AIR BLAST LOADING OF CELLULAR MEDIA
G.A. Gazonas
U.S. Army Research Laboratory
Weapons and Materials Research Directorate
Aberdeen Proving Ground, Maryland 21005-5069 U.S.A.
[email protected]
J.A. Main
National Institute of Standards and Technology
100 Bureau Drive, Mail Stop 8611
Gaithersburg, Maryland 20899-8611 U.S.A
[email protected]
ABSTRACT
Motivated by recent efforts to mitigate blast loading using energy-absorbing materials, this paper investigates the mechanics of
uniaxial crushing of cellular sandwich plates under air blast loading using analytical and computational modeling. This model is
also applicable to the crushing of cellular media in “blast pendulum” experiments. In the analytical model, the cellular core is
represented using a rigid, perfectly-plastic, locking (RPPL) idealization, a simplification that is often observed in the uniaxial
response of honeycomb material. The front and back faces are modeled as rigid, with pressure loading applied to the front
face and the back face unrestrained. Predictions of this analytical model show excellent agreement with explicit finite element
computations, and the model is used to investigate the influence of the mass distribution between the core and the faces on
the response of the system. Increasing the mass fraction in the front face is found to increase the impulse required for
complete crushing of the cellular core but also to produce undesirable increases in back-face accelerations. Optimal mass
distributions are investigated by maximizing the impulse capacity while limiting the back-face accelerations to a specified level.
It is shown that a larger critical reflected impulse can be sustained in a graded core relative to a homogeneous core, prior to
complete crushing, particularly for structures consisting of large core and face-sheet mass fractions. If one, however, is
attempting to minimize back-face accelerations to a fixed level in order to enhance the survivability of electronic components
subjected to ballistic shock, then the maximum reflected impulse that can be sustained is greater in the homogeneous core
than in a core with a linearly increasing yield stress.
Introduction
Much of the early literature on shock and wave propagation in porous media is found in the geophysical literature, but
Riemann [1] was the first to describe how a compressional wave transforms into a shock wave. A century later, shock and
particle velocity measurements in solids enabled McQueen et al. [2] to develop Hugoniots for minerals thought to comprise the
earth’s interior. This naturally led to the study of shock-induced phase transformations in minerals as the possible cause of the
major body-wave velocity discontinuities in the Earth. The interest in shock wave loading of porous geologic media
accelerated with the advent of underground nuclear testing (e.g. at the Nevada test site [3]), and concurrent computational
modeling of such events [4]. Studies of shock and wave propagation in porous and permeable sedimentary media [5] ensued
during the era of oil exploration. Today, the wide-spread use of porous materials such as cellular foams and honeycombs in
structural applications is attributed to their capacity to absorb energy, particularly at low impact velocities. Surprisingly,
however, the use of porous crushable materials subjected to air blast pressure loading has, in many instances, led to the
enhancement rather than mitigation of blast effects [6, 7, 8].
If a cellular or porous medium is subjected to an intense air-blast, shock waves are transmitted from the air to the solid
medium and can be propagated as jump discontinuities in the field variables, e.g. stress, velocity, et cetera. The propagation
of the jump discontinuities are governed by conservation equations in mass, momentum and energy and are known as the socalled Rankine-Hugoniot equations. Visar traces that depict the propagation of a weak shock wave (elastic longitudinal wave),
and a plastic shock wave in a 4 mm thick target of 1100 aluminum that has been subjected to impact by a 2 mm thick flyer of
the same material are illustrated in Figure 1. Also shown is the visar trace of a simultaneously impacted “porous,” laminated
1100 aluminum target. The laminated aluminum target was formed by building up the target, layer by layer through ultrasonic
consolidation of each aluminum layer; each layer is 0.05 mm thick. The speed of the longitudinal elastic waves measured 6.46
km/s and 6.38 km/s respectively, for the solid and laminated (porous) 1100 specimens; plastic shock wave velocities were
determined from the traces to be 5.155 km/s and 4.623 km/s respectively. We note that the propagation of the plastic shock
waves in such porous media might also be viewed as the propagation of an irreversible phase transformation through the
medium, that is, from a less dense porous phase, to a fully dense phase whereby the propagation velocity of the phase
transformation is governed by the Rankine-Hugoniot jump conditions.
250
solid
porous−layered
spall "pullback"
Free surface velocity (m/s)
200
150
arrival of plastic
shock waves
100
50
arrival of elastic waves
0
−50
−1
0
1
2
3
4
5
Time (μs)
Figure 1. Visar traces depicting important features of shock wave propagation in solid and “porous,” laminated targets
composed of 1100 aluminum.
In order to gain a better understanding of the mechanics of shock wave propagation in cellular or porous media, an analytical
model [9] will be presented that investigates the influence of mass distribution on the uniaxial crushing of cellular material
sandwiched between rigid layers. This model is also applicable to the crushing of cellular media in “blast pendulum”
experiments. In the analytical model, the cellular core is modeled as a continuum using a rigid, perfectly-plastic, locking
idealization. The front and back faces are modeled as rigid masses, with pressure loading applied to the front face, and
unrestrained motion of the back face. Predictions of this analytical model show excellent agreement with explicit finite element
computations [9, 10]. The model is designed to investigate the influence of the mass distribution between the core and the
faces on the system response. An increase in the mass fraction in the front face is found to increase the impulse required for
complete crushing of the cellular core but also produces an undesirable increase in back-face accelerations. Optimal mass
distributions are found that maximize the impulse capacity while limiting the back-face accelerations to a specified level [9].
This paper will also extend previous work [9] to include a class of cellular media that exhibits a continuously nonhomogeneous
(graded) plateau yield stress distribution. The plateau yield stress magnitude could be graded by spatially varying the relative
density of the foam. This will influence the microphysical failure mechanisms, either through buckling, plastic yielding, or
crushing of the cell walls [11]. Alternatively, one could grade the yield strength through the thickness by bonding thin layers
using ultrasonic consolidation methods [12]. We find that grading the yield strength of the cellular medium through the
thickness mitigates some of the deleterious effects of the shock as in other continuously nonhomogeneous media [13]. Finally,
the effects of fluid-structure interaction are likely to be significant, particularly for the case of a very light front face, yet the
analysis of such interaction is beyond the scope of this paper, and is addressed in more detail in [9].
Experimental
Uniaxial stress compression tests were conducted on right circular cylinders of aluminum (5052 alloy) honeycomb (hexagonal
-1
-1
crossection) material, of length 19.7 mm, and diameter 18.9 mm at strain rates of 1 s and 110 s . The diameter of a single
hexagonal cell averages 3.28 mm (face to face), and the wall thickness averages 0.343 mm. The specimens were subjected
to uniaxial compression on a high rate material test system (MTS) 810 which consists of a conventional two-pole press with a
servohydraulically actuated ram that operates from quasi-static velocities to a maximum velocity approaching 12 m/s (Figure
2a); the maximum velocity imparts a maximum strain rate of 1200 s-1 on a 10 mm long specimen. Other essential components
include a bell and cone piston assembly which permit fixed amounts of total specimen strain, a lower nitrogen-spring piston
designed to absorb the impact shock, a 60 kN Kistler force gage mounted in the upper moving piston, and an externally
mounted LVDT for displacement measurement. A Thermotron conditioning oven/refrigerator surrounds both upper and lower
o
o
pistons and permits temperature testing from -45 C to 90 C. Arbitrary load and/or displacement histories can be imparted to
the specimen by computer control. The 60 kN Kistler force gage can be calibrated using a Morehouse Ring Dynamometer
which is certified by the National Institute of Standards and Technology (NIST) to have an uncertainty to within 0.003 % of the
applied load. The uncertainty in the displacement measurement is within 1 percent as determined by comparison with a NIST
certified displacement dial gage. The response of the aluminum honeycomb material is relatively rate-independent over the
strain rate range 1 s-1 to 110 s-1 (Figure 2b). After an initial peak force of about 9 kN (5 % axial strain), the structure unloads,
and progressive collapse occurs at a relative uniform load of 6 kN (nominally 21 MPa across the 18.9 mm specimen diameter).
Complete densification of the material is observed at 70 % engineering strain with a concomitant increase in the load level.
We note that the initial hardening and subsequent unloading of the honeycomb material at 5 % axial strain was not observed
to occur during cyclic loading of the material. See also, for example, the reloading behavior of the material at 1 s-1 when the
actuator stopped momentarily, and the force dropped and reloaded again to levels less than that observed during the initial
load cycle (Figure 2b). The relatively rate-independent, elastic perfectly plastic locking behavior is common in honeycomb
materials under uniaxial loading; we have simplified this behavior in our analytical model described in the next section (Figure
3b) by assuming that the initial behavior is rigid, although for the computational finite element model, we model this initial
rigidity using a Young’s modulus that is ten times larger than the actual elastic modulus for aluminum.
(a)
Force (kN)
(b)
14
12
10
8
6
4
2
0
110/s
1.0/s
0
20
40
60
80
100
Strain (%)
Figure 2. (a) MTS 810 servohydraulic test apparatus showing exploded view of upper bell and impact cone assembly, (b)
Uniaxial compression of aluminum honeycomb material showing rate independent response over the strain rate range of 1 s-1
to 110 s-1.
Analytical model
Motivated by these observations, an analytical model is presented that investigates the influence of mass distribution on the
uniaxial crushing of cellular material sandwiched between rigid layers. The cellular core material is represented by the
simplified stress-strain relationship shown in Figure 3b originally proposed by Reid and Peng [14] for modeling the crushing of
wood, and subsequently applied to cellular metals in a number of studies (e.g., [7,9,15]). Arbitrary masses of the front and
back faces are permitted, and a pressure pulse p(t) is applied to the front face with the back face unrestrained. This sandwich
model is a generalization of that in [7], which considered a fixed back face, and of that in [15], which considered front and back
faces of equal mass with blast loading represented by an initial velocity imparted to the front face.
A strip of sandwich panel with unit cross-sectional area is considered, with total mass given by m = m1 + ρ 0 l 0 + m2 , where ρ 0
and l 0 are the uncompressed density and thickness of the cellular core, and m1 and m2 are the areal densities of the front
and back faces. The acceleration of the center of mass, denoted u&&G , follows directly from application of Newton’s second law
to the strip:
p(t ) = m u&&G
(1)
Provided the applied pressure is sufficiently high, densification of the cellular core commences at the front face, and a
densification front propagates through the core. By conservation of mass, the density of the compressed core material is
ρ0 /(1 − ε D ) . According to the simplified model of Figure 3b, the compressed core material moves as a rigid body with the
same velocity as the front face, denoted u&1 , while the uncompressed core material moves as a rigid body with the velocity of
the back face, u&2 . The stress just ahead of the densification front is σ p , and application of Newton’s second law to the
material ahead of the densification front then yields the following equation:
σ p = ( ρ0 x + m2 )u&&2
(2)
where x denotes the thickness of the uncompressed core material, and the thickness of the densification front itself is assumed
to be negligible. By forming and differentiating an expression for xG , the distance of the center of mass from the back face, it
follows that
&&
xG = (ε D / m) {[ m1 + ρ 0 (l 0 − x) ] &&
x − ρ 0 x& 2 }
(3)
xG to yield the following nonlinear ordinary differential
Eqs. (1) - (3) can then be combined through the relation u&&2 = u&&G + &&
equation for x:
−ε D [ m1 + ρ0 (l 0 − x)] &&
x + ε D ρ 0 x& 2 = p (t ) − σ p m / ( ρ 0 x + m2 )
(4)
Eq. (4) can be integrated numerically with initial conditions x(0) = l 0 and x& (0) = 0 . A triangular pressure pulse is considered,
as shown in Figure 3c, with total impulse denoted i0 . The following symbols are introduced to denote the dimensionless peak
pressure and total impulse:
P0 =
p0
σp
; I0 =
i0
ρ0
m σ pε D
(5)
The following symbols denote the dimensionless mass fractions in the core and in the front and back faces:
η0 = ρ0 l 0 / m ; η1 = m1 / m ; η2 = m2 / m
(a)
u2
u1
p( t )
m1
ρ0
1− εD
(b) σ
l
(6)
(c)
p
x
G
ρ0
uG
m2
p0
σP
xG
i0 = 12 p0t0
εD ε
t0
t
Figure 3. Analytical model definition: (a) Strip of sandwich panel with partially compacted core; (b) engineering stress-strain
relationship for cellular core material (RPPL idealization); (c) triangular pressure pulse applied to front face.
Table 1. Parameters of computational simulations.
Case
η0
η1
η2
P0
I0
blast pendulum
0.0125
0.0125
0.975
10
0.015
sandwich plate
0.5
0.25
0.25
10
1
Comparison with Computations
The predictions of the analytical model are compared with explicit finite element computations using AUTODYN 2-D [10]. In
the computations, the cellular core was represented by a single row of solid elements with total thickness l 0 = 5 cm, using the
3
crushable foam material model, with ρ 0 = 250 kg/m , σ p = 1 MPa, and ε D = 0.7. A large elastic modulus of E = 700 GPa
was used to represent the “rigid” portions of the idealized core stress-strain curve shown in Figure 3b, and Poisson’s ratio was
set to zero. Both linear and quadratic viscosity coefficients were set equal to 0.003 in order to “correctly” capture the shock
front propagation through the core which was modeled with 50 finite elements through-the-thickness; 150 finite elements gave
no appreciable improvement in the results shown in Figure 4. The front- and back-face masses were modeled by relatively
rigid media with “large” elastic moduli, and two different mass distributions were considered, as indicated in Table 1. The
“pendulum” case corresponds to the blast pendulum experiments of [7], with the large back-face mass representing the
pendulum. The “sandwich” case corresponds to the sandwich plates of [15] and [16], with equal front-face and back-face
masses. A peak pressure, p0 = 10 MPa (Figure 3c), was applied to both the “sandwich” and “pendulum” structures, for
durations of 0.265 ms (io = 1323 Pa · s) and 0.159 ms (io = 794 Pa · s), respectively. Computational results are compared with
predictions of the analytical model (Figure 4), and good agreement is observed, although prior LS-DYNA computational results
were virtually indistinguishable from the analytical results [9]. Results are plotted against dimensionless time τ = (σ p / i0 )t . The
dimensionless velocities in Figure 4 are defined as v1 = u&1 / v∞ , and v2 = u&2 / v∞ , where v∞ = i0 / m is the final velocity of the
center of mass. Due to the small mass of the front face, much larger dimensionless front-face velocities are observed in the
“pendulum” case, despite the much smaller dimensionless impulse I0 in this case, as shown in Table 1.
Sandwich Structure
Pendulum Structure
2
60
(b)
(a)
v̄1
1.8
v̄1
1.6
dimensionless velocity
dimensionless velocity
50
40
30
20
AUTODYN 2−D
ANALYTICAL
10
0
0.2
0.4
1.2
1
0.8
0.6
v̄2
0.4
0.2
v̄2
0
1.4
0.6
0.8
dimensionless time τ
1
1.2
1.4
0
AUTODYN 2−D
ANALYTICAL
0
0.1
0.2
0.3
0.4
0.5
dimensionless time τ
0.6
0.7
Figure 4. Comparison of AUTODYN 2-D computations ( - - - ) with predictions of analytical model (-○-): Dimensionless frontface and back-face velocities for (a) “pendulum” case; (b) “sandwich” case.
Influence of Mass Distribution
Figure 5a shows contours of the critical dimensionless impulse I0 for which complete densification of the core is first achieved.
These contours correspond to the limiting case of a Dirac delta impulse ( P0 → ∞ ) and were obtained by numerical solution of
Eq. (4). Figure 5a shows that increasing the mass fraction in the core, and in the front face, increases the impulse capacity of
the sandwich system. However, Figure 5b shows that increasing the mass fraction in the core, and in the front face, also leads
to increased back-face accelerations, thus sacrificing a protective function of the cellular core. The dimensionless back-face
accelerations are defined as a2 = (m / σ P )u&&2 . It follows from Eq. (2) that the peak back-face accelerations occur at the instant
of complete compaction (x = 0), for which u&&2 = σ P / m2 or a2 = 1/ η2 . A design optimization problem can be posed by seeking
to maximize the impulse I0 that can be sustained while limiting the back-face accelerations to a specified level. Figure 5b
shows a contour plot of the maximum impulse I0 that can be sustained with accelerations limited to a2 = 5 . The solid curve
(see arrow) that intersects the minima of the impulse contours in Figure 5b corresponds to a2 = 1/ η2 = 5 . Below this curve,
the values of maximum impulse correspond to complete densification of the core and are the same as in Figure 5a. Above this
Critical reflected impulse (homogeneous core)
a)
0.9
1.272
0.7
0.7
1.5
0.6
0
1.2
0.5
1
0.4
0.6
0.3
1.27
0.8
Core mass fraction
η
0
Core mass fraction
η
0.9
2
0.8
0.2
0.1
Maximum reflected impulse with back−face accelerations limited to 5
b)
3.2
2.8
1.25
0.6
1.2
0.5
1
0.4
0.6
0.3
0.4
0.2
0.2
0.1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Face−sheet mass fraction
η /(η +η )
1
1
2
0.8
0.9
0.2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Face−sheet mass fraction
η /(η +η )
1
1
2
Figure 5. Contours with varying mass distribution for a homogenous core material: (a) Critical dimensionless impulse required
for complete compaction of the core for Dirac delta impulse ( P0 → ∞ ) loading, (b) contours of maximum dimensionless
impulse with back-face accelerations limited to the dimensionless value of 5.
curve, a2 > 5 at complete compaction, so only partial compaction is permitted and the values of maximum impulse are less
than in Figure 5a). In the lower right-hand corner of Figure 5b, defined by (η0 + η2 ) −1 > 5 , a2 > 5 at initiation of compaction, so
the maximum allowable impulse is zero. It is evident in Figure 5b that for a given mass fraction in a core η0 with a
homogeneous spatial distribution of yield stress, the allowable impulse is maximized along the solid curve, i.e., by adjusting
the mass distribution so that the acceleration at complete compaction equals the allowable value.
Influence of Functionally Graded Yield Strength
Here we illustrate a means for modifying the critical reflected impulse the structure can sustain using a functionally graded core
material. For example, if the plateau yield stress of the core material is graded linearly, and increases monotonically from the
front face to the back face, while maintaining an average yield stress equal to that in the homogeneous core, i.e.,
σ gr ( x) = σ p (1 − β ( x / l 0 − 0.5))
(7)
we see that for large core and face-sheet mass fractions the magnitude of critical reflected impulse prior to complete
densification of the core is increased relative to the homogeneous core; compare for example the magnitude of the impulse
contour values, 2.8, in the upper right-hand regions of Figure 5a for the homogenous core with contour values of impulse, 3.2,
in Figure 6a for the linearly graded core. Alternatively, if one is attempting to limit back-face accelerations to a fixed level, in
order to enhance the survivability of electronic components subjected to ballistic shock, then the maximum reflected impulse
that can be sustained is greater in the homogeneous core than in the graded core; compare for example Figure 5b for the
homogenous core with Figure 6b for the linearly graded core. As a consequence of the graded yield stress, the dimensionless
acceleration at complete compaction is given by a2 = (1 + β / 2) / η2 , and with β = 2 , the solid curve that intersects the
minima in the impulse contours in Figure 6b (see arrow) corresponds to a dimensionless acceleration of a2 = 2 / η2 = 5 . Below
this curve, the values of maximum impulse correspond to complete compaction of the core and are the same as in Figure 6a.
Above this curve, a2 > 5 at complete compaction, so only partial compaction is permitted and the values of maximum impulse
are less than in Figure 6a. We are currently investigating extending the analytical model to consider a linearly decreasing
plateau yield stress distribution, but this problem is more complex since it may involve generating shock waves that initiate at
an intermediate position within the core. It is even conceivable that shock waves may travel forward towards the front-face of
the structure which is subjected to the blast loading. Although this may at first seem physically counterintuitive, it is interesting
to note that such behavior has been observed to occur during the progressive collapse of buildings [17], in which a “crush
down” phase is followed by a “crush up” phase.
a)
Critical reflected impulse (graded core)
0.9
b)
Maximum reflected impulse with back−face accelerations limited to 5
3.2
Core mass fraction
η
1.5
0.6
1.2
0.5
0
0
Core mass fraction
η
0.8
2
0.7
1
0.4
0.3
0.6
0.2
0.1
1.05
0.9
2.8
0.8
1
0.7
0.6
0.5
0.9
0.4
0.8
0.6
0.3
0.4
0.2
0.2
0.1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Face−sheet mass fraction
η /(η +η )
1
1
2
0.8
0.9
0.2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Face−sheet mass fraction
η /(η +η )
1
1
2
Figure 6. Contours with varying mass distribution for a functionally graded, β = 2, core material: (a) Critical dimensionless
impulse required for complete compaction of the core for Dirac delta impulse ( P0 → ∞ ) loading, (b) contours of maximum
dimensionless impulse with back-face accelerations limited to the dimensionless value of 5.
Acknowledgments
We thank Dr. Dan Casem for providing the visar traces of the shock loaded 1100 aluminum and Mr. Michael Leadore for
providing data for the uniaxial compression tests on the aluminum honeycomb material. Certain trade names or company
products are mentioned in the text to specify adequately the procedure used. Such identification does not imply
recommendation or endorsement by NIST or ARL, nor does it imply that the product is the best available for the purpose.
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