Journal of the Mechanics and Physics of Solids 59 (2011) 940–958
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Journal of the Mechanics and Physics of Solids
journal homepage: www.elsevier.com/locate/jmps
A discrete particle approach to simulate the combined effect of blast
and sand impact loading of steel plates
T. Børvik a,b,n, L. Olovsson c, A.G. Hanssen a,c, K.P. Dharmasena d, H. Hansson e, H.N.G. Wadley d
a
Structural Impact Laboratory (SIMLab), Centre for Research-based Innovation (CRI) and Department of Structural Engineering,
Norwegian University of Science and Technology, Rich. Birkelands vei 1A, NO-7491 Trondheim, Norway
Norwegian Defence Estates Agency, Research and Development Department, PB 405, Sentrum, NO-0103 Oslo, Norway
c
IMPETUS Afea AB, Sördalavägen 22, SE-141 60 Huddinge, Sweden
d
Department of Materials Science and Engineering, University of Virginia, Charlottesville, VA 22904, USA
e
KTH Royal Institute of Technology, Division of Concrete Structures, SE-100 44 Stockholm, Sweden
b
a r t i c l e in f o
abstract
Article history:
Received 8 April 2010
Received in revised form
7 January 2011
Accepted 2 March 2011
The structural response of a stainless steel plate subjected to the combined blast and
sand impact loading from a buried charge has been investigated using a fully coupled
approach in which a discrete particle method is used to determine the load due to the
high explosive detonation products, the air shock and the sand, and a finite element
method predicts the plate deflection. The discrete particle method is based on rigid,
spherical particles that transfer forces between each other during collisions. This
method, which is based on a Lagrangian formulation, has several advantages over
coupled Lagrangian–Eulerian approaches as both advection errors and severe contact
problems are avoided. The method has been validated against experimental tests where
spherical 150 g C-4 charges were detonated at various stand-off distances from square,
edge-clamped 3.4 mm thick AL-6XN stainless steel plates. The experiments were
carried out for a bare charge, a charge enclosed in dry sand and a charge enclosed in
fully saturated wet sand. The particle-based method is able to describe the physical
interactions between the explosive reaction products and soil particles leading to a
realistic prediction of the sand ejecta speed and momentum. Good quantitative
agreement between the experimental and predicted deformation response of the plates
is also obtained.
& 2011 Elsevier Ltd. All rights reserved.
Keywords:
Blast loads
Experimental tests
Sand ejecta
Discrete particles
Numerical simulations
1. Introduction
Blast loads from soil-buried landmines are a major threat to both civilian and military personnel in regions of recent
international conflict. According to Fišerová (2006), 100 million landmines lie strewn in 60 different countries, around
26,000 civilians are killed or maimed every year by detonations of these mines and roughly 30% of recent fatalities in
Afghanistan and Iraq are caused by landmines and related devices.
The basic physics of a shallow-buried mine explosion are quite well understood (see Cummings et al., 2002; Luo et al.,
2004; Fišerová, 2006; Grujicic et al., 2006, 2008a, 2008b; Neuberger et al., 2007; Deshpande et al., 2009, and the extensive
n
Corresponding author at: Structural Impact Laboratory (SIMLab), Centre for Research-based Innovation (CRI) and Department of Structural
Engineering, Norwegian University of Science and Technology, Rich. Birkelands vei 1A, NO-7491 Trondheim, Norway.
Tel.: þ 47 73 59 46 47; fax: þ 47 73 59 47 01.
E-mail address: [email protected] (T. Børvik).
0022-5096/$ - see front matter & 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jmps.2011.03.004
T. Børvik et al. / J. Mech. Phys. Solids 59 (2011) 940–958
941
reference lists in these papers). At detonation, a detonation front propagates through the explosive mixture at radial
speeds of around 8000 m/s creating a shock wave that interacts with the surrounding media, which may be air, soil, water
or combinations of the same. The interaction results in a transfer of a part of the momentum of the combustion products to
the soil causing the soil at the combustion products/soil interface to acquire a velocity that can exceed several hundred
meters per second. The faster moving shock front propagates through the soil and eventually reaches the soil/air interface.
The large acoustic impedance difference between the soil and air results in most of the shock being reflected as an
expansion wave back into the soil, and soil ejecta are emitted to satisfy conservation of momentum considerations. As a
result, the primary loading of an above soil structure near such an event is caused by the soil impact (Deshpande et al.,
2009). The significance of the soil loading also depends upon the depth of burial of the explosion since this controls the
mass of soil that can be directed towards the structure. As the depth of burial decreases, the contribution of the air shock
becomes more significant. An accurate, computationally efficient approach for predicting the combined air shock and soil
impact loads and their coupling to the deforming structure is essential for development of a physics-based design
approach for protective structures against buried mine blasts.
The emergence of numerical tools (such as coupled Lagrangian–Eulerian techniques), combined with the large increase
in computational resources over the last decade, has allowed new insight into the complex, coupled loading processes
associated with soil-blast events (Cummings et al., 2002; Luo et al., 2004). Even so, computer modeling of such phenomena
is far from mature (Grujicic et al., 2006, 2008a). A major limitation is the lack of soil constitutive relations that adequately
capture the basic physics. The relations currently used are restricted to a soil or sand packing density so high that the real
particle–particle contacts are semi-permanent, and their ability to describe soil ejecta is questionable (Deshpande et al.,
2009).
In this study the blast load from a model sand-buried charge against a simple structural component has been
investigated. A discrete particle method (Olovsson et al., 2010) has been developed to model the interactions between the
high explosive detonation products, air and sand containing varying concentrations of water. The method works with rigid,
spherical particles that transfer forces between each other through contact and collisions. This method, which is based on a
Lagrangian formulation, has several advantages over coupled Lagrangian–Eulerian approaches as both advection errors
and severe contact problems are avoided. The method has been validated against experimental tests where a spherical
150 g C-4 charge was detonated at various stand-off distances from a square, edge-clamped 3.4 mm thick AL-6XN stainless
steel plate (Dharmasena et al., in preparation). The experiments were carried out for a bare charge, a charge surrounded by
a concentric sphere of dry sand and a charge surrounded by fully saturated wet sand. It will be shown that the particlebased method is able to describe the physical mechanisms of sand acceleration and impact with a deformable structure.
Good agreement between the available experimental data and the numerical simulations is also obtained for loadings by
each of the three media.
2. Experimental observations
Edge-clamped, square 3.4 mm thick AL-6XN plates (with an areal density of 27.4 kg/m2) were blast loaded using a
model spherical test charge consisting of 150 g C-4. AL-6XN is a nitrogen-strengthened super-austenitic stainless steel
with high strength, very good toughness, ductility and formability, and excellent corrosion resistance in chloride
environments (Nemat-Nasser et al., 2001). The major alloying elements are 20 wt% chromium, 24 wt% nickel, 6.2 wt%,
molybdenum, 0.22 wt% nitrogen and 0.2 wt% copper. The plates were tested in a cold rolled and annealed condition. The
C-4 charge was either bare, or surrounded by a concentric shell of either dry or water-saturated sand. The test apparatus
allowed 613 mm square test plates to be fully edge-clamped using a cover plate and series of bolts. The region exposed to
sand impulse was 406 mm 406 mm. The region below the plate was hollow and shielded from the blast, enabling the
target unrestricted deflection. Three plates for each loading case were tested at varying charge-plate stand-off distances
(defined as the distance from the charge center to the nearest face of the target) of 150, 200 and 250 mm. Thus, a total of
9 blast tests were carried out. Additional details about the experimental study can be found in Dharmasena et al.
(in preparation).
The bare charge was made by packing a 30 mm radius plastic sphere with 150 g of C-4 explosive. For the other charges,
the C-4 ball was positioned at the center of a 76.2 mm radius plastic sphere and the annular gap was filled with dry or fully
water-saturated wet sand. Thus, the C-4 charges were completely surrounded by a 46.2 mm thick shell of either dry or wet
sand. The ‘‘sand’’ used in this study was not real sand, but consisted of silica glass microspheres with a diameter of
200 mm. The mass of dry sand used in the charges was 2.770.1 kg, while the mass of added water when wet sand
charges were tested was 0.6870.1 kg. A detonator was placed at the north pole of the C-4 sphere (furthest away from the
test samples), so that the detonation front in the explosive propagated towards the test samples.
The experimental set-up is shown in Fig. 1(a) and (b), while the measured center plate displacements as a function of
stand-off distance for blast loaded plates are given in Fig. 1(c). Only the permanent central deflection of the plate was
measured in the experiments. Thus, pressure–time and impulse–time curves from these tests are not available. Fig. 2
shows pictures of the final deflection of the AL-6XN steel plates after blast loading at a 150 mm stand-off distance. Note
that none of the plates failed as a result of the blast load, and a treatment of plate failure was not required. A sequence of
high-speed camera images of the detonation of the C-4 charge encased in dry sand is shown in Fig. 3, while corresponding
images for C-4 encased by wet sand are given in Fig. 4. The photographs enabled the surface of the sand to be observed and
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Fig. 1. (a) Sketch of the experimental set-up where a sphere of C-4 is detonated above a square AL-6XN stainless steel plate. The charge may be bare,
surrounded by dry sand or surrounded by fully saturated wet sand, (b) picture of a typical set-up at the test site just before detonation and (c) measured
permanent mid-span deflections as a function of stand-off distance R.
the radial expansion rate of the sand ejecta to be measured. The measured radial position of both dry and wet sand versus
elapsed time after detonation is plotted in Fig. 5.
3. Numerical approach
3.1. A particle-based method to model close-range blast loadings
A discrete particle method (Olovsson et al., 2010), also known as the corpuscular method (http://www.impetus.no/,
2010), has been used to model the interaction between high explosive detonation products, the air and the sand. The
modeling principle is indicated in Fig. 6, while a cross-section of the 3D numerical model is shown in Fig. 7. The method
works with discrete, rigid, spherical particles that transfer forces between each other through contact and collisions. There
are three motivations for using a particle-based approach. Firstly, the method is based on a Lagrangian description of
T. Børvik et al. / J. Mech. Phys. Solids 59 (2011) 940–958
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Fig. 2. Final deflection of plates after blast loading, showing 3/4-section profiles of the 3.4 mm thick AL-6XN plates tested with (a) bare charge, (b) dry
sand charge and (c) saturated wet sand charge at 150 mm stand-off distance.
Fig. 3. Sequence of high-speed camera images showing the detonation of 150 g of C-4 encased in dry sand at stand-off distance of 250 mm.
motion which, in contrast to arbitrary Lagrangian–Eulerian (ALE) and Eulerian methods, is not associated with advection
related numerical errors (see also Børvik et al., 2009). Secondly, the framework allows for a simple, physically clear and
robust treatment of the interaction between the high explosive, air, sand and structural parts, where the latter is
represented by finite elements. The contact treatment is especially important for gas and sand interactions with structural
parts of complex geometry. This interaction is difficult to adequately represent when working with coupled Lagrangian–
Eulerian methods. Thirdly, the corpuscular method can be combined with finite elements for studying fully coupled
structural responses. Here, the method has been implemented in the non-linear finite element code IMPETUS Afea Solver
(http://www.impetus.no/, 2010).
3.2. Modeling of high explosive (C-4) and air
The application of the discrete particle model to high explosive detonation products and air essentially follows the
kinetic theory of gases, originally derived by Maxwell (1860). The inter-particle contacts are assumed to be elastic. In the
numerical model, each particle is assigned to represent many molecules. The fundamentals of the method are described in
Olovsson et al. (2010). The parameters available for defining a discrete particle model of a high explosive are the initial
density r0, the initial internal energy E0, the ratio of heat capacities at constant pressure and volume g ¼CP/CV, and the
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Fig. 4. Sequence of high-speed camera images showing the detonation of 150 g of C-4 encased in saturated wet sand at stand-off distance of 250 mm.
Fig. 5. Measured and predicted radial expansion rate of the dry and wet sand ejecta at stand-off distance 250 mm.
initial solid-fill fraction of the particles b. A significant solid-fill fraction gives rise to a co-volume effect that drastically
increases the pressure at high densities. Co-volume effects have therefore been added in the present description to better
represent gas behavior at extreme pressures (see Clausius (1880) or Baibuz et al. (1986) for the formulation
implemented here).
The parameters used in the discrete particle model of C-4 detonation were determined by fitting to the response of a
simulated cylinder test (Souers et al., 1996; Souers, 2007). The cylinder test consists of a pipe made of OHFC copper that is
filled with the high explosive to be characterized. The explosive is initiated at one end, whereupon a detonation wave
T. Børvik et al. / J. Mech. Phys. Solids 59 (2011) 940–958
945
Fig. 6. Modeling principle of the discrete particle method.
Fig. 7. Cross-section of 3D numerical model.
Fig. 8. Simulation of a cylinder test where an OHFC copper pipe with inner diameter 25.43 mm, wall thickness 2.593 mm and length 300 mm is filled
with 1,000,000 particles of C-4.
travels along the pipe. The pipe wall motion is monitored and its radial velocity at various locations along the pipe axis can
be used to determine the properties of the high explosive. Fig. 8 shows a model simulation where a copper pipe with an
inner diameter of 25.43 mm, a wall thickness 2.593 mm and a length of 300 mm was filled with 1,000,000 particles
representing C-4. Experimental data for this geometry using a C-4 explosive were presented by Souers (2007). Fig. 9 shows
the measured radial velocity of the pipe wall compared to the simulated response using the optimized parameters
r0 ¼1601 kg/m3, E0 ¼8.7 GJ/m3, g ¼ 1.4, b¼0.35, and a detonation velocity D ¼8190 m/s. Good agreement between the
measured and predicted velocity–time curve is obtained. It is to be noted that only g and b were varied in the optimization
process. The density, internal energy and detonation velocity were taken from the optimized C-4 Jones–Wilkins–Lee
equation-of-state (JWL-EOS) parameters in Souers et al. (1996). The JWL-EOS expresses the pressure as a function of
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Fig. 9. Comparison between measured and simulated radial velocity v(t) of the pipe wall versus time from the cylinder test of C-4 in Fig. 8.
Table 1
JWL-EOS constants for C-4 (Souers et al., 1996).
r0 (kg/m3)
D (m/s)
A (GPa)
B (GPa)
R1
R2
o
E0 (GJ/m3)
1601
8190
597.4
13.9
4.5
1.5
0.32
8.7
Table 2
Corpuscular constants for C-4.
r0 (kg/m3)
D (m/s)
g
b
E0 (GJ/m3)
1601
8190
1.4
0.35
8.7
relative volume and internal energy according to
o R1 V
o R2 V
e
e
p ¼ A 1
þ B 1
þ oE
R1 V
R2 V
ð1Þ
where A, B, R1, R2 and o are constants, V ¼ r/r0 is the ratio of the current and initial densities and E is the internal energy
per unit volume. The OHFC copper used in the cylinder-test simulations was modeled using the original Johnson–Cook
constitutive relation (Johnson and Cook, 1983) with material parameters from Johnson and Cook (1985) and Frutschy and
Clifton (1998). An alternative approach would be to fit the parameters in the particle method directly to the JWL-EOS for
C-4 given in Souers et al. (1996), but since the cylinder test in any case was used to extract the JWL-EOS data the proposed
approach is preferred. The JWL-EOS parameters for the C-4 charge are listed in Table 1 (8 constants), while fitted
corpuscular constants are given in Table 2 (only 5 constants).
The surrounding air was modeled as an ideal gas with an initial pressure of 1 atm (100 kPa), density rair ¼1.3 kg/m3, initial
internal energy Eair ¼0.25325 MJ/m3 and a ratio between heat capacities at constant pressure and volume g ¼CP/CV ¼1.4.
Both the C-4 and the air particles are assigned initial velocities in random directions. The magnitude of the velocity had the
Maxwell–Boltzmann distribution (Maxwell, 1860). Hence, the initial velocity distribution matches that of an ideal gas. It is noted
that the C-4 particles were not active at time zero. A particle was released into motion at time trelease ¼L/D, where D is the
detonation velocity and L is the initial distance from the particle to the detonation point.
3.3. Modeling of dry and saturated wet sand
The sand material is modeled differently to the high explosive gases and air. A penalty based contact was used instead
of purely elastic collisions. The penalty contact enables incorporation of both friction and damping. The rheological model
for the interaction between two sand particles each with a mass m is shown in Fig. 10, and is similar to that proposed by
Deshpande et al. (2009). It consists of two linear springs, one acting in the normal direction and one in the tangential
direction. Both springs have the same stiffness k. In addition, a linear dashpot with a damping coefficient c is acting in
parallel with the normal contact spring. Furthermore, the tangential spring force is limited by a Coulomb friction
coefficient m. To reduce the computational cost, the soil particles were only given translational degrees of freedom.
T. Børvik et al. / J. Mech. Phys. Solids 59 (2011) 940–958
947
Fig. 10. Rheological model for sand interaction.
Incorporation of the rotational degrees of freedom proposed in Deshpande et al. (2009) severely reduces the critical time
step size in the central difference time integration scheme. However, this simplification should be acceptable as long as
one can tune the soil parameters (stiffness, damping, friction and initial packing) such that the aggregate behaves
correctly.
The dry sand consists of small silicon based glass spheres with a diameter of 200 mm. The density of the glass material is
2700 kg/m3. The initial solid volume fill fraction was 60%, which gives an initial sand density of 1620 kg/m3. In real sand,
energy is dissipated through both friction and particle fracture. However, the fracture process dissipates very little energy
and is therefore not incorporated in the discrete particle model. Elasticity is also not explicitly represented by the elastic
properties of the silica, but rather by the penalty stiffness k of the contact law.
The dry sand particle distribution and the sand model characterization were dealt with in three steps. Firstly, to
initialize the particle distribution in the numerical model, 1000 equally sized particles were randomly distributed in a unit
cell with periodic boundary conditions. The solid-fill fraction was the same as in the real sand (i.e. 60%). A penalty based
contact with gradually increasing contact stiffness enabled the particles to be moved around until particle–particle
penetrations reached a negligible value. No contact friction was used at this stage. In the second step, the unit cell (with
correct solid-fill fraction and without any initial contact penetrations) was characterized by monitoring stresses during
uniaxial compression. The stress components were extracted by dividing cross-section forces by the unit cell cross-section
area. Fig. 11 shows the unit cell and the sand behavior at different particle–particle contact stiffnesses and coefficients of
friction. No real compression test data is presently available for the sand used in this study. Hence, it was decided to
simply test a few reasonable combinations of contact stiffness and friction in the simulations of the experimental tests.
Thirdly, any given sand geometry can be defined by repeating the unit cell from the first step in the x, y and z directions as
many times as needed. Fig. 12 shows a cross-section of a unit cell used to establish the initial sand-particle interaction and
a cross-section of a charge consisting of sand and C-4 prior to detonation. The grid in the foreground has been added to
visualize the boundaries of the unit cells.
The total number of particles in a model can be adjusted by scaling the size of the unit cell. Scaling also affects the
particle–particle contact stiffness k according to
^ ¼ L k
k ¼ kðLÞ
0
L0
ð2Þ
where L is the scaled size of the unit cell, L0 ¼1 m is the initial (un-scaled) size of the unit cell and k0 is the particle–particle
contact stiffness for the un-scaled unit cell.
The saturated wet sand consisted of the same small silica glass spheres as the dry sand and its initial solid-fill fraction
was the same (60%). The remaining volume was filled by water. The result is an increased initial density of 2020 kg/m3.
Water-saturated sand has no room for compaction before building up pressure, and required a denser packing of particles
than in the dry sand model. However, uniformly sized spheres were unable to achieve a sufficiently high packing and so
the unit cell was filled with 1000 particles having a slight variation in radius ( 72.5%) that allowed for a denser packing of
64%. Note that as a result, the actual water is not modeled with particles. Instead the particle mass, packing, friction and
damping were adjusted to account for the effect of water.
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T. Børvik et al. / J. Mech. Phys. Solids 59 (2011) 940–958
Fig. 11. Pressures obtained in simulations of the unit cell during compression tests of dry sand.
Fig. 12. (a) Unit cell that after several re-runs is used to establish the initial sand-particle size distribution, (b) cross-section of the spherical charge prior
to detonation, where the inner circle (in yellow) shows the particles representing C-4 and the outer circle (in brown) shows the particles representing the
sand surrounding the high explosive. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this
article.)
As for the dry sand, the model for the fully saturated wet sand was characterized by monitoring stresses during uniaxial
compression of a unit cell with 1000 particles and periodic boundary conditions. Fig. 13 shows the wet sand behavior at
different particle–particle contact stiffness. Real observations (Deshpande et al., 2009) indicate that saturated sand ejecta
does not turn into a loose spray as the compressive wave reflects into tension at the free surface (see also Figs. 3 and 4).
For this reason it was decided to work with contact damping instead of friction in the saturated wet sand model.
The damping coefficient x (see Fig. 10) is a fraction of the critical damping. This makes the model sand-particle size
dependent when exposed to a uniform strain rate. However, most energy damping dissipation occurs at the shock front. In
the numerical model, the shock front width is only a few particle diameters and, hence, the strain rate is inversely
proportional to the particle size. This motivated the use of an absolute damping parameter that is proportional to (L/L0)2
and not (L/L0). It should be noted that the damping has no effect on the results of the simulated uniaxial compression test
shown in Figs. 11 and 13, which were conducted assuming quasi-static conditions.
3.4. Modeling of the solid steel plate
The 3.4 mm thick AL-6XN stainless steel target plate was fully clamped along its edges. The finite element mesh for the
plate consisted of 1600 (40 40 1) 64-node 3rd-order hexahedrons giving a total of 58,564 nodes. Note that only one
T. Børvik et al. / J. Mech. Phys. Solids 59 (2011) 940–958
949
Fig. 13. Pressures obtained in simulations of the unit cell during compression tests of fully saturated wet sand.
Fig. 14. Target plate showing the mesh of 3rd-order hexahedron elements and the permanent deflection after simulation of (a) a bare C-4 charge and
(b) a charge surrounded by saturated wet sand at stand-off distance of 150 mm.
element is used over the plate thickness in these simulations since the higher-order (cubic) elements show excellent
behavior in bending and since no shear locking is introduced in the solution. More elements would have been required if
shear localization and fracture were an issue. Fig. 14 shows the element mesh of the plate in the deformed configuration
(as a result of loadings by a bare charge and a charge surrounded by saturated wet sand at stand-off distance of 150 mm).
The plate was modeled using a slightly modified version of the Johnson–Cook constitutive equation (Johnson and Cook,
1983; Børvik et al., 2001). In this formulation, the equivalent stress is expressed as
seq ¼ ðA þ Beneq Þð1 þ e_ eq ÞC ð1T m Þ
ð3Þ
where eeq is the equivalent plastic strain and A, B, n, C and m are material constants. The first term on the right-hand side in
Eq. (3) governs strain hardening, the second term governs strain-rate hardening, while the last term controls thermal
softening of the material. The dimensionless plastic strain rate is given by e_ eq ¼ e_ eq =e_ 0 , where e_ 0 is a user-defined reference
strain rate. The homologous temperature is defined as T* ¼(T Tr)/(Tm Tr), where T is the absolute temperature, Tr is the
ambient temperature and Tm is the melting temperature of the material. The temperature change due to adiabatic heating
is calculated as
Z eeq
s de
w eq eq
ð4Þ
TTr ¼ DT ¼
rCp
0
where r is the material density, Cp is the specific heat at constant pressure and w is the Taylor–Quinney coefficient that
represents the proportion of plastic work converted into heat. Since fracture was not present in the experiments, no
fracture criterion has been introduced into the model.
Modified Johnson–Cook material constants for the strain hardening of AL-6XN were obtained based upon a least
squares analysis of the true stress–strain curve for uniaxial tension tests on 0.56 mm thick sheet specimens (Dharmasena
et al., in preparation). These tests were conducted at a quasi-static strain rate and room temperature. Nemat-Nasser et al.
(2001) presented data from uniaxial compression tests on cylindrical samples (D ¼L¼ 3.8 mm) over a range of strain rates
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Table 3
Material constants for AL-6XN (Dharmasena et al., in preparation; Nemat-Nasser et al., 2001).
Elastic constants and density
3
Yield stress and strain hardening
Strain rate hardening
1
E (GPa)
v
r (kg/m )
A (MPa)
B (MPa)
n
e_ 0 (s
195
0.3
8060
410
1902
0.82
0.001
)
Temperature softening and adiabatic heating
C
Tr (K)
Tm (K)
m
Cp (J/kg K)
w
a (K 1)
0.024
296
1700
1.03
500
0.9
1.5 10 5
(from 0.001 to 8000 s 1) and initial temperatures (from 77 to 1000 K). These data were used to estimate the strain-rate
hardening and thermal softening of the steel. All mechanical and physical (from http://www.al6xn.com/properties.php,
2010) parameters used in the numerical simulations are listed in Table 3. A comparison between the experimental data
and the fitted constitutive relation for AL-6XN is given in Fig. 15. One reason for the difference in yield and flow stress
between the two data sets shown in Fig. 15(a) may be due to cold rolling of the plate. As seen, the fit for the true stress–
strain curve at quasi-static strain rate and room temperature using data from Dharmasena et al. (in preparation) is
excellent, while the fit for the temperature softening at 10% plastic strain using data from Nemat-Nasser et al. (2001) is
rather poor. The fits were carried out with the constraint that the model should match the experimental data exactly at
quasi-static strain rate and room temperature for a plastic strain of 10%. It will be shown in the next section that a
maximum plastic strain in the target plate of around 10–20% was typical for this problem.
All simulations were carried out on a cluster machine using a single AMD Opteron processor. Typical CPU-times for a
simulation with a bare C-4 charge varied from 4000 to 17,000 s, dependent on the number of particles used in the model.
A corresponding fully coupled Lagrangian–Eulerian LS-DYNA simulation took 54,000 s. For a dry sand enclosed C-4 charge,
the CPU-times varied between 1600 and 26,000 s, while for a saturated wet sand C-4 charge the CPU-times were between
2200 and 54,000 s. It should finally be mentioned that the particle-based method is currently being implemented on
graphic cards. It is believed that this will result a considerable speed-up of the simulations.
4. Numerical results
4.1. Simulations with a bare C-4 charge
In the bare charge experiments, 150 g of C-4 was detonated at stand-off distances 150, 200 and 250 mm above a
3.4 mm thick, 406 mm 406 mm, AL-6XN steel plate. A cross-section of the bare charge simulation model at two instances
in time (0 and 150 ms after detonation) is shown in Fig. 16. The volume of air used in the model was 0.9 m3. Note that freeflow boundaries were used in these simulations, so that no significant reflections from the boundaries are present.
A convergence study was used to determine the optimum number of particles for the C-4 charge, Nc-4, and the
surrounding air, Nair, for a converged solution. Simulations were conducted using (Nc-4; Nair)¼ (10,000; 500,000), (20,000;
1,000,000) and (40,000; 2,000,000). The predicted permanent central deflections of the plate are given in Table 4, and
compared to the experimental results. From this study it was concluded that convergence was reached when (Nc-4; Nair)¼
(40,000; 2,000,000). The simulations with particles were terminated after 1 ms. Damping was then added to the plate and
the permanent deformation was computed. Impulse–time curves from these simulations as a function of number of air
particles are given in Fig. 17(a), while the deformed shape of the plate after loading at a stand-off distance of 150 mm is
shown in Fig. 14(a).
The numerical results for a bare C-4 charge at stand-off distance of 150 mm using the particle-based method were also
validated against fully coupled Lagrangian–Eulerian LS-DYNA simulations (LSTC, 2007). In the LS-DYNA models quarter
symmetry conditions were applied. The air and high explosive domain were modeled with 8-node reduced integrated
Eulerian hexahedrons. To ensure convergence, models with different mesh densities were tested, the finest one with
1,134,000 hexahedrons having a characteristic element size of 2.5 mm in the critical regions. The second-order van Leer
advection scheme, with half-index shift for velocities, was applied (Benson, 1992). The quarter symmetry model of
the plate was modeled with 400 fully integrated shell elements with five integration points through the thickness. The
JWL-EOS parameters for the C-4 charge used in the LS-DYNA simulations are given in Table 1. The air was modeled as an
ideal gas with initial internal energy 0.25 MJ/m3 and g ¼1.4. The target plate was modeled using the modified Johnson–
Cook constitutive relation with the same material parameters as in the particle-based simulations (see Table 3). As seen
from Table 4, the result from the LS-DYNA simulation is very close to those obtained using the particle-based approach and
give additional confidence in the proposed discrete particle method. Both the particle-based approach and the fully
coupled Lagrangian–Eulerian simulation over-predict the measured permanent deflection by about 15%.
4.2. Simulations with dry sand and C-4
In the simulations of a dry sand enclosed C-4 charge, the loading by air was assumed to be negligible. Hence, only the
C-4 charge and the surrounding sand were modeled using the discrete particle method. The sensitivity upon the
simulations of the particle contact law parameters (contact stiffness k0 and friction coefficient m) for the sand were
T. Børvik et al. / J. Mech. Phys. Solids 59 (2011) 940–958
951
Fig. 15. Comparison between experimental data from Dharmasena et al. (in preparation) and Nemat-Nasser et al. (2001) and best fits to the modified
Johnson–Cook constitutive relation for AL-6XN for (a) strain hardening, (b) strain-rate hardening and (c) temperature softening.
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T. Børvik et al. / J. Mech. Phys. Solids 59 (2011) 940–958
t = 0 μs
t = 150 μs
Fig. 16. Cross-section from simulation with 40,000 C-4 particles and 2,000,000 air particles at times 0 and 150 ms. Note that free-flow boundaries were
used in these simulations, so that no reflections are present.
Table 4
Simulated center displacement as function of stand-off distance.
Stand-off distance R (mm)
150
200
250
Bare C-4 (150 g) charge – center displacement (mm)
Nc-4 ¼10k, Nair ¼500k
Nc-4 ¼ 20k, Nair ¼1M
Nc-4 ¼ 40k, Nair ¼ 2M
LS-DYNA
Test
19.7
15.8
12.1
19.2
14.7
12.3
19.4
14.4
12.3
19.8
–
–
17.0
12.7
11.3
investigated, namely (k0; m)¼(0.4 GN/m; 0.3), (0.4 GN/m; 0.1) and (0.2 GN/m; 0.3). The assessment began by using
the combination (k0; m)¼(0.4 GN/m; 0.3) as the base-line and performing a convergence study for the number of C-4
particles, Nc-4, and the number of sand particles, Nsand, using (Nc-4; Nsand)¼(10,000; 50,000), (40,000; 200,000), (80,000;
400,000) and (160,000; 800,000). From these simulations it was concluded that the resulting deflection is relatively
insensitive to the number of particles. After finishing the convergence study, simulations with (k0; m)¼(0.4 GN/m; 0.1) and
(k0; m) ¼(0.2 GN/m; 0.3) were performed with (Nc-4; Nsand)¼(160,000; 800,000). All simulations with particles were
conducted for a physical time of 2 ms by which point the impulse transfer from the sand and detonation products to the
plate had ceased. As for the bare charge simulations, a second simulation with damping (but without particles) was
performed until the plate came to rest. The permanent central deflections of the steel plate from these simulations are
given in Table 5, where they are compared with the experimental results, while impulse–time curves as a function of
number of particles are given in Fig. 17(b). It can be seen that good agreement between the simulated and experimentally
measured deflections was achieved (see also the discussion in Section 4.4). It is also noteworthy that the addition of the
dry sand shell almost doubled the momentum transferred to the plate (from around 130 to 250 Ns).
4.3. Simulations with saturated wet sand and C-4
The simulations with saturated wet sand adopted the same procedure as for the dry sand, but with damping x instead
of Coulomb friction m in the sand model. The parameter variations were (k0; x)¼ (4 GN/m; 0.01), (4 GN/m; 0.005) and
(2 GN/m; 0.01). First, the combination (k0; x) ¼(4 GN/m; 0.01) was taken as base-line parameters for a convergence study.
As for the dry sand models, the resulting permanent deflection of the steel plate was found to be relatively insensitive to
the number of particles used in the simulations. After the convergence study, simulations with (k0; x) ¼(4 GN/m; 0.005)
and (k0; x)¼(2 GN/m; 0.01) were run with (Nc-4, Nsand)¼(160,000; 800,000) particles. The permanent central deflections of
the steel plate from the simulations are given in Table 6, where they are compared to the experimental results, and
impulse–time curves as a function of number of particles are given in Fig. 17(c). The encasement of the C-4 by saturated
wet sand more than tripled the impulse (from 130 to above 400 Ns). The deformed shape of the plate after loading at
stand-off distance of 150 mm was shown in Fig. 14(b). Again, good agreement between the simulations and the
experiments is achieved.
4.4. Comparison of predicted and experimental results
Cross-sectional images showing particle locations at various times after detonation are shown in Fig. 18 for the dry
sand and in Fig. 19 for the fully saturated wet sand at stand-off distance of 250 mm. Note that the use of a damping
T. Børvik et al. / J. Mech. Phys. Solids 59 (2011) 940–958
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Fig. 17. Impulse–time curves at stand-off distance of 150 mm from (a) bare C-4 blast, (b) C-4 and dry sand blast and (c) C-4 and saturated wet sand blast
as a function of number of particles (in thousands) used in the simulations.
coefficient in the wet sand model reduces the loose spray spatial distribution of the sand ejecta. If these plots are compared
to the corresponding high-speed video images in Figs. 3 and 4, the overall agreement is seen to be excellent. The predicted
radial expansion rate of the dry and wet sand ejecta from these simulations are compared to the experimental results in
Fig. 5. Again, the agreement is very good. Fig. 20 gives a sequence of 3D plots showing the deformation of the AL-6XN plate
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T. Børvik et al. / J. Mech. Phys. Solids 59 (2011) 940–958
Table 5
Simulated center displacement as function of stand-off distance.
Stand-off distance R
(mm)
150
200
250
C-4 charge surrounded by dry sand – center displacement (mm)
k0 ¼ 0.4 GN/m
m ¼ 0.3
Nc-4 ¼10k
Nsand ¼50k
k0 ¼0.4 GN/m
m ¼0.3
Nc-4 ¼40k
Nsand ¼ 200k
k0 ¼0.4 GN/m
m ¼0.3
Nc-4 ¼ 80k
Nsand ¼ 400k
k0 ¼0.4 GN/m
m ¼0.3
Nc-4 ¼ 160k
Nsand ¼ 800k
k0 ¼0.4 GN/m
k0 ¼0.2 GN/m
m ¼0.1
m ¼0.3
Test
Nc-4 ¼ 160k
Nsand ¼ 800k
Nc-4 ¼ 160k
Nsand ¼ 800k
31.7
23.7
18.1
31.6
23.7
18.2
30.6
23.6
18.5
30.6
23.1
17.7
34.6
27.8
20.8
31.2
22.4
17.4
38.5
26.8
18.5
k0 ¼ 4 GN/m
Test
56.4
41.7
34.5
Table 6
Simulated center displacement as function of stand-off distance.
Stand-off distance R
(mm)
C-4 charge surrounded by saturated wet sand – center displacement (mm)
k0 ¼ 4 GN/m
Nc-4 ¼10k
Nsand ¼ 50k
k0 ¼ 4 GN/m
x ¼ 0.01
Nc-4 ¼ 40k
Nsand ¼ 200k
k0 ¼ 4 GN/m
x ¼ 0.01
Nc-4 ¼80k
Nsand ¼400k
k0 ¼4 GN/m
x ¼0.01
Nc-4 ¼ 160k
Nsand ¼ 800k
Nc-4 ¼ 160k
Nsand ¼ 800k
k0 ¼ 2 GN/m
x ¼ 0.01
Nc-4 ¼160k
Nsand ¼800k
50.9
41.0
31.4
51.2
41.3
32.5
49.1
41.5
33.3
48.0
40.3
31.3
52.4
43.2
33.3
47.7
40.4
31.3
x ¼0.01
150
200
250
x ¼ 0.005
Fig. 18. Sequence of plots showing the deformation of the AL-6XN plate after impact of C-4 and dry sand at stand-off distance of 250 mm (the model is
sectioned through the center for better visibility). (a) t¼ 0 ms, (b) t ¼200 ms, (c) t ¼400 ms, (d) t¼ 800 ms.
for a charge enclosed in saturated wet sand at stand-off distance of 150 mm. Both the sand ejecta and the permanent
deflection of the plate are well described.
Fig. 21 shows energy–time curves using those sand parameters that best match the experimentally obtained
permanent deflections (k0 ¼0.4 GN/m and m ¼ 0.1 for the dry sand model, and k0 ¼4 GN/m and x ¼0.005 for the saturated
wet sand model) at stand-off distance of 150 mm. The figure shows the conversion of C-4 energy into kinetic and
dissipated energy for both dry and wet saturated sand. The numerical results suggest that the sand stiffness is of less
importance than its inter-particle friction. This is evident from Tables 5 and 6, where it was shown that a change in the
contact stiffness k0 has a much smaller effect on the center displacement of the plate than a change in the friction
coefficient m (for dry sand) and the damping x (for wet sand). Most energy dissipation occurs when the initial shock wave
passes through the material. This is clearly displayed by the energy curves in Fig. 21. In the dry sand model the shock wave
reaches the sand–air interface roughly 35 ms after detonation, while in the wet sand model this occurs somewhat earlier.
T. Børvik et al. / J. Mech. Phys. Solids 59 (2011) 940–958
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Fig. 19. Sequence of plots showing the deformation of the AL-6XN plate after impact of C-4 and saturated wet sand at stand-off distance of 250 mm
(the model is sectioned through the center for better visibility). (a) t ¼0 ms, (b) t ¼200 ms, (c) t ¼ 400 ms, (d) t ¼800 ms.
Fig. 20. Sequence of plots showing the deformation of the AL-6XN plate after impact of C-4 and saturated wet sand at stand-off distance of 150 mm
(only half the model is shown for better visibility). (a) t¼ 0 ms, (b) t¼ 200 ms, (c) t ¼400 ms, (d) t ¼ 800 ms.
By that time the energy dissipation process is nearly complete. After reflection, the shock wave is reversed into a tensile
wave. Upon unloading, the elastic compaction energy is released and transformed into sand kinetic energy. This
compaction phase is crucial for the difference in plate response when comparing dry and saturated wet sand. It should
also be mentioned that some of these simulations were run both with and without the surrounding air to reveal how much
of the impulse that was carried by the sand and how much was carried by the air shock. Hardly any differences in results
were obtained when the air was present or not. Thus, the impulse applied by the sand impact causes the primary loading
to the deforming structure. This is as expected since the mass of air is significantly smaller than the mass of sand in the
problem.
The numerically predicted permanent plate deflection is plotted and compared with the experimental data in Fig. 22.
The overall agreement between the measured and predicted permanent deflections is seen to be very good. For a bare
charge, the response is over-predicted by 10–15%, giving a conservative estimate for all stand-off distances within the
experimental limitations of this study. For dry and wet sand, the predicted response is generally closer and within 5–10%.
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T. Børvik et al. / J. Mech. Phys. Solids 59 (2011) 940–958
Fig. 21. Energy–time curves from simulations of (a) C-4 and dry sand and (b) C-4 and saturated wet sand at stand-off distance of 150 mm.
Fig. 22. Comparison between measured and predicted permanent mid-span deflections as function of stand-off distance R.
It is slightly over-predicted for large stand-off distances, while for short stand-off distances the response seems to be
somewhat under-predicted. Thus, non-conservative results are observed at the closest stand-off distances.
In the simulations of a bare charge at stand-off distance of 150 mm, the maximum plastic strain in the plate after blast
loading was found to be around 5% and the maximum strain rate reached 100 s 1. Corresponding values from
simulations involving saturated wet sand were 20% and 250 s 1, respectively. These maximum plastic strains are below
the uniaxial tensile strain to failure justifying our neglect of fracture in the study and consistent with the absence of
fracture in the experimentally tested panels. We do note that the maximum strain rates are in the regime where the
modified Johnson–Cook model best fits the data from Nemat-Nasser et al. (2001), see Fig. 15(b). The temperature increase
due to the plastic work was negligible (of the order of 35 K).
5. Discussion
The primary goal of this study was to test a newly proposed particle-based method to quantitatively describe the
physical mechanisms by which a buried mine blast loads and deforms a structure. The study also provide insights into
energy transfer processes between the rapidly expanding detonation products and sand, the sensitivity of the transfer
process to sand constitutive parameters and the dynamic deformations of a ductile metallic structure loaded by
explosively accelerated sand.
From the numerical studies presented above, good qualitative and quantitative results are in general obtained when the
predicted responses are compared against available experimental results. In the convergence studies, the sand parameters
T. Børvik et al. / J. Mech. Phys. Solids 59 (2011) 940–958
957
were defined to match the bulk modulus suggested by Deshpande et al. (2009) reasonably well. These values are 1.7 GPa
for the dry sand and 10.7 GPa for the saturated wet sand. Not knowing the effect of the friction and damping coefficients
ahead of time, those parameters were picked arbitrarily. Subsequently, based on the resulting deflections in the
convergence study, the friction and damping parameters were adjusted to obtain a better fit with the experimental
results. It still remains to fully validate these assumptions. However, we do note that the measured sand front position
versus time data shown in Fig. 5 were not used to fit the model and are consequently an independent test of its validity.
They show an excellent agreement between the prediction and experiment.
The fundamental modeling strategy for sand was to apply the rheological model of Fig. 10. Within the degrees of
freedom allowed for in this concept, the dry sand behavior was captured by introducing inter-particle friction. For
modeling of wet sand, no inter-particle friction was used. Instead an inter-particle damping coefficient was introduced.
The effect of these assumptions for quasi-static loading can be seen by comparing Fig. 11 (dry sand) and Fig. 13 (saturated
wet sand). It is evident that the lateral pressure in the wet sand model almost equals the pressure applied in the uniaxial
strain direction. This is physically correct as pressure loading in un-drained saturated wet sand will mostly be distributed
hydrostatically by the fluid component. For the dry sand, the inter-particle friction model allows for shear forces to be
carried by the sand, resulting in less lateral pressures in the uniaxial strain set-up.
It is important to re-state that we lack experimental data showing the real sand response to different loading
conditions. Hence, one cannot conclude how well the suggested sand model with the used sets of parameters describes the
real mechanical response of the dry and saturated wet sand. However, based on the simulated target plate deflections it
seems reasonable to believe that the model with the parameters used here predicts the energy dissipation in the sand
relatively accurately. The energy that is not dissipated will be converted to kinetic energy, of which the generated impulse
load is nearly a direct function.
6. Concluding remarks
The blast loading of an edge-clamped plate by a sand-buried charge has been studied using a discrete particle method
to model the high explosive detonation products, the air and the sand. The method utilizes rigid, spherical particles that
transfer forces between each other through contact and collisions. This method, which is based on a Lagrangian
formulation, has several advantages over coupled Lagrangian–Eulerian approaches as both advection errors and severe
contact problems are avoided. The method has been tested against some experimental results where a spherical 150 g C-4
charge was detonated at various stand-off distances from a square 3.4 mm thick AL-6XN stainless steel plate. The
experiments were carried out for a bare charge, a charge enclosed in dry sand and for a charge enclosed in fully saturated
wet sand. It has been shown that the particle-based method is able to predict the primary physical loading mechanisms of
the problem, and the temporal and spatial behavior of the loose sand ejecta. Furthermore, good quantitative agreement
between the available experimental data and the numerical simulations has been obtained.
Acknowledgments
The financial support of this work from the Structural Impact Laboratory (SIMLab), Center for Research-based
Innovation (CRI) at the Norwegian University of Science and Technology (NTNU) and the US Office of Naval Research
(ONR Grant number N00014-07-1-0764) is gratefully acknowledged. The experiments were conducted at the Force
Protection Industries Explosives Test Range (Edgefield, South Carolina) and we are grateful to Keith Williams for his
assistance. The IMPETUS Afea Solver has been developed with financial support from The Competence Development Fund
of Southern Norway, SR-Bank Næringsutvikling and Innovation Norway.
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