Design and characterization of a three-dimensional anisotropic additively manufactured pentamode material^a)

The concept of pentamode material (PM) was proposed by Milton and Cherkaev in 1995¹ as one of a family of materials with names based upon the number of vanishing Kelvin moduli.² For PM materials, five of the six moduli reduce to zero, a situation not unfamiliar in acoustics, since fluids with low viscosity, like water, can be considered as the limit of a solid with zero shear modulus, resulting in five “easy” modes of deformation associated with shear. Unlike water, PMs can display anisotropy because the single “stiff” mode of deformation is not necessarily associated with a simple hydrostatic stress state, but can be biaxial or triaxial. The unique aspect of PM material behavior is that they only support a single wave type, analogous to acoustic wave motion in fluids, with the additional property of acoustic anisotropy.

While the idea of PM is a static one, it is impractical to actually make a material with only one non-zero modulus which is structurally stable. Fortunately, in the development of the two-dimensional (2D) lenses,^3,4 it became apparent that the most interesting and useful PM effects are dynamic rather than static. The 2D structures studied in those works possess nonzero shear rigidity in the static limit, but display broadband one-wave pass bands as first noted by Martin et al.⁵ The quasi-longitudinal wave in the one-wave passband has minimal dispersion, indicating the PMs act as acoustic metafluids. The one wave passband (or shear wave bandgap) is, in practice, difficult to predict for a given unit cell design and requires full computational modeling, see for instance Refs. 6 and 7 for 2D studies. The term PM in this paper is used in the context of its dynamic one-wave property, rather than the purely static idea of achieving a vanishing ratio of shear modulus to bulk modulus,⁸ and is preferred to the ambiguous “metafluid.” Furthermore, our unit cell designs start from the same type of PM microstructure originally proposed by Milton and Cherkaev in 1995¹ but with significant modifications to provide the desired structural stability, mass density, and range of elastic properties suitable for underwater acoustic applications.

Interest in PMs increased when it was shown that they provide the material properties necessary to realize transformation acoustics,⁹ which is the basis for acoustic cloaking and other exotic wave manipulation concepts.¹⁰ Underwater acoustic cloaks using PMs have since been demonstrated.^11–14 Pentamode applications in acoustics extend beyond cloaking and transformation acoustics. The ability to emulate the acoustic properties of water-like isotropic fluids in terms of acoustic impedance and sound speed¹⁵ allows for the design of underwater acoustic lenses, whether based upon classical focusing algorithms³ or using negative index effects.⁴ Other applications of PMs in acoustics include pentamode metasurfaces for directional control of sound,^16,17 devices for bending acoustic waves,¹⁸ and seismic isolation.¹⁹ 

PM acoustic devices to date have relied upon 2D microstructure and manufacturing technology, such as computer controlled machining/cutting.^3,4,15 The 2D honeycomb geometry is easy, in principle, to machine using aluminum, yielding local isotropy for symmetric hexagonal unit cells, or acoustic anisotropy by breaking the symmetry.²⁰ 2D honeycomb PMs have also been demonstrated using additive manufacturing (AM) and static measurements.²¹ Quasi-three-dimensional (3D) hexagonal cone elements, suitable for spherical geometry,²² have been proposed and there have been preliminary realizations. Previous work conducted by Rohde et al.²³ demonstrated a successfully additively manufactured, stainless steel PM and was used to determine limitations in printer capabilities through varying geometric unit cell parameters. However, the material was not mechanically or acoustically characterized and the authors are unaware of any realizations and corresponding acoustic demonstrations of 3D PMs for underwater environments. The main reason for the lack of attention is the difficulty of fabrication: standard machining cannot produce the diamond-like unit cells necessary to achieve the one wave properties and the complexity and fine structure of 3D PM structures paired with a need for large samples for acoustic testing pushing the limits of current AM capabilities.

Our motivation is in underwater acoustic applications, for which metallic PMs provide the ability to yield acoustic metamaterial effects. The lensing effects demonstrated in the work of Su et al.³ and 2D cloaking effects shown in the work of Quadrelli et al.²⁴ rely upon 2D PMs (also known as bimodal materials¹) that can be simulated and fabricated without much difficulty. Long term interests are in 3D designs which will allow much greater capability in terms of acoustical devices, such as passive gradient index lenses, acoustical cloaks, and related devices.

The first fabricated PMs^25,26 did, in fact, employ 3D AM, specifically state-of-the-art optical lithography using polymer feed. This process was successful in achieving very large bulk modulus relative to shear modulus, the B/G ratio, and in fabricating PM samples with tunable mass density and bulk modulus²⁷ as well as elastic anisotropy.²⁸ However, the PM properties are far from the range of interest here; specifically, densities and bulk moduli are orders of magnitude too low to be of practical use in underwater acoustics. Likewise, more readily accessible polymer-based materials would be inexpensive to manufacture, but can only rarely provide the effective material properties necessary for most underwater applications using a pentamode structure.³ Notable recent work by Allam et al.²⁹ and Konarski et al.³⁰ has demonstrated that polymeric materials can indeed be used for underwater acoustic metamaterials for specific applications. Effective density close to that of water requires using a material denser than water, ideally much denser and stiffer so that the structural volume fraction remains low, as in the polymer PMs.^25,27 Hence the idea of “metal water.”¹⁵ With the advent of metal AM techniques it is possible to realize 3D metallic PMs and has been demonstrated in previous work.^31–34 However, these are intended to achieve large B/G ratios, which they do well, but they are consequently light and of low stiffness. For instance, 3D printed metal PMs of Hedayati et al.³² displayed elastic stiffness as large as 7.6 MPa, but still orders of magnitude below our goal in the range of 2.25 GPa, the bulk modulus of water. To achieve such a value while at the same time having effective density of water requires abandoning the static PM concept, but instead using the one-wave bandpass design.

The purpose of this paper is therefore to demonstrate that metal AM can not only provide the means to realize PMs in 3D with acoustic properties similar to water but also show that the fabrication technique can create PM structures that demonstrate significant acoustic anisotropy. One challenge in using AM is to fabricate a sample large enough for acoustic in-water measurements. This requires having enough unit cells in all three directions to yield a sample displaying the predicted effective properties and that is sufficiently large compared to an acoustic wavelength in water to minimize the effects of diffraction around the sample, thereby enabling characterization using acoustical methods. Recent availability of metal AM capabilities has allowed us to take concept and design to fabrication and finally to acoustic measurement.

We present an experimental characterization of an anisotropic pentamode lattice that was additively manufactured using a 3D Systems ProX DMP 320 Metal printer. The process and printer were selected after several initial samples were fabricated using various commercial methodologies. These techniques ranged from metal casting to post-processing of selective laser melting (SLM)-printed samples with brass infusion to improve their overall porosity. Theses attempts were limited in their success due to several drawbacks from their functionality. Specifically, they suffered from a lack of choice in materials, scalability, repeatability, and all displayed unacceptably high amounts of non-uniformity of material properties. The build size and material quality varied based on the choice of manufacturer and the fabrication technique. Metal casting and other SLM printers provided high quality finished parts, but the overall size was too small. Some SLM printers offered large build size but the quality was poor, such as obvious porosity. A brass infusion technique was used to try and correct for the unwanted porosity but was difficult to achieve uniformity throughout the samples. These same printers had samples fail during fabrication because of unavoidable issues with print overhang, or the parts of a print that extend past the previously printed layer without direct support. The samples used in the measurements reported in Secs. IV and V were fabricated using the reported method because it offered the largest build size with consistent material quality. With that said, a few issues presented themselves with the 3D Systems printer where defects such as cracking and missing unit cell faces were not uncommon in the samples made prior to the largest version depicted in Fig. 1(a). Complications like these are most likely associated with large thermal stresses present in the sample during printing and EDM post-processing.

The paper proceeds as follows. The design and the AM process behind the PM are described in Sec. II. This is followed by an account of the simulation and analysis to characterize 3D PM materials in Sec. III. Section IV dives into the general experimental methodology used for the measurement of time of arrival differentials with and without the PM sample. All testing results are outlined in Sec. V. Concluding remarks and future work can be found in Sec. VI.

In this section, we describe the design and fabrication processes for the PM samples. The final design and the sample made using metal AM are shown in Fig. 1.

The initial PM design goal was a structure that emulates the acoustic properties of water. Slight modifications of the initial design were then used to introduce material variation, specifically acoustic anisotropy. The most straightforward parameter to establish is the total volume fraction of metal in the metal-air assembly, which can be determined from the water/metal density ratio. Titanium with a mass density of ρ = 4430 kg/m³ was ultimately used as the metal component, implying a volume fraction of $1000/4430=0.2257$⁠. The next step in the PM design is the creation of a microstructure that is isotropic with an effective bulk modulus that matches the bulk modulus of water. Recall that the adiabatic bulk modulus in a fluid, K, is related to the sound speed, c, and density, ρ, through the relation $K=ρc2$⁠. We therefore seek a material with a modulus of K = 2.25 GPa, assuming the sound speed and density in water are 1500 m/s and 1000 kg/m³, respectively. Isotropy of elastic properties requires a geometry that ensures the fourth order tensor displays this symmetry. This can be achieved to a high precision with a diamond-like unit cell, such as the one shown in Fig. 2.

Previous demonstrations of PM materials have shown that advanced fabrication technology can create macrostructures with a large ratio of bulk modulus to shear modulus.^25,32 This is not our objective. Rather, we seek effective properties displaying the one-wave effect over as large a range of frequency as possible while retaining elastic stability in the static limit.

The only practical way to fabricate isotropic and anisotropic samples for underwater testing is to employ 3D metal AM technology. This approach presents new opportunities to go beyond traditional fabrication techniques such as 2D machining. It also presents new limitations, most significantly in the authors' experience, on (i) feature size, (ii) build size, and (iii) material quality. The fabrication feature size determines the minimum structural dimension, the narrow neck of the model in Fig. 2. Feature size is the most important design parameter because it determines the unit cell size, and from that, the frequency range in which one may observe anisotropic one-wave behavior. It was also the single most important quantity in scaling the computer-aided design (CAD) model for porting to the chosen 3D printer as described in Sec. II B.

A representative PM sample modeled in SolidWorks is shown in Fig. 1(a). The printer noted in Sec. I, employs a specialized titanium powder, Laserform TI Gr23 (A). The powder was selectively melted and fused in layers by a high power-density laser until the part was built, a process referred to as SLM.³⁵ The print direction was along the x₃ axis corresponding to the coordinate representation shown in Fig. 1(a). After fabrication, the build plate was removed from the bottom face by electrical discharge machining (EDM).³⁶ The post-processed sample was 98.9 × 98.7 × 84.6 mm³ with the first two dimensions representing the cross-sectional area of the face normal to the fast wave propagation. Notable dimensions and tolerances of the printed sample include sealing wall thicknesses of 1.33 ± 0.013 mm and the circular faces of the unit cell (Fig. 2) with diameters of 5.44 ± 0.026 mm. The sample size variations were not expected to significantly impact observation of predicted behavior because the tolerances represent an uncertainty of ±1% of the respective average dimension. No other deficiencies were observed by visual inspection. Representative images of the final, as-built sample are shown in Figs. 1(b) and 1(c). In preparation for underwater testing, 1.3 mm thick titanium plates were secured to the open top and bottom of the unit cell array using J-B Weld epoxy to seal the internal structure from water penetration. The final sealed sample has dimensions of 98.9 × 98.7 × 87.2 mm³.

Simulations were performed to design the anisotropic acoustic properties of the PM structure that would later be fabricated and experimentally characterized, as described in Sec. IV). All simulations were done with the finite element method (FEM) software COMSOL Multiphysics, utilizing the acoustics and structural mechanics modules. Initial simulations were done in order to study the band structure of propagating bulk modes. Floquet periodicity boundary conditions are applied to all faces of the tetrahedral unit cell that an adjacent unit cell would be in contact with, as depicted in Fig. 2.

The band structure is determined by performing a parametric sweep of the wavenumber along the x₁- and x₃-components of the Floquet wavenumber vector, $k→$⁠, and solving the resulting eigenvalue problem for the frequencies of modes propagating with that wavenumber. The results yield solutions in wavenumber-frequency pairs, and thus the phase speed of each propagating mode along the axes of the PM structure. The results of the model are then used to identify propagating modes whose phase speeds are either less than or greater than the sound speed in water. Once identified, we classify directions with bulk longitudinal modes less than water as the “slow” direction (x₁- and x₂-directions) and those that are higher than that of water the “fast” direction (x₃-direction). The band diagrams along the fast- and slow-directions for the PM structure designed, built, and tested in this work are provided in Fig. 3. Some key features of Figs. 3(a) and 3(b) are that both possess a range of frequencies at which only a quasi-nondispersive longitudinal mode is present (passband) and a region where no energy propagation is supported (stop band). Further investigation of the band diagrams show qualitatively different sound speeds of the longitudinal modes for each orientation, as indicated by their individual slopes. The phase speeds averaged over the passband for longitudinal-only wave propagation are approximately 2847 and 888 m/s for the fast and slow orientations, respectively.

The reflection and transmission effects of the PM when submerged in water were simulated in anticipation of the measurements. A semi-infinite medium approach was taken in order to help mitigate the issue of computational demands of large degrees-of-freedom (DOFs) problems. Specifically, a rectangular slab that is cut out of the original CAD file that was used for 3D printing, was then imported into COMSOL via the LiveLink feature and given the material parameters associated with Ti Gr23 (A) (ρ = 4430 kg/m³, E = 114 GPa, $ν=0.342$⁠). Four additional domains are extruded from the cross-sections of the PM slab to act as the exterior water domain (⁠$ρw=1000 kg/m3, cw=1500 m/s$⁠). The four domains are two water domains (one on each side of the PM) for propagation of acoustic pressure fields, and two perfectly-matched layer (PMLs) terminating the water domains on each side of the PM structure to minimize reflections from the truncated water domains. The mesh of the PM slab was kept constant for all frequencies, but chosen in such a way that it would be sufficiently smaller than the smallest wavelength of interest. Meshes assigned to the water domains varied with frequency to support several wavelengths per mesh element. The boundaries orthogonal to the incident wave field are assigned periodic (Floquet) boundary conditions in pairs, an example is highlighted in Fig. 4. Note that only normal incidence has been considered.

Results of the reflection-transmission analysis are shown for the fast orientation in the frequency domain for 10–110 kHz. Inspection of Fig. 5(a), which depicts the normalized total displacement field within the PM slab at 20 kHz, suggests that motion inside the PM is dominated by longitudinal motion in the propagation direction. In fact, Fig. 5(b) demonstrates that the in-plane motion (x₃-component) is at least two orders of magnitude greater than the out-of-plane motion (x₁ and x₂-component) at this frequency. This concludes that a dominating longitudinal wave is propagating through the slab, agreeing with the band diagrams of Fig. 3. This holds true for all longitudinal passband frequencies. Note that there is clear flexural motion in the plates separating the water from PM domains seen in Fig. 5(b); however, this motion remains local to the surface of the plate and is not present in the far-field of the water domains. Therefore, it is easy to compute the reflection and transmission coefficients sufficiently far away from the plate surface as only the acoustic pressure needs to be taken at a single point due to a lack of cross-sectional variation in the pressure field. Results are seen in Fig. 6, where reflection $|R|$ and transmission $|T|$ coefficients are evaluated as the magnitudes of the scattered pressure on their respective side of the PM slab. Figure 6(a) shows the results from 10 to 80 kHz where peaks and dips in $|T|$ occur in an oscillating pattern with respect to frequency. Since these simulations are steady-state, standing wave patterns will exist inside the PM, which lend themselves to patterns of constructive and destructive interference associated with Fabry-Perot thickness resonances. Ripple effects present in Fig. 6 are artifacts of the frequency step which was chosen to be 500 Hz to remain computationally efficient. As the frequency of the normal background field is increased to 90 kHz and above, a non-recovering decline in transmission $|T|$ is present suggesting that we have entered a stop band.

In an effort to extract the wave speed of the longitudinal mode inside the PM, we consider purely the in-plane displacement w. By averaging w for every unique x₃-coordinate (direction of propagation), we reduce the system down to a single dimension as seen in Fig. 7. Close inspection shows the existence of a partial waveform which can be further expanded via a half-range cosine expansion,

With this expansion, and therefore the coefficients which correspond to integer multiples of wavelength, one can measure a modal density to evaluate the integer multiple fit of the partial waveform of v. For example, at 22.5 kHz the half-wavelength density $a1/(∑n=1Nan)$⁠, where N is a positive integer, is 0.929 for N = 10. A half-wavelength density of approximately 93% indicates that a pure half-wavelength exists in the PM at or around 22.5 kHz. With that said, as long as the length of the PM slab is known, it is possible to calculate the wave speed with the simple relation $cPM=λf$⁠. Using f = 22.5 kHz and $λ=2LPM$⁠, where $LPM=58.97$ mm is the length of the slab, the longitudinal wave speed is evaluated as $cPM=2654$ m/s. Note that the simulated band diagrams of Fig. 3(a) suggest that the longitudinal mode to be slightly dispersive; however, at frequencies around that which supports a half-wavelength in the PM, dispersion is negligible.

Similar results were seen with the semi-infinite approach of a PM slab oriented in the slow direction. One shared characteristic is the dominant in-plane displacement allowing for wave speed evaluation through Fourier series expansion like that of Eq. (1) (see Fig. 7). In the simulated slow direction, a full wavelength was seen at 24 kHz with a full-wavelength density of $a2/(∑n=1Nan)=0.893$ for N = 10. Using f = 24 kHz and $λ=LPM=43.04$ mm, the longitudinal wave speed is evaluated as $cPM=1033$ m/s for the slow orientation. It is important to note that dispersion effects are more significant in the slow direction. This lends itself to increased uncertainty with respect to the average longitudinal passband value stated in Sec. III A. However, we still observe good agreement between this full-wave simulation of a slab of PM material and phase speed estimates extracted from band diagrams for bulk modes as presented in Fig. 3(b).

The underwater experiments were designed to experimentally confirm the acoustic wave propagation behavior of the as-built PM samples, particularly the designed anisotropy. The large size of the 3D additively manufactured sample enabled acoustic measurements, specifically through-transmission measurements along principle axes in order to quantify anisotropic wave speeds of the sample. Underwater characterization requires that a PM elastic lattice be completely sealed along the entire control volume boundary. The sealed PM was placed in front of a source capable of emitting plane waves over a broad range of frequencies and the changes in time of wavefront arrivals were observed with and without the PM sample present. For a finite sample, the field measured when the sample is present will consist of acoustic waves transmitted through the sample as well as those diffracted around it. Ideally, a large enough sample could allow for time-of-flight measurements to be made that unambiguously measure the transmitted field without contamination from diffraction, in order to clearly characterize the anisotropic sound speeds in the PM. However, physically realizing a PM metamaterial sample of this size was cost-prohibitive and a smaller-than ideal sample was made using AM, which led to some complications interpreting the measurements along the slow direction, as discussed in Sec. II B.

Acoustic pressure measurements in the water outside of the sample were used to quantify the different orthogonal phase speeds in the PM. An experimental apparatus, similar to that used by Su et al.³ was built for use in an indoor water tank. The water tank has an average depth of roughly 4.5 m and large enough lateral dimensions to avoid contamination of results from multipath arrivals associated with reflections from tank walls. The walls of the tank are made of concrete and the bottom is covered in sand. The anisotropic PM sample was suspended in front of a plane wave source using an aluminum support structure mounted to a vertical support column. It should be noted that the front face of the source is 29.5 × 22.9 cm², which is a considerably larger cross-sectional area than any face of the PM. The centerline of the PM sample was approximately aligned with the center of the source and held at constant depth of approximately 70 cm.

Three different experimental arrangements were used: (i) a source-only measurement with no PM material present, (ii) a measurement with the PM oriented such that the source excitation propagates through the PM along the fast direction, and (iii) a measurement where the PM is oriented such that wave propagates through it along the slow direction. The measurement reference frame is defined in Fig. 8 where an “axial scan” is a hydrophone scan in the direction of the wave propagation and an “aperture scan” is a hydrophone scan parallel to the face of the PM.

An Arduino Uno was used to control stepper motors to move a RESON TC4013 hydrophone to desired measurement locations. The stepper motors and belt-driven hydrophone suspension platform are supported by an aluminum frame that was suspended over the water surface. The hydrophone was attached to a 75-cm-long aluminum rod that centered the sensor in front of the source and PM. matlab was used to send positional commands to the Arduino using G-code while simultaneously importing hydrophone data from a Tektronix TDS3014B oscilloscope. A 10-ms-long exponential chirp from 0.1 to 100 kHz was created in matlab and used to excite the acoustic source via a Tektronix AFG 3021B function generator and Krohn-Hite 7500 amplifier. A 100 ms delay time was incorporated between successive chirps to allow system transients to decay below the noise floor. The sampling frequency was 500 kHz resulting in a time step of 0.002 ms. The received signal was averaged 32 times in the time domain at each measurement position and then exported from the oscilloscope to be stored on a laptop for post-processing. The hydrophone was then moved to the next location followed by a 2 s pause before the next measurement was recorded to allow for the noise and vibration created by the motion control system to abate. The process was repeated using a 5 mm step size to complete the desired axial, aperture, and 2D planar scans.

A long-duration exponential frequency-modulated chirp excitation signal was selected to drive the acoustic source in order to increase the time-bandwidth product of the signal, thereby improving the signal-to-noise ratio of the measurements. However, these chirped signals must be compressed in time in order to observe accurate arrivals of wavefronts transmitted through either water-only or water-PM-water paths. This is achieved using the well-known deconvolution method.^37,38 In this method, the exponential chirp used in the experiment can be represented as an idealized impulse that has been spread out over time using a specialized pre-processing filter, $hpre(t)$⁠. Mathematically, we can write $x(t)=hpre(t)*δ(t)$⁠, where x(t) is the chirp signal, $δ(t)$ is the Dirac delta function, and $*$ represents the convolution operation. Likewise, we can define a post-processing filter, $hpost(t)$⁠, that we can apply to the signal obtained at any measurement point, $ym(t)$⁠, that compresses the output from the chirp excitation to the impulse response of the system being measured, i.e., $hIR,m(t)=hpost(t)*ym(t)$⁠. The impulse response of the system will include time delays associated with the propagation through the PM sample and thus can be used to recover the speed of sound in the sample. The signal chain described above can be written as $hIR,m(t)=hpost(t)*hIR,m(t)*hpre(t)*δ(t)$⁠. Recalling that $x(t)=hpre(t)*δ(t)$⁠, we can therefore define the post-processing filter as $hpost(t)≡F−1{1/F{x(t)}}$⁠, where $F{·}$ represents the Fourier transform operation. This filter can be applied to the signals measured in the tank when using chirp excitation in order to recover the impulse response of the system being measured. Because of the form of this expression, the post-processing filter is often called the “inverse” chirp. It should be noted that the resulting impulse response is band limited by the frequency content of the initial filter.

The system for which we are measuring the impulse responses consists primarily of the hydrophone, the source, and the PM sample. To remove the frequency content of the hydrophone and source responses, a finite impulse response (FIR) filter was created from the band limited impulse response of the closest measurement point to the source on the centerline of the source-only measurement. The filter was created by using $hFIR=F−1{F{(t)}/F{ym,w(t)}}$⁠, where $hwin(t)$ is a Hann window and $ym,w(t)$ is the response measured without the PM sample present. Finally, the FIR filter was convolved with the hydrophone data at each position to produce a band limited impulse response that is altered by an PM sample alone, $hIR,m,F=hIR,m*hFIR$⁠. These final FIR filtered IRs are the post-processed data sets that are used for determining the sound speeds of the PM. An example showing the results of three intermediate steps of the signal processing methodology is shown in Fig. 9.

The frequency response was determined by applying a Hann window to the direct arrival of the impulse response to isolate the effect of the PM on the system and to neglect any surface or wall reflections recorded by the hydrophone. The fast Fourier transform (FFT) was taken off the windowed responses for each measurement location in the pressure field. A ratio of the averaged spectras of direct arrivals immediately in front of the PM to the averaged source-only spectra was calculated to isolate the effect of the PM. Although the time domain analysis provided a direct comparison of the predicted sound speeds through the material, the frequency domain analysis provides a means to observe predicted stop- and pass-bands demonstrated through the simulations in Sec. III.

The post-processed time domain results were plotted for the source-only, fast direction, and slow direction orientations to determine the times of wave front arrivals and, consequently, the apparent sound speeds observed due to the presence of the PM sample. The method of plotting for the individual aperture and axial line scans was chosen to display all impulse responses (IRs) for a given scan as a waterfall plot. The IRs for a scan were each normalized to a universal maximum of the entire line scan then scaled and spaced for legibility. As an example of the presentation of the time domain results, the IRs on an aperture line scan of the pressure field were plotted for the source-only configuration in Fig. 10.

The IRs along the axial centerline of the source were measured and used to determine the sound speed of water at the tank temperature by finding the slope of the peaks in the direct arrivals as a function of space time. For this result, the amplitudes of the IRs were not important, but the spacing between each IR must be equivalent to the actual spacing of the hydrophone measurement in the tank. The slope of the IR peaks in Fig. 11 was found to be 1495.6 m/s, which is in agreement with tabulated values for fresh water. This experimentally determined sound speed of water was subsequently used in determining the arrivals of wavefronts in the PM-inclusive experiments. A spatial depiction of the propagating source wave at various instances in time is shown in Fig. 12. The source-only case provided confirmation that the source does produce a plane wave and also proves the functionality of the experimental apparatus and the signal processing approach.

The differences in time of arrivals of the wavefronts with and without the PM present were determined and the sound speed of the material was calculated using the standard expression,

where d₂ is the thickness of the PM in the propagation direction, and $cw$ is the sound speed of water. The difference in time-of-flight between IRs traveling through water-PM-water path and water-only path is given by $Δt=T¯PM−T¯source$ where $T¯PM$ and $T¯source$ and shown in Fig. 13. Equation (2) shows that if the PM wavefront arrives before the water-only path arrival, $Δt$ is negative, which increases the calculated sound speed of the PM. To confirm the functioning system and calculation, a test was executed for a solid aluminum block with approximately the same dimensions as the PM sample. The longitudinal sound speed of aluminum was experimentally determined to be within the accuracy capability of the experiment.

The single aperture scan results for the PM in the fast orientation are shown in Fig. 13 at a location that crossed directly behind the PM. A clear wavefront arrival is visible that is earlier than the arrival times from the source. The groupings of wavefront arrival times are shown in Fig. 13 and were found to be $T¯PM=11.691$ ms and $T¯source=11.714$ ms. Using Eq. (2), the fast PM metamaterial sound speed was found to be 2472 m/s. Other measurements were performed at multiple propagation distances in order to provide statistical information for the experimental data. These results can be summarized as an average fast sound speed of 2499 ± 30 m/s.

Figure 14 is the qualitative result of the wavefronts developing as they propagate through space at select instances in time. The fast wavefront maintains a consistent distance away from the source wave while a diffracted field can be seen converging with increasing distance from the sample.

The frequency response of the PM was determined using the process described in Sec. IV B. The ratio of the fast PM spectra to the source spectra is shown in Fig. 15. One property that is observed in Fig. 15 is the 3–4 dB decrease in amplitude of the PM response when compared to the source-only measurements. This is likely due to several factors: (i) the solid titanium sealing plates that were epoxied to the lattice causing an increased impedance mismatch between water and the PM structure, (ii) inconsistent material properties along the print direction due to the AM process, and (iii) imperfect impedance matching of the unit cells to water. However, spectra features show good agreement with expected broadband behavior. Frequencies indicated by vertical lines labeled (1), (2), and (3) in Fig. 15 are 46.2, 62.1, and 83.5 kHz, respectively. These results compare to 39.9, 62.5, and 86.0 kHz of the unit cell in (i), (ii), and (iii) in Fig. 3(a).

The slow direction results are shown in a waterfall plot in Fig. 16. Notably, the amplitude directly behind the PM sample has decreased by an order of magnitude compared to the source response. The slow direction band diagram in Fig. 3(b) shows that the passband for longitudinal waves is significantly more limited in frequency range than the fast direction in Fig. 3(a). The more limited longitudinal wave passband is likely a major contributor to the energy loss, but unknowns of the varying material properties during the AM process as well as the titanium walls could also be factors in the poor transmission of acoustic energy through the PM sample.

Closer inspection of Fig. 16 suggests that the pressure field is dominated by diffraction around the sample. The development of this diffracted field can be seen in the spatial propagation plots in Fig. 17. Also, of note, the left-most plot shows that there are point-like sources present in the pressure field. These wave fronts are likely line sources due to the radiation associated with extensional wave motion, namely, the s₀ Lamb mode in the plates enclosing the PM structure and excited by the incident pressure field. Extracting the arrival times of the plate radiation fronts and the source wave fronts and plugging the times into Eq. (2), an apparent sealing plate sound speed of 4942 m/s was found. This compares to the analytical sound speed of 5398 m/s using properties of titanium found in Sec. II A and Eq. (8.195) from Graff.³⁹ Although not specifically observed in the plots, it stands to reason that plates on the top and bottom of the sample are also radiating into the measured field.

There are 3–4 measurement locations near the center of the aperture where peaks of the impulse responses appear outside of the main diffraction pattern. The data suggests that this portion of the measurement occurs within the shadow region and could be the wavefront from the PM in the slow direction. Noting the different features in the pressure field and extracting the times of assumed slow wave front in the shadow region (⁠$TPM=11.752$ ms, $Tsource=11.722$ ms), the slow sound speed of the PM was estimated to be 1028 m/s. There were similar results from the other slow direction both in qualitative comparisons in the pressure fields, as well as a quantitative sound speed result from the assumed shadow region of 1050 m/s. Overall, the slow sound speed has very good agreement with the sound speeds of 1033 m/s predicted in Sec. III C, but future work should be dedicated to increasing the confidence of this result.

The fast direction showed a distinct wavefront with a measured longitudinal wave speed having good agreement with its simulated counterpart, Figs. 14 and 15. The slow direction, on the other hand, saw heavy influence from diffraction around the sample and poor transmitted amplitudes, as seen in Fig. 16. The finite element simulation effort eluded to a significant stop band within the frequency range of excitation. However, it also suggested that the passband transmission should not have been as diminished as it was observed in the experiment. With that said, we turn our attention to the sample itself. While the post-processed sample appears defect-free on the surface, it is possible that there exists internal defects that might lend themselves to poor energy transmission. The nature of the layer-by-layer process of SLM makes it so that there is an inherent unknown degree of anisotropy when comparing the build direction (fast) to its orthogonal directions (slow). The size of the sample was limiting in the fact that, although it was the largest underwater 3D PM sample produced to date, it was not sufficiently large in terms of acoustic wavelengths to minimize the effects of diffraction on the measurement of the slow sound speed. This fact, in tandem with poor transmission, prevented definitive conclusions from being made for the determination of the speed of sound in the slow orientation.

Metal AM has progressed to a point where fabrication of 3D PM materials demonstrating a high degree of anisotropy is now a reality. With that said, it is clear that there are still advancements that need to be made in order to mitigate the issues of scalability, print-specific anisotropy, and variability in as-built material properties. These issues are not as overt as cracks, deformities, or damaged structures, but are still important to assess as AM technology improves. Numerical analyses conducted in this work were useful in determining the qualitative extent of the construction variance and the overall quality of the printed sample by comparing the ideal elastic lattice behavior to that observed in the experiment. A semi-infinite medium approach in the frequency domain allowed for accurate prediction of PM behavior without requiring full 3D time domain simulations of the entire PM structure that was evaluated experimentally. The experiment was successful in demonstrating the anisotropic behavior of the as-designed PM structure. The direction-dependent sound speed was shown by measuring the arrival times of wavefronts propagating through the PM sample along the principle axes of the PM structure. Overall, the numerical and experimental results of this work show that 3D PM materials can be created using AM, which is of significant importance for the realization of many of the exotic concepts predicted by transformation elastodynamics.⁹ 

Continued improvements in both time-efficiency and high build cost are also needed in order for these processes to be more widely accessible.

This work was supported by ONR through MURI Grant No. N00014-13-1-0631. Thanks to Professor Haseung Chung and the AM group at MSU for fabricating the test samples. C.W.C. notes useful discussions with Kyle Spratt regarding signal processing theory. C.W.C. acknowledges partial support by the McKinney Fellowship in Acoustics from the Applied Research Labs at the University of Texas at Austin. M.R.H. and C.W.C. partially supported by ONR YIP Award No. N00014-18-1-2335.