experimental and numerical study of blast-structure

Structures Congress 2020 105

© ASCE

Experimental and Numerical Study of Blast-Structure Interaction

Benjamin J. Katko1; Rodrigo Chavez2; Heng Liu3; Barry Lawlor4; Claire McGuire5;

Lingzhi Zheng6; Jane Zanteson7; and Veronica Eliasson, Ph.D.8

1Dept. of Structural Engineering, Univ. of California San Diego, La Jolla, CA. E-mail:

[email protected] 2Dept. of Structural Engineering, Univ. of California San Diego, La Jolla, CA. E-mail:


[email protected] 4Dept. of Mechanical and Aerospace Engineering, Univ. of California San Diego, La Jolla, CA.

E-mail: [email protected] 5Dept. of Mechanical and Aerospace Engineering, Univ. of California San Diego, La Jolla, CA.

E-mail: [email protected] 6Dept. of Mechanical and Aerospace Engineering, Univ. of California San Diego, La Jolla, CA.

E-mail: [email protected] 7Dept. of Structural Engineering, Univ. of California San Diego, La Jolla, CA. E-mail:


[email protected]

ABSTRACT

Blast wave interaction with structures is a complex phenomenon. Not only are the fluid

mechanics and combustion physics involved in the blast wave dynamics difficult to correctly

model, but also the interaction with a structure, the structure’s response to the dynamic loading,

and the structure’s influence on the blast wave propagation itself. Clearly, blast waves

propagating on nearby structures fall under the category of three-dimensional scenarios featuring

highly time dependent fluid and solid mechanics responses. In addition, the high strain rates

imparted onto the structure make it difficult to extrapolate a low strain rate response of the

structure. Here we present a novel small-scale experimental setup to study blast wave-structure

interaction in a three-dimensional setting in which a cityscape is subjected to a dynamic blast

wave event. The experimental turn-around time is very fast: less than two minutes per

experiments. Experimental data is obtained via ultra-high-speed schlieren photography and local

strain gages attached to the structure(s) of interest.

INTRODUCTION

Excellent presentations of the basics of compressible flows and blast waves are presented in

detail in the books by J.D Anderson (2003) and C. Needham (2010). Here, only a short review

will be given. Shock waves are formed when objects move faster than the speed of sound or

when energy is suddenly deposited into a restricted volume. The source of energy can be

mechanical, chemical, nuclear or through electromagnetic radiation. A shock wave is a thin

discontinuity in which flow properties change abruptly from one state ahead of the shock wave

to another state behind the shock wave. The thickness of a shock wave is on the order of a few

mean free paths of the fluid, which is about 200 nm at standard atmospheric conditions. Shock

waves may form in both gases or condensed matter. The mathematical expressions describing

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shock waves are nonlinear by nature, so acoustic theories where superposition is used are not

applicable for shock waves.

The most basic expressions describing a shock wave are the Rankine-Hugoniot conditions;

conservation of mass, momentum and energy. These are clearly derived step by step including

illustrative figures in the book by Apazidis and Eliasson (2018). This system of equations is

closed with an equation of state. One of the simplest equations of state one can use is the ideal

gas law, which is valid for moderate temperatures and low pressures, but for a blast wave

scenario more complex equations of state that take into account the elastic properties of the

explosive must be used. The simplest is the Landau-Stankovich-Zeldovich and Kompaneets

equation of state (C. Needham, 2010). For blast waves, these equations are rewritten using the

Chapman-Jouguet relations, which only changes the conservation of energy term by the addition

of another term that represents the detonation energy per unit mass of explosive.

Real gas effects need to be taken into account for high temperatures or high pressures. One

has to take into account vibrational modes, rotational modes, and chemical reactions including

dissociation and ionization as temperatures increase. For example, oxygen dissociates at about

2000 K, and ionization occurs at about 8000 K. Once these changes occur, cp and cv (specific

heats at constant pressure and constant volume, respectively) are no longer constants, so the gas

constant cp/cv changes. This also means that properties of the fluid, for example pressure, internal

energy and enthalpy, need to be obtained from first principles combined with statistical

mechanics and statistical thermodynamics.

A blast wave can be thought of as a shock wave followed by an exponential decay in fluid

properties. Overpressure is defined as pressure above ambient pressure, and underpressure is

defined as that below ambient pressure. The underpressure can return to ambient pressure

smoothly or through one or more shocks. Pressure-impulse is the area underneath the pressure

curve, and together with the peak pressure are measures of how much potential damage a blast

wave could induce. A note should be made, as explained by C. Needham (2010), that the sign of

dynamic pressure and the direction of the flow associated with a certain Mach number are

important for understanding how the surrounding is influenced by the blast wave. One can

choose to use a definition of dynamic pressure as in which both the magnitude and direction are

preserved, as recommended by C. Needham (2010).

In the 1940s, the mathematical and physics description of point source explosions became

important. This can be described as a Taylor point source meaning that a finite amount of energy

is deposited in a point in space (G. Taylor, 1950). This in turn generates a large pressure jump

compared to ambient conditions, and oftentimes this is denoted as a shock wave with infinite

initial strength. In 1946 L.I. Sedov published solutions for three different types of geometries

(linear, cylindrical and spherical) using three different density distributions (constant pressure,

density as a power function of the radius and vacuum). In our previous work, we have used an

extended Taylor solution (S. Lin, 1954) of cylindrical geometry with a constant density solution

assuming that the energy is released instantaneously and that perfect gas conditions remain all

the time (S. Qiu, 2017).

EXPERIMENTAL APPROACHES TO STUDY CLOSE-BY AND ATTACHED

EXPLOSIONS

Explosion physics in the near field can be studied by investigating pressure and impulse

histories. However, as a result of the flow of the hot gaseous products and the turbulence that

goes along with this, it is never a trivial exercise to correctly measure peak pressures and


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pressure histories in the near field of an explosion. To better describe near-field explosion

effects, some ballisticians and engineers have designed novel systems to measure these

properties of blast waves in the proximity of a fireball.

In 1999 M. Held designed a novel system to measure the momentum transfer of blast waves.

Several 70 mm-diameter aluminum cylinders were arranged around a high explosive (HE) of the

same height on a testing range. The momentum transfer from one-end initiated cylindrical HE

charges was obtained by measuring the flight distances of these cylinders after detonation.

Thanks to the registered movement of individual cylinders, the distribution of certain explosion

properties in terms of the location was recovered.

Held also noticed that when compared to the momentum distribution produced from a

spherical charge in the direction of the detonation wave (axial/longitudinal direction) a large

amount of momentum was transferred to the two cylinders around the charge axis which was

about 10 times higher compared to the rearward direction; whereas in the direction against the

initiation (radial direction) a similar momentum distribution to that of a spherical charge was

measured.

In 2015 S.E. Rigby et al. reported another methodology and experimental setup to measure

the pressure history from near-field explosions. This setup was comprised of a large steel plate

with Hopkinson pressure bars (HPBs) inserted into holes where the end was flush with the face

of the plate. The HPBs had strain gauges attached on the other end on the perimeter, such that

loadings on the exposed face can be measured. The plate was supported by a structure made of

concrete columns. Several tests were conducted with 100 g spherical PE-4 charges detonated 75

mm away from the bottom of the plate. An arbitrary-Lagrangian-Eulerian analysis was

conducted to compare the results for overpressure. The results seemed to be compelling with a

highly accurate model of the impulse-time history and a pressure curve that fit the adjusted data.

There have been several publications with a focus on the influence of the configuration of a

charge on blast waves in the vicinity of the explosion. Since most charges are cylindrical rather

than spherical, the study of blast waves resulted from cylindrical charges has practical relevance.

In 2010 C. Wu et al. published experimental results of air-blast effects from cylindrical

charges. Reflected overpressures and impulses acting on a flat target from spherical and

cylindrical charges were measured. Different spatial and temporal distributions of the reflected

overpressures and impulses were obtained as a function of two variables: charge geometry

(cylindrical, spherical) and orientation of longitudinal axis of cylindrical charges (vertical,

horizontal). Several observations were made. First, if the charge weight was small (0.24 kg),

both spherical and horizontal cylindrical charges produced very close reflected overpressures but

different arrival times. The blast from the cylindrical charge arrived earlier than that from the

spherical charge. Second, if moderate charges were used (0.95 kg), the vertically orientated

cylindrical charge showed a much higher reflected overpressure and earlier arrival time of the

blast wave than those of vertically placed cylindrical charges and spherical charges. Finally, if

the weight of the charges increased to 2.5 kg for both cylindrical and spherical charges placed

horizontally, both charges produced much higher reflected overpressures than if lighter charges

were used. However, such influence was much more significant on the cylindrical charge whose

peak overpressure was almost 2.5 times greater than that of the spherical charge. Wu et at. also

discussed how the detonation location within the cylindrical charge would affect the blast

conditions at the target. It was found that the least reflected overpressure at the target was always

generated by the charge initiated in the middle center; in contrast the detonation initiated at the

middle of one side surface produced the most extreme flow conditions at the target.


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The same topic was also visited by hydrocode simulations. In 2016 P. Shekar et al.

performed a numerical study on charge characteristics including charge shape, charge weight and

the point of detonation within the charge. AUTODYN 2D was applied to compute the

propagation of blast front produced by 10 kg cylindrical charges with different aspect ratios

(L/D). Overpressure and impulse histories were recorded by the sensors at several fixed locations

in both axial and radial directions. Some observations were consistent with those from C. Wu

(2010) and H.D. Zimmerman (1999): a large portion of energy was directed axially for the

cylindrical charge with small aspect ratio while more energy was directed radially for a large

L/D. Shekar et al. also noticed that the overpressure histories generated by the cylindrical

charges included secondary and tertiary shocks that were more pronounced at a greater scaled

distance. Secondary shocks were shown to be the bridge waves traveling behind the primary

wave in the axial direction. This also explains why there was no significant secondary shocks

observed in the overpressure histories generated by the spherical charge. Combining the

observations from pressure contours and overpressure/impulse histories Shekar et al. concluded

that for a charge weight ranging from 10 kg to 1000 kg the effect of charge shape can be ignored

at distances greater than 35 charge diameters (or scaled distance) for peak incident overpressure,

and 26 charge diameters (or scaled distance) for incident impulse. The detonation point within

the cylindrical charge was investigated by initiating the charge to the side of the center. Results

showed that the shape of the pressure field and the magnitudes of the peak overpressures and

impulses were influenced by the point of detonation within the explosive.

In 2002 Schrami et al. reported the blast environments in near field produced by cylindrical

charges. Numerical simulations were performed using arbitrary-Lagrangian-Eularian general

research application (ALEGRA) in 2D. The cylindrical charge used in this study was composed

of LX-14 with an aspect ratio of 1.6. Four scenarios were studied by varying the detonation point

within the charge. The dynamic pressure was selected to compare the results of the four

configurations studied. Detonating the charge at a point in the center of the cylinder produced a

blast field that was uniform along both the axial and radial directions. Detonation of the charge

along the axis in the center of the cylinder produced a blast field that was more concentrated than

that from initiating at a point in the middle center. Initiating the charge at one end of the cylinder

produced a blast environment that was skewed in the direction of the detonation. Simultaneously

detonating the charge at both ends of the cylinder produced a highly focused disk-shaped blast

field at the midplane of the charge.

Held’s experiments on cylindrical charges (M. Held, 1999) was repeated using numerical

simulations by C.Y. Tham in 2009. Hydrocode simulations were performed using AUTODYN

3D to compute the impulse of the aluminum cylinders when they were impinged by blast waves.

The simulations agreed with Held’s earlier experiments in that the momentum transferred to the

cylinder gauges was not spherically distributed and the gauges near the two ends recorded

different values of momentum. Both simulations and experiments identified a higher momentum

in the direction of the detonation wave. Additionally, simulation results revealed three distinctly

high-velocity regions produced by a cylindrical charge detonated at one end, which explains the

expansion of the non-spherical transfer of momentum.

DYNAMICS OF THE EXPLOSION PROCESS

As described in Semenov’s original description of the thermal explosion theory (A.R.

Shouman, 2006; N.N. Semenov, 1928; N.N. Semenov, 1942), an explosion occurs when the state

of the system jumps from the low-temperature equilibrium state to the high-temperature


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equilibrium state as a result of the imbalance between the heat generation within the explosive

and the heat dissipation to the surroundings. In the initial stage of an explosion, along with

extreme pressure and temperature conditions complex wave interactions always take place. The

shock wave front coincides with the fireball until they detach.

The fireball and shock wave dynamics were analyzed by Gordon et al. (2013) using image

processing. Thirteen aluminized novel munitions varying in the liner-HE ratio and total weight

were detonated. All charges shared a similar shape of L/D>2 and were initiated in the center of

the charge. The entire detonation process was recorded by a high-speed camera. Tracking of the

fireball and shock wave front was made possible by image processing. Since in the early stage

the shock wave front was observed to be coincident with the fireball until they detached, the

fireball front can be used to represent the shock wave front to fit the Sedov-Taylor model (L.I.

Sedov, 1993). The calculated energy release in the detonation using the best fitted model

indicated an underestimation of detonation efficiency compared to the theoretical values. For the

fireball behavior, high-speed camera photos registered that the fireball size quickly approached

its maximum value within 30–50 ms after initiation, later a fairly constant fireball size was

observed. The drag model (S.D. Gilev, 2006) was selected to fit the experiment data and the

analysis of the model showed a much smaller initial detonation velocities than the theoretical

values.

Stepanov et al. (2011) developed a complete analytical model to compute the size and

lifetime of the fireball. Their hydrodynamic study revealed that, since the duration of the

hydrodynamic phase of explosion is much shorter than the fireball lifetime, the energy and

characteristic size of the fireball can be realized based on the explosion energy and type, upon

the cessation of the hydrodynamic expansion of explosion products. Therefore, the modeling can

be performed without taking into consideration the hydrodynamic process accompanying the

explosion of chemical explosives. The radiation of the fireball was analyzed by Stepanov et al.

on the basis of the optical properties of the combustion products of the explosion, and a

simplified model was developed, which was capable of estimating the size and lifetime of the

fireball.

The fireball size in an ethyne-air cloud explosion was examined by Huang et al. (2018) using

numerical simulations. As a key factor, the turbulence in the gas explosion mechanism was

considered and a one-step irreversible reaction model was adopted for the gaseous ethyne-air

reaction. The numerical simulations were initiated with an on-ground explosion of a

hemispherical cloud 2 m in diameter. Results revealed the existence of a transient flow of the

unburned premixed gas in front of the flame that was induced during flame propagation in the

cloud. If the flame range is defined by using the lower flammability limit, the ratio of the

combustion area radius to that of the original cloud reached 1.4–2.7 in the radial direction on the

ground and 1.5–4 along the axis of symmetry perpendicular to the ground for an ethyne–air

mixture explosion in unconfined space. Both peak overpressure and peak reaction rates were

reached beyond the original cloud.

Other investigations on the dynamics of the explosion process concentrate on wave

interactions at the initial stage without taking into consideration the fireballs. By modeling the

energy release rate as a simplified kinetic equation considering the chemical reaction in the

detonation wave, D.O. Morozov (2013) investigated the initial phase of explosion with account

for the detonation wave propagation in the HE. It was observed that if the detonation was

initiated in the symmetry plane inside the explosive, when the detonation wave reached the

charge boundary with ambient air it split into two shock waves. One shock wave propagated in


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the air and the other one that propagated in the explosion product from the air interface was

carried to the other direction by the rapidly expanding products. On the other hand, if the

detonation was initiated from its surface, flows were initiated simultaneously in the two media.

The detonation wave propagated into the depth of the charge, and the air shock wave traveled

from the interface to the ambient air. After the detonation wave reached the symmetry plane it

reflected back propagating in the reacted products. Since the pressure behind it was a few orders

of magnitude higher than the pressure induced by the air shock wave, the reflected shock wave

ended up with catching up with the air shock wave and amplified it. Complex interactions

between the waves and the interface for two initiation methods led to the observation that the

pressure at the front of the air shock wave in the case of initiation at the interface was about 1.5

times lower than in the case of initiation in the symmetry plane. Then, it was amplified by the

reflected detonation wave and reached a similar strength as the counterpart. Morozov et al.

noticed that the influence of the site of initiation on the motion of the air shock wave vanished at

larger distances.

Similar to Morozov’s 2013 study that described the evolution of waves in detail in the initial

stage of an explosion, Smetannikov et al. (2010) used a two-dimensional numerical model to

investigate the wave interactions in a near-ground explosion produced by a cylindrical charge.

Smetannikov et al. did not describe the process of detonation but used the approximation of

instantaneous detonation. Three scenarios were investigated that only differed in L/D of the

charges. Detailed descriptions of shock wave evolution and interaction were given for all three

scenarios. In the first case, when the incident shock propagated away from the explosion center,

a rarefaction wave propagated inwards from the surface of the charge. Once the rarefaction wave

reflected back from the symmetry plane, a low-pressure rarefied region, called cavity, formed in

the explosion products. The cavity first gradually expanded then decelerated. Since the pressure

gap between the inside and the outside of the cavity was high, a secondary shock wave arose and

then moved to the center of the charge. When the secondary shock wave was reflected from the

symmetry plane, the incident shock wave had reached the reflective ground. Then, the reflected

secondary shock interacted with the reflected primary shock and the intensity decreased

significantly. For the second and third scenarios, both a rod-shaped charge and a disc-shaped

charge were able to produce a quicker expansion of the primary shock along a smaller-size axis

at the early stage. Smetannikov et al. observed that the flow gradually acquired the shape close to

a semi-spherical one in the far field for independent of the charge shape.

EXPLOSIONS NEAR STRUCTURES IN URBAN ENVIRONMENTS

To understand the potential damage to an urban environment under extreme events such as

blast, it is important to simulate such conditions by varying the parameters that can have

significant effects on the blast wave propagation. There exists many different models for

experimental and numerical investigations mimicking urban environments subjected to blast

loads. Some of the key parameters for an urban environment are street configuration, size of

buildings, the width of the streets, and proximity of buildings to each other etc.

In a study conducted by C. Fouchier et al. (2017), an experimental investigation that sought

to replicate explosive detonation within a typical urban setting was carried out. The authors

performed a laboratory scale experiment to simulate five typical urban environments under blast

loading. This study emphasized the influence of street configuration and building placement with

respect to the blast source changed the structural response. The five types of street configurations

studied were: free-field, straight street, T – Junction, cross-junction, and channeling. The free-


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field investigation was carried out to understand the repeatability, geometry and TNT equivalent

of each explosive. From the above configurations, it was found that the straight street

configuration was the most dangerous in an urban environment since it confined the blast waves

in one direction. Moreover, it was shown that increasing the height of the buildings and

narrowing the streets confined the blast waves thus enhancing their intensity leading to more

severe damage to urban environments.

Figure 1: (a) Model of the two-dimensional test section with PMMA windows (inner test

section dimension). The inside gap in between the two PMMA windows is 19 mm. Each

exploding wire is held in place using two circular brass electrodes with a v-shaped notch.

(b) Model of the three-dimensional test section with polycarbonate windows. Each

exploding wire is held using two tower structures that are equipped with circular brass

electrodes that contain a v-shaped notch at the top of the tower. From W. Mellor et al.,

2019.

The propagation of blast waves and their effect on various objects were investigated by S.A.

Valger et al. (2017). AUTODYN 2D was used to simulate the propagation of a blast wave

formed by an explosion of a spherical charge in a semi-infinite space. The main topic of this

study was to investigate the interaction of blast waves with a set of prims in the domain that

represented structures and buildings in an urban area. Complicated wave interactions were

reported and explained by Valger et al. The numerically predicted loads on the prisms were

compared to the measured data that showed adequate reproduction of the primary pressure peak

and some errors in predicting the secondary peaks. Also, the CONWEP empirical functions were

tested in all cases. Compared to the hydrodynamic codes, CONWEP failed in reproducing the

minor variations and negative phase of the pressure evolution at given locations.

UCSD BLAST WAVE EXPERIMENTS

At UCSD, we have developed an exploding wire experiment setup (W. Mellor et al., 2019) in

which the experiments can be performed in either a 2D or 3D space, while studying shock-shock

interaction by itself, or the effects of a single or multiple shock waves impacting different types

of structures at varying complexity levels -- such as a single flat composite plate, or multiple

scaled “buildings” featuring a city block. In particular, these experiments were designed with

two things in mind: 1) very fast turn-around time between consecutive experiments at an order of

minutes; and 2) highly flexible setup that allows for many different types of scenarios being


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modeled.

The current experiments were performed using the exploding wire system coupled with an

ultra-high-speed camera (Shimadzu HPV-X2) and a z-folded schlieren setup. The exploding wire

setup was designed to pass a controlled amount of energy through a very thin metal wire to

create shock waves of varying strengths. Exploding wire setups have been used since the late

1700s (J. R. McGrath, 1966) in a number of different studies. Interestingly, these types of setups

have also been used in the area of shock dynamic studies. Perhaps most notably are the

experiments performed by Ernst Mach on shock reflections in the late 1870s (J. T. Blackmore,

1972) where he was able to deduce the existence of Mach stems based on the pattern left behind

by the shock waves on sooted plates.

The UCSD experimental setup consists of four main parts: a charging circuit; a load circuit; a

damping circuit; and a triggering mechanism (W. Mellor et al., 2019). The two-dimensional test

section setup is shown in Figure 1(a), and the three-dimensional test section setup is shown in

Figure 1(b).

Four photographs from a three-dimensional experiment is shown in Figure 2. Two 0.05-mm-

diameter nickel chromium wires were exploded at a capacitor charge voltage of 20 kV. The

wires were spaced 76 mm from one another and the resulting shock wave interaction was

observed. The schlieren photographs are taken at a frame rate of 500,000 frames per second. This

experiment validates the ability of the experimental setup to visualize the regular (two-shock

system) to irregular (three-shock system) interaction of two three-dimensional shock waves.

Figure 2: Schlieren photographs taken during an experiment in the three-dimensional test

section where two exploding wires are exploded at the same time. The resulting shock

waves are propagating from the bottom to the top with the intersection point in the middle

of the photograph. The resolution is 400 × 250 pixels and the viewing area is approximately

130 × 80 mm2. From W. Mellor et al., 2019.

A Matlab script is used to remove optical distortions in the photographs after they have been

recorded. Then, the individual shock fronts are tracked and radius versus time information is

obtained. From this, velocity fits can be calculated for all experiments. An example is shown in

Figure 3.

Next, the setup was used for preliminary experiments to obtain the shock dynamics response

during a simulated explosive event in an urban landscape. Therefore, a 2D setup was prepared

with several buildings arranged according to Figure 4. A single exploding wire was located at the

bottom of the experiment, also shown in Figure 4. Ultra-high-speed photography was used to

obtain schlieren images of the shock-structure interaction as the shock propagated through the

cityscape model. This shows the feasibility to perform future experiments in 3D where one can

study parameters like height of structure; width of structure; depth of structure; material

properties; geometry of structure including windows and door openings; location and response of

other structures or objects that are nearby the structure of interest; and location of blast wave


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relative structure of interest and other objects in the surrounding area.

Figure 3. Example for capacitor charges of 13 kV and 21 kV showing tracking of the

vertical position of the Mach stem with curve fitting using a model created by Axpro

Group (Colorado School of Mines). From Eliasson Shock and Impact Group, UCSD, 2019.

The exploding wire setup can also be used to study the structural response and its health

during and after the structure has been subjected to a blast wave. For this, another setup was

developed, and it is shown in Figure 5.

Figure 4. (Left) Drawing of the exploding wire setup experiment, simulating and explosion

in a cityscape. Note that the viewing is marked by the dashed lines. (Right top) Schlieren

images showing the individual shock wave locations at early to late times. (Right bottom)

Shock tracking obtained using the Matlab script developed in this study. From Eliasson

Shock and Impact Group, UCSD, 2019.

Preliminary results from the blast box experiments indicate that weak shock waves, produced


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by the in situ exploding wire apparatus, can force the carbon fiber plate to deflect in the out-of-

plane direction. Furthermore, the results indicate that the out-of-plane deflection varies

depending on the structural health state of the carbon fiber plate.

Figure 5. Blast box with electrode post inserted, carbon fiber reinforced polymer plate, and

pressure sensor wall mounts. The boundary conditions are simply supported parallel to the

transverse axis and free-free parallel to the longitudinal axis. From Eliasson Shock and

Impact Group, UCSD, 2019.

Figure 6. (a) The average out-of-plane response for both the damaged and undamaged

carbon fiber plates. (b) The convolution of the two averages are indicating at 60 𝜇s the out-

of-plane responses begin to deviate, implying that the structure has indeed been altered.

From Eliasson Shock and Impact Group, UCSD, 2019.

The blast box experiments used strain gages to measure the dynamic, out-of-plane response

of the carbon fiber plate. Analysis of the strain gage data indicated that further experimentation

was required. This led to pure out-of-plane blast experiments of the carbon fiber plates, in

differing health states. The purpose being to quantify, more precisely, the carbon fiber’s plate’s

response to the blast waves. This quantification was aided by ultra-high-speed stereo digital

image correlation. Figure 6 shows the average out-of-plane deflections of a pristine carbon fiber


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plate and a moderately damaged carbon fiber plate.

Further experimentation of the carbon fiber plate’s response to the blast loading and how it

relates to the fundamentals of stress wave propagation in an inhomogeneous material, from the

perspective of nondestructive evaluation techniques, such as an impact-echo evaluation, should

be investigated as the modal response of the plate will change with varying degrees of damage.

Furthermore, resolving the spatial location of the damage would provide researchers with the

capability of not only knowing the magnitude of damage, but the location as well, which is

critical to the structural health paradigm.

UCSD NUMERICAL MODELING USING REDUCED ORDER MODELS

Here, our numerical simulations were developed with an end goal of having a fast code to

perform future optimization simulations; thus our choice fell on Geometrical Shock Dynamics

(GSD) (G.B. Whitham, 1974; W.D. Henshaw et al, 1986). Compared to the Euler simulations of

gas dynamics, which solve the equations of conservation of mass, momentum and energy, that

provide a solution to the entire fluid field, geometrical shock dynamics – as a particle method –

only tracks the motion of the shock front. Therefore, the complexity of the problem is lowered

and the simulation turnaround time is significantly reduced. However, such an advantage comes

at the expense of accuracy. A blast wave induces a non-uniform flow state behind the shock front

that leads to the invalidation of the underlying assumption of the Area-Mach number relation

(that the shock motion should be independent of the flow conditions behind the shock front) on

which the original GSD theory depends. In fact, a complete form of GSD was derived by (J.P.

Best 1991). Here, the post-shock flow effect term is incorporated and it models the non-

uniformity behind the shock front. The original Area-Mach number relation is just a special case

of these equations where this post-shock flow effect term is simply set to zero.

There are at least two ways of expressing the post-shock flow effect term in a GSD system.

One way is to expand the term as an infinite series that leads to a new GSD system consisting of

infinite coupled ordinary differential equations that can be truncated at any level to achieve a

certain order of completeness. If a 1st-order complete system is solved by a 3rd-order accurate

TVD Runge-Kutta method, only a negligible improvement over the original GSD can be

observed independent of initial blast strength. The other way is to explicitly express the post-

shock flow effect term as an input obtained from manipulating the existing analytical solution to

the blast propagation. This way, the post-shock flow effect term is complete and the problem is

closed with a system of two coupled ordinary differential equations.

Figure 7 shows a comparison of a Mach number-radius plot obtained from an analytical

equation, the original GSD, and the modified GSD that incorporates a complete post-shock flow

effect term. Here, it is clear that the modified GSD achieved a very good agreement with the

analytical solution.

The reason for the success of the modified GSD in simulating the blast propagation was

investigated by recording the evolution of the post-shock flow effect term. It turned out that the

modified GSD is able to compute a more accurate post-shock flow effect term than the 1st-order

complete GSD when compared to the Euler simulation, which is used as a reference. A

conclusion about the impact of the post-shock flow effect term in GSD can be made: the

completeness of the post-shock flow effect term determines the accuracy of GSD on blast

propagation. If it is not fully expressed, part of information about the interaction between the

blast front and the flow behind it is missing resulting in the loss of accuracy; once it can be

calculated based on a complete form, the non-uniform state behind the blast can be correctly


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accounted for by GSD.

Figure 7. Comparisons of M-R plots of modified GSD, original GSD and G.G. Bach and

J.H.S. Lee’s analytical results. From Eliasson Shock and Impact Group, UCSD, 2019.

Several scenarios were then investigated using the modified GSD model. The simulations

were initialized with a continuous shock front resulted from multiple converging blast waves at

the transition from regular to irregular reflection. Such transition condition was determined by an

analytical model (S. Qiu, 2017) that communicates the sonic criteria [50] to the analytical

solution to blast propagation (J. Von Neumann, 1943) by geometric transformation. The growth

and attenuation of Mach stems generated by the interaction of two adjacent blasts were recorded

and a good agreement with the Euler simulation was observed.

The conclusions can be summarized as follows:

The novel modular design of the experimental setup and its different test sections

successfully allowed for the creation of multiple cylindrical shocks or spherical shocks,

which have decaying properties behind the shock front.

The experimental setup also allows for the study of (1) regular to irregular reflection of

cylindrical or spherical shock waves; (2) study of 2D and 3D cityscape models; and (3)

structural response of plates using strain gages and 3D digital image correlation

techniques.

Geometrical shock dynamics can be used to model an expanding shock wave in two

dimensions featuring a decay of properties behind the shock front if the post-shock term

is incorporated into the solver. Here, a two-dimensional code was developed in C++,

which relies on a third order Runge-Kutta system and mesh regularization procedure.

Different methods to initialize the post-shock flow term were investigated in this work,

and the results show that this term really matters for the results to be accurate if the flow

behind the shock front is not at a constant state for an extended period of time.

One method to obtain the post-shock term is to use a lookup table in which data –

obtained either through detailed simulations (e.g. Euler simulations) or experiments – for

the post-shock flow properties are tabulated to be used at each time step.


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Future work includes a systematic experimental study to find the point in time and space

when regular reflection transitions into irregular reflection for cylindrical and spherical shock

waves. Also, 3D cityscape experiments are planned and will be performed as parametric studies

of e.g. geometrical parameters of the structures. Furthermore, the two-dimensional GSD code is

being extended to three dimensions.

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experimental and numerical study of blast-structure

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