The deformation and shear failure of peripherally clamped centrally supported blast loaded circular plates

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International Journal of Impact Engineering 32 (2005) 92–117 www.elsevier.com/locate/ijimpeng

The deformation and shear failure of peripherally clamped centrally supported blast loaded circular plates T.J. Cloete, G.N. Nurick, R.N. Palmer Blast Impact and Survivability Research Unit (BISRU), Department of Mechanical Engineering, University of Cape Town, Private bag, 7701, Rondebosch, South Africa Received 29 October 2004; received in revised form 20 May 2005; accepted 11 June 2005 Available online 11 August 2005

Abstract An experimental investigation of deformation and shear failure in peripherally clamped centrally supported (PCCS) blast loaded circular plates is presented. In particular, the timing of shear failure events and the associated magnitude of the shear forces are reported. The plates exhibited three distinct modes of failure: mode I (large ductile deformation), mode II (boundary failure after large deformation) and mode III (boundary failure prior to large deformation). Approximate energy analyses for mode I deformation of annular and PCCS plates are presented. An expression for dimensionless impulse, applicable to annular and PCCS plates, is derived. The experimental data and analytical predictions exhibit satisfactory correlation. r 2005 Elsevier Ltd. All rights reserved. Keywords: Blast load; Peripherally clamped centrally supported circular plates; Hopkinson pressure bar; Large inelastic deformation; Failure modes

1. Introduction During the past 20 years there has been phenomenal growth in the sophistication of computational mechanics codes and the power of the work stations. Commercial simulation codes have made these powerful tools available to a broad spectrum of engineering practice. The level of Corresponding author. Tel.: +27 21 6503235; fax: +27 21 6503240.

E-mail address: [email protected] (T.J. Cloete). 0734-743X/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijimpeng.2005.06.002

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Nomenclature Ek H I m r Ri Ro w w~ Wp a d
r l F r sd sy

kinetic energy plate thickness total impulse plate mass radial distance inner plate radius outer plate radius vertical plate displacement maximum plate displacement (or the ave–max displacement in the case where PCCS plate deformation is not axisymmetric) plastic work ratio of the dynamic flow stress to the quasi-static yield stress dimensionless plate deflection radial strain dimensionless parameter describing the effect of plate geometry on energy dissipation through plastic work dimensionless impulse plate density mean dynamic flow stress quasi-static yield stress

detail to which problems can be analysed has surpassed the level of experimental detail generally available in the literature. Over the last decade there has been growing recognition of the need for reliable and detailed experimental data for the purpose of code validation. Such experiments have been termed ‘precision tests’ and criteria for what constitutes a precision test have been proposed [1]. The greater the amount and variety of experimental readings, the more stringent a code validation process will be. Of particular importance in the context of blast loading experiments, is that the blast load must be well characterized and repeatable. Ideally, there should be some way to measure the blast wave intensity for every test. Furthermore, it is preferable that the measurements have some degree of redundancy. In other words, there should be some way to cross-check a given experimental reading. For instance, if a pressure transducer is used to measure the pressure history of a particular blast loading event, the resulting pressure history could be integrated to yield the impulse delivered to the target. This could be compared to the actual impulse obtained by mounting the target on a ballistic pendulum. The ballistic pendulum has yielded valuable data for code validation. According to Nurick and Martin [2], the ballistic pendulum was originally devised to generate experimental results to confirm the predictions of analytical studies of the behaviour of blast loaded plates. Over the past 40 years, this simple and robust technique has yielded results, such as deflection vs. total impulse, with remarkable repeatability. These experimental results have generally been expressed

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in non-dimensional form and tabulated for convenient comparison with analytical solutions and more recently, numerical simulations [2–6]. It is in the latter application that the need for additional experimental measurements has become apparent. Instrumented versions of the basic ballistic pendulum apparatus have been developed that allow for additional experimental parameters, such as the deflection history of the centre of the plate, to be recorded [3,4]. However, there remain some experimental parameters that would be useful in code validation that have not been successfully captured. One phenomenon that has proven difficult to investigate experimentally is shear failure at the boundaries of blast loaded plates [5–8]. Although the energy dissipated by shearing has been estimated [5], the timing of shear failure events and the magnitude of the associated shear forces have not been recorded during conventional ballistic pendulum tests. Conventional high-speed load cells and displacement transducers are not suited to capturing shear failure events. The reason for this is that the contribution that shearing makes to the overall deflection of the plate is small and difficult to discern [9]. Furthermore, any attempt to measure the shear forces at a clamped (or built-in) boundary is obscured by the inertia of the clamps and/or the dynamic interaction between the load transducers and their support structures. Recently, a novel variation of the ballistic pendulum apparatus has been developed in an attempt to address these issues [10]. It incorporates a circular plate specimen mounted in a peripherally clamped centrally supported (PCCS) configuration. This configuration has been developed specifically to capture shear failure events during blast loading experiments. In this paper, a series of blast loading experiments on PCCS mild steel circular plates is reported. In particular, the timing of shear failure events and the associated shear force history is documented.

2. Experimental configuration A schematic diagram of the PCCS configuration is shown in Fig. 1. A thin specimen plate is clamped between two thick plates with circular cutouts. A Hopkinson bar, made from silver steel, is placed against the centre rear face of the specimen plate to act as a central support. The other end of the bar is free to move a short distance before coming into contact with a stopper. An

Fig. 1. Schematic diagram of a peripherally clamped centrally supported (PCCS) configuration for a blast loaded plate.

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Fig. 2. PCCS plate specimen mounted on a ballistic pendulum in a blast chamber.

explosive charge is placed at a specific stand-off distance by means of a polystyrene buffer. The assembly is mounted on a ballistic pendulum as shown in Fig. 2. The PCCS configuration is essentially the same as that used for clamped circular plates [2–5], except for the additional central support boundary. Whereas the plate configurations used in conventional ballistic pendulum are essentially scale models of plates found in practice, the PCCS configuration clearly is not. Rather, the PCCS configuration is specifically designed to provide precision data in a convenient form for purpose of code validation. The addition of the central support enhances the precision of the ballistic pendulum. It performs the role of both structural support and force transducer allowing the reaction force history between the plate specimen and the central support to be deduced from the stress waves captured in the central support during a blast test. In this way the problems associated with clamp inertia and dynamic sensor/structure interaction are circumvented. The resulting force history provides information on the magnitude and duration of the impulsive loading in addition to the overall plate response. In particular, the time to failure at the inner support is captured. This is accomplished without altering the simple axisymmetric nature of the original configuration which made it so amenable to both analytical and numerical treatment. For the test series reported here, the specimen plates were cut from 1.6 mm thick mild steel sheet. Fig. 3 depicts the results of four quasi-static tensile tests, all at a strain rate of 5
104 s1, using specimens cut from the mild steel plates. An average yield stress of approximately 315 MPa was observed. The clamped outer boundary was 100 mm in diameter, while the inner boundary diameter was formed by the central support bar of 22 mm diameter and 1 m length. The explosive used throughout this study was PE4 at a stand-off distance of 30 mm. Some initial experiments were conducted with a central charge, but the results were unsatisfactory. Significant deformation prior to shear failure at the inner boundary could not be induced. It was concluded that the impulsive load was too localized. To spread the blast load, the charge configuration shown in Fig. 4 was adopted. A charge of 1 g was placed in the centre with two cross-leaders, of 1 g each, leading to a 60 mm diameter ring charge of mass ranging between 2 and 7 g. Detonation was initiated in the centre. Blast tests with annular plates were also conducted. The annular plates have a 24 mm diameter central hole, which ensured that the central bar would not be in contact with the plate during a test. This allowed the central bar to record the blast load without being affected by the plate

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engineering stress [MPa]

400 350 300 250 200 150 100 50 0 0

5

10 15 20 25 30 35 plastic engineering strain [%]

40

45

Fig. 3. Graph of four quasi-static tensile tests, all at a strain rate of 5
104 s1, using specimens cut from the mild steel plates used in this test series.

(a)

(b)

Fig. 4. General layout of the explosive charges with and without an outer ring.

response. The term ‘central bar’, as opposed to ‘central support’, will be used in the context of the annular plate tests since the annular plate was not in contact with, and hence not supported by, the central bar. By comparing the central bar stress history for an annular plate test to that of the central support for a PCCS plate subjected to essentially the same charge and impulse, the contribution of various deformation mechanisms to the total impulse transfer were determined. (This will be discussed later in more detail.) All deflection data were obtained a posteriori, by physically measuring the deformed plates. The total impulse imparted to the plate was obtained from the recorded motion of the pendulum. The stress wave induced in the bar by the plate response was recorded via two centrally located strain gauges, diametrically mounted in a half bridge configuration, using a 1 MHz amplifier and data logging card sampling at 10 MHz with 12-bit resolution. Dispersion correction was carried out according to the multi-mode methodology proposed by Lee et al. [11]. Previous results reported by Cloete et al. [10] show that an algorithm incorporating the first four Pochhammer-Cree modes is superior to an algorithm incorporating only the fundamental mode. However, there were still spurious oscillations remaining in the processed signals after dispersion correction, which had to be smoothed. The smoothing algorithm will be discussed in the next section in the context of the

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experimental results. The authors are presently continuing with research aimed at extending the useful frequency range of the Hopkinson bar.

3. Experimental results A test series with 14 PCCS and six annular plates was conducted, the results of which are summarized in Table 1. The test numbering system follows a format where the first two letters stand for the initials of the researcher who conducted the tests, while the next six numbers and the final lower case letter indicate the day and the order in which a group of tests were conducted. In addition to the standard ballistic pendulum measurements, such as the total impulse and final deflection, the portion of the impulse transferred through the central support is reported. A typical central support stress signal recorded is shown in Fig. 5. The reaction force history of the central support is obtained by multiplying the stress signal by the cross-sectional area of the support bar. Integrating the resulting reaction force history provides a measure of the impulse transferred through the central support. The particular stress wave depicted in Fig. 5 was chosen to illustrate a worst case with respect to noise. All the other recorded signals generally contained less severe oscillations. It can be seen that despite the use of dispersion correction methods the signal still required smoothing. Table 1 Summary of experimental data Test no.

Mode

Explosive Total mass (g) impulse (N s)

Bar impulse (N s)

Impulses: Max–max Min–max Ave–max Final hole bar/total deflection deflection deflection diameter (mm) (mm) (mm) (mm) (%)

RP031008b TC040306b TC040306a RP030905a RP030904a RP031007b RP030904b RP030919b RP031009c RP031009a RP030905b RP030923e TC040318a TC040313a TC040304c TC040304b TC040304a RP030905c RP031009b TC040313b

I I I I II* II* II* II II II II III III III Annular Annular Annular Annular Annular Annular

5 5.5 6 7 6 6 6.5 7 7 8 8 9 9.5 10 5 6 7 8 8 10

2.38 2.51 2.62 2.27 2.83 3.09 2.91 2.46 2.32 2.51 2.43 2.25 no rec. 2.55 0.33 0.60 0.78 0.94 0.98 no rec.

39 39 40 38 27 30 26 22 19 18 18 16 no rec. 14 12 9 7 7 7 no rec.

6.2 6.5 6.6 6.0 10.4 10.3 11.2 11.0 12.4 14.1 13.4 14.3 16.9 18.1 2.7 6.9 10.9 12.9 13.6 17.1

2.6 3.5 2.13 1.82 4.58 4.6 4.18 8.08 8.12 13.24 10.98 14.68 18.82 20.2 1.4 8.84 14.2 15.94 16.34 20.82

1.12 1.28 1.52 1.58 3.9 3.7 2.9 6.8 7.86 9.42 8.64 14.52 18.8 19.8 1.14 7.9 13.26 13.44 15.8 20.42

The entry ‘no rec.’ indicates that a usable stress signal was not recorded in the central bar.

1.86 2.39 1.83 1.7 4.24 4.15 3.54 7.44 7.99 11.33 9.81 14.6 18.81 20 1.27 8.37 13.73 14.69 16.07 20.62

— — — — — — — 22.4 22.4 23.5 23 25.5 29 29.5 24 25.5 28.5 30.5 32 33.5

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200

RP031007b - 6g - 10.28Ns weighted smoothing Central Bar Stress [MPa]

150

Stress signal obtained from the central support using 4-mode dispersion correction.

100

50

Stress signal obtained by applying a weighted smoothing algorithm to the dispersion corrected signal from the central support.

0

-50 -50

0

50

100

150

200

Time [µs]

Fig. 5. Graph of a typical central support signal for a PCCS plate test.

Filtering techniques could not be used to smooth the signal since all frequencies up to 1 MHz were required for dispersion correction. Simple averaging type smoothing also proved to be inappropriate. The amount of averaging required to smooth high-frequency oscillations (periods of less than 4 ms) also tended to reduce the magnitude of the longer duration stress spikes (duration greater than 10 ms). The existence of sharp stress spikes was expected since the primary loading was due to the blast waves, which have a characteristic sharp wave form featuring a pressure rise followed by an exponential decay. The effect of ‘blunting’ these stress spikes is therefore inappropriate. Furthermore, it was found that averaging tended to reduce the momentum obtained by integrating the stress signal. This is clearly incorrect since neither dispersion nor noise effects should alter the apparent momentum of a recorded stress wave. A weighted smoothing algorithm was developed that was able to effectively eliminate the highfrequency oscillations, while maintaining the sharp character of the stress spikes and preserving the momentum. The algorithm follows a three-step procedure. In the first step, the signal is smoothed by means of a least-squares fit over an 8 ms sampling window. This step removes the high-frequency oscillations, but also blunts the stress spikes. The second step involves evaluating the second derivative of the smoothed signal. A large absolute value for the second derivative indicates a rapid change in the gradient of the signal. Two particular stress wave features, a stress spike and a sudden drop or rise in the stress signal, are of great relevance to this study and both are associated with rapid changes in gradient. With this in mind, the third step in the algorithm involves a repetition of the original smoothing process, but this time the sampling window is reduced according to the absolute magnitude of the second derivative evaluated in step two. In other words, the amount of smoothing applied to a fluctuating portion of the signal is weighted according to the duration and severity of the fluctuation. Hence, a high-frequency oscillation of

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Central Bar Stress [MPa]

200 RP031009c - 7g - 12.42Ns

150

RP030919b - 7g - 10.99Ns

100 50 0 -50 -50

0

50

100

150

200

Time [µs]

Fig. 6. Graph of the central support reaction force history for two PCCS plate experiments with identical charge masses, but slightly different total impulses.

short duration would be damped out, while pressure spikes resulting from the blast loading, which are of relatively long duration, would not be altered. An example of the resultant weighted smoothing is also plotted in Fig. 5 where it is superimposed upon the dispersion corrected signal. It is evident that the high-frequency components oscillations have essentially been eliminated, while the overall character of the signal has been preserved. Furthermore, the weighting algorithm did not alter the integral of the signal and hence the evaluation of the momentum contained in the stress wave was unaffected. The same weighting parameter has been used on all the signals. To test the repeatability of the present test configuration, two PCCS plate tests with identical 7 g charges were conducted. The smoothed central support stress histories for these two experiments are shown in Fig. 6. The two curves show good repeatability, even though the total impulse values are slightly different. Furthermore, the displacements, as recorded in Table 1, were also similar with the larger impulse resulting in a slightly larger deflection. In the sections to follow, the repeatability of the PCCS and annular test configurations will be discussed with respect to specific features of the central support stress histories and the correlation of the test data with analytical solutions. In the context of blast testing, these results are considered to represent an acceptable level of repeatability, especially considering that the tests were conducted by two different researchers over a period of 7 months.

4. Discussion of results Due to their novelty, the PCCS and annular plate configurations provided results not previously reported, and hence there is no data set in the literature with which they can directly be compared. In lieu of this, the aim of the present section is to show that the data reported here is internally consistent while the next section introduces an analytical derived dimensionless damage number that allows the data to be compared with circular plate results. The discussion of these results is divided into three parts:

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In the first part the central bar/support stress signals for annular and PCCS plate tests are compared. This comparison forms the basis for the interpretation of the various features of the central bar/support stress signal. In the second part the central support stress signals from five PCCS plate experiments are compared with reference to the deformed plates. The observed transitions between various modes of plate failure are discussed and new generalized definitions for the failure modes of blast loaded plates are introduced. In the third part, the variation of the final plate deflections with respect to the total impulse, for both annular and PCCS plate configurations, are discussed. This is in keeping with what is standard practice for circular and rectangular blast loaded plates. In addition, the variations of the final hole size and the fractions of the total impulse transmitted through the central bar with respect to the total impulse are presented. In particular, the transitions between the various failure modes are highlighted. 4.1. Comparison of annular and PCCS plate responses

Central Bar Stress [MPa]

Consider the central bar stress histories plotted in Figs. 7 and 8. Fig. 7 depicts the central bar stress wave for test number RP031007c, where a ring of explosive was omitted from the charge (see Fig. 4b) resulting in a total charge mass of 3 g. Only one stress spike appears in the stress signal. The signal exhibits classic blast wave features of a sudden rise followed by an exponential decay. Fig. 8 depicts the central bar stress wave for test number RP031009b where a ring of explosive was included (see Fig. 4a) bringing the total charge mass to 8 g. Two pressure spikes are observed with the first spike being almost identical to that observed in Fig. 7. This implies that the second pressure spike is due to the blast wave from the ring charge arriving at the central bar. The delay of approximately 20 ms between the two spikes is due to two factors. Firstly, the detonation is initiated in the centre of the charge and thus the detonation of the ring charge would be slightly delayed due to a finite detonation velocity. Secondly, the blast wave from the ring charge has slightly further to travel to the central bar.

200 RP031007c - 3g - 3.2Ns weighted smoothing

150 100 50 0 -50 -50

0

50

100

150

200

Time [µs]

Fig. 7. Graph of the central support reaction force history for an annular plate without the outer ring charge.

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Central Bar Stress [MPa]

200 RP031009b - 8g - 13.63Ns weighted smoothing

150 100 50 0 -50 -50

0

50

100

150

200

Time [µs]

Fig. 8. Graph of the central support reaction force history for an annular plate with an 8 g charge.

Previous researchers have based the load duration on the so-called ‘burn time’ of the explosive charge, or the time required for the detonation front to reach the furthest extremity of the charge [12,13]. The detonation velocity of PE4 is approximately 8190 m/s [14] and thus, for the charge shape used in the test series reported here, the burn time is approximately 6.54 ms. Furthermore, the mean blast pressure is normally obtained by dividing the impulse intensity by the burn time. In this case, assuming a uniform impulse would result in mean blast pressure of 410 MPa. From the pressure duration exhibited in Fig. 8, it is clear that the burn time underestimates the load duration and consequently leads to an overestimate of the mean peak blast load pressure. The uniformity of the impulse intensity distribution can be gauged by calculating the fraction of the total impulse to pass through the central bar during an annular plate test. If the impulse intensity distribution was perfectly uniform, the ratio of the bar area to the total area implies that approximately 5% of the total impulse would pass through the central bar. The results for annular plate tests recorded in Table 1 indicate that between 7% and 12% of the total impulse passed through the central bar. This indicates that the impulse distribution for the charge configuration used in this test series was close to, but not completely uniform. Fig. 9 depicts the central support stress wave for test number RP031009a where a PCCS plate was loaded with a charge of the same configuration as that used in the annular test RP031009b (see Fig. 4a). The stress waves depicted in Figs. 8 and 9 both exhibit two pressure spikes of similar magnitude and timing within the first 30 ms. However, after the second stress spike the signals differ considerably, with the stress wave for the annular plate test decaying exponentially to zero, while for the PCCS plate test a local maximum is observed after approximately 50 ms. Since the two tests differ only with respect to their inner boundary conditions, this difference is attributed to the interaction of the PCCS plate with its central support. Based on this observation, an estimate of the contribution of the plate response to the stress signal for the PCCS plate test was obtained by subtracting the signal for the annular test from the signal for the PCCS plate test. The resulting plate response is depicted by the thick grey curve in Fig. 10, where it is superimposed upon the signals plotted in Figs. 7–9. The plate reponse curve in Fig. 10 is an estimate in the sense that the original curves are obtained from two different tests, and hence some parts of the curve may not accurately represent the plate response. In particular, the dip in the plate response curve between 20 and 25 ms is not considered to represent the plate behaviour. Nevertheless, from 30 ms onward the plate response appears to dominate and the recorded signal is taken to be representative of the plate response.

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Central Bar Stress [MPa]

200 RP031009a - 8g- 14.07Ns weighted smoothing

150 100 50 0 -50 -50

0

50

100

150

200

Time [µs]

Fig. 9. Graph of the central support reaction force history for a PCCS plate with an 8 g charge.

200 Central support reaction force history for an 8g charge acting on a PCCS

Central Bar Stress [MPa]

150

RP031009a - 8g - 14.07Ns RP031007c - 3g - 3.2Ns RP031009b - 8g - 13.63Ns approximate shearing contribution

Central support reaction force history for an 8g charge acting on an annular plate.

100

Contribution of the shear forces to the central support reaction force history calculated by subtracting the annular plate response from that of the complete plate.

50

0 Central support reaction force history for a charge without an outer ring acting on an annular plate. -50 -50

0

50

100

150

200

Time [µs]

Fig. 10. Combined plot of the curves in Figs. 7–9. An estimate of the reaction force history due to the shear forces alone (thick grey line) is obtained by subtracting the 8 g annular response from the full plate response.

The features discussed above appeared, to varying degrees, in the support bar signals for all the PCCS plate tests. Based on the above observations, stress spikes that appear in the support bar signals were taken to be a result of the blast wave propagating through the plate and directly in the central support, while features that appear after approximately 50 ms were taken to be representative of plate behaviour.

4.2. Transitions in failure modes with increasing total impulse In Figs. 11 and 12, a selection of support bar signals and photographs of the corresponding deformed plates show the transition from one mode of failure to another as the impulse increases.

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200 Consistent initial spike.

150 Central Bar Stress [MPa]

Secondary spike increases with increasing ring charge.

RP031008b -5g - 6.16Ns RP031007b -6g - 10.28Ns RP030919b -7g - 10.99Ns RP031009a -8g - 14.07Ns RP030923e -9g - 14.29Ns

Consistent maximum shear forces.

100

No shear separation.

50

0 Partial shear separation.

Sudden and complete shear separation.

-50 -50

0

50

100

150

200

Time [µs]

Fig. 11. Central support reaction force history for five different charge masses.

The three distinct modes of failure that occur in blast loaded plates were first described by Teeling-Smith and Nurick [5], who followed the definitions used by Menkes and Opat [15] for blast loaded beams. The definitions were subsequently refined by Nurick et al. [6,7] to include subcategories that distinguish, for instance, between necking and tearing or complete and partial failure. In the context of this work the relevant failure modes are: Mode Mode Mode Mode

I II* II III

Large inelastic deformation. Partial tearing along the clamped boundary with large inelastic deformation. Complete tearing along the clamped boundary with large inelastic deformation. Complete transverse shearing along the clamped boundary without large inelastic deformation.

The following modifications to the failure modes definitions are proposed to make them applicable to the data reported in this paper: Mode I Large inelastic deformation. Mode II* Partial failure (tearing or shearing) along a boundary subsequent to large inelastic deformation. Mode II Complete failure (tearing or shearing) along a boundary subsequent to large inelastic deformation. Mode III Complete failure (tearing or shearing) along a boundary prior to large inelastic deformation. The revised definitions given above were motivated by the observation that the general behaviour of the PCCS circular plates did not conform to the conventional definitions. In

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RP031008b 5 g - 6.16 Ns

No observable shear failure, i.e. mode I

RP031007b 6 g- 10.28 Ns

Partial shear failure, i.e. mode II*

RP030919b 7 g- 10.99 Ns

Complete shear failure with strong convex plate curvature and final hole size equal to disk size indicates significant membrane action prior to shear failure, i.e. mode II

RP031009a 8 g- 14.07 Ns

Complete shear failure with neutral plate curvature indicates a transition from late to early shear failure, i.e. mode II to mode III.

RP030923 9 g- 14.29 Ns

Complete shear failure with slight concave plate curvature and final hole size larger than disk size indicates significant membrane action occurred after complete shear failure, i.e. mode III.

Fig. 12. Photographs of deformed plate specimens corresponding to the reaction forces histories in Fig. 11.

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particular, tearing failure was never observed whereas large deformations were always observed. Hence, it is apparent that the distinction between the failure modes should not primarily be based on the nature of the failure, i.e. tearing or shearing, or on whether large inelastic deformations occurred. Rather, the distinction should be based on the sequence in which the various failure mechanisms occur. A graphical interpretation of the revised definitions is given in Fig. 13. The revised definitions adequately describe the behaviour of both conventionally clamped and PCCS plates. The proposed revisions make the failure mode definitions more generally applicable. Hereafter, the results depicted in Figs. 11 and 12 will be discussed separately and in detail within the context of the revised failure mode definitions. 4.2.1. RP031008b—5 g (2 g in ring)—6.16 N s—mode I failure An initial stress spike is observed, but no clear second stress spike was discerned. The total impulse of 6.16 N s is approximately double the impulse obtained in a test where the ring charge is omitted (see Fig. 7), indicating that the ring charge did make a contribution despite the lack of a discernable second spike.

Conventional Peripherally Clamped Circular Plate

Peripherally Clamped Centrally Supported Circular Plate

Mode I: Large inelastic deformation

Mode II: Large inelastic deformation followed by tearing and/or shearing at the boundary. In this paper only shearing at the inner boundary occured.

Mode III: Shearing at the boundary prior to large inelastic deformation. In the case of a conventional plate this would result in a mildly deformed disk with high residual velocity. By contrast, a PCCS plate shearing at the inner boundry would continue to deform as an annular plate resulting in a cone with an enlarged central hole.

Fig. 13. Modes of failure for a PCCS plate specimen compared with a conventional clamped circular plate.

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After the first stress spike the signal maintains a stress level of approximately 35 MPa. At 100 ms the signal begins to rise again and reaches a maximum of 40 MPa after approximately 120 ms. By this stage the blast wave has dissipated and the observed signal is due to the plate response. The value of the average transverse shear stress in the plate along the inner boundary was approximately 138 MPa. This value was obtained by multiplying the recorded stress by the crosssectional area of the central support bar and dividing by the transverse shear area of the plate at the inner boundary. Blast loaded plates are reported to experience residual elastic vibrations after the plastic deformation is complete [3,4]. Residual elastic strain energy in the plate at the point of maximum deflection will cause the plate to rebound, initiating the vibrations. In the PCCS configuration a rebounding plate would continue exerting a force on the central support. This force would gradually decrease as the elastic strain decreases until reaching zero, at which point the plate separates from the central support. Based on this reasoning, the drop in the signal after 130 ms represents the end of the plastic deformation phase after which separation of the plate from the central support occurs after approximately 180 ms. The duration of the plastic deformation phase is consistent with the deformation times measured by Nurick [2,4] using a light interference technique. Furthermore, these observations are consistent with the fact that no shear failure occurred along the inner boundary, which is characteristic of mode I failure. 4.2.2. RP031007b—6 g (3 g in ring)—10.28 N s—mode II* failure A clear second stress spike is observed, after which the plate response reaches a maximum of 70 MPa after 57 ms. At this point the average transverse shear stress is approximately 240 MPa. Although this represents the maximum stress experienced by the plate at the central support, it does not appear to be the stress at which shear failure is initiated. In fact, the support stress drops slightly to 60 MPa after 82 ms, which equates to an average transverse shear stress of 206 MPa. At this point a sudden drop occurs indicating that shear failure has taken place. The fact that shear failure does not occur at the point of maximum transverse shear stress indicates that a finite amount of shear sliding occurs at the boundary prior to shear failure [16]. Shear sliding would decrease the effective shear area while the associated plastic work would increase the temperature in the shearing region. These factors lead to a drop in the shear strength to a point where the shear stress cannot be sustained and shear failure occurs by unstable thermoplastic shear. Evidence of some shear sliding is observed in the deformed plates along portions of the inner boundary that did not fracture. After shear initiation the signal does not drop to zero, but levels off at 25 MPa after 105 ms, indicating that the shear failure did not occur around the entire inner boundary. This result is consistent with the partial shear failure observed along the inner boundary of the deformed specimen, which is characteristic of mode II*. The stress plateau of 25 MPa persists for a considerable time before the signal drops to zero after 170 ms. This could be indicative of the stress level required in the support bar to sustain the shear process zone as it moves along the inner support boundary. 4.2.3. RP030919b—7 g (4 g in ring)—10.99 N s—mode II failure The support stress history for this test is similar to that of test number RP031007b, even though this test had an additional gram of explosive in the ring charge. The most distinctive difference is

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that shear failure occured slightly earlier, at 72 ms, even though the stress levels were virtually identical. At this point the signal dropped significantly, with only a small plateau occurring between 87 and 95 ms, indicating that shear failure had occurred along most of the inner boundary. This interpretation was confirmed by the observation that the plate had separated from the central support and a shear disk had formed on the central support. Separation occured after approximately 110 ms, which is significantly earlier than in test number RP031007b. This implies that the plate continued deforming after 110 ms, but the behaviour was not recorded since that plate is no longer in contact with the central support. Confirmation of this was obtained by observing the significant deflection of the hole formed in the plate due to shearing. Nevertheless, the convex curvature of the deformed plate in the region of the hole and the fact that the hole deflection is less than the maximum deflection indicates that significant plastic deformation preceded the shear failure, which is indicative of mode II failure. 4.2.4. RP031009a—8 g (5 g in ring)—14.07 N s—transition from mode II to mode III failure The support stress history for this test was quite distinct from those of the previous tests. The second pressure spike is greater in magnitude than the first spike, indicating the dominant contribution of the ring charge to the total impulse. Once the pressure spikes dissipate, the plate response reaches a maximum of 77 MPa after 53 ms, which is marginally earlier than the corresponding point in test number RP030919b. The signal decreases slightly before a sudden decrease occurs after approximately 56 ms. Shear failure occurs at least 16 ms earlier than in test number RP030919b, which is a significant period in the context of the overall plate response time. The support bar stress at the time of failure was approximately 70 MPa, which equates to an average transverse shear stress of 240 MPa. The early shear failure was reflected in the fact that the plate curvature in the region of the hole was only slightly convex, indicating that little plastic deformation preceded the shear event failure. This result represents a transition between mode II and mode III failure. 4.2.5. RP030923e—9 g (6 g in ring)—14.29 N s—mode III failure The support stress history for this test is similar to that of test number RP031009a, except that the gradient changes are more distinct. No clear plate response could be discerned, but a sudden drop in the signal occurred at a stress of 73 MPa after 53 ms, which is very short interval in the context in the overall plate response. The early shear failure is reflected in the mildly concave curvature of the deformed plate. Furthermore, the final shape of the deformed plate is essentially identical to that of an annular plate, as is evident when comparing the photograph of the deformed plate for test number RP030923e in Fig. 12 to that for an annular plate test shown in Fig. 14. This indicates that shear failure occurred prior to any significant plastic deformation, which is indicative of mode III failure. The stress level in the support at the time of failure equates to an average transverse shear stress of 251 MPa. The absence of significant deformation prior to the shear failure implies that bending and membrane stresses would have been relatively small and that the Von Mises equivalent failure stress can be estimated based on the shear stress alone. The equivalent failure stress was calculated to be 435 MPa.

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Fig. 14. Photograph of the deformed annular plate for test number RP031009b.

4.2.6. General observations In addition to the foregoing analyses, further general observations were made. The magnitude and form of the initial stress spike is consistent. This is to be expected since the mass of the central charge and cross-leaders were the same in every test. This indicates the tests have a high level of repeatability. The increases in the magnitude of the second stress spike corresponded to an increase in the mass of the ring charge, confirming the link between them. The time to failure decreases with increasing impulse, while the transverse shear stress at failure increases slightly from 206 MPa in test number RP031007b to 251 MPa in RP030923e, indicating a certain amount of rate sensitivity in the failure behaviour. A particularly interesting observation is that in every test depicted in Fig. 11, with the exception of the mode I response, the maximum plate response occurs at approximately the same time (53–57 ms) with virtually identical magnitudes (70–77 MPa). In other words, the maximum transverse shear stress that the specimen plates can sustain along the inner boundary is relatively consistent and marks the transition from mode II to mode III failure. This experimental result lends support to the maximum shear stress criterion used by Olson et al. [12,13] for the prediction of mode III failure. 4.3. Displacement vs. total impulse In blast tests on clamped circular plates, the plate geometry ensures that the maximum deflection will occur at the centre of the plate, even if the loading is slightly asymmetric. No such unique point exists for an annular or PCCS plate. In an ideal blast test on a PCCS plate, the maximum deflection will occur along a circumferential line. In other words, each infinitesimal segment of the plate will have one point that experienced that maximum displacement. It is experimentally difficult to achieve perfect symmetry with the charge configuration used in the present series. Hence, the maximum displacement along an infinitesimal segment will differ from segment to segment. The resulting asymmetry was quantified by firstly measuring the absolute maximum displacement of the plate as shown in the max–max column in Table 1. Secondly, the minimum permanent displacement along a line at the same radius as the maximum displacement was measured. This point would be a saddle point since it constitutes a local minimum with respect to the circumferential direction, but a local maximum with respect to the radial direction. The value of this displacement was recorded in the min–max column of Table 1. The mean of the max–max and min–max displacements were recorded in the ave–max column of Table 1.

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109

mode I mode II* mode II mode III annular plates least squares fit variation within one plate thickness

0

2

4

6

8 10 12 14 total impulse [Ns]

16

18

20

Fig. 15. Graph of deflection vs. total impulse. The error bars indicate the uncertainty in the value of the maximum deflection. The solid lines represent least-squares correlations with the dotted lines depicting bounds of 712 plate thickness.

In Fig. 15 the ave–max displacement vs. total impulse results for all the annular and PCCS plates are plotted. The error bars on the data points indicate the difference between the max–max and min–max displacements and give an indication of the asymmetry of the permanent displacements observed in the deformed plates. The solid lines represent least-squares regressions for the annular plate results and for the mode I and mode III PCCS plate results. All the ave–max data points lie within a half plate thickness of the regression line. The dotted lines are not intended to represent any confidence limits, but are included as a visual aid. The regression line for the annular plate results is w~ ¼ 1:305I  1:495,

(1)

where w~ is the ave–max displacement and I is the total impulse. It is interesting to note that the regression does not pass through the origin. This can be attributed to the fact that a small portion of the total impulse is transferred to the ballistic pendulum through the central support and does not contribute to the plate deformation. The values recorded in Table 1 indicate that this impulse is in the order of 1 N s, which correlates with the offset of the regression line. The annular plate results are useful in that they are comparable to mode III deformation of the PCCS plates, except without shearing. In Fig. 15, the regression line of the mode III results lies almost parallel to that of the annular plate results, with an offset in the order of 1 N s and with the lines converging slightly as the impulse increases. This implies that to achieve the same deflection, a PCCS plate undergoing mode III response requires a greater impulse load than an annular plate. The difference can be attributed to the force required to shear the PCCS plate along the inner boundary. PCCS plates undergoing mode I and II* response also fit on a single regression line, given by w~ ¼ 0:456I  0:903.

(2)

The impulse range between 10.5 and 11 N s is associated with a sudden increase in the final deflection as the inner restraint is released. This constitutes a transition region from mode II* to

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mode II. As the impulse is further increased, the data points for the PCCS plate tests tend toward the annular data points. Transition to mode III is taken to be the point where the trend line of the PCCS plate data becomes approximately parallel to the annular plate data, which occurs at an impulse of approximately 14 N s. 4.4. Support bar impulse vs. total impulse

support impulse / total impulse [%]

In Fig. 16 the fraction of the total impulse, expressed as a percentage, passing through the central support is plotted with respect to the total impulse, for all the annular and PCCS plates. In the case of the annular plate results, the data point with the lowest impulse corresponds to a test where the ring charge contained only 2 g of explosive. In other words, the central charge and cross-leaders would have made a significant contribution to the total impulse. Despite this, only 12% of the total impulse was transferred through the central support, indicating that the chosen charge configuration succeeded in preventing localized loading. As the ring charge mass, and hence the total impulse, is increased, the fraction of the total impulse transferred through the central support drops slightly and appears to reach an asymptotic value of 7% for impulses greater than 10 N s. This apparent asymptote is slightly higher than the 4.84% expected from a test with a uniform impulse intensity. Nevertheless, these results indicate that the blast load was reasonably uniform. It is interesting to note that in most cases the ring charge provided the bulk of the total impulse. This suggests that even though the charge configuration is not axisymmetric, using an axisymmetric numerical simulation where the cross leader mass is added to the central charge may provide useful results. In the case of the PCCS plate results, the data points corresponding to mode I failure indicate that approximately 40% of the total impulse passed through the central support. In other words, the reaction force required to arrest the plate motion is divided between the inner and outer boundaries, with the outer boundary making a slightly larger contribution. Considering the greater extent of the outer boundary, this value is reasonable.

45 40

mode I

35

mode II*

30

mode II

25

mode III

20

annular plates

15 10 5 0 0

2

4

6

8 10 12 14 total impulse [Ns]

16

18

20

Fig. 16. Graph of the fraction of the total impulse transferred through the central support vs. the total impulse. The two data points lying on the horizontal axis depict experiments in which a usable stress signal was not captured.

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As the blast load is increased mode II* failure is induced. A portion of the inner boundary is thus unloaded due to the partial shearing and the load must be taken up by the outer boundary. This is reflected in the gradual decrease in the fraction of the total impulse transferred through the central support with increasing total impulse. The transition, shown in Fig. 15, between mode II* and mode II failure occurs at a load of approximately 11 N s. In Fig. 16, this impulse is associated with a distinct drop in the impulse fraction passing through the central support. This sudden drop reflects the fact that the plate has separated from the central support before the plate motion has been fully arrested. As the total impulse is further increased, the corresponding time to separation decreases. Hence, the fraction of the total impulse captured by the support bar decreases since the time available to transfer momentum to the central support is reduced. The mode II data points appear to approach an asymptote just prior to the transition to mode III failure. Furthermore, the transition appears to be marked by a slight but distinct drop in the impulse fraction transferred via the support bar. This drop is due to the shear failure event occurring earlier in mode III than in mode II, and as a result less momentum is transfered to the central support. Overall, Fig. 16 shows that as the total impulse is increased the PCCS plate data tend towards that of the annular plates, with mode III being the most similar. The remaining difference is due to the impulse transferred to the central support during the mode III shear event. 4.5. Final hole diameters vs. total impulse

increase in hole diameter [%]

In Fig. 17, the final hole size in the deformed plates is plotted with respect to the total impulse, for all the annular and PCCS plates. In the case of the annular plates, the final hole size increases steadily with an increase in the total impulse. The data points for mode I and mode II* failure of the PCCS plates have been assigned a value of zero since complete holes were not formed. During mode II failure a complete hole is formed, but the final hole size showed only a slight increase with increasing total 45 40

mode I

35

mode II*

30

mode II

25

mode III

20

annular plates

15 10 5 0 0

2

4

6

8 10 12 14 total impulse [Ns]

16

18

20

Fig. 17. Graph of the increase in the final hole diameter vs. the total impulse. Data points corresponding to mode I and mode II* behaviour, where no hole was formed, are included for the sake of comparison.

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impulse. By contrast, the transition to mode III failure is accompanied by a marked increase in the final hole size. This observation strengthens the conclusion that shear failure occurs early during mode III failure after which the plate behaves as an annular plate. At larger total impulses the mode III data tend towards the annular plate data, the difference, as previously mentioned, being due to the impulse required to produce shear failure along the inner boundary.

5. Theoretical analysis Krauthammer et al. [1] strongly suggest that simple analyses should be integrated with an experimental research program to ensure that usable data are being collected. In particular, the aim of the theoretical work reported here was to provide a means of correlating the results of blast tests on different plate configurations. Due to the novel nature of the PCCS configuration, no analytical solutions appear in the literature. Hence, the following analytical predictions for the maximum deflection of blast loaded annular and PCCS circular plates have been developed by extending the energy analysis reported by Duffey [17]. 5.1. Peripherally clamped circular plates The well-known case of a clamped circular plate will be discussed first, to serve as a point of reference. According to Duffey [17], and subsequently Wen [18], the deformed shape is described by the function "
2 # r , (3) w ¼ w~ 1  Ro where w is the deflection of the plate at a radius r, w~ is the maximum displacement and Ro is the outer radius of the plate. It is assumed that all radial displacements are negligible so that only radial strains are significant and can be expressed as
 1 dw 2 r2 ¼ 2w~ 2 4 . (4)
r ¼ 2 dr Ro Furthermore, it is assumed that membrane stresses dominate during the deformation and that the mean dynamic flow stress can be expressed as a scalar multiple of the quasi-static yield stress sd ¼ asy ,

(5)

where sd is the mean dynamic flow stress, a is a constant and sy is the quasi-static yield stress. The plastic work done during the deformation of the plate is thus
 Z Z Ro 1 dw 2 sd
r dV ¼ asy 2prH dr ¼ pHasy w~ 2 , (6) WP ¼ 2 dr V 0

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where H is the thickness of the plate. The energy dissipated through plastic work is equal to the initial kinetic energy of the plate, induced by the impulsive load, Ek ¼

I2 ; 2m

where m ¼ rpR2o H,

(7)

where r is the density of the plate and I is the impulse transmitted by the blast load. It is assumed that the blast load is uniform. Equating Eqs. (6) and (7) gives d¼

1 w~ ¼ pffiffiffiffiffi F; H 2a

where F ¼

I pffiffiffiffiffiffiffiffi , pRo H 2 rsy

(8)

where d is the dimensionless deflection and F is the dimensionless impulse. This is the same expression for the dimensionless impulse as that used by Nurick and co-workers [2,4–8] who report gradients varying from 0.407 to 0.425 for regression lines through the data for clamped circular plates showing mode I deformation. To satisfy Eq. (8), this implies that the constant a must have a value of 2.8. In other words, the mean dynamic flow stress is almost three times greater than the quasi-static yield stress. The required value of a can be cross-checked as follows: Teeling-Smith and Nurick [5] report mode I deflections of up to 30 mm for 100 mm diameter plates. Taking a moderate value for deflection of 12 mm, Eq. (4) implies a maximum strain of approximately 0.115. Bodner and Symonds [3] and Nurick [4] report that blast loaded plates reach their maximum deflection in approximately 120 ms which implies an average strain rate in the order of 960 s1. Using the wellknown Cowper–Symonds constitutive relation [19], Eq. (5) can be rewritten in the form
1=q
_ sd ¼a¼1þ , (9) D sy where
_ is the strain rate and D and q are empirical constants, which for mild steel have the values 40.4 s1 and 5, respectively. Substituting the value for the average strain rate into Eq. (9) yields an average value for a of 2.884. While this is merely an estimate, it shows that the a value required to correlate the analytical and experimental results is reasonable. 5.2. Peripherally clamped annular plates In a blast test with annular plates, the final shape was found to be essentially a straight sided cone. Hence, the following formula was chosen to describe the deformation: 
 r  Ri , w ¼ w~ 1  Ro  Ri

(10)

where Ri is the inner radius of the plate. Following Eq. (4), the expression for the radial strain is
r ¼

w~ 2 1 . 2 ðRo  Ri Þ2

(11)

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Upon integration, the dissipated plastic work is 1 Ro þ Ri 2 w~ . W P ¼ pHasy 2 Ro  Ri

(12)

In this case the initial kinetic energy of the plate is I2 (13) where m ¼ rpðR2o  R2i ÞH. 2m This leads to the following expression for the relationship between the dimensionless displacement and dimensionless impulse: w~ 1 I (14) d ¼ ¼ pffiffiffi F where F ¼ pffiffiffiffiffiffiffiffi . H a pðRo þ Ri ÞH 2 rsy Ek ¼

The same symbol for the dimensionless impulse has been retained since Eq. (8) is essentially a special case of Eq. (14) where the inner radius is zero. In other words, Eq. (14) represents an extension to the expression for the dimensionless impulse used by Nurick and co-workers [2,4–8] for clamped circular plates. According to Eq. (1), the gradient of the regression line in Fig. 15 for the annular plates has a gradient of 1.305. When the regression is expressed in terms of the dimensionless numbers in Eq. (14) the gradient is 0.627. Using the value for the constant a of 2.8, Eq. (14) implies that the gradient of the dimensionless regression line is predicted to be 0.597. This constitutes an acceptable degree of correlation considering the analysis does not include radial displacement effects. 5.3. Peripherally clamped centrally supported circular plates A similar analysis was conducted for mode I failure of the PCCS specimen plates. In this case the deformed shape of the plate is assumed to be "
# r  Ri r  Ri 2 . (15)  w ¼ 4w~ Ro  Ri Ro  Ri Following Eq. (4), the expression for the radial strain is   8w~ 2 r  Ri 2
r ¼ 12 . Ro  Ri ðRo  Ri Þ2

(16)

Upon integration the dissipated plastic work is 1 Ro þ Ri 2 W P ¼ lpHasy w~ , 2 Ro  Ri where Z

Ro

l¼ Ri

  32 r  Ri 2 16 r dr ¼ . 12 ðRo þ Ri ÞðRo  Ri Þ 3 Ro  Ri

(17)

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Table 2 Comparison of experimental results with the analytical solution

Plate type

circular

annular

PCCS

2

1

16/3

Predicted gradient

0.422

0.597

0.258

Fitted gradient

0.407_0.425

0.627

0.219

Assumed displacement profile λ - derived

The value of l was found to be constant and independent of the choice of Ro and Ri. This leads to the following expression for the relationship between the dimensionless displacement and dimensionless impulse: d¼

w~ 1 I ¼ pffiffiffiffiffi F where F ¼ pffiffiffiffiffiffiffiffi . H pðRo þ Ri Þ H 2 rsy la

(18)

According to Eq. (2), the regression line in Fig. 15 for mode I failure of the PCCS plates has a gradient of 0.456. When the regression is expressed in terms of the dimensionless numbers in Eq. (18) the gradient is 0.219. Using the value for the constant a of 2.8, Eq. (18) implies that the gradient of the dimensionless regression line is predicted to be 0.258. This again constitutes an acceptable degree of correlation. A summary of the comparison of the experimental and analytical results is given in Table 2. The results show remarkable agreement considering the simplicity of the analyses and the range of plate configurations. Furthermore, the fact that a single value for a is able to correlate the results of distinct plate configurations shows that the annular and PCCS results are consistent with previously published results.

6. Conclusions The results of a series of experiments on blast loaded annular and PCCS circular plates mounted on a ballistic pendulum have been presented. This configuration allows the stress history of the central support to be captured, which provides new insight into the deformation and failure of blast loaded plates. During the annular plate tests, the magnitude and duration of the blast loads were captured. The duration of the blast load was typically in the order of 50 ms and showed the classical blast wave characteristics of a sudden rise followed by an exponential decay. The arrival times of the blast waves from different portions of the explosive charge were discernable. Three distinct modes of failure, similar to those that occur during circular plate tests, were observed during PCCS plate tests. However, the failure modes did not conform to the exact definitions used for circular plates. Revised definitions for the various modes of failure of blast

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loaded plates were proposed such that they are applicable to both the circular and PCCS plate configurations. During the PCCS plate tests, the moment of shear failure and the associated shear loads were captured. Correlating these data with observations from the deformed plates provides some interesting new insights. In particular, it was observed that the maximum transverse shear stress that the specimen plates could sustain along the inner boundary was relatively consistent and marked the transition from mode II to mode III failure. These experimental results lend support to the maximum shear stress criterion used by Olson et al. [12,13] for the prediction of mode III failure. Global experimental parameters, such as the maximum final deflection, the final hole size and the fraction of the total impulse transferred through the central support were captured and plotted with respect to the total impulse. The results show good repeatability and provide an extended set of data for the validation of numerical analysis techniques. Approximate analytical solutions were presented, for predicting the deflection vs. total impulse behaviour of annular and PCCS plates. These analyses included the derivation of a new formula for dimensionless impulse that is applicable to circular, annular and PCCS plates. The new formula can be viewed as an extension of that used in previous work. The analytical solutions not only showed good correlation with the experimental results for the specific plate configurations, but were able to relate the results of various plate configurations using a single material parameter.

Acknowledgements The authors acknowledge the valuable contributions of Mr. H. Emrich of the Department of Mechanical Engineering, University of Cape Town, for machining the modified ballistic pendulum and Mr. N. Jacob of the Blast Impact and Survivability Research Unit, University of Cape Town, for assistance with the experiments.

References [1] Krauthammer T, Jensen A, Langseth M. Precision testing in support of computer code validation and verification. Norwegian Defence Construction Service, 1996. [2] Nurick GN, Martin JB. Deformation of thin plates subjected to impulsive loads, Part II: experimental studies. Int J Impact Eng 1989;8:171–86. [3] Bodner SR, Symonds PS. Experiments on viscoplastic response of circular plates to impulsive loading. J Mech Phys Solids 1979;27:91–113. [4] Nurick GN. A new technique measure the deflection-time history of a structure subjected to high strain rates. Int J Impact Eng 1985;3:17–26. [5] Teeling-Smith RG, Nurick GN. The deformation and tearing of thin circular plates subjected to impulsive loads. Int J Impact Eng 1991;11:77–91. [6] Nurick GN, Shave GC. The deformation and tearing of thin square plates subjected to impulsive loads—an experimental study. Int J Impact Eng 1996;18(1):99–116. [7] Nurick GN, Gelman ME, Marshall NS. Tearing of blast loaded plates with clamped boundary conditions. Int J Impact Eng 1996;18(7–8):803–27.

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[8] Nurick GN, Radford AM. Deformation and tearing of clamped circular plates subjected to localized central blast loads. In: Reddy BD, editor. Recent developments in computational and applied mechanics. Barcelona: CIMNE; 1997. p. 276–301. [9] Nurick GN, Teeling-Smith RG. Predicting the onset of necking and hence rupture of thin plates loaded impulsively. In: Bulson PS, editor. Proceedings of second international conference on structures under shock and impact. Portsmouth, UK: Computational Mechanics Publications; 1992. p. 431–45. [10] Cloete TJ, Ahmed R, Nurick GN. Peripherally clamped centrally supported blast loaded circular plates: a precision test for code validation. J Phys IV France 2003;110:507–12. [11] Lee CKB, Crawford RC, Mann KA, Coleman PA, Petersen C. Evidence of higher Pochhammer-Cree modes in an unsplit Hopkinson bar. Meas Sci Technol 1995;6:853–9. [12] Olson MD, Nurick GN, Fagnan JR. Deformation and rupture of blast loaded plates—predictions and experiments. Int J Impact Eng 1993;13:279–91. [13] Rudrapatna NS, Varizi R, Olson MD. Deformation and failure of blast loaded square plates. Int J Impact Eng 1999;22:449–67. [14] Dobratz BM, LLNL explosives handbook: properties of chemical explosives and explosives simulants, UCRL52997. NTIS US Department of Commerce, 1981. [15] Menkes SB, Opat HJ. Tearing and shear failures in explosively loaded clamped beams. Exp Mech 1973;13:480–6. [16] Li QM, Jones N. Formation of shear localization in structural elements under transverse dynamic loads. Int J Solids Struct 2000;37:6683–704. [17] Duffey TA. The large deflection dynamic response of clamped circular plates subjected to explosive loading. Sandia Laboratories research report, SC-RR-67-532, 1967. [18] Wen HM. Deformation and tearing of clamped circular work-hardening plates under impulsive loading. Int J Pressure Vessels Piping 1998;75:67–73. [19] Jones N. Structural Impact. Cambridge: Cambridge University Press; 1989 and 1997.