The Resistance of Clamped

Sandwich Beams to Shock

Loading
N. A. Fleck1
e-mail: naf1@eng.cam.ac.uk A systematic design procedure has been developed for analyzing the blast resistance of
clamped sandwich beams. The structural response of the sandwich beam is split into three
V. S. Deshpande sequential steps: stage I is the one-dimensional fluid-structure interaction problem during
the blast loading event, and results in a uniform velocity of the outer face sheet; during
Engineering Department, stage II the core crushes and the velocities of the faces and core become equalized by
Cambridge University, momentum sharing; stage III is the retardation phase over which the beam is brought to
Trumpington Street, rest by plastic bending and stretching. The third-stage analytical procedure is used to
Cambridge, CB2 1PZ, UK obtain the dynamic response of a clamped sandwich beam to an imposed impulse. Per-
formance charts for a wide range of sandwich core topologies are constructed for both air
and water blast, with the monolithic beam taken as the reference case. These performance
charts are used to determine the optimal geometry to maximize blast resistance for a
given mass of sandwich beam. For the case of water blast, an order of magnitude im-
provement in blast resistance is achieved by employing sandwich construction, with the
diamond-celled core providing the best blast performance. However, in air blast, sand-
wich construction gives only a moderate gain in blast resistance compared to monolithic
construction. 关DOI: 10.1115/1.1629109兴

1 Introduction modeled the core topologies explicitly but ignored the fluid-
A major consideration in the design of military vehicles 共such structure interaction; a prescribed impulse was applied to the outer
as ships and aircraft兲 is their resistance to air and water blast. face of the sandwich beam and was applied uniformly to the
Early work 共at the time of World War II兲 focused on monolithic monolithic beam. A limited number of FE calculations were per-
plates, and involved measurement of blast resistance by full scale formed to identify near-optimal sandwich configurations, and the
testing for a limited range of materials and geometries. Simple superior blast resistance of sandwich beams compared to that of
analytical models were also developed, such as the one- monolithic beams was demonstrated.
dimensional fluid-structure interaction model of Taylor 关1兴. Review of the Characteristics of a Water Blast. The main
Over the last decade a number of new core topologies for sand-
characteristics of a shock wave resulting from an underwater ex-
wich panels have emerged, showing structural advantage over
plosion are well established due to a combination of detailed
monolithic construction for quasi-static loadings. These include
metallic foams, 关2兴, lattice materials of pyramidal and tetrahedral large-scale experiments and modeling over the past 60 years. Use-
arrangement, 关3兴, woven material, 关4兴, and egg-box, 关5兴. The cur- ful summaries of the main phenomena are provided by Cole 关7兴
rent study is an attempt to extend and to synthesize analytical and Swisdak 关8兴, and are repeated briefly here in order to underpin
models for the dynamic response of clamped beams in order to the current study.
optimize the blast resistance of clamped sandwich beams. Explicit The underwater detonation of a high explosive charge converts
comparisons are made between the performance of competing the solid explosive material into gaseous reaction products 共on a
core concepts. time scale, t, of microseconds兲. The reaction products are at an
The clamped sandwich beams, as sketched in Fig. 1, is repre- enormous pressure 共on the order of GPa兲, and this pressure is
sentative of that used in the design of commercial and military transmitted to the surrounding water by the propagation of a
vehicles: For example, the outermost structure on a ship com- spherical shock wave at approximately sonic speed. Consider the
prizes plates welded to an array of stiffeners. While it is appreci- response of a representative fluid element at a radial distance r
ated that the precise dynamic response of plates is different from from the explosion. Upon arrival of the primary shock wave, the
that explored here for beams, the qualitative details will be simi- pressure rises to a peak value p o almost instantaneously. Subse-
lar, and major simplifications arise from the fact that simple ana- quently, the pressure decreases at a nearly exponential rate, with a
lytical formulas can be derived for the beam. time constant ␪ on the order of milliseconds, and is given by
In a parallel study, Xue and Hutchinson 关6兴 have compared the
p(t)⫽p o exp(⫺t/␪). The magnitude of the shock wave peak pres-
blast resistance of clamped sandwich beams to that of monolithic
sure and decay constant depend upon the mass and type of explo-
beams of the same mass via three-dimensional finite element 共FE兲
simulations. In these FE calculations, Xue and Hutchinson 关6兴 sive material and the distance r. After the primary shock wave has
passed, subsequent secondary shocks are experienced, due to the
1
To whom correspondence should be addressed. damped oscillation of the gas bubble which contains the explosive
Contributed by the Applied Mechanics Division of THE AMERICAN SOCIETY OF reaction products. However, these secondary shock waves have
MECHANICAL ENGINEERS for publication in the ASME JOURNAL OF APPLIED ME- much smaller peak pressures, and are usually much less damaging
CHANICS. Manuscript received by the ASME Applied Mechanics Division, May 19,
2002; final revision, July 10, 2003. Associate Editor: R. M. McMeeking, Discussion than the primary shock to a structure in the vicinity of the explo-
on the paper should be addressed to the Editor, Prof. Robert M. McMeeking, Depart- sion than the primary shock.
ment of Mechanical and Environmental Engineering University of California—Santa Experimental data 共and physical models兲 support the use of
Barbara, Santa Barbara, CA 93106-5070, and will be accepted until four months after
final publication of the paper itself in the ASME JOURNAL OF APPLIED MECHAN- simple power-law scaling relations between the mass m of explo-
ICS. sive, the separation r between explosion and point of observation,

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 Fig. 1 Geometry of the sandwich beam

and the resulting shock wave characteristics, p o and ␪. For ex-

ample, for an underwater TNT explosion, the peak pressure is
taken from Table 2 of Swisdak 关8兴 as

p o ⫽52.4 冉 冊
m 1/3
r
1.13
MPa, (1)

where m is in kilograms and r is in meters. Also, the time constant

␪ is

␪ ⫽0.084m 1/3 冉 冊
m 1/3
r
⫺0.23
ms. (2)

These relations have been validated for the domain of m and r

such that p o lies in the range 3–140 MPa, see Swisdak 关8兴 for
further details. Similar scaling relations have been obtained for
other high explosives, and the coefficients in the above relations
hold to reasonable accuracy for them also.
Next consider the case of a blast wave in air due to the deto-
nation of a high explosive. Again, a primary shock wave travels at
near sonic speed, with an exponential pressure-time history at any
fixed location from the explosive. The time constant for the pulse
␪ is similar in magnitude to that in water, but the peak pressure is
an order of magnitude lower 共see Ashby et al. 关2兴 for a recent
discussion, building upon the work of Smith and Hetherington
关9兴兲.
Fig. 2 Sketches of the sandwich core topologies; „a… pyrami-
Scope and Motivation of the Study. The main objective of dal core, „b… diamond-celled core, „c… corrugated core, „d…
this study is to develop analytical formulas for characterizing the hexagonal-honeycomb core, and „e… square-honeycomb core
structural response of a sandwich beam subjected to blast loading
in water or in air. These formulas are of direct practical use for
designing laboratory-scale and industrial-scale blast-resistant
sandwich beams, including the choice of face sheet and core. buckling strength of the struts to exceed their yield strength, the
First, the relevant mechanical response of candidate core to- out-of-plane compressive strength of the pyramidal core also
pologies is reviewed. Second, the dynamic structural response of a scales linearly with ¯␳ . A detailed discussion on the mechanical
clamped sandwich beam is analyzed; it is argued that the response properties of lattice materials such as pyramidal cores has been
can be separated into three distinct stages. Stage I is the response given previously by Deshpande and Fleck 关3兴. For example, the
of the front face sheet to the primary shock wave, including the normal compressive strength ␴ nY of the pyramidal core with the
effects of fluid-structure interaction. Crushing of the core occurs struts making an angle ␻ ⫽45° with the face sheets is
in stage II. And in stage III the sandwich beam is brought to rest
96& ⑀ Y

冦
by plastic bending and stretching. Third, performance charts for a
0.5¯␳ set by yield, if ¯␳ ⬎
wide range of sandwich core topologies are constructed for both
␴ nY ␲2
air and water blast, with the monolithic beam taken as the refer- ⫽
ence case. These performance charts are used to determine the ␴Y ␲2
optimal geometry to maximize blast resistance for a given mass of ¯␳ 2 , set by elastic buckling, otherwise,
96& ⑀ Y
sandwich beam. (3)
where ␴ Y and ⑀ Y are the uniaxial yield strength and strain of the
2 Review of Core Topologies solid material from which the pyramidal core is made. Here we
In recent years a number of micro-architectured materials have have assumed that the core struts are pin-jointed to the face sheets
been developed for use as the cores of sandwich beams and pan- in order to get a conservative estimate of the elastic buckling
els. Here we briefly review the properties of the following candi- strength. The in-plane strength of the pyramidal core in the length
date cores for application in blast-resistant construction: pyrami- direction of the sandwich beam is governed by the bending
dal cores, diamond-celled lattice materials, metal foams, strength of the nodes. Consequently, the in-plane strength scales
hexagonal-honeycombs and square-honeycombs. as ¯␳ 3/2 and at the low relative densities for which these pyramidal
Pyramidal cores, as shown schematically in Fig. 2共a兲, are fab- cores find application, this strength is negligible, ␴ lY ⫽0.
ricated from sheet-metal by punching a square pattern and then by Diamond-celled lattice materials have the geometry shown in
alternately folding the sheet to produce a corrugated pattern. The Fig. 2共b兲, and have recently been proposed as cores of sandwich
core is then bonded to the solid faces by brazing. The pyramidal beams. These lattice materials can be manufactured either by
core has an out-of-plane effective modulus 共and longitudinal shear brazing together wire meshes, 关4兴, or slotting together sheet metal.
modulus兲 which scale linearly with the relative density ¯␳ of the With the diamond-like cells aligned along the longitudinal axis of
core. Provided the struts are sufficiently stocky for the elastic the beam as shown in Fig. 2共b兲, these materials provide high

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 strengths in both the normal and longitudinal directions of the This core is 100% efficient in carrying load in both these direc-
beam. Typically diamond-cells have a semi-angle ␻ ⫽45° and the tions. It is not clear whether such a core is physically realizable:

再
core has a normal compressive strength The diamond-celled core with the diamond cells aligned along the
longitudinal axis of the beam or a square-honeycomb come clos-
4 冑3 ⑀ Y est to this ‘‘ideal’’ performance.
0.5¯␳ , set by yield, if ¯␳ ⬎ ;
␴ nY ␲
⫽ 3 Analytical Models for the Structural Response of a
␴Y ␲2 3
¯␳ , set by elastic buckling, otherwise, Clamped Sandwich Beam to Blast Loading
96⑀ Y
(4a) For the sandwich beam, the structural response is split into a
sequence of three stages: stage I is the one-dimensional fluid-
while the longitudinal strength is given by structure interaction problem during the blast event, and results in
a uniform velocity being imposed on the outer face sheet; stage II
␴ lY
⫽¯␳ . (4b) is the phase of core crush, during which the velocities of the faces
␴Y and core equalize by momentum transfer; stage III is the retarda-
tion phase during which the beam is brought to rest by plastic
Note that the diamond-celled core has identical strength-density
bending and stretching. This analysis is used to calculate the trans-
relations to the single layered corrugated core shown in Fig. 2共c兲.
verse displacement 共and longitudinal tensile strain accumulated兲
However, unlike in a corrugated core, the size of the diamond-
of selected sandwich beams as a function of the magnitude of
cells can be varied independently from the sandwich beam core
blast loading.
thickness and hence made as small as required to prevent wrin-
kling of the sandwich face sheets. 3.1 Order-of-Magnitude Estimate for the Time Scale of
Metal foams are random cellular solids with a highly imperfect Each Stage of the Dynamic Response. The justification for
microstructure. In most cases they are close to isotropic in elasto- splitting the analysis into three distinct stages is the observation
plastic properties. The connectivity of neighboring cell edges is that the time periods for the three phases differ significantly. The
sufficiently small for the cell walls to bend under all macroscopic duration of the primary shock for a typical blast wave in air or
stress states, Ashby et al. 关2兴. Consequently, the modulus scales water due to the detonation of an explosive is of the order of 0.1
quadratically with relative density ¯␳ , while the macroscopic yield ms. In contrast, the period for core crush is approximately 0.4 ms,
strength scales with ¯␳ 3/2 according to, 关2兴, argued as follows. Suppose that a blast wave in water provides an
␴ nY ␴ lY impulse of 104 Nsm⫺2 to a steel sandwich structure, with a 10
⫽ ⫽0.3¯␳ 3/2. (5) mm thick face sheet. Then, the front face acquires an initial ve-
␴Y ␴Y locity v o of 127 ms⫺1 . On taking the core to have a thickness of
Hexagonal-honeycombs are extensively used as cores of sand- c⫽100 mm and a densification strain ⑀ D ⫽0.5, the compression
wich beams in the configuration sketched in Fig. 2共d兲, i.e., with phase lasts for ⑀ D c/ v o ⫽0.39 ms. In contrast, the structural re-
the out-of-plane direction of the honeycomb aligned along the sponse time is on the order of 25 ms: this can be demonstrated by
transverse direction of the beam. Thus, neglecting the elastic considering the dynamic response of a stretched rigid-ideally plas-
buckling of the cell walls we take tic string. Consider a string of length 2L, gripped at each end,
made from a material of density ␳ f and uniaxial yield strength
␴ nY ␴ f Y . Then, the transverse equation of motion for the membrane
⫽¯␳ . (6) state is
␴Y
⳵ 2w
On the other hand, in the longitudinal direction of the beam, ␳ f ẅ⫺ ␴ f Y ⫽0, (9)
hexagonal-honeycomb cores deform by the formation of plastic ⳵x2
hinges at the nodes which results in a negligible strength. Thus, in where w(x,t) is the transverse displacement, the overdot denotes
practical applications it is reasonable to assume ␴ lY ⫽0 for these differentiation with respect to time t, and x is the axial coordinate
honeycombs. from one end of the string. For illustrative purposes, assume the
Square-honeycombs as sketched in Fig. 2共e兲 can be manufac- string is given an initial velocity profile ẇ(t⫽0)
tured by slotting together sheet metal. With the square cells ⫽ẇ o sin(␲x/2L). Then, the solution of 共9兲 is

冉 冑 冊
aligned parallel to the longitudinal axis of the beam as sketched in
Fig. 2共e兲, the square-honeycomb core provides high strength in
both the normal and longitudinal directions. Neglecting elastic w⫽
2ẇ o L
␲
冑 ␳f
␴fY
sin
␲
2L
␴fY
␳f
t sin
␲x
2L
. (10)
buckling of the cell walls in the normal direction, the normal and
longitudinal strength of the square-honeycomb are given by The string attains its maximum displacement and comes to rest
after a time
␴ nY
␴Y
⫽¯␳ , and, (7a)
T⫽L 冑 ␳f
␴fY
. (11)
␴ lY Now substitute representative values for the case of a steel ship
⫽0.5¯␳ , (7b)
␴Y hull: L⫽5 m, ␳ f ⫽7850 kgm⫺3 , and ␴ f Y ⫽300 MPa, gives T
⫽25 ms, as used above.
respectively.
All the cores discussed above have their relative advantages 3.2 Stage I: One-Dimensional Fluid-Structure Interaction
and disadvantages with regards to properties, ease of manufacture Model. Consider the simplified but conservative idealisation of
and cost. For the purposes of judging the relative performance of a plane wave impinging normally and uniformly upon an infinite
the cores described above we define an ‘‘ideal’’ core. The ‘‘ideal’’ sandwich plate. For most practical geometries and blast events,
core has optimal strengths in the normal and longitudinal direc- the time scale of the blast is sufficiently brief for the front face of
tions given by a sandwich panel to behave as a rigid plate of mass per unit area
m f . We adopt the one-dimensional analysis of Taylor 关1兴, and
␴ nY ␴ lY
⫽ ⫽¯␳ . (8) consider an incoming wave in the fluid of density ␳ w , traveling
␴Y ␴Y with a constant velocity c w in the direction of increasing x mea-

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 sured perpendicular to the sandwich panel. The origin is taken at
the front face of the sandwich panel, and the transverse deflection
of the face is written as w(t) in terms of time, t. Then, the pres-
sure profile for the incoming wave can be taken as
p I 共 x,t 兲 ⫽p o e ⫺ 共 t⫺x/c w 兲 / ␪ , (12)
upon making the usual assumption of a blast wave of exponential
shape and time constant ␪ 共on the order of 0.1 ms, as discussed
above兲. The magnitude of the peak pressure p o is typically in the
range 10–100 MPa, and far exceeds the static collapse pressure
for the sandwich plate 共typically on the order of 1 MPa兲.
If the front face were rigid and fixed in space, the reflected
wave would read
p r1 共 x,t 兲 ⫽p o e ⫺ 共 t⫹x/c w 兲 / ␪ , (13)
corresponding to perfect reflection of the wave, traveling in the
⫺x direction. But the front face sheet is not fixed: it accelerates as
a rigid body with a mass per unit area m f , and moves with a Fig. 3 The ratio of the impulse transmitted to the struc-
velocity ẇ(t). Consequently, the fluid elements adjacent to the ture I trans , and the impulse transmitted to a fixed rigid
structure 2 p o ␪ , as a function of the fluid-structure interaction
front face possess the common velocity ẇ(t), and a rarefaction parameter ␺
wave p r2 , of magnitude

p r2 共 x,t 兲 ⫽⫺ ␳ w c w ẇ t⫹ 冉 冊 x
cw
, (14)
impulse decreases substantially with increasing ␺. It is instructive
is radiated from the front face. Thus, the net water pressure p(x,t) to substitute some typical values for air and water blast into rela-
due to the incoming and reflected waves is tions 共19兲 and 共20b兲 in order to assess the knock down in trans-
p 共 x,t 兲 ⫽p I ⫹p r1 ⫹p r2 ⫽p o 关 e ⫺ 共 t⫺x/c w 兲 / ␪ ⫹e ⫺ 共 t⫹x/c w 兲 / ␪ 兴 mitted impulse and the magnitude of the cavitation time in rela-
tion to the blast time constant ␪ due to the fluid-structure

冉 冊
⫺ ␳ w c w ẇ t⫹
x
cw
. (15)
interaction. For the case of an air blast, we take ␳ w
⫽1.24 kgm⫺3 , c w ⫽330 ms⫺1 , ␪ ⫽0.1 ms, and m f ⫽78 kgm⫺2
for a 10 mm thick steel plate. Hence, we find that ␺ ⫽0.052,
The front face of the sandwich panel 共at x⫽0) is accelerated by ␶ c / ␪ ⫽3.1 and I trans /I⬇0.85. In contrast, a water blast, we take
the net pressure acting on it, giving the governing ordinary differ- ␳ w ⫽1000 kgm⫺3 , c w ⫽1400 ms⫺1 , ␪ ⫽0.1 ms, m f ⫽78 kgm⫺2 ;
ential equation for face motion as this implies the values ␺ ⫽1.79, ␶ c / ␪ ⫽0.74 and I trans /I⫽0.267.
m f ẅ⫹ ␳ w c w ẇ⫽2p o e ⫺t/ ␪ . (16) We conclude that a significant reduction in transferred impulse
can be achieved by employing a light face sheet for the case of
Upon imposing the initial conditions w(0)⫽ẇ(0)⫽0, and defin- water blast, while for air blast the large jump in acoustic imped-
ing the nondimensional measure ␺ ⬅ ␳ w c w ␪ /m f , the solution of ance between air and the solid face sheet implies that all practical
共16兲 is designs of solid face sheet behave essentially as a fixed, rigid face
2p o ␪ 2 with full transmission of the blast impulse. We anticipate that
w共 t 兲⫽ 关共 ␺ ⫺1 兲 ⫹e ⫺ ␺ t/ ␪ ⫺ ␺ e ⫺t/ ␪ 兴 , (17) sandwich panels with light faces can be designed to ensure the
m f 共 ␺ ⫺1 兲 ␺ reduced transmission of impulse from an incoming water blast
and the pressure distribution follows immediately via 共15兲. In par- wave.
ticular, the pressure on the front face is In summary, the first phase of the analysis comprises the accel-
eration of the front face to a velocity v o by the incoming 共and
2p o ␺ ⫺t/ ␪ ⫺ ␺ t/ ␪ reflected兲 primary shock wave. The core and back face of the
p 共 t,x⫽0 兲 ⫽2p o e ⫺t/ ␪ ⫺ 关e ⫺e 兴. (18)
␺ ⫺1 sandwich beam remain stationary during this initial stage. It is
instructive to obtain order of magnitude estimates for the initial
For the case of a liquid containing dissolved gases, the pressure velocity of the front face, and its deflection at time t⫽ ␶ c . For an
loading on the front face ceases and the liquid cavitates when impulse of magnitude 103 Nsm⫺2 in air, and 104 Nsm⫺2 in water,
p(t,x⫽0)→0, thereby defining the cavitation time ␶ c . Substitu-
the acquired velocity of the front face is approximately 13 ms⫺1
tion of this condition into 共18兲 provides the simple relation
for the air blast, and 34 ms⫺1 for the water blast 共steel face sheet,
␶c 1 of thickness 10mm兲. Relation 共17兲 reveals that the lateral deflec-
⫽ ln ␺ , (19) tion of the front face is 2.5 mm for the air blast and 1.83 mm for
␪ ␺ ⫺1
the water blast. It is expected that sandwich beams for ship appli-
and the impulse conveyed to the face follows from 共17兲 as cation will be of core thickness c of order 0.1–1.0 m, and so the
I trans⫽ ␨ I (20a) degree of core compression during the initial phase of blast load-
ing is negligible.
where Taylor 关1兴 has modeled the influence of structural support to the
dynamic response of the face sheet by adding the term kw to 共16兲,
␨ ⬅ ␺ ⫺ ␺ / ␺ ⫺1 , (20b)
corresponding to a uniformly distributed restraining force of mag-
and I is the maximum achievable impulse given by nitude kw giving

I⫽ 冕 0
⬁
2p o e ⫺t/ ␪ dt⫽2p o ␪ . (21) m f ẅ⫹ ␳ w c w ẇ⫹kw⫽2p o e ⫺t/ ␪ . (22)
The physical interpretation is that k denotes the structural stiffness
This maximum impulse is only realized for the case of a station- due to an array of supports between the face sheet and the under-
ary rigid front face. The ratio I trans /I is plotted as a function of the lying, motionless structure. By solving 共22兲, and considering rep-
fluid-structure interaction parameter ␺ in Fig. 3; the transmitted resentative values for k for the case of a steel plate on a ship

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 superstructure, Taylor demonstrated that the stiffness term can be
neglected with little attendant loss of accuracy. The main objec-
tive of the current study is to compare the relative performance of
various sandwich panel configurations, and so the simplified
analysis is adequate for our purposes.

3.3 Stage II: One-Dimensional Model of Core Compres-

sion Phase. In the second phase of motion it is envisaged that
the core is crushed by the advancing outer face sheet, and conse-
quently the outer face sheet is decelerated by the core while the
core and the rear face of the sandwich beam are accelerated. For
simplicity, we consider a one-dimensional slice through the thick-
ness of the sandwich beam and neglect the reduction in momen-
tum due to the impulse provided by the supports. This approxi-
mation is motivated by noting that the time period of this phase is
much smaller than the overall structural response time of the
structure. Subsequent retardation of the sandwich beam is due to
plastic bending and stretching in Stage III of the motion. Detailed
finite element calculations carried out recently by Qiu et al. 关10兴
support this assertion. The core is treated as a rigid, ideally plastic
crushable solid with a nominal crush strength ␴ nY up to a nominal
densification strain ⑀ D . After densification has been achieved, it is
assumed that the core is rigid.
Overall considerations of energy and momentum conservation
can be used to determine the final value of core compressive strain
⑀ c (⭐ ⑀ D ) and the final common velocity v f of faces and core at
the end of the core crush stage. The quantities ⑀ c and v f suffice to
proceed with the third stage of analysis to calculate the beam
deflection. However, if additional information on the core crush
phase is to be obtained, such as the time for core crush T c , a Fig. 4 „a… Sketch of the propagation of a one-dimensional
one-dimensional plastic shock wave analysis is required. First, we shock in the sandwich core, „b… the nondimensional core com-
present the immediate results for ⑀ c and v f , and then we outline pression time T̂ c as a function of the nondimensional impulse Î
the shock wave analysis in order to obtain T c . transmitted to the structure
Momentum conservation during core crush dictates that
共 2m f ⫹ ␳ c c 兲v f ⫽m f v o , (23)
and so a direct relation exists between the common velocity of the Plastic Shock-Wave Analysis. The above analysis assumes
sandwich beam v f after core crush and the initial velocity of the that the core compresses uniformly through its thickness at con-
outer face, v o . The ratio of the energy lost U lost in this phase to stant stress. In reality, the core can compress nonuniformly due to
the initial kinetic energy of the outer face sheet is then given by buckling of strut elements within the core and due to inertial ef-
fects. Here, we consider the case of a core which contains a suf-
U lost 1⫹ ␳ˆ ficiently large number of microstructural units 共the cells of a metal
⫽ (24)
m f v 2o /2 2⫹ ␳ˆ foam, or the units of a diamond-celled core兲 for it to be repre-
sented by a porous solid. However, the role of inertia is included,
where ␳ˆ ⫽ ␳ c c/m f . This loss in energy is dissipated by plastic and a plastic shock wave analysis is performed in order to deduce
dissipation in compressing the core and thus we equate the spatial and temporal evolution of strain within the core.
Consider a sandwich structure, with face sheets of mass per unit
U lost⫽ ␴ nY ⑀ c c, (25) area m f , and a core of initial thickness c and relative density ␳ c .
where ⑀ c is the average compressive strain in the core. Combining The front face sheet has an initial velocity v o , while the core and
the two above relation, the core compression strain ⑀ c is given by inner face sheet are initially at rest. As assumed above, we con-
sider a one-dimensional problem as sketched in Fig. 4共a兲 with the
⑀ D ␳ˆ ⫹1 2 core treated as a rigid, ideally plastic solid with a nominal crush
⑀ c⫽ Î , (26) strength ␴ nY up to a nominal densification strain ⑀ D ; at densifi-
2 ␳ˆ ⫹2
cation the core locks up and becomes rigid. After impact of the
in terms of the dimensionless parameter Î⫽I trans / 冑m f c ␴ nY ⑀ D . front face sheet upon the core, a plastic shock wave moves
However, if U lost is too high such that ⑀ c as given by 共26兲 exceeds through the core at a velocity c pl . Suppose that the shock wave
the densification strain ⑀ D , then ⑀ c is set to the value ⑀ D and has advanced by a distance X after a time t has elapsed, as
additional dissipation mechanisms must occur for energy conser- sketched in Fig. 4共a兲. Upstream of the shock wave, the unde-
vation. The above analysis neglects any such additional mecha- formed core and rear face of sandwich beam have a velocity v u ,
nisms. FE calculations by Xue and Hutchinson 关6兴 and Qiu et al. whilst downstream of the shock wave the core has compacted to
关10兴 reveal that the additional mechanism are tensile stretching of the densification strain ⑀ D and shares the velocity v d with the
the outer face near the supports together with additional crushing front face. The propagation behavior of the shock wave can be
of the core under sharply increasing stress. determined by numerical integration as follows.
Now a word of warning. The Stage II analysis neglects the Conservation of momentum dictates
impulse provided by the support reactions during the core com-
pression phase. This assumption breaks down for stubby beams
关 m f ⫹ ␳ c 共 c⫺X 兲兴v u ⫹ 关 m f ⫹ ␳ c X 兴v d ⫽m f v o , (27)
subjected to large impulses; the quality of the approximation is
analyzed in detail by Qiu et al. 关10兴 via a set of dynamic finite
element calculations. while energy conservation states

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 1 1 1 from 2 to &. Thus, it is predicted that the plastic shock wave will
m v 2 ⫽ 关 m f ⫹ ␳ c 共 c⫺X 兲兴v 2u ⫹ 关 m f ⫹ ␳ c X 兴v 2d ⫹ ␴ nY ⑀ D X, arrest before it traverses the core provided Î is less than & for all
2 f o 2 2
(28) ratios of core to face sheet mass.
The dependence of T̂ c ⫽T c v o / ⑀ D c on Î is shown in Fig. 4共b兲
and mass conservation across the shock wave provides for selected values of ␳ˆ . It is clear from the figure that T̂ c in-
c pl ⑀ D ⫽ v d ⫺ v u . (29) creases from zero to a peak value as Î increases from zero to the
transition value Î t . At higher values of Î, T̂ c decreases: at very
Now the compressive stress on the upstream face of the shock
wave is related directly to the mass and acceleration of upstream large values of Î, T̂ c approached a finite asymptote which equals
material, giving unity for the case ␳ˆ ⫽0. It is assumed that the core becomes rigid
after it has densified, and the core and face sheet velocities instan-
␴ u ⫽ 关 m f ⫹ ␳ c 共 c⫺X 兲兴v̇ u , (30) taneously jump in value to v f at T̂⫽T̂ c .
and a similar relation holds for the compressive stress on the Simple analytical expressions for the dependence of T̂ c upon Î
downstream face of the shock wave, can be obtained in the limiting case of a negligible core mass, ␳ˆ
→0. Consider first the case where the impulse is sufficiently small
␴ d ⫽⫺ 关 m f ⫹ ␳ c X 兴v̇ d . (31) for the core to compress by a strain ⑀ c less than the densification
Time differentiation of 共27兲 and the elimination of ( v̇ u , v̇ d ) value ⑀ D . Then, the core provides a constant compression stress
from the resulting expression via 共30兲 and 共31兲 leads to the well- ␴ nY upon the front and back face sheets, so that the front face has
known statement of momentum conservation across the shock the velocity
wave, ␴ nY t
v d⫽ v o⫺ , (37)
␴ u ⫺ ␴ d ⫽ ␳ c c pl 共 v u ⫺ v d 兲 . (32) mf
As the shock wave progresses through the core it slows down, while the rear face has the velocity
and, for a sufficiently low initial value of front face velocity v o ,
the shock wave arrests at a travel X c less than the core thickness ␴ nY t
v u⫽ . (38)
c. Upon noting that Ẋ⫽c pl the crush time T c is calculated via mf
共29兲 to give The core compression time T c is obtained by equating v d and v u ,

T c⫽ 冕 0
Xc dX
c pl
⫽ 冕 0
Xc ⑀D
v d⫺ v u
dX. (33)
to obtain

Î 2
T̂ c ⫽ . (39)
Now ( v d ⫺ v u ) can be expressed as a function of X via 共27兲 and 2
共28兲, and 共33兲 thereby integrated numerically in order to obtain Continuing with the choice ␳ˆ →0, now address the case where
the core crush time, T c . The integral reads in nondimensional the impulse exceeds the transition value Î t ⫽2, so that the core
form, densifies before the front and rear-face sheet velocities have

T̂ c ⫽
T cv o
⑀ Dc
⫽ 冕 0
¯X
c 1
v̄ d ⫺ v̄ u
dX̄, (34)
equalized to v o /2, as demanded by momentum conservation. The
core compression time is set by the time for the face sheets to
undergo a relative approach of ⑀ D c. Upon noting that the front
where X̄⬅X/c, X̄ c ⬅X c /c⫽ ⑀ c / ⑀ D , as specified by 共26兲, v̄ d face sheet displaces by
⬅ v d / v o and v̄ u ⬅ v u / v o . In the above relation v̄ d ⫺ v̄ u depends ␴ nY 2
upon X̄ according to s d ⫽ v o t⫺ t , (40)
2m f
1⫹ ␳ˆ 共 2⫺X̄ 兲 ⫹ ␳ˆ 2 共 1⫺X̄ 兲 while the back face sheet displaces by
共 v̄ d ⫺ v̄ u 兲 2 ⫽
关 1⫹ ␳ˆ 共 1⫺X̄ 兲兴 共 1⫹ ␳ˆ X̄ 兲
2
␴ nY 2
s u⫽ t , (41)
2m f
2 共 2⫹ ␳ˆ 兲 ␳ˆ X̄
⫺ . (35) the core compression time T c is determined by the condition
关 1⫹ ␳ˆ 共 1⫺X̄ 兲兴共 1⫹ ␳ˆ X̄ 兲 Î 2
␴ nY 2
For the case X̄⬅X/c⬍1, T̂ c is calculated as a function of Î by s d ⫺s u ⫽ v o T c ⫺ T ⫽ ⑀ D c. (42)
mf c
evaluating 共34兲, with ( v̄ d ⫺ v̄ u ) expressed by 共35兲, and the upper
limit of integration X̄ c ⫽ ⑀ c / ⑀ D expressed in terms of Î via 共26兲. with solution
However, at sufficiently high values of impulse Î, the plastic
⫽ 关 Î⫺ 冑Î 2 ⫺4 兴 .
shock wave traverses the thickness of the core c without arrest. T c v o Î
T̂ c ⬅ (43)
The period of core compression is again specified by 共34兲, with ⑀ Dc 2
( v̄ d ⫺ v̄ u ) expressed by 共35兲, and the upper limit of integration
3.4 Stage III: Dynamic Structural Response of Clamped
X̄ c ⫽1.1 At the transition value Î t , the shock wave arrests at the
Sandwich Beam. At the end of stage II the core and face sheets
same instant that it traverses the core thickness; Î t is obtained by have a uniform velocity v f as dictated by 共23兲. The final stage of
equating ⑀ c to ⑀ D in 共26兲, to give sandwich response comprises the dissipation of the kinetic energy
2 共 ␳ˆ ⫹2 兲 acquired by the beam during stages I and II by a combination of
Î t2 ⫽ . (36) beam bending and longitudinal stretching. The problem under
␳ˆ ⫹1
consideration is a classical one: what is the dynamic response of a
It is noted in passing that Î t is only mildly sensitive to the mag- clamped beam of length 2L made from a rigid ideally-plastic
nitude of the mass ratio ␳ˆ : as ␳ˆ is increased from zero 共negligible material with mass per unit length m subjected to an initial uni-
core mass兲 to infinity 共negligible face sheet mass兲, Î decreases form transverse velocity v f ? This problem has been investigated
by a number of researchers. In particular, Symmonds 关11兴 devel-
1
Note that in such cases the above analysis conserves momentum but does not oped analytical solutions based on a small displacement analysis
account for the additional dissipation mechanisms required to conserve energy. while Jones 关12兴 developed an approximate method for large dis-

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 Fig. 5 Analysis of stage III of the blast response. „a… Velocity profile in phase I, „b… a free-body
diagram of the half-beam in phase I, with the deflected shape sketched approximately, „c…
velocity profile in phase II, and „d… a free-body diagram of the half-beam in phase II, with the
deflected shape sketched approximately. The accelerations of the beam are shown in „d….

placements using an energy balance method. These methods are In the dynamic analysis we shall assume that displacements
summarized in Jones 关13兴. Here we present an approximate solu- occur only in a direction transverse to the original axis of the
tion that is valid in both the small and large displacement regime: beam and thus stretching is a result of only transverse displace-
it reduces to the exact small displacement solution of Symmonds ments. Moderate transverse deflections are considered, such that
关11兴 for small v f and is nearly equal to the approximate large the deflection w at the mid-span of the beam is assumed to be
deflection solution of Jones 关13兴 for large v f . small compared to the beam length 2L and the longitudinal force
Active plastic straining in the beam is by a combination of N⫽N o can be assumed to be constant along the beam. The motion
plastic bending and longitudinal stretching with shear yielding of the beam can be separated into two phases as in the small
neglected: An evaluation of the magnitude of the transient shear displacement analysis of Symmonds 关11兴. In phase I, the central
force within the face sheet in the dynamic clamped beam calcu- portion of the beam translates at the initial velocity v f while seg-
lation of Jones 关13兴 reveals that shear yielding is expected only for ments of length ␰ at each end rotate about the supports. The bend-
unrealistic blast pressures as discussed above. We assume that ing moment is taken to vary from ⫺M o at the outer stationary
yield of an beam element is described by the resultant longitudinal plastic hinges at the supports to ⫹M o at ends of the segments of
force N and the bending moment M . The shape of the yield sur- length ␰ with the bending moment constant in the central flat
face in (N,M ) space for a sandwich beam depends on the shape portion. Thus, time increments in curvature occur only at the ends
of the cross section and the relative strength and thickness of the of the rotating segments while axial straining is distributed over
faces and the core. A yield locus described by the length of the rotating segments. A free-body diagram for half
of the clamped beam is shown in Fig. 5共b兲; conservation of the
兩M 兩 兩N兩 moment of momentum about a fixed end after a time t gives
⫹ ⫽1, (44)

冉 冊
M o No
L L⫺ ␰ 1
where N o and M o are the plastic values of the longitudinal force 共 mL v f 兲 ⫽m 共 L⫺ ␰ 兲v f ␰ ⫹ ⫹2M o t⫹ N o v f t 2
and bending moment, respectively, is highly accurate for a sand- 2 2 2
wich beam with thin, strong faces and a thick, weak core. It be-
comes less accurate as the beam section approached the mono-
lithic limit. It is difficult to obtain a simple closed-form analytical
⫹ 冕0
␰ mv fx2
␰
dx, (47)

solution for the dynamic beam response with this choice of yield where x is the axial coordinate from one end of the beam, as
surface. Here, we approximate this yield locus to be a circum- shown in Fig. 5共b兲. This equation gives ␰ as a function of time t
scribing square such that
兩 N 兩 ⫽N o (45a) ␰⫽ 冑 3t 共 v f N o t⫹4M o 兲
mv f
. (48)
兩 M 兩 ⫽M o , (45b)
Phase I continues until the traveling hinges at the inner ends of
with yield achieved when one or both of these relations are satis- the segments of length ␰ coalesce at the midspan, i.e., ␰ ⫽L. Thus,
fied. We could equally well approximate the yield locus to be an from 共48兲, phase I ends at a time T 1

冋冑 册
inscribing square such that
Mo mL 2 v 2f N o
兩 N 兩 ⫽0.5N o (46a) T 1⫽ 4⫹ ⫺2 , (49)
N ov f 3M 2o
兩 M 兩 ⫽0.5M o , (46b)
and the displacement of the mid-span w 1 at this time is given by

冋冑 册
with again at yield one or both of these relations satisfied. Jones
关13兴 has explored the choice of circumscribing and inscribing Mo mL 2 v 2f N o
yield surfaces for a monolithic beam and shown that the resulting w 1⫽ v f T 1⫽ 4⫹ ⫺2 . (50)
No 3M 2o
solutions bound the exact response. We proceed to develop the
analysis for the circumscribing yield locus: the corresponding for- In phase II of the motion, stationary plastic hinges exist at the
mulas for the inscribed locus may be obtained by replacing M o by midspan and at the ends of the beam, with the moment varying
0.5M o and N o by 0.5N o . between ⫺M o at the beam end to ⫹M o at the midspan. The

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 velocity profile is triangular, as sketched in Fig. 5共c兲. The equation where ␨ I is the blast impulse transmitted to the structure by the
of motion of the half-beam in phase II follows from the free-body fluid. Consequently, the response time T, as given by 共53兲, can be
diagram sketched in Fig. 5共d兲 as rewritten in the nondimensional form as

2M o ⫹N o w⫽⫺
ẅ
L 冕 L

0
mx 2 dx⫽⫺
mL 2
3
ẅ, (51)
T̄⫽
␣ 2 c̄ 共 2h̄⫹¯␳ 兲
2 Ī ␨
冋冑 1⫹
4 Ī 2 ␨ 2 ␣ 3
3 ␣ 1␣ 2
⫺1 册
冑 冋 冑 册
where x is the axial coordinate from one end of the beam as
shown in Fig. 5共d兲. With initial conditions w(T 1 )⫽w 1 and c̄ 共 2h̄⫹¯␳ 兲 ␣3
ẇ(T 1 )⫽ v f , this differential equation admits a solution of the ⫹ tan⫺1 4 Ī ␨ ,
form 3ĉ 共 2ĥ⫹ ¯␴ l c̄/ĉ 兲 3 ␣ 1 ␣ 2 ⫹4 Ī 2 ␨ 2 ␣ 3

w共 t 兲⫽
vf
␻
sin关 ␻ 共 t⫺T 1 兲兴 ⫹
2M o
No 冉
⫹w 1 cos关 ␻ 共 t⫺T 1 兲兴 ⫺
2M o
No 冊
,
where
(60)

(52a)
␣ 1 ⫽ĉ 3 关共 1⫹2ĥ 兲 2 ⫺1⫹ ¯␴ l c̄/ĉ 兴共 1⫹2ĥ 兲 c̄ 共 ¯␳ ⫹2h̄ 兲 , (61a)
where

␻⫽
1
L
冑 3N o
m
. (52b) ␣ 2⫽
ĉ 关共 1⫹2ĥ 兲 2 ⫺1⫹ ¯␴ l c̄/ĉ 兴
2ĥ⫹ ¯␴ l c̄/ĉ
, and (61b)

The maximum deflection w of the midspan of the beam occurs at

a time T when ẇ(T)⫽0. Upon substituting this termination con- ␣ 3 ⫽ĉ 共 1⫹2ĥ 兲 . (61c)
dition in the velocity equation, as given by the time derivative of The maximum defection 共54兲 of the inner and outer faces at the
共52a兲, the response time T is obtained as midspan can be written nondimensionally as

T⫽T 1 ⫹
1
␻
tan⫺1 冋
N ov f
␻ 共 2M o ⫹w 1 N o 兲
, 册 (53)
w̄⬅
w ␣2
L
⫽
2
冋冑 1⫹
8 Ī 2 ␨ 2 ␣ 3
3 ␣ 1␣ 2
⫺1 , 册 (62a)
and the corresponding maximum deflection of the midspan of the
beam is and

冑 冉 冊
w̄ o ⫽w̄⫹ ⑀ c c̄, (62b)
v 2f 2M o 2
2M o
w⫽ 2⫹ ⫹w 1 ⫺ . (54) respectively. It is emphasized that the deflection of the inner face
␻ No No
of the sandwich beam is due to only stage III of the deformation
The deflected shape of the beam can be obtained using the proce- history, while the deflection of the outer face is the sum of the
dure detailed on p. 81 of Jones 关13兴 but the derivation and result deflections in stage III and the deflection due to core compression
are omitted here as they are not central to the present discussion. in stage II.
We specialize this analysis to the case of sandwich beams. Re- It is difficult to give a precise failure criterion for the beam as it
call that we are considering clamped sandwich beams of span 2L is anticipated that the blast impulse for incipient failure is sensi-
with identical face sheets of thickness h and a core of thickness c, tive to the details of the built-in end conditions of the clamped
as shown in Fig. 1. The face sheets are made from a rigid ideally beams. Here, we state a failure criterion based on an estimate of
plastic material of yield strength ␴ f Y and density ␳ f , while the the tensile strain in the face sheets due to stretching of the beam
core of density ␳ c has a normal compressive strength ␴ nY and a and neglect the tensile strains due to bending at the plastic hinges.
longitudinal strength ␴ lY . The plastic bending moment of the The tensile strain ⑀ m in the face sheets due to stretching is ap-
sandwich beam with the compressed core is given by proximately equal to

M o ⫽ ␴ lY
共 1⫺ ⑀ c 兲 c 2
4
⫹ ␴ f Y h 关共 1⫺ ⑀ c 兲 c⫹h 兴 , (55) ⑀ m⫽
1 w
2 L冉冊 2
. (63)

while the plastic membrane force N o is given by By setting this strain ⑀ m to equal the tensile ductility ⑀ f of the face
N o ⫽2 ␴ f Y h⫹ ␴ lY c. (56) sheet material, an expression is obtained for the maximum nondi-
mensional impulse Ī c that the sandwich beam can sustain without
For simplicity we assume that the plastic membrane force N o tensile failure of the face sheets; substitution of 共63兲 into 共62a兲,
due to the core is unaffected by the degree of core compression; with the choice ⑀ m ⫽ ⑀ f , gives

冑 冋冉 冑 冊 册
while this assumption is thought to be reasonable for all the cores
2
considered, it requires experimental verification. We now intro- 1 3 ␣ 1␣ 2 2 2⑀ f
duce the nondimensional geometric variables of the sandwich Ī c ⫽ ⫹1 ⫺1 . (64)
beam ␨ 8␣3 ␣2
The above analysis, comprising stages I, II, and III for the re-
c h h̄
c̄⬅ , h̄⬅ , ĉ⬅c̄ 共 1⫺ ⑀ c 兲 , and ĥ⬅ , (57) sponse of a clamped sandwich beam to blast loading, gives the
L c 1⫺ ⑀ c deflection w̄, response time T̄, the core compression ⑀ c and the
and the nondimensional core properties maximum tensile strain ⑀ m in the sandwich beam in terms of

␳c ␴ lY ␴ nY i. the loading parameters as specified by the blast impulse Ī ,

¯␳ ⬅ , ¯␴ l ⬅ and ¯␴ n ⬅ . (58) and the fluid-structure interaction parameter ␺,
␳f ␴fY ␴fY
ii. the beam geometry c̄ and h̄, and
The nondimensional structural response time T̄ and blast impulse iii. the core properties as given by the core relative density ¯␳ ,
Ī are its longitudinal tensile strength ¯␴ l , compressive strength ¯␴ n
and its densification strain ⑀ D .
T̄⬅
T
L
冑 ␴fY
␳f
, Ī ⬅
I
L 冑␳ f ␴ f Y
⬅
Îc̄ 冑¯␴ n ⑀ D h̄
␨
, (59) We proceed to illustrate graphically the functional dependence
of w̄, T̄, ⑀ c , and ⑀ m on the blast impulse Ī . Consider a represen-

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 In a typical design scenario, the solid material and length of the
structural element are dictated by design constraints such as cor-
rosion resistance and bulkhead spacing, thus leaving the sandwich
panel geometry, viz. the face sheet and core thickness, and core
relative density and topology, as the free design variables. Two
design problems will be addressed:
1. For a given material combination, beam length and blast
impulse, what is the relation between sandwich geometry
and the inner face sheet deflection?
2. For a given material combination, beam length and allow-
able inner face sheet deflection, what is the relation between
the required sandwich geometry and the level of blast im-
pulse?
4.1 Monolithic Beams. As a reference case we first present
the response of a monolithic beam subjected to a water blast.
Consider a monolithic beam of thickness h and length 2L made
from a rigid-ideally plastic solid material of density ␳ f and yield
strength ␴ f Y subjected to a blast impulse I.
We define a fluid-structure interaction parameter ¯␺

h ␳ wc w␪
¯␺ ⫽ ␺ ⫽ , (65)
L ␳fL

which is closely related to the Taylor 关1兴 fluid-structure interaction

parameter ␺ but written in terms of the specified beam length. The
impulse I trans transmitted to the beam is given by 共20b兲 for a
specified value of ¯␺ and a known beam geometry h/L.
First, we specialize the analysis of Section 3.4 to the case of a
monolithic beam with plastic moment M o ⫽N o h/4, where N o
⫽h ␴ f Y is the plastic membrane force. The nondimensional maxi-
mum deflection of the midspan of the beam w̄⫽w/L and normal-
ized structural response time T̄⬅T/L 冑␳ f / ␴ fY follow from 共54兲
and 共53兲, respectively, as

Fig. 6 Response of a clamped sandwich beam „ c̄ Ä0.1, h̄

Ä0.1… with a pyramidal core „ ␳¯ Ä0.1, ⑀ Y Ä0.002, ⑀ D Ä0.5… for an
w̄⫽
h
2L
冋冑 1⫹
3 冉冊 册
8 Ī 2 ␨ 2 L
h
4
⫺1 and (66a)

冉 冊 冋冑 冉冊 册
assumed ␺ Ä1.78; „a… the normalized response time T̄ and de- 2 4
flection w̄ and „b… core compression ⑀ c , and tensile strain in 1 h 4 L
beam ⑀ m , as a function of the normalized blast impulse Ī T̄⫽ 1⫹ ¯I 2 ␨ 2 ⫺1
2 Ī ␨ L 3 h

tative sandwich beam with c̄⫽h̄⫽0.1 and comprising a pyramidal

core of relative density ¯␳ ⫽0.1 made from the same solid material
as the face sheets 共with ⑀ Y ⫽0.2%). As specified in Section 2, the
core yields rather than elastically buckles, and the normal and
⫹
1
)
tan⫺1 冋冑 2 Ī ␨ 共 L/h 兲 2

3⫹4 Ī ␨ 共 L/h 兲
2 2 4
册 , (66b)

longitudinal strengths of this pyramidal core are ¯␴ n ⫽0.05 and where ␨ I is the impulse transmitted into the structure. For ␨ Ī
¯␴ l ⫽0, respectively. The densification strain of the core is taken as Ⰶ1, the above relations reduce to
⑀ D ⫽0.5. To complete the specification, we assume a fluid-
structure interaction parameter ␺ ⫽1.79 which is representative of
an underwater blast with a time constant ␪ ⫽0.1 ms and 10 mm
thick steel faces as discussed in Section 3.1. The normalized de-
2
w̄⫽ ¯I 2 ␨ 2
3
L
h 冉冊 3
(67)

flection w̄ of the inner face of the sandwich beam and response

time T̄ are plotted in Fig. 6共a兲 as a function of the normalized
blast impulse while the compression ⑀ c and tensile stretch ⑀ m are
T̄⫽ Ī ␨ 冉冊
L
h
2
, (68)

plotted in Fig. 6共b兲. For Ī ⬍0.03, the compressive strain ⑀ c in- which are identical to the small deflection predictions of Sym-
duced in the core in Stage II is less than ⑀ D and w̄ increases monds 关11兴.
approximately quadratically with Ī . At higher impulses the core With the tensile strain in the beam given by 共63兲, the maximum
compression is fixed at the densification limit ⑀ D and w̄ scales impulse Ī c sustained by a monolithic beam made from material of
approximately linearly with Ī . On the other hand, the structural tensile ductility ⑀ f is
response time initially increases linearly with Ī , but at high im-
pulses the beam behaves as a stretched plastic string and T̄ is
almost independent of the magnitude of Ī . Ī c ⫽
1
␨
冑 冉 冊 冋冉 冑 冉 冊 冊 册
3 h
8 L
2
2 2⑀ f
L
h
⫹1
2
⫺1 . (69)

4 Performance Charts for Water Blast Resistance A representative design chart is now constructed for a monolithic
The analysis detailed above is now used to investigate the rela- beam subjected to a water blast. Consider a steel beam of length
tive response of monolithic and sandwich beams to blast loading. 2L⫽10 m subjected to a blast with a decay time ␪ ⫽0.12 ms. The

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 Fig. 8 Design chart for a sandwich beam, with a pyramidal
Fig. 7 Design chart for a monolithic beam of tensile ductility core „ ␳¯ Ä0.1, ⑀ Y Ä0.002, ⑀ D Ä0.5…, subjected to a water blast. The
¯ Ä5Ã10À3 . Contours
⑀ f Ä0.2, subjected to a water blast with ␺ nondimensional impulse is Ī Ä10À2 , and the fluid-structure in-
of the midspan displacement w̄ are given as solid lines and teraction parameter is taken as ␺ ¯ Ä5Ã10À3 . The regime of ten-
contours of dimensionless mass M̄ are shown as dotted lines. sile failure is shown for an assumed tensile ductility of face
sheets of ⑀ f Ä0.2. Contours of w̄ and ⑀ c are included.

fluid-structure interaction parameter ¯␺ then takes the value ¯␺ ⫽5 ¯␺ 共associated with shorter spans, 2L, and with longer values of
⫻10⫺3 . Contours of nondimensional deflection w̄ are plotted in the decay constant ␪兲, tensile failure is less likely. Thus, tensile
Fig. 7 as a function of the normalized blast impulse Ī and beam failure is unlikely to occur for sandwich beams provided ¯␺ ex-
geometry, h/L, for ¯␺ ⫽5⫻10⫺3 . Note that the contours of the w̄ ceeds approximately 0.02.
have been truncated at high impulses due to tensile tearing as The inverse design problem of the relation between the pyra-
dictated by 共69兲, with the choice ⑀ f ⫽0.2. Contours of nondimen- midal core (¯␳ ⫽0.1, ⑀ Y ⫽0.002, ⑀ D ⫽0.5) sandwich beam geom-
etry and the blast impulse for a specified deflection w̄⫽0.1 and for
sional mass M̄ ⫽M /(L 2 ␳ f )⫽2h/L, where M is the mass per unit ¯␺ ⫽5⫻10⫺3 is addressed in Fig. 10. Tensile failure of the steel
width of the beam, have also been added to the figure. As ex-
pected, the beam deflection increases increasing with blast im- faces ( ⑀ f ⫽0.2) is inactive for the choice w̄⫽0.1. For the purposes
pulse, for a beam of given mass. of selecting sandwich beam geometries that maximise the blast
impulse at a given mass subject to the constraint of a maximum
4.2 Sandwich Beams. The blast response of clamped sand- allowable inner face deflection w̄, contours of non-dimensional
wich beams, comprising solid faces and the five types of cores mass M̄ have been added to Fig. 10, where
discussed in Section 2, will be analyzed in this section. We restrict
attention to cores made from the same solid material as the solid
face sheets in order to reduce the number of independent nondi-
mensional groups by one. With the sandwich beam length and
material combination specified, the design variables in the prob-
lem are the nondimensional core thickness c̄⬅c/L and face sheet
thickness h̄⬅h/c.
Figure 8 shows a design chart with axes c̄ and h̄ for a clamped
sandwich beam with a pyramidal core (¯␳ ⫽0.1, ⑀ Y ⫽0.002) and
subjected to a normalized blast impulse Ī ⫽10⫺2 . The fluid-
structure interaction parameter is again taken as ¯␺ ⫽5⫻10⫺3 ; this
is representative for steel sandwich beams of length 2L⫽10 m
subject to a water blast with a decay constant ␪ ⫽0.12 ms. Further,
the densification strain ⑀ D of the core is assumed to be 0.5 and the
tensile ductility of the solid steel is taken as ⑀ f ⫽0.2. Contours of
nondimensional maximum deflection of the mid-span of the inner
face of the beam and contours of the compressive strain ⑀ c in the
core have been added to the chart: both w̄ and ⑀ c increase with
decreasing c̄ and beam failure by tensile tearing of the face sheets
is evident at the top left-hand corner of the chart.
The effect of the fluid-structure interaction parameter ¯␺ upon
the likelihood of tensile failure of the above sandwich beam is
shown in Fig. 9. The figure shows the regime of tensile failure of
Fig. 9 The effect of ␺
¯ upon the magnitude of the tensile failure
the face sheets on a design chart with axes (c̄,h̄). Apart from the regime within the design chart, for face sheets of ductility ⑀ f
choice of ¯␺ , the nondimensional parameters are the same as those Ä0.2. The sandwich beam has a pyramidal core „ ␳¯ Ä0.1, ⑀ Y
used to construct Fig. 8: ¯␳ ⫽0.1 and ⑀ D ⫽0.5 for the pyramidal Ä0.002, ⑀ D Ä0.5… and the nondimensional impulse is taken as
core, ⑀ f ⫽0.2 for the faces and Ī ⫽10⫺2 . With increasing values of Ī Ä10À2 .

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 Fig. 11 A comparison of the maximum blast impulse sus-
Fig. 10 Design chart for a sandwich beam, with a pyramidal tained by monolithic beams and by optimal designs of sand-
core „ ␳¯ Ä0.1, ⑀ Y Ä0.002, ⑀ D Ä0.5…, subjected to a water blast. The wich beams, subjected to the constraints w̄ Ï0.1 and h Õ L
beam deflection is w̄ Ä0.1 and the fluid-structure interaction pa- Ð10À2 . Results are presented for ␺¯ Ä5Ã10À3 and 0.02. The
rameter is taken as ␺ ¯ Ä5Ã10À3 . Contours of Ī and M̄ are dis- core relative density is ␳¯ Ä0.1 and densification strain is
played. The dotted lines trace the paths of selected values of ⑀ D Ä0.5.
hÕL.

M
M̄ ⫽ ⫽2 共 2h̄c̄⫹c̄¯␳ 兲 , (70)
␳ fL2 h
M̄ ⫽4 ⫹2c̄¯␳ . (71)
and M is the mass per unit width of the sandwich beam. The L
figure reveals that geometries that maximize the blast impulse Ī
for a given mass M̄ have h̄→0 at almost constant c̄, implying that
h/L→0. The physical interpretation is as follows. With decreasing The above constraint on the minimum h/L implies a minimum
face sheet thickness 共or face sheet mass兲 the blast impulse trans- value for M̄ of 4h/L. Thus, for the constraint h/L⭓10⫺2 , M̄ has
mitted to the structure reduces: The Taylor analysis gives Ī trans the minimum value of 0.04 as evident in Fig. 11. Similarly, for a
→0 as h→0. This limit is practically unrealistic as a minimum monolithic beam of thickness h, M̄ is given by M̄ ⫽2h/L and so
face sheet thickness is required for other reasons, for example to a constraint on the minimum value of h/L gives directly a mini-
withstand wave loading, quasi-static indentation by foreign ob- mum acceptable mass M̄ . With increasing values of the ¯␺ , the
jects such as rocks and other vessels and fragment capture in a fraction of the blast impulse transmitted into the structure de-
blast event. Consequently, we add the additional constraint of a creases and thus all the beams sustain higher blast. However, the
minimum normalized face sheet thickness h/L into the analysis. relative performance of the various beam configurations remains
Contours of h/L for two selected values of h/L have been added unchanged.
to Fig. 10. These lines represent limits on acceptable sandwich The effect of the constraint on h/L on the performance of the
beam designs, with designs lying above these lines satisfying the above sandwich beams is illustrated in Fig. 12 for the choice ¯␺
constraint on h/L: designs that maximize blast impulse for a ⫽5⫻10⫺3 . As the allowable minimum value of h/L decreases
given mass then lie along the lines of constant h/L. from 10⫺2 to 10⫺3 , the blast impulses sustained by the sandwich
The maximum blast impulse sustained by the sandwich beams beams increase. Further, the rankings of the cores change slightly:
with the five different topologies of the core 共but ¯␳ ⫽0.1, ⑀ Y while the diamond-celled core still performs the best followed by
⫽0.002 and ⑀ D⫽0.5 in all cases兲, subject to the constraints h/L the square-honeycomb core, the metal foam core is now seen to
⬎10⫺2 and the inner face deflection w̄⭐0.1 are plotted in Fig. 11 out perform the pyramidal and hexagonal-honeycomb cores at
as a function of the nondimensional mass M̄ for the choice ¯␺ higher masses. This can be rationalized as follows. Upon impos-
⫽5⫻10⫺3 . For comparison purposes, the blast impulse sustained ing the constraint h/L⭓10⫺3 , a large fraction of the mass of the
by a monolithic beam subjected to the same constraints is also sandwich beam is in the core. Recall that the pyramidal and
included in Fig. 11. It is evident that sandwich beams all perform hexagonal-honeycomb cores have no longitudinal strength while
considerably better than the monolithic beam. This is mainly due the metal foam core gives some additional longitudinal stretching
to the fact that the sandwich beams have a thin outer face sheet resistance to the sandwich beam, and this results in its superior
which results in a small impulse transmitted into the structure performance.
whereas the relatively thick beams in monolithic design absorb a So far we have determined the optimal designs of sandwich
larger fraction of the blast impulse. A comparison of the various beams for a midspan deflection of w̄⭐0.1. But how does the
sandwich cores shows that sandwich beams with a metal foam and relative performance depend upon the allowable value of w̄? The
pyramidal core almost attain the performance of the hexagonal- performance of the sandwich beams with constraints h/L⭓10⫺2
honeycomb core. However, the diamond-celled and square- and ¯␺ ⫽5⫻10⫺3 is illustrated in Fig. 13 for w̄⭐0.1 and w̄
honeycomb core beams, which have high strength in both the ⭐0.4. As expected, the beams can sustain higher impulses when
through-thickness and longitudinal directions, out perform the the constraint on w̄ is relaxed to w̄⭐0.4. However, the rankings
other sandwich beams. The performance of the diamond-celled change for the two levels of w̄ considered in Fig. 13. With the
core approaches that of the ‘‘ideal’’ sandwich core. It is noted that higher allowable deflections, the longitudinal stretching of the
M̄ has minimum achievable values. This is explained as follows. core becomes increasingly important and the metal foam core out
Since h/L⬅h̄c̄, the expression 共70兲 for M̄ can be rewritten as performs the pyramidal or hexagonal-honeycomb cores. The

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 Fig. 12 A comparison of the maximum blast impulse sus-
tained by monolithic beams and by optimal designs of sand-
wich beams, subjected to the constraint w̄ Ï0.1 with ␺ ¯ Ä5
Ã10À3 . Results are presented for constraints h Õ L Ð10À2 and
10À3 . The core relative density is ␳¯ Ä0.1, ⑀ Y Ä0.002 and densi-
fication strain is ⑀ D Ä0.5.

diamond-celled core has a high compressive and an ideal longitu-

dinal strength, and has a blast performance which is nearly indis-
tinguishable from that of the ‘‘ideal’’ core under the constraint
w̄⭐0.4.
In the above analysis the relative density of the core has been
taken to be ¯␳ ⫽0.1, and the yield strain of the core material taken
to be representative of that for structural steels, ⑀ Y ⫽0.002. Con-
sequently, the individual struts of the pyramidal and diamond-
celled cores deform by plastic yield. We proceed to investigate the
blast performance of the pyramidal and diamond-celled core sand-
wich beams at relative densities ¯␳ such that elastic buckling of the Fig. 14 Comparison of the maximum blast impulse sustained
core members can intervene. The optimal performance of by optimal „a… diamond-celled and „b… pyramidal core sandwich
beams for selected core densities, with ␺ ¯ Ä5Ã10À3 and h Õ L
diamond-celled core sandwich beams with the constraints h/L
Ð10À2 , w̄ Ï0.1. The yield strain of the core parent material is
⭓10⫺2 and w̄⭐0.1 is plotted in Fig. 14共a兲 for selected values of
assumed to be ⑀ Y Ä0.002 and densification strain of the core is
core relative density ¯␳ ⫽0.02, 0.05, 0.1 and 0.2. The core is as- taken as ⑀ D Ä0.5.
sumed to be made from a solid of yield strain ⑀ Y ⫽0.002 and
consequently cores of density ¯␳ ⫽0.02 and 0.05 deform by elastic
buckling. While the performance of the low core density beams is

slightly superior to the ¯␳ ⫽0.1 beams, these beams of low core

density have stubby designs with high values of c/2L. Thus, these
optimal designs become impractical for high blast impulses and
the curves in Fig. 14共a兲 have been truncated at c/2L⫽0.2. A simi-
lar analysis was performed for the pyramidal core; these cores
deform by elastic buckling at ¯␳ ⭐0.015. The results for the opti-
mal blast performance of these beams are summarized in Fig.
14共b兲; again the low density cores provide superior performance
but the beams are stubby 共high c/2L) and hence practical designs
of these beams are unable to sustain high blast impulses. A com-
parison of Figs. 14共a兲 and 14共b兲 reveals that over the entire range
of relative densities investigated, the diamond-celled core beams
always out perform the pyramidal core beams.

5 A Comparison of Structural Performance Under Air

and Water Blast Loading
Fig. 13 A comparison of the maximum blast impulse sus-
tained by monolithic beams and by optimal designs of sand- Due to the low acoustic impedance of air, the Taylor fluid-
wich beams, subjected to the constraint h Õ L Ð10À2 with ␺ ¯ Ä5 structure interaction parameter ␺ ⬇0 for an air blast, as discussed
Ã10À3 . Results are presented for constraints w̄ Ï0.1 and 0.4. in Section 3.2. In this section we discuss blast loading in air by
The core relative density is ␳¯ Ä0.1, ⑀ Y Ä0.002 and densification assuming ¯␺ ⬅ ␺ h/L⫽0: The entire blast impulse is transmitted to
strain is ⑀ D Ä0.5. the sandwich structure.

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 Fig. 15 Design chart for a sandwich beam, with a pyramidal
core „ ␳¯ Ä0.1, ⑀ Y Ä0.002, ⑀ D Ä0.5…, subjected to an air blast. The
nondimensional impulse is Ī Ä10À3 . The regime of tensile fail-
ure is shown for an assumed tensile ductility of face sheets of
⑀ f Ä0.2. Contours of w̄ and ⑀ c are included.

Consider the representative case of a sandwich beam with a

pyramidal core (¯␳ ⫽0.1,⑀ D ⫽0.5,⑀ Y ⫽0.002), subjected to an air
blast of magnitude Ī ⫽10⫺3 . The design chart is given in Fig. 15,
with axes of c̄ and h̄, and with contours displayed for the midspan
deflection w̄ of the inner face and through-thickness core com-
pression ⑀ c . The tensile ductility of the face sheet material is
taken to be ⑀ f ⫽0.2 representative of structural steels; despite this
moderately high value of ⑀ f , tensile failure of the face sheets
dominates the chart with less than half the design space of Fig. 15
resulting in acceptable designs. In contrast, for water blast 共Fig. Fig. 17 „a… Comparison of the maximum impulse sustained by
8兲, tensile failure is of less concern even for a higher blast impulse monolithic and sandwich beams for an air blast with the con-
of Ī ⫽10⫺2 ; the underlying explanation is that only a small frac- straint w̄ Ï0.1. The core relative density and densification
strain are, ␳¯ Ä0.1 and ⑀ D Ä0.5, respectively, and ⑀ Y Ä0.002. „b…
tion of the impulse is transmitted into the sandwich structure for
The optimal designs of sandwich beams with pyramidal and
water blast loading. diamond-celled core.

A design map for air blast loading of the above pyramidal core
sandwich beam is given in Fig. 16, with contours of Ī required to
produce a mid-span deflection of w̄⫽0.1. The figure should be
contrasted with the water blast map shown in Fig. 10, again for
w̄⫽0.1; the only difference in the assumed values of the plots is
that ¯␺ ⫽0 in Fig. 16 and ¯␺ ⫽5⫻10⫺3 in Fig. 10. While the con-
tours of M̄ are identical in the two figures, the contours of Ī are of
markedly different shape. For the case of air blast 共Fig. 16兲 there
is no need to impose a constraint on the minimum value for h/L:
The trajectory of (c̄,h̄) which maximizes Ī for a given M̄ no
longer lies along a line of constant h/L and is associated with
h/L⬅h̄c̄ values in the range 0.003 to 0.032. The arrows shown in
Fig. 16 trace the optimum designs with increasing mass. This can
be contrasted with the water blast problem where the optimum
designs lay along the specified minimum value of h/L.
The air blast performance of the optimized sandwich beams is
compared to that of the monolithic beam in Fig. 17共a兲. Specifi-
cally, the maximum sustainable impulse is plotted against the non-
Fig. 16 Design chart for a sandwich beam, with a pyramidal
core „ ␳¯ Ä0.1, ⑀ Y Ä0.002, ⑀ D Ä0.5…, subjected to an air blast. The dimensional mass M̄ , with the deflection constraint w̄⭐0.1 im-
beam deflection is w̄ Ä0.1. Contours of Ī and M̄ are displayed. posed. In contrast to the case of water blast, the performance gain
The arrows trace the path of designs which maximize the im- upon employing sandwich construction instead of monolithic
pulsive resistance with increasing mass. beams is relatively small; at best the diamond-celled core sustains

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 Fig. 18 The normalized deflection of the bottom face of a
diamond-celled core „ ␳¯ Ä0.1, ⑀ Y Ä0.002… sandwich beam with c̄
Ä h̄ Ä0.2 as a function of the normalized impulse, for two se-
lected values of the core densification strain ⑀ D . The response
of a monolithic beam of the same mass M̄ Ä0.2 is included.

an impulse about 45% greater than a monolithic beam of equal

mass. The geometry of the optimal pyramidal and diamond-celled
sandwich core beams of Fig. 17共a兲 are plotted in Fig. 17共b兲. For
both configurations, c̄ increases with increasing mass, with the
optimal pyramidal core beams having a lower c̄ 共and a higher h̄)
as compared to the optimal diamond-celled core beams.
In water blast, the sandwich beam out performs the monolithic
beam mainly due to the fact that the thin 共and therefore light兲
outer face of the sandwich beam acquires a smaller fraction of the
blast impulse compared to the relatively thick monolithic beam.
However, in the case of air blast, the full blast impulse is trans- Fig. 19 Comparison of the analytical predictions and the
mitted to the structure for both the sandwich and monolithic three-dimensional FE predictions of Xue and Hutchinson †6‡ for
the deflection of sandwich beams with a corrugated core. The
beams. The superior air blast resistance of sandwich beams to
beams have a mass M̄ Ä0.04 and are subjected to an impulse
monolithic beams, as seen in Fig. 17共a兲 is attributed solely to the
Ī Ä5Ã10À3 . The effect upon w̄ and w̄ o of „a… core relative den-
shape factor effect of the sandwich construction. To clarify this sity ␳¯ for c̄ Ä0.1 and „b… c̄ with the core relative density held
point, the deflection of a sandwich beam with a diamond-celled fixed at ␳¯ Ä0.04. The solid lines give the analytic solutions and
core (¯␳ ⫽0.1,c̄⫽h̄⫽0.2) is plotted in Fig. 18 as a function of the the dotted lines „with symbols… give the FE results.
air impulse for two assumed values of core densification strain
⑀ D ⫽0.01 and ⑀ D ⫽0.5 along with the response of a monolithic
beam of equal mass M̄ ⫽0.2. Figure 18 reveals that the beam with
the core densification strain ⑀ D ⫽0.01 which maintains the sepa- solid material response, and include elastic buckling of the core
ration of the face sheets and is the strongest while the monolithic members by assuming a solid material yield strain ⑀ Y ⫽0.2%. In
beam is the weakest: it is the shape factor effect that gives the line with experimental data for metal foams, we take the densifi-
sandwich construction structural advantage in air blast. cation strain ⑀ D of the core to be related to the relative density ¯␳
through, 关2兴,
6 Comparison With Three-Dimensional Finite
Element Simulations
⑀ D ⫽0.8⫺1.75¯␳ . (72)
Xue and Hutchinson 关6兴 conducted three-dimensional finite el-
ement 共FE兲 simulations of the dynamic response of clamped sand-
wich beams with the corrugated, square-honeycomb, and pyrami- Xue and Hutchinson 关6兴 investigated the effects of core relative
dal core geometries. In these FE simulations, Xue and Hutchinson density and core thickness for sandwich beams of total mass M̄
关6兴 modelled the core members explicitly including the develop- ⫽0.04 and considered an impulse Ī ⫽5⫻10⫺3 . Comparisons be-
ment of contact between the core members and the face sheets tween the FE and analytical predictions of the maximum face
under increasing through-thickness compressive strain. An im- sheet displacements of the corrugated core sandwich beams are
pulse was applied to the front face of the sandwich beam and thus shown in Fig. 19: In Fig. 19共a兲 the effect of core relative density
their numerical results can be compared directly to our analytical is investigated with c̄⫽0.1, while in Fig. 19共b兲 the effect of c̄ is
predictions for air blast, with ¯␺ ⫽0. studied for a core of relative density ¯␳ ⫽0.04. While the analytical
Xue and Hutchinson 关6兴 modeled sandwich beams made from predictions are within 15% of the FE calculations in all cases, the
304 stainless steel and assumed an elastic, power-law hardening analytical model does not capture the qualitative form of the
stress versus strain response for the solid steel with a yield strain variations as predicted by the FE analysis. A careful comparison
⑀ Y ⫽0.2% and a power law hardening exponent N⫽0.17. In the with the FE results indicates that this is mainly due to the fact that
analytic predictions given below we assume a rigid, ideally plastic for Ī ⫽5⫻10⫺3 , the analytical solutions predict full densification

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 neglected. Despite these approximations, the analysis has been
shown to compare well with three-dimensional FE calculations.
Thus, the analysis presented here is not only adequate to explore
trends and scaling relations but is also expected to suffice to make
approximate predictions for the purposes of selecting core topolo-
gies and sandwich beam geometries. The nondimensional formu-
las presented here bring out the stages of the response clearly and
hence aid the interpretation of more accurate numerical calcula-
tions such as the recent dynamic finite element analysis of Xue
and Hutchinson 关6兴.
Two notes of caution on the model presented here must be
mentioned. Recent numerical fluid-structure interaction calcula-
tions on similar sandwich structures performed by Belytschko关14兴
indicate that the one-dimensional Taylor analysis underestimates
the impulse transmitted into the sandwich structure and thus the
performance gains due to sandwich constructions indicated here
may be somewhat optimistic. Second, the failure of the face sheets
near the supports by dynamic necking have not been addressed
here. Additional investigations are required to establish an appro-
Fig. 20 Comparison of the analytical predictions and the priate failure criterion under dynamic conditions.
three-dimensional FE predictions of Xue and Hutchinson †6‡ for
the deflection w̄ of monolithic beams and sandwich beams with
corrugated, square-honeycomb, and pyramidal cores. The 8 Concluding Remarks
beams have a fixed mass M̄ Ä0.04 and the sandwich beams
have a core of relative density ␳¯ Ä0.04 and aspect ratio c̄ An analytical methodology has been developed to analyze the
Ä0.1. The symbols denote the FE results while the lines are the dynamic response of metallic sandwich beams subject to both air
analytical predictions. and water blasts. The response of the sandwich beams is separated
into three sequential stages: Stage I is the fluid-structure interac-
tion problem, stage II is the phase of core compression, and in
in nearly all cases while in the FE simulations no distinct densi- stage III the clamped beam is brought to rest by plastic stretching
fication limit exits; rather, continued core compression occurs at and bending. The simple analytical formulas presented above are
increasing stress level after contact has begun between the core in good agreement with more accurate three-dimensional FE cal-
members and the face sheets. An improved core constitutive culations given in a parallel study of Xue and Hutchinson 关6兴.
model with continued hardening rather than lockup after some The analysis has been used to construct performance charts for
critical strain ⑀ D may be able to address this deficiency; this is the response of both monolithic and sandwich beams subject to
however beyond the scope of the present study. both air and water borne blasts. For the case of water blast, an
Xue and Hutchinson 关6兴 employed a series of FE calculations to order of magnitude improvement in blast resistance is achieved by
identify a ‘‘near-optimal’’ sandwich configurations with mass M̄ employing sandwich construction. This is mainly due to fluid-
⫽0.04. They concluded that a sandwich beam with a core of structure interaction: The reduced mass of the sandwich outer face
relative density ¯␳ ⫽0.04 and c̄⫽0.1 共giving h̄⫽0.08) is an opti- leads to a reduction in the impulse transmitted to the structure
mal configuration for a moderately large blast. Comparisons be- from the water. In air, the impedance mismatch between air and
tween the FE and analytical predictions 共with the choice ⑀ D the face sheet is comparable to that between air and a monolithic
⫽0.5) of the deflections of the inner face sheet of these ‘‘opti- beam; consequently, the use of sandwich construction gives a
mum’’ sandwich beams as a function of blast impulse are shown more moderate gain in blast resistance compared to monolithic
in Fig. 20. Over the range of impulses considered, the analytical construction. For both air and water blast the diamond-celled core
predictions are within 10% of the three-dimensional FE calcula- sandwich beam gives the best performance due to the longitudinal
tions for the pyramidal, corrugated and square-honeycomb core strength provided by the core. Comparisons of the predictions
sandwich beams as well as for the monolithic beams. Note that the presented here with three-dimensional coupled fluid-structure nu-
FE calculations predict that the monolithic beam out performs the merical calculations and blast experiments need to be performed
pyramidal core sandwich beam 共i.e., smaller deflections at the to validate and extend this analysis.
same impulse兲 for impulses Ī ⬎5⫻10⫺3 . This is due to the wrin-
kling of the face sheets between the nodes of the pyramidal truss.
While this effect is not included in the current analysis, the ana- Acknowledgments
lytical model too will predict that the monolithic beam out per- The authors are grateful to ONR for their financial support
forms the pyramidal core beam at large deflections: at large de- through US-ONR IFO grant number N00014-03-1-0283 on the
flections the degree of axial stretching becomes significant, yet the The Science and Design of Blast Resistant Sandwich Structures.
pyramidal core provides no longitudinal strength. We are pleased to acknowledge Profs. M. F. Ashby, T. Belytschko,
A. G. Evans, J. W. Hutchinson, R. M. McMeeking, and F. Zok for
7 Discussion many insightful discussions during the course of this work.
An approximate analytical methodology has been presented to
predict the dynamic response of sandwich beams to blast loadings
in air and water. A number of approximations have been made to References
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