151 5.15 Graph of the central support reaction force history for an annular plate without the outer. 175 6.5 Comparison of the central support force histories of five short standoff blast tests at 100 mm.

## Theory and Experimentation

### Verification and Validation

Precise analytical solutions are preferred, but if none exist, then "benchmark tests" can be used. A benchmark test amounts to a numerical consensus solution obtained by comparing the results on a code-vs-code basis.

### Precision Experiments

There is general consensus [1, 2] that the process by which the accuracy of a numerical code is evaluated can be separated into two distinct activities, namely verification and validation.

## THEORY AND EXPERIMENTATION 3 1. The structural behaviour must be accurately described and highly repeatable

### Blast and Shock Tube Loading

In addition to the precision test requirements, a topic of great relevance to this thesis was discussed in Session 3 of the Trondheim workshop devoted to laboratory-scale explosion and shock tube experiments. Shock tube experiments lack the impulsive loading that is often of interest, and explosive loading is often difficult to define throughout the structure.. explosive testing introduces inaccuracies such as unreliable pressure-space-time data. [1].

Purpose, Scope and Objectives

## Thesis Outline

*Literature Survey**Literature Critique**Transient Behaviour of PCS Plates**Experimental Method*

In Chapter 3, a selection of experimental data and theoretical models reviewed in Chapter 2 are considered in more detail. 7 Characterization of blast loading techniques, where are the experimental and numerical results.

## THESIS OUTLINE 7 characterization of the blast loading techniques, where both experimental and numerical results are

*Experimental Results**Analytical Modelling of PCCS and PCA Plates**Discussion of Results**Conclusions and Recommendations**Appendices*

In Appendix B, the quasi-static and dynamic material properties of the sample material used in this thesis are reported. In Appendix D, the detailed derivations of the Bessel displacement profiles for PCCS plates, as discussed in Section 7.2.1, are presented.

## Introduction

### A Brief History of Shock Loaded Structure Research

However, the analytical treatment of plate loading by blasting can usually be traced back to the seminal work of Taylor [15]. Richardson & Kirkwood developed an approximate solution to the equations of motion and compared the results with UNDEX tests on small circular plates.

## INTRODUCTION 11 pursued by Hudson [18] in 1951 and Frederick [19] in 1959. However, during this period there was

The early 1970s also marked the beginning of the work of Youngdahl [46], who attempted to eliminate the effect of pressure pulse shape on response predictions for various structures. For most of the 1970s there appears to have been little experimental work on the loading of slabs by blasting, although work continued on other structures.

## INTRODUCTION 13 The early 1970’s marked the initial work of Johnson [54] who proposed a dimensionless number with

### Alternative Impulsive Loading Techniques

However, the impact velocity range is limited by the yield strength of the braking device. This appears to be the first example of the use of a shock tube for structural loading since the work of Munday & Newitt [34].

## INTRODUCTION 15 The interest in shock tube work has continued into the 2000’s [97, 98] including the development of some

### Shock Loading Diagnostic Techniques

Tripwires have been used by Teeling-Smith & Nurick [67] to measure the speed of blast fragments. Although used for impulse calibration, flash X-rays and contact pins do not appear to have been used for deflection diagnostics, streak photography or laser Doppler systems.

## INTRODUCTION 17

### Modes of Failure

Ia Large inelastic deformation with a neck around part of the boundary Ib Large inelastic deformation with a neck around the entire boundary. II* Large inelastic deformation with partial tearing around part of the border II Tensile tearing at the border.

## Uniform Shock Loading of Circular Plates in Air

### Blast Loaded Plate Deflection Data

Additionally, Figure 2.3 shows the regression fit derived by Nurick & Martin [5] and has the form,. These data will be discussed again in Section 3.2.1, where the reasons for the positive y-intercept of the regressions derived by Nurick &.

## UNIFORM SHOCK LOADING OF CIRCULAR PLATES IN AIR 21

### Blast Loaded Plate Response Duration

In fact, under impulsive loading, the plate center has a constant velocity, and thus a linear displacement curve, during most of the response duration, as shown in Figure 2.6, which is due to Nurick [61]. In section 4.2, a closed-form model is presented that captures both the membrane-like behavior and the initial constant velocity of the plate center.

## UNIFORM SHOCK LOADING OF CIRCULAR PLATES IN AIR 23

### Shock Tube Loaded Plate Deflection Histories

The first parts of the curves due to Munday & Newitt were hidden by the clamp attachment. In addition to the deflection history, Munday & Newitt reported the permanent residual shape of the plates.

## UNIFORM SHOCK LOADING OF CIRCULAR PLATES IN AIR 25

### Blast Tube Data and Stand-off Effects

The primary focus of the work of Jacob et al. was to investigate the effect of explosive charge distance on the deformation of mild steel plates. Jacob et al. argued that a lack of localized deformation indicates that from a distance of 75 mm the blast load can be considered to be essentially uniformly distributed over the plate specimen.

## UNIFORM SHOCK LOADING OF CIRCULAR PLATES IN AIR 27

### Shock Loaded plate shapes

It is interesting to note that despite the differences in loading configurations, Figure 2.12 shows that the final plate shapes are reasonably similar. Furthermore, it is also clear that parabolic and cosine curves are reasonable approximations to consider if the use of the Bessel function ever turns out to be mathematically intractable.

## Analytical modelling of Circular Plates

Therefore, before giving a more detailed overview of some of the analytical methods, it is instructive to consider the shapes of recovered plate samples. This is consistent with the view of Taylor [15], as discussed in section 2.2.2, who argued that the response would be analogous to that of a linear vibrating membrane.

## ANALYTICAL MODELLING OF CIRCULAR PLATES 29

### Displacement Profiles

The most rigorous method for determining the plate profile is through solving the corresponding differential equations of motion. Published analytical solutions usually reduce the plate deformation problem to a moving form using various approximations and assumptions to simplify the equations of motion.

## ANALYTICAL MODELLING OF CIRCULAR PLATES 31 relates to the relative magnitude of the maximum deflection expected to occur. This approach has met

Ignoring the inertia term in equation (2.7) gives, . which is the differential equation for a circular diaphragm under a quasi-static compressive load. Equation (2.10) describes the free vibration of an axisymmetric membrane for which the solution is also known [121].

## ANALYTICAL MODELLING OF CIRCULAR PLATES 33 and,

### The Perfectly Plastic Membrane Assumption

Most of the models reviewed in this chapter and those developed as part of this thesis, which will be presented in Chapters 4 and 7, were based on the assumption of a Taylor membrane. However, the models will be used to interpret experimental data obtained using sheet metal specimens where strains and strain rates are neither constant nor homogeneous.

### Ideal Impulsive Loading

He noted that the stress in a soap film or stretched membrane is uniform regardless of the amount of deformation and that this would also be the case for a perfectly plastic diaphragm. It will be shown that these refined models determine the domain within which the membrane assumption is valid and provide simple correction factors that allow various effects, such as strain and strain rate sensitivity, to be accounted for in an average sense.

## ANALYTICAL MODELLING OF CIRCULAR PLATES 35 where I is the impulse transmitted by the load, m is the mass of the plate, V 0 is the initial uniform plate

### Energy Methods

Duffey assumed that the dynamic flow stress is equal to the quasi-static yield stress. However, for a solution that includes time, ˜wand δ will refer to the maximum deflection at a given instant, while the subscript 'max' will be added to denote the final state.

## ANALYTICAL MODELLING OF CIRCULAR PLATES 37 4.2 Bessel Displacement Profile

### Membrane Mode Methods

Obtaining the "best" approximation involves finding the initial maximum velocity of the state shape by minimizing a functional as defined by Martin &. An interesting feature of equation (2.32) is that it does not depend on the magnitude of the initial impulse, but depends on the dynamic flow stress.

## ANALYTICAL MODELLING OF CIRCULAR PLATES 39

### Travelling Hinge Methods

From the literature, it appears that relatively few researchers have considered models similar to Taylor's traveling hinge approach for the large plastic deformation of thin plates. This approach is similar in concept to the two-phase model of Jones [43], who limited the first phase to small displacements to include a bending-type moving hinge.

### Strain Hardening and Strain Rate Sensitivity

Hudson [18] and Frederick [19], who worked in the same era as Taylor, developed more detailed models of travel hinges, but their work does not appear to have continued. Half a century later Wierzbicki & Nurick [63] and Lee & Wierzbicki [125] developed a two-phase model for localized impulsive loading of thin plates, where the first phase used the concept of membrane-type traveling hinges.

## ANALYTICAL MODELLING OF CIRCULAR PLATES 41

The radial strain rate distribution is obtained by taking the temporal derivative of equation (2.37), which yields. 2.39) The maximum value of ˙r will also occur at radial positionrm, but, unlike the maximum strain, will occur at timet= 12T, i.e. halfway through the deformation duration, resulting in. In other words, the maximum strain rate at any time during the deformation is 57% higher than the simple mean estimate obtained from the quotient of the maximum strain and deformation duration.

## ANALYTICAL MODELLING OF CIRCULAR PLATES 43 while the temporal integral is,

These values were then used in equations (2.38) and (2.40) to obtain the maximum strain rate during deformation. The result for mild steel plates subjected to uniform impulse loading is shown in Figure 2.14.

## ANALYTICAL MODELLING OF CIRCULAR PLATES 45

### Bending and Radial Displacement

Equation (2.50) indicates that the membrane strain is always positive, i.e. tensile, but that the flexural regions are above and compressive below the midplane of the plate due to the sense of z. Below the midplane, however, the radial membrane and bending deformation are opposed and the sense of the flow stress will depend on which deformation mechanism dominates.

## ANALYTICAL MODELLING OF CIRCULAR PLATES 47 of perfect plasticity has the form,

The original reason for the neglect of radial displacements is unclear but can be deduced from the literature. Vanzant [130], who reported that radial displacements during the impact loading of plates are significantly less than during static loading.

## ANALYTICAL MODELLING OF CIRCULAR PLATES 49 The aforementioned experimental evidence implies that the strain distribution obtained when assuming

Solving equation (2.58) and noting that the radial displacement is zero at the center and periphery of the plate, Taylor found that the radial displacement distribution is, . 2.59) This result indicates that if the transverse deflection is small, then the radial displacement is negligible. The analysis of Taylor is considered to provide an acceptable prediction for the radial displacement distribution.

## Analytical modelling of Annular Plates

These experimental results agree with those published as part of this work [52, 53], which will be presented in Chapter 6. Furthermore, a theoretical justification for the inclusion of radial displacements in PCA models will be given in Section 3.3.6.

## Miscellaneous Topics

### Finite Duration Loading

Jones reported that although his analysis included finite displacements, it was only valid for maximum displacements on the order of two plate thicknesses. There do not appear to be any published large deflection models for impulsively loaded PCA plates other than those published as part of this work [52].

## MISCELLANEOUS TOPICS 53

### Pressure-Impulse Isodamage Diagrams

In contrast, the quasi-static domain is associated with the horizontal asymptotes in Figure 2.16, where the level of 'damage' depends only on the peak load, i.e. it is independent of the overall impulse and load history. Finally, the dynamic domain is not associated with any of the asymptotes in Figure 2.16, and the level of 'damage' is dependent on the peak load, the load history and the total impulse.

## MISCELLANEOUS TOPICS 55 view is incorrect by presenting analytical solutions for finite duration loads based on energy methods

### Shock-Structure Interaction

The analysis of Kambouchevet al. and Peng et al., although informative, does not lead to a single dimensionless number. Furthermore, Kambouchevet compares al. and Peng et al. not their results with experimental data, i.e. no validation was provided.

Introduction

## Uniform Shock Loading of Circular Plates in Air

### Blast Loaded Plate Deflection Data

The regression was found to have a positive y-intercept, which is physically impossible as it implies a non-zero positive deflection as the load pulse tends to cancel. In fact, you would expect the opposite, ie. the load pulse should be above a certain threshold before permanent deformation occurs.

## UNIFORM SHOCK LOADING OF CIRCULAR PLATES IN AIR 59

### Correlation of Blast Loading Techniques

It is noticeable that the cross section crosses essentially at the origin and that the regression gradient of 1.821 Ns/g is almost identical to the Florence & Wierzbicki result. Nurick [151] does not report the mass of the crossbar nor its contribution to the total impulse.

## UNIFORM SHOCK LOADING OF CIRCULAR PLATES IN AIR 61

### Blast Loaded Plate Response Duration

This result, together with the ease of general applicability of the membrane analogy, is considered sufficient justification for using equations (3.3). UNIFORM SHOCK LOADING OF CIRCULAR PLATES IN AIR 63 From Figure 3.4 an additional trend is evident, namely that most of the dimensionless duration data are points.

## UNIFORM SHOCK LOADING OF CIRCULAR PLATES IN AIR 63 From Figure 3.4 an additional trend is evident, namely that most of the dimensionless duration data points

### Shock Tube Loaded Plate Deflection Histories

Furthermore, the arrival time of the shock relative to the plate response was not reported. Munday & Newitt argued that the divergence shown in Figure 3.5 is the result of a decrease in the reflected pressure due to the rapid response of the plates.

## UNIFORM SHOCK LOADING OF CIRCULAR PLATES IN AIR 65

Given the above, it is concluded that the extension approach produces acceptable correlations for data from Munday & Newitt and Stoffel. In Section 2.2.3 it was shown that the dimensionless time parameter of Munday & Newitt, given in Equation (2.4), is not able to sufficiently correlate the data of Munday & Newitt and Stoffel.

## UNIFORM SHOCK LOADING OF CIRCULAR PLATES IN AIR 67

### Shock-Structure Interaction

In particular, if the size of the plate velocity is a significant part of the velocity of the shocked air particles, i.e. the relationship between the plate velocity and shock pressure is given by equation (3.5), while the plate acceleration is given by equation (3.6). .

## UNIFORM SHOCK LOADING OF CIRCULAR PLATES IN AIR 69

The reason for this is the expansion of the two terms used in equation (3.13), which causes the acceleration to be consistently underestimated. The form of equation (3.14) is not suitable to develop an estimate for the deviation of velocity from constant pressure behavior due to SSI.

## UNIFORM SHOCK LOADING OF CIRCULAR PLATES IN AIR 71

### Blast Tube Data and Stand-off Effect Analysis

The net effect is that the mean deviation for a given pulse appears to decrease with increasing distance. In other words, equation (3.18) states that with respect to the mode I response, a localized load at a remote distance from.

## Analytical models of Circular Plates

### Generalized Energy Method for Ideal Impulsive Loading

This implies that an expression for the kinetic energy of a given plate shape is required. The quantity ˜K will be referred to as the 'plastic stiffness' since the shape of plastic work is reminiscent of the expression for the strain energy in a spring.

## ANALYTICAL MODELS OF CIRCULAR PLATES 75 1.2 Final Central Deflection

### Maximum Displacement

In this section, the three separate PCS deformation profiles considered in this thesis will be considered from the perspective of the general energy method. Furthermore, where necessary, the literature result will be re-derived using the general energy method and the effective mass will be determined.

## ANALYTICAL MODELS OF CIRCULAR PLATES 77 This value of λ leads to a normalized non-dimensional deflection of 0.7071, as reported in Table 3.2

To find an expression for the total kinetic energy, the velocity distribution is obtained by taking the temporal derivative of equation (3.33). To find an expression for the total kinetic energy, the velocity distribution is obtained by taking the temporal derivative of equation (3.44).

## ANALYTICAL MODELS OF CIRCULAR PLATES 81

### Comparison of Non-Energy based Analytical Solutions

These materials are generally sensitive to strain and strain rate, as briefly discussed in Section 2.3.7, implying that the deformation history must be taken into account. The subject of strain rate sensitivity will be discussed in section 3.3.5, while in section 4.2.1 an approximate energy model including transient strain profiles will be presented.

### Displacement Duration

Despite the fundamental difference between the two methods, it was shown in Section 3.3.2.3 that, assuming a conical deflection profile, the energy approximation can give an identical result to that obtained using the moving hinge assumption. As mentioned in section 3.3.2.4, the conic assumption does not correlate with measured deformation profiles, leading to unrealistically high exp values.

## ANALYTICAL MODELS OF CIRCULAR PLATES 83 method can be used to obtain the same result as the membrane mode method and, thereafter, the results

Furthermore, when reversed, Equation (3.59) gives the same expression for the central displacement history as in Equation (2.30). The preceding discussion explains the good correlation of the theoretical strain-strain results given in Table 3.2.

## ANALYTICAL MODELS OF CIRCULAR PLATES 85

### Strain Hardening and Strain Rate Sensitivity

Overall, Symonds & Wierzbicki's iterative solution, incorporating strain rate effects, showed an improved correlation with the raw experimental data from Bodner & Symonds [56]. The comparison shows that the energy approach results in a better correlation with the experimental data and provides some confirmation of the limitations of the membrane mode solution, as discussed in Section 3.3.3.1.

## ANALYTICAL MODELS OF CIRCULAR PLATES 87

### Radial Displacement Effects

Therefore, the purpose of this section is to consider the effect of radial displacements on the prediction of maximum deflection and strain distribution. Whether both terms in the integrand of equation (3.62) are required depends on the choice of the efficiency criterion and the flow rule.

## ANALYTICAL MODELS OF CIRCULAR PLATES 89

Bessel - excluding radial displacement Bessel - including radial displacement Parabolic - excluding radial displacement Parabolic - including radial displacement Nurick et al. Nurick et al.[62] pointed out that the stress distribution obtained by ignoring radial displacements is the inverse of the experimentally observed distribution, as shown in Figure 3.13.

## ANALYTICAL MODELS OF CIRCULAR PLATES 91

### Bending Effects

An expression for the position of the neutral line ¯z(r) is obtained by setting equation (3.68) equal to zero, which gives, . Equation (3.69) indicates that ¯z is always negative, that is, the neutral line is on or below the midplane of the plate.

## ANALYTICAL MODELS OF CIRCULAR PLATES 93

For displacements greater than half the plate thickness, Wen's uncoupled model predicts that buckling effects will result in a constant negative offset compared to the pure Duffey & Key membrane solution. In contrast, a model incorporating coupled yielding predicts that membrane stresses are not only larger but also tend to suppress the contribution of bending stresses, although not completely.

## ANALYTICAL MODELS OF CIRCULAR PLATES 95

### Non-Monotonic Strain Effects

This is accomplished by integrating the product of the yield stress and the maximum compressive stress for each point between the midplane and ¯z, i.e. By integration, which is long but simple and therefore omitted, the final shape of the plastic work is,.

EVALUATION OF PUBLISHED DATA USING PRECISION TEST REQUIREMENTS 97

## Evaluation of Published Data using Precision Test Requirements

As shown in Section 2.1.4, the identification and description of failure modes is one of the traditional strengths of laboratory-scale explosive testing. As discussed in Section 1.1.3, this aspect of laboratory-scale blast loading has received strong criticism.

## EVALUATION OF PUBLISHED DATA USING PRECISION TEST REQUIREMENTS 99 is significant, it can be defined with sufficient precision such that distinct and consistent failure

In general, the reporting accuracy in the explosion load studies discussed in Chapter 2 is poor, i.e. the information is often insufficient to allow correlations with other studies. From the foregoing discussion, it appears that a significant number of classical and modern laboratory scale explosion load studies meet most of the precision test criteria.

Introduction

## Ideal Impulsive Loading

Transient Deformation Profiles

## IDEAL IMPULSIVE LOADING 103

By the same reasoning as described in Section 3.3.4, the instantaneous kinetic energy of the plate must be equal to the difference between the initial kinetic energy and the accumulated plastic work. Furthermore, this means that at a given moment the lateral displacement of the driving course is wh V0t.

## IDEAL IMPULSIVE LOADING 105

### Deflection Duration

The goal is to find a general expression for the deformation history of the center of the plate. In addition, the center of the plate would have behaved as a rigid body with a constant velocity V0 until now.

## IDEAL IMPULSIVE LOADING 107 the constant mode shape approach, are valid provided that the appropriate temporal offset is accounted

### Comparison with Experimental Results

In other words, the two-phase model gives an aτ value between the experimental τmin and τmode values for mild steel of 0.818 and 0.855, respectively, as shown in Figure 3.4. This is an acceptable result given the simplicity of the analysis and sufficient justification for using an energy-based two-phase model.

## IDEAL IMPULSIVE LOADING 109

### Discussion of Hinge Velocity Predictions

However, the theory about a constant hinge speed in a plastic membrane is well established and therefore the results of Section 4.2.3 present an apparent contradiction. As reviewed in Sections 2.1.1 and 2.3.6, moving hinge solutions exist for large membrane deflection and result in a constant hinge velocity that has been shown to overestimate the deformation duration of impulsively loaded thin plates.

FINITE DURATION LOADING 111

## Finite Duration Loading

### Transient Deformation Profiles

In equations (4.21) and (4.22), the first term represents the work done on the inner area of the plate, i.e. within the hinge radius, while the second term represents the outer region. READING OF FINITE DURATION 113Since the inner region of the plate moves like a rigid body, it can be considered isolated.

## FINITE DURATION LOADING 113 Since the inner region of the plate is moving as a rigid body, it may be considered in isolation. Substituting

### Impulsive vs Dynamic Loading

From the previous section, it is clear that the value of TI is characteristic of a certain plate configuration and marks the transition from a transient deformation profile to a constant one. Consequently, an ideal impulse load (IIL) has a duration that tends to vanish, while an ideal dynamic load (IDL) has a constant intensity and a duration equal to or greater than the total response time of the plate8.

### Impulsive Loading - Parabolic Phase I Profile

Therefore, for the purposes of this thesis, an unambiguous and physically meaningful criterion for the transition from impulsive to dynamic loading must be defined. FINITE DURATION LOAD can be adjusted for a constant intensity impulsive load by replacing the initial kinetic energy term with a .

## FINITE DURATION LOADING 115 be adapted for a constant intensity impulsive load by replacing the initial kinetic energy term with an

### Critical Impulsive Loading

The Phase I model presented in Section 4.3.1 is limited to the use of a parabolic displacement profile to provide a convenient closed-form solution for the transient hinge position. However, a more general solution can be found for the specific case where the load duration is equal to the phase I duration, i.e. TL=TI.

## FINITE DURATION LOADING 117

Following the energy approach in the previous section, the duration of phase I is obtained by equating the kinetic energy of the plate with the difference between the compressive work and the plastic dissipation. In the case of the parabolic profile, the value is identical to the corresponding values in table 4.1, i.e. the result obtained in section 4.3.1 is found again.

## FINITE DURATION LOADING 119

### Impulsive Loading - General Profile

Once the value for TI is known, the central deflection at the end of phase I can be obtained by treating the central part of the slab as a rigid mass moving at a constant speed between times TL and TI, which gives, . Substituting the above-mentioned equations into equation (4.48) and introducing the dimensionless deflection and the Taylor duration, we obtain 4.49) The normalized dimensionless deflection at the end of phase I is therefore,.

## FINITE DURATION LOADING 121 Introducing the Taylor duration and rearranging gives the total dimensionless deformation duration as,

### Dynamic Loading

From the above, and including the normalized form of δI given in Equation (4.62), the argument of the arc term in Equation (4.66) becomes,. The energy expression required to obtain the final deflection δII at the end of Phase II is similar to Equation (4.63), except that the kinetic energy term vanishes, giving,.

### Ideal Dynamic Load

Finally, the deflection duration at the end of phase II is obtained by recognizing that from time TL to TII, the strain behavior is essentially identical to the IIL response of a constant displacement profile, that is, a sinusoidal displacement history. Consequently, equating the arguments in equations (4.75) and (4.76) and rearranging yields the total dimensionless deformation duration under DL conditions as: 4.77) The predictions of equation (4.77) for the parabolic and Bessel function type displacement profiles are shown in Figure 4.10.

## FINITE DURATION LOADING 125

### Comparison with Numerical and Experimental Results

However, figure 4.10 shows that solving the dynamic deformation duration based on a parabolic displacement profile provides a better deflection duration correlation. Furthermore, Figure 4.10 shows that the slope of the strain-duration curves tends to 'level off' as the load duration TL approaches the IIL limit.

## FINITE DURATION LOADING 127

Additionally, the strength curves appear to be expressed in terms of engineering strain, as opposed to true strain, and elastic compliance is not removed. These TI values correspond to the inflection point of the curves, where they begin to diverge from the free plate curve.

FINITE DURATION LOADING 129 between these results and the previous comparison of the general dynamic loading predictions with

Introduction

## The Instrumented Ballistic Pendulum

### Conventional Ballistic Pendulum Techniques

In other words, no significant component of the explosion could act directly on the pendulum. Regarding the third assumption, Bonorchis & Nurick's results show that the clamp area can affect the measured pulse.

### Peripherally Clamped Centrally Supported Circular Plates

Instrumented versions of the ballistic pendulum have been developed to allow recording of plate deflection history, as described in Section 2.2.3. Although the energy dissipated by fracture has been estimated [67], the timing of fracture limits and the magnitude of the associated forces have not been recorded during conventional pendulum ballistic tests.

## THE INSTRUMENTED BALLISTIC PENDULUM 135

### Peripherally Clamped Annular Plates

When mounted in the same clamps as the PCCS plates, the deformation of the PCA plate is such that the central hole never contacts the central rod during a test. Therefore, the difference between the PCCS and PCA plate responses under similar loading conditions can only be due to the central support limit condition.

## THE INSTRUMENTED BALLISTIC PENDULUM 137

### The Central Hopkinson Bar

The rapid pressure rise associated with the blast charge tends to excite these natural frequencies, and the transient response of the sensor and its assembly is often visible in the pressure traces. Rather, the HPB technique relies on the tape being long enough to ensure that the stress wave reflections do not return before the full duration of the load signal is captured.

## THE INSTRUMENTED BALLISTIC PENDULUM 139

### Charge Configurations

The blast tube configuration used in this thesis is very similar to that of Jacobet al.[107] and is presented in figures 5.1(b) and 5.3. The charge is mounted on a polystyrene disc and placed in the open end of the blast tube.

### Data Processing and Analysis

Essentially, a blast tube aims to combine the ability of HE charges to produce blast charges of large impulse intensities and short durations with the tendency of a shock tube to produce uniform charges. When mounted on a ballistic sling, a blast tube provides an additional benefit in that it prevents any other portion of the sling from being loaded.

## THE INSTRUMENTED BALLISTIC PENDULUM 141 The former two data types are those of the traditional ballistic pendulum and will be briefly discussed,

THE INSTRUMENTED BALLISTIC PENDULUM 141 The first two data types are those of the traditional ballistic pendulum and will be discussed briefly. Part of the delay was presumably due to the tedious calculations required to evaluate the solution.

## THE INSTRUMENTED BALLISTIC PENDULUM 143 Mode II, and so forth 4 . Note that the 2 nd mode phase velocity curve has an asymptote at a dimensionless

Furthermore, the high-frequency oscillations can be seen to recede from the head of the wave as it travels further down the bar, i.e. they are subject to dispersion. Therefore, the result of incomplete dispersion correction is the inability to resolve the wave head.

## THE INSTRUMENTED BALLISTIC PENDULUM 145

These oscillations are clearly an artifact of the measurement technique and do not represent any physical behavior of the plate samples or blast waves. In addition, averaging was found to tend to decrease the temporal integral of the signal.

THE INSTRUMENTED BALLISTIC PENDULUM 147

## THE INSTRUMENTED BALLISTIC PENDULUM 149

The transient behavior of the signal after the sudden drop, which may represent electromagnetic oscillations in the circuit or subsequent contact of the severed rhythm meter with the steel walls of the blast tube, was not considered important and was not investigated further.

## Blast Load Characteristics

Short Stand-Off Blast Characteristics

## BLAST LOAD CHARACTERISTICS 151

The explosion pressure history captured by the central Hopkinson bar for a PCA plate loaded by a charge configuration without a ring charge, i.e. figure 5.4b, is shown in figure 5.15. The blast pressure history of a PCA plate loaded by a charge configuration including a ring charge, i.e. figure 5.4a, is shown in figure 5.16.

## BLAST LOAD CHARACTERISTICS 153

It is suggested that the asymptotic behavior is due to the localized contribution of the 1 g conductor charge. The correlation is sufficient to suggest that the theoretical interpretation of the experimental data is correct.

## BLAST LOAD CHARACTERISTICS 155

### Blast Tube Load Characteristics

In contrast, the bar impulse seems to decrease with increasing tube length, although there is no clear trend below 200 mm tube lengths. Given the face plate and Hopkinson bar diameters of 100mm and 20mm respectively, an ideally uniform blast pressure distribution should result in a bar impulse that is 4% of total impulse when scaled by area ratio.

## BLAST LOAD CHARACTERISTICS 157

For charge masses less than 10 g, the blast tube data are above and approximately parallel to the short stand. This result was expected since the blast tube limits the explosive event to a certain extent.

## BLAST LOAD CHARACTERISTICS 159

It is noteworthy that the time of arrival (TOA) of the blast wave front is slightly faster in Figure 5.22 than in Figure 5.21. This behavior is also evident in Figure 5.23, which shows five explosion pressure histories from the range of charge masses used in this thesis.

## BLAST LOAD CHARACTERISTICS 161

The quotient I(t)/I is an expression of the fraction of the total impulse I that has been applied to the target plate at the time. The form of equation (5.8), with γsas the subject of the formula, allows the parameters of equation (5.6) to be determined by a simple heuristic procedure.

## BLAST LOAD CHARACTERISTICS 163

Blast Tube Simulations

BLAST LOAD CHARACTERISTICS 165