The scattering of acoustic waves propagating in porous rigid 3D stereolithographical materials that are made of resin, has been investigated. 3D stereolithographical (STL) materials were obtained using a form of rapid prototyping that allows complex solid objects to be manufactured directly from3D computer models in the form of successive layers of light-cured resin. Relative variation of velocity and attenuation of 3D materials in air filled and water filled media has been predicted at normal and oblique directions. The scattering perturbation is similar to diffusion waves at low frequency while dispersion velocity and attenuation change with respect to square root of angular frequency and with respect to angular frequency at high frequency respectively

Aygun, H

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Aygun, Haydar: Scattering of acoustics waves in porous materials                                                                  1   Scattering of acoustic waves in air and water filled 3D stereo-lithographical (STL) porous rigid materials   Haydar Aygun, PhD London South Bank University School of the Built Environment and Architecture, 103 Borough Rd, London, SE1 0AA, UK  ABSTRACT The scattering of acoustic waves propagating in porous rigid 3D stereolithographical materials that are made of resin, has been investigated. 3D stereolithographical (STL) materials were obtained using a form of rapid prototyping that allows complex solid objects to be manufactured directly from 3D computer models in the form of successive layers of light-cured resin. Relative variation of velocity and attenuation of 3D materials in air filled and water filled media has been predicted at normal and oblique directions. The scattering perturbation is similar to diffusion waves at low frequency while dispersion velocity and attenuation change with respect to square root of angular frequency and with respect to angular frequency at high frequency respectively.   1. INTRODUCTION  Poroelastic materials may be employed for noise control in aircraft, buildings and various other engineering applications. According to Rayleigh [1], ‘the fundamental characteristic of porous materials treated to reduce noise is that the produced fluid flow through the poroelastic material is opposed because of the frictional force produced by the fibres or the cell walls on the fluid. This mechanism allows the energy to be absorbed from the acoustic wave and to be converted into heat.’ The most widely-exploited and acknowledged absorption mechanism in porous materials is viscous friction due to relative motion between solid and fluid. If the frame of the porous material is viscoelastic then other dissipative mechanisms are possible.  Biot theory has been used extensively to describe the wave propagation in rigid porous media, specifically developed to describe acoustic wave propagation in fluid-saturated porous elastic media [2, 3]. Biot theory predicts two compressional waves (fast and slow waves), when the waves propagating through the solid frame and fluid are in-phase and out-of-phase respectively, and a shear wave. It allows for an arbitrary microstructure, with separate motions considered for the solid elastic framework and the interspersed fluid, induced by the acoustic wave, and also includes energy loss due to viscous friction between solid and fluid. The anisotropic pore structure and elasticity of cancellous bone cause wave speeds and attenuation in cancellous bone to vary with angle [4].  Previous work on the influence of anisotropic pore structure and elasticity in cancellous bone has been extended by developing an anisotropic Biot-Allard model allowing for angle-dependent Aygun, Haydar: Scattering of acoustics waves in porous materials                                                                  2  elasticity, and angle-and-porosity dependent tortuosity [4]. The extreme angle dependence of tortuosity corresponding to the parallel plate microstructure has been replaced by angle-and-porosity dependent tortuosity values based on data for slow wave transmission through air-filled stereolithography (STL) replicas [5]. It has been suggested that the anisotropic Biot-Allard model could be used to give further insight into the factors that have the most important influence on the angle dependency of wave speeds and attenuation in porous rigid materials. Nevertheless the applicability of Biot-based theories to ultrasonic propagation in porous rigid medium remains in question given the expected role of scattering which is neglected in these theories. Aygün et al. [6, 7, 8] have transmitted ultrasonic signals through water saturated stereolithograpical replica samples in the form of 57 mm cubes with microstructural dimensions that are 13 times real scale at normal angle and oblique angles. Remarkably, it is found that the expected occurrence of scattering does not cause significant discrepancies between predictions and data at 100 kHz, perhaps as a consequence of the fact that the samples behave as low pass filters.  Boutin [9] has investigated low frequency scattering of acoustic waves propagation through heterogeneous media made of air and motionless inclusions. The method developed by Boutin has been used to investigate relative variation of velocity and attenuation of 3D materials in air filled and water filled media at normal and oblique incidences. 2. STEREO-LITHOGRAPHY MANUFACTURING SYSTEM  Stereo-lithography (STL) is a form of rapid prototyping that allows complex solid objects to be manufactured directly from 3D computer models in the form of successive layers of light-cured resin [10]. There are two stages to stereolithography, design and manufacturing. During the design stage the required object is initially created using standard 3D solid modelling techniques and then converted into the stereolithography format consisting of a series of thin slices. The stereolithography manufacturing system consists of a vat of light-sensitive resin with an elevator and computer-controlled scanning laser. At the start of the process, the elevator is positioned just below the surface (typically 0.1 mm) of the liquid stereolithography resin. The laser scanner “prints” the bottom layer onto the resin surface, which solidifies upon exposure to the laser beam. The elevator then moves down by an incremental distance and the stereolithography resin is respread over the surface of the vat prior to the next scan. As successive solid layers are formed, they bond to produce a single solid object. When the model is completed the elevator is raised and the unused resin is allowed to drain. The laser cures the stereolithography resin to approximately 60%. The curing process is completed in an ultraviolet oven. The resolution of the stereolithography process is governed by the laser spot Aygun, Haydar: Scattering of acoustics waves in porous materials                                                                  3  size and the vertical movement of the elevator. Typically, the laser spot diameter will be better than 0.3 mm and the elevator movement resolution will be about 2.5μm.   The primary reason for creating stereolithography samples is that multiple copies may be created. This is particularly valuable for mechanical testing, where measurement in one direction may damage a sample and preclude testing in other orthogonal directions. The models were created at a magnification of x13 to ensure spatial fidelity between the voxel size of the microCT scan of the original natural samples and the minimum wall thickness achievable with STL. Material ‘corrosion’ due to degradation can be simulated physically by corresponding stereolithographical samples.        Sample C                                   Sample F Figure 1: views of 3D stereolithographical samples. 3. SCATTERING OF PLANE WAVES The effects of scattering on plane wave propagation in porous medium has been investigated using a model developed by Boutin [9] to determine wave velocity and attenuation in rigid porous materials. A harmonic plane wave propagating in a given direction of unit vector d is given by  𝑃𝑃 = 𝑃𝑃0𝑒𝑒−(𝑖𝑖𝐻𝐻𝑑𝑑𝑥𝑥.𝑑𝑑+𝑖𝑖𝑖𝑖𝑖𝑖)        (1) where 𝑃𝑃0 is the amplitude of zeroth order pressure, 𝐻𝐻𝑑𝑑is the complex wave number, 𝜔𝜔 is the angular frequency, d is the unit vector, t is the time taken, and x is the distance travelled.   The wave velocity 𝐶𝐶𝑑𝑑, wavelength 𝜆𝜆𝑑𝑑, and attenuation 𝜉𝜉𝑑𝑑 in the propagation direction are related to the wave number  by [9]  𝐻𝐻𝑑𝑑 = 𝑖𝑖𝐶𝐶𝑑𝑑 (1 − 𝑖𝑖𝜉𝜉𝑑𝑑)                         (2) where 𝐶𝐶𝑑𝑑 = 𝑖𝑖𝐻𝐻𝑟𝑟, 𝜆𝜆𝑑𝑑 = 2𝜋𝜋𝐻𝐻𝑟𝑟,  𝜉𝜉𝑑𝑑 = − 𝐻𝐻𝑖𝑖𝐻𝐻𝑟𝑟, 𝐻𝐻𝑖𝑖is the imaginary part of complex wave number and 𝐻𝐻𝑟𝑟is the real part of complex wave number.  Harmonic plane wave shows a strong dispersion due to changes in frequency dependent dynamic permeability leading to that; a) A diffusion wave at low frequency, 𝜔𝜔 < 𝜔𝜔𝑑𝑑 , produces 𝐶𝐶𝑑𝑑 ≈ �𝑖𝑖𝑖𝑖𝑐𝑐 � 𝑃𝑃𝑒𝑒𝜌𝜌𝑒𝑒𝛼𝛼∞  and 𝜉𝜉𝑑𝑑 ≈ 1 −𝑂𝑂�𝑖𝑖𝑖𝑖𝑐𝑐. b) A propagation wave at high frequency, 𝜔𝜔 > 𝜔𝜔𝑑𝑑, produces 𝐶𝐶𝑑𝑑 ≈ � 𝛾𝛾𝑃𝑃𝑒𝑒𝜌𝜌𝑒𝑒∝∞  and 𝜉𝜉𝑑𝑑 ≈ �𝑖𝑖𝑐𝑐𝑖𝑖 . where ρe is the density of water, α∞ is the tortuosity,  𝜔𝜔 is the angular frequency, 𝜔𝜔𝑑𝑑is the diffraction frequency, 𝜔𝜔𝑐𝑐 is the critical frequency,  𝑃𝑃𝑒𝑒 is the equilibrium state pressure, 𝛾𝛾 is the specific heat ratio.   The macro-pressure can be expressed as [9]  Aygun, Haydar: Scattering of acoustics waves in porous materials                                                                  4  𝑃𝑃𝑚𝑚 = 𝑃𝑃0(𝑒𝑒−𝑖𝑖𝐻𝐻𝑑𝑑�𝑥𝑥.𝑑𝑑 + 𝑂𝑂(𝜖𝜖3))                             (3) where 𝐻𝐻𝑑𝑑� = 𝐻𝐻𝑑𝑑 − 𝐻𝐻𝑑𝑑𝑄𝑄𝑑𝑑(−𝑖𝑖𝐻𝐻𝑑𝑑)2 and 𝑂𝑂(𝜖𝜖3) is the correction order given by𝑂𝑂 � 𝑙𝑙3(𝜆𝜆/2𝜋𝜋)3�. Consequently the wave number is modified by diffraction yielding that [9];   𝐻𝐻𝑑𝑑� = 𝑖𝑖𝐶𝐶𝑑𝑑 (1 − 𝑖𝑖𝜉𝜉𝑑𝑑) �1 + |𝑄𝑄𝑑𝑑|(cos𝜒𝜒 + sin𝜒𝜒)𝑥𝑥 �𝑖𝑖(1−𝑖𝑖𝜉𝜉𝑑𝑑)𝐶𝐶𝑑𝑑 �2 �                    (4) Apparent wave velocity can be derived as follow; 𝐶𝐶𝑑𝑑� = 𝐶𝐶𝑑𝑑 �1 − |𝑄𝑄𝑑𝑑| �𝑖𝑖𝐶𝐶𝑑𝑑�2 ((1 − 𝜉𝜉𝑑𝑑2)(cos𝜒𝜒 + 𝜉𝜉𝑑𝑑sin𝜒𝜒) + 2𝜉𝜉𝑑𝑑(cos𝜒𝜒 − 𝜉𝜉𝑑𝑑sin𝜒𝜒)�  (5) Apparent attenuation factor can be derived as follow;  𝜉𝜉𝑑𝑑� = 𝜉𝜉𝑑𝑑 + |𝑄𝑄𝑑𝑑| �𝑖𝑖𝐶𝐶𝑑𝑑�2 (2𝜉𝜉𝑑𝑑(1 + 𝜉𝜉𝑑𝑑2)cos𝜒𝜒 − sin𝜒𝜒 + 𝜉𝜉𝑑𝑑4sin𝜒𝜒)   (6) Apparent wave velocity and attenuation explain the differences between the effects of scattering on poroacoustic waves and on thermo-elastic waves.  Frequency dependent wave velocity and attenuation in viscous regime are given by equations below respectively [9]; 𝐶𝐶𝑑𝑑� = 𝐶𝐶𝑑𝑑 �1 + 2𝑄𝑄𝑑𝑑0 �𝑖𝑖𝐶𝐶𝑑𝑑�2�       (7) 𝜉𝜉𝑑𝑑� = 𝜉𝜉𝑑𝑑 + 4𝑄𝑄𝑑𝑑0 �𝑖𝑖𝐶𝐶𝑑𝑑�2        (8) where 𝜉𝜉𝑑𝑑  is the attenuation, 𝐶𝐶𝑑𝑑 is the wave number, and 𝑄𝑄𝑑𝑑0 is the frequency dependent complex coefficient.  In the inertial regime, the waves will be damped while they propagate through porous rigid media. Wave velocity and attenuation are given by following equations [9];  𝐶𝐶𝑑𝑑� = 𝐶𝐶𝑑𝑑 �1 − 2𝑄𝑄𝑑𝑑0 �𝑖𝑖𝐶𝐶𝑑𝑑�2 (cos𝜒𝜒 + 3𝜉𝜉𝑑𝑑sin𝜒𝜒)�    (9)  𝜉𝜉𝑑𝑑� = 𝜉𝜉𝑑𝑑 − |𝑄𝑄𝑑𝑑| �𝑖𝑖𝐶𝐶𝑑𝑑�2 [sin𝜒𝜒 − 2𝜉𝜉𝑑𝑑cos𝜒𝜒]      (10) where 𝑄𝑄𝑑𝑑is the frequency dependent complex coefficient and 𝜒𝜒is the angle of incidence.  Critical frequency for porous rigid materials is given by: ∞=ακρφµω ec         (11) where ϕ is the porosity, ρe is the density of water, 1000 kg/m3, α∞ is the tortuosity, К is the permeability, and µ is the gas viscosity.   4. RESULTS  Frequency dependent relative variation of wave velocity and attenuation are predicted in viscoinertia regime at 1 MHz and in inertia regime at 1 kHz for both oblique and normal incidences in air and water filled medium. The results of diffraction effects within the viscoinertial regime are shown in Figures 2, 3 and 4. Anisotropic effect can be seen throughout frequency range. In Figure 2a, velocity variation reduces in both direction while attenuation significantly increases in both direction and oblique velocity becomes more attenuated. The variations in air filled medium shown Figure 2b are about 35 times higher than the variations in water filled medium. Diffraction effect on the relative variation of wave velocity and attenuation in inertial regime are shown in Figure 3. Relative variation of attenuation increases throughout frequency range at normal and oblique directions in water filled medium. This means that attenuation is not affected by diffraction in this regime. Wave velocity increases in normal direction while it is effected by diffraction and decreases at oblique direction.   Aygun, Haydar: Scattering of acoustics waves in porous materials                                                                  5  Wave velocity and attenuation for sample F at normal and oblique direction in water and air filled medium are shown in Figure 4. Relative velocity at oblique incidence is decreased by diffraction while others are not affected by diffraction in water filled porous rigid medium.        Figure 2: Relative variation of velocity �𝐶𝐶𝑑𝑑�−𝐶𝐶𝐶𝐶�  and variation of attenuation ( 𝜉𝜉𝑑𝑑� − 𝜉𝜉𝑑𝑑 ) versus frequency ratio in viscoinertial regime at 1 MHz; a) water filled STL sample C and b) air filled STL sample C.   Figure 3: Relative variation of velocity �𝐶𝐶𝑑𝑑�−𝐶𝐶𝐶𝐶�  and variation of attenuation ( 𝜉𝜉𝑑𝑑� − 𝜉𝜉𝑑𝑑 ) versus frequency ratio in inertial regime at 1 kHz; a) water filled STL sample C and b) air filled STL sample C.   5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6Log(w/wc)-0.3-0.2-0.100.10.20.30.4Relative variation of velocity and attenuationVelocity, obliqueAttenuation, obliqueVelocity, normalAttenuation, normal5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6Log(w/wc)-5051015Relative variation of velocity and attenuationVelocity, normalAttenuation, normalVelocity, obliqueAttenuation, oblique2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3Log(w/wc)-0.500.511.522.5Relative variation of velocity and attenuation10-7Velocity, normalAttenuation, normalVelocity, obliqueAttenuation, oblique-1 -0.5 0 0.5 1 1.5 2 2.5 3Log(w/wc)-3-2-101234567Relative variation of velocity and attenuation10-6Velocity, obliqueAttenuation, obliqueAttenuation, normalVelocity, normal(a) (b) (b) (a) Aygun, Haydar: Scattering of acoustics waves in porous materials                                                                  6     Figure 4: Relative variation of velocity �𝐶𝐶𝑑𝑑�−𝐶𝐶𝐶𝐶� and attenuation (𝜉𝜉𝑑𝑑� − 𝜉𝜉𝑑𝑑) versus frequency ratio in inertia regime at 1 kHz; a) water filled STL sample F and b) air filled STL sample F.   5. CONCLUSION  A scattering model developed by Boutin has been applied to 3D STL porous rigid materials to determine the relative variation of velocity and attenuation at normal and oblique incidences. The results in air filled medium has been compared with the ones in water filled medium. In water filled medium, mostly wave velocity is effected by diffraction at oblique direction while in air filled medium, wave is attenuated at normal direction for both studied samples. The anisotropic pore structure and elasticity of STL samples cause wave speed and attenuation in rigid porous samples to vary with angle.    6. ACKNOWLEDGEMENTS This paper has been supported by Leverhulme Grant No.F/00 181/N, and by Acoustics Group of London South Bank University.  7. REFERENCES 1. Lord Rayleigh, “Theory of Sound” (London, 1926) Vol.(I), and (Dover, New York, 1945), Vol.(II). 2. Biot M. A., 1956 Theory of propagation of elastic waves in a fluid saturated porous solid, I. Low frequency range. J. Acoust. Soc. Am. 28 168–1178. 3. Biot M. A., 1956 Theory of propagation of elastic waves in a fluid saturated porous solid, II. High frequency range. J. Acoust. Soc. Am. 28 179–191.  4. Aygün H., Attenborough K., Postema M., Lauriks W., and Langton C. M., 2009, Predictions of angle dependent tortuosity and elasticity effects on sound propagation in cancellous bone. J. Acoust. Soc. Am. 126 (6), 3286-3290. 5. Attenborough K., Qin Q., Fagan M. J., Shin H.-C., and Langton C. M., 2005 Measurements of tortuosity in stereolithographical bone replicas using audio-frequency pulses. J. Acous. Soc Am. 118 2779-2782. 6. Aygün H., Attenborough K., Lauriks W., and Langton C. M., 2010, “Ultrasonic wave propagation in Stereo-lithographical bone replicas.” J. Acoust. Soc. Am. 127 (6), 3781 – 3789. 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3Log(w/wc)-2-101234567Relative variation of velocity and attenuation10-8Velocity, obliqueAttenuation, obliqueVelocity, normalAttenuation, normal1 1.5 2 2.5 3Log(w/wc)-5-4-3-2-10123Relative variation of velocity and attenuation10-6Velocity, normalAttenuation, normalVelocity, obliqueAttenuation, oblique(a) (b) Aygun, Haydar: Scattering of acoustics waves in porous materials                                                                  7  7. Aygün H., Attenborough K., Lauriks W., Rubini P.A., and Langton C. M., 2011, “Wave propagation in stereo-lithographical (STL) bone replicas at oblique incidence.” Applied Acoustics. 72 (7) 458 – 463.  8. Aygün H. and Barlow C. 2014, “Behaviour of ultrasonic waves in porous rigid materials: anisotropic Biot-Attenborough model.” Anglo-French Physical Acoustic Conference, 15 – 17 January 2014, Surrey, UK.  9. Boutin, C. 2007. “Rayleigh scattering of acoustic waves in rigid porous media”. J. Acoust. Soc. Am. Volume (122), 1888.  10.  Langton C M, Whitehead M A, Langton D K, and Langley G, 1997. Development of a cancellous bone structural model by stereolithography for ultrasound characterisation of the calcaneus. Med. Eng. Phys. 19 599-604.  

Scattering of acoustic waves in air and water filled 3D stereo-lithographical (STL) porous rigid materials

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Scattering of acoustic waves in air and water filled 3D stereo-lithographical (STL) porous rigid materials

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