- Email: [email protected]
A transformation elasticity based device for wavefront manipulation
NDT and E International 102 (2019) 304–310
Contents lists available at ScienceDirect
NDT and E International journal homepage: www.elsevier.com/locate/ndteint
A transformation elasticity based device for wavefront manipulation Sai Aditya Raman Kuchibhatla, Prabhu Rajagopal
∗
T
Centre for Nondestructive Evaluation, Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai, India
A R T I C LE I N FO
A B S T R A C T
Keywords: Transformation elasticity Metamaterial plate Wave manipulation GRIN PC Guided waves
The authors propose a metamaterial plate made of gradient refractive index phononic crystals for manipulation of the wavefront of propagating elastic waves in solid media. A square unit cell with through holes is considered as the basis for the proposed metematerial. Guidance on the design of the hole pattern and the choice of materials is obtained with the aid of transformation elasticity principles. Manipulation of a plane wavefront into a cylindrical wavefront is first observed through Finite Element simulations at low frequency (100 kHz), and then practically demonstrated through experiments. Wave propagation with and without the holey region is compared and possible applications at higher frequencies are discussed.
1. Introduction Ultrasonic testing (UT) is a popular technique among the various non-destructive evaluation (NDE) methods used in the industry for the inspection of pipes and tubes. Particularly, Phased Array Ultrasonic Transducers (PAUT) can be used for controlled generation of ultrasonic waves [1]. These are composed of multiple individually controllable transducer elements, allowing for firing of elements with specific electronic time delays. By using various delay schemes, it is possible to steer the beam or alter the shape of wavefront [2–4]. Further, an improved inspection capability of linear phased arrays for irregular surfaces by use of a coupling medium consisting of water encapsulated by a membrane has been reported in literature [5,6]. In recent years, metamaterial devices have attracted attention as a possible alternative for controlled propagation of waves inside a medium. Such devices were first proposed in Optics and were soon followed by their acoustic analogues. A variety of methods have been proposed in the last two decades to control wave propagation in electromagnetic [7,8] as well as acoustic media [9,10]. Particularly, transformation methods have been found to be very effective, for example, in the design of acoustic cloaking devices [11–14]. More recently, much thinner metamaterials [15] have been demonstrated for sound modulation [16–18] in which novel techniques such as spacecoiling etc. have been used. Metascreens made of Helmholtz resonators capable of shaping acoustic wavefronts by controlling their phase have also been proposed [19]. However, for elastic waves, the wave equation is not form invariant under transformation [20]. This limits the usefulness of this method for making wave manipulating devices for elastodynamics, when compared to that in acoustics. Other approaches for
∗
designing devices for wave manipulation in elastic media have been discussed in literature. For example, a device [21,22] consisting of pathways engineered using the concept of drop-channels [23] has been proposed for frequency selective beam steering and collimation of ultrasound waves. Another device based on local resonators, with hyperbolic dispersion curves, has been used for sub-wavelength imaging [24]. Metasurfaces for elastic waves have only been reported very recently [25,26]. Here, we study the possibility of exploiting Transformation Elastodynamics to design a wavefront manipulating device by taking into account some engineering approximations. We propose a method to design customized holey plates which can generate curved wavefronts though the excitation is provided by a single ultrasound transducer, using the concept of gradient phononic crystals (PCs) [27–30]. The proposed approach is graphically illustrated in Fig. 1. In the first demonstration of this approach described here, we have used a wave source at 100 kHz. However, this approach is applicable at higher frequencies where practical UT is typically carried out (please see the discussion in Section 5.2, for demonstration of the approach at higher frequencies using numerical simulation). This paper is organised as follows. Firstly, we describe the method proposed to design holey plates and demonstrate it using Finite Element (FE) simulations and experiments at a low frequency (100 kHz). The application of curved wavefronts so generated using the designed holey plates is then discussed along with the validity of proposed method at higher frequencies. The paper concludes with some directions for future work.
Corresponding author. E-mail address: [email protected] (P. Rajagopal).
https://doi.org/10.1016/j.ndteint.2019.01.006 Received 21 August 2018; Received in revised form 30 November 2018; Accepted 10 January 2019 Available online 14 January 2019 0963-8695/ © 2019 Elsevier Ltd. All rights reserved.
NDT and E International 102 (2019) 304–310
S.A.R. Kuchibhatla, P. Rajagopal
due to the transformation. These influence the stress-strain relationship and hence the wave propagation in the transformed domain. These are given by Ref. [20].
S′pqr =
∂2x r′ 1 ∂x p′ ∂xq′ Cijkl = S′qpr |J| ∂x i ∂x j ∂xk ∂xl
(6)
D′pqr =
2 ∂xq′ ∂x r′ 1 ∂ x p′ Cijkl = S′qrp |J| ∂x i ∂x j ∂xk ∂xl
(7)
In mathematics, a complex function is said to be analytic in a region if it is differentiable at all the points in the region and such a complex differentiable function f (x , y ) → (x ′, y′) satisfies Cauchy-Riemann relations [32]:
∂y′ ∂y′ ∂x ′ ∂x ′ = ; =− ∂x ∂y ∂x ∂y
(8)
An analytic function is said to be conformal at a point if it has a nonzero derivative at that point. Conformal transforms preserve local angles and orientation. The conformal map given in Eq. (9) below was used in this study.
w=
2. Methods 2.1. Transformation elasticity and conformal mapping for holey plate Transformation acoustics/elasticity provides one method for design of devices for cloaking and other wave steering systems. The equation governing propagation of elastic waves is given by (1)
where u is displacement, ω is the frequency of propagating wave, ρ is the density of medium and C is the constitutive property matrix. Unlike the governing equations for electromagnetic and acoustic waves which are form invariant under coordinate transformation [7–9], the equation for elastic waves involves additional terms after transformation [20,31]. The transformed wave equation (with transformed terms indicated by a prime) is given by
∇′. σ ′ = D'∇'u′ − ω2ρ'u′
(2)
σ ′ = C'∇' u′ + S'u′
(3)
z 2 − 4a2 )
(9)
where z and w are complex numbers, z = x + iy represents a point ( x , y ) before transformation and w = x ′ + iy′ denotes the same point with new coordinates ( x ′, y′) after the transformation. ‘a ’ is a transformation parameter. Fig. 2(a) shows the transformation of a finite discrete grid under the mapping given by Eq. (9). It may be noted that horizontal parallel lines in the input plane are transformed into conforming curved lines in the region near the semi-circle in the output plane. For a conformal map, as used in this work, the cross terms in the density tensor are zero. Under the assumption that displacements are time harmonic i.e. varying as exp(iωt ) , the density along principal directions reduces to the isotropic density plus an additional term which is inversely dependent on the square of the frequency [31,33]. This additional term can hence be neglected at ultrasonic frequencies. The device can therefore be made out of a material with isotropic density. The coupling terms D′ and S′ in Eq. (2) and Eq. (3), respectively, are typically dependent on both Lame's parameters. On using a conformal map, these are reduced to depend only on one of the two parameters (μ) which indicates that S’ (or D′) on using a conformal map could be smaller when compared to S’ (or D′) on using a non-conformal transform [33]. Further, these terms occur in combination with strains and displacements which are small quantities to begin with, and hence can be neglected as an engineering approximation in this study (for instance, in Eq. (3), the term S'u' is much smaller when compared to C'∇' u' ). The approximated wave equation can hence be given as
Fig. 1. Schematic illustration of wavefront manipulation using the holey plate concept discussed in this paper.
∇ . (C∇u) = −ω2ρu
1 (z ± 2
∇ '. (C'∇'u′) ≈ −ω2ρ′u′
(10)
where the transformed density is a matrix given by
ρ′pq =
2 ∂2xq′ ρ ∂x p′ ∂xq′ 1 ∂ x p′ Cijkl + |J| ∂x i ∂x i |J| ∂x i ∂x j ∂xk ∂xl
(4)
This indicates the need for a medium with anisotropic mass density. The transformed elasticity tensor is given by
C′pqrs =
∂x ′ ∂x ′ 1 ∂x p′ ∂xq′ Cijkl r s |J| ∂x i ∂x j ∂xk ∂xl
Fig. 2. Illustration of the conformal map in Eq. 9 - (a) Transformation of grid lines under the mapping (b) Jacobian of the transformation (color scale in the figure indicates the value of |J| ).
(5)
The additional terms S' and D' are third order tensors which arise 305
NDT and E International 102 (2019) 304–310
S.A.R. Kuchibhatla, P. Rajagopal
which is of the same form as the wave equation before the transformation. The velocity of the wave in the transformed space is related to the velocity in space before transformation as
v′ (x ′, y′) =
|J| v (x , y )
(11)
where ‘v ’ is the wave velocity in the region before transformation and ‘J ’ is the Jacobian of the transformation given by
⎡ ∂x′ J=⎢ ⎢ ∂y′ ⎣ ∂x
∂x′ ⎤ ∂y
∂x
⎥
∂y′ ⎥ ∂y ⎦
(12)
2.2. Design of the holey plate
Fig. 4. Plot showing dispersion curves of S0-like wave for square unit cells with varying hole diameter.
Since the mapping in Eq. (9) is conformal, and hence analytic, it satisfies Cauchy-Riemann relations given in Eq. (8), using which the |J| can be calculated as in Eq. (13).
|J| = =
1 (1 ± 2
(
∂y′ 2 ∂x ′ 2 ∂w ∂ 1 ) +( ) =| |= ( (z ± ∂x ∂x ∂x ∂x 2 z z 2 − 4a2
)=
w 1 = 2w − z (1 −
a2 ) w2
required wave velocities in the region, PCs of different hole radius were incorporated in the plate. Since the velocity required in the region is less than that of the S0 mode ( |J| < 1 in the region), the choice of air hole is justified. The Jacobian of the transformation in Eq. (9) evaluated in a spatial grid made out of 5 mm square unit cells is shown in Fig. 3(b). An indicative semi-circular geometry representing the curvature desired is also presented in the figure. Dispersion curves (frequency vs. wavenumber) for 5 mm × 5 mm x 3 mm square units cells of Copper were found for various hole diameters ranging from 1.2 mm to 4 mm in steps of 0.4 mm, using an eigen frequency solver in a commercial package [37]. Periodic (or Floquet) boundary conditions were placed on the unit cell (of side ‘a ’) in the direction of wave propagation. Eigenfrequencies were evaluated while varying the wave vector between (0,0,0) and ( π ,0,0) for ΓX direction,
z 2 − 4a2 ))
(13)
Fig. 2(b) shows the variation of |J| , and thus of the required wave velocity, in the proposed device. To achieve such varying wave velocity, here we use the concept of Gradient Refractive INdex Phononic Crystals (GRIN PCs) [30]. The idea of GRIN PCs has been recently reported in literature and is inspired from the analogous idea of Graded Photonic Crystals in Optics [34]. These have been used to cause a gradient in the velocity within the medium and are not constrained by the feature of periodicity required in standard phononic/photonic crystal structures. In our work, in order to achieve better control over the velocity variation in the region, we used a unit cell size of 5 mm. The GRIN PCs used here are made of square unit cells of copper, with air-filled cylindrical through holes (please see Fig. 3(a)). The fundamental symmetric Lamb wave mode S0 [35] was chosen to be generated in the holey plate since the mode has in-plane (axial) displacements which are ideal for the generation of bulk longitudinal ultrasonic waves in the sample to be inspected (In the holey plate, an ‘S0like mode’ was verified to exist based on the mode shape, please see Discussions for details). The reciprocal lattice of a square unit cell is also a square unit cell. The wave speed of the phononic crystal is calculated as the average of wave speeds along two principal orientations ΓX and ΓM, in the reciprocal lattice [34,36]. Thus, corresponding to the
a
π
and (0,0,0) and ( π , a ,0) for ΓM direction. The phase velocity was a calculated as the ratio of angular frequency and wavenumber at various points on the dispersion curve to obtain phase velocity curves. The phase velocity of the S0-like mode reduces with increase in the diameter of the hole in the unit cell for frequencies around 100 kHz (please see Fig. 4). Also, at smaller hole diameters, as the phase velocity around 100 kHz is almost constant, the group velocity was taken to be approximately equal to the phase velocity. The diameter of hole which leads to a calculated velocity closest to that desired at a location was mapped to that position in space. Table 1 lists the different sizes of hole and the total number of such holes used in the design. 3. Procedure for simulations and experiments 3.1. Wave propagation simulations for low frequency A 200 mm × 80 mm x 3 mm plate of Copper material ( ρ = 8960 kg/m3, E = 125 GPa, ν = 0.35) with holes at different locations (obtained as described earlier) was modelled in a commercial package [38] for FE simulation of wave propagation. For simplicity the simulations were carried out considering the holes to be voids instead of being filled with air and accordingly the boundary was taken to be a Table 1 Number of Holes in the holey region for 3 mm thick copper plate for manipulating curvature of S0 Lamb wave mode.
Fig. 3. (a) Illustration of square unit cell along with top view of the unit cell showing principal orientations. (b) Contour map showing variation of desired change of velocity across unit cells (color scale in the figure indicates the value of |J| ). 306
Diameter of Hole (in mm) in unit cell
Number of unit cells in the Holey region
1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0
40 32 18 20 6 6 4 8
NDT and E International 102 (2019) 304–310
S.A.R. Kuchibhatla, P. Rajagopal
Fig. 5. Snapshots showing contour of Y-direction displacement in FE simulation. Color scale adjusted to show wavefront at various time instances (a) Entering (b) While passing through (c) Exiting the holey region in metamaterial plate. The dashed lines illustrate the wavefront curvature.
free boundary. An automatic mesh with C3D8R elements (3D stress element) was used for the FE discretization. The phase velocity of the S0 mode in Copper at 100 kHz is about 3760 m/s. The seeding was set to 0.5 mm which led to an average mesh size of about λ/40 (λ being the wavelength of the wave mode under consideration). The time stepping was calculated based on the CFL criterion to represent the wave phenomena [39] and was set to 10−8 s. An input load, modelled as a 3 cycle (Hanning windowed) pulse at 100 kHz, was applied at one end (face) of this plate to generate Lamb waves. All the nodes in a span of 40 mm about the center of the face were excited to generate a plane wavefront, mimicking patch transduction. The simulation was run in the explicit time domain to evaluate the displacement field in the propagation direction. Fig. 5 shows the snapshots from simulation of wave propagation through the domain. The change in the curvature of the wavefront can be observed which implies that the wave will arrive at different locations at different time instances. In particular, in the present case, the wave will reach the other end of the plate sooner in the sides than in the middle. This was used as an indication of curvature in experimental results.
Fig. 6. Photograph of the specimen in experimental setup indicating the direction of wave propagation.
Fig. 7. B-Scans obtained by interpolating A-Scans (time-traces) collected in experiment a) Plate without holes b) Holey plate.
3.2. Experimental setup For the experiments, a copper sample of size 110 mm × 100 mm x 3 mm was used. Holes of required diameter were made by drilling pilot holes followed by wire electrical discharge machining. Input excitation was provided using a Panametrics-NDT (Waltham, USA) 100 kHz Longitudinal wave probe (setup as shown in Fig. 6) in order to generate the S0-like wave mode (predominantly) in the medium. This is also expected to act as a ‘patch’ source yielding a plane wave like wavefront. Additionally, a spatial buffer of 25 mm, made of the same material as the plate, was placed between the transducer and the specimen to allow the formation of a plane wave. The displacement at various locations, at intervals of 2.5 mm along the breadth of the plate, was measured using a Polytec GmbH (Waldbronn, Germany) laser Doppler vibrometer. For comparison, similar data was also collected for a plate without holes, at intervals of 5 mm along the breadth of the plate.
Fig. 8. A-Scans obtained in the experiment at Scan Locations (SLoc) of 30 mm and 50 mm in a plate with holey region (Holey Plate - solid line) and at similar positions in a plate without holes (Normal Plate - dotted line). Width of both the plates is 100 mm and SLoc is the lateral distance from the side of the plate.
4. Results
μs for the plate with holes. These values in simulation were −0.34 μs and 0.72 μs, respectively. This demonstrates the curvature induced in the wavefront using our proposed approach.
The collected A-Scans (time-traces) were used to obtain B-Scans which are shown in Fig. 7(a) for the plate without holes and Fig. 7(b) for the plate with the holey region. We observe, in the case of the plate with holes (Fig. 7(b)), the arrival of the wave is delayed in the middle of the plate (x = 50 mm) as compared to that in the sides of the plate (x = 30 mm and x = 70 mm). Further, there is some inherent curvature in the wave as can be noted in Fig. 7(a). Fig. 8 shows A-Scans collected at x = 50 mm and x = 70 mm for both the plates in the experiments. The delay in time of arrival in the middle (relative to the side) of the plate was calculated to be −0.43 μs for the plate without holes and 1.05
5. Discussion In this section, the inferences drawn from the results presented in the previous section are discussed. Further, the usefulness of the method for designing holey plates for a higher frequency is demonstrated using FE simulation. 307
NDT and E International 102 (2019) 304–310
S.A.R. Kuchibhatla, P. Rajagopal
Fig. 10. . (a) (FE Simulation) In-plane displacement field of unit cell with hole of diameter 2 mm, at k = 188.5 m-1 and eigenfrequency = 100410 Hz. The curves in black represent undeformed shape. (b) Wave structure of the S0-like mode within the solid region across the thickness of the unit cell shown in (a).
Fig. 9. (a) Plot of wavefronts estimated from results of simulations and experiment on holey plate compared with a semi-circular geometry. Wavefront obtained in simulation results with improved velocity variation in the holey region is closer to the target geometry (b) and (c) (FE simulation) Contour of Ydisplacement magnitude showing wave incidence on curved surface in plate without holey region and with holey region respectively.
transformed as desired when they pass through the holey region. However, drastic changes in the order of the frequency will change the order of wavelength which is undesirable. The curvature will deviate from the circular shape with changes in the velocity of the incoming wave.
5.1. Comparison between experimental and simulation results An estimate of the wavefront was made using time of arrival measured in experiments on holey plate. A similar estimate was also made from results of the simulations discussed earlier. A plot containing these estimated wavefronts along with a reference semi-circular geometry is shown in Fig. 9(a). The radius of curvature calculated from the data obtained from simulations and experiments is closer to the expected curvature in the middle of the holey plate but deviates at the sides (please see Fig. 9(a)). The deviation can be understood from the fact that a continuous spatial variation in the velocity is required ideally, as per the physics of the transformation. However, from a practical viewpoint, only a finite number of cells were used in the design of the holey region. This puts a limit on the achieved curvature as the phase is being adjusted/corrected only over finite and small regions in the wavefront but not continuously. Improvement can be achieved by extending the holey region, refining the hole sizes in the unit cells (as demonstrated using FE simulations) or by changing the unit cell itself, thereby controlling the wave velocity over a larger region in the plate.
5.3. Mode shape of the propagating S0-like mode The wave structure of the propagating S0-like mode in the plate was analysed by considering the in-plane displacement of unit cells for which dispersion curves were obtained in Section 2.2. The deformed shape of the unit cell, with a hole of diameter 2 mm, at a frequency close to 100 kHz is shown in Fig. 10(a). Also, the displacement across the thickness extracted at a point in the solid region of the unit cell is shown in Fig. 10(b). This is similar to the wave structure of S0 Lamb wave mode in plates [35]. Due to its predominant in-plane (axial) displacements, it is expected that the S0-like wave would generate longitudinal bulk waves in an object to be inspected. Since we are exciting the symmetric S0 Lamb mode and the holes, being through the thickness of the plate, are also thickness-wise symmetric, mode conversions upon scattering from holes are not expected.
5.1.1. Improvements to curvature using finer variation in velocity The holey region design was refined by considering hole diameter change in increments of 0.1 mm, keeping the unit cell size same. This allowed for higher control over the variation of velocity within the region. FE simulations carried out on the model with refined holey region showed an improvement in shaping of the wavefront. A wavefront estimated from this FE simulation results which shows a curvature closer to the target semi-circle is included in Fig. 9(a). We find that indeed, as expected, refining the hole radii and thus allowing for smoother variation in wave velocities seems to yield stronger control on wavefront curvature. The match between experiment and simulation enables one to verify/optimise the holey region design through simulations while keeping in mind the practical limitations in manufacturing.
5.4. Simulations demonstrating the application of the holey plate approach at higher frequency Using the method described in Section 2.2, a hole pattern was designed for manipulating S0-like Lamb wave at 1 MHz in a steel plate of 1 mm thickness. The cut-off frequency for higher order modes is above 1 MHz for this plate. As a result, only the fundamental Lamb wave modes are expected. S0 mode with a phase velocity of 5275 m/s was been chosen as it is the faster of the two modes. Wave propagation through the plate was simulated using a 2D plane stress approximation to facilitate faster computation. As the wave in consideration is the S0 Lamb wave, this approximation is valid as described in literature [40]. A plate of size 110 mm × 40 mm was modelled with square unit cells of side 0.5 mm. The hole sizes varied from 0.04 mm to 0.4 mm in diameter. The FE simulation results showing snapshots of wave propagation are shown in Fig. 11. In practice, these holes can be realised in a two-stage process. First the larger holes can be made using conventional drilling or Wire-EDM and then the smaller holes can be made by means of methods like micro-punching. Laser based micro hole drilling and 3D printing are also some of the alternatives for fabrication of such holey regions. Once the wave crosses the holey region, it can be impinged on a curved surface causing in-phase excitation of the boundary. As the wave exits the holey region, due to sudden impedance mismatch a portion of the energy is transmitted and the remaining is reflected as can be seen in Fig. 12. It may also be noted in Fig. 12 that further propagation of the wave leads to a focussing effect.
5.2. Simulations demonstrating the conforming of a wavefront to the circular shape of a target sample FE Simulations were carried out on a geometry having a curved surface as illustrated in Fig. 9(b). For 100 kHz S0-like wave, the presence of designed holey region caused the wavefront to conform to the curvature and improved the area of first incidence (please refer Fig. 9(b) and (c)). Since the design of hole pattern depends on the velocity of the wave in the base material used, the pattern remains the same for all the waves propagating at such velocity (3760 m/s in the simulations described in section 3.1). Thus, for a Copper plate, an S0like wave with frequency thickness of 300 kHz-mm can be manipulated by the hole pattern used in the experiments of Section 3.2. More precisely, only the waves which have velocity of about 3760 m/s will be 308
NDT and E International 102 (2019) 304–310
S.A.R. Kuchibhatla, P. Rajagopal
propagating in an elastic medium can be transformed with the use of holey plates. The approach presented here has been demonstrated via simulations of an S0-like wave at 100 kHz in 3 mm thick copper plate. Based on the curvature desired, a new holey plate can be designed for a different material or frequency of operation as illustrated at high frequency in Steel plate. These holey plates can improve the wave incidence on curved surfaces, such as in case of pipes, without using phased arrays. The close match between experimental results and FE model allows us to verify the curvature due to a new design of holey region, where the parameter ‘a ’ is directly related to the desired radius of curvature. Another important application of the device presented here could be to nullify any inherent curvature present in an ultrasonic wavefront leading to an in-phase excitation of a larger area on an object to be inspected. Finally, although this paper discussed the conversion of plane wavefront to a circular wavefront, the approach described can also be used for more general surface features. Currently, the authors are studying the performance of these holey plates in Non-destructive Evaluation.
Fig. 11. (FE Simulation) Contour plots of displacement in Y direction showing snapshots of 1 MHz S0-like wave propagation in 1 mm Steel holey plate (zoomed in inset: Wavefront at the exit of holey region).
Funding This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. References [1] Drinkwater BW, Wilcox PD. Ultrasonic arrays for non-destructive evaluation: a review. NDT E Int 2006;39:525–41. https://doi.org/10.1016/j.ndteint.2006.03.006. [2] Azar L, Shi Y, Wooh S-C. Beam focusing behavior of linear phased arrays. NDT E Int 2000;33:189–98. [3] Lee Joon-Hyun, Choi Sang-Woo. A parametric study of ultrasonic beam profiles for a linear phased array transducer. IEEE Trans Ultrason Ferroelectrics Freq Contr 2000;47:644–50. https://doi.org/10.1109/58.842052. [4] Von Ramm OT, Smith SW. Beam steering with linear arrays. IEEE Trans Biomed Eng 1983:438–52. [5] Russell J, Long R, Cawley P. Development of a membrane coupled conformable phased array inspection capability. AIP Conf. Proc. 2010;1211:831–8. [6] Russell J, Long R, Duxbury D, Cawley P. Development and implementation of a membrane-coupled conformable array transducer for use in the nuclear industry. Insight Non-Destructive Test Cond Moniz 2012;54:386–93. https://doi.org/10. 1784/insi.2012.54.7.386. [7] Pendry JB, Schurig D, Smith DR. Controlling electromagnetic fields. Science (80 ) 2006;312:1780–2. [8] Leonhardt U. Optical conformal mapping. Science (80 ) 2006;312. 1777 LP-1780. [9] Cummer SA, Schurig D. One path to acoustic cloaking. New J Phys 2007;9. https:// doi.org/10.1088/1367-2630/9/3/045. [10] Cummer SA, Christensen J, Alù A. Controlling sound with acoustic metamaterials. Nat Rev Mater 2016;1. https://doi.org/10.1038/natrevmats.2016.1. [11] Chen H, Chan CT. Acoustic cloaking and transformation acoustics. J Phys D Appl Phys 2010;43. https://doi.org/10.1088/0022-3727/43/11/113001. [12] García-Meca C, Carloni S, Barceló C, Jannes G, Sánchez-Dehesa J, Martínez A. Space-time transformation acoustics. Wave Motion 2014;51:785–97. https://doi. org/10.1016/j.wavemoti.2014.01.008. [13] Garcia-Meca C, Carloni S, Barcelo C, Jannes G, Sanchez-Dehesa J, Martinez A. Analogue transformation acoustics: generalizing transformation techniques to nonform-invariant equations. 2013 7th int. Congr. Adv. Electromagn. Mater. Microwaves opt. METAMATERIALS 2013 2013. p. 235–7. https://doi.org/10.1109/ MetaMaterials.2013.6809011. [14] Craster RV, Guenneau S. A. coustic metamaterials: negative refraction, imaging, lensing and cloaking vol. 166. Springer Science & Business Media; 2012. [15] Assouar B, Liang B, Wu Y, Li Y, Cheng J-C, Jing Y. Acoustic metasurfaces. Nat Rev Mater 2018. https://doi.org/10.1038/s41578-018-0061-4. [16] Memoli G, Caleap M, Asakawa M, Sahoo DR, Drinkwater BW, Subramanian S. Metamaterial bricks and quantization of meta-surfaces. Nat Commun 2017;8:14608. [17] Ghaffarivardavagh R, Nikolajczyk J, Glynn Holt R, Anderson S, Zhang X. Horn-like space-coiling metamaterials toward simultaneous phase and amplitude modulation. Nat Commun 2018;9:1349. https://doi.org/10.1038/s41467-018-03839-z. [18] Li Y, Liang B, Gu Z, Zou X, Cheng J. Reflected wavefront manipulation based on ultrathin planar acoustic metasurfaces. Sci Rep 2013;3:2546. [19] Li Y, Jiang X, Liang B, Cheng JC, Zhang L. Metascreen-based acoustic passive phased array. Phys Rev Appl 2015;4:1–7. https://doi.org/10.1103/ PhysRevApplied.4.024003. [20] Milton GW, Briane M, Willis JR. On cloaking for elasticity and physical equations with a transformation invariant form. New J Phys 2006;8. https://doi.org/10. 1088/1367-2630/8/10/248. [21] Zhu H, Semperlotti F. Metamaterial based embedded acoustic filters for structural applications. AIP Adv 2013;3. https://doi.org/10.1063/1.4822157.
Fig. 12. (FE simulation) Contour of displacement in Y-direction of a 1 MHz S0 mode Lamb wave after passing through the holey region in Steel plate. Reflected and transmitted waves are seen as the wave encounters sudden impedance mismatch at the exit of the holey region. Focussing of the transmitted waves can also be noticed.
5.5. Applications of the proposed method As discussed in Section 5.1 the device demonstrated here can cause an improved in-phase excitation on a curved surface leading to bulk longitudinal ultrasonic waves which are useful, for example, in pipe inspections. Although the approach has been illustrated using fundamental Lamb wave mode in plates, the method can very well be used for bulk waves. As the proposed device can convert a plane wave to a cylindrical wave, the reverse is also true due to reciprocity. Schwarz-Christoffel mapping is a conformal map which transforms the upper half of a complex plane to an arbitrary polygon. Modification of this mapping to achieve polygons with rounded corners have been reported in literature [41]. This opens the possibility of manipulation of the wave to achieve arbitrary wavefronts as well as engineering the wave path as desired. In theory, the method proposed in this article can be used for any features where conformal transformations are available between initial wavefront shape and target wavefront shape. Other types of mappings that could be used with this method are yet to be explored. 6. Conclusions This work showed that the curvature of an ultrasonic wavefront 309
NDT and E International 102 (2019) 304–310
S.A.R. Kuchibhatla, P. Rajagopal
doi.org/10.1016/j.wavemoti.2011.03.002. [32] Nehari Z. Conformal mapping. Courier Corporation; 2012. [33] Lee MK, Kim YY, Kyung M, Young Y. Add-on unidirectional elastic metamaterial plate cloak. Natl Publ Gr 2016:1–17. https://doi.org/10.1038/srep20731. [34] Centeno E, Cassagne D, Albert JP. Mirage and superbending effect in two-dimensional graded photonic crystals. Phys Rev B - Condens Matter Mater Phys 2006;73:1–5. https://doi.org/10.1103/PhysRevB.73.235119. [35] Rose JL. Ultrasonic guided waves in solid media. Cambridge university press; 2014. [36] Ke M, Liu Z, Qiu C, Wang W, Shi J, Wen W, et al. Negative-refraction imaging with two-dimensional phononic crystals. Phys Rev B - Condens Matter Mater Phys 2005;72:1–5. https://doi.org/10.1103/PhysRevB.72.064306. [37] COMSOL multiphysics reference manual (version 5.3). Sweden: COMSOL AB Stockholm; 2015. [38] ABAQUS/Standard User's Manual (version 6.14). Providence, RI, USA: Dassault Systèmes; 2014. [39] Amireddy KK, Balasubramaniam K, Rajagopal P. Deep subwavelength ultrasonic imaging using optimized holey structured metamaterials. Sci Rep 2017;7:7777. https://doi.org/10.1038/s41598-017-08036-4. [40] Rajagopal P, Lowe MJS. Scattering of the fundamental shear horizontal guided wave by a part-thickness crack in an isotropic plate. J Acoust Soc Am 2008;124:2895–904. https://doi.org/10.1121/1.2982410. [41] Andersson A. Modified Schwarz-Christoffel mappings using approximate curve factors. J Comput Appl Math 2009;233:1117–27. https://doi.org/10.1016/j.cam. 2009.09.006.
[22] Zhu H, Semperlotti F. A passively tunable acoustic metamaterial lens for selective ultrasonic excitation. J Appl Phys 2014;116. https://doi.org/10.1063/1.4894279. [23] Pennec Y, Djafari-Rouhani B, Vasseur JO, Larabi H, Khelif A, Choujaa A, et al. Acoustic channel drop tunneling in a phononic crystal. Appl Phys Lett 2005;87:1–3. https://doi.org/10.1063/1.2158019. [24] Lee H, Oh JH, Seung HM, Cho SH, Kim YY. Extreme stiffness hyperbolic elastic metamaterial for total transmission subwavelength imaging. Sci Rep 2016;6:1–12. https://doi.org/10.1038/srep24026. [25] Zhu H, Semperlotti F. Anomalous refraction of acoustic guided waves in solids with geometrically tapered metasurfaces. Phys Rev Lett 2016;117:1–6. https://doi.org/ 10.1103/PhysRevLett.117.034302. [26] Su X, Lu Z, Norris AN. Elastic metasurfaces for splitting SV- and P-waves in elastic solids. J Appl Phys 2018;123. https://doi.org/10.1063/1.5007731. [27] Ma G, Sheng P. Acoustic metamaterials: from local resonances to broad horizons. Sci Adv 2016;2. https://doi.org/10.1126/sciadv.1501595. e1501595–e1501595. [28] Hussein MI, Leamy MJ, Ruzzene M. Dynamics of phononic materials and structures: historical origins, recent progress, and future outlook. Appl Mech Rev 2014;66. https://doi.org/10.1115/1.4026911. 040802. [29] Zhu Q, Jin L, Fu Y. Graded index photonic crystals: a review. Ann Phys 2015;527:205–18. https://doi.org/10.1002/andp.201400195. [30] Lin SCS, Huang TJ, Sun JH, Wu TT. Gradient-index phononic crystals. Phys Rev B Condens Matter Mater Phys 2009;79:1–6. https://doi.org/10.1103/PhysRevB.79. 094302. [31] Norris AN, Shuvalov AL. Elastic cloaking theory. WAMOT 2011;48:525–38. https://
310
Contents lists available at ScienceDirect
NDT and E International journal homepage: www.elsevier.com/locate/ndteint
A transformation elasticity based device for wavefront manipulation Sai Aditya Raman Kuchibhatla, Prabhu Rajagopal
∗
T
Centre for Nondestructive Evaluation, Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai, India
A R T I C LE I N FO
A B S T R A C T
Keywords: Transformation elasticity Metamaterial plate Wave manipulation GRIN PC Guided waves
The authors propose a metamaterial plate made of gradient refractive index phononic crystals for manipulation of the wavefront of propagating elastic waves in solid media. A square unit cell with through holes is considered as the basis for the proposed metematerial. Guidance on the design of the hole pattern and the choice of materials is obtained with the aid of transformation elasticity principles. Manipulation of a plane wavefront into a cylindrical wavefront is first observed through Finite Element simulations at low frequency (100 kHz), and then practically demonstrated through experiments. Wave propagation with and without the holey region is compared and possible applications at higher frequencies are discussed.
1. Introduction Ultrasonic testing (UT) is a popular technique among the various non-destructive evaluation (NDE) methods used in the industry for the inspection of pipes and tubes. Particularly, Phased Array Ultrasonic Transducers (PAUT) can be used for controlled generation of ultrasonic waves [1]. These are composed of multiple individually controllable transducer elements, allowing for firing of elements with specific electronic time delays. By using various delay schemes, it is possible to steer the beam or alter the shape of wavefront [2–4]. Further, an improved inspection capability of linear phased arrays for irregular surfaces by use of a coupling medium consisting of water encapsulated by a membrane has been reported in literature [5,6]. In recent years, metamaterial devices have attracted attention as a possible alternative for controlled propagation of waves inside a medium. Such devices were first proposed in Optics and were soon followed by their acoustic analogues. A variety of methods have been proposed in the last two decades to control wave propagation in electromagnetic [7,8] as well as acoustic media [9,10]. Particularly, transformation methods have been found to be very effective, for example, in the design of acoustic cloaking devices [11–14]. More recently, much thinner metamaterials [15] have been demonstrated for sound modulation [16–18] in which novel techniques such as spacecoiling etc. have been used. Metascreens made of Helmholtz resonators capable of shaping acoustic wavefronts by controlling their phase have also been proposed [19]. However, for elastic waves, the wave equation is not form invariant under transformation [20]. This limits the usefulness of this method for making wave manipulating devices for elastodynamics, when compared to that in acoustics. Other approaches for
∗
designing devices for wave manipulation in elastic media have been discussed in literature. For example, a device [21,22] consisting of pathways engineered using the concept of drop-channels [23] has been proposed for frequency selective beam steering and collimation of ultrasound waves. Another device based on local resonators, with hyperbolic dispersion curves, has been used for sub-wavelength imaging [24]. Metasurfaces for elastic waves have only been reported very recently [25,26]. Here, we study the possibility of exploiting Transformation Elastodynamics to design a wavefront manipulating device by taking into account some engineering approximations. We propose a method to design customized holey plates which can generate curved wavefronts though the excitation is provided by a single ultrasound transducer, using the concept of gradient phononic crystals (PCs) [27–30]. The proposed approach is graphically illustrated in Fig. 1. In the first demonstration of this approach described here, we have used a wave source at 100 kHz. However, this approach is applicable at higher frequencies where practical UT is typically carried out (please see the discussion in Section 5.2, for demonstration of the approach at higher frequencies using numerical simulation). This paper is organised as follows. Firstly, we describe the method proposed to design holey plates and demonstrate it using Finite Element (FE) simulations and experiments at a low frequency (100 kHz). The application of curved wavefronts so generated using the designed holey plates is then discussed along with the validity of proposed method at higher frequencies. The paper concludes with some directions for future work.
Corresponding author. E-mail address: [email protected] (P. Rajagopal).
https://doi.org/10.1016/j.ndteint.2019.01.006 Received 21 August 2018; Received in revised form 30 November 2018; Accepted 10 January 2019 Available online 14 January 2019 0963-8695/ © 2019 Elsevier Ltd. All rights reserved.
NDT and E International 102 (2019) 304–310
S.A.R. Kuchibhatla, P. Rajagopal
due to the transformation. These influence the stress-strain relationship and hence the wave propagation in the transformed domain. These are given by Ref. [20].
S′pqr =
∂2x r′ 1 ∂x p′ ∂xq′ Cijkl = S′qpr |J| ∂x i ∂x j ∂xk ∂xl
(6)
D′pqr =
2 ∂xq′ ∂x r′ 1 ∂ x p′ Cijkl = S′qrp |J| ∂x i ∂x j ∂xk ∂xl
(7)
In mathematics, a complex function is said to be analytic in a region if it is differentiable at all the points in the region and such a complex differentiable function f (x , y ) → (x ′, y′) satisfies Cauchy-Riemann relations [32]:
∂y′ ∂y′ ∂x ′ ∂x ′ = ; =− ∂x ∂y ∂x ∂y
(8)
An analytic function is said to be conformal at a point if it has a nonzero derivative at that point. Conformal transforms preserve local angles and orientation. The conformal map given in Eq. (9) below was used in this study.
w=
2. Methods 2.1. Transformation elasticity and conformal mapping for holey plate Transformation acoustics/elasticity provides one method for design of devices for cloaking and other wave steering systems. The equation governing propagation of elastic waves is given by (1)
where u is displacement, ω is the frequency of propagating wave, ρ is the density of medium and C is the constitutive property matrix. Unlike the governing equations for electromagnetic and acoustic waves which are form invariant under coordinate transformation [7–9], the equation for elastic waves involves additional terms after transformation [20,31]. The transformed wave equation (with transformed terms indicated by a prime) is given by
∇′. σ ′ = D'∇'u′ − ω2ρ'u′
(2)
σ ′ = C'∇' u′ + S'u′
(3)
z 2 − 4a2 )
(9)
where z and w are complex numbers, z = x + iy represents a point ( x , y ) before transformation and w = x ′ + iy′ denotes the same point with new coordinates ( x ′, y′) after the transformation. ‘a ’ is a transformation parameter. Fig. 2(a) shows the transformation of a finite discrete grid under the mapping given by Eq. (9). It may be noted that horizontal parallel lines in the input plane are transformed into conforming curved lines in the region near the semi-circle in the output plane. For a conformal map, as used in this work, the cross terms in the density tensor are zero. Under the assumption that displacements are time harmonic i.e. varying as exp(iωt ) , the density along principal directions reduces to the isotropic density plus an additional term which is inversely dependent on the square of the frequency [31,33]. This additional term can hence be neglected at ultrasonic frequencies. The device can therefore be made out of a material with isotropic density. The coupling terms D′ and S′ in Eq. (2) and Eq. (3), respectively, are typically dependent on both Lame's parameters. On using a conformal map, these are reduced to depend only on one of the two parameters (μ) which indicates that S’ (or D′) on using a conformal map could be smaller when compared to S’ (or D′) on using a non-conformal transform [33]. Further, these terms occur in combination with strains and displacements which are small quantities to begin with, and hence can be neglected as an engineering approximation in this study (for instance, in Eq. (3), the term S'u' is much smaller when compared to C'∇' u' ). The approximated wave equation can hence be given as
Fig. 1. Schematic illustration of wavefront manipulation using the holey plate concept discussed in this paper.
∇ . (C∇u) = −ω2ρu
1 (z ± 2
∇ '. (C'∇'u′) ≈ −ω2ρ′u′
(10)
where the transformed density is a matrix given by
ρ′pq =
2 ∂2xq′ ρ ∂x p′ ∂xq′ 1 ∂ x p′ Cijkl + |J| ∂x i ∂x i |J| ∂x i ∂x j ∂xk ∂xl
(4)
This indicates the need for a medium with anisotropic mass density. The transformed elasticity tensor is given by
C′pqrs =
∂x ′ ∂x ′ 1 ∂x p′ ∂xq′ Cijkl r s |J| ∂x i ∂x j ∂xk ∂xl
Fig. 2. Illustration of the conformal map in Eq. 9 - (a) Transformation of grid lines under the mapping (b) Jacobian of the transformation (color scale in the figure indicates the value of |J| ).
(5)
The additional terms S' and D' are third order tensors which arise 305
NDT and E International 102 (2019) 304–310
S.A.R. Kuchibhatla, P. Rajagopal
which is of the same form as the wave equation before the transformation. The velocity of the wave in the transformed space is related to the velocity in space before transformation as
v′ (x ′, y′) =
|J| v (x , y )
(11)
where ‘v ’ is the wave velocity in the region before transformation and ‘J ’ is the Jacobian of the transformation given by
⎡ ∂x′ J=⎢ ⎢ ∂y′ ⎣ ∂x
∂x′ ⎤ ∂y
∂x
⎥
∂y′ ⎥ ∂y ⎦
(12)
2.2. Design of the holey plate
Fig. 4. Plot showing dispersion curves of S0-like wave for square unit cells with varying hole diameter.
Since the mapping in Eq. (9) is conformal, and hence analytic, it satisfies Cauchy-Riemann relations given in Eq. (8), using which the |J| can be calculated as in Eq. (13).
|J| = =
1 (1 ± 2
(
∂y′ 2 ∂x ′ 2 ∂w ∂ 1 ) +( ) =| |= ( (z ± ∂x ∂x ∂x ∂x 2 z z 2 − 4a2
)=
w 1 = 2w − z (1 −
a2 ) w2
required wave velocities in the region, PCs of different hole radius were incorporated in the plate. Since the velocity required in the region is less than that of the S0 mode ( |J| < 1 in the region), the choice of air hole is justified. The Jacobian of the transformation in Eq. (9) evaluated in a spatial grid made out of 5 mm square unit cells is shown in Fig. 3(b). An indicative semi-circular geometry representing the curvature desired is also presented in the figure. Dispersion curves (frequency vs. wavenumber) for 5 mm × 5 mm x 3 mm square units cells of Copper were found for various hole diameters ranging from 1.2 mm to 4 mm in steps of 0.4 mm, using an eigen frequency solver in a commercial package [37]. Periodic (or Floquet) boundary conditions were placed on the unit cell (of side ‘a ’) in the direction of wave propagation. Eigenfrequencies were evaluated while varying the wave vector between (0,0,0) and ( π ,0,0) for ΓX direction,
z 2 − 4a2 ))
(13)
Fig. 2(b) shows the variation of |J| , and thus of the required wave velocity, in the proposed device. To achieve such varying wave velocity, here we use the concept of Gradient Refractive INdex Phononic Crystals (GRIN PCs) [30]. The idea of GRIN PCs has been recently reported in literature and is inspired from the analogous idea of Graded Photonic Crystals in Optics [34]. These have been used to cause a gradient in the velocity within the medium and are not constrained by the feature of periodicity required in standard phononic/photonic crystal structures. In our work, in order to achieve better control over the velocity variation in the region, we used a unit cell size of 5 mm. The GRIN PCs used here are made of square unit cells of copper, with air-filled cylindrical through holes (please see Fig. 3(a)). The fundamental symmetric Lamb wave mode S0 [35] was chosen to be generated in the holey plate since the mode has in-plane (axial) displacements which are ideal for the generation of bulk longitudinal ultrasonic waves in the sample to be inspected (In the holey plate, an ‘S0like mode’ was verified to exist based on the mode shape, please see Discussions for details). The reciprocal lattice of a square unit cell is also a square unit cell. The wave speed of the phononic crystal is calculated as the average of wave speeds along two principal orientations ΓX and ΓM, in the reciprocal lattice [34,36]. Thus, corresponding to the
a
π
and (0,0,0) and ( π , a ,0) for ΓM direction. The phase velocity was a calculated as the ratio of angular frequency and wavenumber at various points on the dispersion curve to obtain phase velocity curves. The phase velocity of the S0-like mode reduces with increase in the diameter of the hole in the unit cell for frequencies around 100 kHz (please see Fig. 4). Also, at smaller hole diameters, as the phase velocity around 100 kHz is almost constant, the group velocity was taken to be approximately equal to the phase velocity. The diameter of hole which leads to a calculated velocity closest to that desired at a location was mapped to that position in space. Table 1 lists the different sizes of hole and the total number of such holes used in the design. 3. Procedure for simulations and experiments 3.1. Wave propagation simulations for low frequency A 200 mm × 80 mm x 3 mm plate of Copper material ( ρ = 8960 kg/m3, E = 125 GPa, ν = 0.35) with holes at different locations (obtained as described earlier) was modelled in a commercial package [38] for FE simulation of wave propagation. For simplicity the simulations were carried out considering the holes to be voids instead of being filled with air and accordingly the boundary was taken to be a Table 1 Number of Holes in the holey region for 3 mm thick copper plate for manipulating curvature of S0 Lamb wave mode.
Fig. 3. (a) Illustration of square unit cell along with top view of the unit cell showing principal orientations. (b) Contour map showing variation of desired change of velocity across unit cells (color scale in the figure indicates the value of |J| ). 306
Diameter of Hole (in mm) in unit cell
Number of unit cells in the Holey region
1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0
40 32 18 20 6 6 4 8
NDT and E International 102 (2019) 304–310
S.A.R. Kuchibhatla, P. Rajagopal
Fig. 5. Snapshots showing contour of Y-direction displacement in FE simulation. Color scale adjusted to show wavefront at various time instances (a) Entering (b) While passing through (c) Exiting the holey region in metamaterial plate. The dashed lines illustrate the wavefront curvature.
free boundary. An automatic mesh with C3D8R elements (3D stress element) was used for the FE discretization. The phase velocity of the S0 mode in Copper at 100 kHz is about 3760 m/s. The seeding was set to 0.5 mm which led to an average mesh size of about λ/40 (λ being the wavelength of the wave mode under consideration). The time stepping was calculated based on the CFL criterion to represent the wave phenomena [39] and was set to 10−8 s. An input load, modelled as a 3 cycle (Hanning windowed) pulse at 100 kHz, was applied at one end (face) of this plate to generate Lamb waves. All the nodes in a span of 40 mm about the center of the face were excited to generate a plane wavefront, mimicking patch transduction. The simulation was run in the explicit time domain to evaluate the displacement field in the propagation direction. Fig. 5 shows the snapshots from simulation of wave propagation through the domain. The change in the curvature of the wavefront can be observed which implies that the wave will arrive at different locations at different time instances. In particular, in the present case, the wave will reach the other end of the plate sooner in the sides than in the middle. This was used as an indication of curvature in experimental results.
Fig. 6. Photograph of the specimen in experimental setup indicating the direction of wave propagation.
Fig. 7. B-Scans obtained by interpolating A-Scans (time-traces) collected in experiment a) Plate without holes b) Holey plate.
3.2. Experimental setup For the experiments, a copper sample of size 110 mm × 100 mm x 3 mm was used. Holes of required diameter were made by drilling pilot holes followed by wire electrical discharge machining. Input excitation was provided using a Panametrics-NDT (Waltham, USA) 100 kHz Longitudinal wave probe (setup as shown in Fig. 6) in order to generate the S0-like wave mode (predominantly) in the medium. This is also expected to act as a ‘patch’ source yielding a plane wave like wavefront. Additionally, a spatial buffer of 25 mm, made of the same material as the plate, was placed between the transducer and the specimen to allow the formation of a plane wave. The displacement at various locations, at intervals of 2.5 mm along the breadth of the plate, was measured using a Polytec GmbH (Waldbronn, Germany) laser Doppler vibrometer. For comparison, similar data was also collected for a plate without holes, at intervals of 5 mm along the breadth of the plate.
Fig. 8. A-Scans obtained in the experiment at Scan Locations (SLoc) of 30 mm and 50 mm in a plate with holey region (Holey Plate - solid line) and at similar positions in a plate without holes (Normal Plate - dotted line). Width of both the plates is 100 mm and SLoc is the lateral distance from the side of the plate.
4. Results
μs for the plate with holes. These values in simulation were −0.34 μs and 0.72 μs, respectively. This demonstrates the curvature induced in the wavefront using our proposed approach.
The collected A-Scans (time-traces) were used to obtain B-Scans which are shown in Fig. 7(a) for the plate without holes and Fig. 7(b) for the plate with the holey region. We observe, in the case of the plate with holes (Fig. 7(b)), the arrival of the wave is delayed in the middle of the plate (x = 50 mm) as compared to that in the sides of the plate (x = 30 mm and x = 70 mm). Further, there is some inherent curvature in the wave as can be noted in Fig. 7(a). Fig. 8 shows A-Scans collected at x = 50 mm and x = 70 mm for both the plates in the experiments. The delay in time of arrival in the middle (relative to the side) of the plate was calculated to be −0.43 μs for the plate without holes and 1.05
5. Discussion In this section, the inferences drawn from the results presented in the previous section are discussed. Further, the usefulness of the method for designing holey plates for a higher frequency is demonstrated using FE simulation. 307
NDT and E International 102 (2019) 304–310
S.A.R. Kuchibhatla, P. Rajagopal
Fig. 10. . (a) (FE Simulation) In-plane displacement field of unit cell with hole of diameter 2 mm, at k = 188.5 m-1 and eigenfrequency = 100410 Hz. The curves in black represent undeformed shape. (b) Wave structure of the S0-like mode within the solid region across the thickness of the unit cell shown in (a).
Fig. 9. (a) Plot of wavefronts estimated from results of simulations and experiment on holey plate compared with a semi-circular geometry. Wavefront obtained in simulation results with improved velocity variation in the holey region is closer to the target geometry (b) and (c) (FE simulation) Contour of Ydisplacement magnitude showing wave incidence on curved surface in plate without holey region and with holey region respectively.
transformed as desired when they pass through the holey region. However, drastic changes in the order of the frequency will change the order of wavelength which is undesirable. The curvature will deviate from the circular shape with changes in the velocity of the incoming wave.
5.1. Comparison between experimental and simulation results An estimate of the wavefront was made using time of arrival measured in experiments on holey plate. A similar estimate was also made from results of the simulations discussed earlier. A plot containing these estimated wavefronts along with a reference semi-circular geometry is shown in Fig. 9(a). The radius of curvature calculated from the data obtained from simulations and experiments is closer to the expected curvature in the middle of the holey plate but deviates at the sides (please see Fig. 9(a)). The deviation can be understood from the fact that a continuous spatial variation in the velocity is required ideally, as per the physics of the transformation. However, from a practical viewpoint, only a finite number of cells were used in the design of the holey region. This puts a limit on the achieved curvature as the phase is being adjusted/corrected only over finite and small regions in the wavefront but not continuously. Improvement can be achieved by extending the holey region, refining the hole sizes in the unit cells (as demonstrated using FE simulations) or by changing the unit cell itself, thereby controlling the wave velocity over a larger region in the plate.
5.3. Mode shape of the propagating S0-like mode The wave structure of the propagating S0-like mode in the plate was analysed by considering the in-plane displacement of unit cells for which dispersion curves were obtained in Section 2.2. The deformed shape of the unit cell, with a hole of diameter 2 mm, at a frequency close to 100 kHz is shown in Fig. 10(a). Also, the displacement across the thickness extracted at a point in the solid region of the unit cell is shown in Fig. 10(b). This is similar to the wave structure of S0 Lamb wave mode in plates [35]. Due to its predominant in-plane (axial) displacements, it is expected that the S0-like wave would generate longitudinal bulk waves in an object to be inspected. Since we are exciting the symmetric S0 Lamb mode and the holes, being through the thickness of the plate, are also thickness-wise symmetric, mode conversions upon scattering from holes are not expected.
5.1.1. Improvements to curvature using finer variation in velocity The holey region design was refined by considering hole diameter change in increments of 0.1 mm, keeping the unit cell size same. This allowed for higher control over the variation of velocity within the region. FE simulations carried out on the model with refined holey region showed an improvement in shaping of the wavefront. A wavefront estimated from this FE simulation results which shows a curvature closer to the target semi-circle is included in Fig. 9(a). We find that indeed, as expected, refining the hole radii and thus allowing for smoother variation in wave velocities seems to yield stronger control on wavefront curvature. The match between experiment and simulation enables one to verify/optimise the holey region design through simulations while keeping in mind the practical limitations in manufacturing.
5.4. Simulations demonstrating the application of the holey plate approach at higher frequency Using the method described in Section 2.2, a hole pattern was designed for manipulating S0-like Lamb wave at 1 MHz in a steel plate of 1 mm thickness. The cut-off frequency for higher order modes is above 1 MHz for this plate. As a result, only the fundamental Lamb wave modes are expected. S0 mode with a phase velocity of 5275 m/s was been chosen as it is the faster of the two modes. Wave propagation through the plate was simulated using a 2D plane stress approximation to facilitate faster computation. As the wave in consideration is the S0 Lamb wave, this approximation is valid as described in literature [40]. A plate of size 110 mm × 40 mm was modelled with square unit cells of side 0.5 mm. The hole sizes varied from 0.04 mm to 0.4 mm in diameter. The FE simulation results showing snapshots of wave propagation are shown in Fig. 11. In practice, these holes can be realised in a two-stage process. First the larger holes can be made using conventional drilling or Wire-EDM and then the smaller holes can be made by means of methods like micro-punching. Laser based micro hole drilling and 3D printing are also some of the alternatives for fabrication of such holey regions. Once the wave crosses the holey region, it can be impinged on a curved surface causing in-phase excitation of the boundary. As the wave exits the holey region, due to sudden impedance mismatch a portion of the energy is transmitted and the remaining is reflected as can be seen in Fig. 12. It may also be noted in Fig. 12 that further propagation of the wave leads to a focussing effect.
5.2. Simulations demonstrating the conforming of a wavefront to the circular shape of a target sample FE Simulations were carried out on a geometry having a curved surface as illustrated in Fig. 9(b). For 100 kHz S0-like wave, the presence of designed holey region caused the wavefront to conform to the curvature and improved the area of first incidence (please refer Fig. 9(b) and (c)). Since the design of hole pattern depends on the velocity of the wave in the base material used, the pattern remains the same for all the waves propagating at such velocity (3760 m/s in the simulations described in section 3.1). Thus, for a Copper plate, an S0like wave with frequency thickness of 300 kHz-mm can be manipulated by the hole pattern used in the experiments of Section 3.2. More precisely, only the waves which have velocity of about 3760 m/s will be 308
NDT and E International 102 (2019) 304–310
S.A.R. Kuchibhatla, P. Rajagopal
propagating in an elastic medium can be transformed with the use of holey plates. The approach presented here has been demonstrated via simulations of an S0-like wave at 100 kHz in 3 mm thick copper plate. Based on the curvature desired, a new holey plate can be designed for a different material or frequency of operation as illustrated at high frequency in Steel plate. These holey plates can improve the wave incidence on curved surfaces, such as in case of pipes, without using phased arrays. The close match between experimental results and FE model allows us to verify the curvature due to a new design of holey region, where the parameter ‘a ’ is directly related to the desired radius of curvature. Another important application of the device presented here could be to nullify any inherent curvature present in an ultrasonic wavefront leading to an in-phase excitation of a larger area on an object to be inspected. Finally, although this paper discussed the conversion of plane wavefront to a circular wavefront, the approach described can also be used for more general surface features. Currently, the authors are studying the performance of these holey plates in Non-destructive Evaluation.
Fig. 11. (FE Simulation) Contour plots of displacement in Y direction showing snapshots of 1 MHz S0-like wave propagation in 1 mm Steel holey plate (zoomed in inset: Wavefront at the exit of holey region).
Funding This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. References [1] Drinkwater BW, Wilcox PD. Ultrasonic arrays for non-destructive evaluation: a review. NDT E Int 2006;39:525–41. https://doi.org/10.1016/j.ndteint.2006.03.006. [2] Azar L, Shi Y, Wooh S-C. Beam focusing behavior of linear phased arrays. NDT E Int 2000;33:189–98. [3] Lee Joon-Hyun, Choi Sang-Woo. A parametric study of ultrasonic beam profiles for a linear phased array transducer. IEEE Trans Ultrason Ferroelectrics Freq Contr 2000;47:644–50. https://doi.org/10.1109/58.842052. [4] Von Ramm OT, Smith SW. Beam steering with linear arrays. IEEE Trans Biomed Eng 1983:438–52. [5] Russell J, Long R, Cawley P. Development of a membrane coupled conformable phased array inspection capability. AIP Conf. Proc. 2010;1211:831–8. [6] Russell J, Long R, Duxbury D, Cawley P. Development and implementation of a membrane-coupled conformable array transducer for use in the nuclear industry. Insight Non-Destructive Test Cond Moniz 2012;54:386–93. https://doi.org/10. 1784/insi.2012.54.7.386. [7] Pendry JB, Schurig D, Smith DR. Controlling electromagnetic fields. Science (80 ) 2006;312:1780–2. [8] Leonhardt U. Optical conformal mapping. Science (80 ) 2006;312. 1777 LP-1780. [9] Cummer SA, Schurig D. One path to acoustic cloaking. New J Phys 2007;9. https:// doi.org/10.1088/1367-2630/9/3/045. [10] Cummer SA, Christensen J, Alù A. Controlling sound with acoustic metamaterials. Nat Rev Mater 2016;1. https://doi.org/10.1038/natrevmats.2016.1. [11] Chen H, Chan CT. Acoustic cloaking and transformation acoustics. J Phys D Appl Phys 2010;43. https://doi.org/10.1088/0022-3727/43/11/113001. [12] García-Meca C, Carloni S, Barceló C, Jannes G, Sánchez-Dehesa J, Martínez A. Space-time transformation acoustics. Wave Motion 2014;51:785–97. https://doi. org/10.1016/j.wavemoti.2014.01.008. [13] Garcia-Meca C, Carloni S, Barcelo C, Jannes G, Sanchez-Dehesa J, Martinez A. Analogue transformation acoustics: generalizing transformation techniques to nonform-invariant equations. 2013 7th int. Congr. Adv. Electromagn. Mater. Microwaves opt. METAMATERIALS 2013 2013. p. 235–7. https://doi.org/10.1109/ MetaMaterials.2013.6809011. [14] Craster RV, Guenneau S. A. coustic metamaterials: negative refraction, imaging, lensing and cloaking vol. 166. Springer Science & Business Media; 2012. [15] Assouar B, Liang B, Wu Y, Li Y, Cheng J-C, Jing Y. Acoustic metasurfaces. Nat Rev Mater 2018. https://doi.org/10.1038/s41578-018-0061-4. [16] Memoli G, Caleap M, Asakawa M, Sahoo DR, Drinkwater BW, Subramanian S. Metamaterial bricks and quantization of meta-surfaces. Nat Commun 2017;8:14608. [17] Ghaffarivardavagh R, Nikolajczyk J, Glynn Holt R, Anderson S, Zhang X. Horn-like space-coiling metamaterials toward simultaneous phase and amplitude modulation. Nat Commun 2018;9:1349. https://doi.org/10.1038/s41467-018-03839-z. [18] Li Y, Liang B, Gu Z, Zou X, Cheng J. Reflected wavefront manipulation based on ultrathin planar acoustic metasurfaces. Sci Rep 2013;3:2546. [19] Li Y, Jiang X, Liang B, Cheng JC, Zhang L. Metascreen-based acoustic passive phased array. Phys Rev Appl 2015;4:1–7. https://doi.org/10.1103/ PhysRevApplied.4.024003. [20] Milton GW, Briane M, Willis JR. On cloaking for elasticity and physical equations with a transformation invariant form. New J Phys 2006;8. https://doi.org/10. 1088/1367-2630/8/10/248. [21] Zhu H, Semperlotti F. Metamaterial based embedded acoustic filters for structural applications. AIP Adv 2013;3. https://doi.org/10.1063/1.4822157.
Fig. 12. (FE simulation) Contour of displacement in Y-direction of a 1 MHz S0 mode Lamb wave after passing through the holey region in Steel plate. Reflected and transmitted waves are seen as the wave encounters sudden impedance mismatch at the exit of the holey region. Focussing of the transmitted waves can also be noticed.
5.5. Applications of the proposed method As discussed in Section 5.1 the device demonstrated here can cause an improved in-phase excitation on a curved surface leading to bulk longitudinal ultrasonic waves which are useful, for example, in pipe inspections. Although the approach has been illustrated using fundamental Lamb wave mode in plates, the method can very well be used for bulk waves. As the proposed device can convert a plane wave to a cylindrical wave, the reverse is also true due to reciprocity. Schwarz-Christoffel mapping is a conformal map which transforms the upper half of a complex plane to an arbitrary polygon. Modification of this mapping to achieve polygons with rounded corners have been reported in literature [41]. This opens the possibility of manipulation of the wave to achieve arbitrary wavefronts as well as engineering the wave path as desired. In theory, the method proposed in this article can be used for any features where conformal transformations are available between initial wavefront shape and target wavefront shape. Other types of mappings that could be used with this method are yet to be explored. 6. Conclusions This work showed that the curvature of an ultrasonic wavefront 309
NDT and E International 102 (2019) 304–310
S.A.R. Kuchibhatla, P. Rajagopal
doi.org/10.1016/j.wavemoti.2011.03.002. [32] Nehari Z. Conformal mapping. Courier Corporation; 2012. [33] Lee MK, Kim YY, Kyung M, Young Y. Add-on unidirectional elastic metamaterial plate cloak. Natl Publ Gr 2016:1–17. https://doi.org/10.1038/srep20731. [34] Centeno E, Cassagne D, Albert JP. Mirage and superbending effect in two-dimensional graded photonic crystals. Phys Rev B - Condens Matter Mater Phys 2006;73:1–5. https://doi.org/10.1103/PhysRevB.73.235119. [35] Rose JL. Ultrasonic guided waves in solid media. Cambridge university press; 2014. [36] Ke M, Liu Z, Qiu C, Wang W, Shi J, Wen W, et al. Negative-refraction imaging with two-dimensional phononic crystals. Phys Rev B - Condens Matter Mater Phys 2005;72:1–5. https://doi.org/10.1103/PhysRevB.72.064306. [37] COMSOL multiphysics reference manual (version 5.3). Sweden: COMSOL AB Stockholm; 2015. [38] ABAQUS/Standard User's Manual (version 6.14). Providence, RI, USA: Dassault Systèmes; 2014. [39] Amireddy KK, Balasubramaniam K, Rajagopal P. Deep subwavelength ultrasonic imaging using optimized holey structured metamaterials. Sci Rep 2017;7:7777. https://doi.org/10.1038/s41598-017-08036-4. [40] Rajagopal P, Lowe MJS. Scattering of the fundamental shear horizontal guided wave by a part-thickness crack in an isotropic plate. J Acoust Soc Am 2008;124:2895–904. https://doi.org/10.1121/1.2982410. [41] Andersson A. Modified Schwarz-Christoffel mappings using approximate curve factors. J Comput Appl Math 2009;233:1117–27. https://doi.org/10.1016/j.cam. 2009.09.006.
[22] Zhu H, Semperlotti F. A passively tunable acoustic metamaterial lens for selective ultrasonic excitation. J Appl Phys 2014;116. https://doi.org/10.1063/1.4894279. [23] Pennec Y, Djafari-Rouhani B, Vasseur JO, Larabi H, Khelif A, Choujaa A, et al. Acoustic channel drop tunneling in a phononic crystal. Appl Phys Lett 2005;87:1–3. https://doi.org/10.1063/1.2158019. [24] Lee H, Oh JH, Seung HM, Cho SH, Kim YY. Extreme stiffness hyperbolic elastic metamaterial for total transmission subwavelength imaging. Sci Rep 2016;6:1–12. https://doi.org/10.1038/srep24026. [25] Zhu H, Semperlotti F. Anomalous refraction of acoustic guided waves in solids with geometrically tapered metasurfaces. Phys Rev Lett 2016;117:1–6. https://doi.org/ 10.1103/PhysRevLett.117.034302. [26] Su X, Lu Z, Norris AN. Elastic metasurfaces for splitting SV- and P-waves in elastic solids. J Appl Phys 2018;123. https://doi.org/10.1063/1.5007731. [27] Ma G, Sheng P. Acoustic metamaterials: from local resonances to broad horizons. Sci Adv 2016;2. https://doi.org/10.1126/sciadv.1501595. e1501595–e1501595. [28] Hussein MI, Leamy MJ, Ruzzene M. Dynamics of phononic materials and structures: historical origins, recent progress, and future outlook. Appl Mech Rev 2014;66. https://doi.org/10.1115/1.4026911. 040802. [29] Zhu Q, Jin L, Fu Y. Graded index photonic crystals: a review. Ann Phys 2015;527:205–18. https://doi.org/10.1002/andp.201400195. [30] Lin SCS, Huang TJ, Sun JH, Wu TT. Gradient-index phononic crystals. Phys Rev B Condens Matter Mater Phys 2009;79:1–6. https://doi.org/10.1103/PhysRevB.79. 094302. [31] Norris AN, Shuvalov AL. Elastic cloaking theory. WAMOT 2011;48:525–38. https://
310