Fluid–Structure Interaction Effects in the Dynamic Response of FreeStanding Plates to Uniform Shock Loading Nayden Kambouchev Raul Radovitzky1 e-mail:
[email protected] Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA 02139
Ludovic Noels Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA 02139; LTAS-Milieux Continus & Thermomécanique, University of Liège, Chemin des Chevreuils 1, B-4000 Liège, Belgium
The problem of uniform shocks interacting with free-standing plates is studied analytically and numerically for arbitrary shock intensity and plate mass. The analysis is of interest in the design and interpretation of fluid–structure interaction (FSI) experiments in shock tubes. In contrast to previous work corresponding to the case of incident blast profiles of exponential distribution, all asymptotic limits obtained here are exact. The contributions include the extension of Taylor’s FSI analysis for acoustic waves, the exact analysis of the asymptotic limits of very heavy and very light plates for arbitrary shock intensity, and a general formula for the transmitted impulse in the intermediate plate mass range. One of the implications is that the impulse transmitted to the plate can be expressed univocally in terms of a single nondimensional compressible FSI parameter. 关DOI: 10.1115/1.2712230兴 Keywords: uniform shock waves, fluid-structure interaction
1
Introduction
The reduction of impulse transmitted to structures subject to blast loading provided by the fluid–structure interaction 共FSI兲 effect was recognized in the early work of Taylor 关1兴 who studied the reflection of a blast wave with an exponential pressure profile in the case of negligible fluid compressibility. Taylor 关1兴 showed that the impulse transmitted to the plate is reduced as the plate mass decreases because lighter plates acquire velocity quickly thus relieving the pressure acting on the surface of the plate. This peculiar property of the response of light structures to blast loads has been used in the design of sandwich panels with increased resistance to underwater explosions 关2–10兴. As part of the conclusions of the analysis, Taylor showed that the impulse transmission 1 Corresponding author. Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received July 4, 2006; final manuscript received September 16, 2006. Review conducted by Robert M. McMeeking.
1042 / Vol. 74, SEPTEMBER 2007
depends on a single nondimensional parameter representing the relative time scales of the blast overpressure and of the fluid– structure interaction. In previous work 关11兴, we extended Taylor’s results by incorporating the effect of compressibility, which is important in the case of blast waves propagating in air. The contributions included: the analysis of the asymptotic limits when the plate is very light and very heavy, the identification of an extended nondimensional FSI parameter which becomes relevant in the compressible range, and a practical formula for calculating the impulse transmitted to the plate for arbitrary plate weights and blast intensities. Whereas the light plate asymptotic result was exact, the heavy plate asymptotic result was approximate due to lack of an explicit solution for the flow field of an exponentially decaying blast pressure profile reflecting from a fixed boundary. An approximate asymptotic result was obtained by assuming that each level of the incident pressure reflected according to the Rankine–Hugoniot shock jump conditions. In this technical brief, we discuss the simpler case of freestanding plates subject to uniform shock loading. The analysis should prove useful in the design and interpretation of FSI experiments using conventional shock tubes. It is shown that in the case of uniform shocks, the asymptotic case of heavy plates can be obtained exactly. As a first step, the analysis of the acoustic limit is discussed in Sec. 2 resulting in an explicit relationship between the transmitted impulse and the acoustic nondimensional parameter. Section 3 is devoted to the asymptotic analysis for heavy and light plates for arbitrary shock intensities. In Sec. 4 a practical formula interpolating the exact limits and encompassing the intermediate range is also proposed and verified. The analysis reveals that the FSI is governed by a nondimensional parameter which is analogous to the acoustic parameter due to Taylor but incorporates the state of compressibility of the fluid and, thus, the intensity of the shock. Interestingly, it is found that the dependence of the transmitted impulse ratio with the mass of the plate collapses onto a single curve, independently of the shock intensity. It is concluded that significant reductions in impulse transmission are achievable by reducing the mass of the structure facing the shock.
2
Exact Solution for Acoustic Waves
The derivation presented in this section is a direct extension of Taylor’s analysis for exponentially decaying pressure profiles 关1兴 to the case of uniform waves. The problem setup is as follows. An infinite uniform pressure wave of overpressure ps, propagating in the positive x direction in a fluid medium with density 0, sound speed a0, and pressure p0, impinges on an initially stationary plate of mass per unit area m p located at x = 0 m. The wave reaches the plate at time t = 0 s. The location of the plate is denoted by = 共t兲. The pressure on the right side of the plate is assumed to stay constant and to be equal to the atmospheric pressure p0 at all times. Newton’s second law gives the equation of motion of the plate mp
d 2 = p共,t兲 − p0 dt2
共1兲
where p共 , t兲 − p0 is the overpressure acting on the plate. The onedimensional equation of motion for the fluid is
p du =− dt x
共2兲
where the convective derivative is given by d / dt = / t + u共 / x兲 and the density = 共x , t兲, the pressure p = p共x , t兲, and the velocity u = u共x , t兲 are all functions of both the particle location x and the time t. Acoustic waves will cause only small perturbations around the steady-state values, leading to: = 0 + ˜, p = p0 + ˜p, and = 0
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+˜. The small pressure perturbation ˜p must then satisfy the wave equation 2 ˜ 2˜p 2 p − a =0 0 t2 x2
共3兲
and therefore can be expressed as a sum of two waves propagating to the left and to the right at the sound speed a0 ˜p共x,t兲 = f共x − a0t兲 + g共x + a0t兲
共4兲
Using Eqs. 共1兲 and 共2兲, the function g is eliminated leading to mp
d3˜ d2˜ ˜ 3 + 0a0 2 = − 2a0 f ⬘共 − a0t兲 dt dt
共5兲
If the shape f of the incoming wave is known then this equation can be solved for ˜. For uniform shock waves the shape is given by f共˜ − a0t兲 = ps = const. The appropriate boundary conditions of the differential Eq. 共5兲 are ˜共t = 0兲 = 0 d˜ 共t = 0兲 = 0 dt
共6兲
f共0兲 + g共0兲 d2˜ 2 共t = 0兲 = dt mp Since initially the plate is not moving and behaves as a rigid boundary that reflects the wave completely, g共0兲 = f共0兲 and the solution is
共
兲
a ˜ = 2psm p e− m0 0 t − 1 + 2ps t p 2 2 0a 0 0a 0
共7兲
The solution clearly shows that the time scale of the fluid structure interaction is given by the time constant t* = m p / 0a0. As the incident pressure wave lacks an intrinsic time scale, we arbitrarily choose a time scale ti and interpret it as the time elapsed from the time of shock impact t = 0. In what follows we consider the motion of the system comprising the fluid and the plate up to the fixed moment of time t = ti, which can be chosen arbitrarily. Following Taylor 关1兴, one can define a nondimensional parameter 0 = ti / t* which compares the relative time constants of the fluid–structure interaction t* and the incident wave ti. The quantity Ii = psti represents the impulse carried by the incident pressure wave through the point x = 0 up to the moment of interest t = ti. By noting that the acceleration of the plate d2˜ / dt2共t ⬎ 0兲 = 2ps / m pe−共0a0/mp兲t remains positive at all times, one concludes that the maximum velocity and, therefore, the maximum impulse of the plate occur at time t = ti. This maximum impulse when expressed in terms of the nondimensional parameter 0 is 1 − e −0 Ip =2 Ii 0
共8兲
Equation 共8兲 is the uniform-wave analog to Taylor’s result for exponentially decaying pressure waves 关1兴 Ip = 200 Ⲑ 共1−0兲 Ii
共9兲
where 0 is defined as above and ti is the decaying time of the incident exponential pressure wave. In both cases lim0→0共I p / Ii兲 = 2, which represents the fact that for fixed rigid plates the incident wave is reflected completely. Journal of Applied Mechanics
Fig. 1 Incident and reflected shock waves; „a… incident shock wave, „b… reflected shock wave
3
Extension to the Compressible Range
The problem considered in the previous section can be extended to compressible flows by eliminating the assumption that the overpressure of the incident wave is small. By contrast to the acoustic limit, the resulting coupled problem of a compressible nonlinear one-dimensional flow interacting with a plate is not amenable to analytical treatment. Instead we follow the approach in Ref. 关11兴 and find the two asymptotic limits for heavy and light plates. In the intermediate range, we extrapolate a curve from the asymptotic limits and verify these results against numerical computation. 3.1 Heavy Plate Asymptotic Limit. In contrast to the case of exponential incident pressure waves considered in Ref. 关11兴, for uniform incident pressure waves the impulse transmitted to heavy plates may be found exactly. In the heavy plate limit the plate may be considered as a fixed rigid boundary. Figure 1 shows a schematic of the problem where a is the local speed of sound, u is the local particle velocity, is the local particle density, p is the local pressure or overpressure, and U is the shock wave speed. Subscripts s, r, and 0 are used to denote states behind the incident shock, behind the reflected shock, and atmospheric, respectively. Assuming that the fluid is an ideal calorically perfect gas with constant specific heats ratio equal to ␥, which has been shown to be a realistic assumption even for very strong explosions in air 关12兴, and using the Rankine–Hugoniot relations it is found that 关13兴
s = 0
Us = a0
us = a0
共␥ + 1兲ps + 2␥ p0 共␥ − 1兲ps + 2␥ p0
冑
共␥ + 1兲ps + 2␥ p0 2␥ p0
冑 冑 2ps ␥ p0
ps 共␥ + 1兲ps + 2␥ p0
共10兲 共11兲 共12兲
Similarly for the reflected wave, the relations are
␥ ps + ␥ p0 共␥ − 1兲ps + ␥ p0
共13兲
共␥ − 1兲ps + ␥ p0 ps
共14兲
共3␥ − 1兲ps + 4␥ p0 共␥ − 1兲ps + 2␥ p0
共15兲
r = s
Ur = us
pr = ps
A direct consequence of these expressions is that the gas density neither in the incident wave s nor in the reflected wave r can grow unboundedly as the incident overpressure ratio ps / p0 becomes large. In the limit ps / p0 → ⬁, the incident density ratio tends to 共␥ + 1兲 / 共␥ − 1兲, while the reflected density ratio tends to ␥ / 共␥ − 1兲. For air 共␥ = 1.4兲 these two ratios are 6 and 3.5, respecSEPTEMBER 2007, Vol. 74 / 1043
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tively. Another known consequence of the shock reflection is that the pressure reflection coefficient CR defined as 关13兴 CR =
pr 共3␥ − 1兲ps + 4␥ p0 = ps 共␥ − 1兲ps + 2␥ p0
共16兲
is also bounded, 2 艋 CR 艋 共3␥ − 1兲 / 共␥ − 1兲 共or 2 艋 CR 艋 8 for air兲. The lower limit corresponds to weak acoustic waves and the upper limit to very strong shocks. It should be emphasized that for air the reflection coefficient CR departs from 2 for rather small overpressures. For example, CR = 2.75 when ps = p0 = 1 atm, and therefore the use of results from the acoustic limit theory in the evaluation of the effects of blast loads on structures cannot be justified, as it has been pointed out recently 关14,11兴. The impulse per unit area transmitted to the plate is the time integral of the overpressure it experiences Ip =
冕
pr dt = prti
共17兲
Consequently the impulse transmission coefficient is equal to the pressure reflection coefficient I p p rt i = = CR I i p st i
− pr = r共− Ue兲2 − e共u p − Ue兲2
共24兲
the plate velocity is found to be u2p =
冉 冊
2pr pr r pr −1 = r e r 共␥ − 1兲pr + 2␥ p0
3.2 Light Plate Limit. Following Ref. 关11兴, we first consider the acoustic limit. In this case 0 → ⬁ and the plate instantaneously reaches its final velocity 2ps d˜ 共1 − e−0t Ⲑ ti兲 = 2ps = lim 0a 0 0→⬁ dt 0→⬁ 0a0 lim
共19兲
It is interesting to note that in the case of a uniform incident wave, the velocity of a very light plate remains constant in time, whereas in the case of an exponential profile the plate velocity decays exponentially. From Eq. 共19兲 the transmitted impulse is 2 I p 2m p = lim = 0a 0t i  0 0→⬁ Ii
共20兲
and independent of time 0 艋 t 艋 ti. Following Ref. 关11兴, we assume that the maximum transmitted impulse in the nonlinear compressible range can be derived from the plate velocity u p at time 0+. Toward this end, we consider the expansion wave produced by a fluid initially compressed at overpressure pr = CR ps on a free surface which is initially at rest. Instantaneously upon reflection, the fluid state is characterized by the normal shock reflection on a fixed boundary 共ur = u p = 0兲 independently of the plate mass m p. The reflected state can be characterized as 共3␥ − 1兲ps + 4␥ p0 共␥ − 1兲ps + 2␥ p0
共21兲
共␥ + 1兲ps + 2␥ p0 ␥ ps + ␥ p0 共␥ − 1兲ps + 2␥ p0 共␥ − 1兲ps + ␥ p0
共22兲
where Eqs. 共10兲, 共13兲, and 共16兲 have been used. In the limit m p → 0, the motion of the plate is equivalent to that of a free surface acted upon by the reflected fluid overpressure pr on one side and zero overpressure on the other side. An expansion wave propagating at speed Ue is instantaneously formed with the overpressure pe = 0 and the velocity of the fluid particles ue = u p on the right and overpressure pr and velocity ur = 0 on the left. Applying mass and momentum conservation across the expansion wave 1044 / Vol. 74, SEPTEMBER 2007
共25兲
where the ratio r / e has been expressed in terms of the pressure ratio by using the jump conditions Eq. 共10兲. After some algebraic manipulation, the expression for the plate velocity can be written as
共26兲 where f R is a nondimensional factor that depends exclusively on the incident overpressure ratio ps / p0. From Eq. 共26兲, the transmitted impulse ratio is finally obtained lim
共18兲
in direct contrast with the result for exponential pressure profiles 关11兴 in which it is always smaller. Another important difference is that this result is exact, while the result for exponential pressure profiles is approximate 关11兴.
r = 0
共23兲
ti
0
pr = ps
r共− Ue兲 = e共u p − Ue兲
Ip
0→⬁ Ii
=
m pu p m pC R f R = p st i sU st i
共27兲
This equation clearly reveals that ts* = m p / sUs is the time scale of the interaction of the shock wave with the plate and that the interaction process is characterized by the nondimensional parameter s = ti / ts*. In terms of this parameter, Eq. 共27兲 becomes I p CR f R = s 0→⬁ Ii lim
共28兲
It is evident that this expression degenerates to Eq. 共20兲 for weak sonic disturbances as limps→0CR = 2 and limps→0 f R = 1. In the case of air, f R remains close to one, reaches a maximum of 1.26 for ps / p0 ⯝ 3.5, and tends to 冑9 / 7 as ps / p0 → ⬁.
4 Intermediate Plate Weights and Numerical Verification It is useful for the purpose of practical application to devise an expression for the maximum momentum transmission coefficient for arbitrary plate weights and shock intensities. As discussed in Ref. 关11兴, the resulting expression should reduce to: • • •
The acoustic result Eq. 共8兲 for very small overpressures; The heavy plate response Eq. 共18兲 for small s and arbitrary shock intensities; and The light plate limit Eq. 共28兲 for large s and arbitrary shock intensities.
A possible expression satisfying these requirements is 1 − e−s/f R Ip = C RI i s/f R
共29兲
This formula represents the ratio of momentum acquired by the plate for an arbitrary plate weight and shock intensity and the impulse that would otherwise be transmitted to the plate should fluid–structure interaction effects be ignored. It is interesting that in the case of a uniform incident shock considered in this paper, the resulting expression Eq. 共29兲 collapses into a single curve as a function of the parameter s / f R. The main difference between Eq. 共29兲 and the result presented in Ref. 关11兴 is that the expression proposed here is exact in the heavy plate limit. A numerical method has been used for the purposes of verifying the various results of the analysis presented in the foregoing as well as the accuracy of the empirical formula Eq. 共29兲 in the intermediate range of plate masses where exact solutions are not available. The numerical method employed as well as the simulations were reported elsewhere 关15兴. These consisted of generating Transactions of the ASME
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It is found that the relative impulse transmitted to the plate can be described by a single nondimensional parameter which is an extension of Taylor’s acoustic FSI parameter to the compressible range and incorporates the shock intensity. An explicit approximate expression for this dependence is proposed. In the intermediate range of plate masses the proposed general formula is verified numerically. Similarly to what has been found before, the use of lighter plates has the benefit of reducing the transmitted impulse, which potentially can be exploited in structural designs with improved blast resistance, e.g., sandwich plates with light front sheets.
Acknowledgment This research was supported by the U.S. Army through the Institute for Soldier Nanotechnologies, under Contract No. DAAD-19-02-D-0002 with the U.S. Army Research Office. The content does not necessarily reflect the position of the Government, and no official endorsement should be inferred. Fig. 2 Impulse transmission as function of the compressible parameter s for different values of the incident overpressure ps / p0
uniform shocks of varying intensity by applying a piston velocity at one end of the computational grid, followed by a computation of the propagation of the shock and its reflection on plates of varying mass modeled as a concentrated mass at the opposite extreme of the domain. The transmitted impulse I p was extracted from the simulations and compared with the predictions of Eq. 共29兲. The numerical results as well as the comparisons with the theory are shown in Fig. 2 where a plot is given of the normalized transmitted impulse I p / CRIi versus the combination of parameters s / f R. As it can be seen in this figure, an excellent agreement is found between the numerical results and the theory. Specifically, for s → 0 the curve becomes horizontal supporting the correctness of the assumption that heavy plates behave as fixed walls and therefore absorb the same impulse independently of the plate mass. For s → ⬁ the curve has slope −1 which is consistent with the assumption that all plates acquire the same maximum velocity 共specifically I p / Ii ⬀ m p while s ⬀ 1 / m p, so that I p / Ii ⬀ 1 / s兲. In addition and most importantly, the numerical results support the predictions of the proposed formula Eq. 共29兲 in the intermediate range.
5
Conclusions
Earlier work on the influence of compressibility on fluid– structure interaction effects in the case of exponential blast-wave profiles impinging on free-standing plates of varying mass has been extended to the case of uniform shocks. In this simplified problem, the asymptotic limits of very heavy and very light can be derived exactly for arbitrary shock intensities. The linearized problem for uniform shock profiles is also solved exactly.
Journal of Applied Mechanics
References 关1兴 Taylor, G. I., 1963, “The Pressure and Impulse of Submarine Explosion Waves on Plates,” The Scientific Papers of Sir Geoffrey Ingram Taylor, Vol. III: Aerodynamics and the Mechanics of Projectiles and Explosions, G. Batchelor, ed., Cambridge University Press, Cambridge, UK, pp. 287–303. 关2兴 Xue, Z., and Hutchinson, J. W., 2003, “Preliminary Assessment of Sandwich Plates Subject to Blast Loads,” Int. J. Mech. Sci., 45共4兲, pp. 687–705. 关3兴 Fleck, N., and Deshpande, V., 2004, “The Resistance of Clamped Sandwich Beams to Shock Loading,” Trans. ASME, J. Appl. Mech., 71共3兲, pp. 386–401. 关4兴 Xue, Z., and Hutchinson, J. W., 2004, “A Comparative Study of ImpulseResistant Metal Sandwich Plates,” Int. J. Impact Eng., 30共10兲, pp. 1283–1305. 关5兴 Qiu, X., Deshpande, V., and Fleck, N., 2004, “Dynamic Response of a Clamped Circular Sandwich Plate Subject to Shock Loading,” J. Appl. Mech., 71共5兲, pp. 637–645. 关6兴 Qiu, X., Deshpande, V., and Fleck, N., 2005, “Impulsive Loading of Clamped Monolithic and Sandwich Beams Over a Central Patch,” J. Mech. Phys. Solids, 53共5兲, pp. 1015–1046. 关7兴 Hutchinson, J. W., and Xue, Z., 2005, “Metal Sandwich Plates Optimized for Pressure Impulses,” Int. J. Mech. Sci., 47共4-5兲, pp. 545–569. 关8兴 Deshpande, V., and Fleck, N., 2005, “One-Dimensional Response of Sandwich Plates to Underwater Shock Loading,” J. Mech. Phys. Solids, 53共11兲, pp. 2347–2383. 关9兴 Rabczuk, T., Kim, J. Y., Samaniego, E., and Belytschko, B. T, 2004, “Homogenization of Sandwich Structures,” Int. J. Numer. Methods Eng., 61共7兲, pp. 1009–l027. 关10兴 Liang, Y., Spuskanyuk, A. V., Flores, S. E., Hayhurst, D. R., Hutchinson, J. W., McMeeking, R. M., and Evans, A. G., 2007, “The Response of Metallic Sandwich Panels to Water Blast,” J. Appl. Mech., 74共1兲, pp. 81–99. 关11兴 Kambouchev, N., Noels, L., and Radovitzky, R., 2006, “Compressibility Effects in Fluid–Structure Interaction and their Implications on the Air-Blast Loading of Structures,” J. Appl. Phys., 100, p. 063519. 关12兴 Taylor, G. I., 1950, “The Formation of a Blast Wave by a Very Intense Explosion, II The Atomic Explosion of 1945,” Proc. R. Soc. London, Ser. A, 201共1065兲, pp. 175–186. 关13兴 Anderson, J., 2001, Fundamentals of Aerodynamics, McGraw–Hill, New York. 关14兴 Tan, P. J., Reid, S. R., and Harrigan, J. J., 2005, “Discussion: The Resistance of Clamped Sandwich Beams to Shock Loading 共Fleck, N. A., and Deshpande, V. S., 2004, ASME. J. Appl. Mech., 71, pp. 386–401兲,” Trans. ASME, J. Appl. Mech., 72, pp. 978–979. 关15兴 Kambouchev, N., Noels, L., and Radovitzky, R., 2007, “Numerical Simulation of the Fluid–Structure Interaction Between Air Blast Waves and Free-Standing Plates,” Comput. Struct., in press.
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