Stress and Energy Transmission by Inhomogeneous Plane Waves into Dissipative Media
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1 Purdue University Purdue e-pubs Publications of the Ray W. Herrick Laboratories School of Mechanical Engineering Stress and Energy Transmission by Inhomogeneous Plane Waves into Dissipative Media Daniel C. Woods Purdue University, J Stuart Bolton Purdue University, Jeffrey F. Rhoads Purdue University, Follow this and additional works at: Woods, Daniel C.; Bolton, J Stuart; and Rhoads, Jeffrey F., "Stress and Energy Transmission by Inhomogeneous Plane Waves into Dissipative Media" (2015). Publications of the Ray W. Herrick Laboratories. Paper This document has been made available through Purdue e-pubs, a service of the Purdue University Libraries. Please contact epubs@purdue.edu for additional information.
2 Stress and Energy Transmission by Inhomogeneous Plane Waves into Dissipative Media Daniel C. Woods, J. Stuart Bolton, and Jeffrey F. Rhoads School of Mechanical Engineering, Ray W. Herrick Laboratories, and Birck Nanotechnology Center, Purdue University West Lafayette, Indiana, USA November 6,
3 Premise and Motivation Detection of improvised explosive devices (IEDs) Strong dependence of vapor pressure on temperature May improve detection capabilities by selective heating Stand-off distance Low-frequency source Energetic material Increase in temperature Increase in vapor pressure 2
4 Premise and Motivation Optimization of incident wave parameters for maximal stress and energy transmission Theoretical study of incident inhomogeneous plane waves in dissipative media Minimization of reflection coefficient magnitude Air Solid Incident wave Basic representation of acoustical interface 3
5 General Acoustic Plane Waves Homogeneous wave Inhomogeneous wave Ideal fluid or Elastic solid Real fluid or Viscoelastic solid 4
6 Representation in Dissipative Media x Propagation vector, P γ Attenuation vector, A Stress field (Pa) 1-Pa, 1000-Hz wave in Sylgard θ (θ γ) z Wave inhomogeneity angle, 0 γ < 90 5
7 Representation in Dissipative Media Each wave type is characterized by the corresponding material wavenumber: k = k R + jk I Homogeneous waves (γ = 0 ): v H = ω k R, A H = k I Inhomogeneous waves (γ 0 ): v = function γ < v H, A = function γ > A H k x = P sin θ j A sin(θ γ) k z = P cos θ j A cos θ γ 6
8 Creation of Inhomogeneous Plane Waves Selective absorbing geometries (Deschamps, 1994) Phased arrays of sources (Robin et al., 2014) Vibrating plates coupled to piezoelectric transducers (Fujii et al., 2014) 7
9 Boundary conditions at interface (z = 0) σ 1,z x, 0 = σ 2,z (x, 0) v 1,z x, 0 = v 2,z x, 0 σ 2,xz x, 0 = 0 Fluid Solid Interface Trace wavenumber continuity k 1,x = k 2,x = κ 2,x 8
10 Optimal Incident Wave Parameters Optimal incidence angle is the Rayleigh angle Wavefronts of incident wave λ inc λ Ray Wave inhomogeneity: Spatial resonance of induced longitudinal and shear particle motions (image: For low-loss solids, optimal incident wave is inhomogeneous (unique such inhomogeneity) For higher-loss solids, optimal incident wave is homogeneous 9
11 Position along interface (m) Incident decay rate (rad/m) [i.e., inhomogeneity] Results for Ideal Fluid Solid Interface Magnitude of Reflection Coefficient (1000 Hz) Stress Distribution in Elastic Solid (Pa) Incidence angle (deg.) Position normal to interface (m) 10
12 Low-Loss Interface: Water Stainless Steel Magnitude of Reflection Coefficient Effect of Incidence Angle Effect of Wave Inhomogeneity (Near Rayleigh Angle) 11
13 Low-Loss Interface: Water Stainless Steel Magnitude of Reflection Coefficient Effect of Incidence Angle Effect of Wave Inhomogeneity (Near Rayleigh Angle) 12
14 Effect of Increasing Material Dissipation Magnitude of Reflection Coefficient vs. Dissipation Level (θ ) 13
15 R Low-Loss Interface: Water Stainless Steel Transmitted Normal Stress Distribution Water Stainless Steel +σ z,max Incidence Angle, θ 0 σ z,max Incidence Angle, θ 14
16 R Low-Loss Interface: Water Stainless Steel Transmitted Normal Stress Distribution Water Stainless Steel +σ z,max Incidence Angle, θ 0 σ z,max Incidence Angle, θ 15
17 R Low-Loss Interface: Water Stainless Steel Transmitted Normal Stress Distribution Water Stainless Steel +σ z,max Incidence Angle, θ 0 σ z,max Incidence Angle, θ 16
18 R Low-Loss Interface: Water Stainless Steel Transmitted Normal Stress Distribution Water Stainless Steel +σ z,max Incidence Angle, θ 0 σ z,max Incidence Angle, θ 17
19 R Low-Loss Interface: Water Stainless Steel Transmitted Normal Intensity Distribution Water Stainless Steel +I z,max Incidence Angle, θ 0 I z,max Incidence Angle, θ 18
20 R Low-Loss Interface: Water Stainless Steel Transmitted Normal Intensity Distribution Water Stainless Steel +I z,max Incidence Angle, θ 0 I z,max Incidence Angle, θ 19
21 R Low-Loss Interface: Water Stainless Steel Transmitted Normal Intensity Distribution Water Stainless Steel +I z,max Inhomogeneity Angle, γ 0 I z,max Inhomogeneity Angle, γ 20
22 R Low-Loss Interface: Water Stainless Steel Transmitted Normal Intensity Distribution Water Stainless Steel +I z,max Inhomogeneity Angle, γ 0 I z,max Inhomogeneity Angle, γ 21
23 R Low-Loss Interface: Water Stainless Steel Transmitted Normal Intensity Distribution Water Stainless Steel +I z,max Inhomogeneity Angle, γ 0 I z,max Inhomogeneity Angle, γ 22
24 High-Loss Interface: Air Sylgard R Transmitted Normal Stress Distribution Air Sylgard +σ z,max ~0.99 Inhomogeneity Angle, γ 0 σ z,max Inhomogeneity Angle, γ 23
25 R High-Loss Interface: Air Sylgard Transmitted Normal Stress Distribution Air Sylgard +σ z,max Inhomogeneity Angle, γ 0 σ z,max Inhomogeneity Angle, γ 24
26 R High-Loss Interface: Air Sylgard Transmitted Normal Stress Distribution Air Sylgard +σ z,max Inhomogeneity Angle, γ 0 σ z,max Inhomogeneity Angle, γ 25
27 Conclusions Use of general acoustic plane waves for increased transmission in solids Interface Ideal fluid Elastic solid Real fluid Viscoelastic solid Optimal incident wave Inhomogeneous At Rayleigh angle Inhomogeneous for low-loss solids Homogeneous for high-loss solids Attainable energy transmission Total transmission Reflection 0 Narrow domain Less than total transmission Reflection > 0 Wider domain 26
28 Next Steps Further characterization of transmission into viscoelastic materials of interest Polymer-bonded energetic materials Transmission by finite, spatially-distributed waves Bounded wave profiles typically used in practice Various spatial distributions (Van Den Abeele & Leroy, 1993) 27
29 Acknowledgement The authors would like to thank the U.S. Office of Naval Research for its support of this research under ONR Grant No. N
30 Results for Ideal Fluid Solid Interface Transmitted Normal Intensity (W/m 2 ) 1-Pa, 1000-Hz incident wave β = 0 (homogeneous) β = rad/m β = rad/m β = rad/m Sample solid ρ = 1000ρ air c = 10c air b = 7c air I z = 1 T T (σ z v z + σ xz v x )dt 0 29
31 Results for Ideal Fluid Solid Interface Transmitted Normal Stress (Pa) Transmitted Normal Velocity (m/s) Approximate parameters for R = , β rad/m θ 1,r 30
32 Results for Ideal Fluid Solid Interface Transmitted Normal Intensity (W/m 2 ) Compare with ~ W/m 2 for homogeneous waves below the critical angle 31
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