Engineering Failure Analysis 17 (2010) 1221–1229

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Engineering Failure Analysis

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Failure analysis for cylindrical explosion containment vessels

Li Ma a, Yang Hu a, Jinyang Zheng a,*, Guide Deng b, Yongjun Chen c,d

a Institute of Process Equipment, Zhejiang University, Hangzhou, Chinab China Special Equipment Inspection and Research Institute, Beijing, Chinac Institute of Reliable Legality of Components and Systems, University Karlsruhe (TH), Karlsruhe, Germanyd Institute of Materials Physics and Technology, Technical University Hamburg-Harburg, Hamburg, Germany

a r t i c l e i n f o a b s t r a c t

Article history:Received 19 November 2009Accepted 10 February 2010Available online 14 February 2010

Keywords:Explosion containment vesselsRate-dependent failure criteriaImpulsive loadingFailure analysis

1350-6307/$ - see front matter � 2010 Elsevier Ltddoi:10.1016/j.engfailanal.2010.02.009

* Corresponding author. Fax: +86 571 87953393.E-mail address: jyzh@zju.edu.cn (J. Zheng).

The elastic or elastic–plastic dynamical response of the explosion containment vessels(ECVs) subject to the impulsive loading have been studied intensively, however the dam-age mechanism of ECVs is still scarcely investigated. In this work two cylindrical explosioncontainment vessels under the different explosion loads are tested. The overpressure ismeasured and compared with the numerical result. The damage mechanism of adiabaticshear band is successfully applied to explain the failure mode of the ECVs, where the insta-bility analysis for the thermo-viscoplastic constitutive law is conducted to yield a rate-dependent failure criterion. Based on the overpressure analysis and rate-dependent failurecriterion, the shear failure mode of ECVs is studied for the first time, including the potentialinitial flaw in the meso-scale in the vessel. The failure analysis indicated that the rate-dependent failure criterion governs the damage mode of the vessel with the impulsiveloading however the initial flaw is mainly to ignite the shear band, which has minor influ-ence to the final failure mode of the vessel. The simulated fracture profile shows a goodagreement with the experimental result.

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1. Introduction

Explosion containment vessels (ECVs) have been widely used to contain the effects of explosion and detonation products,to protect persons and experimental equipments from the shock waves and contamination. Also, the applications are con-cerned with the storage, transposition and disposal for some explosive materials. In manufacturing field, ECVs offer a closedspace for explosive welding, forming and synthesis.

Since the first ECV’s born in Los Alamos National Laboratory (LANL) in USA in 1945, there are many styles of ECVs havebeen invented including spherical and cylindrical, single and multi-layered, metallic and composite material vessels. The cre-ative research works on the elastic and elastic–plastic response of the spherical and cylindrical vessels to the impulsive load-ing are carried out by Baker et al. [1,2] and Duffey et al. [3–5], in addition to which a special response of ‘‘strain growth” inthe elastic rang has been addressed by Duffey and Romero [6], Li et al. [7,8] and Zhu et al. [9] among others, where somedifferent mechanisms are presented to explain the cause of ‘‘strain growth”.

Comparing with the intensive investigation of the dynamical response of the vessels, the damage mechanism, includingfailure modes and failure criteria are scarcely studied. For the vessels under the severe impulsive loading, the materialbehavior exhibits great difference against the static load condition, and the failure mode transition occurs with the increas-ing of impulse. Some failure criteria have been proposed according to different failure modes. For example, the critical

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effective membrane strain was prescribed as n and 2n/3 for cylindrical and spherical ECVs at the instance of plastic tensileinstability failure by Nakamura et al. [10] (where n is the exponent for material model of reff ¼ Cen

eff ), however Duffey andDoyle [11] suggested the critical effective membrane strain of n for spherical ECVs, in addition, another different critical fail-ure strain was proposed by Nickell et al. [12] for the ductile tearing failure mode.

In this work, a new shear failure mode is considered with the proposed rate-dependent failure criterion. As has been rec-ognized that adiabatic shear band (ASB) is a typical failure mode for most materials subject to the impact or explosive loads[13], in which a high-speed deformation occurs, and the heat energy transformed from the inelastic work contributes to therise in temperature in local region of the material and therefore promotes the initiation and growth of shear band.

Micro-structural observations [14,15] have shown that the evolution of ASB includes the inhomogeneous shear deforma-tion with a distinct occurrence of shear localization and some tiny voids within the shear zone, and the voids growing andcoalescing into a large crack. The relevant numerical simulation of shear band also can be found in [16–18], etc. Howevermost of them focus on the onset of ASB, rather than ASB propagation. The main reason is that the used failure criteria areaptotic, such as the equivalent plastic strain or equivalent stress, which cannot accommodate to the dynamic characteristicof the ASB and so result in inaccurate solutions. In fact, the ASB involves complex multi-physical mechanisms, where the rateeffect and heat effect should be considered when determining the critical strain. Zhou et al. [19,20] proposed a rate-depen-dent failure criterion, and recently, a rate-dependent temperature criterion was proposed by Medyanik et al. [21]. Here thedamage mechanism of adiabatic shear band is attempted to account for the failure of the explosion containment vessels. Tosimplify the computation, only the rate-dependent failure strain is considered. In numerical simulation, material separatesalong the path of shear band propagation to form the final failure profile.

The paper is organized as follows. Firstly, two cylindrical vessels are tested with different explosive loads and the dynam-ical response is measured. Secondly, based on the instability analysis to the thermo-viscoplastic constitutive relation a rate-dependent failure criterion is presented for the evolution of shear band. Finally, the failure analysis model is built, where thepressure boundary condition obtained from the overpressure analysis is applied, and the initial flaw in meso-scale on thesurface of the vessel is involved, which is mainly to ignite the shear band and has been proved with little influence to thefinal failure of the vessel. The simulated result shows a good agreement with the experimental observation.

2. Experiments

2.1. Cylindrical explosion container vessels

The two cylindrical explosion containment vessels are tested under the different explosion loads. The vessel consists of acylindrical shell, bottom head, flange and head cover, as shown in Fig. 1. The bottom head was welded on the shell, and thehead cover was coupled with the shell by bolts. The inner diameter and the thickness of the shell are 150 mm and 18.5 mm,respectively.

2.2. Dynamic response measurements

The high explosive charges TNT are pressed into a hollow cardboard tube and located on the middle of the longitudinalaxis of the cylinder. The applied different charge sizes of TNT are listed in Table 1.

Fig. 2 shows the dynamic response measurements of the vessels. Three pressure rod gauges are used to record the over-pressure on the points A, B and C, and some electric resistance strain gauges are located on the points D–I to measure thehoop and axial strain in the experiments. The longitudinal distance L between the points is 56.25 mm.

The measured overpressure will be discussed with the simulated result in the Section 3.2, however the strain analysisincluding the strain growth observed in the experiments is excluded from this work for the relevant studies have beenaccomplished by our previous research, i.e. Li et al. [7,8] and Zheng et al. [22–25].

Fig. 1. Structure of the vessels.

Table 1Different charge sizes of TNT used in experiments.

Vessels TNT (g)

No. 1 20 55 250 425 500 600No. 2 20 55 250 425 550 –

Fig. 2. Dynamic response measurements.

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3. Failure analysis

3.1. Instability analysis and rate-dependent failure criterion

The failure of material under the impulsive loading can be explained by competitive interaction of strain rate hardeningeffect, strain hardening effect and thermal softening effect, which has been recognized in many investigations and treat-ments, and some thermo-viscoplastic constitutive relations have been presented to describe such behavior of materials[26,27]. Based on the instability analysis for the thermo-visoplastic material, a rate-dependent failure criterion is proposedin this section.

The below thermo-viscoplastic constitutive law is used

s ¼ ðs0 þ E1cÞ 1þ g ln_c_c0

� �1� a

TT0

� �ð1Þ

where s is shear stress, c is shear strain, _c is shear strain rate, and T is temperature, T0 and _c0 are reference temperature andreference strain rate, respectively. r0, E1, g and a are the material constants.

When the material is at the point of instability on the true stress–strain curve, then

dsdc¼ @s@cþ @s@ _c

d _cdcþ @s@T

dTdc¼ 0 ð2Þ

Also we have the energy equation with the heat conduction

bs @c@t¼ qc

@T@t� k

@2T@y2 ð3Þ

where y is the direction of the heat conduction and k is heat conductivity, q is mass density, c is the specific heat and b is thetransformation coefficient of heat energy.

For the adiabatic process, k = 0, thus

dT ¼ bsdcqc

ð4Þ

Substituting Eqs. (1) and (4) into Eq. (2), we obtain

E1

s0 þ E1cþ 1

1þ g ln _c_c0

g_c

d _cdc� ab

qcT0ðs0 þ E1cÞð1þ g ln

_c_c0Þ ¼ 0 ð5Þ

Eq. (5) is a Bernoulli differential equation, and its general solution is

1þ g ln_c0

c0

� �A� abE1

T0qcc

� �s0

E1þ c

� �¼ 1 ð6Þ

where the material constants for the instability analysis are listed in Table 2.

Table 2Material parameters for instability analysis.

g s0 (MPa) E1 (MPa) a/T0 (1/K) b q (g/cm3) c (J/g/K)

3.5 � 10�2 1060 740 1.68 � 10�3 0.9 7.85 0.48

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Eq. (6) represents a new rate-dependent failure criterion, where the critical strain decreases with the increment in strainrate. Fig. 3 shows the critical strain pattern with A = 1.0015.

3.2. Overpressure analysis

The finite element analysis of overpressure of the vessel is accomplished in this section using ALE (Arbitrary Lagrangian–Eulerian) algorithm and decoupled method, respectively. The JWL equation-of-state is used for high explosive:

p ¼ A 1� xR1V

� �e�R1V þ B 1� x

R2V

� �e�R2V þxe ð7Þ

where A, B, R1, R2, x are material parameters of JWL equation, and the initial density q0, initial internal energy e0, detonationvelocity D and Chapman–Jouget pressure PCJ are involved to startup the computation, see Table 3.

The air is taken as ideal gas in the initial time, and the polynomial equation-of-state is used with the initial density of1.293 kg/m3, initial internal energy of 2.53 � 105 J/m3, and polytropic index of 1.4.

In ALE algorithm, the thermo-viscoplastic relation as in Eq. (1) is considered for the vessel shell, while in the decoupledmethod, the vessel is taken as a rigid material. The 1/8 model (see Fig. 4) is constructed according to the test set-up.

Fig. 5 gives the measured and decoupled method calculated overpressure on point A of the vessel with different chargesizes of 250 g and 425 g TNT, from which the simulated overpressure wave is coincident with the measured results, and Ta-ble 4 shows the detailed comparison of peak overpressure and specific impulse between the experiments and simulation,where the experimental data is the average value of two vessels. From Table 4, the maximum error of specific impulse is12.7%. Though the error of peak overpressure reaches 20.6% under the station of 250 g TNT, the corresponding error of spe-cific impulse is only 0.7% and the latter is more decisive to the dynamic response of the structure.

The overpressure in ALE method is a little less than that in decoupled method, it is apparent that the movements of vesselin ALE method consume some extra energy than the decoupled method. Moreover, our intensive study indicates when theeffective plastic strain is less than 17.8%, the difference of peak pressure and specific impulse between two computationmethods does not exceed 5%.

The peak overpressure decreases with the increment in the longitudinal distance, i.e. with 55 g TNT charge, the peak over-pressure is 342.1 MPa, 41.8 MPa and 21.9 MPa on point A, B, and C, respectively. The percentage of peak overpressure onpoint B and C are only 12% and 6.4% of point A, and the percentage of specific impulse on point B and C are 34.1% and9.6% of point A correspondingly. The similar pulse decaying is found in all of detonation charges conditions. From which,the explosive load focus on the middle of the cylinder where the explosive is located, with the longitudinal length of 1.5R(R is the inner radius of the cylindrical shell). The obtained cognition is important to the next failure analysis, for the calcu-lated overpressure and the decoupled method will be applied on the failure analysis model, to greatly decrease the compu-tational cost, at the same time, have sufficient accuracy.

3.3. Failure analysis model

Fig. 6 shows the failure analysis model. Due to the test vessels experienced many times of explosive loading until to rup-ture, the formation of micro-crack can be anticipated. Here an equilateral cross groove with length of 1 mm and depth of185 lm is prenotched on the inner surface of the vessel to imitate the potential micro-crack in the vessel. The cross shapeis chosen to permit the shear band initiates longitudinally or circumferentially with equi-probability, by which the deflecting

Fig. 3. Rate-dependent failure criterion.

Table 3JWL parameters for TNT charge.

q0 (kg/m3) e0 (J/m3) D (m/s) PCJ (GPa) A (GPa) B (GPa) R1 R2 x

1630 7 � 109 6930 21 371.213 3.2306 4.15 0.95 0.30

Fig. 4. Computation model for overpressure analysis.

Fig. 5. Overpressure on point A, (a) 250 g TNT and (b) 425 g TNT.

Table 4Overpressure comparison.

TNT (g) Peak overpressure (MPa) Specific impulse (Pa S)

Experiment Simulation aError (%) Experiment Simulation aError (%)

Decoupled ALE Decoupled ALE

20 223.3 213.7 213.5 4.3 2111.2 1842.9 1824.1 12.755 342.1 391.2 390.7 14.4 5120.3 4766.3 4718.9 6.9

250 846.0 1020.0 1013.7 20.6 18427.5 18293.6 17946.1 0.7425 1556.5 1454.3 1439.9 6.6 27461.8 28556.3 27840.5 4.0500 1747.3 1709.2 – 2.2 33106.2 33892.2 – 2.4

a The error is calculated by the results from experiment and decoupled simulation.

Fig. 6. Failure analysis model.

L. Ma et al. / Engineering Failure Analysis 17 (2010) 1221–1229 1225

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influence to the propagation direction of shear band can be eliminated. The 8-node solid element is used in the model, withthe least mesh size of 74 lm around the cross groove, and 2 mm in the homogenous far-field of the shell.

Based on the discussion in the last section, the decoupled method is used in the failure analysis, where the pressureboundary conditions are directly applied on the inner surface as depicted in Fig. 6. The inner surface with length of225 mm (3R) is divided into 12 parts, with label of S1, S2, . . ., S12, and every part bears different explosive loads. The loadcurves are selected shown in Fig. 7. Using this pressure boundary condition, the fluid–solid interactive effect is ignored togreatly decrease the computation cost.

The rate-dependent failure criterion as shown in Eq. (6) is used in this section. Since the material in the band lost the loadbearing capacity, the corresponding elements are deleted to imitate the crack propagation on the vessel.

3.4. Results and discussion

Fig. 8 shows the failure procedure of the vessel at different times, at the time of 65 ls, the dilatation on the middle of thevessel can be observed, with the maximum effective plastic strain of 0.19. When at the time of 95 ls, a small crack penetratesthe wall of the vessel, and at the time of 105 ls, the crack propagates faster and begins to bifurcate. At the time of 150 ls, thecrack bifurcation is obvious and proximately forms an ‘‘X” shape fracture of the vessel (see Fig. 8e). Fig. 8f is the failed vessel

Fig. 7. Imposed pressure boundary conditions.

Fig. 8. Failure procedure with rate-dependent failure criterion: (a) t = 65 ls, (b) t = 95 ls, (c) t = 105 ls, (d) t = 150 ls, (e) zoomed map, and (f) failed vessel.

L. Ma et al. / Engineering Failure Analysis 17 (2010) 1221–1229 1227

(No. 1) at the explosive loading with 600 g TNT, where the simulated failure mode is in a good agreement with the real pro-file of the rupture. But there still have some difference, the characteristic length and buckling extent of the real fracture islarger than the simulated one, it can be explained that when the crack penetrates the wall of the vessel, most of the highpressure energy will concentrate and release from the rupture at once, which aggravates the damage localization and en-larges the fracture. However in simulation environment, the load acts on the inner surface durably, which leads to more gen-eral plastic deformation of the central ring of the vessel.

Some comparative computations are accomplished to identify the influences from the failure criteria and initial flaws tothe failure mode. Fig. 9 shows the simulated failure procedure with the same load condition and flaw type but the criticalequivalent plastic strain is ef = 0.25. Here we should acknowledge that a definite critical strain has never been proved to exist,and such critical strain is chosen can be referenced in [18]. The termination time is 250 ls to allow the crack develop fully,however the crack extends slowly, no penetration is found until the time of 142 ls, and the profile of the crack is along thelongitudinal direction of the cylinder, which contravenes the experimental observation.

The difference between two failure models is focused in the neighborhood where crack propagates, while in the otherarea far away from the crack, the structural response is very close. Fig. 11 shows that the outward radial displacement ofpoints 1, 2 and 3 on the opposite side (see Fig. 10) against the crack propagation surface highly resembles 10, 20 and 30, where1, 2, 3 and 10, 20, 30 are the result of rate-dependent failure model and constant critical strain model, respectively.

The lost of mass is used to illustrate the propagation extent of the crack. Fig. 12 shows that the drop of mass in the rate-dependent failure model is about three times larger than that of constant critical strain model. In the rate-dependent failure

Fig. 9. Failure procedure with constant critical strain: (a) t = 95 ls, (b) t = 142 ls, and (c) t = 200 ls.

Fig. 10. Opposite side against the crack propagation surface.

Fig. 11. Radial displacement.

Fig. 12. Drop of mass.

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model, the drop of mass decreases sharply, which means the crack propagates quickly; while in the constant critical strainmodel, the crack does not propagate immediately, it holds for a period of time until the preset critical equivalent strain issatisfied then begins to develop slowly and gets arrested at the time of 200 ls.

Also two another different types of cross grooves are used in the failure analysis model, with hoop length of 0.5 mm, lon-gitudinal length of 1 mm, and hoop length of 1 mm with longitudinal length of 0.5 mm, respectively. The simulated resultsexhibit similar shear failure mode as in Fig. 8, only with small variation of the characteristic length of the fracture. It is there-fore conclude that with a certain impulsive loading, the shear failure mode of the vessel is mainly governed by the rate-dependent failure criterion, and the initial flaw is used to ignite the shear band, which has little influence to the final profileof the crack.

4. Conclusions

The failure analysis of the cylindrical vessel to the impulsive loading is conducted in this work, by which the damagemechanism of the vessel can be understood. Firstly, the damage mechanism of shear bands are successfully applied to ex-plain the shear failure mode of the explosion containment vessels, where the thermo-viscoplastic constitutive relation isdecisive to reflect the interactive effects between the strain and strain rate hardening and the heat softening of materialbehavior. Secondly, the proposed rate-dependent failure criterion governs the failure mode as well as the impulsive loading.However the preset initial flaw in-meso-scale is used to ignite the shear band, which has minor influence to the final failureof the vessel. The obtained treatment is helpful to the design and life-span evaluation for the engineering-used explosivecontainment vessels.

Acknowledgements

This work is supported by National Natural Science Foundation of China (Grant No. 50875236) and China PostdoctoralScience Foundation (Grant No. 20080431309). The support from the Science Foundation of Chinese University is also greatlyappreciated.

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