Structural Response of Piping to Internal Gas Detonation

Joseph E. ShepherdGraduate Aeronautical LaboratoriesCalifornia Institute of Technology

Pasadena, CA 91125 USAphone: 626-395-3283

e-mail:jeshep@caltech.edu

Abstract

Detonation waves in gas-filled piping or tubing posespecial challenges in analysis and prediction of struc-tural response. The challenges arise due the nature ofthe detonation process and the role of fluid-structureinteraction in determining the propagation and ar-rest of fractures. Over the past ten years, our lab-oratory has been engaged in studying this problemand developing methodologies for estimating struc-tural response. A brief overview of detonation wavesand some key issues relevant to structural waves ispresented first. This is followed by a summary ofour work on the elastic response of tubes and pipesto ideal detonation loading, highlighting the impor-tance of detonation wave speed in determining flex-ural wave excitation and possibility of resonant re-sponse leading to large deformations. Some issuesin measurement technique and validation testing arethen presented. The importance of wave reflectionfrom bends, valves and dead ends is discussed, aswell as the differences between detonation, shockwave, and uniform internal pressure loading. Fol-lowing this, we summarize our experimental findingson the fracture threshold of thin-walled tubes withpre-existing flaws. A particularly important issuefor hazard analysis is the estimation of loads asso-ciated with flame acceleration and deflagration-to-detonation transition. We give some recent resultson pressure and elastic strain measurements in thetransition regime for a thick-wall piping, and someremarks about plastic deformation.

1 Introduction

The goal of this paper is to summarize the state ofknowledge about gaseous detonation loading of pip-ing and indicate some of the more important issuesthat have to be considered in analyzing these prob-

lems. Detonation of gaseous mixtures inside of pip-ing systems is of interest to the practicing engineerand as a subject of scientific investigation [1, 2]. Itis a hazard that is occasionally encountered in thechemical [3, 4], nuclear [5, 6, 7, 8], and transportationindustries [4]. In some technologies like pulse deto-nation engines [9], detonations are deliberately cre-ated. The coupling between detonations and struc-tures like pipes are a model for fluid-structure inter-action, which has been extensively examined in theclosely related subject of shock (blast wave) or impactloading of structures [10, 11].

The response of pipes to detonations is part of thebroader problem of designing or analyzing the abil-ity of pressure vessels to contain internal gaseous ex-plosions. Broadly speaking, gas explosions can becharacterized into two categories, low-speed flamesor deflagrations and high-speed coupled shock andreaction waves known as detonations [12]. Defla-grations are subsonic, usually turbulent, flames thatcause slow pressurization that is treated as a quasi-static, spatially non-uniform structural load. Deto-nations are supersonic waves that result in dynamicstructural loading that is spatially nonuniform. Thepeak pressures in detonations are usually about twiceas high as for a deflagration and in exceptional cases,may be as much as 10 times higher.

There is some guidance on structural design for de-flagrations contained in NFPA 69 [13] which refers tothe ASME Boiler and Pressure Vessel Code SectionVIII - Division 1, Part H. It is quite clear that deto-nations are excluded from these considerations sincethe pressure loading of a deflagration is restricted tothe quasi-static regime. At present, there is no provi-sion within the ASME Boiler and Pressure Vessel orPiping Codes for designing pressure vessels or pipingto withstand gaseous detonations.

Recently, an ASME Code Case [14] has been devel-oped for designing high-explosive containment vesselsbased on extensive work by Los Alamos [15, 16] to

1

formulate ductile failure criteria. Although focusedon high explosives, the material response aspect ofthe Los Alamos work is quite relevant to the presentstudy. A key idea is the reliance on modern fracturemechanics [17] to design vessels that will plasticallydeform but not catastrophically fail under extremeimpulsive loading. Many of these ideas are directlyapplicable to the situation of gaseous detonation butthere are some crucial distinctions. High explosivesare usually approximated as impulsive loads and thestandoff between the vessel and the explosive is a sig-nificant aspect of the loading magnitude. Gaseousexplosions usually result in a combination of step andimpulse loading and the gas is often assumed to com-pletely fill the vessel or pipe. Deliberate high explo-sive detonations inside containment vessels are con-trolled events while gaseous detonations often are theresult of an uncontrolled ignition process that can re-sult in very high localized pressures if the deflagrationaccelerates to detonation (deflagration to detonationtransition or DDT [18]). In addition, the propagationof the explosion wave in the gas adjacent to the vesselor tube may result in a strong coupling between thepressure wave and the structural response.

2 Detonation Waves

A detonation wave [2, 19] consists of a shock waveclosely followed by a chemical reaction zone in whichthe fuel and oxidizer rapidly react to produce hot(temperatures of 2000-3000 K) combustion products.An ideal detonation travels at a nearly constantspeed close to (usually within 90%) the theoreticalor Chapman-Jouguet (CJ) velocity UCJ , which is be-tween 1500 and 3000 m/s in gases depending on thefuel-oxidizer combination, see Table 1. The reactionzone in a detonation is usually very thin, less than 10mm for most stoichiometric fuel-air mixtures and lessthan 100 µm for stoichiometric fuel-oxygen mixtures.Due to reaction zone instability, the effective widthof the reaction zone (characterized by the detonationcell width S) is typically 10-100 times larger than theidealized reaction zone [20, 21, 22]. Within this reac-tion zone, temperature, pressure and other propertieschange rapidly while just downstream of the reactionzone, a much slower variation occurs due to the gasdynamics of the wave propagation process. The pres-sure just behind the detonation can be as high as 20to 30 times the ambient pressure, depending on thefuel-oxidizer mixture, see Table 1. The computationswere carried out with standard thermochemical equi-librium methods [23, 24, 25].

The values in the table are only representative

Table 1: Measured and computed detonation param-eters for stoichiometric mixtures at standard initialconditions (25◦C, 1 atm).

Fuel % (vol) UCJ PCJ S(m/s) (bar) (mm)

Fuel-air mixtureshydrogen H2 29.6 1971 15.6 6–10acetylene C2H2 7.75 1867 19.1 10–15ethylene C2H4 6.54 1825 18.4 24–26ethane C2H6 5.66 1825 18.0 50–59propane C3H8 4.03 1801 18.3 40–60methane CH4 9.48 1804 17.2 250–350

Fuel-oxygen mixtureshydrogen H2 66.7 2841 19.0 1–2acetylene C2H2 28.6 2425 34.0 0.1–0.2ethylene C2H4 25.0 2376 33.7 2-3ethane C2H6 22.2 2372 34.3 1–2propane C3H8 16.7 2360 36.5 0.5-1methane CH4 33.3 2393 29.6 2-4

of detonation parameters, all values will dependstrongly on composition and the results in the ta-ble are for stoichiometric mixtures only. Detonationvelocities, pressures and cell widths will all depend onthe ratio of fuel to oxygen, and the amount and typeof additional gas components such as nitrogen. Deto-nation properties will also depend on initial pressureand temperature. To evaluate the hazard for a spe-cific mixture and initial conditions, computations ormeasurements for that particular situation should becarried out.

00.5

11.5

22.5

33.5

-0.1 0 0.1 0.2 0.3 0.4 0.5time (ms)

pressu

re (M

Pa)

(a)

Figure 1: Measured pressure vs time for detonationloading [26].

A typical experimental pressure-time trace for

Shepherd JPVT-07-1061 2

a detonation propagating longitudinally from theclosed end of a tube is shown in Figure 1. The almostinstantaneous jump in pressure at time t = 0 corre-sponds to the passage of the detonation wave acrossthe measuring station. The rapid decrease in pres-sure in the first .01 ms is associated with the reactionzone. The more gradual decrease in pressure out to t= 0.25 ms and plateau for longer times is associatedwith the gas dynamics of the flow behind the wave.Superimposed on the general trend are pressure fluc-tuations due to the unstable nature of the couplingbetween chemical kinetics and the leading shock front[19]. This instability consists of weaker shock wavespropagating transversely to the main front and or-ganized in a quasi-periodic structure that creates acellular structure on sooted foils placed with the det-onation tube [27], see Fig. 2.

The spacing S of the transverse waves (detonationcell width) observed on sooted foils (see Table 1) hasbeen measured [31] for many mixtures and can beused as a measure of the sensitivity of a mixture todetonation. The smaller the cell size, the easier it isto initiate and propagate detonations. As shown inTable 1, mixtures of fuel and oxygen have cells thatare much smaller than fuel and air, in agreement withthe observed ease of detonation [32].

The fluctuating pressure is usually ignored in struc-tural models and only the average pressure is consid-ered computing the force (loading) on the tube wall.In terms of a structural load, the net effect of a deto-nation is to produce a spatially nonuniform propagat-ing load as shown in Figure 3. Experimental pressuretraces and gas dynamic models can be used to definean idealized loading profile. For a tube with a closedend, the situation can be characterized by three re-gions (see Figure 3). First, there is the initial mixtureahead of the detonation front. Since the detonationwave propagating supersonically, there is no loadingproduced ahead of the detonation front and the pres-sure at a fixed location jumps up suddenly when thedetonation front arrives there. The detonation frontconsists of the shock wave and reaction zone, whichis not resolved in the loading model. The peak pres-sure just behind the front P2 can be approximated bythe CJ value, computed with the same thermochem-ical equilibrium codes that are used to obtain thedetonation velocities. The detonation is followed byan expansion wave which extends to approximatelymid-way between the wave front and the initiationend of the tube. Behind the expansion wave the gasis stationary and the pressure P3 in this region is ap-proximately 0.4PCJ .

The ideal pressure distribution in such a tube canbe described with an analytical solution known as

the Taylor-Zel’dovich model [33, 34]. The expansionwave region stretches as the wave propagates and theprofile evolves in a self-similar fashion with the sim-ilarity parameter x/UCJ t) and the leading pressurewave propagating at the Chapman-Jouguet detona-tion velocity UCJ . This ideal solution is commonlyused as a base for modeling the pressure distribu-tion required from structural loading computations;a simple approximation in terms of an exponentiallydecaying solution can also be used as discussed inAppendix B of Beltman [26].

For long tubes and high temperature mixtures, sig-nificant heat transfer and frictional effects will modifythe pressure profile [35]. As thermal energy is trans-ferred into the tube, the temperature and pressure ofthe gas will drop more rapidly than due to the isen-tropic expansion of the ideal Taylor-Zel’dovich modeland the tube will heat up. Edwards [36] found that af-ter the detonation had propagated about 70 tube di-ameters, the pressure profile actually became steadyrelative to the detonation front itself, in other words,the expansion wave reaches a fixed extent once thewave has propagated sufficiently far. A second con-sequence is that the thermal energy that is depositedinto the tube will set up thermal stresses [37], in-creasing the strain within the tube in addition to themechanical strain induced by the internal pressureload. In general, for detonations the thermal stressesare much lower than the peak stresses that occur dueto the mechanical load. Since the heat transfer intothe pipe is relatively slow, the peak thermal stressesoccur at long times compared to the detonation load-ing duration. As a consequence, thermal stress is amuch more significant issue for deflagration-type ex-plosions than detonations, and, may result in strainsmuch larger than the mechanical strain alone.

In addition to the main pressure loading shown inFigure 3, idealized models [38] predict the existence ofa pressure peak (Von Neumann spike) at the front ofthe reaction zone with a value approximately doublethat of the Chapman-Jouguet pressure. This pressurespike is usually not resolved in experiments becauseof its localized nature and short duration. Since thereaction zone is of such a short length compared toa typical tube length, the influence on the structuralresponse is small in comparison with the effects of themain loading produced by the Taylor-Zeldovich pres-sure profile behind the detonation front. For thesereasons, the influence of the Von Neumann pressurespike on the structural response is usually neglected.

Shepherd JPVT-07-1061 3

(a) (b)

Figure 2: (a) Shadow image of a detonation front in 2H2-O2+3N2 at 20 kPa, propagation is left to right.The instability of the front is manifested by the curved and kinked leading shock, the fine-scale densityfluctuations behind the front and the secondary shock waves extending into the products at the left of thewave. (b) Cellular pattern on sooted foil created by a detonation in 2H2-O2+2N2 at 20 kPa. The cell sizefor this mixture is approximately 43 mm and the soot foil is about 150 mm wide [28, 29, 30].

3 Elastic Response

From a structural point of view, the tube experiencesa traveling internal load that produces transient de-formation of the tube. This situation is similar tothe case of a gaseous shock wave propagating in atube [39] but with a slightly more complex temporal(Figure 1) and spatial (Figure 2) variation of internalpressure than the simple step function that can beused to represent a shock wave. For an ideal wave,the pressure initially jumps to the CJ value PCJ andthen decays to the plateau value P3 after the Tay-lor wave has passed and the fluid has come to rest.If the CJ pressure is sufficiently small (defined sub-sequently), then the motions of the tube wall willbe elastic and no permanent deformation will occurwhen the detonation passes through the tube. On theother hand, for sufficiently large values of the CJ pres-sure the yield strength of the tube will be exceededand plastic deformation will ensue. In this section wewill review the elastic case and we will consider theproblem of plastic response in Section 6.

Extensive results [26] in the form of measurements,analytical theories based on shell models, and numer-ical simulations are available for the case of elasticmotion created by propagating detonations or shockwaves. The key results are that the wave acts asa traveling load which produces progressive flexuralwaves having a phase speed equal to the load speed.The mode and peak amplitudes of the deformationsdepend on the speed of the wave as compared to thecharacteristic group velocities of the various elasticmodes of tube oscillations.

3.1 Modeling

The simplest approach [40], often used in hazardanalyses for quick estimates of peak deflection, ig-nores the axial distribution of pressure (or equiva-lently, the bending resistance of the tube), and as-sumes that the transient pressure load at a location(Fig. 1) can be used as the forcing function for asingle-degree-of-freedom (radial motion only) of thetube structure. The axi-symmetric radial vibrations[41] of a long (axially unconfined) cylinder have afundamental frequency of

f =1

2πR

√E

ρ(1− ν2), (1)

corresponding to an oscillation period of

T = 1/f (2)

The time required for elastic wave transit timethrough the thickness h is

τwave = h/cl , (3)

where the longitudinal sound speed cl is typically be-tween 5500-6000 m/s for metals, so that

τwave ¿ T . (4)

Therefore, even under gaseous detonation loading itis not necessary to consider [42] elastic wave motionthrough the thickness but only the structural modesthat involve the bulk radial motion of the pipe wall.

Shepherd JPVT-07-1061 4

The single degree of freedom model describes the ra-dial deflections as simple forced harmonic motions

∂2x

∂t2+ ω2x =

∆P (t)ρh

, (5)

where the oscillator natural frequency (radian/s) is

ω = 2πf (6)

and f is given by Eq. 1. Solutions of this model forelastic systems and extensions to the elasto-plasticcase are discussed in great detail by Biggs [43] aswell as Baker et al. [11] for a variety of forcing func-tions ∆P (t). The key result for shock and detonationloading is that the peak deformation is primarily afunction of the characteristic duration of the pres-sure pulse τ as compared to the structural period T .For pulses significantly shorter than the period τ <T/4, the loading is in the impulsive regime and thepeak deformation will scale linearly with the impulse,defined as the time integral of the pressure. For longduration pulses, τ > T , the loading is in the stepfunction regime and the peak deformation will be in-dependent of the pressure history and equal to twicethe static deflection that would be obtained for thesteady pressure equal to the peak value. For interme-diate values of τ/T , the peak deformation will dependon the details of the time history and can be describedin terms of a theoretical dynamic load factor Φ(τ/T ),as defined in Eq. 11.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8distance

pres

sure

P2 = P

P3

P1

stationary reactantsexpansion wavestationary products

detonation

UCJ

CJ

Figure 3: Detonation propagation in tube with closedends, initiation at left-hand side, propagation fromleft to right [26].

In order to go beyond this simple approach, it isnecessary to consider the axial distribution of loadingand the deformation, taking into account the bend-ing stiffness of the tube which results in coupling theradial and axial motion. The first comprehensive the-ories for predicting the elastic response of a tube toa moving load were developed by Tang [44] and Reis-man [45], who used transient shell theory to model

the structural response. By assuming a tube of infi-nite length, the problem reduces to a “steady state”problem and an analytical solution for the shell mo-tion can be obtained. The model presented by Tangincludes the effects of rotatory inertia and transverseshear. He also presented transient results for a fi-nite length shell using the method of characteristics.Reisman [45] developed a model that includes the ef-fect of pre-stress on the structural response and gavean elegant explanation of how the resonant couplingbetween a moving load and the flexural waves comesabout. Simkins [46, 47, 48] extended the analysis tothick-wall tubes and first applied these ideas to ex-plain observations of large strain amplitudes in guntubes.

The Tang model was applied to the case of idealdetonation loading by Beltman [26] and analyticalsolutions for the deformations were given for the caseof a “steady state” wave. One of the key results is theprediction of peak structural deflections as a functionof wave speed and the possibility of resonant responseat four critical speeds. These speeds are found as asolution to the characteristic equation describing thedispersion relationship [44]. The most relevant speedfor detonation problems is lowest critical speed Vc0,which corresponds to the group velocity of flexuralwaves that consist of coupled radial-bending oscilla-tions. A simpler model was given by Simkins [46],whose analysis neglects rotatory inertia and shear de-formation. In this model, there is a closed form forthe first critical speed Vc0, which is useful for estima-tion

Vc0 =[

E2h2

3ρ2R2(1− ν2)

] 14

. (7)

Solutions of the dispersion relation typically give val-ues that are up to 10% lower than Eq. 7.

The other critical velocities are vc1, equal to themodified shear wave speed

Vc1 =

√κG

ρ; (8)

the dilatational wave speed in a bar,

Vc2 =

√E

ρ; (9)

and Vc3, equal to the dilatational wave speed

Vc3 = vd =

√E

ρ (1− ν2)(10)

where E is the Young’s modulus, G is the shear mod-ulus, ν is the Poisson’s ratio, ρ is the density, and κ

Shepherd JPVT-07-1061 5

is the shear correction factor. For a more detaileddiscussion on the formulation and solution of the dis-persion relation, see Tang [44] or Beltman [26]. Twoother relevant velocities are the bulk shear wave speedcs and longitudinal dilatational wave speed cs. Ex-amples of characteristic speeds for two different typesof tubing are given in Table 2.

Table 2: Examples of tubes and computed parame-ters, including critical and characteristic speeds.

Material Al 6061 SS 304 AISI 1010

thickness (mm) 1.5 25.4 1.5radius (mm) 19.9 152 64.5fhoop (kHz) 42 5.4 13.4Xλ (mm) 4.3 48. 7.6

Speed (m/s)

Vc0 1013 1455 614Vc1 2847 2797 2922cs 3055 3070 3208Vc2 4982 4912 5172Vc3 5278 5116 5422cl 6064 5554 6001

Comparing these speeds with the detonation ve-locities in Table 1, it appears that most stoichiomet-ric mixtures have CJ velocities between Vc0 and Vc1.For sufficiently low detonation speeds (lean or dilutedmixtures), the CJ speed could be comparable to Vc0;for mixtures diluted with sufficient amounts of heliumor hydrogen, the detonation velocity may be compa-rable to Vc1. In either case, the possibility of resonantexcitation of the tube motion exists.

The existence of critical velocities and the poten-tial for resonance effects was first recognized in theinvestigation of the response of railroad tracks andbridges to the passage of a train or other heavy load.That physical situation can be modeled as a beam onan elastic foundation with a moving load, which re-sults in a governing equation that is identical to thesimplest thin-cylinder model of a shock or detona-tion wave in a tube. The solution for the radial tubemotion becomes unbounded when the loading travelsat the critical speed of this model. Although a res-onant response is observed in the experiments, vari-ous non-ideal effects such as damping, non-linearities,and ultimately plastic deformation limit the peak am-plitudes to finite values.

In actual practice, detonation tubes have a finitelength and transient effects may be important, par-

ticularly in the near resonant cases. Beltman [26]carried out analytical transient solutions by droppingthe effects of rotary inertia and transverse shear fromthe Tang model. More general cases were treated byBeltman [26] using the finite element method. Theanalytical transient model was extended by Mirzaei[49] to include the effects of rotary inertia and trans-verse shear, yielding improved agreement of the mod-eled hoop strains with the experiments [26].

t (ms)

Pre

ssure

(Mpa)

109876543210

1.0

0.5

0.0

-0.5

(a)

t (ms)

Pre

ssure

(Mpa)

109876543210

1.0

0.5

0.0

-0.5

(b)

t (ms)

Pre

ssure

(Mpa)

109876543210

1.0

0.5

0.0

-0.5

(c)

Figure 4: Measured pressure signals for a detonationpropagating at 1267 m/s in the GALCIT large det-onation tube [50]. a) transducer 1. b) transducer 2.c) transducer 3 [26].

3.2 Experiments

The earliest experiments on detonation-induced vi-brations were carried out by Malherbe et al. [40], whocompared the results of the single-degree-of-freedommodel to experimental values of hoop strain producedby propagating H2-O2 detonations in a 2-ft (.61 m)diameter, 20-ft (6.1 m) long stainless steel tube. Theyfound reasonable agreement between the measured

Shepherd JPVT-07-1061 6

Figure 5: The GALCIT large detonation tube facility [50].

and computed maximum strains and the frequencyof oscillation. Transverse and longitudinal strains inPVC and stainless steel pipes with diameters between16 and 33 mm were measured by Brossard [51] usingdetonations in C3H8-O2-N2 mixtures. The results forthe PVC tubes were found to be quite different thanfor stainless steel, which was attributed in part tothe visco-elastic properties of PVC. Van de Ven etal. [52] analyzed the response of a tube to an inter-nal dust detonation with a non-rotatory symmetricpressure loading. They present dynamic amplifica-tion factors derived from experimentally determinedstrains. Sperber et al. [53] measured strains pro-duced in a thick wall tube by an acetylene decompo-sition detonation. They noted that the peak strainswere under-predicted by a factor of up to 4 whenstatic formulas were used to estimate the maximumdeformation. Thomas [54] carried out experimentsin two types of plastic (GRP and MDPE) pipes us-ing ethylene-air mixtures and detonation initiationby a spark-ignited oxy-acetylene booster. A longitu-dinal strain signal was observed propagating slightlyahead of the detonation wave and the strain signalsappeared to be significantly different than those ob-served in shock wave experiments [39] with metaltubes.

Experiments on gun tubes [46, 47] revealed thatthe propagation speed of the load is an importantparameter. Peak strain amplitudes up to three timeshigher than those predicted by the static Lame for-mula were observed when the propagation speed ofthe load approached a critical value. Further investi-gation [46, 48, 55] showed that the radial motion cre-ated by the traveling load was being resonantly cou-pled into flexural waves when the load propagationvelocity approached the flexural wave group speed.More recently, Beltman [39] observed the same ef-fect in an experimental and analytical investigation

into the structural response of a thin shell to inter-nal shock loading. Subsequently, Beltman [26] inves-tigated the case of detonation loading and showedthat resonance effects can be observed experimen-tally. Not only do the resonance effects amplify thepeak strains, they can also produce uncompensatedaccelerations in piezoelectric pressure gages mountedin tube walls, resulting in artifacts (precursors and su-perimposed oscillations) in the pressure signals thatcan be quite significant in some cases [56].

3.3 Straight tubes

The simplest situation is a detonation initiated atone end of a long tube, resulting in a close approxi-mation to the idealized pressure field shown in Fig. 3.A typical experimental configuration [50] used in thelaboratory at Caltech is shown in Fig. 5. The tubeis made of up of three sections 2.3 m long, each withthe properties shown in the second column of Table 2.The tube was instrumented with piezoelectric pres-sure transducers and bonded strain gages (orientedto measure the hoop component) [26]. A typical setof pressure measurements made at three locations isgiven in Fig. 4. The negative values of pressure ob-served at long times on gages 1 and 2 is an artifactdue to the thermal response of the gages to the hotgases in the detonation products.

Strain signals from gage 10 (located at 4.5 m fromthe initiation end, close to the axial location of trans-ducer 2) are shown in Fig. 6 for three detonationvelocities bracketing the first critical speed of 1455m/s. The main signal is coincident with the arrivalof the detonation at the measurement location andshows characteristic oscillations with a frequency ofclose to 5 kHz. For the high speed wave, Fig. 6c, ahigh frequency precursor is visible. The existence ofthe precursor and the frequency content can be seen

Shepherd JPVT-07-1061 7

more clearly in Fig. 7a. The lower branch of the dis-persion relation corresponds to the main oscillationsat ∼ fhoop and the upper branch corresponds to thehigh-frequency precursor wave. The detailed featuresof the waveforms in Fig. 6 are determined by the in-teractions of the flexural waves with the tube flanges;interference between the incident and reflected wavesproduces the amplitude modulation and beating ob-served in these records.

t (ms)

Str

ain

109876543210

2.e-4

1.e-4

0

-1.e-4

-2.e-4

(a)

t (ms)

Str

ain

109876543210

2.e-4

1.e-4

0

-1.e-4

-2.e-4

(b)

t (ms)

Str

ain

109876543210

2.e-4

1.e-4

0

-1.e-4

-2.e-4

(c)

Figure 6: Measured strain signals [26] from gage 10for three detonation velocities propagating at a) 1400m/s, b) 1478 m/s, c) 1700 m/s.

In order to discuss the amplitude of the signals anddetermine if there are any resonance effects, we needto normalize the signals in order to eliminate the ef-fects of different detonation pressures for the differentvelocities. The normalized elastic response is usuallydefined in terms of the dynamic amplification or loadfactor. The dynamic amplification factor is defined asthe ratio between the maximum dynamic strain andthe equivalent static strain calculated from static for-mulas using the measured peak applied pressure

Φ =εdynamic max

εstatic. (11)

Since the measured pressure is subject to substantialfluctuations, the static strain is computed using thecalculated value of PCJ as long as the detonation ap-pears to be ideal, i.e., wave speed close to UCJ . Fora thin tube that is sufficiently long to be consideredaxially unconstrained, the static strain will be givenby

εstatic = σθ1− ν2

E(12)

where the hoop stress is related to the pressure dif-ference by the simple membrane expression

σθ =R

h∆PCJ , (13)

For thick tubes, the Lame expressions should be usedinstead of the membrane model.

The amplification factor is plotted for gage 10 asa function of the detonation wave speed in Fig. 7b.The critical velocity for the Graduate AeronauticalLaboratory, California Institute of Technology (GAL-CIT) detonation tube is computed to be 1455 m/s inreasonable agreement with the measured peak in theamplification factor. The amplification factor goesfrom about 1 below Vc0, increases to a maximum of3.5 near Vc0, and drops to about 2 above Vc0. The ex-istence of a resonance at UCJ ≈ Vc0 is clearly shownby these results. The general features of the wavespeed dependence are captured by both the Tang andfinite element method (FEM) models but the FEMmodel with clamped end conditions (corresponding torigid flanges) gives the best agreement for this tube.Note that the Tang theory is for a steady-state prop-agation, which assumes that all transient effects havedied out, corresponding to a “large” distance of prop-agation. In addition, the results shown in Fig. 7 areclose to the step-loading regime with the durationof the Taylor wave comparable to or larger than thehoop oscillation period. Just as in the single-degree-of-freedom, the response predicted by the Tang modelwill depend on the duration of the Taylor wave andthe peak deformations will decrease with decreasingTaylor wave duration, reaching a limiting value cor-responding to step loading at the plateau pressureP3, see Beltman [26]. To include transient effectsand account for the Taylor wave duration, FEM so-lutions should be performed [26] with a loading func-tion that corresponds precisely to the situation beingexamined. The evolution of the peak strain ampli-tude with distance is predicted to be slowest for wavespeeds close to Vc0, and this is observed in the exper-iments, which are in reasonable quantitative agree-ment with the FEM model.

Shepherd JPVT-07-1061 8

v (m/s)

f(k

Hz)

200018001600140012001000

20

15

10

5

0

(a)

3000v (m/s)

Am

plifica

tion

30002500200015001000

5

4

3

2

1

0

(b)

Figure 7: Analysis of strain signals from gage 10. a)frequency content compared with Tang model. b)amplification factor from experiments ¦ comparedwith Tang − − − and finite element models withsimply-supported , and clamped ends ,[26].

Further experiments were carried out with thinnertubes and higher wave speeds by Chao [56, 57] in or-der to examine the possibility of resonance at the sec-ond critical speed and also, obtain strain signals thatwould not be contaminated by flange effects. Fivestrain gages were located in the center of a 1.5 mmthick, 40 mm diameter aluminum tube specimen thatwas 1.5 m long (column 1 in Table 2). Detonationswere initiated in a separate thick-wall section 1.5 mlong and connected to the specimen by slip-on con-nections. H2-O2 mixtures diluted with helium wereused to create detonations with speeds up to 3600m/s. An example of the strain gage measurementsare shown in Fig. 8. The resulting dynamic load fac-tors are shown in Fig. 9. From these results it isclear that there is no resonance in hoop strain whenthe detonation speed is equal to either Vc1 or cs. Nu-merical simulations of a finite element model imple-

mented in LS-DYNA (Livermore Software Technol-ogy Corp, Livermore, CA.) revealed that a resonantresponse in the shear strain was predicted but sincedynamic transverse shear strain of a tube cannot bemeasured by any known metrology, this effect cannotbe observed in experiments.

Time (ms)

Str

ain

3.5 4 4.5 5

SG1

SG2

SG3

SG4

SG5

0.01%

Figure 8: Measured strain signals for a detonationpropagating at 2841 m/s in a 40 mm diameter Alspecimen [56].

Ucj (m/s)

Hoo

pS

trai

nD

ynam

icA

mpl

ific

atio

nF

acto

rΦ

0 1000 2000 30000

1

2

3

ExperimentSimulationTang (1965) model

vc0 = 1013 m/s cs = 3055 m/s

vc1 = 2847 m/s

Figure 9: Predicted and measured strain amplifica-tion factors for detonations propagating in a 40 mmdiameter Al specimen [56].

After carrying out these experiments, careful quan-titative comparisons [58] were made of the measuredand computed strains. Despite our best efforts, wewere unable to get agreement to within better thanabout 15-20% for the peak amplitude of the hoopstrains. Through repeated trials, we were finally ableto determine that there were two sources of system-atic error in our measurements. First, the specimentube wall thickness varied around the circumference

Shepherd JPVT-07-1061 9

about 13%. Second, the strain gages showed shot-to-shot variation and evidence of micro-cracking caus-ing artifacts in the signals. The strain measurementswere repeated using an optical vibrometer in additionto the strain gages and the results were much moresatisfactory [58]. A series of replica tests demon-strated that the detonation process and the vibrom-eter measurements were very repeatable with a shot-to-shot variation of about ± 2%. The measured deto-nation pressure and velocity were highly reproducible(within 0.5%) but there is significant high frequencystructure in the pressure signal. Modeling the wallthickness variation and making comparisons with thevibrometer measurements [58] instead of the bondedstrain gages, elastic wave propagation simulations[59, 60] based on approximations to the experimentalpressure loading [61] were able to predict the precur-sor and main wave profile in correct phase and alsothe peak strain amplitudes were predicted within 7%.

3.4 Tubes with closed ends

As part of the study by Beltman [26], measurementswere near the closed end of the tube and more re-cently, we have examined this situation using otherfacilities. When the detonation reaches a closed end,it will reflect as a shock wave that propagates awayfrom the closed end. The peak pressure of the re-flected shock wave [62] is about 2.5PCJ and the pres-sure decays as the wave moves away from the reflect-ing surface. The reflected shock wave will induce flex-ural waves in the pipe which will interfere with thewaves that were created by the incident wave. Anexample of this is shown in Fig. 10 for a detonationinitiated in a H2-N2O mixture within a 316L SS tubeof 70 mm radius and 12 mm wall thickness [63]. Themixture was initiated by a spark but rapidly transi-tioned to a detonation due to the presence of peri-odic obstacles which generated turbulent flow, caus-ing the flame to accelerate quickly. Although theflexural wave train created by the reflection is notdistinct, the peak strains occur after the detonationwave has reflected. The peak dynamic load factor isΦ = 3.1 based on the CJ pressure of 2.6 MPa and thepeak measured strain of 197 µ. This is an increaseof 50% due to the reflection process. If we base thedynamic load factor on the computed peak reflectedshock pressure of 6.46 MPa, then Φ = 1.1. How-ever, no strain gages were located closer than 0.31 mfrom the end wall so that the measured peak strainsare smaller due to the attenuation of the shock as ittravels away from the end wall.

0

10

20

30

40

50

-0.5 0 0.5 1 1.5 2

Pre

ssur

e (M

Pa)

time (ms)

shot 40, 30%H2, P0=1 bar, BR=0.25

P1, 0.305m

P2, 0.432m

P3, 0.559m

P4, 0.686m

P5, 0.813m

P6, 0.940m

P7, 1.067m

end, 1.25m

(a)

0

200

400

600

800

1000

-0.5 0 0.5 1 1.5 2

Sta

in (

mic

ro s

trai

n)

time (ms)

shot 40, 30%H2, P0=1 bar, BR=0.25

S0, 0.432m

S1, 0.559m

S2, 0.686m

S3, 0.813m

S4, 0.940m

(b)

Figure 10: Detonation propagation (rapid DDT nearignition point) and reflection from a closed end. a)pressure measurements. b) strain measurements [63].

3.5 Tubes with bends and tees

Process plant piping is characterized by straight runsof pipe connected by elbows, tees, valves, pumps, re-actor vessels, holding tanks and other features, in-cluding detonation and flame arrestors. In addition,the piping system is suspended or supported from aframework that provides reaction forces and limitsthe motion of the piping. If detonations are possi-ble within the piping, then a comprehensive analysisof the structural response requires considering howthe detonation will interact with these features andwhat structural loads will be created. One genericsituation is the dead-end (closed valve or blank-offflange) that was considered previously. Another isthe elbow or tee connection. Detonation propagationthrough an elbow or tee is an example of detonationdiffraction [64] which, depending on the direction of

Shepherd JPVT-07-1061 10

curvature, may cause the detonation to intensify orweaken [65]. In general, the situation is quite complexsince combinations of these features occur in variousparts of a processing plant. Thomas [54] has carriedout experiments on simple models of plastic pipingruns measuring both the forces on the supports andthe strains on the pipes. Liang et al. [66, 67] havemade measurements of strains and pressures in small-scale specimens of thin-wall metal tubing containing90-degree bends and tees. These studies shows thatit is feasible to obtain useful data on these complexproblems but it is necessary to consider a wide rangeof time scales since the motion of the supports oc-curs over a time of 0.1 to 1 s while the deformationdue to flexural waves occur on a sub-ms time scale.Substantial motion of the pipe supports was observedin the Thomas tests, raising the possibility of pipingcontaining the explosion at early times but failing dueto excessive distortion of the supports.

4 Rupture of Tubes

Recently, two failures occurred in piping systems innuclear power plants that have prompted examina-tion of failure mechanisms due to detonation load-ing. On November 7th, 2001, the Hamaoka-1 NPPin Japan suffered a pipeline rupture accident, appar-ently due to the detonation of a hydrogen-oxygenmixture that accumulated due to radiolysis [7, 68].The Brunsbuttel KBB in Germany had similar failure[5] on December 14, 2001. In both cases, the tubeswere observed to have multiple fractures and frag-mentation occurred, see Fig. 11. The detailed analy-ses carried out after these accidents showed that thefailures could be explained by excessive deformation(hoop strains of 23-27%) caused by the pressure load-ing of a detonation propagating in radiolysis productsat 70 bar initial pressure.

Although similar failures can be produced in thelaboratory with sufficiently thin tubing or high in-ternal pressure, it is difficult to study the failuremechanisms in detail without pre-existing flaws tocreate reproducible fracture initiation sites. The is-sues of fracture threshold, crack propagation speeds,crack branching and the effect of prestress have beenexamined by Chao [57, 69, 70] for thin-walled alu-minum specimens with coin-shaped axial flaws partlythrough the tube wall, for example, see Fig. 12. Fora given tube, flaw, and internal pressure, the crackpropagation and resulting tube deformation is quitedifferent for hydrostatic, pneumatic, and detonationloading due to the very different amount of energystored in these cases. Paradoxically, static pressuriza-

(a)

(b)

Figure 11: Pipe rupture due to overpressure by explo-sions. a) Hamaoka-1 NPP [7]. b) Brunsbuttel KBB[5].

tion with gas (pneumatic case) creates a greater crackdriving force than using a detonation wave when thegas pressure is equal to the CJ value [70]. This isdue to the elastic energy stored in the pre-stressedtube and the lower sound speed in the cold gas versusthe hot detonation products. Since even the slowestgaseous detonations rapidly outrun even the fastestpropagating cracks (the highest crack tip speeds weobserved were less than 350 m/s), for practical struc-tural analysis and simulations in gaseous detonation-fracture interaction, the influence of the venting onthe chemical kinetics within the detonation front ap-pears to be negligible.

Dramatic differences between plastic and metaltubes are observed [71, 51, 54] in elastic response andfracture because detonation waves are typically much(up to an order of magnitude) faster than any of thecritical or characteristic speeds and significant visco-elastic effects can occur. For example, in polycarbon-ate tubes, only very limited oscillatory deformationwas observed in hoop and axial directions with grad-ual increase in strain leading to a single (axial) ordouble (hoop) peak followed by a monotonic decayto a constant strain in the hoop direction. Observa-tions in both polyvinylchloride (PVC) [51] and glass-reinforced plastic (GRP) [54] showed a fast precur-sor in the longitudinal strain, and rapid damping ofthe oscillations, and in PVC, a residual hoop strain.The peak amplitude of the deformations observed inPVC was significantly smaller than predicted by the

Shepherd JPVT-07-1061 11

single-degree-of-freedom model with an ideal deto-nation loading profile, leading Brossard and Renard[51] to speculate about the possible importance ofcoupling between the flexural motion and detonationprocess.

Fracture thresholds and crack propagation in plas-tics can also be expected to be quite different than inmetals due to the much lower (by up to an order ofmagnitude) yield strength and fracture toughness, aswell as the wide range of fracture types. Polycarbon-ate tubes usually exhibited [71] a straight crack thatpropagated upstream and downstream from the flawbefore running in a helical fashion around the tubes,similar to behavior observed for aluminum tubes. Inacrylic tubes [71], the dominant fracture pattern wasthe catastrophic fragmentation, with a possible corre-lation between fragment size and number with deto-nation load strength. PVC [72] and MDPE (mediumdensity polyethylene) [54] also rupture in a brittlefashion if the wall is sufficiently thin or the detona-tion pressure sufficiently high. Thomas reports thatGRP was “remarkably resilient” with no failures ob-served [54] in his trials.

(a)

(b)

(c)

Figure 12: Thin-wall (first column of Table 2) tubeswith pre-existing flaws, rupture due to propagatingdetonations [69]. a) 12.7 mm long flaw. b) 25.4 mmlong flaw. c) 50.8 mm long flaw.

A simple fracture criterion for detonation loadingof pre-flawed tubes has been developed [69] based onlinear elastic fracture mechanics, the critical stressintensity factor, and treating the flaw as a surfacecrack in a wide plate under far field tension. Themagnitude of the tension is obtained from the strainfield predicted by the Tang model of tube responseto the detonation load for tubes without pre-stress.Experiments with various flaw depths and lengthsin aluminum tubes show a reasonable agreement be-

tween the predicted fracture threshold and the ob-served structural failure. Limited data on polycar-bonate tubes [71] indicate that the fracture thresholdis substantially under-predicted by the model.

The effect of torsional pre-stress on crack paths inaluminum tubes was also studied [57]. By applyingtorsion, the initial crack path could be altered andthe crack kinking angle could be correlated to themode mixity (the ratio of the stress intensity factorsfor mode I and mode II loading). Visual observa-tions of the crack motion (Fig. 13) and strain mea-surements revealed a significant influence of the shearwaves created by the release of the torsion upon crackinitiation.

5 Deflagration to DetonationTransition Loads

It has been known for some time [73, 74] that DDTcan produce pressures in excess of the CJ or reflectedCJ pressure. White [73] reports observations of re-flected pressures in stoichiometric H2-air initially at300 K and 1 atm. During flame acceleration in a 3.5× 3.5 in. (89 × 89 mm) tube 32 ft (9.75 m) long, apeak pressure of 170 atm was recorded. This is 4.5times the usual reflected CJ pressure and is probablydue to the overdriven detonation produced during thetransition process. As discussed previously, reflectedpressures of this magnitude can be produced by a det-onation that is 20% faster than a CJ detonation. Ourcomputations indicate that such overdriven detona-tions can be readily generated during the transitionprocess and persist for some time afterward. Thetube used by White was not damaged since it was avery robust design.

Craven and Grieg [74] report the production of highpressures during DDT events in ammonia-nitrous ox-ide mixtures. Static pressures up to 70 atm in thetransition region (the ideal CJ pressure PCJ = 30atm for this mixture) were recorded. More signifi-cant, reflected pressures up to 340 atm were inferred(the ideal reflected CJ pressure PR = 71 atm forthis mixture) from the deformation of the endplate.These pressures were not actually measured. More-over, the velocities reported by Craven and Griegwere recorded in separate experiments without anend plate. The relation between inferred pressure andwave velocity plays a key role in their conclusions butappears suspect.

The very high pressures were only obtained whenthe transition was arranged to occur near the closedend of the tube. However, it is not clear if the dy-namic load factor was properly accounted for in these

Shepherd JPVT-07-1061 12

Figure 13: Crack opening and propagation in thin-wall (first column of Table 2) tube under torsion witha pre-existing flaw and detonation loading [57].

tests. Craven and Grieg do not give any discussionof the structural dynamics. This is a serious problemsince pressures were determined from deformation ofa sacrificial end plate rather than pressure transduc-ers. Despite the relative weakness of the tube usedby Craven and Grieg, no failure of the tube itself wasnoted.

Craven and Greig speculated that these pressures

were created when the detonation occurred withinthe shocked gas ahead of the flame. They proposed a“double discontinuity” model consisting of a detona-tion behind a shock. Their shock interaction calcula-tions suggested that this model could easily accountfor the magnitude of the pressure waves produced indetonation reflection. They obtained calculated peakreflected pressures between 340 and 880 atm. How-ever, these extreme values were based on an idealizedinteraction for the detonation and shock merging justas the end of the tube was reached. These pressuresrepresent extreme upper bounds that may only beachieved in exceptional cases.

Unsteady gas dynamic computations [75] indicatethat the pressures observed by Craven and Greigcould in fact be produced by some variation on mul-tiple detonation–shock interactions. Thermochemi-cal computations and simulations [75] yield similarbounds on the peak pressures of 350 to 540 Po forthe case of H2-air mixtures. However, a very spe-cial set of circumstances is required to produce thesepressures. Values in the range of 150Po, as observedby White and computed in the simulations, appearto be more likely.

The experimental evidence that DDT can result inpressures much higher than direct initiation of det-onation is reviewed by Thibault et al. [3]. The ba-sic notion, as discussed above, is that the flame pre-compresses the mixture prior to the onset of deto-nation. This is sometimes referred to as “pressurepiling” in the process industry. Transition to detona-tion in this pre-compressed mixture results in muchhigher detonation pressures, and consequently pres-sures created by the detonation reflection, than if CJdetonation has simply reflected from the end of thetube. The worst case situation that was identified byCraven and Greig involves transition to detonationwithin the gas processed by the reflected shock pro-duced by a fast flame. Numerical simulations [76, 75]indicate that the peak pressure created by such anevent can be up to an order of magnitude higher thanthe CJ detonation pressure. An example of an exper-imental measurement of such an event is shown inFig. 14 for a detonation initiated in a H2-N2O mix-ture within a 316L SS tube of 70 mm radius and 12mm wall thickness. The mixture was initiated by aspark and transition to detonation did not occur un-til the last 0.25 m of the tube. A precursor shockwave is clearly visible at transducer P7 but not atthe end wall. The peak pressure at the end wall wasso high that it saturated the transducer and a large-amplitude flexural wave associated with the reflectedshock wave can be observed propagating away fromthe end wall after transition has taken place. The

Shepherd JPVT-07-1061 13

peak strain was 270 µ and based on the CJ pressureof 2.5 MPa, the dynamic load factor was Φ = 4.4 ata location 0.3 m away from the end wall.

0

10

20

30

40

50

0 0.5 1 1.5 2

Pre

ssur

e (M

Pa)

time (ms)

shot 52, 17%H2, P0=1 bar, BR=0.25

P1, 0.305m

P2, 0.432m

P3, 0.559m

P4, 0.686m

P5, 0.813m

P6, 0.940m

P7, 1.067m

end, 1.25m

(a)

0

200

400

600

800

1000

0 0.5 1 1.5 2

Sta

in (

mic

ro s

trai

n)

time (ms)

shot 52, 17%H2, P0=1 bar, BR=0.25

S0, 0.432m

S1, 0.559m

S2, 0.686m

S3, 0.813m

S4, 0.940m

(b)

Figure 14: Deflagration to detonation transition neara closed end. a) pressure measurements. b) strainmeasurements [63].

This type of extreme situation, DDT within shockcompressed gas, has been observed in laboratory testsby many others, including [77, 78, 79, 3]. Kogarkowas carrying out initiation and critical tube test-ing and observed that attempts to initiate insensi-tive mixtures with detonations very often resulted inrapid flame propagation and DDT. Very high pres-sures, up to 347 bar, were measured at the closed endof the tube in methane-oxygen-nitrogen experiments.Kogarko’s tube (305 mm diameter, 10 mm wall thick-ness) was destroyed by a DDT event in a mixture with6.8% acetylene in air. A critical tube experiment wasbeing carried out, the detonation failed and an accel-erating flame was produced with velocities up to 880m/s. The consequences were dramatic: “At the end

of the tube there arose, to all appearances, a pres-sure which was exceedingly high, and which actedover a period of time, as a result of which the the endsection of the steel tube was demolished with heavyfragments flying out in all directions (the end cap,flange, valve, etc.). The tube rupture was accompa-nied by an extremely powerful sound effect. It shouldbe noted that in a large number of experiments in-volving the detonation of methane-air mixtures, inwhich the pressure in the tube was over 200 kg/cm2,some sort of mechanical alterations in the end sectionwas not observed.”– [77]. Note that 1 kg/cm2 is equalto 0.981 bar. A series of experiments in propane-airwere carried out to measure the peak pressures inmore detail. For a stoichiometric mixture at an ini-tial pressure of 1 bar, the maximum pressure was anaverage of 461 bar over a time of 4 µs following re-flection.

Chan and Dewitt [78] measured pressures aheadof a “choked” flame that were up to a factor ofthree higher than the initial pressure. In [79], apeak reflected pressure of 250 bar was observed fora hydrocarbon-air mixture at an initial pressure of1 bar. In Thibault et al. [3], DDT was observedduring flame acceleration tests in pure ethylene oxideand they computed that the peak pressure could havebeen as high as 140 times the initial pressure due topre-compression.

6 Plastic Deformation Re-sponse

For pressure vessels or tubes in the plastic defor-mation regime, catastrophic failure associated withductile tearing or plastic instability is usually termed“rupture”. For static loading this occurs at the burstpressure [15] which depends crucially on the detailsof the ductility of the material. For dynamic load-ing, the situation is considerably more complex anddepends on the details of the pressure waveform [10]and not just the peak pressure. For high-ductilitysteels like those used in the present study, substantialdeformation and energy absorption can take place inthe process of stretching the material up to the pointof rupture.

Duffey et al. [15] suggest that for dynamic loads itis this ability of a structure to absorb energy throughplastic deformation that is most important to design-ing fracture safe vessels to contain explosive loading.Therefore, they propose that design criteria based onspecifying safe levels of plastic strain are most rel-evant. For certain metals, plastic strains of up to28% are observed without rupture although we are

Shepherd JPVT-07-1061 14

certainly not advocating this as a design limit. Anextensive discussion of proposed ductile failure crite-ria is given by Duffey et al. [15].

In a static loading situation, the onset of plasticbehavior in the wall of a pressure vessel occurs wheninternal pressure exceeds a critical value, Py and at asomewhat higher pressure, Pp, the entire vessel wallwill be in a state of plastic deformation. If the ma-terial exhibits significant strain hardening, then thepressure that results in the onset of the yielding for astructure can be substantially lower than the pressurePburst that results in rupture or burst of the vessel.

In a dynamic loading situation, it is more useful tofocus on the peak deformations than the pressures.For elastic analyses, we introduced the concept of dy-namic load factor and used this to estimate peak de-formations for a given waveform and peak pressure.For plastic analyses, the details of the waveform aremore important than in the elastic case and it is nec-essary to consider the motion of the material up tothe onset of yielding, the duration of the plastic defor-mation period, and the subsequent elastic oscillationsonce plastic deformation has ceased.

In order to model plastic deformation, there are keyissues that must be addressed. One is the materialmodel - the appropriate description of both strainhardening and strain rate effects must be included.Another is that the loading history must be known.Although it is sufficient in elastic analysis to specifya dynamic load factor and peak pressure, this is notadequate for a plastic analysis. Finally, nonlinear fi-nite element methods are usually required in order tohandle spatially distributed loads and the large defor-mations that may occur in the plastic regime. In somecases, useful results and guidance for the FEM model-ing can be obtained from the single degree of freedommodel [80, 81, 82, 83, 84, 85, 10]. Under certain as-sumptions and loading regimes, analytical solutionsare possible; a typical approximation [81] is to ne-glect the axial stress resultant and bending moment,consistent with a purely radial motion. Comparisonof one and two-dimensional solutions [86] shows thatas long as the load is approximately uniform over anaxial distance exceeding (see Table 2)

Xλ =(

R2h2

3(1− ν2)

)1/4

. (14)

then the single degree of freedom model gives a rea-sonable estimate for the maximum deflection. Thesemodels show reasonable agreement with high explo-sive experiments for which the loading can be approx-imated as impulsive and localized spatially, i.e., not atraveling load. These models can be used to estimatemaximum deformation in the case of gas explosions

resulting in DDT near a closed end, for example, asshown in Fig. 15.

Figure 15: Plastic deformation of 15% produced ina thin-wall tube (column 3 of Table 2) of mild steelby DDT next to closed end located at the right-handside. The peak pressure measured at the end wasapproximately 500 bar, twice the computed value fora reflected CJ detonation [87].

It is possible to observe [88, 87] permanent defor-mation due to propagating and reflected CJ detona-tions. For very ductile metals (for example, dead-softcopper) and detonation peak pressures sufficient tocause yielding but not rupture, plastic deformationwaves can be observed [88] to propagate in phasewith the detonation front. Measurements of strain[87] due to CJ detonations propagating in thin-wallsteel tubes (column 3 of Table 2) under these condi-tion show a mixed plastic-elastic response with elas-tic oscillations of small amplitude superimposed onthe permanent plastic deformation. Single-degree-of-freedom simulations of strain resulting from propa-gating detonations give reasonable agreement withthe observed peak deformation as long as both strain-rate and strain-hardening effects are included in theconstitutive relationship. An additional increment ofplastic deformation is produced in this loading regimewhen the detonation reflects from a closed end. Incontrast to the DDT case shown in Fig. 15, the max-imum plastic strain is only about 2% due to a re-flected CJ detonation in a mixture with comparableCJ properties (a CJ pressure of 110 bar). This con-firms the very significant role that DDT can play instructural loading.

Conclusions

The structural response to detonations inside pipesand tubes has been studied using experiments, ana-lytical methods, and numerical simulation. The elas-

Shepherd JPVT-07-1061 15

tic response depends strongly on the detonation speedas compared to the first critical speed, the flexuralwave group velocity, Vc0. For UCJ < Vc0, the dy-namic load factor Φ ≈ 1; for UCJ > Vc0, the dynamicload factor Φ ≈ 2. For UCJ ≈ Vc0, resonance oc-curs and Φ can reach values as large as 4. The criti-cal velocity concept may be important for design andanalysis if the wave speeds overlap the critical speeds.Finite element simulations of the structural responseshowed fair agreement with the measurements andwere able to predict the transient development of theprofile. Measurements indicate that flanges have asignificant influence on the tube response and thereflection and interference of flexural waves lead tohigher strains. Measurements with different mixturesshow that when the cell size and the structural wavelength are of the same order of magnitude, the flex-ural waves are excited particularly well. This led tothe highest amplification factor (Φ = 4) measured[26]. Detonation interaction with piping componentssuch as dead ends, tees, and elbows can result in sig-nificantly higher and lower pressures than propaga-tion within straight piping runs. Experiments havedemonstrated that substantial loads and deflectionscan be generated in supporting structures due to det-onations in piping runs. Rupture of pipes due to in-ternal detonations has been observed in a number ofsituations. Particularly severe loads can occur if thetransition to detonation takes place near the closedend of a tube. Realistic material models and loadinghistories are required in order to make quantitativepredictions of plastic deformation. Strain rate effectscan be particularly important and difficult to accu-rately capture.

Acknowledgments

The results reported in this paper were obtained bya number of researchers working in my laboratoriesover the last decade. In particular, the bulk of thework was carried out by former students W. Belt-man, T.-W. Chao, (both now at Intel Corporation)and F. Pintgen (currently at GE Global Research,Nisakyuna, NY), and also current student J. Kar-nesky and former postdoctoral scholar Z. Liang, nowat AECL, Chalk River. Term projects by students T.Curran, E. Burcsu, L. Zuhal, A. Lam, M. Zeilonka,M. Kozlowski, and A. Lew contributed important re-sults. While working at the Caltech ASC center, P.Hung, F. Cirak, and R. Dieterding have all carriedout simulations and contributed substantially to ourunderstanding of how to design these experimentsin order to carry out effective validation of simula-

tions. Thanks to my solid mechanics colleagues W.G. Knauss, A. J. Rosakis, and G. Ravichandran fortheir helpful discussions and generous loans of instru-mentation. This research was sponsored over the lastdecade by the US Nuclear Regulatory Commission,the Office of Naval Research (ONR), and by the USDOE (NNSA) through the Caltech ASC project. Fig-ures 1, 3-7 reproduced by kind permission of ElsevierPublishing. Figures 8, 9 , 12 reproduced with kindpermission of Springer Science+Business Media.

References

[1] Shepherd, J. E., 2008, “Detonation in gases,”Proceedings of the Combustion Symposium, 32.

[2] Lee, J. H. S., 2008, The Detonation Phe-nomenon, Cambridge University Press.

[3] Thibault, P., Britton, L., and Zhang, F., 2000,“Deflagration and detonation of ethylene ox-ide vapor in pipelines,” Process Safety Progress,19(3), pp. 125–139.

[4] Grossel, S. S., 2002, Deflagration and Detona-tion Flame Arresters, Wiley.

[5] Kuznetsov, M., Breitung, W., Grune, J., andSingh, R. K., 2005, “Structural response ofDN15-tubes under radiolysis gas detonationloads for BWR safety applications,” 18th Inter-national Conference on Structural Mechanics inReactor Technology, SMiRT 18-J09-1, Beijing,China.

[6] Krieg, R., Dolensky, B., Goller, B., Breitung,W., Redlinger, R., and Royl, P., 2003, “Assess-ment of the load-carrying capacities of a spher-ical pressurized water reactor steel containmentunder a postulated hydrogen detonation,” Nu-clear Technology, 141(2), pp. 109–121.

[7] Naitoh, M., Kasahara, F., Kubota, R., andOhshima, I., 2003, “Analysis of pipe rupture ofsteam condensation line at Hamoaka-1, (I) Accu-mulation of noncondensable gas in a pipe,” Jour-nal of Nuclear Science and Technology, 40(12),pp. 1032–1040.

[8] OECD, 2000, “Flame acceleration and deflagra-tion to detonation transition in nuclear safety,”Tech. Rep. NEA/CSNI/R(2000)7, OECD Nu-clear Energy Agency.

Shepherd JPVT-07-1061 16

[9] Roy, G. D., Frolov, S. M., Borisov, A. A., andNetzer, D. W., 2004, “Pulse detonation propul-sion: challenges, current status, and future per-spective,” Prog. Energy Combust. Sci., 30, pp.545–672.

[10] Florek, J. R. and Benaroya, H., 2005, “Pulse-pressure loading effects on aviation and generalengineering structures,” Journal of Sound andVibration, 284, pp. 421–453.

[11] Baker, W. E., Cox, P. A., Westine, P. S., Kulesz,J. J., and Strehlow, R. A., 1983, Explosion Haz-ards and Evaluation, Elsevier.

[12] Bjerketvedt, D., Bakke, J. R., and van Winger-den, K., 1997, “Gas explosion handbook,” J.Hazardous Materials, 52, pp. 1–150.

[13] NFPA, 2002, “Standard on explosion protectionsystems,” Tech. Rep. NFPA 69, National FireProtection Association.

[14] ASME, 2008, “Case 2564 - Impulsively LoadedPressure Vessels,” Supplement 4 of ASME B &PV Code Cases: Boilers and Pressure Vessels,The American Society of Mechanical Engineers,New York, NY.

[15] Duffey, T., Rodriguez, E., and Romero, C., 2002,“Design of pressure vessels for high-strain rateloading: dynamic pressure and failure criteria.”Bulletin 477, Welding Research Council, P.O.Box 1942, New York, NY.

[16] Rodriguez, E. and Duffey, T., 2004, “Fracture-safe and fatigue design criteria for detonation-induced pressure loading in containment ves-sels,” Bulletin 494, Welding Research Council,P.O. Box 1942, New York, NY.

[17] Pellini, W., 1973, “Design options for selectionof fracture control procedures in the moderniza-tion of codes, rules and standards. Analytical de-sign procedures for metals of elastic-plastic andplastic fracture properties.” Bulletin 186, Weld-ing Research Council, P.O. Box 1942, New York,NY.

[18] Ciccarelli, G. and Dorofeev, S., 2008, “Flame ac-celeration and transition to detonation in ducts,”Prog. Energy Combust. Science, 34(4), pp. 499–550.

[19] Fickett, W. and Davis, W. C., 2001, DetonationTheory and Experiment, Dover Publications Inc.

[20] Tieszen, S. R., Stamps, D. W., Westbrook,C. K., and Pitz, W. J., 1991, “Gaseoushydrocarbon-air detonations,” Combust. Flame,84(3), pp. 376–390.

[21] Shepherd, J. E., 1986, “Chemical kinetics ofhydrogen–air–diluent detonations,” Prog. Astro-naut. Aeronaut., 106, pp. 263–293.

[22] Westbrook, C. K. and Urtiew, P. A., 1982,“Chemical kinetic prediction of critical param-eters in gaseous detonation,” 19th Symp. Int.Combust. Proc., pp. 615–623.

[23] Reynolds, W., 1986, “The element potentialmethod for chemical equilibrium analysis: im-plementation in the interactive program STAN-JAN,” Tech. rep., Mechanical Engineering De-partment, Stanford University.

[24] McBride, B. and Gordon, S., 1996, “Computerprogram for calculation of complex chemicalequilibrium compositions and applications: II.Users manual and program description,” NASAReference Publication 1311, NASA.

[25] Browne, S., Ziegler, J., and Shepherd, J. E.,2004, “Numerical solution methods for shockand detonation jump conditions,” Tech. Rep.FM2006.006, Graduate Aeronautical Laborato-ries, California Institute of Technology.

[26] Beltman, W. and Shepherd, J., 2002, “Lin-ear elastic response of tubes to internal detona-tion loading,” Journal of Sound and Vibration,252(4), pp. 617–655.

[27] Stamps, D. W., Slezak, S. E., and Tieszen, S. R.,2006, “Observations of the cellular structure offuel–air detonations,” Combust. Flame, 144(1-2), pp. 289–298.

[28] Pintgen, F., Austin, J. M., and Shepherd,J. E., 2003, “Detonation front structure: Va-riety and characterization,” Confined Detona-tions and Pulse Detonation Engines, G. Roy,S. Frolov, R. Santoro, and S. Tsyganov, eds.,Torus Press, Moscow, pp. 105–116.

[29] Austin, J., 2003, The Role of Instability inGaseous Detonation, Ph.D. thesis, California In-stitute of Technology, Pasadena, California.

[30] Pintgen, F., 2004, Detonation diffraction inmixtures with various degrees of instability,Ph.D. thesis, California Institute of Technology,Pasadena, California.

Shepherd JPVT-07-1061 17

[31] Kaneshige, M. and Shepherd, J., 1997, “Detona-tion database,” Tech. Rep. FM97-8, GALCIT.

[32] Dorofeev, S., Sidorov, V., Kuznetsov, M., Mat-sukov, I., and Alekseev, V., 2000, “Effect ofscale on the onset of detonations,” Shock Waves,10(2), pp. 137–149.

[33] Taylor, G., 1950, “The dynamics of combustionfronts behind plane and spherical detonations,”Proc. Roy. Soc. London, A200, pp. 235–247.

[34] Zeldovich, Y. B., 1942, “On the distribution ofpressure and velocity in the products of a deto-nation explosion, specifically in the case of spher-ical propagation of the detonation wave,” Jour-nal Experimental Theoretical Physics, 12(1),pp. 389–406.

[35] Radulescu, M. and Hanson, R., 2005, “Effect ofheat loss on pulse detonation engine flow fieldsand pefromance,” J. Propulsion Power, 21(2),pp. 274–285.

[36] Edwards, D., Brown, D., Hooper, G., and Jones,A., 1970, “Influence of wall heat transfer on ex-pansion following a C-J detonation wave,” Jour-nal of Physics. D, Applied Physics, 3(3), pp.365–376.

[37] Noda, N., Hetnarski, R., and Tanigawa, Y.,2002, Thermal Stresses, Taylor and Francis.

[38] Fickett, W. and Davis, W., 1979, Detonation,University of California Press, Berkeley, CA.

[39] Beltman, W., Burcsu, E., Shepherd, J., andZuhal, L., 1999, “The structural response oftubes to internal shock loading,” Journal of Pres-sure Vessel Technology, 121, pp. 315–322.

[40] de Malherbe, M., Wing, R., Laderman, A., andOppenheim, A., 1966, “Response of a cylindricalshell to internal blast loading,” Journal of Me-chanical Engineering Science, 8(1), pp. 91–98.

[41] Blevins, R. D., 1979, Formulas for natural fre-quency and mode shape, van Nostrand ReinholdCompany.

[42] Auslender, F. and Combescure, A., 2000,“Spherical elastic-plastic structures under inter-nal explosion. approximate analytical solutionsand applications.” Engineering Structures, 22,pp. 984–992.

[43] Biggs, J., 1964, Introduction to structural dy-namics, McGraw-Hill, Inc.

[44] Tang, S., 1965, “Dynamic response of a tube un-der moving pressure,” J. Eng. Mech. Div., Proc.ASCE, 91, pp. 97–122.

[45] Reismann, H., 1965, “Response of a pre-stressedcylindrical shell to moving pressure load,” EighthMidwest Mechanics Conference, S. Ostrach andR. Scanlon, eds., Pergamon Press, pp. 349–363.

[46] Simkins, T., 1987, “Resonance of flexuralwaves in gun tubes,” Tech. Rep. ARCCB–TR–87008, US Army Armament Research, Develop-ment and Engineering Center, Watervliet, N.Y.12189–4050.

[47] Simkins, T., Pflegl, G., and Stilson, E., 1993,“Dynamic strains in a 60mm gun tube - anexperimental-study,” Journal of Sound and Vi-bration, 168(3), pp. 549–557.

[48] Simkins, T., 1994, “Amplification of flexuralwaves in gun tubes,” Journal of Sound and Vi-bration, 172(2), pp. 145–154.

[49] Mirzaei, A., Mazaheri, K., and Biglari, H., 2005,“Analytical modeling of the elastic structural re-sponse of tubes to internal detonation loading,”International Journal Of Pressure Vessels AndPiping, 82(12), pp. 883–895.

[50] Akbar, R., 1997, Mach Reflection of GaseousDetonations, Ph.D. thesis, Rensselaer Polytech-nic Institute, Troy, New York.

[51] Brossard, J. and Renard, J., 1979, “Mechani-cal effects of gaseous detonations on a flexibleconfinement,” Gasdynamics of detonations andexplosions: technical papers from the seventh In-ternational Colloquium on Gasdynamics of Ex-plosions and Reactive Systems, J. R. Bowen, ed.,Progress in astronautics and aeronautics, v.75,AIAA, Gottingen, Federal Republic of Germany,pp. 108–121.

[52] Van de Ven, A., Olivier, H., and Gronig, H.,1996, “Dynamic structural response of a dustdetonation tube,” 7th International Colloquiumon Dust Explosions, Bergen, Norway, pp. 4.22–4.32.

[53] Sperber, A., Schildber, H., and Schlehlein, S.,1999, “Dynamic load on a pipe caused by acety-lene detonations - experiments and theoreticalapproaches,” Shock and Vibration, 6, pp. 29–43.

[54] Thomas, G., 2002, “The response of pipes andsupports generated by gaseous detonations,”Journal of Pressure Vessel Technology, 124, pp.66–73.

Shepherd JPVT-07-1061 18

[55] Simkins, T., 1995, “The influence of transientflexural waves on dynamic strains in cylinders,”Journal of Applied Mechanics-Transactions ofthe ASME, 62(1), pp. 262–265.

[56] Chao, T. and Shepherd, J., 2005, “Detonationloading of tubes in the modified shear waveregime,” Proceedings of the 24th InternationalSymposium on Shock Waves, Z. Jiang, ed., vol. 2,Springer, pp. 865–870.

[57] Chao, T. W., 2004, Gaseous detonation-drivenfracture of tubes, Ph.D. thesis, California Insti-tute of Technology.

[58] Shepherd, J. E., Karnesky, J., Pintgen, F., andKrok, J. C., 2008, “Experimental measurementsof strains and blast waves resulting from detona-tions in tubes,” Tech. Rep. FM2006.010, Gradu-ate Aeronautical Laboratories, California Insti-tute of Technology.

[59] Deiterding, R., Cirak, F., Mauch, S., and Me-iron., D., 2007, “A virtual test facility for simu-lating detonation-induced fracture of thin flexi-ble shells,” Int. J. Multiscale Computational En-gineering, 5(1), pp. 47–63.

[60] Cirak, F., Deiterding, R., and Mauch, S. P.,2007, “Large-scale fluid-structure interactionsimulation of viscoplastic and fracturing thinshells subjected to shocks and detonations,”Comput. Struct., 85((11-14)), pp. 1049–1065.

[61] Liang, Z., Karnesky, J., and Shepherd, J., 2008,“Detonations in C2H4-O2. Experimental mea-surements and validation of numerical simula-tions for incident and reflected waves.” Techni-cal Report FM2006-009, Graduate AeronauticalLaboratory, California Institute of Technology.

[62] Shepherd, J. E., Teodorcyzk, A., Knystautas, R.,and Lee, J. H., 1991, “Shock waves produced byreflected detonations,” Progress in Astronauticsand Aeronautics, vol. 134, AIAA, pp. 244–264.

[63] Liang, Z., Karnesky, J., and Shepherd, J., 2006,“Structural response to reflected detonationsand deflagration-to-detonation transition in H2-N2O mixtures,” Technical Report FM2006-003,Graduate Aeronautical Laboratory, CaliforniaInstitute of Technology.

[64] Shepherd, J., Schultz, E., and Akbar, R.,2000, “Detonation diffraction,” Proceedings ofthe 22nd International Symposium on ShockWaves, G. Ball, R. Hillier, and G. Roberts, eds.,vol. 1, pp. 41–48.

[65] Thomas, G. O. and Williams, R. L., 2002, “Det-onation interactions with wedges and bends,”Shock Waves, 11, pp. 481–491.

[66] Liang, Z., Curran, T., and Shepherd, J. E.,2008, “Structural response of piping componentsto detonation loading,” Tech. Rep. FM2006.008,Graduate Aeronautical Laboratories, CaliforniaInstitute of Technology.

[67] Liang, Z., Curran, T., and Shepherd, J. E., 2008,“Structural response to detonation loading in a90-degree bend,” Proceedings of the 26th Inter-national Symposium on Shock Waves, K. Han-nemann and F. Seiler, eds., Springer, in press,presented at 26th ISSW Gttingen, Germany, 15-20 July 2007.

[68] Naitoh, M., Kasahara, F., Kubota, R., andOhshima, I., 2003, “Analysis of pipe ruptureof steam condensation line at Hamoaka-1, (II)Hydrogen combustion and generation,” Journalof Nuclear Science and Technology, 40(12), pp.1041–1051.

[69] Chao, T.-W. and Shepherd, J. E., 2005, “Frac-ture response of externally flawed aluminumcylindrical shells under internal gaseous detona-tion loading,” International Journal of Fracture,134(1), pp. 59–90.

[70] Chao, T. W. and Shepherd, J. E., 2004, “Com-parison of fracture response of preflawed tubesunder internal static and detonation loading,”Journal of Pressure Vessel Technology, 126(3),pp. 345–353.

[71] Lam, A. and Zielonka, M., 2002, “Fracture re-sponse of externally flawed thin-walled plastictubes to gaseous detonation loading,” Ae104cReport, Graduate Aeronautical Laboratories,California Institute of Technology, Pasadena,California 91125.

[72] Brossard, J. and Charpentier de Coysevox, N.,1976, “Effects d’un confinement souple sur ladetonation des melanges gazeux,” Acta Astro-nautica, 3, pp. 971–981.

[73] White, D. A., 1957, “On the existence of higherthan normal detonation pressures,” Journal ofFluid Mechanics, 2, pp. 513–514.

[74] Craven, A. D. and Grieg, T. R., 1968, “Thedevelopment of detonation over–pressures inpipelines,” I. Chem. Eng. Symposium Series, 25,pp. 41–50.

Shepherd JPVT-07-1061 19

[75] Shepherd, J., 1992, “Pressure loads and struc-tural response of the BNL high-temperature det-onation tube,” Tech. Rep. A-3991, BrookhavenNational Laboratory, Upton, New York 11973.

[76] Boyack, K., Tieszen, S., and Stamps, D., 1993,“Internal-pressure loads due to gaseous detona-tions,” Proceedings Of The Royal Society OfLondon Series A-Mathematical Physical AndEngineering Sciences, 443 (1918), pp. 343–366.

[77] Kogarko, S. M., 1958, “Invetigation of pressureat the end of a tube in connection with rapidnonstationary combustion,” Sov. Phys. Tech.Phys., 3(9), pp. 1875–1879.

[78] Chan, C. K. and Dewitt, W. A., 1996, “DDT inend gases,” 27th Symposium (International) onCombustion, vol. 2, The Combustion Institute,Pittsburgh, PA, p. 2679.

[79] Zhang, F., Thibault, P. A., and Murray, S. B.,1998, “Transition from deflagration to deto-nation in multi-phase slug,” Combustion andFlame, 114, pp. 13–24.

[80] Duffey, T. and Mitchell, D., 1973, “Containmentof explosions in cylindrical shells,” Int. J. Mech.Sci., 15, pp. 237–249.

[81] Benham, R. A. and Duffey, T., 1974,“Experimental-theoretical correlation on thecontainment of explosions in cylindrical vessels,”Int. J. Mech. Sci., 16, pp. 549–558.

[82] Hodge, P. G., 1956, “The influence of blastcharacteristics on final deformation of circularcylindrical shells,” Journal of Applied Mechan-ics, 284, pp. 617–624.

[83] Duffey, T., 1971, “Approximate solutions of animpulsively loaded long cylinder governed by anelastic-plastic material law,” Acta Mechanica.,11, pp. 45–57.

[84] Duffey, T. and Krieg, R., 1969, “Effects of strainhardening and strain rate sensitivity on the tran-sient response of elastic-plastic rings and cylin-ders,” Int. J. Mech. Sci., 11, pp. 825–844.

[85] Fanous, F. and Greiman, L., 1988, “Simplifiedanalysis for impulsively loaded shells,” Journalof Structural Engineering, 114, pp. 885–899.

[86] Young, W. and Budynas, R., 2002, Roark’s for-mulas for stress and strain, McGraw-Hill, sev-enth Edition.

[87] Pintgen, F. and Shepherd, J. E., 2007, “Elas-tic and plastic structural response of tubesto deflagration-to-detonation transition,” Tech.Rep. FM2006-005, Explosion Dynamics Labora-tory, California Institute of Technology.

[88] Koslowksi, M. and Lew, A., 1999, “Plasticresponse of thin tubes to gaseous detonationwaves,” Ae104c Report, Graduate AeronauticalLaboratories, California Institute of Technology,Pasadena, California 91125.

Shepherd JPVT-07-1061 20