ARTICLE IN PRESS

International Journal of Impact Engineering 34 (2007) 859–873

www.elsevier.com/locate/ijimpeng

Scaling the response of circular plates subjected to large and

close-range spherical explosions. Part I: Air-blast loading
A. Neubergera,b,, S. Pelesc, D. Rittela
a
Technion, Faculty of Mechanical Engineering, Israel Institute of Technology, 32000 Haifa, Israel
b
MOD, Tank Program Management, Hakirya, Tel-Aviv, Israel
c
IMI, Central Laboratory Division, Ramat Hasharon, P.O. Box 1044, Israel
Received 5 January 2006; accepted 9 April 2006
Available online 12 June 2006

Abstract

This two-part paper addresses scaling of the dynamic response of clamped circular plates subjected to close-range and
large spherical blast loadings.
Full-scale experiments involving actual geometries and charges are quite involved and costly, both in terms of
preparation and measurements. For these reasons, scaled-down experiments are highly desirable. However, the validity of
such experiments remains to be firmly established, and this is the main objective of this paper.
In this study, similarity is obtained by using replica scaling for all geometrical parameters, while the blast effect is scaled
by using the well-known Hopkinson scaling law. We also consider the overall effect of the strain rate sensitivity and
variability of material properties with plate thickness on the response of the scaled model.
This study presents numerical and experimental results from a series of controlled explosion experiments. The first part
of the paper deals with spherical charges exploding in free air, while the second part deals with the same charges flush
buried in dry sand. A good agreement between numerical simulation predictions and test results was obtained, so that the
main result of the two papers is that scaling can be successfully applied to assess the dynamic response of armor plates
subjected to close-range large explosions.
r 2006 Elsevier Ltd. All rights reserved.

Keywords: Close-range; Large blast; Dynamic response; Circular plate; Scaling

1. Introduction

Understanding the dynamic behavior of blast loaded armor steel plates is a key to successful protection
projects. The literature on blast loaded plates is quite abundant. Numerous studies of experimental and/or
numerical nature have been carried out and reported (see, e.g. [1–9]). These studies characterize the different
failure modes and the relationship between deformation and tearing of clamped blast loaded plates. The
failure modes were first defined by Menkes and Opat [10] for the case of impulsively loaded clamped beams.

Corresponding author. MOD, MANTAK, Tank Program Management, Hakirya, Tel-Aviv, Israel.
E-mail address: navidov@gmail.com (A. Neuberger).

0734-743X/$ - see front matter r 2006 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ijimpeng.2006.04.001
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Nomenclature

A Johnson–Cook (J–C) coefficient

B J–C coefficient
C J–C coefficient
Ci polynomial coefficients
D plate diameter
I impulse
m J–C coefficient
n J–C coefficient
P blast pressure
R distance from center of spherical charge
S scaling factor
t plate thickness
T temperature
V relative volume of the gas products to the initial explosive state
W charge weight
Z scaled distance
E energy
d peak deflection
e; e_ strain, strain rate
y angle of incidence
r material density
s stress
t time
z scaled impulse
P characteristic similarity parameter

The classification was as follows: mode I for large inelastic deformation, mode II for tearing at the support,
and mode III for shear failure at the support. These failure modes were also adopted for blast loaded circular
plates, and further subdivisions were observed and defined by other authors. In contrast to the large number
of studies mentioned previously, the literature on scaling (similarity) of the structural response is quite sparse.
Wen and Jones [11] investigated the scaling of metal plates subjected to impact and concluded that geometrical
scaling can be applied. These authors also reported that strain rate sensitivity does not significantly affect
scaling for their experiments. Jacob et al. [12] investigated scaling aspects of quadrangular plates subjected to
localized blast loads. The effects of varying the charges’ weight and plate geometries on the deformation were
described and analyzed. Furthermore, a modified dimensionless number was introduced to represent the
quadrangular plate’s response to a localized disc charge. The validity of the presented results is limited to the
investigated small plate geometries and loading conditions, while numerical simulations and experiments for
larger explosions are still required.
Raftenberg presented a study on close-range small blast loading on a steel disc [13]. This study com-
pared experimental data to several numerical finite elements calculations, based on different con-
stitutive models in order to validate specific dynamic material parameters. The Johnson–Cook model for
RHA steel was found to produce results in reasonable agreement with experiments for the problem at hand.
Hanssen et al. [14] investigated the behavior of aluminum foam panels subjected to close-range blast loading.
This study addressed the performance of additional sacrificial layers as a protecting layer. However, it was
observed that the energy and impulse transfer to the protected structure was not reduced by adding a
sacrificial foam panel, but contrary to the expectations, the impulse increased due to the proximity of the
explosion.
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The combination of scale modeling of the dynamic response of a plate subjected to a very large explosion
from a close-range was not studied yet, to the best of the authors’ knowledge.
Therefore, these papers present scaling aspects of the quantities that characterize the dynamic response of a
plate subjected to close-range large blasts. Part I of the study reports finite elements simulations and
experiments of blast loaded circular plates subjected to explosions in free air. The numerical calculations and
the tests were conducted on different scale factors S (where S ¼ 2 refers to one-half scaling), where the
reference plate (S ¼ 1) is 0.05 m thick with a 2 m diameter. Part II of the study addresses the effects of sand-
buried charges. The main outcome of the two papers is that scaling can be applied to analyze armor plates
subjected to close-range large air blasts as well as buried charges. This study also shows that the variation of
mechanical properties with plate thickness affects the similarity result and should therefore be taken into
account.
The paper is organized as follows: Section 2 presents elements of the physical similarity and scale modeling
used in this work, followed by Section 3 that details the numerical approach. Section 4 describes the test setup
and measuring technique, while Section 5 presents the numerical results, followed by the experimental results.
Section 6 discusses the key points of the study, followed by concluding remarks.

2. Elements of scaling theory

The validation of physical similarity of a specific phenomenon is crucial for proper scaling. Before a large
scale prototype is built, experiments on a small scale model are required. However, one must know how to scale
the model experimental results up to the full scale prototype. The concept of physical similarity was stated by
Barenblatt [15] as a natural generalization of similarity in geometry, so that ‘physical phenomena are called
similar if they differ only in respect of the numerical values of the dimensional governing parameters while the
values of the corresponding dimensionless parameters (P-terms) being identical’. The objective is to obtain
identical relationships between quantities that characterize both, the prototype, regarding the original size, and
the model, regarding the scaled-down phenomena.
The principles of scaling and the relationships between the small scale model parameters and the full-scale
prototype parameters were stated by Jones [16]. Here we will present the relevant parameters for the
investigated problem in terms of proportion between the prototype parameter (superscripts P) and the
corresponding model parameter (superscripts M), as follows:

– Linear dimensions are proportional to the scale factor, xPi ¼ xM i  S.

– Angles are the same, aPi ¼ aM i .
– Densities of materials are the same, rPi ¼ rM
i .
– Stresses of each material are the same, sPi ¼ sMi .
– Characteristic times are proportional to the scale factor, tPi ¼ tM
i  S.
– Strains are identical, ePi ¼ eM
i .
– Loads are the same, and must act at scaled locations, F Pi ¼ F M P M
i at xi ¼ xi  S.
– Deformations at geometrically scaled locations for corresponding scaled times are proportional to the scale
factor, dPi ¼ dM P M
i  S at ti ¼ ti  S.
– Angular deformations are the same, oPi ¼ oM i .

It should be noted that several phenomena may not scale according to these principles. For example
gravitational forces cannot be scaled according to the basic principles of geometrically similar scaling.
However, here high accelerations are involved therefore the gravitational forces are not significant and can
thus be neglected. Strain rate sensitivity in a small-scale model is scale factor times larger than that in a
geometrically similar full-scale prototype. For the case at hand, considering the actual strain rates produced
during the dynamic bulging process, the material properties are taken to be approximately scale-independent.
Finally, fracture cannot be scaled according to the basic principles of geometrically similar scaling. However,
fracture is beyond the scope of the present work.
When scaling spherical blast wave phenomena, the most common scaling method is Hopkinson, or ‘‘cube
root’’ scaling law as shown by Baker [17]. This scaling law states that self-similar blast waves are produced at
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identical scaled distances when two explosive charges of similar geometries and explosive, but of dif-
ferent weight, are detonated in the same atmosphere. For explosions in air, the Hopkinson scaled parameters
are [17]:
R t I
Z¼ 1=3
; t ¼ 1=3 ; z ¼ 1=3 , (1)
E E E
where Z is the scaled distance, t is the characteristic scaled time of the blast wave, z is the scaled impulse, R is
the distance from center of blast source, and E is the source blast energy. This law implies that all quantities
with dimension of pressure and velocity are unchanged through scaling, i.e. for the same value of Z (note that
E can be replaced by the blast source weight W). In this study, Hopkinson’s method was used to calculate the
corresponding charge weight for the scaled down model as follows:
W M ¼ W P =S3 . (2)

3. Numerical simulations

The numerical simulations were carried out using LS-DYNA finite element code [18]. The software has the
ability to simulate dynamic structural response in several ways, including pure Lagrangian, and coupled
Lagrange–Eulerian methods. The purely Lagrangian approach combined with a simplified engineering blast
model is more desirable since it reduces the calculation time. However, both methods were used and compared
with the experimental data. The multi-material Eulerian formulation is part of the Arbitrary-Lagrangia-
n–Eulerian (ALE) solver within LS-DYNA. By combining the ALE solver with an Eulerian–Lagrangian
coupling algorithm, a structural or Lagrangian mesh can interact with the gas products or Eulerian mesh. This
technique was used for the ranges where the simplified blast model produced uncertain impulse duration as a
result of blast proximity.

3.1. Simplified blast model

‘Load-blast’ boundary condition, which is implemented in LS-DYNA, was used to load the tested circular
plate with the appropriate varying pressure distribution. This function is based on an implementation by
Randors-Pehrson and Bannister (1997) of the empirical blast loading functions by Kingery and Bulmash [19]
that were implemented in the US Army Technical Manual ConWep code [20].
The blast loading equation is stated as follows:
PðtÞ ¼ Pr  cos2 y þ Pi  ð1 þ cos2 y  2 cos yÞ, (3)
where y is the angle of incidence, defined by the tangent to the wave front and the target’s surface, Pr is the
reflected pressure, and Pi is the incident pressure. This blast function can be used for the following two cases:
free air detonation of a spherical charge, and ground surface detonation of a hemispherical charge. To
calculate the pressure over certain predefined group of surfaces related to the geometry of the analyzed
structure, the model uses the following inputs: equivalent weight of TNT explosive, the spatial coordinate of
the detonation point, and the type of blast (spherical or hemispherical). The actual impulse, I, corresponding
to the charge’s weight and distance to the target, can be derived from the ConWep database [20] that is
implemented into the ‘load-blast’ function of LS-DYNA software. For example, considering the problem
analyzed here, the spherical TNT charge’s weight is W ¼ 50 kg, and the distance from the target is R ¼ 0:5 m.
Then, the impulses are I(incident) ¼ 1563 kPa ms, and I(reflected) ¼ 45,220 kPa ms.
Note that the model accounts for the angle of incidence of the blast wave, but it does not account for
shadowing by intervening objects or for the confinement effects, causing the blast to focus on a certain zone.

3.2. Material behavior

The ALE model involves three different material types. The first material model represents the rolled
homogeneous armor (RHA) steel plate, followed by the TNT spherical explosive charge, and finally the
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surrounding air media, which is used to fill the space affected by the explosive charge. Note that in the purely
Lagrangian approach, only one type of material model is needed, namely for the steel plate.
The circular steel plate is represented by a finite element mesh created by FEM-PATRAN preprocessor
software. By using the inherent symmetry of the studied problem calculation time can be saved. Thus, only a
quarter of the circular plate was modeled with the appropriate boundary conditions applied along the
symmetry planes as shown in Fig. 1. The entire model was constructed from constant stress hexagonal solid
elements formulation with one integration point.
To represent the dynamic mechanical behavior of the RHA steel two constitutive material models were
compared. The first model represents a strain rate insensitive bilinear material. The second model represents a
strain rate sensitive material, represented by the Johnson–Cook (J–C) constitutive model [21]. The J–C model
is stated as follows:
 
sy ¼ A þ B  ēnp  ð1 þ c  ln e_ Þ  ð1  T m Þ, (4)

where A, B, c, n and m are the J–C material coefficients, as listed in Table 1 [20,21], ep is the effective plastic
p
strain, e_ ¼ ē_ =_e0 is the effective plastic strain rate at a reference strain rate e_0 ¼ 1 s1 , and the homologous
temperature T ¼ ðT 0  T 0room Þ=ðT 0melt  T 0room Þ where T0 is the material’s temperature, T0 room is the room
temperature, and T0 melt is the material’s melting temperature.
The J–C constitutive material model is incorporated with a polynomial equation of state which is linear in
internal energy E. The pressure P is given by
P ¼ C 0 þ C 1 m þ C 2 m2 þ C 3 m3 þ ðC 4 þ C 5 m þ C 6 m2 ÞE, (5)
2 2
where the terms C2m and C6m are set to zero if m ¼ ðr=r0 Þ  1o0, and r/r0 is the ratio of the current to initial
density.

Fig. 1. Finite elements model: (a) symmetry B.C.; (b) constrained B.C.; (c) ‘load-blast’ B.C.; (d) entire model including measurement
points.

Table 1
RHA class 1 mechanical properties and the corresponding Johnson–Cook coefficients [24]

t (mm) sy (MPa) sUTS (MPa) eL (%) E (GPa) n Hardness (HBW) A (MPa) B (MPa) n c m

3–20 950 1250 9 380–430 950 560

21–40 900 1150 10 210 0.28 340–390 900 545 0.26 0.014 1
41–80 850 1050 11 300–350 850 355
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The TNT explosive charge was modeled via Jones–Wilkins–Lee (JWL) equation of state (EOS) for explosive
detonation products [22]. The pressure field is given by
   
o o oE 0
P¼A 1 eR1 V þ B 1  eR2 V þ , (6)
R1 V R2 V V

where A, B, C, R1, R2, and o are constants that can be found in the literature [23], V ¼ v=v0 is the relative
volume of the gas products to the initial explosive state, and E0 is the energy per unit volume.
The medium in which the blast wave propagates (air) was modeled with a linear polynomial EOS for linear
internal energy, which is given by


P ¼ C 0 þ C 1 m þ C 2 m2 þ C 3 m3 þ C 4 þ C 5 m þ C 6 m2 E  , (7)

where m ¼ r=r0  1 with r=r0 being the ratio of the current to initial density, C1,y,6 are polynomial
coefficients, and E* has units of pressure. To model gases, the gamma law EOS can be used. This implies that
C 0 ¼ C 1 ¼ C 2 ¼ C 3 ¼ C 6 ¼ 0 and C 4 ¼ C 5 ¼ g  1, where g is the ratio of specific heats. Therefore Eq. (7)
becomes
r 
P ¼ ðg  1Þ E . (8)
r0

4. Test setup

Two different scaled-down similar test rigs (of scale-down factors S ¼ 2 and 4) were built in order to
experimentally assess the applicability of scale-down modeling of the studied problem. The experimental test
setup of each structure is shown in Fig. 2. The target plate was supported by two thick armor steel plates with
circular holes that were tightened together with bolts and clamps. The thick plate that faces the charge has a
hole with inclined side walls to prevent reflection of the blast wave to the tested plate as shown schematically in
Fig. 3.
The measurement of the maximum dynamic deflection of the plate was achieved by means of a specially
devised comb-like device. The teeth of the comb possess a gradually decreasing height, as shown in Fig. 4.
When positioned under the dynamically deflecting plate, the long teeth are permanently bent while those that
are shorter than the maximum deflection remain intact. Therefore, a direct estimation of the maximum
deflection is immediately available after the blast test. We found that this inexpensive device provides a
sufficiently accurate measure of the dynamic deflection. Note that the accuracy of the measurement is related
to the height difference between the teeth.
The spherical TNT charges were hanged in air using fisherman’s net and were ignited from the center of the
charge.

Fig. 2. Experimental setup at 2 different scales (S ¼ 4, left, and S ¼ 2, right)

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Spherical TNT
charge

Clamp W Clamp
Target plate R t

Bridge
D Measuring comb

Fig. 3. Schematic drawing of the test rig and the measurement setup.

Fig. 4. Comb-like device for dynamic deflection measurement.

5. Results

5.1. Numerical results

The numerical calculations comprised 4 distinct stages, and the same problem was calculated each time at
different scale factors, namely S ¼ 1, 2, 4, 10, and 0.5.
The first stage deals with the boundary conditions and their influence on the deflection profile of the plate.
Three different boundary conditions were investigated: ‘free’, ‘constrained’ (or clamped), and a characteristic
‘structure’ as shown in Fig. 5. The characteristic structure enables us to analyze the effect of supporting side
walls on the dynamic response of the plate. The objective of this preliminary study was to investigate the
differences between the three dynamic responses and select the most appropriate boundary conditions for the
following calculations.
The second stage involves selection of a simple rate-insensitive bilinear material model and calculation of
the deflections and stresses at different scaling factors. Here the objective is to explore scalability of the
dynamic response with ‘‘ideal’’ parameters.
The third stage introduces rate-sensitivity of the mechanical properties of the steel plate through the
Johnson–Cook model [21]. The objective of this stage is to study the effect of the strain rate sensitivity on the
dynamic response at different scale factors for the problem at hand.
Finally, stage 4 addresses variability of the material properties with the rolled plate thickness as a result of
the manufacturing process. This stage was carried out at three different scale factors. The results of each stage
will now be presented and discussed.
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Fig. 5. Comparison between different boundary conditions: (a) free, (b) constrained; (c) structure.

2.5
Normalized maximum deflection, δ / t

1.5

0.5
Free
Constrained
Structure
0
0 0.5 1 1.5 2 2.5 3 3.5 4
Scaled time, τ·S msec

Fig. 6. Midpoint normalized deflection vs. scaled time for free, constrained, and structure boundary conditions.

5.1.1. Stage 1: influence of the boundary conditions

The effect of the three different boundary conditions on the dynamic response (i.e. deflection–time history)
at the midpoint of the plate is shown in Fig. 6. The following parameters were taken: plate thickness
t ¼ 0:05 m, plate diameter D ¼ 2 m, charge’s weight W ¼ 50 kg TNT, and distance from the plate’s surface to
the center of the charge R ¼ 0:5 m. The RHA class 1 material properties were represented via bilinear model
with the following parameters: yield strength sy ¼ 1000 MPa, Young’s modulus E ¼ 210 GPa, and hardening
plastic modulus E p ¼ 2 GPa. The deflection at the midpoint of the plate is calculated as the difference between
the midpoint and the circumference vertical displacements. Fig. 6 shows the deflection time history of the blast
loaded plate for each type of boundary condition. It can be seen that the peak deflection occurs at scaled times
of the order of 2 ms. The ‘free’ and ‘structure’ types of boundary conditions produce relatively equal peak
deflections while the ‘constrained’ type has a peak deflection which is approximately 20% smaller than the
others. Furthermore, even though the ‘free’ and ‘structure’ boundary conditions are quite similar in their peak
deflection, it can be seen that they yield a different duration of the bulging process, where the deflection in the
‘structure’ boundary condition lasts longer than in the ‘free’ type as a result of the added inertia of the side
walls. From this point further on, we applied the ‘constrained’ boundary conditions only, since they were
found to be closer to the actual experimental boundary conditions.

5.1.2. Stage 2: scaling with bilinear material model

The objective of the second stage is to validate the scale modeling using the combined Hopkinson-Replica
method for ideal material properties of the blast loaded plate, namely bilinear material properties as shown
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1.8

Normalized maximum deflection, δ / t

1.6

1.4

1.2

0.8

0.6
S = 10
0.4 S=4
S=2
0.2 S=1
S = 0.5
0
0 0.5 1 1.5 2 2.5 3 3.5 4
Scaled time, τ·S msec

Fig. 7. Normalized midpoint deflection vs. scaled time at different scales (bilinear material model).

S = 10
S=4
1 S=2
Normalized effective stress, σeff / σy

S=1
S = 0.5
0.8

0.6

0.4

0.2

0
0 0.5 1 1.5 2 2.5 3 3.5 4
Scaled time, τ·S msec

Fig. 8. Normalized stresses vs. scaled time at different scales (bilinear material model).

previously. Fig. 7 shows the normalized midpoint deflection scaled time history at various scaling factors
ranging from S ¼ 0:5 to 10. The plate thickness at scale factor S ¼ 1 is t ¼ 0:05 m, plate diameter D ¼ 2 m,
charge weight W ¼ 50 kg TNT, and the distance from center of charge R ¼ 0:5 m. This is equivalent to
selecting a scaling factor of S ¼ 4 for instance, so that t ¼ 0:0125 m, D ¼ 0:5 m, and R ¼ 0:125 m. In that case
the corresponding charge is scaled down to W ¼ 0:781 kg TNT. This figure shows that all the curves merge
together, showing an excellent scaling of the plates midpoint deflection. It should be mentioned that the same
procedure was applied for different ‘‘measurement’’ points on the plate (Fig. 1) at all scaling factors, and
identical results were obtained. Fig. 8 shows the scaled time history of the normalized stresses at the midpoint
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2
S = 10
1.8 S=4
S=2

Normalized maximum deflection, δ / t

1.6 S=1
S = 0.5
1.4

1.2

0.8

0.6

0.4

0.2

0
0 0.5 1 1.5 2 2.5 3 3.5 4
Scaled time, τ·S msec

Fig. 9. Normalized midpoint deflection vs. scaled time at different scales (Johnson–Cook constitutive model).

of the plate for the corresponding conditions shown above. One should again emphasize that the material is
considered as rate-insensitive in these calculations.

5.1.3. Stage 3: scaling with strain-rate sensitive material model (J– C)

The objective of the third stage is to assess the influence of the material’s strain-rate sensitivity on the scale
modeling of blast loaded plate. Here, the steel plate was modeled using Johnson–Cook model [21] with the
following coefficients: A ¼ 1000 MPa, B ¼ 500 MPa, c ¼ 0:014, n ¼ 0:26, and m ¼ 1. Fig. 9 shows the
normalized midpoint deflection scaled time history for various scaling factors. Here too, it can be seen that all
the curves merge almost perfectly to a single curve, thus showing excellent scaling of the problem. The
numerical calculations show that for S ¼ 1, the strain rate at the maximum deflected elements is of the order
of e_ S¼1  200 s1 , whereas for S ¼ 10, it reaches e_S¼10  1000 s1 . In this range of strain rates, the
Johnson–Cook model [21] predicts a very small change in the yield or flow stress. Therefore, one may conclude
that, given the actual strain rates produced during the dynamic bulging process, strain-rate sensitivity has a
negligible effect on the scalability of the problem at hand, for the range of scaling factors investigated here.
However, this may not always be the case when highly strain rate sensitive materials are considered.

5.1.4. Stage 4: scaling with material properties variability with thickness (J– C)
The objective of this stage is to assess the extent to which a certain variability of the mechanical properties
of the rolled plate, for each plate’s thickness, may affect the validity of the present scaling procedures. To be
more specific, let us consider the RHA class 1 armor steel material. Due to manufacturing limitations its
properties vary with the thickness, as shown in Table 1. For example, a plate of t ¼ 0:01 m thickness has a
yield strength sy ¼ 950 MPa and UTS strength sUTS ¼ 1250 MPa. When the thickness becomes t ¼ 0:04 m the
corresponding material properties are reduced to following values: yield strength sy ¼ 850 MPa and UTS
strength sUTS ¼ 1050 MPa [24]. The corresponding Johnson–Cook coefficients in this table were taken as
follows: A is the yield strength of the material, B is calculated to obtain the UTS strength at the given
maximum elongation, while n, c, and m remain unaffected and can be found in literature [21]. This
investigation was performed at three different scale factors S ¼ 1, 2, and 4 for the same problem as before.
The simulations results are shown in Figs. 10 and 11, where the differences between the dynamic responses at
each of the three scale factors are shown. In order to further study the influence of the material properties,
larger plastic strains were investigated. This was achieved by modeling a thinner plate and a larger charge from
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closer range as follows: at scale factor S ¼ 1 the plate thickness is t ¼ 0:04 m, plate diameter D ¼ 2 m, charge
weight W ¼ 70 kg TNT, and the distance from center of charge R ¼ 0:26 m. Fig. 12 shows the normalized
midpoint deflection scaled time history at the three scaling factors S ¼ 1, 2, and 4. It can be seen that, due to
the difference between the material properties for each material thickness, the resulting curves of each scale
factor diverge from each other to approximately 7% peak deflection, starting with scale factor S ¼ 1 up to
scale factor S ¼ 4. Therefore, scale modeling of the present problem is distorted, so that the deviation should
be taken into account when scaling the results from the model up to the prototype. Fig. 13 shows the scaled

2
S=4
1.8 S=2
S=1
Normalized maximum deflection, δ / t

1.6

1.4

1.2

0.8

0.6

0.4

0.2

0
0 0.5 1 1.5 2 2.5 3 3.5 4
Scaled time, τ·S msec

Fig. 10. Normalized deflection at different scales vs. scaled time (material properties vary with thickness).

S=4
1.2 S=2
S=1
Normalized effective stress, σeff / σy

0.8

0.6

0.4

0.2

0
0 0.5 1 1.5 2 2.5 3 3.5 4
Scaled time, τ·S msec

Fig. 11. Normalized effective stress at different scales vs. scaled time (material properties vary with thickness).
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Normalized maximum deflection, δ / t

7

2
S=4
1 S=2
S=1
0
0 0.5 1 1.5 2 2.5 3 3.5 4
Scaled time, τ·S msec

Fig. 12. Normalized deflection at different scales vs. scaled time (material properties vary with thickness).

S=4
1.2 S=2
S=1
Normalize deffective stress, σeff / σy

0.8

0.6

0.4

0.2

0
0 0.5 1 1.5 2 2.5 3 3.5 4
Scaled time, τ·S msec

Fig. 13. Normalized effective stress at different scales vs. scaled time (material properties vary with thickness).

time history of the normalized stresses at the midpoint of the plate for the corresponding conditions shown in
the last example.

5.2. Experimental results

Experiments were performed at two selected scales, S ¼ 2 and 4. Three pairs of cases were compared
experimentally. First, a case where the structural response is mostly dynamic elastic, while the second and
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Table 2
Experimental and numerical results

S t (m) D (m) W (kg TNT) R (m) d/t Experimental d/t Numerical

2 0.02 1 3.75 0.2 2.70 2.62

4 0.01 0.5 0.468 0.1 2.60 2.59
2 0.02 1 8.75 0.2 5.35 5.24
4 0.01 0.5 1.094 0.1 4.85 4.98
2 0.02 1 8.75 0.13 8.25 8.15
4 0.01 0.5 1.094 0.065 7.45 7.72

Fig. 14. Experimental result at S ¼ 4 (left) and S ¼ 2 (right).

third cases treat increasing plastic strains in the structural dynamic response. Table 2 lists the various
experimental parameters compared to the numerical predictions of the experiments.
For example, let us select the first case where the model has scale factor S ¼ 2, plate thickness t ¼ 0:02 m,
plate diameter D ¼ 1 m, charge weight W ¼ 3:75 kg TNT, and the plate distance from center of charge
R ¼ 0:2 m. This represents a charge of 30 kg TNT for the full scale prototype, where S ¼ 1, t ¼ 0:04 m,
D ¼ 2 m, and R ¼ 0:4 m. In that case the experimental normalized peak deflection d=t ¼ 2:7, while the
numerical matching parameter d=t ¼ 2:62. This is equivalent to selecting a scaling factor of S ¼ 4 for instance,
so that t ¼ 0:01 m, D ¼ 0:5 m, W ¼ 0:468 kg TNT, and R ¼ 0:1 m. In that case the corresponding normalized
peak deflection remains similar d=t ¼ 2:6 for the experimental data and match the numerical parameter
d=t ¼ 2:59.
Consider now the third case where large dynamic plastic deformations are achieved. The model has scale
factor S ¼ 2, plate thickness t ¼ 0:02 m, plate diameter D ¼ 1 m, charge weight W ¼ 8:75 kg TNT (that
represents 70 kg TNT for the full scale prototype, S ¼ 1), and the plate distance from center of charge
R ¼ 0:13 m. In that case the experimental normalized peak deflection d=t ¼ 8:25 while the numerical matching
parameter d=t ¼ 8:15. This is equivalent to selecting a scaling factor of S ¼ 4 for instance, so that t ¼ 0:01 m,
D ¼ 0:5 m, W ¼ 1:094 kg TNT, and R ¼ 0:065 m. In that case the corresponding normalized peak deflection
remains similar d=t ¼ 7:45 for the experimental data and match the numerical parameter d=t ¼ 7:72. Fig. 14
compares the residual plastic deformations for both scaling factors, S ¼ 2 and 4, regarding the third case.
From this table, it appears that the experimental and numerical results are in excellent agreement. It should
be noted that, as long the dynamic response is mostly elastic the varying material properties with thickness
have no effect on the scaling, as expected. However, when the dynamic deformations are mainly plastic, the
scaling is affected, as discussed previously.

6. Discussion

This paper addresses the delicate problem of scaling the dynamic response of circular RHA steel plates
to large bare spherical air blast explosions (high explosive without fragmentation, not confined, and
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non-localized). The scaling employed here is geometrical (replica) for the structure and based on Hopkinson’s
law for the explosive charge. While this concept is not new, it has not been previously applied and verified to
the specific problem of large charges that are initiated from a close-range.
Three different types of boundary conditions were investigated in the preliminary stage. For each case, a
different response was observed, past an initial period. While some similarity was observed between the ‘‘free’’
and ‘‘structure’’ conditions, it was decided to consider further only the ‘‘constrained’’ condition that is more
practical from an experimental point of view. In any case, care should be exercised to replicate the actual
boundary conditions of the prototype for the scaled problem. One should also keep in mind that the present
simulations do not address fracture. If a critical strain or stress is adopted as the failure criterion, its value may
well be reached during the inertial phase for which all boundary conditions yield identical results.
Scaling model is affected when the ‘‘same’’ material has different mechanical properties at different scaling
factors. Thus, scaling up the results should take into account the effect of the varying mechanical properties
with thickness according to the calculated numerical results. Note that the problem treated here is not affected
by strain rate sensitivity of the RHA steel. The observed variability is of the order of 5–10% for scaling factor
S ¼ 2, which should be taken into account for engineering design purposes.
As a final remark, it may seem surprising that plasticity, as a nonlinear phenomenon, scales so well.
However, this observation has already been made, e.g. in the work of Rosenberg et al. [25], who studied the
geometric scaling of long-rod penetration, and observed that geometrical scaling holds for ductile penetrators
and that any deviation from this scaling should be attributed to brittle failure mechanisms at the penetrator’s
head.
To summarize, the excellent agreement observed between the numerical simulations and the actual
explosion experiments provide a powerful engineering tool, when an accurate estimate of the plate’s maximal
deflection is needed. The scaling down technique employed here is deemed to reduce both the design time and
high experimental costs involved in the investigation of these problems.

7. Conclusions

A scaling procedure for the dynamic deformation of constrained steel circular plates subjected to large
explosions of spherical charges has been assessed with respect to experimental results. The comparisons
indicate very good agreement.
The experimental and numerical results of this investigation show that the structural response of the plate
can be efficiently modeled and scaled down using geometrical (replica) together with Hopkinson’s (cube root)
scaling.
Scaling the actual problem appears to be unaffected by the rate-sensitivity of the RHA steel. However, one
should take into account the variability of mechanical properties with plate thickness, with their effect on the
normalized peak deflection. Thus, when transforming the scaled model result up to the prototype, the test
result should be corrected according to the variability of yield stress with plate thickness used in the numerical
calculations.

Acknowledgement

This research was partly supported by Col. Asher Peled Memorial Research (Grant #2005650). The authors
would like to thank Dr. David Touati from IMI, Central Laboratory Division, for his support and useful
suggestions during the study. Useful discussions with Prof. S.R. Bodner are acknowledged.

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