# Multidimensional Phononic Bandgaps in Three-Dimensional Lattices for Additive Manufacturing

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{xyz}) for vibration isolation purposes, but did not model the dispersion curves (DCs) of the lattice, and did not report on the effect of the lattice volume fraction on achieving vibration isolation. Lu et al. [13] and Hsieh et al. [45] independently predicted the DCs of multimaterial BCC lattice designs, but to date there have been no reports on the manufacturability or performance of these designs. In comparison to single-material AM, which is well-established, multimaterial AM currently requires manual assembly (such as in the lattice work of Matlack et al. [15]), requires support structures that constrain the design of the part, necessitates post-processing (such as in the work of Choi et al. [46]), and is limited to a small range of materials.

_{xyz}, the network gyroid (gyroid TPMS) and a modified BCC

_{xyz}lattice with internal resonators (res-BCC

_{xyz}). The existence of multidimensional phononic BGs would add vibration isolation to the existing panoply of controllable mechanical performance of the examined lattice structures [34,37,38,43,44]; thus enabling them to simultaneously fulfill various mechanical and vibrational functions. The BGs of the examined lattices were identified from their respective DCs and predicted using a multidimensional finite element (FE) wave propagation modelling method.

## 2. Methods

#### 2.1. Lattice Design

_{xyz}unit cell, shown in Figure 2, was designed using the strut-based lattice design equations presented in our previous work [47]. In designing BCC

_{xyz}lattice structures for this study, we considered a range of volume fractions from 5% to 30%. The corresponding ratios of strut diameter d to cell width L are provided in Table 1. A change in the $d/L$ ratio leads to a change in the volume fraction of the lattice.

_{xyz}unit cell to create the res-BCC

_{xyz}unit cell, as shown in Figure 4. The outer scaffold of the res-BCC

_{xyz}is a 5% volume fraction BCC

_{xyz}cell. Although different scaffolds can be considered using the same concept, the 5% volume fraction BCC

_{xyz}lattice features a central void of sufficient size to host spherical masses with a wide range of sizes. The design information for the res-BCC

_{xyz}unit cells at different volume fractions is presented in Table 3.

#### 2.2. Bandgap Prediction

_{xyz}and res-BCC

_{xyz}unit cells were meshed in ANSYS simulation software using tetrahedral elements, and gyroid TPMS unit cells were meshed using hexagonal elements. Mesh convergence was determined through examination of the structure’s first natural frequency, which in each case was found to be well converged with respect to the mesh density (see Figure 5a). To ensure convergence of high frequency results (particularly above a normalised frequency of 0.3), a high frequency vibration mode of the converged mesh was compared to that of a finer mesh. The results, shown in Figure 5b, showed minimal discrepancies in the vibration mode and frequency.

## 3. Results and Discussion

#### 3.1. Verification of the Dispersion Curve Calculations

#### 3.2. Wave Dispersion in Lattices with Infinite Periodicity

_{xyz}and res-BCC

_{xyz}lattices. The gyroid TPMS lattice did not show a multidimensional BG, although it is known to exhibit a 1D BG [18]. Below a normalised frequency of 0.2, the res-BCC

_{xyz}lattice was the only lattice that showed a BG. Above a normalisd frequency of 0.2, both the BCC

_{xyz}and res-BCC

_{xyz}lattices possess one BG. The first BG of the res-BCC

_{xyz}lattice is wider by 57% and has a BG starting frequency lower by 68.5% than that of the BCC

_{xyz}lattice. The BG frequency width (bandwidth) of the BCC

_{xyz}lattice is eight times that of the second BG of the res-BCC

_{xyz}lattice.

_{xyz}and res-BCC

_{xyz}lattices to those of the BCC-inspired multimaterial lattices of Lu et al. [13] and Husieh et al. [45]. A relative BG width is the quotient of the bandwidth and the BG mean frequency (BGMF) and is independent of the unit cell size; a high relative BG width is more desirable as it indicates a wide BG with low frequency. The BG of Lu et al. had a relative width of ~60%, while that of Husieh et al. was ~40%, both at a volume fraction of ~23%. Interpolation of our BG results for the BCC

_{xyz}and res-BCC

_{xyz}lattices at 23% volume fraction showed a relative BG width of 30% and 98.7%, respectively. This indicates that the res-BCC

_{xyz}lattice has the ability to provide wide BGs of low starting frequencies using single material lattices.

_{xyz}lattice is compared to the 1D BG of the gyroid TPMS, which was studied by Elmadih et al. [18]. At similar volume fraction, the BCC

_{xyz}lattice shows BGs at higher frequencies than the gyroid TPMS lattice. For example, at 20% volume fraction, several BGs are present below a normalised frequency of 0.2 in the gyroid TPMS lattice. However, the bandwidth of this BCC

_{xyz}lattice is almost five times wider than that of the 1D gyroid TPMS BG.

_{xyz}lattices of various materials and unit cell sizes. For example, BGs of Ti-6Al-4V BCC

_{xyz}lattices can be predicted. Ti-6Al-4V is used in the aerospace and the biomedical sectors due to its high corrosion resistance, biocompatibility and high fracture toughness [60]. The phononic properties of Ti-6Al-4V strut-based lattices have been studied previously [30]. For the purpose of comparison with the BCC

_{xyz}and res-BCC

_{xyz}lattices presented here, a unit cell 10 mm in size and 20% volume fraction, based on the design of Warmuth et al. was considered. Using the BG tuning tool in Equation (1) from [30], the BG properties were calculated. The BG starting frequency and BG ending frequency of the BCC

_{xyz}, res-BCC

_{xyz}and the Warmuth et al. lattice of 10 mm unit cell size and 20% volume fraction are presented in Figure 11. The relative BG width of the BCC

_{xyz}and res-BCC

_{xyz}lattices at 20% volume fraction were 30% and 94%, respectively, while the relative BG width of the Warmuth et al. structure is 28.8% at the same volume fraction. The BG of the BCC

_{xyz}lattice has a lower bandwidth than the lattice of Warmuth et al.; the BG of the BCC

_{xyz}lattice spanned 99.2 kHz to 134.2 kHz, which is approximately 52% of the bandwidth of the lattice of Warmuth et al. However, the BCC

_{xyz}lattice had the ability to provide BGs at frequencies lower by 50.2% than those of Warmuth et al. at similar cell size and volume fraction.

#### 3.3. Tuning of Multidimensional BGs

_{xyz}and res-BCC

_{xyz}lattices. The 5% and 10% volume fraction BCC

_{xyz}lattices showed two BGs below a normalised frequency of 0.4. The lowest frequency BG of the 5% volume fraction BCC

_{xyz}lattice spanned a normalised bandwidth of 0.014, from 0.151 to 0.165. This BG is the narrowest in width and is formed by an acoustic waveband (wave cutting-on at zero frequency) and an optical waveband (wave cutting-on at a non-zero frequency). However, 20% and 30% volume fraction BCC

_{xyz}lattices had no second BGs.

_{xyz}lattice of 30% volume fraction showed the highest predicted BG, which spanned from 0.25 to 0.34. The res-BCC

_{xyz}lattice of 30% volume fraction showed the lowest predicted BG, which spanned from 0.067 to 0.187. The res-BCC

_{xyz}lattice has a BGMF lower by an average of 43% than the BGMF of the BCC

_{xyz}lattice, as calculated from the BGMF of lattices with volume fractions of 5% to 30%. The bandwidth increased approximately fivefold, and tenfold upon increasing the volume fraction of the BCC

_{xyz}and res-BCC

_{xyz}lattices, respectively, from 5% to 30% as can be seen in Figure 12.

_{xyz}) [14], while internal resonance BGs (res-BCC

_{xyz}) occur around the natural frequency of the internal resonance mechanism [61]. Thus, the two BG mechanisms can be explained by referring to the natural frequency ${f}_{n}$ equation, ${f}_{n}\propto \sqrt{k/m}$. Above a volume fraction of 5%, additional material uniformly enlarges the struts of the BCC

_{xyz}lattices, which results in stiffer lattices of higher mass. It has been shown that the BG frequency increases with the increase in volume fraction of Bragg-scattering BGs lattices [13,18]; indicating that the stiffness increases at a greater rate than the rate at which the mass increases. The BG mechanism for the res-BCC

_{xyz}lattice is different since the mass of the resonance mechanism is dictated by the mass of the solid sphere, while the stiffness is dictated by the stiffness of the resonance struts. Since above a volume fraction of 5% additional material enlarges the size of the sphere of the res-BCC

_{xyz}lattices, but does not increase the diameter of the struts, an overall reduction of the natural frequency of the resonance mechanism is achieved, thus, reducing the internal resonance BG frequency.

#### 3.4. Evolution of the Wave Transmission in Lattices with Finite Periodicity

_{xyz}lattice at 20% volume fraction was selected for this study. This lattice was found to have a wide BG spanning normalised frequencies from 0.067 to 0.187, as seen in Figure 12. The examined lattice structures, shown in Figure 13, had periodicities of one, three and six (i.e., they contained a single unit cell, 3 × 3 × 3 and 6 × 6 × 6 cells, respectively).

## 4. Conclusions

- Single material BCC
_{xyz}and res-BCC_{xyz}lattices can provide BGs that are tunable with the volume fraction of the lattice. - BCC
_{xyz}and res-BCC_{xyz}lattices have BGs of high width and intrinsically low frequency compared to results reported for similar structures in the literature. - Although gyroid TPMS lattices are known to have 1D BGs, they do not exhibit multidimensional BGs.
- An increase in the finite periodicity of the lattice leads to an increase in the bandwidth and to a decrease in transmissibility within the BG.
- The attenuation of longitudinal waves reaches a minimum of −103 dB within the BG.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Qi, X.-L.; Zhang, S.-C. Topological insulators and superconductors. Rev. Mod. Phys.
**2011**, 83, 1057–1110. [Google Scholar] [CrossRef][Green Version] - Hasan, M.Z.; Kane, C.L. Colloquium: Topological insulators. Rev. Mod. Phys.
**2010**, 82, 3045–3067. [Google Scholar] [CrossRef] - Roldán, R.; Castellanos-Gomez, A. A new bandgap tuning knob. Nat. Photonics
**2017**, 11, 407. [Google Scholar] [CrossRef] - Zoorob, M.E.; Charlton, M.D.B.; Parker, G.J.; Baumberg, J.J.; Netti, M.C. Complete photonic bandgaps in 12-fold symmetric quasicrystals. Nature
**2000**, 404, 740. [Google Scholar] [CrossRef] [PubMed] - Slobozhanyuk, A.; Mousavi, S.H.; Ni, X.; Smirnova, D.; Kivshar, Y.S.; Khanikaev, A.B. Three-dimensional all-dielectric photonic topological insulator. Nat. Photonics
**2016**, 11, 130. [Google Scholar] [CrossRef] - Cheng, X.; Jouvaud, C.; Ni, X.; Mousavi, S.H.; Genack, A.Z.; Khanikaev, A.B. Robust reconfigurable electromagnetic pathways within a photonic topological insulator. Nat. Mater.
**2016**, 15, 542. [Google Scholar] [CrossRef] [PubMed] - Lu, L.; Fu, L.; Joannopoulos, J.D.; Soljačić, M. Weyl points and line nodes in gyroid photonic crystals. Nat. Photonics
**2013**, 7, 294. [Google Scholar] [CrossRef] - Abueidda, D.W.; Jasiuk, I.; Sobh, N.A. Acoustic band gaps and elastic stiffness of PMMA cellular solids based on triply periodic minimal surfaces. Mater. Des.
**2018**, 145, 20–27. [Google Scholar] [CrossRef] - Khelif, A.; Hsiao, F.L.; Choujaa, A.; Benchabane, S.; Laude, V. Octave omnidirectional band gap in a three-dimensional phononic crystal. IEEE Trans. Ultrason. Ferroelectr. Freq. Control
**2010**, 57, 1621–1625. [Google Scholar] [CrossRef] - Zhou, X.Z.; Wang, Y.S.; Zhang, C.Z. Three-Dimensional sonic band gaps tunned by material parameters. Appl. Mech. Mater.
**2010**, 29, 1797–1802. [Google Scholar] [CrossRef] - Phani, A.S.; Woodhouse, J.; Fleck, N.A. Wave propagation in two-dimensional periodic lattices. J. Acoust. Soc. Am.
**2006**, 119, 1995–2005. [Google Scholar] [CrossRef] [PubMed][Green Version] - Trainiti, G.; Rimoli, J.J.; Ruzzene, M. Wave propagation in periodically undulated beams and plates. Int. J. Solids Struct.
**2015**, 75, 260–276. [Google Scholar] [CrossRef] - Lu, Y.; Yang, Y.; Guest, J.K.; Srivastava, A. 3-D phononic crystals with ultra-wide band gaps. Sci. Rep.
**2017**, 7, 43407. [Google Scholar] [CrossRef] [PubMed] - Phani, A.S. Elastodynamics of lattice materials. In Dynamics of Lattice Materials; Phani, A.S., Hussein, M.I., Eds.; John Wiley and Sons: Hoboken, NJ, USA, 2017; pp. 53–59. [Google Scholar]
- Matlack, K.H.; Bauhofer, A.; Krödel, S.; Palermo, A.; Daraio, C. Composite 3D-printed meta-structures for low frequency and broadband vibration absorption. Proc. Natl. Acad. Sci. USA
**2015**, 113, 8386–8390. [Google Scholar] [CrossRef] [PubMed] - Richards, D.; Pines, D.J. Passive reduction of gear mesh vibration using a periodic drive shaft. J. Sound Vib.
**2003**, 2, 317–342. [Google Scholar] [CrossRef] - Ampatzidis, T.; Leach, R.K.; Tuck, C.J.; Chronopoulos, D. Band gap behaviour of optimal composite structures with additive manufacturing inclusions. Compos. Part B Eng.
**2018**, 153, 26–35. [Google Scholar] [CrossRef] - Elmadih, W.; Wahyudin, S.; Maskery, I.; Chornopolous, D.; Leach, R. Mechanical vibration bandgaps in surface-based lattices. Addit. Manuf.
**2019**, 25, 421–429. [Google Scholar] [CrossRef] - Jensen, J.S. Phononic band gaps and vibrations in one- and two-dimensional mass-spring structures. J. Sound Vib.
**2003**, 266, 1053–1078. [Google Scholar] [CrossRef] - Maldovan, M. Phonon wave interference and thermal bandgap materials. Nat. Mater.
**2015**, 14, 667. [Google Scholar] [CrossRef] - Raghavan, L.; Phani, A.S. Local resonance bandgaps in periodic media: Theory and experiment. J. Acoust. Soc. Am.
**2013**, 134, 1950–1959. [Google Scholar] [CrossRef] - Zhou, X.; Jun, W.; Wang, R.; Lin, J. Band gaps in grid structure with periodic local resonator subsystems. Mod. Phys. Lett. B
**2017**, 31, 1750225. [Google Scholar] [CrossRef] - Yu, D.; Liu, Y.; Wang, G.; Zhao, H.; Qiu, J. Flexural vibration band gaps in Timoshenko beams with locally resonant structures. J. Appl. Phys.
**2006**, 100, 124901. [Google Scholar] [CrossRef] - Liu, Y.; Yu, D.; Li, L.; Zhao, H.; Wen, J.; Wen, X. Design guidelines for flexural wave attenuation of slender beams with local resonators. Phys. Lett. A
**2007**, 362, 344–347. [Google Scholar] [CrossRef] - Wang, Y.-F.; Wang, Y.-S. Complete bandgap in three-dimensional holey phononic crystals with resonators. J. Vib. Acoust.
**2013**, 135, 41009. [Google Scholar] [CrossRef] - Lucklum, F.; Vellekoop, M.J. Bandgap engineering of three-dimensional phononic crystals in a simple cubic lattice. Appl. Phys. Lett.
**2018**, 113, 201902. [Google Scholar] [CrossRef] - D’Alessandro, L.; Belloni, E.; Ardito, R.; Corigliano, A.; Braghin, F. Modeling and experimental verification of an ultra-wide bandgap in 3D phononic crystal. Appl. Phys. Lett.
**2016**, 109, 221907. [Google Scholar] [CrossRef][Green Version] - Zhang, H.; Xiao, Y.; Wen, J.; Yu, D.; Wen, X. Flexural wave band gaps in metamaterial beams with membrane-type resonators: Theory and experiment. J. Phys. D Appl. Phys.
**2015**, 48, 435305. [Google Scholar] [CrossRef] - Liu, Z.; Zhang, X.; Mao, Y.; Zhu, Y.Y.; Yang, Z.; Chan, C.T.; Sheng, P. Locally resonant sonic materials. Science
**2000**, 289, 1734–1736. [Google Scholar] [CrossRef] - Warmuth, F.; Wormser, M.; Körner, C. Single phase 3D phononic band gap material. Sci. Rep.
**2017**, 7, 3843. [Google Scholar] [CrossRef] - Wormser, M.; Warmuth, F.; Körner, C. Evolution of full phononic band gaps in periodic cellular structures. Appl. Phys. A
**2017**, 123, 661. [Google Scholar] [CrossRef] - Lucklum, F.; Vellekoop, M.J. Design and fabrication challenges for millimeter-scale three-dimensional phononic crystals. Crystals
**2017**, 7, 348. [Google Scholar] [CrossRef] - Kruisová, A.; Ševčík, M.; Seiner, H.; Sedlák, P.; Román-Manso, B.; Miranzo, P.; Belmonte, M.; Landa, M. Ultrasonic bandgaps in 3D-printed periodic ceramic microlattices. Ultrasonics
**2018**, 82, 91–100. [Google Scholar] [CrossRef] - Yoo, D.-J. Advanced porous scaffold design using multi-void triply periodic minimal surface models with high surface area to volume ratios. Int. J. Precis. Eng. Manuf.
**2014**, 15, 1657–1666. [Google Scholar] [CrossRef] - Maskery, I.; Sturm, L.; Aremu, A.O.; Panesar, A.; Williams, C.B.; Tuck, C.J.; Wildman, R.D.; Ashcroft, I.A. Insights into the mechanical properties of several triply periodic minimal surface lattice structures made by polymer additive manufacturing. Polymer
**2017**, 152, 62–71. [Google Scholar] [CrossRef] - Panesar, A.; Abdi, M.; Hickman, D.; Ashcroft, I. Strategies for functionally graded lattice structures derived using topology optimisation for additive manufacturing. Addit. Manuf.
**2018**, 19, 81–94. [Google Scholar] [CrossRef] - Aremu, A.O.; Maskery, I.; Tuck, C.; Ashcroft, I.A.; Wildman, R.D.; Hague, R.I.M. A comparative finite element study of cubic unit cells for selective laser melting. In Proceedings of the Solid Freeform Fabrication Symposium, Austin, TX, USA, 4–6 August 2014; pp. 1238–1249. [Google Scholar]
- Khaderi, S.N.; Deshpande, V.S.; Fleck, N.A. The stiffness and strength of the gyroid lattice. Int. J. Solids Struct.
**2014**, 51, 3866–3877. [Google Scholar] [CrossRef][Green Version] - Al-Ketan, O.; Rowshan, R.; Abu Al-Rub, R.K. Topology-mechanical property relationship of 3D printed strut, skeletal, and sheet based periodic metallic cellular materials. Addit. Manuf.
**2018**, 19, 167–183. [Google Scholar] [CrossRef] - Sundén, B.; Fu, J. Heat Transfer in Aerospace Applications; Academic Press: Cambridge, MA, USA, 2016. [Google Scholar]
- Daliri, A.; Zhang, J.; Wang, C.H. 6-Hybrid polymer composites for high strain rate applications. In Lightweight Composite Structures in Transport; Njuguna, J., Ed.; Woodhead Publishing: Cambridge, UK, 2016; pp. 121–163. ISBN 978-1-78242-325-6. [Google Scholar]
- Syam, W.P.; Jianwei, W.; Zhao, B.; Maskery, I.; Elmadih, W.; Leach, R. Design and analysis of strut-based lattice structures for vibration isolation. Precis. Eng.
**2017**, 52, 494–506. [Google Scholar] [CrossRef] - Hussein, A.Y. The Development of Lightweight Cellular Structures for Metal Additive Manufacturing. Ph.D. Thesis, University of Exeter, Devon, UK, 2013. [Google Scholar]
- Leary, M.; Mazur, M.; Elambasseril, J.; McMillan, M.; Chirent, T.; Sun, Y.; Qian, M.; Easton, M.; Brandt, M. Selective laser melting (SLM) of AlSi12Mg lattice structures. Mater. Des.
**2016**, 98, 344–357. [Google Scholar] [CrossRef] - Hsieh, P.-F.; Wu, T.-T.; Sun, J.-H. Three-dimensional phononic band gap calculations using the FDTD method and a PC cluster system. IEEE Trans. Ultrason. Ferroelectr. Freq. Control
**2006**, 53, 148–158. [Google Scholar] [CrossRef] - Choi, J.-W.; Kim, H.-C.; Wicker, R. Multi-material stereolithography. J. Mater. Process. Technol.
**2011**, 211, 318–328. [Google Scholar] [CrossRef] - Elmadih, W.; Syam, W.; Maskery, I.; Leach, R. Additively manufactured lattice structures for precision engineering applications. In Proceedings of the 32nd Annual Meeting of American Society for Precision Engineering, Charlotte, NC, USA, 29 October–3 November 2017; pp. 164–169. [Google Scholar]
- University of Nottingham and Added Scientific FLatt Pack Modelling Software. Available online: www.Flattpack.com (accessed on 24 April 2019).
- Sigalas, M.M.; García, N. Theoretical study of three dimensional elastic band gaps with the finite-difference time-domain method. J. Appl. Phys.
**2000**, 87, 3122–3125. [Google Scholar] [CrossRef] - Yan, Z.; Wang, Y. Wavelet-based method for computing elastic band gaps of one-dimensional phononic crystals. Sci. China Ser. G.
**2007**, 50, 622–630. [Google Scholar] - Yan, Z.; Wang, Y.; Zhang, C. Wavelet method for calculating the defect states of two-dimensional phononicx crystals. Acta Mech. Solida Sin.
**2008**, 21, 104–109. [Google Scholar] [CrossRef] - Liu, Z.; Chan, C.T.; Sheng, P. Three-component elastic wave band-gap material. Phys. Rev. B Condens. Matter Mater. Phys.
**2002**, 65, 1651161–1651166. [Google Scholar] [CrossRef] - Marwaha, A.; Marwaha, S.; Hudiara, I.S. Analysis of Curved Boundaries by FDTD and FE Methods. IETE J. Res.
**2001**, 47, 301–310. [Google Scholar] [CrossRef] - Qian, D.; Shi, Z. Using PWE/FE method to calculate the band structures of the semi-infinite beam-like PCs: Periodic in z-direction and finite in x–y plane. Phys. Lett. A
**2017**, 381, 1516–1524. [Google Scholar] [CrossRef] - Collet, M.; Ouisse, M.; Ruzzene, M.; Ichchou, M.N. Floquet–Bloch decomposition for the computation of dispersion of two-dimensional periodic, damped mechanical systems. Int. J. Solids Struct.
**2011**, 48, 2837–2848. [Google Scholar] [CrossRef] - Maurin, F.; Claeys, C.; Deckers, E.; Desmet, W. Probability that a band-gap extremum is located on the irreducible Brillouin-zone contour for the 17 different plane crystallographic lattices. Int. J. Solids Struct.
**2017**, 135, 26–36. [Google Scholar] [CrossRef] - Craft, G.; Nussbaum, J.; Crane, N.; Harmon, J.P. Impact of extended sintering times on mechanical properties in PA-12 parts produced by powderbed fusion processes. Addit. Manuf.
**2018**, 22, 800–806. [Google Scholar] [CrossRef] - King, W.E.; Anderson, A.T.; Ferencz, R.M.; Hodge, N.E.; Kamath, C.; Khairallah, S.A.; Rubenchik, A.M. Laser powder bed fusion additive manufacturing of metals; physics, computational, and materials challenges. Appl. Phys. Rev.
**2015**, 2, 41304. [Google Scholar] [CrossRef] - Materialise PA 12 (SLS): Datasheet. Available online: http://www.materialise.com/en/manufacturing/materials/pa-12-sls (accessed on 31 January 2018).
- Agius, D.; Kourousis, K.I.; Wallbrink, C. A review of the as-built SLM Ti-6Al-4V mechanical properties towards achieving fatigue resistant designs. Metals
**2018**, 8, 75. [Google Scholar] [CrossRef] - Yilmaz, C.; Hulbert, G.M. Dynamics of locally resonant and inertially amplified lattice materials. In Dynamics of Lattice Materials; Phani, A.S., Hussein, M.I., Eds.; John Wiley and Sons: Hoboken, NJ, USA, 2017; p. 233. [Google Scholar]

**Figure 1.**Illustration of the bandgap mechanism in (

**a**) Bragg-scattering lattices and (

**b**) internal resonance lattices.

**Figure 4.**Res-BCC

_{xyz}unit cell as designed in CAD with strut diameter $d$, spherical mass of diameter $s$ and cell size $L$.

**Figure 5.**(

**a**) Convergence results of the first natural frequency with respect to the mesh density of a 2 × 2 × 2 BCC

_{xyz}lattice (converged mesh density is highlighted) and (

**b**) comparison of high frequency vibration modes (existing above a normalised frequency of 0.3) of converged mesh (bottom) and finer mesh (top).

**Figure 6.**Selections of sets of nodes in a cubic unit cell: (

**a**) top left edge nodes, (

**b**) front left edge nodes, and (

**c**) top face edge nodes.

**Figure 9.**Multidimensional dispersion curves (DCs) of the lattice proposed by Wang et al. [25] as remodelled using our finite element (FE) modelling technique. The shaded grey area in the DC plot represents the identified bandgap (BG).

**Figure 10.**DCs of (

**a**) BCC

_{xyz}, (

**b**) gyroid TPMS, and (

**c**) res-BCC

_{xyz}lattice structures with 20% volume fraction.

**Figure 11.**BG properties of the BCC

_{xyz}, res-BCC

_{xyz}and Warmuth et al. [30] lattices of 20% volume fraction and 10 mm unit cell size, as predicted using the material properties of Ti-6Al-4V.

**Figure 12.**Attributes of the BGs as identified from the DCs of the BCC

_{xyz}(dashed lines) and res-BCC

_{xyz}(dotted lines) lattices at different volume fractions.

**Figure 13.**Res-BCC

_{xyz}lattice structures of finite periodicities of (

**a**) one, (

**b**) three and (

**c**) six.

**Figure 14.**Transmissibility of longitudinal waves in 20% volume fraction res-BCC

_{xyz}lattices of finite periodicity. The shaded area represents the BG region as depicted by the DCs with infinite periodicity assumptions. The percentage values denote the bandwidth of the finite lattice to that of the infinite one.

Volume Fraction (%) | $\mathit{d}/\mathit{L}$ |
---|---|

5 | 0.084 |

10 | 0.121 |

20 | 0.178 |

30 | 0.226 |

Volume Fraction (%) | $\mathit{d}/\mathit{L}$ |
---|---|

5 | 0.138 |

10 | 0.175 |

20 | 0.250 |

30 | 0.325 |

Volume Fraction (%) | $\mathit{d}/\mathit{L}$ | $\mathit{s}/\mathit{L}$ |
---|---|---|

10 | 0.084 | 0.480 |

20 | 0.084 | 0.680 |

30 | 0.084 | 0.796 |

**Table 4.**Properties of L-PBF Nylon-12 used for modelling lattice structures in this work [59].

Tensile Modulus | Density | Poisson’s Ratio |
---|---|---|

1500 MPa | 950 kg·m^{−3} | 0.3 |

**Table 5.**BG properties as identified in this work and reported by Wang et al. [25].

BG Property | Remodelled Structure (This Work) | Wang et al. ([25]) |
---|---|---|

Normalised BG start frequency | 0.19 | 0.19 |

Normalised BG end frequency | 0.22 | 0.23 |

Normalised BG frequency width (bandwidth) | 0.03 | 0.04 |

**Table 6.**Summary of the evolution of the BG as obtained from studying the transmissibility of longitudinal waves in BG lattices of different periodicity.

Periodicity | Mean Transmissibility (dB) | Lowest Transmissibility (dB) |
---|---|---|

One | −23 | −56 |

Three | −24 | −63 |

Six | −66 | −103 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Elmadih, W.; Syam, W.P.; Maskery, I.; Chronopoulos, D.; Leach, R. Multidimensional Phononic Bandgaps in Three-Dimensional Lattices for Additive Manufacturing. *Materials* **2019**, *12*, 1878.
https://doi.org/10.3390/ma12111878

**AMA Style**

Elmadih W, Syam WP, Maskery I, Chronopoulos D, Leach R. Multidimensional Phononic Bandgaps in Three-Dimensional Lattices for Additive Manufacturing. *Materials*. 2019; 12(11):1878.
https://doi.org/10.3390/ma12111878

**Chicago/Turabian Style**

Elmadih, Waiel, Wahyudin P. Syam, Ian Maskery, Dimitrios Chronopoulos, and Richard Leach. 2019. "Multidimensional Phononic Bandgaps in Three-Dimensional Lattices for Additive Manufacturing" *Materials* 12, no. 11: 1878.
https://doi.org/10.3390/ma12111878