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INDUSTRIAL AND ENGINEERING CHEMISTRY
(58) Losanitsch, S. M., Ber., 42, 4394 (1909). (59) McLean, D. A., and Egerton, L., ~ N D .ENG.CHEM.,37, 73 (1945). (60) McLean, D. A., Egerton, L., and Houtz, C. C., Zbid., 38, 1110 (1946). (61) McLean, D. A., Egerton, L., Kohman, G. T., and Brotherton, M., Ibid., 34, 101 (1942). (62) Manion, J. P., and Burton, M., J. Phus. Chem., 56, 560 (1952). (63) Mark, H., in “Elastomers and Plastomers” (R. Houwink, editor), vol. 1, p. 155, Elsevier, New York, 1950. (64) Mayo, F. R., Gregg, R. A., and Matheson, AI. S., J. Am. Chem. Soc., 73, 1691 (1951). (65) Melville, H. W., and Valentine, L., Trans. Faraday SOC.,46, 210 (1950). (66) Melville, H. W., and Watson, W. F., Zbid., 44, 886 (1948). (67) Michaelis, L., Ann. N . Y . Acad. Sci., 40, 39 (1940). (68) Michaelis, L., and Fetcher, E. S., Jr., J . Am. Chem. Soc., 59, 1246 (1937). (69) Michaelis, L., Schubert, M. P., Reber, R. K., Kuck, J. A,, and Granick, S., Ibid., 60, 1678 (1938). (70) Minder, W., and Heydrich, H., Disc. Faraday Soc., No. 12, 305. 1952. (71) illorgan, C. G., and Harcombe, D., Proc. Phys. SOC.(London), 66B, 665 (1953). (72) Nozaki, K., and Bartlett, P. D., J . Am. Chem. Soc., 68, 1686 (1946). (73) Pollitt, A. A., J . Inst. Blec. Engrs. (London), 90, Pt. 11, 15 (1 943). (74) Preckel, R., and Selwood, P. W., J. Am. Chem. Soc., 63, 3397 (1941). (75) Price, C. C., “Mechanism of Reactions at Carbon-Carbon Double Bonds,” pp. 85-7, Interscience, New York, 1946.
Vol. 47, No. 9
(76) Rollefson, G. K., and Burton, M., “Photochemistry and the Mechanism of Chemical Reactions,” p. 364, Prentice-Hall, New York, 1939. (77) Sauer, H. A., McLean, D. A., and Egerton, L., IND.EXQ. CHEM.,44, 135 (1952). (78) Schoepfle, C. S., and Connell, L. H., Ibid., 21, 529 (1929). (79) Schoepfle, C. S., and Fellows, C. H., Zbid., 23, 1396 (1931). (80) Scholl, R., Ber., 54, 2376 (1922); Ibid., 56, 918 (1923); Ibid., 64,1158 (1931). (81) Smyser, H. F., and Smallwood, H. M., J . Am. Chem. Soc., 55, 3498 (1933). (82) Stanley,”. M., and Nash, A. W., J . SOC.Chem. Znd. (London), 48, 238T (1929). (83) Stern. M., and Uhlig, H. H., J . Electrochem. Soc., 100,543 (1953). (84) Thomas, C. L., Egloff, G., and Morrell, J. C., Chem. Reus., 28, 1 (1941). (85) Vance, J. E., and Bauman, W. C . , J . Chem. Phus., 6,811 (1938). (86) Waters, W. A., “Chemistry of Free Radicals,” 2nd ed., p. 92, Oxford Univ. Press, 1948. (87) Waters, W. A., Disc. Faraday Soc., 14, 247 (1953). (88) Watson, P. K., and Higham, J. B., Proc. Inst. Elec. Engrs. (London), 100, Pt. IIA, 168 (1953). (89) Werner, J. K., Spector, L. J., McLean, D. A.. presented at Confeience on Electrical Insulation, Div. Eng. and Ind. Research, Nat’l Research Council, Pocono Manor, Pa., Oct. 19, 1953. (90) Wheland, G. W., “Advanced Organic Chemistry,” 2nd ed., p. 657, Wiley, New York, 1949. (91) Wheland, G. W., “Theory of Resonance,” p. 199, Wiley, New York, 1944. (92) Worner, T., ErdoE u. Kohle, 3, 427 (19.50). (93) Ziegler, K., Eimers, E., and Wilnes, H., Ann., 567, 62 (1950). RECEIVED for review March 26, 1954.
ACCEPTED March 9, 1955.
Lead-Block Test for Explosives WILLIARZ E. GORDON, F. EVERETT REED, AND BESSIE A. LEPPER Arthur D . Little, Inc., Cambridge, Mass.
R
ESEARCH workers in explosives in recent years have been largely concerned with making a theoretical synthesis of the primary explosive act and its secondary effects. Although they have made considerable progress in the technical development of explosives, they must still rely partly on long-established empirical tests. These tests, such as plate-dent, sand-crush, ballistic-mortar, and lead-block, have themselves been studied at some length to determine what they mean in terms of explosive properties. The work reported here has been concerned with clarifying the meaning of the so-called Trauzl lead-block test. (Because of the semantic confusion surrounding the terms “power” and “brisance,” which have traditionally been associated with several of these empirical tests, these words are not used in the discussions.) The Trauzl test, specifications for which were established by international convention in 1903 (IO), has been used widely in Europe for comparing mining explosives. I n the standard test 10 grams of explosive are placed with a detonator in a central cavity in a cylindrical block of lead (Figure l),and sand is tamped down on top to cover the charge preparatory to firing. After the shot, the volume of the enlarged cavity is measured by filling it with water, and the net expansion due to the explosion is computed. The figure of merit is taken as the ratio of the net expansion for the explosive t o that for 10 grams of a standard explosive, multiplied by 100. Many investigations (2, 3, 6, 7 , 9, I d ) have been made of the effect of the loading density of the explosives, the fineness of the sand used in tamping, the temperature of the molten lead a t the time of casting, and the temperature of the block. None of these variable factors, except the temperature of the block, has a pronounced effect on the result of the test; and by standardization of the test procedure, i t is possible to get good precision. The standard deviation of duplicate tests is usually about 1%.
With some of the more energetic explosives, such as RDX, cracks are produced in the block. The cracks follow a characteristic pattern, and in severe cases, a conical section at the base separates completely (see Figure 2). Therefore, the standard test is very limited in its application to modern military explosives. Haid and Koenen ( 3 ) eliminated the cracks in some cases by using a larger than standard block. I n the present work, a
L
s
Figure 1.
b
-
Standard Trauzl test
September 1955
INDUSTRIAL AND ENGINEERING CHEMISTRY
1795
where a crack has just started, it appears a t a point about midway between the corner and the cavity. The direction of the cracks has been measured in several photographs of blocks containing various explosives and charge weights. The results are summarized in Table I. The cracking phenomena may be readily interpreted with the aid of Figure 3, which represents a simplified picture of the shock waves set up in the block by the explosion. It is considered that a spherical shock front originates at the center of the charge (Figure 3,a). Upon reflection at the surface, the compression wave becomes a rarefaction or tension wave (Figure 3 . b ) . When the tension waves from the side and end surfaces meet (Figure 3,c),
Figure 2.
Cylindrical block w i t h RDX/TNAI, showing severe cracks
9-Gram sample RDX 6.921 g., TNAM2.079 g.
T = 3‘ C .
spherical block was used, and cracks were completely absent. The spherical shape suggested itself when the cause of cracking in the cylindrical block was ascribed to the convergence of tension waves reflected from the sides and ends of the cylinder. The interpretation of lead-block measurements in terms of the properties of the explosive has been considered by a number of investigators. Naoum (7) correlated lead-block values with a calculated quantity called the “specific pressure.” Haid and Koenen ( 3 ) considered that the phenomena in the block were more closely related to another quantity involving dentiity, detonation velocity, and temperature, the physical meaning of which seems rather obscure. I n addition to the variety of opinions concerning the meaning of the test, there have been alternative views on the manner in which the results of the test should be expressed ( 1 3 ) . This paper considers first, from a general point of view, the problem of deriving from any empirical test of t,his sort a figure of merit which relates only to the properties of the explosive and not to those of the test device. Secondly, by deriving an expression for the energy of plastic deformation of the lead, the authors have attempted to demonstrate that the figure of merit from this test gives a direct measure of an energy quantity characteristic of the explosive itself. Finally, they correlated results of lead-block tests for several explosives with values of the perfect gas expression nRT, calculated in a formal manner with the aid of thermochemical data.
a
b
.-
48.8’ SPHERICAL LEAD BLOCK
-4good example of the cracks produced in the standard cylindrical lead blocks by certain explosives is shown in Figure 2. The particular explosive used in this case was a mixture of RDX and tetranitromethane in the ratio by weight of 77 to 23; homever, the cracking effect is observed with pure RDX, tetryl, and a number of other explosive compounds. Particularly striking is the fact that the cracking is not closely related to the expansion of the cavity; it depends, rather, upon the nature of the explosive. For example, a charge of 6 grams of an RDX-tetranitromethane mixture causes cracks, although i t produces a cavity of only 290 cc.; however, 10 grams of a TXTtetranitromethane mixture do not give cracks, although the cavity is 596 cc., more than twice as large. Examination of photographs such as Figure 2 reveals a remarkable uniformity in the nature of the cracks. The line of direction of a crack always passes through the corner, although in most cases the crack itself does not extend to the corner. I n cases
41.2’
1
Figure 3.
Configuration of shock and
tension waves in cylindrical block
t f Direction of propagation of wave front
0+ Direction particle velocity - - - Wave frontconvergence tension waves of
Locus of
of
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Figure 4.
Block w i t h lower half hemispherical 10 Grams of pure RDX.
T = 19’ C.
the tensile stresses combine. Cracks presumably start a t a point where the tensile yield strength is exceeded, and may be further propagated to some extent by stress concentration a t the ends of the crack. The regions of high tensile stress will lie along the loci of convergence of tension waves on conical surfaces passing through the corners (edges), and making angles of 48.8”and 41.2’ with the cylindrical surface of the block (Figure 3 , c ) . These angles are determined, of course, by the geometry of the block and charge location. The agreement of the angles measured from photographs with the predicted values (Table I) is excellent.
T a b l e I.
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plosives on the basis of their ability to cause cracks in a cylindrical block suggests interesting possibilities for further study. A few experiments were done to confirm the explanation for the cracking effects. A most convincing result is illustrated in Figure 4. The lower h9lf of the block, in this case, was machined to a hemisphere, and a charge wae fired in it which would normally cause cracks in both the upper and lower halves of the block; no cracks were produced in the hemispherical section, I n another experiment, a longer-than-standard cylindrical block was cast; however, it had the normal diameter and depth of charge cavity. No cracks developed at the lower corners in this case, presumably because the wave was more attenuated in the longer distance of travel. I n another experiment, one of the standard blocks, loaded in the usual way, was submerged in water and fired; the idea was that because of the closer im’pedance match between water and lead than between air and lead, the tension in the reflected wave would be reduced. However, the effect of water was not great enough to prevent cracks, which, in the experminent, were just about the same as those obtained with air. A4ccordingto the foregoing experiments, cracking can be eliminated by using a completely spherical block [first reported as being used b y Lodati (S)]. Subsequent experiments bore out this conclusion, and a mold was constructed for casting spherical blocks on a routine basis. Some 300 tests have been carried out with spherical blocks, and no cracks have been found, except in blocks which had flaws in the casting. Figure 5 shows the result of a typical test with a spherical block, in this case with a charge of 10 grams of Medina, which would be expected to cause severe cracks in a cylindrical block. The spherical block was 8 inches in diameter, and the charge well was 1 inch in diameter and 4.5 inches deep. The sprue a t the base of the casting was 2.5 inches in diameter; this was cut off to leave a flat surface on which the block could rest. A diagram of the loaded spherical block is shown in Figure 6, and one of the cast-steel mold in Figure 7.
Direction of Cracks in Standard Lead Block
Angle with Vertical Side, Degrees Average Theoretical Location anglea value (Fig. 3c) 5 49 2 (*to 7) 48 8 Upper 41 3 ( 5 0 5 ) 41 2 17 Lower Estimated stapdard deviation of mean.
No. of Measurements
Rinehart and his associates have studied the phenomena of cracking and “scabbing” in metals, and from these and other effects have been able to derive considerable information about the actual pressure contours and particle velocities in metals (11). Most of the details of the cracking phenomena in cylindrical lead blocks can be interpreted, a t least qualitatively, on the basis of Rinehart’s work. For example, the corners are still in compression when the tension waves first intersect there (Figure 3 , b ) ; therefore, the tendency t o crack is not as great in the corner as it is further in along the convergence locus, even though the magnitude of the tension diminishes as the waves advance. A similar phenomenon is observed in scabbing of a flat plate, which occurs a t some distance from the surface (11). Also, there is a smaller tendency for crack formation in the upper corners of the block than in the lower corners, because of the greater attenuation suffered by the waves in traveling the longer distance to the upper corners. A most interesting aspect of the cracking phenomenon is the fact that the tendency to form cracks is largely dependent on the kind, and not the amount, of explosive. As pointed out above, the occurrence of craclrs seems to be scarcely related to the size of the cavity produced by an explosive charge. One is led to conclude that the peak pressure (apd tension) in shock waves from different explosives is largely dependent on the kind, and not on the amount, of explosive. This result is quite in accord with the theory of detonation. The sharp differentiation of ex-
FIGURE OF MERIT
An important question in any empirical test concerns the manner of deriving a figure of merit. I n the standard lead-block procedure ( 7 ) , the expansion volumes for equal weights of two explosives are compared. Logically, there is no reason why the depth of a cavity, or any other parameter, might not be used instead of the volume. However, a number derived in an arbitrary manner has not necessarily any meaning except in terms of the actual device and test conditions to which it refers. An alternative way of deriving a figure of merit in certain cases is to find the weights of two explosives which are equivalent in the
Spherical block w i t h charge that would produce cracks in cylindrical block
Figure 5 .
LO Grams of Medina T = 23’ C
INDUSTRIAL AND ENGINEERING CHEMISTRY
September 1955
test concerned, as, for example, Tolch and Perrott ( 1 3 )have done i n the lead-block test, to find the weights which give equal expansion volumes, Such a method is essentially a null method. Cole ( 1 ) points out that when explosives are compared on the basis of equal effects in underwater work, concordant results are obtained from several different kinds of tests. The meaning and validity of this equivalent-weight representation in explosive testing are examined in the following analysis. Consider a test which yields some measurable indication, I , such as a length, a volume, etc. The quantity I may be a function of several properties of the explosive charge, such as heat of explosion, available work, detonation velocity, or detonation pressure. (In addition, of course, I is functionally dependent on various parameters of the test device; but, because the test is standardized, these parameters appear as constant factors in the functional expression.) Heat and Ivork are examples of what have been termed extensive properties; the quantity G representing such a property is proportional to the mass of the explosive, m-i.e., G = gm. Detonation velocity and pressure, on the other hand, are examples of intensive properties, which are independent of the amount of explosive (aside from certain nonideal effects which disappear under limit conditions). Representing the intensive properties by the symbol g', we may write, in general,
I
1792
whether only a single extensive property is affecting the test indication.
A corollary result from Equation 4 is that if the same extensive property is alone involved in two different tests, the equivalent weights of explosives will be the same. It has often been conjectured, for example, that the ballistic-mortar and lead-block tests measured the same property of explosives, but it was commonly observed that the more energetic explosives gave relatively higher values in the lead block than in the ballistic mortar by the equal-weight testing method. This question could be settled by applying the equivalent-weight technique to both tests. (Comparisons made by the authors indicate that there is a good correlation between lead-block and ballistic-mortar equivalent weights, except in the case of explosives with a positive oxygen balance, where there is some evidence that reaction of explosive products with combustible materials in the detonator rap was taking place in the ballistic mortar.)
(1)
f(gmJ g')
I n the case where I is a function of only a single extensive property or linear combination of such properties, we can write €or two different explosives 1 and 2 I1
(2)
= f(91rnl)
I2 = f(9zmz) If I , is made equal to I?, then
SPHERlE 8
Figure 6.
D
Spherical lead block
Thus in this case the ratio of the equivalent weights is a conetant, independent of the actual weights and equal to the ratio of the specific coefficientsg for the two explosives. If the comparison is made in terms of equivalent weights, the properties of the test device implicit in the function f are eliminated. Only if I were a linear function of the charge weight would a direct comparison of the test indications for equal weights of explosives give such a result. I n general, then, the equivalent-weight comparison is to be preferred. If, for a given test, the relative equivalent weights are found to be constant over a considerable range of charge weight, it can be concluded that relative values of some intrinsic property of the explosives are being obtained; this constancy of the equivalent weights is, in fact, a criterion for determining
Figure 7.
Steel mold for spherical block
If it is found, in a given test, that the equivalent weights vary with the actual weights used, it can be concluded that more than one property is aff'ecting the test indication. Tests of this sort are probably less useful than ones in which unambiguous equivalent-weight values can be assigned. I n a test which involves intensive properties only, the test indication will be independent of the weight of the charge. Often this condition of charge-weight independence appears as a limiting phenomenon as the charge weight or dimensions are increased, just as, for example, the detonation wave velocity ap'proaches a limiting value as the charge diameter is increased. Unlike the situation for an extensive property, there is no way to measure an intensive property which is independent of the functional behavior of the test device. ENERGY OF PLASTIC DEFORMATION
A plausible intuitive assumption is that the expansion volume in the lead block is a measure of the work of plastic deformation. However, the relationship between energy and volume is not linear; as the cavity expands and the walls become thinner, less work is required for succewive increments of volume. Therefore, when a series of increasing weights of explosive is fired in the lead block, the volume of expansion per gram increases significantly. This is illustrated in Figure 8, which represents the data for a series of TST charges fired in the spherical lead block.
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INDUSTRIAL AND ENGINEERING CHEMISTRY
1798
In these experiments, the average values of the quantities vo, v , and V were 10, 400, and 4000 cc., respectively. Using these values in order to evaluate the expression in square brackets in Equation 11, a value close to unity is obtained. Throughout the range of these experiments, the total variation in this bracketed quantity is about 0.1; and as the quantity log V log v ) has an average value of about 2.3, it is permissib e to write, within a sufficient degree of approximation,
5
-
If we assume that for a given explosive the energy of plastic deformation is proportional to the weight of charge, we obtain as the relationship between weight w and net volume of expansion 21,
(v,
=
v
- vo)
W -
= a
- blnv,
(13)
Vn
VOLUME (CC.) Figure 8. Plot of data for series of TNT charges (8 to 18 grams)
where a = 2- s o (In V f I ) and b = 32- s-,o c being the proportional3 c c ity constant connecting the charge weight and the energy.
Showing increase of expansion per gram with increasing volume
The exact relationship between the work of plastic deformation and the volume of expansion can be derived for a spherical body with a completely enclosed spherical cavity a t the center. Consider the force on a spherical polar element (Figure 9). The radial stress is u,; the tangential stress is ut, From considerations of symmetry, it is apparent that the tangential stress is uniform in all tangential directions. The net force on a volume element in the radial direction may be equated to zero in a quasi-static process (acceleration is zero). This leads to the condition:
(5) Applying the von Mises flow equation to this case, we obtain the condition necessary for flow: UT
-
gt
=
so
(6)
where SOis the yield stress of the material. Eliminating u cfrom Equation 5 and integrating, with boundary conditions of zero pressure a t the outer radius, a, and pressure, p , a t the radius of the cavity, b, we obtain the expression for p , the pressure in the cavity just necessary to cause flow,
(7) where V is the volume of material in the body (a constant) and u is the volume of the cavity. Following from Equation 7 , the work of expansion, E, from an initial cavity of volume vo to a final volume v is
E
=
;SO 1 1 ” (1
+ !)
dv
Equation 13 is the relationship to be expected, on the basis of this somewhat idealized analysis, between weight of charge and volume of cavity produced. The analysis has, of course, neglected several important factors, such as the effect of the opening in the lead block provided by the charge well, the effect of the sand used for tamping, and dynamic effects associated with the exploding charge and the shock wave. I n fact, the analysis is not in any sense a “theory” for the phenomenon occurring in the lead-block test; simply, an expression has been obtained for computing the energy of plastic deformation associated with a given size of cavity, and the assumption is made-to be verified by experiment-that this derived energy quantity varies linearly with the charge weight, Accordingly, if the assumption is correct, a linear relationship between w/v, and log v, mill be found for any given explosive. Furthermore, the relative values of the proportionality constant, c, for different explosives will then provide a direct equivalent-weight type of comparison, as described above. An examination of test results on the basis of Equation 13 is made in the following section. TST was chosen as the “standard” explosive and values of a and b were obtained from a least-mean-squares treatment of the data for this explosive. The data for other explosives were then fitted to the curve for T N T by s u i t a b l e choice of the constant k
Integration of Equation 8 gives =
a - b log v n (14)
ViI
