t grit-grit or grit-HE points of contact. This thermal transport theory of initiation was used by Yoffe (1949) to explain the initiation of H E by adiabatic compression of entrapped gas in a hammer-anvil sensitivity apparatus. In the skid test, the thermal transport theory suggests that the primary mechanism of initiation is thermal generation and transport a t the billet-target interface. Thermal generation is determined by the sliding velocity, surface roughness, and mechanical properties of the billet; thermal transport is influenced by the thermal diffusivities of billet and target. Definitive answers to the questions of thermal generation and transport a t sliding unlubricated surfaces are not available. Furey (1969), in a recent review article on friction and lubrication, describes the current lack of detailed experimental information regarding the magnitude and distribution of surface temperatures in real systems. Holm (1948) presented a theoretical study of transient temperature rise in a sliding circular contact, based on simplifications of earlier work of Jaeger. Average temperatures a t the sliding interface were deduced by measuring the thermocouple emf generated a t the bimetallic surface. Fair agreement was obtained between theoretical and experimental results, but numerous simplifications were used in estimating the thermal flux generated a t the surface. Archard 2
Ind. Eng. Chem., Fundam., Vol. 15, No. 1, 1976
(1958) presented a comprehensive theoretical study of the temperature of rubbing surfaces. Small localized circular points of true contact between the two materials were assumed, and the highest transient temperatures developed a t these localized points of contact were calculated. Two limiting cases were considered: high and low sliding velocity. In the former case a small contact sees a stationary unheated surface a t the boundary, while in the latter case temperatures have time to accommodate to steady profiles on both sides of the interface. In both cases the total thermal flux was partitioned between the two surfaces so that the appropriate equations for heat flow in the two media gave the same surface temperature. In the present study thermal generation and transport a t the billet-target surface was described by considering the total measurable contact area between the deformed billet and rigid target as the true contact area. No concentration of energy due to localized contact was assumed. Thermal flux was calculated from the work done a t the sliding surface by direct measurement of tangential forces. These forces were expressed in terms of an apparent coefficient of sliding friction defined as the ratio of tangential to normal contact forces. The deformed area on the billet is much greater than the additional area covered by sliding displacement during contact. Physically, this corresponds to the low-velocity case described by Holm. A necessary consequence of the thermal transport theory of initiation applied to the skid test with smooth target surfaces is that thermal diffusivities in both explosive and target should play an important role in determining H E response. This thermal transport theory of initiation was tested by a series of oblique impacts with P B X 9404 hemispherical billets on three smooth target surfaces having widely divergent thermal diffusivities. These target surfaces were fused quartz, aluminum oxide, and gold. The time-dependent thermal flux generated a t the target-billet interface was calculated from measurements of billet impact dynamics and these fluxes were used in a numerical analysis of the mechanical-thermal dynamics occurring in this H E sensitivity test. Experimental data were adequately explained by this computer simulation.
Development of Friction Model A. Billet Dynamics. The heat flux generated a t the sliding billet-target interface is calculated as g = WUVTIJ
(1)
where fi is an apparent coefficient of friction equal to the ratio of tangential to normal contact forces, u is the normal contact pressure (in this case equal to the plastic flow stress of the explosive), VT is the sliding velocity, and J is the conversion factor from mechanical to thermal units. This heat flux occurs over the contact area A ( t ) during the contact time t,. ‘rhus analysis of thermal transport in the skid friction test must begin with an analysis of impact dynamics. The three pertinent equations of motion for the center of mass of an impacting plastic hemisphere on a rigid plane surface can be written as
WX = F; W Y = F4
I,& = M,,
(3) (4)
If some functional relationships between force and normal deformation, F, = f(X) and Fi.= g ( X ) ,are known, eq 2-4 can be integrated with the following initial conditions a t the instant of impact: X ( 0 ) = 0, deformation zero; Y ( 0 )
Table I. Summary of Dynamics for Oblique Impact of Plastic Hemisphere on Rigid Plane Target Quantity
-___
Predicted from impact model ~
t,
( n / 2 ) (W / n u D ) % X ( V n o / wsin ) w t where w = (ni~D/W)"2 V, V," cos w t and Vno= V ocos cp A , =n-DVn0/w 0 = [(a'p+ b')DVnoW/21cg](1-cos u t ) where p = y / u a' = 1 - 0 . 3 7 5 COS €' ' b' = 0.375 sin \Ir d,, = ( u ' p + b ' ) D V n 0 W / 2 I c g Y c g= ( V T ) ,=~V' [sin + p cos p (cos w t - I)] VT = V" sin cp - V' [ p + ( u ' p + b')DWX,/BI,] cos (1-cos u t ) where X , = ( D / ~ ) [ ( U +' )(*b ' ) ' l x
Contact time Normal deformation Normal velocity Maximum contact area Rotational velocity
Terminal rotational velocity Tangential velocity Sliding velocity
= 0, translation zero; O ( O ) = 0, rotational angle zero; X ( O )= vO, cos cp, normal velocity; Y ( o ) = V O sin cp, tangential velocity; 8 = 0, rotational velocity zero. Solution of these equations, with suitable force-deformation relationships, yields the normal and tangential deformation-time and velocity-time profiles during impact as well as rotation of the billet. Sliding velocity a t the billettarget interface is obtained by correcting the tangential velocity a t the center of mass for motion a t the outer radius due to rotation. Trigonometry relates the hemisphere diameter, D , radius of deformation, a , and normal deformation X (for the case of small deformations) as u 2 = D X , giving the dynamic area of contact as A ( t ) = a a 2 = a D X ( t ) . The stress-strain relationships for the billet were assumed to be those of constant normal and tangential stress, i.e., plastic flow a t the plastic flow stress with a constant ratio of tangential to normal flow stress. Thus, the force-deformation relationships can be written as
F, = f(X) = u a D X ( t )
(5)
F, = g(X) = /.waDX(t)
(6)
and
The dynamics of oblique impact of a plastic hemispherical billet on a rigid plane target were then obtained by solution of eq 2-6. Pertinent quantities of interest from the solution of these equations are listed in Table I. The impact was assumed to be plastic, and thus weak forces during rebound were neglected in the above analysis of motion. The initial drop velocity, V o , is proportional to the square root of drop height, H112.Thus, as maximum area of contact, adC2/4,is proportional to V o ,then maximum contact diameter is given as
d , = kH1I4
(7)
where k is a constant that depends on drop angle and plastic stress. Equation 7 provides a simple way of verifying the constant stress impact model (Figure 3) as well as determining quantitative values for the normal plastic flow stress under actual drop conditions. Solving eq 7 for the (constant) plastic flow stress u gives u
= g,.WDH cos2 ~ / p l r d , ~
(8)
B. Thermal Transients at Billet-Target Interface. The preceding analysis of mechanical impact dynamics permits a quantitative evaluation of thermal flux generated a t the billet-target interface from calculated motions of the impacting billet. Heat flux generated on the surface is equal to the rate of work done a t the surface expressed in thermal units. In the calculation of the heat flux from eq 1, VT is given in Table I; and u are obtained by analysis of impact motion from "no go" drops. Equation 1,partitioned between billet and target, provides a time-dependent
= = =
0.1
0.I
1.0
Drop
D
Hdomh)
Figure 3. Correlation of deformation diameter to drop height for hemispherical billet. boundary condition to the reactive heat transfer equation, The use of eq 1 as a boundary condition is equivalent to assuming that all the frictional work is generated as a line source between the two sliding surfaces rather than as viscous dissipation in the flowing plastic. The one-dimensional heat equation in two media, with the reaction term present for HE, was solved with the flux partitioned between the billet and target according to Fourier's law written a t each instant of time. The one-dimensional heat equation is stated for H E (Zinn and Mader, 1960) and target as
and (10)
where the subscripts 1 and 2 refer to the H E and the target, respectively. R and iUl are the rate and heat of reaction for the HE, and y is the fraction reacted, which is given by
(Because of the critical behavior of the reaction term for secondary explosives (AI3 typically 45 to 55 kcallmol), no appreciable error in calculating explosion time is caused by neglecting the fraction of H E reacted. Equation 11 is merely added to complete the formalism and as a computational convenience t o define the point of thermal explosion.) The reaction rate R can be adequately represented for PBX9404 explosive by the Arrhenius reaction rate expression, A exp(-EIRT). The boundary conditions used for solution of eq 9 and 10 are continuity of temperature a t the billet-target interface
Tl(0,t ) = T2(0,t ) = T B ( t )
(12)
Ind. Eng. Chem., Fundam., Vol. 15, No. 1, 1976
3
and a partitioning of flux as given by Fourier’s law
Table 11. Experimental 50% Drop Heights Thermal conductivity, calisec cm “C
Target material Equations 9-13 were finite-differenced with an explicit forward difference scheme and were made dimensionless for efficient digital computation. Computations were carried out on LASL’s CDC-6600 computer. Each separate run stimulates the thermal events after impact from a vertical height H . Three normal exits from the program are provided: “thermal event,” “hemisphere rebounds,” or “hemisphere stops sliding.” A “thermal event” is defined when the fraction reacted exceeds the value 0.99 a t any computational mesh point. The critical height to develop a thermal event is found by direct search of the height space between upper and lower bounds. The search algorithm chooses the next height for evaluation as the geometric mean of the lowest “go” and highest “no go” heights, thus converging in 12 to 15 calculations to the true critical height.
Experimental Section Hemispherical billets of PBX 9404 were impacted on rigid targets of fused quartz, aluminum oxide, and gold to determine the effect of the thermal conductivity of the targets on the 50% drop height. (The height a t which there is a 50-50 chance, statistically calculated, that an event will occur. The statistics in the “up-down” testing and analysis procedure assume that the probabilities for an explosive event are distributed log-normally about the median height. Thus, the “50% drop height” is in fact the estimated median height where the probability for an event is onehalf (Dixon and Massey, 1957).) Each billet (25.4-cm diameter) is composed of a small inner hemisphere of inert material that weighs 3.4 kg with a 5-kg outer shell of PBX 9404 glued to the inert. A 1.27cm thick Micarta plate is glued to the top of the billet for attachment to the drop apparatus. Total weight of the assembly is 8.67 kg. Two drop configurations were used to impact the billets on the targets. A pendulum drop configuration (Figure 1) was used to impact the PBX on the quartz and aluminum oxide targets. This configuration was chosen because a 15 x 15-cm target could be hit consistently. The angle of impact was 30’ to the horizontal target. The pendulum configuration had a vertical limitation of 6 m. Because of this limitation, a vertical drop configuration (Figure 2) was used for the gold target experiments. The impact angle was 45O. In both configurations the targets were rigidly secured to a 10-cm thick steel boiler plate. The fused quartz targets were 15 cm X 15 cm by 0.64 cm thick. The surfaces of these targets were ground with an intermediate grit on a grinding wheel to give a roughness of 1.27 to 2.03 wm as measured by a Cutler Hammer Surf-Indicator. The targets were attached to the 10-cm thick steel boiler plate with epoxy adhesive. The aluminum oxide targets were purchased from Coors Ceramics and were nominally 85% alumina and 15% silica, magnesia, and calcia. The targets were 25 X 25 cm X 0.64 cm thick. Three of the targets were ground with boron carbide to give a surface roughness of 1.27 to 2.03 pm, and nine others were ground with aluminum oxide to give a surface roughness of 0.51 to 0.89 pm. The ceramic targets were rigidly mounted with epoxy adhesive to the 10-cm thick steel boiler plate. The gold targets were prepared by electroplating a 7-mil thick layer of copper on a sheet of Dural 25 X 25 cm and 0.64 cm thick. A 3-mil layer of gold was then electroplated on the copper and polished to a surface roughness of 1.27 to 2.03 pm. Four of these 25-cm targets were mounted with 4
Ind. Eng. Chem., Fundam., Vol. 15, No. 1, 1976
Exptl drop height, m
7.03 x lo-’ 2.10 x 10-2
Gold Aluminum oxide (0.51-0.89 pm surface roughness) Aluminum oxide (1.27-2.03 pm surface roughness) Quartz
2.10 x
> 46
- 5.8 - 3.6
10-2
4.05 x 10-3
0.55
Table 111. Drops on Quartz Targets Drop number
Height, m
1
0.46
N
2
3
4
N
0.65
E
5
6
7
N E
8
9
N
1
0
N
E
E
N
Table IV. Drops on Aluminum Oxide Targets Height, m 1
2
3
4
Drop number 5 6 7 8
9
101112
5.64 E N 4.75 Ea N N 3.99 N N 3.35 E E N N N 2.82 a High-speed motion pictures showed drop 7 to be invalid. epoxy adhesive on a sheet of aluminum as a 50 X 50-cm target. This target was then bolted to the 10-cm thick steel boiler plate. The apparent coefficients of friction and the plastic flow stress in the billet were determined experimentally for use in the mechanical-thermal initiation calculations. Highspeed motion picture films of the “no go” impacts on the three targets were analyzed to determine the tangential forces under the conditions of impact. The angular velocity after rebound was found by plotting the angle of the HE billet as a function of time. The angular velocity imparted to the billet during contact is expressed in terms of moment of inertia, billet mass and geometry, and ratio of tangential to normal forces (apparent coefficient of friction) in Table I. This relationship was used to calculate the apparent coefficient of friction from the measured terminal angular velocity together with the geometrical and mechanical properties of the billet. Measured apparent coefficients of friction were about 0.1 for the smooth targets used in this work. Plastic flow stress was determined by dropping PBX 9404 billets vertically onto an 8’ inclined boiler plate target covered by a sheet of vellum paper backed with carbon paper. Drop heights ranged from 13 to 91 cm. The maximum circle of deformation on the billet a t impact could be measured accurately from the circle imprinted on the vellum by the carbon paper. From the deformation diameter, weight and diameter of the billet, and the target angle, the plastic flow stress can be determined for the billet from eq 8. Reproducible values of the plastic flow stress calculated for different drop heights verified that the billets impacted with constant stress and plastic flow (see Figure 3). The permanently deformed area on the billets after impact was found to be 85 to 90% of the maximum deformed area (measured by the carbon paper imprint); therefore, the impact was assumed to be plastic and normal forces during
Table V. Apparent Coefficients of Friction Apparent coefficient of friction
Target material Gold Aluminum oxidea (0.51-0.89 pm surface roughness) Aluminum oxideb (1.27-2.03 pm surface roughness) Quartz a Average of three “no go” drops. go” drops.
0.117
0.0576 0.0808
b
0.132 Average of two “no
Table VI. Physical and Thermal Properties Used in Calculations
Material
Heat Thermal Density, capacity, conductivity, g/cm3 cal/g “C calisec cm “C
Thermal diffusivity, cm2/sec
9.2 X 10-4a 1.65 X HMX 1.91 0.295 4.05 X 8.11 X Quartz 2.67 0.188 2.1 X lo-* 3.45 X lo-* Aluminum 3.40 0.18 oxide 1.16 Gold 19.3 0.0316 7.03 X 10-I a No value of thermal conductivity for crystalline HMX was available; this measured value for PBX 9404 was used in all calculations. b Based on thermal conductivity of PBX 9404. rebound were neglected in the idealized impact model used in thermal calculations. The apparatus indicating vertical drop height was calibrated for both pendulum and vertical drops, Thermal events were defined as violent thermal explosions in which the billet was destroyed and >99% of the H E consumed. No attempt was made to characterize the violence of the events, although overpressure measurements were made for some drops. No partial events were observed with the PBX 9404 and target surfaces of this study. Verification of impact accuracy on the target was made by examination of high-speed motion picture films of each drop. Results Table I1 summarizes the experimental 50% drop heights and the thermal conductivities of the targets used in the experiments. The experimental 50% drop height for PBX 9404 dropped on the quartz targets was determined by the up-down method of statistical analysis (Dixon and Massey, 1957). Because the aluminum oxide targets had two different surface finishes, data were insufficient to allow a determination of the 50% drop height by the up-down method. The values given in Table I1 are estimates based on the available data. Drops from 8, 15, 30, and 46 m produced no events on the gold targets. The height of the drop tower is 46 m. Table I11 is a summary of the drops on quartz targets with N being no event, and E being event. These data on quartz targets indicate not only a markedly decreased 50% drop height (increased sensitivity), but also a significantly lowered variance for the test. Table IV is a summary of the drops on the aluminum oxide targets; in drops 1 through 3 the surface roughness of the targets was 1.27 to 2.03 pm, and in drops 4 through 12 the surface roughness of the targets was 0.51 to 0.89 pm. Table V gives the apparent coefficients of friction for the gold, aluminum oxide, and quartz targets as obtained from analysis of drop films. The plastic flow stress for PBX 9404 was determined to
Table VII. Thermal-Kinetic Properties of HMX Used in Critical Drop Height Calculations Property Heat of reaction (burning) Heat of melting Arrhenius frequency factor Arrhenius activation temperature, E t / R Crystal melting temperature
Value 6.66 x l o 5 50 5.0 x l o t 9 2.63 x 104 553
Units cal/(g-mol) cal/g sec-’ K K
Table VIII. Experimental and Calculated 50% Drop Heights Drop height, m Target material
Calculated
Gold Aluminum oxide (0.51-0.89 pm surface roughness) Aluminum oxide (1.27-2.03 pm surface roughness) Quartz
157 6.95 4.3 0.38
Experimental
> 46 -5.8
- 3.6 0.55
be 2.0 X 10’ kg/m2, which was assumed to be constant over the entire regime studied and was used in all calculations. For constant stress and plastic flow, the deformation diameter, corrected for angle of impact, should vary with the 0.25 power of the drop height, as in eq 7. The deformation diameter vs. drop height, corrected for differing angles, is plotted in Figure 3. These data plot on log-log paper with a least-squares slope of 0.25, as indicated by eq 7. Thus, the assumption of constant stress and plastic flow seems justified. Table VI summarizes the various physical and thermal properties used to calculate the critical drop heights expected under the given experimental conditions. Table VI1 summarizes the thermal-kinetic properties of pure crystalline HMX used in these calculations. These values were obtained from LASL’s Group WX-2 (Rogers, 1970). An initial billet temperature of 295 K was used in all calculations. Table VI11 gives the calculated drop height of P B X 9404 dropped on various targets and the corresponding experimental values. Conclusions This series of drops on controlled target surfaces substantiates the proposed thermal transport mechanism of initiation in the skid test. The extremely wide range in 50% drop heights on the three different target materials indicates the importance of target thermal conductivity; this is a necessary consequence of the thermal transport mechanism. Materials with high thermal conductivities are much less likely to produce events when HE is dropped on them than are materials with lower thermal conductivities. During the series of experiments with the gold target, one of the drops from 46 m missed the gold target and impacted on a freshly sand-blasted aluminum surface; no event occurred. The thermal conductivity of aluminum is 76% that of gold. Excellent agreement was obtained between the critical drop height calculations and experimental 50% drop heights, where sufficient experimental data could be obtained for comparison. The calculated critical drop height for PBX 9404 on gold targets is out of the range of testing with the LASL skid test; the fact that no events could be produced on gold targets is consistent with these calculations. Insufficient drops were made on the two aluminum Ind. Eng. Chem., Fundam., Vol. 15, No. 1, 1976
5
oxide surfaces t o determine 50% drop heights precisely; however, calculations indicated critical heights close to the estimated 50% heights and also distinguished between the results obtained with the two surface finishes. These experiments were not extensive enough to demonstrate conclusively that surface roughness of the target can be adequately accounted for in the thermal transport model by the measured apparent coefficient of friction, although this is probably true for smooth surfaces. However, in the experiments cited, the thermal model adequately accounted for the effect of surface roughness by the differences in work produced a t the surfaces. The effect of surface finish on drop height deserves further study. In summary, these studies indicate that the primary mechanism in initiation by oblique impact is thermal generation and transport a t the billet-target interface. Factors such as billet stiffness, sliding velocity, and target roughness (up to the limit where machining occurs) increase the generation of heat a t the interface and sensitize the test. High target thermal conductivity transports the heat away from the H E billet, thus desensitizing the test. Sliding velocity '(determined by drop height and impact angle), plastic flow stress (billet stiffness), coefficient of friction (surface finish), and target conductivity are thus the strongest variables affecting skid test sensitivity. When the experimental drop conditions conform to the idealizations of constant stress impact and thermal generation a t a well defined target-billet interface, then the friction heating model quantitatively predicts skid test drop heights within the limits of uncertainty of the input data and the experimental drop heights. Acknowledgments The mechanical impact model developed in this study was based on earlier unpublished work by Charles A. Anderson, Group WX-3, Los Alamos ,Scientific Laboratory. The writers are deeply indebted to Dr. Anderson for many discussions concerning the impact model and subsequent heat transfer calculations. This work was carried out under the auspices of the U S . Atomic Energy Commission. Nomenclature a = radius of deformation circle on H E billet, cm A = area of contact deformation circle on H E billet, cm2, or Arrhenius rate constant for thermal decomposition of explosive, sec-l c = specific heat of HE, cal/g K D = diameter of H E billet, cm E = Arrhenius activation energy for decomposition of explosive, cal/g-mol d = deformation diameter on H E billet, cm F = force acting on billet, dyn g, = gravitational acceleration constant, gf cm/g, sec2 H = vertical drop height, m AH = heat of reaction (burning) for HE, cal/g-mol I = moment of inertia of billet, cm2 g k = thermal conductivity, cal/sec cm "C
6
Ind. Eng. Chem., Fundam., Vol. 15, No. 1, 1976
k = constant relating d, to H1I4 M.W. = molecular weight of HE, g/g-mol M = rotational moments due to contact forces, dyn-cm q = thermal flux generated a t billet-target interface, cal/ cm2 sec R = H E reaction (burning) rate, mol/sec cc t = time,sec T = temperature of billet, K V = billet velocity, cm/sec W = mass of billet, g X = deformation of billet normal to impact target, cm Y = distance coordinate of H E center of gravity parallel to impact target, cm y = fraction of H E reacted Greek Letters = thermal diffusivity, cm2/sec y = tangential shear stress, dyn/cm2 p = angle of target plane to horizontal for vertical drop, or complement of angle of approach for pendulum drop, deg = coefficient of sliding friction defined as y/a u = normal plastic flow stress, dyn/cm2 p = HEdensity, g/cm3 0 = angular velocity of hemisphere billet after rebound, rad/sec 0 = angle with horizontal of diameter of hemisphere billet after rebound, rad $ = angle of target plane to horizontal for vertical drop, or angle of billet tip from horizontal in pendulum configuration, deg (maintained approximately zero)
iy
Subscripts c = relating to maximum deformed area on H E billet and contact time x = forces or motions in x direction y = forces or motions in y direction cg = moments or moment of inertia about center of gravity n = normal to target T = tangential to target Supercripts 0 = relating to initial conditions a t instant of impact L i t e r a t u r e Cited Archard. J. F., Wear, 2, 438 (1958). Bowden, F. P., Gurton, 0. A,, Proc. Roy. SOC. London, Ser. A, 198, 337 (1949). Bowden, F. P.,Yoffe, A. D., "Initiation and Growth of Explosion in Liquids and Solids," Cambridge, 1952. Dixon, W. J., Massey, F. J., "introduction to Statistical Analysis," pp 319327, McGraw-Hill. New York, N.Y., 1957. Dyer, A. S.,Taylor, J. W., "Initiation of Detonation by Friction on a High Explosive Charge," The Fifth Symposium on Detonation, Pasadena, Calif., Aug 1970. Furey, M. J.. Ind. Eng. Chem., 61, (3), 12 (1969). Holm, R., J. Appl. Pbys., 19, 361 (1948). Rogers, R., private communication, Group WX-2. Los Alamos Scientific Laboratory, Los Alamos, N.M., 1970. Yoffe, A,, Proc. Roy. SOC.London, Ser. A, 198, 373 (1949). Zinn, J., Mader, C. L.. J. AppI. Pbys., 31, 323 (1960).
Received for review March 28, 1975 Accepted September 24,1975
