role of the friction coefficient in the frictional heating ignition of explosives

Propellants and Explosives 6, 17-23 (1981)

Role of the Friction Coefficient in the Frictional Heating Ignition of Explosives

W. H. Andersen

Shock Hydrodynamics Division, Whittaker Corporation, North Hollywood, California 91602 (USA)

Die Funktion des Reibungskoeffizienten bei der Ziindung von Spreng- stoffen durch Reibungswarme

Eine mathematische Beziehung wird entwickelt zwischen dem Rei- bungskoeffizienten eines Materials und den Parametern, die die Rei- bung beim Schervorgang beeinflussen. Diese Gleichung beschreibt in Verbindung mit der Gleichung fur die Reibungswarme die bei Rei- bung auftretende Hot spot-Temperatur. Die Hot spot-Temperatur steigt an bei der Zunahme der TeilchengroRe und der Scherfestigkeit des Sprengstoffmaterials, bei der Zunahme des (StoR) Belastungs- druckes und der Reibgeschwindigkeit und bei der Abnahme der War- meleitfahigkeit des Materials. Die Gleichungen werden diskutiert unter Berucksichtigung der Faktoren, die die Empfindlichkeit eines Spreng- oder Treibstoffs gegen Reibungswarme bei unterschiedlichen Bedingungen beeinflussen und werden rnit experimentellen Ergebnis- sen aus der Literatur verglichen.

Summary

An expression is developed for the friction coefficient of a material in terms of the parameters that control the friction shear. This expression in conjunction with the frictional heating equation describes the hot spot temperature produced in a friction event. Hot spot temperature increases with an increase in the particle size and shear strength of the explosive material, with an increase in the (shock) loading pressure and friction velocity, and with a decrease in material thermal conductivity. The equations are discussed in terms of their implica- tions regarding the factors that control the sensitivity of an explosive or propellant to frictiona1 heating under various conditions, and the results of experimental studies given in the literature.

1. Introduction

Explosives and propellants can be initiated to burning or detonation by friction under suitable conditions. The initiation can be brought about by various means including intercrystalline friction, material friction with the confinement medium or grit particles, and (possibly) material friction with an impacting projectile. Early studies by Bowden and co-workers(’) determined many of the factors involved in the initiation process. It was shown that ignition results from the thermal decomposition of the explosive at suitable hot spots produced at the friction surface. Certain factors control the hot spot temperature. Under appropriate conditions the ignition sites may grow into an explosive reaction. There has also been some continuing speculation that the propagation of detonation in heterogeneous materials may involve microshear and frictional heating (in addition to compressional heating) at load-bearing contact points in the material (for early com- ments see Ref. 2). However, due to the complexity of the processes involved, there is yet much to be learned before the importance of frictional heating in a detonation event can be reasonably assessed.

0 Verlag Chemie, GmbH, D-6940 Weinheim, 1981

17

Influence du coefficient de frottement sur I’amorgage des explosifs sous I’effet de la chaleur de friction On ttablit une relation methtmathique entre le coefficient de frotte-

ment d‘un mattriau et les paramktres qui determinent la friction dans une sollicitation par cisaillement. En liaison avec I’tquation qui fournit la chaleur de friction, cette relation permet de dtterminer la temptrature du point chaud rtsultant du frottement. Cette temptrature du point chaud croft avec la grosseur des grains et avec la rtsistance au cisaillement de I’explosif, elle est proportionnelle a la pression de sollicitation (par choc) et 21 la vitesse de frottement et inversement proportionnelle a la conductivitt thermique du mattriau. On examine ces tquations en tenant compte des facteurs qui dkterminent la sensi- bilitC d’un explosif ou d’une poudre au frottement sous difftrentes conditions et on compare les valeurs qu’elles fournissent avec des rtsultats exptrimentaux tirks de la litttrature.

As an effort in this direction, this paper develops an approximate expression for the friction coefficient of a material in terms of the parameters that control the friction shear. The results are then used in conjunction with the frictional heating equation and thermal decomposition kinetics of the material to discuss the factors that control the sensitivity of the material to frictional heating under various conditions. Because of the general complexity of the subject, the spirit of the paper is to provide a qualitative basis of understanding of the entire event, upon which further improvements can be made.

2. Frictional Formation of Hot Spots

The sliding frictional force, Fxy, between the surface of two materials in relative motion is known experimentally to be proportional to the normal force, F, pressing the two materials together, i.e.,

F x y = P F

~1 is the coefficient of friction, and its value depends on various factors and parameters. On a microscopic scale, the friction surface between two materials is not plane, but rather contains micro-corrugations and irregularities, due to the non-uniform nature of most surfaces. The major load-bearing contact area between the materials thus occurs at the peaks of the irregularities (asperities), and depending on conditions may be relatively small. The friction at any point in the surface varies with time. The frictional energy produces heat, and the energy dissipation is concentrated largely in the localized contact areas, which become hot spots. The hot spots have various sizes and temperatures, and last for various lengths of time depending on conditions. Indirect evidence indicates that hot spots in explosives often range in size from about cm to lo-’ cm, with durations of from to 0.1 seconds and tem-

0340-746218 110102-00 17$02.50/0

18 W. H. Andersen Propellants and Explosives 6, 17-23 (1981)

peratures of up to several(‘) hundred degrees Celsius. For sliding substances with high melting points, or the sliding of a metal on a heat insulator (e.g., metal on glass), the temperatures can be much greater.

2.1. Hot spot temperature

The accurate estimation of the hot spot temperatures produced by frictional heating is difficult. The problem has been discussed by Jaeger(’)), based on certain idealized models. Under conditions that a parameter L = rv/2al is small, the average surface (hot spot) temperature, T,, over a circular surface contact area with radius r is given approximately by

0.266 pvF r (Ki + Kz)

T, = To +

where To is the initial (ambient) temperature, v is the relative velocity between the friction surfaces of the two materials (designated 1 and 2), K is thermal conductivity coefficient and a is thermal diffusivity (a = K/eC, where e is density and Cis heat capacity). Material 1 is assumed to be at rest with respect to the relative motion, and in many cases would thus correspond to the explosive or propellant. Material 2 might be an impacting projectile, a grit particle, a metal casing, or in the case of intercrystalline friction, the same as material 1. Equa- tion (2) also assumes that there is no interaction between the hot spots (the equation applies to a single contact area), that transient effects on the temperature rise are negligible, and that no melting occurs. In view of the assumptions, it is not presently possible to make accurate calculations of hot spot temperature using this equation, but an understanding of the effects of its parameters is of importance, For example, it has been shown experimentally that increased v and F, and decreased K will enhance the hot spot temperature(’). It was also shown that if the materials can melt, then the interface (hot spot) temperature will under normal conditions be limited by the lowest melting point material.

Under conditions that L is large, Eq. (2) is replaced by the approximate equation

(3)

For intermediate values of L, the temperature rise can be estimated using methods discussed by Jaeger. Thus the temperature rise is proportional to v/r at small L, and v1’2/r”’2 at large L. Under both conditions it is proportional to the friction coefficient. An understanding of the effect of this parameter is thus necessary for further progress in understanding initiation by frictional heating, and will ‘Sow be examined.

2.2. Evaluation of the friction coefficient

For the purposes of derivation, a metal (or grit) friction element will be assumed to be interacting with (sliding against) an explosive material, but the results also apply to intercrystalline friction. Assume that the friction element contains surface irregularities, as discussed before. The explosive surface also contains irregularities due to the heterogeneous nature of the individual particles (for intercrystalline friction a slip plane may form a fracture surface). The loading force pressing the

friction element against the explosive is F, and the sliding frictional (shear) force is Fxy. By definition,

F,.. = P,, A . (4)

A is the load-bearing contact area between the friction element and the explosive, and P,, is the shear (frictional) stress necessary to break the contacts between the explosive and friction element by deformation or fracture of the friction surface of the material.

The value of A is dependent on F. At zero load consider A to be zero. At very large load, A approaches a limiting area Al, which is essentially the area of the friction element. In between these load limits the value of A depends on how deeply the load-bearing points penetrate into the friction surface. The intimacy of contact of the friction surface (and hence the value of A) thus also depends on (in addition to F) the effective compressibility, Pe, of the bulk material in the friction surface. For a non-porous homogeneous material Pe will essentially correspond to the normal compressibility, p, of the material at various loads. For a heterogeneous explosive material however, the loading pressure initially causes a plastic deformation of the material which tends to fill the voids. In this case, under ordinary loading conditions Pe will depend largely on factors that control the void content and particle geometry in the friction surface, such as the initial porosity content and particle size of the material. At larger loads p will make an increasing contribution to Pe.

Let Vo be the initial (no load) volume of the corrugations and irregularities in the surface of the friction element, and let V be the volume remaining (i.e., not within the friction surface) at load F. Assume that p, is given by an equation of the same form as for normal compressibility, i.e., fie = ( - IN) . (dV/dP)T, where P is the loading pressure. Integrating gives

V/Vo = exp(- Pep)

The general validity of this equation is supported by the fact that the porosity, p = 1 - V,/V,, of a porous material at pressure P obeys the same form of equation at ordinary

and is derivable from the same assumption (V, is the apparent specific volume and V, is the specific volume of the solid (nonporous) material). The value of Pe for porous materials depends experimentally (often weakly) on the size and shape of the particles.

In order to establish the proper boundary conditions, the following identity is assumed (to be discussed)

Combining Eqs. (4), (5) and (6) gives

Equation (7) is of the same form that was deduced empiri- cally and found experimentally by Sieglaff and Kucsma(’) for polyvinyl chloride, and hypothesized to be valid also for other polymers and non-metals. The preceding approximate derivation accounts for the presence of the shear strength and effective compressibility of the material (whose values depend on the temperature). In the studies of Sieglaff and Kucsma the corresponding parameters were found to correlate with the tensile strength and bulk modulus, respectively. The assumption of Eq. (6) thus appears to be justified, since the resulting

Propellants and Explosives 6, 17-23 (1981) Role of the Friction Coefficient of Explosives 19

equation (Eq. 7) was found experimentally to be essentially valid.

Substitution of Eq. (7) in Eq. (1) gives the coefficient of friction. Under ordinary conditions the exponential in Eq. (7) can often be expanded and only the linear term retained. Since F = P Al there is then obtained

P = PePxy ' (8)

The values of both Be and P,, will depend on conditions, and their detailed evaluation can be quite difficult. For orientation purposes here it may be noted that for a typical explosive material, Be = 1.5 . cm2/dyn and P,, = 4 . (lo8) dyn/cm2, giving p = 0.6. This is of the general magnitude measured experimentally under normal experimental conditions. For example, Dyer and Taylor found 1-1 = 0.4 for cast 98/2 HMX/ TNT in contact with a revolving sandepoxy resdsteel friction surface@). Reference 9 gives values of p that range from about 0.25 to 1.1 for several explosive mixtures in sliding contact with themselves or an aluminum element under various conditions of loading pressure and friction velocity.

The preceding value of Pe was estimated as follows. Huffine and Bonilla found that elastic recovery in typical compressed powders does not start to occur until pressures of about 20000 psi (1379 bar) are attained@). Thus at this pressure the porosity of the compressed material is largely removed. Using the porosity analog of Eq. (5) it can then be estimated for a typical cast material containing about 5% porosity that p, = 1.5. cm2/dyn. However, this general magnitude has also been obtained experimentally for the compression of various inorganic and organic powders(4, '), granulated poly- mer@, and clays and shales(5).

Values of the normal shear strength of explosives are not available, but on the basis of certain empirical relations and the values obtained on related materials they are estimated to be several hundred bars, giving P,, = 4 . (lo8) dyn/cm2 used in the example. Afanas'ev and Bobolev used an indirect impact method and obtained the approximate correlation(")

P,, = 2.46 T, (9)

where T, is the melting point of the material in "C, and P,, is in bars. On this basis the estimated shear strength of TNT, PETN, RDX and HMX is 199,347, 502 and 672 bars, respectively. It is of interest to note that the predicted increased shear strength of HMX over TNT (factor of 3.5) is roughly consistent with the increased magnitude of the friction coefficient of HMX over TNT-based explosives given in Ref. 9.

2.3. Friction shear mechanisms

The force required to pull the friction element through the explosive depends on the shear stress that is necessary to break the contacts between the sliding surfaces (move the element asperites through the explosive). For small friction velocities and loads the friction surface of the material may merely undergo plastic deformation. The shear stress would then correspond to the stress necessary to deform the material under the local conditions. However, at higher velocities and loads the shearing material would be expected to largely undergo surface fracture, since the material would not have time to deform to the contours of the element. The shear stress in this case would correspond to the shear strength of the material (as used in the preceding example). Moreover, depending on the

material and conditions, the shear strength may be a function of the shear velocity, since it is well known that the strength of a material usually increases with decreased loading times (in a certain time range). For example, it has been found(") that the time, t, required to fracture many materials depends on the tensile stress, o, according to the empirical equation

t = to exp [(E - Bo)/RT] (10)

where T is material temperature, R is the gas constant, and to, E and B are experimental constants. The shear strength should likewise depend on the loading time. However, after surface melting sets in the shear strength would be controlled by the fluid viscosity, rather than the strength properties of the solid material.

2.4. Effect of friction coefficient on hot spot temperature

The substitution of Eqs. (1) and (7) into Eqs. ( 2 ) or (3) shows the functional effect of the friction coefficient on the hot spot temperature. For the frictional heating of an explosive arising from shock or impact (as is largely under consideration here), the loading pressure P in Eq. (7) can in most cases be considered to be roughly the same as (or proportional to) the shock wave pressure. For shock pressures greater than several kbar fie would most likely be controlled by the ordinary compressibility, 6, of the material. For a typical explosive material, p = 3 . (lo-") cm2/dyn, and decreases with increasing pressure. Thus the exponential in Eq. (7) can for the present illustrative purposes be expanded and only the linear term retained. As noted before, r in Eqs. ( 2 ) and (3) corresponds to the radius of a single contact (hot spot) area between the sliding surfaces. Thus Al must be redefined in terms of a single contact. From the definition of r and Eqs. (5) and (6)

where A and Al now correspond to the area (and limiting area) of a single contact point. For a heterogeneous material whose individual particles are all the same size and shape, the friction points will arise at the particle surface, and the limiting friction area, A*, should be roughly equal (or proportional) to the area of the particles, i.e.,

A1 = fnd2/4 (12)

where d is particle diameter and f is a constant. Substituting the preceding equations into Eqs. (2) and (3) then gives, respectively

0.418 * (fpP)0.5 . P,, . vd Ts = To f

(K1 + K2)

2.5. Discussion

The two preceding equations show the effect of the control- ling factors and material properties on the hot spot temperatures produced in a material during a friction event. Since a higher hot spot temperature generally enhances the ignition


and reaction ability of an explosive or propellant material, the equations give qualitative information regarding the effect of the various parameters on the frictional sensitivity of the material. This will now be discussed. Some factors relating to the ignition and reaction buildup processes in the material will then be considered.

To begin with, it should be noted that Eqs. (13) and (14) apply to a single surface friction contact, and that in a real material there would be various sized contacts (due to the presence of different particle sizes and shapes, and perhaps composition variations). The values of PX,and v may also vary in different localized regions of the shearing material. Thus a friction event would produce various size hot spots with different temperatures, but not all hot spots would be effective in initiating reaction in the material. For orientation purposes, the hot spot temperature computed by these two equations is 8506 K and 2782 K, respectively, using the following values for the parameters: P = 10" dyn/cm2, P,, = lo9 dyn/cm2, v = 300 cmis, b = 3 . (lo-'') cm2/dyn, d = 100 pm, K1 = Kz = cal/(cm . s K), el = 1.6g/cm3, C1 = 0.4cali (g K), f = 1. Based only on the magnitude of the numbers, the friction temperature would be sufficient to induce instantaneous reaction in the hot spot material in this case. The early studies using simple experiments showed that in frictional heating the hot s ot temperature was limited by the lowest

and in any event do not include the effect of the heat absorp- tion by melting.

melting material(' P . Thus the preceding values may not apply,

2.5.1. Effect of shock pressure on friction parameters

In addition to its direct effect on friction temperature as given in Eqs. (13) and (14), the shock pressure can have indirect effects. For example, the high loading pressure in shock waves would increase the melting temperature. This effect can be estimated in many cases using the Simon equation("). A more approximate equation is given(") by

T, = T, + 20 P

where T, is the normal melting temperature ("C), and T, is the melting temperature at pressure P (in kbar). This equation shows that for shock pressures of the order of a kbar the increase in melting temperature is relatively small (20 "C), but at higher pressures the effect would be much more significant (200 "C at 10 kbar).

It is also known that high pressure increases the shear strength of solid materials, and high temperature decreases the strength(13). These effects are in addition to the potential effect that velocity may have (mentioned earlier) on the shear strength, which is related to shear rate effects. They further complicate the accurate estimation of hot spot temperature in a friction event. Towle and Riecker(13) found that the shear strength of many materials at temperature T and pressure P is given by an equation of the form

Pxy = Pxy0 + (1 + a P ) exp(bT/T,) (16)

where Pxyo is shear strength at ambient conditions, T, is melting temperature at pressure P, and a, b are experimental constants. Bridgman measured the shear strength of several explosives as a function of pressure and obtained values close to 4 kbar for all materials at a pressure of about 50 kbar(14). He stated that the effect of pressure was much greater than for metals, but did not give any values except fur the preceding

number. Studies have shown that the shear strength of polymers often increases by a factor of up to 5 to 10 or more at pressures of 5-10 kbars(").

If the hot spot temperature is limited by melting, then the temperature could be estimated under various shock loading conditions by the effect of the pressure on the melting point. However this limitation probably does not hold under shock loading conditions, since viscous heating should further increase the temperature.

The material compressibility also depends on the pressure, and is determined by the shock Hugoniot of the material. The compressibility decreases with increased pressure. According to Cook and Rogers(16) the compressibility f i at pressure P of a material is often given an equation of the form

where Po is the normal compressibility at atmospheric pressure, a = G + 1, and G is the Gruneisen parameter (the value of a is usually determined by fitting the integrated form of this equation to experimental shock Hugoniot data). It should be noted that the effect of temperature on compressibility is contained directly in this equation.

2.5.2. Effect of the friction parameters

The most interesting effect obtained by including the functional form of the friction coefficient in the heating equations is that the resulting equations (Eqs. 13, 14) show that the hot spot temperature (hence the ignition sensitivity of the material) increases with an increase in the particle size of the material. Early studies had shown that larger grit particles are more effective in producing hot spots than smaller particles('), but Eqs. (2) and ( 3 ) suggest the opposite effect. Studies by Seely (quoted in Ref. 17) on the shock sensitivity of tetryl showed that large particles have a lower initiation threshold than small particles, and this effect was also found by Roth(17) for RDX, and by others. More recent studies by Taylor and Ervin("), Howe(") et al. and others have shown that the ignition and subsequent reaction buildup are independent processes; and that large particles enhance the ignition reaction, and small particles enhance the reaction buildup. That Eqs. (13) and (14) predict this particle size effect on the ignition sensitivity gives some support to the hypothesis that frictional heating may be involved in the shock initiation of materials. However, it is not presently known whether the observed effect is actu- ally due to frictional heating, or rather is the result of other heating mechanisms such as shock compression.

Equation (11) shows that the area of a hot spot increases with increased loading (shock) pressure, up to a limiting value that is determined by the particle size. This result is of some interest, since it is known that ignition and buildup of reaction are enhanced by a larger hot spot size. Thus, both and the size and temperature of the hot spots produced by friction are increased by an increase of the shock pressure.

The equations show that increased material shear strength increases the hot spot temperature. Data on the effect of this parameter are not available for explosives, but studies with other materials have shown that hot spot temperatures are higher for harder materials than for softer ones(','').

Experimental data on the effect of the compressibility coefficient are likewise not available. The independent effect of this parameter may be difficult to evaluate, since its value and that of the shear strength are generally interrelated. A lower


compressibility usually implies a higher shear strength. Thus the variational effects of these parameters may be partially cancelling, the shear strength predominating.

The predicted effect of friction velocity, loading pressure and thermal conductivity on hot spot temperature has been demonstrated experimentally in the literature many times; increased velocity, increased load and decreased thermal conductivity increases the hot spot temperature. There is some interrelation between some of the parameters, as just noted. Thus in addition to the relation between p and Pxy, it would be expected that an increase in shock pressure would generally also cause an increase in friction velocity. The interpretation of friction effects can also sometimes be difficult, if more than one parameter can cause the effect. For example, the use of hard grit particles on rubbing surfaces is more effective in causing the initiation of an adjacent explosive than the use of softer grit particles. This has ordinarily been considered to be the result of the more concentrated load of the hard particles. However, depending on conditions, the increased shear strength of the harder particles could also be involved. In any event, it would appear that the predicted effects of the various

The curve for PETN also applied to nitroglycerin, as the two curves are essentially the same.

The reaction time of the heated material decreases rapidly with increasing temperature, the time changing by two or more orders of magnitude for only a hundred degree change in temperature in some cases. The temperature necessary to produce a specified reaction time is much smaller for a sensitive explosive such as PETN than for a relatively insensitive explosive such as TNT.

3.2. Hot spot cooling

The time required for the material to react is of considerable consequence, since the temperature at a localized point will only last for a very short time period, and then the hot spot undergoes conductive cooling - which may quench the reaction. The depth that the heating penetrates into the material surface during the friction process depends on various factors. For a surface temperature of T,, the temperature T produced by heat conduction at a distance y beneath the surface at time t

material properties and friction parameters on hot spot temperature are in qualitiative agreement with the known experimental evidence.

is given by

T = T, - (T, - To) erf [y/(4a1t)”’] .

Application of this equation shows that in time periods of the order of a microsecond, appreciable temperature rise is limited to conductive depths of only several micrometres at the most in the surface of the material. However, the actual depth may be greater due to variation in the size of the friction

3. Ignition and Reaction Buildup Considerations

3.1. Adiabatic reaction time

Ignoring transient effects, the contact points in the friction surface are heated instantaneously to the hot spot temperature, T,, and the heated explosive material then undergoes simultaneous cooling and reaction. A variety of processes can be involved prior to the major exothermal reaction, including melting, pyrolysis and multistage reaction. However, little information is generally available regarding most of these processes. For orientation purposes here, if it is assumed that the induced reaction is adiabatic in nature, then the heated material undergoes a reaction delay time, t,, followed by very rapid reaction to completion, where

t, = - CRT’ exp(E/RT,) ZQeE

E is activation energy, Z is frequency factor, C is heat capacity, Qe is effective heat release and R is the gas constant. Since the hot spot reaction occurs under essentially constant pressure conditions, Q, = Q/y, where Q is the conventional constant volume heat release and y is the adiabatic exponent.

The reaction time according to Eq. (18) is a sensitive function of temperature. The top curves in Fig. 1 illustrate the calculated values for different explosives, using the following values of the parameters:

contacts. Small hot spots may be roughly spherical in shape, but large diameter spots are probably more disk-like in nature.

Subsequent to its formation, a hot spot in the friction surface will undergo simultaneous reaction and cooling. For sufficiently high temperatures the reaction will be very rapid (instantaneous) but for smaller temperatures the delay time will be longer. Near the critical (minimum) conditions just sufficient to initiate the explosive, the delay time will be com- parable (but not necessarily equal) to the time that the material is subjected to the shock pressure. Sufficiently small hot spots will undergo conductive cooling and consequently not react, but larger hot spots will undergo rapid reaction after the reaction delay. For illustration, according to Zir~n(’~) the critical (minimum) spherical hot spot diameter, D, at which reaction will just occur if its temperature is T, is given by

D = 10. (altJ1” . (20)

The lower curves in Fig. 1 show the values computed for the preceding explosives, using a1 = cm2/s. The computed critical hot spot diameters are small, and decrease rapidly with increasing hot spot temperature. The critical size at a specified temperature is smaller for a sensitive explosive like PETN than for a relatively insensitive explosive like TNT. Hot spots smaller than the critical size undergo conductive cooling and hence do not react. One dimensional conductive cooling (which may be more appropriate for a disk-like hot spot) has been discussed by Friedmad2’).

C [cal/(g. K)] 0.45 Q rcalld 1027

0.45 0.45 1326 1383

3.3. Ignition and reaction buildup - I-

2.73 2.74 2.81 The adiabatic reaction of a hot spot can aptly be described Y E [cal/mol] 41 100 52 700 47 000 Z[SC’] 1.585. l O I 3 5.01 . lo19 6.31. 10’9 as a micro-explosion. If this reaction can initiate a propagating

reaction in the adjacent material, then the adiabatic decom-


600 700 800 900 1000 1100

HOT SPOT TEMPERATURE [Kl - Figure 1. Computed adiabatic reaction time (tr) and critical hot spot diameter (D).

position reaction can be considered to constitute the ignition or initiation reaction of the friction event. Thus in the case of homogeneous liquid explosives, the thermal explosion produced by shock compressional heating initiates detonation in the liquid. The formation of hot spots by the addition of defects to the liquid before it is shocked can also lead to detonation under suitable conditions. The properties and initiation characteristics of shock-produced hot spots have been discussed by Mader(26).

On the other hand, for solid explosives (which on a small scale are always heterogeneous in nature) the present evidence suggests that there are additional factors (besides adiabatic hot spot reaction) that control the shock initiation process. Whether this is also true for a friction initiated reaction is not known, but is expected to be under conditions that the shock creates the friction process. As noted before, the adiabatic decomposition reaction time (Eq. 18) is very sensitive to temperature, and hence to pressure. Thus small changes in shock pressure should be sufficient to change the reaction time by a large factor. In actuality, however, the shock initiation time of a solid explosive is much less sensitive to pressure than implied by Eq. (18), the apparent activation energy of the overall reaction being only a few kcal, rather than the large values found for the decomposition reaction(’’).

It has been shown by Walker and Wasley(”) and others that the shock initiation to detonation of an explosive charge appears to require that a certain critical energy per unit area, E,, be delivered to the charge, where

E, = Put = P2t/e&J = K (21)

P, u and U are the pressure, particle velocity and propagation velocity of the shock, t is the shock duration, eo is charge density and K is an experimental constant for a particular charge. The effect of pressure on reaction time as given by this equation is much less than that implied by Eq. (18). Although it has been found that Eq. (21) is in many cases only an approximation to experimental data, the pressure dependence of the initiation reaction given by this equation is usually much more realistic than that given by Eq. (18). The meaning of Eq. (21) is not presently known.

The fundamental meaning of what constitutes ignition in a friction initiated reaction is thus somewhat in doubt. It is clear that an exothermic decomposition reaction must occur, but other factors are also involved that help control the overall ignition event. For example, the size and number of hot spots are probably important. Thus although the hot spot temperature may be sufficient to induce immediate reaction, the size of the hot spots may be too small for the reaction to induce a propagating reaction in the adjacent material. It has been shown experimentally that very small isolated hot spots produced by friction can undergo reaction without inducing reaction in the adjacent material(’). It has also been shown that numerous hot spots can lead to initiation by cooperative effects@).

The length of time for which the surface is maintained at the hot spot temperature is another factor, i.e., the time for which a portion of a contact surface is maintained in contact with a sliding surface. For a hot spot with diameter D, a portion (one edge) of the surface will be maintained at the full friction temperature for a time period given by

t = D / v . (22)

Thus a higher friction velocity (higher shock pressure) decreases the contact time. According to Eqs. (13) and (14), a higher friction velocity gives a higher hot spot temperature, which leads to faster reaction. Therefore the friction velocity has two opposing effects on the initiation process, and depending on conditions there will be an optimum value (the friction velocity may also affect the shear strength, as discussed before). Also related to the surface contact time and hot spot cooling are the effects of multiple friction points of various sizes, and the time dependent nature of the event.

4. Conclusions

In this paper an approximate expression was developed for the friction coefficient of a material in terms of the parameters that control the friction shear. The results were used in conjunction with the frictional heating equations and thermal decomposition kinetics to discuss the factors that control the sensitivity of explosive materials to frictional heating under various conditions. The predicted effects are in good qualitative agreement with known experimental evidence. On a more quantitative basis, however, it was shown that the processes involved can be quite complex. It is evident that the more closely the frictional initiation event is examined, the more closely one arrives at the proverbial can of worms.

5. References

(1) F. P. Bowden and A. D. Yoffe, “Initiation and Growth of Explo- sion in Liquids and Solids”, University Press, Cambridge 1952, Chapt. 11.


(2) H. Eyring, R. E. Powell, G. H. Duffey and R. B. Parlin, Chem.

(3) J. C. Jaeger, ‘Proc. Roy. SOC. 76, 203 (1943). (4) I. Shapiro and I. M. Kolthoff, J . Phys. Colloid Chem. 51, 493

( 5 ) R. S. Spencer, G. D. Gilmore and R. M. Wiley, J . Appl. Phys.

(6) C. L. Huffine and C. F. Bonilla, AlChE Journal 8, 490 (1962). (7) C. L. Sieglaff and M. E. Kucsma, J . Appl. Phys. 34, 342 (1963). (8) A. S. Dyer and J . W. Taylor, “Initiation of Detonation by Fric-

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Acknowledgements

This work was supported by the Army Research Office. The writer also whishes to acknowledge the encouragement of this work by Dr. L. Zernow of Shock Hydrodynamics.

(Received April 25, 1980)

role of the friction coefficient in the frictional heating ignition of explosives

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