International Journal of Impact Engineering 38 (2011) 837e848
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International Journal of Impact Engineering
journal homepage: www.elsevier.com/locate/ijimpeng
Wet-sand impulse loading of metallic plates and corrugated core sandwich panels
J.J. Rimoli a, B. Talamini a, J.J. Wetzel b, K.P. Dharmasena b, R. Radovitzky a, H.N.G. Wadley b, *
a
b
Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Department of Materials Science and Engineering, University of Virginia, Charlottesville, VA 22904, USA
a r t i c l e i n f o
a b s t r a c t
Article history:
Received 11 November 2010
Received in revised form
17 May 2011
Accepted 20 May 2011
Available online 30 May 2011
We have utilized a combination of experimental and modeling methods to investigate the mechanical
response of edge-clamped sandwich panels subject to the impact of explosively driven wet sand.
A porthole extrusion process followed by friction stir welding was utilized to fabricate 6061-T6
aluminum sandwich panels with corrugated cores. The panels were edge clamped and subjected to
localized high intensity dynamic loading by the detonation of spherical explosive charges encased by
a concentric shell of wet sand placed at different standoff distances. Monolithic plates of the same alloy
and mass per unit area were also tested in an identical manner and found to suffer 15e20% larger
permanent deflections. A decoupled wet sand loading model was developed and incorporated into
a parallel finite-element simulation capability. The loading model was calibrated to one of the experiments. The model predictions for the remaining tests were found to be in close agreement with
experimental observations for both sandwich panels and monolithic plates. The simulation tool was then
utilized to explore sandwich panel designs with improved performance. It was found that the performance of the sandwich panel to wet sand blast loading can be varied by redistributing the mass among
the core webs and the face sheets. Sandwich panel designs that suffer 30% smaller deflections than
equivalent solid plates have been identified.
2011 Elsevier Ltd. All rights reserved.
Keywords:
Sandwich panels
Blast loading
Impulse mitigation
1. Introduction
Metallic sandwich panel structures with cellular cores made
from ductile, ferritic, and in some cases corrosion resistant stainless
steels have attracted significant interest for protecting structures
from impulsive loading. The performance and advantages of
sandwich panel construction have been thoroughly studied in the
case of water blast, and to a lesser extent for air blast. In each of
these two cases the sandwich structure performance is controlled
by the design of the panels (i.e., core topology and geometry) [1,2],
the mechanical properties and density of the material [3], the
integrity of the joints [4], and the nature of the impulse transfer
from the surrounding medium to the structure [5e9]. There has
been relatively less literature on impulsive loading with soil as the
medium.
In the case of water and air blast, efforts have focused on
exploiting the so-called fluid structure interaction (FSI) effect to
reduce the impulse imparted to the structure. For explosive loading
in water and air (in the acoustic weak shock limit), the impulse
transferred to a massive plate is twice the incident impulse due to
* Corresponding author.
E-mail address: [email protected] (H.N.G. Wadley).
0734-743X/$ e see front matter 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ijimpeng.2011.05.010
complete shock reflection. However, when low inertia (thin and
light) plates are impacted by these shocks, the reflection factor can
be decreased and the impulse acquired by the structure is then less
than that of the high inertia system [5,3]. As deduced in Taylor’s
seminal work [5], the ratio of transferred to incident impulse, I/I0, is
controlled by a dimensionless parameter b ¼ rwcwt0/mf, where rw
and cw are the density and sound speed of the fluid respectively, t0
is the characteristic decay time for the pulse, and mf is the mass per
unit area of the front face sheet. The acoustic limit for the impulse
transfer ratio derived by Taylor is given by I/I0 ¼ 2bb/(1eb). In water,
sandwich panels may exploit the FSI effect by using light face sheets
supported by soft cellular structure cores [3]. Reductions of impulse
as high as 25% have been experimentally reported for steel face
sheets that are several millimeters thick supported by cores with
a compressive strength of 5e10 MPa [10]. Improvements to Taylor’s
expression have been proposed by Hutchinson et al. [11] to account
for the resistance to the motion of the front face sheet offered by
the core in sandwich panels. The level of impulse reduction
depends upon the time and location of cavitation after the shock
reflection, which in turn is affected by panel design [12].
Panel designs that seek to exploit the FSI effect in water have
utilized a variety of core topologies and geometries including:
Y-truss [2], corrugated [13], triangular honeycomb [14], square
honeycomb [15], and lattice truss cores [16], among others. These
838
J.J. Rimoli et al. / International Journal of Impact Engineering 38 (2011) 837e848
concepts use a tough outer face sheet that is impacted by the
waterborne shock and a relatively low-strength core. The impulsive
loading causes the outer face sheet to accelerate and to crush the
core. At larger deflections, additional resisting forces are activated by
membrane stretching of the front face sheet, and for severe impulsive loading, by stretching of the back face sheet and core [17,18].
The FSI effect for air blasts has been studied theoretically by
Kambouchev et al. [8,9,19]. They extended Taylor’s analysis to the
nonlinear regime where the compressibility of the fluid is pressure
dependent [8]. Their approach allowed a decoupling of the structural loading from the fluids dynamics. It was found that nonlinear
compressibility effects further decrease the transmitted impulse in
comparison to the acoustic case. However, the effect is only present
for very high intensity loading and very low inertia face sheets. In
practice, the requirement of high intensity and light face sheets
may limit the practicality of exploiting FSI effects in the design of
air blast-mitigating systems [7,19]. Even so, it is important to
recognize that sandwich panels have still demonstrated improved
performance over equivalent mass per unit area solid plates by
their increased plastic bending resistance [18].
If an explosive charge is buried in soil, the detonated explosive
creates a shock that propagates through the soil [20]. The shock
front is reflected at the soileair interface causing soil spallation. The
majority of impulse that loads a structure arrives by impact of soil
particles [20]. The success of sandwich panels in blast protection
applications has naturally extended interest to their use for soil
blast. However, engineering design tools for computing the structural loading due to the impacting soil are not well developed.
Soil blasts can be numerically modeled by explicitly considering
the detonation process, the soil response, and its interaction with
the structure and surrounding air. In general, creating constitutive
models for the soil response is difficult due to wide variations in soil
particle properties and level of water saturation [20,21]. Soil
constitutive models incorporating some of these effects have been
proposed [20e23] and used with coupled codes, e.g. [24e26]. This
method has the drawback that it is computationally expensive,
requiring significant effort to compute the fluid fields even when
the primary interest is the structural response. Recent experimental and numerical studies [27], indicate that the FSI effect
during soil blast is small (and of secondary importance to the
design of protective structures), suggesting that decoupled soil and
structural simulation may be a viable and efficient panel design
option.
An alternative to coupled simulation for determining the soil
blast loading is to use empirical models, such as that of Westine el al
[28]. This model provides a (spatially varying) specific impulse
given the problem geometry and the size of the charge. Load is
applied as an initial velocity presuming that the structural response
time is long compared to impulsive loading from the soil. The
approach is simple, although its validity is restricted to the range of
the original experimental conditions. Furthermore, when
compared to other loading models that incorporate a pressure time
history, the initial velocity approach can significantly overestimate
the core crushing of sandwich panels [7].
The objectives of the present study are to (i) understand the
mechanical response of edge-clamped sandwich panels subject to
the impact of explosively driven wet sand, (ii) develop a modeling
approach that enables the prediction of such response as a function
of the impulse spatio-temporal distribution, the geometry of the
panels and their material properties, and (iii) explore a model-based
optimization of the panel design. To address these objectives,
a combined experimental and modeling approach was adopted.
For the experimental section of the study, a set of 6061-T6
aluminum sandwich panels with corrugated cores were fabricated.
The panels were subsequently edge clamped and subjected to
localized high intensity dynamic loading by explosively accelerated
wet sand.1 Spherical test charges were detonated at different
standoff distances from the target panels, and a post-mortem
analysis of the specimens was performed and their permanent
deformation measured. Monolithic plates with same mass per unit
area and made from the same alloy were also tested in an identical
manner.
A decoupled loading model was developed and incorporated
into a parallel finite-element simulation capability to analyze the
experiments. The loading model was calibrated to match the results
of one experiment of the test matrix, and the values obtained
through the calibration procedure were then adopted for all
subsequent calculations. The simulation tool was then utilized to
explore alternative sandwich panel designs.
The remainder of this article is organized as follows: In Section 2
we describe the experimental test setup. In Section 3 we present
the computational model. In Section 4 experimental results are
presented and compared to simulation results as a means for model
assessment and experimental interpretation. The model is then
used to investigate the panel design space and potential improvements to the panel design are presented. It is shown that significant
panel deflection reductions can be achieved by redistributing panel
mass from the core to the back face sheet.
2. Experimental procedures
Experiments were conducted on a simplified system to understand the structural response of metallic sandwich panels to wet
sand blast (in particular as a function of blast standoff). The
experimental design, similar to that of Deshpande et al. [20], aims
to produce a controlled loading environment that reproduces the
relevant physical interactions. We note that the experimental
design is not intended to reproduce the conditions of any particular
explosive device or scenario.
The experiments were conducted on aluminum alloy sandwich
panels with a triangular corrugated and I-core and 1.7 cm thick
solid plates of the same alloy and areal density (mass per unit
area). Spherical test charges were placed at different standoff
distances from the center of the target panels and plates and
detonated to evaluate the effects of the impulsive loading on the
structures. After testing, the plates and panels were water-jet
sectioned and their permanent deflections and deformation
modes were evaluated.
A detailed explanation of the panel fabrication processes, the
preparation of the test charge, the panel testing configuration, and
the test matrix adopted for this work is given in the subsections
below.
2.1. Panel fabrication
The sandwich panels were fabricated from a 6061 aluminum
alloy using a combination of porthole die extrusion and friction
stir welding. The panels were extruded by Mid East Aluminum
(Mountaintop, PA) into 0.143 m wide, w 3 m long stick extrusions
with the corrugated structure shown in Fig. 1, using a 17.8 cm
diameter, 300 ton direct extrusion press operating at 482 C. The
panels were heat-treated to the T6 (peak strength) condition after
extrusion. The core of the extruded structures had a web
1
The choice of studying wet sand over dry sand is motivated by the following
conclusions of Deshpande et al. [20]: (i) wet sand blast applies higher average
pressures and impulses to the structure, hence restricting attention to this case is
conservative, and (ii) wet sand encapsulates the expanding explosive gases and
therefore avoids the complication of considering separate air and soil blasts.
J.J. Rimoli et al. / International Journal of Impact Engineering 38 (2011) 837e848
839
Fig. 1. (a) The aluminum extrusion process. (b) The extruded aluminum panel.
inclination angle of 60 , an inclined web thickness of 3.2 mm,
a core height of 19.3 mm, and 5.2 mm thick top and bottom face
sheets and (Fig. 1). The relative density of the corrugated core was
25.2%. The core’s areal density was 13 kg/m2 while that of each of
the faces was 14 kg/m2. The edges of the extrusions were terminated with vertical solid webs to facilitate the fabrication of wider
square panels. These vertical webs were machined to a thickness
of 4.75 mm.
To create panels of the required width for testing, five extrusions
were friction stir welded side-by-side at Friction Stir Link, Inc.
(Waukesha, WI). The process was performed using a 17 mm
diameter welding tool turning at a speed of 1000 RPM as schematically illustrated in Fig. 2. The 4.75 mm wide vertical trusses at
the two sides of the extrusion (Fig. 1) were retained to minimize the
distortion of the structure due to the high pressures generated
during the stir welding process. The vertical trusses also shielded
the welds from some stress during subsequent testing and
compensated for the local decrease in mechanical properties
associated with the welding process.
The resulting test panels were 0.632 m wide and 0.610 m long
(Fig. 2). The relative density of the hybrid core including the vertical
webs was 29% and the areal density of the entire sandwich panel
was 46 kg/m2 (equivalent to a solid 6061-T6 aluminum panel of
1.7 cm thickness).
Since the sandwich panel face sheets were vulnerable to pullout at the bolts used to edge clamp the panels during testing,
four 10.6 cm wide strips along the panel edges were filled with
a Crosslink Epoxy (CLR1061 resin/CLH6930 hardener) to create an
outer strengthened panel border.
2.2. Charge preparation
Model test charges were made by packing nominally 200 mm
diameter glass microspheres (Mo-Sci Corp, Rolla, MO) in a 152 mm
840
J.J. Rimoli et al. / International Journal of Impact Engineering 38 (2011) 837e848
Fig. 2. (a) Joining of extruded panels by Friction Stir Welding (b) The test panel.
diameter, thin-walled polyester glycol plastic sphere, concentric with
an internal 80 mm diameter, thin-walled acrylic shell filled with 375 g
of C-4 explosive. The test charge assembly sequence is shown in Fig. 3.
First, a measured mass of explosive was tightly packed inside
the 80 mm diameter plastic sphere. A thin wooden rod was then
inserted through the explosive sphere to facilitate subsequent
suspension of the charge. To enable later insertion of a detonator,
a thin-walled polymer tube was then inserted into a hole cut in the
north pole of the plastic sphere and hot glued to the surface
(Fig. 3a). This 80 mm plastic sphere was then centered inside
a hemispherical, 152 mm diameter plastic shell. The hemispherical
shell was partially filled with 200 mm diameter glass microspheres
(artificial “sand”), periodically vibrating the hemisphere so that the
microspheres were tightly packed. The top hemispherical half of
the outer sphere shell was then positioned on the bottom hemisphere, and the seam joint sealed with hot glue (Fig. 3b). The
internal volume of the outer sphere was then completely filled
with more glass microspheres (Fig. 3c) and the weight of sand
measured. The mass of sand used in the charges was 2.466 kg.
Finally, water was poured into the larger sphere to fill the void
space between the glass microspheres (Fig. 3d). The mass of water
added to the charges was 0.617 kg and the total mass of the “water
saturated sand” was 3.083 kg.
2.3. Panel testing configuration
The impulsive loading experiments were conducted at the
Force Protection Inc. Explosives Test range located near Edgefield,
SC using a test geometry shown schematically in Fig. 4. Approximately square (61 cm 61 cm) test panels were mounted horizontally to a 3.8 cm thick steel plate bolted to a welded I-beam
frame resting on a concrete support base. A 19.5 mm diameter,
32-hole pattern in the steel plate allowed the test panels to be
securely edge clamped in the test configuration. The 3.8 cm thick
steel plate contained a 41 cm 41 cm square center opening to
enable the unrestricted deformation of the test panel when
subjected to a centrally applied impulse load. The sand-encased
spherical charges were suspended above the center of the test
panels at varying standoff distances, defined as the distance from
the center of the charge to the impact surface of the test panels,
Fig. 5.
2.4. Test matrix
A series of 10 experiments were conducted. Five of these were
performed at standoff distances of 15, 19, 22, 25 and 30 cm
(measured from the front face) with an equivalent weight
J.J. Rimoli et al. / International Journal of Impact Engineering 38 (2011) 837e848
841
Fig. 3. Spherical “soil” encapsulated explosive charge preparation.
monolithic 6061-T6 plate of thickness 1.7 cm, Fig. 5a, and the
remaining five by replacing the solid plates with five corrugated
core sandwich panels, at the same standoff distances, Fig. 5b. The
panels were water-jet sectioned after testing and the panel
deflections and deformation modes were evaluated.
3. Computational modeling
We conducted computer simulations of the plates and sandwich
panels subjected to impulsive wet sand loading (i) to study details
of the experiments that are difficult to measure; and (ii) to identify
potential improvements to the panel geometry.
In the course of this endeavor, we utilized a simplified, decoupled method for applying the soil impact loads to the structures.
This spatio-temporal distribution of the loading was calibrated
against one experiment from the testing matrix based upon
a recently proposed soil model developed by Deshpande et al. [20].
In the final step, we compared two strategies for improving the
sandwich panels (measured in terms of core crush and maximum
back face deflection) with numerical simulations.
motion [30]. The spatial discretization of the plate and panel geometries utilized 10-node quadratic tetrahedral elements. The mesh
used for the analysis of the solid plate comprised approximately
160,000 elements while that for the extruded panel contained
approximately 200,000 elements. Time integration was performed
with the explicit, second order accurate, central-difference scheme,
which falls within the Newmark family of time stepping algorithms.
3.2. Constitutive modeling
The constitutive response of 6061-T6 aluminum was modeled
using an isotropic, finite deformation variation of the Johnson-Cook
viscoplastic model [31], formulated within the framework of variational constitutive updates developed by Radovitzky and Ortiz
[32], Ortiz and Stainier [33], Yang et al. [34], and Ortiz and Pandolfi
[35]. To avoid complications from fracture, we disregarded the
experimental results that display large-scale failure (the sandwich
panel at 15 cm standoff) since we did not attempt to model fracture.
We adopted a modified JohnsoneCook flow stress Y defined as
"
ep
#n
2
0
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 13
u
2
p
u
_
e_ p C7
6
B e
41 þ Clog@ p þ t1 þ
A5
p
2e_ 0
2e_ 0
3.1. Finite element modeling
Y ¼ s0 1 þ
The responses of the plates and sandwich panels were simulated
using a parallel finite-element computational framework [29] based
on a Lagrangian finite deformation formulation of the equations of
where ep is the equivalent plastic strain, e_ p is the equivalent plastic
strain rate,
ep0
(1)
842
J.J. Rimoli et al. / International Journal of Impact Engineering 38 (2011) 837e848
Fig. 4. Schematic of experimental test setup.
ep0 ¼
s 1
0 n
B
(2)
p
is a reference plastic strain, and e_ 0 is a reference plastic strain rate.
The choice of this flow stress avoided singularities in the traditional
Johnson-Cook hardening as e_ p /0 and in the hardening modulus as
ep /0, in accordance with the variational update method requirements. The constitutive model is supplemented with an isotropic
Hencky hyperelasticity extension of linear elasticity.
The strain-rate effect parameter C was obtained from the literature [36], fitted at a reference strain rate e_ p ¼ 1 s1 . The Young’s
modulus E and the strain hardening parameters n, B, and s0 were
calibrated to quasi-static tensile tests, conducted on coupons cut
from the faces of the finished panels and measured in the direction
of extrusion. Fig. 6 compares the measured uniaxial true stressstrain data to the constitutive law prediction. The derived constitutive parameter values are shown in Table 1, along with the
density r.
Fig. 5. Variable standoff test arrangement of (a) equivalent weight solid plate (b)
extruded and friction stir welded sandwich panel.
3.3. Wet sand blast loading model
Traditionally, the modeling of blast loads on solids due to
explosions have been based on empirical formulas, such as the
Westine model [28]. This method has proven fruitful in design;
however, it is limited in applicability to conditions similar to the
original calibration experiments. Another approach has been to
develop constitutive models for soils, e.g. Grujicic et al. [21] and
Grujicic and Cheeseman [23]. Such models may then be used in
coupled FSI codes to simulate virtually any scenario, within the range
of validity of the constitutive model. In the present context, this
approach has the disadvantage that significant computational effort
is expended on the soil mechanics, which here is of less interest than
the response of the structure. Since the FSI effect during impulsive
Fig. 6. Modified Johnson-Cook material model fit against uniaxial test data.
J.J. Rimoli et al. / International Journal of Impact Engineering 38 (2011) 837e848
is a function of the distance r to the charge center, and takes
the form
Table 1
Material parameters used in numerical simulations.
E
n
s0
B
n
C
e_ p0
r
72 GPa
0.343
250 MPa
365 MPa
0.04
0.002
1
2700 kg/m3
soil loading is small, we created a decoupled loading model with
a priori determined tractions that incorporates information from
a constitutive model applicable to impulsively loaded soil [20]. This
approach may be viewed as a compromise between the two previous
methods, with some of the advantages of both.
Deshpande et al. [20] performed simulations of dry and wet
sand impact from test charges similar to those investigated here.
These simulations provided values for the time dependence of
the impulse I and the average pressure p on a target panel or
plate as a function of the radial distance r. Our loading consists of
a time-dependent surface traction based upon the impulse and
pressure data obtained from this model. The introduced coefficients were calibrated by fitting to one of the sandwich panel
experiments (at 30 cm standoff). A brief description of this
procedure follows; additional mathematical details are contained
in an appendix.
Consider a point P on the target surface located a distance r from
the center of the charge O as depicted in Fig. 7. As shown in the
Appendix, the load at the point P can be reduced to the application
of a time-dependent pressure pP ðtÞ given by
pP ðtÞ ¼
pðrÞsin2 ðqÞ; if t˛ taP ; taP þ tdP
0 otherwise
(3)
where p(r) is the pressure as a function of distance to the charge
center, and the arrival time taP and the duration of the pulse tdP are
given by
"
r0 V0
1
1
4pað1 þ bÞ r 1þb r 1þb
0
taP ¼
arb
tdP ¼
pðrÞ
#
(4)
(5)
In these equations r0 and V0 are the initial average density and
volume of the sand-water mix surrounding the explosive charge. In
addition, we assumed that the impulse, defined as
Z
IðrÞh
pðtÞdt
843
IðrÞ ¼ arb
(7)
The expressions for the impulse I(r) and the pressure p(r) were
obtained by a procedure which utilizes calculated wet sand blast
data for a similar experimental setup (from [20]), followed by
calibration against the current experimental data. The calibration
procedure is discussed in Section 4.2.
4. Results and validation of the model
4.1. Experimental results
The effect of changing the standoff distance upon the deflection
of a 17 mm thick monolithic aluminum 6061-T6 plate (with a mass
per unit area identical to the sandwich panels) is shown in Fig. 8. All
of the tests caused significant bending and plastic stretching of the
panels. This increased in severity as the standoff distance was
decreased. A small crack formed on the loaded surface near the
mid-point of the gripped edge of the most intensely loaded panel.
The effect of the same impulsive load upon the sandwich panel
structures is shown in Fig. 9. Sandwich panel permanent deflection
were 15e20% less than those of the solid plates with the exception
of the most intensely loaded panel, Fig. 9a. In this case, face sheet
fracture at the edge and center of the panel occurred. Edge cracking
was also evident in the front and back faces of the panel tested at
a standoff of 19 cm (Fig. 9b) and facilitated additional panel
deflection over that due to face sheet stretching and panel bending.
Face sheet stretching and deflection between corrugation web
nodes in the most intensely loaded regions of the panels is also
evident (see Fig. 9a and b). At lower impulses, the core was only
modestly compressed by inelastic buckling of the corrugation webs
and sandwich action was preserved for a significant fraction of the
loading event.
4.2. Model analysis
The three loading parameters were obtained by incorporating
results obtained from Deshpande’s calculations [20] with a calibration against one experimental test result. For wet-sand blasts,
Deshpande’s model predicts that the pressure is approximately
(6)
tdP
Fig. 7. Sand blast front schematics. The dashed lines represent the position of the wet
sand blast front for times t ¼ taP0 and t ¼ taP .
Fig. 8. Sectioned photos of equivalent weight solid plates tested at standoff distances of,
(a) 15 (b) 19 (c) 22 (d) 25 and (e) 30 cm. A small crack formed at the region circled in (a).
844
J.J. Rimoli et al. / International Journal of Impact Engineering 38 (2011) 837e848
Fig. 10. Plate displacement vs. time for 30 cm standoff distance.
deflection of the front and back face sheets is estimated as the
average of the peak values of the corresponding oscillation:
d ¼
Fig. 9. Sectioned photos of sandwich panels tested at standoff distances of, (a) 15 (b)
19 (c) 22 (d) 25 and (e) 30 cm.
constant for the range of standoff distances considered in this work.
Additionally, the impulse decay with distance is well characterized
by a power law:
IðrÞ ¼ arb pðrÞ ¼ p0
(8)
In our calculations, the exponent b is derived from published
simulation data in Eq. (20) for a similar experimental setup; the
impulse versus standoff distance data for fully saturated wet sand
were fitted to the power law (Eq. (8)) and the resulting exponent was
adopted for the current simulations. The remaining loading parameters p0 and a were calibrated through simulation by matching the
values of the core strain and the permanent deflection at the center
of the back face of the corrugated panel for a standoff of 30 cm.
The loading parameters found through this procedure are listed
in Table 2. From Eq. (8), the predicted peak impulses ranged from
3.28 kN s/m2 at r ¼ 30 cm to 6.00 kN s/m2 at r ¼ 19 cm. We note that
a b value less than 2 implies a weaker than inverse square law
dependence of impulse upon standoff distance for fully saturated
wet sand. Further assessments of this finding await fully coupled
simulations that account for the direction of the explosives detonation wave front (in this case towards the target), impulse transfer
from the explosive to the wet sand and the interaction of the radially
stretching wet sand shell with a dynamically deforming target.
Fig. 10 shows the displacements at the centers of both panel face
sheets as a function of time for the calibration case. The application
of the wet sand blast load to the panel produces an initial rapid
deflection, peaking at tp ¼ 0.36 ms. The initial peak is followed by
a series of oscillations with period T z 0.61 ms. The permanent
dmax dmin
2
(9)
The last oscillation of the simulation window t ¼ [0, 0.002] s is used
for this calculation. From Fig. 10, it can be observed that the front
face exhibits a larger permanent deflection than the back face. This
difference, divided by the core height defines the permanent core
strain:
epcore ¼
dfront dback
hcore
(10)
where hcore ¼ 19.1 mm.
After calibrating the loading parameters, the other experiments
were simulated using the same loading parameters.2 In total, the
simulation suite comprises both plates and corrugated panels
subjected to wet sand blast loads at standoffs of 19 cm, 22 cm,
25 cm and 30 cm. Fig. 11 shows the deformed shape of sectioned
panels after the simulation concluded.
Fig. 12 shows the numerical results for the corrugated panels
together with their experimental counterparts. As expected, the
deflection for the 30 cm standoff distance case exactly matched the
experimental observation since this was the calibration case. The
remaining three cases (at standoffs of 25 cm, 22 cm and 19 cm) all
are in good agreement.
Fig. 13 compares the numerical deflection results and the
experimental observations for the monolithic plates. Good agreement is again observed. The worst case, at 19 cm standoff, exhibits
an 8% difference with the corresponding experiment. It bears
emphasis that the good agreement observed in both the plate and
sandwich panel simulations was obtained with identical loading
parameters a, b, and p0 for every simulation, using only one case for
calibration (the sandwich panel at 30 cm standoff).
Comparisons of the center permanent deflections of the sandwich panel, Fig. 12, and monolithic plate, Fig. 13, show that the
sandwich panel suffered 15e20% smaller deflections than
the monolithic plates. Examination of Fig. 11 shows that the
Table 2
Calibrated loading parameters.
a
b
p0
0.665 kN s/m2
1.325
90 MPa
2
Simulation results for the standoff distance of 15 cm are not reported since the
current model neglects failure. The experimental results indicate that extensive
failure occurs in the sandwich panel at this standoff, see Fig. 9.
J.J. Rimoli et al. / International Journal of Impact Engineering 38 (2011) 837e848
Fig. 11. Simulation results for panels at standoff distances of, (a) 19 (b) 22 (c) 25 and
(d) 30 cm. A quadrant of the panel has been removed (on the left) to show the panel
deflections at the middle of the panel.
simulations predicted modest core compression, in agreement with
experimental observations, cf. Fig. 9.
4.3. Exploration of the design space
The validated model provides a simple means for exploring the
panel design space by calculating the response of corrugated panels
with variations of the original design. We have restricted our
analysis to cases where the core compression strain did not exceed
50% (whereupon the panels’ sandwich action is lost). Panel modifications explored two different strategies:
e Case 1: core mass fraction variation. The overall core area was
reduced by evenly decreasing the thickness of both vertical and
inclined webs. The total panel mass was kept constant by adding
the subtracted mass to the back face sheet. In addition to the
original panel, two cases were considered, with 10% and 20%
core density reduction.
Fig. 12. Sandwich panel center displacement (back face) vs. standoff distance. The
incident impulse used in the simulations is also shown.
845
Fig. 13. Plate center displacement (back face) vs. standoff distance.
e Case 2: vertical web variation. The core mass was reduced
by decreasing the thickness of the I-core (vertical) webs only.
The total panel mass was again kept constant by adding the
subtracted mass to the back face sheet. In addition to the original panel, three cases were considered, with 20%, 60% and 100%
reduction of the vertical web thickness. The latter case corresponds to a panel without vertical webs.
Fig. 14 shows the permanent deflection results obtained for case
1. A reduction of the total core mass produced a slight decrease of
the permanent back face deflection for each standoff distance.
Larger reductions in core mass result in decreasing back face
deflections but significantly more core crushing, Fig. 17a.
Fig. 15 summarizes the permanent deflection trends for case 2.
Here, reduction of the total core mass decreased the permanent
back face deflection for all standoff distances. The limiting case of no
vertical webs exhibited the best performance of any case considered
resulting in 30% reduction in deflection compared to the solid plate.
No interpenetration was observed for any of these simulations.
Fig. 14. Panel center displacement (back face) vs. standoff distance for different values
of core web thickness.
846
J.J. Rimoli et al. / International Journal of Impact Engineering 38 (2011) 837e848
Fig. 17. Panel effective plastic strain for 19 cm standoff distance for different designs.
Fig. 15. Panel center displacement (back face) vs. standoff distance for different values
of vertical web thickness.
The successful reduction of back face deflection relative to the
original design is tempered by increments in core crush strain at all
standoff distances, particularly for case 1. Fig. 16 compares the core
crush strain of the original design to the best performing panels
from cases 1 and 2. Fig. 17 illustrates this effect for the 19 cm
standoff distance. The difference in the core crush strains can be
explained by the occurrence of different failure mechanisms.
In case 1, the stubby vertical webs imposed significant constraint
upon front face motion, limiting the buckling of neighboring inclined
webs. This constraint resulted in an inhomogeneous buckling
(crushing) of the core. As a consequence, failure localized between
the innermost vertical webs, and energy was not dissipated
uniformly (See Fig. 17a). The effective plastic strain attained
a maximum value ep ¼ 0.48 in the inclined webs, which is significantly greater than the value reached in the original design, ep ¼ 0.25.
This localized buckling in turn caused greater bending and stretching
in the top face sheet, implying an inefficient use of material.
The response is markedly different when no vertical webs are
present. In this scenario buckling proceeded more homogeneously
throughout the core; see Fig. 17b. Consequently, energy was
initially dissipated though distributed plastic buckling of the core
Fig. 18. Center displacement (back face) vs. standoff distance for the monolithic plate,
original panel design and optimized panel design.
webs across the entire panel, and later through membrane
stretching of both face sheets. The effective plastic strain in the core
webs attained a maximum value of ep ¼ 0.28, which was similar to
the original design. Strains throughout the face sheets were more
uniform in comparison to cases with vertical webs.
In all cases considered, high effective plastic strains were formed
in the face sheets near the clamped edges. Their location corresponded to the region where face sheet failure occured in the most
intensely loaded experiments. The deformation in this region was
little affected by design modifications. The maximum effective plastic
strain was ep ¼ 0.33 in the original design, ep ¼ 0.34 for the design
shown in Fig. 17a, and ep ¼ 0.37 for the design depicted in Fig. 17b.
Fig. 18 summarizes the gains realized through the design space
exploration. The limited space design exploration led to the identification of panel designs that suffer about 30% smaller center deflections
than equivalent solid plates under localized wet-sand blast loading.
5. Summary and conclusions
Fig. 16. Panel core crush strain vs. standoff distance for different designs.
We have used a combined experimental and modeling approach
to investigate wet sand blast loading response of 6061-T6
aluminum plates and sandwich panels with corrugated cores of the
same mass per unit area.
The corrugated core panels suffered 15e20% smaller maximum
permanent deflections than equivalent mass per unit area monolithic plates of the same alloy. The sandwich panel deflection was
accommodated by a small (5e10%) crushing of the core and by face
sheet stretching. The permanent center deflection of the sandwich
panels increased with load intensity and failed by face-sheet fracture at the gripped panel edges and in the panel center for the most
intensely loaded panels.
J.J. Rimoli et al. / International Journal of Impact Engineering 38 (2011) 837e848
A parallel finite-element simulation capability based on
a Lagrangian large-deformation formulation of the equations of
motion was used to simulate the response of plates and panels
under wet sand blast loading. A modified Johnson-Cook constitutive model was adopted and its material constants were fitted to
match the experimentally obtained stress-strain response of the
aluminum. We developed a decoupled loading model informed
with Deshpande’s results which predict that the pressure on the
target is approximately constant for the range of standoff distances
considered in this work. By calibrating the loading model for one
single experiment (corrugated panel, 30 cm standoff distance) we
were able to predict the remaining 7 experiments with excellent
accuracy.
The validated computational model provided the means for
performing a brief exploration of the design parameter space of the
structures. Modifications to the original panel design were studied
by pursuing two different strategies: (i) core mass fraction variation, and (ii) vertical web variation. Different structural failure
mechanisms were observed during this exploration: heterogeneous core crushing for panels with vertical webs, and membrane
deformation of the panels as a whole for the case with no vertical
webs. The latter case corresponds to the best design obtained from
our brief exploration of the design space. In this case, the energy
dissipation is distributed evenly as elasticeplastic buckling of the
diagonal webs followed by a subsequent membrane stretch of the
panel front and back sheets as a whole.
rðrÞ4pr 2 vðrÞIðrÞ
r0 V0 ¼
rðrÞ ¼
vðrÞ ¼
r0 V0 2 pðrÞ 1
4p
IðrÞ2 r 4
Let us consider a point P on the target surface at a distance r
from the explosive charge center O as depicted in Fig. 7. The wet
sand velocity, pressure and density at the arrival point are
related by
rðrÞ ¼ rðrÞvðrÞ2
(11)
IðrÞ ¼ arb
(18)
This assumption, combined with eq. (17), implies a representation of the velocity of the form
vðrÞ ¼
4pa bþ2
r
r0 V0
taP
r0 V0
1
1
¼
4pað1 þ bÞ r 1þb r 1þb
0
tdP ¼
IðrÞ
pðrÞ
(13)
In the previous equation, w(r) is the thickness of the expanded
wet sand shell, related to the velocity and duration of the load td by
vðrÞIðrÞ
pðrÞ
(14)
Combining the last three equations, the conservation of mass
reduces to
(20)
(21)
At this point, we know that the point P is loaded by a constant
pressure from time taP to taP þ tdP , but we must be careful regarding
the definition of such a pressure. The data provided by Deshpande
et al. considers the point where the target surface is perpendicular
to the sand velocity, namely P0. Therefore, we must only consider
the normal component of the sand velocity at the arrival point:
vt ¼ v sinðqÞ
(22)
In this way, the load at the point P is reduced to the application
of a time-dependent pressure pP ðtÞ given by
References
VðrÞ ¼ 4pr2 wðrÞ
#
Furthermore, since p(r) is the average pressure, the duration tdP
of the pressure pulse is, by definition, given by
r0 V0 ¼ rðrÞVðrÞ
wðrÞ ¼ td ðrÞvðrÞ ¼
(19)
Assuming that t ¼ 0 for r ¼ r0 and integrating the previous equation
we obtain the position of the blast front as a function of time. In this
manner, by simple inversion, the arrival time can be obtained as:
pP ðtÞ ¼
where r0 and V0 are the initial average density and volume of the
explosive charge respectively and V(r) is the volume of the
expanded wet sand shell, approximated by
(17)
Since v is not constant, the arrival time taP must be obtained
through integration of equation 17. In order to solve for the arrival
time, let us assume that the impulse can be fit with a power law:
In addition, let us consider the mass conservation equation
assuming constant density across the sand blast
(12)
(16)
4p
IðrÞr 2
r0 V0
"
Appendix. Derivation of simplified wet sand blast loading
model
(15)
pðrÞ
Solving equations (11) and (15) simultaneously we obtain
standoff-dependent expressions for the arrival velocity v and the
density r:
Acknowledgments
We are grateful to Vikram Deshpande for helpful discussions of
this work. The experiments were conducted at the Force Protection
Industries Explosives Test Range (Edgefield, South Carolina) with
the considerable assistance of Keith Williams. Research was supported by the Office of Naval Research (ONR grant number N0001407-1-0764) as part of a Multidisciplinary University Research
Initiative (David Shifler, Program manager).
847
pðrÞsin2 ðqÞ
0
if t˛ taP ; taP þ tdP
otherwise
(23)
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