Fracture Behavior of Tubular BombsChester R. Hoggatt and Rodney F. Recht Citation: Journal of Applied Physics 39, 1856 (1968); doi: 10.1063/1.1656442 View online: http://dx.doi.org/10.1063/1.1656442 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/39/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Plastic Behavior and Fracture of Aluminum and Copper in Torsion Tests AIP Conf. Proc. 907, 521 (2007); 10.1063/1.2729566 Erratum: Molecular dynamics simulations of bending behavior of tubular graphite cones [Appl. Phys.Lett.85, 1778 (2004)] Appl. Phys. Lett. 85, 4538 (2004); 10.1063/1.1821651 Fracture toughness estimation of thin chemical vapor deposition diamond films based on thespontaneous fracture behavior on quartz glass substrates J. Appl. Phys. 82, 6056 (1997); 10.1063/1.366473 Nonlinear quasifracture behavior of polymers J. Appl. Phys. 57, 170 (1985); 10.1063/1.334838 Fracture Phys. Today 19, 49 (1966); 10.1063/1.3048102

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1856 MILLER, BRUCE, AND KREGEL

the shoulder rather than farther out. It is reasonable to suggest that Hatch's probe was itself surrounded by a sheath, thereby causing some degradation of positional resolution. No really quantitative theoretical pre­dictions for the plasmoid case, with which comparisions can be made, have been published, although the work of Taillet does indicate that the relatively flat central region and the steep fall off are the result of negative-ion accumulation within the central core of the discharge.

It is clear that the rather high fields found in the dark sheath region are intimately associated with the occurrence of a diminished optical emission rate. There is some evidence to suggest that only the emission of ionic species is diminished in the dark sheath, while

excited neutral emitters show no strong intensity decreases.1O The combination of these results suggests that either the formation of excited ions or else the emission from such species is inhibited by the strong static electric field and that this effect gives rise to the dark sheath.

ACKNOWLEDGMENTS

The authors would like to acknowledge the con­tributions of the late H. B. Williams who initiated the research reported here. This work was sponsored by the Office of Naval Research.

10 M. D. Kregel and A. Miller, Appl. Phys. Letters 11, 316 (1967).

JOURNAL OF APPLIED PHYSICS VOLUME 39, NUMBER 3 15 FEBRUARY 1968

Fracture Behavior of Tubular Bombs

CHESTER R. HOGGATT AND RODNEY F. RECHT Denver Research Institute of the Uniwsity of Denw, Denl1ef, Colorado

(Received 20 October 1967)

The plastic deformation behavior and modes of fracture exhibited by tubular bomb casings are greatly influenced by the stress state imposed by explosive and inertial forces. These forces combine to produce triaxial compression over a varying inner portion of the tube wall. Noting that compressive hoop stresses would exist over a portion of the wall, Taylor has previously developed a hypothesis for prediction of frac­ture radius, assuming a radial fracture mode. This paper introduces hypotheses related to the influences of stress state and thermoplasticity upon fracture mode as well as fracture radius. The resulting prediction model closely predicts fracture radius and explains the development of commonly observed shear-lip frac­tures. It illustrates why radial fractures are typical only when detonation pressures are relatively low, and why shear fractures are typical when detonation pressures are high.

Tubular steel bomb casings, driven radially outward by high explosive detonation pressures, commonly expand to about twice their original diameter prior to fracture and breakup. Historically, three names stand out as investigators of the fragmenting bomb problem. Gurney hypothesized concerning the partitioning of available explosive energy to derive the form of an empirical expression widely used in the prediction of casing fragment velocities.l Mott developed probability statements concerning the distribution of incipient radial fractures occurring within ring-type tubular bombs, incorporated the effects of local stress reduction due to release waves propagating from radially frac­tured surfaces, and, thereby, developed models for the prediction of fragment size due to radial breakup.2 Taylor devised models to describe bomb dynamics and analyzed the effects of stress state on the radius asso­ciated with casing fracture.3 ,4

1 R. W. Gurney, BRL Rept. No. 405, (Sept. 1943). 2 N. F. Mott, Proc. Roy. Soc. (London) 189 (1947). 3 G. I. Taylor, in Scientific Papers of G. I. Taylor (Cambridge

University Press, Cambridge England, 1963), Vol. III. 4 G. I. Taylor, in Scientific Papers of G. I. Taylor (Cambridge

University Press, Cambridge, England 1963), Vol. III.

This paper is concerned with the influence of stress state on the deformation and fracture behavior of bomb casings. It attempts to extend Taylor's work by analyzing the state of stress in the casing wall as a function of radial expansion, and by examining the influence of the stress state on observed modes of fracture.

PREDICTION HYPOTHESES

Taylor developed a model to predict the radius associated with fracture based upon the assumption that fractures would be radia1.3 This model observed the fact that compressive hoop stresses exist over an inner portion of the tube wall. The portion of the wall subjected to compressive hoop stresses is governed predominately by explosive pressure. Radial fractures cannot propagate into this compressive zone. He con­cluded that fractures would initiate at the outer surface where hoop stresses are tensile and penetrate to a depth ITT / P defining the boundary between tensile and compressive hoop stresses (CT, T, and P are tensile strength, wall thickness, and internal pressure). Hence, cracks would penetrate through the

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F R ACT U REB E H A V lOR 0 F TUB U L .\ R B 0 '}1 II S 1857

FIG. 1. Typical steel frag­ment from tube expanded by a high-order explosive detonation.

wall when u=P. Since P is a function only of rlrQ (ratio of instantaneous to initial internal radius), the fracture radius is defined as the radius associated with P=u. That his theory is essentially correct is well supported by experimental observations.3,s-7 However, observations of fracture modes do not generally support his assumption of radial fracture. Fracture surfaces more commonly lie along planes of maximum shear stress as illustrated by Fig. 1. Only when the explosion

FIG. 2. Typical steel frag­ment from tube expanded by a low-order explosive detonation.

is of low order (rapid combustion rather than detona­tion of the explosive) do radial fractures predominate, as illustrated in Fig. 2.

The sequence of photographs displayed in Fig. 3 clearly illustrates the dynamics of an explosively expanded steel tube. These photos were obtained by investigators at the Ballistics Research Laboratories of the U. S. Army who devised the front lighting and high-speed camera techniques required to obtain

5 M. Famiglietti, U.S. Army Ballistics Research Laboratory Memo Rept. No. 597 (March 1952). 6 F. E. Allison and J. T. Schriempf, J. App!. Phys. 31, 846 (1960). 7 C. R. Hoggatt, W. R. Orr, and R. F. Recht, Final Rept. Contracts DA-23-017-AMC-1399(A) and 1400(A) (August 1965).

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1858 C. R. HOGGA.TT A.ND R. F. RECHT

FIG. 3. Fragmen­tation of an explo­sively expanded steel tube. Sequential photographs ob­tained by the Solid 11echanics Branch, Terminal Ballistics Laboratory, U.S. Army Ballistic Re­search Laborato­ries. (Time between frames is 61 !-,sec.)

pictures of this type.s The puffs of smoke observable in the photos are detonation products indicating that complete fracture has occurred. During expansion, the casing material is subjected initially to triaxial com­pression. Until the local hoop stresses become tensile, deformation proceeds by shear-flow processes and no fractures (separations) are possible. The radial com­pressive stress is the principal stress and causes the material to extrude so as to produce wall thinning. Recovered fragments typically exhibit reductions in wall thickness of 40%-60%.5.6

8 Private communication with Joseph Regan, Seymour Kronman, and Jules Simon, Solid 11echanics Branch (Dr. C. M. Glass, Chief), Terminal Ballistics Laboratory, U.S. Army Ballistic Research Laboratories, 1965.

The hypotheses upon which this paper is based acknowledge the early appearance of small radial cracks in the tensile hoop stress region near the outer surface of the tube. Within the compressive hoop stress zone, tube expansion is accomplished by extru­sion which activates shear planes rotated approx. 45° from the radial direction. The radial cracks developing in the outer tensile region lead to shear stress concen­trations along certain shear planes in the inner com­pressive hoop stress region. Unstable thermoplastic shear (adiabatic shear) eventually transfers the entire burden of strain to a finite number of these shear planes during the later deformation stages.9

Unstable thermoplastic shear occurs when the local flow stress decreases with increasing strain; this results when the rate of thermal softening (due to internally generated heat) exceeds the rate of isothermal work­hardening. lO This phenomenon is illustrated by the photomicrographs of mild steel machining chips pre­sented as Fig. 4. The chip removed at low speed shows the uniform shear deformation which results when work-hardening predominates. Removed at very high

(a)

(b)

'FIG. 4. 11ild steel machining chips. (a) Cutting speed: 140 surface ft/min. (b) Cutting speed: 16000 surface ft/min. Chips provided by H. ]. Siekmann, Carbolloy Division, Gereral Electric Co.

9 Clarence Zener, AS11 3 (1948). 10 R.F. Recht, J. App!. 11ech., Trans AS11E, 31, E, (1964).

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FRACTURE BEHAVIOR OF TUBULAR BOMBS 1859

speed (16 000 surface ft/min) unstable thermoplastic shear zones appear in the second chip due to the predominance of thermal softening.ll

When the component of stress normal to the shear direction changes from compressive to tensile, the preferentially weakened material in the shear zone fractures, leaving the shear lips which are characteristic of recovered fragment fracture surfaces. The hoop stress changes from compressive to tensile before this normal stress component does; however, the material in the shear zone has been thermally softened by the heat of plastic deformation resulting in fracture along shear planes and the appearance of shear lips rather than radial fractures. The component of stress normal to the shear direction will become zero when the tensile hoop stress becomes equal in magnitude to the com­pressive radial stress.

Consequently, it is assumed that fractures will penetrate from the outside inwardly to a radial posi­tion no smaller than that at which these stresses have equal magnitudes. Complete fracture will occur when the internal pressure equals the tensile hoop stress at the internal surface of the tube.

COMPRESSIVE RADIAL STRESS IN THE ~ WALL

Fractures will appear only in the tensile hoop stress region of the wall. The hypothetical maximum depth of these fractures extends from the outer tube surface to the stress radius defined by rJ'6 = pa (rJ'8 and Pa are circumferen tial stress and radial pressure at radius a). Fracture in the tensile zone reduces the hoop stresses to zero. Consequently, hoop stress components affect­ing radial motion derive almost entirely from com­pressive hoop stresses and produce positive accelera­tions. These radial components of the hoop stresses are small compared to the radial internal pressure and

1.4,----------------,

1.2

r.o .. g 0.8

Vl a. a. 0.6

0.4

0.2

Po~ 2.28 x 106 psi

y>3

O~ __ L_ __ L_ __ L_ __ ~====~ 10 1.2 r.4 16 1.8 2.0 2.2

rlro

FIG. 5. Typical isentropic expansion curve for a high explosive.

11 H. J. Siekmann, ASTE 58 (10 ,';En.

fall to zero when fractures extend through the wall. Neglecting these insignificant effects of hoop stress on wall acceleration leads to the following equation of motion for the wall:

(1)

where P is internal explosive pressure acting on wall, r is internal radius of the tube, ro is initial internal radius of the tube, Ro is initial external radius of the tube, and p is density of. the tube material. The pressure (radial compressive stress) acting to produce the radial acceleration, ii, at any radius a within the wall is defined by the equation of motion for the portion of the wall external to this radius, written in the same form as Eq. (1).

(2)

where a is radius to a point within the wall, and R is external radius of the tube. Dividing one motion equation by the other and rearranging yields

Pa= (r/a) [(R2-a2) /(R02_ r02) JP. (3)

Neglecting elastic strains and considering axial strain f. to be zero, equations for a and Rare

a= (r2+ao2-r;)1/2

R= (r2+R02-r02)1/2.

(4)

(5)

Substituting Eqs. (4) and (5) into Eq. (3) provides the required expression for the radial pressure in the wall at any radius a as a function of the internal explo­sive pressure P.

Pa= [Pr/(r2+aoLr02)1/2J[(R02-a02) /(R02_ r02) J. (6)

EXPLOSIVE PRESSURE

The internal explosive pressure can be represented by an isentropic expansion without significant error. Thus,

(7)

where Po is the effective detonation pressure acting on the wall when r=ro and "I is the expansion exponent.

The gas expansion exponent "I actually varies during expansion, typically approaching a value of five during the early stages, and dropping as the expansion proceeds. A constant value of about "1=3 is suitable for computations.3,6,7 The Chapman-Jouquet detonation pressure for RDX high explosive is 2.28X 106 psi. Using this value for Po and "1=3 in Eq. (7) results in the curve shown on Fig. 5.

STRESS-STATE DERIVATION

If a cylindrical differential element of initial radius ao and thickness dao is expanded plastically to a radius ap (assuming axial strain, f., to be zero) the natural strain in the radial direction is negative and given by

(8)

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1860 C. R. HOGGATT AND R. F. RECHT

TABLE I. Comparison of predicted and observed fracture radii. Steel tubing. ,

Explosive Normalized fracture radius: r/ro

Data Specific Taylor Predicted source Steel Type gravity Observed prediction herein

BRL Elastuf (300 BHN)

DRI SAE 1020 RDX

DRI SAE 1020 RDX

Taylor Mild steel 50/50 Pentolite

• Minimum value (i.e., strength of material neglected).

Under uniaxial stress conditions, compressive plastic strain can usually be represented by12

(9)

where Pae is the compressive uniaxial stress or pressure, k is the strength coefficient, n is the work hardening exponent.

During plastic expansion of the cylindrical element the stress state is triaxial. An octahedral shear stress (flow stress) equivalent to Pa. in the uniaxial case is produced by the triaxial octahedral stress function13

Pa.= - (1/v2) [( - Pa -0"8) 2+ (0"8-O'z) 2+ (O'z+pa) 2]112

= (-3T./v2) , (10)

where pa, 0"8, 0'. are the radial pressure, the hoop stress, and the axial stress acting on the differential element at expanded radius a and Te is the octahedral shear stress under uniaxial stress conditions.

If the ring is now expanded elastically, the elastic

1.2 r-~-----------::;:;=----------,

1.0 O'a = Po

0.2

OL-___ -L ___ ~~ _____ ~----~~~

10 20 30 40 50

alaa

FIG. 6. Relationships between radial presssure, circumferential (hoop) stress, and radius for given conditions on circumferential stress. Steel tube.

12 Joseph Marin, Mechanical Behavior of Engineering Materials (Prentice-Hall, Inc., Englewood Cliffs, New Jersey 1963), pp. 38-41.

13 A. Nadai, Theory of Flow and Fracture of Solids (McGraw­Hill Book Co., Inc., New York, 1950), Vol. I, p. 104.

1.7 1.8 1. 74&

1.01 1.6 1.8 1. 74

1.24 1.9 1.8 1. 74

1.8-2.1 1.8 1. 74

strains in the radial, circumferential, and axial direc­tions within the differential element are

Er = - (Pal E) - (vi E)( 0"8+0'.) (11)

E8=0'8IE- (viE) (O'z-Pa) = (alap ) -1 (12)

E.=O'.IE- (viE) (0'8-Pa) =0, (13)

where v is Poisson's ratio, E is Young's Modulus, and Er is the total strain in the radial direction.

In accordance with the prediction hypotheses, frac­ture will not occur until the tensile hoop stress is equal in magnitude to the compressive radial stress (0'8 = Pa) . The radial position within the wall which separates the compressive and tensile hoop stress zones is defined by the condition 0"9=0.

Substitution of 0"9=pa into Eqs. (12) and (13) leads to a relationship between the plastic and elastic radii associated with this stress condition.

ap =a/[l+(l+v) PalE] (14)

Similarly, substitution of 0'9=0 into Eqs. (12) and (13) yields the relationship

ap = a/[l + (v2+v) PalE] (0'9=0). (15)

For the two conditions, Eq. (10) reduces to

Pa.=Pa (0"9 = pa) (16)

Pae = pa (1- v+v2) 1/2 (0'9=0) • (17)

20r-------------------,

en w :I:

18

~ 1.6

I f/)

~

0 14 <[ a::

12

R FIG. 7. Tube wall

1.2 14 1.6 1.8 20

. INTERNAL RADIUS - r INCHES

dimensions during expansion and po­sitions at which cir­cumferential (hoop) stress has a given value (in terms of both initial and ex­panded radial posi­tions within the wall) .

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FRACTURE BEHAVIOR OF TUBULAR BOMBS 1861

FIG. 8. Deformation and fracture mechanisms during explosive expan­sion of a cylindrical tube.

Equations (8) and (9) can be combined to obtain

(18)

Substituting Eqs. (14) and (16) for ap and Pae in Eq. (18) and rearranging, yields

Similar substitution of Eqs. (15) and (17) gives

alao= [1 + (palE) (v2+v) ]

Equations (19) and (20) provide the means for com­puting the radii a at which the specified stress condi­tions exist [where Pa is determined using Eqs. (6) and (7)]. These equations are plotted for mild steel on Fig. 6, using values of the constants as given by Marin (E=30X106 psi, v=0.25, k=97 800 psi, n=0.325).12

STRESS STATE IN THE TUBE WALL DURING EXPANSION

Consider a mild steel tube of internal radius, ro= 1.00 in., and external radius, Ro= 1.25 in., packed with RDX explosive. After detonation, the positions in the expanding wall at which (1) The tensile hoop stress is equal to the radial compressive stress (U6=pa) and (2) The tensile hoop stress is zero (U6=0), can be predicted as a function. of radial expansion in the following manner:

A value for Eq. (6) can be obtained by assuming a value for ao at a selected value of internal radius r. Recall that ao is the radial distance to a point in the unexpanded tube wall. The corresponding radial posi-

tion, a, of this point in the expanded tube is given by Eq. (4). Equation (7) or Fig. 5 is used to determine internal explosive pressure P. The computed value for Pa is used in Eq. (19) or (20) (or with Fig. 6) to determine a value for al ao. Only when this value for al ao is equal to the value of al ao as determined using Eq. (4) does the assumed value of ao define the position in the wall at which the specified stress condition exists. Consequently, an iterative trial and error process is used to determine the radial position defining the stress states of interest as a function of radial expansion. Due to the nature of the equations, three trials are usually sufficien t.

This procedure was used to develop the graph presented as Fig. 7, which concerns the steel tube described above. Equation (5) was used to determine the relation between external and internal radii Rand r. Wall thickness at any radius r is the difference between these two radii. The curves labelled a show the positions within the wall, where U6= Pa and UB=O. (Curves labelled ao show these same positions in the unex­panded tube.) Hypothetically, wall fracture will be complete no earlier than when a=r for the condition where UB = pa. This occurs when r = 1. 74 in. Values of radius could all have been divided by ro and plotted as a function of rlro. Since, in the example, ro= 1.00, the same numbers would apply. Consequently, breakup is predicted at a minimum value of r I ro = 1. 74. Tensile hoop stresses begin to appear at the internal radius when rlro=1.71.

COMPARISONS WITH EXPERIMENTAL RESULTS

The Taylor hypothesis and the hypothesis presented herein lead to essentially the same prediction of fracture

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1862 C. R. HOGGATT AND R. F. RECHT

radius, the predictions being almost independent of fracture mode. Table I illustrates the degree to which the prediction represents experimental data for mild steel tubes obtained from three sources (Taylor's paper, Ballistics Research Laboratory, and the Denver Re­search Institute).

The development of prediction models included an explanation of observed modes of fracture, which represents the major contribution of this paper. It should be noted that the tensile hoop stress zone (above curve labelled "a" on Fig. 7) remains very thin out to expansions of about r/ro= 1.4 when RDX is used. Consequently, radial cracks cannot penetrate very far and the plastic shear extrusion process is well developed in the compressive zone before the stress radius a, defined by 0'8 = Pa, sweeps across the wall. This leads to the type of fragmentation illustrated on Fig. 8, which was developed from Fig. 7. Starting at r=ro, the tube begins to expand. Radial cracks develop in the tensile hoop stress zone near the outside surface. (The radius a, shown on Fig. 8 is the position of zero hoop stress.) Shear extrusion thins the wall as illus­trated in the r= 1.2 segment of the figure. The small radial cracks at the outer circumference effectively decrease the shear areas along certain shear planes as shown. This leads to localized unstable thermoplastic shear which thermally softens the material in these zones. The appearance of a tensile component normal to the slip direction fractures the preferentially weak­ened material and causes cracks to develop in the direction of the shear plane. As these cracks proceed, they reduce shear area in active shear zones even further, thereby promoting the continuation of the fracture along the shear plane. Stress relief due to fracture on another local plane either parallel or perpendicular to the plane being considered can de­activate it. When a=r for the condition 0'8= pa, the fracture is hypothesized to be completed. Note the similarity between the predicted typical fragment shape and the fragment shown in Fig. 1.

Use of an explosive which has a low detonation pressure results in a much larger tensile hoop stress zone initially. Consequently, radial cracks can become relatively deep before the shear extrusion process is well developed. In this case, radial fracture surfaces would be expected, shear lips being developed only near the inner surface of the tube. This mode of failure is illustrated by Fig. 2.

CONCLUSIONS

Bomb casings commonly expand to nearlv twice their initial diameter before fragmenting because of the stress state induced in the wall by explosive ex­pansion. The high explosive pressures, which accelerate the tubular wall, produce compressive hoop stresses extending from the internal radius to a radius which depends upon internal pressure, initial tube dimensions, and expansion radius. Hypotheses used to predict frac­ture radius are based upon the appearance of specified tensile stress conditions at the inside surface of the tube. While the hypotheses presented in this paper differ from the Taylor hypothesis in their presumption on failure mode, predictions of fracture radius are essentially identical. Predictions agree quite well with experimental results.

The characteristic shear-lip fractures observed when casings are fragmented by explosives having high detonation pressures can be explained. The fracture mode model illustrates the role of small radial cracks, which first appear at the outer surface, on the develop­ment of unstable thermoplastic shear zones within the plastically extruding material in the inner region of triaxial compression. As tensile hoop stresses develop their components normal to these zones fracture the thermally softened material causing cracks to propa­gate in the direction of maximum shear.

Explosives which generate low detonation pressures do not produce compressive hoop stresses over as large a portion of the wall. Hence, deep radial cracks develop from the outside before unstable thermoplastic shear zones begin to develop. Consequently, fragments are observed to possess radial fracture surfaces, shear lips being confined to the inner region of the wall.

ACKNOWLEDGMENTS

The analysis of tubular bomb dynamics was accom­plished during the performance of U. S. Army Con­tracts DA-23-017-AMC-1399(A) and 1400 (A) for Picatinny Arsenal (Ted Stevens and Jack Brooks Technical Monitors). The willingness of the Solid Mechanics Branch, Terminal Ballistics Laboratory, U. S. Army Ballistics Research Laboratories to provide the authors with copies of the excellent high-speed photographs of tube fragmentation is greatly appre­ciated.

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