Shock wave interactions with nano-structured materials - Springer Link

Shock Waves (2013) 23:69–80 DOI 10.1007/s00193-012-0397-4

ORIGINAL ARTICLE

Shock wave interactions with nano-structured materials: a molecular dynamics approach A. K. Al-Qananwah · J. Koplik · Y. Andreopoulos

Received: 14 January 2012 / Revised: 16 June 2012 / Accepted: 25 June 2012 / Published online: 17 July 2012 © Springer-Verlag 2012

Abstract Porous materials have long been known to be effective in blast mitigation strategies. Nano-structured materials appear to have an even greater potential for blast mitigation because of their high surface-to-volume ratio, a geometric factor which substantially attenuates shock wave propagation. A molecular dynamics approach was used to explore the effects of this remarkable property on the behavior of traveling shocks impacting on solid materials. The computational setup included a moving piston, a gas region, and a target solid wall with and without a porous structure. The materials involved were represented by realistic interaction potentials. The results indicate that the presence of a nano-porous material layer in front of the target wall reduced the stress magnitude and the energy deposited inside the solid by about 30 %, while at the same time substantially decreasing the loading rate. Keywords Shock waves · Blast mitigation · Nano porous materials · Nano-shock tube · Molecular dynamics · Shock reflections · Shock attenuation

Communicated by A. Hadjadj. A. K. Al-Qananwah · Y. Andreopoulos (B) Experimental Fluid Mechanics and Aerodynamics Laboratory, Department of Mechanical Engineering, The City College of New York/CUNY, New York, NY 10031, USA e-mail: [email protected] J. Koplik Levich Institute and Department of Physics, The City College of New York/CUNY, New York, NY 10031, USA

1 Introduction Nanotechnology offers researchers the potential for creating unprecedented new materials with enhanced properties and devices that operate at length scales where classical Newtonian physics breaks down. Materials such as microframe structures with nano-trusses and nano-structure porous materials are seen as potentially revolutionizing the field of blast and ballistic protection. Structured nano materials because of their high surface-to-volume ratio offer an unprecedented platform to manipulate shock or blast waves propagating through them and mitigate their effect by dissipation or phase cancellation. Understanding and predicting the behavior of material under the effects of dynamical thermomechanical extremes requires knowledge of interactions at the atomic, molecular, and microstructural levels. This understanding may lead to better ways of controlling them and design new materials for future applications. At the continuum level, a shock is assumed to be a discontinuity with unknown properties since the shock thickness is not resolved and its reflection occurs instantaneously. Realistically, for typical shock thicknesses and propagation velocities of 100 nm and 400 m/s, respectively, the interaction time is 0.25 ns. Thus the fast time—and small length—scales of this interaction necessitate the use of atomistic-level computer simulations. The present work, which is an outgrowth of our experimental work with periodic composite structures and functionally graded materials subjected to shock loading [1–5], is aiming at a better understanding of the interaction of propagating shock waves with structured porous materials at the nanoscale level. The novelty of the present work is in the nanoscale resolution of the mutual interaction between a shock wave and structured materials which includes, for the first time, a simultaneous and integrated/coupled treatment of multiple

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wave phenomena in both gas and solid phases. The objective of our current research is to develop a physical understanding and quantitative predictive capability to deal with the complex interactions between nanoscale materials and shock waves. This is accomplished using molecular dynamics (MD) simulations. The present work in addition to extending our physical understanding has the potential to bridge the gap between atomistic and meso-scale modeling. Previous work describes shock interactions only at a much coarser level by averaging over atomic variables and can only address this issue by trial and error. The use of MD in flows with shocks is not new. In the past few decades, several studies were carried out to investigate the behavior of flows in the presence of a shockwave through MD. Tsai and Trevino [6] studied the propagation of a planar shock in a dense Lennard-Jones fluid. Holian et al. [7] used 4,800 particles to simulate a dense-fluid shock wave and found differences in the predictions of mean temperature density and velocity to be relatively small when compared with the Navier–Stokes continuum mechanics by using non-equilibrium constitutive relations. Holian [8] demonstrated that the constitutive model of Navier–Stokes hydrodynamics was accurate even on time and distance scales of MD. In all these continuum computations using Navier– Stokes the gradients across the shock were approximated by hyperbolic tangent distributions. Woo and Greber [9] have simulated piston-driven shock wave in hard sphere gas at Mach 1.5, 3, and 10. Hoover [10] simulated the structure of a moving shock wave front in a dense Lennard–Jones fluid and found a good agreement with the solutions of the Navier–Stokes equation for strong shocks. Schlamp et al. [11] simulated shock waves in dense nitrogen via non-equilibrium molecular dynamics (NEMD) and the steady-state shock structure is resolved with 0.25 Å resolution, corresponding to 0.1 mean—free path or 1/30th of the shock thickness. Wave propagation in nonlinear granular columns was studied by Sinkovits and Sen [12] who used mesoscale resolution concepts of molecular dynamics that involved a potential based on repulsive forces between grains which consists of many atoms. Although the authors called their method atomistic, its resolution is clearly not at the atomic level where resolution at the nanoscale is required. An extensive but non-exhaustive review on the subject of propagation of nonlinear waves in porous media at the macroscopic level can be found in [13]. Of particular interest is the interaction of shock waves with gas–solid mixtures where a substantial amount of experimental theoretical and computational work has been reported. It is reasonably well documented that shock waves propagating in foams or porous materials are attenuated [14,15] and such accurately controllable shock attenuations are widely used in various engineering and industrial applications.

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Modeling these phenomena is based on microscopic conservation laws (mass, momentum, and energy) from which macroscopic evolution equations (ordinary and partial differential equations) are derived, complemented by appropriate closures. Single and multi-phase approaches have been developed [16,17]. The first fundamental study on the subject is that of Biot [18] on compression wave propagation using poro-mechanics. A large number of papers have appeared in the literature following Biot’s pioneering work, referring to microscopic representations of the phase balance equations within the framework of the theory of mixtures, but applying these concepts to nonlinear waves led to poor results. A more accurate representation of the momentum and energy processes at the internal surfaces of the porous medium was introduced [19] by including the Forchheimer term in the model of [20]. The basic paradox of these equations is that although they are inviscid in nature, so that the shock wave motion is adequately captured and resolved, the Forchheimer term, which approximately accounts for energy losses inside the pores, contains viscous effects. It has been shown that upon the shock wave loading of a saturated porous material, the initial step-wise perturbation is split into two parts [21,22]: the first wave in which pore fluid and porous material are compressed simultaneously, i.e. the solid and fluid particles have in phase motion, and a second wave in which the pore fluid is compressed again, while the porous material relaxes, i.e. the two phases have out of phase motion. In some papers these two waves are known as deformation (fast, mode 1) and filtration (slow, mode 2) waves. Grinten [22] found experimentally that the pressure inside the porous specimen increases gradually and no wave structure is evident in the pressure signals, suggesting that the wave propagation through the material is fully dispersed. In addition, the fluid–solid interaction has a non-linear nature as a non-linear drag law known as Forchheimer term in the momentum transport equation was shown to best fit the data. In similar experiments, Levy et al. [24] found that the compaction wave broadens as it travels through the specimen and becomes dispersed. In general, waves through porous materials will be attenuated. However, substantial amplification can occur when the material is placed in the vicinity or in contact with a backing plate [14,15]. If porous materials are used as barriers to protect structures, measurement of the total stress transmitted to the backing wall is more important than the pressure in the fluid phase. Such measurements have been reported in cases of polymeric (flexible) foams and granular materials under shock compaction [21,25]. Yasuhara et al. [26] and Kitagawa et al. [29] investigated experimentally the effect of porosity and the internal structure on the total stress at the back wall. For all foams tested in these experiments the maximum transient total stress at the back wall is larger than the pressure behind the reflected

Shock wave interactions with nano-structured materials

shock off a rigid wall. It was also observed that there is a delay time for the total stress rise compared with the rise of the pressure upon the reflection of the incident shock wave off a rigid wall suggesting that the stress wave transmitting through the foam decelerates as it moves within the foam. Britan and Ben-Dor [28] conducted a series of experiments on different types of granular materials where pressures in front, behind, and within the granular material were recorded during the impact with weak shock waves. The stress within the granular materials and the total stress at the back wall were measured. It was observed that the pressure behind the shock wave reflected off the material front face reaches the value of the pressure behind a reflected shock over a solid surface. The experiments of Ben-Dor et al. [27] concluded that the gas flow in the granular material due to the impact of the shock on the specimen is the main reason for the formation of the total stress at the end-wall which increases its peak value due to the additional compressive drag force that is applied on the granular particles. These authors argue that filtration process causes energy loss in the transmitted stress wave in the granular material, thus reducing the stress value. The recent work by Kazemi-Kamyab et al. [13], which has focused on the effects rigid open-cell metallic (or ceramic) foams have on the transmission of stresses to a structure impacted by a shock wave, has shown that by carefully selecting a porous material and its size it is possible to cancel out the contributions from the gas and solid phases and therefore avoid the stress wave overshot inside the back wall. The measured time-dependent signals were decomposed into two components, one with low frequency content which is argued that it is associated with Biot’s slow wave and one with high frequency contributions which corresponds to the fast wave propagation. It is also argued that this decomposition can differentiate the effects of gas pressure on the pores from the stress in the matrix of the porous medium. 2 Molecular dynamics computations Our computational component is based on molecular dynamics (MD) simulations. These calculations are capable of resolving changes in bonding that occur under pressure, the melting of materials at high pressures and temperatures, and identifying nanostructural evolution and deformation. In an MD simulation [30,31], one integrates Newton’s equations for individual atoms, using a specified interaction potential, and then averages over atomic degrees of freedom to obtain the continuum fields relevant to larger-scale calculations. For example, local density, velocity, and temperature fields are found by dividing the simulation domain into sampling bins and computing the average number, velocity, and kinetic energy of the atoms contained in each bin. The stress tensor and heat flux are obtained by a more complicated average,

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and operating restrictions such as fixed temperature or pressure may be imposed if needed. In the present simulations, an in-house numerical code was developed and tested, in which a boundary (piston) would simply be translated through a region containing gas molecules at a sufficiently high speed to produce the shock wave, which would in turn impinge on the desired solid material. The piston is simulated as a solid body by stiffening the forces between the atoms while a bulk velocity is assigned to it. 2.1 Mathematical model The behavior of the gas and the target are the focus of this calculation and these materials require realistic interaction potentials, while the piston and holding wall play only a secondary role in the process and a more efficient but semi-realistic description suffices. The gas molecules and the atoms in the piston and holding wall interact via a basic, two-body, short-range attractive potential (“SHRAT”) of Lennard–Jones (LJ) form [33,34]. The short-range attractive potential takes the form 3 512 r r 1− 3−2 φ , r ≤ 1.5ψ ψ ψ (1) U SHRAT (r ) = 27 0 0, r > 1.5ψ whereψ, φ0 are length and energy scales, respectively, while r is the separation distance between two gas molecules/atoms. The potential U SHRAT (r ) is negative in the range 1.0ψ < r ≤ 1.5ψ and positive in the range r < 1.0ψ. Its first derivative yields the force/acceleration, while subsequent integrations can provide the velocity and the position. Here, the cutoff is rather short ranged and smooth, such that not only the potential but also its first and second derivatives vanish at the cutoff distance. For temperatures below 10φ0 /kB , where kB is the Boltzmann constant = 1.3806503 × 10−23 m2 kg/s2 K, this is of no practical concern since the Boltzmann factor exp(−φ0 /kB T ) governing the fraction of particles that can reach this distance is less than 6 × 10−23 . The solid particles in the target interact according to a standard potential for solid nickel, the generic embedded atom method (GEAM) potential [32]. This potential is the sum of two contributions to the total potential energy E: a conventional binary interaction term through a two-body interaction potential U and a term stemming from an embedding functional , which models the effect of the electronic “glue” between atoms E=

N i=1

(ρiemb ) +

N N

U (ri j )

(2)

i=1 i= j

where ri j denotes the norm of the relative vector rij = ri − rj between atoms i and j.

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The embedding functional is a nonlinear function of the (local) embedding densities ρiemb of atoms i = 1, 2 . . . N . The local embedding density ρiemb is constructed from the radial coordinates of surrounding atoms and requires the choice of a weighting function ρiemb =

N N

(w(ri j ) + w(0))

(3)

i=1 i=j

Here, w(0) is the local embedding density of a solitary atom. The (effectively multi-body) model potentials introduced above serve to model a variety of metal properties. For the binary potential function U a radially symmetric short-ranged attractive (SHRAT) potential is used U (r ) = φ0 ψ −4 [3(rcut − r )4 − (rcut − rmin )(rcut − r )3 ] (4) where rcut = 1.6ψ and rmin = 21/6 ψ. The normalized Lucy’s weight function in the definition of the embedding density is chosen r r 3 1− w(r ) = w0 1 + 3 rcut rcut for r ≤ rcut and 0 otherwise with a pre-factor obtained by normalizing the weight function, w0 = w (0) = 3 ). 105/(16πrcut The embedding potential in polynomial form is (ρiemb ) = φ0

i=1 i=j

i=1

where λ is the well depth. In this work, the system of solid and gas molecules was equilibrated at normalized temperature = 1, i.e. Temp = φ0 /κB and λwas set to 100 to ensure that the Argon was in a condensed gaseous state. The rest of the parameters used in the present work, which involved a nickel wall, are shown in Table 1. Values for Ag and Cu are also given for comparison. 2.2 Reference values

emb k emb k 3k αk [(ρiemb − ρdes ) − (w0 − ρdes ) ]r

k=2,4,...

(5) emb is the desired embedding number density and α where ρdes k are the embedding strengths being part of the model. Odd terms in the sum are excluded since their contribution would be always repulsive in nature; the linear term (k = 1) could be adsorbed in a modified pair potential U . The desired density emb = ψ −3 , the embedding in the model equals roughly ρdes density and particle number density n = VN = ψ −3 . The polynomial format of the embedding functional is computationally less expensive than the standard logarithmic form. In addition, the piston and holding wall atoms are tethered to the lattice location with elastic springs, so as to provide rigidity.

Table 1 GEAM and physical parameters for various metals

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The GEAM potential involves both local two-body terms and a collective “embedding” term based on a sum over neighbor atoms within a cutoff. The potential was developed for a fully periodic material where all sites are equivalent, and the potential maintains rigidity of the solid, but the geometry of this problem breaks periodicity in the y-direction, and there are no neighbor atoms at the two edges of the target. Without modification the target would fall apart due to insufficient binding, and a simple remedy is to add layers of “ghost” atoms to either side—the ghost and holding walls— which provide a uniform embedding density and maintain the target’s rigidity. The holding wall, additionally, prevents the target from simply sliding downstream when the shock reaches it. The resulting GEAM potential reads ⎞ ⎛ N N N (ρiemb ) + U (ri j )⎠ (6) E = λ⎝

The determined model parameters and reference values are not unique in the sense that it is possible to find similar sets which would as well resemble the properties of real materials. In the following, reference values used to translate between dimensionless simulation quantities and experimental values of real materials. Any measurable quantity Q with a dimension [Q] specified in SI units kg, m, and s is made dimensionless by a reference quantity −γ /2

Q ref = m α+γ /2 ψ β+γ φ0

for [Q] = kgα mβ sγ

such that Q = Q dimless Q ref ; quantities m; ψ and φ0 provide the scales via the interaction potential (1) and the equations of motion. Quantities in the code were normalized by the length and energy scales as well as the mass per molecule of the solid particles. These values are shown in Table 1 while

Material

rcut /1.6

rmin /21/6

α2

Mass/atom (kg)

ψw (m)

φ0w (J)

Cu

1.010

1.00

0.42

1.06E−25

2.27E−10

5.62E−21

Ag

1.006

1.00

0.70

1.790E−25

2.58E−10

4.74E−21

Ni

1.017

1.02

0.20

9.75E−26

2.28E−10

7.15E−21


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Derived quantity

Reference quantity

Reference value

molecules. For computational purposes, a “ghost wall” identical to the holding wall is placed to the left of the target.

Time

ψw (m w /φ0w )1/2

8.42E−13 s

m w /ψw3

8.23E+03 kg/m3

2.4 System equilibration

Density Temperature

φ0w /κB

5.18E+02 ◦ K

Pressure

φ0w /ψw3

6.03E+08 Pa

Table 2 Reference values for dimensionless derived quantities

Velocity

√ φ0w /m w

Heat flux

−1/2 φ0w ψw−3 m w

270.8239508 m/s 1.63E+11 W/m2

Fig. 1 Schematic view of atomistic shock tube model

Table 2 shows the values used to make the derived quantities dimensionless. 2.3 Shock tube model setup The physical computational domain is shown in Fig. 1: a long channel of square cross section which reading from left to right contains a piston, a gas region, a target solid material, and a holding wall. The size of the domain is L x × L y × L z = 29ψw × 133.7ψw × 29ψw , where ψw is the characteristic length in the gas potential, a few Angstroms or roughly an atomic diameter. The piston and the holding wall are composed of two layers of face-centered cubic lattice with thickness a = 3.22ψw and 2,592 nickel atoms. The target solid is placed at the right end of the domain, and consists of a thicker slab of 12,960 nickel atoms occupying an axial length of 16.1ψw . The gap between the piston and target (of length L y,fluid = 107.93ψw ) is filled with 2,708 argon

The fluid molecules are initially located at face-centered cubic lattice sites within the fluid region of the domain. The time step is set to 10 fs and a cut–off radius of rc = 1.5ψg or ψgw for all interactions except the solid–solid EAM interaction where it requires a cut-off radius of 1.6ψw . The system is then equilibrated at a temperature T = 1 for 10,000 time steps by rescaling the atomic velocities. (We use MD units based on ψw , the argon mass, and a characteristic MD time related to the SHRAT potential of about 2 ps.) At the end of the equilibration phase, the gas molecules fill their region randomly and uniformly on average, while the solid atoms have small thermal fluctuations about their original lattice positions. This equilibrated configuration provides the initial condition for shock motion. Local quantities such as density, velocity, temperature and stress can be computed by subdividing the computation domain into 2D cells or “sampling bins” in the Y–X plane, which fill the computational domain without gaps and overlaps (see Fig. 2). Each cell has a size of 1.61ψw or one layer of a cubic face-centered lattice. In each cell, one accumulates the value of the field of interest, given by a local function of atomic positions and velocities, r some time interval, and then computes the average. One-dimensional profiles are computed by averaging over the cells in transverse-direction, reducing the 2D cell configuration to a one-dimensional array of vertical slabs. 2.5 Solution method and boundary conditions Periodic boundary conditions are used in the lateral directions but were abandoned in the normal direction since the atomistic is symmetric in the lateral directions only. The Lorentz– Berthelot mixing rules are used for estimating intermolecular potential parameters between pairs of non-identical molecules; for the interaction of heteroatomic pairs, the effective values of ψ and φ0 are calculated from those for the homo-atomic pairs using the mixing rules: arithmetic mean

Fig. 2 Flow domain with 2-D bins

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Fν = −λ

N i i = j

∇ν U (ri j )+

∂(ρ emb ∂(ρiemb ) j ) + w(r ) ×∇ ν i j ∂ρiemb ∂ρ emb j

(7) The Greek subscript ν stands for Cartesian components associated with the x, y, z directions. Finally, correct the predicted values of the first-step approximations by a scaled term which represents the discrepancy between the second derivative obtained from Taylor series expansion and the acceleration calculated explicitly from force. 2.6 Shock wave formation and propagation Results are presented below for a flow with compression ratio ρ0 /ρ1 = 0.38, a corresponding piston velocity u p = 4.0(φ0w /m w )1/2 , and a shock wave velocity u s = 6.452(φ0w /m w )1/2 . Under these initial conditions Ar remains in the gas regime (Holian et al. [7]). The shock thickness was evaluated as = (1 − 0 )/(∂/∂ y|max ) is = 6.614ψw ; the volumetric strain in the shock was defined as ζ = ρ0 /ρ1 − 1 = −u p /u s is ζ = −0.62, and ·

the volumetric strain rate defined as ζ = ζ u s / = −u p / ·

is ζ = −0.605 s−1 . After equilibration, the piston was set into motion by giving its atoms an extra prescribed velocity to the right, which forces the gas to compress and move in that direction. When this additional velocity exceeds the sound speed a shock wave is formed, which moves towards and eventually impacts the solid wall. The motion of the piston is then stopped but the shock continues to move through the solid, and reflects at the gas-target and target-holding wall boundaries. Properties of both solid- and gas-phase regions were averaged locally over the cell size every 100 time steps. These properties were stored in a separate file for post-processing. 3 Results Fig. 3 shows the temperature profiles in the flow along the gas channel behind the moving shock wave at various instances.

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9 8

t=3 t=6.0 t=9.0 t=12.0 t=14.0 t=16.0 t=18.0 t=20.0 t=22.0

7

Temperature (φ 0w/κB)

for ψ : ψab = 0.5(ψa + ψb ) and geometric mean for φ0 : √ φ0ab = φ0a φ0b . Length and energy scales for solid particles are shown in Table 1 and length and energy scales for Argon particles are 3.405E−10 (m) and 1.65E−21(J), respectively. The well-known gear 5th order predictor corrector algorithm was used to solve the equation of motion F = ma. The algorithm is built as follows: first, predict positions and up to fifth-order derivatives at time t + dt via Taylor expansions. Force is then computed at time t + dt from the derivative of the pair potential

6

Incoming shock front

5 4 3 2 1 0 -1

Piston end position 0

20

40

60

80

100

120

140

Location (ψ w)

Fig. 3 Temperature profiles in the gas flow domain generated by the propagating shock wave at various times

The shock and the induced flow behind appear to be developing up to the distance of 40ψw where the temperature reaches a constant value. The shock impacts the end wall (target) and it is subsequently reflected moving in the opposite direction, towards the piston. This reflection is associated with a considerable increase in temperature and a temperature ratio of about 4 across the incoming shock wave is been observed; however, across the reflected shock the temperature is about 8 times greater than that of the un-shocked gas. At the solid wall, the shock wave is partially reflected and partially transmitted into the solid phase with energy being exchanged, so that part of the shock wave energy is reflected back into the gas channel and the remaining part is exchanged with the solid molecules. The transmitted shock starts propagating inside the solid structure till it hits the holding wall. Unlike the microscopic behavior, the gas temperature behind the reflected shock is not constant at different times. Initially the temperature is high and is subsequently decreased due to the energy exchange with the solid wall particles. The reflected shock off the end wall propagates toward the piston which has been stopped some time earlier and secondary reflections occur. The simulations were terminated at this time. As shown in Fig. 4 the pressure profiles in the gas channel exhibit a behavior similar to that of the temperature. The pressure ratio across the incoming shock wave is about 10 and about 5 across the reflected shock. According to the classical shock tube theory, the corresponding pressure ratio in the case of a rigid wall reflection P5 /P2 is more than 33 for the same pressure ratio across the incoming shock of 10. This comparison clearly demonstrates the striking differences between the present case with an elastic end wall and the classical shock reflection off a rigid body planar surface. In addition, due to the energy exchange with the solid molecules, there is a pressure drop behind the reflected shock as it propagates back into the gas channel.


75 0.25

t=3 t=6.0 t=9.0 t=12.0 t=14.0 t=16.0 t=18.0 t=20.0 t=22.0

Pressure (φ 0w/ψ w3)

2.5 2.0 1.5 1.0

t=3 t=6.0 t=9.0 t=12.0 t=14.0 t=16.0 t=18.0 t=20.0 t=22.0

0.20

Density (mW/ψ 3)

3.0

0.15

0.10

0.05

0.5 0.00

0.0 0

20

40

60

80

100

120

140

0

20

Location (ψ w)

5 t=3 t=6.0 t=9.0 t=12.0 t=14.0 t=16.0 t=18.0 t=20.0 t=22.0

Velocity (φ 0w/mw)1/2

4 3 2 1 0 -1 -2

0

20

40

60

80

100

60

80

100

120

140

Location (ψ w)

Fig. 4 Pressure profiles in the gas flow domain generated by the propagating shock wave at various times

-3

40

120

140

Location (ψ w)

Fig. 5 Velocity profiles in the gas flow domain generated by the propagating shock wave at various times

Figure 5 shows the velocity profiles of the shocked gas particles ahead of the piston which is the same as that of the piston. The results show that the distance between the shock front and the piston is getting greater as the simulation time increases since the shock speed is higher than the induced velocity of the molecules behind. Once it hits the target, reflection causes deceleration of the gas flow and eventually the flow changes direction towards the piston. When the shock hits the solid wall, not only energy is exchanged, but also the solid structure deforms, initially pushed and displaced in the downstream direction and subsequently in the opposite direction, which causes the gas flow to change direction. In that respect, the solid-phase wall behaves like a vibrating elastic wall which is initially displaced in the y-positive direction during which the flow is decelerated followed by a displacement in the y-negative direction which results in a push of the gas molecules in the same direction. This behavior shows some strong similarities with the corresponding continuum case of shock waves impacting elastic

Fig. 6 Density profiles in the gas flow domain generated by the propagating shock wave at various times

plates which has only recently been studied in [2]. In such situations the plate starts to vibrate while at the same time it affecting the flow behind the reflected shock wave in a fully coupled manner under a two-way interaction. This strong coupling changes the dynamics of the plate by reducing the vibrational frequencies due to increased damping in the combined fluid-structure system. Figure 6 shows the density profiles in both the un-shocked and the shocked gas regions as the shock wave propagates in the gas channel. At the solid surface reflection takes place and density builds up gradually. At t = 18ψw (m w /φ0w )1/2 , the first profile after the shock refection, the maximum density observed is 0.185. It builds up further to a maximum value of 0.185 almost twice that of the shocked gas. As the shock reflection goes on, a drop in gas density is observed in the region close to the solid-phase wall which is due to its elastic behavior. The drop in density in the left part of each of the density profiles after the shock reflection is associated with the arrest of the piston motion, which generated a rarefaction wave moving towards the end wall. The piston was stopped at about the 66ψw location just before the reflected shock was about to impact upon it. Profiles of the heat flux in the gas channel during the shock propagation are shown in Fig. 7. The instantaneous heat flux in a given direction is evaluated from the energy associated with each atom in the simulation

N 1 ∂ ri E i (8) q= ∂t V i=1

where q is the heat flux vector, ri the position vector of atom i and E i is the energy associated with atom i, and the sum is over all N atoms. The energy associated with each atom is the sum of its kinetic energy and potential energy Ei =

N 1 j=1

1 U (ri j ) + m i (˙ri − (vi ))2 2 2

(9)

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76


Heat flux (φ 0wψ -3mW-1/2)

2

t=3 t=6.0 t=9.0 t=12.0 t=14.0 t=16.0 t=18.0 t=20.0 t=22.0

1

0

-1

-2

0

20

40

60

80

100

120

140

Location (ψ w)

Fig. 7 Heat flux profiles in the gas flow domain generated by the propagating shock wave at various times 275

Bulk Modulus (φ 0W/ψ 3)

Wall 270

265

260

255

250

0

5

10

15

20

25

Time (ψ w(mW /φ0W )1/2)

Fig. 8 Variation of average bulk modulus of the solid-phase wall during shock impact

where m i is the mass of atom i, vi denotes the (macroscopic) flow velocity on position of particle, r˙ i is the velocity vector of atom i, and U (ri j ) the pair-wise interaction between atoms i and j when separated by a distance ri j . Substituting into the above equation we get ⎤ ⎡ N N N 1 1 q= ⎣ r˙ i E i + rij (Fij · (˙ri − (˙vi )))⎦ (10) V 2 i=1

i=1 i= j

where rij = ri − rj and Fij is the force exerted on atom i by atom j. The first term within the square brackets is related to convection and the second to conduction. However, solid particles with embedded atom interaction will have also embedding energy that needs to be added when calculating the heat flux in the solid structure. From the data in Fig. 8 it can be seen that distribution of the heat flux is mostly affected by the behavior of the gas molecular velocity and therefore some obvious similarities are evident between the heat flux and the velocity profiles.

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The velocity reverses direction at t = 20ψw (m w /φ0w )1/2 and so does the heat flux. The heat flux increases across the propagating shock front. During the shock wave reflection at the solid wall, the heat flux reaches a maximum value which is about 25 % higher than that of the shocked gas behind the incoming shock. This jump in heat flux is due to the conversion of kinetic energy into heat during the reflection as well as to heat conduction that takes place at the wall. The heat flux decreases at different times after the shock reflection takes place and then changes sign as the flow changes direction. Figure 8 shows the bulk modulus of the solid-phase wall spatially averaged over the slabs as a function of time. It should be noted here that no constitutive law between the stress and the corresponding strain has been introduced here. The stresses, the strains and the moduli have been evaluated directly and independent of each other. At equilibrium, the front slabs have the same average value since they are surrounded by solid molecules that interact only according to the ‘GEAM’ potential. However, solid particles at the end slab of the target are interacting with the holding wall particles according to the ‘SHRAT’ potential that does not possess any embedding contribution, so the average value of the bulk modulus at the back edge is less than that of the front slabs. As the shock wave impinges on the wall, the front slabs will have the most impact and will have higher increase in the bulk modulus than that in the back slabs. In the front slabs, the shock impact results in a primary jump in the modulus before it starts decreasing, and when the shock is reflected back at the holding wall it causes the bulk modulus to have another rise. However, the back slabs are closer to the holding wall and the shock reflection occurs earlier and the primary rise in the bulk modulus due to the shock impact is immediately followed by another rise due to the shock reflection. Therefore one jump in the bulk modulus is observed. A strain is a normalized measure of deformation representing the displacement between particles in the body relative to a reference length. The average strain shown in Fig. 9 is obtained by averaging the local strain over the whole length of the solid wall. During equilibration and before the shock wave reaches the target, the solid particles are vibrating about an average local position and the distance between any two successive layers of solid particles is, on average, constant. This state represents an unstrained structure. During and after the shock impact, the solid structure is deformed and the spacing between layers of solid molecules in the normal direction is changed. Strains of the order of 0.15 are observed during the passage of the transmitted wave. 3.1 Energy considerations According the present model the energy in the system is contributed by


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1. Gas molecules through their potential and kinetic energy. 2. Wall molecules through their potential, kinetic and embedded energy. 3. Piston molecules through potential energy ‘SHRAT’ and kinetic energy. 4. Holding wall molecules through potential energy ‘SHRAT’ and kinetic energy. 5. Ghost molecules through potential energy, kinetic energy and embedded energy. Fig. 11 Schematic view of porous material structure with straight poles attached to solid structure

The total energy of the systems remains constant but the amount of energy in each component of the shock model varies differently. In the following, focus is provided to the change in the energy of the gas and the solid wall molecules rather than to the absolute value of energy. The GEAM potential describes the interaction of the solid–solid molecules and solid–ghost molecules, whereas, SHRAT potential describes the interaction of solid–gas and solid–holding wall particles. There are three components of energy in the solid structure: Potential, kinetic, and embedding. Change in energy can be found by evaluating each energy component at each time step and finding the difference in their magnitudes during and after the shock impact and reflection. However, the total change in energy can be also found by calculating the work done by the solid particles during their displacement. The magnitude of the change in energy by either way is the same. Figure 10 shows the energy change on the solid wall associated with the shock impact and the traveling of the transmitted wave. There is a rise in the energy in the solid structure due to the energy exchange between gas particles and solid particles which results in an increase in stress and strain in the solid structure. As the shock reflection takes place and the shock travels upstream, the structure tends to restore its unstrained state.

4 Shock waves impacting porous materials A porous structure was designed by removing some of the solid particles to form circular poles that are attached to full layers of particles. The dimensions of the gas channel remained the same but only the solid wall structure (target) is changed. The removed particles are compensated by the ghost particles in a way that the total number of particles of the structure remains the same regardless of the number and size of the poles. In the structure shown in Fig. 11 there are three rows of poles with three poles in each row for a total of nine poles. The first and third rows are in line with each other and the second is offset by half the distance between any two rows. All the poles have the same diameter and depth. Porosity is controlled by varying the pole diameter. In all cases the depth is set to 9.66ψw . The volume occupied by the pole particles is the crosssectional area multiplied by the pole length. The poles are attached to a solid wall (target) that has an axial length of 16.1ψw . When the poles are formed a desired value of the pole diameter was set. However, porosity cannot be calculated based on this diameter since the position vector has dis-

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Fig. 13 Gas normal stress on poles face and pole base area for η = 0.65 and comparison of with the non-porous wall case with η = 1.00

crete values and there always will be a deviation between the desired diameter value and actual size of the produced poles. Porosity was calculated based on the number of particles in the volume occupied by the pole particles only divided by the number of solid and ghost particles in that volume, η = Nsolid /(Nsolid + Nghost ). The shock wave has been generated by the sudden forward motion of the piston and the bulk flow parameters remained the same as in the previous case where the shock impacted a solid wall. Figure 12 shows the pressure distribution along the gas channel at various times. A shock wave was formed and developed as it was discussed earlier and it propagated in the gas channel till it reached the porous wall. Part of the shock impacts the pole face and then is reflected back towards the piston: this is the primary reflection, while the other part of the shock keeps propagating in the space between the poles, impacting the base of the wall and then reflected backwards in the opposite direction a process designated as secondary reflection. The results in Fig. 12 show that the secondary reflection results in higher pressure rise than the primary one at t = 0.20. However, if a comparison with the data in Fig. 4 is made one can conclude that in the case of porous wall, the pressure behind the reflected shock is always less than that of a solid wall. As porosity increases primary reflection strength weakens and secondary reflection becomes stronger. The outcome of this split reflection depends on the frontal reflecting area ratio. For the same depth of each test case lower porosity is equivalent to higher frontal area associated with the primary reflection. On the other hand higher porosity results in higher reflecting area in the back wall. Figure 13 shows the gas-phase normal stresses exerted on the poles face and on the pole base area during the primary and secondary shocks impulsive load on the solid structure: the primary shock impacts the poles face first followed by

secondary shock impact on the base area. The normal stress at the poles face area rises with time due to the primary shock impact which results in a partial reflection and partial transmission into the porous material. It decreases subsequently as the shock travels upstream and the flow changes direction. Besides, the normal stress at poles base area also rises due to the secondary shock impact. Then it decreases as the shock is fully reflected and the flow changes direction within the pores. It is also observed that the normal stress rise at the face is smaller in magnitude from that at the base and it occurs earlier than that at the base. The reason for this behavior is most probably due to a leak of high pressure formed at the pole face into the lower pressure inside the pores behind the shock traveling towards the base. As a result of this not only the peak value at the face is lower than that at the base but also the rate of stress build up (slope) is lower. Of interest is a comparison of these data to the corresponding data obtained in the case of a shock impacting a pure solid wall only without any porous front structure. This case is simply designated as a case with η = 1.00 porosity. This is also shown in Fig. 13 where the gas and solid stresses are plotted as a function of time at the interface between these two phases. First, there is a continuation of the stress from the gas to the solid phases as expected since the gas load stress is completely transmitted into the wall. The effect of the porous wall now becomes clear. There is an approximate attenuation of the maximum transmitted stress by 30 %. In addition, the rate of loading the target appears to be slower in the case of a porous wall with η = 0.65 than in the case of η = 1.00. Finally, the change in the energy of the solid wall (target) particles (shown in Fig. 14) associated with the transmitted stress in the case of porous wall with η = 0.65 is by 35 % lower than that shown in Fig. 10 with η = 1.00.

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5 Conclusions The interaction of shock waves with nano-structured materials was investigated using molecular dynamics. This work has been motivated by the promise of these materials to dissipate drastically fast traveling shock or blast waves at the nano-scales because their surface-to-volume ratio is substantially higher than in the case of macro-scale materials. The interaction was setup by a piston in a stagnant gas moving with a velocity large enough to generate a shock wave upstream traveling through the gas molecules towards a solid wall target that is covered with a nano-structured porous material in front which provides the required protection. This setup allowed a preliminary evaluation of the effectiveness of the material in blast mitigation strategies. In order to provide a reference for comparison, the case of the shock impacting the target wall in the absence of a protective material was extensively investigated and documented first. This case also provided details of the fluid-structure interaction which involved the shock reflection at the front interface of the target wall, its transmission into the solid material as well as the deformation of the target. Although the setup of the interaction in this case may show similarities with the shock tube flow at the macro-scale level, there are notable differences between them, the most significant being the flow reversal immediately after the impact of the shock on the target wall as shown in the velocity profiles of Fig. 5. In the classical shock tube flow, the end wall is considered rigid and the flow behind the shock reflection is suddenly decelerated to full stagnation. In the present case, the wall is elastic and its time-dependent deformation and motion are the reasons for the flow reversal. In MD the stress tensor and heat flux were evaluated using relations which do not assume or invoke any constitutive law between stress and strain or heat flux and temperature. Shear

and bulk moduli were evaluated independently of constitutive relations. Our results suggest the presence of high-strainrate effects in the solid wall material which show a moderate increase of its strength during the shock impact. Another advantage of MD calculations is that the flow inside the moving shock can be resolved, a capability which is not evident in computations using classical Navier–Stokes equations. The presence of nano-porous material layers in front of the target wall appears to have beneficial effects on its survivability. First, it reduces the stress magnitude observed inside the target wall by about 30 % while the deposited energy is also decreased by about the same amount when compared with the non-protected target case which is equivalent to a fully porous material with 100 % porosity. Second, it decreases the loading rate of the target wall and thus further protecting the integrity of the structure. Additional work is required to fully document the behavior of such materials during shock impact in different geometric arrangements, sizes, properties, and parameters. Acknowledgments The financial support provided by NSF under Grant CCI-0800307 is greatly appreciated.

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